Bragg-MOKE and Vector-MOKE Investigations: Magnetic Reversal of Patterned Microstripes

Transcription

1 Bragg-MOKE and Vector-MOKE Investigations: Magnetic Reversal of Patterned Microstripes DISSERTATION zur Erlangung des Grades eines Doktors der Naturwissenschaften der Fakultät für Physik und Astronomie an der Ruhr-Universität Bochum vorgelegt von Till Schmitte Bochum 22

2 Mit Genehmigung des Dekanats vom wurde die Dissertation in englischer Sprache verfasst. Eine deutschsprachige Zusammenfassung befindet sich am Ende der Arbeit. Mit Genehmigung des Dekanats vom wurden Teile dieser Arbeit vorab veröffentlicht. Eine Zusammenstellung befindet sich am Ende der Dissertation. Dissertation eingereicht am Erstgutachter: Prof. Dr. H. Zabel, Bochum Zweitgutachter: Prof. Dr. W. Kleemann, Duisburg Disputation am

3 Contents I. Introduction 5 1. Introduction 7 2. Magnetism of thin films and thin film elements Free energy Domains Observation of domains and magnetic hysteresis in magnetic stripes Conclusion II. Methods Magneto-optical Kerr effect of thin films and thin film grating structures Theory of the Kerr effect Kerr effect - basics Electro-magnetic theory of the Kerr effect Second order contributions to the longitudinal Kerr effect Vector-MOKE Diffraction gratings Bragg-MOKE Review of Bragg-MOKE literature Some simulations of Bragg-MOKE effects Interference between stripe and substrate MOKE setup Standard setup Measurement method Extensions of the standard setup Sample preparation Thin film preparation Molecular beam epitaxy rf-sputtering Lithography Electron-beam lithography

4 Contents Other Lithography techniques Image transfer Imaging Scanning electron microscopy AFM and MFM Microscopy and Kerr microscopy III. Results and discussion Anisotropy of Fe(1) Introduction Measurements and discussion Fe-nanowires Introduction Sample preparation and experimental setup Experimental results Magnetic properties of the continuous Fe film Magnetic properties of the Fe nanowire array: longitudinal component Magnetic properties of the Fe nanowire array: transverse component Analysis and discussion Conclusions CoFe grating Introduction Sample preparation Remagnetization process of the CoFe-grating Results from MOKE measurements Results from Kerr-microscopy Bragg-MOKE measurements at the CoFe grating sample Summary Ni-gratings Introduction Experimental setup Results and Discussion Bragg-MOKE MFM measurements Summary and Conclusion Fe-gratings Introduction Sample Preparation Results

5 Contents Single crystal film, sample A Polycrystalline Fe-gratings Discussion Saturation Bragg-MOKE signal Shape of Bragg-MOKE curves of the single crystalline sample Shape of Bragg-MOKE curves of the polycrystalline sample Summary and Conclusions Co gratings on a Fe-film Introduction Experimental details Results Standard MOKE measurements Bragg-MOKE measurements Intensity measurements Discussion Increasing Kerr effect in the spin valve region Shape of Bragg-MOKE curves Bragg-MOKE amplitude Summary and Conclusion Further measurements Diffuse Kerr effect Fe grating with giant Kerr rotation Conclusions 147 Bibliography 153 Zusammenfassung 161 Publications 168 Acknowledgments 169 Lebenslauf 17 3

6 Contents 4

7 Part I. Introduction 5

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9 1. Introduction Motivation The understanding of the magnetization reversal process of artificially structured magnetic islands and wires is important both from a fundamental point of view and also for potential magneto-electronic device applications [1] or mass storage devices. Of particular interest for the design of magnetic thin film devices such as read-heads and magnetic random access memories (MRAM) is the magnetic domain structure within these microor nano-structured elements, their remanent magnetization, and the shape of their magnetic hysteresis loop. On the one hand, these parameters primarily depend on both the shape and the aspect ratio of the magnetic elements, and on the other hand, they depend on the intrinsic magnetic anisotropy constants of the magnetic material used [2]. Particularly, if magnetic islands or wires are separated by only small distances, long-range magnetic dipole interaction between the elements also has to be taken into account [3]. Generally, the interest in this new materials raises new questions in the field of experimental techniques for the measurements of the micromagnetic properties. Magnetic domain structures as well as the magnetization reversal process of nanostructured magnetic elements may be investigated by a number of experimental methods. On the one hand, magnetic domains of single magnetic elements may be imaged in real space by various techniques such as Kerr-microscopy [4], Lorentz-microscopy [5], scanning electron microscopy with polarization analysis (SEMPA) [6], X-ray magnetic circular dicroism (XMCD) microscopy [7] or magnetic force microscopy (MFM) [8]. Hysteresis loops of magnetic elements are derived by evaluating the total size of magnetic domains having a particular direction of the magnetization vector with respect to the direction of the applied magnetic field. Also, the hysteresis loop of single domain magnetic elements may be measured with magnetic force microscopy, by using a calibrated MFM-tip [9]. On the other hand, hysteresis loops of magnetic elements may as well be measured via the magneto-optical Kerr effect (MOKE), superconducting quantum interference device (SQUID) magnetometry or vibrating sample magnetometry (VSM) for which the magnetization reversal process may as well be identified from the shape of the corresponding hysteresis loop. However, resolution and accuracy of the latter techniques ask for a large number of identical elements to be investigated in parallel, such that the obtained hysteresis loop yields information upon the average magnetization reversal process of all elements, and not just upon the magnetization reversal of a single element. Nevertheless, such techniques more easily allow for hysteresis loop measurements taken at various angles e.g. in orthogonal directions, from which the vector of the magnetization can be reconstructed. 7

10 1. Introduction In the present study two new techniques based on the magneto-optical Kerr effect (MOKE) are explored and used to investigate the remagnetization process of arrays of magnetic stripes or wires. First, the MOKE can easily be operated as a vector-magnetometer. In addition to the longitudinal MOKE geometry, also the perpendicular component of the magnetization of nanowires is measured by applying an external magnetic field in a direction normal to the plane of incidence. Both the longitudinal and the perpendicular field orientation allows to derive a vector model for the magnetization process, following previous work by Daboo et al. [1]. Second, a diffraction technique is introduced: Whereas the magneto-optical Kerr effect is a well-established method for the investigation of thin film magnetism, the application to samples with lateral structures of the order of the wavelength of the illuminating laser light is new and challenging, promising to be a powerful technique. Here the laterally structured sample acts as an optical grating leading to interference effects in the reflected laser light. Principally scattering-techniques on periodic arrays of stripes or dots can provide valuable information for the study of micromagnetism. When laser light is reflected from these samples, Kerr hysteresis loops cannot only be measured in specular reflection but also at diffraction spots of different order. This technique has been named Bragg-MOKE. For instance, for ferromagnetic line gratings, the combination of diffraction and the magneto-optical Kerr effect (MOKE) can yield information about the mean lateral magnetization distribution [11]. The technique can also be used to change the sign and the amplitude of the MOKE signal [12]. Whereas measurements of the Bragg-MOKE exist for the polar [12] and transverse [11, 13, 14, 15] MOKE configuration, Bragg-MOKE hysteresis measurements in the longitudinal geometry have not been published so far 1. Aim of this thesis This thesis has two main goals: One is to investigate the remagnetization process of micro- and nano-patterned grating arrays and systematic studies of the remagnetization process will be presented. Geometrical factors as aspect ratio and angle between stripes and magnetic field as well as material parameters are varied. Especially Fe is in the focus of this thesis. This material can be prepared in polycrystalline and single crystalline states each with different magnetic anisotropy and hence completely different remagnetization processes. A variety of stripe arrays will be analyzed mainly using the MOKE, and thus revealing integrated information of the magnetic properties. The second goal is to to explore the potential of the magneto-optical techniques Bragg- MOKE and vector-moke. In particular Bragg-MOKE needs further investigations as the observed effects are rather intriguing. Three main effects will be of interest: The influence of interference effects, e.g. between light reflected by the grating structure and the surrounding substrate, may amplify the observed Kerr rotation. In diffraction geometries the incident angle is not identical to the reflecting angle. 1 There is one exception: In [16] the authors report on Kerr-spectra in longitudinal geometry but in a different diffraction geometry (conical diffraction), no Bragg-MOKE hysteresis curves are discussed. 8

11 Therefore the question arises for off-specular Fresnel coefficients and how this will influence the longitudinal Bragg-MOKE effect. During the remagnetization process domains will occur inside the magnetic stripes. Correlated domain structures will influence the shape of the measured Bragg- MOKE curves. This thesis will demonstrate each of these manifestations of the Bragg-MOKE effect and qualitative explanations will be given. The two main subjects of this thesis, namely remagnetization processes of magnetic thin film elements and magneto-optics in diffraction geometry, are both discussed from the experimentalist point of view: Different parameters of the systems under investigation are varied systematically and the effects are recorded. The explanations given follow this phenomenological approach rather than an analytical or numerical description from first principles. Outline The thesis on hand is organized in three parts. The first part gives an introduction to the subject and a brief discussion of the domain structures observed in thin film elements. The second part deals with the methods used here to investigate ferromagnetic gratings in the nano- or micrometer scale. These are mainly the magneto-optical Kerr effect and sample preparation techniques. The MOKE and the particular detection techniques will be explained in detail and the actual state of research in the field of Bragg-MOKE is discussed. The theoretical section of the second part also consists of a section in which some basic models of the Bragg-MOKE effect are calculated analytically. The experimental results are reported in the third part of this thesis. The chapters in this part are organized following the different samples and series of samples prepared and analyzed. In addition, the chapters are partial extensions of previously published work. The third part ends with a conclusion and outlook. 9

12 1. Introduction 1

13 2. Magnetism of thin films and thin film elements The phenomenon of ferromagnetism in the 3d-metals (Fe, Co, Ni) is essentially due to a quantum mechanical exchange energy, resulting from the Pauli principle and the almost localized electrons of the 3d orbitals. The exchange energy leads to a spin asymmetry in the 3d sub-band and thus a permanent moment of the metal. In this chapter some basic facts about thin-film magnetism are summarized. The origin and the theory of magnetism itself is not discussed any further, as many textbooks on this subject are available [17, 18, 19] and a decent discussion would be beyond the scope of this thesis. In the first section ferromagnetic thin films are discussed in a thermodynamic context and qualitative arguments for the existence of domains are given. Subsequently, typical domain-structures of thin-film elements are reviewed and a summary of relevant literature on magnetic stripes and grating structures is given Free energy By definition, in thermodynamic equilibrium, any system will always be in a state of minimum total free energy. The magnetic fields acting on the magnetic moments of electrons create a local magnetization. In this field of micromagnetics the ferromagnetic film is described by a vector-field m( r), where m is the reduced magnetization, m = M/M s. In a phenomenological approach several energy-terms are contributing to the total free energy: Exchange stiffness This term expresses the preference of a ferromagnet for a uniform magnetization direction: E ex = A (grad m) 2 dv, (2.1) where A is the exchange constant, a material parameter. This is the phenomenological description of the quantum mechanical exchange energy. From this equation follows that a infinite ferromagnet is in its energetic minimum if all magnetic moments are aligned parallel. For non equilibrium cases (non-uniform magnetization distribution) the free energy depends on the exchange constant A. Hard magnetic material (e.g. Co) has a higher exchange constant than soft magnetic material (e.g. permalloy). Crystalline anisotropy The energy of the ferromagnet depends on the relative orientation of the magnetization vector and the crystalline axes of the lattice. This is an 11

14 2. Magnetism of thin films and thin film elements effect of the spin-orbit coupling and the crystal symmetry. Three types of crystalline anisotropy can be distinguished: cubic, uniaxial and hexagonal. In this thesis mainly Fe with a fourfold, cubic anisotropy is considered. The energy density of a magnetic moment in polar coordinates is: E K = (K 1 + K 2 sin 2 θ) cos 4 θ sin 2 φ cos 2 φ + K 1 sin 2 θ cos 2 θ, (2.2) where φ and θ are the in-plane angle and out-of-plane angle, respectively. K 1 is the cubic anisotropy constant. In two dimensions and the case of a (1) oriented thin film this reduces to E K = K 1 4 sin2 (2φ). (2.3) The easy axes of Fe are aligned along the [1] directions of the crystal lattice. Surface anisotropy Several reasons can lead to a twofold, uniaxial anisotropy. A common example is the out-of-plane surface anisotropy leading to a strong easy axis perpendicular to the surface of a thin film. This is found in several thin film systems, such as Co/Pd. This effect is important for the technical implementation of magnetooptical storage devices, but will not be discussed in this thesis. Instead, many samples display an in-plane uniaxial anisotropy due to steps at the surface or due to the artificial structuring of the surface. In this cases a phenomenological expression is: E U = K U sin 2 (φ φ U ), (2.4) where K U is the uniaxial anisotropy constant and φ U is the angle between the coordinate axis and the easy axis of the uniaxial anisotropy. Stray field energy The magnetized specimen produces a magnetic field itself, the stray field H d. The systems tries to minimize the energy density of this field. The stray field energy is given by: E d = µ H d mdv. (2.5) 2 sample The stray field (also called demagnetizing field, the corresponding anisotropy is also called shape anisotropy) depends on the shape of the specimen, a good approximation for many situations is to assume a general ellipsoidal shape for the sample. Than the demagnetizing field is H d = N M S, (2.6) with the symmetric demagnetizing tensor N. For the three axes of the ellipsoid, a, b, c, the x component of N is [2]: N a = abc 2 [(a 2 + η) (a 2 + η)(b 2 + η)(c 2 + η)] 1 dη, (2.7) analogous expression are valid for the other directions. This expression can be evaluated numerically to gain the demagnetizing tensor elements for an arbitrary ellipsoid. 12

15 2.1. Free energy For the case of an infinitely extended plate the magnetization depends only on the z-coordinate and N c = 1. The stray field energy density is E d = µ 2 M s 2 cos 2 θ. (2.8) This model is a very good approximation for thin magnetic films. If no other anisotropy favors out-of-plane magnetization the magnetization will remain in the film plane. The stray field energy contribution takes the form of an uniaxial anisotropy with an anisotropy constant K U,d = µ M 2 2 s. From the measurement of the polar (out-of-plane) magnetic hysteresis of a thin film the anisotropy energy can be calculated by integrating the hysteresis curve. Another important geometry are small magnetic stripes with dimensions l, w, h, length, width and height, respectively. It can be shown that for l w h the demagnetizing factor N w for a magnetization in the plane, but perpendicular to the stripe is and therefore the stray field energy is N w = h h + w, (2.9) E d,w = µ h 2 h + w M S 2 sin 2 φ, (2.1) which is again of the form of a uniaxial (in-plane) anisotropy with the anisotropy constant K U,w = µ M 2 h 2 s. In this case the easy axis of the anisotropy is in-plane and along the h+w stripes (φ = ). Zeeman energy The energy of the magnetic moment in an external field is given by: E Z = µ M S H cos(φ φ H ), (2.11) where H is the (homogenous) external field and φ H the angle between the field and the coordinate axis. There are other contributions to the complete free energy function, like magnetoelastic and magnetostrictive contributions. These contributions are neglected throughout this thesis. Sum of energies The complete energy function is the sum of all terms discussed above. For the case of homogenously magnetized samples the exchange stiffness is always zero and for the case of in-plane magnetized samples one finds: E(φ, H) = µ O M S H cos(φ φ H ) + K 1 4 sin2 (2φ) + K U sin 2 (φ φ U ) (2.12) where the uniaxial anisotropy may be due to the shape anisotropy of magnetic elements, to the (in-plane) surface anisotropy of a continuous film or a combination of both. In this model the magnetization rotates from one direction into the other during remagnetization, discontinuities are not described. However, several practical cases can be evaluated using the above formulas, as will be shown in the experimental sections. If the exchange constant is small, such that the magnetization is not always homogenous, domains are formed which is discussed in the next section. 13

16 2. Magnetism of thin films and thin film elements Figure 2.1.: Landau pattern of a square magnetic element Domains The most simple situation is found if only the exchange energy and the stray field energy is taken into account (these two contributions always exist). If the stray field energy is dominating, as for soft magnetic material, magnetization patterns are formed that prevent stray fields completely. An example is a magnetic disc. The magnetic moments will form a closed circular structure, however, paying an energy-penalty by increasing the exchange energy. These kind of structures have been observed in circular magnetic dots [2, 2]. Domains in the more common sense are established if additionally magnetic anisotropy is taken into account. Instead of smooth magnetization patterns domains with sharp boundaries (domain walls) occur. Inside a domain the magnetization is homogenous and parallel to an easy axis of the magnetic anisotropy. The occurrence and the shape of domains are thus depending on the relative strengths of the three energy terms: exchange energy, stray field energy and anisotropy. If an external field is applied it couples to the system via the Zeeman energy. Domains with discontinuous domain boundaries are also established in magnetic par- Figure 2.2.: Equilibrium domain states of different permalloy elements, together with the van-den-berg construction of the domains, taken from [21] 14

17 2.2. Domains Figure 2.3.: Construction of domain states in rectangular elements, see main text. ticles like square or rectangular dots even without the existence of anisotropy axes. If the magnetic pattern has sharp corners the demand for a zero stray field can only be fulfilled by forming lines of discontinuous magnetization distribution. Therefore magnetic dots with square and rectangular shape display the typical Landau pattern, as depicted in Fig An extensive study of domains in thin film elements was performed by van Berg [22]. He invented a geometrical procedure to construct the equilibrium domain structure of magnetic thin film elements. The algorithm is as follows: draw circles inscribed in the magnetic element which touch the edges at least at two points. the centers of these circles form lines which correspond to the domain walls the magnetization is oriented perpendicular to the radius which runs to the point of contact of the circle and the edge. if one circle touches the edges in more than two points the center of this circles marks an intersection of domain walls. As an example for this construction some measurements and schematic domain patterns from [21] are reproduced in Fig In addition to the lowest energy pattern constructed as explained above other higher energy patterns are also observed. If for instance a magnetic rectangular element is divided into two virtual halves the domain pattern can be constructed for the two halves separately using the van-den-berg algorithm, see Fig The resulting pattern is of higher energy but may be also stable depending on the demagnetization process. Comparable domain states have been observed in [23]. If additional anisotropy energy is taken into account, the domain pattern may get even more complex. An additional uniaxial anisotropy with the easy axis perpendicular to the long side of the rectangular element in Fig. 2.3 would result in a stabilization of the domains along the easy axis. The depicted state may than be the lowest energy state. The most important result of van den Berg is that in most cases domains along the edges (closure domains) will form in order to minimize the stray field. Closure domains were experimentally observed in [24] for stripes of permalloy with an external 15

18 2. Magnetism of thin films and thin film elements field perpendicular to the stripe axis: The internal region of the stripes is magnetized along the field but depending on width and thickness more and more edge domains are formed when the external field is reduced. Two cases can be distinguished: first the domain state is dominated by the shape of the specimen. This is the case for soft magnetic material. The domains can be constructed with the van-den-berg method. In this case the angle of the magnetization between two domains is arbitrary reflecting the angle of the geometric shape of the element (e.g. 9 walls for square elements). Another case is a film or an element with anisotropy. In this case the domains will be magnetized along an easy axis of the anisotropy. This leads two 18 domain walls in uniaxial and 9 walls in fourfold anisotropy material. Wether a material is dominated by the anisotropy or by the stray field is given by the parameter Q = K/K d, where K is a general anisotropy constant taking four- or two-fold crystal anisotropy into account. For soft magnetic material Q Observation of domains and magnetic hysteresis in magnetic stripes Several measurements of the domain structure and the hysteresis of magnetic stripes and wires can be found in the literature: Ebels et al. [25] have investigated Fe stripes on GaAs with the rather large width of 15 µm. They found an induced uniaxial anisotropy due to edge effects and a two step magnetization process with two different domain types, due to the combination of four-fold anisotropy of Fe and the patterning. Shearwood et al. [3] report upon magneto-resistance and magnetization loops of arrays of sub-micron sized Fe stripes. The stripes were held constant in shape (.5 µm width) but arrays with different separations between the elements where produced. The result is that again an uniaxial anisotropy is induced and hints of dipolar interactions depending on the separation were found. Hausmanns et al. [26, 27] show in combined work of experiment and simulation the remagnetization behavior of Co nanowires (width: 15 to 4 nm). They show an increase of the coercive field with decreasing width of the wires proportional to 1/w. In addition, the coercive field for different in-plane angles of the external field was examined, showing a simple behavior consistent to a model where the magnetization first rotates into the wire direction and afterwards switches by 18. Because of the shape anisotropy 18 domain walls in small wires are expected to be of the head-to-head type. This was theoretically confirmed by McMichael et al. [28]. The head-to-head domain wall consist of additional intermediate domains with a magnetization perpendicular to the wire axis or forming vortex-like structures. McCord et al [23] performed an intensive Kerr microscopy study on rectangular permalloy elements. Very different domain structures were detected depending on 16

19 2.4. Conclusion the magnetic history of the element. Typically closure domains with large internal domains having perpendicular direction were observed. Mattheis et al. [24] also measured large edge domains aligned with the edge of the stripes using Kerr microscopy for an external field direction perpendicular to the wire axis Conclusion The measurement and interpretation of domains in thin film elements is a very important subject in the field of magneto-electronics and general research on magnetism. Therefore there exist a large amount of publications on the subject. There are several approaches to the problem: Today s computer-power enables to calculate the domain structure of thin film element. Several commercial and non-commercial programs are available. However, only in combination with the experiment the real domain structure can be concluded. For special cases, like the spin structure inside of domain walls, the numerical simulation is almost the only possibility to gain insight due to difficulties observing very small magnetic structures. Measurements of integral physical properties like the hysteresis curves, transport phenomena or dynamic properties gain important information on the magnetic system and can be used together with numerical simulations of the domain structure. General features of the domain structure can be concluded. These kind of measurements provide important parameters of the complete system like remanence, saturation magnetization, time constants or the magnetization vector. Direct observation of the domain structure using Kerr microscopy, MFM or other techniques has the obvious advantage of directly imaging the domains, no simulation or assumptions are needed. However, every method has its specific limitations such as resolution or problems with the contrast. In addition, it is often difficult to obtain integral quantities such as remanence or coercive fields. Only a small portion of a sample may be visualized and the overall behavior may not be detected. The most comprehensive review of the subject is found in [29], where many methods, the domain theory and a vast amount of examples are given. The present thesis contributes to this field. The combination of diffraction and MOKE will be shown to yield information about the domain structure and the hysteresis simultaneously. 17

20 2. Magnetism of thin films and thin film elements 18

21 Part II. Methods 19

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23 3. Magneto-optical Kerr effect of thin films and thin film grating structures This chapter covers the theory and the experimental realization of Kerr effect measurements of thin films and ferromagnetic grating structures. The first sections explain the theory of the Kerr effect, the fundamentals of vector-moke and the theory of diffraction gratings. The next section provides a review of the literature of the Bragg-MOKE effect. This section closes with simulations of some of the Bragg-MOKE effects described in the literature. This simulations are very important for a comparison of the experimental results reported in the third part of this thesis. In the last section of this chapter the experimental setup is introduced which was used to measure the standard MOKE hysteresis curves, the vector-moke results and the Bragg-MOKE curves Theory of the Kerr effect Kerr effect - basics In general the magneto-optical Kerr effect is the change of polarization and/or intensity of a light beam reflected by a ferromagnetic surface. The measured quantity, e.g. the rotation of the polarization, is a linear function of the magnetization of the ferromagnetic material. A corresponding effect which has a quadratic dependence on the magnetization is called Voigt or Cotton-Mouton effect. A special case of this will be discussed later in Sec Another magneto-optical effect is the Faraday effect: the polarization of light is rotated by transmitting light through dielectric material in the presence of a magnetic field. In this case the rotation is proportional to the applied field. This effect is used in the experimental setup (Sec. 3.5). The simplest model of MOKE is to consider a Lorentz-Drude model of a metallic film. The incident light wave causes the electrons in the metal to oscillate parallel to the plane of polarization. In the absence of any magnetization the reflected light is polarized in the same plane as the incident light, this is the regular component with an amplitude R N. If a magnetization is thought to be acting on the oscillating electrons like an internal magnetic field, the electrons exhibit a second motion due to the Lorentz force. This second component is perpendicular to the direction of the magnetization and 21

24 3. Magneto-optical Kerr effect of thin films and thin film grating structures Figure 3.1.: Geometry of the three magneto-optical Kerr effects, see main text. perpendicular to the primary motion. The second component, R K, generates a secondary amplitude of the reflected light which has to be superimposed onto the primary beam [4]. In this framework one can understand easily the three general geometries of the magneto-optical Kerr effect [4], which are displayed in Fig. 3.1: a) In the polar geometry the magnetization is perpendicular to the reflecting surface. A linear polarized wave generates a second component, which is strongest if the angle of incidence is zero (perpendicular incidence, α i = ). In addition, the effect is independent of the direction of the polarization for α i =. b) In the longitudinal configuration the magnetization is parallel to the reflecting surface and parallel to the plane of incidence. The effect generates a polarization rotation of the reflected beam in both cases, perpendicular (s-) and parallel (p-) polarized light with respect to the plane of incidence. The sign of the Kerr rotation in the two cases is different. The special case of perpendicular incidence generates no Kerr rotation in either case, because R K points along the beam (s-polarization) or R K is zero (p-polarization). Thus the measured Kerr effect increases with α i. c) For the transverse configuration the magnetization is oriented perpendicular to the plane of incidence and parallel to the surface. For p-polarized light this configuration causes a change of the amplitude of the reflected beam, but no Kerr rotation. The three cases can be combined to yield a formula of the Kerr effect for an arbitrary magnetization and polarization of the electromagnetic wave. However in the present work only the longitudinal configuration is used. The samples under investigation usually exhibit no out-of-plane magnetization components, thus excluding the use of the polar Kerr effect. The longitudinal case is preferred over the transverse case because of the easier detection of polarization rotations than intensity shifts, as will be explained in Sec The longitudinal measurements are performed using s-polarized light. A polarization parallel to the plane of incidence would add an intensity modulation due to the transverse Kerr effect. The theory of the MOKE in the framework of a free electron gas as discussed above has several shortcomings and the complexity of band structures of the ferromagnetic materials demands a quantum-mechanical treatment. The magnetic moment of the 22

25 3.1. Theory of the Kerr effect Figure 3.2.: Definition of the coordinate system used in the discussion of the Kerr effect. ferromagnetic material is caused by the spin-asymmetry of the spin-up and spin-down sub-bands of the 3d-band structure of Fe, Co or Ni. An incoming polarized light wave interacts through its electric field with the electrons, and changes their orbital momentum. Because of the weak spin-orbit interaction this results in an interaction between the electric field of the light wave and the magnetization. The incoming electromagnetic wave can be split into left- and right-circular polarized eigenmodes, which have different quantum-mechanical probabilities to excite spin-up or spin-down electrons of the 3dband near the Fermi-level. The exited electrons will emit electromagnetic waves, with different circular polarization depending on their spin-state. Effectively this mechanism results in matrix elements of a 2 2 reflection matrix R c relating the Jones vector of the incoming wave in a circular-polarization basis to the Jones vector of the emerging wave (for a discussion of Jones vectors see Sec. 3.5 and [3]). The calculation of magnetooptical effects from first principles is a complicated task (see [31, 32, 33, 34]), however, for the present investigations a theory in the framework of electro-magnetism is sufficient and will be discussed in the next section. For an introduction to the theory of MOKE see [35, 36] Electro-magnetic theory of the Kerr effect As discussed in the above section the reflection of a electromagnetic wave by a ferromagnetic surface can be split into the regular reflection, which is described by standard Fresnel formulas and Fresnel reflection coefficients, and into a second component which adds a small contribution polarized perpendicular to the regular component. The superposition of the two components leads to a rotation of the polarization axis. The goal of the electromagnetic theory is to generalize the Fresnel formulas in order to yield magneto-optical reflection coefficients. In the following the situation depicted in Fig. 3.2 is assumed: A linear polarized wave is incident on a ferromagnetic surface under the angle α i and reflected at α f = α i. The magnetic medium has the refractive index n 1. For the reflection at a magnetic medium the use of a dielectric constant is not sufficient. In stead of this, in the dielectric law, 23

26 3. Magneto-optical Kerr effect of thin films and thin film grating structures D = ɛe, ɛ is a complex tensor, which can be written as [37, 38, 35] 1 : 1 iqm z iqm y ɛ = ɛ xx iqm z 1 iqmx, (3.1) Qm y iqm x 1 where the m i are the components of the magnetization vector M and the material constant Q is the magneto-optical constant, also called Voigts constant 2. In order to take the absorption of the electromagnetic wave into account, the regular dielectric constant ɛ xx is a complex number, the imaginary part corresponding to the absorption coefficient. Equivalently, the refractive index n is complex. With the above dielectric tensor the Maxwell equations have to be solved [38], which leads to a reflection matrix ( ) rpp r R = ps. (3.2) r sp Eq. 3.2 relates the p- and s-polarized components of the incoming wave to the respective components of the reflected wave. The coefficients r ij are the ratio of the incident j polarized electric field and reflected i polarized electric field. r ss is the standard Fresnel reflection coefficient, r pp is the standard Fresnel coefficient plus a term depending on m x Q (transverse MOKE) and both the off-diagonal elements of R are functions of m z Q and m y Q (polar and longitudinal MOKE: see Fig. 3.2 for the definition of the coordinate system). All coefficients are functions of the refractive index and the refraction angle inside the ferromagnetic material. Explicit formulas can be found e.g. in [38]. The complex Kerr angles are defined as following: r ss Θ p K = θ p K + iɛ p K = r sp r pp, (3.3) Θ s K = θ s K + iɛ s K = r ps r ss. (3.4) Here θ K and ɛ K are the Kerr rotation and ellipticity, respectively, and the superscripts denote whether the incoming light wave is polarized in the p- or s-state. In Ref. [38] simplified formulas for different MOKE geometries are derived. As this thesis deals with the longitudinal MOKE with s-polarized light, only this case is discussed. In general, there are two situations which have to be distinguished. If the ferromagnetic film is thick compared to the wavelength in the material, the MOKE signal is independent of the thickness t F M. If the layer is thin, the MOKE signal is a function of t F M, for ultrathin layers this is a linear function. For the laser wavelength used in the present investigations (λ = nm) bulk iron magneto-optical parameters are given in Tab 3.1 [39]. This leads to a wavelength inside of iron of 22 nm. The longitudinal Kerr effect for the case of thick ferromagnetic films (i.e. the Kerr effect is independent of the thickness) is derived from Eq. 3.3 and the formulas of r ij for m z = m x = [38]: θk s = cos α i tan α 1 in n 1 m y Q cos(α i α 1 ) (n 2 1 n 2 ), (3.5) 1 Different sign conventions are used in literature, the discussion in this thesis follows the proposed scheme in [37] 2 The product QM is also called the gyromagnetic vector 24

27 n Q α β i i Theory of the Kerr effect Table 3.1.: Magneto-optical parameters of Fe. The refractive index and the Voigt constant were taken from [39]. The parameters for second order contributions, α, β, were measured in [4] for a small angle of incidence (13 ) and for single crystalline Fe(1)/GaAs films. where n is the refractive index of the medium above the ferromagnetic film, for air n = 1 is assumed. The angle of the refracted beam, α 1, has to be calculated using Snell s law: n sin α i = n 1 sin α 1. The interesting case of thin ferromagnetic films is difficult to solve as multiple reflections and interference have to be taken into account. However, a method described in [41, 38] implements a matrix formalism which allows to calculate the MOKE signal for complicated film structures and superlattices. A simplified formula for the case of a thin ferromagnetic film on a non-magnetic substrate with the refractive index n sub (angle of the refracted light in the substrate: α sub ) is given in [38] as θ s K = 4πn n 1 n sub Qd F M cos α i sin α 1 λ(n cos α sub + n sub cos α i )(n cos α i n sub cos α sub ). (3.6) In more realistic situations also a layer on top of the ferromagnetic film, e.g. an oxide layer, has to be taken into account. In the present study magnetic films are considered with a typical thickness of d F M = nm, for which the approximation of a thick ferromagnetic film turned out to be satisfactory. In Fig. 3.3 the Kerr rotation as given by Eq. 3.5 is plotted for the bulk Fe parameters as a function of the incident angle. The function exhibits a maximum at α i = 55. For most experimental situations an incident angle of 45 is realistic for which a Kerr rotation of θ K =.68 can be expected. The ellipticity shows a maximum for the same angle of incident, which is ɛ K = rad Second order contributions to the longitudinal Kerr effect The dielectric tensor in Eq. 3.1 is linear in the magnetization. As already mentioned, there are cases where a second order contribution to ɛ are important. Especially single crystalline Fe often exhibits strong second order effects. The second order effects are quadratic in M and a second order term is added to the dielectric tensor in Eq. 3.1, which is given by [4]: B 1 m 2 x B 2 m x m y B 2 m x m z B 2 m x m y B 1 m 2 y B 2 m y m z B 2 m x m z B 2 m y m z B 1 m 2 z. (3.7) From this tensor expressions for the MOKE can be derived, as is shown in [42]. Experimental examples for the case of Fe can be found in [43, 4]. In Ref. [4] fits to MOKE data in the longitudinal geometry show that the Kerr effect can be described effectively by θ K m y + αm y m x + βm 2 x. (3.8) 25

28 3. Magneto-optical Kerr effect of thin films and thin film grating structures Re(θ K s ) [ ] α i [ ] Figure 3.3.: Plot of the Kerr rotation as a function of the incident angle α i, as described in Eq. 3.5 for bulk Fe and s-polarized light. The two phenomenological parameters α and β were estimated in [4] for the case of a single crystalline Fe(1) film on GaAs and are given in Tab As discussed in [42] the second order contributions depend on the angle of incidence. The values in Tab. 3.1 were taken at α i = 13, for larger angles as were used in this thesis smaller second order effects are expected. For the longitudinal configuration the magnetization components m x and m y in Eq. 3.8 can be identified with the two orthogonal magnetization components m t and m l along the transverse and longitudinal in-plane direction, respectively. Therefore the second order effects lead to a contribution in the longitudinal MOKE of transverse magnetization components. If the re-magnetization process of the sample under investigation is dominated by magnetization rotation processes, the orthogonal magnetization component increases around zero field and will lead to strong asymmetries in the measured hysteresis loop. Vice versa, if the re-magnetization process involves only 18 domain wall movements, the magnetization in any domain is always oriented parallel or antiparallel to the external field, thus no second order contributions are detected. For Fe, tending to 9 domain walls, a combination of the two limiting cases is expected Vector-MOKE It is often advantageous to measure not only the component of the magnetization along the applied field, but also the orthogonal magnetization component in order to reconstruct the magnetization vector from the measurement. The longitudinal MOKE can be used as a vector-magnetometer in the following manner: Magnetic hysteresis measurements were performed using a high resolution magneto- 26

29 3.2. Vector-MOKE Figure 3.4.: Definition of the sample rotation χ and the angle φ of the magnetization vector M for the case of the longitudinal setup (a) and the perpendicular setup (b). In order to measure the transverse magnetization component m t the field and the sample are rotated by 9, such that the angle χ is held constant, but the magnetization component m t is in the scattering plane. optical Kerr effect setup (MOKE) in the longitudinal configuration with s-polarized light, which is able to measure the exact Kerr angle as a function of the applied magnetic field. Details of the experimental setup can be found in Sec Here the magnetic field lies in the scattering plane and the resulting Kerr angle is proportional to the component of the magnetization vector along the field direction, θ l K m l, where m l is the longitudinal component of M projected parallel to H. Additionally, the design of the setup enables one to rotate the sample around its surface normal (angle χ), in order to apply a magnetic field in different in-plane directions. This kind of measurement cannot distinguish between a magnetization reversal via domain rotation and/or via domain formation and wall motion. Therefore measurements were performed with the external magnetic field oriented perpendicular to the scattering plane and the sample rotated by 9 with respect to the scattering plane, keeping the rest of the setup constant. In this perpendicular configuration MOKE detects the magnetization component parallel to the scattering plane and perpendicular to the magnetic field, θ t K m t, as has been shown by [1]. The geometry of the setup is sketched in Fig Both components, m l and m t, yield the vector sum for the average magnetization vector M sampled over the region, which is illuminated by the laser spot. This area is 1mm 2. The magnetization vector can be written as M = ( ml m t ) = M ( cos φ sin φ ). (3.9) The proportionality constant between the Kerr angle θ K and the two magnetization components is a priori unknown. For the samples under investigation it was found that in saturation the Kerr angle does not dependent on the sample rotation χ. Furthermore, the angle of incidence of about 4 was kept constant for both set-ups. Therefore the 27

30 3. Magneto-optical Kerr effect of thin films and thin film grating structures error - if at all - is tolerable by assuming the same proportionality constant for both configurations. A source of error may be a contribution from the polar MOKE effect, which would add a signal proportional to a magnetization component perpendicular to the sample surface. In addition second order magneto-optical effects [42] (see Sec ) could interfere with the following analysis. However, neglecting these potential problems, one can write: m l = cos φ m t sin φ = θl K, (3.1) θk t from which follows the rotation angle of the magnetization vector: ( ) θ t φ = arctan K. (3.11) Furthermore one can express M, normalized to the saturation magnetization: M M sat = θl K θ l,sat K θ l K 1 cos φ. (3.12) Another possibility of yielding magnetic vector information from MOKE measurements is to use a combination of the transverse and longitudinal Kerr effect, as has been shown by [44]. In the case of the transverse Kerr effect the magnetic information is obtained from an intensity shift of the reflected light, which is proportional to the magnetization along the applied magnetic field perpendicular to the scattering plane. In this geometry obviously a rotation of the polarization can be attributed to the longitudinal Kerr effect which is then sensitive to the magnetization component perpendicular to the magnetic field. Thus, by measuring both, the rotation and the intensity one can extract information of two orthogonal magnetization components. Details of the procedure can be found in [44]. The advantage here is that the magnetic field and the sample stay in the same position and the two components can be measured simultaneously, as opposed to the geometry used in this work. The drawbacks are that the detection is more complicated and the two signals yielded are not directly comparable concerning their magnitude, because two different physical quantities are measured. The results of vector-moke measurements provide important information which allow to distinguish between different magnetization reversals. Two limiting cases can easily be separated (see Fig. 3.5): Coherent rotation (Fig. 3.5(a)): If the length of the magnetization vector, M, is constant during the reversal, the magnetization rotates from one direction into the other. Domain formation (Fig. 3.5(b)): If only domains are formed, the angle of the magnetization stays always aligned with the external field but the magnitude changes. In this case the transverse component is zero. It is instructive to plot the transverse component and the angle φ as given by Eq and Eq as a function of the longitudinal magnetization component m l (the component parallel to the external field) For the case of coherent rotation the transverse component is increased if the longitudinal component is decreases and and vice versa. If no transverse component can be detected no rotation of the magnetization takes place and the reversal is governed by domain processes. 28