Sato and H Kata amaY oshida ISIR Osaka University 5670047 Osaka Jap an H Dederic hs IFF ese ar ch Center ulich D52425 ulich Germany Abstract describ exc hange in teractions in dilute magnetic semiconductors DMS based on abinitio calculations Electro ID: 23766 Download Pdf

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Sato and H Kata amaY oshida ISIR Osaka University 5670047 Osaka Jap an H Dederic hs IFF ese ar ch Center ulich D52425 ulich Germany Abstract describ exc hange in teractions in dilute magnetic semiconductors DMS based on abinitio calculations Electro

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SCIENTIFIC HIGHLIGHT OF THE MONTH: Diluted Magnetic Semiconductors Dilute Magnetic Semiconductors K. Sato and H. Kata ama-Y oshida ISIR, Osaka University, 567-0047 Osaka, Jap an H. Dederic hs IFF, ese ar ch Center ulich, D-52425 ulich, Germany Abstract describ exc hange in teractions in dilute magnetic semiconductors (DMS) based on ab-initio calculations. Electronic structure of DMS is calculated on the basis of the densit functional theory using the Korringa-Kohn-Rostok er coheren oten tial appro ximation (KKR-CP A). will sho that there are classes of DMS with ery

dieren prop erties. In systems with lo calised ma jorit -states deep in the alence band, the ferromagnetism is induced Zener’s exc hange in teraction. This in teraction is eak but long ranged. or systems with impurit y-bands in the gap, the ferromagnetism is driv en Zener’s dou- ble exc hange mec hanism. This in teraction is ery strong but short ranged. Sophisticated Mon te Carlo metho ds sho that for small concen trations the ercolation eect should in- cluded to estimate Curie temp eratures of DMS. In particular, the ferromagnetism is strongly suppressed in double exc hange

systems due to the absence of the ercolation for lo concen- trations. In tro duction Half-metals are considered to the ideal materials for spin tronics. They are particular ferro- magnets whic from electronic structure oin of view can considered as ybrids et een metals and semiconductors, since the ma jorit densit of states is metallic, i.e., nite at the ermi lev el while the minorit bands exhibit gap at Therefore at 100 spin olarisation exists, whic is ideal for the eciency of spin dep enden devices [1, 2]. Since the disco ery of half-metallicit in the Heusler allo ys de Gro

ot [3 ], man other materials lik certain manganites, e.g., La Sr MnO [4], double ero vskites, e.g., Sr eReO [5 as ell as the transition metal xides CrO [4 ha een sho wn to half-metals. this class also elong dilute magnetic semiconductors (DMS), suc as In Mn As and Ga Mn As disco ered Munek ata et al. [6] and Ohno et al. [7]. In these systems small concen tration of 93

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Mn atoms, or in general of transition metal atoms (TM), with ypical concen trations of %, are randomly distributed on the cation sites. Due to the small concen trations the systems eha structurally as

semiconductors and can easily gro wn on the corresp onding paren substrate, i.e., (Ga, Mn)As on GaAs. Moreo er they can dop ed and manipulated as semiconductors, whic oers large prosp ect for applications. Ho ev er problem of these DMS-systems is, that the Curie temp eratures are ell elo ro om temp erature, e.g., 170 for (Ga, Mn)As, represen ting the est in estigated system. This is the ma jor obstacle for applications [1, 2, 8]. In this pap er, will discuss the basic electronic structure of dilute magnetic semiconductors. will concen trate on the magnetic prop erties, in particular

the exc hange mec hanism whic con trol the ferromagnetism in these systems. Moreo er presen calculations of the Curie temp eratures based (i) on the most simple mean-eld appro ximation and (ii) on sophisticated Mon te Carlo metho ds. The ab-initio calculations are erformed within the densit functional formalism using the Korringa-Kohn-Rostok er (KKR) metho together with the coheren oten tial appro ximation (CP A) to describ the disorder in these systems. As result will sho that there are classes of DMS, one, in whic the ma jorit -states are ell lo calised elo the alence band, and

second one, where impurit -bands in the gap exist. In the former class the in teraction is dominated Zener’s exc hange eing relativ ely eak, but longer ranged, while in the latter one Zener’s double exc hange prev ails, eing strong but short ranged. Both ha imp ortan consequences for the Curie temp eratures. Ab-initio Calculations for Dilute Magnetic Semiconductors The results presen ted in this review are obtained ab-initio calculations based on densit functional theory (DFT). Exc hange and electronic correlation eects are describ ed the lo cal densit appro ximation, the standard

orking horse in the eld. As calculational metho use the KKR-Green function metho d. Green function metho ds oid the calculation of eigenfunction and eigen alues of the Kohn-Sham equations of DFT. The Green function ), dened as the causal solution of the Kohn-Sham equation with unit source term at the osition (1) allo ws to determine the harge densit directly from its imaginary part in tegrating er all ccupied states dE Im (2) and the densit of states (DOS) in certain olume olume in tegral Im (3) The KKR metho is based on ultiple scattering theory whic is of strong adv an tage

for the description of the disorder in tro duced the transition metal impurities lik Mn, whic are 94

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randomly distributed on the cation sites, i.e., on the Ga sites in GaAs. Therefore this disorder corresp onds to the disorder in random allo where denotes the concen tration of atoms and the one of atoms. This disorder problem can ell describ ed the coheren oten tial appro ximation (CP A) [9], in whic the atoms and are em edded in an eectiv ‘CP A’-medium whic is determined selfconsisten tly If denote the atomic -matrices of the A- and B-atoms and of the CP medium and CP

then the CP selfconsistency condition, whic determine CP leads in the ultiple scattering KKR description to (4) where describ es the total single-site -matrix of an atom or em edded in the CP medium on site CP 00 CP CP (5) where 00 CP are the on-site elemen ts of the CP Green function CP at site 0. These elemen ts can calculated from CP Brillouin zone in tegration CP BZ BZ CP (6) where are the free space structure constan ts. According to eq. (5) the CP medium, i.e., the CP A-scattering matrix CP has to hosen suc that on the statistical erage the insertion of an and atom at the considered site

in to the CP medium do es not hange the scattering whic is the condition of eq. (4). In our calculations used the KKR-CP co de MA CHIKANEY AMA 2000 pro duced H. Ak ai of Osak Univ ersit [10 ]. All the ab considerations can easily generalised to the case of spin olarised system where ha to distinguish harge densities and ), where and are the harge- and magnetisation-densities. All quan tities G; etc. ha then an additional spin index. or dilute magnetic semiconductors it is ery imp ortan to distinguish states, the ferromagnetic one, where all momen ts of the magnetic atoms are aligned in one

direction, or the disordered-lo cal-momen (DLM) state (or spin-glass state) where the directions of all lo cal momen ts are randomly distributed, so that the erage magnetisation anishes. While the description of the ferromagnetic states is straigh tforw ard in the ab CP formalism, the DLM state can considered as three comp onen allo where in addition to the Ga atoms with concen tration the Ga sites are ccupied either with Mn atoms with lo cal momen ‘up (Mn or with Mn atoms with momen ‘do wn (Mn ), oth with equal concen trations c= [9 ]. ev aluate the thermo dynamic prop erties, describ the

system classical Heisen erg mo del ij (7) 95

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where and denote the lo cal momen ts, in particular their directions, of the magnetic impurities and and ij the exc hange in tegral et een these atoms. This calculated the form ula of Liec tenstein [11 ], whic describ es the energy hange due to small hange of the angle et een oth momen ts within the frozen oten tial appro ximation. ij Im dE ij )( (8) is the atomic -matrix of atom for ma jorit y(+) and minorit y(-) spin directions and ij is the Green function of the system. The disorder of the other atoms is describ ed iden tifying

the Green function ij et een and the Green function of the ferromagnetic CP medium. The exc hange coupling constan ts ij describ not only the sign and the strength of the coupling, but also their spatial exten t, whic is particular imp ortan for the considered diluted systems. Giv en the ij the thermo dynamic prop erties and in particular the Curie temp erature can describ ed standard statistical metho ds. In the simplest mean-eld appro ximation (MF A) the action of all neigh ors of impurit is calculated an erage eld ij (9) where is the erage momen and the ccupation probabilit

for site The condition of anishing then yields as Curie temp erature in MF MF cM ij (10) Note that in MF only the sum of all ij en ters, but not the spatial exten t. Therefore the mean-eld alue MF can also calculated directly from the CP total energies for the ferromagnetic ground state FM and from the disordered lo cal momen state DLM In the mean-eld appro ximation of the Heisen erg mo del the ground state energy DLM anishes for the DLM states DLM ij (11) since the erage momen ts anish due to the erage er all directions. On the other hand for the ferromagnetic ground state one

obtains: FM ij (12) Since orien tational degrees of freedom should describ ed ell the Heisen erg mo del, can iden tify the dierence DLM FM CP DLM CP FM (13) 96

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-3 -2 -1 (a) (Ga, Mn)N -3 -2 -1 (b) (Ga, Mn)P -3 -2 -1 (c ) (Ga, As -3 -2 -1 (d ) (G , Mn) Sb Energy rel ati e to i e y ( (1 l, At om) Figure 1: Densit of states of dilute magnetic semiconductors with Mn impurities: (a) (Ga, Mn)N, (b) (Ga, Mn)P (c) (Ga, Mn)As and (d) (Ga, Mn)Sb. The full curv giv es the erage DOS of the whole system, the dotted curv the lo cal DOS of the Mn atoms. the total energy

dierence for the ferromagnetic system, e.g., Ga Mn As, and the DLM system with 50 Mn momen ts up and 50 do wn, i.e., Ga Mn c= Mn c= As. By comparison with the ab result for MF obtain then MF CP DLM CP FM (14) Th us in MF the Curie temp erature is determined the total energy dierence er Mn atom et een the DLM and FM state [12 ]. Often the MF do es not giv reliable results. In this case Mon te Carlo sim ulations oer an (n umerically) exact metho to calculate the thermo dynamic prop erties. or details see Sect. 5. Lo cal Densit of States and Curie emp eratures in MF Here

presen results of ab-initio calculations for series of I-V DMS with Mn impurities. ha hosen the sequence (Ga, Mn)N, (Ga, Mn)P (Ga, Mn)As and (Ga, Mn)Sb, where only the anions N, As and Sb are dieren t. or the results it is most imp ortan that the ma jorit -lev el of Mn has lo er energy than the atomic -lev el of Sb, but higher energy than the -lev el of N, while the -lev els of and As are in termediate. Fig. sho ws the densit of states (DOS) for the considered systems with Mn on the Ga sites. The upp er curv es refers to the ma jorit DOS, the lo er in erted ones to the minorit DOS, oth

for the ferromagnetic conguration. The full curv es sho the erage total densit of states of the DMS with Mn. Due to the small concen tration of Mn this is roughly the DOS of the pure semiconductors, consisting of the ccupied alence band, dominated the anion -states and the empt conduction band, formed mostly the Ga -states. The dotted lines sho the lo cal DOS of the Mn atoms. consider only the neutral harge state without additional dopan ts. Since Mn has alence electrons and substitutes for Ga atom, of the electrons can replace the Ga electrons in the alence band. The remaining

electrons ha to put in new lo calised -states in the band gap. Therefore the electronic structure of transition metal impurities in semiconductors is dominated -states in the gap, whic for nite concen trations dev elop in to impurit bands. Since Mn has large momen t, only the ma jorit states are ccupied leading 97

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CB VB CB VB Figure 2: Impurit lev els of magnetic transition metal impurities in semiconductors: or Mn on the I-site in I-V semiconductors the double degenerate state and of the three degenerate states are ccupied (left gure); the same states are

ccupied for Cr impurities on I-sites in I-VI semiconductors. On the other hand for Mn impurities in I-VI and impurities in I-V semiconductors all v ma jorit states (righ gure) are ccupied. to so-called ‘high-spin state’. The impurit lev els are sc hematically indicated in Fig. 2. Tw dieren impurit lev els ha to distinguished: ofold degenerate -state ), the functions of whic for symmetry reasons ybridize ery little with the alence band -states, and threefold degenerate -state xy whic strongly ybridizes with the states, resulting in onding- and an tib onding ybrides.

While the onding ybrides are lo cated in the alence band, the an tib onding ybrides form the impurit -states in the gap, whic are due to the ybridization shifted to higher energies than the -states. In the neutral conguration only the -states and of the three -states in the ma jorit band are ccupied, while the minorit gap states are empt In Fig. oth the and -states can ery ell seen for the GaN comp ound with Mn. Since the -states around the individual Mn atoms erlap and form an impurit band, the higher and broader band corresp onds to the more extended -states, and the lo er narro one

to the more lo calised -states. Within the alence band there is also some ybridised-in Mn DOS from the onding -h ybrides. The ermi lev el falls in to the ma jorit -impurit band, suc that er Mn atom exactly -states and -states are ccupied, lea ving one ma jorit -state and all minorit -states empt Therefore the considered system is half-metallic ferromagnet, with momen of er Mn atom. When mo from Mn in GaN to Mn in GaP and GaAs notice that the Mn -lev el is shifted to lo er energies. or (Ga, Mn)P the -state has fully mo ed in the alence band, while the -state forms with the alence -states of the

atom resonance at Most of the lo cal -in tensit of the Mn atom is no lo cated at the onding -states within the alence band. or (Ga, Mn)As these trends are ev en somewhat stronger. Finally for (Ga, Mn)Sb, the resonance at the ermi lev el has more or less disapp eared, suc that at the lo cal Mn DOS agrees ell with the DOS of the Sb atoms. Since the minorit -lik gap states are in all cases uno ccupied, the total momen is xed to er Mn. Ho ev er in the case of (Ga, Mn)Sb the situation is ery dieren from (Ga, Mn)N, since in GaSb all ma jorit -states are ccupied, while hole exist in

the Sb ma jorit -states at the ermi lev el. Therefore the lling of the v -resonances leads to total momen of whic is, ho ev er, reduced to er Mn atom the empt states in 98

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100 200 300 400 10 12 14 (Ga, M n)N (Ga, M n)P (Ga, M n)As (Ga, M n)Sb te tur e (K) co Figure 3: Curie temp eratures of Mn dop ed I-V semiconductors, as calculated in the mean eld appro ximation as function of the Mn concen tration. the ma jorit -band. Th us in the CP description the Sb atoms are eakly and homogeneously olarised, with an erage momen of -1 er Mn atom, eing an

tiferromagnetically coupled to the Mn momen ts, suc that the total momen er Mn atom is still In summary the eha viour of Mn in GaN and GaSb is completely dieren t. In fact, oth systems represen extremes: in (Ga, Mn)N the -states are in the gap and form impurit bands at while in (Ga, Mn)Sb the -states are at the lo er end of the alence band and fully ccupied, while hole exist in the ma jorit alence band. The eha viour of Mn in GaP and GaAs lies et een these extremes. In all cases the minorit -states are uno ccupied. The Curie temp eratures calculated in MF for these systems,

reect this strongly dieren eha viour. Fig. sho ws the calculated MF alues for the four systems as function of the concen tration of Mn impurities. or (Ga, Mn)Sb nd linear dep endence on the concen- tration, but in the other cases strong non-linear dep endence is obtained, whic is particular pronounced for (Ga, Mn)N. As will demonstrate elo w, in this case MF scales as the square ro ot of the concen tration leading to ery large alues already for small concen tra- tions of Mn. The eha viour of (Ga, Mn)As is in termediate et een these extremes: eak er -dep endence for

smaller concen trations and linear dep endence for larger concen trations. As will discuss in the follo wing, the dieren concen tration dep endences are caused dieren exc hange mec hanisms, whic stabilize the ferromagnetism, i.e., double exc hange in the case of impurit bands in the gap and kinetic or exc hange in the case of nearly lo calised -lev els elo the alence band [13 ]. Double Exc hange, Sup er Exc hange and Kinetic Exc hange iden tify the exc hange mec hanism, whic stabilises ferromagnetism or an tiferromagnetism, is an imp ortan problem in magnetic materials. or

dilute magnetic semiconductors this means iden tifying the mec hanism whic stabilises the ferromagnetism already for small concen trations [12 13 14 ]. 99

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va e p- d s d s Figure 4: Double exc hange: Due to the broadening of the impurit -band with increasing Mn concen tration states are transferred to lo er energies, leading to an energy gain, if the ermi energy lies in the band. As explained in the text, the band width increases as or (Ga, Mn)N the haracteristic features of the DOS of the ferromagnetic state is sk etc hed in Fig. 4. The ermi energy lies in the ma jorit impurit

band of -symmetry The imp ortan energy gain arises from the broadening of the impurit band with increasing concen tration If increase the concen tration from lo er alue, with the DOS giv en the full line, to larger alue corresp onding to the broader DOS as giv en the dashed line, transfer DOS-w eigh from around to lo er energies, leading to an energy gain, whic h, as will sho w, stabilises the ferromagnetic state. This energy gain is prop ortional to the band width of the impurit band, whic scales as the square ro ot of the concen tration. The energy gain due to band broadening is kno wn as

Zener’s double exc hange [14 ]. This can pro en theorem for tigh t-binding mo del. The square of the band width is giv en the energy ariance =0 (15) whic itself is determined the sum of the hopping probabilities from site to an other site is the so-called hopping matrix elemen t. Consider no disordered lattice with random distribution of Mn atoms. Starting from giv en Mn atom at site 0, an electron can only hop from the state cen tered at to the site if this site is ccupied another Mn atom. If denote the Mn-Mn hopping in tegral nd in the disordered system (if Mn atom at m) or

0(otherwise) (16) By eraging er all congurations of Mn atoms therefore obtain conf =0 hj =0 (17) since the probabilit to nd Mn atom at site is giv en the atomic concen tration Th us the eectiv band width e scales as This eha viour of band width can clearly 100

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10 12 14 W (eV) Mn concentration (%) -2 -1 1% 2% 4% 8% PDOS (1/eV/Ato m) Energy (eV) W (eV ) Figure 5: Impurit band width and its square for the impurit -band in (Ga, Mn)N as function of Mn concen tration. The inset sho ws the lo cal densit of Mn gap states. seen in calculated DOS as is sho wn in

Fig. 5. In the gure and its square are plotted as function of Mn concen tration. The linear dep endence of on Mn concen tration indicates dep endence of Th us, the energy gain of the ferromagnetic state with increasing concen tration scales as whic explains the strong increase of the Curie temp erature in MF as sho wn in Fig. for (Ga, Mn)N. The double exc hange mec hanism is only imp ortan t, if the ermi energy lies in the band. If the band is completely ccupied or empt no energy can gained band broadening. Let us no consider the stabilit of the disordered lo cal momen (DLM) (or

spin-glass) state as compared to the ferromagnetic one. In the CP A-description of the DLM state, for giv en Mn atom 50 of the neigh oring Mn atoms ha momen eing parallel aligned to the cen tral momen t, and 50 are an tiferromagnetically aligned. The parallel aligned pairs lead, as in the ferromagnetic case, to broadened impurit band, but with reduced band width scaling as c= since only 50 of the pairs are parallel aligned. Therefore in total the double exc hange due to band broadening alw ys fa ors the ferromagnetic conguration. The 50 an tiferromagnetically aligned pairs gain energy

sup er-exc hange. The densit of states in the gap of Mn impurities with an tiparallel aligned momen ts is sc hematically sho wn in Fig. 6. Note that the minorit and ma jorit eaks are exc hanged for the atoms. Since the functions with the same spin directions ybridise with eac other, co alen onding and an tib onding ybrides are formed. rom the energetic oin of view it is imp ortan t, that the lo er onding states are shifted to lo er energies, while the higher an tib onding states are shifted to higher energies. Th us energy is gained sup er-exc hange, if the ermi energy is lo cated et een the

eaks or in the eaks, ho ev er not, if is elo or ab oth eaks. As can sho wn, the energy gain is giv en where is the eectiv hopping matrix elemen and is the exc hange splitting, giv en the exc hange in tegral times the lo cal momen It is linear in since the eects of sev eral an tiparallel aligned neigh ours on the cen tral atom sup erimp ose on eac other. Th us in the case of impurit bands in the gap, double exc hange fa ors the ferromagnetic cong- 101

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d s d s Figure 6: Sup er exc hange: Sho wn are the lo cal densities of states for impurities with momen

ts ‘up and ‘do wn’. Due to ybridisation of the ma jorit and the minorit efunctions the lo er energy lev els are shifted to lo er and the higher lev els to higher energies as indicated the dashed lines. Due to ybridisation also small eaks ccur lo cally for the ‘wrong spin direction. The do wn ard shift stabilises the an tiferromagnetic alignmen of the lo cal momen ts, pro vided the ermi lev el falls et een the eaks, but not elo or ab e. uration and alw ys wins, if the ermi energy lies (w ell) in the band. Then the energy gain due to double exc hange, scaling as is alw ys larger than the energy

gain due to sup er exc hange, scaling lik =I Ho ev er if the ermi energy lies et een the bands or lies close to the band edges, sup er exc hange wins stabilising the disordered lo cal momen state. Th us the system (Ga, Mn)N is exp ected to ferromagnet, while (Ga, e)N should disordered system, since the -band is completely lled. or the same reason, in the I-VI comp ounds Cr impurities should fa our the ferromagnetic state, but Mn impurities the DLM state. Ho ev er these considerations are only alid for impurit band systems. If the ma jorit -lev el lies elo the alence -states, as it is

the case for (Ga, Mn)Sb in Fig. 1, then ha dieren situation whic is sc hematically sk etc hed in Fig. 7. In the ma jorit band the lo lying -states of Mn ybridise with the alence band -states of Sb, and eectiv ely push the ma jorit alence band to higher energy The opp osite is true for the minorit alence states eing pushed the empt minorit -states to lo er energies. Th us the alence band ecomes spin olarised, with momen of er Mn atom, i.e., an tiparallel to the Mn momen t. The other Mn atoms gain energy also aligning an tiparallel to this host olarisation, th us leading to an

indirect ferromagnetic coupling of the Mn-atoms. Due to the strong lo calisation of Mn efunctions the direct in teraction is ery small. This kind of exc hange in teraction is called Zener’s kinetic or exc hange and fa ours ferro- magnetism. No energy gain is obtained in the DLM state, since the host olarisations induced non-aligned Mn momen ts cancel eac other. All ab-initio calculations presen ted in the previous section are based on the lo cal densit ap- 102

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d s va e p- co ion d s Figure 7: Kinetic exc hange: The lo w-lying ‘lo calised ma jorit -states ybridise with the ma

jorit alence -band, pushing it up to higher energies as indicated the dashed line. Anal- ogously the empt minorit -lev el pushes the minorit alence -states to lo er energies. Since due to harge neutralit the alence band ust ha one hole er Mn atom, this hole is conned to the ma jorit band, leading to an Sb momen of er Mn eing an tiferromagnetically aligned to the Mn-momen ts. 100 200 300 400 500 600 2.5 7.5 10 12.5 15 T ( K) Mn-concentration (%) LDA LDA+U -2 -1 -5 -2.5 2.5 Mn-LDOS (states/ spin/ eV) E (eV) majority minority LDA+U LDA Figure 8: Meaneld Curie temp erature of (Ga,

Mn)As ev aluated in the LD and in the LD A+ appro ximation (with eV). The inset sho ws the lo cal densit of states of the Mn atoms. Due to the Hubbard of eV, the Mn ma jorit -states are shifted to lo er energies, while the resonance at the ermi lev el is diminished. This increases the imp ortance of kinetic exc hange and reduces the double exc hange, so that aries linear with concen tration. 103

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pro ximation (LD A). This appro ximation orks in most cases ery ell, but has its limit for correlated systems. One ypical error is, that the spin splitting is usually to small. The

error can partially remo ed the LD A+ metho d, where stands for the Hubbard param- eter of the Hubbard mo del. Fig. sho ws the results of an LD and LD A+ calculations for (Ga, Mn)As system with Mn. The inset sho ws the lo cal Mn DOS in LD and LD A+ using parameter of eV. As one sees, the parameter of eV shifts the ma jorit eak ab out 1.3 eV to lo er energies, suc that it is in go agreemen with photo emission measuremen ts [15 16 ]. Since the -states are no lo cated in the lo er region of the alence band, one exp ects that the exc hange ecomes more dominan t. The calculated Curie tem- eratures

MF indeed sho this eect. The LD results sho -b eha vior resulting from double exc hange, while the LD A+ results are more or less linear in indicating that in (Ga, Mn)As the kinetic exc hange is most imp ortan [13 ]. This exc hange mec hanism underlies all mo del calculations based on the Kondo Hamiltonian, whic describ the ph ysics of (Ga, Mn)As rather ell. Ho ev er the ph ysics of the impurit band systems is ery dieren and cannot describ ed suc Hamiltonian (although this is ery often done). Exc hange Coupling Constan ts and Curie emp eratures The exc hange coupling constan ts

ij giv according to eq. (7) the information ab out the ori- en tational coupling et een the lo cal momen ts and or the DMS-systems they ha een calculated em edding the magnetic impurities and in the CP medium of the ferromagnetic state, in this including the substitutional disorder of all other impurities in an erage As result, the coupling constan ts are strongly concen tration dep enden due to the magnetic screening of the other impurities. The calculated ij constan ts for (Ga, Mn)N, (Ga, Mn)P (Ga, Mn)As and (Ga, Mn)Sb are sho wn in Fig. for three dieren concen trations, i.e., %, and

15 of Mn impurities. The results sho qualitativ ely ery dieren eha viour, in particular for the extreme cases of (Ga, Mn)N and (Ga, Mn)Sb. In (Ga, Mn)N the in teraction is ery strong for the nearest neigh ors, but the coupling of the further atoms is ery small. This is ypical for the double exc hange mec hanism mediated the impurit band. The coupling arises from the erlap of the impurit states on neigh oring sites. Since these states are relativ ely ell lo calised, the coupling is strong, but short ranged. (Ga, Mn)Sb represen ts the other extreme, the coupling of whic is dominated exc

hange. Here the coupling is eak, but ery long ranged. This arises from the large spatial exten of the Sb- states near the alence band maxim um, since the in teraction is basically transferred the spin olarised hole states at the oin t. The estimation of the Curie temp erature the mean eld expression =0 is ery problematic for dilute systems with lo concen trations, since it do es not require an information on the in teraction range. This simplication leads to signican errors in the calculations of for lo concen trations [17 18]. It can easily understo and is kno wn as

the ercolation problem [19 ]. Let us consider Heisen erg mo del with ferromagnetic exc hange in teraction only et een nearest neigh ors (nearest neigh or Heisen erg mo del), and see what happ ens when the system is diluted with non-magnetic sites as sc hematically sho wn in Fig. 10-(a). 104

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10 12 14 1% 5% 15% (Ga, Mn)N Exchange interactions (mRy) Distance (lattice constant) -0 -0 3% (G , M Exch ct (mR y) ce e c t) 5% 0.2 0.4 0.6 0.8 1.2 1.4 1% 15% Mn) As han ge in ac ions (mR tance ( ns ant) 5% 1% (G , M Exchange interactions (mRy) ce e c t) 5% Figure 9: Exc hange coupling

constan ts ij et een Mn atoms as function of the distance for three dieren concen trations. The concen tration dep endence arises from the screening eects of the other impurities, eing describ ed the em edding of the impurities in the CP medium. When the concen tration of magnetic sites is 100 ha erfect ferromagnetic net ork. Due to the dilution, the net ork is eak ened, and for concen tration elo ercolation threshold the ferromagnetism cannot spread all er the system leading to paramagnetic state since due to missing longer ranged in teractions the momen ts can no longer

align. Ob viously this eect is not coun ted in the mean eld equation for ecause the dilution eect is included only as concen tration factor in the equation. In case of the nearest neigh or Heisen erg mo del, the ercolation threshold for the fcc structure is 20 (note that the impurities sit on the fcc Ga sublattice of the zinc blende structure). In real systems suc as (Ga, Mn)N the in teraction reac hes ey ond the nearest neigh ors and the real ercolation threshold should lo er. Ho ev er, elo 20 the strong nn coupling is not so imp ortan an ymore, since only the uc eak

er longer ranged in teraction induces the ferromagnetism, so that the Curie temp erature is exp ected to drop considerably and to uc smaller than the mean eld alue, eing determined to large exten the strong nn coupling 01 In order to tak the ercolation eect in to accoun t, erform Mon te Carlo sim ulation (MCS) for the classical Heisen erg mo del. The thermal erage of magnetization and its ers are calculated means of the Metrop olis algorithm [20 ]. Due to the nite size of sup er cells used in the sim ulation, it is dicult to determine from the temp erature dep

endence of In particular, when considering dilute systems, nite size eects and appropriate nite size scaling are of particular imp ortance for correct and ecien ev aluation of Mon te Carlo sim ulations. oid this dicult use the cum ulan crossing metho prop osed Binder [20 ]. This metho uses the nite size scaling in the forth order cum ulan whic is dened as is calculated for arious cell sizes and plotted as function of 105

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(b) fcc n n ighbor H is mo el k T = 4 c MCS : k T = k k T 01 concent c c 20 (a) 2D-sq lattice (c = 9)

c = 1 c = 0 .7 c = 0.3 ne tic s te no gn etic si te Figure 10: (a) Sc hematic picture of dilute 2-dimensional nearest neighgor Ising mo del in square lattice. The ercolation thereshold is 0.59 in this case. (b) Curie temp eratures of the classical nearest neigh our Heisen erg mo del for the fcc lattice as function of the concen tration. The full line giv es the mean eld results, eing linear in The crosses connected the dashed line giv the exact alues as obtained Mon te Carlo sim ulations (MCS), whic anish elo the ercolation threshold of 20 %. The nn coupling constan 01 has een

xed at constan alue. temp erature. If the cell size is larger than the correlation length, it can sho wn that the curv es for dieren sizes cross eac other at three haracteristic temp eratures. Tw of them are and and the other is use cell sizes (6 10 10 10 and 14 14 14 con en tional fcc cells) to carry out the cum ulan crossing metho for calculations. First, as edagogical example sho the calculated for the dilute fcc nearest neigh our Heisen erg mo del as calculated MF and MCS in Fig. 10-(b). or MCSs for dilute systems, tak 30 dieren random congurations of

magnetic sites for the ensem ble erage. As sho wn in Fig. 10-(b), it is found that the MF giv es reasonable, but to high estimation of for 1. Ho ev er, with decreasing oth curv es decline with nearly the same slop and elo the ercolation threshold, 20, the Curie temp erature anishes. Th us in the dilute concen tration range elo 20 %, whic is most relev an for DMS systems, the failure of the MF is eviden [17 18 ]. Next, sho the calculated alues of (Ga, Mn)N (Fig. 11-(a)) and (Ga, Mn)As (Fig. 11-(b)) as obtained the MCS from the ij alues in Fig. 9. Thirt congurations of Mn atoms are

considered for eraging and ij in teractions up to 15 shells are included. As sho wn in Fig. 11-(a), ery small alues are predicted for lo concen trations in (Ga, Mn)N. The MF alues are almost orders of magnitude to large. Th us nd that the magnetism is 106

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0 100 200 300 0 2 4 6 8 10 12 14 16 Cur ie t em peratur e (K) MF MCS (a) (Ga, Mn)N Mn Co nce ntrat io n (%) 0 100 200 300 0 2 4 6 8 10 12 14 16 e t tu re (K MF MCS (b) (Ga, Mn)As Mn Co nce ntrat io n (%) 0 0 5 Curie erature co ce n (% MF MC )S 0 0 5 (K co ce n (% MF MC Figure 11: Curie temp eratures of (a) (Ga, Mn)N,

(b) (Ga, Mn)As, (c) (Zn, Cr)S and (d) (Zn, Cr)T as ev aluated in the mean eld appro ximation (MF A) and Mon te Carlo sim ulations (MCS) from the ij alues obtained in the LD (see Fig. 9). Due to the ercolation problem the Curie temp erature of (Ga, Mn)N is strongly reduced for small concen trations. This eect can also seen in (Zn, Cr)S and (Zn, Cr)T e. Due to the longer in teraction range the reduction of Curie temp eratures eect is more mo derate in (Ga, Mn)As. strongly suppressed due to the missing ercolation of the strong nearest neigh our in teractions. Only the eak,

longer ranged in teractions satisfy the ercolation requiremen t, leading to small but nite Curie temp eratures for 5, 10 and 15 of Mn. As sho wn in Fig. 11-(b), due to the longer ranged in teraction in (Ga, Mn)As, the reductions from the MF are not ery large, but still signican t. Naturally these hanges are larger for smaller concen trations. The alues of 103 obtained for Mn is in go agreemen with the exp erimen tal alues of 118 rep orted Edmonds et al. [21 ]. This alues refers to measuremen ts in thin lms whic are free of Mn-in terstitials represen ting double donors.

Including in teractions ey ond 15th shell, MCS could giv sligh tly higher alues for lo concen trations. ery high concen trations exp ect our results to increase to ards the MF alues. The exp erimen tal situation for in (Ga, Mn)N is ery con tro ersial. There are man rep orts, where ery high Curie temp eratures, ell ab ro om temp erature, ha een observ ed, but also man observ ations of no ferromagnetism or only ery lo Curie temp eratures. The ab calcu- lations suggest, that homogeneously ferromagnetic phase with Curie temp erature around or ab ro om temp erature can excluded. Therefore the exp

erimen tally observ ed ery high alues ha to attributed to small ferromagnetic MnN clusters and segregated MnN phases, where the strong ferromagnetic nn in teraction ecomes fully eectiv e. The same metho for calculating is applied to (Zn, Cr)S and (Zn, Cr)T as ypical examples of I-VI DMS systems [22 ]. Results are sho wn in Fig. 11-(c) and -(d). In these comp ounds, impurit -bands app ear in the gap and 2/3 of the impurit bands are ccupied (namely they 107

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are equiv alen to Mn-dop ed I-V DMS suc as (Ga, Mn)N from electron ccupation oin of view), therefore the double

exc hange is dominan mec hanism. As result, MF alues of sho dep endence. or oth cases, MCS alues of deviate ery uc from MF alues due to the same reason in the case of (Ga, Mn)N. The suppression of the ferromagnetism in (Zn, Cr)S at small concen trations is as signican as in (Ga, Mn)N. The eect is sligh tly mo derate in (Zn, Cr)T than in (Zn, Cr)S, ecause ZnT has smaller band gap than ZnS and the Cr -band is ery near to the host alence band in (Zn, Cr)T [22 ]. Calculated exc hange in teractions of (Zn, Cr)T are short ranged but not as short as in (Ga, Mn)N. The observ ation of

alue of 300 for (Zn, Cr)T with 20 Cr is in go agreemen with MCS results. The linear scaling in MCS alues of in Fig. 11-(d) has een observ ed in recen exp erimen ts [23 24 Summary Due to their half-metallicit and structural similarit to semiconductors dilute magnetic semi- conductors are hop eful materials for future spin tronics. Ho ev er the Curie temp eratures are in general ery lo w. In this review ha discussed the origin of ferromagnetism in these materials and presen ted ab-initio calculations for the electronic and magnetic prop erties of (Ga, Mn)N, (Ga, Mn)P (Ga, Mn)As and (Ga, Mn)Sb.

The results oin of the existence of classes of DMS with ery dieren prop erties: (i) In systems with lo calised ma jorit -states deep in the alence band suc as (Ga, Mn)As and (Ga, Mn)Sb, the ferromagnetism is induced Zener’s exc hange, leading to holes in the ma jorit -v alence band. This in teraction is relativ ely eak, but long ranged. small concen trations the Curie temp erature is only mo derately reduced the ercolation eect. (ii) In systems with impurit y-bands in the band gap suc as (Ga, Mn)N and (Zn, Cr)T e, the ferromagnetism is driv en Zener’s double exc hange. Here the

magnetic coupling is strong, but short ranged. Therefore, in the dilute limit the ferromagnetism is strongly suppressed, since ercolation of the strong nearest neigh or in teractions cannot ac hiev ed. to ac hiev higher Curie temp eratures migh to try to increase the impurit concen tra- tion. or the systems this should help, since the Curie temp erature basically scales linearly with the concen tration. or the impurit band systems the ercolation eects ecome less im- ortan at higher concen trations, so also here higher concen trations ould help. The observ ation of the of 300 for 20 Cr

dop ed ZnT supp orts this argumen [23 24 ]. References [1] F. Matsukura, H. Ohno and T. Dietl, Hand ok of Magnetic Materials 14 (Elsevier, Amsterdam, 2002). [2] Sp ecial issue: Semic onductor Spintr onics in J. Semicond. Sci. ec hnol, 17 275-403 (2002). [3] R. A. de Gro ot, F. M. Mueller, G. an Engen and K. H. J. Busc ho w, Ph ys. Rev. Lett. 50 2024 (1983). 108

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[4] R. J. Soulen Jr., J. M. By ers, M. S. Osofsky B. Nadgorn T. Am brose, S. F. Cheng, R. Broussard, C. T. anak a, J. No ak, J. S. Mo dera, A. Barry and J. M. D. Co ey Science 282 85 (1988). [5] H. Kato, T. Okuda, Y.

Okimoto, Y. omiok a, K. Oik a, T. Kamiy ama and Y. okura, Ph ys. Rev. 69 184412 (2004). [6] H. Ohno, H. Munek ata, T. enney S. on Molnar, L. L. Chang, Ph ys. Rev. Lett. 68 2664 (1992). [7] H. Ohno, A. Shen, F. Matsukura, A. Oiw a, A. Endo, S. Katsumoto and Y. Iy e, Appl. Ph ys. Lett. 69 363 (1996). [8] S. A. olf, D. D. Awsc halom, R. A. Buhrman, J. M. Daugh ton, S. on Molnar, M. L. Rouk es, A. Y. Ch tc helk ano and D. M. reger, Science 294 1488 (2001). [9] H. Ak ai and H. Dederic hs, Ph ys. Rev. 47 8739 (1993). [10] H. Ak ai, Departmen of Ph ysics, Graduate Sc ho ol of Science, Osak Univ ersit

Mac hik aney ama 1-1, onak 560-0043, Japan, ak ai@ph ys.sci.osak a-u.ac.jp (2000) [11] A. I. Liec tenstein et al., J. Magn. Magn. Matter 67 65 (1987). [12] K. Sato, H. Dederic hs and H. Kata ama-Y oshida, Europh ys. Lett. 61 403 (2003). [13] K. Sato, H. Dederic hs, H. Kata ama-Y oshida and J. Kudrno vsky J. Ph ys. Condens. Matt. 16 S5491 (2004). [14] H. Ak ai, Ph ys. Rev. Lett. 81 3002 (1998). [15] J. Ok aba ashi et al., Ph ys. Rev. 59 R2486 (1999). [16] O. Rader et al., Ph ys. Rev. 69 075202 (2004). [17] K. Sato, W. Sc eik a, H. Dederic hs and H. Kata ama-Y oshida, Ph ys. Rev. 70 201202

(2004). [18] L. Bergqvist, O. Eriksson, J. Kudrno vsky V. Drc hal, Korzha vyi and I. urek, Ph ys. Rev. Lett. 93 137202 (2004). [19] D. Stauer and Aharon Intr duction to Per olation The ory (T ylor and rancis, Philadel- phia, 1994). [20] K. Binder and D. W. Heerman, Monte Carlo Simulations in Statistic al Physics (Springer, Berlin, 2002). [21] K. W. Edmonds et al., Appl. Ph ys. Lett. 81 4991 (2002). [22] T. ukushima, K. Sato, H. Kata ama-Y oshida and H. Dederic hs, Jpn. J. Appl. Ph ys. 43 L1416 (2004). [23] H. Saito, V. Za ets, S. amagata and K. Ando, J. Appl. Ph ys. 95 7175 (2004). 109

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[24] N. Ozaki, N. Nishiza a, K.-T. Nam, S. Kuro da and K. akita, Ph ys. Status Solidi 957 (2004). 110

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