Da Silva, JLF

Walsh, A

Wei, S-H

UCL Discovery

Theoretical Description of Carrier Mediated Magnetism in Cobalt Doped ZnOAron Walsh, Juarez L. F. Da Silva, and Su-Huai WeiNational Renewable Energy Laboratory, Golden, Colorado 80401, USA(Received 4 March 2008; published 23 June 2008)Substitutional cobalt in ZnO has a weak preference for antiferromagnetic ordering. Stabilization offerromagnetism is achieved through n-type doping, which can be understood through a band couplingmodel. However, the description of the transition to a ferromagnetic ground state varies within differentlevels of band theory; issues arise due to the density functional theory underestimation of the band gap ofZnO, and the relative position of the nominally unfilled Co t2d states. We examine these limitations,including approaches to overcome them, and explain the contradictions in previous studies, whichdrastically overestimate the doping threshold for magnetic ordering.DOI: 10.1103/PhysRevLett.100.256401 PACS numbers: 71.20.Nr, 71.15.Mb, 75.10.b, 75.50.PpThe potential to simultaneously tune both charge andspin in solid state materials has lead to great interest in thefield of spintronics [1]. The ultimate aim is a controllableroom temperature semiconducting ferromagnet, whichcould be used in magnetoelectric and magnetotransportdevices. Intrinsically magnetic semiconductors generallypossess relatively low Curie temperatures (TC); however, ithas been proposed that incorporating transition metal orrare earth ions into a nonmagnetic semiconductor hostlattice, forming a dilute magnetic semiconductor (DMS),may help raise TC above room temperature [2,3].In DMS materials, magnetism is influenced and con-trolled by the presence of charge carriers in the form ofholes (GaAs:Mn) or electrons (GaN:Gd). There is stillgreat debate over the fundamental mechanism behind theorigin of the observed high temperature magnetic align-ment in many of these systems. The situation is not helpedby large variation in both experimental and theoreticalstudies [4–8].Cobalt doped ZnO has become a focus of attention dueto its reported high TC of up to 700 K and, most promis-ingly, the reversible cycling of FM ordering [9]. Coupledwith existing optical and electrical properties of ZnO, theaddition of controllable magnetism could make it a tech-nologically essential material. It has been reported that atCo concentrations of less than 10%, the solubility is goodand Co occupies a substitutional Zn site in a 2 chargestate [9,10].The stability of the Co(II) 3d74s0 configuration in four-fold tetrahedral coordination sites is demonstrated byCo3O4 [11]. The roughly Td crystal field splits the Co 3dstates into lower ed and upper t2d states. This results in theoccupation of all the majority spin 3d states and minorityspin ed states, with a local Co magnetic moment close to 3B. The splitting between filled ed and empty minority t2dstates for tetrahedral Co is typically on the order of 1.5–2 eV [12]. In a wide gap semiconductor such as ZnO (Eg 3:4 eV), this would place the minority spin Co t2d stateswithin the band gap, as has been reported from both opticaland magnetic measurements for ZnO:Co [13–15]. Surpris-ingly, this is not the case in previous theoretical studies [5–8].From consideration of Co 3d band coupling, FM inter-actions between occupied majority t2d states results in nonet energy gain due to filled bonding and antibondingcombinations, while AFM interactions can result in a smallenergy gain (Fig. 1). Partial occupation of the emptyminority t2d states will result in a transition to FM orderingthrough the energy gained from the filling of the bondingstate. Within such a band coupling model [16,17], theenergy difference between the FM and AFM phases canbe described as EFM-AFM  2me2dd  23me1;2dd ; (1)where me is the additional number of electrons per Co(II)ion and 2dd and 1;2dd are the direct exchange and super-exchange parameters, respectively (Fig. 1).FIG. 1. Coupling of the Co t2d levels for CoZn substitution inZnO for (left) the pure host system and (right) an n-type dopedsystem.PRL 100, 256401 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending27 JUNE 20080031-9007=08=100(25)=256401(4) 256401-1 © 2008 The American Physical SocietyCoupling between the relatively localized Co 3d states ispossible through mediation of O p orbitals, which are ofthe same t2 symmetry and can facilitate superexchangeinteractions. This is consistent with experiment, wherereducing conditions, in the form of the addition of Znvapor or low O2 partial pressure, are needed to stabilizeFM ordering [9]. Extrinsic n-type doping (e.g., Al) has alsobeen shown to have a similar effect [15]. In contrast, bandstructure studies of ZnO:Co have reported preference forboth FM [5] and AFM [6] ordering, in addition to nopreferred magnetic alignment [7].In this work, we demonstrate that careful considerationof the electronic structure description is needed for accu-rate modeling of this system. In particular, the relativepositions of the intrinsic ZnO conduction band and unfilledminority Co t2d levels are shown to determine the magneticbehavior and hence control the onset of ferromagnetismwith respect to carrier concentration.Calculations were performed on a 72 atom wurtzite ZnOsupercell using gradient corrected density functional the-ory [18–20] (PBE-DFT) as implemented in VASP [21]. A2 2 2 Monkhorst Pack k-point grid and a 500 eVplane-wave cutoff were employed. To overcome errorsassociated with the DFT self-interaction, calculationswere also performed using DFT corrected for on-siteCoulombic interactions (DFTUd), and DFT with exactFock exchange (hybrid-DFT). The latter were performedusing a unreleased version of VASP [22]. For the DFTUdcalculations, the Co 3d states were corrected by U J 2 eV, which has been shown to result in Co d-d splittingconsistent with experiment for Co3O4 [11], and U J 5 eV was applied to the Zn 3d states. The ZnO band gaps atthe DFT, DFTUd and hybrid-DFT levels are 0.8 eV,1.4 eV and 3.4 eV (PBE0), respectively. For bulk ZnO, thecell size and atomic coordinates were fully relaxed. For Cocontaining systems, the lattice constant was fixed, but theinternal positions were allowed to relax. Carriers wereintroduced by partially filling the lowest unoccupied bands,with a uniform background charge employed to ensure thatthe net charge of the periodic cell remains zero.For nearest neighbor CoZn pairs, the calculated groundstate magnetic ordering at both the DFT and DFTUdlevels shows enhanced AFM stabilization energies of37 and 31 meV per Co pair, respectively. This is in agree-ment with the absence of ordered magnetism at roomtemperature, as the local moments will randomly orientateunder the influence of thermal fluctuations.Experimentally, reducing conditions are needed to sta-bilize FM ordering. Partial filling of the nominally emptyminority spin Co t2d states is expected to result in atransition to FM ordering (Fig. 1). We have simulated theeffect of donors by calculating both FM and AFM orderingof nearest neighbor Co pairs as a function of electrons. Thisapproximates that the Co ions are more directly affected bythe addition of charge carriers than the structural effects ofthe underlying donor defects.Surprisingly, for DFT, the addition of up to one electronper Co does not significantly alter the energetic differencebetween AFM and FM ordering, EFM-AFM (Fig. 2). Thisalso holds true for DFTUd. This effect originates fromerrors in the relative band level positions as summarized inFig. 3. Absorption measurements indicate that the occupiedminority ed states lie approximately 1.5 eV above the ZnOvalence band maximum (VBM) [13]. At the DFT level, thisseparation is reduced to 0.8 eV, and as this coincides withthe underestimated conduction band minimum (CBM) ofZnO, there is partial occupation of both the ZnO CBM andthe Co ed states producing an incorrect metallic electronicstructure. In this case, adding extra electrons to the unoc-cupied ed state reduces the magnetic moment and theexchange splitting between the majority and minorityspin states, which slightly enhances AFM ordering throughthe increase of 1;2dd [Eq. (1)].DFTUd helps by partially correcting the band gapunderestimation, but fails to resolve the conflict with theexperimental electronic structure: while the VBM-Co edseparation is good, and the Co minority ed  t2d splittingof 1.9 eV is in agreement with available data, the ZnO bandgap is still too small, such that the unoccupied t2d states liewell above the calculated CBM. This results in the occu-pation of Zn 4s bands before Co t2d, and hence a highconcentration of carriers are required to stabilize FM or-dering. Once the initial threshold is overcome, FM cou-pling between the Co ions becomes rapidly stabilized asexpected from our initial band coupling considerations.The addition of 1.5 electrons per Co results in aEFM-AFM of 120 meV (291 meV) for DFT (DFTUd);however, such high doping levels corresponding to 41021 cm3 are physically unrealistic for the initial onset offerromagnetism (Fig. 2).Another widely used approach to compensate for theDFT band gap underestimation is hybrid-DFT, which con-FIG. 2 (color online). Calculated magnetic ordering energiesin ZnO:Co as a function of carrier electrons.PRL 100, 256401 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending27 JUNE 2008256401-2tains the proportion of exact nonlocal Fock exchange as anadjustable parameter; i.e., the exchange-correlation func-tional becomes EHybridXC  1 xEPBEX  xEFockX  EPBEC : (2)Indeed the B3LYP functional has been applied to treatZnO:Co, but again unphysically high levels of electrondoping are needed to stabilize FM ordering [6]. From ouranalysis, this originates from a strong system dependenceof the exact exchange, which cannot be overcome withinthis methodology. For example, to reproduce the experi-mental band gap of ZnO, 25% of Fock exchange is re-quired, while for Co3O4 less than 5% exchange is adequate(Fig. 4). This is due to the localized nature of the Co3O4band edges which contain a stronger self-interaction incomparison to ZnO whose CBM is a delocalized cation-anion s state. At the high levels of exchange needed to treatthe band gap of ZnO, the Co d-d splittings are overesti-mated in the order of 300%. Recently, self-interaction-corrected (SIC) DFT has also been applied to ZnO:Co,but to similar avail, with the Co t2d levels situated wellabove the ZnO CBM [8,23]. In all such approaches, theCBM states are preferentially filled and act as a strongbarrier to ferromagnetism.An alternative solution can be found through applicationof an orbital dependent potential to control the band gap ofZnO and the Co d-d splitting independently. This could bedone by either adding a Christensen potential [24,25] or byapplying a Coulomb U on both the s and d orbitals tofurther raise the CBM level [26,27]. Through the applica-tion of DFTUd=s we can obtain an electronic structuredescription consistent with experiment; however, otherapproaches can also be used to correct the DFT banderrors, and the magnetic physical picture discussed inthis paper is not affected by the method used. As shownin the calculated density of states (Fig. 5), at high bindingenergy are the Zn 3d states (9 eV), which are followed bythe main O p valence band. The topmost occupied statesare the minority Co ed states, with the unoccupied t2d statespresent around the onset of the ZnO conduction band. Thismore realistic description radically lowers the thresholdfor the onset of ferromagnetism. On electron doping, eventhe addition of 0.5 electrons to the system results in astrong stabilization of FM ordering with EFM-AFM 283 meV (Fig. 2). From our calculations, a maximumin the FM interaction is reached at the addition of about oneelectron per Co ion (354 meV), corresponding to a carrierconcentration of 2 1021 cm3. This maximum is some-what lower than the 1.5 electrons predicted from the idealband coupling model [Eq. (1)], where the minority spinbonding combination would be fully occupied (Fig. 1). Thedifference is due to the effects of partially filling of theZnO CBM and the enhancement of the superexchange 1;2ddterm in Eq. (1) at higher carrier concentrations.From our results we can clearly state that the electronicstructure of ZnO:Co is consistent with carrier mediatedferromagnetism. Both the band coupling interpretation andthe calculated transition to FM ordering, on the addition ofa small number of carriers, indicate more general magneticinteractions in contrast to hypothesized defect complexes[8,28]. While our calculations do not rule out the existenceor contributions of such mechanisms, they do show that thetransition to FM ordering should be independent of thenature of the carriers.We have demonstrated that the fundamental magneticinteractions in Co doped ZnO can be understood from bandcoupling of the Co 3d states. The system is intrinsicallyAFM, but the addition of charge carriers (electrons) will0 5 10 15 20 25 30    % Hartree-Fock exchange1.02.03.04.05.06.0Band gap (eV)ZnOCo3O4EgEgFIG. 4 (color online). Calculated   valence and conduc-tion band separations in wurtzite ZnO and spinel Co3O4 as afunction of Fock exchange in hybrid-DFT. The correspondingexperimental band gaps are indicated by dashed lines [30].Eg3.4 eVExperiment DFT Hybrid/SIC-DFTCBVBCo e1.9 eV1.5 eV0.8 eV1.2 eV> 3.0 eV1.5 eVCo t2FIG. 3 (color online). Schematic band level diagram compar-ing the minority Co d levels in ZnO derived from experiment[13] and DFT calculations. The highest occupied state is indi-cated by the dashed horizontal line.PRL 100, 256401 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending27 JUNE 2008256401-3result in a transition to FM ordering through enhancedcoupling between the Co t2d states. Within DFT, theground state of weak AFM behavior is correctly described,but the transition to FM ordering is hampered by the severeband gap underestimation for ZnO. The onset of ferromag-netism is incorrectly described due to the preferentialoccupation of the ZnO conduction band before unoccupiedminority Co 3d bands. This failure cannot be overcomewithin the realms of hybrid-DFT, due to the sensitivedependence of Co on the addition of exact Fock exchange,in comparison to Zn: for all values of exchange, the Cominority ed  t2d splitting is larger than the ZnO band gap.Through the application of an orbital dependent potential(DFTUd=s), a reasonable description of the band struc-ture and level occupations can be obtained, which revealsthe correct origins of ferromagnetic phase stabilization inthis system.Improvement in the treatment of exchange-correlationeffects in DFT is required to accurately treat magneticdoping of semiconductors within a fully ab initio frame-work, i.e., without the introduction of multiple systemdependent parameters. Higher-order methods such asself-consistent GW theory [29] and dynamical mean fieldtheory (DMFT) with total energy may offer a solution inthe future. We expect that electron mediated DMS systemswith calculated band gaps smaller than the typical transi-tion metal d-d splittings will be prone to these DFT bandfilling errors, including GaN, In2O3 and SnO2. The limi-tations of the method adopted should be carefully consid-ered in future theoretical studies.We thank G. Kresse for the provision of VASP 5.1. Thework is supported by the U.S. Department of Energy(Contract No. DE-AC36-99GO10337).[1] D. D. Awschalom and M. E. Flatte, Nature Phys. 3, 153(2007).[2] T. Dietl et al., Science 287, 1019 (2000).[3] K. Sato and H. Katayama-Yoshida, Semicond. Sci.Technol. 17, 367 (2002).[4] R. Janisch, P. Gopal, and N. A. Spaldin, J. Phys. Condens.Matter 17, R657 (2005).[5] P. Gopal and N. A. Spaldin, Phys. Rev. B 74, 094418(2006).[6] E.-J. Kan et al., J. Appl. Phys. 102, 033915 (2007).[7] C. H. Patterson, Phys. Rev. B 74, 144432 (2006).[8] C. D. Pemmaraju et al., arXiv:0801.4945v1.[9] K. R. Kittilstved et al., Phys. Rev. Lett. 97, 037203 (2006).[10] Y. Z. Liu et al., Appl. Phys. Lett. 90, 154101 (2007).[11] A. Walsh et al., Phys. Rev. B 76, 165119 (2007).[12] H. A. Weakliem, J. Chem. Phys. 36, 2117 (1962).[13] K. R. Kittilstved, W. K. Liu, and D. R. Gamelin, Nat.Mater. 5, 291 (2006).[14] A. Barla et al., Phys. Rev. B 76, 125201 (2007).[15] A. J. Behan et al., Phys. Rev. Lett. 100, 047206 (2008).[16] G. M. Dalpian and S.-H. Wei, Phys. Stat. Sol. B 243, 2170(2006).[17] G. M. Dalpian et al., Solid State Commun. 138, 353(2006).[18] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133(1965).[19] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864(1964).[20] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett.77, 3865 (1996).[21] G. Kresse and J. Furthmu¨ller, Phys. Rev. B 54, 11 169(1996).[22] J. Paier et al., J. Chem. Phys. 124, 154709 (2006).[23] M. Toyoda et al., J. Phys. B 376–377, 647 (2006).[24] N. E. Christensen, Phys. Rev. B 30, 5753 (1984).[25] S.-H. Wei and A. Zunger, Phys. Rev. B 48, 6111 (1993).[26] H. Raebiger, S. Lany, and A. Zunger, Phys. Rev. B 76,045209 (2007).[27] Here we apply DFTUd with an additional repulsivepotential (Us  56 eV) on O s. The delocalized O sstates lie to high binding energy and are mildly effected bythe application of U (shifting by 	3 eV). This has a weakeffect of raising the ZnO CBM state (40% O s). SeeEPAPS Document No. E-PRLTAO-101-028852 for furtherdiscussion and background on the band gap correctionmethod adopted. For more information on EPAPS, seehttp://www.aip.org/pubservs/epaps.html..[28] M. Venkatesan et al., Phys. Rev. Lett. 93, 177206 (2004).[29] M. van Schilfgaarde et al., Phys. Rev. Lett. 96, 226402(2006).[30] The Co3O4 fundamental gap is found at X 	1:2 eV,corresponding to a  separation of 1.7 eV (Ref. [11]).Energy (eV)n (electrons)Zn dO pZn sCo t2Co t2Co e-10 -8 -6 -4 -2  0   2   4FIG. 5 (color online). DFTUd=s calculated density of statesfor Zn34O36:Co2 in an antiferromagnetic configuration. The Costates are 2 for clarity and the Fermi level is indicated by thedashed vertical line.PRL 100, 256401 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending27 JUNE 2008256401-4

Theoretical description of carrier mediated magnetism in cobalt doped ZnO

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Theoretical description of carrier mediated magnetism in cobalt doped ZnO

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