Effect of coupling strength on magnetic properties of exchange spring magnets

OURNAL OF APPLIED PHYSICS VOLUME 94, NUMBER 10 15 NOVEMBER 2003 Effect of coupling strength on magnetic properties of exchange spring magnets Vamsi M. Chakka, Z. S. Shan a), and. P. Liu b) Department of Physics, The University of Texas at Arlington, Arlington, Texas 76019 Received 31 March 2003; accepted 2 September 2003 The main feature of an exchange spring magnet can be characterized by ferromagnetic exchange coupling between a magnetically soft phase and a magnetically hard phase. The coupling constant, a measure of the coupling strength between the two phases, is a key parameter in controlling the spring magnet properties. A ferromagnetically coupled hard soft bilayer FCB has been used in this article to develop analytical expressions for a spring magnet that correlate the magnetic properties of a FCB and its layer parameters. These analytical expressions have been developed by solving a modified Stoner Wohlfarth model. A set of analytical solutions describing the magnetic properties of the FCB at the different stages of have been derived. The evolution of the magnetization reversal in a FCB, as a function of the coupling constant, and an applied field H, is analyzed in detail. As a result, the approach to enhance the maximum energy product (BH) max is revealed. 2003 American Institute of Physics. DOI: 10.1063/1.1621712 I. INTRODUCTION Exchange spring magnets are nanocomposites of magnetically soft and hard phases. These composites have the potential of very high (BH) max, since they take advantage of high saturation magnetization from the soft phase and high coercivity from the hard phase. For enhanced (BH) max values the soft and hard phases should be effectively exchange coupled without any kink in the hysteresis loop. 1 3 Thus, the coupling strength between the two phases is an important parameter in controlling the magnetic properties of a spring magnet. A ferromagnetically coupled hard soft bilayer FCB has the main feature of an exchange spring magnet. 4 In this article, the energy-barrier analysis based on the Stoner Wohlfarth model 5 of a FCB system is done to study the magnetization reversal mechanism and to derive the analytical expressions correlating the magnetic properties and the structural parameters of the hard soft bilayer, with emphasis on the effects of the coupling constant. II. FCB PROPERTIES The main parameters of the FCB are the coupling constant between the two phases, and the magnetizations, the anisotropy constants, and the thickness of the hard and soft layers denoted as M, K, and t with subscripts S and H representing the soft and hard phases respectively Fig. 1a. The angles between the easy axis and M S, M H, and the applied field H are denoted by,, and, respectively Fig. 1b. In this article, is treated as a varying parameter, and the effects of on the hysteresis and the magnetic properties of the FCB are analyzed in detail. The magnetization inhomogeneity along the interface of the soft and hard phases in the a Present address: iatai University, 102 Moganshan Rd, Hangzhou 310005, People s Republic of China. b Electronic mail: pliu@uta.edu FCB has been ignored and thus, is considered to be constant across the bilayer. The parameter has a well-defined physical meaning when expressed as a function of the geometric and atomic scale magnetic parameters. 6,7 III. EVOLUTION OF MAGNETIZATION REVERSAL IN FCB Energy E of the FCB system depends on the angles and and can be expressed as E,E S E H E exchange,, 1 where E S (), E H (), and E exchange (,) are the soft phase, hard phase, and the coupling energy between them. These energy terms can be expressed as E S HM S t S cosk S t S sin 2, 2 E H HM H t H cosk H t H sin 2, E exchange, cos. 4 It is known that a stable magnetization state corresponds to a low-energy state. Also, magnetization reversal is a 180 change in the angle or that M S and M H make with the easy axis. The coercive field H C corresponds to the applied field H required for the reversal of magnetization. The analysis presented in Ref. 8 shows that, at the coercive field H C, the system becomes unstable. This is because the energy barrier which prevents the magnetization from reversing vanishes and therefore the magnetization reversal occurs. 3D energy surfaces Fig. 2 were used to study the magnetization reversal mechanism in FCB as a function of and H. As the applied field H varies along the hysteresis loop from a positive value to a negative value Fig. 1c, 9 M S and M H reversal occurs as follows. Theoretical analysis 10 and experimental works 11,12 have confirmed that both M S and M H are aligned with H for large values of H i.e., 0 ) Fig. 1c. When the applied field is reduced to H0, the 3 0021-8979/2003/94(10)/6673/5/$20.00 6673 2003 American Institute of Physics

6674. Appl. Phys., Vol. 94, No. 10, 15 November 2003 Chakka, Shan, and Liu FIG. 1. a FCB. b schematic of orientations of the magnetization vectors with respect to the easy axis. c Schematic of hysteresis loop for a FCB with and without a kink. 9 No kink condition here is obtained by increasing the coupling strength. H C is the critical value of coercivity and H C H C (I) or H C (II) depending upon the type of reversal mechanism. 3D energy surface E(,) in Fig. 2 shows that the energy is at a minimum at 0 position a. This indicates that the magnetizations M S and M H of the FCB system are stable at position a. A minimum energy condition also exists at 180, marked as position c in Fig. 2 not directly visible in Fig. 2. There are three main energy barriers controlling the reversal mechanism: E B-S is the energy barrier that needs to be overcome for the reversal of M S fromatob position b: 180, 0 ), E B-H is the energy barrier for the reversal of M H from b to c after the reversal of M S from atob,ande B-SH is the energy barrier for the reversal of both M S and M H directly from a to c. As the applied field H is varied, the energy surface changes and thus the minimum energy positions and energy-barrier heights vary. As the applied field H is reversed and its magnitude is increased (180 and H0), the reversal of M S and M H for a FCB takes place in one of the following two ways. A. Type I reversal: Energy barrier E B-S vanishes before E B-SH vanishes When H increases in the reverse direction, energy E(,) at position a increases. At the coercivity of the soft phase H CS, E B-S vanishes (E B-SH exists, preventing the reversal of M S and M H directly from a to c and thus M S reverses from a to b i.e., angle changes to 180, 0 ). With a further increase in H in the reverse direction, at the coercivity of the hard-phase H CH, E B-H vanishes and M H reverses from b to c i.e., 180, angle changes to 180. This is a two-step reversal process, and the hysteresis loop would show a kink Fig. 1c. In a physical sense, it means that the soft-phase magnetization M S is not tightly coupled to the hard-phase magnetization M H. Therefore, M S and M H reverse in two steps with increasing H magnitude a to b and b to c, two-step reversal. At a certain value of H above H CS, E B-SH would vanish, but this would have no effect on the reversal mechanism. Let be the actual coupling constant value between the hard and the soft phases of a FCB. When is increased, due to better coupling between the phases, H CS increases and H CH decreases. 8 When is increased to a critical value denoted as C (I)] both the energy barriers E B-S and E B-H vanish simultaneously at a particular reverse applied H field denoted as H C (I)]. Simultaneous reversal of M S and M H from a to b and b to c occurs, which is a one-step reversal process, and the hysteresis loop would not show a kink. Thus, when C (I), there will be no kink and the resultant coercivity is H C (I) Fig. 1c. B. Type II reversal: Energy barrier E B-SH vanishes before E B-S vanishes When H is increased in the reverse direction, depending upon the parameters M, K, t, and the energy barrier E B-SH may vanish before the energy barrier E B-S vanishes, which will be discussed in more detail with Eq. 11. Since the energy barrier E B-SH has vanished, both M S and M H reverse simultaneously from a to c i.e., the angles change to 180 ), which is a one-step reversal process and the hysteresis loop would not show a kink Fig. 1c. The coupling at which this kind of reversal starts is denoted as C (II), and the corresponding applied field is denoted as H C (II). For a FCB with C (II), E B-SH exists and the reversal mechanism is similar to a type I two-step reversal process and the hysteresis loop would show a kink character Fig. 1c. Thus, when C (II), this kink will be eliminated Fig. 1c. The difference between C (I) and C (II) is that the C (I) is determined under the condition when the energy barrier E B-S vanishes before E B-SH would vanish, and C (II) is determined under the condition when the energy barrier E B-SH vanishes before E B-S would vanish, which will be discussed in more detail in Sec. IV. FIG. 2. Typical 3D energy surface E(,) for a FCB system at H0Oe, showing the energy barriers E B-S between a E(0,0 ) and b E(180,0 ) and E B-SH between a and c (180,180 ). 9 A low energy condition exists at point c. Point c is hidden behind the barrier E B-SH, and is not directly seen in this 3D plot.

. Appl. Phys., Vol. 94, No. 10, 15 November 2003 Chakka, Shan, and Liu 6675 IV. ANALYTICAL SOLUTIONS Analytical solutions for the coercivities of the soft and hard phases, the critical values of the coercivity and the coupling constant are determined by finding the maxima and minima of the energy E(,) as follows: Q0 and 2 E 2 or 2 E 2 gives minimum energy point, Q0 and 2 E 2 or 2 E 2 gives maximum energy point, Q0 and 2 E 2 or 2 E 2 0 for 0, 5 6 0 gives H CS. After the soft-phase reversal at H CS 180, Q0 and 2 E 2 and 0 gives H CH, or 2 E 2 0 for 180 where Q 2 E 2 2 E 2 E 2 2. 7 from 0 to After a lengthy operation, the coercivity H CS for the softphase and H CH for the hard-phase were determined to be as follows: 8 9 H CS 1 2 2K H M H M H t H 2K S M S M S t S 2K H M H M H t H 2K S M S M S t S H CS 1 2 2K H M H M H t H 2K S M S M S t S 2K H M H M H t H 2K S M S M S t S 2 4 4K SK H M S M H 2 4 4K SK H M S M H 2 M S t S M H t H K S t S K H t H 2 M S t S M H t H K S t S K H t H,. 10 11 Let C denote the expression under the square root in Eq. 11: C 2K H M H M H t H 2K S 2 M S M S t S 4 4K SK H 2 K M S M H M S t S M H t S t S K H t H. 12 H In type I reversal, even under the critical condition C (I), where the kink disappears, the reversal is from a to bandbtocfig. 2. Thus, the condition (180 and 0 ) assumed in Eq. 8 is always valid for type I reversal and C0 for all values of. However, the condition is true for type II reversal mechanism only with C (II). For type II reversal with C (II), since both M S and M H reverse simultaneously from a to c Fig. 2, the condition ( 180 and 0 ) assumed in Eq. 8 for deriving the coercivity of the hard-phase H CH is not valid. For C (II), C0 and for C (II), the H CH value obtained from Eq. 11 is invalid since C0. It can also be seen from the expression for C that when the properties of the soft and the hard phase layers of the FCB i.e., values of the parameters M S and M H, K S and K H, t S and t H ) are very close to each other then C0 for any value of. However, when these parameters differ by large values and is very large C (II), it turns out that C0. Typically, the layer parameters differ by large values in exchange-spring magnets and hence type II reversal is more common. For the type I reversal, the critical coupling constant C (I) and the critical coercivity H C (I) are determined by H CS H CH, which is calculated numerically because it is too complicated to have an analytical solution. However, C (I) and H C (I) for small can be determined as 8 C I 2K H 13 M H 2K S M S 1 M S t S 1 M H t H, H C I 2K St S K H t H M H t H M S t S 2K eff M eff, where K eff K St S K H t H t S t H and M eff M Ht H M S t S. 14 t S t H For the type II reversal mechanism, C0 is the condition for determining the critical coupling constant C (II) and the critical coercivity H C (II), which are obtained as follows:

6676. Appl. Phys., Vol. 94, No. 10, 15 November 2003 Chakka, Shan, and Liu TABLE I. (BH) max calculated for FCB films. The following parameters are assumed for the calculations: K S 510 5 erg/cm 3, M S 2000 emu/cm 3, M H 500 emu/cm 3, t H 400 Å. Film K H No. t S (10 6 ) C (II) H C (I) a (10 3 ) M eff (10 3 ) C II 2K H 2K S M H M S 1 M H t H 1 2 M S t S 1 1 2 M H t H M S t S, 15 H C II 1 2 2K H M H CII M H t H 2K S CII M S M S t S. (BH) max (10 6 ) 1 120 1 2.5 2.1 846 3.2 28 2 120 3 6.8 5.7 846 0.4 28 3 120 30 65.9 54.8 846 49.5 28 4 50 10 13.9 26.8 667 22.6 18 5 200 10 27.8 13.6 1000 7.4 39 6 400 10 36.0 8.4 1250 0.5 62 7 800 100 437.2 44.9 1500 35.4 89 a H C (I)2M eff. Units: t S Å, K H erg/cm 3, C (II) erg/cm 2, H c (I) (Oe), M eff emu/cm 3, Oe, (BH) max erg/cm 3. V. EXPRESSION FOR BH max AND APPROACH TO IMPROVE BH max 16 The maximum energy product (BH) max is the figure of merit characterizing the magnetic properties of a permanent magnet. Since BH(H4M eff ) H and (BH) max is determined by d(bh)/dh0, (BH) max can be expressed as for 2 BH max 4 2 M eff 2 4 M 2 St S M H t H t S t H, 17 H C 2M eff 2 M St S M H t H t S t H. 18 Here H C is the coercivity of the FCB system and H C (I) or H C (II) may be used as H C to estimate the obtainable values of (BH) max. Equations 14, 16, and 18 indicate that larger effective magnetization M eff may raise the (BH) max but will reduce H C. Therefore, a set of optimized parameters of M S, K S, t S, M H, K H, and t H has to be chosen to get the maximum value for (BH) max. Table I shows an example for the calculated (BH) max values using Eqs. 17 and 18. The following important conclusions can be derived from Table I: i As K H increases from 110 6 to 3 10 7 erg/cm 3 film No. 1 to No. 3, both C (II) and H C (I) increase and H C (I)2M eff changes from negative to positive. ii As t S increases from 50 to 400 Å film No. 4 to No. 6, the coupling C (II) required increases. The energy product (BH) max, and M eff increase, but H C (I) and decrease. may be negative if t S increases further. iii Film No. 5 has high values for t S and K H and this film shows a large (BH) max value. Thus, we can conclude that high values of t S and K H along with high values are important to enhance the energy product (BH) max. It should be kept in mind that the coercivity predicted by the Stoner Wohlfarth model is usually 2 times larger than the experimental value, and thus the experimental value of (BH) max will be lower than the data shown in Table I. It is suggested that the energy product of the FCB is based on the optimization of the whole set of parameters, which we believe is a useful conclusion for improving the energy products in nanocomposite permanent magnets in general. TABLE II. A summary of the reversal mechanisms. Reversal mechanism Critical Condition Kink Comments Two-step Type I C (I) Yes reversal of soft-phase magnetization from a to Occurs when the soft bath CS, reversal of hard-phase magnetization and hard phase C (I) frombtocath CH ) properties a are very close to each other in One-step their values C (I) No reversal of soft-phase magnetization from a to Always C0) b and reversal of hard-phase magnetization from b to c take place simultaneously at H C (I)] Two-step Type II b C (II) Yes reversal of soft-phase magnetization from a to Occurs when the soft bath CS, reversal of hard-phase magnetization and hard phase C (II) frombtocath CH )(C0) properties a differ by a very large range One-step C (II) No reversal of soft and hard phase magnetizations take place directly from a to c at H C (II)] (C0) a The soft and hard phase properties are the values of the parameters M S and M H, K S and K H, t S, and t H. b Type II is a typical reversal mechanism seen in exchange spring magnets.

. Appl. Phys., Vol. 94, No. 10, 15 November 2003 Chakka, Shan, and Liu 6677 The solutions above have been derived by assuming that all spins in the soft-layer rotate coherently. We have not taken into account thermal activation, lateral magnetic structure and the moment orientation distribution in the quasi- Bloch wall nucleated in the case of thick soft layers. Indeed, experimental, analytical and numerical analyses have shown that the nucleation field is significantly reduced when the soft-layer reverses incoherently via the formation of a spiral spin structure. 13 15 However, within the thickness regime where the Stoner Wohlfarth model does apply, this modeling analysis provides an intuitive and yet a systematic way to estimate the magnetic properties of exchange spring magnets. VI. CONCLUSIONS Table II summarizes the important points of the two reversal mechanisms. The energy-barrier analysis shows two distinct types of magnetization reversal mechanisms in a FCB. Based on the values of the layer parameters (M S, K S, t S, M H, K H, and t H ) of the FCB, either the type I or type II reversal mechanism would be followed. The critical coupling constant value C (I) for type I reversal and C (II) for type II reversal plays an important role in controlling the magnetization reversal process of the FCB. The hysteresis loop of a FCB will show a kink if the actual coupling constant for a FCB is lower than C (I) or C (II), and there is no kink if C (I) or C (II), which is desirable for enhanced (BH) max. Type II is a typical reversal mechanism seen in exchange spring magnets. Analytical expressions for C (II), the resultant coercivity H C (II) and (BH) max have been derived and they can be used to choose the optimal set of parameters of soft and hard layers to enhance the (BH) max value. The analysis done here clearly shows that a higher value of is desirable to obtain a high (BH) max value. ACKNOWLEDGMENT This work is supported by DOD/DARPA through ARO under Grant No. DAAD-19-03-1-0038. 1 E. F. Kneller and R. Hawig, IEEE Trans. Magn. 27, 3588 1991. 2 R. Skomski and. M. D. Coey, Phys. Rev. B 48, 15812 1993. 3 T. Schrefl, H. Kronmüller, and. Fidler,. Magn. Magn. Mater. 127, L273 1993. 4 R. Röhlsberger, H. Thomas, K. Schlage, E. Burkel, O. Leupold, and R. Rüffer, Phys. Rev. Lett. 89, 237201 2002. 5 E. C. Stoner, F. R. S. Wohlfarth, and E. P. Wohlfarth, IEEE Trans. Magn. 27, 3475 1991. 6 R. Skomski,. P. Liu,. M. Meldrim, and D.. Sellmyer, in Magnetic Anisotropy and Coercivity in Rare-Earth Transition Metal Alloys, edited by L. Schultz and K. H. Müller Werkstoffinformationsgesellschaft, Frankfurt, 1998, p. 277. 7 R. Skomski, H. Zeng, and D.. Sellmyer, IEEE Trans. Magn. 37, 2549 2001. 8 Z. S. Shan,. P. Liu, Vamsi M. Chakka, H. Zeng, and. S. iang, IEEE Trans. Magn. 38, 2907 2002. 9 Care should be taken while dealing with the sign for H. Figure 1c uses Cartesian coordinates. Figure 2 and all the equations use polar coordinates. In Fig. 1c, since H is a Cartesian coordinate, H in the second quadrant is negative. However, the same H is positive with 180 in the case of polar coordinate system. In this article, the or sign of the H field follows the polar coordinate system. 10 Z. S. Shan, D. in, H. B. Ren,. P. Wang, S. N. Piramanayagam, S. I. Pang, and T. C. Chong, IEEE Trans. Magn. 37, 1500 2001. 11 I. A. Al-Omari and D.. Sellmyer, Phys. Rev. B 52, 3441 1995. 12. P. Liu, Y. Liu, Z. Shan, and D.. Sellmyer, IEEE Trans. Magn. 33, 3709 1997. 13 E. E. Fullerton,. S. iang, M. Grimsditch, C. H. Sowers, and S. D. Bader, Phys. Rev. B 58, 12193 1998. 14 L. Leineweber and H. Kronmüller, Phys. Status Solidi B 201, 2911997. 15 T. Nagahama, K. Mibu, and T. Shinjo,. Phys. D 31, 431998.