arX

iv:0

904.

1565

v1 [

cond

-mat

.mtr

l-sci

] 9

Apr

200

9

Magnetic anisotropy in (Ga,Mn)As: Influence of epitaxial strain and holeconcentration

M. Glunk, J. Daeubler, L. Dreher, S. Schwaiger, W. Schoch, R. Sauer, and W. Limmer∗

Institut für Halbleiterphysik, Universität Ulm, 89069 Ulm, Germany

A. Brandlmaier and S. T. B. GoennenweinWalther-Meissner-Institut, Bayerische Akademie der Wissenschaften,

Walther-Meissner-Strasse 8, 85748 Garching, Germany

C. Bihler and M. S. BrandtWalter Schottky Institut, Technische Universität München, Am Coulombwall 3, 85748 Garching, Germany

We present a systematic study on the influence of epitaxial strain and hole concentration on themagnetic anisotropy in (Ga,Mn)As at 4.2 K. The strain was gradually varied over a wide rangefrom tensile to compressive by growing a series of (Ga,Mn)As layers with 5% Mn on relaxed graded(In,Ga)As/GaAs templates with different In concentration. The hole density, the Curie temperature,and the relaxed lattice constant of the as-grown and annealed (Ga,Mn)As layers turned out to beessentially unaffected by the strain. Angle-dependent magnetotransport measurements performedat different magnetic field strengths were used to probe the magnetic anisotropy. The measurementsreveal a pronounced linear dependence of the uniaxial out-of-plane anisotropy on both strain andhole density. Whereas the uniaxial and cubic in-plane anisotropies are nearly constant, the cubicout-of-plane anisotropy changes sign when the magnetic easy axis flips from in-plane to out-of-plane. The experimental results for the magnetic anisotropy are quantitatively compared withcalculations of the free energy based on a mean-field Zener model. An almost perfect agreementbetween experiment and theory is found for the uniaxial out-of-plane and cubic in-plane anisotropyparameters of the as-grown samples. In addition, magnetostriction constants are derived from theanisotropy data.

PACS numbers: 75.50.Pp, 75.30.Gw, 75.47.–m, 61.05.–aKeywords: (Ga,Mn)As; (In,Ga)As; Strain; Magnetic anisotropy; Magnetotransport

I. INTRODUCTION

Spin-related phenomena in semiconductors, such asspin polarization, magnetic anisotropy (MA), andanisotropic magnetoresistance (AMR), open up new con-cepts for information processing and storage beyond con-ventional electronics.1,2 Being compatible with the stan-dard semiconductor GaAs, the dilute magnetic semicon-ductor (Ga,Mn)As has proven to be an ideal playgroundfor studying future spintronic applications.3,4 In particu-lar, the pronounced MA and AMR, largely arising fromthe spin-orbit coupling in the valence band,5,6 potentiallyapply in novel non-volatile memories and magnetic-field-sensitive devices. Ferromagnetism is implemented in(Ga,Mn)As by incorporating high concentrations (&1%)of magnetic Mn2+ ions into the Ga sublattice. The fer-romagnetic coupling between the S=5/2 Mn spins is me-diated by itinerant holes provided by the Mn acceptoritself. Curie temperatures TC up to 185 K, i.e. wellabove the liquid-N2 temperature, have been reported

7,8

and there is no evidence for a fundamental limit to highervalues.9

The magnetic properties of (Ga,Mn)As are stronglytemperature dependent and can be manipulated to agreat extent by doping, material composition, and strain.In (Ga,Mn)As grown on GaAs substrates, however, holedensity p, Mn concentration x, and strain ε are intimately

linked to each other and cannot be tuned independentlyby simply varying the growth parameters. p and ε sen-sitively depend on the concentration and distribution ofthe Mn atoms which are incorporated both on Ga latticesites (MnGa) and, to a lower extent, on interstitial sites(MnI), where they act as compensating double donors.Post-growth treatment techniques such as annealing orhydrogenation are frequently used to increase or decreasethe hole concentration due to outdiffusion and/or rear-rangement of MnI

10,11,12,13,14 or due to the formationof electrically inactive (Mn,H) complexes,15,16,17 respec-tively. In both cases, however, the treatment concur-rently leads to a decrease or increase of the lattice pa-rameter, respectively, and thus to a change of the strain.

The epitaxial strain in the (Ga,Mn)As layers, aris-ing from the lattice mismatch between layer and sub-strate, can be adjusted by tailoring the lattice parame-ter of the substrate. While (Ga,Mn)As grown on GaAsis under compressive strain, tensily strained (Ga,Mn)Ascan be obtained by using appropriate (In,Ga)As/GaAstemplates.4,18,19,20 Experimental studies addressing thisissue, however, have so far been restricted to merely alimited number of representative samples.

In this work, the influence of epitaxial strain and holeconcentration on the MA at 4.2 K is analyzed in a sys-tematic way by investigating a set of (Ga,Mn)As layersgrown on relaxed (In,Ga)As/GaAs templates with differ-

http://arxiv.org/abs/0904.1565v1

2

ent In concentration. Keeping the Mn content at ∼5%and changing the maximum In content in the (In,Ga)Asbuffer layers from 0% to 12%, the vertical strain εzz inthe as-grown (Ga,Mn)As layers could be gradually variedover a wide range from εzz=0.22% in the most compres-sively strained sample to εzz=−0.38% in the most tensilystrained sample without substantially changing p. Post-growth annealing leads to an increase in p, yielding asecond series of samples with nearly the same range ofεzz but higher hole concentrations. The strain depen-dence of the anisotropy parameters for the as-grown andthe annealed samples was determined by means of angle-dependent magnetotransport measurements.21,22 Part ofthe experimental data has already been published in con-ference proceedings.23 Here, we combine the earlier withthe present extensive experimental findings advancing aquantitative comparison of the intrinsic anisotropy pa-rameters with model calculations for the MA, performedwithin the mean-field Zener model introduced by Dietl etal.5 Note that several samples analyzed in Ref. 23 havebeen substituted by new samples grown under optimizedconditions and that the sample series has been expandedby one specimen with εzz=-0.38%.

II. EXPERIMENTAL DETAILS

A set of differently strained (Ga,Mn)As layers withconstant Mn concentration of ∼5% and thickness of∼180 nm was grown by low-temperature molecular-beamepitaxy (LT-MBE) on (In,Ga)As/GaAs templates withdifferent In content in a RIBER 32 MBE machine.Indium-mounted semi-insulating VGF GaAs(001) waferswere used as substrates. After thermal deoxidation, a30-nm-thick GaAs buffer layer was deposited at a sub-strate temperature of Ts≈580

◦C. Then the growth wasinterrupted, Ts was lowered to ∼430

◦C, and a graded(In,Ga)As buffer with a total thickness of up to ∼5 µmwas grown. Starting with In0.02Ga0.98As, the temper-ature of the In cell was first continuously raised to in-crease the In content up to a value of ≤12% and was thenkept constant until the required thickness of the bufferlayer was reached. The growth was again interrupted,Ts was lowered to ∼250

◦C, and the (Ga,Mn)As layerwas grown in As4 mode at a growth rate of ∼200 nm/h.The growth was monitored by reflection high-energy elec-tron diffraction showing no indication of a second-phaseformation. The use of a graded (In,Ga)As buffer24 min-imizes the deterioration of the (Ga,Mn)As layer causedby threading dislocations in the relaxed (In,Ga)As/GaAstemplate. The resulting (Ga,Mn)As layers exhibit nearlythe same quality as conventional samples directly grownon GaAs.20 After the growth, the samples were cleavedinto several pieces and some of the pieces were annealedin air for 1 h at 250 ◦C. The structural properties ofthe (Ga,Mn)As layers were analyzed by means of high-resolution x-ray diffraction (HRXRD) measurements per-formed with a Siemens D5000HR x-ray diffractometer us-

ing the Cu-Kα1 radiation at 0.154 nm. Hall bars withcurrent directions along the [100] and [110] crystallo-graphic axes were prepared from the samples by stan-dard photolithography and wet chemical etching. Thewidth of the Hall bars is 0.3 mm and the longitudinalvoltage probes are separated by 1 mm. The hole densitieswere determined by high-field magnetotransport mea-surements (up to 14.5 T) at 4.2 K using an Oxford SMD10/15/9 VS liquid-helium cryostat with superconductingcoils. The Curie temperatures were estimated from thepeak positions of the temperature-dependent sheet re-sistivities at 10 mT.8,25,26 The MA of the samples wasprobed by means of angle-dependent magnetotransportmeasurements at 4.2 K using a liquid-He bath cryostatequipped with a rotatable sample holder and a standardLakeShore electromagnet system with a maximum fieldstrength of 0.68 T. To determine the saturation magne-tization, we employed a Quantum Design MPMS-XL-7superconducting quantum interference device (SQUID)magnetometer using the Reciprocating Sample Option(RSO). The measured SQUID curves were corrected forthe diamagnetic contribution of the substrate.

As discussed in detail in Ref. 27, our samples ex-hibit spin wave resonances which are most pronouncedfor the external magnetic field oriented perpendicularto the sample plane. These spin wave excitations havebeen traced back to an inhomogeneous free-energy den-sity profile, or more precisely to a linear variation of theMA parameters along the growth direction, presumablyarising from a vertical gradient in the hole density.14,28

Therefore, all physical parameters derived via magneto-transport in this study have to be considered as effec-tive parameters representing the averaged electronic andmagnetic properties of the layers.

III. THEORETICAL CONSIDERATIONS

In the present context, MA represents the dependenceof the free-energy density F on the orientation m of themagnetization M = Mm.29 In the absence of an ex-ternal magnetic field, m is determined by the minimumof the free energy. Since for reasons of crystal symme-try F usually exhibits several equivalent minima, morethan one stable orientation of M exists. This symmetry-induced degeneracy of F can be lifted by the applicationof an external magnetic field H .

The theoretical considerations on the AMR and theMA in this paper are based on a single-domain modelwith a uniform magnetization M . While the directionof M is controlled by the interplay of F and H , itsmagnitude M is assumend to be constant under thegiven experimental conditions. For sufficiently high fieldstrengths H , this assumption can be considered as agood approximation. Hence, the normalized quantityFM=F/M is considered instead of F , allowing for a con-cise description of the MA.

3

A. Phenomenological description of the MA

There are several contributions to FM which we referto as intrinsic (magnetocrystalline) or extrinsic:

FM = FM,int + FM,ext . (1)

The intrinsic part FM,int=FM,c+FM,S originates from theholes in the valence band (FM,c) (see Sec. III B) and fromthe localized Mn spins (FM,S).

5 Whereas FM,c is stronglyanisotropic with respect to the magnetization orienta-tion, reflecting the anisotropy of the valence band, the

localized-spin contribution FM,S=∫M

0 dM′µ0H(M

′)/Mis isotropic and therefore irrelevant for the following con-siderations. In a phenomenological description, FM,intcan be expressed in terms of a series expansion in ascend-ing powers of the direction cosines mx, my, and mz ofthe magnetization with respect to the cubic axes [100],[010], and [001], respectively. Considering terms up tothe fourth order in m, FM,int for cubic systems withtetragonal distortion along the [001] growth direction isgiven by22

FM,int(m) = B0+B2⊥m2z+B4‖(m

4x+m

4y)+B4⊥m

4z . (2)

In the case of a perfect cubic crystal, symmetry requiresB2⊥=0 and B4‖=B4⊥.The extrinsic part FM,ext comprises the demagneti-

zation energy due to shape anisotropy and a uniaxialin-plane anisotropy along [1̄10]. The origin of the lat-ter anisotropy is controversially discussed. It is tracedback either to highly hole-concentrated (Ga,Mn)As clus-ters formed during the growth,30 to the anisotropy ofthe reconstructed initial GaAs (001) substrate surface,31

or to a trigonal-like distortion which may result from anonisotropic Mn distribution, caused, for instance, by thepresence of surface dimers oriented along [1̄10] during theepitaxy.32 Approximating the (Ga,Mn)As layer by an in-finite plane, we write the total extrinsic contribution as

FM,ext(m) = Bdm2z +B1̄10

1

2(mx −my)

2, (3)

where Bd=µ0M/2.In the presence of an external magnetic field H=Hh,

the normalized Zeeman energy −µ0Hm has to be addedto the total free-energy density. This corresponds to atransition from FM to the normalized free-enthalpy den-sity

GM (m) = B0 + (

B001︷ ︸︸ ︷

B2⊥ + Bd)m2z

+B4‖(m4x +m

4y) +B4⊥m

4z

+B1̄101

2(mx −my)

2 − µ0Hhm. (4)

The anisotropy parameters B2⊥ and Bd are both relatedto m2z and are therefore combined into a single parameterB001. Given an arbitrary magnitude and orientation of

H , the direction of m is determined by the minimum ofGM .All anisotropy parameters introduced above are in

SI units. Expressed by the anisotropy fields incgs units as defined, e.g., in Ref. 33, they readas B1̄10=−µ0H2‖/2, B2⊥=−µ0H2⊥/2, B4‖=−µ0H4‖/4,and B4⊥=−µ0H4⊥/4. Note also that the magneticanisotropy field used in Ref. 27 and the anisotropy pa-rameters used here are related via µ0H

001aniso = 2(K

001eff +

K⊥c1)/Msat = −2B001 − 4B4⊥.

B. Microscopic theory

For a microscopic description of the intrinsic partFM,int, we adopt the mean-field Zener model of Dietl etal. introduced in Ref. 5. The objective of the microscopiccalculations discussed below is first, to justify the approx-imation in Eq. (2), made by considering only terms upto the fourth order, and second, to compare the experi-mentally found dependence of the intrinsic anisotropy pa-rameters B2⊥, B4‖, and B4⊥ on εzz and p (see Sec. IVC)with that predicted by the mean-field Zener model.According to the k · p effective Hamiltonian theory

presented in Ref. 5, the Hamiltonian of the system isgiven by

H = HKL +Hε +Hpd. (5)

Here, HKL represents the 6×6 Kohn-Luttinger k · pHamiltonian for the valence band and Hε=

∑

i,j D(ij)εij

accounts for the strain εij in the (Ga,Mn)As layer via the

deformation potential operatorD(ij). Hpd=−N0βSs de-scribes the p-d hybridization of the p-like holes and thelocalized Mn d-shell electrons, which results in an in-teraction between the hole spin s and the Mn spin Scarrying a magnetic moment SgµB. Here, g=2 is theLandé factor, µB the Bohr magneton, and β and N0 de-note the p-d exchange integral and the concentration ofcation sites, respectively. In terms of the virtual crystaland mean-field approximation, the exchange interactioncan be written as Hpd=Msβ/gµB. Explicit expressionsfor the individual contributions in Eq. (5) can be found inRef. 5. As an approximation, the values of the Luttingerparameters γi (i=1,2,3), the spin-orbit splitting ∆0, andthe valence band shear deformation potential b are chosenas those of GaAs. Explicit values are γ1=6.85, γ2=2.1,γ3=2.9, ∆0=0.34 eV and b=−1.7 eV, respectively.

5 Thequantity parameterizing the exchange splitting of the va-lence subbands is given by

BG =AFβM

6gµB, (6)

with the Fermi liquid parameter AF. In contrast toRef. 5, we restrict our calculations to zero temperature(T=0) and zero magnetic field (H=0). In this approxi-mation, the Fermi distribution is represented by a step

4

function and the Zeeman as well as the Landau split-ting can be neglected. These simplifications are justified,as our measurements were carried out at T=4.2 K andµ0H < 0.7 T, where the Zeeman and Landau splittingsof the valence band are expected to be much smaller thanthe splitting caused by the p-d exchange coupling. Diag-onalization of the Hamilton matrix H yields the sixfoldspin-split valence band structure in the vicinity of theΓ point, depending on the magnetization orientation mand the strain εij in the (Ga,Mn)As layer.Them- and εij-dependent normalized free-energy den-

sity of the carrier system FM,c(εij ,m) is obtained by firstsumming over all energy eigenvalues within the four spin-split heavy-hole and light-hole Fermi surfaces and thendividing the resulting energy density by M . The twosplit-off valence bands do not contribute to FM,c becausethey lie energetically below the Fermi energy for commoncarrier concentrations. For biaxially strained (Ga,Mn)Aslayers grown pseudomorphically on (001)-oriented sub-strates, the tetragonal distortion of the crystal latticealong [001] can be fully described by the εzz componentof the strain tensor using continuum mechanics.In order to compare the microscopic theory with the

phenomenological description of the MA in Sec. III A,we consider the dependence of FM on m with respect tothe reference directionmref=[100]. Accordingly, we writethe anisotropic part ∆FM,int of the intrinsic contributionFM,int as

∆FM,int = FM,c(εzz ,m)− FM,c(εzz,m = [100]). (7)

In terms of the anisotropy parameters from Eq. (2),∆FM,int reads as

∆FM,int = B2⊥m2z +B4‖(m

4x +m

4y − 1) +B4⊥m

4z. (8)

For m rotated in the (001) and the (010) plane, Eq. (8)can be rewritten as

∆FM,int(ϕ) = B4‖(cos4 ϕ+ sin4 ϕ− 1) (9)

and

∆FM,int(θ) = B2⊥ cos2 θ +B4‖(sin

4 θ − 1) +B4⊥ cos4 θ,(10)

respectively, where we have introduced the azimuthangle ϕ and the polar angle θ with mx=sin θ cosϕ,my=sin θ sinϕ, and mz=cos θ. We proceed by calculat-ing ∆FM,int numerically in the microscopic model as afunction of ϕ and θ with εzz varied in the range −0.4% ≤εzz ≤ 0.3%, using typical values for p, BG, andM . Equa-tions (9) and (10) are then fitted to the resulting angulardependences using B2⊥, B4‖, and B4⊥ as fit parameters.

For the hole density we use the value p=3.5×1020 cm−3,for the exchange-splitting parameter BG=−23 meV, andfor the magnetization µ0M=40 mT. Inserted into Eq. (6),the latter two values yield AFN0β=−1.8 eV, in goodagreement with the parameters used in Ref. 5.The results of the microscopic calculations are depicted

by the solid symbols in Figs. 1(a) and 1(b). Since the

0 10 20 30 40 50 60 70 80 90

-200

-150

-100

-50

0

50

100

150

200

-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3-200-100

0100200

Microscopic calculations

Fit using Eq.(9)

B 2

(mT)

= -0.4%zz

(degree)

M in (001) plane (a)

FM

,int(

) (m

T)

Microscopic calculations Fit using Eq.(10)

M in (010) plane

(%)zz (b)

0.2

-0.4

-0.3

0.0

-0.1

-0.2

0.1

0.3

FM

,int(

) (m

T) (degree)

compressive straintensile strain

(c)

zz (%)

0 10 20 30 40 50 60 70 80 900

4

8

12

16

[001] [100]

[010][100]

FIG. 1: ∆FM,int calculated as a function of the magnetizationorientation and the strain within the microscopic model (solidsymbols) for M (a) in the (001) plane and (b) in the (010)plane. ϕ and θ denote the azimuth and polar angles of M ,respectively. The solid lines are least-squares fit curves using(a) Eq. (9) and (b) Eq. (10) with B2⊥, B4‖, and B4⊥ as fitparameters. (c) The anisotropy parameter B2⊥ obtained fromthe fit shows a pronounced linear dependence on εzz.

variation of ∆FM,int(ϕ) with εzz is found to be marginal,only one representative curve calculated with εzz=−0.4%is shown in Fig. 1(a). It clearly reflects the fourfoldsymmetry of ∆FM,int within the (001) plane. By con-trast, ∆FM,int(θ) in Fig. 1(b) strongly depends on εzz.The solid lines in Figs. 1(a) and 1(b) are least-squaresfits to the calculated data using Eqs. (9) and (10), re-spectively. The perfect agreement between the micro-scopic results and the fit curves demonstrates that theintrinsic part of the free energy calculated within themicroscopic theory can be well parameterized by B2⊥,B4‖, and B4⊥. It thus justifies the phenomenological ap-

5

proach in Eq. (2), taking into account only terms up tothe fourth order in m. Whereas the cubic anisotropy pa-rameters B4‖≈B4⊥≈−30 mT obtained from the fit arenot substantially affected by εzz, the uniaxial parameterB2⊥ exhibits a pronounced linear dependence on εzz, asshown in Fig. 1(c). The slope of B2⊥(εzz) is stronglyinfluenced by the exchange-splitting parameter BG andthe hole density p, as will be discussed in more detail inSec. IVC, Fig. 8(b).The microscopic calculations show that for εzz < 0

(tensile strain) ∆FM,int exhibits two equivalent minimafor m oriented along [001] and [001̄], which become morepronounced with increasing tensile strain. In contrast, inthe regime of compressive strain (εzz > 0), the minimaoccur for m along [100], [1̄00], [010], and [01̄0]. Thus,the theoretical model correctly describes the well knownexperimental fact that for sufficiently high hole densitiesand low temperatures the magnetically hard axis along[001] in compressively strained layers turns into an easyaxis in tensily strained layers. Note, however, that for aquantitative comparison between experiment and theorythe extrinsic contributions to ∆FM have also to be takeninto account.

IV. RESULTS AND DISCUSSION

In the following, the experimental data obtained forthe (Ga,Mn)As samples under study are discussed.

A. Lattice parameters and strain

In order to study the structural properties of the(Ga,Mn)As/(In,Ga)As/GaAs samples, reciprocal spacemaps (RSM) of the asymmetric (224), (2̄2̄4), (2̄24), and(22̄4) reflections were recorded using HRXRD. Figure 2exemplarily shows an RSM contour plot of the (224) re-flection for a nearly unstrained (Ga,Mn)As layer withεzz=−0.04%, depicting separate peaks for the GaAs sub-strate, the (In,Ga)As buffer, and the (Ga,Mn)As layer.From the peak positions (h,l) and their shifts (∆h,∆l)relative to that of the substrate the lateral and verticallattice parameters of (Ga,Mn)As and (In,Ga)As can bedetermined using the relations

a‖ = as(1 −∆h/h), a⊥ = as(1−∆l/l), (11)

where as denotes the lattice constant of the GaAs sub-strate. h and l are the coordinates in k-space referringto the reciprocal lattice vectors of GaAs along the [100]and [001] directions, respectively. In Fig. 2, the peaksof (Ga,Mn)As and (In,Ga)As are centered at the samevalue of h, confirming that the (Ga,Mn)As layer has beengrown lattice matched to the (In,Ga)As buffer. The lat-eral lattice parameters a‖ are therefore the same in the(In,Ga)As and (Ga,Mn)As layers. The shift ∆h from thesubstrate peak at h=2 to lower values is due to the strain

1.985 1.990 1.995 2.000 2.0053.975

3.980

3.985

3.990

3.995

4.000

4.005

4.010

l

h

Substrate

GaMnAs

InGaAs

h

l

FIG. 2: Reciprocal space map around the asymmetric(224) reflections of a nearly unstrained (Ga,Mn)As layer(εzz=−0.04%) grown on relaxed (In,Ga)As/GaAs template.h and l are the coordinates in k-space in units of the recipro-cal lattice vectors along [100] and [001] in the GaAs substrate,respectively. ∆h and ∆l denote the shift of the peak positionrelative to that of the substrate. In the plot, ∆h and ∆l areonly depicted for the (Ga,Mn)As layer.

relaxation in the buffer layer (a‖>as). The lattice param-eters a‖ of the (In,Ga)As templates in the as-grown andannealed samples are plotted against the In content inFig. 3(a). Apparently, post-growth annealing had no sig-nificant influence on a‖, which linearly increases with theIn content.

For both (Ga,Mn)As and (In,Ga)As, the HRXRD mea-surements yielded different values of (h,l) for the (224)and (2̄2̄4) reflections, revealing a tilt of the lattice to-wards the [110] direction.34 The tilt of the (Ga,Mn)Aslayer originates from an equal tilt in the (In,Ga)As bufferpointing to an anisotropic relaxation of the (In,Ga)Astemplates, typically found in layers grown on vicinalsubstrates.35 The samples under investigation, however,were grown on non-miscut (001) wafers. As can be seen inFig. 3(b), the measured tilt angles tend to higher valueswith increasing In fraction. It should be emphasized thatit is imperative to take the tilt into account when deter-mining the lattice parameters in order to avoid erroneousresults. This can be done by inserting into Eq. (11) theaveraged values of the peak positions and shifts obtainedfrom the (224) and (2̄2̄4) reflections. For the angle-dependent magnetotransport measurements, the influ-ence of the tilt is negligible since the tilt angles observedare smaller than 0.06 degree.

The relaxed lattice parameter arel of a biaxiallystrained layer on (001)-oriented substrate is obtained

6

0 2 4 6 8 10 12In content (%)

0.00

0.02

0.04

0.06

Tilt

ang

le(d

egre

e)

as grownannealed

(b)

5.65

5.66

5.67

5.68

5.69

a ||(Å

)(a) InGaAs

FIG. 3: (a) Lateral lattice parameter a‖ and (b) tilt angletowards the [110] direction of the (In,Ga)As buffer layer, plot-ted against the In content. The solid line (as grown) and thedashed line (annealed) are regression lines.

0 2 4 6 8 10 12

In content (%)

-0.4

-0.2

0.0

0.2

ε zz(%

)

(b)

compressive strain

tensile strain

5.65

5.66

5.67

5.68

5.69

a rel

(Å)

as grownannealed(a) GaMnAs

FIG. 4: (a) Relaxed lattice parameter arel and (b) verticalstrain εzz of the (Ga,Mn)As layer plotted against the In con-tent in the (In,Ga)As buffer. Post-growth annealing onlyleads to a slight decrease of the values.

from the relation

arel =2C12

C11 + 2C12a‖ +

C11C11 + 2C12

a⊥ , (12)

where C11 and C12 are elastic stiffness constants. Asan approximation, we use the values C11=11.90×10

10 Paand C12=5.34×10

10 Pa of GaAs for both the (Ga,Mn)As

and the (In,Ga)As layers.36 In Fig. 4(a), arel is shownfor the (Ga,Mn)As layers as a function of the In con-tent in the (In,Ga)As buffer. It is found to be nearlyunaffected by the (In,Ga)As template underneath. Thefluctuations in the values of arel are mainly attributedto slight variations of the growth temperature. For all(In,Ga)As templates under study, the degree of relax-ation defined by R=(a‖ − as)/(arel − as) was above 80%.Once the vertical and relaxed lattice parameters of the(Ga,Mn)As layers are known, the vertical strain εzz canbe calculated from the relation

εzz = (a⊥ − arel)/arel. (13)

In Fig. 4(b), εzz is plotted against the In content. Theslight decrease of arel and εzz upon annealing is supposedto arise from the outdiffusion and/or rearrangement ofthe highly mobile MnI.

10,11,12,13,14

B. Hole density and Curie temperature

Determination of the hole concentrations p in(Ga,Mn)As is complicated by a dominant anomalous con-tribution to the Hall effect proportional to the normalcomponent of the magnetization M (anomalous Halleffect). To overcome this problem, magnetotransportmeasurements were performed at high magnetic fieldsup to 14.5 T. Assuming the magnetization to be satu-rated perpendicular to the layer plane at magnetic fieldsµ0H&4 T, the measured transverse resistivity was fittedusing the equation

ρtrans(H) = R0µ0H + c1ρlong(H) + c2ρ2long(H) (14)

for the ordinary and anomalous Hall effect with R0, c1,and c2 as fit parameters. Here R0=1/ep is the ordinaryHall coefficient and ρlong the measured field-dependentlongitudinal resistivity. The second term on the righthand side arises from skew scattering37,38 and the thirdterm from side jump scattering39 and/or Berry phaseeffects.40 As mentioned in Sec. II, the Curie tempera-tures TC were inferred from the peak positions of thetemperature-dependent resistivities ρlong.

25,26 Consider-ing that the TC values thus obtained generally differ fromthose determined by temperature-dependent magnetiza-tion measurements,8 we estimate an error margin of upto 20%. Similar to arel, neither p nor TC are signifi-cantly influenced by the strain as shown in Figure 5. Thevalues scatter around pag=3.5×10

20 cm−3 and TC=65 Kfor the as-grown samples and pann=5.8×10

20 cm−3 andTC=91 K for the annealed samples. It is well known thatthe lattice constant, the hole density, and the Curie tem-perature strongly depend on the concentration of MnGaacceptors, MnI double donors, and other compensatingdefects such as AsGa antisites. The insensitivity of arel,p, and TC to strain in the as-grown and annealed samplesunder study suggests the assumption that strain has nosignificant influence on the incorporation of MnGa, MnI,

7

-0.4 -0.2 0.0 0.2

εzz (%)

5

10

15p

(102

0 cm

-3)

0

50

100

TC

(K)

p

TC

FIG. 5: Hole density p and Curie temperature TC of(Ga,Mn)As plotted against the strain εzz for the as-grown(solid symbols) and the annealed samples (open symbols).The fluctuations in the values are mainly attributed to slightvariations of the growth temperature. For p we estimate anerror margin of about ±10% and for TC an error margin ofup to ±20%.

and AsGa. At least the sum of the changes caused by thedifferent constituents seems to be unaltered. Moreover,the insensitivity of TC with respect to εzz supports theo-retical predictions that the magnetic coupling should beunaffected by strain, since the corresponding deforma-tion energies are expected to be too small to significantlyenhance or reduce the p-d kinetic exchange interaction.4,5

C. Anisotropy parameters

Experimental values for the anisotropy parametersB001=B2⊥+Bd, B4‖, B4⊥, and B1̄10 were determinedby means of angle-dependent magnetotransport measure-ments. A detailed description of the corresponding pro-cedure is given in the Refs. 21 and 22. It can be brieflysummarized as follows. The longitudinal and transverseresistivities ρlong and ρtrans, respectively, are measuredas a function of the magnetic field orientation at fixedfield strengths of µ0H=0.11, 0.26, and 0.65 T. At eachfield strength, H is rotated within three different crys-tallographic planes perpendicular to the directions n, j,and t, respectively. The corresponding configurations,labeled I, II, and III, are shown in Fig. 6. The vectors

FIG. 6: The angular dependence of the resistivities wasprobed by rotating an external magnetic field H within thethree different planes (a) perpendicular to n, (b) perpendic-ular to j, and (c) perpendicular to t. The correspondingconfigurations are referred to as I, II, and III.

form a right-handed coordinate system, where j definesthe current direction, n the surface normal, and t thetransverse direction. For current directions j ‖ [100] andj ‖ [110], the resistivities can be written as22

ρlong = ρ0 + ρ1m2j + ρ2m

2n + ρ3m

4j + ρ4m

4n + ρ5m

2jm

2n,

(15)

ρtrans = ρ6mn + ρ7mjmt + ρ8m3n + ρ9mjmtm

2n, (16)

where mj , mt, and mn denote the components of malong j, t, and n, respectively. At sufficiently high mag-netic fields, the Zeeman energy in GM (m) dominatesand the magnetization direction m follows the orien-tation h of the external field. The resistivity param-eters ρi (i=1,...,9) are then obtained from a fit of theEqs. (15) and (16) to the experimental data recorded at0.65 T. With decreasing field strength, the influence ofthe MA increases and m more and more deviates fromh. Controlled by the magnetic anisotropy parameters,the shape of the measured resistivity curves changes andB001, B4‖, B4⊥, and B1̄10 are obtained from a fit to thedata recorded at µ0H=0.26 and 0.11 T. In the fit pro-cedure, m is calculated for every given magnetic fieldH by numerically minimizing GM with respect to m.Figure 7 exemplarily shows the angular dependence ofρlong and ρtrans for a nearly unstrained (Ga,Mn)As layer(εzz=−0.04%) with H rotated in the (001) plane (con-figuration I) and j ‖ [100]. The experimental data aredepicted by red solid circles and the fits by black solidlines.Applying the procedure described above to the whole

set of (Ga,Mn)As layers under study, the resistivity pa-rameters ρi (i=1,...,9) and the anisotropy parametersB001=B2⊥+Bd, B4‖, B4⊥, and B1̄10 were determinedas functions of the vertical strain εzz. The results forthe resistivity parameters were extensively discussed inRef. 22. In the present work, we exclusively focus on theanisotropy parameters.Figure 8(a) shows the values of the parameter

B001=B2⊥+Bd describing the uniaxial out-of-planeanisotropy. For both the as-grown and the annealed sam-ples, a pronounced linear dependence on εzz is found inqualitative agreement with the microscopic model calcu-lations presented in Fig. 1(c). For zero strain, the cubicsymmetry requires B2⊥=0 and the extrinsic parameterBd≈60 mT is inferred from the intersections between theregression lines (dotted lines) and the vertical axis. Ifshape anisotropy was the only extrinsic contribution tothe m2z term of FM , as assumed in Sec. III A, the valueBd≈60 mT would correspond to a sample magnetizationof µ0M=2Bd≈120 mT. This value, however, exceeds thesaturation magnetization determined by SQUID mea-surements by a factor of ∼3. Figure 9 shows as an ex-ample the SQUID curves obtained for the as-grown andannealed samples with εzz≈0.2%. At the moment, thereason for this discrepancy is not yet understood. We sus-pect, however, that it might be related to the vertical gra-dient in the MA mentioned at the end of Sec. II. Assum-ing the value of Bd to be nearly the same for all samples

8

0 100 200 300 400

Angle of rotation (degree)

9.3

9.6

9.9

10.2

10.5

-0.2

0.0

0.2

0.4

0.65 T ρlongρtrans

9.6

9.9

10.2

10.5

ρ lo

ng

(10-

3Ω

cm)

-0.2

0.0

0.2

0.4

ρ tra

ns(1

0-3

Ωcm

)

0.26 T

H in (001) plane

9.6

9.9

10.2

10.5

10.8

-0.2

0.0

0.2

0.4

0.11 T

[100]_

[010]_

[100] [010] [100]_

FIG. 7: (Color online) Resistivities ρlong and ρtrans recordedfrom a nearly unstrained (Ga,Mn)As layer with εzz=−0.04%at 4.2 K and j ‖ [100] (red solid circles). The measurementswere performed at fixed field strengths of µ0H=0.11, 0.26and 0.65 T with H rotated in the (001) plane correspondingto configuration I. The black solid lines are fits to the exper-imental data using Eqs. (15) and (16), and one single set ofresistivity and anisotropy parameters.

under consideration, the strain-dependent intrinsic pa-rameter B2⊥ is obtained by subtracting Bd from the mea-sured B001 data. In Fig. 8(b), the values of B2⊥ derivedin this way are shown together with model calculationsperformed within the microscopic theory (see Sec. III B).Using for p the mean values pag=3.5×10

20 cm−3 andpann=5.8×10

20 cm−3 (see Sec. IVB), the calculated val-ues of B2⊥ are found to be in quantitative agreementwith the experimental data if BG=−23 meV is chosenfor the as-grown samples and BG=−39 meV for the an-nealed samples. The corresponding curves are depictedby the solid and dashed lines in Fig. 8(b). Comparing theexchange-splitting parameters with the respective holedensities, we find

BG,annBG,ag

≈pannpag

≈ 1.7 . (17)

For the annealed samples, theoretical results are alsoshown for −32 meV and −23 meV, demonstrating thatthe slope of B2⊥(εzz) is drastically reduced with decreas-ing BG. Remarkably, normalization of the experimen-tally derived values of B2⊥ to the corresponding hole con-centration p yields the same linear dependence of B2⊥/p

-0.4

-0.2

0.0

0.2

0.4

B0

01(T

)

as grownannealed Bd

(a)

-0.4

-0.2

0.0

0.2

0.4

B2_

|_(T

)

BG= -23 meV, as grownBG= -39 meV, annealedBG= -32 meV, "BG= -23 meV, "

(b)

-0.4 -0.2 0.0 0.2εzz (%)

-80

-60

-40

-20

0

20

40

60

B2

_|_/p

(10-

23

Tcm

3 ) as grownannealed

(c)

FIG. 8: (a) Dependence of the uniaxial out-of-planeanisotropy parameter B001=B2⊥+Bd on εzz for the as-grown (solid circles) and annealed samples (open circles).Bd=60 mT is inferred from the intersections between the cor-responding regression lines (dotted lines) and the verticalaxis. (b) Anisotropy parameter B2⊥ obtained by subtract-ing 60 mT from the measured B001 data. The lines are modelcalculations for B2⊥ performed within the microscopic theorydescribed in Sec. III B using values for the parameter BG asshown and the averaged p values from Fig. 5. (c) Anisotropyparameter B2⊥=B001 −Bd normalized to the hole density p.Experimentally, the same linear dependence of B2⊥/p on εzzis obtained for both the as-grown and the annealed samples.The dotted line represents a linear regression.

on εzz for both the as-grown and the annealed samplesas shown in Fig. 8(c). We thus find, at least for the rangeof hole densities and strain under consideration, the ex-perimental relationship

B2⊥ = Kpεzz , (18)

with K=1.57×10−19 Tcm3. If we assume a linear rela-tion BG=AFβM/6gµB ∝ p, in accordance with Eq. (17),Eq. (18) can be reproduced by the microscopic theory.Note however, that this is not trivial, since ∆FM,int ex-plicitly depends on both BG and p. Anyhow, the relation

9

-5 0 5

µ0H (T)

-40

-20

0

20

40

µ 0M

(mT

)

as grownannealed

εzz 0.2%

H || [001]

FIG. 9: SQUID curves of the as-grown and annealed sampleswith εzz≈0.2%, measured at 5 K for H oriented along [001].The saturation magnetization of ∼40 mT is representative forthe whole set of (Ga,Mn)As samples. The values of the co-ercive fields, derived from the hysteresis loops (not shown),were found to be below 10 mT.

-100

-80

-60

-40

-20

0

20

B4|

|(m

T)

as grownannealed

(a)

-0.4 -0.2 0.0 0.2εzz (%)

-120

-100

-80

-60

-40

-20

0

20

B4_

|_(m

T)

BG= -23 meV, as grownBG= -39 meV, annealedBG= -32 meV, "BG= -23 meV, "

(b)

FIG. 10: Dependence of the cubic anisotropy parameters (a)B4‖ and (b) B4⊥ on εzz for the as-grown (solid circles) andannealed samples (open circles). The lines are model calcula-tions within the microscopic theory using the same values forp and BG as in Fig. 8(b).

BG ∝ p demands a future detailed investigation.The experimental values of the fourth-order parame-

ters B4‖ and B4⊥ are presented in Figs. 10(a) and 10(b),respectively. Whereas B4‖ only slightly varies between−40 and −10 mT, B4⊥ exhibits positive values close to10 mT for εzz.−0.15% and negative values of about−20 mT for εzz&−0.15%. The lines depicted in Fig. 10were calculated using the same values for p and BG asin Fig. 8(b). Obviously, the change of sign of B4⊥ does

-0.4 -0.2 0.0 0.2εzz (%)

-30

-20

-10

0

10

20

30

B-1

10(m

T)

as grownannealed

FIG. 11: Dependence of the uniaxial in-plane parameter B1̄10for the as-grown (solid circles) and annealed samples (opencircles).

not appear in the theoretical curves. Apart from this dis-agreement, the experimental data of the as-grown sam-ples are again well reproduced for BG=−23 meV. In thecase of the annealed samples, now the splitting param-eter BG=−32 meV yields a much better description ofthe measured data than BG=−39 meV, found to fit thestrain dependence of B2⊥.

In view of the perfect quantitative interpretation ofB2⊥ by the mean-field Zener model, this small discrep-ancy and the fact that the change of sign of B4⊥ is notreproduced by the calculations, should not be overesti-mated. The experimentally observed change of sign ofB4⊥ may be caused by extrinsic influences, not accessibleby the model, such as the increasing density of threadingdislocations in the (Ga,Mn)As/(In,Ga)As layers with in-creasing In concentration. Moreover, we cannot rule outthat the change of sign is an artifact of the experimentalmethod for determining the relatively small anisotropyparameter B4⊥. The same problem has also been re-ported in Ref. 19, where the authors did not extractinformation on the fourth-order out-of-plane anisotropyparameters from their ferromagnetic resonance measure-ments, since the corresponding contributions to the MAwere masked by the much larger contributions of thesecond-order out-of-plane term. As to the differing val-ues of −39 meV and −32 meV for BG in the case of theannealed samples, it should be pointed out that for highhole densities the Fermi energy shifts deep into the va-lence band. Therefore, the values of the free energy ob-tained by using a 6×6 k ·p effective Hamiltonian becomeincreasingly unreliable when analyzing higher order con-tributions to the free energy. One should not jump to theconclusion that this has to be interpreted as a deficiencyof the Zener model itself.

In agreement with the data presented in Refs. 19, 41,and 42, the experimental values of the extrinsic uni-axial in-plane parameter B1̄10 are much smaller thanthose of the cubic in-plane parameter B4‖ at 4.2 K.As shown in Fig. 11, they scatter around zero with−10 mT≤B1̄10≤10 mT.

10

Due to the strong dependence of B001 on εzz, the out-of-plane axis [001] becomes magnetically harder with εzzincreasing from −0.4% to 0.2%. Neglecting the smallinfluence of B1̄10 in our samples at 4.2 K, the criticalstrain εcritzz , where a reorientation of the easy axis fromout-of-plane to in-plane occurs, can be estimated fromthe condition B001+B4⊥=B4‖. We obtain for the as-

grown (Ga,Mn)As layers under study εcritzz =−0.13% andfor the annealed layers εcritzz =−0.07%. Remarkably, thesevalues are very close to the εzz values in Fig. 10 whereB4⊥ changes sign.Since the MA sensitively depends on the individual

growth conditions, care has to be taken when compar-ing the values of anisotropy parameters published bydifferent groups. Keeping this restriction in mind, thedata presented in this work are in reasonable agreement,e.g., with the results obtained by Liu et al.19 for a rep-resentative pair of compressively and tensily strained(Ga,Mn)As samples with 3% Mn.

D. Magnetostriction constant

Changing the magnetization of a ferromagnet, e.g. byan external magnetic field, leads to a variation of its ge-ometrical shape. For crystals with cubic symmetry, therelative elongation λ in a given direction β can be ex-pressed in terms of the magnetization orientation m andthe magnetostriction constants λ100 and λ111 along [100]and [111], respectively, according to43

λ =3

2λ100

(

m2xβ2x +m

2yβ

2y +m

2zβ

2z −

1

3

)

+3λ111 (mxmyβxβy +mymzβyβz +mxmzβxβz) .(19)

Starting from Eq. (18), we are able to determine λ100defined by43

λ100 =2

9

a1C12 − C11

, (20)

where a1 denotes the magnetoelastic coupling con-stant. First-order expansion of the free-energy densityF (εij ,m) with respect to εij and continuum mechanicsyield the relation B2⊥M=εzza1(1 + C11/2C12). Thus,Eq. (20) can be rewritten as

λ100 =2KMp

9(C12 − C11)(1 + C11/2C12). (21)

Inserting the experimental values for p, µ0M , and K,we obtain λ100≈−3 ppm for the as-grown samples andλ100≈−5 ppm for the annealed samples. In Ref. 44,we already deduced an approximately constant value ofλ111≈5 ppm below 40 K decreasing to zero at higher

temperatures (40 K

11

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