Using Standard Prb S

PHYSICAL REVIEW B, VOLUME 63, 054418

Theory of magnetic anisotropy in III1ÀxMnxV ferromagnets

M. Abolfath,1 T. Jungwirth,2,3 J. Brum,4 and A. H. MacDonald2 1Department of Physics and Astronomy, University of Oklahoma, Norman, Oklahoma 73019-0225 2Department of Physics, Indiana University, Bloomington, Indiana 47405 3Institute of Physics ASCR, Cukrovarnicka´ 10, 162 00 Praha 6, Czech Republic 4Department of Physics, UNICAMP, Campinas, Brazil ͑Received 8 June 2000; published 8 January 2001͒

We present a theory of magnetic anisotropy in III1ϪxMnxV-diluted magnetic semiconductors with carrier- induced ferromagnetism. The theory is based on four- and six-band envelope function models for the valence- band holes and a mean-ﬁeld treatment of their exchange interactions with Mnϩϩ ions. We ﬁnd that easy-axis reorientations can occur as a function of temperature, carrier density p, and strain. The magnetic anisotropy in strain-free samples is predicted to have a p5/3 hole-density dependence at small p,apϪ1 dependence at large p, and remarkably large values at intermediate densities. An explicit expression, valid at small p, is given for the uniaxial contribution to the magnetic anisotropy due to unrelaxed epitaxial growth lattice-matching strains. Results of our numerical simulations are in agreement with magnetic anisotropy measurements on samples with both compressive and tensile strains. We predict that decreasing the hole density in current samples will lower the ferromagnetic transition temperature, but will increase the magnetic anisotropy energy and the coercivity.

DOI: 10.1103/PhysRevB.63.054418 PACS number͑s͒: 75.50.Pp, 75.30.Gw, 73.61.Ey

I. INTRODUCTION do not include effects due to interactions among the itinerant holes, and ͑iii͒ we do not account for correlations between The discovery of carrier-mediated ferromagnetism1–3 in localized spin configurations and itinerant hole states. The 4 III1ϪxMnxV and doped II1ϪxMnxVI-diluted magnetic semi- importance of each of these deficiencies is difficult to judge conductors ͑DMS’s͒ has opened up a broad and relatively in general, and probably depends on adjustable material pa- unexplored frontier for both basic and applied research. rameters. In our view, it is likely that there is a substantial 2,5 Experiments in Ga1ϪxMnxAs and In1ϪxMnxAs have dem- range in the parameter space of interest where the predictions onstrated that these ferromagnets have remarkably square of the present theory are useful. We expect that important hysteresis loops with coercivities typically ϳ40 Oe, and that progress can be made by comparing this simplest possible the magnetic easy axis is dependent on epitaxial growth theory of carrier-induced DMS ferromagnetism with experi- lattice-matching strains. In this paper we discuss the mag- ment. netic anisotropy properties of III1ϪxMnxV DMS ferromag- This work has two objectives. Most importantly, we have nets, predicted by a mean-field theory6 of the exchange in- attempted to shed light on how various adjustable material teraction coupling between localized magnetic ions and parameters can influence magnetic anisotropy. Second, we valence band free carriers. We use phenomenological four- have made an effort to estimate the magnetic anisotropy en- or six-band envelope function models, depending on the car- ergy in those cases where experimental information is pres- rier density p, in which the valence-band holes are charac- ently available. Our hope here is to initiate a process of care- terized by Luttinger, spin-orbit splitting, and strain-energy ful and quantitative comparison between mean-field theory parameters. and experiment, partially to help judge the efficiency of this The physical origin of the anisotropy energy in our model approximation in predicting other physical properties. Even is spin-orbit coupling in the valence band. Our work is based in the mean-field theory, we find that the magnetic anisot- in part on theoretical descriptions developed by Gaj et al.7 ropy physics of these materials is rich. We predict easy-axis and Bastard et al.8 to explain the optical properties of un- reorientations as a function of hole density, exchange inter- doped, paramagnetic DMS’s. As the critical temperature is action strength, temperature, and strain and identify situa- 6 approached, the mean-field theory we employ reduces to an tions under which III1ϪxMnxV ferromagnets are remarkably earlier theory4 that invokes generalized RKKY carrier- hard. mediated interactions between localized spins. The two ap- In Sec. II we detail our mean-field theory of the ordered proaches differ, however, in their description of the magneti- state. The theory simplifies in the limit of low-temperature cally ordered state. As this work was nearing completion, we and low-hole densities. Our results for this limit, presented in learned of a closely related study9,10 that uses the same Sec. III, predict a ͗111͘ easy axis in the absence of strain, mean-field theory to address critical temperature trends in and a magnetic anisotropy energy that is approximately 10% this material class and that also addresses magnetic anisot- of the free-carrier band-energy density. This value is ex- ropy physics. We are aware of three elements of the physics tremely large for a cubic metallic ferromagnet; typical ratios Ϫ of these materials that make the predictions of our mean-field in transition metal ferromagnets are smaller than 10 6, for theory uncertain: ͑i͒ we do not account for the substantial example. The anisotropy energy in this limit varies as the disorder that is usually present in these ferromagnets; ͑ii͒ we free-carrier density to the 5/3 power and is independent of

11 Ͼ the exchange-coupling strength. Explicit results for the strain ments is believed to be antiferromagnetic, i.e., J pd 0. For 14–17 dependence of the magnetic anisotropy in the same limit are GaAs, experimental estimates of J pd fall between presented in Sec. IV. We find that unrelaxed lattice-matching 0.04 eV nm3 and 0.15 eV nm3, with more recent work sug- strains due to epitaxial growth contribute a uniaxial anisot- gesting a value toward the lower end of this range. ropy that favors magnetization orientation along the growth The form of the valence band for Bloch wave vectors near direction when the substrate lattice constant is larger than the the zone center in a cubic semiconductor follows from k•p ferromagnetic semiconductor lattice constant and an in-plane perturbation theory and symmetry considerations.18 The orientation in the opposite case. Unfortunately, perhaps, the four-band ( jϭ3/2) and six band ( jϭ3/2 and 1/2͒ models are simple low-density limit does not normally apply in situa- known as Kohn-Luttinger Hamiltonians and their explicit tions where high critical temperatures are expected. The form is given in the Appendix. The eigenenergies are mea- more complicated, and more widely relevant, general case is sured down from the top of the valence band, i.e., they are discussed in Sec. V. We find that magnetic anisotropy has a hole energies. The Kohn-Luttinger Hamiltonian contains the ⌬ nontrivial dependence on both temperature and exchange- spin-orbit splitting parameter so and three other phenom- ␥ ␥ ␥ coupling strength and that easy-axis reversals occur, in gen- enological parameters, 1 , 2, and 3. These are accurately eral, as a function of either parameter. According to our known for common semiconductors. For GaAs and InAs, the theory, anisotropy energy densities comparable to those in two materials in which III1ϪxMnxV ferromagnetism has been ⌬ ϭ ␥ ␥ ␥ typical metallic ferromagnets are possible when the ex- observed, so 0.34 eV and 0.43 eV, and ( 1 , 2 , 3) change coupling is strong enough to depopulate all but one ϭ(6.85,2.1,2.9) and (19.67,8.37,9.29), respectively. Most of of the spin-split valence bands, even with saturation magne- the specific illustrative calculations discussed below are per- tization values smaller by more than an order of magnitude. formed with GaAs parameters. In the limit of large-hole densities, we find that the anisot- Our calculations are based on the Kohn-Luttinger Hamil- ropy energy of strain-free samples is proportional to hole tonian and on a mean-field theory in which correlations be- Ϫ1 4 density p and exchange coupling J pd . We find that in tween the local-moment configuration and the itinerant car- ϳ typical situations a strain e0 of only 1% is sufficient to rier system are neglected. We comment later on limits of overwhelm the cubic anisotropy of strain-free samples. We validity of this approximation. There are a number of equiva- conclude in Sec. VI with a discussion of the implications of lent ways of developing this mean-field theory formally. In these calculations for the interpretation of present experi- the following paragraphs we present a view that is conve- ments, and with some suggestions for future experiments that nient for discussing magnetic anisotropy. could further test the appropriateness of the model used here. In the absence of an external magnetic field, the partition function of our model may be expressed exactly as a II. FORMAL THEORY weighted sum over magnetic impurity configurations speci- fied by a localized spin quantization axis, Mˆ , and azimuthal Our theory is based on an envelope-function description spin quantum numbers m : of the valence-band electrons, and a spin representation for I their kinetic-exchange interaction11 with localized d 12 ϩϩ ϭ ͑Ϫ ͓ ͔ ͒ ͑ ͒ electrons on the Mn ions: Z ͚ exp Fb mI /kBT , 2 mI ͓ ͔ HϭH ϩH ϩ ជ ជ ␦ ជ Ϫ ជ ͒ ͑ ͒ where Fb mI is the valence-band-free energy for holes that m b J pd͚ SI si ͑ri RI , 1 i,I • experience an effective Zeeman magnetic field where i labels a valence-band hole and I labels a magnetic ជ͑ជ͓͒ ͔ϭϪ ˆ ␦͑ជϪ ជ ͒ ͑ ͒ ͑ ͒ H h r mI J pdM ͚ mI r RI . 3 ion. In Eq. 1 , m describes the coupling of magnetic ions I with total spin quantum number Jϭ5/2 to an external field ជ ជ ͑ ͒ ជ ជ The mean-field theory consists of replacing h(r)͓m ͔ by its if one is present , SI is a localized spin, si is a hole spin, and I H spatial average for each magnetic impurity configuration, b is either a four- or six-band envelope-function Hamiltonian13 for the valence bands. In this paper we do not thereby neglecting correlations between spin distributions in H → local-moment and hole subsystems. The effective Zeeman consider external magnetic fields so m 0. The four-band Kohn-Luttinger model describes only the total angular mo- magnetic field experienced by the holes then depends only mentum jϭ3/2 bands, and is adequate when spin-orbit cou- on Mˆ , the direction of the local-moment orientation, and the pling is large and the hole density p is not too large. As mean azimuthal quantum number averaged over all local discussed later, in the case of GaAs, a four-band model suf- moments, M: Շ 18 Ϫ3 fices for p 10 cm .InIII1ϪxMnxV semiconductors, the ϭ ⌬ ជ ͑ ͒ϭ ˆ ϵ ˆ ͑ ͒ four j 3/2 bands are separated by a spin-orbit splitting so hMF M J pdNMnMM hM, 4 from the two jϭ1/2 bands. In the relevant range of hole and ϭ ϩϩ where NMn NI /V is the number of magnetic impurities per Mn densities, no more than four bands are ever occupied. unit volume. The mean-field partition function is Nevertheless, mixing between jϭ3/2 and jϭ1/2 bands does occur, and it can alter the balance of delicate cancellations Z ͑M ͒ϭexp͕͓N Ts͑M ͒ϪF ͑hជ ͔͒/k T͖, ͑5͒ that often controls the net anisotropy energy. The exchange MF I b B interaction between valence-band holes and localized mo- where the entropy per impurity is defined by

054418-2 THEORY OF MAGNETIC ANISOTROPY IN III1ϪxMnx . . . PHYSICAL REVIEW B 63 054418

Note that ln͚ͫ ␦ͩ ͚ m ϪN M ͪͬ I I mI I s͑M ͒ϭk lim , ͑6͒ ͑ ˆ ͒ B dFb hM ជ N →ϱ NI ϭ͗S Mˆ ͘, ͑15͒ I dh tot• and F (hជ ) is the free energy of a system of noninteracting b where S is the total hole spin, and the angle brackets indi- fermions with single-particle Hamiltonian H ϪhMˆ sជ. tot b • cate a thermal average for the noninteracting valence-band Following standard ‘‘large number’’ arguments, s(M)is system. Since the valence-band system experiences an effec- readily evaluated by considering an auxiliary system consist- tive Zeeman coupling with strength proportional to h and ing of magnetic impurities coupled only to an external mag- Ϫ ˆ ͑ ͒ netic field H. For this model problem, a familiar exercise19 direction M, it is clear that the right-hand side of Eq. 15 gives the result is negative in sign and that its magnitude increases mono- tonically with h, making it easy to solve Eq. ͑14͒ numeri- ͒ϭ ͒ ͑ ͒ M͑H JBJ͑x , 7 cally. To simplify the calculations presented in subsequent sec- ϭ ␮ where x gL BHJ/kBT, gL is the Lande´ g factor of the ion, tions, we take advantage of the fact that temperatures of ␮ B is the electron Bohr magneton, and interest are almost always considerably smaller than the itin- ជ 2Jϩ1 1 erant carrier Fermi energy. This allows us to replace Fb(h) B ͑x͒ϭ coth͓͑2Jϩ1͒x/2J͔Ϫ coth͑x/2J͒ ͑8͒ ជ ͑ ͒ J 2J 2J by the ground-state energy Eb(h). Then, using Eq. 13 and ͑ ͒ ˆ Eq. 11 , a single calculation of Eb(hM) over the range from is the Brillouin function. The Brillouin function is a one-to- hϭ0tohϭN J J may be used to determine the local- ͓ ϱ͔ Mn pd one mapping between reduced fields x in the interval 0, moment magnetization M(T) and the free energy F(T) ͓ ͔ and reduced magnetizations M/J in the interval 0,1 ; the ϭF ͓M(T)͔ at all temperatures. Ϫ1 MF inverse function BJ maps M/J to x. Since the magnetiza- The mean-field theory critical temperature can be identi- ϩ ␮ tion maximizes s(M) gL BHM/kBT, fied by linearizing the self-consistent equation at small h.We find that ds͑M ͒ ϭϪ ␮ ͑ ͒ gL BH/kBT. 9 dM 2 J͑Jϩ1͒ N J d2F ͑hMˆ ͒ ͑ ˆ ͒ϭϪ Mn pd b ͯ ͑ ͒ ͑ ͒ kBTc M . 16 Equation 9 can be used to eliminate H and arrive at the 3 V dh2 following explicit expression for s(M): hϭ0

ϱ ͒ The second derivative of the valence-band free energy with dBJ͑x s͑M ͒ϭk ͵ dxx . ͑10͒ respect to field is proportional to its Pauli spin susceptibility, B Ϫ1 dx BJ (M/J) which is, in turn, proportional to the valence-band density of 1/3 The entropy per impurity vanishes for MϭJϭ5/2 because states at the Fermi energy, and to p at small p.Inthe ͚ ϭ absence of strain, it follows from cubic symmetry that the there is a single configuration with ImI NIJ, and ap- proaches ln(2Jϩ1)Ϸ1.79 for M→0. right-hand side of Eq. ͑16͒ is independent of Mˆ . Below the The mean polarization of the localized spins at a given critical temperature, however, the mean-field free energy temperature and for a given orientation of the local moments does depend on Mˆ ; this dependence is the magnetic anisot- is determined by minimizing the mean-field free energy ropy energy we wish to calculate. We will see that the dependence of the anisotropy energy on hole density is very ͑ ͒ϭϪ ͑ ͒ FMF M kBTlnZMF M different from that of the critical temperature. ϭ ជ ϭ ˆ ͒Ϫ ͒ ͑ ͒ Fb͑h NMnJ pdMM kBTNIs͑M , 11 III. MAGNETIC ANISOTROPY IN THE STRONG with respect to M. Setting the derivative to zero gives EXCHANGE COUPLING LIMIT ͒ ˆ ͒ ds͑M J pd dFb͑hM Our mean-field theory simplifies at low temperatures and, ϭ . ͑12͒ dM k TV dh for the four-band model, simplifies further when h is much B larger than the characteristic energy scale of occupied Kohn- ͑ ͒ Comparing with Eq. 9 , it follows that FMF(M) is mini- Luttinger states. A convenient typical energy scale is the h ϭ ␮ ϭ ϭ ⑀ mized by M JBJ(gL BHeffJ/kBT) JBJ(xeff), where 0 hole Fermi energy F0. For a given value of NMnJ pd the ϭ largest value of h is reached at T 0. Then, since Heff is g ␮ H J dF ͑hMˆ ͒ always nonzero, x