JOURNAL OF APPLIED PHYSICS 108, 043921 共2010兲
Interpretation of hysteresis loops of GaMnAs in the framework of the Stoner–Wohlfarth model A. Winter,1 H. Pascher,1,a兲 H. Krenn,2 X. Liu,3 and J. K. Furdyna3 1
Experimentalphysik I, Universität Bayreuth, D-95440 Bayreuth, Germany Institut für Physik, Karl-Franzens-Universität Graz, A-8010 Graz, Austria 3 Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556, USA 2
共Received 8 April 2010; accepted 18 June 2010; published online 31 August 2010兲 We have used the magnetooptical Kerr effect to investigate the shape of the hysteresis loops of thin GaMnAs films grown on substrates with different buffer layers. Depending on whether the easy axis of magnetization is in the plane of the thin film or out of the plane, and depending on the orientation of the external magnetic field with respect to the crystallographic axes, a great variety of hysteresis loops is observed. Because magnetooptical effects depend linearly on specific components of the magnetization, it has been possible to determine the orientation of the magnetization with varying magnetic field. The experimental findings are very well described by the Stoner–Wohlfarth model of coherent magnetization rotation, yielding precise values for the anisotropy constants. We present this model and its use in the context of magnetooptical measurements as a relatively simple and straightforward method for establishing magnetization parameters of ferromagnetic semiconductors in thin film form. © 2010 American Institute of Physics. 关doi:10.1063/1.3466771兴 I. INTRODUCTION
The diluted magnetic semiconductor GaMnAs has been studied intensively since ferromagnetism has been discovered in this material.1 It attained interest due to its possible applications in spintronic devices as well as due to interesting basic physics.2 It is well known that GaMnAs films show rather strong magnetic anisotropy which is controlled by strain in these epitaxial films.3–6 The easy axis of magnetization is oriented out of plane under tensile strain and in plane under compressive biaxial strain. More recent investigations demonstrated that the material also exhibits a strong in-plane uniaxial magnetic anisotropy.7 Magnetooptical effects as Kerr effect 共MOKE兲 and magnetic circular dichroism 共MCD兲 are very well suited to observe hysteresis loops of ferromagnetic III-Mn-Vsemiconductors. These effects are proportional to the magnetization of the sample and use reflected light, which is important because of the high free carrier absorption of the material. By different orientations of the external magnetic field and different angles of light incidence, it is possible to determine the reorientation of the magnetization during a varying magnetic field. Hence these measurements provide the evaluation of the relevant magnetic anisotropy constants for strained magnetic epilayers. MOKE or MCD measurements are quite easily performed 共see below兲. In systems such as GaMnAs, it is possible to engineer a wide variety of sample types simply by changing a few basic growth parameters. The ability to determine the anisotropy constants of these specimens then constitutes a very useful method for monitoring sample properties after the growth process. Furthermore, having a a兲
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method of establishing the behavior of the sample magnetization in an external field is also expected to be of use for spintronic applications. II. EXPERIMENTAL METHOD
Our GaMnAs samples have been grown by low temperature molecular beam epitaxy on 共100兲 GaAs substrate with different buffer layers. By choosing a specific buffer material 共GaAs or GaInAs兲, it is possible to select the kind of strain acting on the GaMnAs film due to the mismatch of lattice constants. Thus, the resulting uniaxial magnetic anisotropy is uniquely determined by the strain-status of the epilayer. With GaAs buffer the easy axis of magnetization is in plane whereas GaInAs causes the easy axis to be perpendicular to the sample surface. The GaMnAs/GaAs sample with easy axis in the sample plane investigated in the present work contains 1.4% of Mn, is 300 nm thick and has been grown on low-temperature-GaAs. Measurements with a superconducting quantum interference device 共SQUID兲 magnetometer yield a value for the saturated magnetization of 13300 A/m. The GaMnAs/GaInAs sample with easy axis perpendicular to the sample plane contains 3% of Mn and exhibits a saturated magnetization of 34000 A/m 共the In-content in the buffer layer is 20%兲. Because of the high free carrier absorption of GaMnAs, reflected light provides information just on the top layer. The magnetooptical Kerr effect 共MOKE兲 manifests itself in a rotation of the plane of polarization of linearly polarized light which is reflected from the surface of a magnetized sample. Restricted by the geometry of the split-coil magnet possible angles of incidence of the light beam on the sample surface are 90°, 45°, and less than 6°, respectively. The external magnetic field is in most cases parallel to the direction of the incoming beam. Depending on the easy axis and on the orientation of the sample with respect to the direction of the
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magnetic field a specific orientation of the magnetization is enforced. The polar MOKE 共incident beam parallel to sample normal兲 measures the component of the magnetization perpendicular to the sample surface whereas in the other two geometries 共see above兲 in-plane magnetization components are detectable. Yang and Scheinfein8 gave a detailed discussion how different orientations of the sample magnetization contribute to the Kerr rotation 共see, for example, Eqs. 12 and 17 in Ref. 8兲. Thus by MOKE the magnetization reversal following the magnetic field can be investigated. The connection of the Kerr rotation with the material properties is described in various textbooks.9,10 Our experimental arrangement using a photoelastic modulator to achieve sensitivities up to rotation angles of 0.002° is described in Refs. 11 and 12. The article by Thurn et al.12 describes magnetooptical measurements on semimagnetic semiconductors as a function of wavelength. The present work can be considered to be supplementary, in that it shows what information can be extracted while doing measurements as a function of the strength and orientation of the external magnetic field.
III. STONER–WOHLFARTH MODEL
Stoner and Wohlfarth13 calculated the free energy density F of a homogeneously magnetized ellipsoid in an external magnetic field. The minima of F determine the equilibrium orientations of the magnetization. F is a sum of several contributions as follows: 共i兲
the magnetostatic contribution given by a magnetizaជ relative to an external magnetic tion orientation M ជ 共0H ជ = Bជ ext; 0 being the vacuum permeabilfield H ity兲
ជM ជ, Fstatic = − 0H 共ii兲
a shape anisotropy term describing the demagnetizing effect due to the geometry of the thin film sample Fshape =
共iii兲
0 2 M cos2共兲, 2
共2兲
ជ and the “hard” axis as共 is the angle between M sumed perpendicular to the sample surface兲; a contribution of the crystallographic anisotropy described in leading order by: Fcrystal =
共iv兲
共1兲
Kc 关sin2共2兲 + sin4共兲sin2共2兲兴, 4
共3兲
共 is the angle between the in-plane easy axis and the ជ on the sample plane兲; and projection of M there is also an uniaxial stress between the ferromagnetic GaMnAs layer and the buffer layer, modifying the lattice constant and thereby the magnetic hard axis, fixing the easy axis either along the sample nor mal 共Ku ⬎ 20 M 2兲 or in plane 共Ku ⬍ 20 M 2兲
Funiaxial = − Ku cos2共兲,
共4兲
共Note that, because 20 M 2 is always positive, negative values of Ku can only be found in samples with inplane easy axes.兲 The total free energy density is given by Fges = − 0H ⫻ M ⫻ 关sin sin H cos共 − H兲 + cos cos H兴 + +
0 2 M cos2 − Ku cos2 2
Kc 关sin2共2兲 + sin4 sin2共2兲兴 4
共5兲
H is the angle between the external magnetic field and the sample normal. H is the angle between the in-plane easy axis and the projection of the external magnetic field onto the sample surface. The absolute value of the magnetization is considered to be constant and only the orientation changes during a sweep of the external magnetic field. ជ , one has to determine all the For a given value of H ជ at all relative minima of F. possible orientations 共 , 兲 of M ជ for a Starting with a magnetization oriented parallel to H saturating field and gradually reducing H, it is possible to determine 共 , 兲 for the minimum of F which is closest to its high field value. For very high fields, there exists only one possible orientation, but for low fields there are different minima of F and the actual orientation corresponding to one specific minimum depends on the history of external field sweeps and on the preceding magnetization. This leads to the appearance of hysteresis loops. If more than one minimum of the free energy density exists usually the one lying closest to ជ is occupied. the previous orientation of M In ferromagnets, it is possible that different domains are formed to lower the energy due to reduction in the stray fields. The Stoner–Wohlfarth model usually does not take domains into account. In the investigation discussed here we have used a 755.3 nm laser diode in combination with a lens in front of the sample resulting in a small laser spot on the sample surface with a diameter of about 50 m which means that the experiments only yield information about this small area. In the case of GaMnAs grown on GaAs 共easy axis in plane兲 Welp et al.14 have shown that in thin films the domains are usually quite large 共up to several hundred microns兲. This is a proof for the assumption that samples of GaMnAs grown on GaAs show single-domain behavior over the laser spot. If a magnetic field is applied perpendicular to the surface of thin film samples of GaMnAs grown on GaInAs 共easy axis out of plane兲 we can also assume that the magnetization over the sample 共and certainly over the laser spot兲 acts as a single domain due to the combined effects of magnetic field and strain. The fact that we obtain signals from a single domain is supported by the following considerations: in a sample that contains many domains, an initial magnetization curve will be different from the curve observed in a complete cycle of magnetization from negative to positive external field and back. Such initial magnetization would arise from the generation and reorientation of different domains. We never ob-
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FIG. 2. Simulated orientation of the magnetization in relation to the crystallographic axes; the five different pictures represent five different situations during the magnetic field sweep from ⫺2 kG to +2 kG 共see Fig. 1 for comparison兲. FIG. 1. Comparison between measurement and calculation of MOKE for a magnetic field perpendicular to the sample surface at 1.8 K, measured at 755.3 nm.
served such an initial magnetization directly after cooling the samples below the Curie temperature. From this we can conclude that the laser focus always illuminates a spot that lies completely inside a single domain. Nevertheless, there are different domains present in the sample. Specifically, there are critical external magnetic fields at which the magnetization reorients itself quite rapidly. If larger spots of light are used—such as those illuminating the whole sample with light of a single wavelength generated by a halogen lamp and a monochromator—the magnetooptical curves in the vicinity of critical fields do not change as rapidly as they do in experiments carried out with a laser. This can be attributed to the reorientation of different domains at slightly different critical fields. One should note that the existence of domains and the possibility of creating new ones can be incorporated into the Stoner–Wohlfarth model as follows: while the external magnetic field is altered, a small area of the sample can temporarily form a new domain. If the free energy density of this nucleated domain is smaller than that of the original domain by an amount ⌬F necessary for domain wall formation, then the nucleated domain is stable and its size can increase while the remaining domain共s兲 decrease in size. ⌬F depends on the difference of orientations of the magnetization on both sides of the domain wall denoted by the angle ␥. ⌬F0 is a characteristic property of the samples ⌬F = ⌬F0 ⫻ 关1 − cos共␥兲兴.
共6兲
This reorientation of the magnetization can be quite fast. Hrabovsky et al.15 also introduced this domain-switching energy density ⌬F0 as a phenomenological energy gap into their calculations. In this augmented model the domain-creation has to be nearly instantaneous to justify the initially postulated homogeneous magnetization of the sample 共at least of the small area in the laser spot that we look at兲. In reality, different domains may be present in the vicinity of the laser spot and the domain walls may also move through the spot 共thereby increasing the size of a nearby domain兲, and thus effectively acting as a reorientation of the magnetization of the sample area that we look at. This process is not instantaneous, and can be identified in some of our measurements by the fact that some peaks in the magnetooptical measurements are weaker and broader than predicted by the model. We attribute this to the temporary presence of more than one do-
main in the laser focus which leads to discrepancies between measurements and theoretical prediction. These, however, are exceptions which are easy to recognize, and the approach which we describe appears to work well in explaining the general process of the reorientation of the magnetization in GaMnAs films. IV. RESULTS AND DISCUSSION A. Sample with easy axis in plane
At an angle of light incidence of ⬇0°, the component of the magnetization perpendicular to the surface is detectable, i.e., one measures the projection of the magnetization on the axis perpendicular to the sample surface. The resulting hysteresis loops do not contain information about the in-plane anisotropy. Therefore, measurements at different angles of incidence and with different orientations of the external field ជ ext = 0 · H ជ with respect to the “polar” easy axis of the B samples are necessary to figure out reliable anisotropy constants. Figure 1 shows a hysteresis loop of the sample with easy axis in plane, measured by polar MOKE 共external magnetic field and direction of light incidence parallel to sample normal兲. The following discussion of this loop starts at position ជ ext兩 greater than 1.3 kG the magneti共1兲. At external fields 兩B ជ ជ ext 关see positions 共1兲 and 共5兲 zation M is oriented parallel to B in Figs. 1 and 2兴. The axis perpendicular to the sample surface being the crystallographic 关001兴 axis is one of the easy axes. However, due to the compressive strain between buffer and GaMnAs layer and due to the demagnetizing field the 关100兴 and 关010兴 axes lying in the sample surface are favored
FIG. 3. Comparison between measurement and calculation of MOKE for a magnetic field parallel to the sample surface at 1.8 K, measured at 755.3 nm with grazing incidence.
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FIG. 4. Comparison between measurement and calculation of MOKE for a magnetic field orientated under 45° with respect to the sample surface at 1.8 ជ ext on the surface is parallel to the easy axis 关100兴; K, the projection of B measured at 755.3 nm with light incidence under 45° to the sample surface as indicated by the symbolic representation of the sample on the right hand side; parts 共a兲 and 共b兲 different directions of light propagation 共see insets on the upper left side兲.
FIG. 6. Comparison between measurement and calculation of MOKE for a magnetic field orientated under 45° with respect to the sample surface at 1.8 ជ ext on the surface is parallel to the hard axis 110; measured K, projection of B at 755.3 nm with light incidence under 45° to the sample surface as indicated by the symbolic representation of the sample on the right hand side; parts 共a兲 and 共b兲: different directions of light propagation 共see insets on the upper left side兲.
at low magnetic fields. With 共from large negative to positive values兲 increasing field at a critical value of about ⫺1 kG the magnetization flips rapidly to an alignment nearly parallel to the sample surface. As noted above at this field domains are formed since the domain-switching-energy density is exceeded by the gain of magnetostatic energy. If at all fields ⌬F is greater than the energy gain, then the flips occur at higher magnetic fields where the minimum in F merges into a saddle point. ជ out of Between ⫺0.9 kG and +0.9 kG the rotation of M the plane is proportional to the strength of the magnetic field which manifests itself as a linear increase in the MOKEsignal 关positions 共2兲 to 共4兲 in Figs. 1 and 2兴. At a critical field of about 1.25 kG, the magnetization flips again collinear to ជ ext. For low fields the preferred direction of M ជ is as close as B ជ possible to the easy axis 关100兴 in the sample surface with M exactly in plane at Bext = 0 as can be seen in Fig. 1. No reជ has zero commanence and coercitivity is present since M ponent perpendicular to the sample surface 关see position 共3兲 in Figs. 1 and 2兴. In order to observe the in-plane magnetization one can ជ ext using orient the sample with its normal perpendicular to B grazing incidence of the optical beam 共see Fig. 3兲. In this case, the Kerr signals are very small; therefore, we used the geometry with the sample normal oriented 45° with respect to Bជ ext to determine in-plane anisotropy by fitting the meaជ ext onto sured hysteresis loops. In Fig. 4, the projection of B
the surface is parallel to the easy axis of the sample 共关100兴 for low temperatures兲. In this configuration the magnetic field is oriented along 关101兴 which is a “middle-hard” axis. Therefore, one needs ជ into that direction. extremely high external fields to force M In Fig. 4, this can be seen by the still increasing Kerr-signal at high positive and negative fields 关positions 共1兲 and 共5兲 in ជ parallel to Bជ ext is Figs. 4共a兲 and 5兴. The orientation of M ជ saturating for high values of 兩Bext兩 共saturation fields not shown in Fig. 4兲. For lower fields, the magnetization tends to orientate itself parallel to the easy axis 关100兴 in the sample ជ ext is increased to small positive values 关position surface. If B ជ is rotated by a small amount out 共3兲 in Figs. 4共a兲 and 5兴, M of the surface which means an angle of nearly 135° with ជ ext resulting in a high magnetostatic energy. 关Berespect to B ជ tween positions 共1兲 and 共2兲 in Fig. 4共a兲, the angle between M and Bជ ext is smaller than 45°. At position 共3兲, the direction of ជ ext is inverted and the angle between M ជ and Bជ ext is close to B ជ 135°.兴 M switches quite fast from position 共3兲 to positions 共4兲 and 共5兲. Position 共4兲 represents the transverse MOKE geometry which results in a quite high signal with our measurement technique. This flip of magnetization is visible as a ជ does sharp peak in Fig. 4共a兲. It seems to be the case that M not rotate from 共3兲 to 共5兲 as a whole but in steps over a broadened interval of B fields resulting in lower peaks than expected by the simulation. ជ ext is also oriented under Figure 6 shows the case where B 45° to the sample surface but the projection on the surface is
FIG. 5. Simulated orientation of the magnetization in relation to the crystallographic axes; the five different pictures represent five different situations during the magnetic field sweep from ⫺1 kG to +1 kG 共see Fig. 4 for comparison兲.
FIG. 7. Simulated orientation of the magnetization in relation to the crystallographic axes; the five different pictures represent five different situations during the magnetic field sweep from ⫺4 kG to +4 kG 共see Fig. 6 for comparison兲.
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FIG. 9. Parameters describing the anisotropy of the GaMnAs sample for different temperatures as obtained by comparison between measurement and simulation in Fig. 8.
FIG. 8. Measurement and calculation of MOKE for different temperatures, on the left hand side the magnetic field is orientated under 45° with respect to the easy axis and the surface and on the right hand side the polar MOKE is shown.
parallel to 关110兴 which is a hard axis at low temperatures. In this configuration two steps are observed. The first one 关poជ sition 共2兲 to 共3兲 in Figs. 6 and 7兴 is due to a rotation of M from 关100兴 to 关010兴, i.e., between two neighboring easy axes occurring at low external fields 共⬇0.25 kG兲. For the second one 关position 共4兲 to 共5兲 in Figs. 6 and 7兴 the magnetization rotates via a hard axis. As a consequence, this second step occurs at a relatively high external field 共⬇2.5 kG兲. The different shapes of the hysteresis loops for different directions of the incident light 关compare Figs. 4共a兲 and 4共b兲 and, respectively, Figs. 6共a兲 and 6共b兲兴 are a result of the ជ different modes of detection geometry: for components of M parallel to the sample normal the signal does not change its sign, when incident and reflected light beams are reversed. For the other 共in plane兲 components the signal changes its sign and therefore these components contribute Kerr intensities of different signs when the direction of the light beam is reversed. A remarkable effect in GaMnAs systems is the reorientation of the easy axis with increasing temperature.7 This can be detected by observing the different shapes of the hysteresis loops at different temperatures 共see Fig. 8兲. In all Figs. 1–8, measured hysteresis loops are compared with model calculations. It is seen, that the agreement is excellent for all configurations with a unique set of anisotropy constants given in Table I. The magnetization is measured using a SQUID magnetometer, the anisotropy constants are fitted to our experimental data. In Fig. 9, the dependence of the anisotropy constants on the temperature is
plotted. In this figure it can be seen, that the energy density ⌬F0 which is necessary to build new domains, decreases rapidly with temperature and is nearly zero at about 14 K. In contrast, the magnitude of the strain-induced anisotropy Ku decreases slowly with temperature and, by remaining below zero, allow the thin film samples to retain their easy in-plane anisotropy. Because the in-plane crystal-anisotropy 共Kc兲 decreases nearly linearly and is at some temperature below the strain-induced anisotropy 共Ku兲 the easy axis in the sample plane rotates from the 具100典 to the 具110典 axes. Our extracted anisotropy constants are in good agreement with values published in Ref. 16 although these authors used a different, but more complicated method 共ferromagnetic resonance兲.
B. Sample with easy axis out of plane
In case of a sample with easy axis parallel to the sample normal and a magnetic field perpendicular to the surface, the measurements yield rectangular hysteresis loops 共see Fig. 10兲. In such a geometry, there is only one critical magnetic field value. Therefore, within the Stoner–Wohlfarth-model not enough information is available to figure out all anisotropy constants. We used the parameters from Ref. 16 in which the same sample has been investigated. We only adjusted the domain-switching energy density ⌬F0. In Table I, all anisotropy constants are summarized. It is seen that the strain-induced uniaxial constant Ku can even reverse its sign for samples with in-plane easy axes 共as compared to those with an out-of-plane easy axis兲—a behavior that is characteristic for distinct in-plane anisotropy. The magnetocrystalline anisotropy is very similar to the value published by Bihler et al.17 and an order of magnitude greater than the value for Fe-layers on GaAs found by Pulwey et al.18
TABLE I. Parameters for the calculations presented in Figs. 1–10 at a temperature of 1.8 K
Sample type
x 共%兲
M 共A/m兲
Kc 共kJ/ m3兲
In plane Out of plane
1.4 3.0
13 300 34 000
1.862 3.74
2Kc M
共T兲
Ku 共kJ/ m3兲
2Ku M
共T兲
⌬F0 共kJ/ m3兲
0.28 0.22
⫺1.15 6.12
⫺0.18 0.36
0.22 1.1
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surements described in this paper appears to be a straightforward and convenient method that is particularly well suited for determining anisotropy constants of ferromagnetic semiconductors in thin film form. ACKNOWLEDGMENTS
Work at Notre Dame supported by the National Science Foundation Grant No. DMR06-03762. 1
H. Ohno, A. Shen, F. Matsukura, A. Oiwa, A. Endo, and S. Kutsumotu, Appl. Phys. Lett. 69, 363 共1996兲. H. Ohno, Science 281, 951 共1998兲. 3 T. Dietl, H. Ohno, and F. Matsukura, Phys. Rev. B 63, 195205 共2001兲. 4 T. Dietl, H. Ohno, F. Matsukura, J. Cibert, and D. Ferrand, Science 287, 1019 共2000兲. 5 M. Abolfath, T. Jungwirth, J. Brum, and A. H. MacDonald, Phys. Rev. B 63, 054418 共2001兲. 6 M. Sawicki, F. Matsukura, A. Idziaszek, T. Dietl, G. M. Schott, C. Ruester, C. Gould, G. Karczewski, G. Schmidt, and L. W. Molenkamp, Phys. Rev. B 70, 245325 共2004兲. 7 M. Sawicki, K.-Y. Wang, E. W. Edmonds, R. P. Campion, C. R. Staddon, N. R. S. Farley, C. T. Foxon, E. Papis, E. Kaminska, A. Piotrowska, T. Dietl, and B. L. Gallagher, Phys. Rev. B 71, 121302 共2005兲. 8 Z. J. Yang and M. R. Scheinfein, J. Appl. Phys. 74, 6810 共1993兲. 9 K. Shinagawa, Faraday and Kerr Effects in Ferromagnets 共Springer, Berlin, 2000兲, Chap. 5, pp. 137–174. 10 A. K. Zvezdin and V. A. Kotov, in Modern Magnetooptics and Magnetooptical Materials, Studies in Condensed Matter Physics, edited by A. Zvezdin and V. Kotov 共IOP, Bristol, 1997兲, Chap. 3.4, pp. 40–51. 11 R. Lang, A. Winter, H. Pascher, H. Krenn, X. Liu, and J. K. Furdyna, Phys. Rev. B 72, 024430 共2005兲. 12 C. Thurn, V. M. Axt, A. Winter, H. Pascher, H. Krenn, X. Liu, J. K. Furdyna, and T. Wojtowicz, Phys. Rev. B 80, 195210 共2009兲. 13 E. C. Stoner and E. P. Wohlfarth, Philos. Trans. R. Soc. London, Ser. A 240, 599 共1948兲. 14 U. Welp, V. K. Vlasko-Vlasko, X. Liu, J. K. Furdyna, and T. Wojtowicz, Phys. Rev. Lett. 90, 167206 共2003兲. 15 D. Hrabovsky, E. Vanelle, A. R. Fert, D. S. Yee, J. P. Redoules, J. Sadowski, J. Kanski, and L. Ilver, Appl. Phys. Lett. 81, 2806 共2002兲. 16 X. Liu, Y. Sasaki, and J. K. Furdyna, Phys. Rev. B 67, 205204 共2003兲. 17 C. Bihler, H. Huebl, M. Brandt, S. Goennenwein, M. Reinwald, U. Wurstbauer, M. Döppe, D. Weiss, and W. Wegscheider, Appl. Phys. Lett. 89, 012507 共2006兲. 18 R. Pulwey, M. Zölfl, G. Bayreuther, and D. Weiss, J. Appl. Phys. 91, 7995 共2002兲. 2
FIG. 10. Comparison between measurement and calculation of MOKE for a magnetic field orientated perpendicular to the sample surface at 1.8 K measured at 755.3 nm with light incidence parallel to the magnetic field.
V. CONCLUSION
The magnetooptical Kerr effect is very well suitable to probe hysteresis loops in ferromagnetic III-Mn-Vsemiconductor epitaxial films. Polar Kerr effect provides rectangular loops for samples with easy axis parallel to the sample normal as expected for a single-domain arrangement. The component of the magnetization oriented in plane of the film can be observed by orienting the sample surface collinear to the external field and using grazing incidence of the optical beam propagating almost parallel to the external field. In this configuration, the Kerr rotation is very small, therefore, we preferred an oblique geometry with the sample normal oriented 45° off the magnetic field. In the framework of a slightly extended Stoner–Wohlfarth model under coherent rotation of the magnetization vector, the free energy density F in an external field is calculated and minimized where the dominant contributions to anisotropy are considered: magnetostatic energy, demagnetizing anisotropy, crystallographic anisotropy, and uniaxial stress. The model gives perfect fits to the observed hysteresis loops with one set of anisotropy constants for different magnetic field and light beam geometries. Therefore, the simple method of magnetooptical mea-
