Magnetic anisotropy of epitaxial (Ga,Mn)As on (113)A GaAs

Magnetic anisotropy of epitaxial (Ga,Mn)As on (113)A GaAs

Wiktor Stefanowicz,^1,2Cezary Śliwa,¹ Pavlo Aleshkevych,¹Tomasz Dietl,^1,3Matthias Döppe,⁴Ursula Wurstbauer,⁴ Werner Wegscheider,⁴ Dieter Weiss,⁴and Maciej Sawicki¹

1Institute of Physics, Polish Academy of Sciences, al. Lotników 32/46, PL-02-668 Warszawa, Poland

2Laboratory of Magnetism, University of Białystok, ul. Lipowa 41, PL-15-424 Białystok, Poland

3Institute of Theoretical Physics, University of Warsaw, PL-00-681 Warszawa, Poland

4Department of Physics, University Regensburg, 93040 Regensburg, Germany

共Received 8 February 2010; revised manuscript received 30 March 2010; published 13 April 2010兲 The temperature dependence of magnetic anisotropy in共113兲A共Ga,Mn兲As layers grown by molecular-beam epitaxy is studied by means of superconducting quantum interference device magnetometry as well as by ferromagnetic resonance共FMR兲 and magnetooptical effects. Experimental results are described considering cubic and two kinds of uniaxial magnetic anisotropy. The magnitude of cubic and uniaxial anisotropy constants is found to be proportional to the fourth and second power of saturation magnetization, respectively. Similarly to the case of共001兲samples, the spin reorientation transition from uniaxial anisotropy with the easy axis along the关¯110兴direction at high temperatures to the biaxial具100典anisotropy at low temperatures is observed around 25 K. The determined values of the anisotropy constants have been confirmed by FMR studies. As evidenced by investigations of the polar magnetooptical Kerr effect, the particular combination of magnetic anisotropies allows the out-of-plane component of magnetization to be reversed by an in-plane magnetic field. Theoretical calculations within thep-dZener model explain the magnitude of the out-of-plane uniaxial anisotropy constant caused by epitaxial strain but do not explain satisfactorily the cubic anisotropy constant. At the same time the findings point to the presence of an additional uniaxial anisotropy of unknown origin. Similarly to the case of 共001兲films, this additional anisotropy can be explained by assuming the existence of a shear strain. However, in contrast to the共001兲samples, this additional strain has an out of the共001兲plane character.

DOI:10.1103/PhysRevB.81.155203 PACS number共s兲: 75.50.Pp, 75.30.Gw

I. INTRODUCTION

Since many decades, a lot of attention has been devoted to ferromagnetic semiconductors. More recently, the intense re- search has been triggered by the synthesis of the共III,Mn兲V diluted magnetic semiconductor共Ga,Mn兲As,¹ which has be- come the canonical example of a dilute ferromagnetic semi- conductor, DFS 共Refs. 2 and 3兲. It has been demonstrated that a number of pertinent properties of this material can be explained by thep-dZener model.^3–6Magnetic anisotropy of strained 共Ga,Mn兲As layers can be calculated within this theory and many experimental studies^7–15 were devoted to verify its predictions. However, despite these intense studies, some important features of magnetic anisotropy in this sys- tem are at present not completely understood.

An example of such a property is a rather strong in-plane uniaxial magnetic anisotropy of epitaxial 共Ga,Mn兲As layers grown on GaAs substrates of共001兲orientation. Owing to the presence of the twofold symmetry axes关100兴and关010兴, the in-plane zinc-blende directions关110兴and关1¯10兴are expected to be equivalent. Yet, as implied by the character of magnetic anisotropy, the symmetry is lowered fromD_2d toC_2v, possi- bly due to the growth-induced lack of symmetry between the bottom and the top of the layer,^9,10,16which can be phenom- enologically described by introducing a shear strain.^10,17,18

Since these symmetry considerations are limited to共001兲 layers, investigation of layers grown on substrates of other orientations may not only allow to compare experimental observations with predictions of the p-d Zener model in a more general situation but also provide information from which conclusions on the nature of the additional anisotropy can be drawn.

In this paper we present results of studies on Ga_1−xMn_xAs layers grown by low-temperature molecular-beam epitaxy 共MBE兲on GaAs substrates with the共113兲Aorientation. Until now, magnetic anisotropy in such films was probed at low temperatures by magnetoresistance,^19–22scanning Hall probe microscopy,²³ and ferromagnetic resonance measurements.^21,24,25 Experimental techniques employed here include superconducting quantum interference device 共SQUID兲 magnetometry, ferromagnetic resonance 共FMR兲, and polar magnetooptical Kerr effect. Our measurements are carried out over a wide temperature and magnetic field range.

We find that magnetic anisotropy can be consistently de- scribed taking into account three contributions: a uniaxial anisotropy with the hard axis tilted from 关113兴 toward the 关001兴direction, an in-plane uniaxial anisotropy with the easy axis along the 关¯110兴 direction, and a cubic anisotropy with easy 具100典 directions. The general form of anisotropy is, therefore, similar to the case of共001兲films, but the direction of the hard axis is found to be neither along 关001兴nor per- pendicular to the film plane in the 共113兲case. The accumu- lated experimental results allow us to determine how the three relevant magnetic anisotropy constantsKas well as the tilt angle depend on the temperature. We find that the mag- nitudes of energies corresponding to the competing cubic and uniaxial anisotropies in the共001兲plane depend, as could be expected, as the fourth and second power of spontaneous magnetizationM共T兲, respectively. In contrast, a complex de- pendence on M共T兲 is observed in the case of the energy characterizing the out-of-plane uniaxial anisotropy. We as- sign this behavior to the spin-splitting-induced and, hence, temperature-dependent redistribution of holes between the 1098-0121/2010/81共15兲/155203共11兲 155203-1 ©2010 The American Physical Society

valence-band subbands that are characterized by different di- rections of the angular momentum and, hence, of the easy axes.

In the theoretical part, we present a theory of magnetic anisotropy in epitaxially strained layers of 共Ga,Mn兲As and related systems within the p-d Zener model. Our approach generalizes earlier theories developed for 共001兲 films4,5,17,18,26 by allowing for an arbitrary crystallographic orientation of the substrate. Similarly to previous studies,^10,17 in order to explain the experimental findings, we introduce an additional shear strain whose three components constitute adjustable parameters. We also take into account the Hamil- tonian terms linear inkand find that they give a minor con- tribution to the magnitude of magnetic anisotropy constants.

II. SAMPLES AND EXPERIMENT

We study a 50-nm-thick Ga_1−xMnxAs layer which has been grown on a共113兲AGaAs substrate共see Fig.1兲by low- temperature MBE.²⁷The total Mn concentration of x= 6.4%

has been determined by means of secondary ion mass spec- trometry, however a more than twice lower value of an ef- fective Mn concentration x_effcan be inferred from the low- temperature experimental saturation magnetization, M_exp. This reduction ofxis primarily caused by a presence of Mn interstitials. These point defects act as double donors and form strongly coupled spin singlet pairs with neighbor sub- stitutional Mn cations.^6,28,29These pairs neither participate in the ferromagnetic order nor do they contribute to M. Thus, the effective concentration of Mn ions which generatesM_exp gets reduced tox_eff=x− 2x_I, wherex_IN₀is the concentration of the Mn interstitials and N₀ is the cation concentration.

However, the experimentally measured M_exp is further re- duced by the hole magnetization, M_h, which is oppositely oriented to magnetization of Mn spins, M_Mn, and so M_Mn

=M_exp+兩Mh兩 should be used to calculatex_eff, withMhbeing computed in the framework of the mean-field p-d Zener model.^4,5 We perform these calculations in a self-consistent way taking the hole concentration as p=N₀共x− 3x_I兲

=N₀共3xeff−x兲/2, that is neglecting other charge compensat- ing defects.

The open air post growth annealing at temperatures below or comparable to the growth temperature^30,31is a fre- quently used procedure for improving material parameters of 共Ga,Mn兲As since the corresponding out-diffusion and passi- vation of Mn interstitials³² increases x_eff, p, and eventually the Curie temperature T_C. Therefore in order to widen the parameter space employed here to study the magnetic aniso-

tropy we investigate both the as-grown material共sample S1兲 and the samples annealed at 200 ° C for 1.5 h 共sample S2兲 and 5 h 共sample S3兲. Taking the determined values ofM_exp we end up withx_eff= 2.7%, 3.1%, and 3.3%, andp= 2.0, 3.3, and 3.8⫻10²⁰ cm⁻³ for which calculated values of T_C= 46, 73, and 85 K compares favorably with the experimentally established values of 65, 77, and 79 K, for samples S1, S2, and S3, respectively.

Magnetic properties referred to above and described fur- ther on have been obtained by utilizing a Quantum Design MPMS XL-5 magnetometer. A special demagnetization pro- cedure has been employed to minimize the influence of para- sitic fields on zero-field measurements. The temperature de- pendence of remnant magnetization, TRM, serves to obtain an overview of magnetic anisotropy as well as to determine T_C共Sec.III A兲. After cooling the sample acrossT_Cdown to 5 K in an external magnetic field of 0.1 T, the field is re- moved, allowing the magnetization to align along the closest easy axis. The magnitude of the magnetization component along the magnet axis, TRM_i, is then measured while heat- ing, where i indicates one of the three mutually orthogonal directions of the magnetizing field, corresponding to the sur- face normal n₁=关113兴 and the two edges n₂=关332¯兴 andn₃

=关¯110兴, as depicted in Fig.1. Since, except in the immediate vicinity of the spin reorientation transition, magnetization of 共Ga,Mn兲As films tend to align in a single domain state, the measurements performed for the three orthogonal axes pro- vide the temperature dependence of the magnetization mag- nitude and direction.

To study magnetic anisotropy in a greater detail, magnetic hysteresis loops Mi共H兲have been recorded in external mag- netic field in the range of⫾0.5 T along the three directions i. The measurements have been carried out at various tem- peratures and the parameters of the anisotropy model 共Sec.

III B兲 have been fitted to reproduce the magnetization data.

To crosscheck magnetic anisotropy constants obtained from SQUID studies, FMR measurements have been performed at

␻/2␲= 9.3 GHz and T= 10 K. We have performed angle- dependent measurements of the resonance field in the four different crystallographic planes 共1¯10兲, 共332¯兲, 共113兲, and

1

2^冑11共3 −

冑

11, 3 +

冑

11, −2兲. As discussed in Sec. III C, the FMR data are in a good agreement with the anisotropy model, employing parameters determined from the SQUID measurements.

III. EXPERIMENTAL RESULTS A. Overview of magnetic anisotropy

The TRM studies of all three samples are summarized in Fig.2. We immediately see that the TRM_关1^¯_10兴 component of TRM is the strongest for all of the samples and that at el- evated temperatures its magnitude is nearly equal to the satu- ration magnetization M共T兲, established by the measurement in␮0H= 0.1 T. Since the magnitude of the other two mag- netization components is vanishingly small, we find that in this temperature range the in-plane uniaxial anisotropy with the easy axis along关1¯10兴direction dominates. This perfectly uniaxial behavior at T→TC allows us to use TRM_关1^¯_10兴 to [113]

[001]

[010]

[332]_

[110]_ [100]

FIG. 1. 共Color online兲 Crystallographic directions for a GaAs substrate of共113兲orientation.

precisely determineT_Cin the studied samples共already given in the previous section兲. On the other hand, below a certain temperatureT^ⴱ共marked by arrow for every sample in Fig.2兲 TRM_关1^¯_10兴 gets visibly smaller than M共T兲, and the other in- plane TRM component, TRM_关332^¯_兴, acquires sizable values, followed at still lower temperatures by the out-of-plane com- ponent TRM_关113兴. This clearly indicates a departure of the easy direction from the 关1¯10兴 direction below these charac- teristic temperatures. Such a scheme turns out to be fully equivalent to the general pattern of magnetic anisotropy in 共001兲 共Ga,Mn兲As under compressive strain.7,8,10,11,16,33,34 In such films uniaxial anisotropy between 关110兴 and关11¯0兴 di- rections, dominating at elevated temperatures, gives way at low T to biaxial anisotropy with in-plane 具100典 easy axes.

This spin reorientation transition共SRT兲takes place at a tem- perature, at which uniaxial and biaxial anisotropy constants equilibrate,¹¹and is corroborated numerically in our samples from analysis of the magnetization processes presented in Sec. III B.

In an analogy to 共001兲 共Ga,Mn兲As, let us assume for a moment that M of a共113兲 sample remains共without a mag- netic field兲in the共001兲plane. Then, a similar description in terms of two in-plane anisotropies 共one biaxial and one uniaxial兲is possible. Furthermore, assuming that the uniaxial anisotropy constant is proportional to M共T兲², the biaxial an- isotropy constant is proportional toM共T兲⁴共Ref.11兲, and that both are equal at T^ⴱ we are able to model qualitatively the temperature-induced rotation of magnetization in the sample and calculate all three components of magnetization that would be measured by SQUID. The thick solid lines in Fig.

2show the results for sample S1 and we find them reproduc- ing the experimental findings reasonably well. Therefore we identifyT^ⴱas the temperature at which the spin reorientation transition from a biaxial anisotropy along 具100典 to uniaxial one along 关11¯0兴 takes place in this system. On the other

hand, the discrepancies seen in Fig. 2 indicate that a more elaborated model is needed. In particular, we can infer from the low temperature TRM data that the orientation ofMat 5 K moves actually away from具100典on annealing. The angle between M and具100典 is increasing from 9, through 19° to 26° for samples S1, S2, and S3 respectively. At the same time the angle betweenMand共113兲plane is dropping from 13° to 7°. This indicates that the plane in which both easy orientations of Mreside at lowT is tilting away from共001兲 toward the 共113兲 plane. This observation is fully confirmed from the comprehensive analysis of the magnetic anisotropy presented in the next section. We remark here that the origin of the symmetry breaking between 关110兴and关11¯0兴 in共001兲 共Ga,Mn兲As is still unknown and it is very stimulating to see a preferred in-plane关11¯0兴orientation also in layers of differ- ent surface reconstruction than共001兲GaAs.

B. Experimental determination of anisotropy constants In order to build up a more complete anisotropy descrip- tion we analyze the full magnetization curves M共H兲. It was shown by Limmer et al.共Refs. 20 and21兲that an accurate description of the magnetic anisotropy in共113兲A共Ga,Mn兲As requires at least four components: a cubic magnetic aniso- tropy with respect to the具001典axes, uniaxial in-plane aniso- tropy along the 关1¯10兴 direction, and two uniaxial out-of- plane anisotropies along the 关113兴and关001兴directions. The first two anisotropy components are commonly observed in

FIG. 3.共Color online兲Examples of magnetization curves for the as-grown sample measured along three, mutually orthogonal, major sample directions关1¯10兴,关332¯兴, and关113兴:共a兲below spin reorienta- tion transition at 10 K and 共b兲 above, at 38 K. Symbols indicate measurement points; lines represent the best fit of the model de- scribed by Eq.共1兲.

0 20 40 60 80 100

0 5 10 15 20 25 30

S1 calc. S2 S3 calc.

[110]µ₀H = 0.1 T [110] TRM [332] TRM [113] TRM

Magnetization(kA/m)

Temperature (K)

FIG. 2. 共Color online兲 Temperature dependence of remnant magnetization components in all three samples 共points兲. Black lines:M共H兲at 0.1 T along关¯110兴—the in-plane uniaxial easy axis.

Solid colored lines共color online兲: the same magnetization compo- nents calculated according to a model of only two: uniaxial and biaxial magnetic anisotropies operating at共001兲plane and undergo- ing a spin reorientation transition at temperatures T^ⴱ 共marked by arrows兲, as in共001兲 共Ga,Mn兲As.

共001兲-oriented共Ga,Mn兲As samples, and, as shown in the pre- vious section, they are sufficient to provide a semiquantita- tive description in the 共113兲case. The other two arise from the epitaxial strain and demagnetizing effect, both of which depend on the orientation of the substrate. In our approach we combine the two out-of-plane magnetic anisotropy con- tributions into a single one, with its hard axis oriented be- tween the关001兴and关113兴directions. Accordingly, we write the free energy in the form

F= −␮0H·M+K_C共w_x²w_y²+w²_yw_z²+w_z²w_x²兲 +K_u1^¯₁₀sin²⌰sin²⌽+K_u1共cos⌰Acos⌰

− sin⌰Asin⌰cos⌽兲². 共1兲

Here,K_C,K_u1^¯₁₀, andK_u1are the lowest order cubic, in-plane uniaxial and out-of-plane uniaxial anisotropy energies, re- spectively;⌰Adescribes the angle between theK_u1hard axis and关113兴direction;w_x,w_y, andw_zdenote direction cosines of the magnetization vector with respect to the main crystal- lographic directions具100典;⌰is the angle betweenMand the 关113兴direction, and⌽is the angle between the projection of M onto the sample plane and the关332¯兴 direction.

By numerical minimizing of the free energy with respect to⌰ and⌽we are able to trace the rotation of M, starting from a given orientation, while sweeping or rotating the ex- ternal magnetic field. Adjusting the obtained “trace” to the experimental data we get the values of the four parameters of the model. We perform this procedure numerically for every sample for all three orientations and for all temperatures the Mi共H兲 curves have been recorded. Figure 3 shows an ex-

ample of the measured and fittedMi共H兲for sample S1 at two different temperatures.

The temperature dependence of the three magnetic aniso- tropy constants and the angle ⌰A is presented in Figs.4共a兲 and4共b兲, respectively. AllKi’s monotonically decrease with temperature, and, like in 共001兲 共Ga,Mn兲As, the cubic aniso- tropy constant KC 关with具100典 easy axes, see Fig.4共c兲兴and in-plane uniaxial constant K_u1^¯₁₀ 关with 关11¯0兴 easy axis, see Fig. 4共d兲兴 are proportional to M⁴ and M², respectively, so confirming the validity of the single domain approach used to analyze the observed magnetization rotations. The KC共T兲 andK_u1^¯₁₀共T兲 data point to the presence of the spin reorien- tation transition in the 共001兲 plane. This already inferred from TRM data magnetic easy axis changeover must take place as KC共T兲 and K_u1^¯₁₀共T兲 swap their intensities in our samples. The relevant temperatures are marked in Fig. 4共a兲 by arrows. Importantly, we find these temperatures to agree within 2–3 K with those indicated in Fig. 2, strongly under- lining the correctness of the approach we employ here to describe the magnetic anisotropy in our samples. We note that the SRT shifts to lower temperatures on going from sample S1–S3 since on annealing the in-plane uniaxial an- isotropy gets strongly enhanced relative to the cubic one 关compare Figs.4共c兲and4共d兲兴.

In contrast, K_u1 shows a more complex dependence on M², see Fig. 4共e兲. A proportionality of the out-of-plane an- isotropy constant toM²is seen only at lowM, i.e., at highT.

On lowering temperatureK_u1departs from this trend and the effect is strongest for sample S1. We ascribe this behavior to the proximity of the system to another spin reorientation transition, the transition from the hard to easy out-of-plane axis of theK_u1 uniaxial magnetic anisotropy. This switching

0.0 0.5 1.0

0 1 2 3 4

Ku1(kJ/m3 )

M²(10⁹A²/m²) K_u1

0.0 0.5 1.0

0.0 0.2 0.4 0.6 0.8

-Ku110(kJ/m3 )

M²(10⁹A²/m²) K_u110

0.0 0.5 1.0

0.0 0.5 1.0 1.5

KC(kJ/m3 )

M⁴(10¹⁸A⁴/m⁴) K_C

0 20 40 60 80

0 5 10 15 20 25

ΘA(deg)

Temperature (K)

S1S2 S3

[113]

[001]

(e)

0 20 40 60 80

0 1 2 3

4 _{S1 S2 S3}

K_u1 K_C K_u110

Anisotropyconstants(kJ/m3 )

Temperature (K) SRT

(a) (b)

(c) (d)

FIG. 4. 共Color online兲 共a兲and共b兲points: temperature dependence of共a兲K_C,K_u1^¯₁₀andK_u1and共b兲angle⌰Aobtained from numerical fitting of Eq.共1兲to experimental magnetization curves for all three samples considered in this study. Solid, dashed, and dotted arrows in共a兲 indicate the spin reorientation temperature in samples S1, S2 and S3, respectively.关共c兲–共e兲兴:K_C,K_u1^¯₁₀, andK_u1dependence onM⁴,M², and M², respectively. In all panels, the various lines are guides for eye only.

of the magnetic easy axis, already inferred from TRM data, must take place in compressively strained共001兲 共Ga,Mn兲As on lowering T, and was already observed in samples with moderate or high xbut rather low hole density.⁹ The effect depends on the ratio of valence-band spin splitting to the Fermi energy. Therefore the S1 sample, the one with the lowest p is expected to show the strongest deviations from the expected functional form. Then on annealing, along with the increase inp, we expect the so called in-plane magnetic anisotropy共for the compressively strained layers兲to become more robust关less dependent on the magnitude of the valence band splitting, i.e. on M共T兲兴, as experimentally observed.

Finally, we comment on⌰A, the angle between an “easy plane” with respect toK_u1 共perpendicular兲hard axis and the sample face. As indicated in Fig. 4共b兲 this angle remains nearly constant at elevated temperatures and shows a weak but noticeable turn towards关001兴below temperatures which can be associated to theK_u1^¯₁₀⇔KCSRT. This behavior again indicates the departure of the easy direction ofMfrom关11¯0兴 共direction兲in the共113兲plane. However, the maximum value of⌰A⬵15° indicates, that the rotation ofMactually neither takes place in the 共001兲 plane, nor is it directed exactly to- wards具100典directions.Mrather follows a complex route in between 共001兲 and共113兲 planes, a conclusion that is a nu- merical confirmation of the results of the simple analysis of the TRM data presented in the previous section.

C. Ferromagnetic resonance

A tool widely used to study magnetic anisotropy is ferro- magnetic resonance spectroscopy. Magnetic anisotropy in thin 共Ga,Mn兲As films on共113兲A GaAs was recently studied by Bihler²⁴ and Limmer.^20,21 In a ferromagnetic resonance experiment the magnetization vector M of the sample pre- cesses around its equilibrium direction in a given external magnetic field H with Larmor frequency ␻L. The resonant condition at a fixed microwave frequency␻ is given by

冉

^␻␥

冊

^2=sin12⌰

冋

^{⳵⳵⌰2F2⳵⳵2⌽F2−}

^冉

^{⳵⌰⳵2⳵F⌽}

^冊

²

册

^{. 共2兲}

Here,␥^=g␮Bប⁻¹is the gyromagnetic ratio,gis thegfactor,

␮B the Bohr magneton, and ប is the Planck constant. The resonance field is obtained by evaluating Eq.共2兲at the equi- librium position ofM 共⳵^F/⳵⌰= 0 and ⳵F/⳵⌽= 0兲.

In Fig. 5the dependence of the measured resonant fields on the orientation of the applied magnetic field is shown for the sample S2 along with the results of a calculation made according to Eq.共2兲with the magnetic anisotropy parameters obtained from SQUID magnetization curves. The agreement between the calculation and the measured data is very good, indicating that Eq.共1兲captures the main features of magnetic anisotropy and that the numerical procedure, employed to extract the anisotropy constants, is correct.

D. Magnetization reversal

We end the experimental part demonstrating an interesting reversal mechanism of the out-of-plane magnetization com- ponent by an in-plane magnetic field. Below the spin reori-

entation transition 共about 20–30 K兲 the magnetization easy axes are moving close to the 具100典 directions, i.e., they are tilted up from the sample face, the共113兲plane, and soM is acquiring a sizable nonzero component M_关113兴. These two axes define a plane lying between the共113兲and共001兲planes, which share the common 关1¯10兴 direction with those two planes. Therefore, any sweep of an external field, except that along the 关¯110兴 direction, will result in a magnetization ro- tation across the 关1¯10兴line from one half of that plane共say that one “above” the sample face兲to the other one共say “be- low”兲resulting in M_关113兴 reversal.

Such a process is illustrated in Fig.6in the most interest- ing case, when the field is swept in the sample plane, along 关332¯兴. We record theM_关113兴 magnetization component using the polar magneto-optic Kerr effect共MOKE兲technique and a

0.0 0.3 0.6

0 30 90 60

120 150

210

240 270 300 330 0.0

0.3 0.6

[332]_ [110]_

H [113]

measured calculated [110]

[110]

[113]

µ0Hres(T)

a)

0.0 0.3 0.6

0 30 90 60

120 150

210

240 270 300

330 0.0

0.3 0.6

[332]_ [110]_

H [113]

[113]

µ0Hres(T)

0.0 0.3 0.6

0 30 90 60

120 150

210

240 270 300

330 0.0

0.3 0.6

[332]_ [110]_

H [113]

[332]

[3 32]

[113]

µ0Hres(T) 0.0

0.3 0.6

0 30 90 60

120 150

210

240 270 300

330 0.0

0.3 0.6

[113]

[332]_ [110]_ H

[110]

µ0Hres(T)

[110]

[332]

(a) (b)

(c) (d)

FIG. 5. 共Color online兲Angular dependence of the ferromagnetic resonance fields for external magnetic field rotating in four different crystallographic planes: 共113兲, 共¯110兲, 共332¯兲, and _2冑¹₁₁共3 −

冑

11, 3 +

冑

11, −2兲. Points show the measured resonance field values. The lines show resonant field values calculated using the anisotropy energy described by Eq.共1兲and magnetic anisotropy constants ob- tained from hysteresis loops.

-40 -20 0 20 40

-2 -1 0 1 2

M M M[113]

H

M M

M[113]

H

M[113](arb.units)

µ₀H_[332](mT)

T= 15K

FIG. 6. 共Color online兲The out-of-plane magnetization compo- nent dependence on the in-plane magnetic field共swept along关332¯兴兲. The measurement is performed on the as-grown sample S1 at T

= 15 K. The cartoons inserted in the figure illustrate the process.

clear change of sign of the signal shows the reversal ofMby the application of an in-plane magnetic field. The cartoons inserted in this figure visualize the mechanism of this rever- sal.

IV. THEORY OF MAGNETIC ANISOTROPY A.k·pHamiltonian

The current theory describing the properties of the 共Ga,Mn兲As ferromagnetic semiconductor is the p-d Zener model.⁴In this model, the thermodynamic properties are de- termined by the valence-band carriers contribution to the free energy of the system, which is calculated taking the spin- orbit interaction into account within the k·p theory^4–6,17 or tight-binding model³⁵ with thep-d exchange interaction be- tween the carriers and the localized Mn spins considered within the virtual-crystal and molecular-field approxima- tions. Within this approach, magnetic anisotropy depends on the strain-tensor components.

The six-band Luttinger k·p Hamiltonian of a valence- band electron in a zinc-blende semiconductor is a block ma- trix共cf. Ref.36兲

H_6⫻6=

冉

^HH^{vvsv H}H^vsss

冊

^{, 共3兲}

where H^vv= −ប²

m

再

^{12␥1k2−␥2}

^{冋 冉}

^Jx2−13J2

^冊

^kx2+c.p.

^册

^{− 2␥3关兵Jx,Jy其}

⫻兵kx,ky其+c.p.兴

冎

^{, 共4兲}

H^vs= −ប²

m关− 3␥2共U_xxk_x²+c.p.兲− 6␥3共U_xy兵k_x,k_y其+c.p.兲兴, 共5兲

H^ss= −

冉

^⌬0+2m^ប2␥1k²

冊

^共6兲

共we use the notation of Ref.36:mis the free electron mass,

␥iare the Luttinger valence-band parameters,⌬0is the split- ting of the valence band at the ⌫ point,Ja are the angular momentum matrices for spin 3/2, and U_abare the Cartesian components of a rank-2 tensor operator for the cross space^36,37兲. Our basis is related to that of Ref.5 as follows:

u₁= −兩³₂,₂³典, u₂= −i兩³₂,¹₂典, u₃=兩³₂, −¹₂典, u₄=i兩³₂, −³₂典, u₅=

−兩¹₂,¹₂典, and u₆=i兩¹₂, −¹₂典, i.e., we use the standard basis of angular momentum eigenvectors 共notice the change of sign in兩¹₂,¹₂典and兩¹₂, −¹₂典with respect to Ref.36that accounts for the difference in the sign ofH^vs,H^sv兲. In this basis the p-d exchange Hamiltonian is

Hpd=BG

冋

^{− 6共T2共J··w兲w兲 − 6共U−共}␴^··w兲^w兲

册

^{, 共7兲}

wherew=M/M, the bold symbolsJ,␴^{,U, andTdenote the} vectors of matricesJa, Pauli matrices ␴a, rank-1 tensor op-

eratorsUa, andTa=U_a^†, respectively, andBGis given by Eq.

2 of Ref. ⁵

BG=AF␤M/共6g␮B兲 共8兲 while the strain Hamiltonian

H_⑀^vv= −b

关共

^Jx2−13J2

兲

^⑀xx+c.p.

兴

⁻

_冑

^d

3关2兵J_x,J_y其⑀xy+c.p.兴, 共9兲

H_⑀^vs= − 3b共Uxx⑀xx+c.p.兲−

冑

3d共2Uxy⑀xy+c.p.兲, 共10兲

H_⑀^ss= 0 共11兲 is given in terms of the symmetric strain tensor⑀ab and pa- rametrized by the deformation potentials b and d. For nu- merical values of the material parameters we refer the reader to Refs.5 and38.

Since the strain tensor for the 共113兲substrate orientation features nonzero nondiagonal components, it is necessary to include in the k·p Hamiltonian the so-calledk-linear terms, i.e., terms linear in kand⑀^comingviasecond-order pertur- bation 共Ref. 39, paragraph 15兲 from the terms in the 8⫻8 Kane Hamiltonian⁴⁰that mix the conduction and the valence bands. The corresponding Hamiltonian is

Hk⑀=C₄

冤

³²

^冉

^{1 −J·␩2␸}

^冊

^{T·␸ 32共}

^冉

^{1 −1 −␩␩2兲␴}

^冊

^U·␸·␸

冥

^{, 共12兲}

where ␩⁼⌬0/共Eg+⌬0兲and the components of the vector ␸ are␸z=⑀zxkx−⑀zyky共c.p.兲. The numerical value given in Ref.

40isC₃/ប= 8⫻10⁵ m s⁻¹, hence for C₄= −C₃/共2␩兲we ob- tain C₄/ប= −2.2⫻10⁶ m/s.

B. Strain tensor

Determining the components of the strain tensor for an unrelaxed epitaxial layer grown on a lattice mismatched sub- strate can be considered a classical topic. The two possible approaches to this problem are共i兲to solve a system of linear equations for the strain and stress components assuming that some components of those tensors vanish 共this is our ap- proach兲 or 共ii兲to determine the strain of the layer by mini- mizing the elastic energy共this is the approach formulated in Ref. 41兲. Our approach involves a transformation of the co- ordinate system that is feasible in general only using a com- puter algebra system. Using one we arrive to the form of the symmetric strain tensor that is in a perfect agreement with that of Ref. 41.

In Ref. 41, the deformation of the layer is decribed by a matrix of coefficients␣which relates the lattice vectors after the deformation to those before one. For a 关k,k,n兴-oriented substrate this matrix is

␣= f

A+B+C

冢

^{B−−+AEC B−−+AEC B−−− 2CDD}

冣

^{, 共13兲}

where

A= 3k²关共2k²+n²兲d₃+n²d₄兴d₁, 共14兲 B= 2关共2k²+n²兲²d₂− 3k⁴d₄兴d3, 共15兲 C=共n²−k²兲共2k²+n²兲d1d₃, 共16兲 D= 3kn关共2k²+n²兲d3+n²d₄兴d1, 共17兲 E= 3kn关共2k²+n²兲d3+k²d₄兴d1 共18兲 with d₁=c₁₁+ 2c₁₂, d₂=c₁₁−c₁₂, d₃=c₄₄, d₄=c₁₁−c₁₂− 2c₄₄, and

f= −⌬a/a=共a0−a兲/a 共19兲 is the relative difference between the lattice constant of the substrate,a₀, and that of the layer,a.

Using the values c₁₁= 119. GPa, c₁₂= 53.8 GPa, and c₄₄

= 59.5 GPa 共Ref. 42, p. 105兲 we obtain the strain compo- nents⑀ij

epi=共␣ij+␣ji兲/2 that enter thek·p Hamiltonian

⑀epi= −⌬a

a

冢

^0.9488 − 0.0512 − 0.3478

− 0.0512 0.9488 − 0.3478

− 0.3478 − 0.3478 − 0.6260

冣

^.

Here, the components of the epitaxial strain tensor are given with respect to the coordinates 共x,y,z兲 associated with the crystallographic axes,x=关100兴,y=关010兴, andz=关001兴.

To determine the strain components from x-ray diffraction data, the components of the strain tensor in the coordinate system associated with the epitaxial film are needed. We take as the coordinate system: x⬘^{=关n,n, −2k兴, y}⬘⁼关−1 , 1 , 0兴, z⬘

=关k,k,n兴. The relative difference of the lattice constants along the 关k,k,n兴 direction between that layer and the sub- strate is

⌬d d =⌬a

a +␣z⬘^z⬘= 3共A+C兲 A+B+C

⌬a

a = 1.7284⌬a

a . 共20兲 For the sake of completeness we notice that there is also a shear strain component

␣x⬘^z⬘=

冑

2共D−E兲 A+B+C

⌬a

a = − 0.5492⌬a

a , 共21兲 which corresponds to a superposition of a deformation with

⑀x⬘^z⬘=⑀z⬘^x⬘=␣x⬘^z⬘/2 and a rotation by the angle ␣x⬘^z⬘/2 around the axisy⬘^.

Following Ref. 10, to account for the mechanism which generates the in-plane uniaxial anisotropy in共001兲samples, we incorporate in thep-d Zener model an additional Hamil- tonian term corresponding to shear strain⑀⬘

⑀⁼⑀epi+⑀⬘^{. 共22兲} In case of a共001兲-oriented substrate the additional strain⑀⬘ has a nonzeroxy component,⑀xy⬘. The corresponding aniso-

tropy is of the form K_xyw_xw_y 共as in Ref. 43兲, hence it is a difference of uniaxial anisotropies along the关110兴and关1¯10兴 directions, and the anisotropy field is Hu= 2Kxy/共␮0M兲 共this is the field required to align the magnetization along the hard axis, e.g.关110兴; onlyH_u/2 is required to align the magneti- zation along the zdirection兲. In the case of a共113兲-oriented substrate the additional strain may have more nonzero com- ponents. We assume that the mirror symmetry with respect to the 共1¯10兲plane is preserved, hence⑀xz⬘ ⁼⑀yz⬘^.

C. Numerical procedure

The numerical procedure to determine the magnetic an- isotropy from the Hamiltonian matrix is described in Ref.5.

Let us note that including the k-linear terms in the Hamil- tonian leads to a tenfold increase in the processing time, although in specific cases it is possible to generate a sym- bolic expression for the characteristic polynomial of the 6

⫻6 Hamiltonian matrix. Moreover, since numerical interpo- lation of the dependence of the hole concentration on the Fermi energy may lead to uncontrollable inaccuracies, an alternative procedure that avoids those inaccuracies is to di- rectly integrate the energy of the carriers in momentum space. However, the integration has to be done separately for each hole concentration共this is an advantage if a single hole concentration is specified兲. Moreover, one still needs to solve the inverse eigenvalue problem to find the discontinuities of the integrand.

In a numerical calculation, it is possible to determine the full magnetic anisotropy by computing the free energy of the carriers for a number of directions of magnetization. In our case we choose a grid of directions that is rectangular in the spherical coordinates 共w_x= sin␪^cos␾^{, w}y= sin␪^sin␾^{, and} wz= cos␪兲, i.e. ␪=␪i and ␾=␾j, where cos␪i, i

= 1 , 2 , . . . ,N_␪ are the nodes of a Gaussian quadrature and

␾j= 2␲^j/N_␾, j= 0 , 1 , . . . ,N_␾− 1 are equally spaced. Then, following the method used in the software packageSHTOOLS

共Ref. 44兲 共routine SHExpandGLQ兲we expand the magnetic anisotropy 共free energy兲 into a sum of low-order spherical harmonics. Since the free energy is even, choosing evenN_␾ allows to restrict the grid to a half of a sphere. We use the standard quantum mechanics 共orthonormalized兲 spherical harmonics Ylm共␪,␾兲, and denote the coefficients of this ex- pansionrlm,m= 0 , 1 , . . . ,l, andslm,m= 1 , 2 , . . .l

Fc=

兺

l

冋

^rl0Yl0+m=1

^兺

^{l 共rlmRYlm+slmIYlm兲}

册

^{, 共23兲}

where the outer sum is over l= 0 , 2 , . . . ,N_␪− 1. The scalar product in this representation is diagonal, with a weight of 1 for r_l0and 1/2 for r_lmands_lm,m⬎0.

D. Magnetic anisotropy

There are a few sources of magnetic anisotropy in epitax- ial Ga_1−xMnxAs: the cubic anisotropy of the valence band, epitaxial strain, the additional off-diagonal strain⑀⬘^{, and the} shape anisotropy caused by the demagnetization effect. Since

⑀⬘ is unknown, it is inevitable to parametrize the anisotropy

in a manner that separates the components affected by ⑀⬘ from what is predictable. If we measure共␪,␾兲in the spheri- cal harmonic representation of the carriers’ free energy with respect to the crystallographic axes, the above spherical har- monic representation allows to separate the components due to a nonzero value of⑀⬘from the remaining sources of mag- netic anisotropy. Indeed, as far as l= 2 is concerned,⑀⬘ ^af- fects primarily only Kxy, Kyz, and Kxz, where Kxy=

冑

_8␲¹⁵s₂₂, Kyz= −

冑

_8␲¹⁵s₂₁, andKxz= −

冑

_8␲¹⁵r₂₁. The remaining components are r₂₀ and r₂₂. As r₂₂= 0 due to the mirror symmetry, they can be collected into one term K_u001共w_z²−₃¹兲, with K_u001

=³₄

冑

_␲⁵r₂₀. Thus, we describe thel= 2 anisotropy共without the demagnetization contribution兲 by Kxy, Kxz=Kyz, and K_u001. Finally, the cubic anisotropy corresponds to 共r₄₀,r₄₄兲=

−²₁₅^冑␲KC共1 ,

冑

10/7兲.

We have to relate now the components of the spherical harmonic representation, r_lm and s_lm, to the experimentally determined magnetic anisotropy constants KC,K_u1,K_u1^¯₁₀, and⌰A, as specified in Eq.共1兲and presented in Fig.4. Since the demagnetization effect adds to 共r₂₀,r₂₁,s₂₁,r₂₂,s₂₂兲 the contribution ^4冑₁₆₅^10␲Kd共4

冑

2 , −3

冑

3 , −3

冑

3 , 0 ,

冑

3兲, with Kd

=␮0M²/2, the constants are related to those of Eq. 共1兲 as follows:

K_u001=1 + 3 cos 2⌰A⬘ 4 K_u1−1

2K_u1^¯₁₀− 8

11Kd, 共24兲

Kxy=1 − cos 2⌰A⬘

2 K_u1−K_u1^¯₁₀− 2

11Kd, 共25兲

Kxz= −

冑

2 sin 2⌰A⬘ 2 K_u1− 6

11Kd, 共26兲 where⌰A⬘⁼⌰A− arccos共3/

冑

11兲.

We carry out numerical calculations with band-structure parameters and deformation potentials specified previously.^5,10 We include hole-hole exchange interactions via the Landau parameter of the susceptibility enhancement, AF= 1.2共Ref. 5兲. This parameter, assumed here to be inde- pendent of the hole density and strain, enters into the relation betweenMandB_Gbut also divides the anisotropy constants.

More specifically, we make the calculation withBGenhanced by the factor AF, and divide the resulting anisotropy con- stants by A_Fⁿ⁻¹, where n is the power of magnetizationM to which a given anisotropy constant is proportional. The result is proportional toAF. We haven= 2 for the uniaxial anisotro- pies andn= 4 for the lowest-order cubic anisotropy共the pro- portionality holds forBGsmaller than a few meV兲. We note that the cubic anisotropy field shown in Fig. 9 of Ref.5was divided byAFrather thanA_F³.

To evaluate the effect of thek-linear terms, we use a non- zero value ofC₄and calculate the difference of the resulting anisotropy with respect to theC₄= 0 case. This difference has only one noticeable component, ⌬r20=r₂₀共C4⫽0兲−r₂₀共C4

= 0兲, which corresponds to a uniaxial anisotropy with a关001兴 axis 共or with 关100兴 and 关010兴 axes for ⑀yz⫽0 and ⑀xz⫽0, respectively兲. A plot of⌬K_u001⬀⌬r₂₀is shown in Fig.7 for

⑀xy= 0.05% 共as implied by the symmetry, ⌬Ku001 is second

order in⑀xy兲. The values are rather small. In fact, assuming

⌬a/a= 0.5%, we have ⑀xz=⑀yz⬇0.17%, and the magnitude of ⌬Ku100=⌬Ku010is below 10 J/m³if we consider the ep- itaxial strain only. This estimate appears to remain valid in case of a general strain of a similar magnitude although other anisotropy components are affected as well and the depen- dence on strain components is nonlinear. However, we stress that this estimate depends on the value of the parameterC₄, which is somewhat uncertain and may be different for the ordinary strain and ⑀⬘. Considered this, it is justified to set C₄= 0 in the remaining part of this paper.

Before we proceed to the calculations specific to the par- ticular samples, we make a remark that the data originally shown in Fig. 6 of Ref.10were not correct due to a numeri- cal error in the form of the strain Hamiltonian. We show corrected results forHu in Fig.8. The present results are in agreement with Fig. 17 of Ref.17共remember that our model includes the Landau parameterA_F, neglected in Ref.17兲.

As discussed previously,^9,10 owing to sign oscillations of the anisotropy constants, the direction of magnetization can be changed by temperature 共BG兲or hole concentration, par- ticularly in the vicinity of p= 6⫻10²⁰ and 1⫻10²⁰ cm⁻³, according to the results displayed in Fig.8 The correspond- ing in-plane spin reorientation transition has indeed been ob- FIG. 7. 共Color online兲The contribution of thek-linear terms to the 关001兴 uniaxial anisotropy for C₄/ប= −2.18⫻10⁶ m/s and⑀xy

= 0.05%.

FIG. 8. 共Color online兲 Hole concentration dependence of the in-plane uniaxial anisotropy field due to shear strain⑀xy= 0.05% for various values of the valence-band spin splitting.

served by some of us either as a function of temperature¹⁰or the gate voltage in metal-insulator semiconductor structures^45,46 in these two hole concentration regions in 共Ga,Mn兲As, respectively.

V. COMPARISON BETWEEN EXPERIMENT AND THEORY

We detailed above a microscopic model of the magnetic anisotropy in a DFS. In order to assess the applicability of this model to an arbitrarily oriented DFS we compare its predictions with experimental findings for共113兲 共Ga,Mn兲As.

First, we specify the magnitude of the lattice mismatch to establish the components of the strain tensor. According to Fig. 2 of Ref.47, for a共113兲sample containing 6.4% of Mn we expect ⌬d/d= 5.6⫻10⁻³ which, employing Eq. 共20兲, translates into ⌬a/a= 0.323%. We assume this value throughout this section.

Then, using the values of x_effalready established in Sec.

II, we calculate

M_Mn=x_effN₀Sg␮B 共27兲 and obtain BG for each of our samples from Eq. 共8兲. It is worth repeating here, that the established共upon total x and M_exp兲values ofx_effandpreproduce, within the same model, experimental values ofT_Cremarkably well. This reconfirms our confidence regarding the accuracy of the material param- eters used here for computing the magnetic anisotropy and gives a solid ground for the presented conclusions.

The calculations are performed as a function of p, em- ploying for each sample the corresponding value of BG:

−16.4, −18.7, and −19.7 meV for samples S1, S2, and S3,

respectively. The results are presented in Fig. 9 as curves while full symbols represent experimentally established val- ues of the anisotropy constants共a square for the sample S1, a triangle for S2, and a circle for S3兲. The experimentalK_u001, Kxy,Kxz, and KC are obtained from K_u1^¯₁₀, K_u1,KC, and ⌰A

using Eqs.共24兲–共26兲. We take theT= 5 K experimental an- isotropy data, as this is what is consistent with theT= 0 limit, implicitly assumed in Eq.共27兲.

We start by discussing the strongest component of the magnetic anisotropy, the K_u001 term. The calculated curves are presented in Fig. 9共a兲. The calculations have been per- formed without introducing the fictitious shear strain⑀⬘^共i.e.

⑀⬘= 0兲. However, since we know that the magnitude ofK_u001 is negligibly affected by⑀⬘, the results should already match the experimental data, and indeed they do. Although the spread of the experimental points in Fig.9共a兲is significantly larger than that of the theoretical curves, the correspondence between the computed and experimental values is good, and it has been achieved without introducing any adjustable pa- rameters into the model. This has been only possible by in- cluding the hole liquid magnetization in the calculation of x_eff. When Mh is disregarded, the experimental values of K_u001are systematically above themaximaof the theoretical curves.

As already mentioned, in the case of 共001兲 共Ga,Mn兲As films an additional low-symmetry term has to be introduced into the Hamiltonian in order to reproduce the experimen- tally observed uniaxial in-plane magnetic anisotropy. In the general case of an arbitrarily oriented substrate, there are three anisotropy constants of this kind, K_xy,K_yz, andK_xz. In the case of 共113兲 共Ga,Mn兲As, the symmetry requires that Kxz=Kyz, an assumption confirmed by experimental results.

FIG. 9. 共Color online兲 Lines: theoretical dependence of the anisotropy constants 共a兲 K_u001, 共b兲 K_xy, 共c兲 K_xz, and 共d兲 K_C on the hole concentrationp, calculated within mean-field Zener model. The values of the exchange parameterB_Gand the lattice constant mismatch⌬a/a are specific to the investigated samples and are specified in the text. Symbols depict values determined from experiment.⑀⬘= 0 is assumed here except for共c兲, where also the case of⑀_xz⬘= −0.1% is included.