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Theory of Magnetic Anisotropy in III1−xMn

xV Ferromagnets

M. Abolfath1, T. Jungwirth2,3, J. Brum4 and A.H. MacDonald21Department of Physics and Astronomy, University of Oklahoma, Norman, OK 73019-0225

2Department of Physics,Indiana University, Bloomington, Indiana 474053Institute of Physics ASCR, Cukrovarnicka 10, 162 00 Praha 6, Czech Republic

4Department of Physics, UNICAMP, Campinas, Brazil

(October 25, 2018)

We present a theory of magnetic anisotropy in III1−xMnxV diluted magnetic semiconductors withcarrier-induced ferromagnetism. The theory is based on four and six band envelope functions modelsfor the valence band holes and a mean-field treatment of their exchange interactions with Mn++

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-densitydependence at small p, a p−1 dependence at large p, and remarkably large values at intermediatedensities. An explicit expression, valid at small p, is given for the uniaxial contribution to themagnetic anisotropy due to unrelaxed epitaxial growth lattice-matching strains. Results of ournumerical simulations are in agreement with magnetic anisotropy measurements on samples withboth compressive and tensile strains. We predict that decreasing the hole density in current sampleswill lower the ferromagnetic transition temperature, but will increase the magnetic anisotropy energyand the coercivity.

I. INTRODUCTION

The discovery of carrier-mediated ferromagnetism1–3

in III1−xMnxV and doped4 II1−xMnxVI diluted mag-netic semiconductors (DMS’s) has opened up a broadand relatively unexplored frontier for both basic andapplied research. Experiments2,5 in Ga1−xMnxAs andIn1−xMnxAs have demonstrated that these ferromagnetshave remarkably square hysterisis loops with coercivitiestypically ∼ 40Oe, and that the magnetic easy axis is de-pendent on epitaxial growth lattice-matching strains. Inthis paper we discuss the magnetic anisotropy proper-ties of III1−xMnxV DMS ferromagnets, predicted by amean-field-theory6 of the exchange interaction couplingbetween localized magnetic ions and valence band freecarriers. We use phenomenological four or six band en-velope function models, depending on the carrier densityp, in which the valence band holes are characterized byLuttinger , spin-orbit splitting, and strain-energy param-eters.The physical origin of the anisotropy energy in our

model is spin-orbit coupling in the valence band. Ourwork is based in part on theoretical descriptions devel-oped by Gaj7 and Bastard8 to explain the optical prop-erties of undoped, paramagnetic DMS’s. As the criti-cal temperature is approached, the mean-field-theory weemploy6 reduces to an earlier theory4 that invokes gener-alized RKKY carrier-mediated interactions between lo-calized spins. The two approaches differ, however, intheir description of the magnetically ordered state. Asthis work was nearing completion we learned of a closelyrelated study9 which uses the same mean-field theoryto address critical temperature trends in this material

class and which also comments on the mean-field the-ory’s ability to address magnetic anisotropy physics. Weare aware of three elements of the physics of these mate-rials which make the predictions of our mean-field theoryuncertain: i) we do not account for the substantial dis-order which is usually present in these ferromagnets; ii)we do not include effects due to interactions among theitinerant holes and iii) we do not account for correla-tions between localized spin-configurations and itineranthole states. The importance of each of these deficien-cies is difficult to judge in general, and probably dependson adjustable material parameters. In our view, it islikely that there is a substantial range in the parameterspace of interest where the predictions of the present the-ory are useful. We expect that important progress canbe made by comparing this simplest possible theory ofcarrier-induced DMS ferromagnetism with experiment.This work has two objectives. Most importantly, we

have attempted to shed light on how various adjustablematerial parameters can influence magnetic anisotropy.Secondarily, we have made an effort to estimate the mag-netic anisotropy energy in those cases where experimen-tal information is presently available. Our hope here isto initiate a process of careful and quantitative compari-son between mean-field theory and experiment, partiallyto help judge the efficiency of this approximation in pre-dicting other physical properties. Even in the mean-fieldtheory, we find that the magnetic anisotropy physics ofthese materials is rich. We predict easy axis reorienta-tions as a function of hole density, exchange interactionstrength, temperature, and strain and identify situationsunder which III1−xMnxV ferromagnets are remarkablyhard.

1

In Section II we detail our mean-field-theory of theordered state. The theory simplifies in the limit of low-temperature and low hole densities. Our results for thislimit, presented in Section III, predict a 〈111〉 easy axisin the absence of strain, and a magnetic anisotropy en-ergy which is approximately 10% of the free carrier band-energy density. This value is extremely large for a cubicmetallic ferromagnet; typical ratios in transition metalferromagnets are smaller than 10−6 for example. Theanisotropy energy in this limit varies as the free-carrierdensity to the 5/3 power and is independent of the ex-change coupling strength. Explicit results for the strain-dependence of the magnetic anisotropy in the same limitare presented in Sec. IV. We find that unrelaxed lattice-matching strains due to epitaxial growth contribute auniaxial anisotropy which favors magnetization orienta-tion along the growth direction when the substrate latticeconstant is larger than the ferromagnetic semiconductorlattice constant and an in-plane orientation in the oppo-site case. Unfortunately, perhaps, the simple low densitylimit does not normally apply in situations where highcritical temperatures are expected. The more compli-cated, and more widely relevant, general case is discussedin Section V. We find that magnetic anisotropy has a non-trivial dependence on both temperature and exchangecoupling strength and that easy axis reversals occur, ingeneral, as a function of either parameter. Accordingto our theory, anisotropy energy densities comparable tothose in typical metallic ferromagnets are possible whenthe exchange coupling is strong enough to depopulate allbut one of the spin-split valence bands, even with satura-tion magnetization values smaller by more than an orderof magnitude. In the limit of large hole densities, wefind that the anisotropy energy of strain-free samples isproportional to hole density p−1 and exchange couplingJ4pd. We find that in typical situations a strain e0 of only

∼ 1% is sufficient to overwhelm the cubic anisotropy ofstrain-free samples. We conclude in Section VI with adiscussion of the implications of these calculations forthe interpretation of present experiments, and with somesuggestions for future experiments which could furthertest the appropriateness of the model used here.

II. FORMAL THEORY

Our theory is based on an envelope function descrip-tion of the valence band electrons, and a spin representa-tion for their kinetic-exchange inteaction10 with localizedd electrons11 on the Mn++ ions:

H = Hm +Hb + Jpd∑

i,I

~SI · ~si δ(~ri − ~RI), (1)

where i labels a valence band hole and I labels a mag-netic ion. In Eq. (1), Hm describes the coupling of mag-netic ions with total spin quantum number J = 5/2

to an external field (if one is present), ~SI is a local-ized spin, ~si is a hole spin, and Hb is either a four ora six-band envelope-function Hamiltonian12 for the va-lence bands. In this paper we do not consider exter-nal magnetic fields so Hm → 0. The four-band Kohn-Luttinger model describes only the total angular momen-tum j = 3/2 bands, and is adequate when spin-orbit cou-pling is large and the hole density p is not too large. Asdiscussed later, in the case of GaAs, a four-band modelsuffices for p <∼ 1018 cm−3. In III1−xMnxV semiconduc-tors, the four j = 3/2 bands are separated by a spin-orbitsplitting ∆so from the two j = 1/2 bands. In the rele-vant range of hole and Mn++ densities, no more thanfour bands are ever occupied. Nevertheless, mixing be-tween j = 3/2 and j = 1/2 bands does occur, and itcan alter the balance of delicate cancellations which oftencontrols the net anisotropy energy. The exchange inter-action between valence band holes and localized momentsis believed to be antiferromagnetic10, i.e. Jpd > 0. ForGaAs, experimental estimates13–16 of Jpd fall between0.04eVnm3 and 0.15eVnm3, with more recent work sug-gesting a value toward the lower end of this range.The form of the valence band for Bloch wavevec-

tors near the zone center in a cubic semiconductorfollows from k · p perturbation theory and symmetryconsiderations.17 The four band (j = 3/2) and six band(j = 3/2 and 1/2) models are known as Kohn-LuttingerHamiltonians and their explicit form is given in the Ap-pendix. The eigenenergies are measured down from thetop of the valence band, i.e., they are hole energies.The Kohn-Luttinger Hamiltonian contains the spin-orbitsplitting parameter ∆so and three other phenomenolog-ical parameters, γ1, γ2 and γ3. These are accuratelyknown for common semiconductors. For GaAs and InAs,the two materials in which III1−xMnxV ferromagnetismhas been observed, ∆so = 0.34 eV and 0.43 eV, and(γ1, γ2, γ3) = (6.85, 2.1, 2.9) and (19.67, 8.37, 9.29) re-spectively. Most of the specific illustrative calculationsdiscussed below are performed with GaAs parameters.Our calculations are based on the Kohn-Luttinger

Hamiltonian and on a mean-field theory in which corre-lations between the local-moment configuration and theitinerant carrier system are neglected. We comment lateron limits of validity of this approximation. There area number of equivalent ways of developing this mean-field theory formally. In the following paragraphs wepresent a view which is convenient for discussing mag-netic anisotropy.In the absence of an external magnetic field, the par-

tition function of our model may be expressed exactly asa weighted sum over magnetic impurity configurationsspecified by a localized spin quantization axis, M , andazimuthal spin quantum numbers mI :

Z =∑

mI

exp(−Fb[mI ]/kBT ), (2)

where Fb[mI ] is the valence band free energy for holeswhich experience an effective Zeeman magnetic field

2

~h(~r)[mI ] = −JpdM∑

I

mI δ(~r − ~RI). (3)

The mean-field theory consists of replacing ~h(~r)[mI ] byits spatial average for each magnetic impurity config-uration, thereby neglecting correlations between spin-distributions in local-moment and hole subsystems. Theeffective Zeeman magnetic field experienced by the holesthen depends only on M , the direction of the local-moment orientation, and the mean azimuthal quantumnumber averaged over all local moments, M :

~hMF (M) = JpdNMnMM ≡ hM, (4)

where NMn = NI/V is the number of magnetic impuri-ties per unit volume. The mean-field partition functionis

ZMF (M) = exp((NITs(M)− Fb(~h))/kBT ), (5)

where the entropy per impurity is defined by

s(M) = kB limNI→∞

ln[∑

mIδ(∑

I mI −NIM)]

NI, (6)

and Fb(~h) is the free-energy of a system of non-interacting

fermions with single-particle Hamiltonian Hb − hM · ~s.Following standard ‘large number’ arguments s(M) is

readily evaluated by considering an auxiliary system con-sisting of magnetic impurities coupled only to an exter-nal magnetic field H . For this model problem, a familiarexercise20 gives the result

M(H) = JBJ (x), (7)

where x = gLµBHJ/kBT , gL is the Lande g-factor of theion, µB is the electron Bohr magneton, and

BJ(x) =2J + 1

2Jcoth

[

(2J + 1)x/2J]

− 1

2Jcoth(x/2J),

(8)

is the Brillouin function. The Brillouin function is a one-to-one mapping between reduced fields x in the interval[0,∞] and reduced magnetizations M/J in the interval[0, 1]; the inverse function B−1

J maps M/J to x. Sincethe magnetization maximizes s(M) + gLµBHM/kBT ,

ds(M)

dM= −gLµBH/kBT. (9)

Eq. (9) can be used to eliminate H and arrive at thefollowing explicit expression for s(M):

s(M) = kB

∫ ∞

B−1

J(M/J)

dx xdBJ (x)

dx. (10)

The J = 5/2 result for s(M) is illustrated in Fig. 1. Theentropy per impurity vanishes for M = J = 5/2 becausethere is a single configuration with

∑

I mI = NIJ , andapproaches ln(2J + 1) ≈ 1.79 for M → 0.

0 1 2 3M

0

0.5

1

1.5

2

s(M

) / k

B

FIG. 1. Entropy per localized spin as a function of averagepolarization.

The mean-polarization of the localized spins at a giventemperature and for a given orientation of the local mo-ments is determined by minimizing the mean-field freeenergy

FMF (M) = −kBT lnZMF (M)

= Fb(~h = NMnJpdMM)− kBTNIs(M), (11)

with respect to M . Setting the derivative to zero gives

ds(M)

dM=

JpdkBTV

dFb(hM)

dh. (12)

Comparing with Eq. (9), it follows that FMF (M) is min-imized by M = JBJ(gLµBHeffJ/kBT ) = JBJ (xeff )where

xeff ≡ gLµBHeff

kBT= − Jpd

V kBT

dFb(hM)

dh. (13)

It follows that h = NMnJpdM is determined by solvingthe self-consistent equation

h = NMnJpdJBJ

[

xeff (h)]

. (14)

Note that

dFb(hM)

dh= 〈~Stot · M〉, (15)

where Stot is the total hole spin and the angle bracketsindicate a thermal average for the non-interacting valenceband system. Since the valence band system experiencesan effective Zeeman coupling with strength proportionalto h and direction −M , it is clear that the right handside of Eq. (15) is negative in sign and that its magnitudeincreases monotonically with h, making it easy to solveEq. (14) numerically.

3

To simplify the calculations presented in subsequentsections, we take advantage of the fact that temperaturesof interest are almost always considerably smaller thanthe itinerant carrier Fermi energy. This allows us to re-

place Fb(~h) by the ground state energy Eb(~h). Then, us-

ing Eq. (13) and Eq. (11), a single calculation of Eb(hM)over the range from h = 0 to h = NMnJpdJ may be usedto determine the local-moment magnetization M(T ) andthe free energy F (T ) = FMF (M(T )) at all temperatures.The mean-field theory critical temperature can be

identified by linearizing the self-consistent equation atsmall h. We find that

kBTc(M) = −J(J + 1)

3

NMnJ2pd

V

d2Fb(hM)

dh2|h=0. (16)

The second derivative of the valence band free-energywith respect to field is proportional to its Pauli spin-susceptibility, which is in turn proportional to the va-lence band density of states at the Fermi energy, and top1/3 at small p. In the absence of strain, it follows fromcubic symmetry that the right-hand-side of Eq. (16) is

independent of M . Below the critical temperature how-ever, the mean-field free energy does depend on M ; thisdependence is the magnetic anisotropy energy we wishto calculate. We will see that the dependence of theanisotropy energy on hole density is very different fromthat of the critical temperature.

III. MAGNETIC ANISOTROPY IN THE

STRONG EXCHANGE COUPLING LIMIT

Our mean-field theory simplifies at low temperaturesand, for the four-band model, simplifies further when h ismuch larger than the characteristic energy scale of occu-pied Kohn-Luttinger states. A convenient typical energyscale is the h = 0 hole Fermi energy ǫF0. For a given valueof NMnJpd the largest value of h is reached at T = 0.Then, since Heff is always non-zero, xeff → ∞ and thesolution to the mean-field equations is M = J , implyingthat h = NMnJpdJ for every orientation M . At T = 0there is no entropic contribution to the free-energy and

FMF (T = 0) = Eb(~h = NMnJpdJM). (17)

We note in passing that the magnetization density atT = 0 has contributions from the localized spins and theitinerant spins:

Ms(T = 0) = 2µBJNMn +κ

3V〈~Stot · M〉, (18)

where κ is an additional parameter of the Luttingermodel. Because of the antiferromagnetic exchange in-teraction, the two terms here will tend to have oppositesigns with the first term, which is independent of thehole density, typically very dominant. For h ≫ ǫF0 twofurther simplifications occur. When the splitting of the

hole bands by the effective Zeeman coupling is sufficientlylarge, or the hole density p is sufficiently small, only thelowest energy hole band will be occupied. Furthermore,the effective Zeeman-term will dominate the mean-fieldsingle-particle Hamiltonian and, as we detail below, theenvelope function spinor for this occupied hole state hasa simple analytic expression. In this section we assumethat the spin-orbit splitting energy ∆so is much largerthan all other energies so that we can work with a fourband model. More generally, the anisotropy will dependon h/∆so, even in the limit of small hole densities.To judge whether or not this limit can be achieved

in practice, we estimate the Fermi energy of holes usingthe spherical approximation18 in which the doubly de-generate h = 0 bulk heavy hole and light hole bands areparabolic with masses mh = m/(γ1 − 2γ) ∼ 0.498m andml = m/(γ1+2γ) ∼ 0.086m respectively. An elementarycalculation then gives

ǫF0 =h2

2m

(3π2p

2

)2/3γ0, (19)

where n = Nh/V is the free-carrier density and

γ0 =[ (γ1 − 2γ)−3/2 + (γ1 + 2γ)−3/2

2

]−2/3. (20)

For GaAs γ0 ∼ 3.05: the Fermi energy is the same as thatof a system with four identical effective massm/γ0 bands.Typical high Tc Ga1−xMnxAs ferromagnetic semiconduc-tor samples have p ∼ 0.1nm−3 and NMn ∼ 1.0nm−3

(x ∼ 0.05), although these parameters can presumablybe varied widely. Choosing a Jpd value in the mid-rangeof estimated values (∼ 0.006Ry nm3) these parametersimply that h ∼ 0.015Ry and ǫF0 ∼ 0.01Ry. h is nei-ther large compared to ǫF0, nor small compared to ∆so.Thus, the simple expressions discussed in this section arenot accurate for current high Tc systems. As our abil-ity to engineer materials improves it should, however, bepossible to grow samples which are in the limit discussedhere. Since h is comparable to ǫF0, we know, even be-fore performing detailed calculations, that valence bandquasiparticle spectra in paramagnetic and ferromagneticstates will differ qualitatively.In the large h limit the lone occupied spinor at each

wavevector ~k will be the member of the j = 3/2 mani-fold for which the total angular momentum is aligned inthe direction −M ; the origin of the minus sign here isthe antiferromagnetic nature of the interaction betweenlocalized spins and hole spins. Explicit expressions forthe expansion of such a spin coherent states in terms ofthe eigenstates of jz are known21:

| − M〉 = u3|3/2〉+√3[u2v|1/2〉+ uv2| − 1/2〉]

+ v3| − 3/2 > . (21)

In Eq. (21) u = −i exp(−iφ/2) sin(θ/2) and v =i exp(iφ/2) cos(θ/2) where θ and φ are the spherical co-

ordinates which specify the unit vector −M . In the large

4

h/ǫF0 limit, the band term in the single-particle Hamil-tonian may be treated using first order perturbation the-ory. Taking the expectation value of the Kohn-LuttingerHamiltonian in the spin coherent state we find that

ǫ(~k) = −h

2+ 〈−M |HL(~k)| − M〉

≡ −h

2+

h2k2

2mγ(M, k). (22)

The first term on the right hand side of Eq. (22) reflects

the spin coherent state property19, M · s| − M〉 = −| −M〉/2. In Eq. (22) we have noted that for any M and any

k, the dependence of hole energy on k = |~k| is quadratic.Using this property, it follows that the Fermi energy

ǫF (M) =h2

2m(6π2p)2/3 γ(M), (23)

and that the ferromagnetic ground state energy densityis

Eb( ~M)

V= −hp

2+

3

5p ǫF (M), (24)

where

γ(M) ≡[

∫

dk

4π(γ(M, k))−3/2

]−2/3. (25)

In analogy with the h = 0 quantity γ0 defined inEq. (20), γ(M) is an average of the band energy curva-

ture over reciprocal space directions k, with the smaller

values of γ(M, k) weighted more heavily. Note that thefactor 3π2/2 in Eq. (19) is replaced by 6π2 in Eq. (23)because only one band is occupied in this limit, insteadof the four which are occupied at h = 0. m/γ(M) maybe thought of as a spin-orientation dependent effectivemass, which is readily evaluated as a function of M , giventhe Luttinger parameters of any material. Although themagnetic condensation energy has a term proportionalto Jpd, only the band-energy contributes to the M de-pendence of the ferromagnetic ground state energy. Themagnetic anisotropy energy in this limit is independentof Jpd and proportional to the hole density p to the 5/3power.We have evaluated γ(M) as a function of angle for

the Luttinger parameters of GaAs and InAs. As dis-cussed in more detail later, we always find that mag-netic anisotropy in the absence of strain is well describedby a cubic harmonic expansion truncated at sixth order,an approximation commonly used in the literature22 onmagnetic materials. The corresponding cubic harmonicexpansion for γ(M) is

γ(M) = γ(〈100〉) + γca1 (M2

xM2y + M2

yM2z + M2

xM2z )

+ γca2 M2

xM2y M

2z . (26)

In our calculations we find that γ(M) < γ1 for all direc-

tions M . This property reflects the fact that small cur-vature (large Fermi wavevector) directions are weightedmore highly in calculating the total hole energy. For bothInAs and GaAs we find that the dominant fourth ordercubic anisotropy coefficient, γca

1 < 0, indicating nickel-

type anisotropy with easy axes along the 〈111〉 cube di-agonal directions. At a qualitative level, the source ofthe higher total hole energy when the moment orienta-tion is along a 〈001〉 (cube edge) direction is easy to un-derstand. With such a moment orientation, the occupiedhole orbital has jz = −3/2 and energy dispersion given

by Hhh(~k) in Eq. (46). It follows that γ(M = 〈100〉, k)has the relatively large value γ1 + γ2 for all orientations

of k in the x− y plane. These large values of γ(M, k) are

important in the average and cause the average over k toreach its maximum for this orientation of M .The cubic magnetic anisotropy energy coefficients in

this limit are given by

Kcai = +

3

5ph2

2m(6π2p)2/3 γca

i . (27)

Values of γcai for GaAs and InAs are listed in Table

I. We will see later that these expressions apply up top ∼ 1018cm−3. Inserting this value for the hole densitygives coefficients ∼ 2kJm−3 for GaAs host material and∼ 4kJm−3 for InAs host materials; magnetic anisotropyis twice as strong in InAs in the strain free case. Theseanisotropy energy coefficients are not so much smallerthan those of the cubic metallic transition metal ferro-magnets, despite the much higher carrier densities inthe metallic case. We will see below that the scale ofthe semiconductor magnetic anisotropy energy does notchange as the carrier density increases from 1018cm−3

to ∼ 1021cm−3. The relatively large anisotropy energiesoccur despite the fact that the saturation moments Ms

of III1−xMnxV ferromagnets are more than an order ofmagnitude smaller than their cubic metal counterparts.It follows from these values that the magnetic hardnessparameters of the III1−xMnxV ferromagnets,

κ ∼[ Kca

1

µ0M2s

]1/2, (28)

will typically be larger than one. This is unusual in cubicmaterials and occurs because spin-orbit coupling has amuch stronger influence on semiconductor valence bandsthan on transition metal d-bands.As we discuss at length in Section V, magnetic

anisotropy does not continue to increase rapidly withhole density once two or more bands are occupied inthe metallic state. In current high Tc samples, we willfind that several bands are always occupied, even at zerotemperature. The simple limit discussed in this sectiondemonstrates that anisotropy energies, T = 0 saturationmoments, and critical temperatures will have radicallydifferent dependencies on engineerable parameters.

5

IV. STRAIN DEPENDENCE OF MAGNETIC

ANISOTROPY: LOW HOLE DENSITY LIMIT

Because of the low soluability of Mn in III-V semi-conductors, III1−xMnxV materials with x large enoughto produce cooperative magnetic effects cannot be ob-tained by equilibrium growth. The MBE growth tech-niques which have been successfully developed23 produceIII1−xMnxV films whose lattices are locked to those oftheir substrates. X-ray diffraction studies3 have estab-lished that the resulting strains are not relaxed by dislo-cations or other defects, even for thick films. Strains inthe III1−xMnxV film break the cubic symmetry assumedin the previous section. Fortunately, the influence ofMBE growth lattice-matching strains on the hole bandsof cubic semiconductors is well17,24 understood. For the〈001〉 growth direction used to create III1−xMnxV films,strain generates a purely diagonal contribution to thefour band single-particle envelope function Hamiltonianin the representation we use in this paper, adding contri-butions δǫh and δǫl respectively to jz = ±3/2 heavy holeand jz = ±1/2 light hole entries. The energy shifts arerelated to the lattice strains by17

δǫh =e0C11

[

− 2a1(C11 − C12)−a22(C11 + 2C12)

]

, (29)

δǫl =e0C11

[

− 2a1(C11 − C12) +a22(C11 + 2C12)

]

, (30)

where e0 is the in-plane strain produced by the substrate-film lattice mismatch:

e0 =aS − aF

aF. (31)

In Eqs. (29–31), the Cij are the elastic constants of theunstrained III1−xMnxV film, which we will assume to beidentical to those of the host III-V material, aS is the lat-tice constant of the substrate on which the III1−xMnxVfilm is grown, aF is the unstrained lattice constant ofbulk III1−xMnxV, and a1 and a2 are phenomenologicaldeformation potentials whose values for common III-Vsemiconductors are known. For the six band model thestrain Hamiltonian includes off-diagonal elements and isgiven up to a constant times the unit matrix by

Hstrain = Γe0

0 0 0 0 0 0

0 1 0 0 0 1√2

0 0 1 0 − 1√2

0

0 0 0 0 0 0

0 0 − 1√2

0 12 0

0 1√2

0 0 0 12

, (32)

where Γ = ǫl−ǫh = a2e0(C11+2C12)/C11. For GaAs andInAs, Γ = −0.2382 Ry and −0.2762 Ry respectively.17

As in the previous section, we can derive an explicitexpression for the strain contribution to the magneticanisotropy energy when h ≫ ǫF0 and h ≫ Γ, allowingband and strain terms to be treated as a perturbativecorrection to the effective Zeeman coupling, and h ≪∆so, allowing a four-band model to be used. In this waywe obtain

Estrain(M)

V=

p

4[3δǫh + δǫl] +

3p cos2 θ

4[δǫh − δǫl]. (33)

Strain produces a uniaxial contribution to the mag-netic anisotropy of III1−xMnxV films, that favors ori-entations along the growth direction when strain shiftsheavy holes down relative to the light holes and orienta-tions in the plane in the opposite circumstance. UsingEq. (29) and Eq. (30) and deformation potential elas-tic constant values17, we find that a contribution to theenergy density given by Kstrain sin

2(θ) where the uniax-ial anisotropy constant is Kstrain = −0.36Ry e0 p forGaAs and Kstrain = −0.41Ry e0 p for InAs. Sincea2 < 0, compressive strain (e0 < 0) lowers the heavyhole energy relative to the light hole energy and favorsmoment orientations in the growth direction while tensilestrain (e0 > 0) favors moment orientations perpendicu-lar to the growth direction. Since the lattice constant ofGa1−xMnxAs is larger2 than that of GaAs, while that ofIn1−xMnxAs is smaller than that of InAs, Ga1−xMnxAson GaAs is under compressive strain and In1−xMnxAson InAs is under tensile strain. Assumming3 Vergard’slaw, e0 = .0004 and e0 = −.0028 for InAs and GaAsrespectively at x = 0.05. For p = 10−3nm−3, a densityfor which this low-density result is still reasonably reli-able, we find that Kstrain = −0.36kJm−3 for InAs andKstrain = 2.2kJm−3 for GaAs. At these densities, strainand cubic band contributions are comparable in GaAs,but the latter contribution is dominant in the InAs case.Strain-induced anisotropy energies can be modified by

growing the magnetic film on relaxed buffer layers. Thiseffect has been demonstrated25 by Ohno et al. whoshowed that the easy axis for Ga1−xMnxAs films changesfrom in-plane to growth-direction when the substrate ischanged from GaAs to relaxed (In,Ga)As buffer layers.For 15% In, the magnetic film strain changes from com-pressive e0 = −0.0028 to tensile e0 = 0.0077 when thischange is made. We note that the sense of this changeis opposite to that predicted by our large h, small p ana-lytic result which predicts that compressive strains favorgrowth direction orientations. Similarly, In1−xMnxAs onInAs is under a small tensile strain but is observed tohave a growth direction easy axis. (Recall that in thelarge h limit the band anisotropy makes the growth di-rection the hard axis.) This distinction may be takenas an experimental proof that several hole bands are oc-cupied in the magnetic ground state of these materials.The strain anisotropy energy does change sign at smallervalues of h partly because the first band to be depop-ulated has primarily heavy-hole character. We will seein the next section that the mean-field does predict the

6

correct sign for the strain anisotropy energy at exper-imental hole densities. We conclude from the presentconsiderations that strain can play a strong role in band-structure-engineering of III1−xMnxV ferromagnet mag-netic properties.

V. PARTIALLY POLARIZED HOLE BAND

STATES: MAGNETIC ANISOTROPY IN THE

GENERAL CASE

The results for Fb(hM) discussed in the previous sec-tion become accurate when the effective Zeeman cou-pling h is large enough to reduce the number of occu-pied hole bands to one, and the Fermi energy remainssafely smaller than the spin-orbit splitting. The situa-tion is much more complicated at smaller h and largerhole densities. Then several bands are occupied, even atT = 0, and these usually give competing contributionsto the magnetic anisotropy. It is not true in general thatband and strain contributions to the magnetic anisotropyare simply additive. We expect that it will eventually bepossible to realize a broad range of material parameters,and hence a broad range of h values, in III1−xMnxV fer-romagnets. The range of possibilities is immense andaccurate modeling of a particular sample will require ac-curate values for p, Jpd, and NMn for that material.In this section we discuss a series of illustrative cal-

culations, starting with ones performed using a four-band model of strain-free Ga1−xMnxAs at hole densityp = 0.1nm−3. The valence band energy density is

Eb

V=

1

V

∑

k

Nb∑

j=1

θ(ǫF − ǫj(k,~h))ǫj(k,~h), (34)

where ǫF is the Fermi energy, ~k is the Bloch wavevector,

and the mean-field-theory quasiparticle energies ǫj(k,~h)are eigenvalues of the Nb × Nb single-particle Hamilto-nian (Nb is the number of bands included in the envelopefunction Hamiltonian)

Hb = HL + ~h · ~s+Hstrain. (35)

For a strain free model we set Hstrain = 0. In Fig. 2 weplot the calculated spin-polarization per hole as a func-tion of h for a growth direction field orientation; recallthat this quantity can be obtained by differentiating theenergy per hole with respect to h and that the effectivefield seen by the localized spins is obtained by multiply-ing this quantity by Jpd p. The hole spin polarizationincreases linearly at small h with a slope proportional tothe valence band Pauli susceptibility. We see that for thehole density of Fig. 2, complete hole spin-polarization isapproached only at values of h comparable to or largerthan the spin-orbit splitting of GaAs, so that the fourband model is not physical in this regime. For any givenmodel, and a given moment orientation, a single calcula-tion of this type provides all the microscopic information

required to solve the mean-field equations at all temper-atures.

0 0.2 0.4 0.6h (Ry)

−0.5

−0.4

−0.3

−0.2

−0.1

0

<S

tot>

/ N

h

FIG. 2. Valence band spin per hole as a function of effectiveZeeman field strength h.

For a fixed values of Jpd, NMn, and p, h must be eval-uated as a function of temperature by solving the self-consistent mean-field equation, Eq. (14), and using nu-merical results like those plotted in Fig. 2. Results forh(T ) calculated for Jpd = 0.15eVnm3, NMn = 1nm−3,and p = 0.1nm−3 are illustrated in Fig. 3 for high sym-metry moment orientation directions. For these param-eters the critical temperature Tc (h(T ) = 0 for T > Tc)is ∼ 100K, in rough agreement with experiment. Thereare, however, other choices of parameters which are alsoconsistent with the measured critical temperature. Inaddition, as we discuss further in the next section, it isnot clear that this level of theory should always yieldaccurate results for Tc.

0.0 50.0 100.0 150.0T (K)

0.000

0.010

0.020

0.030

h (R

y)

<100><110><111>

7

FIG. 3. The dependence of the effective Zeeman couplingstrength seen by the magnetic impurities h on temperaturefor the high symmetry directions < 100 >, < 110 >, and< 111 >. These results were calculated using a strain freefour-band model for Jpd = 0.15eVnm3 and NMn = 1.0nm−3.h(T ) is determined by solving the self-consistent mean-fieldequation.

At T = 0, h reaches its maximum value, JpdNMnJ ∼0.0275Ry. The dependence of energy on magnetizationorientation at this value of h is illustrated in Fig. 4 andcompared with the cubic harmonic expansion truncatedat 6th order. The coefficients of this expansion are fixedby energy per volume calculations in 〈100〉, 〈110〉, and〈111〉 directions. In Fig. 4, and in all other cases we havechecked, the truncated cubic harmonic expansion is veryaccurate. It is therefore sufficient to evaluate the energyper volume in the high-symmetry directions and to use

Kca1 =

4(Eb〈110〉 − Eb〈100〉)V

Kca2 =

27Eb〈111〉 − 36Eb〈110〉+ 9Eb〈100〉V

. (36)

Results for Kcai (h) obtained for the four band model in

this way are summarized in Fig. 5. These results can becombined with those in Fig. 3 to obtain the model’s cu-bic anisotropy coefficients as a function of temperature,hole density and h. Here we see explicitly the anisotropyreversals which commonly accompany hole band depop-ulations. We note in Fig. 5 that the analytic results ofSection III are recovered only for very large values of hat this density.

0 20 40 60−0.0216

−0.0213

−0.021

−0.0207

−0.0204

Eb/

h

<001> <100> <110> <001>

FIG. 4. Hole energy per particle in units of h as a functionof magnetization orientation for h = 0.0275Ry). The solid lineis a cubic harmonic expansion fit to these results truncatedat 6th order. At this value of h, three hole bands are partlyoccupied and the easy axes are the cube edge directions.

0.00 0.01 0.02 0.03 0.04 0.05 0.06h (Ry)

−2.0

−1.5

−1.0

−0.5

0.0

0.5

1.0

anis

otro

py (

mR

y pe

r pa

rtic

le)

K1

ca

K2

ca

FIG. 5. The dependence of the crystalline anisotropy co-efficients, Kca

1 (h) and Kca2 (h) on h for a four-band model with

p = 0.1nm−3.

The valence band Fermi surfaces of III1−xMnxV fer-romagnets will be strongly dependent on both tempera-ture and moment direction orientation. Four-band modelFermi surface intersections with the kz = 0 plane areillustrated in Figs. 6–8 for p = 0.1nm−3 at h = 0,h = 0.01Ry, and h = 0.0275Ry respectively. All figuresare for moments oriented in the 〈100〉 direction. Thelargest of these three values of h is the T = 0 effectivefield (JpdNMnJ) for x = 0.05 and Jpd at the high endof literature estimates. (Jpd = 0.15eVnm3.) These threefigures represent the mean-field-theory Fermi surfaces atthree different temperatures. In the spherical approxi-mation, the h = 0 Fermi energy at this hole density isǫF0 ∼ 0.01Ry, so a strong distortion of the bands is ex-pected in the magnetic state. At h = 0, the hole bandsoccur in degenerate pairs; we refer to the two less dis-

persive bands which occupy the larger area in k space asheavy-hole bands, although this terminology lacks precisemeaning in the general case. As h increases, both heavyand light hole bands split. For small h, the minority-spinheavy-hole band occupation decreases rapidly and allother band occupations increase. The heavy-hole minor-ity spin band is completely depopulated for h ∼ 0.04Ry.Once this band is empty, the light-hole minority-spinFermi radii begin to shrink rapidly. As seen in Fig. 8,this second band is nearly depopulated at h = 0.0275Ry.At still stronger fields, the majority-spin light-hole bandis depopulated and the single-band limit addressed inpreceding sections is achieved. For p = 0.1nm−3, thesingle-band limit is achieved only at h values for whichthe four-band model breaks down.

8

−0.10 −0.05 0.00 0.05 0.10kx (a0

−1)

−0.10

−0.05

0.00

0.05

0.10k y (

a 0−1 )

FIG. 6. Fermi surface intersection with the kz = 0 planefor a four-band model with p = 0.1nm−3 and h = 0. Doublydegenerate heavy-hole (larger contour) and light-hole (smallercontour) bands are occupied.

−0.10 −0.05 0.00 0.05 0.10kx (a0

−1)

−0.10

−0.05

0.00

0.05

0.10

k y (a 0−

1 )

FIG. 7. Intersection of the Fermi surface and the kz = 0plane for a four-band model with p = 0.1nm−3 andh = 0.01Ry. This value of h solves the mean-field equationsequation at T = 0K, NMn = 1nm−3, and Jpd = 0.05eVnm3,which is near the lower experimental estimate16 for the ex-change coupling constant. (For Jpd = 0.15eVnm3 this valueof h solves the mean-field equations at T = 85K). The mag-netization orientation is along the 〈100〉 direction. Heavy-holeand light-hole bands are split at non-zero h.

−0.10 −0.05 0.00 0.05 0.10kx (a0

−1)

−0.10

−0.05

0.00

0.05

0.10

k y (a 0−

1 )

FIG. 8. Intersection of the Fermi surface and the kz = 0plane for a four-band model with p = 0.1nm−3 andh = 0.0275Ry. This value of h solves the mean-field equa-tion at T = 0, NMn = 1nm−3, and Jpd = 0.15eVnm3, whichis the upper experimental estimate16 for the exchange cou-pling constant. The magnetization orientation is along the〈100〉 direction. The minority heavy-hole band is empty atthis value of h.

Finally we turn to a series of illustrative calcula-tions intended to closely model the ground state ofGa.95Mn.05As. For this Mn density and the smaller val-ues of Jpd favored by recent estimates, h(T = 0) =JpdNMnJ ∼ 0.01Ry. This value of h is not so muchsmaller than the spin-orbit splitting parameter in GaAs(∆so = 0.025 Ry), so that accurate calculations requirea six band model. Even with x fixed, our calculationsshow that the magnetic anisotropy of Ga.95Mn.05As fer-romagnets is strongly dependent on both hole densityand strain. The hole density can be varied by changinggrowth conditions or by adding other dopants to the ma-terial, and strain in a Ga.95Mn.05As film can be alteredby changing substrates as discussed previously. The cu-bic anisotropy coefficients (in units of energy per vol-ume) for strain-free material are plotted as a function ofhole density in the inset of Fig. 9; the main plot showsthe coefficients in units of energy per particle. Over thedensity range p < 0.05nm−3, four and six band mod-els are in good agreement. We see from this result thatthe asymptotic low density region where the anisotropyenergy varies as p5/3 holds only for p < 0.005nm−3 atthis value of h. The easy axis is nearly always deter-mined by the leading cubic anisotropy coefficient Kca

1 ,except near values of p where this coefficient vanishes.As a consequence the easy-axis in strain free samples isalmost always either along one of the cube edge direc-tions (Kca

1 > 0), or along one of the cube diagonal di-rections (Kca

1 < 0). Transitions in which the easy axismoves between these two directions occur twice over the

9

range of hole densities studied. (Similar transitions occuras a function of h, and therefore temperature, for fixedhole density.) Near the hole density 0.01 nm−3, bothanisotropy coefficients vanish and a fine-tuned isotropy isachieved. The slopes of the anisotropy coefficient curvesvary as the number of occupied bands increases from1 to 4 with increasing hole density. This behavior isclearly seen from the correlation between oscillations ofthe anisotropy coefficients and onsets of higher band oc-cupations, plotted in Fig. 10.Six-band model Fermi surfaces are illustrated in

Figs. 11-13 by plotting their intersections with the kz = 0plane at p = 0.1nm−3 for the cases of 〈100〉, 〈110〉, and〈111〉 ordered moment orientations. Comparing Fig. 7and Fig. 11, which differ only in the band model em-ployed, we see that there is a marked difference betweenthe majority-spin heavy hole bands in four and six-bandcases. For the six band model, quasiparticle dispersion isparticularly slow, leading to large Fermi radii along the〈110〉 directions. The large and more anisotropic massis a consequence of mixing with the split-off hole bands.As we see from Figs. 12 and 13, this effect occurs forall ordered moment orientations, although the details ofthe small minority band Fermi surface projections changemarkedly. The dependence of quasiparticle band struc-ture on ordered moment orientation, apparent in compar-ing these figures, should lead to large anisotropic mag-netoresistance effects in III1−xMnxV ferromagnets. Wealso note that in the case of cube edge orientations, theFermi surfaces of different bands intersect. This prop-erty could have important implications for the decay oflong-wavelength collective modes28.

0.00 0.05 0.10 0.15p (nm

−3)

−0.4

−0.3

−0.2

−0.1

0.0

0.1

anis

otro

py (

mR

y pe

r pa

rtic

le)

0.0 0.1 0.2 0.3p (nm

−3)

−5.0

−2.5

0.0

2.5

5.0

anis

otro

py (

kJ m

−3 )

K1

ca

K2

ca

FIG. 9. Six-band model, h = 0.01Ry, results for the cu-bic magnetic anisotropy coefficients in units of Rydberg perparticle (main plot) and in kJ per m3 (inset).

0.00 0.05 0.10 0.15p (nm

−3)

0.000

0.002

0.004

0.006

0.008

0.010

dens

ity (

nm−

3 )

2nd band3rd band4th band

FIG. 10. Six-band model band densities as a function ofthe total hole density; h = 0.01Ry.

−0.10 −0.05 0.00 0.05 0.10kx (a0

−1)

−0.10

−0.05

0.00

0.05

0.10k y (

a 0−1 )

FIG. 11. Six-band model Fermi surface intersections withthe kz = 0 plane for p = 0.1nm−3 and h = 0.01Ry. This figureis for magnetization orientation is along the 〈100〉 direction.

10

−0.10 −0.05 0.00 0.05 0.10kx (a0

−1)

−0.10

−0.05

0.00

0.05

0.10k y (

a 0−1 )

FIG. 12. Six-band model Fermi surface intersections withthe kz = 0 plane for the parameters of figure 11 and magne-tization orientation along the 〈110〉 direction.

−0.10 −0.05 0.00 0.05 0.10kx (a0

−1)

−0.10

−0.05

0.00

0.05

0.10

k y (a 0−

1 )

FIG. 13. Fermi surface intersections with the kz = 0 planefor the parameters of figure 11 and magnetization orientationalong the 〈111〉 direction.

In Fig. 14 we present mean-field theory predictionsfor the strain-dependence of the anisotropy energy ath = 0.01Ry and hole density p = 0.35nm−3. Accord-ing to our calculations, the easy axes in the absence ofstrain are along the cube edges in this case. This cal-culation is thus for a hole density approximately threetimes smaller than the Mn density, as indicated by re-cent experiments. The relevant value of e0 depends onthe substrate on which the epitaxial Ga.95Mn.05As filmis grown, as discussed in Section IV. The most important

conclusion from Fig. 14 is that strains as small as 1% aresufficient to completely alter the magnetic anisotropy en-ergy landscape. For example for (Ga,Mn)As on GaAs,e0 = −.0028 at x = 0.05, the anisotropy has a rel-atively strong uniaxial contribution even for this rela-tively modest compressive strain, which favors in-planemoment orientations, in agreement with experiment. Arelatively small (∼1 kJ m−3) residual plane-anisotropyremains which favors 〈110〉 over 〈100〉. For x = 0.05(Ga,Mn)As on a x = 0.15 (In,Ga)As buffer the strain istensile, e0 = 0.0077, and we predict a substantial uni-axial contribution to the anisotropy energy which favorsgrowth direction orientations, again in agreement withexperiment. For the tensile case, the anisotropy energychanges more dramatically than for compressive strainsdue to the depopulation of higher subbands, as shown inFig. 15. At large tensile strains, the sign of the anisotropychanges emphasizing the subtlety of these effects and thelatitude which exists for strain-engineering of magneticproperties.

−0.10 −0.05 0.00 0.05 0.10e0

−5

−3

−1

1

3

5

anis

otro

py (

10−

5 Ry

per

part

icle

)

−0.01 0.00 0.01e0

−1.0

−0.5

0.0

0.5

1.0

anis

otro

py (

10−

5 Ry)

Eb<001> − Eb<100>

Eb<110> − Eb<100>

Eb<111> − Eb<100>

FIG. 14. Energy differences among 〈001〉, 〈100〉, 〈110〉, and〈111〉 magnetization orientations vs. in-plane strain e0 ath = 0.01 Ry and p = 3.5 nm−3. For compressive strains(e0 < 0) the systems has an easy magnetic plane perpendic-ular to the growth direction. For tensile strains (e0 > 0) theanisotropy is easy-axis with the preferred magnetization ori-entation along the growth direction. The anisotropy changessign at large tensile strain.

11

0.002

0.004

0.006

0.008

0.010

0.012

dens

ity (

nm−

3 )

<100>

<111>

<001>

−0.10 −0.05 0.00 0.05 0.10e0

0.000

0.001

0.002

0.003

0.004

0.005

dens

ity (

nm−

3 )

<100>

<111>

<001>

3rd band

4th band

FIG. 15. 3rd and 4th band densities for 〈100〉, and 〈111〉,and 〈001〉 magnetization orientations vs. in-plane strain e0 ath = 0.01 Ry and p = 3.5 nm−3. Curves for the 〈110〉 magne-tization orientation (not shown here) are similar to those forthe 〈100〉 orientation.

VI. DISCUSSION

We first comment on the implications of the consider-ations described in this paper for the interpretation ofcurrent experiments. The hysteretic effects which re-flect magnetic anisotropy have been studied most ex-tensively for the highest Tc samples currently avail-able. These mean-field-theory predictions depend onthree phenomenological parameters, NMn which is sam-ple dependent but accurately known, Jpd which shouldbe nearly universal for a given III-V host compound butis less accurately known, and the hole density p whichis sample dependent and not accurately known. Valuesof Jpd and p must be inferred from experiment, some-times by comparison with theoretical pictures which arenot yet fully developed. The reliability of Jpd and p esti-mates is improving and presumably will continue to im-prove. It now seems clear that the values of Jpd andp in current high Tc samples are such that several holebands are partially occupied in the ferromagnetic ground

state. In this case, we see from Fig. 5 that our calcu-lations predict cube edge easy axes which include thegrowth direction. The anisotropy energies are typically∼ 10−6Rynm−3 ∼ 1kJm−3. Similar anisotropy ener-gies are produced by strains as small as e0 ∼ 0.001 andmore typical strains produce larger anisotropy energies.An important conclusion from this work is that straincontributions to the anisotropy will not normally be neg-ligible.We believe that the in-plane easy-axis observed in

Ga1−xMnxAs films grown on GaAs is a consequence ofcompressive strain in the magnetic film which dominatesthe cubic band anisotropy energy. When Ga1−xMnxAsis grown on (In,Ga)As, the lattice-matching strain is ten-sile, reinforcing the growth direction easy-axis anisotropyof strain free samples. (As discussed earlier, the signs ofboth strain and cubic band contributions to strain changewhen the light hole bands are depopulated.) We notethat in our calculations, the cubic band anisotropy isalmost always dominated by the fourth cubic harmoniccoefficient. Given this, it follows that only the cube edgeeasy axes, which includes the growth direction axis, andcube diagonal axes, which are not in the film plane, arepossible without strain.Since the local moments are fully polarized in the fer-

romagnetic ground state, it is easy to estimate the sat-uration moment Ms ∼ NMngLµBJ , leading to the rel-atively small numerical value µ0Ms ∼ 0.05T. It followsthat the growth direction orientation magnetostatic en-ergy ∼ µ0M

2s ∼ 0.1kJm−3, considerably smaller than

the magnetocrystalline anisotropy coefficient. Even whenseveral hole bands are occupied with competing spin-orbit interactions, these materials have relatively largemagnetic hardness parameters. Unlike the case of metal-lic thin film ferromagnets which have much larger satura-tion moments, the magnetostatic shape anisotropy playsa minor role in the dependence of total energy on momentorientation. The small saturation moment, will also tendto lead to large domain sizes and square easy-axis direc-tion hysterisis loops, as seen in experiment.Coercivities can be estimated29 from the anisotropy

fields defined by

µ0Ha ∼ µ0MsK

µ0M2s

. (37)

For hard magnetic materials, anisotropy fields are muchlarger than saturation magnetizations. The itinerantfield places an upper bound on and is expected to scalewith the coercivity. Our calculations suggest that thecoercivity in ferromagnetic samples with a single par-tially occupied hole band will be immensely larger thanthe coercivity of current samples. Such samples couldbe fabricated, for example, by adding donors such as Sito current samples, further compensating the Mn accep-tors. According to mean-field-theory, this modificationin the sample preparation procedure will lower the fer-romagnetic critical temperature, and at the same time,increase the anisotropy energy.

12

Finally we conclude with a few words of caution. Thistheory of magnetic anisotropy has three principle limita-tions: i) it is based on a mean-field theory description ofthe exchange interaction between localized spins and va-lence band holes, ii) it neglects hole-hole interactions, andiii) it doesn’t account for disorder scattering of the itin-erant holes. Confronting these weaknesses would in eachcase considerably complicate the theory and we feel itis appropriate to seek progress by comparing the presentrelatively simple theory with experiment. Nevertheless itis worthwhile to speculate on where and how the theorymay be expected to fail.Mean-field theory should be reliable when the range of

the hole mediated interaction between localized spins islong and the spin-stiffness parameter that characterizesthe energy of long-wavelength spin-fluctuations is suffi-ciently large. Considerations of this type suggest26 thatmean-field theory will fail at high temperatures unlessthe ratio of the hole density to the localized spin densityis small and the field h is not too large compared to theFermi energy. Since p is typically smaller than NMn be-cause of anti-site defects in low-temperature MBE growthsamples, mean-field theory is likely to be reasonable atleast at T = 0 in many (III,Mn)V ferromagnets.Hole-hole interactions will clearly tend to favor the fer-

romagnetic state by countering the band-energy cost ofthe spin polarization. Because of strong spin-orbit cou-pling in the valence band, estimates based on many-bodycalculations for electron gas systems may be of little usein estimating the importance of this effect more quan-titatively. Work is currently in progress which shouldshed more light on this matter27. Nevertheless, it seemslikely that these interactions will not have an overridingimportance, at least in large hole density samples.Finally we come to disorder. It is clear from experi-

ment, that disorder does not have a qualitative impacton free-carrier mediated ferromagnetism even when thosefree carriers have been localized by a random disorderpotential. It seems likely that disorder will destroy theferromagnetic state, only when the localization lengthbecomes comparable to the distance between localizedspins. On the other hand, since elastic disorder scatter-ing will mix band states with different orientations onthe Fermi surface, it also seems clear that a reduction inmagnetic anisotropy energy must result. Indeed the co-ercivities that follow from our anisotropy energy resultsappear to be larger than what is observed. As far as weare aware, no theory of this effect exists at present.

ACKNOWLEDGMENTS

We gratefully acknowledge helpful interactions withW.A. Atkinson, T. Dietl, J. Furdyna, J.A. Gaj, J. Konig,B.-H. Lee, E. Miranda, Hsiu-Hau Lin, and Hideo Ohno.Work at the University of Oklahoma was supported bythe NSF under grant No. EPS-9720651 and a grant

from the Oklahoma State Regents for Higher Education.Work at Indiana University was performed under NSFgrants DMR-9714055 and DGE-9902579. Work at theInstitute of Physics ASCR was supported by the GrantAgency of the Czech Republic under grant 202/98/0085.AHM gratefully acknowledges the hospitality of UNI-CAMP where project work was initiated.

APPENDIX

In the literature, different representations are used forthe four band and six band model Kohn-Luttinger Hamil-tonians. In the interest of completeness and clarity, thisappendix specifies the expressions on which our detailedcalculations are based. Detailed derivations of k · p per-turbation theory for cubic semiconductors can be foundelsewhere.17

The k = 0 states at the top valence band have p-likecharacter and can be represented by the l = 1 orbitalangular momentum eigenstates |ml〉. In the coordinaterepresentation we can write

〈r|ml = 1〉 = − 1√2f(r)(x + iy)

〈r|ml = 0〉 = f(r)z

〈r|ml = −1〉 = 1√2f(r)(x − iy). (38)

The Kohn-Luttinger Hamiltonian for systems with nospin-orbit coupling, HL, is written in the representationof the following combinations of |ml〉

|X〉 = 1√2

(

|ml = −1〉 − |ml = 1〉)

|Y 〉 = i√2

(

|ml = −1〉+ |ml = 1〉)

|Z〉 = |ml = 0〉 . (39)

It reads

HL =

Ak2x +B(k2y + k2z) Ckxky Ckxkz

Ckykx Ak2y +B(k2x + k2z) Ckykz

Ckzkx Ckzky Ak2z +B(k2x + k2y)

,

(40)

where

A = − h2

2m(γ1 + 4γ2),

B = − h2

2m(γ1 − 2γ2),

C = −3h2

mγ3, (41)

m is the bare electron mass, and γ1, γ2, and γ3 arethe phenomenological Luttinger parameters. To include

13

spin-orbit coupling we use the basis formed by total an-gular momentum eigenstates |j,mj〉:

|1〉 ≡ |j = 3/2,mj = 3/2〉|2〉 ≡ |j = 3/2,mj = −1/2〉|3〉 ≡ |j = 3/2,mj = 1/2〉|4〉 ≡ |j = 3/2,mj = −3/2〉|5〉 ≡ |j = 1/2,mj = 1/2〉|6〉 ≡ |j = 1/2,mj = −1/2〉 (42)

The basis (42) is related to the orbital angular momen-tum (ml = 1, 0,−1) and spin (σ =↑, ↓) eigenstates by

|1〉 = |ml = 1, ↑〉

|2〉 = 1√3|ml = −1, ↑〉+

√

2

3|ml = 0, ↓〉

|3〉 = 1√3|ml = 1, ↓〉+

√

2

3|ml = 0, ↑〉

|4〉 = |ml = −1, ↓〉

|5〉 = − 1√3|ml = 0, ↑〉+

√

2

3|ml = 1, ↓〉

|6〉 = 1√3|ml = 0, ↓〉 −

√

2

3|ml = −1, ↑〉 (43)

or

|1〉 = − 1√2

(

|X, ↑〉+ i|Y, ↑〉)

|2〉 = 1√6

(

|X, ↑〉 − i|Y, ↑〉)

+

√

2

3|Z, ↓〉

|3〉 = − 1√6

(

|X, ↑〉+ i|Y, ↑〉)

+

√

2

3|Z, ↑〉

|4〉 = 1√2

(

|X, ↓〉 − i|Y, ↓〉)

|5〉 = − 1√3

(

|X, ↓〉+ i|Y, ↓〉)

− 1√3|Z, ↑〉

|6〉 = − 1√3

(

|X, ↑〉 − i|Y, ↑〉)

+1√3|Z, ↓〉 (44)

The six band model Kohn-Luttinger Hamiltonian, HL,in the representation of vectors (42) is

HL =

Hhh −c −b 0 b√2

c√2

−c∗ Hlh 0 b − b∗√3√

2−d

−b∗ 0 Hlh −c d − b√3√2

0 b∗ −c∗ Hhh −c∗√2 b∗√

2

b∗√2

− b√3√2

d∗ −c√2 Hso 0

c∗√2 −d∗ − b∗

√3√

2b√2

0 Hso

(45)

In the matrix (45) we highlighted the four band modelHamiltonian. The Kohn-Luttinger eigenenergies aremeasured down from the top of the valence band, i.e.,they are hole energies and we use the following notation:

Hhh =h2

2m

[

(γ1 + γ2)(k2x + k2y) + (γ1 − 2γ2)k

2z

Hlh =h2

2m

[

(γ1 − γ2)(k2x + k2y) + (γ1 + 2γ2)k

2z

Hso =h2

2mγ1(k

2x + k2y + k2z) + ∆so

b =

√3h2

mγ3kz(kx − iky)

c =

√3h2

2m

[

γ2(k2x − k2y)− 2iγ3kxky

]

d = −√2h2

2mγ2[

2k2z − (k2x + k2y)]

. (46)

The four band Kohn-Luttinger Hamiltonian can be di-agonalized analytically and yields a pair of Kramers dou-blets with eigenenergies

εk =Hhh +Hlh

2∓√

1

4(Hhh −Hlh)2 + |b|2 + |c|2.

In the spherical approximation18 (γ2, γ3 → γ ≡ 0.6γ2 +0.4γ3), the top of the valence band consists of two doublydegenerate parabolic bands with effective masses mh =m/(γ1−2γ) ∼ 0.498m and ml = m/(γ1+ 2γ) ∼ 0.086m.(An approximation in which the light hole bands, whichhave a much smaller density of states, are ignored is ad-equate for some purposes.)The four band and six band representations for the

hole spin-operator components read

sx =

0 0 12√3

0 1√6

0

0 0 13

12√3

− 13√2

0

12√3

13 0 0 0 1

3√2

0 12√3

0 0 0 − 1√6

1√6

− 13√2

0 0 0 − 16

0 0 13√2

− 1√6

− 16 0

sy = i

0 0 − 12√3

0 − 1√6

0

0 0 13 − 1

2√3

− 13√2

0

12√3

− 13 0 0 0 − 1

3√2

0 12√3

0 0 0 − 1√6

1√6

13√2

0 0 0 16

0 0 13√2

1√6

− 16 0

14

sz =

12 0 0 0 0 0

0 − 16 0 0 0 −

√23

0 0 16 0 −

√23 0

0 0 0 − 12 0 0

0 0 −√23 0 − 1

6 0

0 −√23 0 0 0 1

6

(47)

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3 For a thorough recent review see Hideo Ohno, J. Magn.Magn. Mater. 200, 110 (1999).

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7 J. A. Gaj et al., Solid State Comm. 29, 435 (1979).8 B. Bastard et al., J. Physique 39, 87 (1978).9 T. Dietl, H. Ohno, F. Matsukura, J. Cibert, and D. Fer-rand, Science 287, 1019 (2000).

10 J.K. Furdyna, J. Appl. Phys. 64, R29 (1988); Semi-

magnetic Semiconductors and Diluted Magnetic Semicon-

ductors, edited by Michel Averous and Minko Balkanski(Plenum, New York, 1991); Tomasz Dietl in Handbook on

Semiconductors, Chap. 17 (Elsevier, Amsterdam, 1994).11 We assume free carrier screening makes the central cell cor-

rections required for isolated Mn acceptors in GaAs unim-portant. See A.K. Bhattacharjee and C. Benoit a la Guil-laume, Solid State Commun. 113, 17 (2000) and work citedtherein.

12 J.M. Luttinger, and W. Kohn, Phys. Rev. 97, 869 (1955).13 H. Ohno, N. Akiba, F. Matsukura, A. Shen, K. Ohtani,

and Y. Ohno, Appl. Phys. Lett. 83, 6548 (1998).14 J. Okabayashi, A. Kimura, O. Rader, T. Mizokawa, A. Fu-

jiimori, T. Hayashi, and M. Tanaka, Phys. Rev. B 58, 4211(1998).

15 F. Matsukura, H. Ohno, A. Shen, and Y. Sugawara, Phys.Rev.B 57, 2037 (1998).

16 T. Omiya, F. Matsukura, T. Dietl, Y. Ohno, T. Sakon, M.Motokawa, and H. Ohno, Physica E, to appear (2000).

17 See W. W. Chow and S. W. Koch Semiconductor-Laser

Fundamentals, (Springer-Verlag, Berlin, 1999).18 A. Baldereschi and N.O. Lipari, Phys. Rev. B8, 2697

(1973)). See also M. Altarelli and F. Bassani, Chapter 5,in Handbook on Semiconductors, ed. by T.S. Moss, vol. 1,ed. by W. Paul, North-Holland (1982), M. Altarelli in Het-

erojunctions and Semiconductor Superlattices, edited by

G.Allan, G. Bastard, N. Boccara, M. Lannoo, and M. Voos(Springer-Verlag, Berlin, 1985) and work cited therein.

19 See for example E. Merzbacher, Quantum Mechanics (Wi-ley, New York, 1970) p. 400.

20 See for example Neil W. Ashcroft and N. David MerminSolid State Physics (Saunders, Orlando, 1976).

21 See for example Assa Auerbach, Interacting Electrons and

Quantum Magnetism (Springer-Verlag, New York, 1994)p.73.

22 See for example R. Skomski and J.M.D. Coey, Permanent

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Endo, S. Katsumoto, and Y. Iye, in Proceedings of the 23rd

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Lett. 84, issue 24 (2000); cond-mat/0001320.27 T. Jungwirth and A.H. MacDonald, in preparation.28 M. Abolfath, and A.H. MacDonald, in preparation.29 R. Skomski and J.M.D. Coey Permanent Magnetism (IOP

Publishing, Bristol, 1999).

Host γ < 100 > γ < 110 > γ < 111 > γca1 γca

2

GaAs 5.965 5.088 4.639 -3.509 -4.24

InAs 13.207 10.854 9.705 -9.412 -9.84

TABLE I. High symmetry direction moment-orientationdependent average Luttinger parameters and their cubic har-monic expansions for GaAs and InAs based III1−xMnxV fer-romagnetic semiconductors. These parameters specify theT = 0 magnetic anisotropy energy in the limit of largespin-orbit splitting and large exchange coupling parameter(Jpd) or small hole density p.

15