Theory of optical conductivity for dilute Ga1−xMnxAs

Cătălin Paşcu Moca,1,2 Gergely Zaránd,1 and Mona Berciu3

1Department of Theoretical Physics, Budapest University of Technology and Economics, Budapest H-1521, Hungary2Department of Physics, University of Oradea, 410087 Oradea, Romania

3Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, Canada V6T1Z1Received 21 April 2009; revised manuscript received 2 July 2009; published 7 October 2009

We construct a semimicroscopic theory, to describe the optical conductivity of Ga1−xMnxAs in the dilutelimit, x1%. We construct an effective Hamiltonian that captures inside-impurity-band optical transitions aswell as transitions between the valence band and the impurity band. All parameters of the Hamiltonian arecomputed from microscopic variational calculations. We find a metal-insulator transition within the impurityband in the concentration range, x0.2–0.3 % for uncompensated and x1–3 % for compensated samples,in agreement with the experiments. We find an optical mass moptme, which is almost independent of theimpurity concentration except in the vicinity of the metal-insulator transition, where it reaches values as largeas mopt10 me. We also reproduce a mid-infrared peak at 200 meV, which redshifts upon doping at afixed compensation, in quantitative agreement with the experiments.

DOI: 10.1103/PhysRevB.80.165202 PACS numbers: 75.50.Pp, 78.66.Fd, 78.20.e

I. INTRODUCTION

The diluted magnetic semiconductor Ga1−xMnxAs hasemerged as one of the most promising materials due to itspotential applications in spin-based technology. This materialis, however, equally interesting from the point of view offundamental research for its unique properties and the richphysics it displays; Ga1−xMnxAs is a strongly disordered fer-romagnet, where the interplay of ferromagnetism, localiza-tion, magnetic fluctuations and the presence of strong spin-orbit coupling lead to many interesting properties such as astrong magnetoresistance,1,2 resistivity anomalies,2–5 or thepresence of a large anomalous Hall effect.6

Although Ga1−xMnxAs has been the subject of intense the-oretical and experimental investigation, even its most basicproperties are still debated. One of these fundamental andunresolved issues is the existence or nonexistence of an im-purity band in this material. There is a general consensus thatfor very small magnetic concentrations Ga1−xMnxAs is de-scribed in terms of an impurity band. For higher concentra-tions, however, there is no general consensus yet. On onehand, essentially all spectroscopic measurements such asangle-resolved photoemission spectroscopy ARPES,7 scan-ning tunneling spectroscopy,8 optical conductivity data,9–12

and ellipsometry13 seem to favor the presence of impuritystates even up to moderate concentrations x3–5 %, andeven high-quality samples with a high Curie temperature anda clearly metallic behavior have surprisingly small values ofkFl1.2,12 On the other hand, many properties of these ma-terials can be even quantitatively understood in terms of adisordered valence-band picture, and, in fact, from a theoret-ical point of view, it is hard to understand how an impurityband could survive up to concentrations as high as x5%.14

In the present paper we shall not attempt to resolve thisdiscussion, rather, we would like to focus on the optical con-ductivity. Typical optical conductivity measurements inGa1−xMnxAs display two rather remarkable features: i amid-infrared resonance peak at approximately 200 meV thatredshifts with increasing hole concentration, p more pre-

cisely, as a function of effective optical spectral weight, Neff,defined later; ii a large optical mass of the order mopt0.7–1.5me, implying a mobility which is orders of magni-tude smaller than the one of GaAs doped with similar con-centrations of nonmagnetic impurities.15

In this paper we aim at developing a semimicroscopictheory for the optical conductivity of Ga1−xMnxAs in the verydilute limit, x1%. There are several theoretical studies ofthe optical conductivity of Ga1−xMnxAs by now.16–20 Maybethe most realistic calculations have been carried out in Refs.16 and 17, however, the conclusions of these works aresomewhat conflicting. In Ref. 16 a disordered valence banddescribed in terms of the Luttinger model has been investi-gated, and the mid-infrared peak was studied. In these calcu-lations, rather surprisingly, the mid-infrared peak appeared atthe right location even in a simplified parabolic model,where a completely isotropic valence-band mass was as-sumed, and only the amplitude of the mid-infrared signalwas modified when a more complete Luttinger model calcu-lation was performed. In another, very thoroughcalculation,17 realistic tight-binding models have been used,but the results were conflicting in that features associatedwith an impurity band have or have not appeared dependingon the particular tight-binding scheme applied. In the tight-binding scheme which provided optical conductivity curvesin line with the experimentally observed mid-infrared peak, aclear impurity-band feature appeared in the density of statesfor x1%.

Unfortunately, none of these previously applied methodsis appropriate to study accurately the dilute limit, x1%.This small concentration limit is of particular interest, sincethe metal-insulator transition may take place at concentra-tions as low as x0.5%,21 and furthermore, it is in this limitwhere features associated with the impurity band should bepresent.

In the present paper, we shall follow the lines of Ref. 22and develop a semimicroscopical approach to capture thephysics of this dilute limit. At the microscopic level, wedescribe the valence-band holes as spin F=3 /2 carriers,which interact with the external electromagnetic vector po-

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tential and the Mn ions, as described by the following Hamil-tonian:

H = i

1

2m0pi − qAri2 +

i,mVri − Rm

+ i,m

Jri − RmFi · Sm. 1

Here ri and pi i=1, . . . ,Nh denote the valence hole coordi-nates and momenta, and the Mn atoms are located at randompositions, Rm i=1, . . . ,NMn. In the first term we assume anisotropic effective mass m0=0.56me. This approximation,which corresponds to setting the coefficient of the F·p2

term in the spherical approximation of Ref. 23 to zero, issomewhat uncontrolled. However, according to the results ofRef. 16, inclusion of a more realistic band structure influ-ences only slightly the structure of the optical spectrum forfrequencies 800 meV, we shall therefore ignore ithere. The second term describes the random potential createdby the Mn atoms, and it incorporates the Coulomb attractionas well as the central-cell correction.23 Finally, the last termdescribes the local exchange interaction between the manga-nese magnetic moments, Sm and the spin of hole i, Fi. Theparameters of the “microscopic” Hamiltonian Eq. 1 areall well known.24

In our work we use Hamiltonian 1 as a starting point,and derive a microscopic effective Hamiltonian from it thatnot only captures the impurity-band physics in the very di-lute limit, but also accounts for impurity-band to valence-band transitions for details see Sec. II. This effectiveHamiltonian can then be used to compute the optical conduc-tivity using field theoretical methods, as detailed in Sec. III.With this “multiscale approach,” we are then able to deter-mine the optical properties of Ga1−xMnxAs accurately in thesmall concentration regime, while keeping Nh200.

One of the key elements of our approach is the assump-tion that, for these small concentrations, the Fermi level liesinside the impurity band. This is indeed supported by trans-port measurements,21 which clearly show that the activationenergy associated with localized impurity states to valence-band transitions remains finite as one approaches the metal-insulator transition.21 Based on these activation-energy data,Woodbury and Blakemore21 concluded that the metal-insulator transition takes place within the impurity band, al-though they observed that their samples were inhomoge-neous and exhibited filamentary conduction. We remark thatsuch filamentary conduction would appear quite naturally incase the metal-insulator transition in Ga1−xMnxAs has apercolation-like structure.

If the metal-insulator transition indeed takes place withinthe impurity band then one must be able to capture the prop-erties of small concentration metallic Ga1−xMnxAs samplesusing our impurity-band-based method. Indeed, our resultsalso support this picture; a detailed analysis of the hole statesreveals a metal-insulator transition at about x0.2–0.3 %,where delocalized states appear in the middle of the impurityband. Details of the computation are described in later sec-tions. As shown in Fig. 1, this is approximately the criticalconcentration for uncompensated samples. The Anderson

transition thus takes place inside the impurity band, in agree-ment with the conclusions of Ref. 22, and also in good agree-ment with activation energy values observed for samplesgrown at high temperature “equilibrium conditions”.21,25

Typically, Ga1−xMnxAs samples grown under nonequilibriumconditions are compensated even after annealing: in additionto the substitutional Mn ions of concentration xS there isalso a finite concentration xI of interstitial Mn ions, whichbehave as double donors, and are also believed to make afraction of the Mn ions inactive by simply binding to them.26

As a result, the effective concentration of “active” Mn ions isreduced to xxeffxS−xI and the concentration of holes isalso suppressed compared to xeff by the hole fraction fxS−2xI / xS−xI. Remarkably, even if only 20% of theMn ions goes to interstitial positions, that reduces the effec-tive concentration to 60% percent of the total Mn concentra-tion and amounts in a hole fraction f 0.66. Unannealedsamples tend to have even larger interstitial Mn concentra-tions and thus much smaller hole fractions.24,26,27 As a result,depending on the precise annealing protocol, ferromagneticGa1−xMnxAs samples grown under nonequilibrium condi-tions tend to show the phase transition at higher concentra-tions. According to our calculations, for xI /xtotal=30% f=0.3 the metal-insulator transition takes place at about xS3.2%, while for xI /xtotal=23% f =0.55 the transition oc-curs at xS0.7%. These values are only indicative but con-sistent with the experimental data.21,28–31 In the rest of thepaper, we shall consider only active substitutional Mn ions,and the concentration x shall refer to these. The effect ofinterstitional Mn ions only appears through the reduction inthe hole fraction parameter, f 1.

We also find that the optical mass increases to very largevalues mopt10me at the critical concentration, where the dcconductance vanishes. However, apart from this, the opticalspectrum changes rather continuously. For larger Mn concen-trations, x2–3 %, we obtain an optical mass moptme,

FIG. 1. Color online Density of states for x=0.2% and 0.3%.Both the impurity-band and valence-band contributions are shown.At these concentrations the impurity band is clearly separated fromthe valence band. The gray area indicates the region of localizedstates, while delocalized states are shown as white regions. Forthese calculations we used a hole fraction f =0.5. The arrow indi-cates the position of the mobility edge corresponding to f = fc=0.8.

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which is almost independent of the hole concentration p, andis in surprisingly good agreement with the experimentalvalues.11

Our approach is designed to work in the limit of smallconcentrations, and it should break down for large concen-trations. Where exactly this breakdown takes place, is notquite clear. Our calculations show that the impurity band isnot completely merged with the valence band even for con-centrations as large as x3%.32 The impurity-band picturemay make sense for even larger concentrations if the Fermilevel is inside the gap as indicated by ARPESmeasurements,7 where states are expected to exhibit a stron-ger impurity state character. We are not able to convincinglydetermine the upper limit of our approach. However, ifwe blindly extend it to the regime x3%, without justifyingtheir applicability, rather surprisingly, our calculations quali-tatively as well as quantitatively reproduce all important fea-tures of the experiments.

One of the typical optical conductivity results is displayedin Fig. 2, where, for comparison, we also show some typicalexperimental data.11 In our calculations the Drude contribu-tion originates entirely from carriers residing within the im-purity band, and the “Drude peak” appears as a plateau atsmaller frequencies, just as in the optical conductivity ex-periments. The mid-infrared peak, on the other hand, is dueto transitions from the impurity to the valence band. Noticethat the position as well as the overall size of the signal arequantitatively reproduced. This latter may just be a coinci-dence, since we assumed a simplified band structure in thiswork.

Having the Fermi level within the impurity band naturallyexplains the redshift of this resonance with increasing holeconcentration. In Fig. 3 we show the computed peak positionas a function of the effective optical spectral weight, Neffdefined later through Eq. 28 for a variety of concentra-tions x and hole fractions, f =Nh /NMn. Our calculations showsurprising agreement with the experimental results ofRef. 11.

The paper is organized as follows. First, in Sec. II weoutline the calculations that lead from the microscopicHamiltonian Eq. 1 to the effective Hamiltonians used.

Some of the details of these rather technical variational cal-culations are given in Appendices A and B. Section III pro-vides the basic expressions for the optical conductivitywithin a mean-field approach, while the results of our nu-merical calculations are given in Sec. IV. In Sec. V we showhow to incorporate the effects of magnetic fluctuations. Fi-nally, we conclude in Sec. VI.

II. THEORETICAL FRAMEWORK

With current day computer technology, it is essentiallyimpossible to treat the Hamiltonian of Eq. 1 accurately forsmall Mn concentrations, x1%. We therefore describe thisregime in terms of an effective Hamiltonian, which we de-rive from Eq. 1, and which consists of an impurity-bandpart and a valence-band part. The first part of the presentsection focuses on this mapping. However, to compute theoptical conductivity, we also need to determine how the elec-tromagnetic field in Eq. 1 couples to impurity states withinthe effective Hamiltonian. This is discussed in the Sec. II B.In the main body of the text we only outline and summarizethe most important steps. Details on these simple but ratherlengthy calculations are given in Appendices A and B.

A. Effective Hamiltonian

1. Impurity-band Hamiltonian

Since for very small concentrations the Fermi energy is inthe impurity band, we first need to describe the impuritystates. In the small concentration limit, hole wave functionsare localized at the Mn sites, providing a strong and attrac-tive potential for them. Therefore the impurity band can bedescribed using a tight-binding Hamiltonian of the form,22,33

Himp = − i,j,

tijci† cj +

i,Eici

† ci + G i,,

Si · ci† Fci.

2

Here ci† creates a bound hole at Mn position Ri in a spin

state F=3 /2,Fz=, and tij denotes the hopping betweenMn sites Ri and R j. Since the hole mass is isotropic in our

101 102 103 104

Frequency [cm -1]

0

50

100

150

σ[Ω

−1cm

-1]

101 102 1030

50

100

150

1% Mnf= 0.5

Experimental Data

Theory

2.8 % Mn

FIG. 2. Comparison between our theory and experimental data.The optical conductivity data presented in the inset were extractedfrom Ref. 11.

0.0 5.0×1020

1.0×1021

Neff

[cm-3

/me]

1000

1500

2000

2500

3000

Peak

posi

tion

[cm

-1] Experimental data

f = 0.3f = 0.4f = 0.5f = 0.7f = 0.9

FIG. 3. Color online Mid-infrared peak position extractedfrom the optical conductivity curves vs the effective optical spectralweight, Neff defined through Eq. 28. Open symbols representtheoretical results while the solid dots are the experimental dataextracted from Ref. 11. Calculations for different hole fractions fand concentrations x x=1–5 % fall onto a single curve.

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microscopic Hamiltonian, the hopping is independent of thehole spin .34 The on-site energy Ei contains the bindingenergy of the hole, E0=−110 meV, but it also accounts forthe Coulomb and kinetic-energy shifts generated by neigh-boring Mn ions. Finally, the last term describes the localantiferromagnetic exchange with the core Mn spin Si.

The parameters appearing in Eq. 2 can be determinedfrom experiments and from the microscopic Hamiltonian,Eq. 1. The coupling G is known from electron paramag-netic resonance experiments to be G5 meV for a singleMn impurity.35,36 However, the hopping parameters and theon-site energy, Ei need to be computed from Eq. 1. Similarto Ref. 22, we determined them from a variational calcula-tion of the molecular orbitals for an Mn2 “molecule,”described by the Hamiltonian

H2site = −

22 −

1

r1−

1

r2+ Vccr1 + Vccr2 , 3

where we used atomic units, r1,2= r−R1,2, =12.65 is thedielectric constant of GaAs, and =me /m0=1.782. The so-called central-cell correction Vcc accounts for the local inter-action at the Mn core, and is given by23

Vr = Vccr = − V0e−r/r0. 4

The parameters V0 and r0 must be chosen to reproduce theexperimentally observed impurity state at E0110 meV. Inour calculations we have used V0=1.6 eV and r0=2 Å, butour results do not depend on this particular choice as long asr0 is sufficiently small. Details of this calculation are pre-sented in Appendix B. In these calculations we neglectedthe coupling G, since it is much smaller than the bindingenergy of the holes. The final result is that for G=0 thelow-lying spectrum of two Mn ions at a separation R can bedescribed by the following simple effective Hamiltonian:

H = − tR

c1† c2 + H.c. + ER

i=1,2ci

† ci. 5

The parameters tR and ERER−E0 are shown inFig. 4. As the inset of the lower panel of Fig. 4 shows, muchof the energy shift ER originates from a simple long-ranged Coulomb shift due to the Coulomb potential of theneighboring Mn site, ECoulombR=−1 /R. The remaining

kinetic shift, ERER+1 /R is relatively large forsmall separations, but vanishes exponentially for large valuesof R.

Having determined the spectrum and the effective Hamil-tonian of the two-Mn ion complex, we can now express theparameters of the effective Hamiltonian, Eq. 2. The hop-ping parameter between sites i and j depends just on theirseparation, Rij Ri−R j, and is simply tij = tRij, the hop-ping obtained from the two-Mn problem, while the on-siteenergies in Eq. 2 are given as

Ei = E0 + Ei,

Ei = ji

ERij . 6

The sum in this equation should be evaluated carefully: aswe discussed above, for large separations, the shifts ERi−R j are dominated by the long-range Coulomb contri-butions, and therefore the sum in Eq. 6 formally diverges.This Coulomb potential is, however, screened by the boundvalence holes and other charged impurities in the system,with a screening length of the order of the typical Mn-Mndistance. Therefore we shall compute the sum in Eq. 6 withan exponential cutoff, ERij→ERije−Rij/RSC, RSC=2rMn,with 4rMn

3 /3=nMn the Mn concentration.Also, our calculation for the hopping matrix elements

makes sense only for nearest-neighbor sites. Correspond-ingly, in the hopping part of the Hamiltonian we introduce arigid cutoff, R0=2rMn, to keep only hopping to the first“shell” of atoms. Our results do not depend too much onthese cut-off parameters; changing the Coulomb energy cut-off results in an overall shift of the impurity and valence-band energies, and does not influence the optical spectrum.Similarly, the results are almost independent of R0 as long asit is in the range of R0=2rMn.

2. Valence band

In the tight-binding approach of the Sec. II A 1 only thebound hole states appear. However, to describe optical tran-sitions, we also need to account for extended states within

R [ ]Å

t(R

)[e

V]

0.0 10.0 20.0 30.0 40.0 50.00.0

0.1

0.2

0.3

0.4

single parameter calculation

full variational calculation, s-waves

full variational calculation, s and p-waves

R [ ]Å

∆Ε[e

V]

0.0 10.0 20.0 30.0 40.0-0.10

-0.05

0.00

0.05

0.10

0.0 10.0 20.0 30.0 40.0

R [Å]

0.00

0.05

0.10

0.15

0.20

∆E[e

V]

single parameter calculation

full variational calculation, s-waves

full variational calculation, s and p-waves

~

(b)

(a)

FIG. 4. Color online Top: variationally computed hopping pa-rameter t as function of the relative distance between the Mn sites,R. Bottom: the energy shift ER. The inset shows the correction

ER to the classical Coulomb shift.

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the valence band. These states are involved in local opticaltransitions, where a hole localized at site m absorbs a rela-tively high-energy photon E200 meV to make a transi-tion relatively deep into the valence band.

To account quantitatively for these transitions, we makethe following crucial observations. i The transitions involvehole states of relatively high energy, and correspondingly,states of a relatively short wavelength. ii Since the initialhole states are localized at the Mn site, the created valence-band hole will also be localized close to it. There the majoreffect of neighboring Mn sites is to create a Coulomb poten-tial, which is smooth at the scale of the wavelength of thecreated hole. As a result, we can describe the final hole statein terms of the following simplified “semiclassical” Hamil-tonian:

Hvali −

22 −

1

ri+ Vccri + Ei,val, 7

where ri= r−Ri is the electron coordinate measured fromimpurity i, and the shift Ei,val is given by

Ei,val = − j

1

Rij. 8

Notice that Ei,valEi since most of the shift Ei comesfrom the Coulomb shift in Eq. 6 see Fig. 4. However,these two energies are not exactly the same, Ei,valEi,

since Ei also contains the kinetic shift Ei, shown in theinset in Fig. 4. It is precisely this kinetic shift, which isresponsible for the redshift of the mid-infrared optical peakwith increasing Mn concentrations, x.

Furthermore, we observe that local optical transitions onlyinvolve scattering states in the l=1 p angular momentumchannel. These states vanish at the origin, and do not reallyfeel the central-cell correction. Therefore, to describe the lo-cal optical transitions, it is reasonable to include locally thesel=1 states only, and describe the valence band in a second-quantized formalism as

Hvalband =

m,i,

0

dk

k2

2m0+ ERi akm,i

† akm,i, 9

where akm,i† creates a scattering state in the p channel around

impurity site i in angular momentum channel m=x ,y ,z, withband index Fz= =−3 /2, . . . ,3 /2, and radial momentumk. The operators akm,i

† are normalized to satisfy the anticom-mutation relation

akm,i† akm,j = mmijk − k . 10

As mentioned above, scattering states in the p channel do notfeel the central-cell correction, therefore the akm,i

† ’s create toa very good approximation Coulomb scattering states. Noticethat in Eq. 9 an independent local band is associated witheach impurity site, i.e., propagation between various impu-rity sites within the valence band is ignored. This is a rea-sonable approximation for the high-energy, fast processesconsidered in this work.

B. Coupling to the electromagnetic field

At the microscopic level, the vector potential couples di-rectly to the momenta of the valence holes. However, withinthe impurity band, these valence holes reside on specific or-bitals localized at the Mn sites, and therefore, as a first step,we need to determine how the vector potential A couples tothese impurity-band states. To this end, we determined thematrix elements of the momentum operator p between thelowest lying states of our Mn2 molecule see Appendix B.We find that the effective coupling of a homogeneous vectorpotential to the impurity band is given by

Hextimp =

i,j,pRij

e

m0cA · nijici

† cj + H.c. , 11

where the unit vector nij = Ri−R j /Rij specifies the directionof the bond, and pR is the momentum matrix element ex-tracted from the variational calculation. This matrix elementis plotted in Fig. 5. As we also verified, for R8 Å, thistediously obtained matrix element is quite well approximatedby the simple Peierls substitution,

pR 1

iR tRm0/ . 12

The term 11 describes intraband transitions between statesinside the impurity band. It is this term that generates theDrude peak and which is responsible for the dc conductance.

However, we also need to account for interband transi-tions, i.e., transitions from the impurity band to extendedstates in the valence band. In our approach, these transitionsare local, and are described by the Hamiltonian,

Hexttr =

i,,m

0

dk

pk

e

m0cAmiakm,i

† ci + H.c. . 13

Here Am denote the m=x ,y ,z components of the vector po-tential A, and the on-site optical matrix elements pk canbe computed from the variational wave function of a singleMn site. We determined pk for Coulomb scattering statesas well as for free-electron states details of this calculationare presented in Appendix A. The final results are shown inFig. 6.

0.0 10.0 20.0 30.0 40.0 50.0R [Å]

0.0

0.1

0.1

|p(R

)|[Å

-1]

Full variational calculation, s-waves

Single parameter calculation

Single parameter calculation, Peierls substitution

FIG. 5. Color online Momentum matrix element in units =1 as a function of the separation R between the two Mn sites. ForR8 Å the matrix elements are very well approximated by thePeierls substitution dashed line.

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For free-electron scattering states the single-parametermatrix element takes on a particularly simple form,

kp =25/2

3

5/2k2

2 + k22 , 14

where =0.091 Å−1. The momentum matrix elements van-ish for k→0 since p states have a node at the origin, but theyalso vanish for very large momenta, where the valence holewave functions oscillate much faster than the characteristicscale of the bound state. This results in a maximum opticaltransition rate for valence states at about 100 meV below thevalence-band edge. It is this momentum dependence of theoptical matrix element that is ultimately responsible for themid-infrared peak in the optical spectrum at frequencies200 meV. The Coulomb and free-electron matrix ele-ments behave qualitatively the same way and are in bothcases well approximated by the expression obtained for asingle hydrogenlike variational wave function, . How-ever, the more realistic Coulomb matrix elements have asomewhat smaller amplitude and the maximum transitionrate occurs at slightly higher energies. In the rest of thispaper we shall use these more realistic Coulomb matrix ele-ments to compute the optical conductivity.

III. OPTICAL CONDUCTIVITY AT T=0: MEAN-FIELDAPPROXIMATION

In the rest of the paper we shall focus on the calculationof the optical properties of Ga1−xMnxAs at low temperatures.First, we shall discuss the case of T=0 temperature and treatthe Mn spins within the mean-field approximation, justifiedby the relatively large value of the Mn spins, SMn=5 /2. Laterin Sec. V we shall discuss how one can go beyond this ap-proximation and compute spin-wave corrections. These cor-rections, however, turn out to be small, and the mean-fielddescription turns out to be quite accurate. This is related tothe fact that, in the concentration range studied here, thecoupling G is much smaller than the binding energy of theholes GE0 as well as the width of the impurity band, andtherefore it hardly influences the optical spectrum at theseenergies.

A. Mean-field Hamiltonian

In our model Hamiltonian, Mn spins couple directly onlyto states in the impurity band. Therefore, as a first step, weonly need to treat the coupled Mn spin-impurity-band sys-tem, described by Eq. 2. At the mean-field level, the inter-action part is rewritten as

GSici†Fci = GSici

†Fci + Sici†Fci − Sici

†Fci + Hflucti ,

15

Hflucti = GSi − Sici†Fci − ci

†Fci , 16

and the fluctuating part, Hfluct is neglected. With this approxi-mation, Eq. 2 becomes exactly solvable at any temperature.For given values of Si, the hole part of the Hamiltonian isdiagonalized by the unitary transformation

a† =

i,ici

† , ci† =

ian

†, 17

where wave functions ni can be found by solving a rela-

tively simple eigenvalue equation. The mean-field Hamil-tonian can then be rewritten in terms of the operators a

† andthe corresponding eigenenergies, E, as

HimpMF =

Ea† a −

i

Sihi + i

Sihi, 18

where the local fields hi=−Gci†Fci can be expressed in

terms of the new basis as

hi = − G

Fi fE , 19

Fi

iFi , 20

with f the Fermi function. Trivially, hi determines the expec-tation value Si, that enters the eigenvalue equation ofi. This field must therefore be determined self-consistently. At T=0 temperature the self-consistency equa-tions become rather simple, since then the spins are com-pletely polarized in the direction hi, Si=SMnhi / hi, and theFermi function simply becomes a step function. However,for small concentrations and/or finite temperature the Mnspins are not fully polarized, and the numerical solution ofthe mean-field equations becomes time consuming.

B. Optical conductivity

The mean-field solution of the Hamiltonian provides usthe equilibrium state of the coupled spin-impurity-band sys-tem. To compute the optical spectrum, however, we alsoneed to transform the coupling to the electromagnetic field tothe “canonical” basis, Eq. 17.

We can express the coupling of the impurity band to atime-dependent vector potential At in terms of the canoni-cal operators, a as Hext

imp=,A ·a† ja, where the current

operator’s matrix element is defined as

0.0 1.0 2.0 3.0-1

k [Å ]

0.00

0.10

0.20

0.30

0.40

0.50|p

(k)|

[Å-1

/2]

Full calculation, free scattering states

Single parameter calculation, free scattering states

Full calculation, Coulomb scattering states

FIG. 6. Color online Wave-vector dependence of the on-siteoptical matrix element computed for free and Coulomb scatteringstates =1. We also compare this with the result obtained for asimple Hydrogenlike variational wave function and free scatteringstates.

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j = i,j,

pRije

mc· niji

ij − iji .

21

Similarly, we rewrite the term 13 in the Hamiltonian in thiscanonical basis as

Hexttr =

k,m,,i,AmtJk,i,akm,i

† a + H.c. , 22

with Jk,i,= pk emc ii.

The optical conductivity can be computed by meansof the Kubo formula, which expresses it in terms of the re-tarded current-current correlation function,

= −1

Im JJ

ret . 23

Since, corresponding to Eqs. 21 and 22, our current op-erator consists of an interband and an intraband part, thecurrent-current correlation function can also be expressed interms of two contributions: an intraband contribution, intra,corresponding to transitions within the impurity band, and aninterband contribution, inter, describing transitions from theimpurity band to the valence band. In our approach the low-frequency behavior arises from impurity-band scatteringalone, while the valence band plays practically no role there.The high-frequency conductivity, on the other hand, is typi-cally dominated by interband transitions to the valence band.In the present, impurity-band approach, these transitions giverise to a mid-infrared peak close to 2000 cm−1.

The previously mentioned two contributions can be easilycomputed within the mean-field approach, using the dia-grammatic formalism presented in Sec. V, and the interbandcontribution can be simply expressed as

inter = −1

k,

i,

Jk,i,2 Im fE − fki − ki + E + i

.

24

Notice that the transition being local, it is also the local en-ergy of a valence-band hole, that appears in the denominatorof this expression, ki= k2

2m0+Ei.

The intraband contribution can be written in a similar wayas

intra = −1

,

je02 Im fE − fE + E − E + i

, 25

with e0 the polarization of the external light.Clearly, both the energy and the precise structure i.e.,

localized or extended character of the impurity-band statesenter the matrix elements in the above expressions. However,apart from the value of the dc conductance, which clearlymust vanish as one approaches the localization transition, thegross high-frequency features will turn out to not be verysensitive to the localization transition itself. This is not verysurprising, since the localization transition involves mostlystates at the Fermi energy, while optical transitions are domi-nated by states deep below or high above the Fermi energy.

IV. MEAN-FIELD RESULTS

A. Density of states

Let us start first by discussing some details of the numeri-cal solution of the mean-field equations described in the pre-vious section, and the structure of the hole states obtained.To start the numerical calculation, we first generate an initialdistribution for the substitutional Mn impurities on an fcclattice with lattice constant a05.6 Å. Nearest-neighbor Mnsites are, however, not favored during the growth process,since Mn ions act as charged impurities. To simulate thiseffect, we let the Mn ions relax through a simple classicalMonte Carlo diffusion process while assuming a screenedCoulomb interactions between them.22 Typical values of thisMonte Carlo time used are in the range of tMC1 step/atom.In all our calculations we use periodic boundary conditionsto suppress surface effects.

Once the configuration of the ions is determined, we canconstruct the effective Hamiltonians Eqs. 2 and 9 as de-scribed in Sec. II A. Having constructed the Hamiltonian, wethen solve the mean-field equations iteratively and determinethe equilibrium spin configuration. Here we only focus onT=0 temperature, where Si=Si, with i acting simply asa classical variable. In the end of the mean-field self-consistency loop the states in the impurity and valence bandare fully characterized, i.e., the corresponding eigen energiesE and ki as well as the wave functions are available. Wecan then compute the necessary matrix elements of the cur-rent operator using Eq. 11, and apply Eqs. 23 and 38immediately to compute the contributions to the opticalconductivity.

For a good accuracy, we consider system sizes L as largeas possible, and we average over many configurations. Usu-ally for a given Mn concentration and hole fraction, we av-erage over 100 different impurity configurations, and weconsider systems with the number of impurities in the rangeof 200.

In Fig. 7 we present results for the impurity-band densityof states DOS for different Mn concentrations but for afixed hole fraction, f =0.5. We also indicate the average DOSfor the valence band. Shaded regions, denoting localizedstates, were determined by analyzing the finite-size scalingof the participation ratio PR

PR = i

i2 −1. 26

For extended states PR scales with the system size, while forthe localized states remains finite for L→ . Figure 8 showsthe details of such a finite-size scaling analysis.

As we expect, increasing the Mn concentration, the impu-rity band gets broader and starts to overlap with the valenceband for Mn concentration as low as 0.5% see also Fig. 7.However, at the same time increasing Coulomb disordershifts the tail of the impurity band deeper inside the gap thetop of the valence band is fixed at EV

top=0. States inside thistail are strongly localized and have an overwhelming local-ized impurity state character.

We clearly observe a metal-insulator transition MIT at acritical concentration x0.2%, which corresponds to a hole

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concentration p4.51019 cm−3 for f 1. In our approachthis metal-insulator transition is simply an Anderson local-ization transition, since electron-electron interaction effectsare neglected: for x0.2% all the states in the impurity bandare localized by disorder and the system is an insulator, whilefor x0.2% the metallicity of the system still depends on theposition of the Fermi level inside the impurity band. As anexample see Fig. 8, right panel for x=0.3% and hole frac-tions less than f =0.8 the system is still an insulator, while for

a hole fraction f 1.0, the Fermi level is already inside thedelocalized region and we find a metallic state. Increasingthe Mn concentration to larger values, the extended regiongrows and the system has a metallic behavior for typicalvalues of the hole fraction, f 0.5–1, see also Fig. 7. Forhole fractions f 0.6, e.g., we find that the ground state ismetallic for x0.5%. These numbers are in good agreementwith experimental results, where the MIT has been reportedto occur in a concentration range x0.2–0.3 %.14,25

At this point we must make two remarks. First, we com-pletely neglected electron-electron interactions. Electron-electron interactions can obviously modify many of thephysical properties close to the MIT and lead to the appear-ance of a Coulomb gap as well as Altshuler-Aronov-typeanomalies.37 On the other hand, even annealed Ga1−xMnxAssamples are typically compensated, and the electron-electroninteractions have been found to be less relevant for suchsystems. Also, we can argue that our results are in a wayself-consistent, and probably not very sensitive to Coulombinteraction; for typical concentrations x3% and hole frac-tions in the range f 0.3–1, we find that approximately60–80 % of the sites have less then one electrons on them,and less then 5% of sites are more then doubly occupied.Furthermore, even for the most localized orbitals, the partici-pation ratio is rather large 6–10. Therefore, a typical site isoccupied by a single electron and the Coulomb correlationsare not too relevant.22 Treating the Coulomb interaction ap-propriately is, however, a real theoretical challenge even formuch simpler model systems, and is beyond the scope of thepresent work.

Second, we also need to discuss the issue of conservingthe total number of states within our approach. In reality,impurity-band states are formed from valence-band statesand therefore, formation of the impurity band also depletesthe valence band. In contrast, in our scheme, the total num-ber of states is not conserved by construction, and the mixingof valence-band states and impurity-band states is ignored.However, a careful spectral analysis shows that the boundacceptor states are formed from high-energy valence-bandstates: about 90% of the bound state is composed fromvalence-band states of energy 150 meV below the

FIG. 7. Color online Ground-state density of states for differ-ent Mn concentrations and a fixed hole fraction, f =0.5. The solidlines represent the DOS for the impurity band while dashed linesdenote the valence-band DOS. The Monte Carlo relaxation time isfixed to tMC=1 per Mn ion. Shaded areas indicate localized states,while unshaded regions correspond to extended states. For theseconcentrations we find two mobility edges in the impurity band: oneseparating states in the tail of the impurity band, while the otherseparating localized states in the impurity-band—valence-band gap.

(b)(a)

FIG. 8. Color online Ground-state density of states and participation ratios for different Mn concentrations but a fixed hole fraction,f =0.5. The Monte Carlo relaxation time is fixed to tMC=1. The shaded regions indicate the position of the mobility edge for eachconcentration.

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valence-band edge. As a result, in the concentration rangeconsidered here depletion of the valence-band states byforming the impurity band is not essential. From this spectralanalysis of the acceptor state we estimate that for x1–2 % the valence-band density of states at 100 meV and thus the optical signal at 200 meVshould be reduced only by about 10–20 % at maximum.Possible matrix element effects and incorporation of a morerealistic band structure probably have a more serious effecton the overall amplitude of the signal.

Let us close this section by presenting a rather curiousquantity that we termed “shifted local density of states”shifted LDOS, where at every site we computed the localdensity of states by subtracting the binding energy of a hole,E0, as well as the value of the average Coulomb shift Fig.9. While the unshifted LDOS is rather featureless, this quan-tity displays a sharp peak around zero energy. A more de-tailed analysis reveals that this sharp peak is associated withlocalized states having a small participation ratio, i.e., statesin the tail of the impurity band. To show this, we also plottedthe contribution of states with PRs smaller than a given valueto the shifted DOS. Clearly, the peak observed at zero energyis entirely due to localized states deep inside the gap. Thephysical interpretation of this peak is thus plausible. States inthe tail of the impurity band are generated by random andlarge fluctuations of the Coulomb disorder. These states aresimply localized impurity states deep inside the gap that arebeing mostly occupied, and they give rise to local opticaltransitions which have a strong impurity state transitioncharacter.

B. Optical conductivity

Having diagonalized the mean-field Hamiltonian, it isstraightforward to compute the optical conductivity. As weshall see, our theory, which incorporates interimpurity-bandtransitions as well as intraband transitions, accounts qualita-tively as well as quantitatively for all major features of theexperimentally observed optical spectrum. We shall also

present a scaling analysis of the optical conductivity close tothe metal-insulator transition and compute the dynamicalcritical exponent.

The “Drude” contribution, i.e., the →0 feature is con-trolled by excitations close to the Fermi level. Since inthis small concentration limit the Fermi level resides in theimpurity band, the Drude peak and also the dc conductivityare expected to be dominated by the intraimpurity-bandcontribution. The contribution of extended valence-bandstates is negligible for E0, since these states are too farfrom the Fermi level to be of any relevance for the Drudecontribution.38

The interband contribution, on the other hand, is gener-ated by transitions between the impurity and valence bands.As can be seen in Fig. 1, the two bands start to overlap forMn concentrations x0.5%. However, although it also de-pends on the hole fraction f , the distance between the Fermilevel and the top of the valence band is always in the range=0.1 eV. We therefore expect to see a broad mid-infraredfeature in the energy range 2.

These expectations are indeed met by our numerical re-sults shown in Fig. 10. In panel d we also display theintraband and interband contributions separately for a typicalGa1−xMnxAs sample in the metallic regime, with x=1% anda hole fraction f =0.5. The intraband contribution has aDrude-type behavior, while the interband signal presents apeak at the energy 2500 cm−1. However, the Drude contri-bution is rather broad, and in fact, already for this relativelysmall concentration, it almost completely merges with theinterband contribution. The overall behavior is in goodagreement with the experimental data presented in Refs.

-0.2 -0.1 0 0.1 0.2

0

10

20

30

40

50

0

10

20

30

40

-0.2 -0.1 0 0.1 0.2

0

10

20

30

40

-0.2 -0.1 0 0.1 0.2

0

10

20

30

40

0

10

20

30

-0.2 -0.1 0 0.1 0.2

0

10

20

30

All statesPR < 5PR < 10PR < 20

0.2 % Mn

0.1 % Mn

0.3 % Mn

0.5 % Mn

1 % Mn

2 % Mn

Energy (eV)

Shif

ted

LD

OS

(eV

-1)

FIG. 9. Color online Shifted local density of states for differ-ent Mn concentrations. The Monte Carlo relaxation time is fixed totMC=1.

102

103

104

0.1

1

10

f = 0.3f = 0.5f = 0.9

102

103

104

1

10

100

1000

f = 0.3f = 0.5f = 0.9

102

103

104

0.01

0.1

1

10

Total contributionIntraband contributionInterband contribution

102

103

104

1

10

100

1000

Total contributionIntraband contributionInterband contribution

0.03 % Mn 1 % Mn

σ(ω

)(Ω

−1cm

-1)

Frequency (cm-1

)

f = 0.50.03 % Mn

1 % Mnf = 0.5

a) b)

c) d)

FIG. 10. Color online The real part of the optical conductivityfor different Mn concentrations and hole fractions. In a and c theMn concentrations is 0.03%, while in b and d it is x=1%. Inc and d the intraband and interband contributions are presentedfor a hole fraction f =0.5. The sample in panel a is insulatingwhile the one in panel b is metallic. The peak observed at2000–2500 cm−1 for the metallic sample is due to the valence toimpurity-band transitions while the broad Drude feature at smallerenergies is generated by the intraimpurity-band contributions.

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9–12. Remarkably, not only the peak position but also theoverall magnitude work out to have values in the experimen-tal range.

As already shown in Fig. 3, increasing the number ofcarriers leads to a redshift of the mid-infrared peak in agree-ment with the experimental observations; more metallicsamples tend to have a strongly redshifted mid-infrared peakat a frequency 1500 cm−1 rather than at 2500 cm−1. Wefind that in the concentration range considered here, to agood approximation, the redshift only depends on the effec-tive optical spectral weight, Neff, defined through Eq. 28.Surprisingly, there is an almost perfect match between ourtheoretical results and the experimental values for intermedi-ate concentrations. The decreasing trend observed in the po-sition of the mid-infrared peak as function of hole fractioncan be understood simply from Fig. 3: for a given Mn con-centration, increasing the carrier numbers shifts the Fermilevel closer to the top of the valence band. As a consequence,the optical gap decreases and the mid-infrared peak movestoward lower energies.

A first look at the results presented in panels c and d ofFig. 10 can drive us to the conclusion that there is not muchdifference between the overall optical response of the metal-lic and insulating samples. However, the effective-massanalysis and spectral weight analysis presented in Sec. IV Cclearly shows the difference between these phases.

However, before turning to the effective-mass analysis, letus discuss the low-frequency properties of the optical re-sponse close to the MIT transition. The scaling theory ofAbrahams et al.39 has been extended to the dynamical con-ductivity by Shapiro et al.,40 who showed that at the mobilityedge the ac conductivity of a d-dimensional system obeys apower law, d−2/d. They also found that on the me-tallic side the conductivity goes as −0d−2/2

while in the insulating phase 2.To our knowledge, the first numerical evaluation of the

optical conductivity close to the localization transition basedon the Kubo formula was done in Ref. 41, where indeed itwas shown that the dynamical exponent for the optical con-ductivity of a three-dimensional Anderson model at the criti-cal point is 1/3. More recently, the analysis was extended tounitary and symplectic systems42 where the same exponentwas found. In the lower panel in Fig. 11 we present thelow-frequency limit of the intraband optical conductivityclose to the MIT transition for a fixed hole fraction f =0.8,where the critical carrier concentration is found to be pc41019 cm−3 x=0.2 by a participation ratio analysis.For p pC the low-energy optical data can be nicely de-scribed by a power law, =0+A. Since we use pe-riodic boundary conditions, we observe a continuous metal-insulator transition similar to the case of the Andersonmodel.43 For an infinite system, the dc conductivity 0approaches zero at the transition point and vanishes in theinsulating phase. Of course, for a system of a finite size suchas ours, 0 is not strictly zero even in the insulating phase,and only a crossover is observed between these two regimessee the upper left panel of Fig. 11.

The most relevant quantity is the exponent . In the me-tallic state a best fit gives 0.5 in good agreement with theanalytical predictions. Although the error bars are rather

large, seems to slightly decrease down to 0.35 at the MITwhere it starts to increase again in the insulating regime. Thetwo approaches, the participation ratio and the scaling analy-sis of the dynamical conductivity based on which we haveidentified the MIT point are in good agreement. In both casesthe critical point is associated with approximately the samehole carrier concentration.

C. Effective-mass analysis

Experimentally, optical conductivity data are often used toextract important information on the charge carriers such astheir concentration or effective mass, or the value of kFl.From the analysis of the experimental data, two types ofeffective masses can be extracted: i the effective mass m

that enters the Drude formula for the resistivity, i.e., the dclimit of the optical conductivity and ii the optical mass moptthat is related to the spectral sum rule. These two masses areoften quite different. Also, they are typically extracted basedon some assumptions isotropical mass and the quadraticHamiltonian, etc., which are violated in the real system.Nevertheless, they both contain important information, and itis therefore interesting to determine them from our theoreti-cal results and compare to the experimental data.

The effective mass m can only be roughly estimated fromthe data. Following the procedure of Ref. 11, we typicallyfind it to be in the rage of m10–30me. The mass m isalso related to the metallicity parameter, kFl, with kF theFermi momentum and l the mean free path. In fact, this

1018

1019

1020

1021

0

50

100

150

σ(0

)(Ω

−1cm

-1)

1018

1019

1020

10210

0.2

0.4

0.6

0.8

1

Cri

tical

expo

nent

α

0.001 0.01 0.1 1 10 0.001 0.01 0.1 1 10

p (cm-1

)

σ (ω) = σ(0) + Α ωα

a) b)

Mn concentration ( %)

MIT ?

MIT ?

0 10 20 30 40 50

Frequency1/2

[cm-1/2

]

0

0.5

1

1.5

2

σ intr

a(ω)/

σ intr

a(ω=

100

cm-1

)

0.3 %0.5 %1 %2 %

f = 0.8

(a) (b)

FIG. 11. Color online Upper panels: a the dc limit of theintraband optical conductivity for concentrations between 0.02%and 1%. b The scaling exponent for the conductivity in thelow-energy limit as function of the carrier concentration. The MITtransition is indicated by a drop of the critical coefficient down to0.35. Bottom panel: frequency dependence of the intraband opticalconductivity in the small frequency limit presented for different Mnconcentrations. In all three figures the hole fraction was fixed to f=0.8.

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dimensionless parameter can be extracted much more reli-ably from the experimental data than m, and it has a muchmore obvious interpretation. Small values of kFl correspondto samples in the insulating phase or close to the metal-insulator transition, while good metals are characterized bylarge values of kFl. We shall therefore focus on this param-eter.

The value of kFl can be estimated from the hole concen-tration and the dc conductivity using the Drude formula, re-written as

kFl = h

e2 03

2sF. 27

Here s is the spin degeneracy of the band. In an s=4 fourband model we found that for carrier concentrations betweenp=2.01020–9.01020 cm−1 our optical conductivitywould correspond to values 0.3kFl3.0. In other words,for these high resistances the mean free path is already lessthan the Fermi wavelength F=2 /kF. Thus the Drudeanalysis clearly leads to a inconsistency, and shows that theoptical conductivity does not originate from a weakly per-turbed valence band. While in the present work, where weuse an impurity-band approach, this is obvious, the sameanalysis can also be performed for the experimental data toshow, that a weakly perturbed valence-band picture is inap-propriate to describe most Ga1−xMnxAs samples.2 The dcconductivity values obtained within our impurity-band pic-ture are, on the other hand, perfectly consistent with the ex-perimentally extracted values of kFl.

The optical mass, mopt, can be defined through the effec-tive optical spectral weight, Neff, defined by the integral

p

mopt Neff

2

e20

c

intrad , 28

with c a somewhat arbitrary energy cutoff, which has beenset to 800 meV in Ref. 11. The optical mass mopt definedthrough this formula can be quite different from the micro-scopic mass of the carriers, m0, and it provides a useful mea-sure of the “heaviness” of the charge carriers. To comparewith the experiments, we extracted mopt from our data, and inFig. 12 we plotted it as a function of carrier concentration.Deep in the metallic regime, we have obtained an opticalmass that remains approximately constant, mopt /me1, forcarrier concentrations larger than p21020, i.e., for Mnconcentrations larger than about x1%. Our kFl values aswell as our optical mass results are in good agreement withthe experimental data,11 where the optical mass was found tobe in the range 0.7mopt /me1.4. Notice that these valuesare about twice as large as the bare valence-band mass, m0=0.56me. As we approach the MIT transition, mopt increasesrapidly, and can reach values as large as mopt10me. Thislarge optical mass renormalization is characteristic of the vi-cinity of the critical concentration, see Fig. 12. In this re-gime we find a metallicity parameter kFl that is always lessthan 0.5.

V. SPIN FLUCTUATIONS

So far we have treated Mn spins at the mean-field level. Inthe present section we shall discuss how one can go beyondmean field by systematically including spin fluctuations. Todo that, we shall make a 1 /S expansion around the classicallimit, and represent spin fluctuations using Holstein-Primakoff bosons.44 The ground state is, however, generallynoncollinear,45,46 leading to some computational complica-tions.

Here we shall consider again only the T=0 temperaturelimit. There the mean-field equations imply that Si=SMnei

z,with the unit vector ei

z=hi / hi being parallel to the effectivefield created by the valence holes. We shall use this directionas a quantization axis of the Mn spin at site i, and introducetwo additional unit vectors, ei

x,y perpendicular to eiz. Using

Holstein-Primakoff bosons, we can then represent the Mnspin operators at site i as follows:

eiz · Si = S − bi

†bi

eix · Si =S

2bi + bi

† + ¯ ,

eiy · Si = iS

2bi

† − bi + ¯ , 29

where we have neglected terms that are subleading in 1 /S.In this language, the mean-field Hamiltonian, Eq. 18

simply becomes

HimpMF =

Ea† a +

i

hibi†bi, 30

while corrections to the mean field the arise from the fluc-tuation part of the interaction, Eq. 16, neglected so far

Hflucti − Gbi†bi:ci

†eizFci:+ GS

2bi

†:ci†ei

+Fci:

+ bi:ci†ei

−Fci: , 31

where we defined eiei

x ieiy, and :¯ : denotes normal or-

1.0×1018

1.0×1019

1.0×1020

1.0×1021

p (cm-3

)

0

5

10

15

mop

t/me

f = 0.3f = 0.5f = 0.9

Criticalregion

Metallicregime

Insulatingregime

FIG. 12. Color online The optical mass as function of numberof carriers. Only the impurity-band contribution was used to com-pute mopt. Deep in the metallic regime, for carrier concentrationslarger than 21020 cm−3 the effective mass remains practicallyconstant, while in the critical regime as approaching the MIT tran-sition p1019–1020 cm3 a powerlike increase in the effectivemass is observed.

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dering with respect to the mean-field ground state. Thus theneglected fluctuation terms just describe the interaction ofspin waves with spin excitations of the impurity band.

We can also rewrite the interaction part of the Hamil-tonian using the proper eigenstates of the mean-field Hamil-tonian and the corresponding creation and annihilation op-erators as

Hfluct = i

,

z ibi

†bi:a† a:+

i,

+ ia

† abi†

+ − ia

† abi 32

with the couplings m defined from Eq. 20 as

z = GS

2ei

zFi ,

= GS

2ei

Fi . 33

Having introduced a bosonic representation for the spins, wecan now use standard diagrammatic methods to computespin-fluctation corrections to the optical conductivity pertur-batively in the small coupling, G. In the noninteractingtheory, defined by Eqs. 9 and 30, we have three fields,and, correspondingly, we have three Green’s functions. Thenoninteracting Green’s functions

Gimp0 ,n =

1

in − E

34

Gval0k, j,n =

1

in − ki35

describe single-particle excitations in the impurity and va-lence bands. We shall denote them by dashed and continuouslines, respectively. Spin waves are described by the noninter-acting bosonic propagator,

D0j,n =1

in − hj. 36

This propagator is site diagonal, and accounts for the spinprecession created by the local exchange field of theimpurity-band holes. The interaction part, Eq. 32, intro-duces then three types of vertices between these Green’sfunctions, which we depicted in Table I.

To compute the current-current response function, weshall use a random-phase approximation-type RPA-type ap-proximation in the bosonic line, which mediates the interac-tion between the valence-band holes: in the diagrammaticlanguage this means that one neglects self-energy correc-tions, and sums up only the diagrams shown in Fig. 13 toplayer. Notice that at T=0 the vertex z does not give acontribution to the series in leading order in G. The RPAseries can be converted into an integral equation Bethe-Salpeter equation, as shown in Fig. 13 bottom layer, whereone formally defines a renormalized current vertex function,. Expressed analytically, the vertex equation reads

− i

+ D− in,Hi,

0 in

− i

− Din,Hi

0 in = j 37

Here in stands for in=ine0 with e0 thepolarization of the electric field, and similarly, j= je0.The intraband polarization bubble is then expressed in termsof the bare j and full vertex as

JJintrain =

,jin,

0 in , 38

with the bare polarization bubble defined as

,0 in =

in

Gimp0 ,inGimp

0 ,in + in 39

Immediately, this gives for the intraband optical conductivitythe expression

intra =i

2,

je0e0fE − fE

1

+ E − E + i−

1

− E + E + i .

40

The mean-field limit is recovered if one sets →0 in theBethe-Salpeter equations, Eq. 37. Then Eq. 40 reduces toEq. 25.

We have solved the Bethe-Salpeter equations Eq. 37 nu-merically, and computed the resulting corrections to the op-tical conductivity. However, we found that, at typical opticalfrequencies, they do not give an important correction to theoptical conductivity as compared to the previously presentedmean-field results. This is not very surprising: magnetic ex-citations have energies in the range of TC, i.e., in the 5–6meV range, which is very small compared to the opticalfrequencies studied here. In the frequency range

TABLE I. Vertices functions within the interacting field theory.Dashed lines represent impurity hole propagators, while wavy linesdenote bosonic propagators.

ν

µ j

,+ j

ν

µ j

,− j

ν

µ j,

z j

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5 meV they are, however, expected to result in interestingeffects, and they may lead to some additional features in theoptical spectrum.

The present calculation has been carried out at T=0 tem-perature. In this case the diagrammatic derivation of Eqs.37 and 40 is very transparent and straightforward. How-ever, we also derived these equations using the much moretedious equation of motion method of Ref. 47, which holdsalso at finite temperatures. It turns out that Eqs. 37 and 40are very general, and in the same form, they carry over tofinite temperatures too. The only modification is that theoriginal mean-field Hamiltonian must be diagonalized usingfinite temperature expectation values. Our formalism thusprovides a way to study the effects of finite temperature onthe optical conductivity and the interplay of ferromagnetism.The study of these finite temperature small frequency effectsis, however, beyond the scope of the present work.

VI. CONCLUSIONS

In this paper, we presented a calculation of the opticalproperties of Ga1−xMnxAs in the very dilute limit, where itcan be described in terms of an impurity-band picture. Ourapproach consisted of constructing an effective Hamiltonianwith parameters determined from microscopic variationalcalculations. The effective Hamiltonian obtained this waynot only accounts for the impurity band, but it also describestransitions from the impurity band to the valence band.

Our mixed approach captures correctly the most essentialfeatures of the experiments. It predicts a rather wide Drudepeak, originating from interimpurity-band transitions, whichis partly merged with a mid-infrared peak at 200 meV.The overall conductivity values as well as the positions ofthese features are well reproduced by our theory. Remark-ably, our calculations give a quantitative description of theconcentration-induced shift of the mid-infrared peak, and

give optical mass and residual resistivity values that agreewell with the experimentally observed values. The redshiftfor a fixed x and increasing hole fraction is an obvious con-sequence of the shift of the Fermi energy within the impurityband. The redshift of the peak for increasing x and fixed holefraction is less obvious: it is related to the shift of “kineticcontribution” see Fig. 4 under decreasing Mn-Mn separa-tion. We think that the peak position is not very sensitive tothe assumption of local transitions Eq. 9. Incorporatingmore realistic band structure may, on the other hand, modifythe precise value of the peak position.

Furthermore, we are also able to capture the metal-insulator transition, which, depending on the level of com-pensation, occurs in the range of x=0.1–0.3 % for uncom-pensated samples while it is in the range of x=1–2 % formoderate compensations. This is in good agreement with theexperiments. Our results thus promote the theoretical picturethat the metal-insulator transition takes place within the im-purity band, well before it completely merges with the va-lence band.22,48 This picture is indeed supported by the ob-servation of a finite activation energy as one approaches themetal-insulator transition.21 Further indirect evidence for thisscenario is given by Fig. 8 of Ref. 14: at very small concen-trations, where the impurity band is expected to be insulat-ing, and the conduction is due to activated behavior to thevalence band, the mobility of holes agrees with that of manyother alloys, that are known to have valence-band conduc-tion. However, right after the metal-insulator transitionxxC0.2%, the mobility clearly drops to a much smallervalue. The strong deviation from other “valence-band” com-pounds could naturally be explained by the fact that conduc-tion is due to holes within the impurity band, and that themetal-insulator transition also occurs there, in agreementwith our calculations.

We have also investigated the scaling of the conductivityand the optical mass close to the metal-insulator transition.We find that the intraband optical conductivity scales as

i Ω n + + ...

i ωn + i Ω n

i

ω

i

ν, i ω n

µ,

Ωn i Ω n i Ω n

iωn+ i Ωn

Ωn

Γ

ni

j,j,

ν,

k,j,a) b)

Ωi nω ni +

= + +

ν

µ

ν ν

µµ, µ,

ν,

~ ~

~

µ

ν

µ

~ν,

i nj,

ni

i nj,

ω ni + nΩi

ni

Ω

ω

Ω

ω

FIG. 13. Top: RPA series giving the leading correction to the optical conductivity. The dot represents the current vertex. Middle: ainterband contribution to the optical conductivity and characterize excitations, b intraband contribution. Bottom: Bethe-Salpeter equation

for the renormalized current vertex , indicated as a filled triangle.

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=0+A1/2 on the metallic side, close to thetransition.40 Close to the transition point the optical mass isstrongly renormalized and takes values mopt10me, whilefar from the metal-insulator transition we find a mass in therange moptme.

Although our calculations were performed at the mean-field level, we extended it to incorporate spin-wave fluctua-tions too. While the formalism discussed here is only appli-cable at T=0 temperature, we can also derive the sameintegral equations using an equation of motion method notdiscussed here that carries over to finite temperatures. Thefinal equations presented here thus apply for finite tempera-tures too. Nevertheless, we find that for typical optical fre-quencies spin waves do not give a substantial contribution.They may, however, influence the low-frequency opticalproperties 20 meV.

Our approach is designed to work in the limit of smallconcentrations, and it should break down for large concen-trations. Where exactly this breakdown occurs, is not quiteclear. In Ref. 14 it was argued that the breakdown shouldappear at some active Mn concentration in the range of x1%.49 However, surprisingly, all spectroscopic data seemto favor a picture in terms of impurity-band physics even atintermediate concentrations.7–11,35

Also, our results are in very good agreement with theexperimental optical data, if we just blindly extrapolate themto x3–4 %. Theoretically, this is rather mysterious: onepossible explanation would be that Ga1−xMnxAs is rather in-homogeneous, and some spectroscopic data are dominatedby regions of small Mn concentration. The other possibilityis that although the impurity-band density of states may al-ready completely merge with the valence band at larger con-centrations, most of the occupied states are in the tail of thevalence band, which has a strong impurity-band character.This scenario is sketched in Fig. 14. Since optical transitionsare local for these states, and since the valence band and thedeep acceptor states are Coulomb shifted by approximatelythe same amount, these levels deep in the gap give rise to animpurity state transition at a frequency close to 200 meV.This is indeed supported by our calculations, where we findthat the shifted local density of states has a large peak com-ing from localized states in the tail of the impurity band. Athird possibility would be that the properties of Ga1−xMnxAson its surface, tested by essentially all spectroscopicalprobes, differ from those of the bulk, and thus the effectiveconcentration of Mn ions is somehow reduced there.

ACKNOWLEDGMENTS

This research has been supported by Hungarian OTKAGrants No. NF061726 and No. K73361, Romanian CNCSISGrants No. 2007/1/780 and CNCSIS No. ID672/2009, andby NSERC and CIfAR Nanoelectronics.

APPENDIX A: GROUND-STATE WAVE FUNCTION OF ASINGLE Mn ION

In this appendix we present some details on the varia-tional calculation of the bound acceptor state of the Mn im-purities and the computation of the corresponding opticalmatrix elements.

1. Acceptor wave function

We describe the Mn acceptor by the following Hamil-tonian =me=e=1:

H = −

22 −

1

r+ Vccr . A1

Here the factor =1.782 describes the mass renormalizationterm, and =12.65 is the dielectric function for GaAs. Theexplicit form of the central-cell correction, Vccr was givenin Eq. 4.

To compute the ground-state wave function of Eq. A1,we used the following variational Ansatz:

r = i=1

n

Aii,r , A2

,r = r =1

3/2e−r, A3

In this expression we fixed the coefficients i and used onlythe Ai’s as variational parameters. The variational equationfor the latter is simply a linear equation of the form

j=1

n

HijAj

= Ej=1

n

SijAj

. A4

For the Hydrogenic wave functions used, the matrix ele-ments H= H as well as the overlap parameters S

= can be computed analytically.The full variational solution computed using 50 param-

eters i is plotted in Fig. 15, where it is also compared to asingle-parameter variational solution. The single-parametersolution with =0.091 gives a remarkably accurate approxi-mation for the wave function, excepting the regime rr0,where small deviations appear due to the central-cell correc-tion.

2. Calculation of on-site optical matrix elements

Once the variational wave function A3 is known athand, we can compute the optical matrix elements tovalence-band states. We have two obvious choices: the sim-plest way to estimate these matrix elements is to neglect the

Valence band

FIG. 14. Deep, strongly localized occupied levels in the tail ofthe valence band move together with the valence-band edge, andcan give rise to impurity transitions at about 200 meV, indicated bythe wavy line. Empty circles denote holes.

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1 /r term as well as the central-cell correction which doesnot interact too much with scattering states in the 1=1 chan-nel, and to use free-electron states

lmkfreer = 2k2jlkrYlm, , A5

with jlkr the spherical Bessel functions and Ylm ,spherical functions. Alternatively, we can neglect only thecentral-cell correction and use Coulomb scattering states,

lmkCoulr =

2

rFlkrYlm, , A6

where Fl is related to the confluent hypergeometric function,

1F1abx, as

Flkr = cleikrkrl+11F1l + 1 + i2l + 2 − 2ikr ,

A7

with =1 /k, the constant cl=2le−/2 l+1+i2l+2 , and

z the gamma function. Both scattering states are normal-ized to satisfy l ,m ;k l ,m ;k=llmmk−k.

The matrix elements of pz can be computed analytically inboth cases. For free valence electrons we obtain

1,0;kpzfree =i

325/2

i=1

n

Ai

i5/2k2

i2 + k22 , A8

while for Coulomb scattering states we have

1,0;kpzCoul =i

325/2e−/2l + 1 + i

j=1

n

Aj

j5/2k2

j2 + k22 j − ik

j + ik i

.

The matrix elements obtained this way were presented inFig. 6. The single-parameter ground-state wave functiongives a very good approximation in both cases. The overallenergy dependence of the transition matrix elements is quali-tatively similar in Coulomb and the free scattering-state ap-proximations. However, the height of the peak is about afactor of two larger in the free-electron approximation.

APPENDIX B: TWO-SITE PROBLEM

In this appendix we determine the energies of the molecu-lar orbitals of an Mn2 system and use them to calculate the

parameters of an effective second-quantized Hamiltonian.We consider two Mn ions, located at positions r=R1,2= R /2, and described by the Hamiltonian of Eq. 3. Simi-lar to the Mn ion case, we first construct the lowest lying“molecular states” using variational wave functions of theform,

r = =s,p

i=1,2

j=1

N

Ai,,ji j,r , B1

constructed from Hydrogenlike s- and p-type wave functionscentered at the Mn sites, R1,2

is,r ri,s, =

1

3/2e−ri, B2

ip,r ri,p, =

1

425/2xie

−ri/2. B3

Here x is the direction connecting the two sites, and ri=r−Ri denote the position of the valence hole relative to R1,2.We considered only states that do not have a mirror planethat contains x, and therefore included only px orbitals. Simi-lar to Appendix A, we fix the parameters j in Eq. B1, andconsider only the Ai,,j as variational parameters.

As in case of the single Mn ion problem, the coefficientsAi,,j satisfy a set of linear equations,

0.0 10.0 20.0 30.0 40.0r [Å]

0.000

0.010

0.020

0.030

0.040

0.050ψ

(r)

[Å-3

/2]

Single parameter

Full calculation

FIG. 15. Color online Normalized ground-state wave function:comparison between the single parameter and the full calculations.

0.0 50.0 100.0-0.2

0.0

Ene

rgy

leve

ls(e

V)

0.0 50.0 100.0-0.2

-0.1

0.0

R [Å]

even

odd

even

odd

even

odd

even

FIG. 16. The lowest few energy levels of the Mn2 molecule as afunction of distance. The left figure presents the case where only swaves were considered 18 parameters were used. The right panelshows the results when both s and p waves are considered14 parameters were used.

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=s,p

j=1,2

l=1

N

Hi,j,k,l − E Si,j

,k,lAj,,l = 0,

B4

with the overlap matrix elements and the Hamiltonian matrixelements defined as

Si,j,, = i,,j,, ,

Hi,j,, = i,,Hj,, .

The above integrals can be performed analytically using el-liptic coordinates.

We remark here that Eq. B4 not only provides an accu-rate estimate for the ground-state energy, but it also accountsfor the first few excited states. In Fig. 16 we show the firstfew lowest energy states obtained in this way as a function ofMn separation, R. As demonstrated in the figure, the inclu-sion of p orbitals does not result in a major improvement forthe first five states, though it introduces a level crossing be-tween the first and the second excited states at a distance R6 Å where the tight-binding approximation fails.

The hopping t and the energy E appearing in Eq. 5 andshown in Fig. 4 can be extracted from this spectrum as

tR =1

2EoddR − EevenR ,

ER =1

2EevenR + EoddR ,

where Eeven and Eodd represent the lowest lying even stateof the Mn2 molecule and that of the first odd excited state.

The electric field induces transitions between the previ-ously mentioned even and odd states, and thereby generatesa coupling of the form, Eq. 11. To determine the parametersappearing in this equation, one simply needs to compute themomentum matrix elements oddrpevenr that enter thecoupling to the external electromagnetic field. Since wealigned the two Mn ions along the x direction the only non-vanishing component of the momentum matrix elements isoddrpxevenr. It is possible to express this matrix ele-ment analytically for our variational wave functions, thoughthe final expressions are rather cumbersome. The final resultswere plotted in Fig. 5.

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