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Transport in the metallic regime of Mn-doped III-V semiconductorsLouis-Franois Arsenault,* B. Movaghar, P. Desjardins, and A. YelonDpartement de Gnie Physique and Regroupement Qubcois sur les Matriaux de Pointe (RQMP), cole Polytechnique de Montral,Case Postale 6079, Succursale Centre-Ville, Montral, Qubec, Canada, H3C 3A7Received 7 August 2007; revised manuscript received 8 January 2008; published 26 March 2008The standard model of Mn doping in GaAs is subjected to a coherent potential approximation CPAtreatment. Transport coefficients are evaluated within the linear response Kubo formalism. Both normal andanomalous contributions to the Hall effect are examined. We use a simple model density of states to describethe undoped valence band. The CPA band structure evolves into a spin split band caused by the p-d exchangescattering with Mn dopants. This gives rise to a strong magnetoresistance, which decreases sharply withtemperature. The temperature T dependence of the resistance is due to spin-disorder scattering increasingwith T, CPA band-structure renormalization, and charged impurity scattering decreasing with T. The calcu-lated transport coefficients are discussed in relation to the experiment, with a view of assessing the overalltrends and deciding whether the model describes the right physics. This does indeed appear to be the case,bearing in mind that the hopping limit needs to be treated separately, as it cannot be described within the bandCPA.DOI: 10.1103/PhysRevB.77.115211 PACS numbers: 75.50.Pp, 72.10.d, 72.25.Dc, 75.47.mI. INTRODUCTIONMagnets, which can be made using semiconductors bydoping with magnetic impurities, are potentially very impor-tant new materials because they are expected to keep some ofthe useful properties of the host system. One important ques-tion is how much of the doped semiconductor band struc-ture does the system still have? Is it a completely new alloyor does the system still behave like a doped semiconductor,which can, for example, be described using Kane theory, astaught in standard textbooks.1 Many of these questions havebeen seriously investigated in extensive recent reviews.25The objective of this paper is to present a simple and mostlyanalytically tractable description of the magnetism and trans-port properties of GaMnAs. We need a theory which capturesthe essential physics, can be parametrized, and can be ap-plied to magnetic nanostructures without having to carry outab initio calculations each time some parameter has changed.Previous theories of GaMnAs and related materials havebeen either computational or semianalytical. However, thetreatment has often been split into three parts: magnetism,resistivity, and magnetotransport have been treated sepa-rately. Here, we show that the magnetism and transport canbe treated on the same footing using a one-band model. Theone-band model is of course not enough to explain the opti-cal properties. However, we show that magnetism, spin, andcharged impurity scattering can be formulated in the sameframework, using the coherent potential approximationCPA and a generalization thereof. In this paper, we focus,however, only on the magnetism and spin scattering aspectsand leave the computation of the charged impurity scatteringand the full quantitative comparison to a later paper. One ofthe most difficult and controversial aspects for all conductingmagnets is the origin of the anomalous Hall effect AHE.This is why we have devoted a section to remind the readerof the basic definitions and results. The full history of theAHE is given in Ref. 3.We focus on GaMnAs as a much-studied prototype. It isnow generally accepted that the observed ferromagnetic or-der of the localized spins in Mn-doped GaAs is due to Mnimpurities acting as acceptor sites, which generate holes.These are antiferromagnetically coupled to the local 5 /2 Mnspins, lowering their energy when they move in a sea ofaligned moments. It is the configuration of lowest energybecause spin-down band electrons move in attractive poten-tials of spin-up Mn and vice versa. Given that the spins arerandomly distributed in space, the magnetic state is also thestate of maximum possible order. Thus, free and localizedspins are intimately coupled. These materials exhibit strongnegative magnetoresistance because aligning the spins re-duces the disorder, and this lowers the resistance. The degreeto which the magnetization of the spins affects the scatteringprocess is dependent on the degree to which the magneticspin scattering is rate determining for resistance. We shall, inthis paper, only include the spin scattering process in order tofirst gain an intuitive understanding of the processes in-volved.The hole mediated magnetism point of view is notshared by everyone. Mahdavian and Zunger6 argue that themobile hole induced magnetism cannot explain the magne-tism in high band gap, strongly bound, hole materials such asMn-doped GaN. Their model, based on first principles super-cell calculations, predicts that the Mn-induced hole takes ona more d-like character as the host band gap increases, mak-ing the magnetism a d-p coupling property.The material of this paper is structured as follows. Wefirst present a short review of the phenomenology of the Halleffect in magnets. The Hamiltonian which describes theproperties of Mn-doped semiconductors is then introduced.We then discuss, in more detail, how to formulate transportin magnetically doped semiconductors. Following that, werecall how one can calculate the self-consistent CPA self-energy caused by the spin dependent term in the generallyaccepted Hamiltonian, for a one-band system. Once the self-energy is known, we compute the longitudinal and transverseconductivities using the Kubo formulas. When the Fermilevel is just above the mobility edge of the hole band, theKubo formula is still valid and describes the strong-PHYSICAL REVIEW B 77, 115211 20081098-0121/2008/7711/11521113 2008 The American Physical Society115211-1http://dx.doi.org/10.1103/PhysRevB.77.115211scattering random phase limit. The localized hopping regimeis treated in another paper.7 The CPA equations only need thedensity of states of the active free hole band as input. We willtherefore consider the semicircular Hubbard band which re-produces the correct free hole band edges and is mathemati-cally convenient for illustrating the effects of band inducedmagnetism.II. HALL EFFECTThe experimentally measured Hall coefficient RH is oftenwritten asRH = RN + RA,RA = axx + bxx2 M/Bz, 1where a, b are constants and xx is the resistivity. The firstterm, RN, is the normal Hall coefficient and scales with re-sistivity in the usual way and the second, RA, is the anoma-lous term, which, in general, can have two components, onelinear and the other quadratic with resistivity2 and is propor-tional to the magnetization M. The general relation for theHall coefficient isRH =RexyBzBzRexxBz2, 2where xy and xx denote the transverse and normal conduc-tivities, respectively, and Bz is the magnetic field.Karplus and Luttinger8 pointed out that the B field in-volves the magnetic moment of the material via the internalmagnetization M,Bz = 0Hzext + 1 NM Bzext + 01 NM , 3where N is the demagnetizing factor. Thus, the magnetizationterm is implicit in the normal contribution as a shift in themagnetic field. However, this form is not normally sufficientto explain the much larger magnetization contribution ob-served in ferromagnets.8 Throughout, we shall, for simplic-ity, suppose a thin film with perpendicular-to-plane magneticfield and thus take N=1, unless otherwise mentioned.III. MICROSCOPIC DESCRIPTIONA. HamiltonianIn this section, we formulate the basic model. The Hamil-tonian for this Mn-doped alloy, which most workers in thefield accept as the correct description,2 is given byHtot = Hp + Hloc + Hd + Hpd + HpB + HdB + Hso. 4In Eq. 4, the first term describes the free band holes. In thelocal Wannier or tight-binding representation, it isHp = m,n,stmncms cns, 5with tmn denoting the overlap or jump terms from sites m ton, cns , cns are creation and annihilation operators for a carrierof spin s at site n, and where the indices include multibandtransfers, if any. In the presence of a magnetic field, theoverlap has a Peierls phase, so that we write it as assumingthin film geometry, no internal field effect for a field perpen-dicular to planetmn = tmn0 eie/2BextRnRm, 6where ext stands for external, e is the electronic charge, andRn the position vector of site n. The second term in Eq. 4describes the diagonal atomic orbital energy that the va-lence band p hole experiences when it is sitting on an impu-rity site which may or may not be magnetic,Hloc = m,sEm,cms cms, 7where stands for magnetic M or nonmagnetic NM.The third term is the direct exchange coupling betweenthe Mn d-localized spins, which we neglect here becausewhen mobile carriers are present, they mediate the exchangecoupling and in the case of a low concentration, the Mn spinsshould be, on average, far away from each other. The fourthterm is the antiferromagnetic spin exchange coupling be-tween the valence p hole and the local d-Mn and ultimatelythe reason for magnetism in these materials,911Hpd =Jpd2 mNS s,scms ss Smcms, 8where ss is a vector containing Paulis matrices, i.e.,x ,y ,z. The indices s and s indicate which terms of the22 matrix we are considering. Finally, Sm is the Mn spinoperator at site m. The fifth and sixth terms are the Zeemanenergies of the holes and Mn spins in an external magneticfield along the z axis, Bzext.HpB = g*2BmsBz,mext cms ssz cms, 9HdB = 2B mNsBz,mext Smz , 10where g* is the effective g factor. Finally, we have the spin-orbit couplingHso = m,ns,ss,mhson,scms cns, 11wherehso =4m2c2 Vr p 12and where Vr is the total potential acting at a point r andwill include both normal crystal host sites, and impuritysites, and p is the momentum operator. Finally, c is the speedof light. In the tight-binding representation, disorder is nor-mally in the diagonal energies. For the spin-orbit term, dis-order will enter the Hamiltonian through variations in the sitepotential.ARSENAULT et al. PHYSICAL REVIEW B 77, 115211 2008115211-2B. Spin-orbit couplingIt is generally believed that the spin-orbit coupling is thecause of the anomalous Hall effect. This is one of the mostfascinating and universal observations made in conductingmagnets. In this section, we briefly discuss the basic phe-nomenology of spin-orbit coupling. It is useful to recall thebasic premises.When the Bloch wave function ansatz is inserted into theHamiltonian, the spin-orbit coupling produces two newterms which enter the Hamiltonian for the periodic part ofthe wave function unk nkr=unkeikr,hso =4m2c2Vr p + Vr k .13The first is the usual term and enters the band-structure cal-culation. The second is called the Rashba term12 and in acrystal is, in general, less important than the first because inthe nucleus, momenta are larger than the lattice momenta k.The Rashba term is not normally treated in the compu-tation of unk. However, in a lattice, it may be enhanced. Inthe Kane model, Chazalviel13 and De Andrada e Silva et al.14have shown how to handle the spin-orbit terms in a latticeand the Rashba term in the presence of a triangular potentialproduced by a gate in an inversion layer. In particular, Ref.14 has shown how to renormalize the coefficient of theRashba term and that this term acts when inversion symme-try is broken in an external field or triangular potential, forexample. The origin of the enhancement of both these spin-orbit effects can perhaps be understood as similar to the ori-gin of the change of the effective mass in the lattice. Thespin-orbit energy can be enhanced by the presence of thelattice, but the enhancement can be very different from caseto case, and needs to be examined in each case separately.For example, naively speaking, one may consider the spin-orbit force acting on a particle moving in a slowly varyingpotential of longer range than the lattice spacing as a spincurrent of an effective mass electron. One can think of thescattered particle as having an effective mass m* and gener-ating a spin-orbit interaction which scales as 4m*c2insteadof 4mc2 . When the particle is scattered from an atomic sizeimpurity in a lattice, it is forced to come back many timesand spends a longer time on the impurity than in free space.Leaving aside these intuitive pictures, one can, in any case,make the analysis quantitative using the Kane Hamiltonian,which takes the lattice modulation into account in a kind ofrenormalized perturbation theory and gives explicit resultsfor the unk part of the wave function.1 The Kane Hamiltoniancan then be used to treat scattering from an impurity. Thus,for example, the matrix elements of the position operatorkrk are very different for plane wave states and for theKane solutions.8,13In tight binding TB, the spin-orbit matrix elements arecalculated using atomic orbitals, with two terms: the intra-and the interatomic contributions. The magnitude of theintra-atomic term is just what it would be for the correspond-ing orbitals on the atoms in question. The interatomic term isnormally small and not included in tight-binding band-structure calculations. When an electron jumps from site tosite, it experiences a net magnetic field produced via thecross product of its velocity and the electric field gradient ofthe neighboring ions. This field then couples to its spin.Since it is related to the intersite momentum or velocity, itdepends on the transfer rate tmn where the indices m, n in-clude a band index m=m ,. Remember that the velocity xdirection operator is given byvx =im,n,sRm,nxtmncms cns, 14where Rm is the position vector of site m and Rm,n=RmRn. An enhancement of this spin-orbit energy, if any, has tobe calculated by taking the expectation value of the spin-orbit term using the calculated Bloch states as one would forthe dispersion relation ks and for the effective masses. Weshall examine the spin-orbit coupling in more detail below.IV. KUBO TRANSPORT EQUATIONSIn order to compute the transport properties of the mag-netically doped semiconductors in the linear response re-gime, we need to introduce the Kubo formulas. In this sec-tion, we show how to compute the transport coefficients.A. General considerationsThe conductivity in linear response to an electric field isusually written as1517 =ie2lim0,vv + + if f,15where the v are the velocity operators in the respectivedirection . For a given Hamiltonian, they are derived fromthe Heisenberg relation iv= x ,H with x the positionoperator. and are the exact wave functions and energylevels. The wave functions include any disorder and mag-netic and spin-orbit couplings. f is the Fermi function and is the frequency of the applied electric field; e is the elec-tronic charge. This formula can now be written in the par-ticular representation selected, e.g., TB or Bloch states.Within the band-structure approach for semiconductors, onewould substitute the eight-band k p wave functions and en-ergies into Eq. 15, include spin splitting through a Weissfield and spin-orbit coupling, and treat disorder as a lifetimecontribution in the energy levels. In this picture, the dopedsemiconductor retains the pure band-structure features, apartfrom a complex lifetime shift. We shall use the one-bandCPA for the coupling of the holes to the magnetic impurities.For the calculation of the ordinary transport coefficients, wemay neglect the spin-orbit coupling. We obtain, for the lon-gitudinal conductivity of the one-band TB model for =0and per spin s,TRANSPORT IN THE METALLIC REGIME OF Mn-DOPED PHYSICAL REVIEW B 77, 115211 2008115211-3xxs =e2 dE fEE dXImGsE,2,16whereGsE, =1E s, 17s = sR isI, 18andX =1Nkvx2 k . 19sE is the CPA self-energy for spin state s, to be calculatedself-consistently using the CPA condition and where the ve-locity and effective masses of the free electron band aregiven byv =1kk, 20M1 =122kkk, 21where M is the effective mass tensor. The CPA self-energycontains the information on the net spin splitting producedby the magnetic impurities and the scattering time. It definesthe effective medium which is to be discussed below.The Hall conductivity, given by the antisymmetric part ofthe transverse conductivity,18,19 is somewhat more compli-cated because the magnetic field has to be dealt with first. Tofirst order in magnetic field and with one TB band, we findfor the antisymmetric part referred to by the index a for=0,15,17,18xya =2e32B3 s dE fEE dYImGsE,3, 22whereY 1Nk vx2Myy+vy2Mxx2vxvyMxy k . 23In Eq. 22, the dispersion relation k can be any one-bandstructure any choice of tmn0 where k is defined throughtmn0 = 1NkeikRmnk. As was shown by Matsubara andKaneyoshi,18 for a weak magnetic field, the Peierls phase canbe factored out of the Green function and this is why we onlyneed the Green function Eq. 17 that does not include theorbital effect of the field. The effects of the different phasesfrom the tmn and from the Green functions, after an expan-sion to first order in the field, are reflected in the topologydependent term Y. Furthermore, Eq. 22 does not containthe spin-orbit scattering contribution. It can be included as anextrinsic effect. In the spirit of the skew scattering Bornapproximation1921 and in the weak scattering limit, we can,at the end of the calculation, introduce an extra internal mag-netic field so that the total field entering Eq. 22 isB = Bz +m*e 1sz , 24where Bz includes the internal magnetization. However, fromEq. 3, for a field perpendicular to plane, N=1. This result,Eq. 24, is written in the notation of Ballentine.19 The effec-tive spin-orbit field depends on the thermally averaged spinpolarization z, with its sign to be discussed in Sec. IV B,and 1s is the so-called skew scattering rate. The secondterm of Eq. 24 involves, in addition to the sign of thecarrier, the sign of the spin orientation.Engel et al.21 claimed that the impurity spin-orbit cou-pling in the lattice environment can be 6 orders of magni-tude larger than in vacuum. The large enhancement canmean that, in some cases, the skew scattering dominateswhere there should exist a substantial intrinsic contributionas well.21,22 This is why developing a way to calculate theextrinsic contributions to the anomalous Hall effect, as we dohere, is important and useful.B. Anomalous Hall effectLet us now show how one can derive such an anomalousHall effect, in our formalism TB approximation, from theexistence of a spin-orbit coupling and how one can arrive atthe concept of an effective magnetic field, as given inEq. 24.In tight binding, the spin-orbit coupling is given by Eq.11 and Vr is the total potential experienced by the chargeat point r, with p the momentum operator. The spin-orbitcoupling can be separated into an intra- and an inter-atomic contribution by splitting the momentum operator asp=pa+pinter. For spherically symmetric potentials, the twoterms are given byVso = iirli + nr Rnnr pinter ,25nr =4m2c2Vnr Rnr Rn , 26where nr is related to the spin-orbit coupling strength.Here, pinter is the usual zero-field interatomic momentum op-erator and li is the atomic orbital operator and is oftenquenched so that its expectation value is zero, but nondiago-nal same site interorbital matrix elements can be veryimportant.2325The second contribution in Eq. 25 is due to the inter-atomic motion and must be evaluated using pinter=mv, wherev is given by Eq. 14 and involves the intersite transferenergy t and position operatorr = m,sRmcms cms. 27Here, we have a spin-orbit coupling only because the particlecan jump to another orbital. When substituting the site rep-ARSENAULT et al. PHYSICAL REVIEW B 77, 115211 2008115211-4resentation in the second term of Eq. 25 and assumingspherical symmetric potentials, either impurity with effectivelocal charge eZn=eZimp or host eZh, we haveiVso2j =4mc2in,l eZn40r Rn3 r Rnl i Rljtlj , 28where i is the imaginary number, while i is the orbital atsite i. The analysis of this term is not trivial and depends onthe lattice and potential distribution in question. If one hasdecided that the extrinsic contributions are dominant, thenone only needs to sum around the impurities. In general, thesimplest approach is to take only the largest terms in thesum, i.e., the diagonal terms l= i. In this case, we getiVso2j = iijtij ,iij =4mc2in eZn40r Rn3 r Rni i Rij . 29It is convenient to treat this effect as a spin dependent phasein the transfer, in analogy to the Peierls phase, to first order.Indeed, to first order ex1+x, so that we can write iij=eiij 1 and Hp+Hso asHp + Hso = i,j,tijeiijci cj. 30Then, we also note that the Rn=Ri and Rn=R j terms in Eq.29 are zero after the vector product, leaving the gradientfield contributions due to the nearest neighbors as the largestof the remaining terms in the n sum. A similar result has beenderived in the context of the Rashba coupling by Damkeret al.26With Eq. 29, we can rewrite the spin-orbit matrix ele-ment ashijso = ie2tij n,ni,jR j Rin Bn,so, 31where Rin=RiRn and where we have introduced a spin-orbit magnetic field defined byBn,soi j =2mc2i Zn40Ri Rn3i . 32In summary, let us write in this approximation a new totalphase for Eq. 6,ij =e2 Bz R j Ri + n,ni,j R j Rin Bn,so .33We thus see what the spin-orbit coupling does to the TBHamiltonian. It is, in terms of energy, a small effect. Itseffect on the band structure and magnetism will be neglected.In order to obtain Eq. 22 with B defined as Eq. 24, onehas to decouple the trace over the spin polarization from theremaining expression and this automatically makes z theoverall mobile spin polarization. In reality, the skew scatter-ing is due to carriers near the Fermi level and therefore zshould be the average weighted at the Fermi level. We willreturn to this point when we attempt to fit Eqs. 22 and 24to the experiment in Sec. VI B.C. Transport coefficients for a simple cubic bandTo perform a calculation, one needs to specify a latticetopology tmn0 . The simplest one is a simple cubic latticewith nearest neighbor hopping. The dispersion relation isgiven, for a d-dimensional system, by k=2t=1d coska,where a is the lattice constant of the simple cubic. With thisparticular form, we can show27 that the longitudinal and Hallconductivities at zero frequency can be written, using Eqs.16 and 22, with no approximations, asxx =e2dad2s fEE ImGsE,2 zD0zdzddE , 34xya =4e3B3dd 1ad42s fEE ImGsE,3 zD0zdzddE ,35where GsE , is given by Eq. 17, d is the number ofdimensions, a stands once again for antisymmetric, andD0z is the density of states of the pure crystal. In three-dimensions, D0z can be found knowing that, for the abovek, for a three-dimensional cubic lattice, the Green functionis28G0E =122t0dxKx , 36where Kx is the complete elliptic integral of the first kind,with x= 4tE+i2t cos .Now, we can return to the calculation of the new bandsgenerated by the hole-spin coupling, and for this, we use theCPA. Later, we use Eq. 24 to estimate the magnitude of theskew scattering Hall effect.V. COHERENT POTENTIAL APPROXIMATIONEQUATIONS FOR A ONE-BAND MODEL WITHCOUPLING TO LOCAL SPINSIn this section, we show how to compute the new energybands in the presence of a high concentration of dopants,magnetic and nonmagnetic. The magnetic coupling betweenthe Mn spin and charge and the holes in the valence bandcreates a new valence band structure. This new Mn-inducedTRANSPORT IN THE METALLIC REGIME OF Mn-DOPED PHYSICAL REVIEW B 77, 115211 2008115211-5band will be magnetic at low enough temperatures. Since theMn are randomly distributed, we cannot use Blochs theo-rem. One way forward is to use the powerful self-consistentsingle site approximation known as the CPA. The CPAself-energy29 E is determined by the condition that if at asingle site, the effective medium is replaced by the true me-dium, then the configurationally averaged t matrix producedby scattering from the difference between the true mediumand effective medium potentials must vanish.29In the present case, we have localized Mn spins of mag-nitude S=5 /2 so that there are six possible states Sz=5 /2, . . . ,5 /2 and thus seven possible sites in the systemincluding the nonmagnetic ones. We have for a spin-upcarrier the normal nonmagnetic sites with concentration 1x with local site energy ENM and magnetic field interaction g*2BBz,mext and the magnetic sites with concentration x withlocal site energy EM and magnetic couplingJpd2 Smz g*2BBz,mext . However, there is also the possibility of a spinflip of both the carrier and the impurity with interactionJpd2 Sm+ . For a spin-down carrier, we have ENM at a nonmag-netic site and magnetic field interaction g*2BBz,mext . For a mag-netic site, we still have EM as the local site energy but themagnetic coupling is now Jpd2 Smz + g*2BBz,mext and the spin-flipinteraction is given byJpd2 Sm . The concentration of spin car-rying impurities is defined by x and isNSNL, where NS is thenumber of sites with an impurity, while NL is the total num-ber of sites in the system. The CPA conditions are givenby27,301 xtmNM th + xtmM th = 0,1 xtmNM th + xtmM th = 0, 37and the thermal average of an operator isOmSzth =TrOmSzeHSTreHS=SzOSzehmSzSzSzehmSzSz SzOSzPSz . 38In order to find the spin dependent t-matrix operator for theCPA, it is useful to rewrite the Hamiltonian as a 22 matrixin spin space. The t and t operators are the diagonal ele-ment of the t matrix. The expressions for the t matrix areessentially the same as the ones obtained previously by Ta-kahashi and Mitsui.30 The minor difference is that in thepresent case, we include the magnetic field, so that the Zee-man term enters the t operator. The inclusion of the Zeemanterm is straightforward: one has to add, when N=1, g*2BBz,mextwith the appropriate sign in the expressions of Ref. 30. Tocalculate the CPA t matrices, we need the effective mediumGreens function which is no longer dependent on the site m.GsE GmmsE =1Nk1E ks sE. 39In the previous equation, GijsE refers to the Greens func-tion of a carrier of spin s between sites i and j for oneparticular configuration of disorder. The average is over allpossible realization of disorder.The local density of states isDsE = 1ImGmmsE , 40and the hole-spin concentration with spin s at site m can bewritten aspms = dEfEDsE . 41Assuming mean field theory for the energy entering the Bolt-zmann factor, the probability that the local spin has a valueSz isPSz =ehmSzSzehmSz , 42wherehm =Jpd2pm pm 2BBz,mext . 43To compute the CPA self-energy sE, we need to specifythe lattice dispersion of the hole band. In effect, it suffices tospecify the bare pure lattice density of states D0E sincethe real and imaginary parts are simply related. The Fermilevel is determined by the condition that the total hole con-centration p is known and fixed relative to the total impurityconcentration with x as a maximum value. In general, therewill be fewer holes than dopants, but the number is notknown and remains a fit parameter, so we havep = dEfEDE + DE . 44This now allows us to compute the CPA self-energy self-consistently and then to determine the new density of statesvia Eq. 40 and the local spin polarization and mobile spinconcentration via Eqs. 38 and 41. Thus, we can determinethe magnetization and the transport coefficients, the resistiv-ity and the conductivity as a function of B via Eq. 34, andthe magnetoresistance as the relative change with B. Finally,the normal and anomalous Hall RH can be obtained viaEq. 35.VI. APPLICATIONS OF THE COHERENT POTENTIALAPPROXIMATIONFor numerical calculations, to simplify the analysis andfor proof of principle, instead of the D0E found with Eq.36, we use the Hubbard function, given byARSENAULT et al. PHYSICAL REVIEW B 77, 115211 2008115211-6D03DE =2W EW2W2 E2, 45where W is half the bandwidth and is the Heaviside stepfunction. Equation 45 has the same band edge behavior asthe D0E calculated with Eq. 36. With the simple form ofEq. 45, the first integration in Eqs. 34 and 35 is analyti-cally tractable.A. dc conduction and magnetoresistivityHere, we calculate the dc conductivity xx from Eq. 34and the magnetoresistance. The calculations have been doneassuming N=1 thin film. The calculations for the magne-toresistance have been performed without taking into ac-count the spin-orbit interactions.Figures 1 and 2 are plots of the CPA density of states fortwo values of p and Jpd assuming that the scattering is solelydue to the spin potential, that is, when EM =ENM =0. Thediagrams shows the evolution of the spin dependent densityof states as a function of temperature for four different tem-peratures in each figure. The temperature is measured inunits of the bandwidth W. In Figs. 1d and 2d, one can nolonger see the spin splitting. The position of the Fermi levelis also shown. Note that all energy parameters are normal-ized by W E=E /W. From Figs. 1 and 2, one can see thatwhen the magnetic coupling Jpd is very small, the split banddisappears see Ref. 27 for details at low Jpd, whereas in theopposite limit, the spin bands split off and a pseudogap ap-pears.Figure 3 shows the resistivity in zero B field as a functionof temperature with p as a parameter. Results for two valuesof Jpd are presented. The results for lower Jpd than 0.35W aresimilar to Fig. 3a.27 When the impurity bands completelysplit from the valence band for example, for Jpd=0.5W, thequalitative behavior changes, as there are now two well de-fined bands, above and below the Fermi level impurity andvalence separated by a gap see Fig. 2 above and Ref. 27.At high hole concentration p0.8x, the system is metallicand resistance increases with spin disorder, at first rapidly,and then decreases again above Tc. This is intuitively to beexpected because at first, as temperature increases, the disor-der increases, and then in the paramagnetic phase, thermalbroadening overcomes the potential scattering disorder andthe lifetime averages out. We should remember that in thesimplest Boltzmann approach the conductivity is given byxx = e2s DsE fEE vs2EsEdE 46and at low temperatures, depending on the product of thescattering time and the density of states at the Fermi level. Inan alloy, either quantity can change with B and T and deter-mine the conductivity. Above Tc, the spin splitting disappearsand the quantities involved are, in the absence of chargedimpurity scattering, only weak functions of temperature. Atvery low hole concentration, the Fermi level is in a region ofsmall density of states where we expect localization. How-ever, CPA is a mean field method and does not produce lo-calization. Even though the resistance increases with de-1.5 1 0.5 0 0.5 1 1.500.10.20.30.40.50.60.7ED(E)1.5 1 0.5 0 0.5 1 1.500.10.20.30.40.50.60.7ED(E)1.5 1 0.5 0 0.5 1 1.500.10.20.30.40.50.60.7ED(E)1.5 1 0.5 0 0.5 1 1.500.10.20.30.40.50.60.7ED(E)(b)(a)(c)(d)FIG. 1. Density of states DOS for different temperatures, akBT=0, b kBT=5103W, c kBT=7.99103W, and d kBT=8.1103W, when x=0.053, ENM =EM =0, BB=0, Jpd=0.4W,and p=0.8x. The full line curve is the DOS for the spin-down car-rier, the dashed one is the DOS for the spin up, and the vertical lineshows the Fermi level. One can see that the density of states is,relative to a nonmagnetic system, a strong function of temperature.TRANSPORT IN THE METALLIC REGIME OF Mn-DOPED PHYSICAL REVIEW B 77, 115211 2008115211-7creasing density of states at the Fermi level, the conductionprocess continues to be a band conduction albeit with a shortrelaxation time. Even if the mean free path reaches the latticespacing random phase limit, the conductivity is still fargreater than hopping conductivity between localized levels.31The effect of temperature on resistivity above Tc is not toosignificant in the nonlocalized intermediate cases. The tem-perature dependence of the resistance is, in general, a com-plex interplay of density of states, velocity, and relaxationtime imaginary part of the self-energy. The temperatureinduced lowering of the charged impurity screening lengthsee Eqs. 4951 is not included here. In this range ofexchange coupling, Jpd does not seem to strongly influencethe structure of the resistivity versus T curves but doeschange their magnitude. In Fig. 3, for p=0.1x, a low carrierconcentration, one may see that the resistance increases withJpd. In contrast, in the high hole concentration limit p=x,there is only a relatively weak variation of resistance behav-ior with Jpd. The complex but regular behavior of resistancewith temperature is a manifestation of Eq. 46, reflecting thedifferent elements which determine the value of resistancefor a given set of parameters.1.5 1 0.5 0 0.5 1 1.500.10.20.30.40.50.60.7ED(E)1.5 1 0.5 0 0.5 1 1.500.10.20.30.40.50.60.7ED(E)1.5 1 0.5 0 0.5 1 1.500.10.20.30.40.50.60.7ED(E)1.5 1 0.5 0 0.5 1 1.500.10.20.30.40.50.60.7ED(E)(b)(a)(c)(d)FIG. 2. Density of states DOS for different temperatures, akBT=0, b kBT=2.5103W, c kBT=4.5103W, and d kBT=5.4103W, when x=0.053, ENM =EM =0, BB=0, Jpd=0.5W,and p=0.3x. The solid line curve is the DOS for the spin-downcarrier, the dashed one is the DOS for the spin up, and the verticalline shows the Fermi level. One can see that the density of states is,relative to a nonmagnetic system, a strong function of temperature.0 0.005 0.01 0.0150246810121416kBT/W(mcm)p = 0.1xp = 0.3xp = 0.5xp = 0.6xp = 0.7xp = 0.8xp = 0.9xp = x0 0.005 0.01 0.0150510152025kBT/W(mcm)p = 0.1xp = 0.3xp = 0.5xp = 0.8xp = 0.9xp = x(b)(a)FIG. 3. Resistivity as a function of temperature for two Jpd, aJpd=0.35W and b Jpd=0.4W for an impurity concentration x=0.053, ENM =EM =0, and BB=0, for two values of Jpd, with p asa parameter.ARSENAULT et al. PHYSICAL REVIEW B 77, 115211 2008115211-8Figure 4 shows the resistivity at one temperature as afunction of impurity concentration for varying degrees ofhole doping. The results of Figs. 3 and 4 are based on thespin scattering model with no additional potential scatteringterms and no electron-phonon interaction. A fit to experimen-tal data must take these other mechanisms into account aswell. Thus, the complete CPA self-energy should includeother sources of random potential scattering nonzero valuesof ENM, EM, in particular, the charged impurity scatteringterms discussed below. We should also add, when necessary,the electron-phonon self-energy epE. The imaginary partof the charged impurity self-energy will contribute anothersource of temperature dependent lifetime broadening, andthe real part will enter the density of states.Figure 5 shows the resistivity as a function of temperaturefor various magnetic fields, for one value of x, p, and Jpd,and again with EM =ENM =0. Apart from very low and veryhigh temperatures, increasing the magnetic field decreasesthe resistivity, by favoring the alignment of the impurityspins. The magnetic field also eliminates the sharp metal-insulator transition around Tc, replacing it with a smoothtransition.Figure 6 shows the relative magnetoresistivity defined byMR xxBext xxBext = 0xxBext = 0, 47as a function of B, for various value of T. The overall trendin this exclusively spin scattering model is the strong nega-tive magnetoresistivity, as one would expect, since increas-ing the Mn spin alignment reduces disorder and thus reducesthe scattering lifetime, and no other sources of scattering areconsidered. However, lifetime is not the only quantity enter-ing the conductivity. As shown in Eq. 46, the density ofstates DEF at EF also plays an important role. There arealso regimes of positive magnetoresistivity. From Fig. 6a,we see that it is also possible for the resistance to increasewith B at low T when the hole density is very low. This isprobably because in this limit, the density of states at theFermi level decreases with magnetic field and this effect isstronger than the concomitant increase of the carrier lifetimedue to the suppression of spin disorder with B. However, wealso know that for low concentrations, when the Fermi levelis at the band edge or in the region of localized states, otherchanges arise which are not due to spin-disorder scatteringand which require another approach which is based on local-ization. In the hopping regime, not describable by CPA, the0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1100101102x(mcm) p = 0.1xp = 0.3xp = 0.5xp = 0.8xp = xFIG. 4. Resistivity as a function of impurity concentration x, forvarying degrees of hole doping. The other parameters are Jpd=0.35W, ENM =EM =0, kBT=3.5103W, and BB=0.0 0.005 0.01 0.01501234567891011kBT/W(mcm)FIG. 5. Resistivity as a function of temperature for various mag-netic fields. Other parameters are x=0.053, ENM =EM =0, Jpd=0.4W, and p=0.8x. The solid line is the zero-field result, thedashed line is for BB=1105W, the dotted line is for BB=1104W, and the dashed-dotted line is for BB=3104W.0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 516141210864202BB/W ( 10 4 )MR(%)kBT = 3.64x10 4WkBT = 1.31x10 3WkBT = 1.80x10 3WkBT = 0.015W0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5504540353025201510505BB/W ( 10 4 )MR(%)kBT = 3.64x10 4WkBT = 4.03x10 3WkBT = 5.78x10 3WkBT = 0.015W(a)(b)FIG. 6. Relative magnetoresistivity for two carrier concentra-tions, a p=0.1x and b p=0.8x, for impurity concentration x=0.053, ENM =EM =0, and Jpd=0.35W, with T as a parameter.TRANSPORT IN THE METALLIC REGIME OF Mn-DOPED PHYSICAL REVIEW B 77, 115211 2008115211-9magnetic field can, for example, squeeze the localized wavefunctions and increase the resistance by reducing overlap.However, it also shifts the energy and the mobility edge suchas to reduce resistance.32 When the calculated resistance ishigh and when the Fermi level is in a region of small densityof states 1019 /cm3 eV, we should therefore not trust theCPA, which gives a good description only of the metallicregime.B. Hall effectBy definition, the Hall resistance is given byyx = ReyxRexx2 ReyxRexy. 48Equation 48, being a general definition, the normal and theanomalous spin-orbit generated Hall resistances are included.In the CPA, the sign of the normal Hall effect is not de-termined by a simple relationship but depends on the disper-sion of the lattice and the behavior of the real part of theGreens function at the Fermi level.15,27 A full analyticalanalysis of the CPA Hall sign is beyond the scope of thispaper but we can state that there is no simple rule. The clos-est to one is that RH is electronlike when the density of statesincreases with energy and holelike when it decreases. In theapproximation of skew scattering, the sign of the anomalouseffect follows the sign of the normal effect as long as thecarriers at the Fermi level are polarized in the same directionas the total magnetization. This is true here, as can be seenfrom Fig. 1. The majority spin band at the Fermi level hasthe same sign as the overall magnetization, which is domi-nated by the localized spins. Thus, in the phase addition ap-proximation of Eq. 33, in order to get the right sign, it isessential to keep the correlation between the spin and energy.The decoupling of the spin magnetization out of the Kuboformula can give rise to the wrong sign. Here, the overallpolarization, found using Eq. 41, is opposite to the direc-tion of the field as it should be. We correct for this effect byassuming that z is in the same direction as the overallmagnetization.Figure 7 shows, for a selected class of parameters, theoverall behavior of the Hall resistivity in the skew spin scat-tering model. In order to explain the experimental data, weneed s12t =0.2, which means that the skew scattering energyhas to be 0.4 times the tight-binding overlap. This is a verytoo strongly enhanced skew scattering rate. It suggests thatthe correct interpretation of the AHE in GaMnAs is mostlikely the intrinsic mechanism proposed by Sinova et al.3 Inthis work, the AHE is due to the intrinsic spin-orbit field andrelies on the multiple band nature of this class of semicon-ductors. The usual simple nearest neighbor tight-bindingmodel only gives a skew scattering contribution. The univer-sality and order of magnitude of the AHE in ferromagnetssuggest that the intrinsic process dominates in most cases.At high temperature, the magnetism disappears. The nor-mal Hall conduction, linear in B field, is recovered see thekBT=0.015W results Figs. 7a and 7b. Note that for sim-plicity of notation, we refer to the applied field in the figureas B, even if it was defined otherwise previously.VII. EXPERIMENTAL RELEVANCEFigure 8 shows results of the CPA calculations for thethree transport coefficients: resistance, magnetoresistance,and Hall effect in a range of parameters for which a behaviorclose to the ones observed experimentally by Ruzmetov etal.33 and Ohno et al.34 is observed. We have added a constantcontribution to the resistivity so that the relative change inresistivity, such as discussed in connection with Fig. 3, be-tween Tc and T=0 is of the same order of magnitude, asobserved in the experiment. The constant term is chosen tobe of the same order of magnitude as the spin scattering rate.The qualitative temperature structure is satisfactory in themetallic regime of the material, but because we have ne-glected the charged impurity scattering, we are underestimat-ing the temperature drop of the resistivity with temperature,at higher temperatures. The latter is due to a reduction inscreening length when the density of states at the Fermi levelincreases see Eq. 51. Figure 8 also illustrates the strongtemperature dependence of the magnetoresistance, whichreaches 40%50% at kBT0.0057W this is 60 K for W=1 eV, and how it is intimately connected to the magneticorder, as observed experimentally. For an even clearer pic-5 4 3 2 1 0 1 2 3 4 52.521.510.500.511.522.5BB/W H/W(m/J)(x1014)kBT = 3.64x104WkBT = 1.31x103WkBT = 1.80x103WkBT = 0.015W( 10-4 )5 4 3 2 1 0 1 2 3 4 51.510.500.511.5BB/W ( 10 4 ) H/W(m/J)(x1014)kBT = 3.64x10 4WkBT = 4.03x10 3WkBT = 5.78x10 3WkBT = 0.015W(a)(b)FIG. 7. Hall resistivity as a function of the applied magneticfield for two carrier concentrations, a p=0.1x and b p=0.8x, foran impurity fraction x=0.053, ENM =EM =0, and Jpd=0.35W, withT as a parameter.ARSENAULT et al. PHYSICAL REVIEW B 77, 115211 2008115211-10ture of the connection, see Fig. 5. The reader should note thatVan Esch et al.35 have observed magnetoresistances of500% at T=4 K and 50% at T=20 K in their more resis-tive samples with 1 /cm. In this sample, the magneti-zation dropped by 100% between 4 and 20 K.Hwang and Das Sarma36 and Lopez-Sancho and Brey37quite rightly pointed out that when charged impurities arepresent, they will normally dominate the scattering rate andthe temperature dependence of resistance in a wide range ofparameters. They then decided to use two different conduc-tivity models for the transport in Mn-doped GaAs, one toexplain the temperature dependence of the resistance whichemphasizes charged impurity scattering and one for the mag-netoresistance which emphasizes spin scattering. We believethat this is not necessary. It is enough to add the configura-tionally averaged charged impurity self-energy to the basisband dispersion ksks+ks, and then to neglect, in thesimplest limit, the real part of the impurity scattering self-energy ks. Thus, we replace the CPA self-energy s, whichappears in Eq. 16, byImsE + ImksE = ImsE+ dk 123Vkk21 cos kkE ks ,49where = 2 Ni, with Ni the impurity concentration, andVkk =4Zimpe20k k211 + qsc/k k2. 50In Eq. 50, Zimp is the impurity charge number, andqsc2 = 4e2s dEDsE fEE 51is the inverse screening length. Since the results of Sec. VIshow that DsE at the Fermi level change with temperatureand field, the screening length will change as well. Note thatusually, in Boltzmann transport, one does not take into ac-count the effect of the real part of the self-energy the imagi-nary part gives the scattering on the actual band structurebecause the concentration of dopants is low. Here, the con-centration is not low. The real part should, in principle,modify the band gap and effective masses and can be in-cluded in our formalism.We agree with Hwang and Das Sarma36 that the completeproblem in parameter space is enormously complex. In ef-fect, we should want the impurity scattering to only representan additional self-consistent, density of states dependent,lifetime process. A fully self-consistent Coulomb potential spin, multiband CPA, is far too complex and of little value.In addition, the CPA band theory cannot account for the hop-ping regime, as pointed out in the previous section.VIII. PROBLEM OF THE INTRINSIC AND EXTRINSICHALL EFFECTIn the Kane-Luttinger theory of Jungwirth et al.,2 the Mndopants only cause a lifetime broadening and no change tothe Kane-Bloch band structure. In this formalism, the intrin-sic spin-orbit interaction is Bloch invariant and only causesband mixing. All disorder is treated only as a lifetime effect.0 0.005 0.01 0.01534567891011121314kBT/W 0+(T)(mcm)0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5504540353025201510505BB/W ( 10 4 )MR(%)kBT = 3.64x10 4WkBT = 4.03x10 3WkBT = 5.78x10 3WkBT = 0.015W5 4 3 2 1 0 1 2 3 4 51.510.500.511.5BB/W ( 10 4 ) H/W(m/J)(x1014)kBT = 3.64x10 4WkBT = 4.03x10 3WkBT = 5.78x10 3WkBT = 0.015W(b)(a)(c)FIG. 8. Transport coefficients which can be directly compared tothe experiment, a =0+T, b MRB, and c HB, forENM =EM =0, Jpd=0.35W, and p=0.8x. Two possible constant con-tributions are shown in a, the solid line curve has 0=10 m cm, and the dashed line curve has 0=3.6 m cm.TRANSPORT IN THE METALLIC REGIME OF Mn-DOPED PHYSICAL REVIEW B 77, 115211 2008115211-11This is not so in the CPA where the dopant scattering causesa substantial renormalization of the band structure via thereal part of the self-energy and indeed gives an explanationof the magnetism. The spin-orbit coupling produces newterms in the Hamiltonian which allow the electrons jumpingor tunneling from one site to the next site to experience theelectric field of the neighboring atoms. The three-site pro-cesses we invoke are not normally included in the tight-binding modeling of semiconductors. They represent onlysmall modifications of the band structure, much smaller thanthe on-site spin-orbit admixture. The anomalous Hall mecha-nism in one-band tight binding is basically one of skewscattering, except that the jump is always from orbit to or-bit, and the effective magnetic field can only come fromanother third site. We have seen that the one-band TB de-scription is inadequate to describe the AHE in diluted mag-netic semiconductors. On the other hand, the multiple bandKane and/or Kohn-Luttinger38 approach seems to work well.It gives the right order of magnitude with an intrinsic AHE.This suggests that a TB modeling of the AHE must includethe multiple band aspect and at least a second nearest neigh-bor overlap. Since, for example, a 5% GaMnAs alloy cannothave a true Bloch band structure, in the orbital description, itis the short range many orbital aspect which must give rise tothe intrinsic AHE.IX. CONCLUSIONWe have presented a CPA theory which could explain themagnetism in Mn-doped semiconductors in the metallic re-gime and allows one to calculate the transport parameters.The numerical evaluation of resistance, magnetoresistance,and Hall coefficient, normal and anomalous, show that theoverall trends observed experimentally are reproduced by theCPA. The power of the method lies in its simplicity. All theinformation is contained in a spin and energy dependent self-energy sE. In good approximation, we may add the effectof the charged impurity and electron-phonon lifetime correc-tions to the calculated CPA self-energy. Of the two, the firstis the most important addition because it explains why theexperimental resistance decreases at high temperatures ingeneral, more strongly than shown in Fig. 8a, when theferromagnetism has reduced to paramagnetism, and spin-disorder scattering should have reached its highest value.The small resistance drop at high temperature seen in Fig.8a is due to the CPA band-structure renormalization. TheCPA and t-matrix methods can be extended to treat magneticclusters and evaluate effective localized spin-spin couplingmediated by the band.We have also demonstrated how to include charged impu-rity scattering within the same formalism. The central aim ofthis paper was to achieve an understanding of the importantmechanism which determines the conductivity behavior. Forthis purpose, we have used a one-band approach which isadequate to understand the conductivity. For optical proper-ties and the intrinsic AHE, the many band aspects are essen-tial.The AHE has been modeled as an extrinsic effect in theframework of skew scattering. We did this using the tight-binding language. The intrinsic AHE, as invoked by Jung-wirth et al.,2 cannot be derived using a nearest neighbor TBformalism, and it is not clear how to recover it in the TBformalism. The problem of the intrinsic AHE and the TBmodel is an interesting one. It should be reexamined in detailbecause so far, the nearest neighbor tight-binding methodshave proved useful as band-structure descriptions of semi-conductors.ACKNOWLEDGMENTSThe authors gratefully acknowledge financial supportfrom the Natural Sciences and Engineering Research Councilof Canada NSERC during this research. 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