single-band model for diluted magnetic semiconductors...

PHYSICAL REVIEW B 68, 045202 ~2003!

Single-band model for diluted magnetic semiconductors: Dynamical and transport propertiesand relevance of clustered states

G. Alvarez and E. DagottoNational High Magnetic Field Lab and Department of Physics, Florida State University, Tallahassee, Florida 32310, USA

~Received 18 March 2003; published 29 July 2003!

Dynamical and transport properties of a simple single-band spin-fermion lattice model for~III,Mn !V dilutedmagnetic semiconductors~DMS’s! is here discussed using Monte Carlo simulations. This effort is a continu-ation of previous work@G. Alvarezet al., Phys. Rev. Lett.89, 277202~2002!# where the static properties of themodel were studied. The present results support the view that the relevant regime ofJ/t ~standard notation! isthat of intermediate coupling, where carriers are only partially trapped near Mn spins, and locally orderedregions~clusters! are present above the Curie temperatureTC . This conclusion is based on the calculation ofthe resistivity vs temperature, that shows a soft metal-to-insulator transition nearTC , as well on the analysis ofthe density-of-states and optical conductivity. In addition, in the clustered regime a large magnetoresistance isobserved in simulations. Formal analogies between DMS’s and manganites are also discussed.

DOI: 10.1103/PhysRevB.68.045202 PACS number~s!: 75.50.Pp, 75.10.Lp, 75.30.Hx

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I. INTRODUCTION

Diluted magnetic semiconductors~DMS’s! are attractingmuch attention lately due to their potential for device appcations in the growing field of spintronics. In particular,large number of DMS studies have focused on III-V copounds where Mn doping in InAs and GaAs has beachieved using molecular beam epitaxy~MBE! techniques.The main result of recent experimental efforts is tdiscovery1–4 of ferromagnetism at a Curie temperatureTC;110 K in Ga12xMnxAs, with Mn concentrationsx up to10%. It is widely believed that this ferromagnetism is ‘‘caried induced,’’ with holes donated by Mn ions mediatingferromagnetic interaction between the randomly localizMn21 spins. In practice, antisite defects reduce the numof holes n from its ideal valuen5x, leading to a ratiop5(n/x) substantially smaller than 1.

Until recently, theoretical descriptions of DMS materiacould be roughly classified in two categories. On one hathe multiband nature of the problem is emphasized as acial aspect to quantitatively understand these materials.5–8 Inthis context the lattice does not play a key role and a ctinuum formulation is sufficient. The influence of disorderconsidered on average. On the other hand, formulatbased on the possible strong localization of carriers atMn-spin sites have also been proposed.9,10 In this context asingle impurity-band description is considered sufficientthese materials. Still within the single-band framework, bwith carrier hopping not restricted to the Mn locations, rcent approaches to the problem have used dynammean-field11,12 or reduced-basis13 approximations. An effec-tive Hamiltonian for Ga12xMnxAs was derived in the dilutelimit and studied in Ref. 14. All these calculations are imptant in our collective effort to understand DMS materials.

However, it is desirable to obtain a more general viewthe problem of ferromagnetism induced by a diluted setspins and holes. To reach this goal it would be better totechniques that do not rely on mean-field approximatioand, in addition, select a model that has both the continu

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and impurity-band formulations as limiting cases. Suchapproach would provide information on potential proceduto further enhanceTC and clarify the role of the many parameters in the problem. In addition, these general consiations will be useful beyond the specific detailsGa12xMnxAs, allowing us to reach conclusions for othDMS’s. An effort in this direction was recently initiated bMayr and two of the authors.15 A detailed Monte Carlo studyof a simple model for DMS already allowed us to argue thTC could be further enhanced—perhaps even to rotemperature—by further increasingx andp from current val-ues in DMS samples. These predictions seem in agreemwith recent experimental developments since very receGa12xMnxAs samples withTC as high as 150 K wereprepared,16 a result believed to be caused by an enhanfree-hole density. Also samples withTC;127 K and x.0.08 were reported in Ref. 17, and, very recently, aTC of140 K was achieved on high quality GaMnAs films growwith arsenic dimers.18 There seems to be plenty of roomfurther increase the critical temperatures according to ththeoretical calculations.

The model used in Ref. 15 was a single-band lattHamiltonian, also studied by other groups, which doeshave the multiband characteristics and spin-orbit coupliof the real problem. However, it contains spin and holesinteraction and it is expected to capture the main qualitaaspects of carrier-induced ferromagnetism in DMS materiThe choice of a single-band model allows us to focus onessential aspects of the problem, leaving aside the numecomplexity of the multiband Hamiltonian, which unfortunately cannot be studied with reliable numerical techniqat present without introducing further approximations19

Note that a quite similar philosophy has been followed inrelated area of manganites for many years. In fact, it is wknown that the orbital order present in those compoundsonly be studied with a two-band model. However, singband approaches are sufficient to investigate the competbetween ferromagnetic and antiferromagnetic states, anfect recently argued to be at the heart of the colossal mag

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G. ALVAREZ AND E. DAGOTTO PHYSICAL REVIEW B68, 045202 ~2003!

toresistance phenomenon.20,21 As a consequence, a qualitatively reliable study of the single-band DMS model—wiitinerant holes and localized classical spins—is expectebe important for progress in the DMS context as well.

In our previous publication15 the phase diagram of thsingle-band model for DMS’s was already sketched.22 Thisphase diagram is reproduced in Fig. 1. The only parametethe Hamiltonian is the ratioJ/t, whereJ is the local antifer-romagnetic coupling between the spins of carriers and loized Mn21 ions, andt is the carrier-hopping amplitude~fordetails, see the full form of the Hamiltonian in the next setion!. Due to its large value 5/2, the Mn spin is assumed toclassical for numerical simplicity. At smallJ/t, individualcarriers are only weakly bounded to the Mn spins. Evensmall realistic values ofx, the carrier wave functions strongloverlap and a large-bandwidth itinerant band dominatesphysics. In this regime,TC grows withJ/t. This portion ofthe phase diagram is referred to as ‘‘weak coupling’’ in F1, and in our opinion it qualitatively corresponds to tcontinuum-limit approach pursued in Refs. 5–7. In the otextreme of largeJ/t, the previous effort15 found aTC that ismuch suppressed, converging to zero asJ/t increases~atleast at small values ofx). The reason for this behavior is thstrong localization of carriers at Mn spin locations, to taadvantage of the strongJ coupling. The localization suppresses the mobility and the carrier-induced mechanism ilonger operative. However, when several Mn spins are cto one another small regions can be magnetized efficieAs a consequence, a picture emerges where small islanferromagnetism are produced at a fairly large temperascaleT* (.TC) that grows withJ, but a global ferromag-netic state is only achieved upon further reducing the teperature such that the overlaps of wave functions inducpercolationlike process that aligns the individual preformclusters. The proper procedure to distinguish between thtwo regimes is through an analysis of short- and londistance spin correlations. In our opinion, the effort of Re9 and 10 belongs to this class, and it corresponds to‘‘strong coupling’’ region of Fig. 1. Between the weak- anstrong-coupling domains, anintermediate regionprovides anatural interpolation between those two extreme cases.Curie temperature is optimal~i.e., the largest! in this inter-mediate zone, at fixed values ofp andx. In this regime, there

FIG. 1. Phase diagram of the single-band model Eq.~1!, asdiscussed in Ref. 15. The figure shows schematically the threepling regions discussed in the text:~I! weak,~II ! intermediate, and~III ! strong coupling, as well as theTC andT* dependence on thecoupling J/t. The region betweenT* and TC contains FM ‘‘clus-ters,’’ with magnetic moments that are not aligned.

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is sufficient interaction between the clusters to become gbally ferromagnetic, and at the same time theJ is sufficientlystrong to induce a robustTC . If chemical control overJ/twere possible, this intermediate region would be the mpromising to increase the Curie temperature~again, at fixedxandp).

However, it may even occur that the DMS materials aalready in the intermediate optimal range ofJ/t couplings.An indirect way to test this hypothesis relies on calculatioof dynamical and transport properties, which are presentethis paper, and its comparison with experiments. In partilar, the experimentally measured dc resistivityrdc of DMScompounds has a nontrivial shape with a~poor! metallic(drdc/dT.0) behavior belowTC , which turns to insulating(drdc/dT,0) at higher temperatures. This nontrivial profiresembles results reported in the area of manganites, walso have a metal-insulator transition as a functiontemperature,21 although in those materials the changes insistivity with temperature are far more dramatic thanDMS. The formal similarities DMS manganites have alreabeen remarked in previous literature,15 and suggest a common origin of therdc vs temperature curves. In Mn oxidesis believed that above the Curie temperature, preformedromagnetic clusters with random orientations contributethe insulating behavior of those materials.20,21 For DMS’s, asimilar rationalization can be envisioned if the state of revance aboveTC has preformed magnetic clusters. From tprevious discussion15 it is known that the intermediateJ/tregion can have clusters without collapsingTC to a verysmall value. At the Curie temperature, the alignment of thpreformed moments leads to a metallic state. An explicit cculation of the resistivity—using techniques borrowed frothe mesoscopic physics context—is provided in this papand the results support the conjecture that clustered scan explain the transport properties of DMS’s. Theseveiled analogies with manganites are not just accidenClearly, approaches to DMS’s that rely on mobile carrieand localized spins in interaction have close similarities wthe standard single-band double-exchange model whereoneeg orbital is considered. The key difference is the preence of strong dilution in DMS’s as compared with mangnites.

It is interesting to remark that in agreement with t‘‘clustered’’ state described here, recently Timm and von Open have shown thatcorrelateddefects in DMS’s are needeto describe experimental data.23 Their simulations with Cou-lombic effects incorporated lead to cluster formation, wsizes well beyond those obtained from a random distributof Mn sites as considered here. In this respect, the resultRef. 23 provide an even more dramatic clustered state tours. If the Mn spins were not distributed randomly in osimulations but in a correlated manner, the state aboveTCwould be even more insulating than reported below andphysics would resemble much closer that observed in mganites. Note also that Kaminski and Das Sarma haveindependently arrived to a polaronic state24 that qualitativelyresembles the clustered state discussed here. It is alsoesting that in recent ion-implanted~Ga,Mn!P:C experimentsthat reported a high Curie temperature, the presence of

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SINGLE-BAND MODEL FOR DILUTED MAGNETIC . . . PHYSICAL REVIEW B 68, 045202 ~2003!

romagnetic clusters were observed and they were attribto disorder and the proximity to a metal-insulattransition.25 Even directly in~Ga,Mn!As, the inverse mag-netic susceptibility deviates from the Curie law at tempetures aboveTC @Fig. 2~b! of Ref. 2#, which may be an indi-cation of an anomalous behavior. In manganites, these sanomalies in the susceptibility are indeed taken as evideof the formation of clusters.20

The paper is organized as follows: In Sec. II the detailsthe method of simulation, definitions, and conventionspresented. Section III describes the results for the densitstates~DOS! of this model, the appearance of an ‘‘impuriband’’ at largeJ/t, and the relation of the DOS with thoptimal TC . The temperature and carrier dependence ofoptical conductivity and Drude weight are presented in SIV, with results compared with experiments. In Sec. V, tresistance of a small cluster is calculated and also compwith experimental data. The intermediate coupling regimethe only one found to qualitatively reproduce the availaresistivity vs temperature curves. From this perspective, bthe weak- and strong-coupling limits are not realistic to adress DMS materials. In Sec. VI, the influence of a magnfield is studied by calculating the magnetoresistance. Laeffects are observed. Section VII describes the characteriof the clustered-state regime aboveTC at strong coupling.Finally, in Sec. VIII the ‘‘transport’’ phase diagram is presented, and it is discussed in what region it appears to bemost relevant to qualitatively reproduce resultsGa12xMnxAs MBE-grown films.

II. METHOD AND DEFINITIONS

A. Hamiltonian formalism

There are two degrees of freedom in the DMS probldescribed here:~i! the local magnetic moments corresponing to the five electrons in thed shell of each Mn impurity,with a total spin 5/2, and~ii ! the itinerant carriers produceby the Mn impurities. Since these Mn impurities substituGa atoms in the zinc-blende structure of GaAs, there isprinciple, one hole per Mn. However, it has been found tthe system is heavily compensated and, as a consequthe actual concentration of carriers is lower. In the presstudy, both the density of Mn atomsx and the density ofcarriersn are treated as independent input parameters. Mover, the ratiop5n/x is defined, which is a measure of thcompensation of the system, e.g., forp50 the system istotally compensated and forp51 there is no compensation

The Hamiltonian of the system in the one-band appromation can be written as

Ĥ52t (^ i j &,s

ĉis† ĉ j s1J(

ISW I•sW I , ~1!

whereĉis† creates a carrier at sitei with spin s. The carrier-

spin operator interacting antiferromagnetically with thecalized Mn spinSW I is sW I5 ĉIa

† sW a,bĉIb . Through nearest-neighbor hopping, the carriers can hop toany site of thesquare or cubic lattice. The interaction term is restricted trandomly selected but fixed set of sites, denoted byI. Note

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that in addition to the use of a single band, there are otapproximation in the model described here:~i! The HubbardU/t is not included. This is justified based on the low-carrconcentration of the problem, since in this case the probaity of double occupancy is small. In addition, Mn-oxidinvestigations20,21have shown that an intermediate or largeJcoupling acts similarly asU/t, also suppressing double occupancy.~ii ! A nearest-neighbors antiferromagnetic couplibetween the Mn spins is not included. Again, this is justifias long asx is small. ~iii ! Finally, also potential disorder isneglected~together with the spin, the Mn sites shouldprinciple act as charge trapping centers!. This approximationis not problematic when the Fermi energy is larger thanwidth of the disorder potential~i.e., largep andx). However,in the (p, x) range analyzed in this paper it becomes moquestionable to neglect this effect. Its influence will be stuied in a future publication.

In the present study the carriers are considered to be etrons and the kinetic term describes a conduction band. Hever, in Ga12xMnxAs the carriers are holes and the valenband has to be considered instead. Although the lattemore complicated, the simplified treatment followed heshould yield similar results for both cases.9,10,15

The local spins are assumed to be classical which allothe parametrization of each local spin in terms of sphercoordinates: (u i ,f i). The exact-diagonalization method fothe Fermionic sector is described in this section, followiRef. 20. The partition function can be written as

Z5)i

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p

du isinu iE0

2p

dfZg~$u i ,f i%! D , ~2!where Zg($u i ,f i%)5Tr(e

2bK̂) is the partition function ofthe Fermionic sector,K̂5Ĥ2mN̂, N̂ is the number operatorandm is the chemical potential.

In the following, a hypercubic lattice of dimensionD,length L, and number of sitesN5LD will be considered.Since K̂ is a Hermitian operator, it can be representedterms of a Hermitian matrix which can be diagonalized byunitary matrixU such that

U†KU5S e1 0 . . . 00 e2 . . . 0A A � A0 0 . . . e2N

D . ~3!The basis in which the matrixK is diagonal is given by theeigenvectorsu1

†u0&, . . . ,u2N† u0&, where the Fermionic opera

tors used in this basis are obtained from the original opetors throughum5( j sUm, j s

† cj s , with m running from 1 to2N.

Defining um† um5n̂m and denoting bynm the eigenvalues

of n̂m , the trace can be written

Trg~e2bK̂!5 (

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5 (n1 , . . . ,n2N

^n1•••n2Nue2b(l512N

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since in the$um† u0&% basis, the operatorK̂ is (leln̂l , and

the number operator can be replaced by its eigenvalues. TEq. ~2! can be rewritten as

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~11e2bel!, ~5!

which is the formula used in the simulations.

B. Monte Carlo method

The integral over the angular variables in Eq.~5! can beperformed using a classical Monte Carlo simulation.26 Theeigenvalues must be obtained for each classical spin conration using library subroutines. Finding the eigenvaluesthe most time consuming part of the numerical simulatiThe integrand is clearly positive. Thus ‘‘sign problems,’’which the integrand of the multiple integral under considation can be nonpositive, are fortunately not present instudy.

Although the formalism is in the grand-canonical esemble, the chemical potential was adjusted to give thesired carrier densityn. To do so, the equationn(m)2n50was solved form at every Monte Carlo step by using thNewton-Raphson method.27 This technique proved very efficient in adjusting with precision the desired electronic dsity and, as a consequence, our analysis can be considerin the canonical ensemble as well, with a fixed numbercarriers.

Usually, 2000–5000 Monte Carlo iterations were usedlet the system thermalize, and then 5000–10 000 additiosteps were carried out to calculate observables, measuevery five of these steps to reduce autocorrelations.

Since for fixed parameters there are many possibilitiesthe random location of Mn impurities, results are averagover several of these disorder configurations. Approxima10–20 disorder configurations were generally used for smlattices (43 and 10310) and 4–8 for larger lattices (63 and12312). The inevitable uncertainties arising from the usea small number of disorder configuration does not seemaffect in any dramatic way our conclusions below. Thisdeduced from the analysis of results for individual disordsamples. The qualitative trends emphasized in the prepaper are present in all of these configurations.

C. Observables

Quantities that depend on the Mn degrees of freedomu iandf i in the previous formalism! are calculated simply byaveraging over the Monte Carlo configurations. Note tany observable that does not have the continuous symmof the Hamiltonian will vanish over very long runs. Thus itstandard in this context to calculate the absolute value of

magnetization,uM u5A( i j SW i•SW j , as opposed to the magnetzation vector. Another useful quantity is the spin-spin corlation, defined by

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N~x! (y SW

y1x•SW y , ~6!

whereN(x) is the number of nonzero terms in the sum. Tcorrelation at a distanced is averaged over all lattice pointthat are separated by that distance, but since the systediluted, the quantity must be normalized to the numberpairs of spins separated byd, to compare the results for different distances.

The observables that directly depend on the electronicgrees of freedom can be expressed in terms of the eigenues and eigenvectors of the Hamiltonian matrixK̂. TheDOS,N(v), is simply given by(ld(v2el). However, themajority and minority DOS,N↑(v) and N↓(v), were alsocalculated in this study.N↑(v) indicates the component thaaligns with the local spin, i.e.,N↑(v) is the Fourier trans-form of ( i^c̃i↑

† (t) c̃i↑(0)&, where c̃i↑5cos(ui/2)ci↑1sin(ui/2)e

2 if ici↓ . Then

N↑~v!5(l

2N

d~v2el!S (i

N

Ui↑,l† Ul,i↑cos2~u i /2!

1Ui↓,l† Ul,i↓sin2~u i /2!1@Ui↑,l

† Ul,i↓exp~2 if i !

1Ui↓,l† Ul,i↑exp~ if i !#cos~u i /2!sin~u i /2!D , ~7!

where for sitesi without an impurityu i5f i50 is assumed.A similar expression is valid forN↓(v). The optical conduc-tivity was calculated as

s~v!5p~12e2bv!

vN E2`1` dt

2peivt^ jWx~ t !• jWx~0!&, ~8!

where the current operator is

jWx5 i te(j s

~cj 1 x̂,s†

cj ,s2H.c.!, ~9!

with x̂ the unit vector along thex direction. ForvÞ0, s(v)can be written as

s~v!5 (lÞl8

pt2e2~12e2bv!

vN

3

U(j s

~U j 1 x̂s,l†

U j s,l82U j s,l† U j 1 x̂s,l8!U2

~11eb(rl2m)!~11e2b(rl82m)!

~10!

3d~v1rl2rl8!. ~11!

Both N(v) and s(v) were broadened using a Lorentziafunction as a substitute to thed functions that appear in Eqs~7! and ~11!. The width of the Lorentzian used wase50.05in units of the hoppingt.

The optical conductivity ind dimensions obeys the sumrule:

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pe2^2T̂&2Nd

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whereD is the Drude weight andT̂ is the kinetic energy:

2T̂5t (^ i j &,s

~cis† cj s1H.c.!. ~13!

Although the Drude weight gives a measure of theconductivity properties of the cluster, the ‘‘mesoscopic’’ coductance was also calculated in order to gather additioinformation. The details are explained in Sec. V.

III. DENSITY OF STATES

In this section, the density of states calculated at sevcouplings is shown and discussed. A few basic facts abthe DOS of the model, Eq.~1!, are presented first. If carrielocalization effects are not taken into account and a femagnetic state is considered at strongJ/t coupling, then twospin-split bands~‘‘impurity bands’’! will appear for thismodel, each with weight proportional tox, at each side of theunperturbed band which would have weight 2(12x) ~Fig.2!. For partially compensated samples,p,x, the chemicalpotential will be located somewhere in the first spin-spband, and, as a consequence, only this band would beevant in the model Eq.~1!.

In practice, Monte Carlo simulations show that, in tregime of interest whereTC is optimal, the impurity band isnot completely separated from the unperturbed band. Forreason, both the unperturbed band and the impurity bhave to be considered in quantitative calculations. To illtrate this, Fig. 3 showsN(v) vs v for different J/t ’s in twodimensions. With growingJ/t, the impurity band begins toform for J/t>3.0, and atJ/t56.0 it is already fairly sepa-rated from the main band. However, note that for thistreme regime ofJ/t, TC is far lower than forJ/t53.0, asdiscussed before in Ref. 15. In fact, the optimalJ/t for the(x,p) parameters used here was found to be in the ra2.0–4.0. This implies that the model shows the highestTCwhen the impurity band is about to form, but it is not yseparated from the main band. This introduces an impordifference between our approach to DMS materials, andtheoretical calculations presented in Ref. 9. Figures 4~a!–~c!indicate that a similar behavior is found in three dimensioIn this case, once again, the optimalTC occurs for interme-

FIG. 2. Schematic representation of the DOS, for a ferromnetic configuration and strong enoughJ/t coupling. The impurityband has weightx, and the chemical potentialm lies within it.

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diateJ/t, as seen in Fig. 4~d! where the phase diagram obtained from a 63 lattice with x50.25 andp50.3 is shown.

The physical reason for this behavior is that at very laJ/t, the states of the impurity band are highly localizwhere the Mn spins are. Local ferromagnetism can be eaformed, but global ferromagnetism is suppressed by the ccomitant weak coupling between magnetized clusters.

Both Figs. 3 and 4 represent results obtained at particvalues of parametersx50.25, p50.3, andT/t50.01. How-ever, results for several other sets$(x,p,T)% were gatheredas well~not shown!. Thex dependence of the DOS is simplincreasingx produces a proportional increase in the numbof states of the impurity band, and a corresponding decreof the main band weight~see Fig. 2!. In terms ofp, increas-ing the effective carrier concentration by a small amount wfound to simply shift the chemical potential to the right. Cocerning the temperature dependence of the DOS, at low tperature the states contributing to the impurity band arelarized, i.e., the system is ferromagnetic~FM!, as shown bythe different weights of the majority and minority band~Figs. 3 and 4!. As the temperature increases, spin disorgrows due to thermal fluctuations, and whenT;T* , i.e., inthe paramagnetic regime, the bands become symmetric.

It appears that the only change that a small increase incarrier concentration produces is an increase in the chempotential. However, experimentally the appearance o

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FIG. 3. N↑(v) andN↓(v) vs v2m for a 10310 periodic sys-tem with 26 spins (x;0.25) and 10 electrons (p;0.4) at T/t50.01. Results are shown for~a! J/t52.0, ~b! J/t53.0, ~c! J/t54.0, and~d! J/t56.0. v2m is in units of the hoppingt. Theresults are averages over eight configurations of disorder~but thereis no qualitative difference in the results from different configutions!.

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pseudogap at the Fermi energy has been reported.29 Althoughthe finite-size effects in the present theoretical calculationnot allow for a detailed study ofN(v) at or very near theFermi energy with high enough precision, in certain casepseudogap was indeed observed in the present study achemical potential, particularly when the system is in tclustered regime. While this result certainly needs confirmtion, it is tempting to draw analogies with the more detailcalculations carried out in the context of manganites, whthe presence of a pseudogap both in theoretical modelsin experiments is well established.20,21,30Pseudogaps are alspresent in underdoped high-temperature superconducGiven the analogies between DMS materials and transitmetal oxides unveiled in previous investigations,15 it wouldnot be surprising that DMS presents a pseudogap in the ctered regime as well. More work is needed to confirm thspeculations.

Results presented in this section for the DOS qualitativagree with those found in Ref. 12 using the dynamical mefield technique~DMFT! in the coupling regime studied thereHowever, the DMFT approach is a mean-field approximatlocal in space and, as a consequence, the state emphasithis paper—with randomly distributed clusters—cannotstudied accurately with such a technique.

FIG. 4. N↑(v) and N↓(v) vs v2m for a 63 periodic system

with 54 spins (x;0.25) and 16 electrons (p;0.4) at T/t50.01.Results are shown for~a! J/t52.0, ~b! J/t54.0, ~c! J/t56.0. v2m is in units of the hoppingt. The results are averages over eigconfigurations of disorder~but there is no qualitative difference ithe results from different configurations!. ~d! TC andT* vs J/t fora 63 lattice with x50.25 andp50.4. These temperatures wedetermined from the spin-spin correlations at short and largetances, as explained in Ref. 15.

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Experimental evidence of the formation of the impuriband for Ga12xMnxAs has been provided by Okabayaset al. in Refs. 31 and 32~see also Ref. 33!. It is interesting toremark that half the total width of the impurity band obtainin our simulations is about 2t ~see, e.g., Figs. 3 and 4!, andusingt50.3 eV,15 this is equal to 0.6 eV, in good agreemewith the width 0.5 eV estimated from the photoemission eperimental measurements mentioned before.31

The local DOS is studied next. The purpose of this anasis is to further understand the inhomogeneous stateforms as a consequence of Mn-spin dilution and concomicarrier localization. The local DOS is shown in Fig. 5 on838 lattice atx50.25, p50.4, J/t52.5, andT/t50.05. Inthis particular case only, an arrangement of classical spwas introduced ‘‘by hand’’ such that the clusters are cleaformed. In this way, sites can easily be classified in thgroups, as discussed below. Despite using a particularly csen spin configuration, the resulting DOS is similar to thoobtained previously~Fig. 3! using a truly random distributionof spins. Lattice sites are classified as follows: the first grois defined to contain lattice sites that have a classical sand belong to a spin cluster. The second class is composesites that have an isolated classical spin, i.e., nearby sitenot have other classical spins. Finally, lattice sites withclassical spins in the same site or its vicinity belong tothird class.NI(v), NII(v), and NIII (v) denote the localDOS at sites corresponding to each of the three classesspectively. It can be observed thatNI(v) contributes to statesinside the impurity band~note the sharp peak near21), butalso contributes to the main band since theJ/t used is inter-mediate.NII(v) contributes a sharp peak near zero, and ahas weight in the main band. This can be explained aslows: the electron is weakly localized at an isolated syielding a state atv II'2J or, sincem'2J for p'0.4,v II2m'0 @see Fig. 5~a!#. This interesting result shows thamany of the states near zero in Figs. 3 and 4 may belocal-ized, and they do not contribute to the conductivity. Finalin NIII (v) the empty sites contribute weight only in near

s-

FIG. 5. Local DOS for three different classes of sites~see text!on an 838 lattice with J/t52.5, T/t50.01, x;0.2, andp50.4.This coupling regime corresponds to the intermediate optimalgion, where the impurity band is not fully formed. ‘‘FM CLUSTER’’ is NI(v) in the text notation, while ‘‘ISOLATED SPIN’’ and‘‘EMPTY SITE’’ are NII(v) andNIII (v), respectively.

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unperturbed states, i.e., states that belong to the main bIn conclusion, the local carrier density is very inhomogneous and this fact has an important effect on the form ofsite-integrated DOS. Our results show that scanning tuning microscopy~STM! experiments would be able to revethe clustered structure proposed here, if it indeed exwhen applied to DMS materials.

IV. OPTICAL CONDUCTIVITY

In the previous section, it was argued that at the optimcouplingJ/t – whereTC is maximized—the impurity band isnot completely separated from the main band. This consion is supported by results obtained from the optical cductivity as well, which is shown in Fig. 6~a! for a two-dimensional lattice. This optical conductivity has two mafeatures:~i! a zero-frequency or Drude peak and~ii ! a finite-frequency broad peak which is believed to correspondtransitions from the impurity band to the main band, asgued below. From Fig. 6~a!, it is observed that interbantransitions are not much relevant at weak coupling (J/t51.0), but they appear with more weight at intermedicouplings (J/t52.0–3.0), whereTC is optimal. It is worthremarking that atJ/t strong enough, e.g.,J/t56.0, when theimpurity band is well formed, the finite-frequency peakweaker in strength than for the optimalJ/t due to localiza-tion. In fact, if J/t were so large that the impurity bandcompletely separated from the unperturbed band, then calocalization would be so strong that interband transitiowould not be possible. The Drude peak is plotted separain Fig. 6~b! as a function ofJ/t at low temperatures. Asexpected, whenJ/t increases and localization sets in, tconductivity of the cluster decreases. Similar results foroptical conductivity and Drude weight are found in thrdimensions~see Fig. 7!.

The rise in absorption that appears forJ/t>2.5 at inter-mediate frequencies is due to interband transitions, i.e., tsitions from the impurity band to the unperturbed band. Tfrequency of this peak,v inter , depends uponJ/t as well ascarrier concentrationp, but for J/t>2.5 ~which is an inter-

FIG. 6. Coupling dependence of the conductivity at low teperature in two dimensions.~a! s(v) vs v for a 10310 periodicsystem with 26 spins (x;0.25), 10 electrons (p;0.4), T/t50.01, and for differentJ/t ’s as shown.~b! Drude weightD vs J/tfor the same lattice and parameters as in~a!.

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explained before in two different ways:~i! as produced bytransitions from the impurity band to the GaAs valenband12 or ~ii ! as caused by intervalence band transition6

Due to the frequency range and temperature behaviorserved, our study supports the first possibility, i.e., thatfinite-frequency peak observed for Ga12xMnxAs at around0.2 eV is due to transitions from the impurity band to tmain band. Recent observations34 also appear to support thiview.

The Drude weight increases in intensity with increascarrier density, as can be seen from Fig. 10. In fact, the rp of carriers to Mn concentration, which is a measure ofcompensation of the samples, could explain, withinframework of this model, the different behavior observedlow frequencies for GaMnAs and InMnAs. In the formcase, no tail of the Drude peak is found. However, for InnAs, a clear Drude tailing is observed,3,28 with increasingintensity as the temperature increases. At large enoughing, our calculations based on model Eq.~1! predict a Drude-like peak for s(v) which could correspond to the regim

FIG. 9. Absorption coefficienta(v) spectra for a metallicsample prepared by low temperature annealing. The temperafrom down to top are 300, 250, 200, 150, 120, 100, 80, 60, 40,10, and 4.2 K~from Katsumotoet al., Ref. 3!.

FIG. 10. p dependence of the optical conductivity. Showns(v), including the Drude weight, vsv for a 10310 lattice,x50.25, J/t52.5, T/t50.01, and different values ofp. In the direc-tion of the arrow,p takes the values 0.08, 0.12, 0.20, 0.30, 0.38, a0.5. Inset: Drude weightD vs fraction of carriersp.

04520

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valid for InMnAs, while for low doping this study predictsvery small Drude peak, which is the case for GaMnAs. Ufortunately, the precise carrier concentrations for both marials are not known experimentally with precision.

V. CONDUCTANCE AND METAL-INSULATORTRANSITION

The conductanceG is calculated using the Kubo formuladapted to geometries usually employed in the contexmesoscopic systems.35 The actual expression is

G52e2

hTr@~ i\ v̂x!ImĜ~E!~ i\ v̂x!ImĜ~E!#, ~14!

where v̂x is the velocity operator in thex direction andImĜ(E) is obtained from the advanced and retarded Grfunctions using 2i ImĜ(E)5ĜR(E)2ĜA(E), whereE is theFermi energy. The cluster is considered to be connectedideal contacts to two semi-infinite ideal leads, as represenin Fig. 11. Current is induced by an infinitesimal voltagdrop. This formalism avoids some of the problems associawith finite systems, such as the fact that the conductivitygiven by a sum of Diracd functions, since the matrix eigenvalues are discrete. A zero-frequency delta peak with finweight corresponds to an ideal metal—having zeresistance—unless an arbitrary width is given to that depeak. In addition, in numerical studies sometimes it occthat the weights of the zero-frequency delta peak are netive due to size effects.36 For these reasons, calculationsdc resistivity using finite close systems are rare in the liteture. All these problems are avoided with the formulatidescribed here.

The entire equilibrated cluster as obtained from the Msimulation is introduced in the geometry of Fig. 11. The ideleads enter the formalism through self-energies at theright boundaries, as described in Ref. 35. In some casvariant of the method explained in Ref. 35 was used in tpaper to calculate the conductance~this modification wassuggested to us by J. Verge´s!. Instead of connecting the cluster to an ideal lead with equal hoppings, a replica of tcluster was used at the sides. This method, although sligslower, takes into account all of the Monte Carlo data forcluster, including the periodic boundary conditions. In adtion, averages over the random Mn spin distributionscarried out. The physical units of the conductanceG in thenumerical simulations aree2/h as can be inferred from Eq~14!. In Fig. 12 the inverse of the conductance, which ismeasure of the resistivity, is plotted for a three-dimensiolattice at weak, intermediate, and strong coupling, at fixx50.25 and p50.3. For the weak-coupling regime (J/t51.0) the system is weakly metallic at all temperatures.

res0,

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FIG. 11. Geometry used for the calculation of the conductanThe interacting region~cluster! is connected by ideal contacts tsemi-infinite ideal leads.

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the other limit of strong couplings, 1/G decreases with in-creasing temperature, indicating a clear insulating phase,result of the system being in a clustered state at the temptures explored,10,15with carriers localized near the Mn spinAt the important intermediate couplings emphasized ineffort, the system behaves like a dirty metal forT,TC ,while for TC,T,T* , 1/G slightly decreases with increasintemperature, indicating that a soft metal to insulator trantion takes place nearTC . For T.T* , where the system isparamagnetic, 1/G is almost constant. Note that for stronenough J/t, TC→0 and therefore no metallic phasepresent.

Similar qualitative behavior is found for the twodimensional case~Figs. 13 and 14!. Furthermore, in Fig. 14

FIG. 12. Dependence of the theoretically calculated resistivitrwith temperature in three dimensions. Shown isr5L/G vs T on 43

lattices, 16 spins (x50.25), and 5 carriers (p50.3) for theJ/t ’sindicated. An average over 20 disorder configurations has beenformed in each case. Units are shown in two scales,ah/(2e2) onthe left and mV cm on the right, withL54 and assuminga55.6 Å. The estimated critical temperatures are also shown~thearrows indicate the current accuracy of the estimations!.

FIG. 13. Inverse of the conductance 1/G vs T calculated on a10310 lattice with 26 spins (x;0.25), 8 electrons (p'0.3), and 2values ofJ/t as indicated. Shown is an average over three disoconfigurations. 1/G has units ofh/(2e2) in two dimensions.

04520

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the spin-spin correlations have been plotted to show thecation ofTC andT* and their relation to the resistivity.

It is interesting to compare our results with experimenFor Ga12xMnxAs, data similar to those found in our investigations have been reported.2,3 A typical result for resistivityversusT can be found in Ref. 4. The qualitative behaviorthe resistivity in these samples agrees well with the theoical results presented in Fig. 12 if intermediate couplingsconsidered. On the other hand, stronger or weaker couplare not useful to reproduce the data, since the result is einsulating or metallic at all temperatures, respectively~defin-ing metallic and insulator regimes by the slope of the restivity vs temperature curves!.

Furthermore, even the experimental numerical valuesthe resistivity are in agreement with the results presenhere@even though model Eq.~1! does not include a realistictreatment of the GaAs host bands#. This can be shown asfollows. The conductivity of a three-dimensional samplerelated to the conductance bys5G/L, whereL is the sidelength of the lattice. Hence the resistivity isr5L/G. Theunits ofG are, as explained before,h/(2e2), and in our caseL54a, wherea is the lattice spacing. Assuminga55.6 Å,then the values shown in Fig. 12 are obtained. Note thaFig. 12 the minimum resistivity forJ/t52.5 is 3.3 mV cm,whereas the minimum possible value for that carrier dopand cluster size used is 1.5 mV cm, which corresponds tothe caseJ/t50. In spite of the label ‘‘metallic’’ for theseresults, the absolute values of the resistivity are high. Silarly, in the metallic phase of the sample shown in the eperimental results of Ref. 4, the minimum resistivity is on;3 –6 mV cm. Both in theory and experiments, the metalphase appears to be ‘‘dirty,’’ which is likely due to the rduced number of carriers, and localizing effect of the disder.

er-

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FIG. 14. MagnetizationuM u, spin-spin correlationsC(d), atmaximum and minimum distance, and inverse of the conducta1/G vs T on a 12312 lattice with 22 spins (x'0.15), 6 electrons(p'0.3), andJ/t51.0. The approximate vanishing of spin corrlations at different temperatures, depending on the distanced stud-ied, allowed us to obtain an approximate determination ofTC andT* , similarly as carried out in previous studies~Ref. 15!.

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For J/t intermediate or strong, localization of the wavfunction is observed at intermediate temperatures.15 This im-plies that carriers tunnel between impurity sites without viting the main band, leading to insulating behavior. BeloTC conduction is favored by the ferromagnetic order anda consequence, as temperature decreases conductancreases. To further understand why the conductance~resis-tance! has a minimum~maximum! aroundTC it is helpful toconsider the three states: FM, clustered, and paramag~PM! as depicted for a special spin configuration in Fig.on an 838 lattice. This configuration is not truly random bit was chosen by hand for simplicity so that the spin clustcan easily be recognized. In the case of a random configtion of spins a similar reasoning can be drawn. For thestate of Fig. 15, conduction is possible and indeed the msured conductance isG'2.8(2e2/h). For the two possibleclustered states represented in Figs. 15~b! and ~c!, the con-duction is reduced due to the different alignment of spinsdifferent clusters. This is verified by calculating the condutance which isG'0.0 in those cases. For a paramagnespin arrangement with spin pointing in random directio~not shown!, the conductance is small but finite,G'0.5(2e2/h). It follows that the clustered state has the minmum conductance, and this explains the observed behaof the resistivity with a maximum nearTC , i.e., when thesystem is clustered.

The inverse of the Drude weightD21 also shows a peakaroundTC as seen in Fig. 16, where the same Mn-spin c

FIG. 15. Qualitative explanation of the transport propertiesclustered states. Shown are three clusters created by hand on38 lattice and with spin configurations also selected by handillustrate our ideas. For a~a! FM state atJ/t52.5 andp50.4,conduction is possible due to the alignment of magnetic momand the conductance was found to beG52.8(2e2/h). For the clus-tered state regime, two typical configurations~b! and~c! are shownwhere the conduction channels are broken and as a consequG'0. For the same clustered state but with randomly selectedorientations~not shown! the conductance is small but finiteG'0.5(2e2/h). The radius of the solid circles represent the loccharge density.

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figuration as in Fig. 14 was used. This provides further edence for the metal-insulator transition nearTC describedhere.

It is also interesting to remark that the optimal valueJ/t changes withx ~e.g., for smallx, J/t optimal is alsosmall!. Since J/toptimal approximately separates the metfrom the insulator, working at a fixedJ/t ~as in real materi-als! and varyingx, then an insulator at high temperaturesfound at smallx, turning into a metal at largerx. This is inexcellent agreement with the experimental results of Reusing a careful annealing procedure~while results of previ-ous investigations with as-grown samples had foundinsulator-metal-insulator transition with increasingx2!. Theissue discussed in this paragraph is visually illustrated in F17, for the benefit of the reader.

VI. MAGNETORESISTANCE

The magnetoresistance percentage ratio was calculusing the definitionMR51003@r(0)2r(B)#/r(B). Figure18 shows the magnetizationuM u and the resistivity as a function of the applied fieldB on a 12312 lattice,x50.15, p50.4, J/t51.0, and low temperature. Note that in two d

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FIG. 16. Inverse of the Drude weight vsT for a 12312 latticeand parameters as in Fig. 14. Note that both the inverse contance in Fig. 14 and Drude21 provide evidence for a metal-insulatotransition at approximately the same temperatureTC .

FIG. 17. Dependence ofJ/toptimal with x, approximately separating the metal from the insulator~standard notation!. At fixed J/t~horizontal solid line!, as in experiments, the system should tranform from an insulator to a metal with increasingx, in agreementwith experiments~Ref. 4!.

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mensions atx50.25 andp50.3, the intermediate couplincorresponds toJ/t52.5, but atx50.15, nowJ/t51.0 cor-responds to optimal coupling since asx decreases the optimaJ/t also decreases. For these parameters,TC was estimated tobe 0.02t andT* '0.06t. The value in T of the magnetic fieldwas calculated assumingg52.0 andS55/2 for the localizedspins. The units of the resistivity areh/(2e2) for the two-dimensional lattice.

In our simulations it was observed thatuM u increases withincreasing magnetic field. At zero magnetic fielduM u is 60%of its maximum value, while at 12 T it has reached;80%.This is in agreement with the experimental results in Figof Ref. 2 where the magnetization is 50% of its maximuvalue at zero field, but it reaches 70% of that maximuvalue at 4 T. In our studies it was also observed that 1Gdecreases with increasing magnetic field, showing atfields a negative magnetoresistance. NearTC the resistivitydecreases by 20–30% increasing the field from zero to 1while the decrease in resistivity at higher temperaturesmuch smaller. The present computational results agreewell with the experimentally measured dependence ofresistivity and magnetization on magnetic fields. For eample, in Fig. 2~b! of Ref. 2, the decrease in resistivity25% nearTC'60 K when increasing the field from zero t15 T. A much smaller decrease in resistivity is observedhigher temperatures (T.120 K). It is also indicated in thaexperimental reference that the magnetoresistance is bet30% and 40% at 12 T.

It is interesting to remark that substantial magnetoretance effects have been reported in thin films consistingnanoscale Mn11Ge8 ferromagnetic clusters embedded in alute ferromagnetic semiconductor matrix.37 The characteris-tics of these materials are analogous to our clustered sthat also show a robust MR effect due to magnetizationtation of spontaneously formed clusters.

VII. MORE ABOUT CLUSTER FORMATION

It was discussed before in this paper and in previous plications that a clustered state is formed aboveTC for inter-

FIG. 18. MagnetizationuM u and inverse conductance 1/G vsmagnetic fieldB on a 12312 lattice with PBC,x50.15, p50.3,J51.0, andT/t50.01 ~the same parameters as in Fig. 14!. Theunits ofB are T, assumingg52.0 andS55/2. The units of 1/G areh/2e2. The approximate values of the magnetoresistance are 184 T and 30% at 12 T.

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mediate and largeJ/t couplings. This state is a candidatedescribe DMS materials since it explains both the resistivmaximum aroundTC as well as the decrease in resistiviwith increasing applied magnetic field. In addition, it prvides an optimalTC . This clustered state is formed only fointermediate or largerJ/t and when the compensationstrong,p,0.4. Since for largeJ/t the carriers are localizedthe problem becomes one of percolation theory, whichalready been treated using different approachesapproximations.24 Here, only a very simple way of visualizing this state will be presented. First, consider a twdimensional lattice with 5% Mn spins represented as bldots, as shown in Fig. 19~a!. The carrier wave functionc(r )is considered to be localized around a Mn spin and itassumed to be a step function for simplicity, i.e.,c(r ) isnonzero only ifr ,r l wherer l is the localization radius in-troduced by hand. Sites where the wave function is not zare the black areas of Fig. 19~b!. In this case, sites can bclassified into connected regions, and that feature is indicain Fig. 19~c! using different shades. Each region is correlaand will correspond to a FM cluster. In this case all spbelong to some large cluster. As the concentrationx grows,the clusters will tend to merge. Asx decreases, these clustewill become more and more isolated.

Recent experimental work on~Ga,Cr!As have revealedunusual magnetic properties which were associated withrandom magnetism of the alloy. The authors of Ref. 38plained their results using a distributed magnetic polamodel that resembles the clustered-state ideas discussedand in Refs. 15, 23, and 24.

VIII. CONCLUSIONS

In this paper, dynamical and transport properties osingle-band model for diluted magnetic semiconductors h

at FIG. 19. Percolation picture for the formation of the clusterestate regime.~a! Two-dimensional lattice representing randomly lcated classical spins as black dots withx50.05. ~b! Black areasrepresent nonzero carrier wave function, assuming a step-funcprofile for the wave functions with radius equal to two sites,explained in the text.~c! Same as~b! but now showing connectedregions~which could in practice correspond to the FM clusters dcussed in the text! indicated with different shades of gray.

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been presented. The calculations were carried out on a laand using Monte Carlo techniques. The optical conductivdensity of states, and resistance vs temperature agreeexperimental data for Ga12xMnxAs if the model is in a re-gime of intermediate J/t coupling. In this region, the carrierare neither totally localized at the Mn sites nor free, asprevious theories. The state of relevance has some charaistics of a clustered state, in the sense that upon cooling fhigh temperatures first small regions are locally magnetiat a temperature scaleT* –causing a mild insulatingbehavior—while at an even lower temperatureTC the align-ment of the individual cluster moments occurs—causing mtallic behavior. This is in agreement with previouinvestigations.10,15,23,24

One of the main conclusions of the paper is summariin Fig. 20 where the ‘‘transport’’ phase diagram is sketch

FIG. 20. Sketch of the phase diagram indicating the conducand insulating regions, as obtained in the present investigationsdashed area in the paramagnetic phase indicates the crossovgion from a mild metallic weak-coupling regime to a mild insulaing strong-coupling regime.

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The low-temperature ferromagnetic phase is metallic~al-though often with poor metallicity!. The clustered state between T* and TC has insulating properties, and the samoccurs in a good portion of the phase diagram above thtwo characteristic temperatures. Note that here the termstallic and insulating simply refer to the slope of the resistity vs temperature curves. The values of the resistivitiesthe two regimes are not very different, similarly as observexperimentally.

This and related efforts lead to a possible picture of DMmaterials where the inhomogeneities play an important rIn this respect, these materials share characteristicsmany other compounds such as manganites and cuprwhere current trends point toward the key importancenanocluster formation to understand the colossal magnetsistance and underdoped regions, respectively. ‘‘Clusterstates appear to form a new paradigm that is useful to unstand the properties of many interesting materials.

ACKNOWLEDGMENT

The authors acknowledge the help of J. A. Verge´s in thestudy of the conductance. The subroutines used in this ctext were kindly provided by him. Conversations with AFujimori are also gratefully acknowledged. G.A. and E.were supported by NSF Grant No. DMR-0122523. Adtional funds have been provided by Martech~FSU!. Part ofthe computer simulations have been carried out on the CIBM SP3 system at the FSU School for Computational Sence and Information Technology~CSIT!.

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.

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66, 045207~2002!; Mona Berciu and R.N. Bhatt, Phys. ReLett. 87, 107203~2001!; see also C. Timm, F. Scha¨fer, and F.von Oppen,ibid. 90, 029701 ~2003!; Mona Berciu and R.N.Bhatt, ibid. 90, 029702~2003!.

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~2002!.16K.C. Ku, S.J. Potashnik, R.F. Wang, M.J. Seong, E. Johns

Halperin, R.C. Meyers, S.H. Chun, A. Mascarenhas, A.C. Gsard, D.D. Awschalom, P. Schiffer, and N. Samarth, Appl. PhLett. 82, 2302~2003!.

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2-12

T.

Sinrot

re

.

ne

hee

55

ere

-

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19In Ref. 7, a hybrid MC study of the multiband model for DMwas presented. In that reference the model was treatedkspace, approximating the Fermionic trace for its zetemperature value. On the contrary, our present work treatsFermionic trace exactly, and calculations are performed inspace.

20E. Dagotto, T. Hotta, and A. Moreo, Phys. Rep.344, 1 ~2001!; seealso J. Burgy, M. Mayr, V. Martin-Mayor, A. Moreo, and EDagotto, Phys. Rev. Lett.87, 277202~2001!.

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22The term ‘‘sketched’’ is used instead of ‘‘established’’ due to tinevitable uncertainties of the numerical effort, which is bason nearly exact results but obtained in small clusters.

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single-band model for diluted magnetic semiconductors...

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