Fundamental Aspects of Exchange Bias Effect in AF/F Bilayers and Multilayers

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2 Fundamental Aspects of Exchange Bias Effect in AF/F Bilayers and Multilayers DISSERTATION zur Erlangung des Grades eines Doktors der Naturwissenschaften der Fakultät für Physik und Astronomie an der Ruhr-Univärsität Buchum vorgelegt von Florin RADU Bochum 2005

3 Mit Genehmigung des Dekanats vom wurde die Dissertation in englisher Sprache verfast. Mit Genehmigung des Dekanats vom wurden Teile dieser Arbeit vorab veröffentlicht. Eine Zusammenstellung befindet sich am Ende der Dissertation. Erstgutachter Zweitgutachter Prof. Dr. Dr. h.c. Hartmut Zabel Prof. Dr. Werner Keune Tag der mündlichen Prüfung:

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5 Contents Introduction I Fundamentals of Exchange Bias 5 1 Exchange Bias Models Stoner-Wohlfarth model Discovery of the Exchange Bias effect Ideal model of the exchange bias Phenomenology The ideal Meiklejohn-Bean model The sign of the exchange bias The magnitude of the EB The 1/t F dependence of the EB field Meiklejohn and Bean model Analytical expression of exchange bias field Azimuthal dependence of the exchange bias field Magnetization reversal Rotational hysteresis Néel s AF domain wall-weak coupling Malozemoff Random Field Model Domain State Model Mauri model Analytical expression of exchange bias field Azimuthal dependence of the exchange bias field Kim-Stamps Approach-Partial wall The Spin Glass model of exchange bias Hysteresis loops as a function of the conversion factor f Asymmetry of the hysteresis loop Phase diagram of exchange bias and coercive field within the spin glass model Dependence of exchange bias field on the thickness of the antiferromagnetic layer The blocking temperature for exchange bias Azimuthal dependence of exchange bias and coercive field within the spin glass model i

6 Contents Dependence of the exchange bias field on lateral size of the AF domains Interface disorder and the training effect Empirical expression for the training effect Experiments and Simulations for IrMn/CoFe EB system II Polarized Neutron Reflectometry 73 2 Theory Fundamental Properties of the Neutron Discovery of the neutron The neutron mass Magnetic moment of neutron The neutron spin β decay of the neutron The magnetic interaction Interaction of neutrons with matter Specular Polarized Neutron Reflectometry Generalized Matrix Method (GMM) Recurrence Method (RM) Spin Asymmetry Reflectivity of a semi-infinite magnetic media Hysteresis Loops Spin states of neutrons in magnetic thin film Introduction Sample characterization by MOKE and PNR Rotation experiment Experimental determination of magnetization orientation Multilayers Final Remarks Neutron Resonances in Layered Structures Bound State Neutron Resonances Wave function in the resonant layer: generalized case Neutron density in neutron resonators Bound State Neutron Resonator for non-polarized Neutrons Bound State Neutron Resonator for Polarized Neutrons Interface and Interface-Specific Sensitivity of PNR Resonance splitting Bright-Wigner Resonances in a Quasi Bound State Neutron Resonator115 4 Off-specular Neutron Scattering from Magnetic Domain Walls The polarized 3 He gas spin filter technique for polarization analysis Polarized 3 He Gas Spin Filter on EVA Polarized 3 He Gas Spin Filter on ADAM Off-specular scattering due to refraction at domain walls ii-

7 Contents Introduction Experimental Procedures Kerr microscopy Polarized neutron reflectivity from a thin Co film Neutron data analysis Final remarks III Exchange Bias in CoO/FM Bilayers and Multilayers Negative, positive and perpendicular exchange bias in CoO/Co bilayers Sample preparation and characterization Anisotropy at room temperature measured by MOKE Temperature dependence of the exchange bias and coercive fields Positive exchange bias Field cooling dependance of positive exchange bias Perpendicular exchange bias Conclusions Magnetization reversal in CoO/Co exchanged biased bilayers Characterization by MOKE Asymmetric magnetization reversal as revealed by PNR Interfacial domain formation during magnetization reversal AF magnetic state after the field cooling Conclusions Exchange bias in [CoO/Co] n Multilayers Introduction Sample characteristics Magnetic Characterization Magnetization reversal in CoO/Co multilayers Thermal relaxation of magnetization in CoO/Co multilayers Conclusion Exchange bias properties of Fe(110)/CoO(111) bilayers Sample Preparation and Characterization Magnetic characterization Temperature dependence of exchange bias and coercive fields Azimuthal dependence of the exchange bias and coercive fields The dependence of the exchange bias and coercive fields on the field cooling orientation Anisotropy orientation of the AF layer Conclusions iii-

8 Contents 9 Soft X-ray Magnetic Scattering of Fe/CoO Exchanged Biased Bilayers Introduction Sample growth and characterization Soft x-ray resonant magnetic scattering studies Reflectivity and asymmetry ratio Temperature dependence of the element-specific hysteresis loops XRMS comparison between CoO single layer and CoO/Fe bilayer system Discussion Conclusions Conclusions 195 -iv-

9 Introduction 1

10 Introduction In this thesis the fundamental aspects of exchange bias effect are studied using polarized neutron reflectometry (PNR), magneto-optical Kerr effect (MOKE), superconducting quantum interference device (SQUID) magnetometry, and soft-xray magnetic scattering (XRMS). The thin film deposition techniques, rf-sputtering, as well as molecular beam epitaxy and, characterization methods like x-ray diffraction, atomic force microscopy, reflection high energy electron diffraction (RHEED), and low energy electron diffraction LEED, were extensively used in obtaining the necessary high quality samples relevant for the studies performed in this thesis. The emphasis of the thesis is, however, not on the description of well documented effects in the literature, but on discussion of new methods and effects, surrounding the exchange bias phenomenon. The exchange bias effect was discovered more that 50 years ago by Meiklejohn and Bean, and it essentially manifest itself as a shift of the hysteresis loop to negative or positive applied fields. Its origin is related to the magnetic coupling across the common interface shared by a ferromagnetic (F) and an antiferromagnetic (AF) layer. Extensive research is being carried out to unveil the details of this effect. It has resulted in more then 600 publications in the last five years. An exchange bias bilayer consist of three key elements, with rather different magnetic and structural properties: the ferromagnetic layer, the antiferromagnetic layer and the interface between former two. While the ferromagnetic layer can be studied in detail by using laboratory equipment like SQUID, MOKE and MFM, this is not the case for the magnetic interface and for the antiferromagnetic layer. The interface embedded in between the F and AF layer has low volume, therefore it is difficult to separate its contribution from the F layer. Still, for the exchange bias effect the interfacial magnetic properties are essential for understanding the effect. For this purpose polarized neutron scattering and soft-xray magnetic scattering techniques can reveal some of the key magnetic properties of the interface. The AF layer has (should have) no macroscopic magnetization, even so the magnetic moment of individual atoms are rather high. The magnetic properties of the AF materials are traditionally studied by neutron diffraction. However, in thin films due to the reduced AF volume available for scattering, this approach is rather difficult to follow. Here, the soft-xray magnetic scattering through the linear dichroism can reveal information about the magnetic properties of the AF layer, therefore providing useful insights into the origin of the EB effect. The situation from the application point of view appears to be, however, less complex. The effect is being used in building functional elements like pinning heterostructures which are embedded in devices such as storage media, readout sensors, and magnetic random access memory(mram). Nevertheless, robust, reliable and easy to control functional elements based on exchange bias phenomenon require more understanding of the fundamentals of the effect which is further motivating research in this field. This thesis is organized in three parts, Fundamentals of Exchange Bias, Polarized Neutron Reflectometry, and Exchange Bias in CoO/ferromagnet Bilayers and Multilayers, were detailed studies on the fundamentals of the exchange bias effect are summarized and discussed. As AF layer we have mostly used the CoO because its convenient magnetic and structural properties: strong anisotropy, the Néel temperature is about T N =291 K, it can -2-

11 Introduction be grown both in polycrystalline and crystalline phases and, it has a simple cubic structure. As ferromagnetic layers we have used both Fe and Co. In the first part we review some of the current models for exchange bias. We focused on the numerical calculations and analytical treatment of those models which are based on the Stoner-Wohlfarth model. The behavior of the F and AF spins during the magnetization reversal, as well as the dependence of the critical fields on the parameters of the F and AF layer are analyzed in detail. Furthermore, we have extended the Meiklejohn and Bean model by assuming a realistic state of the interface between the F and AF layer. Surprisingly, this model describes well the experimental observation like azimuthal dependence of exchange bias, AF and F thickness dependence, coercivity, the inverse linear dependence on the lateral structures dimensions, training effect, etc. In the second part of the thesis we describe fundamental aspects of polarized neutron reflectometry and its applications to magnetic thin films and heterostructures. We describe two generalized models of PNR, Generalized Matrix Method (GMM) which is proper for the numerical calculations and Recurence Method (RM) which is particularly successful for analytical analysis. Based on the GMM method we have build two numerical codes. One is called POLAR, a user friendly code made available to the scientific community. Another one is a fitting code used for the fits shown in this thesis. Analytical analysis using the RM allowed us to develop the technique of measuring hysteresis loops by neutrons. This technique is particular useful in discriminating between different magnetization reversals like domain wall movement and rotation, and also in identifying the appearance of magnetic roughness and/or magnetic domains at the interface, through the slight increase of the spinflip probability. Off-specular scattering is an important tool to monitor the appearance of magnetic domains. It is now used in connection with a polarized 3 He gas spin-filter which allows broad angle analysis. Off-specular scattering can discriminate between ripple domains and 180 degrees magnetic domains through scattering asymmetries, as well as, being sensitive to magnetically disordered interfaces. In the third section we describe experimental results on Co/CoO bilayers, Co/CoO multilayers and Fe/CoO bilayers. The Co/CoO bilayers were used to study the dependence of EB and coercive fields on the thickness of the ferromagnet, to study the asymmetry of the hysteresis loops and the temperature dependence of the critical fields. Correlations between the training effect and the magnetic state of the interface were studied in detail by PNR. The multilayers were alternatively grown to provide increased sensitivity to magnetic roughness due to increased number of interfaces which provides higher scattering volume. The Fe/CoO bilayers were studied systematically, as well. The replacement of the ferromagnetic Co by Fe was mandatory for the XRMS studies. Soft X-ray magnetic scattering yields complementary information with respect to PNR. Through its element specific property we have particularly studied the induced ferromagnetic components in the AF CoO layer. Strikingly, all experiments show ferromagnetic induced moments in the AF CoO layer. -3-

12 INTRODUCTION -4-

13 Part I Fundamentals of Exchange Bias 5

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15 Chapter 1 Exchange Bias Models 1.1 Stoner-Wohlfarth model The term anisotropy refers to the orientation of the magnetic moments with respect to given geometrical directions. In bulk materials the crystal axes are the reference directions, while in thin films other reference systems become important. In order to account for the orientation of the magnetic moments in magnetic materials, the minimum energy state is provided by analysis of the free energy. This evaluation is performed by minimization of the free energy with respect to various parameters. Figure 1.1: Definition of angles and vectors used in Stoner-Wohlfarth model calculations. The reference direction of the film is along the unidirectional anisotropy. The anisotropies and their properties are extensively discussed in literature. They can be divided in two categories[1] intrinsic contributions which includes the crystal anisotropy, shape anisotropy, and surface anisotropy; and extrinsic anisotropy like uniaxial anisotropy to vicinal surfaces, finite size effects and magneto-elastic energy due to pseudomorphic 7

16 Stoner-Wohlfarth model growth. In the following we use the simplest possible expression for the free energy of a ferromagnetic thin film and calculate the magnetic hysteresis loops. We assume that all the spins are confined in the film plane and the film has an uniaxial anisotropy. The response of the magnetization to an applied magnetic field is then uniform, therefore the spins will coherently rotate during the variation of the external field. The direction of the magnetization can be described by only one variable parameter, namely the (θ β) angle defining the direction of the magnetization with respect to the applied field. Many complexities of the magnetization reversal are neglected in this approach. Nevertheless, the Stoner-Wohlfarth (SW) model [2, 3, 4], named after the investigators who developed it for treating the magnetization reversal of a small single domain, is used with considerably success for various magnetic thin films and heterostructures. Figure 1.2: a) The azimuthal dependence of the normalized coercive field of a ferromagnetic film with uniaxial anisotropy. The curve is calculated with Eq b) The azimuthal dependence of the normalized irreversible switching field of a ferromagnetic film with uniaxial anisotropy. The curve is calculated with Eq The total energy per unit volume of a ferromagnetic film with in-plane uniaxial anisotropy reads: E(β) = µ 0 H M F cos(θ β) + K F sin(β) 2, (1.1) where the first term is the Zeeman energy contribution and the second term is the uniaxial anisotropy energy. The other parameters are: H the applied field, M F the saturation magnetization of the ferromagnet, K F the volume anisotropy constant of the ferromagnet, θ the orientation of the applied magnetic field with respect to the uniaxial anisotropy direction, and β the orientation of the magnetization vector during the magnetization reversal. The minimization of free energy with respect to the β angle and the stability equation: E(β) β = 0, 2 E(β) β 2 > 0 (1.2) -8-

17 Stoner-Wohlfarth model Fundamentals of Exchange Bias Figure 1.3: The longitudinal (left column) and transverse (right column) components of the magnetization for a film with in-plane anisotropy. The curves are delivered by the numerical evaluation of the Eq leads to the following equations: µ 0 H M F sin(θ β) + K F sin(2 β) = 0 (1.3) µ 0 H M F cos(β θ) + 2K F cos(2 β) > 0 (1.4) By solving Eq. 1.3 with the condition imposed by the Eq. 1.4 one obtains the β angle, which determines the longitudinal component (m = cos(β θ)) and the transverse component (m = sin(β θ)) of the magnetization. Both components are plotted in Fig. 1.3 for different measuring in-plane orientations (θ). The evolution of the hysteresis loops as function of the angle between the applied magnetic field and the orientation of the uniaxial anisotropy is shown in the left column of Fig. 1.3 and corresponds to typical behavior of the thin films with in-plane uniaxial anisotropy. Along the easy axis the hysteresis loop is square shaped and the transverse component is zero. When the applied field is perpendicular to the anisotropy axis (hard axis), the hysteresis loop has a linear shape whereas the transverse component is ovally shaped. -9-

18 Stoner-Wohlfarth model The expression for the coercive field can be easily inferred from the Eq. 1.3: H c (θ) = 2 K F µ 0 M F cos θ (1.5) For the hysteresis loops shown in Fig. 1.3, the coercive field follows in details the expression above. At the position of the easy axis (θ = 0, π) the coercive field is equal to the anisotropy field H a = 2 K F µ 0 M F, whereas along the hard axis (θ = ±π/2) the coercive field is zero. Experimentally, this is an often encountered situation. For instance, polycrystalline films grown on a-plane sapphire substrates show such uniaxial growth induced anisotropy due to the steps at the substrate surface(see Fig 5.3). The analysis of the Eq. 1.3 and Eq. 1.4 further leads to the so called asteroid curve [3] which defines stability conditions for the magnetization reversal. Besides the coercive field dependence as a function of the azimuthal angle θ, another critical field can be noticed in Fig It is the field where the magnetization changes irreversibly (where the hysteresis opens). This field can be extracted by solving both equations Eq. 1.3 and Eq By replacing the applied magnetic field H with its components along the easy axis direction: H = H cos θ +H sin θ = H x +H y, the solution of the system of equations Eq. 1.3 and 1.4 will give: H x = H a cos 3 β and H y = + H a cos 3 β. By eliminating β from the previous two equations we obtain the asteroid equation: Hx H y = 3 H a (1.6) Now, introducing back into the equation above the expression of the field components we obtain for the irreversible switching field the following expression[4]: H irr = K F 2 µ 0 M F 3 sin θ cos θ 23 (1.7) This field is plotted in Fig At the position of the easy axis (θ = 0, π) the irreversible field is equal to the anisotropy field H a, whereas at θ = π/4, 3π/4, the irreversible field is equal to half of the anisotropy field (H irr (π/4) = H a /2). The irreversible switching field can be experimentally extracted from the transverse components of the magnetization, whereas the coercive field is extracted from the longitudinal component of the magnetization(see Fig. 1.3). -10-

19 Discovery of EB effect 1.2 Discovery of the Exchange Bias effect Fundamentals of Exchange Bias The exchange bias (EB) effect, also known as unidirectional anisotropy, was discovered in 1956 by Meiklejohn and Bean [5, 6, 7], when studying Co particles embedded in their native antiferromagnetic oxide CoO. It was concluded from the beginning that the displacement of the hysteresis loop is brought about by the existence of an oxide layer surrounding the particles of Co. This implies that the magnetic interaction across their common interface is essential in establishing the effect. Being recognized as an interfacial effect, the studies of the EB have been performed mainly on thin films consisting of a ferromagnetic layer in contact with an antiferromagnetic one. Figure 1.4: a) Hysteresis loops of Co-CoO particles taken at 77 K. The dashed line show the loop after cooling in zero field. The solid line is the hysteresis loop measured after cooling the system in a field of 10 koe. b) Torque curve for Co particles at 300 K showing uniaxial anisotropy. b) Torque curve of Co-CoO particles taken at 77 K showing the unusual unidirectioal anisotropy. d) The torque magnetometer. The main component is a spring which measures the torque as a function off the θ angle on a sample placed in an magnetic field. [5, 6] In Fig. 1.4 the original figures from Ref. [5, 6] show the shift of a hysteresis loop of Co- CoO particles. The system was cooled down from room temperature to 77 K through the Néel temperature of the CoO (T N (CoO) = 291 K). The magnetization curve is shown in Fig. 1.4a) as a dashed line. It is symmetrically centered around zero value of the applied field, which corresponds to a general behavior of ferromagnetic materials. When, however, the sample is cooled in a positive magnetic field, the hysteresis loop is displaced to negative values (see continuous line of Fig. 1.4a). Such displacement did not disappear even when extremely high applied fields of Oe were used. In order to get more insight into this unusual effect, the authors studied the behavior of anisotropy by using a self-made torque magnetometer shown schematically in the Fig. 1.4a). It essentially consists of a spring connected to a sample placed in an external magnetic field. Generally, the torque magnetometry is an accurate method for measuring the magnetocrystalline anisotropy (MCA) of single crystal ferromagnets. The torque on a sample -11-

20 Discovery of EB effect is measured as a function of the angle θ between certain crystallographic directions and the applied magnetic field. In strong external fields, when the magnetization of the sample is parallel to the applied field (saturation), the torque is equal to: T = E(θ) θ, where E(θ) is the MCA energy. In the case of Co which has a hexagonal structure, the torque of the c-axis follows a sin(2θ) function as seen in Fig. 1.4b). The torque and the energy can be written as: T = K 1 sin(2θ) E = K 1 sin(2θ) dθ = K 1 sin 2 (2θ) + K 0, where K 1 is the MCA anisotropy and K 0 is an integration constant. It is clearly seen from the energy expression that along the c-axis, at θ = 0 and θ = 180, the particles are in stable equilibrium. This typical case of uniaxial anisotropy is seen for the Co particles at room temperature, where the CoO is in a paramagnetic state. At 77 K, after field cooling, the CoO is in an antiferromagnetic state. Here, the torque curve of the Co-CoO system looks completely different as seen in Fig. 1.4c). The torque curve is a function of sin(θ): T = K u sin(θ), (1.8) hence, E = K u sin(θ) dθ = K u cos(θ) + K 0, (1.9) The energy function shows that the particles are in equilibrium for one position only, namely θ = 0. Rotating the sample to any angle, it tries to return to the original position. This direction is parallel to the field cooling direction and such anisotropy was named (unidirectional anisotropy). Now, one can analyze if the same unidirectional anisotropy observed by torque magnetometry is responsible for the loop shift. In Fig. 1.1 is shown schematically the vectors involved in writing the free energy for a ferromagnetic layer with uniaxial anisotropy having the magnetization oppositely oriented to the field. It reads: F = µ 0 H M F cos(β) + K F sin 2 (β) (1.10) where H is the external field, M F is the saturation magnetization of the ferromagnet per unit volume and, K F is the MCA of the F layer. The two terms entering in the formula above are the Zeeman interaction energy of the external field with the magnetization of the F layer, and the MCA energy of the F layer, respectively. Now, writing the stability conditions and assuming that the field is parallel to the easy axis, we find that the coercive field is: H c = 2 K F /µ 0 M F Next step is to cool the system down in a field (Fig. 1.5b)) and introduce in the Eq 1.10 the unidirectional anisotropy term. The expression for the free energy becomes: E = µ 0 H M F cos(β) + K F sin 2 (β) K u cos(β) (1.11) -12-

21 Discovery of EB effect Fundamentals of Exchange Bias One observes that the solution is identical to the previous case with the substitution of an effective field: H = H K u /M F. This causes the hysteresis loop to be shifted by K u /µ 0 M F. Thus, Meiklejohn and Bean concluded that the loop displacement is equivalent to the explanation for the unidirectional anisotropy. Besides the shift of the magnetization curve and the unidirectional anisotropy, Meiklejohn and Bean have observed another effect when measuring the torque curves. Their experiments revealed appreciable hysteresis of the torque (see Fig. 9 and Fig. 10 of Ref [6] and Fig. 2 of Ref [7] ), indicating that irreversible changes of the magnetic state of the sample take place when rotating the sample in an external magnetic field. As the system did not display any rotational anisotropy when the AF was in the paramagnetic state, this provided the evidence for the coupling between the the AF CoO shell and the F Co core. Such irreversible changes were supposed to take place in the AF layer. -13-

22 Phenomenology 1.3 Ideal model of the exchange bias Phenomenology The macroscopic observation of the magnetization curve shift due to unidirectional anisotropy of a F/AF bilayer can be qualitatively understood analyzing the microscopic magnetic state of their common interface. Phenomenologically, the onset of exchange bias is depicted in Fig A F layer is in close contact to an AF one. Their critical temperatures should satisfy the condition: T C > T N, where T C is the Curie temperature of the ferromagnetic layer and T N is the Néel temperature of the antiferromagnetic layer. At a temperature which is higher than the Néel temperature of the AF layer and lower than the Curie temperature of the ferromagnet (T N < T < T C ), the F spins align along the direction of the applied field, whereas the AF spins remain randomly oriented in a paramagnetic state. The hysteresis curve of the ferromagnet is centered around zero, not being affected by the proximity of the AF layer. Next, we saturate the ferromagnet by applying a high enough external field H F C and then, without changing the applied field, the temperature is being decreased to a finite value smaller then T N ( field cooling procedure). After field cooling the system, due to the exchange interaction at the interface, the first monolayer of the AF layer will align ferromagnetically (or antiferromagetically ) to the F spins. The next monolayer of the antiferromagnet will have to align antiparallel to the previous layer as to complete the AF order, and so on (see Fig 1.5-2). Note that the AF interface spins are uncompensated, leading to a finite net magnetization of this monolayer. It is assumed that both the ferromagnet and the antiferromagnet are in a single domain state and that they will remain single domains during the remagnetization process. When reversing the field, the F spins will try to rotate in-plane to the opposite direction. Being coupled to the AF, which is considered to be rigid, it takes a stronger force and therefore a stronger external field is needed to overcome this coupling and to rotate the ferromagnetic spins. As a result, the first coercive field is higher than the similar one at T > T N, where the F/AF interaction is not yet active. On the way back from negative saturation to positive one (Fig 1.5-4), the F spins will need a smaller external force in order to rotate back (Fig 1.5-5) to the original direction. A torque is acting on the F spins for all other angles, except the stable direction which is along the field cooling direction (unidirectional anisotropy). As a result, the magnetization curve is shifted to negative values of the applied field. This displacement of the center of the hysteresis loop is called exchange bias field, and is negative in relation to the orientation of the F spins after field cooling (negative exchange bias ). -14-

23 Phenomenology Fundamentals of Exchange Bias Figure 1.5: Phenomenological model of exchange bias for an AF-F bilayer. 1) The spin configuration at a temperature which is higher than T N and smaller than T C. The AF layer is in a paramagnetic state while the F layer is ordered. Its magnetization curve (top-right) is centered around zero value of the applied field. Pannel 2): the spin configuration of the AF and F layer after field cooling the system through the T N of the AF layer in a positive applied magnetic field (H F C ). Due to the uncompensated spins of the AF interface, the F layer is coupled to the AF. Panel 3): the saturated state at the negative fields. Panel 4) and 5) show the configuration of the spins during the remagnetization, assuming that this takes place through in-plane rotation of the F spins. The center of magnetization curve is displaced at negative values of the applied field by H eb. (The description is in accordance with Ref. [5, 6, 8]) -15-

24 The ideal M&B model 1.4 The ideal Meiklejohn-Bean model Based on their observation about the rotational anisotropy, Meiklejohn and Bean proposed a model to account for the magnitude of the hysteresis shift. The assumptions made are the following [5, 8, 4]: The F layer rotates rigidly, as a whole Both the F and AF are in a single domain state The AF/F interface is atomically smooth. The AF layer is magnetically rigid, meaning that the AF spins remain unchanged during the rotation of the F spins The spins of the AF interface are fully uncompensated: the interface layer has a net magnetic moment The F and the AF are coupled by an exchange interaction across the F/AF interface. The parameter assigned to this interaction is the interfacial exchange coupling energy per unit area J eb The AF layer has an in-plane uniaxial anisotropy As general formalism for the coherent rotation of the magnetization one uses the Stoner- Wohlfarth [2, 3] model to which different energy terms can be added to best account for quantitative and qualitative behavior of the macroscopic reversal of the magnetization. In Fig.1.6 is shown schematically the geometry of the vectors involved in the ideal Meiklejohn and Bean model. H is the applied magnetic field, which makes an angle θ with respect to the field cooling direction denoted by θ = 0, K F and K AF are the uniaxial anisotropy directions of the F and the AF layer, respectively. They are set to be oriented parallel to the field cooling direction. M F is the magnetization orientation of the F spins during the magnetization reversal. It is assumed that AF spins do not deviate from their orientation defined during the field cooling procedure (rigid AF). The applied field is assumed to be parallel to the field cooling direction in the analysis below (θ = 0). Last condition refers to the direction along which the hysteresis loops is measured, whereas θ 0 is used for torque measurements or azimuthal dependence of the exchange bias field. Within this model the energy per unit area assuming coherent rotation of the magnetization, can be written as [5, 8, 9]: E = µ 0 H M F t F cos( β) + K F t F sin(β) 2 J eb cos(β) (1.12) where J eb [J/m 2 ] is the interfacial exchange energy per unit area. The stability condition E / θ = 0 has two types of solutions: one is β = cos 1 [(J eb µ 0 H M F t F )/(2 K F )] for µ 0 H M F t F J eb 2 K F, ; the other one is β = 0, π for µ 0 H M F t F -16-

25 The ideal M&B model Fundamentals of Exchange Bias Figure 1.6: Schematic view of the angles and vectors used by the ideal Meiklejohn and Bean model. The AF layer is assumed to be rigid and no deviation from its initially set orientation is allowed. K AF and K F are the anisotropy of the AF layer and F layer, respectively, which are assumed to be parallel oriented to the field cooling direction. β is the angle between F magnetization vector M F and the anisotropy direction of the F layer. This angle is variable during the magnetization reversal. H is the external magnetic field which can be applied at any direction θ with respect to the field cooling direction at θ = 0 (see Ref. [5, 6, 8]). J eb 2 K F, corresponding to positive and negative saturation, respectively. The coercive fields H c1 and H c2 are extracted from the stability equation above for β = 0, π: H c1 = 2 K F t F + J eb µ 0 M F t F (1.13) H c2 = 2 K F t F J eb µ 0 M F t F (1.14) Using the expressions above, the coercive field H c of the loop and the displacement H eb can be calculated according to: H c = H c1+h c2 2 H eb = H c1+h c2 (1.15) 2 which further gives: and H c = 2 K F µ 0 M F (1.16) H eb = J eb µ 0 M F t F (1.17) -17-

26 The ideal M&B model The equation Eq is the master formula of EB effect. It gives the expected characteristics of the hysteresis loop for an ideal case, therefore it serves as a standard estimation expression to which the experimental values are compared. In the next section I discuss some the predictions of the model above The sign of the exchange bias The Eq predicts that the sign of exchange bias is negative. Almost all the hysteresis loops shown in literature are shifted oppositely to the field cooling direction. The positive or negative exchange coupling across the interface produces the same sign of the exchange bias field. There are, however, exceptions. Positive exchange bias was observed for CoO/Co systems when the measuring temperature was close to the blocking temperature[10, 11]. At low temperatures positive exchange bias was observed in Fe/FeF 2 [12] and Fe/MnF 2 [13] bilayers. Specific of the last two systems is the low anisotropy of the antiferromagnet and the antiferromagnetic type of coupling between the F and AF layers. It was proposed that, at high cooling fields, the interface layer of the antiferromagnet aligns ferromagnetically with the external applied field and therefore ferromagnetically with the F itself. As the preferred orientation between the interface spins of the F layer and AF layer is the antiparallel one (antiferromagnetic coupling), the EB becomes positive. More theoretical end experimental details of the positive exchange bias mechanism are presented in Ref. [14, 15]. In the original Meiklejohn and Bean model the interaction of the cooling field with the AF spins is not taken into account. However this interaction can be easily introduced in the model. The positive exchange bias could also be accommodated in the M&B model, by simply changing the sign of J eb in Eq from negative to positive The magnitude of the EB Generally the exchange coupling parameter is taken as being the exchange constant of the AF layer (J AF ), or for various calculations a random value ranging for J AF to J F. For CoO J AF = 21.6 K = 1.86 mev [16]. Using this value, the expected exchange coupling constant J eb of a CoO(111)/F layer can be estimated as [17, 18]: J eb = NJ AF /A = 4 mj/m 2, (1.18) where N = 4 is the number of Co 2+ atoms of the CoO uncompensated interface per unit area A= (3) a 2, with a being the lattice parameter a = 4.27 A. With this number we would expect for a 100 Å thick Co layer which shares an interface with a CoO AF layer, an exchange bias of 2700 Oe: H eb [Oe] = J eb [J/m 2 ] M F [ka/m] t F [A ] 1011 (1.19) H eb = = 2740 Oe So far an ideal magnitude of the EB field predicted by the Eq was not yet observed experimentally, even so for some bilayers high EB fields were measured (see Table 1.1). -18-

27 The ideal M&B model Fundamentals of Exchange Bias We encounter here two problems: first is that we do not know how to evaluate the real coupling constant J eb at the interfaces with variable degrees of complexity and the second is that we might have not prepared interfaces which are atomically smooth. The interface unknown was nicely labelled by Kiwi [19] as a hard nut to crack. Indeed, the features of the interfaces may be complex regarding the structure and the magnetic properties. We usually characterize interfaces by their roughness parameter, but even this parameter is rarely given. One attempt to provide more information about the interface structure is described in Chapter 8. In Table 1.1 is summarize some EB data of systems with CoO as the AF layer. We focus on experimental interfacial exchange coupling constants extracted as J eb = H eb µ M F t F. One observes that the exchange coupling constant is usually smaller then the expected value of 4 mj/m 2 for CoO/Co bilayers by a factor ranging from 3 to several orders of magnitude. One anomaly is seen for the multilayer system Co/CoO which is actually 3 times higher then the expected value of 4 mj/m 2, which to our knowledge is the highest value observed experimentally. Such a variation of the experimental values for the interfacial exchange coupling constant is motivating further consideration of the mechanisms for describing the EB effect The 1/t F dependence of the EB field Qualitatively, Eq predicts that the variation of the EB field is direct proportional to the inverse thickness of the ferromagnet: H eb 1 t F (1.20) This law was subjected to a large number of experimental investigations [8], because it is associated with the interfacial nature of the exchange bias effect. For the CoO/Co bilayers no deviation was observed [24], even for very low thicknesses (2 nm) of the Co layer[10]. For other systems with thin F layers of the order of several nanometers it was observed that the 1/t F law is not closely obeyed [8]. It was suggested that the F layer is no longer laterally continuous [8]. In Chapter 5 we show experimental data on CoO/Co where the very low thickness of the F layer give rise to a curious shapes of the hysteresis loops from which the EB field cannot be precisely extracted. The deviations for the other extreme regime when the F layer is very thick were observed as well [8]. For this regime it is assumed that, when the F layer is thicker than the domain wall thickness (500nm for permalloy), the F spins vary appreciably across the film upon the magnetization reversal [25]. -19-

28 The ideal M&B model Table 1.1: Experimental values related to Co/CoO exchange bias systems. The symbols used in the table are: ebe-electron-beam evaporation, rsp-reactive sputtering, mspmagnetron sputtering, mbe-molecular beam epitaxy, F-ferromagnet, AF antiferromagnet, t AF -the thickness of the AF, t F -the thickness of the F, H eb -measured exchange bias field, H c -measured coercive field, T mes -measured blocking temperature, T mes -the measuring temperature, J eb -the coupling energy extracted from the experimental value of exchange bias field (J eb = H eb (µ 0 M F t F )). AF F t AF t F H eb H c T B T mes J eb Ref [Å] [Å] [Oe] [Oe] [K] [K] [mj/m2 ] CoO (air) Co(rsp) NA [20] CoO (air) Co(rsp) [10] CoO (air) Co(rsp) [10] CoO (air) Co(rsp) [10] CoO (air) Co(rsp) [10] CoO (air) Co(rsp) [10] CoO (air) Co(rsp) [10] CoO (air) Co(rsp) [10] CoO (air) Co(rsp) [10] CoO (air) Co(msp) [21] CoO (air) Co(msp) NA [22] CoO (rsp) Co(rsp) [11] [CoO (rsp) Co(rsp)] x ! [23] CoO (in-situ) Co(ebe) [24] CoO(111)(mbe) Fe(110)(mbe) sec

29 M&B model 1.5 Meiklejohn and Bean model Fundamentals of Exchange Bias In Ref. [6] a new degree of freedom for the AF spins was allowed: the AF is still rigid, but it can slightly rotate during the magnetization reversal as shown in Fig This assumption of the rotation of the whole AF layer was introduced in order to account for the rotational hysteresis observed during the torque measurements. Allowing the AF layer to rotate is not in contradiction to rigid state of the AF, because it is allowed only to rotate as a whole. Therefore, the fourth assumption of the ideal M&B model in section 1.4 is removed. The new condition for the AF spins is: α 0. With this new assumption, the equation Eq reads [7, 8]: E = µ 0 H M F t F cos(θ β) + K F t F sin 2 (β) + K AF t AF sin(α) 2 J eb cos(β α), (1.21) where t AF is the thickness of the antiferromagnet and M F is the saturation magnetization of the ferromagnet and, K AF is the MCA of the AF layer per unit area. The new energy term in the equation above as compared to Eq is the anisotropy energy of the AF layer. The Eq above can be analyzed numerically by minimization of the energy in respect to the α and the β angles. Below, we will perform a numerical analysis of Eq and highlight a few of the conclusions discussed in Ref. [6, 7, 4]. The minimization in respect to α and β leads to a system of two equations: H sin(θ β) + sin(β α) = 0 J eb µ 0 M F t F K AF t AF J eb sin(2 α) sin(β α) = 0, (1.22) where the unknown variables α and β are numerically extracted as a function of the applied field H. In order to simplify the discussion, the parameters involved in the equations above are grouped as: Heb J eb µ 0 M F t F and R K AF t AF J eb, where Heb is the value of the exchange bias field when the anisotropy of the AF is infinity large and the R-ratio is the parameter which defines the strength of the AF layer with respect to the exchange interfacial energy J eb. Note that for clarity reasons the anisotropy of the ferromagnet was neglected (K F = 0) in the system of equation above. As a result the coercivity, which will be discussed further below, is not related to the F layer but to the AF behavior. Also, in order to simplify the discussion we consider first the case θ = 0, which corresponds to measuring a hysteresis parallel to the field cooling direction. Numerical evaluation of the Eqs delivers: α angle of the AF spins as a function of the applied field during the hysteresis measurement β angle of the F spins which rotates coherently during their reversal -21-

30 M&B model Figure 1.7: Schematic view of the angles and vectors used by the Meiklejohn and Bean model. The AF layer is assumed to be rigid but an α deviation from its initially set orientation is allowed. M AF is the sublattice magnetization of the AF layer. K AF and K F are the anisotropy of the AF layer and F layer, respectively, which are assumed to be parallel oriented to the field cooling direction. β is the angle between F magnetization vector M F and the anisotropy direction of the F layer. This angle is variable during the magnetization reversal. H is the external magnetic field which can be applied at any direction θ with respect to the field cooling direction at θ = 0 ( Ref. [5, 6, 7, 8]). The β angle defines fully the hysteresis loop which delivers the coercive fields H c1 and H c2, which further defines the coercive field of the hysteresis loop H c and the exchange bias field H eb (see Eq. 1.15). The α angle influences the shape of the hysteresis loops when the R-ratio has low values, as we will see below. For high R ratios the rotation angle of the antiferromagnet is close to zero, giving a maximum exchange bias field equal to H eb. The properties of the EB systems originate from the properties of the AF layer, which are accounted for by one parameter, R-ratio. We will consider the influences of the R-ratio on the β and α angles which, as stated above, they define the macroscopic behavior and critical fields of the EB systems. Numerical simulations of Eq as a function of R-ratio are shown in Fig. 1.8 and Fig.1.9. The exchange bias is finite when the condition R 1 is satisfied, meaning that the MCA energy of the AF layer should be strong enough to hold the AF order during the magnetization reversal. During the rotation of the ferromagnet, the AF spins rotate by an angle α without breaking the AF order. The process is reversible, therefore no additional -22-

31 M&B model Fundamentals of Exchange Bias Figure 1.8: Left: The phase diagram of the exchange bias field and the coercive fields given by the Meiklejohn-Bean formalism. Right: Typical behavior of the antiferromagnetic angle α for the three different regions of the phase diagram. Only region I can lead to a shift of the hysteresis loop. The other two regions would lead to a coercivity but not to an exchange bias field. coercivity is predicted. When, however, the condition R < 1 is satisfied, there will be no displacement of the hysteresis loop. During the switching of the F layer, the AF spins will follow the F spins irreversibly due to the strong interfacial coupling. In this case the system will behave as a single F layer with an increased coercive field characterized by an effective anisotropy. Within the Stoner-Wohlfarth model the resultant coercive field can be roughly approximated as as [4]: H c 2 K AF t AF /(µ 0 M F t F ). For an exact numerical solution see Fig It is quite interesting to stress that for many AF/F bilayers and multilayers both situations above may take place, therefore the EB shift is accompanied by an increased coercive field. In Fig. 1.8 is shown the phase diagram of the H c and H eb as a function of R ratio. We distinguish three physically distinct regions [6, 7, 26, 4]: I. R 1 In this region the coercive field is zero and the exchange bias field is finite, decreasing from the asymptotic value Heb to the lowest finite value at R = 1. The AF spins are rotating reversibly during the complete reversal of the F spins. The α angle has a maximum value as a function of the R-ratio, ranging from approximatively zero for R = to α = 45 at R = 1. Notice that an increased maximum angle of AF spins causes a slight decrease of the exchange bias field. When the R-ratio approaches the critical unity value, the exchange bias is minimum. In the phase diagram, only this range of R-ratio can cause a shift of the magnetization curve. II. 0.5 R < 1 The specific of this region is that the AF spins are not anymore reversibly. They -23-

32 M&B model Figure 1.9: Several hysteresis loops and antiferromagnetic spin orientation during the magnetization reversal. For the simulation we used the Meiklejohn-Bean formalism. Top row shows three hysteresis loops calculated for different R ratios of the region I. The left graph of top row shows the α angle of the antiferromagnetic layer for the three R parameters of the AF layer. The middle raw is the corresponding hysteresis loops and α angles for the region II. The bottom row shows the simulations for region III. follow the F spins and they change irreversibly causing coercive field at the expense -24-

33 M&B model Fundamentals of Exchange Bias of the exchange bias field, which is zero. One observes that, depending on the field sweeping direction, there is a hysteresis-like behavior of the AF spins rotation. There is a critical value of the F spins orientation when the AF spins cannot withstand the torque force caused by the coupling to the F spins and they are simply jumping in an discontinuous modus to another angle (jump angle). III. R < 0.5 This region preserves the features of the previous with one exception, namely that the AF spins follow reversibly the F spins, without any jumps, therefore no hysteresis-like behavior of the α angle is seen. The exchange bias field is zero and the coercive field is finite, depending on the R-ratio. Table 1.2: The experimental magnetic parameters of Co,Fe and Ni. The listed parameters are: Curie temperature T C, paramagnetic Curie temperature Θ C, saturation magnetization M s, Curie constant C, and the effective magnetic moment µ eff. T C [K] Θ C [K] M s [ka/m] C[K] µ eff [µ B ] Co Fe Ni CoFe Table 1.3: Antiferromagnets and their properties. AF T N [K] µ eff [µ B ] J AF [mev] K AF J/m 3 Magnetic Axis Ref CoOk [16] [117] [27] FeO 198 2[27] x - (111) [27] NiO [27] [28] (111) [27] MnO [29, 30] (111) [27] FeF MnF FeMn compensated 1 IrMn compensated [31] It is easy to recognize that allowing the AF to rotate as a whole leads to an impressive rich phase diagram of the EB systems as a function of the parameters of the AF layer (and F layer). The R-ratio can be varied across the whole range from zero to infinity by changing the thickness of the AF layer [8], by varying its anisotropy (dilution of the AF layers with non-magnetic impurities [32]), or by varying the interfacial exchange energy J eb (low dose ion bombardment [33, 34, 35]). -25-

34 M&B model Analytical expression of exchange bias field Firstly, let us calculate analytically the expression of the exchange bias field for θ = 0. The solution is obtain by solving analytically the system of equations Eq for β = 0, which reads 1 : J eb µ 0 M F t F 1 J eb 2 K AF t AF 4KAF 2 H eb = t2 AF J eb 1 (1.23) 0 K AF t AF J eb < 1 The equation above retains the 1/t F dependence of the exchange bias field, and it provides new features. The most impressive one is an additional term, which effectively lowers the exchange bias field when the R-ratio approaches the critical unity value. The R-ratio has three terms. One of them includes the thickness of the AF layer. The analytical expression for exchange bias field (Eq. 10.1) predicts that there is a critical AF thickness t cr AF below which the exchange bias cannot exist. It is: t cr AF = J eb K AF (1.24) Below this critical thickness the interfacial energy is transformed in coercivity. Above the critical thickness the exchange bias increases as a function of the AF layer thickness, reaching the asymptotic (ideal) value Heb when the t AF is infinity Azimuthal dependence of the exchange bias field In the following we consider the exchange bias field for the region I, where it acquires non vanishing values. In the regime II and III it is zero, as resulting from Eq and from the numerical simulations shown in Fig The coercive fields and the exchange bias field are extracted from the condition β = θ + π/2 for both H c1 and H c2. It gives H c1 = H c2 = J eb µ 0 M F t F cos(α(r, θ + π/2) θ), where α(r, θ + π/2) is the value of the rotation angle of the AF spins at the coercive field. With the notation: α 0 α(r, θ + π/2), and using the expression 1.15, the angular dependence of the exchange bias field becomes: H eb (θ) = J eb µ 0 M F t F cos( α 0 θ) (1.25) This equation is the most general expression for an exchange bias field. It includes both the influence of the rotation of the AF layer and influence of the azimuthal orientation of the applied field. To illustrate its generality we consider below two particular case: θ = 0 For the particular case, when the hysteresis loop is measured along the field cooling direction (θ = 0), the Eq 1.25 is equivalent to the Eq It might be that this expression is obtained here for the first time. -26-

35 M&B model Fundamentals of Exchange Bias R When R is very large (R 1), then the α can be taken as being approximatively zero. For this case the exchange bias field as function of θ can be written as: H α=0 eb (θ) = J eb µ 0 M F t F cos(θ) (1.26) This equation is valid for very large values of the R-ratio (R 1), when the rotation of the AF layer is negligible. It is actually the original assumption of the Meiklejohn and Bean model. For instance, such a condition (R ) is satisfied for large thicknesses of AF layer. Solving the system of equations 1.22, the expression for the EB field can also be written as 2 : H eb (θ) = K AF t AF µ 0 M F t F sin(2 α 0 ) (1.27) Interestingly, the exchange coupling parameter J eb disappeared, leaving instead an explicit dependence of the exchange bias field on the parameters of the antiferromagnet and ferromagnet. The exchange coupling constant and the θ angle are accounted for by the AF angle α 0. In order to get more insight into the azimuthal dependence of the exchange bias, we show in Fig the normalized exchange bias field H eb (θ) / Heb (0) as a function of the θ angle, where Heb (0) J eb µ 0 M F t F. We have plotted the results as given by Eq and Eq. 1.27, for three different values of the R ratio (R = 1.1, R = 1.5, R = 20). The α 0 angle (see Fig. 1.9) was obtained by numerically solving the system of equations Eqs We see in Fig that, for large values of R, the azimuthal dependence of the exchange bias field follows closely a cos(θ) unidirectional dependence. When, however, the R-ratio takes small values but higher then unity, the azimuthal behavior of exchange bias field deviates from the ideal unidirectional characteristic. There are two distinctive features: one is that the at θ = 0 the exchange bias field is reduced, and the other one is that the maximum of exchange bias field is shifted from zero. In other words, the exchange bias field is not maximum along the field cooling direction. Another striking feature is that the shifted maximum of the exchange bias field with respect to the azimuthal angle θ does not depend on thickness and anisotropy of the AF layer: H MAX eb = J eb µ 0 M F t F. (1.28) Summarizing we may state that, within the Meiklejohn and Bean model, a reduced exchange bias field is observed along the field cooling direction and it depends on the parameters of the AF layer (K AF and t AF ). However, the maximum value for the exchange bias field which is reached at θ 0 does not depends on the anisotropy constant (K AF ) and thickness (t AF ) parameters of the AF layer, for R 1. The azimuthal characteristic 2 It might be that this expression is obtained here for the first time. -27-

36 M&B model of the exchange bias allows to extract all tree essential parameters defining the exchange bias field: J eb, K AF and, t AF. The condition for extracting the H c1 and H c2 from the same β angle hides an important property of the magnetization reversal which we describe it further below. Figure 1.10: Azimuthal dependence of exchange bias as a function of the θ angle. The curves are delivered by the Eq.1.25 and Eq Magnetization reversal A distinctive feature of exchange bias phenomena is the magnetization reversal mechanism. In Fig is shown the parallel component of the magnetization m = cos(β) versus the perpendicular component m = sin(β) for several R-ratio and for θ = 30. The geometrical conventions are the ones shown in the Fig We see that for R < 1 the reversal of the F spins is symmetric, similar to the typical ferromagnets with uniaxial anisotropy. There is, practically, no difference with respect to the F spin rotation between the regions 0.5 R < 1 and R < 0.5, therefore they can be assumed being similar from the ferromagnetic reversal point of view. When R 1 another reversal mechanism is observed. The ferromagnetic spins are rotating first towards the unidirectional axis as lowering the field from positive to negative values, and then the rotation proceeds continuously until the negative saturation is reached. On the return path, when the field is swept from negative to positive values the ferromagnetic spins follow the same path towards the positive saturation. The rotation is continuous without any additional steps or jumps. Similar behavior was observed theoretically within the domain state model [36]. -28-

37 M&B model Fundamentals of Exchange Bias Figure 1.11: Magnetization reversal for several values of the R ratio. The parallel component of the magnetization vector m = cos(β) is plotted as a function of the perpendicular component of the magnetization m = sin(β). The reversal for R < 1 is resembling the typical reversal of ferromagnets with uniaxial anisotropy. For R 1 the reversal proceeds along the same path for increasing and decreasing branch of the hysteresis loop Rotational hysteresis I shortly discus here the the rotational hysteresis deduced from torque measurements [5, 6, 7]. The torque measurements were carried out in strong applied magnetic fields, therefore the applied field H and the magnetization M F can be assumed to be parallel (β = θ). The torque force is: T = E(θ) = J eb sin(θ α(θ)) θ The new correction from the ideal model (Eq. 1.8) is that the torque curve is affected by the rotation of the AF spins through the α angle, which enters in the formulae above. However, this does not explain the energy loss during the torque measurements. The torque curve would look a bit distorted but completely reversible. The integration of the energy curve will give rotational hysteresis W rot = 0. Nevertheless, the model leaves an open window, and that is to assume that a f fraction of the F-AF particle are uniaxial, behaving as in region II and the rest fraction (1 f) of the F-AF particles are unidirectional, having the ideal behavior of regime I. As seen in Fig. 1.8Left, when the R-ratio of the uniaxial particle will be in the range 0.5 R < 1, the AF spins will rotate irreversibly, showing hysteresislike behavior given by the α jumps indicated in the Fig. 1.8Right. The rotational hysteresis is not expected for unidirectional particles with R 1 because the AF structure changes reversibly with θ. With this assumption the uniaxial particle will contribute to the energy loss during the torque measurements, while the unidirectional particles are responsible for the unidirectional feature of the torque curve. This argument was used by Meiklejohn and -29-

38 M&B model Bean [7] when studying the exchange bias in core-co/shell-coo. A fraction f = 0.5 was inferred from the torque curves shown in Fig

39 Néel s AF domain wall 1.6 Néel s AF domain wall-weak coupling Fundamentals of Exchange Bias The rigid AF and rigid rotating AF concepts impose a restriction on the behavior of the antiferromagnetic spins, namely that the AF order is preserved during the magnetization reversal. Such restriction implies that the interfacial exchange coupling is found almost entirely in the hysteresis loop either as a loop shift or as coercivity. Experimentally however, the size of the exchange bias does not agree with the expected value, being several orders of magnitude lower then predicted. In order to cope with such loss of coupling energy, one can assume that a partial domain wall develops in the AF layer during the magnetization reversal. This concept was introduced by Néel [37] when considering the coupling between a ferromagnet and a low anisotropy antiferromagnet. The AF partial domain wall will store an important fraction of the exchange coupling energy, lowering the shift of the hysteresis loop. Néel has calculated the magnetization of orientation of each layers through a differential equation. The weak coupling is consistent with a partial AF domain wall which is parallel to the interface (Néel domain wall). His model predicted that a minimum AF thickness is required to produce hysteresis shift. More importantly the partial domain wall concept forms the basis of further models which incorporate either Néel walls or Bloch walls formation as a way to reduce the observed magnitude of exchange bias. -31-

40 Malozemoff model 1.7 Malozemoff Random Field Model Malozemoff (1987) proposed a novel mechanism for exchange anisotropy postulating a random nature of the exchange interactions at the F-AF interface[17, 38, 39]. He assumes that the chemical roughness or alloying at the interface which is present for any realistic bilayer system, causes lateral variations of the exchange field acting on the F and AF layers. The resultant random field causes the AF to break up into magnetic domains due to the energy minimization. By contrast with other theories, where the unidirectional anisotropy is treated either microscopically [40, 41, 42] or macroscopically [5, 6, 43], the Malozemoff approach belongs to the mesoscopic scale models for the surface magnetism. Figure 1.12: Schematic side view of a F/AF bilayer with a ferromagnetic wall driven by an applied field H [17]. The general idea for estimating the exchange anisotropy is depicted in Fig. 1.12, where a domain wall in an uniaxial ferromagnet is driven by an applied in-plane magnetic field H [17]. Assuming that the interfacial energy in one domain (σ 1 ) is different from the energy in the neighboring domain σ 2, then the exchange field can be estimated by the equilibrium condition between the applied field pressure 2 H M F t F and the effective pressure from the interfacial energy σ: H eb = σ, (1.29) 2 M F t F where M F and t F are the magnetization and thickness of the ferromagnet. When the interface is treated as ideally compensated, then the exchange bias field is zero. On the other case, if the AF/F interface is ideally uncompensated there is an interfacial energy difference σ = 2J i /a 2, where J i is the ferromagnetic exchange constant across the interface, and a is the lattice parameter of a simple cubic structure assigned to the AF layer. The EB field is H eb = J i /a 2 M F t F (see Fig 1.13). (For this example the energy is calculated as E kl = J i S k S l per pair of nearest neighbor spins kl at the interface ) Estimating numerically the size of the EB field using the equation above for an ideally uncompensated interface, results in a discrepancy of several orders of magnitude with respect to the experimental observation. Therefore, a novel mechanism based on random fields at the interface acting on the AF layer is proposed as to drastically reduce the resulting exchange bias field. By simple and schematic arguments Malozemoff describes how roughness on the atomic scale of a compensated AF interface layer can lead to uncompensated spins required -32-

41 Malozemoff model Fundamentals of Exchange Bias Figure 1.13: Schematic view of possible atomic configurations in a F-AF bilayer with ideal interfaces. Frustrated bonds are indicated by crosses. Compensated configuration a) will result in configuration b) by reversing the ferromagnetic spins through domain wall movement. It gives an exchange bias field of H eb = J i /a 2 M F t F. The compensated configuration c) will result in the compensated configuration d). The exchange bias field for this case is zero (H eb = 0). [17] for the loop to shift. An atomic rough interface depicted in Fig 1.14a) containing a single mono-atomic bump in a cubic interface gives rise to 6 net antiferromagnetic deviations from a perfect compensation. A bump shifted by one lattice spacing as shown in Fig 1.14b), which is equivalent to reversing the F spins, gives a 6 net ferromagnetic deviation away from perfect compensation. Thus a net energy difference of z i J i with z i = 12 acts at the interface favoring one domain orientation over the other. Note that for an ideally uncompensated interface the energy difference is only 8J i when reversing the F spins. This implies that, actually, such an atomic step roughness of a compensated roughness gives a higher exchange bias field as compared to the ideally compensated interface. The estimates of this local field can be further reduced assuming a more detailed model. For example, by inverting the spin in the bump shown in Fig. 1.14c, the interfacial energy difference is reduced by 5 2J i at the cost of generating one frustrated pair in the AF layer just under the bump. This frustrated pair increases the energy difference by 2J A, where J A is the AF exchange constant. Thus the energy difference between the two domains becomes 2J i + 2J A or roughly 4J if J i J A J. If one allows localized canting of the spins, one expects the energy difference to be reduced somewhat further. Each interface irregularity will give a local energy difference between domains whose sign depends on the particular location of the irregularity and whose magnitude is on the average 2zJ, where z is a number of order unity. Further, for an interface which is random on the atomic scale, the local unidirectional interface energy σ l = ±zj/a 2 will also be random and its average σ in a region L 2 will go down statistically as σ σ l / N, where N = L 2 /a 2 is the number of sites projected onto the interface plane. Therefore the effective AF-F exchange energy per unit area is given by: J eb 1 N J i 1 L J i, where J i is the exchange energy of a fully uncompensated AF-surface. Given the random field and assuming a region with a single domain of the ferromagnet, -33-

42 Malozemoff model Figure 1.14: Schematic side view of possible atomic moment configurations for non-planar interface. The bump should be visualized on a two-dimensional interface. Configuration c) represents the lower energy state of a). The configuration b) is energetically equivalent to flipping the ferromagnetic spins of a). The x signs represents frustrated bonds. [17] it is energetically favorable for the AF to break up into magnetic domains, as shown schematically in Fig A perpendicular domain wall is the most preferable situation. This perpendicular domain wall is permanently present in the AF layer, oppositely to the Mauri model where a domain wall parallel to the interface develops temporarily during the rotation of the F layer. Figure 1.15: Schematic view of a vertical domain wall in the AF layer. It appears as an energetic favorable state of F/AF system with rough interfaces. [17] By further analyzing the stability of the magnetic domains in the presence of random fields, a characteristic length L of the frozen-in AF domains and their characteristic height are obtained: L π A AF /K AF and h = L/2, where A AF is the exchange stiffness and h is the characteristic height of the AF domains. Once these domains are fixed, flipping the ferromagnetic orientation causes an energy change per unit area of σ = 4zJ/πaL, which further leads to the following expression for the EB field: H eb = 2 z AK π 2 M F t F (1.30) Assuming a CoO/Co(100 A ) film, the calculated exchange bias using the Eq gives: H eb = [J] [m] 7 [J/m 3 ] 10 = 580 Oe (1.31) π [ka/m]

43 Malozemoff model Fundamentals of Exchange Bias For the estimations above we used for the exchange stiffness the following value: A AF = J AF /a, where a is the lattice parameter of CoO (a = 4.27A ) and J AF is the exchange constant for CoO J AF = 1.86 mev[16]. The characteristic length of the AF domains is for CoO: L = π A AF /K AF = π 19 [J] [m] [J/m 3 ] = 16.6 (1.32) A The height of the AF domains is h = L/2 = 8.3 A. Comparing this value to the experimental data on CoO(25A )/Co studied in this thesis, we notice that the calculated EB bias field agrees well with the value observed experimentally. For example, the exchange bias field for CoO(25 A )/Co(119 A ) is 557 Oe and the theoretical value calculated with Eq is 487 Oe. Also, the length and the height of the AF domains have enough space to develop. The difference is that, experimentally we observed that the AF domains inferred from the appearance of the interface disorder take place after the very first magnetization reversal, whereas within the Malozemoff model the AF domains are assumed to develop during the cooling down procedure. Nevertheless, the agreement appears to be excellent. -35-

44 DS model 1.8 Domain State Model The Domain State model (DS) [32, 44, 45, 36, 46, 47, 48] is a microscopic model in which disorder is introduced via magnetic dilution not only at the interface but also in the bulk of the AF layer as in Fig The key element in the model is that the AF layer is a diluted Ising antiferromagnet (DAFF) which exhibits a phase diagram like the one shown in Fig [45]. In zero field the system undergoes a phase transition from a disordered, paramagnetic state to a long-range-ordered antiferromagnetic phase at the dilution dependent Néel temperature. In the low temperature region, for small fields, the long-range interaction phase is stable in three dimensions. When the field is increased at low temperature the diluted antiferromagnet develops a domain state phase with a spinglass-like behavior. The formation of the AF domains in the DS phase originates from the statistical imbalance of the number of impurities of the two AF sublattices within ay finite region of the DAFF. This imbalance leads to a net magnetization which couples to the external field. A spin reversal of the region, i. e., the creation of a domain, can lower the energy of the system. The formation of a domain wall can be minimized if the domain wall passes through nonmagnetic defects at a minimum cost of exchange energy. Figure 1.16: Sketch of the domain state model with one ferromagnetic layer and three diluted antiferromagnetic layers. The dots mark defects [45]. Figure 1.17: Schematic phase diagram of a three-dimentional diluted antiferromagnet [45]. Nowak et al. [45] further argue that during the field cooling below the irreversible line T i (B), in an external field and in the presence of the interfacial exchange field of the ferromagnet, -36-

45 DS model Fundamentals of Exchange Bias the AF develops a frozen domain state with an irreversible surplus of magnetization. This irreversible surplus magnetization controls then the exchange bias. The F layer is described by a classical Heisenberg model with the nearest-neighbor exchange constant J F. The AF is modelled as a magnetically diluted Ising system with an easy axis parallel to that of the F. The Hamiltonian of the system is given by [45]: H = J F J AF J INT (d z S 2 iz + d x S 2 ix + µbs i ) S i.s j <i,j>ɛf iɛf ɛ i ɛ j σ i σ j µb z ɛ i σ i iɛaf <i,j>ɛaf <iɛaf,jɛf > ɛ i σ i S jz, (1.33) where the S i and σ i are the classical spin vectors at the ith site of the F and AF, respectively. The first line contains the energy contribution of the F, the second line describes the diluted AF layer, and the third line includes the exchange coupling across the interface between F and DAFF, where it is assumed that the Ising spins in the topmost layer of the DAFF interact with the z component of the Heisenberg spins of the F. In order to obtain the hysteresis loop of the system, the model 1.33 is treated by Monte Carlo simulations. Typical hysteresis loops are shown in Fig [47], where both the magnetization curve of the F layer and of the interface monolayer of the DAFF are shown. The coercive field extracted from the hysteresis curve depends on the anisotropy of the F layer but it is also influenced by the DAFF. It, actually, depends on the thickness and anisotropy of the DAFF layer. The coercive field decreases with the increasing thickness of the DAFF layer [47] which can be understood as following: the interface magnetization tries to orient the F along its direction. The coercive field has to overcome this barrier, and the higher the interface magnetization of the DAFF the stronger is the field required to reverse the F. Since the interface magnetization decreases with increasing DAFF thickness, the coercive field becomes smaller. The strength of the exchange bias field can be estimated from the Eq using simple ground state arguments. Assuming that all spins in the F remain parallel during the field reversal and some net magnetization of the interface layer of the DAFF remains constant during the reversal of the F, a simple calculation gives the usual estimate for the bias field [45]: lµb eb = J INT m INT, (1.34) where l is the number of the F layers and m INT is the interface magnetization of the AF per spin. For an ideal uncompensated interface (m INT = 1) the exchange bias is too high, whereas for ideally compensated interface the exchange bias is zero. Within the DS model the interface magnetization m INT = 1 is neither constant nor is it a simple quantity. Therefore, it is replaced by m IDS, which is a measure of the irreversible domain state magnetization of the DAFF interface layer and is responsible for the EB field. With this, -37-

46 DS model Figure 1.18: Simulated hysteresis loops within the domain state model. The top hysteresis belongs to the ferromagnetic layer and the bottom hysteresis belongs to an AF interface monolayer [47]. an estimate of exchange bias field for l = 9, J INT = J, and µ = 1.7 µ B gives a value of about 30 mt. The exchange bias field depends also on the bulk properties of the DAFF layer as shown by Miltényi et al.[32]. There the AF layer was diluted by introducing non-magnetic defects in the bulk part, away from the interface. It was shown experimentally that the EB field depends strongly on the dilution of the AF layer. As a function of concentration of the non-magnetic Mg impurities, the EB evolves as following: at zero dilution the exchange bias has finite values, whereas by increasing the Mg concentration, the EB field increases first, showing a broad peak-like behavior, and then, when the dilution is further increased the EB field decreases again. Simulations within the DS model showed an overall good qualitative agreement. The peak-like behavior of the EB as a function of the dilution is clearly seen in the simulations. However, it appears that at zero dilution, the DS gives vanishing exchange bias whereas experimentally finite values are observed. The exchange bias is missing at low dilutions because the domains in the AF cannot be formed as they would cost too much energy to break the AF bonds. This discrepancy [45] is thought to be explained by other imperfections such as e.g., grain boundaries in the twinned AF layer which is similar to dilution and which can also reduce the domain-wall energy, thus leading to domain formation and EB even without dilution of the AF bulk. An important property of the dynamics of the DAFF is the slow relaxation of the remanent magnetization, i.e., the magnetization obtained after switching off the cooling field. It is known that the remanent magnetization of the DS relaxes nonexponentially on extremely -38-

47 DS model Fundamentals of Exchange Bias long-time scales after the field is switched off or even within the applied field. In the DS model EB is related to this remanent magnetization. This implies a decrease of EB due to slow relaxation of the AF DS. The reason for the training effect can be understood within DS model from Fig bottom panel, where it is shown that the hysteresis loop of the AF interface layer is not closed on the righthand side. This implies that the DS magnetization is lost partly during the hysteresis loop due to a rearrangement of the AF domain structure. This loss of magnetization clearly leads to a reduction of the EB. The blocking temperature within the DS model can be understood by considering the phase diagram of the DAFF shown in Fig The frozen DS of the AF occurs after cooling in a field below the irreversibility temperature T i (b). Within this interpretation, the blocking temperature corresponds to T i (b) where in an EB system the role of the cooling field is complemented or replaced by the interface exchange field of the F. Since T i (b)<t N, the blocking temperature should be at least slightly below the Neél temperature and should be dependent on the strength of the interface exchange field. The simulations within the DS model shows that EB depends linearly on the temperature, as observed experimentally in some Co/CoO systems, but no reason is given for this behavior [45]. Overall, it is believed that strong support for the DS model is given by experimental observations where the nonmagnetic impurities are added to the AF layer in a systematic and controlled way [32, 49, 34, 50, 51]. Also, it appears that a good agreement is observed with the experiments shown in Ref. [52], where the dependence of the EB as a function of AF thickness and temperature for IrMn/Co is shown. The asymmetry of the magnetization reversal modes[36] is shown to be dependent on the angle between the easy axis of the F and DAFF layers. It was found that either identical or different F reversal mechanisms (domain wall movement or coherent rotation) can occur as the relative orientation between the anisotropy axis of the F and AF is varied. -39-

48 Mauri model 1.9 Mauri model The model of Mauri et al. [43] renounces the assumption of a rigid AF layer and proposes that the AF spins develop a domain wall parallel to the interface. The motivation to introduce such an hypothesis was to explore a possible reduction of the exchange bias field resulting from the Meiklejohn and Bean model. The assumptions of the Mauri model are: both the F and AF are in a single domain state the F layer rotates rigidly, as a whole the AF layer develops a domain wall parallel to the interface the AF interface layer is uncompensated (or fully compensated) the AF layer has an uniaxial anisotropy the cooling field is parallelly oriented to the uniaxial anisotropy of the AF layer the AF and F spins rotates coherently, therefore the Stoner-Wohlfarth model is used to describe the system Figure 1.19: Mauri model for the interface of a thin ferromagnetic film on a antiferromagnetic substrate [43]. Schematically the spin configuration within the Mauri model is shown in Fig The F spins rotate coherently, when the applied magnetic field is swept as to measure the hysteresis loop. The first interfacial AF monolayer is oriented away from the F spins making an angle α with the direction of the field cooling direction and with the anisotropy axis of the AF layer. The next AF monolayers are oriented away from the interfacial AF spins as to form a domain wall parallel to the interface. The spins of only one AF sublattice are depicted, the spins of the other sublattice being oppositely oriented as to complete the AF order. At a distance ξ at the interface, a ferromagnetic layer of thickness t F follows. Using the Stoner-Wohlfarth model, the energy can be written as: E = µ 0 H M F t F cos(θ β) + K F t F sin 2 (β) J eb cos(β α) 2 A AF K AF (1 cos(α)) (1.35) -40-

49 Mauri model Fundamentals of Exchange Bias where the first term is the Zeeman energy of the ferromagnet in an applied magnetic field, the second term is the anisotropy term of the F layer, the third term is the interfacial exchange energy and, the forth term is the energy of the partial domain wall [43]. The new parameter in the equation above is the exchange stiffness A AF. As in the case of Meiklejohn and Bean model, the interfacial exchange coupling parameter J eb [J/m 2 ] is again left undefined. One assumes that its value ranges from exchange constant of the F layer to the exchange constant of the AF layer divided by and effective area(see section: 1.4.2). The free energy can be written in units of 2 A AF K AF, which is the energy per unit surface of a 90 domain wall in the AF layer: e = k (1 cos(β)) + µ cos(β) 2 + λ [1 cos(α β)] + (1 cos(α)), (1.36) where λ = J eb /( 2 A AF K AF ), is the interface exchange, with J eb being redefined as J eb A 12 /ξ, µ = K F t F / 2 A AF K AF is the reduced ferromagnet anisotropy, and k = µ 0 H M F t F / 2 A AF K AF is the reduced external magnetic field. Mauri et al. [43] have calculated the magnetization curves by numerical minimization of the reduced free energy Eq Several values of the λ and µ parameters were considered providing quite realistic hysteresis loops. Their analysis highlighted two limiting cases, which delivers the following expressions for the exchange bias field: (A 12 / ξ)/ µ 0 M F t F for λ 1 H eb = 2 (1.37) A AF K AF / µ 0 M F t F for λ 1 For limiting case λ 1 (strong coupling) the value of exchange bias field is similar to the value given by the Meiklejohn and Bean model. For this situation, practically no important differences between the predictions of the two models exist. When the coupling is weak λ 1, the Mauri model delivers a reduced exchange bias field which, practically is independent of the interfacial exchange energy. It depends on the domain wall energy and the parameters of the ferromagnet. The 1/t F law is preserved by the Mauri model Analytical expression of exchange bias field In order to compare the predictions of the Mauri model and the Meiklejohn and Bean approach, we reconsider the analysis of the free energy. Starting from the expression of the free energy Eq. 1.35, the minimization with respect to the α and β angle leads to the following system of equations: H sin(θ β) + sin(β α) = 0 J eb µ 0 M F t F 2 A AF K AF J eb sin(α) sin(β α) = 0 (1.38) Similarly to the Meiklejohn and Bean model we define the parameters M 2 A AF K AF J eb and H eb J eb µ 0 M F t F. Also we set θ = 0 meaning that the applied field is swept along -41-

50 Mauri model the easy axis od the AF layer. Also, we do not take into account the anisotropy of the ferromagnet (K F = 0), for two reasons: one is that usually the coercive fields of the exchange bias systems are much higher then the coercive field given by the anisotropy of the ferromagnet, and secondly it is easier to compare the results within the Mauri model to the M&B model, where also the anisotropy of the ferromagnet was disregarded. After a short inspection of the equations Eq to the Eq one can clearly see that the first equations of the two systems are identical, while the second ones are different in two respects. First difference is related to the term M which includes the domain wall energy instead of the AF anisotropy term in the R ratio. The second difference is that instead of a sin(2 α) term in the second equation of Eq. 1.22, the Mauri model has a sin(α) term, which influences strongly the phase diagram shown in Fig Figure 1.20: Left: The phase diagram of the exchange bias field and the coercive fields given by the Mauri formalism. Right: Typical behavior of the antiferromagnetic angle α for the two different regions of the phase diagram. In both regions I and II a shift of the hysteresis loop can exist. The coercive field is zero in both regimes. The analytical expression for the exchange bias are obtained by solving the system of equations First step in solving the Eq is to extract the α angle (α = ± arccos(± from the second equation and to introduce it in the first equation. Using the condition that at the coercive field β = π/2, we obtain both coercive fields H c1 = H c2. Then, inserting them into the general expressions Eq. 1.15, the coercive field H c is zero and the exchange bias field becomes: M cos(β) 1+M )) 2 2 M cos(β) H eb = J eb 2 AAF K AF = J eb M µ 0 M F t F J 2 eb + 4 A AF K AF µ 0 M F t F 1 + M 2 (1.39) This equation is plotted as a function of M-ratio ranging from M = 0 to M = 5 in Fig left. The behavior of the EB field according to the Eq is monotonic with respect to the stiffness and anisotropy of the AF spins. At M 1 the exchange bias is equal -42-

51 Mauri model Fundamentals of Exchange Bias Figure 1.21: Several hysteresis loops and antiferromagnetic spin orientation during the magnetization reversal. For the simulation we used the Mauri formalism. Top row shows three hysteresis loops calculated for different M ratios of the region I. The left graph of top row shows the α angle of the antiferromagnetic layer for the three M parameters of the hysteresis loop. The bottom raw is the corresponding hysteresis loops and α angles for the region II. to H M eb = J eb µ 0 M F t F, which is the well known expression given by M&B model. When, however, the M-ratio approaches the low values, the exchange bias decreases, vanishing at M = 0, provided that the thickness of the AF layer is sufficiently thick to allow a 180 wall. With some analytical analysis of the Eq one can easily reach the limiting cases of weak coupling (M 1) and strong coupling (M 1) discussed by Mauri et al. [53] (see Eq. 1.37). In Fig right is shown the representative behavior of the α angle of the first interfacial AF monolayer as a function of β orientation of the F spins during the magnetization reversal. It is clear that for M 1 (region I) the α angle behaves differently as compared to the region II where M < 1. In the former case the first AF monolayer deviates slightly from the original cooling direction during the rotation of the F spins, whereas in the region II the α angles can follow a complete 360 rotation. However, it does not rotate 360 because of the spring-like interaction. In this region (II) the EB field is reduced as compared with the M&B model. This reduction is more clearly seen further -43-

52 Mauri model below, when analyzing the azimuthal dependence of the EB field within the Mauri model. In Fig the hysteresis loops (m = cos(β)) and the corresponding AF angle rotation during the magnetization reversal are plotted for several M-ratios. They were obtained by solving numerically the system of equations Eqs For all the values of the M-ratio the magnetization curves are shifted to negative values of the applied magnetic field. We distinguish two different regions with respect to the behavior of the α angle of the first AF monolayer. In the first region, for M 1 (region I), the AF monolayer in the proximity of the F layer behaves similar to the Meiklejohn and Bean, namely the α angle deviates reversibly from the anisotropy direction as function of β. The maximum value of the α angle acquired during the rotation of the F layer is two times higher for the Mauri model as compared to the M&B model, reaching a maximum value of π/2 at M = 1. The coercive field in this region is zero. The angle α in the region II where M < 1 has a completely different behavior. It rotates with the ferromagnet following the general behavior depicted in Fig Notice that α follows monotonically the rotation of the F spins, with no jumps or hysteresis-like behavior in contrast to the M&B model(see Fig. 1.8). Very importantly, the exchange bias field does not vanish in this region and therefore no additional coercive field related to the AF is observed, given that the AF layer is sufficiently thick to allow a domain wall as shown in Fig Comparing the phase diagram of the Mauri model Fig left to the corresponding one given by the M&B model Fig. 1.8 left one can clearly see that region I of both models is very similar with respect to the qualitative behavior of the exchange bias field as a function of the R-ratio and, respectively, M-ratio. However, we can compare those curves only when accounting for the variation of the EB field as a function of the anisotropy of the AF layer. Both models predict that the EB field depends on the anisotropy of the AF layer in a similar qualitative manner. Additionally, within the M&B model the EB field includes also the dependence on the thickness of the AF layer, which is not visible in the Mauri model. The other regions of both phase diagrams are completely different. Within the Mauri model, the exchange bias does not vanishes at M < 1 but it continuously decreases, whereas the M&B model predicts that the exchange bias field vanishes for R < 1 leading to enhanced coercivity. Also note that for the weak coupling region (II) of the Mauri model, the exchange bias would strongly depend on the temperature through the anisotropy constant of the AF layer [54] Azimuthal dependence of the exchange bias field Further, the azimuthal dependence of the EB field is analyzed. First the analytical expression is obtained and plotted as a function of the rotation angle θ. By solving the second equation of the system of equation Eqs with respect to β, one finds the angle α as function of β. Using the condition for the coercive field as β = θ π/2, and introducing it in the first equation of 1.38, one obtains the coercive fields H c1 = H c2. It follows that the coercive field H c (θ) = 0 and the exchange bias field as function of the azimuthal angle is: -44-

53 Mauri model Fundamentals of Exchange Bias H eb (θ) = J eb 2 AKAF cos(θ) µ 0 M F t F Jeb 2 + 4AK AF 4J eb AKAF sin(θ) (1.40) In Fig is plotted the exchange bias field calculated by the expression above (which was also checked numerically) for different values of the M-ratio. In the region I, the EB field is maximum for the field cooling directions only for very large M-ratios. When M approaches the unity value, the maximum of exchange bias field is shifted away from θ = 0, to higher azimuthal angles. This maximum value of exchange bias is equal to H MAX,M 1 eb = J eb µ 0 M F t F (1.41) and is identical to the similar situation encountered for the M&B model The shape of the curves evolves from an ideal unidirectional shape at M to somewhat distorted shapes at M 1. Figure 1.22: Azimuthal dependence of exchange bias as a function of the θ angle. The curves are delivered by the Eq In the region II (Fig. 1.22left) a drastic change as compared to region I is seen for the maximum of the exchange bias as a function of azimuthal angle. Its value is equal now to H MAX,M<1 eb = M = 2 A AF K AF J eb. (1.42) In this region the shape of the curves is also distorted for M-ratio values close to unity, but as the M decreases towards zero, the curves acquire an ideal unidirectional behavior. Similarly to the region II, the exchange bias is not maximum along the field cooling direction, but its maximum is shifted away at higher azimuthal angles. Experimental support for the Mauri model, is provided by the detection of domain walls and characterization of domain wall angles in in Ref. [55]. -45-

54 Kim-Stamps approach 1.10 Kim-Stamps Approach-Partial wall The approach of Kim and Stamps [9, 56, 57, 58, 59, 60] follows from the work of Néel and Mauri et al. extending the model of planar domain wall to the partial domain wall in the AF layer. Biquadratic (spin-flop) and bilinear coupling energies are used to describe exchange biased system, where the bias is created by the formation of partial walls in the AF layer. The model applies to compensated, partially compensated, and uncompensated interfaces. Figure 1.23: (a) Magnetization curve for the ferromagnet/antiferromagnet system. (b) Calculated spin structure at three different points of the magnetization curve. The creation of a partial antiferromagnet domain wall can be seen in (iii). Only the spins close to the F/AF interface are shown [60]. A typical hysteresis loop indicative for a partial domain wall in the AF layer is shown in Fig [60]. The F spins are aligned with the external field while the AF spins are in a perfect Néel state, collinear with the easy axis. The interface spins are antiparallel to the F due to a presumably antiparallel coupling. As the field is reduced and reversed, the AF pins the F due to interfacial coupling until the critical value of the reverse field where the magnetization begins to rotate. When it is energetically more favorable to deform the AF, rather than breaking the interfacial coupling, a partial wall twits up as the F rotates. The winding and unwinding of the partial domain wall in the AF is reversible, therefore the magnetization is reversible (no coercivity). This mechanism is only possible if the AF is thick enough to support a partial wall. The magnitude of the exchange bias is similar -46-

55 Kim-Stamps approach Fundamentals of Exchange Bias to the one given by the Mauri model. The partial-wall theory (also the Mauri model), however, does not account for the coercivity enhancement that accompanies the hysteresis loop shift in single domain materials. The enhanced coercivity observed experimentally, is proposed to be related to the domain wall pinning at magnetic defects. The presence of an attractive domain-wall potential in the AF layer, arising from magnetic impurities can provide an energy barrier for domainwall processes that controls coercivity. Following the treatment of pinning in magnetic materials by Braun et. al.[61], Kim and Stamps examined the influence of a pointlike impurity at an arbitrary position in the AF. As a result, the AF energy acquires, besides the domain wall energy, another term which depends on the concentration of the magnetic defects. These defects decrease the anisotropy locally and lead to an overall reduction of the AF energy. This reduction of the AF energy gives rise to a local energy minimum for certain defect positions relative to the interface. The domain walls can be pinned at such positions and contribute to the coercivity. Figure 1.24: Defect-induced asymmetry in hysteresis loops. The hysteresis loops are shown for a reduced exchange defect at x L = 5 for three concentrations: (a) ρ J = 0.15, (b) ρ J = 0.45, and (c) ρ J = The components of magnetization parallel (M ) and perpendicular (M ) to the field direction are shown. The arrows indicate the directions for reversal and remagnetization. The spin configuration near the interface is shown for selected field values below the hysteresis curves [60]. Kim and Stamps argue that irreversible rotation of the ferromagnet due to a combination of wall pinning an depinning transitions, give rise to asymmetric hysteresis loops. Some examples are given in Fig [60]. The loops are calculated with an exchange defect at x L = 5, for three different values of defect concentration ρ J, where x L denotes the defect -47-

56 Kim-Stamps approach positions in the antiferromagnet, with x L = 0 corresponding to the interface layer and x L = t AF 1 being the free surface. At low defect concentrations, the pinning potential is insufficient to modify partial-wall formation. The resulting magnetization curve, as shown in Fig. 1.24(a), is reversible and resembles the curve obtained with the absence of impurities. Pinning of the partial wall occurs during reversal for moderate concentrations, which appears as a sharp rotation of the magnetization at negative fields, as shown in Fig. 1.24(b). During remagnetization the wall is released from the pinning center at a different field, thus resulting in an asymmetry in the hysteresis loop. The release of the wall is indicated by a sharp transition in M. The energy barrier between wall pinning and release increases with defect concentration, resulting in a larger coercivity and reduced bias as in Fig. 1.24(c). Within this model, the asymmetry of the hysteresis loops is interpreted in terms of domainwall pinning processes in the antiferromagnet. This explanation appears to be consistent with some recent work of Nikitenko et al. on the NiFe/FeMn system [62] who concluded the presence of an antiferromagnet wall at the interface is necessary to explain their hysteresis measurements. -48-

57 Spin-Glass model Fundamentals of Exchange Bias 1.11 The Spin Glass model of exchange bias The experimental results of this thesis allow to suggest that the magnetic state of the interface between the F layer and AF layer is magnetically disordered behaving similar to a spin glass system. The assumptions of the spin glass (SG) model are: the interface F/AF is frustrated spin system (spin-glass) frozen-in uncompensated AF spins are responsible for EB shift low anisotropy AF spins contributes to the coercivity Soft X-ray magnetic scattering studies shown in Chapter 9 show that some AF spins belonging to the AF layer rotate reversibly with the F spins. Due to the shift of the hysteresis loop it is obvious that another part of the AF layer is frozen. Therefore, the AF layer can be seen as containing, in a first approximation, two types of AF states. One part has a large anisotropy which holds the AF spins blocked and another part with a weaker anisotropy which allows the spins to rotate together with the F spins [63, 64, 65, 66]. This part of the AF is the frustrated region (spin-glass) and gives rise to an increased coercivity. We argue that the observation of rotating spins belonging to the AF layer is a direct indication of a spin-glass behavior. How such a low anisotropy AF region can be formed in an AF layer can be argued as follows: the interface between the F and AF layer is never perfect, therefore we may assume chemical intermixing, stoichiometry deviations, structural inhomogeneities, etc, at the interface to take place. This leads to the formation of a transition region from the pure AF state to a pure F state. On average, the anisotropy of such an interfacial region is reduced, leading to a spin-glass behavior. Besides the chemical intermixing and stoichiometry deviations, the roughness can provide a weak AF interface region. Furthermore, AF grains which through border inhomogeneities may also contribute to this behavior. Therefore, we assume that the fraction of interfacial spins which rotate almost in phase with the F spins are frustrated and that they lead to enhanced coercivity. We describe them by an effective uniaxial anisotropy K eff SG, because they are coupled to the presumably uniaxial AF layer. Polarized neutron scattering[67, 10, 68] revealed three effects related to the magnetic state of the CoO/Co interface during the remagnetization processes: a) the interface is a disordered system containing ferromagnetic domains and domain walls even in saturation; b)the interfacial ferromagnetic spins are not collinear to the applied field direction during the magnetization reversal; c) through neutron resonance splitting [68] stray fields emerging from the sample surface were clearly inferred. Such stray fields were imaged by Welp et. al. [22]. The last results point to the existence of interfacial spins which prefers the perpendicular orientation to the surface of the sample which might also be accommodated into Malozemoff model [17]. Overall, the interface becomes magnetically disordered as revealed by the presence of the magnetic domains and domain walls even in saturation and by the non-collinearity between the spins at the interface and the cooling field direction. This was observed to take place after the first reversal, for the case of thin CoO layers [10]. Therefore, we are entitled -49-

58 Spin-Glass model to ascribe an average orientation (γ) to the effective uniaxial anisotropy of the spin-glass interface. This angle would have to be in close relation to the anisotropies of the AF layer. Its value is to be determined from the azimuthal dependence of the exchange bias and coercive field, as we will see later on. The simplest description of a spin glass system is that it is a collection of spins (i.e. magnetic moments) whose low-temperature state is a frozen disordered one, rather than the kind of uniform or periodic pattern we are accustomed to find in conventional ferromagnets. In order to achieve such a state, two ingredients are necessary: a) there must be competition among the different interactions between the moments, in the sense that no single configuration of the spins in uniquely favored by all interactions (this is commonly called frustration ); b) these interactions must be at least partially random. Experimental observations discussed above suggest that the interfacial spins are fulfilling the conditions of a spin-glass state. These low anisotropy spins are in contact with the ordered (uniaxial) AF state which result in a partially random nature of the interactions at the interface. This partial random state will be introduced in the M&B model as an effective uniaxial anisotropy. It is now easy to imagine that after the cooling in a field the energy state of a spin glasslike interface is not energetically the lowest one. Therefore, by consecutively cycling the external field, the system evolves to the lowest energy state through the many possible energy states specific to the spin glass systems (Training Effect). Based on the experimental results mentioned above we retain two properties of the EB systems, namely the existence of low anisotropy spins which cause an induced K eff SG anisotropy for the F layer inferred from experimental XRMS studies on CoO/Fe bilayers, and the noncollinear anisotropies at the interface (average angle γ ) revealed by the interface disorder observed for CoO/Co bilayer (Polarized Neutron Reflectometry experiments). Adding this effective anisotropy to the M&B model, the free energy reads: E = µ 0 H M F t F cos(θ β)+k F t F sin(β) 2 +K AF t AF sin 2 (α)+k eff SG sin(β γ)2 J eff eb cos(β α) (1.43) where, K eff SG is an effective SG anisotropy related to the AF spins with reduced anisotropy at the interface, J eff eb is the reduced interfacial exchange energy, and γ is the average angle of the effective SG anisotropy. From now on, the MCA anisotropy of the ferromagnetic layer (K F = 0) will be neglected as it does not contribute essentially to the properties of the EB systems. Next, we write the system of equations resulting from the minimization os the Eq with respect to the α and β angles: H sin(θ β) + K eff SG Jeff eb J eff eb µ 0 M F t F K AF t AF J eff eb sin(2 (β γ)) + sin(β α) = 0 sin(2 α) sin(β α) = 0 (1.44) -50-

59 Spin-Glass model Fundamentals of Exchange Bias where, Heb J eff eb µ 0 M F t F is the maximum value of the exchange bias and R K AF t AF, is J eff eb the R-ratio defining the strength of the AF layer. Next, we will evaluate numerically the resulting hysteresis loops and azimuthal dependence of the exchange bias within the SG model. When the K eff SG parameter is zero, the system behaves ideally as described by the M&B model Sec.1.5, namely the coercive field is zero and the exchange bias is finite. In the other case, when the interface is disordered we relate the SG effective anisotropy to the available interfacial coupling energy as follows: Keff = (1 f) J eb J eff eb = f J eb (1.45) where J eb is the total exchange energy of an ideal system which shows no additional coercivity. With this notations the system of equations Eqs simplifies to: { h sin(θ β) + (1 f) sin(2 (β γ)) + f sin(β α) = 0 R sin(2 α) f sin(β α) = 0 (1.46) J eff eb where, h = H/ µ 0 M F t F is the reduced field. The system of equations above can be easily solved numerically, but it cannot deliver a simple analytical expression for exchange bias. Numerical evaluation provides the α and β angles as a function of the applied magnetic field H. The reduced longitudinal component of magnetization is m = cos(β) and the transverse component is m = sin(β). It is exactly what can be measured by MOKE techniques. Note that the anisotropic magnetoresistance (AMR) and PNR hides the chirality of the ferromagnetic spin rotation as they provide sin 2 (β) information whereas MOKE does not hide the chirality as it provides sin(β) information. With this assumption the absolute value of the exchange bias field is reduced, direct proportionally to f, as compared with the M&B model(see Fig 1.27). The parameter f can be called conversion factor. It describes the conversion of interfacial energy in coercivity. For example, in the M&B phase diagram in the region II and III corresponding to reduced R-ratio, the exchange bias field is zero and the coercive field is enhanced as a result of such a conversion of the interfacial energy into coercive field. This concept can also be introduced in the Mauri equations. In the limit of strong R and M-ratios (R, M 1) the Mauri and M&B models give similar results. The differences appear for the R and M-ratios which are close to but higher than one. This region (0 < R, M 5) can be experimentally explored in order to decide in favor of one model. In order to distinguish between the Mauri and M&B model the azimuthal dependence of the exchange bias offers an excellent tool because it is visibly different for the two models. -51-

60 Spin-Glass model Hysteresis loops as a function of the conversion factor f If Fig we have simulated several hysteresis loops as a function of the conversion factor f. We have assumed a strong antiferromagnet in contact with a ferromagnet, where the interface has different degrees of disorder depicted in the right column of Fig For the R ratio we have taken the following value: R = K AF t AF fj eb = 62.5/f which corresponds to a 100 Å thick CoO antiferromagnetic layer. The field cooling direction and the measuring field direction are parallel to the anisotropy axis of the AF. The anisotropy of the ferromagnet is neglected in the evaluations below. For the interface we have chosen a SG anisotropy oriented 10 degrees away from the unidirectional anisotropy orientation (γ = 10 ). On the abscissa the reduced exchange bias field h = H/ Heb is plotted and it was chosen so to easily compare the reduction of the exchange bias in respect to the value given by the M&B model. With this assumption the system of equations Eqs was solved numerically. Figure 1.25: Longitudinal (m = cos(β)) and transverse (m = sin(β)) components of the magnetization for a F(K F = 0)/AF(R = 62.5/f, γ = 10 ) bilayer for different values of the conversion factor f (f = 80 %, 60 %, 20 %). We observe, that when the AF layer is strong (R 1) the hysteresis loops are symmetric when measured along the field cooling direction. The hysteresis loops are simulated by solving numerically the Eqs In the right column is schematically depicted the layer structure, here the emphasis is given to the disorder state at the interface. -52-

61 Spin-Glass model Fundamentals of Exchange Bias The left column shows the longitudinal component of the magnetization (parallel to the measuring field direction) (m = cos(β)) whereas the middle column shows the transverse component of the magnetization (m = sin(β)). They can both be measured via MOKE. We observe that as a function of the conversion factor f the exchange bias can be reduced linearly. The reduction of the EB field is accompanied by an increased coercivity. The shape of the hysteresis loop is close to the results found in literature, For instance the hysteresis loop with f = 60% is similar to the data shown in Ref [69, 24]. The hysteresis loop with f = 60% is similar to the data shown in Ref.[10]. The longitudinal and transverse components of the magnetization show that the reversal mechanism is symmetric. The symmetry is directly related to the strength of the AF layer, when no training effect is involved. For the examples depicted in Fig. 1.25, the R-ratio is much larger than 1 (R 1), and therefore the hysteresis loops are symmetric when measured along the field cooling direction. The asymmetry of the hysteresis loops for R > 1 is discussed further below Asymmetry of the hysteresis loop A curios characteristic of the exchange bias systems is that the two branches of the hysteresis loop are different: the descending part is steeper and the ascending one is more rounded. Such asymmetry is observed in exchange biased bilayers with thin antiferromagnetic layers or for systems containing low anisotropy antiferromagnets. We call these antiferromagnets weak antiferromagnets and characterize them by the R-ratio. When the R-ratio is slightly higher that one (weak AF layers), then the asymmetry of the hysteresis loop can be reproduced within the SG model. When the R-ratio is much higher than one (strong AF layers), then the hysteresis loops are symmetric as show in Fig We consider the following example where it is essential that the AF layer is weak but higher than 1 (R > 1): R = 1.1, f = 60% and γ = 5. For these values the minimization of the free energy is evaluated numerically. The results are plotted in Fig In the left part, the longitudinal component of the magnetization vector is shown as a function of the h = H/ Heb parameter. We clearly distinguish that the hysteresis loop is asymmetric. The asymmetry is that the descending leg of the hysteresis loop is steeper than the ascending part. The reason of this asymmetry is that the AF layer is weak, therefore the rotation of the AF spins reach high values. Due to the unidirectional property of the torque acting on the AF spins, the evolution of the α angle is asymmetric during the magnetization reversal. For instance, the irreversible switching of the magnetization for the ascending and descending legs of the hysteresis loops occur at different values of the β angle. The asymmetry of the hysteresis loops is a hallmark property of the exchange bias systems. It is a direct evidence for the unidirectional nature of the interaction at the F/AF interface. The asymmetry is also seen in the transverse component (see Fig. 1.26right) of the magnetization, where the maximum value of the transverse component of the magnetization vector is higher for the descending part then the maximum value of the ascending curve. Note that for the asymmetry described in this section, the reversal mechanism is a coherent rotation for both hysteresis legs and no training effect is involved. -53-

62 Spin-Glass model Figure 1.26: Longitudinal (left) and transverse components (right) of the magnetization vector for an week antiferromagnet: R = 1.1. The hysteresis loop is asymmetric: the descending part is steeper than the ascending part. The asymmetry is clearly seen also in the transverse component of the magnetization. We mention that this asymmetry described above is different from the effects observed in Ref. [70, 67, 71, 68, 72]. There, one side of the hysteresis loop reverses by rotation while the other side reverses by domain wall movement. In Ref. [67, 68] the asymmetry of the hysteresis loops is due to training effects. The asymmetry observed in Ref. [70] is related to the twined state of the AF layer, and no training effect was observed. Strong asymmetric hysteresis loops are reported in Ref. [73, 74]. The last examples appear to be in close relation to the asymmetry of the hysteresis loops shown in Fig Nevertheless, what appears to be a common feature o all EB system showing asymmetry of the magnetization reversal and no training effect, is that their R-ratios are small (R 1), therefore the roots of the effect are given by the properties of the AF layer. When R-ratio of the AF layer is close to, but higher than unity, then the hysteresis loops are asymmetric, irrespective of the reversal mechanism (domain wall movement or coherent rotation) Phase diagram of exchange bias and coercive field within the spin glass model In this section phase diagrams of the exchange bias and coercive field as a function of the conversion factor f are discussed. The parameters are plotted in Fig They are the reduced exchange bias field and the reduced coercive field as a function of R-ratio for several values of the conversion factor f. This allow us to compare directly the behavior of exchange bias field within the SG model to the predictions of the model of Meiklejohn -54-

63 Spin-Glass model Fundamentals of Exchange Bias and Bean. The reduced exchange bias field plotted in the Fig. 1.27left is: h eb = H eb, J eff eb µ 0 M F t F where the H eb is the absolute value of the exchange bias within the SG model and denominator term J eb µ 0 M F t f is the exchange bias field within the ideal M&B model. Figure 1.27: Left: The dependence of the reduced exchange bias field (h eb = H eb / J eff eb / µ t F M F ) and the reduced coercive field (h c = H c / J eff eb / µ t F M F ) as a function of the R-ratio (R = K AF t AF / (fj eb ). There are 4 curves corresponding to different values of the conversion factor f. The curve for f = 98 % can be easily compared to the M&B model phase diagram shown in Fig The observation is that while the exchange bias field decreases with increasing conversion factor, the coercive field has an opposite behavior, it increases with increasing f. Close to the critical value of R = 1, the exchange bias field shows opposite behavior as compared to M&B model. The reduced coercive field shown in the Fig. 1.27right is: h c = H c, J eff eb µ 0 M F t F where H c is the absolute value of the coercive field within the SG model. It has no relation to a similar coercive field of M&B model because the coercive field of the the M&B model is zero when the exchange bias is finite. -55-

64 Spin-Glass model The abscissa of the both graphs is the R-ratio which is reads: R = K AF t AF J eff eb = K AF t AF f J eb, where f is the conversion factor and J eb is the exchange energy for an ideal exchange biased bilayer. By an ideal exchange bias bilayer we understand the system for which there is no additional coercivity related to the interface or to the AF layer Dependence of exchange bias field on the thickness of the antiferromagnetic layer In Fig top the exchange bias field is plotted as a function of the R-ratio. The curves are shown in the phase diagram figure (Fig. 1.8), and they are normalized as to highlight the qualitative differences between them close to the critical value of the R-ratio. Bearing in mind that R is directly proportional to the thickness of the AF layer, the qualitative dependence of the EB field on the t AF, which is plotted in Fig. 1.28bottom has the following behavior: when f has high values close to unity, the EB field decreases with decreasing AF thickness. However, when f is reduced, a completely different behavior of the EB is observed. The EB field increases as the thickness of the AF decreases. This is one of the most important feature of the SG model because such dependence is observed experimentally [75, 52]. So far, only the DS microscopic model model [45] can provide the qualitative dependence of the EB on the t AF observed experimentally. It is remarkable that within the SG model this thickness dependence can be reproduced. A specific feature of the SG model is the critical thickness of the AF layer for which the EB can exist. Within the M&B and SG models the EB is non-vanishing when the R-ratio is higher or equal than unity: R 1. With this condition the critical thickness of the antiferromagnet reads: t AF,cr = f J eb K AF (1.47) As the conversion factor is always smaller than unity it follows that the critical thickness of the AF layer is smaller than the value predicted by the M&B model. The reduction factor is simply the conversion factor f: t SG AF,cr = f t MB AF,cr, (1.48) where the t SG AF,cr is the AF critical thickness predicted by the SG model while the MB t cr AF is the AF critical thickness predicted by the M&B model. An example of such reduced critical thickness can be found in the Ref. [76] The blocking temperature for exchange bias It is observed experimentally that the temperature where the exchange bias sets in is lower then the Néel temperature of the AF layer (T N ). This temperature is called the blocking -56-

65 Spin-Glass model Fundamentals of Exchange Bias Figure 1.28: Top: The normalized exchange bias h eb /f = H eb /Heb as a function of R-ratio for different values of the conversion factor f. An asymmetric peak like behavior of the exchange bias develops for high values of the conversion factor. Bottom: The normalized exchange bias h eb /f = H eb /Heb as a function of the thickness of the AF layer. The abscissa is normalized to the critical thickness as given by the M&B model. This helps to see relative differences between the SG and M&B model. temperature (T B ). Experimentally it is observed that: for thick AF layers the blocking temperature is very close but smaller than the Néel temperature of the AF layer. b) for thin AF layers the blocking temperature could be much lower than the T N ; c) the coercive field increases starting below T N in contrast with the EB field, which appears only below T B. These three feature are qualitatively explained within the SG model (and M&B model). In order to have a non-vanishing EB field, the following condition (R 1) must be fulfilled: K AF > fj eb t AF = K AF,cr, (1.49) -57-

66 Spin-Glass model where K AF,cr is the critical AF anisotropy for the onset of the EB field. The condition above sets a critical value for the AF anisotropy for which the EB can exist. When the thickness of the AF layer is very large, then the exchange bias can appear for blocking temperature which are almost equal but smaller then the Néel temperature. If however, the thickness of the AF layer is small, then the critical anisotropy of the AF could become large, resulting in a low blocking temperature. It is clear from the phase diagrams in Fig and Fig that there is a region of anisotropy 0 < K AF < KAF c where the EB is zero and coercive field is enhanced. It follows that the enhancement of the coercive field should be observed above the blocking temperature and below the Néel temperature of the AF. This situation is being observed experimentally. For the case of CoO(25 Å)/Co layers shown in this thesis the coercive field increases from the TN CoO = 291 K and the exchange bias field appears at T B = 180 K. Further possible causes for a reduced blocking temperature and for the behavior of the EB and coercive fields as a function of temperature are discussed elsewhere: finite size effects[77], stoichiometry [78] or multiple phases [79], AF grains [80] and diluted AF [48] Azimuthal dependence of exchange bias and coercive field within the spin glass model. In this section, the azimuthal dependence of exchange bias and the coercive fields within the SG model is discussed. The hysteresis loops are delivered by numerical minimization of Eqs.1.46, The parameters used in the model are :f = 80%, R = 62.2/f, γ = 20. It is supposed that the AF layer has an uniaxial anisotropy. The MCA anisotropy of the ferromagnetic layer is neglected in the model (K F = 0). Therefore, the coercivity which appears in the simulations below is not related to the F properties, but to the interfacial properties of F/AF bilayer. In order to suppress asymmetries (see Fig. 1.26) related to the reduced strength of the AF layer, a high R-ratio (R 1) is used in the model. In Fig the normalized coercive fields H c1, H c2 and H c and H eb are shown as a function of the azimuthal angle theta. They are extracted from simulated hysteresis loops (few of them are plotted in Fig. 1.29). First we discuss the hysteresis loops shown in Fig The system is cooled down in a field oriented parallel with the AF anisotropy direction. Then, hysteresis loops are simulated for different orientations of the applied field (θ) in respect to the field cooling orientation. In Fig longitudinal (m ) and transverse (m ) representative hysteresis are shown. At θ = 0, the magnetization curves are symmetric and shifted at negative fields. At θ = 180, the magnetization curves are also symmetric but shifted to positive fields. The transverse hysteresis at θ = 0 and θ = 180 are the same (which is not the case when the R-ratio is small, see Fig ). At θ = 3, however, the longitudinal hysteresis loop becomes asymmetric. The first reversal at H c1 is sharp and the reversal at H c2 is more rounded. This asymmetry is also seen in the transverse component of the magnetization. The F spins rotate asymmetrically: the values of the β angle depend on the external field scan direction, being different for swaps from negative to positive saturation as compared with swaps from positive to negative saturation. As the azimuthal angle increases, the coercive -58-

67 Spin-Glass model Fundamentals of Exchange Bias Figure 1.29: Simulated hysteresis loops for different azimuthal angles. The curves are delivered by the Eq.1.46 with the following parameters: f = 80%, R = 62.2/f, γ = 20. field becomes zero. For instance, at θ = 20, 90 and 160 there is almost no coercivity. Also, the transverse component of the magnetization shows that the F spins do not follow a 360 path, but they rotate within the 180 angular space. These asymmetries of the hysteresis loops in Fig are different from the asymmetry described in section Here the asymmetry is due to the azimuthal orientation of the external field with respect to the uniaxial anisotropy of the AF layer, whereas in section the reduced strength of the AF layer is the cause of the asymmetric magnetization reversal. Second we discuss the coercive field dependence as a function of the azimuthal angle, shown in Fig The coercive fields H c1 and H c2 are extracted from the simulated hysteresis loops shown in Fig We see in Fig. 1.30left that the coercive fields H c1 and H c2 follow closely the unidirectional behavior, as given by the M&B model. Additionally, we observe that close to θ = 0 and θ = 180, the coercive fields deviate one from each other, displaying different qualitative modulations. -59-

68 Spin-Glass model Figure 1.30: Azimuthal dependence of exchange bias as a function of the θ angle. The curves are delivered by the Eq.1.44 with the following parameters: f = 80%, R = 62.2/f, γ = 20. In Fig. 1.30right the coercive field and the exchange bias field are assembled from H c1 and H c2 using Eq We distinguish the following characteristics of the H c and H eb : the unidirectional behavior ( cos(θ)) of the H eb as a function of the azimuthal angle is (see Fig. 1.10) clearly visible; additionally, the behavior of the H eb as a function of the azimuthal angle shows sharp modulations close to the orientation of the AF uniaxial anisotropy; the coercive field H c has a peak-like behavior close to the orientation of the AF uniaxial anisotropy, at θ = 0 and θ = 180. Experimentally the azimuthal dependence of the exchange bias field was first explored for NiFe/CoO bilayers [81]. It was suggested that the experimental results can be better described with a cosine series expansions, with odd and even terms for H eb and H c, respectively, rather than being a simple sinusoidal function as initially suggested by Meiklejohn and Bean [5, 6]. The simulations shown in this section are different with respect to the previous reports on the angular dependence of exchange bias field[81, 82, 83, 84, 85]. One difference is that the MCA anisotropy of the F is supposed to be negligible when compared with the coercive fields obtained experimentally, and the sharp features of the H eb are reproduced numerically rather then being described by cosine series expansions. In order to resolve such sharp features of the exchange bias field, a large number of hysteresis loops as a function of the azimuthal angles should be recorded. These experiments are shown in section Dependence of the exchange bias field on lateral size of the AF domains The relation between the exchange bias and reduced size effects due nano-structuring of the AF-F systems is important from both fundamental and technological point of view. From a fundamental point of view, the reduced lateral size of both F and AF objects induces -60-

69 Spin-Glass model Fundamentals of Exchange Bias significant changes of the exchange bias field, coercive fields and also the asymmetry of the hysteresis loops [86, 87, 73, 88, 89, 90, 91, 92, 93, 94, 95, 96]. In the following we consider an AF dot in contact with uniform F layer. Due to nanostructuring it is natural to expect that at the edges of the dot there are AF spins with reduced anisotropy. These spins will contribute to the coercivity at the expense of the ef f interfacial exchange energy. The effective interfacial exchange energy is: Jeb = f Jeb. It is easy to estimate the f -parameter, as the fraction of the outer shell area divided by the total dot area: f A1/A2 = (π(d d)2 )/(πd2 ) = 1 2d/D + d2 /D2, where the d is the lateral thickness of the outer shell and D is the aria of the dot itself. Assuming that d << D and that d is constant we obtain the general expression: 1, (1.50) D where D is the average lateral diameter of nano-structured AF objects or the lateral size of AF domains or grains in uniform AF film. Notice that deviations from 1/tF law is expected when the film becomes discontinues[8] due to the additional (D d)2 /D2 contributions. Heb f Figure 1.31: Left: Bias field distribution as a function of the inverse AF domain diameter. The line is a linear fit to the data[93].right: The dependence of bias field as a function of inverse grain size. The graph is a re-plot of Fig.1d of Ref. [94]. The line is a linear fit to the data. The variation of the hysteresis loops as a function of the f -parameter is shown in Fig We see that for high f corresponding to a large diameter dot or AF domain, the coercive field is small and the exchange bias is high. When, however, the lateral size of a AF dot or AF domain becomes smaller, the f ratio decreases leading to an increased coercive field and reduced exchange bias field. Transition from the top hysteresis to the bottom hysteresis of Fig are usually observed due to nano-structuring of the AF layers. Experimental observation of the linear dependence on the lateral diameter of the AF domains in uniform film was observed by x-ray microscopy [93](see Fig. 1.31left). The exchange bias dependence on the lateral size of AF grains can be deduced from the Fig.1d -61-

70 Spin-Glass model of Ref. [94](see Fig. 1.31right). By re-plotting the H eb as a function of the inverse grain diameter one obtains a linear dependence. The interpretation given in this section for the linear dependence of exchange bias field as a function of the inverse diameter of the AF dots or magnetic domains is close to the interpretation given by Scholl et. al[93]. There, the large ratio of uncompensated to compensated spins leads to a widening of the bias field distribution towards smaller fields. Here, we assume that the coercivity is given by low AF anisotropy spins whereas increased exchange bias field is given by a surplus of frozen-in uncompensated spins located at magnetic (domain walls) or structural defects (dot edges, grains, steps). The distinction between compensated spins and low anisotropy spins is suggested by the observation of AF hysteresis loops using element specific soft-xray magnetic scattering. A compensated spin, as well as a non-compensated spin would not rotate in phase with the F spins, whereas the low anisotropy AF spins would do it. Such microscopically complicated situation, can, surprisingly, be accounted for by a linear dependence of the EB field as a function of the lateral object size shown in Fig Interface disorder and the training effect The training effect refers to the dramatic change of the hysteresis loop when sweeping consecutively the applied magnetic field of a system which is in a biased state. The coercive fields and the resulting exchange bias field versus n, where n is the n th measured hysteresis loop, displays a monotonic dependence. The absolute value of H c1 and of the EB field decreases from an initial value at n = 1 to an equilibrium value at n =. The The absolute value of the coercive field H c2, however, display an opposite behavior, it increases with n. These features of the training effect was named as Type I by Zhang et. al.[97]. The other case when both H c1 and H c2 decrease is called Type. II. In this section we deal only with the so-called Type I training effect. Several mechanisms were suggested as a possible cause of the effect. While it is widely accepted that the training effect is related to the unstable state of the AF layer prepared by field cooling procedure, it is not yet well established what mechanisms are contributing to the effect. Néel [37] discussed the training effect as a tilting of the superficial magnetization of the AF domains. This would lead to a Type I training effect. Néel also discussed that a creeping effect could lead to a Type II training effect. Micromagnetic simulations within the DS model [32, 45] shows that the hysteresis curve is not closed after a complete loop. The lost magnetization is directly related to a partial loss of the superficial magnetization of the AF domains, which further leads to a decreased exchange bias. Zhang et. al.[98] suggested that the training effect can by explained by incorporating into the Fulcomer and Charap s model [99] positive and negative exchange coupling between the grains constituting the AF layers. In Ref. [100], the authors found direct evidence for the proportionality between the exchange bias and the total saturation moment of the heterostructure. The findings were -62-

71 Spin-Glass model Fundamentals of Exchange Bias related to the prediction of the phenomenological Meiklejohn Bean approach, where a linear dependence of the exchange bias on the AF interface magnetization is expected. Binek [101] suggested that the phenomenological origin of the training effect is a deviation of the AF interface magnetization from its equilibrium orientation. Analytical calculations in the framework of non-equilibrium thermodynamics leaded to a recursive relation accounting for the dependence of the H eb field on n. Hoffmann [102] suggested that only biaxial AF symmetry can lead to training effects, reproducing important features of the experimental data, while simulation with uniaxial AF symmetry show no difference between the first and second hysteresis loops. Figure 1.32: Simulations of training effect within the SG model. Experiments performed by PNR [67, 10] support the irreversible changes taking place in the AF layer. It has been observed that after the very first reversal at H c1, interfacial magnetic domains are formed and they do not disappear even in positive or negative saturation. The interfacial domains serve as seeds for the subsequent magnetization reversals. These ferromagnetic domains at the interface have to be intimately related to the AF domain state [103]. The experiments suggest that irreversible changes of the AF domain state is responsible for the training effect. Furthermore, perpendicular to the sample plane spins were observed by PNR [68] hinting at the existence of the perpendicular domain walls in the AF layer, as originally suggested by Malozemoff. Therefore, irreversible changes of the AF magnetic domains and of the interfacial domains during the hysteresis loops play an important role for the training effect. In the following we analyze only the interface contribution to the training effect. We assume that a gradual increase of the interfacial disorder of the F/AF system leads to a training effect. Within the SG model, the magnetic state of the F/AF interface is accounted for by a unidirectional induced anisotropy K eff, which is allowed to have an average direction γ, where γ is related to the spin disorder of the interface. -63-

72 Spin-Glass model In Fig we show first and second hysteresis loops calculated with the help of Eq In these calculations we consider that the AF is strong, R = 62.5/f. For the conversion factor we take a value of f = 60%. Following closely the experimental observations[67, 10], before the first reversal γ is zero, while just after the first reversal, γ increases towards an equilibrium value. The first branch of the hysteresis loop appears rather sharp, therefore we assume that the AF spins and F spins are collinear right after cooling down in a field (γ = 0). For the second branch of the 1st hysteresis loop we consider that γ = 10, therefore the second leg appears rather rounded. This is in accordance with the observation that for thin CoO layers [10] the disordered interface appears first during the first reversal at H c1. Now, the first branch of the second hysteresis loop is simulated with γ = 10. At the third reversal, we again assume that the disorder of the interface increases. Therefore, the second branch of the hysteresis loop is simulated assuming a new value of γ = 20. The simulations above implies a viscosity-like behavior to the disordered interface. For example, when the applied field is opposite to the unidirectional anisotropy, the torque created on the K eff spins will drag them away from the initial direction set by the field cooling. Reversing the field back to positive directions the K eff spins will not follow (completely), they remain in this position (viscosity). This is because the maximum torque exerted by the F spins was already acting at negative fields, while for positive fields it is much reduced. When measuring again the hysteresis loop, at negative coercive field, the K eff spins will rotate even further and so on. Therefore, after n hysteresis loops, the angle of the K eff anisotropy will have a certain value, which will cause a decreased exchange bias field. Comparing the curves shown in Fig. 1.32a) with the experimental curves[63, 98, 97, 21, 24, 67, 23, 100] we observe a striking qualitative agreement. One of the similarities is the asymmetric characteristic of the hysteresis loops. The very first reversal is sharp, while the second reversal is more round. The second hysteresis loop also has similar characteristics seen experimentally. In Fig. 1.32b) is plotted the transverse components of the magnetization for the first and second loop. A quick inspection of the experimental data shown in Ref. [67, 11, 72, 104] already shows an excellent qualitative agreement. The transverse component exhibits more clearly that the saturation moment of system appears to be reduced. This reduction of the saturation moment [100] is due to (within SG model) deviation of the interfacial spins from their original cooling, direction caused by a gradual increase of the interfacial disorder. Note that the asymmetric curve of Fig is fundamentally different from the one discussed above. Here the asymmetry is related to the training effect and is characteristic also for strong antiferromagnets (R 1), whereas the asymmetry in Fig is not related to training effects and it is specific only for weak antiferromagnets (R > 1). -64-

73 Training effect Fundamentals of Exchange Bias 1.12 Empirical expression for the training effect The very first empirical expression for training [105, 106] effect suggested a power law dependence of the coercive fields and exchange bias as a function of cycle index n: H n eb = H eb + k n, (1.51) where k is an experimental constant. This expression follows well the experimental dependence of EB field for n 2, but when the very first point is included to the fit, then the agreement becomes poor. Figure 1.33: Exchange bias as function of the loop index n. The blue line is the best fit to the data using Eq The black line is the best fit to the data using Eq More recently, Binek [101] has shown that using a recursive relation, the evolution of the EB field as a function of n, can be well reproduced for all cycle indexes (n 1). The recursive expression reads: H n+1 eb H n eb = Γ (H n eb H eb ) 3, (1.52) where Γ is a physical parameter which, for n 1, was directly related to the k parameter of Eq It was shown that a satisfactory agreement between the Eq.1.51 and Eq.1.52 is achieved for n 3, therefore the approach of Binek appears to provide the phenomenological origin of the hitherto unexplained power-law decay of the EB field with increasing loop index n >

74 Training effect Figure 1.34: Temperature dependence of the training effect. The lines are the best fit to the data using Eq Table 1.4: The parameters for the training effect shown in Fig as a function of temperature. The Eq was used for fits to the data. T [K] Heb 0 [Oe] H eb [Oe] A f [Oe] P f A i [Oe] P i ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 0.11 In the following we analyze another type of expression which reproduces the dependence of the coercive field and exchange bias field as a function of the loop index n. It is based on the simulations shown in the previous section. There, an additional origin of the training affect is identified as being related to the interfacial spin disorder. Practically, the evolution of the interfacial disorder during the measurements of the hysteresis loops causes a decrease of the exchange bias field. The other effect related to the AF domain dynamics affects also the magnitude of the coercive fields and exchange bias field. Both contributions affect the gradual decrease of exchange bias as a function of n. They can be treated probabilistically. -66-

75 Training effect Fundamentals of Exchange Bias We propose the following relation to simulate the decrease of the EB as a function of n: H n eb = H eb + A f exp( n/p f ) + A i exp( n/p i ), (1.53) where, H n eb is the exchange bias of the nth hysteresis loop, A f and P f are parameters related to the change of the frozen spins, A i and P i are parameters related to the evolution of the interfacial disorder. The A parameters have dimension of Oersted while the P parameters have no dimension but they are similar to a relaxation time, where the continuous variable time is replaced by a discrete variable n. We expect that the interfacial contribution sharply decreases with n as the anisotropy of the interfacial spins is reduced (low AF anisotropy spins), while the contribution from the frozen AF spins belonging to the AF domains ( frozen-in uncompensated spins ) appear as a long decreasing tail as they are intimately confined in to a much stiffer environment. In the following we show fits to the trained exchange bias field. In Fig exchange bias field thirteen consecutive hysteresis loops recorded at T = 10 K are plotted as a function of loop index. The sample is a polycrystalline CoO(40 Å)/Fe(150 Å)/Al 2O 3 bilayer grown by RF sputtering techniques as described in Sec The exchange bias field is plotted as a function of cycle index n. Two fits are shown: one using the empirical relation Eq and another one using Eq We observe that the fit using the Eq (Heb = Oe, k = 44 Oe) follows well the experimental curve for n 2. However, the best fit is obtained using Eq The parameters obtained from this fit are: Heb = 158 Oe, A f = Oe, P i = 4.33, A i = Oe, P i = We distinguish, indeed, a sharp contribution due to low anisotropy AF spins at the interface and a much weaker decrease from the frozen-in uncompensated spins. In the following we describe a study of the training effect as a function of temperature. The sample is a monocrystalline CoO(300 Å)/Fe(150 Å)/Al 2O 3 bilayer grown by rf-sputtering and MBE techniques as described in Sec The sample was field cooled in saturation to the measuring temperature where 31 consecutive hysteresis loops were measured. The experimental data and the fits to the data using Eq are shown in Fig The fit parameters are listed in Table 1.4. We distinguish three main characteristics related to the temperature dependence of the training effect: each curve shows two regimes, a fast changing one and a slow decreasing tail; the relaxation times (P i and P f ) do not visibly depend on the temperature; the interface transition towards the equilibrium state is approximatively ten times faster then the transition of the frozen spins towards their stable configuration. -67-

76 Spin-Glass model: IrMn/CoFe 1.13 Experiments and Simulations for IrMn/CoFe EB system The first experimental study of the angular dependence of the H c and H eb was performed on a NiFe/CoO bilayer [81] where it was suggested that it can be best described with a cosine series expansions, with odd and even terms for EB and coercive field, respectively. In an another approach, Mewes. et all [83, 107] showed that the azimuthal dependence of the H c and H eb can be well described within the Stoner-Wohlfarth model. The coercivity was accounted for by including higher order anisotropy contributions. A recent study on IrMn/CoO bilayers[108] showed that even so a good agreement between the data and the simulations based on the cosine functions can be achieved, still some disagreement exists. In this section I show longitudinal and transverse magnetization curves from IrMn/CoO exchange bias bilayers and simulations based on the Spin-Glass model for exchange bias. An exchange-biased bilayer Si/SiO2/Cu(30 nm)/ir 17 Mn 83 (15 nm)/co 70 F e 30 (30 nm)/t a(5 nm) was prepared by magnetron sputtering with a base pressure below Torr at an Ar pressure of Torr. The initial EB direction is set through an annealing step for 1 h at 548 K in a magnetic in-plane field of H ann = 1 koe after film deposition [34, 109].The sample was prepared as reference sample for other research [110], but we find it helpful in testing the SG model. The sample was measured using the newly commissioned vector-moke [111] setup in Bochum. A number of 360 pairs of longitudinal and transverse components of the magnetization were measured for a external field orientation with respect to the field cooling direction ranging from 0 to 360 and at room temperature. In this geometry, the sample was kept fixed during the measurements, whereas the orientation of the applied external field was varied. Characteristic longitudinal and transverse magnetization curves are shown in Fig In order to observe fine modulation of the EB field, the one degree step variation of the azimuthal angle is required. The hysteresis loops provides the coercive fields H c1 and H c2 shown in Fig left which further provides the coercive field (H c = ( H c1 + H c2 )/2) and the EB field (H eb = (H c1 + H c2 )/2) shown in Fig right. We discuss first the experimental observations denoted as open symbols in Fig. 1.35, Fig a), and Fig c). In Fig some representative hysteresis loops and the transverse components of the magnetization are shown. A distinct feature of the system is that the remagnetization process occurs via coherent rotation as seen from the nonvanishing transverse loops, being completely reproduced by the numerical simulations (solid lines), which will be discussed in the next section. The maximum exchange bias field is H eb = 90 Oe and it is achieved at θ = 30 (see Fig c)), where θ is the azimuthal angle with respect to the field cooling direction defined as θ = 0. This is already a striking feature, as usually the maximum of exchange bias is believed to be achieved along the field cooling direction. The longitudinal hysteresis loop at θ = 0 is completely symmetric. This is also seen in the transverse magnetization, where the amplitudes of forward and revers components are equal in absolute value, but double mirrored with respect to m = 0 and H = 0, -68-

77 Spin-Glass model: IrMn/CoFe Fundamentals of Exchange Bias Figure 1.35: Experimental (open circles)and simulated hysteresis loops (black lines) for different azimuthal angles. The simulated curves are delivered by the Eq and 1.44 with the following parameters: f = 80%, R = 5.9/f, γ = 20. where m is the perpendicular magnetization with respect to the direction of the external field H. This is not the case few degrees forward at θ = 3 where both longitudinal and perpendicular loops are asymmetric. The first branch of the hysteresis loop is steep while the reverse branch is more rounded. It is remarkable that within an azimuthal angle of 3 such strong asymmetry of the hysteresis loop is achieved. This asymmetry is different from the asymmetry observed due to the training effect [67, 71]. There the first reversal occurs via domain wall movement [10], whereas the second reversal, on the way to positive -69-

78 Spin-Glass model: IrMn/CoFe Figure 1.36: a) Azimuthal dependence of the coercive fields H c1 (filled symbol) and H c2 (open symbols) extracted from the experimental hysteresis loops. At θ = 0, 180 which corresponds to the field cooling direction the coercive fields deviates one from each other within a 20 angular range; c): The experimental coercive field and exchange bias field as a function of the azimuthal angle θ. The field cooling orientation corresponds to θ = 0. The lines in the panel b) and d) are simulated coercive fields and exchange bias field as a function of the azimuthal angle. The curves are delivered by the Eq and 1.44 with the following parameters: f = 80%, R = 5.9/f, γ = 20 saturation is occurring mostly via coherent rotation. Here no training effect is observed during the remagnetization process. As the azimuthal angle is further increased, the coercivity disappears at about θ = 20 and reappears again, in a symmetrical manner, close to the opposite direction of the external field orientation with respect to the field cooling direction. The vanishing coercive field can be understood from the transverse loops where it is clearly seen that the rotation of the ferromagnetic spins do not span over the complete 360 available angular space, but almost reversibly rotate within the 180 angular range. Therefore the angle of the magnetization orientation from which the coercive fields are extracted take the same value for both H c1 and H c2. In Fig a) the coercive fields extracted from the 360 longitudinal hysteresis loops are -70-

79 Spin-Glass model: IrMn/CoFe Fundamentals of Exchange Bias plotted versus the azimuthal angle θ. We observe that, globally, they behave according to unidirectional behavior [5], but strikingly fine deviations occurs. They appear as peak-like features close to the field cooling direction. While the coercive fields H c1 and H c2 are behaving similarly for a wide range of angles, this is not the case for the direction of the external field close to the field cooling direction. Here they deviate one from each other, leaving behind an open space which appears as coercive field in Fig c). The coercive field H c of the system shows a peak like behavior as seen in Fig c). Its maximum value (Hc MAX = 20 Oe) is about four times lower than the maximum value of the exchange bias field. Finite values are observed within a 20 degree range centered at θ = 0 and θ = 180 and almost vanishes outside this range. The exchange bias field dependence on the azimuthal angle (solid symbols in Fig c)) shows peculiar sharp but low amplitude modulations. They affect strongly the magnetization reversal as seen in Fig While the unidirectional behavior is clearly visible, there are fine features over-imposed on a overall sinusoidal curve. They appear close to the field direction of the unidirectional anisotropy and cannot be reproduced with the empirical description based cosine series expansion as suggested in Ref. [81]. In order to reproduce the experimental results we need a more realistic model, which is the SG model described before. In Fig longitudinal and transverse components of the magnetization are plotted together with the experimental data. In Fig b) and d) the azimuthal dependence of the coercive fields and exchange bias field are plotted. The simulated curves are in excellent agreement with the experimental curves. It is remarkable, that the azimuthal dependence of exchange bias field and the coercive field observed experimentally are completely reproduced by the SG model. The parameters used for the simulations are: f = 20%, γ = 20 and R = 5.9/f. For calculating the value of the R-ratio we used the anisotropy constant (K AF ) measured in Ref. [84]. R is calculated as: H eb = 90Oe, t AF = 150 A, K AF = 10 4 J/m 2 R = K AF t AF = 104 J/m m = 5.9/f f J/m 2 where: fj eb J eb = H eb /µ M F t F = J/m 2 The conversion factor is related to the magnitude of the coercive field with respect to the shift of the loop. The γ angle plays an important role. For instance, when γ is zero, the coercive field are much enhanced as compared to the experimental and the coercive peak width is reduced. Summarizing, we have studied experimentally the azimuthal dependence of exchange bias field and coercive field along with the longitudinal and transverse components of the magnetization. Striking fine features and complex behavior of the remagnetization process were observed and simulated numerically by using the Spin Glass model. Not only the magnetization reversal mechanism is well accounted by the SG model, but also the coercive field and exchange bias field as a function of the azimuthal angle follows closely the experimental curves. In the following we present the Polarized Neutron Reflectometry method and some of the experimental results obtained by studying CoO/(Co, Fe) bilayers and multilayers. The first -71-

80 Spin-Glass model: IrMn/CoFe part of the thesis is a synthesis based fully on the experimental results presented further on. -72-

81 Part II Polarized Neutron Reflectometry 73

82

83 Polarized Neutron Reflectometry PNR has proven in the past to be an essential technique for the analysis of magnetic thin films and heterostructures. The neutron reflectivity technique goes back to 1946 when it was first used by Fermi and Zinn for determining the coherent neutron-nuclei scattering lengths [112]. The first polarized neutron experiments were carried out by Hughes and Burgy[113] in 1950, when a neutron reflection apparatus together with a polarizer and analyzer was used to produce a polarized neutron beam. It was concluded that beams of completely polarized neutrons, which can now be produced with high intensity (of the order of 10 5 per minute), will be useful in the study of magnetic and nuclear properties. PNR can be divided in two categories, one is specular PNR, and corresponds to reflection experiments for which the incident neutrons are specularly reflected from magnetic heterostructures, and the other one is off-specular PNR, and corresponds to reflection measurements for which the emerging neutrons are scattered away from the specular beam. While specular PNR is widely recognized as a powerful tool for the investigation of depth magnetization profiles in magnetic heterostructures, the off-specular PNR yields information about the in-plane magnetization inhomogeneities like magnetic domains, domain walls and magnetic roughness. In this part some fundamental aspects of both, specular and off-specular PNR are described. -75-

84 -76-

85 Chapter 2 Theory 2.1 Fundamental Properties of the Neutron Discovery of the neutron In 1920, E. Rutherford discussed the existence of the simplest neutral nucleus containing a proton and an electron that are strongly coupled to each other. Such an atom would have very novel properties. Its external field would be practically zero, except very close to the nucleus, and in consequence it should be able to move freely through matter. Its presence would probably be difficult to detect by the spectroscope, and it may be impossible to contain it in a sealed vessel [114]. The neutron was discovered by James Chadwick, who spent more than a decade searching for it. The discovery followed an experimental breakthrough made in 1930 by Bothe and Becker. They have observed that bombardment of beryllium with alpha particles from a radioactive source produced neutral radiation which was penetrating but non-ionizing. They presumed it was gamma rays. In 1932 Irène and Frédéric Joliot-Curie let this radiation hit a block of paraffin wax, and found it caused the wax to emit protons. They measured the speed of these protons and found that the gamma rays would have to be incredibly energetic to knock them from the paraffin wax. Chadwick reported the Joliot-Curie s experiment to Rutherford, who did not believe that gamma rays could account for the protons from the wax. Then, Chadwick worked day and night to prove the neutron theory, studying the beryllium radiation with an ionization counter and a cloud chamber. He found that the wax could be replaced with other light substances, even beryllium, and that protons were still produced. Within a month Chadwick had conclusive proof of the existence of the neutron. He published his findings in Nature, on February 27, 1932 [115]. 77

86 Fundamental properties of the neutron Chadwick proposed the following reaction: 9 4Be He 12 6 C n where 1 0n represents the new particle, the neutron [116] The neutron mass The neutron mass is deducted from the energy conservation of particular nuclear reactions. Chadwick had obtained the mass of neutron [116] to be u. The first accurate measurement of the neutron mass came two years later. Using the new observation of the deuteron desintegration, Goldhaber approached Chadwick and suggested the way to measure the neutron mass[117]. Now, the most precise measurement of the neutron mass [118] gives : m(n) = (82)u., where u is one atomic mass unit Magnetic moment of neutron Estermann and Stern [119] showed that the magnetic moment of neutron should be about -2 µ N, where µ N = e /2m p c is the nuclear magneton and m p is the proton mass. Tamm and Altshuler were able to make a similar suggestion by analysing the nuclear magnetic moment data. The first experiment to measure directly the magnetic moment of the neutron was performed by Alvarez and Bloch [120] using nuclear magnetic resonance. Presently, its most precise value is: µ n = ( ± ) µ N Its negative value means that the magnetic moment of the neutron is directed against to its spin. Eaven so, uptodate there is no satisfactory theory explaining the existence of a magnetic moment of a zero charged particle, the ratio between magnetic moment of the proton and neutron (µ p /µ n = 1.46) is about 3/2, which is in acordance with the quark structure of the neucleons (p=uud, n=udd) The neutron spin The neutron spin is 1/2 as demonstrated by Schwinger[121]. Thus, it is a fermion and obeys the Fermi-Dirac statistics. Its baryon number is B = 1 and has a positive intrinsic parity. Together with the proton, it forms an isotopic doublet of nucleons with isospin T = 1/2 The isospin projection of the neutron is 1/

87 Fundamental properties of the neutron Polarized Neutron Reflectometry β decay of the neutron As an effect of the weak interaction, the free neutron is not stable but decays according to the following reaction: with a life-time of about 900 s. n p + e + ν e ( 780eV ), The magnetic interaction The neutron magnetic moment interacts with homogeneous fields B, according to the magnetic potential: V m = µb (V m [ev ] = ± B [G]), where the minus sign reveals the antiparallel orientation between the spin and the magnetic field Interaction of neutrons with matter The reflectivity experiments fall in the neutron optics category, where the interaction of neutrons with layers is described by the optical potential. It includes a real part which is related to the coherent elastic scattering interaction, an imaginary part describing the absorption of neutrons (and incoherent scattering), when interacting with matter and nuclei, and the magnetic potential. Here we would like to emphasize that the origin of the interaction potential in neutron optics stems from the interaction of neutrons with nuclei. Therefore we shortly discuss the interaction of particle with the nuclei and then extend our analysis to the interaction of neutrons with layers Fermi Potential Ideally, a flux of particles approaching the target nucleus, which is modelled by plane waves (Fig. 2.1), would feel the nucleus as a hard sphere. Such interaction causes the nucleons to be scattered away according to the trivial process of elastic collisions of billiard balls. In order to describe this interaction using the quantum mechanics formalism one should propose an interaction potential and calculate the cross-sections. It was first proposed in an elegant manner by Fermi and it is called Fermi pseudopotential [122]. It reads: V (r) = ( 2π 2 ) a δ(r) (2.1) M where a is the scattering length and r is the vector position of the neutron relative to the nucleus and M is the reduced mass of the neutron-nucleus system. The value of the -79-

88 Fundamental properties of the neutron Figure 2.1: Scattering geometry. A incident neutron flux is hitting the probe under investigation. The detector measures an integrated scattered particule flux over the solid angle dω. scattering length depends on the nucleus and on the nuclear spin of the nucleus. For most of the elements it has a positive value, but for others it is negative. Using the Fermi pseudo-potential it is possible to calculate the scattering amplitude which describes the scattering probability. To do that one writes the Schrödinger equation of the neutron-nucleus system: ( 2 + [E V (r)]) Φ(r) = 0 (2.2) 2M The total wave function which satisfy the Schrodinger equation is: Φ(r) = exp(ikr) + f(θ) exp(ikr) (2.3) r One distinguishes that the total wave function is a sum of incoming plane wave function and a scattered Green wave function. The amplitude f(θ) is determined by the boundary condition at the r = R, where R is the radius of the hard sphere The Weisskopf-Feshbach model of nuclear reactions Even so the Fermi pseudo-potential allows to describe formally the scattering of particle in the Born approximation, it hides an important phenomenon of interaction, namely the absorption of particles by the target nuclei. In order to account for the absorption one needs to propose a structure of the nuclei and reconsider the interaction processes. When considering an assemble of nuclei, the imaginary part of the interaction potential V could be related to the total cross-section (σ tot ) by the optical theorem: Im(f(0)) = k 4π σ tot, where f(0) is the forward scattering amplitude. Physically, Im(f(0)) represents the attenuation of the incident beam which, in general, is proportional to the total crosssection. -80-

89 Fundamental properties of the neutron Polarized Neutron Reflectometry Figure 2.2: The different stages of nuclear reactions according to Weisskopf-Feshbach representation [123]. The interactions of nucleons with the nuclei has been a subject of extensive theoretical and experimental investigations since 1930 s. The different stages of nuclear reactions according to Weisskopf and Feshbach are shown in Fig In the initial stage (independent particles stage) the particle approaching the target nucleus would feel the nucleus, as a hard sphere. This could be described by the Fermi potential. Now, we consider that the nucleus is best described as a finite potential well. The nucleons move in a self-consistent potential and they can have only discrete energies. The resonance structure of the well can be calculated using quantum mechanics providing that the selfconsistent potential is known. Then, we consider that the incident particle enters in the nucleus and forms the so called Compound Nucleus (C. N.). One very important feature of this process is that the incident particles looses memory of its initial state (lost memory). The cross-section of such process will be isotropic, namely the particle after such capture will be emitted with equally probability in a solid angle of 4π. It was, however, observed that in some reactions, the particles are emitted preferentially in the forward direction. This feature suggests that the nucleons may be caught by nuclei forming a Compound System (C. S) but not necessarily a C. N. Examples of interactions through C. S. are direct and multiple collisions of the incident particle with the nucleons inside the target nucleus, excitations of collective oscillations, etc. Nevertheless, compound nucleus formation is assumed to predominate in many cases. The discussion above lead es to the Optical Model of the interaction of particles with nuclei. The key characteristic of the this model is that the interaction potential can be written as: V = V 0 + iv 1, where V 0 is the real part (shape elastic) and V 1 is the imaginary part of the interaction. Similarly, interaction of neutron with layers is described within the optical interaction -81-

90 Fundamental properties of the neutron potential. This model will be used later when describing the neutron reflectivity formalism. The Optical Model is a hybrid between the potential well and compound nucleus models. -82-

91 Specular PNR Polarized Neutron Reflectometry 2.2 Specular Polarized Neutron Reflectometry We analyze the Polarized Neutron Reflectivity on a somehow advanced level because it is not the main topic of the thesis. The method is well documented in literature, in books, as well as in review papers. For instance, the reader may find it useful to consult the following literature and the references therein: [124, 125, 126, 127, 128, 129, 130, 131]. The general scattering geometry for neutron optics is depicted in Fig. 1 of Ref.[125]. A flux of neutrons impinges on the surface of a multilayer system under a grazing angle α. It is then specularly reflected back into the air and hits the detector. The intensity is described as I = r 2 = R, where r is the reflection coefficient which depends on the thicknesses, optical potential, and roughnesses of each individual layer. For polarized neutron reflectometry four intensities are measured: I + +, I +, I +, I corresponding to the relative transition probabilities between the two incoming and outgoing spin states of neutrons. They are related to the coefficients of reflection as: I + + = r + + 2, I = r 2, I + = k0 /k 0 + r + 2, I + = k 0 + /k0 r + 2, where k 0 ± are the wave numbers of the neutron in the measuring medium (air). In the following we show two spinorial methods to calculate the four reflectivities from multilayered magnetic systems [132] Generalized Matrix Method (GMM) Let us consider the one dimensional multilayer system shown in Fig We shall suppose that the i-th layer has the nuclear potential u i = 2 /2m(4πN i b i iw), and magnetic potential µb i, where µ is the neutron magnetic moment, N i is the atomic density, b i is the coherent scattering length, and B i is the magnetic induction inside the i-th layer ( 2 /2m = 1). We suppose that B i in the adjacent layers are noncollinear and their direction, in principle, can be arbitrary. In the i-th layer, which is to the right of the i-th interface, the wave function can be represented as: ψ i = exp(iˆk + i (x x i))ξ + i + exp( iˆk + i (x x i))ξ i, (2.4) where x i is the left interface of the i-th region, ˆk ± i = k0 2 (u i ± µb i ), k 0 is the neutron wave number in vacuum without magnetic field, and ξ ± i are the spinors for the waves going to the right and left respectively. Let us introduce the generalized spinor Ξ i = ( ξ + ) i ξ. i Then, the wave function (2.4) can be represented as: ψ i = ˆT i (x)ξ i, where ˆTi (x) = ( exp(iˆk+ i (x x ) i)) 0 0 exp( iˆk + i (x x i)) (2.5) -83-

92 Specular PNR Figure 2.3: The potential profile of a multilayer system. The wave function to the left of the interface 1 can be represented as exp(iˆk 0 + x) 0 ( ) ψ 0 = ξ0, (2.6) 0 exp( iˆk 0 + ˆρξ x) 0 and to the right of the last n-th interface as exp(iˆk n + (x x n )) 0 ) (ˆτξ0 ψ n =, (2.7) 0 exp( iˆk n + 0 (x x n )) where ξ 0 determines the incident polarization, and ˆρ, ˆτ are the reflection and transmission matrices which should be found. The wave functions (2.6) and (2.7) are related to each other via some matrix ˆM. ( ) ˆM Ξ 0 = ˆMΞ 11 ˆM12 ) ξ0 (ˆτξ0 n =, (2.8) ˆρξ 0 0 ˆM 21 ˆM22 where ˆM ij are 2 2 matrices. From this relation it follows: ˆτ = ˆM 1 11, and ˆρ = ˆM 21 ˆM (2.9) Thus, to find ˆρ and ˆτ it is necessary to find the 4 4 matrix ˆM. For this we will match the wave function (2.5) at all interfaces. The continuity of the function and its first derivative gives two matrix equations: exp(iˆk + i 1 d i 1)ξ + i 1 + exp( iˆk + i 1 d i 1)ξ i 1 = ξ+ i + ξ i ˆk + i 1 [exp(iˆk + i 1 d i 1)ξ + i 1 exp( iˆk + i 1 d i 1)]ξ i 1 = ˆk + i [ξ+ i ξ i ], where d i is the width of the i-th region. In the generalized matrix representation it looks as follows: ˆQ i 1 ˆTi 1 Ξ i 1 = ˆQ i Ξ i, (2.10) -84-

93 Specular PNR Polarized Neutron Reflectometry where I ˆQ i = ˆk + i I e iˆk + i d i 0, ˆTi =. (2.11) ˆk + i From (2.10) it immediately follows: Ξ i 1 = 0 e iˆk + i d i 1 1 ˆT i 1 ˆQ ˆQ i 1 i Ξ i. (2.12) Repeating the matching over all the boundaries from right to left, we obtain ˆM = n 1 1 ˆQ 0 i=1 [ ˆQ i ˆT 1 i ˆQ 1 i ] ˆQ n. (2.13) Thus, we need the reciprocal matrices of (2.11). They are ˆT 1 i = exp( iˆk + i d i) 0 0 exp(iˆk + i d i) For a single interface we can easily check the result. In that case 1 ˆM = ˆQ 0 ˆQ 1 = (ˆk 0 + ) 1ˆk+ 1 1 (ˆk 0 + ) 1ˆk (ˆk 0 + ) 1ˆk (ˆk 0 + ) 1ˆk+ 1 and from (2.9) we immediately obtain, ˆQ 1 i = 1 I (ˆk + i ) 1. (2.14) 2 I (ˆk + i ) 1 ˆM 11ˆτ 01 = 1 ˆτ 01 = 2(1 + (ˆk + 0 ) 1ˆk+ 1 ) 1 = (k k + 1 ) 1 2k + 0, (2.15) ˆρ 01 = ˆM 21ˆτ = [1 (ˆk + 0 ) 1ˆk+ 1 ][1 + (ˆk + 0 ) 1ˆk+ 1 ] 1 = [1 + (ˆk + 0 ) 1ˆk+ 1 ] 1 [1 (ˆk + 0 ) 1ˆk+ 1 ] = (ˆk ˆk + 1 ) 1 (ˆk + 0 ˆk + 1 ). (2.16) Recurrence Method (RM) In the RM [133, 132] we start from calculation of the reflection and transmission matrix for a single layer with identical environment on the both sides: ˆτ 1 = ˆτ 01 e 1 [1 ˆρ 01 ê 1 ˆρ 01 ê 1 ] 1ˆτ 10, (2.17) where and ˆρ 1 = ˆρ 10 + ˆτ 01 ê 1 ˆρ 01 ê 1 [1 ˆρ 01 ê 1 ˆρ 01 ê 1 ] 1ˆτ 10, (2.18) ê 1 = exp(iˆk + 1 d 1 ), ˆρ 10 = ˆρ 01 = (ˆk ˆk + 1 ) 1 (ˆk + 0 ˆk + 1 ) (2.19) ˆτ 10 = (ˆk ˆk + 1 ) 1 2ˆk + 0 ˆτ 01 = (ˆk ˆk + 1 ) 1 2ˆk + 1. (2.20) -85-

94 Specular PNR The matrices ˆρ 01, ˆτ 01 and ˆρ 10, ˆτ 10 describe the reflection and transmission of interface for a plane wave incident from inside and outside of the layer respectively. For the two last layers of the system shown in fig. 1 we get ˆτ n 1,n = ˆτ n [1 ˆρ n 1 ˆρ n ] 1ˆτ n 1 ˆρ n 1,n = ˆρ n 1 + ˆτ n 1 ρ n [1 ˆρ n 1 ˆρ n ] 1ˆτ n 1 Now, we are ready to write the transmission and reflection of the all n layers as recurrent relations: ˆτ 1,2,...,n = ˆτ 2,3...,n [1 ˆρ 1 ˆρ 23...,n ] 1ˆτ 1 ˆρ 1,2,...,n = ˆρ 1 + ˆτ 1 ˆρ 23...,n [1 ˆρ 1 ˆρ 23...,n ] 1ˆτ 1 These equations are easy to be implemented in computer codes and obtain the four reflectivities for non-collinear multilayers. More importantly, the RM is an excellent tool for analytical description of magnetic layered structures, like neutron resonators [132, 134, 135]. Numerically, the GMM method is about 10 times faster as compared to the RM formalism. -86-

95 Specular PNR Polarized Neutron Reflectometry Spin Asymmetry Aside from spin dependent neutron reflectivities, often another useful experimental parameter is measured, which is indicator of the non-vanishing magnetization of the films. Spin Asymmetry (SA) is defined, in the most general case, as: SA(k z, θ, B) = R+ R R + + R = R++ + R + R R +, (2.21) R ++ + R + + R + + R Experimentally, one measures the reflected intensities which include a geometrical factor and background. Assuming a gaussian distribution of the incident beam we can write the intensities as: I = I 0 Erf( l s sin(θ) 2 2W DB )R + backgr (2.22) where, I 0 is the incoming incident intensity, l s is the length of the sample, W DB is the full width at half maximum of the direct beam, θ is the incident angle and backgr is the background. Now, assembling the background corrected intensities as spin asymmetry we get: SA exp = I+ I I + + I = R+ R R + + R = SA theor (2.23) With this technique we have eliminated the geometrical corrections. As clearly seen from Eq. 2.21, there are two experimental ways to obtain SA: one by measuring only R + and R reflectivities and the other one is by measuring all 4 components R ++, R +, R +, and R. When not special cases like off-specular scattering are involved, it is efficient to measure only the R + and R components. The advantage is that both measuring time and therefore the error bars of SA are reduced. Moreover, the spin-flip reflectivity can be inferred from SA as follows: R + = R + = 1 2 ( R ++ R SA ) (R ++ + R ), (2.24) where the SA is measured in saturation through R + and R. This has the advantage that in the non-collinear state only R ++ and R are to be measured. Again this method of obtaining the SF reflectivity can be used in low applied magnetic field and when no off-specular scattering is involved. SA is particularly useful when trying to observe a very small magnetic signal from samples, when one aims at determining the magnetic moment through fitting, when measuring hysteresis loops, and when trying to measure the coupling sign between two ferromagnetic layers. We will show examples for all four instances below. -87-

96 Specular PNR EXAMPLE: Single ferromagnetic Co layer A Co layer was deposited on a glass substrate by rf-sputtering techniques. Due to the exposure in air a thin oxide layer is formed on top of the ferromagnetic layer. Thus, the real structure consists of a thick layer covered by a thin non-ferromagnetic one. However, we neglect the CoO layer as it not contributes to the observations to be described below. The measurements were performed using the ADAM reflectometer installed at the ILL. The reflectivities were measured in saturation. Figure 2.4: Left: Reflectivities R + and R measured in saturation. Right: Experimental SA (open black circles) and the theoretical fit to SA (black line). Figure 2.5: Magnetic profile obtained after fitting both reflectivities and SA. In Fig 2.4 R + and R are plotted as function of wave vector Q. There are several particularities of this reflectivities: -88-

97 Specular PNR Polarized Neutron Reflectometry there are well defined critical edges: one is the Q + c corresponding to the + state and the lower one corresponds to the substrate potential. As the critical edge of the - state is imaginary the Q c cannot be seen. there is a cross-over of the reflectivities. It is well seen in the Fig. 2.4 right that a change of sign occurs from positive to negative SA at Q = A 1 below the total reflection region of the substrate there are some oscillations in the R reflectivities. This features will be explained in the resonance sections. Note that the fit reproduces well also those features. The parameters obtained from the fit are shown in Table 2.1. They are going to be used in determining other parameters discussed in section 4. The SA played an important Table 2.1: Parameters of the Co sample. Layer d [Å] σ[å] SLD B[Oe] µ B CoO e Co e x substrate non e role in obtaining this parameters. When fitting the experimental reflectivities one should account for geometrical parameters shown in Eq When fitting the SA, the geometrical parameters I 0, l s, and W DB do not enter into the fit. Thus, the confidence of the magnetic profile is greatly improved. Although fitting simultaneously the R + and R is already giving a satisfactory results, the best confidence is given by fitting SA. -89-

98 Specular PNR Reflectivity of a semi-infinite magnetic media Now, having developed the generalized algorithms, several particular situations will be considered. As the formulae are quite complicated when expanding them, we take the simplest possible example, which is a single magnetic substrate and derive the analytical formulae for transmission and reflection coefficients. Using the expression ˆµ = µˆσ and the well known properties of the Pauli operator (ˆσ) one can write the expression of the wave vector operator as: ˆk + i = k+ i + k i 2 + (ˆσe i ) k+ i k i 2, (2.25) where k ± i = kz 2 u i µb i 2m are the eigenvalues of the operator ˆk + 2 i and e i = B i /B i is a unit vector which defines the direction of the magnetic induction in the i th media. The matrix representation of ˆR i = ˆσe i reads: cos(θ)) R = sin(θ)e iφ sin(θ)e iφ, (2.26) cos(θ) where θ and φ are the polar and the axial angle in the spherical coordinates with the polar axis being the quantization axis. Choosing the quantization axis to be parallel to the polarization axis, the matrix representation of the reflection operator ˆρ 01 becomes: r ++ r + ˆρ 01 = (2.27) r + r with: r ++ = k 1 k k1 k k 0 + k 1 + k0 k1 k k1 k k 0 + k k0 ( ) ( ) ( ) k 1 2 k k k 0 + k 0 + k 1 k 1 + cos(θ) ( ) ( ) ( ) k k k 1 + k 0 k 0 + k 1 k 1 + cos(θ) (2.28) r = k 1 k k1 k k 0 + k 1 + k0 k1 k k1 k k 0 + k k0 ( ) ( ) ( ) k k k k 0 + k 0 + k 1 k 1 + cos(θ) ( ) ( ) ( ) k k k 1 + k 0 k 0 + k 1 k 1 + cos(θ) (2.29) r + = ( ) 2 k 0 + k 1 k 1 + e i φ sin(θ) ( ) ( ) ( k1 k k1 k k 0 + k k0 k k k 1 + k 0 k 0 + k 1 k 1 + ) cos(θ) (2.30) r + = 2 k 0 k 1 k k 1 k k + 0 k k 0 ( ) k 1 k 1 + e i φ sin(θ) ( ) ( ) ( k k k 1 + k 0 k 0 + k 1 k ) cos(θ) (2.31)

99 Specular PNR Polarized Neutron Reflectometry It is easy to check that the expressions above are the same as ones obtained by Pleshanov [136]. In order to understand how the reflectivities depend on the magnetization reversal in an ferromagnetic media let us consider the case when k 2 z = 4πNb. Choosing just a single value for the incident wave vector and sweeping the applied magnetic field is a key ingredient for measuring a complete hysteresis loop via PNR. We neglect the neutron absorption into the sample and consider the case when the applied magnetic field is much smaller then the internal fields into the ferromagnetic media. With these assumptions and for B x 0 the reflectivities become: R ++ = r ++ 2 = 1 + V mn V mn (1 + V mn ) cos(θ) + 2 V mn cos(θ) 2 ( 1 + Vmn ) 2 (1 + Vmn ) R = r ++ 2 = 1 + V mn 2 2 V mn (1 + V mn ) cos(θ) + 2 V mn cos(θ) 2 ( 1 + Vmn ) 2 (1 + Vmn ) R + = R + = r ± 2 = 2 V mn sin(θ) 2 ( 1 + Vmn ) 2 (1 + Vmn ), where V mn = V m /V n = µb /( 2 4πNb 2m coh) is the ratio of the magnetic to the nuclear potential. The conclusions from the formulae above are that at the critical scattering vector for non-polarized neutron, the reflectivities: - depend on the absolute value of the magnetic induction B; - depend on the relative orientation between the polarization axis and the direction of the magnetic induction into the media; -do not depend on the polar angle φ. We consider below two cases: one when the magnetization reversal proceeds by rotation and another case where the magnetization changes through domain wall movement. -91-

100 Specular PNR Hysteresis Loops The PNR can distinguish between different magnetization reversals as depicted in Fig Moreover, SA measured at fixed incident angles provides similar hysteresis as MOKE and SQUID, but contains more information about the re-magnetization processes. For instance, neutron loops reveal clearly coherent rotation, domain wall movement and combination of both domains and rotation, for buried layers and interfaces Coherent rotation of the magnetization Let us consider that the magnetic moments will rotate during the magnetization reversal in such a way that B = B sat with the components B y = B cos(θ) and B x = B sin(θ). The easiest way to imagine such a rotation is to rotate the sample instead of rotating the magnetic moment, keeping the remanent magnetization equal to the saturation magnetization (M rem /M sat = 1). With this assumption the spin asymmetry SA(k z, θ, B ) becomes: SA(k z, θ, B) = (R + R )/(R + + R ) = 2 V mn cos(θ) = B x 2 V mn 1 + V mn B 1 + V mn The spin flip reflectivity is: SA(k z, θ, B) = SA(k z, 0, B ) B x B R + = 2 V mn sin(θ) 2 ( ) Vmn (1 + Vmn ) = B2 y 1 B 2 sat 2V mn (1 + V mn ) 2 (1 + V mn ) R + = B 2 yf(b sat, V mn ) Measurements of R + and R at any k z value as a function of the applied magnetic field provide the same hysteresis loop as obtained by MOKE or SQUID measurements. Moreover, fitting the non-spin flip reflectivity curves taken in saturation (R + or R or both) one can accurately evaluate the absolute magnetic moment. In comparison, for MOKE one would have to make a calibration for the Kerr angles and for SQUID one needs to determine the volume of the magnetic layer Domain Wall Movement Now we discuss a magnetization reversal by domain wall movement. For simplicity we consider a situation where a magnetic film consists of ferromagnetic domains with the following configuration: a fraction of n domains have their spins oriented parallel to the neutron polarization and a fraction of 1 n domains have the spin oriented antiparallel. The measured reflectivities would not be simply R + or R but given by I + = nr + + (1 n)r and I = (1 n)r + + nr. The measured spin asymmetry becomes: SA(k z, θ, B ) = I+ I I + + I = (2n R 1)(R+ R + + R ) = B x B sat Vmn 1 + V mn,

101 Specular PNR Polarized Neutron Reflectometry where B x = (2n 1) B with B being the magnetic induction into the magnetic domains. Thus, the shape of the hysteresis loop is completely defined. The spin flip scattering will result only from domain walls. In a more general case when for instance the magnetic domains are rotated with respect to the neutron polarization axis, there will also be spin flip scattering. For this case the SA expression will still be valid and, in addition, from the spin-flip scattering their angle relative to the neutron polarization axis can be measured. We mention that the formulae above are only correct in the absence of off-specular scattering. The values of SA and the spin-flip reflectivity will be affected by the loss of the specular signal. This can be seen when comparing SQUID or MOKE based magnetic hysteresis loops to the one extracted from the SA. Regions where the loops do not overlap indicate the presence of magnetic domains. This can be clearly seen in Fig. 6.3(a). A different approach for evaluating the average magnetization vectors inside of a film in a domain state was taken by Lee et al. [137]. Instead of measuring the magnetization reversal at one specific wave vector, complete reflectivity curves are recorded and the average magnetization as well as the mean square dispersion of the domains as function of k z is evaluated. They use similar reflectivity formulae as shown above, but do not derive magnetization loops. For a quantitative comparison of hysteresis loops the method presented here is faster and more effective. In Fig. 2.6 is depicted schematically how the four reflectivities would change as a function of the applied magnetic field for different remagnetization processes. When for coherent rotation, one would expect that at the coercive field the spin-flip reflectivities (R + and R + ) would increase dramatically at the expense of the non-spin flip ones (R + + and R ) which will have a minimum value at the coercive field. The sum of all for reflectivities should be equal to unity for the whole applied fields (see the black dotted line). When the magnetization reverses by domain wall movement, then the spin-flip reflectivities are practically negligible. Only small spin-flip scattering is expected from the domain wall itself. The non-spin flip reflectivities will cross at the coercive field. In case of more complicated situations, like coexistence of magnetic domains and rotations the spin-flip intensities do not vanish but have instead reduced intensity. Moreover, through the offspecular scattering one clearly may decide upon the presence of the magnetic domains for different applied fields. -93-

102 Specular PNR Figure 2.6: Schematic view of the expected hysteresis loops measured by PNR. Several re-magnetization processes are depicted: coherent rotation, domain wall movement and Landau domains. The dotted black line is the sum of all four specular reflectivities.[138] -94-

103 Spin states Polarized Neutron Reflectometry 2.3 Spin states of neutrons in magnetic thin film Introduction Neutron reflectivity goes back to 1946 when it was first used by Fermi and Zinn for the determination of coherent scattering lengths [112]. Subsequently inserting a polarizer and analyzer to produce a polarized neutron beam, Hughes and Burgy started to perform polarized neutron experiments as early as 1951 [113]. It was foreseen by the authors of this work that beams of completely polarized neutrons will be useful in the study of magnetic and nuclear properties. The next major achievement of Polarized Neutron Reflectometry (PNR) was the prediction of spin-flip scattering by Ignatovich (1978) [139] and the pioneering experiments on magnetic surfaces by Felcher (1981) [140]. While specular PNR is widely recognized as a powerful tool for the investigation of magnetization profiles in magnetic heterostructures [130], the description of off-specular scattering from magnetic domains is still under development [128]. In spite of these important developments there is still a confusion concerning the quantum states of neutrons in a magnetic sample. Here we show unambiguously that the neutron has to be treated as a spin 1/2 particle [136, 132] in each homogeneous magnetic layer. This is at variance with the conventional description of neutron reflectivity, which often considers the neutron magnetic potential as a classical dot product [141, 142, 126, 125, 143]. Neutrons interact with a magnetic thin film via the Fermi nuclear potential and via the magnetic induction. Thus, the neutron - film interaction hamiltonian includes both contributions: V = V n + V m = ( 2 /2m)4πNb µb, where m is the neutron mass, N is the particle density of the material, b is the coherent scattering length, µ is the magnetic moment of the neutron, and B is the magnetic induction of the film. Unconventionally, however, neutron reflectivity treats the dot product between the magnetic induction and neutron magnetic moment classically: V m = µb = ± µ B cos(θ), where θ is the angle between the incoming neutron polarization direction and the direction of the magnetization inside the film. Writing the magnetic potential as a classical dot product implies that the neutron energies in the magnetic layer have a continuous distribution from - µ B to + µ B. This predicts that the critical angle for total reflection depends on the angle between the direction of polarization and the direction of the magnetic field inside the layer: 4π sin(α ± c ) λ = Q ± c = 2m 2 (V n ± µ B s cos(θ)), (2.32) where α is the glancing angle to the surface, λ is the wavelength of the neutrons, and Q ± c is the critical scattering vector. There are experimental data [144, 142, 126] on magnetic multilayers, which apparently confirm this behavior. Therefore, the classical representation appears to provide a convenient and transparent way to describe the experimental observations [141, 142, 126, 125, 143]. From the Stern-Gerlach experiment we know that there are only two eigen states for the spin 1/2 particles in a magnetic field. Therefore, the eigen wave number of a neutron in a magnetic thin film has two proper values. After solving the Schrodinger equation one -95-

104 Spin states obtains two eigen wave numbers for neutrons in a magnetic film: k 2 ± = 2m 2 (V n ± µ B ). They correspond to two possible states of spin orientation: one for the case, when the spin is parallel to the magnetic induction, and the other one for the antiparallel orientation. It follows that there are only two possible energies and consequently only two values for the index of refraction corresponding to the spin-up and spin-down states of the neutrons. Therefore, QM predicts that there are only two critical angles for the total reflection: one corresponding to the R + and one to the R reflectivity 4π sin(α ± c ) λ = Q ± c = 2m 2 (V n ± µ B s ) (2.33) Obviously there is a contradiction between the quantum mechanical prediction (Eq.2.33) and the prediction based on the classical representation of the magnetic potential(eq.2.32): quantum mechanics predicts that the spin states of the neutron is determined by the magnetic induction in the sample, whereas classical representation of the magnetic potential, supported by experiments on magnetic multilayers, assert that the spin states of the neutrons is fixed by the incident polarization axis. Here we describe an experiment which provides direct and unambiguous evidence for the spin states of neutrons in magnetic media. The goal is to find a system where the angle between the neutron polarization and direction of the magnetization inside of the film can be fixed and controlled. Then we measure the R + and R reflectivities and determine whether the position of the critical edges changes as a function of the angle θ, or whether the critical edges stay fixed, and only intensity redistributes between reflections R + and R with change of θ. The easiest way to control the angle θ is to rotate the magnetic film and therefore the magnetization direction with respect to the neutron spin polarization, which remains fixed in space outside of the sample. This requires that the film should have a high remanent magnetization. Additionally, the film thickness should exceed the average neutron penetration depth [145]. The last requirement is essential in order to avoid neutron tunnelling effects which will hinder the precise determination of the critical edges Sample characterization by MOKE and PNR To fulfill the aforementioned requirements, we have chosen a 100 nm thick polycrystalline Fe film deposited by rf-sputtering on a Si substrate. The thickness of the Fe films was about 4 times larger than the average penetration depth 1/ 2mV N / 2. The Fe film was covered with thin Co and CoO layers, the latter one protecting the Fe film from oxidation. For sample characterization at room temperature we first recorded hysteresis loops with the magneto-optical Kerr effect (MOKE). A series of hysteresis loops were taken with the field parallel to the film plane but with different azimuth angles of the sample. A typical hysteresis loop is shown in Fig The coercive field is about 20 Oe and the remanence is high. A plot of the ratio between the remanent magnetization and saturation magnetization M rem /M sat versus the rotation angle about the sample normal is shown in Fig We conclude that the system has no macroscopic anisotropy and the remanent magnetization is 97.5% of the saturation magnetization. -96-

105 Spin states Polarized Neutron Reflectometry Figure 2.7: Bottom: hysteresis loop of the polycrystalline Fe/Si sample measured by MOKE. Top: the behavior of the remanent magnetization as a function of the rotation angle extracted from hysteresis loops. Neutron reflectivity experiments were performed using the angle dispersive neutron reflectometer ADAM installed at the Institut Laue-Langevin, Grenoble, which operates at a fixed wavelength of 4.41 Å. The R+ and R reflectivities and the spin asymmetry (R + - R )/(R + + R ) in a saturation field of 2000 Oe are plotted in the top and bottom panel, respectively, of Fig The solid lines are fits to the data points using the PolarFit code based on the general matrix method (GMM) [132]. The fit and sample parameters are listed in Table 2.2. In order to obtain a high confidence of the fit parameters, all reflectivities were fittted together and with the same parameter set. In general it is useful to fit first the spin asymmetry, for which geometrical and normalization parameters drop out. Table 2.2: Parameters of the Fe/Si sample obtained by fitting to the R + an R data shown in Fig.2.8. d is the layer thickness, σ is the rms roughness, SLD is the scattering length density, and B is the magnetic induction in the ferromagnetic films. Layer d [Å] σ[å] SLD B[Oe] Co y O 1 y e-6 0 Co x Fe 1 x e Fe e substrate non e-06 0 From Table 2.2 and the magnetic characterization (Fig. 2.7) we conclude that the sample -97-

106 Spin states fulfills the requirements as concerns thickness, anisotropy, and remanence as required for our experiment. Figure 2.8: (Color online). Top: Polarized neutron reflectivity curves R + (solid black circles) and R (open black circles) of the Fe/Si sample. The blue (dark gray) line is the simulated R+ reflectivity and the red (light gray) line is the simulated R- reflectivity. The applied magnetic field was 2000 Oe. In the inset, the magnetic profile obtained from fitting the data is shown. Bottom: Experimental (open black symbols) and simulated (black line) spin asymmetry ((R + R )/(R + + R )) are plotted for the same sample in saturation. All lines in the top and bottom panels are fits to the data points using the GMM (for more details see text). The abscissa is the wave vector transfer: Q = 4π sin(α)/λ Rotation experiment The rotation experiment was performed as follows: the Fe layer was magnetized parallel to the neutron polarization direction and then the magnet was removed. A small guiding field (H c ) is still present at the sample position in order to maintain the neutron polarization. Subsequently a series of R + and R reflectivities, shown in Fig. 2.9, were measured -98-

107 Spin states Polarized Neutron Reflectometry for several in-plane rotation angles of the sample. We observe two characteristics of the reflectivities: (1) the critical edges are fixed and independent of the in-plane rotation angle θ (see Eq. 2.33), and (2) the R intensity continuously increases at the expense of the R + intensity as a function of the θ angle. Using the parameters obtained from the fit to the saturation data (see Table 2.2) and using the rotation angle θ as set during the experiment, we have simulated [146] the reflectivities using the approach [132] which obviously predicts the behavior of the critical edges described by Eq The simulated curves are plotted together with the experimental data in Fig There are no free parameters for these simulations, providing an excellent description of the experimental results. The fixed critical edges Q c + and Q c can easily be interpreted in the context of the neutron spin states in homogenous magnetic media as discussed in the introduction. Figure 2.9: (Color online). Experimental (symbols) and simulated (lines) reflectivity curves (R + and R ) from Fe(1000 Å)/Si sample. The abscissa is the wave-vector transfer. The two sets of R + (solid black symbols) and R (open black symbols) reflectivity curves were measured for four different θ angles between the neutron polarization along B 0 and the direction of the magnetic induction (B) which lies in the sample plane. The guiding field is B 0 10 Oe. The blue (thick dark gray) lines are the simulated R+ reflectivities and the red (thin light gray) lines are the simulated R- reflectivities. In the right side the experimental geometry is shown. The figure shows that the critical edges Q c + and Q c are not sensitive to the θ angle. -99-

108 Spin states Experimental determination of magnetization orientation The sensitivity to the in-plane rotation angle of the magnetization is seen very clearly seen in the reflected intensities R + and R plotted in Fig It has been shown theoretically [10] that, for a single magnetic layer, the normalized spin asymmetry (nsa(θ)) is directly related to the θ angle through the following expression: nsa(θ) = SA(θ) SA(0) = cos(θ) (2.34) Now, we use our experimental data shown in Fig. 2.9 to confirm the validity of this equation. In Fig is shown the experimental normalized spin asymmetry and the cosine of the experimental angles. The agreement between the experimental normalized spin asymmetry (symbols) and the cosine of the θ angles (lines) set during the experiment is excellent over the whole wave vector transfer range. It follows that the magnetization orientation of a Figure 2.10: (Color online). Solid black symbols: The experimental normalized spin asymmetries (nsa(θ) = SA(θ)/(SA(0))), plotted as a function of the wave-vector transfer Q. Lines: The three lines are the cosines of the corresponding angles set during the experiment. From top: cosine of 48 (thin gray (red) line), 68 (thick gray (blue) line), and 88 (thin black line), respectively. The angles are the experimental rotation angles θ used during the experiment shown in Fig The experimental normalized spin asymmetries are assembled from the R + and R reflectivities shown in Fig 2.9. This figure shows that the equation nsa(θ)=cos(θ) is valid over the whole Q range for a single magnetic layer. It can be used to extract the angle θ directly from the experimental reflectivities. single magnetic layer with respect to the neutron polarization outside the layer can be -100-

109 Spin states Polarized Neutron Reflectometry easily extracted experimentally using Eq For more complicated systems a numerical fitting is still necessary. The nsa is an important measure of hysteresis loops. It was shown in Ref. [10] that nsa can be written, generally, as: nsa = M /M sat for both, magnetization reversal via coherent rotation and via domain wall movement. This implies that nsa reproduces the hysteresis loops as measured by SQUID or MOKE. Here we confirm the validity of the nsa for determining the magnetization reversal via coherent rotation. We mention that this equation is valid for samples which contain a single magnetic layer. Comparing MOKE or SQUID hysteresis loops with nsa is a very useful tool for the evaluation of magnetic domain state and/or a reduced magnetization within the layer Multilayers Our next topic is to investigate the neutron spin state in multilayers with noncollinear magnetization of adjacent layers. We have simulated the reflectivity profile of a [Fe(60Å)/Cr(8Å))] 40/Si superlattice, with thicknesses of the Fe and Cr layers which are typical for many real superlattices [144]. For the simulation we used the freeware code PolarSim [146] based on GMM for the calculation of the reflection and transmission coefficient together with a full quantum mechanical description of the spin states [132]. In the simulation the choice of a Si substrate has the advantage that it does not obscure the critical edge of the (-) neutron state. In the top panel of Fig we show simulations of R + and R reflectivities for three angles γ between the magnetization vectors of adjacent Fe films: γ = 0 (or ferromagnetic alignment); γ = 100 ; and γ = 170 (close to antiferromagnetic alignment). Our focus is on the behavior of the critical scattering vector for total reflection. We observe that for γ = 0 the (+) and (-) critical scattering vectors are well separated and that they contain information about the saturation magnetization. When the γ value increases, the critical edges approach each other. For an angle γ = 180 (not shown here) there is no difference between the R+ and R- reflectivities. The main result from this simulation is the observation that the separation of the critical edges is a continuous function of the angle γ between the in-plane adjacent magnetization vectors. The critical edge positions satisfy the following relation: 4π sin(α ± c ) λ = Q ± c = 2m 2 eff (Vn ± µ B s cos(γ/2)), (2.35) where Vn eff is an effective nuclear potential. Clearly, for this geometry the angles θ and γ/2 coincide if the neutron polarization is parallel to the average field (B 1 + B 2 )/2, where B 1 and B 2 are the magnetic field inductions of adjacent layers. Therefore, numerically, the eqs and 2.35 are almost identical. However, there is a fundamental difference: similarly to the single layer, the θ angle does not influence the position of critical edges, whereas the γ angle is solely responsible for the continuous shift. To shed more light on how the θ and γ angles affect the critical edges for polarized neutron reflectivity at the multilayers we simulated numerically the rotation experiment performed on the single layer. For a fixed coupling angle of γ = 90, as it can be achieved also experimentally via biquadratic exchange coupling, the reflectivities R + and R are plotted -101-

110 Spin states Figure 2.11: (Color online). Top: Simulation of polarized neutron reflectivities (R+ (dashed lines), R (solid lines)) for a Fe/Cr multilayer as a function of coupling angle (γ) between the magnetization vectors of adjacent Fe layers. Bottom: Simulations of R+ (solid lines) and R (dashed lines) as a function of rotation angle θ for γ = 90 as a function of the θ angle. Here the θ angle is the angle between the incoming neutron polarization and the direction of the average magnetization vector of two adjacent ferromagnetic layers. The results are shown in the bottom panel of Fig We observe a similar behavior of the critical edges and intensities as for the single layer. While the -102-

111 Spin states Polarized Neutron Reflectometry positions of the critical scattering vectors Q c + and Q c remain fixed for a constant coupling angle γ, the R intensity increases on the expense of the R + intensity with increasing θ angle. With this simulations we lift the contradiction stated in the introduction by showing that Eq is a particular case of Eq. 2.35, which, in turn, is in agreement with the QM description of the neutron spin states in magnetic media Eq The different behaviors of the critical edges for the case of a single homogeneous ferromagnetic layer and for a multilayer with alternating directions of the layer magnetization vectors now becomes obvious: in the multilayer the neutrons are affected by an average magnetic potential which depends on the relative orientation of the magnetic induction in the individual layers. However, in both cases, single film as well as multilayer, the magnetic potential of the individual layers (V m = µ B s ) enters the algorithm for calculating the reflectivities. It should be noted that the dependence of Q c + and Q c on the angle γ in a multilayer is a general property of the periodic potential with different field orientation and magnitude. It is natural to expect that such a sample is a noncollinear ferrimagnet with ferromagnetic field B f = (B 1 + B 2 )/2, and with antiferromagnetic field B af = (B 1 B 2 )/2, B f = B 1 + B 2 /2 = B cos(γ/2), (2.36) B af = B 1 B 2 /2 = B sin(γ/2). (2.37) Then, the critical edges can be expected to be given by Eq To further stress the origin of the effective nuclear potential Vn eff term in Eq , let us consider the critical edge for non-polarized neutrons when scattered at a [Fe(x Å)/Cr(y Å))] multilayer. Naively, we may expect that the critical edge to be given by the Fermi interaction potential of Fe as it is higher than the potential of Cr. This is, however, not the case. For a finite thickness x of the Fe layer and zero thick Cr layer, indeed the critical edge is equal to the critical edge of a single thick Fe layer. Vice versa, for zero thickness of Fe layer and finite thickness y for the Cr layer the critical edge is given by the Fermi potential of Cr. However, when both layers have finite thicknesses the critical edge of the multilayer will vary from the value for pure Fe to the value for pure Cr. Therefore, the critical edge for non-polarized neutrons reflected from a multilayer not only depends on the Fermi potential of the two separate layers, but also on their individual thicknesses Final Remarks In summary, we have analyzed the behavior of the critical scattering vectors Q c + and Q c for total external reflection of a polarized neutron beam for the case of homogeneous ferromagnetic films and for antiferromagnetically coupled multilayers. For a single film we have observed experimentally and shown theoretically that the critical edges do not change as a function of the angle between the neutron polarization and the direction of the magnetic spins inside the film. They fulfill the relation Eq. 2.33: Q ± 2m c = (V 2 n ± µ B s ), which directly reflects the spin states of the neutron beam in magnetic thin films. For -103-

112 Spin states multilayers we found that the critical edges for total external reflection move towards each other as a function of the coupling angle. Their position is well reproduced by the 2m 2 Eq. 2.35: Q ± c = ± µ B s cos(γ/2)). The cos(γ/2) dependence is not related to the neutron spin states in the magnetic media, but it is the result of the presence of a ferromagnetic field direction along the average field in the noncollinear ferrimagnetic. By choosing a fixed coupling angle γ between the magnetization vectors of adjacent layers and rotating the sample, the critical edges behave again in accordance with the neutron spin states in homogeneous magnetic media. Practically, the coupling angle in non-collinear superlattices can be inferred directly from the experimental data through the separation of the critical edges. For a single layer the orientation of the magnetization can be extracted experimentally from the spin asymmetry. (V eff n -104-

113 Chapter 3 Neutron Resonances in Layered Structures 105

114 Neutron Resonances in Layered Structures 3.1 Bound State Neutron Resonances Polarized Neutron Reflectometry uses an experimental setup where the wave vector component of neutrons normal to the surface is higher then the critical normal wave vector of the layer having the highest nuclear potential. In the past the total reflection region was not much explored, except for ultracold ultracold neutron applications, such as the investigation of neutrons in a gravitational field. We shall pay attention to this region in the frame of PNR and reveal some basic physical properties of the neutron resonances which occur in multilayer systems. In order to achieve resonances, a typical configuration of the sample is required. The theoretical description of PNR from multilayers follows the same principles for polarized and for non-polarized neutrons beams. In the next section we will analyze some properties of neutron resonances in multilayered systems, as well as, define an optimized configuration of the sample. We consider two types of neutron resonators, one which is a typical potential well and another one where the resonances are formed in a single layer. Also, it is shown that monolayer sensitivity for one magnetic interface is comfortably achieved by using neutron resonators. A potentially useful feature of resonances is that PNR can be tuned to be interface sensitive and interface specific Wave function in the resonant layer: generalized case The neutron behavior in a layered system is dictated by the solution of the one-dimensional Schrödinger equation, in which every layer is described by a rectangular potential V j. Let us consider a one-dimensional multilayer system shown in Fig We use the recurrence method for the analytical calculations and GMM for the numerical simulations. The j-th layer has an optical potential ˆV + j = u j +µb j, where the first term u j = 4πN j b j is the nuclear and the second one is the magnetic potential, µ = µσ, where µ is the absolute value of the neutron magnetic moment, σ are the Pauli matrices, N j is the atomic density, b j is the coherent scattering length of the j th layer, and B j is the magnetic induction inside j th layer. The B j in adjacent layers can be noncolinear and their directions, in principle, can be arbitrary. The neutron incident from the left can be described by a plane wave exp(iˆk 0 + x) ξ 0 +, where ˆk ± j = k 2 ˆV ± ± j is the neutron wave number in the j-th layer, ˆV j = u j ± µσb j, k is the neutron wave-number in vacuum in the absence of the external magnetic field, and ξ 0 + denotes the spinors, which for a coliniar case are ( 1 0 ) and (0 1 ). In the j th layer, which is to the right of j th interface, the wave function can be represented as ψ j = exp(iˆk + j (x x j)) Âj ξ exp( iˆk + j (x x j)) ˆB j ξ 0, (3.1) where x j is the left interface of the j-th region, and Âj and ˆB j are the matrix coefficients for the waves going to the right and left, respectively. To calculate the wave function (3.1) in the RM approach, we use the equations for [134, 135] -106-

115 Neutron Resonances in Layered Structures  j and ˆB j :  j = ˆτ 1,...j 1 + ˆρ j 1,...1 e iˆk + j dj ˆρ j+1,...n e iˆk + j d j  j, ˆBj = e iˆk + j dj ˆρ j+1,...n e iˆk + j d j  j. (3.2) where ˆτ 1,...j 1 is the matrix transmission amplitude of the whole potential from the left to inside of the j-th layer, ˆρ j 1,...1, ˆρ j+1,...n are the amplitudes of the reflection from the left and right walls, respectively, while the factors e iˆk + j d j describe the phase accumulation because of particle propagation through the j-th layer of thickness d j. From (3.2) it follows  j = [1 ˆρ j 1,...1 e iˆk + j dj ˆρ j+1,...n e iˆk + j d j ] 1ˆτ 1,...j 1, ˆBj = e iˆk + j dj ˆρ j+1,...n e iˆk + j d j  j. (3.3) The matrix coefficients above correspond to the general case of a non-collinear multilayer system. Below we continue the discussion for the particular cases shown in Fig. 3.2 and Neutron density in neutron resonators For simplifying the analytical analysis we consider the + state of neutrons shown in Fig For this case the optical potential of the structure forms a quantum well. In the resonant layer the neutron density have resonant properties. For the collinear case the matrix nature of the amplitudes collapses in 1-dimensional scalar amplitudes: The equations Eq. 3.3 can be written as:  ± j = A ± j, ˆB± j = B ± j. A 2 = τ 02 1 ρ 20 ρ 24 e 2ik 2d 2, B 2 = e2ik2d2 ρ 24 τ 02 1 ρ 20 ρ 24 e 2ik 2d 2, (3.4) where τ 02 is the transmission amplitude through the whole layer 1, and ρ 24 and ρ 20 are the amplitude of reflection from the right and left walls of layer 2, respectively. k 2 and d 2 are the normal component of the wave vector of the neutrons in the resonant layer-2 and the thickness of the layer 2, respectively. Using the notations Φ = Im[2ik 2 d ]+φ 20 +φ 24 and ρ j = ρ j exp(iφ j ), where φ j is the phase of the reflection amplitude ρ j, the square modulus of the amplitudes A 2 and B 2 becomes: A 2 2 = B 2 2 = τ 02 2 (1 ρ 20 ρ 24 e Re[2ik 2d 2] ) ρ 20 ρ 24 e Re[2ik 2d 2] sin 2 ( Φ 2 ) (3.5) τ 02 ρ 24 2 e 2Re[2ik 2d 2 ] (1 ρ 20 ρ 24 e Re[2ik 2d 2] ) ρ 20 ρ 24 e Re[2ik 2d 2] sin 2 ( Φ 2 ) (3.6) We see that the coefficients have the resonant character caused by the denominator, which have a minimum when the phase Φ = 2πn (resonance condition). The number n is an -107-

116 Neutron Resonances in Layered Structures integer (n > 0) called resonance order. The relationship above determines all the properties that the resonant layer should have in order to achieve the optimum pattern of the neutron resonances required for the specific investigation. The resonance condition is fulfilled for discrete values of the neutron energy inside the resonant layer. If ρ 24 = 1, i.e. there is total reflection from the potential to the right of the 2nd layer, and there is no absorption in the resonant layer, we have at resonance: A 2 2 n = B 2 2 n = τ 02 2 (1 ρ 20 ) 2 (3.7) The enhancement factor defined as M n = ( ψ 2 2 ) ψ 2 n = ψ 2 n, where ψ is the wave function of the incident neutrons ( ψ 2 = 1), and ψ 2 is the wave function in the resonant layer, is represented by the following expression: M n = 4 A 2 2 = 4 B 2 2 = 4 1 ρ 02 2 (1 ρ 20 ) = ρ 02 2 (1 ρ 20 = τ , (3.8) 1 τ 02 2 if τ One should notice that the enhancement factor is specific to the resonance orders (or to the discrete wavelengths ). As the transmission coefficient through the layer 1 τ 02 increases with the increasing of the wave-vector transfer Q we may draw the conclusion that the enhancement factor is highest for the lowest resonance order n. The full width at half maximum (FWHM) of the resonance is determined from (1 ρ 20 ) 2 / ρ 20 2 = (d 2 k) 2, which gives E = 2k k = 2k 1 ρ 20 ρ 20 d 2 = 2k d 2 A 2 2 τ 02 ρ 20. (3.9) The equation above shows clearly that the FWHM decreases as the enhancement factor increases. Also, the tunnelling layer plays an important role for the width of the resonances through the transmission and reflection probabilities τ 02 and ρ 20. A thick tunnelling layer will cause narrow resonances and a high enhancement factor, whereas a thin one will have an opposite effect on the width and enhancement factor. This leads to limitation in experiments, i.e. the configuration of the sample have to be correlated with the maximum achievable k-resolution of the machine Bound State Neutron Resonator for non-polarized Neutrons Let us consider a particular case and show the basic properties of neutron resonances in a trilayer system depicted in Fig We will calculate the neutron density and the reflectivity of a Cu(100 A )/X(2000 A )/Cu(1000 A )/substrate neutron resonator for nonpolarized neutrons. Generally, the thick Cu layer is called reflector layer and its role is to provide a broad total reflection region. The next layer is the resonant layer where the neutron resonances occur. The top layer is the tunnelling layer and it is most often used -108-

117 Neutron Resonances in Layered Structures as the layer to be studied. We focus our attention on the total reflection region. The real part of the nuclear potential of the resonant layer was chosen to be V = This value provides the whole range of the resonances (n=1,2,3...) in the resonant layer. First we discuss the neutron density in the resonant layer and in the tunnelling layer shown in Fig The neutron density is the modulus square of the wave function inside the layers Ψ 2 and it has been calculated by using the GMM method. We observe that in the resonant layer all the resonance orders are present. The enhancement factor varies from M = 150 for n = 1 to M = 10 for n = 6. The highest value of M is achieved for the lower energetic state at n = 1. The absorption part of the potential was taken zero for this calculation. Experimentally the absorption in the resonant layer reduces drastically the enhancement factor. In the tunnelling layer we observe that the neutron density is also enhanced at the incoming neutron wave vector which satisfy the resonant condition. The neutron reflectivity is plotted as a function of perpendicular component of the wave vector k 0 = 2π sin(α)/λ. The black curve is the calculated reflectivity for the case when the absorption of the resonant layer is neglected. It exibits no dips in the total reflection region, even so the neutron resonances occur in the resonant layer. All the neutrons are reflected back into the air. When, however we make use of the imaginary part of the resonant layer, the reflectivity curve is completely different. Down peaks appears at the positions where the resonance condition is fulfilled. This is clearly seen for the reflectivity denoted as red line in Fig For calculating this reflectivity we used the following potential for the resonant layer: V = 10 8 i10 10 ev. Another possibility to visualize the neutron resonances in the reflectivity curve is to use a high absorbing material for the tunnelling layer as reported in Ref. [147]. When the absorption is negligible, the neutron channelling at the edge of the sample reveals also resonances [135, 148] Bound State Neutron Resonator for Polarized Neutrons In this section we discuss a particular case of neutron resonator proper for studying magnetic interfaces. Usually, the information about roughness is retrieved from reflectivities at high wave-vector transfer, which is several times higher than its critical value for total reflection. This is due to the inverse variation of the wavelength with the wave-vector. Larger perpendicular components of the wave vector are sensing better smaller length scales such as the interfaces. However, at high grazing angles, the intensity of the reflected neutrons is weak. Therefore it is rather time consuming for neutrons to measure roughness at high wave vectors. The other classical approach of retrieving information about roughness is by measuring transverse scans. For this case, again, the intensity is limited. Therefore, small effects could remained hidden in the background. A method to overcome the low reflected intensity is to study multilayers and superlatices. The large number of interfaces provide high enough intensity to study roughness with monolayer sensitivity [149]. Here, we should bear in mind that the neutrons average over all interfaces providing mean parameters. Therefore growth techniques should provide identical interfaces for an easy and transparent analysis. Even so for the case of polarized neutron off-specular scattering, the analysis of the data is rather cumbersome, as pointed out in Ref. [128], the multilayers -109-

118 Neutron Resonances in Layered Structures Figure 3.1: Top panel: The potential well structure of a neutron resonator. Middle panel: Neutron reflectvity from the neutron resonator with (red line) and without (black line) absorption in the resonant layer. Bottom panel: The neutron density in the resonant and tunnelling layer plotted as a function of perpendicular component of the incident wave vector k 0. Notice that the tails of the neutron resonances extends also in the tunnelling (Cu) layer

119 Neutron Resonances in Layered Structures are attractive for studying magnetic interfaces. An alternative is given by the neutron resonances and standing waves, which can provide sensitivity to single interfaces [10]. Figure 3.2: Right: The layer structure of a magnetic neutron resonator. Left: The potential profile for the resonator. The + state (black line) forms a potential well leading to the formation of neutron resonance3 inside the resonant layer. The potential for the - state (dashed red line) does not allow resonant behavior of the neutron density, but leads to the formation of standing waves. A particular case of neutron resonator is discussed in connection with the sensitivity to the magnetic interfaces. The system depicted in Fig. 3.2 consists of a reflector layer (Cu), a resonant layer (Ti) and a tunnelling layer(co). The tunnelling layer and its interfaces (interface 1 and interface 2) are magnetically active and they are, for this case, the systems to be investigated. The nuclear potential for the + state forms a quantum well, whereas for the - state the potential profile allows only the formation of standing waves below the critical wave-vector of the reflector layer. The neutron density in both resonator and tunnelling layers are shown in Fig It has been calculated for both + and - states. Also, it was assumed the magnetization inside the Co layer is oriented 45 degrees away from the initial polarization of the neutrons. Thus, the wave-function is calculated as a coherent superposition of the + and - states inside the layers: Ψ ±2 = Ψ ± + + Ψ ± 2. The resonant behavior is clearly seen for the + state. There are some differences when comparing to the previously discussed resonator. There all resonance orders were present in the resonant layer, whereas here the resonances for n=1,2 and 3 are missing. This is due to the optical parameter of the RL which does not fulfill the resonance condition for low orders. Also, similar to the previous case, the tail of the resonances extends over the interfaces of the tunnelling layer. Therefore, sensitivity to the magnetic properties of the interface is enhanced. The neutron density for the - state is completely different. The resonances are not formed because the negative potential of Co does not lead to the formation of a potential well. For this case the maximum neutron density inside the layers is about 4 which is consid

120 Neutron Resonances in Layered Structures Figure 3.3: The neutron density inside both resonant layer and tunnelling layer. For the + state the minimum resonant order is n=4. The lower orders are missing due to the optical parameters of the system. For the - state standing waves occur. For both regimes the sensitivity to the interface is enhanced. erably lower then the enhancement factor for the resonances, which is about 80 for n=4. Nevertheless, this regime is also very useful for studying the interfaces. Notice that the maximum of the neutron density is formed on the interface 1, opposite to the resonant case when the neutron density is higher on the interface 2. Exploiting these properties, it is possible to perform interface specific PNS studies. In order to see the resonances and the standing waves in the reflectivity curve one should, for the collinear case, set an angle between the neutron polarization and the orientation of the magnetization inside the Co layer. This leads to the reflectivity curves shown in -112-

121 Neutron Resonances in Layered Structures Fig An angle of 45 between the neutron polarization and the magnetization of the Co layer was assumed for calculation of these reflectivities. It is easy to distinguish sharp dips and broad dips, corresponding to the resonances and standing wave regimes, respectively. When studying off-specular scattering, the same features appear both in the specular and in the off-specular channels Interface and Interface-Specific Sensitivity of PNR The questions we would like to address here are: a) How sensitive is PNR to interfaces when using neutron resonators? b) What does interface-specific sensitivity of the PNR means when using neutron resonators. In order to answer this questions we perform numerical calculations for 3-situations and for a particular film structure depicted in Fig We consider a neutron resonator substrate T i(2000 A )/Cu(1000 A )/Al 2 O 3. On top of it we place a three layer system: interface1 (2 A )/Co(150 A )/interface2 (2 A ). The interface1 and interface2 also consist of Co but they are only 2 A thick, which is approximatively one monolayer. Next, we calculate R + and R + reflectivities for two different situation: 1) R ± (θ interface1 = 90, θ Co = 0, θ interface2 = 0 ) R ± (90, 0, 0 ), and 2) R ± (θ interface1 = 90, θ Co = 0, θ interface2 = 0 ) R ± (0, 0, 90 ), where θ is the angle between the incident neutron polarization and the magnetization orientation inside the specific layer or interface layer. The two R + and R + reflectivities for each of the two situations described above are plotted in Fig One notices that the spin-flip reflectivities are equal, R + =R + for each case taken separately. The curves are plotted against the wave vector transfer Q. In order to answer the question about the sensitivity to the interfaces we concentrate on the maximum value of the reflectivities. For both cases the spin-flip reflectivities are more than 10 times higher then the actual sensitivity of the neutron reflectivity sensitivity which is, pessimistically Therefore, we may comfortably state that the PNR from neutron resonators is sensitive to monolayer thick magnetic interfaces. The intensity distribution as a function of Q reflects the enhancement factor of the neutron density inside the layers. For instance, the intensity for the R + (0, 0, 90 ) is higher for the highest enhancement factor, which occurs at lower Q. Also, it is clearly seen that the maximum intensity of the R + (90, 0, 0 ) is much higher then the maximum intensity of R + (0, 0, 90 ), because the enhancement factor of the neutron resonances is higher then the enhancement factor of the neutron standing waves. Therefore, the enhancement factor of the neutron density inside the layers is essential for increasing the sensitivity to interfaces. The second question, we stated at the beginning, can be easily answered by analyzing the Fig It is clearly seen that when the interface1 is not collinear to the neutron polarization, the spin-flip reflectivity shows broad peaks specific to the neutron standing waves. When, however, the interface2 is not collinear to the neutron polarization, then the reflectivity is completely different from the previous case. The width of the peaks are smaller and their position is different. Therefore, we may state that PNR is interfacespecific and monolayer sensitive when used with neutron resonators

122 Neutron Resonances in Layered Structures Figure 3.4: Interface sensitivity and interface-specific sensitivity of PNR. To our knowledge these two effects have not been fully explored yet, even so adequate systems were studied experimentally and the interfacial sensitivity has been proven [10, 68]. In order to fully reveal experimentally these effects, one can prepare two systems: Co(100)/CoO(20)/N R and CoO(20)/Co(100)/N R and study them comparatively, as in the references above Resonance splitting The question we would like to address here is: What does resonance splitting mean? In order to answer this question, we calculate R + (0, 0, 90 ) and R + (0, 0, 90 ) when a field of 2000 G is applied parallel to the sample plane. These two reflectivities are shown in Fig Opposite to the previous case, when the external field was assumed to be very small, now the reflectivities are not equal (R + R + ) and the resonance positions are shifted. This shift of the resonances corresponding to + and - neutron states is called resonance splitting. Its origin stems in the sensitivity of the resonance condition to neutron energy inside the resonator. The Zeeman energy of the neutron in a field causes different resonance condition for + and - neutrons in the resonator. Therefore, the peaks in the reflectivities are shifted. These effect was observed in Ref [68]. More analytical details can -114-

123 Neutron Resonances in Layered Structures Figure 3.5: Spin-flip reflectivities for an applied magnetic field of B=2000 G. The resonance splitting is clearly visible as relative shifts of the reflectivity peaks. be found in Ref.[131]. The non-equality of the spin-flip reflectivities for high external fields was also observed for a conventional system in Ref. [150] Bright-Wigner Resonances in a Quasi Bound State Neutron Resonator In this section we provide the theoretical background and describe the experimental conditions necessary to realize either Bright-Wigner resonances or neutron standing waves in single magnetic films. Both conditions are being used in the domain wall experiment (described in the next section) to enhance the signal from domain wall scattering. Let us consider the structure we used for the present experiment, which was a thick Co ferromagnetic film. Due to the exposure in air a thin oxide layer is formed on top of the ferromagnetic layer. Thus, the real structure consists of a thick layer covered by a thin non-ferromagnetic one. However, we neglect the CoO layer as it does not contribute to the observations to be described below. We shall use for the analytical calculation the recurrence method (RM) [133, 132] and for the numerical calculation we use Generalized -115-

124 Neutron Resonances in Layered Structures Matrix Method (GMM) [132]. First we will calculate the neutron wave function inside of the Co layer, and for the sake of simplicity we consider the magnetically saturated state of the Co layer (layer between z = z 0 and z = z 1 ). The neutrons experience nuclear and magnetic potential barriers u 1 for the Co layer, and u 2 for the substrate. u 1 may be positive or negative, depending on the relative orientation of the neutron spin with respect to the layer magnetization vector. The neutron wave function[132] can be written as: Here and ψ ± 1 = [exp(ik ± 1 [z z 0 ])A ± 1 + exp( ik ± 1 [z z 0 ])B ± 1 ]. (3.10) A ± 1 = t ± 01 1 r ± 10r ± 12 exp(2ik ± 1 d 1 ), (3.11) B ± 1 = r ± 12 exp(2ik ± 1 d 1 )A ± 1, (3.12) where t 01 is the transmission coefficient from vacuum into the layer u 1, r 10 is the reflection coefficient from the layer u 1 in vacuum, and r 12 (k z ) is the reflection amplitude from the substrate u 2 inside the layer u 1. The reflection and transmission coefficients (ρ and τ) of the structure can be written as function of the amplitude A ± 1 as: ρ ± = r ± 01 + t ± 10r ± 12 exp(2ik ± 1 d 1 )A ± 1 τ ± = t ± 12 exp(ik ± 1 d 1 )A ± 1 For convenience, from now on we shall drop the ± superscript which refers to parallel or antiparallel orientation of the neutron spin to its quantization axis (which is always parallel to the magnetic induction in each media). The amplitudes t and r are complex numbers, thus they can be written as: t j = t j exp(iξ j ), r j = r j exp(iφ j ). Introducing the representation from above into eq we obtain: A 1 = t 01 1 r 10 r 12 exp( 2k 1 d 1 ) exp(iφ), (3.13) where Φ = 2k 1d 1 + φ 10 + φ 12. k j and k j are the real and imaginary part of the wave-vector. From this we recognize that the amplitudes A and B have resonant behavior due to the denominator, which is minimal, when the phase fulfils the resonance condition Φ = 2πn, where n is an integer (n > 0) called resonance order [134]. In order to emphasize the Breit-Wigner line shape of the resonance, we shall use the following approximation [135] near the n-th order of the resonance: exp(iφ) 1 + i(dφ/dk 1)(k 1 k n ) = 1 + i2d 1 (k 1 k n ). (3.14) Here we took into account that the phases φ 10 and φ 12 slowly vary with the energy. With this approximation the amplitudes A 1 can be represented in the Breit-Wigner form: A 1 = C 1 (k 1 k n ) + iγ/2, B 1 = r 12 e 2ik 1d 1 A 1, (3.15) -116-

125 Neutron Resonances in Layered Structures where the width Γ and the amplitude C 1 are: Γ = exp(2k 1 d 1 ) r 10 r 12 r 10 r 12 d 1 (3.16) t 01 C 1 = i exp(2k 1 d 1 ). 2 r 10 r 12 d 1 From the equations 3.10, 3.15 and 3.16 we can calculate the enhancement factor corresponding to the definite resonance order: M n = ψ 1 2 ψ 0 = 2 A 1 2 = 4 C 1 2 t 01 2 = 4 2 Γ 2 (1 r 10 r 12 ), (3.17) 2 where ψ 0 is the wave function in the vacuum. absorption in the Co layer by setting k = 0. In the last equation we have neglected At this point we should make a clear distinction between standing waves and neutron resonances in magnetic layers. Standing waves refer to cases where the incoming neutron kinetic energy is smaller than positive potential steps or positive potential barriers, whereas neutron resonances occur for incoming neutron kinetic energies slightly higher than positive potential barriers. Standing waves are achieved, for example, if we consider that r 10 = 0. For this case the approximation in eq is not valid and the enhancement factor is at most 4. Vice versa, for the case of a neutron resonator the width Γ is finite and the enhancement factor is considerably higher than 4. Experimentally enhancement factors between 30 and 40 have been realized so far [147]. In Fig. 3.6c and d the neutron densities in the Co film are are calculated for the profile shown in Fig. 3.6a. In these figures the scale of the x-axis is identical to the one in Fig. 3.6b, the y-axis is the film thickness from the vacuum/film interface (0) to the film/substrate interface (2500). The color scale refers to the neutron density in the Co film for the polarization state (Fig. 3.6c) and the + polarization state (Fig. 3.6d) ( state and + state refers to the antiparallel or antiparallel orientation between the neutron magnetic moment and magnetic induction). As the nuclear potential for the state is negative, the critical angle for total reflection is given by the substrate. The enhancement factor in the total reflection region is slightly higher than 4. Therefore we are close to the standing wave regime. One clear indicator for the standing wave regime is the fact that the wave function forms antinodes at the interfaces of the Co layer. This property of the standing waves can be exploited to increase the polarized neutron scattering (PNS) sensitivity to magnetic (nonmagnetic) properties of the interface. For the case shown in Fig. 3.6, there are additional neutron density maxima within the layer. Thus, the sensitivity is not only to the interfaces. However, reducing the thickness of the layer to about 500 Å, the maxima within the layer will disappear, while the maxima of the standing waves will be preserved at the bottom and top interfaces. In this way one can maximize the PNS sensitivity to spin-flip and off-specular scattering from magnetic interfaces. For the + state the nuclear potential of Co is higher than that for the substrate. Therefore the critical regime where total reflection occurs is different from the state. First, we notice that the neutron resonance conditions are fulfilled for energies which are higher -117-

126 Neutron Resonances in Layered Structures Figure 3.6: Top panel (a): The scattering length density (sld) profile for a Co layer deposited on a substrate. The black line is the positive sld which corresponds to the + state. The dashed line is the sld for the - state. + or - state refers to the case when the neutron magnetic moment is parallel or antiparallel, respectively, to the magnetic induction. The dotted lines on top of the potential profile are several resonances which occurs just above the sld of the + state. Panel (b): Reflectivity calculations for a 2500 Å thick Co layer on a SiO 2 substrate for all four cross sections, R++, R, R+, and R +. The critical scattering vector for R++ is given by the + state of Co film, whereas for R it is given by the SiO 2 substrate. The oscillations below the critical edges are due to standing waves inside of the Co film. The resonances which occur above the + critical edge of Co are not distinguishable from the so called thickness oscillations. They are also visible in the off-specular scattering regime shown in Fig.4.7, Fig.4.8, and Fig.4.9. Panel (c): Standing wave pattern for R in the Co film. Panel (d): Resonance pattern superposed on Kiessig fringes for the R++ reflectivity. M n stands for the enhancement factor of different resonance orders. For the calculations above we used the generalized matrix formalism presented in Ref. [132]. than the nuclear potential. The enhancement factor M n is about 40 for n=1, 10 for n=2, and 5 for n=3. We notice as well that the maximum of neutron density is now concentrated within the layer. Nevertheless, the tail of the wave-function extends over the interfaces. As the enhancement factor of neutron resonances is much higher than the one of neutron standing waves, the amplitude of the wave function tail on the interfaces could be higher than 4. Thus, the sensitivity of the neutron resonances on the interfaces provides a higher enhancement for the spin-flip and off-specular scattering. For the domain wall experiment we used both the standing waves and neutron resonances -118-

127 Neutron Resonances in Layered Structures to enhanced off-specular scattering from magnetic domains, which occurs not only at the interface but in the whole Co layer. As can be seen from the Fig. 3.6 and from the experimental data, the standing waves regime occurs for the - neutron state from q 0 = 0, where q 0 = 2π sin θ / λ is the wave vector transfer, λ the wavelength of the incoming neutrons, and θ is the incident angle) up to the critical angle for total reflection while the neutron resonances occur for the + neutron state. The neutron resonances are taking place only for + neutrons at incident angles which are slightly higher than the critical angle of total reflection for + neutrons. The spin-flip off-specular scattering reveals clearly both, the enhancement due to standing waves and the enhancement due to neutron resonances. In Fig. 3.6b we have calculated the reflectivities for the system described above, which are based on the generalized matrix formalism presented in Ref. [132]. For the calculation of the neutron reflectivity we have assumed an angle of 45 between the neutron polarization axis and the direction of the magnetic induction inside the layer. This allows for neutron spin flip processes inside of the sample which than signify more clearly the q 0 values for the resonance conditions. One can clearly see that the minima of the neutron non-spin flip intensity and the maxima of the spin-flip intensity occur at the same position as the maxima of the respective neutron densities (compare Fig. 3.6c and 3.6d)

128 Off-specular Neutron Scattering from Magnetic Domains -120-

129 Chapter 4 Off-specular Neutron Scattering from Magnetic Domain Walls 121

130 Off-specular Neutron Scattering from Magnetic Domains 4.1 The polarized 3 He gas spin filter technique for polarization analysis In the following we describe the experimental set-up for the investigation of off-specular scattering, which requires a polarized incident neutron beam and a complete polarization analysis of the exit beam. For specular neutron reflectivity complete neutron spin analysis is a well documented technique and is frequently applied for the analysis of magnetic thin films and superlattices. In this case, supermirrors in the incident and exit beam, used either in reflection or in transmission mode, together with Mezei spin-flippers providing all four cross-section R +,+, R,, R +,, R,+. However, off-specular scattering requires the spin analysis of the exit beam over a large solid angle. This can be achieve either with a stack of supermirrors or with a spin filter technique in the exit beam. We have carried out most of our measurements using the ADAM [151, 152] and EVA [153] reflectometers at the Institut Laue-Langevin, Grenoble, EU. The EVA and ADAM reflectometers were used in combination with the wide angle 3 He spin filter analyzer. The principle of the 3 He spin filter has been described in detail in Refs. [154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168]. Here we provide a brief outline and show that the initial polarization power of a 3 He spin filter can be as high as that of a supermirror. This was achieved for first time during the experiments performed in November 2002 at EVA-ILL (Fig. 4.9 ) Polarized 3 He Gas Spin Filter on EVA The scheme of the experimental set-up is shown in Fig After the polarizer the spin polarization vector is aligned with the guide-field. Subsequently, it passes through a spinflipper (SF1), which rotates the neutron spins by 180 when activated. Depending on the initial polarization and the scattering potentials, the neutron beam scattered from the sample may change its polarization vector. The component of final neutron polarization parallel to the applied magnetic field is measured by a combination of a second spin flipper and a polarized gaseous 3 He neutron spin filter. Using this particular set-up one is able to accomplish a longitudinal polarization analysis over a wide enough solid angle to collect the four spin-resolved off-specular maps as shown in Fig. 4.8 and Fig In Fig. 4.2 the 3 He spin filter is sketched. As the 3 He nuclei are polarized, the absorption cross section of spin up and spin down neutrons are vastly different: σ a =5 bn and σ a =16500 bn for the up and down polarized neutrons, respectively. The combination of a proper container thickness and gas pressure provides the optimal compromise between polarization and transmission. At present a gas polarization of about 70% is achieved which leads to a neutron polarization efficiency comparable to the one provided by the supermirrors, which is about 97%. For the experiment shown in Fig. 4.8, the reflectivity was high enough to deliberately sacrifice some transmission (35% for the unflipped configuration) in order to have a high analyzing power (96%). As a consequence, the raw data shown in Fig. 4.8 for specular and off-specular regions correspond to well resolved in and out spin-states

131 Off-specular Neutron Scattering from Magnetic Domains Figure 4.1: Schematic outline of the experimental set up for measuring polarized specular and off-specular neutron intensities. Monochromatic and polarized neutrons enter from the left. Their orientation can be flipped by the first spin flipper SF1, and after scattering at the sample, again in spin flipper SF2. The guiding magnetic field is perpendicular to the scattering plane. The 3 He gas spin filter in the exit beam allows only transmission of spin up neutrons. The wide acceptance angle of the spin filter warrants an efficient spin polarization analysis of diffuse magnetic scattering. The detector and the He cell were kept fixed during the experiment. The analyzing power (P ) of the polarizer and the analyzing efficiency (F ) of the 3 He spin filter are: P3 He = T + T T + +T F3 He = T + T = 1 P 3 He 1 P 3He (4.1) Experimentally, however, the analyzing power of the spin filter is not directly measured but the analyzing power of the total instrument, which includes four optical elements: polarizer, two spin flippers and the analyzer. The analyzing power of the 3 He filter can be extracted using the following relationship: P3 He = P instrument /P polarizer, (4.2) where the polarizing power (shown in Fig. 4.3) was measured using another 3 He cell which provides 100% analyzing power. The neutron transmissions T + +, T +, T +, T of the instrument (by moving the sample out of the beam) were measured during the experiment shown in Fig Inserting these values into the equations of Ref.[126], we have calculated the evolution of the 3 He analyzing power as a function of time shown in Fig It is clearly seen that the analyzing power P3 He at the beginning of the experiment (96%) is very close to the polarizing power provided by supermirrors, and then it decays exponentially in time. Nevertheless, after 20 h of measurements the analyzing power is still high (86%) and simple data corrections provides accurate experimental cross-sections. By fitting P3 He with an exponential decay function (P (t) = P (0)exp( t/t 1/2 )) we have obtained a half-life time of T 1/2 = 180 h. From this curve and using the expressions (4.1) -123-

132 Off-specular Neutron Scattering from Magnetic Domains Figure 4.2: Sketch of the 3 He gas spin-filter. Neutrons polarized parallel to 3 He experience an absorption cross section of 5 barn, those of antiparallel polarization an absorption cross section of barn. The thickness of the 3 He gas spin-filter has to be optimized for maximum polarization and transmission. In the present experiments the best polarization was 96% and the transmission of (+) neutrons was 35%. These values are valid for Fig For the experiment shown in Fig. 4.9, the polarization of the 3 He bottle was 94% with a 40% transmission of (+) neutrons. we deduce an analyzing efficiency of the 3 He optical device of about 50% at the beginning of the experiment. The measured analyzing efficiency of the instrument is also shown. Its starting value was about 27% Polarized 3 He Gas Spin Filter on ADAM The polarized 3 He gas spin filter provided at ILL is being used for polarized neutron reflectivity experiments [155, 68, 169] to measure diffuse scattering from thin magnetic films. The device itself is a cylindrical cell of 100 mm in length and 50 mm in diameter, filled with 1 bar 3 He gas. After filling, the neutron analyzing power of the cell P3 He = P instrument /P p (where the P instrument is the measured neutron polarization through both the polarizer and analyzer and P p = 96.4% is the measured polarization of the polarizer) was 94 % and it decayed to about 92 % in 24 hours. The neutron polarization is comparable to the one provided by the supermirror analyzers. The transmission of the neutrons through the filter cell (40 %) is spin dependent with the following cross-sections: σ =5 bn and σ =16500 bn. In order to record the scattered intensities corresponding to all four neutron scattering cross-sections (R + +, R +, R +, R ), the experimental set-up includes, in addition, two Mezei spin flippers placed upstream and downstream of the sample. The advantage of using the 3 He spin analyzer is that it is possible to register the spin-resolved -124-

133 Off-specular Neutron Scattering from Magnetic Domains Figure 4.3: The time dependence of experimental parameters related to the 3 He spin filter are plotted. As a general rule, the points and circles are experimentally measured data while the lines are fits. Exception is the green line which is the measured (see text) polarization power of the polarizer which is constant in time with P polarizer = 97%. The experimental (full black circles) and fitted (black line) analyzing power of the 3 He neutron spin filter P3 He is recalculated as P instrument /P polarizer. The blue line is the calculated He bottle analyzing efficiency and the red full squares are the measured flipping ratio of the instrument(f instrument ) diffuse scattering for a broad range of exit angles, and simultaneously it is free of additional spin-flip or small angle scattering, as it occurs for broad angle supermirror analyzers. In order to assure a good stability for the 3 He gas polarization, the 3 He analyzer is placed in an homogeneous magnetic field. The homogeneous field (30 Oe) is provided by a magnetic box ( magic box ) which is placed on the detector arm of the ADAM instrument as in Fig The neutron polarization stability was tested for measuring fields ranging from 0 to 7 KOe. No decrease of the 3 He gas filter polarization was observed for high applied magnetic fields on the sample. Another important development is the relaxation time of the neutron polarization of the 3 He spin filter. While for previous reflectivity experiments [169] the relaxation time was 180 hours, now the measured relaxation time for the neutron polarization of the 3 He analyzer is 1000 hours. Practically, the spin filter can be used for several days, before being replaced. In the present experiments carried out at ADAM reflectometer of ILL we used a AF ordered [CoO/Co] 30 /Al 2 O 3 multilayer grown by rf-sputtering. All multilayers we have grown over the time are in an antiferromagnetic state. This state disappears after the sample is -125-

134 Off-specular Neutron Scattering from Magnetic Domains Figure 4.4: The 3 He analyzer exchange on ADAM. A). The 3 He bottle is transported in a permanent magnet magic box. The polarized 3 He is preliminary prepared by HomoSapiens 3 (HS3) using the TYREX machine(not shown). The other 2 Homo Sapiens HS1 and HS2 are rotating the electromagnetic magic box for insertion of the analyzer. HS4 is assisting the whole process. B) HS3 inserts carefully and quickly the 3 He analyzer in the electromagnetic magic box. C) The analyzer is in the magic box and the experiment can begin. magnetically saturated once and it can not be recovered anymore. It appears that a Ne el orange peel coupling [171, 172] causes the AF state to occur during the growth process. The measurements were performed at room temperature and in a remanent magnetic field equal to 10 Oe. The four intensity maps (R++, R+-,R-+, R ) shown in Fig. 4.5 have the following characteristics: the 1st order ferromagnetic peak at about αin = 1 is clearly visible in all maps and is accompanied by weak off-specular scattering, which appears to have the same intensity for all channels. The 1/2 and 3/2 AF peaks are clearly seen along with the off-specular Bragg sheets. The intensity of the off-specular magnetic scattering is much stronger for the spin flip-channels as compared with the non spin-flip maps. The intensity of the spin-flip scattering on the AF peaks depends on the relative orientation between the external neutron polarization and the magnetization orientation within the magnetic domains. There are several origins for off-specular scattering in thin films and multilayers: coherent scattering from magnetic domains, (correlated) structural and magnetic roughness, and spin-flip at domain walls. In the example shown here, practically, all the effects listed above play a major or a minor role, which makes the analysis quite difficult. At αin = 0.35 an additional peak is present in the spin-flip maps. Its origin is the formation of neutron resonances in single layer film (see Sec ), where the film is, actually, the whole multilayer. Its position is close to the angle for total reflection and can -126-

135 Off-specular Neutron Scattering from Magnetic Domains Figure 4.5: Off-specular scattering maps from [CoO/Co] 30. The measurements were performed at room temperature in the remanent state of the sample. The data shows well resolved features of AF ordered magnetic domains [170]. be used for accurate magnetization reversal studies. In conclusion, the 3 He gas spin filter was successfully used at the ADAM reflectometer to measure spin-analyzed off-specular maps from AF ordered magnetic multilayers. The stability of the 3 He polarization in high external field, the high relaxation time of the analyzing power, and the lack of additional off-specular scattering from the analyzer itself, recommend the 3 He as an excellent tool for neutron reflectivity experiments. 4.2 Off-specular scattering due to refraction at domain walls Introduction Magnetic domain imaging at remanence and during magnetization reversal has become an extremely important discipline in the area of magnetic thin films. For magnetoelectronic and spintronic device applications, the domain state and switching behavior of various -127-

136 Off-specular Neutron Scattering from Magnetic Domains vertically and laterally structured magnetic systems requires detailed investigations in the space and time domain. The experimental techniques available for domain imaging provide either real space information, such as magnetic force microscopy (MFM)[173, 174], Kerr microscopy (KM)[72], Lorentz microscopy (LM)[175, 176, 177], polarized electron emission microscopy (PEEM)[178, 179], and secondary electron microscopy with polarization analysis (SEMPA)[180], or reciprocal space information, such as resonant soft x-ray magnetic small-angle scattering (SAS)[181]. In any case, magnetic domain information is obtained via magnetic stray fields emanating from the sample (MFM) or via the local magnetic polarization. Most of the methods presently available do not concentrate on the magnetic domain walls, their thickness, and their orientation. In this section we report on the observation of magnetic domain walls, using polarized neutron reflectivity (PNR). PNR has proven in the past to be an essential tool for the analysis of magnetic thin films and heterostructures, including their interfaces and domain structures [129, 130, 182, 10]. Here we will go one step further and show that magnetic domain walls can be characterized via pronounced streaks in the off-specular spin-flip (SF) scattering regime. Usually, off-specular SF neutron reflectivity is symmetric (R + = R+ ) and is discussed in terms of the Fourier transform of the magnetic domain distribution in the film plane and/or in terms of interfacial magnetic roughness [183, 128, 184, 185, 137, 186]. Off-specular asymmetric SF reflectivity (R + R+ ) can be observed, if the magnetic induction in the sample and the external magnetic field to the sample are not collinear[139, 187, 188, 189, 190]. This leads to SF processes at the sample/vacuum interface, which are always accompanied by a Zeeman energy change of the neutrons. Consequently, the SF neutrons follow a different path in the external applied field (off-specular) than the non-spin flip (NSF) neutrons (specular). The effect we report here has to be clearly distinguished from either one of these cases. We observe a striking new and asymmetric off-specular neutron reflectivity in the remanent state of a Co film, which is bunched into pronounced streaks. We argue that the asymmetry is caused by breaking the translational symmetry for the neutrons travelling in the film plane, when crossing domain walls with different magnetization orientation on either side of the wall. Neutron scattering at Bloch walls in bulk ferromagnets has been reported in the literature [191, 191, 192, 193, 194, 195]. In the present work, we consider neutron scattering from Néel domain walls in thin films with an incident angle to the domain walls close to normal. Analysis of the characteristic features of the off-specular SF scattering reveals the magnetic induction of the film at remanence, the average domain wall width, the average domain size, and domain wall angle. Since neutron scattering at domain walls is weak, enhancement factors have been employed, using neutron standing wave properties below the critical angle of total reflection, and neutron resonance conditions above the critical angle. A detailed theoretical background and description of the experimental conditions necessary to realize either Bright-Wigner resonances or neutron standing waves in thin magnetic films, is provided in Ch. 3. Furthermore, the method requires a full spin analysis of the off-specular spin-flip intensity over a large solid angle. This has been achieved with a supermirror in the incident beam and a 3 He spin filter in the exit beam. The polarized gas was provided by TYREX 3rd generation 3 He filling station recently commissioned at ILL [164, 168]

137 4.2.2 Experimental Procedures Kerr microscopy Off-specular Neutron Scattering from Magnetic Domains The Co(2500 Å)/SiO 2 sample was grown by rf-sputtering. Due to exposure to air a thin 25 oxide layer is formed on top of the Co layer. Growth conditions and substrate choice provided a polycrystalline Co film. The magnetization reversal was characterized by magnetooptical Kerr effect (MOKE) magnetometry and wide field Kerr microscopy in the longitudinal mode. MOKE hysteresis measurements did not reveal any signs of a macroscopic magnetic anisotropy in the film. Few characteristic domain images obtained during the reversal process are shown in Fig After saturating the sample at Oe, the field was reduced to the specified value where the image was recorded. A strongly modulated low angle domain structure is seen in the Kerr images. This ripple-like pattern is due to the polycrystalline nature of the film combined with the intrinsic magnetocrystalline anisotropy of Co. The magnetization within the small angle domains is partly tilted to the left and right, perpendicular to the vertical magnetic field direction. The local deviation of magnetization increases strongly close to the coercive field, where the reversal of magnetization proceeds by domain wall motion. The domain structure and the magnetic induction in the domains will next be analyzed via polarized neutron scattering. -90 Oe -45 Oe -22 Oe 0 Oe 11 Oe 15 Oe 22 Oe 24 Oe H ext 27 Oe 45 Oe 90 Oe CoO(2.5 nm)-co(250 nm)-glass Figure 4.6: Kerr microscopy images taken at several applied fields of the hysteresis loop. The magnetization reversal mechanism proceeds first with the ripple-like domain structure as decreasing the field from the saturation field of H ext = 2000Oe. At the coercive field the domain wall movement is the dominant mechanism of reversal

138 Off-specular Neutron Scattering from Magnetic Domains Polarized neutron reflectivity from a thin Co film The neutron measurements were performed for two magnetic states of the sample. One state is the saturation state shown in the top row of Fig. 4.7 and Sec The other is the remanent state. For the remanent state (H ext = 10 Oe) two geometries were used. First the sample was saturated in an applied field of Oe, then the field was reduced to 10 Oe. The sample in this orientation we refer to as the pristine remanent state. Then the sample was rotated by 90 about the film normal, which we call the rotated state. The pristine state is shown in the bottom row of Fig. 4.7 and Fig. 4.8, while the rotated state is shown in Fig The top row of the Fig. 4.7 shows two cross-sections taken in saturation: R + and R. The maps are recorded without using spin analysis. Here the intensity is plotted as a function of the incident and exit angles α in and α out, respectively, of the neutron beam with respect to the sample surface. One can easily distinguish the specular lines as well as the transmitted neutron cross-sections. This is not the case when the field is reduced from saturation (+2000 Oe) to remanence ( 10 Oe). Striking new features appear. For instance, in the R map two additional lines of intensity show up. One line is above the specular line and another one is below (even below the direct beam direction). They are due to off-specularly reflected intensities. The same features appear in the R + map, but the lines are localized at different positions. Both, R + and R maps contain flipped and non-flipped neutrons in the exit beam. Therefore, without spin analysis we can not distinguish between the different spin states in the exit beam. For understanding the origin of the off-specular intensity lines and to obtain quantitative information related to the magnetic state of the sample, it is important to record the intensity maps with a complete spin analysis. In order to measure maps for all four cross sections we have used the 3 He spin filter and the EVA instrument. All four off-specular maps are shown in Fig. 4.8 for the pristine remanent state and in Fig. 4.9 for the rotated remanent state. The diagonal intensity ridge stretching from the lower left to the upper right corner is the specularly reflected R + + and R intensity. We will discuss first these non-spin-flip (NSF) maps R + + and R. They show no off-specular signal, but only the specular scattering. The specular ridge is rich of features, confirming our discussion in the section3.1.7 on standing wave and resonance features of the neutron beam in the Co film. First, we recognize clearly the critical edge for the saturated Co film in the R + + map together with oscillations corresponding to both, standing waves and Kiessig fringes. Correspondingly, we recognize in the R map the critical edge from the substrate and standing wave features below the critical edge of the substrate. The situation is dramatically different for the spin-flip(sf) maps, R + and R +. Aside from the specular ridges there is additional off-specular intensity following a well defined curvature. In the R+ map the additional line of intensity appears at lower exit angles than the specular beam, while in the R + map this line is located at higher exit angles than the specular beam. Moreover, both off-specular intensity lines exhibit a banana shape type curvature, and, in addition, their intensities are modulated similar to the specular spin-flip reflectivity. These two features can be explained as we will show in the next section

139 Off-specular Neutron Scattering from Magnetic Domains Figure 4.7: The top row shows intensity maps for the cross sections I+ (left) and I (right) taken in saturation (2000 Oe). The bottom row shows the cross sections I+ (left) and I (right) taken in a remanent field of +10 Oe. The intensities are plotted in terms of exit angles (y-axis) versus incident angle (x-axis). The specularly reflected intensity is highlighted by a magenta colored line. In remanence (bottom row) the off-specular scattering is clearly distinguishable. The data are not spin analyzed, therefore it is not possible to distinguish between spin-flip and non spin-flip scattering. The data were taken with the ADAM reflectometer at the Institut Laue-Langevin Neutron data analysis In the following we discuss the different features of the neutron maps and their peculiar features. We will discuss these maps in terms of the paths taken by neutrons inside the sample, considering reflection and refraction effects at each boundary and at the domain walls. This is equivalent to geometric optics of electromagnetic waves, which explains the propagation of the wave field but not the intensity. Similarly, in our discussion we will explain the silent features of the maps from a geometrical point of view using Snell s law for polarized neutrons. Clearly, a complete analysis using the distorted wave approximation -131-

140 Off-specular Neutron Scattering from Magnetic Domains Figure 4.8: The left column shows intensity maps for the non spin-flip cross sections I++ (top) and I (bottom), the right column for the spin-flip cross sections I+ (top) and I + (bottom).the sample was saturated in Oe and subsequently the field was reduced to remanence( 10 Oe) before recording the maps. The intensities are plotted in terms of exit angles (y-axis) versus incident angle (x-axis). The specularly reflected intensity runs along the diagonal from the lower left corner to the upper right corner. The critical angles for total reflectivity for the up and down polarized neutrons are clearly visible in the I++ and I maps. Furthermore, the standing wave features can easily be recognized in the I map. The spin flip maps exhibit Kiessig fringes along the specular ridge as well as characteristic streaks in the off-specular regions. The streaks are due to reflection and refraction from perpendicular magnetic domain walls, as described in more detail in the text. should explain both, the geometry and the intensity. Nevertheless, the present analysis has the advantage that it provides a clear and lucid geometric interpretation with quantifiable parameters about the magnetic induction in the Co film (even at remanence) and the orientation of the domain walls

141 Off-specular Neutron Scattering from Magnetic Domains Figure 4.9: The maps refer to the same cross sections as described in Fig The only difference is that the sample after return to the remanent state was rotated by 90 degrees (in plane) such that the average layer magnetization vector is perpendicular to the neutron polarization Intensity modulation The maxima below the angle of total reflection define the incident angles for which the maxima of the standing waves occur. This situation is easily seen in Fig. 3.6, where the maxima of the neutron spin-flip reflectivities correspond to the maxima of the neutron density within the Co layer. For the neutron resonances occurring just above the angle of total reflection for (+) neutrons (in the region of the Kiessig fringes) the expected intensity oscillations (see R + map in Fig. 4.8 and Fig. 4.9) are not visible as the resolution does not allow a separation of the narrow resonances. However, the increase of the off-specular scattering intensity at incident angles, which fulfill the resonance conditions is a direct indication of neutron resonances occurring in this region

142 Off-specular Neutron Scattering from Magnetic Domains Banana shape curvature Imagine that (-) polarized neutrons enter into a magnetic domain as schematically indicated in Fig We assume that the magnetic induction vector in the domain is tilted with respect to the polarization direction of the incident neutrons as shown in Fig.4.6 and Fig Let us follow the path (red line) of the neutrons, which are off-specularly scattered into the ( +) channel. The (-) neutrons (polarized opposite to the field direction) will meet the vacuum/sample interface at position 1 and they are transmitted 1 without spin flip into the layer. Their quantization axis will turn parallel to the magnetic induction into the magnetic domain (B D ) These (-) neutrons will then travel deep into the layer (white shade) and will be reflected at the bottom interface without spin-flip. The (-) neutrons will then cross a perpendicular magnetic domain boundary and pass into the next magnetic domain (grey shade) through the domain wall. The orientation of the magnetic induction inside of this domain is tilted away from the magnetic induction in the previous domain by a tilt angle γ. Now we make a basic and fundamental assumption, which is essential for the interpretation of the banana shaped off-specular diffuse intensity. We assume that the neutrons, which cross a perpendicular domain wall, are being spin-flipped and reflected off-specularly 2. The spin-flip cross-section depends on the tilt angle γ. Due to the Zeeman energy change, the spin-flipped neutron will be refracted and they will follow a different path with respect to the non spin-flipped one. Next, these neutrons will meet the interface No. 3 and will DW SF exit into the air at an angle θ f. The same process is also valid for neutrons transmitted through the substrate (blue lines). In Fig is shown schematically the case when the neutrons are coming straight from the interface No. 1. Those neutrons will impinge on the domain wall from above and will then be transmitted through the back side of the sample. Another possibility is that neutrons bounce back and forth inside of the magnetic film and finally leave the sample again through the back side. This effect is characteristic for incident angles below the angle of total reflection (dashed blue line). The transmitted neutrons can be recognized below the horizon of the sample, which is seen as a dark line in bottom R- map of Fig The discussion above concerns the paths taken by down or ( ) neutrons which are scattered away from the specular ridge, while the specularly reflected neutrons either meet no magnetic domain wall or will pass through it without being spin-flipped. The description above is valid also for incident up or (+) neutrons. Calculations which are based on this simple model are shown as black curves in the Fig. 4.7, Fig. 4.8, and Fig It describes well the mean average position of the off-specular spin-flip intensity. The present situation is similar to the refraction effect of light from a plan-parallel thick transparent plate (thick with respect to the coherence length of the beam). The directly reflected beam and the refracted/reflected beam from the backside are parallel as long as 1 a fraction of neutrons will be specularly reflected back into the air. 2 Certainly, a fraction of neutrons will not be spin-flipped. In this case there is no Zeeman energy change and thus these neutrons will not change their path direction and, consequently, they are not scattered offspecularly -134-

143 Off-specular Neutron Scattering from Magnetic Domains the plate has a homogeneous density and composition in the lateral direction. However, it the plate is graded or exhibits a density modulation, the beams will not be parallel any more. From the beam divergence of the exit beams the difference of the refractive indices can be calculated. Figure 4.10: Top: top view, Bottom: side view. Schematic outline of the polarized neutron path in a magnetic film containing domain walls. The red arrows indicate neutrons oriented anti-parallel to the magnetic field induction inside and outside of the magnetic film, while blue arrows indicate neutrons oriented parallel to the magnetic field induction. Thick black lines show the neutron path before and after entering the film, which contains two magnetic domains, separated by a perpendicular magnetic domain wall. For more details see the main text. In this graph it is assumed that spin flip occurs only once at the top surface when the neutron enters the white magnetic domain. At the lower interface the neutron is reflected. At the domain wall between the white and grey domain, refraction occurs with spin-flip. The neutron leaves the top interface without spin-flip Calculation of the banana shaped off-specular scattering The qualitative description outlined above can be casted into a set of equations for the reflection and refraction effects. Let us write the refraction in terms of Snell s law for neutron optics at each interface 1, 2, and 3 as indicated in Fig. 4.10, yielding the following system of equations: -135-

144 Off-specular Neutron Scattering from Magnetic Domains k ± 0 cos(θ 0 ) = k ± 1 cos(θ ± 1 ) k ± 1 sin(θ ± 1 ) = k ± 2 sin(θ ± 2 ) k ± 2 cos(θ ± 2 ) = k ± 0 cos(θ ± f ) (4.3) The solutions of this system of equations are: cos(θ DW NSF cos(θ DW SF f f ) = cos(θ 0 ) ) = ±2µB D + ( 2 2m k2 0 ) cos2 (θ 0 ), (4.4) DW NSF where the θf is the exit angle of the neutrons, which have not been spin-flipped DW SF by passing through a domain wall (DW), θf is the exit angle of the neutrons, which were spin-flipped by passing through a domain wall, B D is the magnetic induction in the magnetic domain, and k 0 = 2π/λ is the incident wave number. The formulae above describe very well all features related to the banana shaped off-specularly reflected neutrons as observed experimentally. For example, the (+ ) off-specular spin flip intensity appears only when the incident angle satisfies the condition cos 2 (θ 0 ) 2µB D /( 2 2m k2 0), while for the ( +) curve there is no limiting incident angle: the equation cos 2 (θ 0 ) 2µB D /( 2 2m k2 0) is always satisfied. This is the case for Co which has a negative scattering length density for (-) polarized neutrons. The streaks can be reproduced by the following set of value (shown lines in Fig. 4.8 and 4.9) and : B D1 = B D2 = Gauss, γ = 30, and perpendicular domain walls. B D1,2 is reduced compared to the measured saturation value of Gauss, which we believe it is due to spin canting within the domains. Furthermore, the intensity of the streaks contains additional information, from which the average domain wall width [192], the average domain size D D nm, and the distribution of domain wall angles can be determined. For other magnetic materials, such as Fe, both spin states have a positive scattering length density and therefore all neutrons which pass through a domain wall must first enter into the film from the top surface with a positive potential barrier. This implies that additional optical limiting angles have to be taken into account, which are the critical angles for total reflection for up and down polarized neutrons: (cos(θ ± 0 ) u ± µ B D ). A special case is when the neutrons exit through the substrate. Such scattering can be calculated using the following equation: (±2µB D + V substrate ) DW SF cos(θf ) = ( 2 2m k2 0) + cos 2 (θ 0 ), (4.5) where V substrate is the nuclear potential of the substrate material. In the R map of Fig. 4.7 the above formulae were used to calculate the position of the scattering at negative exit angles (below the sample horizon). Next we discuss the R++ and R maps in Fig In these maps the non-spin flip specular scattering is seen as diagonal line stretching from the lower left corner to the -136-

145 Off-specular Neutron Scattering from Magnetic Domains upper right corner. In addition, one should expect off-specular diffuse scattering from double spin-flip processes. Imagine that (+) neutrons arrive at the interface No. 3 from inside of the gray shaded magnetic domain after being spin-flipped at the perpendicular domain wall (red path in Fig. 4.10). Those neutrons may again be spin-flipped by passing through the film/vacuum interface, giving rise to ( ) off-specular scattering. However, in Fig. 4.9 the R and R++ maps show very week off-specular intensity. Therefore we conclude from an experimental point of view that spin-flip processes at interface No. 3 must be negligible. The missing off-specular scattering in the R++ and R maps can be rationalized by calculating the transmission probability (++) in relation to the transmission probability (+ ) through the interface No. 3 from the inside into vacuum, using the formalism described in Ref [132]. The transmission operator from medium a (gray shaded magnetic domain) to medium b (vacuum) reads: ˆτ ab = ( ˆk a + ˆk b ) 1 2 ˆk a, where ˆk x is the wave vector operator in the medium xɛa, b with the following eigenvalues: k x ± = k0 2 + u x µb D. The proper values of ˆτ ab are: τ ++ τ + τ ab =, with: τ + τ ) cos(θ) )) τ ++ = ( ( ka k a + + (ka k a + ) cos(θ)) ( ka + k a + + k b + k+ b + ( ka k a + + k b k+ b 2 ( ) ( ) ka + k b k + a + k + b ) )) cos(θ) τ = ( (ka k a + + (ka k a + ) cos(θ)) ( ka k a + k b k+ b + ( ka k a + + k b k+ b 2 ( ) ( ) ka + k b k + a + k + b τ + = τ+ = (k a k a + ) ( ka k a + + k b ) k+ b sin(θ) 2 2 ( ) ( ) ka + k b k + a + k + b Since we consider a remanent state, the + and - values of the wave vectors in the vacuum are equal (k0 2 = k+ b = k b ). Assuming the angle φ between the magnetization direction inside of the magnetic domain to be 90 3 with respect to the external field, as it is the case for the data shown in Fig. 4.9, the ratio τ ++ 2 / τ + 2 becomes: ( ) ( k τ ++ 2 / τ + 2 = a k a + )(2k 0 + ka + k a + 2 ), (ka k a + ) 2 where k 0 = 2π sin(α out )/λ. Using for the magnetic field induction in the layer a value B=15500 Oe, and for the nuclear potential of the Co layer a value of u Co = ev, 3 for this angle the spin-flip probability reaches the maximum possible value. Therefore, the off-specular scattering in the R++ and R due to double spin-flip effects would be maximized

146 Off-specular Neutron Scattering from Magnetic Domains the probability ratios can be estimated for different outgoing angles for the off-specular intensities. For instance, for α out = 0.4 deg, the probability ratio is 300 and for α out = 0.6 deg, the probability ratio becomes Thus, the non spin-flip transmission probability is at least two orders of magnitude higher then the spin-flip transmission probability. This implies that an up neutron passing from the magnetic domain to the vacuum side through the interface No. 3 has a very low probability to be flipped. Thus, in the frame of the refraction model we can explain the observed very low non spin-flip off-specular intensity in Fig Intensity Here, we discuss the key elements which define the intensity of spin-flip scattering at domain walls (DW). For one large domain wall D DW, the neutron will adiabatically follow the magnetic induction and no SF scattering occurs. In contrast, for very small D DW the spin-flip probability is very small. Therefore, the thickness of the domain wall affects the scattered off-specular intensity. Second, the transmitted intensity through the DW is affected by the angle γ between the magnetization vectors in neighboring magnetic domains. For γ = 0 no SF scattering is expected, for γ = 90 the SF scattering should have a maximum. Also, for γ = 180 one should observe off-specular scattering because, even so the neutron spin do not flips, the magnetic field flips instead, leading to non-sf (in respect to the incident neutron polarization) off-specular signal. Third, the number of domain walls that the neutron passes through before being reflected or transmitted by the next interface affects the off-specular intensity. This gives information about the average lateral size of the magnetic domains. Numerical calculations of the transmission coefficient as function of D DW show that for SF scattering at domain walls a finite DW width and an angle γ > 0 is required Discussion The solutions (eq. 4.4) imply that at the domain walls a spin-flip process is required. A change of the direction of the spins with respect to the quantization axis gives rise to the Zeeman energy term. This phenomenon takes place only if neutrons pass from one magnetic domain to another by changing their polarization relative to its quantization axis (magnetic induction) from parallel to antiparallel or vice versa. This spin-flip scattering will depend on the relative orientation of the magnetization vector in the neighboring magnetic domains. If, for instance, no spin-flip scattering takes place by passing through a domain wall, then the neutrons just follow adiabatically the direction of the magnetic induction. This implies that the neutrons experience no Zeeman energy change and that there will be no banana shape off-specular scattering. We have derived Eq. 4.4 were derived under the assumption that the domain wall is perpendicular to the sample surface and that the interfaces are magnetically smooth. In case of an angular distribution of the orientation of the domain walls and/or magnetic roughness at the interfaces, the banana shaped off-specular spin-flip diffuse scattering will streak -138-

147 Off-specular Neutron Scattering from Magnetic Domains out towards the specular ridge. This feature has indeed been observed previously in an exchange biased CoO/Co heterostructure [68]. The most important characteristic of the here reported type of off-specular scattering is that the position of the scattered neutrons is only a function of the magnetic induction inside of the magnetic domains. This is to be distinguished from the traditional interpretation, which describes off-specular scattering as a result of coherent scattering from magnetic domains. In the latter case the rocking curve is symmetric and its widths depends on the size distributions of the magnetic domains[196, 137, 183]. The present situation is completely different as we observe here an asymmetric rocking curve and the position of the off-specular signal does not depend on the size of magnetic domains. The intensity of the streaks, however, depends on the magnetic domain size and on the width of the magnetic domain walls, as well as on the angle between the magnetic induction of the. We believe that the magnetic refraction at the domain walls is a general phenomena which occurs in magnetic thin films and heterostructures and is the underlying effect of the magnetic roughness. Let us, for example consider the off-specular scattering observed in Ref [10]. It was observed that in saturation the spin-flip off-specular signal was still measurable. It was clearly shown that it comes from the interface between the Co and CoO layer. Interpreting this scattering as originating from the coherent scattering from the magnetic domains would be rather unrealistic. Instead, assuming that refraction at the domain walls takes place the effect becomes accountable. It requires a disordered magnetic state consisting of domain walls. This simple argumentation together with the data presented above lead us to the suggestion that the magnetic roughness observed in thin magnetic films is caused by the refraction at the domain walls. We believe that the effect described in the present paper applies directly also to other data reported in the literature such as Ref [197] and Ref [198]. Another outcome of this effect relates to the ferromagnetism itself, namely to the surprising remark in Ref. [199], stating that there is no way to measure the spontaneous magnetization M s (T ) at H=0, with any reasonable accuracy. We suggest that the off-specular scattering from magnetic domain walls can indeed provide this information by sensing the magnetic induction inside of the domains. Since our method requires a measurement of angles in the range of a few degrees and a determination of the incoming wave vector, which are both easily carried out, the accuracy for describing the remanent state of a magnetic film could be in fact very high. 4.3 Final remarks In conclusion we have measured with polarized neutron reflectivity the specular and offspecular magnetic scattering from a CoO/Co multilayer and a single ferromagnetic Co film in the remanent magnetic domain state. For single film, the specular neutron reflectivity exhibits clear features which are due to quasi-bound neutron resonances for the + state and neutron standing waves for the state of the polarized neutrons. Furthermore, we have analyzed the off-specular magnetic neutron scattering using a 3 He gas spin-filter. The 3 He -139-

148 Off-specular Neutron Scattering from Magnetic Domains spin filter provided the highest analyzing power of 96% ever achieved before, comparable to the polarization of supermirrors usually used for neutron polarization and analysis. The off-specular spin-flip scattering is dominated by streaks above and below the specular ridge. They appear due to spin-flip transmission at the domain walls. These streaks at well defined angles are clearly to be distinguished from magnetic diffuse scattering centered symmetrically at the specular ridge, which are due to the size distribution of the magnetic domains. In the former case we could show that the streaks can be explained by refraction and reflection of spin polarized neutrons at domain walls, similar to the rainbow effect being the result of refraction and reflection at a water drop. We suggest that this is a new method for determining the magnetization M(T,H=0) of thin films with domains

149 Part III Exchange Bias in CoO/FM Bilayers and Multilayers 141

150

151 Chapter 5 Negative, positive and perpendicular exchange bias in CoO/Co bilayers In this chapter we describe experimental result on exchange bias obtained by studying a CoO/Co wedge sample. The results are partially published in the Physical Review B journal [10]. 5.1 Sample preparation and characterization The polycristalline Co layers are grown by rf-sputtering technique. Taking advantage of the natural gradient in the sputtering rate, the samples acquire an increasing thickness from 30 Å (s1) to 400 Å (s9) (see Fig. 5.1 ). Subsequently, the samples were exposed to air at room temperature which results in a 25 Å thick CoO layer on top of the Co film. Figure 5.1: The design of the wedge type CoO/Co(111) sample grown on a sapphire substrate. Before growth, the Al 2 O 3 substrate is cut in 9 pieces and placed on the sample holder at equal distances away from each other. The gradient growth rate provides samples s1 to s9 of different thickness. 143

152 Experimental Results on Exchange Bias A series of nine samples were prepared by rf-sputtering in 99.99% Ar (samples s1-s9 in Table. 5.1). The a-plane Al 2 O 3 (11 20) substrates were kept at 300 C during the Co deposition, which is the optimized growth temperature as concerns mosaicity, structural coherence length, and surface morphology, as determined in Ref. [200]. The substrate for sample s9 was placed in the middle of the sample holder while the other 8 substrates were placed at equal distances away from each other. Taking advantage of the natural gradient in the sputtering rate, the samples acquire an increasing thickness from 30 Å (s1) to 400 Å (s9) (see Fig. 5.1 ). Subsequently, the samples were exposed to air at room temperature which results in a 25 Å thick CoO layer on top of the Co film. The characterization of the surface morphology by Atomic Force Microscopy (AFM) shows an exceptionally low roughness of 1 Å for all samples, which has also been confirmed by x-ray reflectivity measurements performed at the W1.1 beamline of the HASYLAB. As a representative example in Fig. 5.2 the reflectivity is shown for the thickest sample (s9) together with a best fit to the data points. The thickness parameters obtained for all samples from the fits are listed in Table 5.1. High angle x-ray diffraction measurements (Fig. 5.2) show that the Co layer consists mainly of the cubic fcc phase with the (111) growth direction yielding a strong peak at Q = 3.08 Å 1, and a minority hcp phase with a tilted (10 11) growth responsible for the peak at Q = 3.25 Å 1. The observed fcc peak position is closer to the ideal fcc position at Q = Å 1 than to the ideal hcp position at Q = Å 1, which is in good agreement with earlier measurements of Co films on sapphire substrates grown at the same substrate temperature and similar thicknesses[200]. The full width at half maximum (FWHM) of the rocking curve of the pseudo Co(111) Bragg peak is 0.01 degrees for an incoming wavelength of Å, confirming a very good texture of the Co film. Natural oxidation of the Co(111) films leads to a preferential growth of CoO with (111) orientation. We have studied, in addition, epitaxial Co(111) layers, which have been thermally oxidized in-situ. Synchrotron diffraction studies of the in-plane CoO structure showed that [111] is the surface normal of the 2.5 nm thick CoO film. By analogy we assume that for the highly textured CoO/Co bilayer used for the present study, the same orientational relationship holds. If the interface were ideal, this orientation would provide uncompensated spins and therefore would exhibit an exchange bias effect. This has been proven in the work of Gökemeijer et al.[69], who showed that for CoO/Co single crystalline films an exchange bias occurs only in the (111) orientation but not in the (110) and (100) orientations. The presence of uncompensated spins at the interface is one of the key requirements for achieving a shift of the hysteresis loop [69, 32, 51, 45], other important factors are AF exchange constant and AF magnetocrystalline anisotropy [17, 9]. 5.2 Anisotropy at room temperature measured by MOKE We first show magnetic characterizations of the Co films in the unbiased state at room temperature by taking hysteresis loops with the magneto-optical Kerr effect (MOKE). Several hysteresis loops were taken with the field parallel to the film plane but with different azimuth angles of the sample. A plot of the ratio between the remanent magnetization and -144-

153 Experimental Results on Exchange Bias Figure 5.2: Top panel: Reflectivity measurement of sample s9 (CoO(25Å)/Co(400Å)).using synchrotron radiation (open dots). The solid line shows a best fit to the data points. Bottom panel: X-ray radial Bragg scan parallel to the surface normal of CoO(25Å)/Co(150Å). The main peak at Q =3.08 Å 1 is close to the ideal fcc Co(111) position, indicating a majority fcc phase. The second peak is from a minority tilted hcp component with [10 11] orientation. The inset shows a transverse scan through the Co(111) peak. saturation magnetization M re /M sat, versus the rotation angle φ, reveals a two-fold in-plane anisotropy, which is observed for all samples and which is induced by the sapphire substrate. In addition a small contribution from a four-fold anisotropy can be recognized by closer inspection. These results are in complete agreement with earlier hysteresis measurements of Co(111) films on Al 2 O 3 (11 20) substrates [201]. A typical example for an unbiased hysteresis loop recorded at 270 K is plotted in Fig. 5.4 for sample s4. It is characterized by a completely symmetric shape and a small coercive field of 60 Oe. Additional temperature dependent studies of the coercive field showed that the Néel temperature of a 25 Å thick CoO layer is about the same as in the bulk (T N =291 K). In fact, it might even be slightly higher than in the bulk as shown in Ref.[202]

154 Experimental Results on Exchange Bias Figure 5.3: Remanent magnetization divided by the saturation magnetization versus the rotation angle of the sample CoO(25Å)/Co(26 Å) (full circles) and CoO(25Å)/Co(400 Å) ( open triangles) showing. The two fold anisotropy was observed for all samples. 5.3 Temperature dependence of the exchange bias and coercive fields The temperature dependence of the hysteresis loops for samples s1-s9 was measured by a superconducting quantum interference device (SQUID) magnetometer. The exchange bias field H EB, the coercive fields H c1 and H c2, and the half width of the total coercive field H c = (H c1 H c2 )/2 ) were extracted from the hysteresis loops measured at temperatures between 310 and 5 K. The samples were cooled from above the Néel temperature to below different target temperatures in a magnetic field of H F C = Oe applied closely along the easy axis of the sample magnetization. The procedure was repeated for each target temperature. In Fig. 5.4 typical hysteresis loops are shown for the sample s4 with a Co thickness of 87 Å and for the target temperatures 270 K, 160 K, 80 K, and 5 K. The hysteresis curves were taken directly after cooling and thus before training, unless otherwise stated. The hysteresis loops show the following typical and general features: H c1 increases strongly with decreasing temperature, while H c2 remains almost constant at a field value of about 300 Oe. The slope of the hysteresis loops at H c1 is steeper than at H c2 on the return path. Fig. 5.5 summarizes the analysis of the temperature dependence of H c1, H c2, H c, and H EB for the samples s2 and s3. The top panel reproduces the coercive field of the first reversal H c1 and of the second reversal H c2 for the samples s3. Both coercive fields start to slightly increase just below the Néel temperature with the same rate of 0.21 Oe/K. At the blocking temperature of about T B 186 K, the slope increases drastically. Below T B a bifurcation for the temperature dependence of H c1 and H c2 develops. While H c1 keeps rising with a rate of 11.1 Oe/K, H c2 levels off and reaches saturation at the lowest -146-

155 Experimental Results on Exchange Bias Figure 5.4: Hysteresis loops for different temperatures of sample s4 (CoO(25Å)/Co(120Å)). For each hysteresis the bilayer was cooled in a field of Oe from 320K to the respective temperature. The hysteresis curves were taken after cooling and before training. temperature. Thus, there are three distinguishable temperature regions: (1) from T N to T B the coercive fields are equal and increase slowly; (2) close to T B the slopes increase drastically and H c1 is slightly smaller than H c2 ; (3) below T B both coercive fields develop linearly but with different slopes such that the absolute value of H c1 is bigger than H c2. Only in this last region a strong negative EB is observed. The different temperature dependencies of both coercive fields are consistent with the different magnetization reversal processes, which are recognized via polarized neutron reflectivity measurements at these points to be discussed further below. Previous studies of CoO/Co thin film systems have shown an inverse proportionality between the exchange bias field and the Co layer thickness [24]. Our experiments completely confirm these results and extend the linearity down to a thickness of 27 Å (see Table. 5.1 and Fig. 5.7). 5.4 Positive exchange bias In Fig. 5.5 the exchange bias and coercive fields for samples s2 and s3 are shown. At T B =186 K the H EB becomes positive and reaches a maximum of H EB = +20 Oe. The temperature region of the positive EB is shown on an enlarged scale in the insert of Fig This surprising feature has been observed for all our samples. After changing sign, H EB decreases steadily as the temperature is lowered. Because of the sign change we define the blocking temperature T B as the temperature where H EB first deviates from zero. This definition coincides with a linear extrapolation of H c1 to zero, shown by the solid line in the bottom panel

156 Experimental Results on Exchange Bias Figure 5.5: Top panel: Coercive fields H c1 and H c2 of sample s3. The line is a fit to the linear region of H c2. It intersects the abscissa at the blocking temperature T B = 186 K. Bottom panel: temperature dependence of the coercive field and the exchange bias field for samples s2 and s3. In the inset the positive exchange bias occurring close to the blocking temperature is shown. The lines are guides to the eye. The linear dependence of H c and H EB on the temperature is in good agreement with reports from other CoO/Co bilayers [203, 24, 32]. A positive H EB close to T B is seen in our bilayers for the first time. Positive exchange bias fields have been reported in the literature, but only for high cooling fields, when the external field exceeds the interfacial coupling, breaking the parallel alignment between the F and AF layer [12, 14]. Recently, -148-

157 Experimental Results on Exchange Bias Figure 5.6: First (solid circles) and second hysteresis (open squares) loop taken at 175 K. For the second hysteresis loop H c1 decreased while H c2 remained constant, implying a reduction of the positive exchange bias after training. three further papers have reported about the observation of a positive exchange bias effect [11, 15, 204]. In contrast to those papers we observe a weakening of the positive exchange bias after training. This is shown in Fig The positive exchange bias is considered as a proof for an antiparallel alignment of the spins at the AF/F interface in moderate cooling fields [12]. For parallel alignment of the magnetic moments in the F and AF layers the bias field would not change sign even in high fields. Therefore, the change of sign from negative to positive exchange bias as a function of applied field hints to antiferromagnet interfacial coupling requiring a superexchange mechanism working at the F/AF interface[205]. 5.5 Field cooling dependance of positive exchange bias In order get more insight into the origin of the positive exchange bias, we have measured its field cooling dependence shown in Fig The sample s2(see Table. 5.1) was cooled down from T=320 K to 175 K in different applied fields, ranging from -20 Oe to 40 koe (see the inset of (Fig. 5.8). First observation is that the exchange bias is positive for all cooling fields. Second the bias field is constant and about +20 Oe between 1 koe and 40 koe, but below 1 koe the bias field increases dramatically to twice its former value. This increase is solely due to a shift of H c1 to smaller values with decreasing cooling field, while H c2 remains constant. We also note that the increasing EB field correlates with a decreasing remanence at 320 K as the cooling field is lowered. Thus for small cooling fields the initial magnetic state of the sample is not fully saturated. At these small cooling fields -149-

158 Experimental Results on Exchange Bias (< 1000 Oe) we observe a surprisingly strong increase of the EB field. Figure 5.7: H c1 (open squares), H c2 (full squares), H EB (full circles), H c (open circles) are plotted as a function of the Co thickness. The samples were cooled down through the Néel temperature of CoO to 10 K in an applied magnetic field of Oe. The lines are linear fits to the data points. This behavior can be explained assuming two contributions to the interfacial exchange coupling. Normally the interface exchange coupling is ferromagnetic between CoO and Co as already mentioned above, being the result of a direct exchange of Co monolayers on either side of the ferro- and antiferromagnetic layer[205]. For a positive exchange bias effect we have to assume that some parts at the interface are, however, antiparallel coupled. Then, an antiparallel alignment may be caused by a superexchange type of mechanism mediated by an oxygen monolayer at the interface. It is reasonable to assume that due to roughness or thickness fluctuations, small fractions of the CoO film may be oxygen terminated instead of metal terminated at the interface. Next we discuss the training effect in the region below the blocking temperature [21, 67], which is different from the training for the positive exchange bias Fig.5.6. After closing a complete loop the hysteresis becomes symmetric and assumes an S-like shape. The H c1 field for all subsequent loops and the EB-field is smaller than for the first reversal. Thus the trained hysteresis has a different shape and a smaller EB field than the hysteresis of the virgin sample. Obviously the first reversal is dominated by a different process than the following reversals, transforming the sample from a unique and irreversible state to a different state with reproducible and symmetric branches of the descending and ascending parts of the hysteresis loop. When the sample is first cooled in a positive field, the Co layer -150-

159 Experimental Results on Exchange Bias Figure 5.8: Top: The field cooling dependence of the positive exchange bias for sample s2 (CoO(25Å)/Co(56Å)). The bilayer was cooled in different fields from above the Néel temperature of CoO to 175 K, where hysteresis loops were measured. The inset shows a hysteresis loop at T=320 K (full dots), together with the cooling field values (down triangles) as reference of the initial state of the sample. Bottom: The dependence of the coercive field values ( H c1, H c2 and H c = ( H c1 + H c2 )/2) as a function of the applied cooling field. The lines are guides to the eyes. consists of a single ferromagnetic domain. Upon cooling below the blocking temperature, the orientation of antiferromagnetic domains in CoO is affected by the exchange coupling at the AF/F interface. To optimize the interfacial energy and to avoid spin frustration it is favorable for the CoO film to develop a multi-domain state. However, after cooling in the saturation field of the ferromagnetic layer, the frustrated spins at the interface may very well be preserved, which would yield a CoO layer in a single domain state. The lower the interface roughness, the higher will be the probability for a single domain state. The first reversal at H c1 changes the magnetic state of the interface, leaving behind a more disordered spin structure. A more precise analysis of the processes which occur at -151-

160 Experimental Results on Exchange Bias Table 5.1: Values for in-plane coercive and exchange bias fields and for Co layer thicknesses of a wedge type Co/CoO sample are listed. The thickness of the CoO layer is constant and about 25 Å. The coercive fields H c1 and H c2 and the exchange bias field H EB are for samples which were cooled in a field of 2000 Oe from above the Néel temperature( 290 K) to 10 K. Sample index d Co [Å] H c1[oe] H c2 [Oe] H EB [Oe] H c s s s s s s s s the CoO/Co interface is possible with polarized neutron reflectivity measurements, to be discussed next. 5.6 Perpendicular exchange bias Next we discuss in more detail the exchange bias and magnetization reversal for our thinnest sample s1, which has a Co thickness of d Co = 27 Å. For Co-films a perpendicular anisotropy occurs at a critical thickness depending on the structure and crystallinity of the film. For a (111) oriented Co film, the critical thickness is 24 Å, whereas for poly-crystalline films perpendicular anisotropy occurs below 12 Å[206]. Our samples exhibit a strong (111) texture and the thinnest sample may therefore be predestinated for perpendicular anisotropy. For this sample we measured the magnetic hysteresis and the exchange bias effect after applying a cooling field in both directions, parallel and perpendicular to the film plane (Fig. 5.9). After cooling in a perpendicular field of Oe, the magnetic hysteresis develops a typical shift below the blocking temperature, as shown in the left panel of Fig In contrast to the samples discussed previously, the hysteresis appears to be symmetric from the beginning, indicating that no drastical irreversible changes of the interface take place during the first magnetization reversal. The training effect at T =3 K, however, causes a decrease of the exchange bias field from H EB = Oe to H EB = Oe. Therefore, a difference in the magnetization reversal at the first and second coercive fields is likely to occur [207]. The rounded S-like shape is observed at the coercive fields in decreasing as well as in increasing fields, indicative of a magnetization reversal via perpendicular domain rotation. The remanence is only about 20% of the saturation magnetization. Thus only a small fraction of the Co domains have an easy axis perpendicular to the plane. In contrast, -152-

161 Experimental Results on Exchange Bias Figure 5.9: Magnetic hysteresis loops of sample s1 (CoO(25Å)/Co(26Å)) after field cooling to 10 K in an external field of Oe perpendicular (left panel) and parallel to the film plane (right panel). Left panel: hysteresis loop for an out-of-plane field sweep. Right panel: hysteresis loop for an in-plane field sweep. after field cooling the same sample in a longitudinal field of Oe, the shape of the hysteresis is completely different, as shown in the right panel of Fig The shape is similar to the thicker samples as concerns the asymmetry at H c1 and H c2. In addition a plateau region occurs, extending over approximatively 4000 Oe at 10 K with a nearly vanishing magnetization, followed by a sharp reversal at H c1. One possible explanation for the plateau region would be a compensated domain state. However, this appears to be quite unlikely from an energetic point of few, since it would require a pinning of the domain walls up to about 6000 Oe. In order to gain further hints about the origin of this peculiar shape, we have measured minor loops starting from saturation (point A in Fig. 5.9) to fields corresponding to about half of the plateau value (point B) and back again. Along this minor loop the hysteresis exhibits features which are typical for the loops taken in a perpendicular field. Thus, we are led to the assumption that during the magnetization reversal from A B spins rotate out-of plane due to an inherent perpendicular anisotropy of the FM layer. After completing a minor loop, we have continued to measure the full hysteresis loop. The full loop recovers the complete exchange bias as determined after the first reversal without taking a minor loop. This shows that the minor loop does not affect the exchange bias and that there is essentially no training effect along the minor loop, which otherwise is strong when crossing H c1. The experimental facts lead us to the following conclusions for the longitudinal field cooling. The thin Co films appears to be decomposed into two subsystems: One subsystem reveals a coherent rotation with no exchange bias and no training effect. The other is strongly affected by exchange bias and shows a pronounced training effect. Both subsystems appear to have about the same size, since they compensate each other within the plateau region. In the plateau region these two subsystems exhibit features similar to a spin valve system. While one of the subsystem is strongly pinned to the AF CoO layer, the other one is free to -153-

162 Experimental Results on Exchange Bias rotate. In fact, closer inspection of the hysteresis reproduced in the right panel of Fig.5.9 indicates a superposition of two independent loops for longitudinal and for perpendicular cooling fields. Also the thicker films show signs for the presence of a free layer. In negative fields often a slight drop of the magnetization is observed. Obviously in thicker films the free layer is a minority component which can not compensate the magnetization in the plateau region as is the case for sample s1. In Fig we show the exchange bias fields and the coercive fields of sample s1 as a function of temperature and for perpendicular cooling fields. The perpendicular exchange bias shows the usual behavior. The EB field increases starting from the blocking temperature (T B = 186 K) and reaches a value of 300 Oe at 10 K. However, the increase is not linear as is the case for the thicker Co films, instead it follows more a Brillouin function like, as observed in Ref. [54]. The coercive field has a peculiar temperature dependence, showing a strong peak like enhancement in the region of the blocking temperature. Figure 5.10: Temperature dependence of H c1 and H c2 for the perpendicular measurement of the same sample. Figure 5.11: Temperature dependence of the perpendicular exchange bias and coercive field for sample s1 (CoO(25Å)/Co(26Å)). In the lower panel of Fig the coercive fields H c1 and H c2 are plotted separately to reveal the origin of the peak like structure. H c1 exhibits a stepwise increase at T B and then increases linearly with decreasing temperature, similar to the temperature dependence of H c1 for thicker films. H c2 mimics the peak like structure at T B and then drops to almost zero at lower temperatures. Thus there is a clear evidence that the origin of the H c peak is related to the temperature dependence of the second magnetization reversal. A similar peak like structure has often been observed [8] (and references therein) and was attributed to a weakening of the AF anisotropy close to T B, allowing the F spins to pull more AF spins upon reversal, thus increasing the coercivity. With decreasing temperature, the AF anisotropy increases and therefore the coercivity decreases again. The width of the peak is seen as an indicator for inhomogeneities in the AF layer. In Ref. [208] the broad peak in the coercivity is observed only for a specific value of the layer thickness ratio of the F to AF layers, whereas in Ref. [203] the coercivity enhancement is observed at the blocking temperature for a 1000 thick Co film. Such behavior is suggested to arise from the relative dominance of the F or the AF layer [209]

163 Experimental Results on Exchange Bias It is interesting to compare the H EB and H c fields for longitudinal (see Table 5.1) and perpendicular cooling fields (see Fig. 5.11). The longitudinal EB-field and the coercive field are larger by a factor of 4.5 and 15, respectively, as compared to the perpendicular case. Maat et. all [210] have proposed a phenomenological model in order to explain the difference between the longitudinal and perpendicular biasing assuming that the interfacial AF CoO spins freeze to the spin anisotropy axes that are closest to the applied field. This provides the out-of-plane AF spin components necessary for the perpendicular EB. This model would explain also the perpendicular anisotropy seen for the in-plane cooling direction considering that it is generated at the interface between the FM (Co) and the AFM(CoO) due to the presence of the hcp tilted phase in the system. 5.7 Conclusions In summary, using a wedge type sample with a Co thickness gradient we have determined the coercive fields and the exchange bias field after field cooling for different Co layer thicknesses. The exchange bias field at low temperature follows a 1/d F M behavior. In addition, the temperature dependence of the exchange bias field exhibits a small positive value just below the blocking temperature, which is most likely due to a superexchange mechanism with antiferromagnetic exchange coupling at the AF/F interface. For thin CoO(25Å)/Co(27Å) bilayers with a weak perpendicular anisotropy we observed an exchange bias for a perpendicular and a longitudinal orientation of the cooling field. Surprisingly, for the longitudinal orientation of the cooling field, the FM layer appears to be decoupled into two subsystems with distinguishable magnetization reversals, one subsystem being pinned to the AF layer and showing a large exchange bias effect and the other subsystem being essentially free to rotate. This leads to a magnetization compensation in a plateau region which stretches over about 4000 Oe. Overall, our data reveal clear differences between the coercive fields of the very first reversal H c1 and the second reversal at H c

164 Experimental Results on Exchange Bias -156-

165 Chapter 6 Magnetization reversal in CoO/Co exchanged biased bilayers In order to study magnetization reversal and interfacial domain formation using PNR, we have designed the sample as neutron resonator(see sec ). It provides monolayer sensitivity (see sec ) to the magnetic interface between ferromagnetic and antiferromagnetic layer. First, we show thermal relaxation of exchange bias using MOKE and then we continue with describing PNR experiments. The results presented in this chapter are partially published in Refs. [67, 10, 68] The sample is a Co(200 Å) layer deposited on a Ti(2000 Å)/Cu(1000 Å)/Al 2O 3 template. The CoO(25 Å) layer is formed on top of the Co layer by oxidation in air. The sample was characterized by X-ray diffraction at the HASYLAB, by AFM, MOKE, and by polarized neutron reflectometry (PNR). The sample is polycrystalline with a strong (111) texture growth along the growth direction. The surface roughness measured by AFM is about 3 Å, which has been confirmed by X-ray reflectivity measurements. 6.1 Characterization by MOKE Fig.6.1a) shows the magnetic hysteresis loop measured by MOKE at T=50 K, after cooling in an applied field of Oe. Upon descending the field for the first time to negative values, an abrupt magnetization reversal is observed at the coercivity field H c1. Ascending again to positive field values, the magnetization curve at H c2 is more rounded. In subsequent cycles the magnetization curves at H c1 and H c2 are of about the same shape characterized by H EB =60 Oe and H c =200 Oe. From these measurements, we conclude that the first magnetization reversal of the virgin sample after field cooling is conspicuously different from any subsequent trained reversals. This difference also becomes obvious by a strong thermal relaxation process, which is observable only in the virgin state of the sample at constant field value H<H c1, i.e. just before the sharp magnetization reversal. This is described further below. The Fig.6.1 is giving a flavor of how the thermal fluctuations affects H c1 of the system at 157

166 Experimental Results on Exchange Bias Figure 6.1: Thermal relaxation of coercivity field H c1 of sample A at T=50 K. The cooling field was about FC=2000 Oe. The inset shows the dependence of the time spent to reach the zero magnetization versus the waiting fields. T=50 K. The time relaxation measurements were performed by stopping the field driving above the magnetization reversal point (see P1, P2, P3 on Fig.6.1a).) and counting the Kerr rotation evolution. The result for several waiting fields is shown in Fig.6.1b). One notices that H c1 field is not stable in time, it decreases towards zero. Plotting the waiting field versus the time when the magnetization reverses, one obtains an exponential type of relaxation shown in Fig.6.1c). This data shows that the thermal processes lead to changes in the magnetic configuration by overcoming energy barriers. Moreover, the magnetization reversal at H c2 (not shown) after continuing the field sweep has the same behavior as for the trained hysteresis loop. The thermal relaxation could be related to irreversibly changes at the F/AF interface and/or in the AF layer. More results on the analysis of the thermal relaxation for Co/Co samples shown in the previous chapter are analyzed within the Fulcomer and Charap formalism [211] in Ref. [212]. An activation energy per spins of 3 mev was deduced from data analysis

167 Experimental Results on Exchange Bias 6.2 Asymmetric magnetization reversal as revealed by PNR MOKE is a fast method for determining hysteresis loops but cannot reveal the spin configuration at the interface. Therefore we have characterized the sample by PNR at T=10 K. The measurements were performed at the ADAM reflectometer [151, 152] installed at the ILL. The method we used to study the magnetization reversal by PNR includes the following concepts: (1) the dependence of spin asymmetry measured at one specific value of the wave vector transfer depends only on the cosine of the angle (θ) between the magnetic induction in the magnetic layer and the neutron polarization direction; (b) the spin-flip reflectivity is proportional to sin 2 (θ); (c) the off-specular scattering shows the existence of magnetic domains within the magnetic layer; (d) the enhancement of the neutron density achieved at the magnetic interface, using the neutron resonator, increases the off-specular scattering from the interfacial magnetic domain walls. Figure 6.2: Polarized neutron reflectivity measurements of a CoO/Co/Ti/Cu/Al 2 O 3 resonator sample taken at room temperature in a magnetic field of about 60 Oe (=H c1 ). The arrows for Q NSF and Q SF denote the points where non-spin flip and spin flip intensities were recorded at low temperature after field cooling. In order to find the optimized scattering vectors for measuring neutron hysteresis loops, we have first recorded a complete set of polarized neutron reflectivities for all four crosssections. The raw data without experimental corrections (spin flip efficiency and footprint) is shown in Fig The measurements were carried out at room temperature and close to the coercive field. Since at room temperature without exchange bias, reversal is dominated -159-

168 Experimental Results on Exchange Bias by coherent rotation, the resonant peaks in the spin-flip reflectivities are strong and well defined. We used this information to choose the proper wavevector (Q SF = ) for recording the spin-flip intensity as a function of the applied magnetic field at low temperatures. Similar, for the non-spin flip intensities we have fixed the scattering vector just above the critical edge for total external reflection of the unpolarized beam (Q NSF = ). SF and NSF refer to spin flip and non-spin flip scattering, respectively. We have chosen this Q NSF value, because it provides the highest contrast for non-spin flip reflectivity from the magnetic layer combined with high intensity, whereas Q SF provides the maximum intensity. Figure 6.3: (a) MOKE hysteresis loop (full squares) and neutron hysteresis loop (open squares) of a CoO/Co bilayer after field cooling to 50K (MOKE) and to 10 K (neutrons) in an external field of Oe. (b) and (c) The specular non spin-flip intensities I ++ and I and the specular spin-flip intensities I + and I + are plotted as a function of external magnetic field. The intensities are measured at special scattering vector values of the reflectivity curves (see text). The magnetic hysteresis loop can be measured by PNR using the Spin Asymmetry (SA) (see section 2.2.5). It is shown in Fig. 6.3a) together with the magnetic hysteresis obtained by MOKE from the same sample. The field dependencies of the non-spin flip (NSF) reflectivities, from which the magnetic hysteresis is derived, are shown in panels (b) and (c). The non-spin flip reflectivities were measured at a wave vector transfer Q corresponding to the inflection point of the non-polarized neutron reflectivity (near the critical edge for total external reflection) and by sweeping the magnetic field in the usual manner. Note that for technical reasons the MOKE hysteresis curve was taken at 50 K while the neutron measurements were performed at 10 K. Both curves show the same asymmetric shape -160-

169 Experimental Results on Exchange Bias at H c1 and H c2 as discussed before. The different H c1 values for the MOKE and PNR measurements are solely due to the different sample temperatures. Nonetheless, we find very good agreement between both loops during the first reversal. On the return path through H c2 the agreement is by far not as good as one would expect if no specular intensity loss were occurring due to off-specular diffuse scattering. On the other hand, this discrepancy between MOKE and PNR is an indication for the presence of magnetic domains in the ferromagnetic layer on the return path. In addition to the standard magnetization curves, PNR is capable of distinguishing between different magnetization reversal processes and to provide information on magnetic domains by analyzing the specular and the off-specular spin flip (SF) intensities I + and I +. Specular SF scattering is sensitive to magnetization components in the sample plane, which are perpendicular to the applied field direction. The off-specular SF signal reveals the presence of magnetic domains. We first discuss the specular spin flip intensities shown by triangles in panels (b) and (c) of Fig They were measured at the scattering vector Q corresponding to the resonance peak near the critical edge. The magnetization reversal at H c2 exhibits strong spin-flip intensities I + and I +. This is always observed for rounded or trained hysteresis loops and is characteristic for a magnetization reversal via domain rotation. Magnetization reversal by rotation provides a large magnetization component perpendicular to the field or polarization axis, giving rise to neutron spin-flip process. Vice versa, the rather low spinflip intensities, which are observed during the first magnetization reversal in the virgin state at H c1 are indicative of pure 180 domain wall movement. The step like intensity change at H c1 =-750 Oe is followed by a steady decrease as the system approaches saturation (see small SF reflectivity step Fig. 6.3b). The slightly enhanced specular spin-flip scattering takes place in a very narrow field range of not more than 15 Oe [FWHM=9 Oe]. From these features it is obvious that domain wall nucleation and propagation let the magnetic spins at the interface canted away from the applied field direction. 6.3 Interfacial domain formation during magnetization reversal In order clarify the reason for the small SF reflectivity increased at H c1, we have measured rocking curves at several points on the hysteresis loops. They are sensitive to the ferromagnetic domains formation and interfacial magnetic disorder. The off-specular spin flip scattering (I + ) close to H c1 is reproduced in Fig The offspecular diffuse intensities taken at H c1 = -750 Oe, at H= Oe, and upon return to H= -230 Oe are rather strong. Their asymmetric shape is attributed to the specific wavevector where they have been measured. Note that the off-specular intensity appears only after the magnetization reversal at H c1 has taken place for the first time and is negligible before. The green line (2) in Fig. 6.4bottom is the SF scattering taking in a descending field at -530 Oe (before first magnetization reversal ), representing the instrumental resolution. After reversal the diffuse scattering is large, then decreases towards saturation in negative -161-

170 Experimental Results on Exchange Bias Figure 6.4: Top: MOKE hysteresis loop (full squares) and neutron hysteresis loop (open squares) of a CoO/Co bilayer after field cooling to 50K (MOKE) and to 10 K (neutrons) in an external field of Oe. Bottom: Off-specular diffuse spin-flip scattering ( +) taken before magnetization reversal at -530 Oe (green line 1), at H c1 =-750 Oe (blue line 2), in saturation at Oe (red line 3) and before the second magnetization reversal at -230 Oe (black line 4) fields and increases again on the way back towards H c2. The diffuse spin flip scattering reveals the existence of magnetically disordered interface consisting of domain walls. We argue that those domain walls are located at the interface between the ferromagnetic and antiferromagnetic layer and that they have to be distinguished from ferromagnetic domains in the Co layer. If the diffuse scattering were originating from domain walls inside of the Co-layer, then the off-specular scattering should vanish in saturation, which is clearly not -162-

171 Experimental Results on Exchange Bias the case here. The creation of interfacial domains is crucial for the change of the reversal character from domain wall motion at H c1 to domain rotation at H c2. The breakdown occurs already during the very first reversal and does not heal anymore even in saturation. We argue that the observed domain formation is the reason for the trained hysteresis curve AF magnetic state after the field cooling. In the previous section we have shown off-specular scans measured at ADAM. The data suggests that before the very first reversal the interface is not disordered because we did not see any off-specular signal. This is an important point when discussing the state of the AF layer. Micromagnetic simulations [45, 31] suggest that during the field cooling the AF develops a domain state. These domains, should create disorder at the interface to which we are sensitive through off-specular scattering. The data above did dot show this disorder before the first reversal at H c1. It appears only after the very first reversal, which is consistent with a AF break-up into domains. Therefore, we show full spin-analyzed off-specular maps which strengthen the observations in the previous section. They also help in understanding the origin of the asymmetric shape of the off-specular signal observed in the rocking curves. The experiments were performed at the EVA reflectometer of the Institut Laue-Langevin (Grenoble, France) [153] with a neutron wavelength of 5.5 Å, using a 1D position sensitive detector (PSD). We have used the polarized 3 He gas-filter, which allows to analyze wide angle off-specular scattering. We discuss first the Fig ). The system (the same sample as before) is cooled down in an applied magnetic field of H a = Oe through the Néel temperature of the antiferromagnet to 10 K. Then, the field is reduced to -370 Oe, which is about two times smaller then the coercive field H c2, where the magnetization would first reverse. Here, we rotated the sample by 45 about the sample normal and record the four the spinanalyzed reflectivities, R + +, R +, R +, and R. (The reason for rotation is related to some scattering details. ) In this orientation the magnetization is oriented also 45 with respect to the neutron polarization, therefore we observe scattering in non-spin flip and in the spin-flip channels. The diagonal scattering is the specular reflected intensity. Important for these maps, is that we do not observe any off-specular scattering, which, if existed, would appear away from the specular reflectivities, at higher or smaller exit angles. Therefore, we suggest that after field cooling the AF state appears to be in a single domain state. This is not the case after the first magnetization reversal and further on, during the subsequent hysteresis measurements which is discussed further below. ( After measuring these four maps (Fig )), the sample was rotated back to its original orientation.) After the second magnetization reversal ath c2 = +230 Oe, the system is supposed to approach the original magnetic configuration. In order to show that this is not the case for our exchange biased bilayer, we have measured four off-specular neutron reflectivity maps R++, R+, R +, R shown in Fig ), at H +370 Oe, where the magnetic spins were mostly reversed back to the direction of the applied field. They show a striking behavior in the total reflection region: while the non-spin-flip scattering exhibits no diffuse reflectivity, the spin-flip scattering shows strong diffuse scattering at incident angles which -163-

172 Experimental Results on Exchange Bias Figure 6.5: Several spin-analyzed maps recorded during the magnetization reversal. 1) This map was recorded in an applied magnetic field of -370 Oe. The sample was cooled in a field of Oe. Then the field was reduced to the measuring field. Before recording the map, the sample was rotated by 45 around the sample normal. By doing so we observe signal in all channels. The key observation is that no off-specular scattering is observed before very firs reversal. After measuring the map the sample was rotated back to the field original field cooling orientation. 2) Next we reverse the magnetization at H c1, saturate the sample in negative fields, and raise the field to the second field which is +370 Oe. Here, we observe off-specular signal due to magnetic domains. 3) We saturate the sample in positive field, and reduce the field to -370 Oe. Here we record again a map where we observe again off-specular scattering due to domains in the Co layer. The spin-flip observed in the map 2) and 3) denotes that the magnetization in the domains is rotated as compared with the field cooling direction. satisfy the resonance conditions. Moreover the spin-flip off-specular part of the reflectivity is asymmetric. The R + diffuse intensity occurs at higher exit angles α out than the specularly reflected neutrons, and R+ intensity is shifted to lower angles. Their intensities -164-