Inter-impurity and impurity-host magnetic correlations in semiconductors with low-density transition-metal impurities
aa r X i v : . [ c ond - m a t . s t r- e l ] J un Inter-impurity and impurity-host magnetic correlations in semiconductorswith low-density transition-metal impurities
Yoshihiro Tomoda , Nejat Bulut , , and Sadamichi Maekawa , Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan CREST, Japan Science and Technology Agency (JST), Kawaguchi, Saitama 332-0012, Japan (Dated: May 31, 2008)Experiments on (Ga,Mn)As in the low-doping insulating phase have shown evidence for the pres-ence of an impurity band at 110 meV above the valence band. The motivation of this paper is toinvestigate the role of the impurity band in determining the magnetic correlations in the low-dopingregime of the dilute magnetic semiconductors. For this purpose, we present results on the Haldane-Anderson model of transition-metal impurities in a semiconductor host, which were obtained byusing the Hirsch-Fye Quantum Monte Carlo (QMC) algorithm. In particular, we present results onthe impurity-impurity and impurity-host magnetic correlations in two and three-dimensional semi-conductors with quadratic band dispersions. In addition, we use the tight-binding approximationwith experimentally-determined parameters to obtain the host band structure and the impurity-host hybridization for Mn impurities in GaAs. When the chemical potential is located between thetop of the valence band and the impurity bound state (IBS), the impurities exhibit ferromagnetic(FM) correlations with the longest range. We show that these FM correlations are generated by theantiferromagnetic coupling of the host electronic spins to the impurity magnetic moment. Finally,we obtain an IBS energy of 100 meV, which is consistent with the experimental value of 110 meV,by combining the QMC technique with the tight-binding approach for a Mn impurity in GaAs.
PACS numbers: 75.50.Pp, 75.30.Hx, 75.40.Mg, 71.55.-i
I. INTRODUCTION
The discovery of dilute magnetic semiconductor(DMS) materials is important because of possible spin-tronics device applications [1, 2, 3, 4]. The electronicstate of the alloy (Ga,Mn)As, where Mn substitutesfor Ga, has been investigated by various experimentalmethods including transport measurements, optical andphotoemission spectroscopy, and scanning-tunnelling mi-croscopy [5, 6, 7, 8, 9, 10]. In the low-doping regime( ≪ µ is located between the valence band and the IBS, theferromagnetic (FM) correlations between the impuritiesexhibit the longest range. We also show that the inter-impurity FM correlations are generated by the antiferro-magnetic (AFM) correlations between the impurities andthe host electrons. In addition, we use the tight-bindingapproximation to determine the host band structure andthe impurity-host hybridization for the three t g orbitalsof Mn in GaAs. Using tight-binding parameters con-sistent with photoemission measurements, we obtain anIBS energy which is in agreement with the experimentalvalue of 110 meV. Because of these results, we think thata microscopic understanding of the low-doping insulat-ing phase of (Ga,Mn)As is possible within the Haldane-Anderson model.The nature of the magnetic correlations in theHaldane-Anderson model of transition-metal impuritiesin semiconductors was studied using the Hartree-Fock(HF) approximation [14, 15]. It was shown that, whenthe chemical potential is located between the top of thevalence band and the IBS located at ω IBS , long-rangeFM correlations develop between the impurities medi-ated by the AFM coupling of the valence electrons tothe impurity magnetic moments. On the other hand,when the IBS becomes occupied, the spin polarizationof the host split-off state cancels the polarization of thevalence band, reducing the range of the FM correlations.The QMC calculations for the Haldane-Anderson modelsupport the HF picture for the role of the IBS in produc-ing the FM correlations [16]. The QMC results on therange of the FM correlations are in agreement with theHF predictions, however, the HF approximation under-estimated the value of ω IBS . These results on the mag-netic correlations for 0 . µ . ω IBS are different thanin the metallic case, µ <
0, where the inter-impuritymagnetic correlations exhibit Ruderman-Kittel-Kasuya-Yosida (RKKY) oscillations with period determined bythe Fermi wavevector k F . We note that the Haldane-Anderson model was also studied within HF by Krstaji´c et al. [17] for DMS materials and by Yamauchi et al. [18]for hemoprotein. The role of IBS for DMS materials wasalso discussed by Inoue et al. [19] within HF.In this paper, we are particularly interested in how themagnetic properties are influenced by the host electronicstate, which we model using two different approaches.First, we consider the simple case where the magneticimpurity has one orbital only and the host band struc-ture consists of quadratic valence and conduction bands.Here, in addition, we treat the impurity-host hybridiza-tion as a freely adjustable parameter. This single-orbitalmodel is described in Section 2.A, and the QMC resultsfor this case are presented in Sections 3 and 4 for twoand three-dimensional semiconductors, respectively. Westudy the inter-impurity and impurity-host magnetic cor-relations, the induced electron-density around the impu-rity, and show how these quantities depend on the param-eters of the Haldane-Anderson model and the dimension-ality of the semiconductor. These results show that theIBS and the magnetic correlations depend sensitively onthe model parameters.In order to develop a more realistic model for(Ga,Mn)As, we next use the tight-binding approxima-tion to obtain the band structure of the bare host GaAsand the hybridization matrix elements with the three t g orbitals of Mn substituted in place of Ga. In this ap-proach, the tight-binding parameters required for calcu-lating the hybridization are taken from an analysis ofthe photoemission data on Mn 2 p core level. This modelis introduced in Section 2.B and the QMC results ob-tained for this case are presented in Section 5. TheseQMC calculations keep all three of the Mn t g orbitals,and hence the multi-orbital effects are included except forthose due to Coulomb repulsion between different Mn or-bitals. Within this approach, we obtain an ω IBS whichis close to the experimental value of 110 meV. We alsoshow that the FM correlations between the impuritiesweaken as the IBS becomes occupied. These results em-phasize the importance of the IBS in the low-density limitof (Ga,Mn)As.Finally, we note that the approaches taken in this pa-per can be extended to the case of finite concentrationof Mn impurities in order to study the insulator-metaltransition and the metallic state of (Ga,Mn)As observedat higher Mn concentrations.
II. IMPURITY MODEL
The general model for describing transition-metal im-purities in a semiconductor host is given by Haldane-Anderson Hamiltonian [12], H = X k ,α,σ ( ε k α − µ ) c † k ασ c k ασ + X i,ξ,σ ( E dξ − µ ) d † iξσ d iξσ + X k ,α,i,ξ,σ ( V k α,iξ c † k ασ d iξσ + H . c . )+ U X i,ξ d † iξ ↑ d iξ ↑ d † iξ ↓ d iξ ↓ , (1)where c † k ασ ( c k ασ ) creates (annihilates) a host electronwith wavevector k and spin σ in the valence or conduc-tion bands denoted by α , and d † iξσ ( d iξσ ) is the creation(annihilation) operator for a localized electron at impu-rity orbital ξ located at site i . The first term in Eq. (1)represents the kinetic energy of the host electrons, andthe second term is the bare energy of the localized impu-rity orbitals, while the third term is due to the impurity-host hybridization. The last term represents the onsiteCoulomb repulsion at the impurity orbitals. We note thathere we are neglecting the Coulomb repulsion among thedifferent impurity orbitals. The effects of the Hund cou-plings will be considered in a separate paper. As usual inEq. (1), U is the onsite Coulomb repulsion, µ the chem-ical potential and E dξ is the bare energy of the impurityorbital ξ . In addition, the hybridization matrix elementis V k α,jξ = V k α,ξ e i k · R j , (2)where R j is the coordinate of the impurity site j and V k α,ξ is the value when the impurity is located at theorigin. We use the Hirsch-Fye QMC technique to studythe Haldane-Anderson impurity Hamiltonian, Eq. (1),for the the single- and two-impurity cases [13].In this paper, we perform the QMC calculations us-ing two different approaches for incorporating the hostband structure and the impurity-host hybridization. InSections 3 and 4, we consider a simple model where theimpurity site contains a single impurity orbital and thesemiconductor bands have quadratic dispersion, whilein Section 5 we use the tight-binding approximation fortreating the three t g orbitals of Mn impurities in GaAs.In the remainder of this section, we describe these twodifferent approaches for modelling the transition-metalimpurities in semiconductors. A. Single-orbital case
In Sections 3 and 4, we consider the simple casewhere the impurity site contains a single orbital in twoand three-dimensional semiconductor hosts, respectively.Hence, in this case Eq. (1) reduces to Eq. (1) of Ref. [16].
FIG. 1: (Color online) Schematic drawing of the semicon-ductor host bands ε αk (solid curves) and the impurity boundstates (thick arrows) obtained with HF in the semiconductorgap. The dashed line denotes the chemical potential µ . Furthermore, in these sections, we assume that the hostband structure consists of one valence ( α = v ) and oneconduction ( α = c ) bands with quadratic dispersionsgiven by ε k ,v = − D ( k/k ) (3) ε k ,c = D ( k/k ) + ∆ G . (4)Here, D is the bandwidth, k the maximum wavevectorand ∆ G the semiconductor gap. The energy scale is de-termined by setting D = 12 .
0. Figure 1 shows a sketchof the host band structure. For the Coulomb repulsionwe use U = 4 .
0, and set the bare value of the impurityorbital energy to E d = µ − U/
2, so that the impurity sitesdevelop large magnetic moments. We report results forsemiconductor gap ∆ G = 2 .
0, and inverse temperature β ≡ /T from 4 to 32. We also use a constant V for V k α,ξ , and treat V as a free parameter.For the single orbital cases of Sections 3 and 4, we showQMC results on the equal-time impurity-impurity mag-netic correlation function h M z M z i for the two-impurityHaldane-Anderson model. Here, the impurity magneti-zation operator is given by M zi = d † i ↑ d i ↑ − d † i ↓ d i ↓ , (5)and the fermion creation and annihilation operators actat a single orbital at the impurity site. In addition,we present results on the impurity-host correlation func-tion h M z m z ( r ) i for the single-orbital and single-impurityHaldane-Anderson model. Here, the host magnetizationat a distance r away from the impurity site is given by m z ( r ) = X α = v,c ( c † α ↑ ( r ) c α ↑ ( r ) − c † α ↓ ( r ) c α ↓ ( r )) . (6)For the metallic case, the correlation functions h M z M z i and h M z m z ( r ) i were previously studied by using QMC[13, 20, 21, 22]. In the single impurity case, we also present QMC data on the square of the impurity moment, h ( M z ) i , and the impurity susceptibility χ defined by χ = Z β dτ h M z ( τ ) M z (0) i . (7)The effects of IBS are clearly visible in these quantities.We also present QMC data on the charge distributionof the host material around the impurity. These QMCresults are obtained using Matsubara time step ∆ τ =0 .
25. At β = 16, h M z M z i varies by a few percent as ∆ τ decreases from 0.25 to 0.125. B. Tight-binding model for Mn in GaAs
In order to construct a more realistic model of(Ga,Mn)As in the dilute limit, we use the tight-bindingapproach to calculate ε k α and V k α,iξ of Eq. (1) for theMn t g orbitals. In Section 5, we will present QMC dataobtained using these results on ε k α and V k α,iξ .In this approach, the tight-binding band structure ε k α of GaAs host is obtained by keeping one s orbital andthree p orbitals at each Ga and As site of GaAs withthe zincblende crystal structure. We consider only thenearest-neighbor hoppings between the Ga and As sites.Figure 2 shows the resulting band energies ε k α versus k along various directions in the face-centered-cubic (FCC)Brillouin zone for pure GaAs. The Slater-Koster parame-ters [23] which have been used for obtaining these resultswere taken from Ref. [24]. In Fig. 2, the top of the va-lence band is located at the Γ point, where the energygap is 1.6 eV, which is consistent with the experimentalvalue. This simple approximation, with nearest-neighborhopping among the eight sp orbitals, reconstructs thefour valence bands reasonably. They are important forthe impurity-host magnetic coupling in (Ga,Mn)As, be-cause the chemical potential is located near the top ofthe valence band.When a Mn impurity is substituted in place of Ga inGaAs, the five 3 d orbitals of the Mn ion are split by thetetrahedral crystal field into the three-fold degenerate t g orbitals and the two-fold degenerate e g orbitals. Sincethe e g orbitals have bare energies which are much lowerthan the t g orbitals, we neglect the e g orbitals and keeponly the three t g orbitals of Mn in Eq. (1). In order tocalculate V k α,iξ for the Mn t g orbitals, we assume thatthe sp orbitals of Mn are the same as those of Ga, andthis way the periodic boundary conditions are also sat-isfied. In other words, at the Mn impurity site, we onlyadd the three Mn t g orbitals to the GaAs host. This is asimple model for a transition-metal impurity substitutedinto GaAs. However, this approach allows us to take intoaccount the effects of the host band structure beyond thequadratic dispersion. Furthermore, in Sections 3 and 4,we treat the hybridization matrix element V as a freeparameter. However, here, we perform the calculationsusing realistic parameters for the hybridization of the 3 d -12-10-8-6-4-20246810L G X W L ea [ e V ] FIG. 2: (Color online) Band structure of GaAs within thetight-binding approximation. orbitals with the host semiconductor bands. We also notethat in Ref. [25] the tight-binding and QMC techniqueshave been combined to study how the crystal structureof ZnO host affects the magnetic properties when Mn im-purities are substituted. However, in these calculationsonly one of the Mn 3 d orbitals is taken into account andthe multi-orbital effects are not included.Within the tight-binding approximation, the hy-bridization matrix element V k α,ξ , Eq. (2), is obtainedfrom V k α,ξ = h ψ k α | H | ϕ ξ i , (8)where | ψ k α i is the Bloch eigenstate of the host electrons, | ϕ ξ i the localized orbital eigenstate of an impurity lo-cated at the origin, and H is the Hamiltonian of thesystem with the Coulomb repulsion turned off. In orderto evaluate V k α,ξ , it is necessary to determine the valuesof the Slater-Koster parameters between the sp orbitalsof GaAs and the d orbitals of Mn, which are denoted bythe notation ( spσ ), ( pdσ ) and ( pdπ ) [23]. In the follow-ing, we determine the value of ( pdπ ) by the general equa-tion ( pdπ ) = ( pdσ ) / ( − .
16) [26]. We also set ( sdσ ) = 0for the following reason: When the chemical potentialis near the top of the valence band, the hybridizationmatrix elements near the Γ point become important forthe magnetic couplings. However, the s orbitals do notcontribute to the three degenerate valence bands at thetop of the valence band [27]. This is why we ignore thecontribution of the s orbitals to hybridization and set( sdσ ) = 0. The experimental estimate for the remain-ing Slater-Koster parameter is ( pdσ ) = − . p core level with a configuration-interaction analysis based (a) G X W L | V a = v . xy | [ e V ] (b) G X W L | V a = c . xy | [ e V ] FIG. 3: (Color online) Hybridization matrix element V k α,xy of the ξ = xy orbital of a Mn impurity with (a) the valenceand (b) the conduction bands of GaAs versus k along variousdirections in the FCC Brillouin zone obtained using the tight-binding approximation. on a cluster model [28]. In Section 5, we will presentQMC data obtained using ( pdσ ) = − .
14 eV yielding an ω IBS = 100 meV, which is in reasonable agreement withthe experimental value of 110 meV for (Ga,Mn)As in thelow-density limit.Figure 3 displays tight-binding results on V k α,ξ versus k for the ξ = xy orbital, and (a) the valence and (b)conduction bands along various cuts in the FCC Bril-louin zone. The values of V k α,ξ at the top of the valenceand at the bottom of the conduction band are particu-larly important for the IBS. We note that V k α,ξ = xy takeslarge values near the Γ point for the top valence bands,while it is weaker for the conduction bands. In fact, thehybridization of the xy orbital with the lowest-lying con-duction band vanishes at the Γ point. We also notice that V k α,ξ can be discontinuous around the high-symmetrypoints. In Section 5, we will present QMC data obtainedusing these values for ε k α and V k α,ξ . In addition, forthe bare energies of the impurity orbitals, we will use E dξ = µ − U/ U = 4 .
0, so that large magnetic mo-ments develop at the impurity sites. The QMC resultsdo not depend sensitively on small changes in the valuesof E dξ and U .After obtaining ε k α and V k α,iξ within the tight-bindingmodel described above, we perform the QMC calculationskeeping all three Mn t g orbitals. Hence, the QMC resultswhich will be presented in Section 5 include the multi-orbital effects except for the Hund coupling. However,we will present these QMC results only for the ξ = xy orbital in order to make comparisons with the the single-orbital cases of Sections 3 and 4. In particular, definingthe magnetization operator for ξ = xy as M zi = d † iξ ↑ d iξ ↑ − d † iξ ↓ d iξ ↓ , (9)we will present data on h ( M z ) i and the magnetic sus-ceptibility χ , Eq. (7), for the xy orbital within the single-impurity Haldane-Anderson model. For the two-impuritycase, we will present data on the inter-impurity magneticcorrelation function h M z M z i between the xy orbitals lo-cated at the impurity sites R and R . In addition, wewill discuss the magnetic correlation function betweenthe magnetic moment of the xy orbital and the host elec-trons h M z m z ( r ) i , where the host magnetization operatorat lattice site r is m z ( r ) = X α ( c † r α ↑ c r α ↑ − c † r α ↓ c r α ↓ ) (10)with α summing over the eight semiconductor bands, forthe single impurity case. In Section 5, we will see that theexperimental value of ω IBS is reproduced reasonably bycombining the tight-binding approach with the QMC cal-culations. In addition, we will see that the magnetic cor-relations weaken as the IBS becomes occupied, in agree-ment with the results of Sections 3 and 4.In the following sections, we will find that the quanti-tative results depend on the dimensionality and the bandstructure of the host material. In Sections 3 and 4, weshow results for the two and three dimensional host ma-terials with simple quadratic band dispersions. Then, inSection 5, we discuss the case for a GaAs host using thetight binding approximation.
III. QMC RESULTS FOR A 2DSEMICONDUCTOR HOST
In this section, we show results for the 2D case. Here,the density of states of the pure host, ρ , is a constantwith a sharp cutoff at the semiconductor gap edge, whichleads to stronger impurity-host coupling compared to the3D case. Here, we present results for hybridization ∆ ≡ πρ V varying from 1 to 4. A. Magnetic correlations between the impurities
Figures 4(a) and (b) show h M z M z i versus k R , where R is the impurity separation, for the two-impurity An-derson model for half-filled metallic ( µ = − .
0) and semi-conductor ( µ = 0 .
1) cases. In the metallic case, h M z M z i exhibits RKKY-type oscillations with both FM and AFMcorrelations depending on the value of k R . On the other FIG. 4: (Color online) Inter-impurity magnetic correlationfunction h M z M z i vs k R at various β for (a) µ = − . µ = 0 . -1 -0.5 0 0.5 10.60.70.80.9 b =4 b =8 b =16 b =32 D =1 m < ( M z ) > D =2 D =4 FIG. 5: (Color online) Impurity magnetic-moment square h ( M z ) i vs µ at various β for ∆ = 1, 2 and 4 for the single-impurity Haldane-Anderson model. hand, for µ = 0 .
1, we observe FM correlations of whichrange increases with β .The remainder of the data shown for the 2D case inthis section are for the single-impurity Anderson model.In Fig. 5, we show results on h ( M z ) i vs µ for variousvalues of ∆. As T is lowered, a step discontinuity de-velops in h ( M z ) i near the gap edge. The location ofthe discontinuity coincides with the location of IBS de-duced from data on h M z M z i , the impurity single-particlespectral weight A ( ω ), and the inter-impurity susceptibil-ity χ discussed previously [16]. Hence, Fig. 5 shows -1 -0.5 0 0.5 100.20.40.60.8 b =4 b =8 b =16 b =32 D =1 m T c FIG. 6: (Color online)
T χ vs µ at various β for ∆ = 1 forthe single-impurity Haldane-Anderson model. that the local moment increases rapidly as the IBS be-comes occupied. Here, we also observe that the momentsize decreases with increasing hybridization, as expected.Figure 6 shows T χ vs µ for ∆ = 1, where we observe thata step discontinuity develops at the same location as in h ( M z ) i shown in Fig. 5(a). For µ < ∼ ω IBS , T χ decreaseswith decreasing T due to the screening of the impuritymoment by the valence electrons. However, for µ > ∼ ω IBS ,the impurity susceptibility exhibits free-moment behav-ior in agreement with the role of the IBS discussed above.We note that determining the location of the IBS from A ( ω ) is costly in terms of computation time. For thisreason, in the single-impurity case, it is more convenientto determine ω IBS from data on h ( M z ) i versus µ . Inthe remaining sections, we will use this approach for de-termining ω IBS .Within the HF approximation and in 2D, the range ofthe FM correlations is given by (16 πρ ω IBS ) − / for 0 <µ < ω IBS . This implies that the range decreases withincreasing ω IBS and, hence, ∆. We find that the QMCresults on the maximum range of the FM correlationsare in quantitative agreement with the values from theHF calculations [16]. However, we also find that the HFapproximation underestimates the value of ω IBS . Forexample, for ∆ = 1, HF yields ω IBS = 0 . µ . B. Impurity-host correlations
In this section, we discuss the magnetic correlationsbetween the impurity and the host. In addition, we showresults on the induced charge oscillations around the im-purity site.Figures 7(a)-(c) show the impurity-host magnetic cor- s (r) k r D =1(a) m =-6.0 b =4 b =8 b =16 b =32 s (r) k r D =1(b) m =0.1 b =4 b =8 b =16 b =320 5 10 15 20-0.6-0.4-0.20 s (r) k r D =1(c) m =0.5 b =4 b =8 b =16 b =32 FIG. 7: (Color online) Impurity-host magnetic correlationfunction s ( r ) vs k r at various β for (a) µ = − .
0, (b) µ = 0 . µ = 0 . relation function s ( r ) defined by s ( r ) = 2 πk rn h M z m z ( r ) i (11)versus k r for the single-impurity Anderson model for µ = − . µ = 0 . µ = 0 . n is the electron density and r isthe distance from the impurity site. For µ = 0 .
1, thecoupling between the impurity and host spins is AFMfor all values of k r , while, for µ = − . s ( r ) exhibitsRKKY-type 2 k F oscillations. We also note that, for themetallic case, the magnetic correlations saturate beforereaching β = 32. Comparison of Fig. 7(b) and Fig. 4(b)for µ = 0 . µ is increased to 0.5, theAFM correlations between the impurity and host spinsbecome much weaker. This is because the IBS becomes -1 -0.5 0 0.5 1-0.8-0.6-0.4-0.20 b =4 b =8 b =16 b =32 D =1 m S ( k r = ) FIG. 8: (Color online) S ( k r = 25) vs µ at various β for thesingle-impurity Haldane-Anderson model. occupied for µ > .
1. Within the HF approximation[14, 15], the FM interaction between the impurities ismediated by the impurity-induced polarization of the va-lence electron spins, which exhibit an AFM coupling tothe impurity moments. The impurity-host hybridizationalso induces host split-off states at the same energy asthe IBS. When the split-off state becomes occupied, thespin polarizations of the valence band and the split-offstate cancel. This causes the long-range FM correlationsbetween the impurities to vanish. These QMC and HFresults emphasize the role of the IBS in determining therange of the magnetic correlations for a semiconductorhost.The total magnetic coupling of the impurity magneticmoment to the host is obtained from S ( k r ) = Z k r d ( k r ′ ) s ( r ′ ) (12)Figure 8 shows S ( k r = 25) vs µ for ∆ = 1 at various β . Here, we see that the impurity becomes magneticallydecoupled from the host when µ > ∼ ω IBS .Next, in Fig. 9, we show the modulation of the chargedensity around the impurity. Here, we plot p ( r ) vs k r ,where p ( r ) is defined by p ( r ) = X α = v,c πk rn ( n α ( r ) − n α ( ∞ )) (13)with n α ( r ) = P σ h c † ασ ( r ) c ασ ( r ) i . For the metallic caseof µ = − .
0, we observe long-range RKKY-type of os-cillations in p ( r ). When µ = 0 .
1, the charge densityaround the impurity is depleted up to k R ≈
20 at thesetemperatures. This depletion represents the extendedvalence hole which forms around the impurity. However,for µ = 0 .
5, the induced charge density decreases signifi-cantly as T is lowered, because now the IBS is occupied.We next integrate p ( r ), P ( k r ) = Z k r d ( k r ′ ) p ( r ′ ) , (14) p (r) k r D =1(a) m =-6.0 b =4 b =8 b =16 b =32 p (r) k r D =1(b) m =0.1 b =4 b =8 b =16 b =320 5 10 15 20-0.8-0.6-0.4-0.20 p (r) k r D =1(c) m =0.5 b =4 b =8 b =16 b =32 FIG. 9: (Color online) p ( r ) vs k r for ∆ = 1 and (a) µ = − .
0, (b) 0.1 and (c) 0.5 for the single-impurity Haldane-Anderson model. and plot P ( k r = 25) vs µ in Fig. 10. We observe thatthe total charge density around the impurity is most de-pleted when 0 < ∼ µ < ∼ ω IBS , which is due to the valencehole induced around the impurity. In the metallic case,the induced charge density is oscillatory and has a longrange as we have seen in Fig. 9(a). As µ approachesthe gap edge, the electron density around the impurityis depleted. However, when µ > ∼ ω IBS , we see that thisdepletion is cancelled by the extended charge density ofthe split-off state.
IV. QMC RESULTS FOR A 3DSEMICONDUCTOR HOST
In this section, we discuss the three-dimensional case,where the hybridization parameter πV N (0) vanishes atthe gap edge because of the vanishing of N (0) of thepure host. Hence, the impurity-host coupling near the -1 -0.5 0 0.5 1-1-0.8-0.6-0.4-0.20 b =4 b =8 b =16 b =32 D =1 m P ( k r = ) FIG. 10: (Color online) P ( k r = 25) vs µ at various β forthe single-impurity Haldane-Anderson model. gap edge is much weaker compared to the 2D case. Con-sequently, the FM correlations between the impuritieshave a shorter range. We find that the dimensionality ofthe host material strongly influences the magnetic corre-lations. In particular, we see that the IBS does not existin 3D unless the hybridization is sufficiently large.Here, we define hybridization as ∆ = πρ ∗ V where ρ ∗ is the density of states when the valence band is half-filled, ρ ∗ = k / (4 √ π D ). This choice allows us to usecomparable values for the hybridization matrix element V when we compare the 2D and 3D results. In the fol-lowing, we present results for ∆ = 1, 2 and 4.Figures 11(a)-(c) show h M z M z i vs k R at various val-ues of µ for ∆ = 1, 2 and 4. These results are for µ = − . µ = − . µ = 0 .
0. We observe that,for ∆ = 1, the FM correlations between the impuritiesweaken as µ approaches the top of the valence band. Onthe other hand, for ∆ = 2 and 4, the FM correlations arestronger for µ = 0 .
0. This is because, for ∆ = 1, the IBSdoes not exist in a 3D host, as we will see later in Fig.13, which shows results on h ( M z ) i versus µ .Figure 12 shows the impurity-host magnetic correla-tion function s ( r ) vs k r for the single-impurity case. In3D, s ( r ) is defined by s ( r ) = 4 π ( k r ) n h M z m z ( r ) i . (15)We see that the impurity-host coupling weakens rapidlyfor µ > ∼ .
0. These figures show that, in 3D and for ∆ = 2,the IBS is located at ω IBS ≈ .
0, which is consistentwith the results on h ( M z ) i shown in Fig. 13. In Fig.13, for ∆ = 1, we do not observe the development ofa discontinuity for temperatures down to β = 32. For∆ = 2, we observe the development of a step centered at µ ≈ .
0, as β increases. For ∆ = 4, a step discontinuity at µ ≈ . V . This is consistent with the dependence ofthe IBS on the dimensionality in the U = 0 case. These k R < M z M z > b =16(a) D =1 m =0.0 m =-1.0 m =-6.00 5 10 15 2000.10.20 5 10 15 2000.10.2 k R < M z M z > b =16(a) D =1 m =0.0 k R < M z M z > b =16(a) D =1 m =0.0 m =-1.0 m =-6.00 5 10 15 2000.10.2 k R < M z M z > b =16(b) D =2 m =0.0 m =-1.0 m =-6.00 5 10 15 2000.10.2 k R < M z M z > b =16(c) D =4 m =0.0 m =-1.0 m =-6.0 FIG. 11: (Color online) h M z M z i vs k R at β = 16 andvarious µ for hybridization (a) ∆ = 1 .
0, (b) 2.0 and (c) 4.0for the two-impurity Haldane-Anderson model in the 3D case. results show that the dimensionality of the host materialstrongly influences the magnetic properties.The results presented in Sections 3 and 4 show thatthe density of states of the pure host at the gap edge andthe value of the hybridization matrix element are crucialin determining presence and the location of the IBS. TheIBS is in turn important in determining the magneticproperties of the systems when transition metal impu-rities are substituted into a semiconductor host. Thismeans that the electronic state of the pure host materialwill also be crucial in determining the magnetic proper-ties. In the next section, we explore the consequences ofa more realistic band structure for a GaAs host using thetight-binding approximation. s (r) k r D =2(a) m =-0.1 b =4 b =8 b =16 b =320 5 10 15 20-2-10 s (r) k r D =2(b) m =0.0 b =4 b =8 b =16 b =32 s (r) k r D =2(c) m =0.1 b =4 b =8 b =16 b =32 FIG. 12: (Color online) s ( r ) vs k r at various β for ∆ = 2 andfor (a) µ = − .
1, (b) 0.0 and (c) 0.1 for the single-impurityHaldane-Anderson model in a 3D host.
V. QMC RESULTS FOR THE TIGHT-BINDINGMODEL OF A Mn IMPURITY IN GaAs
In this section, we present QMC data obtained withinthe tight-binding model of a Mn impurity in GaAs, whichwas introduced in Section. 2.B. Here, we have per-formed the QMC calculations keeping all three of theMn t g orbitals in Eq. (1), hence this approach includesthe multi-orbital effects except for the Hund coupling.In addition, we use a more realistic description of thesemiconductor bands ε k α for GaAs compared to thatof Sections 3 and 4. Furthermore, the hybridization V k α,ξ is determined by the tight-binding approach us-ing parameters consistent with photoemission measure-ments on Mn in GaAs, instead of being a free parameter.The following QMC results are obtained for the Slater-Koster parameters ( pdσ ) = − .
14 eV, ( sdσ ) = 0 eV, and( pdπ ) = ( pdσ ) / ( − . h ( M z ) i and T χ versus µ for the xy orbital near the top of the valence band for the single- -1 -0.5 0 0.5 10.50.60.70.80.9 b =4 b =8 b =16 b =32 D =1 m < ( M z ) > D =2 D =4 FIG. 13: (Color online) Impurity magnetic-moment square h ( M z ) i vs µ at various β for (a) ∆ = 1, 2 and 4 for thesingle-impurity Haldane-Anderson model in a 3D host.(a) -0.3 -0.2 -0.1 0 0.1 0.2 0.30.750.80.850.9 T=1500 KT=730 KT=360 KT=180 K m [eV] < ( M z ) > (b) -0.3 -0.2 -0.1 0 0.1 0.2 0.300.20.40.60.81 m [eV] T c T=1500 KT=730 KT=360 KT=180 K
FIG. 14: (Color online) (a) h ( M z ) i and (b) T χ versus µ for a Mn xy orbital obtained using the tight-binding model.Here, the top of the valence band is located at µ = 0. impurity Haldane-Anderson model. Here, the develop-ment of a step discontinuity in the semiconductor gap isclearly seen as T decreases down to 180 K. At low T andfor ( pdσ ) = − .
14 eV, the inflection point of the disconti-nuity occurs at 100 meV, which implies that ω IBS = 100meV. This is in good agreement with the experimentalvalue of 110 meV, especially if we note that the estimateof ( pdσ ) from the photoemission experiments is − . ω IBS dependson the Slater-Koster parameters, we repeated these cal-culations for different values of ( pdσ ) keeping ( sdσ ) = 0and ( pdπ ) = ( pdσ ) / ( − . pdσ ) = − . ω IBS ≈
300 meV, while for ( pdσ ) = − . | R |/a < M z M z > T=1500 KT=730 K(a) m =0.00 eV0 0.5 1 1.500.050.1 | R |/a < M z M z > T=1500 KT=730 K(b) m =0.05 eV0 0.5 1 1.500.050.1 | R |/a < M z M z > T=1500 KT=730 K(c) m =0.10 eV FIG. 15: (Color online) Inter-impurity magnetic correlationfunction h M z M z i vs R/a at various T for (a) µ = 0 . M zi isthe magnetization operator at the Mn xy orbital. tained a more smeared discontinuity in h ( M z ) i versus µ centered at µ ≈ h M z M z i versus the im-purity separation in lattice units, | R | /a , for the xy or-bital in the case of two Mn impurities substituted intoGa sites. Since we keep three orbitals at each Mn site,these calculations are more costly in terms of computertime. In this case, we present data for T = 1500 K and730 K. The Hirsch-Fye algorithm for the single and two-impurity Anderson model does not have the fermion signproblem. Hence, by using sufficient amount of computertime it is possible, in principle, to extend the calcula-tion of h M z M z i to lower temperatures. Even thoughthe data on h M z M z i are at high temperatures, the ef-fect of the IBS is observable. For T = 730 K and µ = 0 .
0, the FM correlations extend up to 1 . a , how-ever the FM correlations weaken as µ approaches ω IBS .We note that the spatial extent of the Mn-Mn interac-tion in low-density (Ga,Mn)As was also considered by | r |/a < M z m z ( r ) > As Ga As Ga As Ga Ga (a) m = 0.00 eVT=1500 KT=730 KT=360 K0 0.5 1 1.5-0.06-0.04-0.020 | r |/a < M z m z ( r ) > As Ga As Ga As Ga Ga (b) m = 0.05 eVT=1500 KT=730 KT=360 K0 0.5 1 1.5-0.06-0.04-0.020 | r |/a < M z m z ( r ) > As Ga As Ga As Ga Ga (c) m = 0.10 eVT=1500 KT=730 KT=360 K FIG. 16: (Color online) Impurity-host magnetic correlationfunction h M z m z ( r ) i vs r/a at various T for (a) µ = 0 . M z isthe magnetization operator at the Mn xy orbital. using the tight-binding approximation and perturbativetechniques in Ref. [29].Finally, we discuss the correlations between the hostelectronic spins and the magnetic moment of the xy or-bital at the Mn site for the single-impurity case. InFig. 16, we plot h M z m z ( r ) i versus the impurity distance | r | /a for various values of µ and T . Here, we denote thefirst-neighbor As site by As , the second-neighbor Ga siteby Ga , and so on. These results show that the devel-opment of the AFM correlations between the magneticmoment of the xy orbital and the neighboring host elec-trons as T is lowered down to 360 K. We also observethe weakening of the AFM correlations as µ approaches ω IBS .Using the tight-binding results for ε k α and V k α,ξ asinput, in this section, we have performed the QMC sim-ulations to study the magnetic properties (Ga,Mn)As inthe low-density limit. These results are similar to thosepresented in Sections 3 and 4 for the single-orbital casewith simple band structure. In particular, we saw that,1by using a realistic model for the host band structure andhost-impurity hybridization of (Ga,Mn)As, it is possibleto obtain an accurate value for ω IBS . Such quantitativeagreement supports the physical picture described in thispaper for the origin of the FM correlations in DMS in thelow-density limit.
VI. DISCUSSION
The computational approaches taken in this paper canbe extended to study the case of finite density of Mn im-purities in GaAs. Here, we have reported results for thesingle- and two-impurity cases. However, the Haldane-Anderson model can also be studied for a finite den-sity of impurities using the Hirsch-Fye QMC algorithm.This type of investigation could shed light on the metal-insulator transition encountered in the DMS materials atfinite density of impurities. In addition, it could help tounderstand the nature of the metallic state observed in(Ga,Mn)As for more than 2% doping.In the QMC results for the single-impurity case, wehave observed the extended nature of the induced chargedensity and the spin polarization around the impuritysite with a range ℓ determined by model parameterssuch as the hybridization, the bare band structure of thehost, the Coulomb repulsion, etc. In the two-impuritycase, when the polarization clouds overlap, the QMC re-sults showed that FM correlations develop between theimpurities. Then, the interesting question is what hap-pens for finite density of impurities as the average sepa-ration of the impurities becomes comparable to ℓ . Doesthe Haldane-Anderson model capture the insulator-metaltransition observed in (Ga,Mn)As? In addition, is it pos-sible to describe the electronic properties of the metal-lic state bordering the insulator-metal transition as thatof a disordered valence band with weak correlation ef-fects? Or, does the metallic state retain more propertiesof the single-impurity case so that the IBS and the split-off states continue to play roles even though they emergewith the valence band? In this case, what is the natureof the magnetic correlations among the impurities? Itwould be interesting to study this problem even in thesimplest case where each impurity has one orbital andthe density of states of the bare host and the hybridiza-tion are constants as discussed in Section 3. By extend-ing the QMC calculations to the finite-density case, wethink that it would be possible to address the question ofwhether the metallic state bordering the insulator-metal transition has unusual electronic properties. VII. SUMMARY
In this paper, we studied the single- and two-impurityHaldane-Anderson models using the Hirsch-Fye QMCtechnique. Our purpose was to develop a microscopicunderstanding of the low-doping insulating phase of theDMS material (Ga,Mn)As. We first discussed the sim-ple case where each impurity consists of one localizedorbital and the host band structure is described by twoand three-dimensional quadratic bands. We have pre-sented QMC results on the inter-impurity and impurity-host magnetic correlations and the charge density aroundthe impurity site. We observed that the presence andthe occupation of the IBS is important in determiningthe magnetic correlations. We saw that long-range FMcorrelations, which are induced by the AFM coupling ofthe valence electrons to the impurity moment, developbetween the impurities when the chemical potential isbetween the valence band and the IBS. We have alsoshowed that, in the low-doping limit, the magnetic cor-relations between the impurities do not exhibit RKKY-type oscillations in a semiconductor. These calculationsalso displayed how the model parameters determine themagnetic correlations.In order to develop a more realistic model of thelow-doping insulating phase of (Ga,Mn)As, we used thetight-binding approximation to map the system to theHaldane-Anderson Hamiltonian. In this approach, weused tight-binding parameters determined from the pho-toemission experiments and kept all three of the Mn t g orbitals in the QMC calculations. The resulting valuefor ω IBS is close to the experimental value of 110 meV.We also observed that the magnetic correlations weakenas the IBS becomes occupied. These results are usefulfor developing a microscopic understanding of the low-doping insulating phase of (Ga,Mn)As.
Acknowledgments
This work was supported by the NAREGI NanoscienceProject and a Grant-in Aid for Scientific Research fromthe Ministry of Education, Culture, Sports, Science andTechnology of Japan, and NEDO. [1]
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