Momentum-resolved spin splitting in Mn-doped trivial CdTe and topological HgTe semiconductors
Carmine Autieri, Cezary Śliwa, Rajibul Islam, Giuseppe Cuono, Tomasz Dietl
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J un Momentum-resolved spin splitting in Mn-doped trivial CdTeand topological HgTe semiconductors
Carmine Autieri, ∗ Cezary ´Sliwa, Rajibul Islam, Giuseppe Cuono, and Tomasz Dietl
1, 3 International Research Centre MagTop, Institute of Physics,Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02668 Warsaw, Poland Institute of Physics, Polish Academy of Sciences,Aleja Lotnikow 32/46, PL-02668 Warsaw, Poland WPI-Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
A series of experiments demonstrated that exchange-induced splitting of magnetooptical spectraof Cd − x Mn x Te at the L points of the Brillouin zone is, unexpectedly, more than one order ofmagnitude smaller compared to its magnitude at the zone center and show an unexpected sign ofthe effective Land´e factor. We have determined spin-splitting of the valence and conduction bandsin the whole Brillouin zone in Cd − x Mn x Te and in topologically-nontrivial Hg − x Mn x Te by meansof relativistic first-principles density functional calculations. We find that spin splitting of bandsis relatively large at the L points but, in contrast to the Γ point, effective exchange integrals havethe same sign and similar magnitudes at the L points. This results in comparable energies of theoptical transitions for two circular light polarizations leading to small splitting of optical spectrain Cd − x Mn x Te. Our results substantiate also previous suggestions that the antiferromagnetic signand a relatively high magnitude of the effective exchange integral in the conduction band awayfrom the Γ point results from an admixture of anion p -type wave functions and the proximity toupper Hubbard band of Mn d electrons. We use the obtained results to determine parameters of aminimal tight-binding model that describe rather accurately the band structure, including spin-orbitand exchange splittings of bands in the whole Brillouin zone of Cd − x Mn x Te and Hg − x Mn x Te.
I. INTRODUCTION
Dilute magnetic semiconductors (DMSs), such asCd − x Mn x Te and Hg − x Mn x Te, have played a centralrole in the demonstrating and describing a strong andintricate influence of the sp - d exchange interactions uponeffective mass states in semiconductors , paving theway for the rise of dilute ferromagnetic semiconductors and magnetic topological insulators . One of the keycharacteristics of DMSs in a giant spin splitting ofbands proportional to the field-induced and tempera-ture dependent magnetization of paramagnetic Mn ions, M ( T, H ). In the case of high electron mobil-ity modulation-doped Cd − x Mn x Te/Cd − y Mg y Te het-erostructures, the exchange splitting leads to crossings ofspin-resolved Landau levels, at which the quantum Hallferromagnet forms at low temperatures . It has been re-cently proposed that magnetic domains of this ferromag-net, if proximitized by a superconductor, can host Ma-jorana modes . Similarly, Hg − x Mn x Te/Hg − y Cd y Tequantum wells of an appropriate thickness and Mn cationconcentration x . .According to the present insight there are two ex-change mechanisms involved in the interaction H sp − d = − J s · S between effective mass electrons in the vicinityof Γ and localized spins residing on the half-filled Mn d -shells . The first on them is the ferromagnetic di-rect (potential) exchange J sd between band carriers withwave functions derived from Mn s orbitals and electronslocalized on the open Mn d shells, usually denoted N α , typically of the order of 0.2 eV. The second one is theantiferromagnetic kinetic exchange between band carri-ers with anion p -type wave functions and d electrons, ofthe order of J pd ≡ N β ≈ − kp Hamilto-nian allows one to describe satisfactorily various spectac-ular magnetotransport and magnetooptical phenomenafor carriers near the Γ point of the Brillouin zone as afunction of M ( T, H ) , particularly if effects of strongcoupling are taken into account .However, in contrast to the Γ point, the physics ofexchange splittings at the L points of the Brillouinzone is challenging: a series of magnetoreflectivity andmagnetic circular dichroism (MCD) studies, notably forCd − x Mn x Te , has revealed that the magnitudes ofspectra splittings for two circular light polarizations atthe L points ( E and E + ∆ transitions) are smallerby a factor of about seventeen compared to the valueat the Γ point, the effect not explained by tight-bindingmodeling . Furthermore, effective Land´e factors cor-responding to these transitions show unexpected signs .The situation is also unsettled in Hg − x Mn x Te, in whicha large magnitude of spin-orbit-driven spin-splittings ac-counts for a controversy concerning the actual magni-tudes of the sp - d exchange integrals at the Γ point ,making their comparison to spin-splitting values at the L points not conclusive. Accordingly, it has been pointedout that the electronic structures of II–VI DMSs have notbeen as well clarified as we previously believed . Amongother issues, this fact may preclude a meaningful evalua-tion of the role played by interband spin polarization inmediating indirect exchange interactions between mag-netic ions. This Bloembergen-Rowland mechanism isknown to play a sizable role in p -type dilute ferromag-netic semiconductors, in which it involves virtual transi-tions between hole valence subbands . Moreover, thisspin-spin exchange is expected to be particularly impor-tant in the absence of carriers in the inverted band struc-ture case (such as Hg − x Mn x Te), in which both the va-lence and conduction bands are primarily built of anion p -type wave functions .In the last years, several ab initio studies ofCd − x Mn x Te have been carried out . However,these works were not attempted to elucidate the originof the anomalously exchange-induced splittings of op-tical spectra corresponding to transitions at the Bril-louin zone boundary. In this paper, we present re-sults of our relativistic first-principles density functionalcomputations for both Cd − x Mn x Te and Hg − x Mn x Te.Within the model that neglects the spin-orbit coupling(SOC) we discuss the origin of inverted band structurein Hg − x Mn x Te. Furthermore, we trace the evolution ofexchange-induced splittings of the valence and conduc-tion band with the k vector. These results, together withthe determined orbital components of the wave functions,explain qualitatively the origin of a reduction of exchangesplittings at L points of the Brillouin zone compared tothe Γ point as well as substantiate the origin of the changeof sign of the conduction band splitting with the k vectorin Cd − x Mn x Te. We then include SOC and use the ob-tained ab initio band dispersion to determine parametersof a versatile tight-binding model that describe rather ac-curately the band structure and sp - d exchange splittingof bands in the whole Brillouin zone of Cd − x Mn x Te andHg − x Mn x Te taking SOC into account. This model al-lows us to determine optical transition energies and mag-netic circular dichroism for any point of the Brillouinzone, and for an arbitrary orientation of the magnetiza-tion vector in respect to crystal axes. The model is val-idated by its good agreement with hitherto unexplainedmagnetic circular dichroism data corresponding to opti-cal transition at the L points of the Brillouin zone inCd − x Mn x Te and Hg − x Mn x Te. An important outcomeof our work is the determination, by combining DFTand experimental information, of a tight-binding modelthat describes quantitatively the electron band structureof CdTe, HgTe, Cd − x Mn x Te, and Hg − x Mn x Te in thewhole Brillouin zone.
II. COMPUTATIONAL METHODOLOGY
We have performed first-principles density functionaltheory (DFT) calculations by using the relativistic VASPpackage based on plane wave basis set and projector aug-mented wave method (PAW) . We perform a fully rel-ativistic calculation for the core-electrons while the va-lence electrons are treated in a scalar relativistic approx-imation considering the mass-velocity and the Darwinterms. Spin-orbit coupling of the valence electrons is in-cluded using the second-variation method and the scalar- relativistic eigenfunctions of the valence states .A plane-wave energy cut-off of 400 eV has been used.For the bulk, we have performed the calculations using8 × × k -point Monkhorst-Pack grid with 176 k -points inthe absence of SOC and with 512 k -points in presence ofSOC in the irreducible Brillouin zone. We use the exper-imental lattice constants corresponding to a = 6 . .For the treatment of exchange-correlation, Perdew-Burke-Ernzerhof (PBE) generalized gradient approxima-tion (GGA) and the the modified Becke-Johnson ex-change potential (MBJLDA) have been applied. Ac-cording to the computed band structures in GGA, themagnitudes of the band gap E = E (Γ6) − E (Γ8) are0 .
77 eV and − .
50 eV for CdTe and HgTe, to be com-pared to experimental values at 4.2 K E = 1 .
60 eV and − .
30 eV, respectively. These discrepancies reflect thewell-known inaccuracies of the GGA in the evaluationof the band-gap. Thus, in order to improve the tight-binding parametrization of CdTe and HgTe band struc-tures, the MBJLDA have been employed for the deter-mination of the hopping parameters. Our results, ob-tained within this computationally more demanding ap-proach, confirm that the determined magnitudes of theband gaps , as well as of spin-orbit splittings, are closeto experimental values.The effect of Mn doping in Cd − x Mn x Te andHg − x Mn x Te has been studied using a 4 × × × × k -point grid. We usethe special quasi-random structure (SQS) to model thedistribution of cation-substitutional Mn atoms in the su-percell. To create a large SQS model, we used the mcsqsalgorithm within the framework of alloy theoretic au-tomed toolkit (ATAT) . The mcsqs method is based onthe Monte Carlo simulated annealing loop with an ob-jective function that search for a perfectly match max-imum number of correlation functions for a fixed shapeof the supercell along with the occupation of the atomicsite by minimizing the objective function. The clustersof doublet, triplet and quadruplet are generated usingthe cordump utility of the ATAT toolkit. We use the pa-rameters -2, - 3, and -4 for which the longest pair, triplet,and quadruplet correlation distance to be matched is 2.0,1.5, and 1.0 lattice constants, respectively. To create thebest SQS structure, we produce all possible structuresand choose the one for which the correlation difference inrespect to random structure is closest to zero.In our work, we focused on Mn content x = 2 / , / /
64. Since we look for magnitudes of sp - d exchangesplittings, the Mn magnetic moments are always ferro-magnetically aligned. The Hubbard U effects for Mnopen d shell have been included. We use the valuesof U Mn = 3 , 5 and 7 eV and J H = 0 . U eV forthe Mn-3 d states. After obtaining the Bloch wave func-tions in density functional theory, the maximally local-ized Wannier functions (MLWF) are constructed us-ing the WANNIER90 code . To extract the character of -4-2 0 2 4 Γ X W K Γ L U W L K E ne r g y [ e V ] Cd Mn x Tespin up -4-2 0 2 4 Γ X W K Γ L U W L K E ne r g y [ e V ] Cd Mn x Tespin down
FIG. 1. DFT band structure obtained using the MLWF for Cd . Mn . Te, U Mn = 5 eV, and assuming ferromagneticalinement of Mn spins without spin-orbit coupling. The spin up (down) channel is shown in the left (right) panel with the grey(red) dashed line. The interpolated Cd- s , Te- p and Mn- d Wannier bands are shown with solid green line. Zero energy is set atthe valence band top. the electronic bands at low energies, we used the Slater-Koster interpolation scheme based on Wannier functions.Quantities of interest here are effective exchange ener-gies J c ( k ) and J v ( k ) calculated from k -dependent split-tings of the lowest conduction and highest valence bands,generated by exchange interactions with Mn spins S =5 /
2, aligned by an external magnetic field, J c ( k ) = ∆ E c xS = E ↓ c − E ↑ c xS , J v ( k ) = ∆ E v xS = E ↓ v − E ↑ v xS . (1)According to this definition, in the weak coupling limitand for the normal band ordering, i.e., for Cd − x Mn x Te, J c ( k = 0) ≡ N α and J v ( k = 0) ≡ N β , where N is the cation concentration, whereas α and β are s - d and p - d exchange integrals according to the DMSliterature . The same situation takes place in thecase of Hg − x Mn x Te with x & . . However, at lower x , Hg − x Mn x Te is a zero-gap semiconductor with an in-verted band structure (topological zero-gap semiconduc-tor) for which the s -type Γ band is below the Γ j = 3 / bandbelow the Fermi level as J c . We note also that becauseof antiferromagnetic interactions between Mn spins, aneffective Mn concentration x eff that contributes to the sp − d exchange splitting of bands in a magnetic fieldis much smaller than x , typically x eff .
5% for any x in relevant magnetic fields µ H . . For ran-dom distribution of Mn over cation sites, these antiferro-magnetic interactions result in spin-glass freezing at lowtemperatures . III. RESULTSA. DFT band structure for Cd − x Mn x Te andHg − x Mn x Te without spin-orbit coupling
We discuss first the electronic structure ofCd − x Mn x Te and Hg − x Mn x Te computed with rel-ativistic corrections in the scalar approximation, i.e.,taking into account the Darwin and mass-velocity terms(essential in Hg − x Mn x Te) but neglecting spin-orbitcoupling (SOC). This allows us to extract spin splittingssolely due to the exchange interactions between host andMn spins, i.e., effective exchange integrals J c and J v forrelevant bands and arbitrary k vectors in the Brillouinzone.Figure 1 presents the electronic structure ofCd . Mn . Te for spin up and spin down evalu-ated assuming U Mn = 5 eV. The Mn lower and upperHubbard 3 d -bands reside around 4.6 eV below and2.5 eV above the valence band top, respectively. Hence,in agreement with photoelectron spectroscopy , aneffective Hubbard energy of Mn-3 d electrons is 7.1 eVfor U Mn = 5 eV and, of course, would increase withthe increasing U Mn . At the same time, experimentaldata indicate that the Mn d -bands reside by about1 eV higher in respect to host bands than implied byour DFT results. In the whole Brillouin zone and forboth spin channels, the lowest unoccupied states consistmainly of Cd-5 s states, whereas Te-5 p states give adominant contribution to the highest occupied bands.To give an estimation of the orbital contribution inDFT, we evaluate the system at the Γ point, where the s -states are decoupled from the p and d -states. For the(Mn,Cd)Te, the conduction band is composed roughlyby 70% Cd- s states and 30% Te- s states with a minorcontribution from the impurity Mn- s states for the low -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 Γ X W K Γ L U W L K E ne r g y [ e V ] Hg Mn x Te Γ Γ FIG. 2. DFT band structure of Hg . Mn . Te with fer-romagnetically aligned Mn spins without spin-orbit couplingand for U Mn = 5 eV. Zero energy is set at the valence bandtop. Bands for two spin orientations in respect to the direc-tion of Mn spins are shown by solid grey lines for the spin upand red dashed lines for spin down. Mn concentration x in question. The conduction bandat the Γ point is composed roughly by 80% Te- p statesand 20% Cd- d states with a minor contribution from theimpurity Mn-d states.From the electronic structure of Hg . Mn . Te at U Mn = 5 eV without SOC, the effective Hubbard en-ergy of Mn-3 d electrons at the Γ point is 7.8 eV for U Mn = 5 eV. As shown in Fig. 2, the Γ bands and the Γ are inverted in Hg − x Mn x Te resulting in the topologicalcharacter of the compound. The relativistic Darwin termgives a weak positive contribution to the energy of the for s -bands in heavy atoms like Hg, whereas the relativisticmass-velocity term gives a strong negative contribution,and accounts for the band inversion. We can clearly seein Fig. 2 that the Γ band at 0.5 eV below the Fermi levelhas the spin up component at lower energies indicatingthe ferromagnetic sign of the exchange interaction withMn spins. Instead, the carrier spins in the Γ bands areantiferromagnetically coupled to Mn spins. B. Spin splitting along the k path withoutspin-orbit coupling
Figure 3 shows exchange energies J c ( k ) and J v ( k ) com-puted for Cd − x Mn x Te with various Mn concentrations x and U Mn = 5 eV. In agreement with the experimen-tal results , the values determined for the Γ point donot depend on x , and their DFT values, N α = 0 .
28 eVand N β = − .
63 eV, describe the sign, and also reason-ably well the experimental magnitudes, N α = 0 .
22 eVand N β = − .
88 eV , depicted by horizontal lines inFig. 3. The exchange splittings at Γ means that thereis a large energy difference between transitions from the -0.8-0.4 0 0.4 Γ X W K Γ L U W L K J v [ e V ] FMAFM Cd Mn x Te -0.8-0.4 0 0.4 Γ X W K Γ L U W L K J c [ e V ] FMAFM Cd Mn x Te FIG. 3. The values of effective exchange integrals for the va-lence and conduction band, J v in the top panel (green lines)and J c in the bottom panel (red lines), for Cd − x Mn x Tecompared to experimental values at the Γ point, determinedby exciton magnetospectroscopy, represented by horizontallines . The experimental effective exchange integral is neg-ative for the valence and positive for the conduction bandat the Γ point. The solid, dashed, and dotted lines repre-sent the band spin splitting for entirely spin-polarized Mnions with concentrations x = 2 /
64 = 0 . , /
64 = 0 . /
64 = 0 . U Mn = 5 eV. two heavy hole subbands (or for the creation of the heavyhole excitons). This giant Zeeman splitting is describedby ∆ E ≃ xS ( N α − N β ).As seen, J v remains negative (antiferromagnetic) forall k -values, and its magnitude slightly oscillates alongthe k -path, it reaches a maximum at Γ and a minimumat L for small Mn concentrations, and between the Γ and X points at the highest studied x = 8 /
64 = 0 . J v , J c changes sign and is highly oscil-lating along the k -path: The sign of J c is positive (ferro-magnetic potential s - d exchange) at the Γ point, in whichthe conduction band wave function has the s -type char-acter, but becomes antiferromagnetic away from the Γpoint. This behavior originates from (i) an admixture ofanion p wave functions to the Bloch amplitudes u k and,thus, from a significant role of antiferromagnetic p - d ki-netic exchange and (ii) hybridization of exp( i k · r ) com-ponent of the Bloch wave function with d shell resultingin antiferromagnetic exchange, known from the Kondophysics in dilute magnetic metals . In the antiferro-magnetic sign region, the absolute value of J c reachesa maximum at the X points and a minimum at the U points at small concentrations and at the L points for x = 0 . p - d hybridization and, thus, of the kinetic exchangeif a given state approaches the 3 d Mn shell, in the J c case, the upper Hubbard 3 d band. In agreement withthis interpretation, except for J c at the Γ point, the spinsplitting gets reduced when we increase U Mn because theMn d -states are pushed away from bands, and the elec-tronic hybridization between the host bands and the 3 d shells of Mn-impurities is suppressed.According to the ab initio results, the exchange split-tings J c and J v at the L points have the same sign andsimilar magnitudes. This qualitatively explains why theexperimentally observed splitting of optical spectra is rel-atively small at L compared to Γ . In view of ourresults, the previous attempt to interpret this large re-duction of ∆ E at L was quantitatively unsuccessful be-cause a strong dependence of J c on k was disregarded .At the same time, our data suggest a relatively strongdependence of J c and J v at L on x . This is not corrob-orated by experimental results accumulated so far. Inparticular, defining a splitting reduction factor p by p = J c ( L ) − J v ( L ) J c (Γ) − J v (Γ) , (2)the data in Fig. 3 imply p = − .
082 and 0.055 for x = 3 .
125 and 0.0625, respectively. The experimentallydetermined ratio of splittings at L and Γ was found tobe in the range 0.051-0.082, i.e., p is small in accord withour results but stays positive independently of x .Finally, we mention the relevance of our ab initio results to experimental and theoretical studies of N α as a function of quantum well thickness t in a seriesof n-Cd . Mn . Te quantum wells sandwiched betweenCd . Mn . Mg . Te barriers . Our evaluation indi-cates that the decrease of J c with k (see Fig. 3) togetherwith an increase in the penetration of the electron wavefunction into barriers can explain an experimentally ob-served decrease of N α with decreasing t .Figure 4 shows J c ( k ) and J v ( k ) extracted fromthe band structure computations without SOC forHg − x Mn x Te with different values of x . In the vicinityof the zone center we present single data points corre-sponding to the exchange energy of the Γ band, i.e., N α . The trends in k dependencies are similar to theMn-doped CdTe. In particular, J v stays negative in thewhole Brillouin zone and J c ( k ) becomes negative awayfrom the zone center. -1-0.8-0.6-0.4-0.2 0 0.2 0.4 Γ X W K Γ L U W L K J v [ e V ] FMAFM Hg Mn x Te -1-0.8-0.6-0.4-0.2 0 0.2 0.4 Γ X W K Γ L U W L K J c [ e V ] FMAFM Hg Mn x Te FIG. 4. The values of effective exchange integrals for the va-lence and conduction band, J v in the top panel (green lines)and J c in the bottom panel (red lines), for Hg − x Mn x Te com-pared to experimental values at the Γ point represented byhorizontal lines . The experimental effective exchange in-tegral is negative for the valence and positive for the conduc-tion band at the Γ point. The solid, dashed, and short-dashedrepresent the spin splitting for x = 2 /
64 = 0 . , /
64 =0 . /
64 = 0 . bandfor x = 2 /
64 = 0 . , /
64 = 0 . /
64 = 0 . On the experimental side, there are two sets of the de-termined N α and N β values, differing by more than afactor of two, in the case of Hg − x Mn x Te . Our computa-tional results point to the lower values, i.e., N α = 0 . N β = − . . Furthermore, according to ex-perimental findings, magnetic circular dichroism at L hasthe same sign for Hg − x Mn x Te as found for Cd − x Mn x Teat L and at Γ, independently of Mn content x . Ourdata suggest the opposite sign since, according to the re-sults in Fig. 4, J c ( L ) − J v ( L ) < − x Mn x Te in therelevant effective Mn concentrations, x . C. Effects of spin-orbit coupling
The DFT results presented in Figs. 3 and 4, obtainedwithout taking SOC into account, have qualitativelyshown how exchange spin-splitting of bands evolves withthe k vector spanning the whole Brillouin zone. Thisdependence reflects (i) the k -dependent mixing betweencation and anion wave functions, which affects a relativecontribution of the potential and kinetic components tothe sp - d exchange and (ii) the energy position of a given k state in respect to the open Mn d shells, which con-trols the magnitude of the k -dependent kinetic exchange.Quantitatively, however, the position of bands and, thus,their exchange splitting depends significantly on SOC.Moreover, in the presence of SOC, exchange splitting ofa given band state changes with the orientation of its k vector in respect to the direction of M ( T, H ). Thismeans that, in general, exchange splitting of particular L valleys differs, depending on the angle between k L and M ( T, H ). Furthermore, under non-zero magnetization M ( T, H ), degeneracy of states with different projectionsof the orbital momentum is removed in the presence ofSOC. This results in magnetic circular dichroism (MCD),i.e., different transition probabilities for two circular lightpolarizations σ + and σ − .We are interested in interpreting magnetooptical re-sults, particularly concerning MCD, for Cd − x Mn x Teand Hg − x Mn x Te taken at photon energies correspond-ing to free excitons at the fundamental gap at the Γ point( E and E +∆ transitions) and at the L points ( E and E + ∆ transitions) , where ∆ and ∆ are thespin-orbit splitting of the valence band at the Γ and L points of the Brillouin zone, respectively. Our theoreticalapproach considering SOC involves four steps. First, weuse the DFT calculations with SOC taken into account inorder to determine parameters of a tight-binding modelfor CdTe and HgTe. Second, we consider the Mn-dopedcase and obtained from DFT on-site and hopping energiesfor Mn d shell and its coupling to band states in CdTe andHgTe. Third, these parameters are incorporated into sp - d exchange hamiltonian that takes into account the pres-ence of k -dependent kinetic and potential exchange inter-actions in the molecular-field and virtual-crystal approx-imations suitable for Cd − x Mn x Te and Hg − x Mn x Te.
D. DFT with spin-orbit coupling and minimaltight-binding model for CdTe and HgTe
As mention in Sec. II, we use MBJLDA to determinethe relativistic band structure of CdTe and HgTe withexperimental lattice constants 6.48 ˚A for CdTe and 6.46˚A for HgTe. To extract energy dispersions E ( k ) of theelectronic bands, the Slater-Koster interpolation schemeis employed. The obtained results are shown in Fig. 5.The extracted values of energy gaps and spin splittingsfor CdTe and HgTe are summarized in Table I, and showgood agreement with experimental values. Our aim is to use the ab initio results in order to deter-mine parameters of the one-electron Hamiltonian in thetight-binding approximation (TBA), which will properlydescribe the band structure and sp - d exchange splittingsof bands at an arbitrary k -point of the Brillouin zonewith SOC taken into account. Similarly to the previ-ous descriptions of CdTe and HgTe within TBA , weconsider sp orbitals per atom and the nearest neighborhopping. In particular, from positions of the electronicbands at Γ we obtain the TBA on-site energies and thespin-orbit splittings. Then we use as constraints the DFTvalues of the band energies at the Γ, X and L points. Wecreate an equation system and search for the values of thehopping energies V . If this procedure results in multiplesolutions, we select the value of V which has the samesign as the first-neighbor hopping energy among Wan-nier functions. The TBA parameters obtained in thisway are shown in Table II. Since the atomic radius ofCd is smaller than of Hg, whereas the bond length isgreater in CdTe compared to HgTe, there are no system-atic differences in the magnitudes of the hybridizations V between these two compounds. A comparison of theband structures resulting from MBJLDA and our TBAis shown in Fig. 5.By construction, the TBA parameters collected in Ta-ble II lead to similar bandgaps and spin-orbit splittings asdetermine be DFT, and displayed in Table I and Fig. 5.For comparison, we present in Table I the magnitudesof bandgaps and spin-orbit splittings computed by usingthe tight binding parameters determined by Tarasenko etal. in reference to experimental data. As seen, this em-pirical tight-binding (ETB) model provides, by designed,the energies in accord with the experimental values. E. Tight-binding parameters from DFT forCd − x Mn x Te and Hg − x Mn x Te We are interested in evaluating Slater-Koster parame-ters associated with the presence of open 3 d shells of Mnin Cd − x Mn x Te and Hg − x Mn x Te, i.e. , hopping ener-gies between Mn 3 d orbitals and 5 sp states of the near-est neighbor Te anions as well as energetic positions ofMn d levels. For this purpose we use supercells with2 × × U Mn = 5 eV andJ Hund = 0 .
75 eV as well as with the PBE exchange func-tional. Since in such alloys no E ( k ) dependencies canbe derived, we extract the Slater-Koster parameters V directly from the hopping energies among the relevantWannier functions, which means that their accuracy ispresumably of the order of 20%. The magnitudes of de-termined parameters are shown in Table III. A lower po-sitions (by about 0.3 eV) of d levels in HgTe comparedto CdTe originates from the valence band offset betweenthese two compounds . The spin-up Mn states aremore localized and the hopping energies V related to d ↑ are smaller. On the other hand, noticeable dissimilarities FIG. 5. Comparison between the Wannier bands obtained by MBJLDA (solid green line) and our tight-binding model (dashedblue lines) for CdTe (left panel) and HgTe (right panel) taking spin-orbit coupling into account.TABLE I. Energy gaps and spin-orbit splittings (in eV) at the Γ and L points, where E = E (Γ ) − E (Γ ) and ∆ = E (Γ ) − E (Γ ) and at the L points of the Brillouin zone, where E = E ( L ) − E ( L ) and ∆ = E ( L ) − E ( L ) for CdTe andHgTe, as determine from ETB, our TB model, MBJLDA and experimentally. The notation at L is shown in Fig. 5 and it isthe same of reference . CdTe HgTeETB TB MBJLDA expl ETB TB MBJLDA expl E E ∆ . ± . TABLE II. Values of the on-site E , hopping V , and spin-orbit splitting ∆ energies (in eV) of our minimal tight-bindingmodel for CdTe and HgTe, which includes sp orbitals of an-ions a and cations c , and the nearest-neighbor hopping. Zeroenergy is set at the top of the valence band.CdTe HgTe E s ( a ) -8.7752 -9.1555 E s ( c ) -0.9526 -3.1156 E p ( a ) -0.2669 -0.1742 E p ( c ) V ss σ -1.2431 -1.2569 V s ( a ) p ( c ) σ V s ( c ) p ( a ) σ V pp σ V pp π -0.9875 -0.9830∆ a c in hopping energies of the two compounds are caused bydifferences in the bond length and in the participation of TABLE III. Values of on-site and hopping energies (in eV)for Mn 3 d orbitals ( t g and e g ) and the nearest-neighbor Te5 s and 5 p states for Cd − x Mn x Te and Hg − x Mn x Te. Zeroenergy is set at the top of the valence band.Cd − x Mn x Te Hg − x Mn x Te E t g ↑ -4.702 -4.997 E t g ↓ E eg ↑ -4.525 -4.858 E eg ↓ V sd ↑ σ -1.081 -1.232 V sd ↓ σ -1.949 -1.957 V pd ↑ σ -0.488 -0.364 V pd ↓ σ -0.957 -0.987 V pd ↑ π V pd ↓ π cation orbitals to the s -like and p -like wave functions. F. Tight-binding model with sp - d exchangeinteraction We now present and discuss the tight-binding Hamil-tonian with four sp orbital per atom containing a termdescribing giant Zeeman splitting of bands in the pres-ence of spin polarized Mn ions. This splitting is broughtabout by: (i) direct (potential) exchange coupling of elec-trons residing on the open Mn 3 d shell to band carriersvisiting Mn 4 s or 4 p orbitals (ii) the kinetic exchange re-sulting from spin-dependent hybridization between Mn3 d shells and band states derived from the 5 s and 5 p or-bitals of the four neighboring Te anions. Our approach isdeveloped within the molecular-field and virtual-crystalapproximations, and generalizes previous descriptions ofDMSs within TBA by taking into account the k -dependence of the kinetic exchange according to, H ( k ) = H TBA ( k ) + H sp − d ( k ) . (3)Within our model H TBA ( k ) is a 16 ×
16 matrix, with on-site energies on the diagonal and k -dependent hopping t mn ( k ) between the orbitals labeled m, n , h m, s |H TBA ( k ) | n, s ′ i = E n δ mn + t mn ( k )+ ∆ a ( c ) X α I αmn σ αss ′ . (4) In this equation, if the orbitals labeled ( m, n ) are locatedeither on the anion and the cation or vice versa, t mn ( k )is the total of the hoppings to the nearest neighbors inthe zinc-blende lattice; momentum k enters the phasefactors: t mn ( k ) = X R m ∈ n.n.( R n ) V mn ( R m − R n ) exp[ i k · ( R m − R n )] . (5)The Slater-Koster interatomic matrix elements (depen-dent on the direction cosines of the vector from the loca-tion R n of the orbital n to the location R m of the orbital m ) are denoted as V mn ( R m − R n ), with parameters forthe various combinations of orbitals ( ssσ , spσ , ppσ , ppπ )given in Table II. The spin-orbit splitting of the anion(cation) p states are denoted by ∆ a ( c ) , respectively, thespin-1 orbital momentum operator I αβγ in the Cartesianbasis ( α, β, γ = x, y, z ) has been written using the Levi-Civita symbol ǫ αβγ as I αβγ = − iǫ αβγ , and ( σ α ) α = x,y,z stand for the set of Pauli matrices. The exchange inter-action is taken into account in the molecular-field andvirtual-crystal approximations: spin polarization of Mnions is described by a vector X = x eff S , where x eff is themolar fraction of Mn and S = 5 / sp - d exchange Hamiltonianassumes the form: h m, s |H sp − d ( k ) | n, s ′ i = − X α X α σ αss ′ " S X d ( 1 E d ↑ − E k − E d ↓ − E k ) t md ( k ) t dn ( k ) + D m (cid:12)(cid:12)(cid:12) J s − d ˆ P sc + J p − d ˆ P pc (cid:12)(cid:12)(cid:12) n E (6)The first term in the brackets was given by Schriefferand Wolff , and accounts for the kinetic exchange; thiscontribution is k -dependent and may be non-diagonal. Inthis term the matrix of hoppings is defined as in (5), withparameters given in Table III; the d label runs over the t g and e g orbitals of Mn, and the d orbitals are consideredto be located on the cation ( t dn may be non-zero only of n is located on the anion). The second term describesintra-Mn direct (potential) exchange J s − d and J p − d between electrons residing on 3 d and 4 s or 4 p states, re-spectively; ˆ P sc and ˆ P pc are the projectors on Mn cation4 s and 4 p states. These projectors restrict the exchangeinteraction to the site on which the Mn ion is located,and the corresponding terms resemble the frequently en-countered expression J sp − d S · s ( r ). According to spectro-scopic studies, J s − d = 0 .
392 eV and J p − d = 0 .
196 eVfor free Mn +1 ions . IV. CONCLUSIONS
We have shown, by combining appropriate DFT andTBA approaches, that it is possible to determine, in satisfactory agreement with magnetooptical data, fun-damental gaps and exchange splittings of bands at theΓ and L points of the Brillouin zone in Mn-doped topo-logically trivial CdTe and topologically non-trivial HgTewith no adjustable or empirical parameters. In particu-lar, a strong reduction of the splittings at the L pointscompared to the Γ point, hitherto regarded as not un-derstood, originates–according to our results–from (i) k -dependent hybridization between Bloch states and Mnopen d shells, which controls the magnitude of the ki-netic exchange; (ii) k -dependent changes in the orbitalcomponents of the Bloch functions, which affects the rel-ative magnitudes of antiferromagnetic kinetic exchangeand ferromagnetic potential exchange, and (iii) the im-portant role played by spin-orbit coupling in these sys-tems and magnetooptical phenomena employed to de-termine exchange splittings. Furthermore, we have dis-cussed the form of the empirical tight-binding model thatcan serve, by design, to even more accurate descriptionof phenomena relevant to these magnetic and topologicalcompounds. ACKNOWLEDGMENTS
We acknowledge Marcin M. Wysoki´nski for useful dis-cussions. The work is supported by the Foundation forPolish Science through the International Research Agen- das program co-financed by the European Union withinthe Smart Growth Operational Programme. We ac-knowledge the access to the computing facilities of theInterdisciplinary Center of Modeling at the University ofWarsaw, Grant No. G73-23 and G75-10. ∗ [email protected] in Introduction to the Physics of Diluted Magnetic Semi-conductors , edited by J. Kossut and J. A. Gaj (Springer,Heidelberg, 2010). T. Dietl, “(Diluted) Magnetic Semiconductors,” in
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