A theory for the reliable prediction of the EPR g-tensor for paramagnetic
defects in solids is presented. It is based on density functional theory and on
the gauge including projector augmented wave (GIPAW) approach to the
calculation of all-electron magnetic response. The method is validated by
comparison with existing quantum chemical and experimental data for a selection
of diatomic radicals. We thenperform the first prediction of EPR ${\rm g}
$-tensors in the solid state and find the results to be in excellent agreement
with experiment for the $E'_1$ and substitutional P defect centers in quartz.Comment: 5 pages, 4 table

Mauri, Francesco

Pickard, Chris J.

English

arXiv

Pickard, Christopher James

Mauri, F

University of St. Andrews - Pure

First-principles theory of the EPR g tensor in solids: Defects in quartz

University of St Andrews Research Portal

A theory for the reliable prediction of the EPR g tensor for paramagnetic defects in solids is presented. It is based on density functional theory and on the gauge including projector augmented wave approach to the calculation of all-electron magnetic response. The method is validated by comparison with existing quantum chemical and experimental data for a selection of diatomic radicals. We then perform the first prediction of EPR g tensors in the solid state and find the results to be in excellent agreement with experiment for the E-1' and substitutional phosphorus defect centers in quartz

Pickard, CJ

UCL Discovery

VOLUME 88, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 25 FEBRUARY 2002086403-1First-Principles Theory of the EPR g Tensor in Solids: Defects in QuartzChris J. PickardTCM Group, Cavendish Laboratory, Madingley Road, Cambridge, CB3 0HE, United KingdomFrancesco MauriLaboratoire de Minéralogie-Cristallographie de Paris, Université Pierre et Marie Curie,4 Place Jussieu, 75252, Paris, Cedex 05, France(Received 17 September 2001; published 11 February 2002)A theory for the reliable prediction of the EPR g tensor for paramagnetic defects in solids is presented.It is based on density functional theory and on the gauge including projector augmented wave approachto the calculation of all-electron magnetic response. The method is validated by comparison with existingquantum chemical and experimental data for a selection of diatomic radicals. We then perform the firstprediction of EPR g tensors in the solid state and find the results to be in excellent agreement withexperiment for the E01 and substitutional phosphorus defect centers in quartz.DOI: 10.1103/PhysRevLett.88.086403 PACS numbers: 71.15.–m, 61.72.Bb, 76.30.–vElectron paramagnetic resonance (EPR), also known aselectron spin resonance, is the most powerful spectroscopictechnique for the study of paramagnetic defects in solids.Indeed, defect centers are often named directly after theirEPR spectra. Applications of EPR extend to any situationwhere there are unpaired electrons, including the under-standing of reactions involving free radicals in both bio-logical and chemical contexts or the study of the structureand spin state of transition metal complexes.EPR spectra of spin 12 centers are made up of two con-tributions: (i) the hyperfine parameters, which can be com-puted from the ground state spin density, and have beenused to connect theoretical studies of defects to availableexperimental data [1–6], and (ii) the g tensor. Only re-cently have there been attempts to calculate the g tensorin molecules from first principles using density functionaltheory (DFT) [7,8]. However, these approaches are validonly for finite systems and, thus, are not useful for the cal-culation of the g tensor for paramagnetic defects in solids,except possibly within a cluster approximation. In the ab-sence of a predictive scheme, experimentally determinedg tensors are, of necessity, interpreted in terms of theirsymmetry alone, leaving any remaining information unex-ploited. A reliable, first-principles approach to the predic-tion of g tensors in solids, in combination with structuraland energetic calculations, would access this information,and could be used for an unequivocal discrimination be-tween competing microscopic models proposed for defectcenters. In this Letter we describe an approach for the cal-culation of the g tensor in extended systems, using periodicboundary conditions and supercells.In a previous paper [9] we have shown how to com-pute the all-electron magnetic linear response, in finite0031-90070288(8)086403(4)$20.00and extended systems, using DFT and pseudopotentials.To achieve this we introduced the gauge including projec-tor augmented wave (GIPAW) method, which is an exten-sion of Blöchl’s projector augmented wave (PAW) method[10]. In Ref. [9], we used GIPAW to compute the NMRchemical shifts in molecules and solids. Here we applythe GIPAW approach to the first-principles prediction ofEPR g tensors for paramagnetic defects in solids. We vali-date our theory and implementation for diatomic radicals,for which both all-electron quantum chemical calculationsand experimental data exist. As, until now, there have beenno first-principles calculations of g tensors in solids, wevalidate our method in the solid state by a direct compari-son with experiment. In particular, we interpret, from firstprinciples, the EPR spectrum of the well characterized andtechnologically important defects in quartz, the E01 andphosphorus substitutional centers.The g tensor is an experimentally defined quantity, aris-ing from the recognition that the EPR spectrum can bemodeled using the following effective Hamiltonian, bi-linear in the total electron spin S, and the applied uniformmagnetic field or nuclear spins, B and II , respectively:Heff a2S ? g ? B 1XIS ? AI ? II . (1)Here, and in the following, atomic units are used, a isthe fine structure constant, and the summation I runs overthe nuclei. The tensors AI are the hyperfine parameters (aPAW based theory for its calculation has been describedelsewhere by Van de Walle and Blöchl [1,3]), and thetensor g is the EPR g tensor.In order to calculate the g tensor we start from the elec-tronic Hamiltonian which includes terms up to order a3, inthe presence of a constant external magnetic field B [7,11]:H Xi(pi 1 aAri 222XIZIjri 2 RI j 1Xjﬁi1jri 2 rjj)1 HZ 1 HZ-KE 1 HSO 1 HSOO . (2)© 2002 The American Physical Society 086403-1VOLUME 88, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 25 FEBRUARY 2002The summations over i and j run over the electrons and HZ, HZ-KE, HSO, and HSOO are the electron Zeeman, the electronZeeman kinetic energy correction, the spin-orbit, and the spin-other-orbit terms, respectively:HZ age2XiSi ? B, HZ-KE  2a3ge2Xip2i2Si ? B ,HSO a2g04XiSi ?√XIZIri 2 RIjri 2 RI j3 2Xjﬁiri 2 rjjri 2 rjj3!3 pi 1 aAri ,HSOO  a2Xi,jﬁiSi ?ri 2 rjjri 2 rjj3 3 pj 1 aArj .(3)The constant g0 is related to ge, the electronic Zeemang factor in vacuum, by g0  2ge 2 1, and Ar 12B 3 r is the vector potential.Starting from the Hamiltonian of Eq. (2), we can expandthe total energy in powers of a, up to Oa3, using pertur-bation theory. In the resulting expansion, the term bilinearin Si and B is identified as the first term of Eq. (1). Thisterm can be rewritten within the formalism of spin polar-ized DFT to obtain an explicit expression for the g tensor:g  ge 1 DgZ-KE 1 DgSO 1 DgSOO  ge 1 Dg ,(4)where ge  geI, I being the identity matrix, andDgZ-KE  2a2geT0" 2 T0# I , (5)DgSO ? B a2g0Zd3r  j1" r 3 =V0ks,"r2 j1# r 3 =V0ks,#r , (6)DgSOO ? B  2Zd3r B1r r0" r 2 r0# r . (7)Here " denotes the majority spin channel and r0" r,T0" , and V0ks,"r are the unperturbed electron probabilitydensity, kinetic energy, and Kohn-Sham potential of the"-spin channel, respectively. j1" r is the electronic chargecurrent linearly induced by the constant magnetic field Bin the "-spin channel. Finally, B1r is the magnetic fieldproduced by the total induced current, j1" r 1 j1# r,which we correct for self-interaction by removing thecontribution from the current of the unpaired electron,j1" r 2 j1# r.We can interpret the physical origin of deviation of theg tensor from its value in vacuum. The spin-other-orbitcorrection, DgSOO, describes the screening of the externalfield B by the induced electronic currents, as experiencedby the unpaired electron. The unpaired electron itself isnot at rest and in the reference frame of the unpaired elec-tron the electric field due to the ions and to the other elec-trons is Lorentz transformed so as to appear as a magneticfield. The interaction between the spin of the unpairedelectron and this magnetic field results in the spin-orbit cor-086403-2rection, DgSO [12]. Finally, the electron Zeeman kineticenergy correction, DgZ-KE, is a purely kinematic relativis-tic correction.Equations (5)–(7) show that the evaluation of the gtensor requires, besides ground state quantities, the linearmagnetic response currents j1r. Mauri, Pfrommer, andLouie [13] showed how to calculate the magnetic responseof a system of electrons in an infinite insulating crystal, andour recent paper [9] reformulated this so as to be strictlyvalid for nonlocal pseudopotentials, and to reproduce thevalence all-electron currents even within the pseudizationcore region. An accurate description of the all-electroncurrents in the core regions is essential for the evaluationof the SO term, Eq. (6). Indeed, the dominant contributionto the integral in Eq. (6) comes from the core region as aresult of the divergence of V 0ks r at the nuclei.Using our GIPAW approach to the calculation of theall-electron magnetic response using pseudopotentials,described in detail in Ref. [9], we break the SO term intothree parts which derive from the three GIPAW contribu-tions to the induced current, Eq. (34) of Ref. [9]:DgSO  DgbareSO 1 DgDdSO 1 DgDpSO . (8)The DgbareSO term is evaluated from Eq. (6) using a spindependent version of the j1barer of Ref. [9] and a localKohn-Sham potential consisting of a sum of the self-consistent contribution to the local potential and the localparts of the pseudopotentials.The diamagnetic correction term DgDdSO can be evalu-ated from the ground state valence pseudo wave functionsjC¯0o," using the following expression:DgDdSO ? B XI,o,n,mC¯0o," j p˜I,neIn,mp˜I,m j C¯0o,"2XI,o,n,mC¯0o,# j p˜I,neIn,mp˜I,m j C¯0o,# .(9)The summation o is over occupied states. The projec-tor functions jp˜I,n are defined in Ref. [9] and satisfy p˜I,n j f˜I0,m  dI,I 0dn,m, where jf˜I,n are a set of pseudopartial waves corresponding to the all-electron partialwaves jfI,n. The projector weights eIn,m are given by the086403-2VOLUME 88, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 25 FEBRUARY 2002following atom centered integrals:eIn,m  2a2g04fI,nj B 3 r 3 =V r jfI,m2 f˜I,nj B 3 r 3 =V˜ r jf˜I,m .(10)The potentials V r and V˜ r in Eqs. (10) and (11) are thescreened atomic all-electron and local channel pseudopo-tentials, respectively.The evaluation of the paramagnetic correction termDgDpSO is more involved as it requires the first order linearresponse wave functions. However, the required evaluationcan be described by analogy with the calculation of para-magnetic correction to the NMR chemical shifts, sDpGIPAW,replacing the weights fIn,m in Eq. (60) of Ref. [9] byfIn,m  fI,njg02Lr≠V r≠rjfI,m2 f˜I,nj g02Lr≠V˜ r≠rjf˜I,m , (11)where L is the angular momentum operator.The electron Zeeman kinetic energy correction termDgZ-KE is evaluated by combining a straightforward PAWcorrection with the quantity evaluated from the groundstate pseudo valence wave functions using Eq. (5). In thiswork the SOO term is evaluated from the induced fieldB1r derived from the bare induced current, and the spindensity due to the pseudo wave functions. It is expectedthat a full GIPAW treatment would result in only minorcorrections since (i) the SOO term is small in comparisonto the SO term, and (ii) both the induced field and the spindensity do not diverge at the nuclei.To validate our new expressions for the evaluation ofthe g tensor, and our implementation of them into a paral-lelized plane-wave pseudopotential code, we compare withthe all-electron gauge including atomic orbital (GIAO)DFT results obtained by Schreckenbach and Ziegler [7]for a series of diatomic radicals. We use their calculatedbond lengths for the dimers, but approximate the isolateddimers by using large supercells. Troullier-Martins pseu-dopotentials [14] and the (spin polarized) generalized gra-dient approximation due to Perdew et al. [15] (GGA-PBE)are used throughout our calculations. Table I shows theexcellent agreement between our two approaches. The ex-ception is the AlO radical, for which we obtain much closeragreement with experiment. The otherwise close agree-ment between these two very different approaches sug-gests a technical rather than fundamental problem in theGIAO calculation for AlO. Comparison with experimentis made through Table 2 of Ref. [7], while acknowledgingthat most measurements are performed in solid matrices,which strongly influence the g tensor (most notably theDgk components), and that the experimental errors are ofthe order of several hundred parts per million (ppm).086403-3TABLE I. Calculated Dg tensors, in parts per million (ppm),for diatomic molecules. For comparison with Ref. [7] we omit(in this table only) the SOO contribution to our calculations. A100 Ry plane-wave cutoff is used.Dgk DgMolecule GIPAW SZ [7] GIPAW SZ [7]H12 239 239 241 242CO1 2134 2138 23223 23129CN 2138 2137 22577 22514AlO 2141 2142 22310 2222BO 269 272 22363 22298BS 280 283 29901 29974MgF 249 260 22093 22178KrF 2340 2335 61 676 60 578XeF 2333 2340 157 128 151 518Finally, by analyzing the different contributions, includ-ing the SOO term, we found that in all dimers apart fromH12 , the SO term accounts for more than 90% of TrDg3,and that the paramagnetic correction term DgDpSO accountsfor the overwhelming majority of the SO term.To further validate our approach to the calculation of theg tensor and to apply it for the first time in the solid state,we study two defects of a quartz.The E01 center is associated with a positively chargedoxygen vacancy, with the unpaired electron on a Si dan-gling bond. As in previous calculations [2,3,17], we modelthe defect with a 71 atom (24 Si and 47 O), positivelycharged (11) hexagonal supercell. We use the theoreti-cal GGA-PBE lattice parameters (which are 1% largerthan in experiment) and relax the atoms. For the struc-tural optimization we use a G only k-point sampling anda plane-wave cutoff of 50 Ry. The resulting relaxed struc-ture is very close to that of Ref. [3]. The EPR g tensor iscalculated using our relaxed structure, a plane-wave cutoffof 70 Ry and 4 inequivalent k points. In Table II we com-pare our theoretical g tensor with the experimental results[16], finding excellent agreement.The P2 defect center is neutral and associated with afourfold coordinated P atom, substituting for a Si atom.The center exists as two variants at low temperature(,140 K) in quartz, labeled P2I in the ground stateand P2II in the excited state [18]. Only recently havethese P defect centers been examined using DFT basedtotal energy approaches [4,6]. However, up until now,the connection with experiment has been made using theTABLE II. Calculated Dg tensors for our model E01 defect, andcorresponding experimental data [16].Principal values Principal directionsGIPAW (ppm) Expt. (ppm) GIPAW Expt.u f u f2651 2530 110.0± 223.5± 114.5± 227.7±22255 21790 142.3± 341.6± 134.5± 344.4±22481 22020 120.4± 121.1± 125.4± 118.7±086403-3VOLUME 88, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 25 FEBRUARY 2002TABLE III. Calculated total energies, with respect to ourlowest energy configuration. For nontetrahedral configurations,OPO indicates the largest O-P-O angle after the relaxation,and OSiO the corresponding angle in the unrelaxed a-quartzstructure. With l and s we specify whether the two SiO bondsforming in the OSiO angle are short or long in the unrelaxedstructure. The number of symmetry equivalent structures is Nc .We report our assignment of the experimental centers based onthe comparison of Table IV.SiO OSiO OPO Nc Energy (meV) SpeciesTetrahedral configuration 1 730ls 108.8± 147.4± 2 243ss 108.9± 155.4± 1 178ll 109.5± 157.6± 1 79 P2Ils 110.4± 155.1± 2 0 P2IIhyperfine parameters alone, and the two variants of thedefects, P2I and P2II, have not been distinguishedtheoretically.Using the method described above, and a supercell of72 atoms, 5 distinct total energy local minima are foundas a function of the initial configuration, Table III. Theconfiguration with the highest energy corresponds to asymmetric relaxation with the P remaining tetrahedrallycoordinated, and O-P-O angles of about 109±. In the 4other configurations the P atom moves off-center, open-ing up one of the 6 O-P-O angles, which reaches a valueof about 150±. We computed the EPR g tensors for ourtwo lowest energy structures and compare them with theexperimental results [18] in Table IV. Again, we obtainan excellent agreement between theory and experiment,which confirms that the two lowest energy theoreticalstructures correspond to the two lowest energy experimen-tal structures. However, the comparison between g tensorsshows that the energy ordering between the P2I andP2II species is not correctly described by theory. This isnot surprising given the small energy separation betweenthe two configurations. This is expected to be sensitive toboth the size of the supercell and the use of approximatedDFT functionals.To summarize, we have calculated the EPR g tensor fora paramagnetic defect in an extended solid for the firsttime and find our results to be in excellent agreement withexperimental results for the E01 defect. On applying themethod to the P2 defect, we show that comparison withexperimental g tensors can provide structural informationwhere the accuracy of DFT for energetics is insufficient.Combined with the calculation of hyperfine parameters[1–6], we expect that our GIPAW based first-principlesapproach to the prediction of EPR g tensors will be of greatuse in the assessment of models proposed for less wellcharacterized paramagnetic defects, and add significantlyto the tools available to the electronic structure community.C. J. P. thanks the Université Paris 6 and the UniversitéParis 7 for support during his stay in Paris. The calcula-086403-4TABLE IV. Calculated Dg tensors for our model P2 defectcenters, and corresponding experimental data [18].Principal values Principal directionsGIPAW (ppm) Expt. (ppm) GIPAW Expt.u f u fConfiguration P2I1249 900 65.0± 90.0± 64.8± 90.0±2980 21100 90.0± 0.0± 90.0± 0.0±23414 23200 25.0± 270.0± 25.2± 270.0±Configuration P2II1146 1100 47.1± 13.1± 46.5± 11.8±2824 21000 99.0± 94.7± 101.1± 91.1±23454 23200 135.7± 355.4± 134.4± 350.1±tions were performed at the IDRIS supercomputing centerof the CNRS and on Hodgkin (SGI Origin 2000) at theUniversity of Cambridge’s High Performance ComputingFacility.[1] C. G. V. de Walle and P. E. Blöchl, Phys. Rev. B 47, 4244(1993).[2] M. Boero, A. Pasquarello, J. Sarnthein, and R. Car, Phys.Rev. Lett. 78, 887 (1997).[3] P. E. Blöchl, Phys. Rev. B 62, 6158 (2000).[4] G. Pcacchioni, D. Erbetta, D. Ricci, and M. Fanciulli,J. Phys. Chem. B 105, 6097 (2001).[5] T. Uchino, M. Takahashi, and T. Yoko, Phys. Rev. Lett. 86,4560 (2001); 86, 5522 (2001).[6] J. Lægsgaard and K. Stokbro, Phys. Rev. B 61, 12 590(2000); 65, 075208 (2002).[7] G. Schreckenbach and T. Ziegler, J. Phys. Chem. A 101,3388 (1997).[8] O. L. Malkina et al., J. Am. Chem. Soc. 122, 9206 (2000).[9] C. J. Pickard and F. Mauri, Phys. Rev. B 63, 245101 (2001).[10] P. E. Blöchl, Phys. Rev. B 50, 17 953 (1994).[11] J. E. Harriman, Theoretical Foundations of Electron SpinResonance (Academic Press, New York, 1978).[12] On transforming back to an inertial frame from that of theunpaired electron, a factor of 12 is introduced into theinteraction energy, a result known as Thomas precession:L. H. Thomas, Philos. Mag. 3, 1 (1927).[13] F. Mauri, B. G. Pfrommer, and S. G. Louie, Phys. Rev. Lett.77, 5300 (1996).[14] N. Troullier and J. L. Martins, Phys. Rev. B 43, 1993(1991).[15] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett.77, 3865 (1996).[16] M. G. Jani, R. B. Bossoli, and L. E. Halliburton, Phys.Rev. B 27, 2285 (1983).[17] C. M. Carbonaro, V. Fiorentini, and F. Bernardini, Phys.Rev. Lett. 86, 3064 (2001).[18] Y. Uchida, J. Isoya, and J. A. Weil, J. Phys. Chem. 83, 3462(1979).086403-4

Chris J. Pickard

Francesco Mauri

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Physical Review Letters

First-Principles Theory of the EPR<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="italic">g</mml:mi></mml:math>Tensor in Solids: Defects in Quartz

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First principles theory of the EPR g-tensor in solids: defects in quartz

First principles theory of the EPR g-tensor in solids: defects in quartz

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University of St. Andrews - Pure