PHYSICAI. RE V'IEVf 8 yoI, UME .11, NUMBER 6 15 MARCH 1975

point, defects in hexagonal boron nitride. II. Theoretical studies

A. ZungerDepartment of Theoretical Physics, Soreq Nuclear Research Center, Favne, Israel

and Department of Chemistry, Tel-Aviv University, Rarnat Aviv, Israel

A. KatzirDepartment of Solid State Physics, Soreq Nuclear Research Center, Yavne, Israel

(Received 4 April 1974)

Substitutional and displaced carbon impurities and an isolated nitrogen vacancy in hexagonal boronnitride are theoretically investigated by the "small periodic cluster" approach. The perfect-solid bandstructure is calculated from the solution of the eigenvalues of a finite and periodic cluster of atoms

arranged according to the known crystal structure. The linear combination of atomic orbitalsrepresentation of the crystal orbitals is adopted and semiempirical MO (molecular orbital) methods

(extended Huckel, iterative extended Huckel) are used for the solution of the electronic eigenvalue.problem. Point-defect problems are then treated by introducing the impurity atom or the vacant siteinto the otherwise perfect periodic cluster, and repeating the solution. Lattice relaxations are introducedaround the defect site and charge redistribution among the cluster atoms is allowed for viaself-consistent MO treatment. For a substitutional carbon impurity defect, it is observed that two deepdefect levels, mainly localized on the carbon atom, appear (3.2—4.9 eV below the conduction band).Another level splits from the conduction band as the carbon atom is raised from the layer plane in aperpendicular direction. This level, 1.0—1.3 eV below the conduction band, has a symmetrical chargedistribution on the three boron atoms surrounding the impurity site. As the distance of the carbonatom from the layer plane is increased to infinity, a nitrogen vacancy is formed. It is characterized bya defect level 1, 1—1.4 eV apart from the conduction-band edge which also possesses a three-boroncharacter. Lattice relaxations were shown to stabilize these defects. The findings agree' semiquantitativelywith the experimental results on these defects.

I. INTRODUCTION

In view of the accumulating experimental evi-dence indicating the participation of nitrogen va-'cancies and carbon impurities in determining op-tical, electron-paramagnetic-resonance (EPR),thermolumines cence, and photolumines cence prop-erties of boron nitride (Paper I of this set), atheoretical calculation of these defects was under-taken, Based on known experimental data and pre-vious theoretical studies of deep defects in cova-lent solids, the main requirements for such a the-oretical defect model are the following: (i) Itshould locate the position of the defect levels withrespect to the band edges. The energy separationbetween the defect level and the band edges shouldthen be compared with EPR and thermolumines-cence activation energies and with the temperaturedependence of photoluminescence. (ii) It shouldprovide the electronic wave function of the defectlevel, and thus the charge distribution around thedefect site, to be compared with both the EPR as-signment and the hyperfine splitting. (iii) Due toboth the presence of ionic-covalent bonding in theperfect crystal and to charge redistribution inducedby the vacancy or the impurity in the defect struc-ture, the model should provide a self-consistentpicture of the electron distribution relative to theseparated atoms and unperturbed lattice. (iv) Sinceo-w coupling effects were shown to modify the band-

width' and the self-consistent atomic charges~*'significantly, the model should incorporate all thevalence electrons on the same level of approxima-tion. The o electrons do not behave as an unpolar-ized core and thus the treatment of m electronsalone can lead to erroneous results. 5 (v) Sincelattice distortions in covalent solids around the de-fect site were shown to modify the deep defect lev-els'7 significantly, the model must provide a wayof introducing lattice relaxations. (vi) In view ofthe relative paucity of experimental data on the per-fect boron-nitride crystal, a reliable model basedon parametrizing the theoretical variables in ac-cord with experimental band structure" cannot beemployed. (vii) Due to significant charge delocal-ization in planar m-electron systems, the modelshould account for the coupling of the defect elec-trons with a relatively large portion of the bulk ofthe crystal. '

Theories that start from the periodic Bloch statesof the perfect crystal and introduce the defect per-turbationally" "satisfy conditions (i), (iv), and

(vii) but supply little information on requirements(ii) and (v). On the other hand, theories that con-struct the defect levels from the orbitals of the lo-cal environment (usually nearest neighbors only) ofthe defect '0'4"~ meet conditions (ii)-(v) but are de-ficient in satisfying conditions (i), (vi), and (vii).Effective-mass theories ' '", that have been shownI;o be useful in describing shallow traps, fail to ac-

POINT DE FE CTS IN HEXAGQNAI BQRQN NITRIDE. II.

count for deep levels in semiconductors mainly dueto the neglect of lattice relaxations. SimpleHuckel-type m-electron calculations ' ' satisfyconditions (ii), (v), and (vii), but grossly neglectconditions (i), (iii), (iv) and (vi). Point defects inhexagonal BN have been treated so far. only by thismethod.

Truncated crystal models ' ' which use eithersemiempirical linear-combination-of-atomic-or-bitals (LCAO) methods 2 as or the Xn scattered-wavetechnique, ~' automatically satisfy conditions (iv)and (vi), and the practical possibility of incorporat-ing a relatively large number of atoms in the mod-el with moderate computer time, satisfies condi-tion (vii). Its major disadvantage lies in the factthat the energy bands are not well defined"' andthat due to the presence of many "dangling" bondson the cluster's surface significant charge inhomo-geneity is present over each sublattice. 4'

In recent works of %atkins and Messmer"'" ondiamond and Zunger"" on graphite and boron ni-tride, different approaches were suggested toavoid the problems of charge inhomogeneity andlack of straightforward correlation between theperfect crystal and defect energy levels. Thesemethods use a superlattice representation to thedefect structure by either using a reciprocal-spacetight-binding approach with the defect placed at thecenter of a, large unit cell, "' or by consideringa finite periodic cluster of atoms with a defectplaced in its center, in a real-direct-space ap-proach. '" These methods could be implementedin practice with the aid of large computers andsatisfy the requirements previously stated.

In this paper we treat the problems of substitu-tional ck,rbon impurity, displaced carbon impurity,and nitrogen vacancy in hexagonal boron nitride,using both the truncated-crystal and the "small-periodic- cluster" methods. The essential featuresof both methods are discussed in Sec. II. A crit-ical examination of them is provided in Sec. III,where compa, rison with experimental data is made.

Due to the complexity involved with ab initio cal-culations of such models, semiempirical all-va-lence-electron methods are usually employed. Inview of the experience gained with these methodsin molecular calculations, it seems dangerous toadopt exclusively only one of these approximationsin calculating solid- state properties. Consequentlywe examine some of the common semiempiricalmethods for both the perfect crystal and the defectstructures. Problems regarding the convergence ofthe obtained electronic structure are also discussed.

II. METHODS OF CALCULATION

A. Truncated-crystal method

In the truncated-crystal approach, a cluster of

X atoms arranged spatially according to the crys-tal geometry is considered, A crystal orbital Q;is represented in the I.CAQ approach as a linearcombination of q atomic orbitals (usually va, lenceorbitals are used) centered on the N atoms of thecluster:

p=g n"-1

where y„(r -R„) represents an atomic orbital v

centered on site n. . The coefficients C„„;are thesolutions of the one-electron Hartree-Fock equa-tions, given by

Q(F~„„„S„„,—„„&;)C„„,=0, (2)

where the matrix elements in the atomic orbitalbasis are given by

(3)

and I" is the Hartree- Fock one-electron operator.The perfect-cluster eigenvalues are calculated bysolving Eq. (2) with an atomic valence basis set()v)='2s, 2P„, 2P„, 2P,) for several values of N, inorder to obtain the convergence limit of the one-electron eigenvalues. The resultant eigenvaluesfor a small cluster generally do not correspond toenergy levels in the Brillouin zone(BZ) of the in-finite periodic crystal. It was previously shownhowever, ~' that for layered hexagonal lattices ofD6„or D» point symmetry, such as in two-dimen-sional graphite and boron nitride, respectively, itis possible to select planar geometries in such away that each eigenvalue of its electronic seculardeterminant corresponds to an energy level in theBZ of the infinite lattice. This is true providedthe matrix elements of Eq. (3) have already as-sumed their bulk value due to introduction of suffi-cient nearest-neighbor interactions, and that sur-face effects can be neglected. Thus one is able tocalculate approximately the band edges and bandwidth of the ideal infinite crystal as well as otherhigh-symmetry K-points, by considering finiteclusters. P oint-defect levels are then calculatedby substituting a guest atom for a particular clusteratom (substitutional impurity) or rejecting one atomfrom the cluster (point vacancy) and repeating theeigenvalue calculation, The new eigenvalue spec-trum is analyzed and the defect levels appearing inthe forbidden gap (which have no counterpart in theideal-cluster spectrum) are investigated as a func-tion of lattice distortions around the defect site.The charge associated with such a level is calcu-lated from the coefficients (C„„&Jof the correspond-

A. ZUNGER AND A. KATZIR

FEG. j.. Fifty-atommolecular cluster (sur-rounded by dashed lines)and the extracluster atoms(atoms between dashed andheavy lines) used to es-tablish a periodic inter-action geometry. Thecentral atom (atom 26,marked by heavy circle)has no interaction with anidentical atom in the ex-tracl uster region.

ing level pq through standard' or modified" popu-lation analysis. The dangling bonds that appear onthe surface of the cluster, are either saturated byattaching hydrogen atoms~'~4 or treated as free

2& 23, 36

B. Small-periodic-cluster method

In the small-periodic-cluster (SPC) method, oneconstructs a Periodic cluster of fi atoms (referedhere as the molecular cluster) that is arrangedspatially according to the lattice symmetry of thechosen structure. Since the matrix elements ofEq. (2) appearing in the Hartree-Fock equations[Eq. (2)] depend only on the relative orientations ofpairs of atoms, it is sufficient to specify an inter-atom-distance matrix D' "„' for the X-atom clusterand three interatom direction- cosine matricesE„'"„»(x), E'"„(y), and E'"» (z) relative to arbitraryx, y, and z directions. The absolute atomic co-ordinates, with reference to a fixed origin, neednot be specified. in order to obtain the solution tothe eigenvalue problem of an isolated molecularsystem. This freedom in describing the crystalstructure can be used to construct these matricesas cyclic matrices appropriate for a "pseudomole-cule" that is Periodic in space, and keeps the cor-rect geometry of a given lattice. The range of in-teraction X(N) in such an X-atom periodic clusteris determined solely by the symmetry of the latticeandbyN, e.g. , aclustermadeupof N=2X+1atomscan be used to represent a periodic one-dimensionallattice with interaction range X(X), and clusters ofD» or Ds„symmetry with 8, 18, 32, 50 etc. atomsfor example, could be used to present a periodichexagonal structure with 1, 2, 3, and 4 orders ofinteractions, respectively. Figure 1 shows a 50-atom hexagonal molecular cluster (surrounded bythe dashed parallelepiped) that is made of 5x 5 hex-agonal unit cells with 2 atoms per cell. Extension

of this cluster by two rows of hexagonal cells inthe two planar directions (the cells between thedashed and heavy outer lines) is denoted as the extra-cluster region. Truncation of the interaction be-tween each two atoms, either inside the centralcluster or between an atom in the cluster and anatom in the extra-cluster region, to a range ofX(50)&4, defines a periodic hexagonal "interactiongeometry. " The distance matrix D'"' is construct-ed to define this interaction geometry in terms ofthe nearest-neighbor distance A» while the direc-tion-cosine matrices define the hexagonal pointsymmetry in terms of the hexagonal angle. Thesematrices are used to construct the Hartree-Pockmatrix elements E„„„andS„„, for a minimalbasis set of atomic orbitals, generating thereby aHermitian eigenvalue problem. The crystal orbit-als of such periodic clusters are thus constructedas in Eq. (1), but the lattice vectors R„are nowrestricted to represent the periodic finite struc-ture of interaction range X(N).

The eigenvalues of Eq. (2) are first solved for aregular perfect lattice. At this limit, the eigen-values are simply analyzed according to the wavevector K~ in the BZ by recognizing that the solutionscould be usually represented (by decomposing thecrystal orbital index i to its components i =—ep,where o. = 1...»i; p=O. ..N —1) as: .

i,K ~ RC~n) =—Cvnep= C~ope

where K = (2z/Nb)P and b is a primitive lattice vec-tor. The band structure c (K&) is obtained as adiscrete subset of the infinite crystal spectrum atX/I» K-values, evenly spread in the BZ, where I»

is the number of atoms in the basic unit cell (I» = 2for graphite and boron nitride).

The total ground-state energy is obtained in the

POINT DE FE CTS IN HEXAGONAL BORON NITRIDE. II.

simple one-electron picture as

~Dc N

E„„=:2ZQ &,(Kq)n Kp

(5)

where 0 denotes the highest occupancy. The sta-ble crystal conformation under static equilibriumis then obtained by minimizing E~,~ with respect tothe nearest-neighbor distance R» and by exam-ining the stability of the total energy against dis-placement of a specific atom.

The resulting equilibrium value of R» is com-pared with crystallographic data while the cohesiveenergy at equlibrium is compared with the experi-mental thermochemical determination. Thestretching force constant is obtained by performingnumerical derivatives of the total energy at thestatic equilibrium conformation with respect to sym-metry-stretching coordinates and comparing withthe experimental data from band spectra. The re-sulting wave functions are subjected to a populationanalysis to determine orbital and net atomic chargesto be compared with nuclear quadrupole resonance(NQR) da, ta.

The convergence problems encountered in themodel are (a) convergence of the matrix elementsin Eq. (3) as a function of the interaction orderX(N) employed [Eq. (5)], (b) convergence of the to-tal energy as a function of the K grid used for eachcluster size, (c) convergence of the self-consistent-field (SCF) iteration cycle when the energies arecomputed with a self-consistent LCAO method.

Due to numerical difficulties the convergence ofthe band structure as a function of the size g of theatomic set employed will not be examined, and avalence basis set will be used throughout.

The fact that the 3s orbital energies of free boronand nitrogen atoms are well separated from the va-lence orbital energies suggests that no substantialmixing should occur in the valence bands while theconducting states could be affected by introducingvirtual orbitals.

The convergence problems (a) and (b) could bereduced to one, if we use the maximum interactionrange allowed for a cluster of a given size N. Inthis case both X (N) and the size of the K grid areuniquely determined by N. The size of the clusteris chosen so that one will satisfy the convergenceproblems (a) and (b) within a prescribed toleranceand that the K subset will contain high-symmetrypoints that are of interest in analyzing optical and

energy-loss data concerning band-to-band transi-tions. It was previously shown~ that a 18-atom pe-riodic cluster arranged in Ds„or D6„geometry con-tains in its eigenvalue spectrum the I' and P points(notation of Bassani and Parravicini ) while a 32-atom periodic cluster also contains the Q saddle

point. Optically allowed transitions in both graph-ite and boron nitride are related to these high-sym-metry points.

Employing a self-consistent solution to the N-atom periodic cluster j,s equivalent to sampling amaximum of N/h K points in computing the crystalpotential in a self-consistent approach for infinitecrystals. ' ' It has been previously shown' ' thatconstructing the crystal potential as a superpositionof free-atom potentials without recompiling it onthe basis of the calculated band structure at sever-al selected K points, result in a poor approximationto the self-consistent results in systems exhibitingsome polari. ty (deviations of 1-5 eV in the band en-ergies of binary II-VI crystals).

In self- consistent orthogonalized-plane-wave(OPW) calculations, the contribution of only 2-4 statesof the entire BZ to the charge density were shown tobe sufficient to correct in some cases the non-self-consistent potential. In the small-periodic- cluster(SPC) approach a larger number of states (in thenondegenerate case, N/h different states) are in-corporated automatically in the evaluation of thepotential at each iteration. Moreover, these statesare evenly spread in the BZ and are not subjectedto somewhat arbitrary choice of the K grid, as inmethods that factorize the secular equations utiliz-ing explicitly the lattice translational symmetry inconstructing the crystal functions.

Since in the formulation of the eigenvalue problemof the central molecular cluster we do not imposeexplicit translational symmetry on the eigenfunc-tions inside the large Born-von Karman cell, theSPC method could be easily extended to treat point-defect structures in the representation of a ratherdilute defect superlattice. This is done by remov-ing the central cluster atom (e. g. , atom 26 in Fig.1) to infinity, creating thereby a point vacancy, orby changing its chemical identity and generating apoint impurity. It should be noted that when themolecular cluster is composed of an odd number ofhexagonal unit cells (e. g. , 18-atom cluster with3x 3 or 50-atom cluster with 5x 5 unit cells), thecentral atom in this cluster has no direct interactionwith an equivalent atom in the extra cluster regionsince the periodicity of the structure is alreadysatisfied by applying only intracluster interactionsfor this atom. This suggests that the superlatticerepresentation of the defect problem in this model .

should not place a too severe restriction when com-parison is made with experimental data concerningdilute defects, since the dispersion of the defectband is induced only by the relatively small polari-zation effects of the guest atoms on the charge den-sity of the host atom. These polarization effects,being generally of short range, could then be easilysuppressed by increasing the cluster size, thus re-ducing the interaction range.

A. Z UNGE R AND A. KAT Z IR

The defect levels, the charge distribution on them,and their location relative to the band edges are thuscalculated on the same level of approximation asthe perfect crystal bands and charges, enablingthereby a straightforward comparison betweenthem,

The main advantages of the small-periodic-clus-ter approach over the truncated-crystal method are(a) The one-electron levels of the perfect periodiccluster are related in a simple way to well definedpoints in the band structure of the infinite solid;(b) Charge homogeneity over each sublattice in theperfect periodic lattice is guaranteed by the factthat each atom in the cluster experiences the samecrystalline environment as the others, and no dan-gling bonds appear. Ad Aoc treatment of the sur-face is not needed; (c) A self-consistent evaluationof the crystal potential, involving a relatively largenumber of crystal states, is feasible.

The model is preferable to the "defect-molecule"approach' '" since it allows coupling of the defectwith a relatively large portion of the bulk solid,thereby allowing for a larger delocalization spacefor the defect electrons, and it relates the defectenergy levels to the band levels. The advantagesover the methods that start from perfect periodiclattice states" ' lie in the fact that lattice relaxa-tions can easily be introduced here, while in theprevious methods these will cause a significant in-crease in the perturbative effect induced by the de-fect on the periodic states. Also the charge dis-tribution in the defect level is simple to calculateby this method using straightforward population an-alysis on the resulting wave functions.

The main disadvantages of the small periodiccluster approach are the following:

(i) It involves solutions of large electronic secu-lar determinants (for a 32-atom cluster with fouratomic valence orbitals on each atom, 128x128 ma-trices are treated) since translational symmetry isnot explicitly utilized, but in turn, a more realisticdescription of deviation from periodicity is possi-ble.

(ii) It does not yield results that are continuousin the wave vector K, but rather a discrete subsetis obtained, Thus, the density of states cannot bereliably calculated.

(iii) The defects electronic properties are de-scribed in a superlattice representation with a clus-ter-size dependence of the indirect defect-defect in-ter actions.

Since the SPC method is more useful than thetruncated-crystal method, the latter will be usedonly to a limited extent.

C. Semiempirical LCAO computation schemes

Due to the complexity of ab initio calculations ofall the E„„, elements, some semiempirical meth-

ods are used to calculate these elements approxi-mately. In the extended Huckel (EXH) approxima-tion the matrix elements are given by

F„„„„=0. 5G(E o+ E„o,„o)Sv„,„,where S„„„arethe atomic overlap intergrals cal-culated from Slater atomic orbitals, E„„oand

E„o,„,are atomic orbital energies of orbitals p, and

p, respectively, taken from Hartree- Fock calcula-tions on free atoms or alternatively from atomicspectra. 6 is a semiempirical constant chosen tofit some electronic properties of medium to largemolecules, and usually taken to be 1.75. ~6

The derivation of the EXH approximation fromthe Hartree- Fock equations was given by Blyholderand Coulson and by Gilbert, ' For systems withmoderately uniform charge distribution (a differ-ence of less than 1.3 in the electronegativities ofthe atoms in Pauling's scale, compared with a dif-ference of 1.0 between X and 8), this was shownto be a reasonable approximation to the rigorousHartree- Pock scheme.

Since in general the Hartree-Fock operator Ecouples electronic contributions to the potentialfrom all atoms and orbitals in the cluster (and notonly from a particular p. n, vm pair), it depends onthe charge distribution of all electrons and nuclei.A semiempirieal self-consistent approach to Eqs.(2) and (3) [iterative extended Huckel (IEXH)27'"],which is a refinement over the non-self-consistentEXH approach and does not contain a free param-eter is characterized by an iterative solution, inwhich the diagonal E„„„„elements are taken to beexplicitly charge dependent:

E„„„„=E„'„,,„+Q„hF„,

+vn, vm= ~vn, vm(+vn, vn+ ~vm, vm)( 0 5~ ~on, vol )'

E„'„„„is the atomic orbital energy of the free atom,Q„ is the net atomic charge on the atom n, calcu-lated from the solutions (C„„,j of Eqs. (1) and (2)for all occupied states. AF„ is the change in thev-orbital potential due to deviation from chargeneutrality (ioni city), as calculated from the Hartree-Fock orbital energies of free ions of different ion-ization states or taken from atomic spectra. Ac-cording to this procedure one guesses Q„values forall the atoms (usually the initial guess for systemsthat are only partially ionic is the free atom valueQ„=O). Then the matrix elements of Eq. (3) areconstructed and solved for the coefficients jC„„;)ofEq. (2), and the atomic charges are calculated fromthe wave functions, A damping procedure is usedto faciliate the convergence of the iteration cycle.

The atomic orbital energies were taken from theoriginal work of Hoffman and the &E„values weretaken from the interpolation scheme of Cusachs and

POINT DE FE CTS IN HEXAGONAL BQRQN NITRIDE. II. 2383

Reynolds. ' The atomic orbitals were taken asSlater orbitals with the usual Slater exponents. In-teraction and overlap matrix elements between theN atoms are evaluated up to three to four orders ofneighbors in the SPC method and between all atomsin the cluster in the truncated-crystal approach.No adjustment of the semiempirical constants to ob-tain better agreement with experiment ' is per-formed. The iteration cycle in IEXH calculation areterminated when the difference between atomiccharges on successive iterations does not exceed10 e. Under these conditions, the band structureand total energy per BN pair satisfy the first threeconvergence requirements previously mentioned towithin less than 0. 1 eV.

D. Electronic properties of the perfect crystal

The calculated band structure of the perfect hex-

agonal lattice of boron nitride using IEXH method

and an 18-atom cluster P. = 2; P, and 1" points in-cluded) was previously3 compared with both experi-mental data and with other non-self-consistent tight-binding calculations. ' ' To examine the conver-gence properties of the results, we have performedthese calculations on an eight-atom cluster P. = 1; Q„and ~ points included) and on a 32-atom clusterp. =3; Q, and ~ points included). Relatively smalldifferences occur upon increasing the interactionrange: the m-m* band-to-band transition at the Qz

points in the BZ is shifted from 5. 8 eV at ~= 1 to6. 0 eV at X=3 [experimental electron energy-lossdata indicate =6.2 eV(Ref. 47)], th'e bottom of the

lowest valence band occurring at the I", point is18. 46 eV away from the center of the band gap forthe X = 1 calculation, 18.81 eV for the X = 2 calcula-tion, and 18~ 86 eV for the X=3 calculation [electron-scattering data give 19.4eV (Ref. 48]. The lowestv-o~ transition (Q:-Q', ) occurs at 13.9 eV in theX= 1 calculation and at 14.1 eV in the X = 3 calcula-tion [energy-loss data indicate 14.1eV (Ref. 47) to10.1eV(Ref. 49)]while thelowestmixedo- m* transi-tion (Qz Q~) occurs at 8. 8 eV in the X=1 and 8. 9eVin the X=3 calculation [energy-loss data give-9. 4 eV(Ref. 47)]~ The band gap is obtained only in the 18-atom cluster calculation as 3. 7 eV [experimentaloptical and photoelectron data are scattered between5. 4 to 3. 6eV (Refs. 50—52)]. The binding energychanges from 6. 6 eV in the X=1 case through 7. 2 eV

in the X = 2 calculation to 6. 7 eV in the ~ = 3 calcula-tion (compared with 6. 6 eV obtained in thermo-chemical determination"). This nonmonotonic be-havior results from the fact that the 8- and 32-atomclusters include the Q saddle point that contributesa, large part of the cohesive energy due to its highdegeneracy, while the 18-atom cluster includesband-edge points with low degeneracy. Optimiza-tion of the total energy with respect to the nearest-neighbor BN distance yields equilibrium values of

1.441 A in the X = 2 case and 1.449 A in the X = 3case [compared with the crystallographic determina-tion of 1.446 A (Ref. 54)]~ The stretching-forceconstant is calculated to be 10.1X10' dyn/cm and10. 8&& 10' dyn/cm for the X=2 and X=3 calculations,respectively (compared to-8. 3x 10' dyn/cm ob-tained from band spectra"). The boron v charge iscalculated to be 0. 52e and 0. 508 in the X= 2 andX= 3 calculations, respectively [NQR results yield0. 45e (Ref. 56)]. The convergence of these elec-tronic and structural features seems thus to be ob-tained at about X= 2-3.

Repeating these calculations with the EXHscheme, a similar convergence rate is obtained.The results exhibit however some pronounced dif-ferences relative to the IEXH results. The resultsobtained with the largest clusters are: a w- m*

transition at 6. 29 eV, the I", point at 19.0 eV, aQm-Q', transition at 15.0 eV, a. Qm-Q~ transitionat 9.4 eV, a gap of 5. 49 eV, binding energy of6.04 eV, equilibrium B-N bond length. of 1~ 443 A,boron m charge of 0. 31e and a stretching-force con-stant of 13.1x10' dyn/cm. Truncated-crystal re-sults for boron nitride have been previously pub-lished~ and seem to be in poorer agreement with ex-periment for reasons discussed above.

It is quite unfortunate that in view of the large un-certainties in optical and energy-loss data for bo-ron nitride and the paucity of experimental data ontransitions at higher energy and on effective char-ges, it is difficult to decide at this stage whichLCAO scheme should be preferred. It seems how-ever that the extended-Huckel method is better fordescribing energies while the iterative-extended-Huckel method is better for determining charges andatomic orbital coefficients. Similar conclusionswere previously suggested from extensive compar-ison of these methods in predicting various prop-erties of molecules. The IEXH is more diffi-cult to implement in practice due to higher compu-ter time requirements. Consequently, in the fol-lowing work we use EXH and IEXH methods for 18-and 32-atom clusters while a limited number of cal-culations on the 50-atom cluster will be performedonly with the EXH method.

III. RESULTS

A. Substitutional carbon impurity in hexagonal boron nitride

Table I summarizes the calculated results for asubstitutional carbon impurity in two-dimensionalhexagonal boron nitride as computed by the trun-cated-crystal and small-periodic-cluster methodswith the EXH approximation. The results for anideal cluster without a defect and for the clusterwith the defect are compared. The truncated clus-ters used for the calculation were 8foN]QH fp,

2384 A. ZUNGE R AND A. EAT ZIR

TABLE I. Calculated results for a substitutional C impurity in hexagona1 boron nitride, obtained using the EXH meth-od. E~z, Ec, and Ec' represent energy levels appearing due to the carbon impurity. DE~d represents the energy differ-ence between the edge of the conduction band and the highest defect level. Q& denotes the carbon ~ charge. Cluster I:BfpNf2Hf2 cluster II: Bf2Nf2Hfo, cluster III: Bf4Nf4Hf4 (see Fig. 2).

Property

Valence edge (eV)C ond uctio n edge (eV)Band gap (eV)E', (ev)E' (eV)E2s (eV)ZE„, (ev)

Vp C character.ln Ecin Ecin E2sc

—12.675—7. 210

5 465

—12. 921—7. 210

—12, 042—12, 140—23. 340

4. 832—1.468

40. 0966. 1298. 39

Cluster. INo. Carbon

defect impurity

—12. 762—7. 269

5. 493

—12, 983—7. 269

—11.931—12. 140—23. 285

4. 871—l. 467

40. 1065. 3186. 65

Truncated crystalCluster II

No. Carbondefect impurity

Cluster IIINo. Carbon

defect impurity

—12. 598 —12. 832—7. 155 —7. 155

5. 443—11.913—12. 135—23. 199

4. 758—1, 467

40. 1264. 4596. &35

—11.960-6.470

5.490

—12. 731—6, 470

—11.340—12. 129—23. 150

4. 879—1.467

—11.962-6.471

5. 491

—12. 735—6. 471

—11.345—12. 127—23. 150

4. 874—1.467

40, 1265. 2197. 10

40 1564. 9297. 12

Small-period clusters18 atoms 32 atoms

No. Carbon No. Carbondefect impurity defect impurity

BqaNtaHtq, and B)4Nt4Ht4(Fig. 2).It is seen that upon substitutionally introducing a

carbon atom instead of a nitrogen atom in the hex-agonal network, three new one-electron levels ap-pear. The lowest of them (Eca') is a level mainlycomposed of the 2s atomic orbitals of the carbonatom and highly localized (more than 96% carboncharacter, Table I) at the carbon site. This levelis slightly stabilized when compared with the freecarbon 2s state, and behaves actually as a corestate, The three degenerate 2P orbitals of the freecarbon atom are split in the crystal (which has aD3„point symmetry) into two crystal-field compo-nents: a doubly degenerate o level (Ec) spread inthe layer plane, and a, m level (Ec) perpendicular tothe layer plane. The Ec state contains 40%%up of car-bon 2P„and 2P character, and due to mixing withthe corresponding o orbitals of the boron and nitro-gen atoms, its energy is stabilized by more thanQ. 5 eV compared with the corresponding level inthe free atom. The E& level is slightly more sta-bilized than the Ec level, due to a relatively higherdelocalization of m electrons in the crystal plane.In the perfect lattice the larger delocalization of them electrons results in higher electrical conductivityin this direction.

The edge of the conduction band remains un-changed upon introducing the impurity, while theedge of the valence band, corresponding to the workfunction of the solid, decreases slightly. The dif-ference ~„~between the highest impurity levelEc (which is only singly occupied in the neutralstate) and the edge of the conduction band is onlyslightly affected by increasing the cluster size andassumes a value of about 4. 8 eV, as can be seenfrom Table L

The carbon impurity level Ec acts therefore as adeep acceptor level that can capture free carriers,thereby decreasing their degree of delocalization.It also decreases the optical gap, shifting the wave-

4

length of the absorbed light towards the visible (col-oring effect). Such a trend was experimentally ob-served in carbon-doped boron nitride. Analysisof the ideal crystal band structure2' reveals thatthe edge of the conduction band (Pa point) is com-posed of boron m orbitals (99. 43% boron m charac-ter in cluster III, and 100%%up boron m character inSPC representations). The optical transition fromthe carbon impurity level to the conduction bandedge is therefore an internal charge-transfer tran-sition similar to those encountered in molecularspectroscopy. ' Such a transition carries the elec-tron from a level localized to a large extent on thecarbon atom, to a level spread over the boron sub-lattice. This should be contrasted with the lowesttransition of the pure crystal which carries the sys-tem from a level delocalized on the nitrogen sub-

FIG. 2. BN clusters used in the truncated-crystal cal-culations. Each cluster is surrounded by H atoms to sat-isfy the free valence. Ra„= l. 446 A, Ra„=l.2 A, and

RNH =1.1 A. Cluster I: BfpNf2Hf2 r Bf2NfpHf2 ClusterII: Bf2Nf2Hf2 Cluster III Bf4Nf4Hf4. ~: boron; 0: nitro-gen.

POINT DE FE CTS IN HEXAGONAL BORON NITRIDE. II.

lattice (the P, edge of the valence band) to a leveldelocalized on the boron sublattice (the Pz state).

Comparison between the results obtained by thetruncated- crystal and the small-periodic- clustermethods indicates that in the former method theedges of the valence and conduction bands arechanged nonmonotonically with increasing clustersize, and these edges can thus be only approxi-mately determined by this method. A similar be-havior was observed by Larkins~4 and by Messmerand Watkins~ in calculating the electronic eigen-value spectrum of diamond and siliconlike atomicclusters, and by Zunger~ in a truncated-crystal cal-culation for graphite. The calculation of atomiccharges in the perfect periodic cluster reveals com-plete homogeneity of charges on each sublattice,while in the truncated-crystal model, strong per-turbations in the charge are evident near the bound-aries of the cluster.

In both truncated-crystal and SIC calculations,the carbon atom accumulates a large negativecharge (Qc' =0. 88e) at the expense of the surround-ing boron atoms, due to the greater electronega-tivity of carbon. This charge transfer is carriedout in a substantial way through the m system, re-sulting in a carbon m charge of 1.467e (comparedto 1.222e in a similar SP C EXH calculation ongraphite); The extent of this effect is probablyoverestimated in a non- self-consistent treatmentsuch as EXH, and a charge iterative calculationshould be applied to achieve more accurate results.We recalculated the 18- and 32-atom periodic clus-ter problem now employing the self- consistent IEXHmethod. The highest carbon impurity level Ec ob-tained when the carbon impurity calculation is per-formed is now 2. 2 eV and 2. 1 eV below the con-duction edge for the 18- and 32-atom clusters, re-spectively. The carbon atom is shown to have anet atomic charge of —0. 28e. The general featuresof the non-self-consistent treatment are preserved,but considerable quatitative differences occur due

to the requirement of self-consistent charge redis-tribution. Similar differences between non- self-consistent and self-consistent band energies werecalculated by Stukel et al. 4' on binary II-VI crys-tals, by the OPW method.

It should be noted that the level splitting and or-dering obtained here for the C-impurity model in-dicate that such a system will exhibit dielectriclosses, as argued by Sussman. ~

We next allow for model lattice relaxationsaround the C impurity, using the self-consistentIEXH method. An inward relaxation of 18% of thebond length, along the bonding distance, of thethree boron atoms surrounding the impurity, low-ered the energy of the clusters to a minimum. The

gap between the highest defect level and the con-duction edge increases now to 3. 2 eV. This large

effect demonstrates the importance of such pro-cesses in deep defect levels in covalent and partlyionic crystals. Thd fact that the IEXH method,when applied to the 18- and 32-atom perfect BNclusters, reproduces the equilibrium interatomdistance and stretching- force constant reasonablywell, justifies its use in investigating model relax-ations. The negative net atomic charge on the cen-tral carbon atom increases to —0. 35e upon relax-ing the boron atoms, the carbon thus assumes agreater stabilizing electrostatic interaction withthe boron atoms.

A similar relaxation calculation with EXH methodreveals a change inE„~ from 4. 8 to 4. 9 eV and achange in carbon net charge from —0. 88e to0. 908. When the carbon atom impurity lying in

the lattice plane is moved from the proper D» site(occupied by a nitrogen atom in the ideal cluster),the twofold degeneracy of the Ec level is lifted,The total lattice energy increases, however, rel-ative to the D» impurity, and therefore no de-tailed mapping of the potential surface was made.The separation E „ofthe defect level from the conduc-tion band is affected only slightly by cluster size.A calculation of an unrelaxed 50-atom cluster withthe EXH method yields E~„=4. 882 eV compared withthe value E„„=4. 879 eV for the 18-atom cluster.

B. Displaced carbon impurity in hexagonal BN

We next investigated the nonsubstitutional carbondefects. These are characterized by a carbonatom replacing a nitrogen atom and located at dis-tance Bc above the vacant site and perpendicularto the lattice plane (Rc = 0 is thus the planar im-purity case). The calculations are performed onthe 18-atom periodic cluster with the EXH approxi-mation for the matrix elements. Figure 3 showssome of the relevant one-electron energy levels asa function of Bc,

The pronounced features are the following.(a) A conduction-band level (denoted E,) splits

from the conduction band as the distance of the car-bon atom from the vacant site increases. Thecharge carried by this level is mainly located sym-metrically with respect to the carbon atom and itsthree equivalent nearest-neighbor boron atoms.The distance of this level from the conduction edgetends to level off for large Ac. For Rc =1.0 A itis at 1.30 eV from the conduction band. The con-tribution of the defect orbital to the charge on thethree boron atoms surrounding the carbon impuritydecreases with increasing Rc. For Bc = 1.0 A thischarge is 0. 368.

(b) The crystal-field splitting between the Ec and

Ec levels decreases with increasing Rc. Both lev-els are destabilized as A~ increases, thus ap-proaching their free atom degenerate state.

2386 A. ZUNGE R AND A. KAT ZIR

Ecb*

Et

-10

Ec

Evb

-15 I

0.0 0.2 0.4, 0.6 0.8Rc(i

1

IO

FIG. 3. Dependence of one-electron energies on theperpendicular distance Rc of the carbon impurity fromthe vacant nitrogen. site. E,& and E„b denote conductionand valence band edges, respectively. (E,*„denotes ahigher conduction state. ) E& denotes the trapping leveland originating from the I' —Q line in the perfect solid,and Ec and E~z denote the carbon impurity 0. and 7t levels,respectively.

(c) The edges of the conduction band (Z,„) and va-lence band (E„b) are slightly shifted as Rc increases.

Similar results are obtained with IEXH method,the defect level lying now at 1.Q eV and contribu-tion a charge of G. 298 to the boron atoms for Rc=1.G A.

In the neutral impurity state, the level Ec isdoubly occupied, Ec is singly occupied, and E, isempty. This last level could act as an electrontrap and would then exhibit a characteristic threeboron EPR signal due to the even distribution ofcharges on the three boron atoms surrounding thevacant site. Its energy below the conduction bandsuggests that it should also give rise to thermolu-minescence effects with an activation energy of-Et -E,b= 1.Q-1. 3 eV. This is consistent withthe model proposed to explain the role of carbonimpurities in BN (see Paper I).

When the distance Bc is further increased, thebinding energy of the carbon atom to the hexagonallattice (0. 64 eV for Bc=0) decreases. At the limitof large displacement the defect structure repre-

sents a nitrogen vacancy. The defect level is of m'

character and its separation from the conductionedge is calculated to be 1.39 and 1.38 eV for the17- and 31-atom clusters, respectively, for theEXH calculation and 1.12 to 1.16 eV for the IEXHcalculation. Detailed comparison of this vacancyproblem with the similar defect in graphite is de-scribed eleswhere. 33

It is interesting to compare the energy and chargedistribution of such a defect as revealed by thetruncated-crystal vs the SP C methods. Table IIsummarizes the results for the energy of the de-fect level and boron charges, as revealed by thesemethods. It is evident by inspection to Table IIthat the truncated crystal approach does not yieldthe proper location of the defect level with respectto the conduction band, and only approximate re-sults are obtained. Due to the differences in ge-ometry of the truncated clusters employed, theatomic charges carried by the nearest three boronatoms in the defect crystal orbital are not identi-cal, as they should be. The perturbative effect in-troduced by the vacant site is propagating some1-2 neighboring shells and introduces changes inthe charge distribution. of these coordination shells.Truncated-crystal models of the size used here areprobably too small to account properly for such ef-fects since the perturbations introduced by the edgeatoms are also extended within 1-2 shells.

Similarly, a defect molecule approach consider-ing only the first coordination shell is perhaps in-sufficient to account for the coupling of this levelwith the bulk crystal (compare the 1-shell and4-shell results in Table II).

When model lattice relaxations are introduced onthe three nearest boron atoms, the defect levelshifts to 1.51 and 1.2G eV for inward and outwardrelaxations of 10% of the equilibrium perfect latticebond length, respectivej. y.

It should be mentioned that both in a nitrogen va-cancy and in displaced carbon-impurity structures,the distribution of charge in the defect level is syrn-metrical with respect to the nearest three boronatoms, and the unpaired spin density in such levelsshould reveal an EPR signal with the same splittingpattern (10 peaks for "Bcrystal). The energy sep-

TABLE II. Calculated results for a nitrogen vacancy in hexagonal boron. nitride, obtained usingthe EXH method. E&,„denotes the energy difference between the conduction edge and thedefectlevel. Qz&, QB2, and Q~3 denote the boron 7( charge in the defect level. 7- 17- 31- 49-atom clus-ters correspond to 1, 2, 3, 4 coordination shells around the defect site.

Truncated crystal

Cluster I Cluster II Cluster III 7 atoms

SPC

17 atoms 31 atoms 49 atoms

E] ~b (eV)

Qai (e)Qa2 «)Qs3 «)

0. 7630. 4820. 4820. 482

0. 9330. 4810. 4810. 475

1.1020. 4350. 5270. 527

0. 8140. 7210. 7210. 721

1.3900. 5720. 5720. 572

l. 3800. 5720. 5720. 572

1.3810. 5710. 5710. 571

PGINT DE FE CTS IN HEXAGONAL BORON NITRIDE. II. 2387

aration between the def ect level and the conductionband is somewhat higher for the vacancy (1.38 and1.16 eV in EXH and IEXH calculations, respective-ly) than for the displaced carbon impurity (1.30and 1.0 eV in EXH and IEXH calculations, respec-tively). Also, higher charge is accumulated on theboron atoms in the vacancy case (0. 57e and 0.48ein EXH and IEXH calculations, respectively) thanin the displaced carbon impurity case (0. 36e and

0. 29e in EXH and IEXH calculations for Rc = 1.0 A).Due to the uncertainties in the semiempirical LCAOmethods used, the absolute values of these calcu-lated properties should not be given too much im-portance. The general features of these model de-fects are however believed to be correctly repro-duced.

IV. ELECTROSTATIC POTENTIALS

The total electrostatic potential V(r) is given bythe sum of the electronic contribution tEg. (6)] and

the contribution from all the nuclei

V(r, ) = Va„(r;)+Q, — (9)

where the wave function 4 is formed from all oc-cupied MO's. It has been recently demonstrated6'that the electronic wave functions obtained from theself-consistent-field molecular-orbital (SCF MO)

semiempirical approximations intermediate ne-glect of differential overlap (INDO), ~' reproducesquite accurately the electrostatic potential calcu-lated from ab initio wave functions for medium-size molecules. When the atomic basis set em-ployed by these LCAO methods is expanded in aGaussian basis, the Poisson equation can be an-alytically integrated. Previous numerical meth-ods ' involved substantial errors in the potentialnear the nucleus and exceedingly large computa-tion times. With the analytic method, we expandthe Slater orbitals used by the INDO method as an

atomic basis set in a series of gaussians (the 6-Gexpansion). With this representation of the atomicbasis set, .the charge distribution p(r) takes theform of a sum of contributions from the individual

The electrostatic effects introduced by either a

estrogen vacancy or a carbon impurity into the oth-erwise unperturbed lattice can be evaluated bysolving the Poisson equation for the charge density'p(x) obtained from the electronic wave function 4'

of a finite BN cluster.

v'V„„(r) = 4mp(r),

where p(x) is obtained from the X-electron wave

function 4

gaussians even in the multicenter case. ' Compu-tatiens of the electrostatic potentials for the unper-turbed clusters I-III (Fig. 2) shows that in the re-gion of. the central BN bond the potential is un-changed within 0. 1% when increasing the clustersize from cluster II to cluster III and changes onlyb$0. 15% when passing from cluster I to cluster II.

In the case of neutral vacancy and neutral carbonimputity, the electronic eigenvalue problem neces-sitates open-shell SCF calculation, and the elec-tronic density p(r) is computed from contributionsof both o'. and P spins, while in the unperturbedcluster this reduces to a closed-shell problem.

The use of many-electron self-consistent wavefunctions that already incorporate quantum ex-change effects, for all the electrons in the cluster(including the "trapped" electrons that are treatedas all other electrons in the neutral defect struc-tures), suggests that the resulting potential shouldpresent a meaningful picture of the affinities of thevarious defects.

The electronic eigenvalue problem was solvedfor cluster I (Fig. 2) by sta.ndard INDO method forthe perfect, N-vacancy and C-impurity structures.Charge density was then computed from the re-sulting wave function and the latter expanded in the6-G expansion and used to solve the Poisson equa-tion. The resulting electrostatic potentials alongthe line marked in Fig. 2, are presented in Fig. 4.

It is seen that upon introducing a carbon atomsubstituting at the site of a nitrogen atom, the elec-trostatic potential along the newly formed B-Cbond is less attractive towards a negative pointcharge than the corresponding potential in the un-perturbed structure, due to the lower affinity ofcarbon (electronegativity 2. 5 on Pauling's scale) toattract electrons than nitrogen (electronegativity3. 0). The electrostatic effect is introduced by thecarbon impurity is especially marked at the vicin-ity of the newly formed 8-C bonds, while at thecenter of the hexagonal rings surrounding the im-purity (R = +4. 1 a. u. ) the potential is already veryclose to its value in the unperturbed structure.

. When a nitrogen vacancy is formed, the electro-static potential around the vacant site becomes veryshallow, increasing thereby the affinity of this cen-ter towards capture of negative charge. The asym-metry of the potential around the center of the bo-ron-vacancy bond (R =0. 0) is increased. Again, theperturbative effect introduced by the vacancy is al-ready small at the center of the closest hexagons.Both the nitrogen vacancy and the carbon impurityin hexagonal boron nitride create thus acceptorstates.

When a similar calculation is performed with theperiodic 32-atom cluster the perturbative effect ofboth defects is seen to extend to a larger radius,and only after 2. 5-3.0 A away from the defect site

A. ZUNGE R AND A. EAT ZIR

0 I

2

0E

-10-O

l-12-

//

II

N

I I I

--- Regu10r—--C- impurity" ."N-vacancy

-5.0 -4.0-18 I

-7.0 -6.0 -5.0

I I

/I

Ilt

BftI

II

Ili,—2.0 -1.0 0.0 1.0 20 30 40 50 60 70

R (a.u}

L I I«~ ~ ~ ~

~ ~ ~

I /]I I()ji

B( ~~ N!t1l Il

1i!~I

1'I)

iL'h

8.0

FIG. 4. Electrostatic potentials computed from INDO wave function f 1 t I 1 hFI ' ' 'ns o c us er a ong the vertical axis denoted in

Fig. 2. R given in a. u. The potential describes electrostatic interactions with a pos't' ' t hposi ive poin c arge. The vacancor the C impurity are placed instead of the nitrogen at 1.37 a. u.

the electrostatic potential approaches its value inthe unperturbed lattice. Due to the presence ofperturbations of edge atoms at this distance fromthe defect in the truncated-crystal models employed,the results obtained by these methods deviate sig-nificantly (Tables I and II) from those of the SH."

model.

V. SUMMARY AND CONCLUSIONS

The electronic eigenvalue problem of a two-di-mensional hexagonal D» lattice was represented ba minimal basis set of LCAO equations for a finiteperiodic array of hexagonally arranged atoms, ex-hibiting 2 —4 orders of neighboring interactions,SemiempiricaJ. EXH and IEXH Mo LCAO methodswere used to facilitate the numerical solutions.For the ideal-lattice problem one thus obtains asubset of the eigenva, lue spectrum of the infinitelattice that is sufficient to account for the variouscrystal band widths, band gaps, band-to-band tran-sitions, equilibrium interatom distances, and atomiccharge distributions. These calculated propertiesagree with the available experimental data.

The same method was then applied to the study ofcarbon impurity states. The main features obtainedare (a) the appearance of a, doubly degenerate v lev-el (4. 9-3.2 eV below the conduction edge) and a.

singly degenerate m level in the forbidden gap, witha large degree of localization on the carbon site,and (b) formation on an extra defect v level thatsplits from the conduction band, when increasingthe vertical distance of the impurity atom from thelayer plane. This trapping defect level is associ-ated with an even charge distribution on the threenearest boron atoms, and its sepa. ration from theconduction band is ca.lculated to be 1.0—1. 3 eV.Lattice relaxations around the ca,rbon impurity de-fect were shown to stabilize the system by lower-ing the trapping level. Upon increasing the dis-tance of the carbon atom from the layer to infinity,a nitrogen vacancy is formed. It was shown to be

associated with a level that splits from the conduc-tion band, and is situated 1.16—1.38 eV below it.An even charge distribution on the three nearest bo-ron atoms is also manifested by this level. The ef-fects of lattice relaxations were investigated aswell as its coupling with the bulk crystal states.

In Paper I of this study a model was proposed toexplain the results of EPB, luminescence, and glow-curve measurements on hexagonal boron nitride.The main features of the model were as follows:carbon impurities introduce an energy level at about4. 1 eV below the conduction band. Electrons maybe excited from this level, pass through the conduc-tion bands, and fall into traps. One such trap is anitrogen vacancy. An electron trapped in the vacan-cy forms a three boron center (TBC), i. e. , thecharge is distributed equally on the three neighbor-ing boron nuclei. This electron is trapped at about1.0 eV below the conduction band, as revealed inglow curve measurements and in a "knee" at 600 Kin the isochronal annealing curves of the EPR»g-nals (Paper I, Fig. 3). A second "knee" appearedat 800 'K, and it reveals the existence of anothertype of TBC, which was not observed in other mea-surements. In these centers electrons are trappedat about l. 4 eV below the conduction band (as canroughly be estimated from the "knee" in the iso-chronal annealing curve). This type of TBC maywell be related to electron trapped in nitrogen va-cancies, in the vicinity of carbon atoms,

The main features of the theoretical calculationsare in good agreement with the model proposed toexplain EPR, thermoluminescence, and thermallystimulated current measurements (see Paper I).

From the investigation of the characteristics ofthe calculation methods, it is concluded that: (a)truncated-crystal models with relatively small num-ber of atoms should be treated with great cautionwhen defect models are considered. The radius ofthe perturbative effects introduced by the defectshould be estimated and kept well within the rangewhere the edge effects of the truncated surface are

POINT DE FE CTS IN HEXAGONAL BORON NITRIDE. II.

not important. (h) Defect-defect interactions pres-ent in the superlattice representation can be effi-ciently suppressed by using small periodic clustersof an odd number of primitive unit cells, keepingthe interaction radius smaller than the molecularcluster radius, and seeking the convergence limitof the investigated electronic properties as a func-tion of the interaction radius. This approach isuseful in describing covalent defect structures ex-hibiting a reasonably short range electrostatic tail.A comparable interaction range is also sufficientto assure stability of the electronic and structuralproperties of the perfect lattice.

(c) The electronic affinity of various defectstructures and its perturbation radius could be es-timated by computing the Poisson potential gener-

ated by the quantum mechanically calculated chargedensity. This representation was previously suc-cessfully used in determining chemical reactivityof various molecules ' ' and seems a promisingtechnique for describing self-consistent potentialrear rang ement in defe ct proble ms.

(d) Due to the uncertainty underlying the semi-empirical LCAQ methods, one should examine theresults yielded by several such methods since com-mitment to a single LCAQ approximation might bemisleading. The characteristics of both perfectlattice and defect structures are only semiquanti-tatively determined. However, the general featuresof these systems are clearly demonstrated. Formore accurate results, ab initio methods are un-

avoidable.

R. Taylor and C. A. Coulson, Proc. Phys. Soc. Lond.65A, 834 (1952).

A. Zunger, J. Phys. C 7, 76 (1974).3A. Zunger, J. Phys. C 7, 96 (1974).4C. A. Coulson and R. Taylor, Proc. Phys. Soc. Lond.

A 65, 815 (1954).~M. B. Khusidman and V. S. Neshpor, Poroshk. Metall.

10 (8), 72 (1970).G. D. W'atkins, in Radiation Effects in Semiconductors,edited by F. L. Vook (Plenum, New York, 1968), p. 67.

~E. L. Elkin and G. D. Watkins, Phys. Hev. 174, 881(1968).

E. Doni and G. P. Parravicini, Nuovo Cimento A 63,117 (1969).

F. Bassani and G. P. Parravicini, Nuovo Cimento 50,95 (1967).

~OC. A. Coulson, M. A. Herraez, M. Leal, E. Santos,and S. Senent, Proc. R. Soc. Lond. A 274, 461 {1963).G. F. Koster and J. C. Slater, Phys. Rev. 95, 1167(1954); 96, 1208 (1954).

~ F. Bassani, G. Iadonisi, and B. Preziosi, Phys. Rev.186, 735 {1969}.K. H. Bennemann, Phys. Rev. A 137, 1497 (1965).

~4B. S. Gourary and A. E. Fein, J. Appl. Phys. 33, 331(1962).C. A. Coulson and M. J. Kearsley, Proc. R. Soc.Lond. 241A, 433 (1957).

~6W. Kohn, in Solid State Physics, edited by F. Seitzand D. Turgull (Academic, New York, 1957), Vol. 5,p. 258.W. Kohn and J. M. Luttinger, Phys. Rev. 97, 1721(1955); 98, 915 (1955).

~8J. D. Levine and S. G. Davison, Phys. Rev. 174, 911(1968).M. B. Khusidman and V. S. Neshpor, Theor. Expt.Chem. 3 (2), 270 (1967).E. B. Moore and C. M. Carlson, Solid State Commun.

4, 47 {1965).R. P. Messmer and G. D. Watkins, phys. Hev. Lett.25, 656 (1970).R. P. Messmer and G. D. Watkins, Phys. Rev. B 7,2568 {1973).C. Weigel, D. Peak, J. W. Corbett, G. D. Watkins,and R. P. Messmer, Phys. Hev. B 8, 2906 (1973).F. P. Larkins, J. Phys. 4, 3065, 3077 (1971).

5M. R. Hayns, Phys. Rev. B 5, 697 (1972).

26R. Hoffman, J. Chem. Phys. 39, 1392 (1963).R. Rein, N. Fukuda, H. Win, and G. A. Clark, J.Chem. Phys. 45, 4743 (1966)~

J. A. Pople and D. L. Beveridge, Approximate M. O.Theories (McGraw-Hill, New York, 1970).

9K. H. Johnson, J. Phys. C 3, 195 (1972).E. B. Moore and C. M. Carlson, Phys. Rev. B 4,2063 {1971).G. D. Watkins and R. P. Messmer, in ComputationalMethods for Large Molecules and I ocalized States inSolids, edited by F. Herman, A. D. McLean and R. K.Nesbet (Plenum, New York, 1973), p. 133.R. P. Messmer and G. D. Watkins, in Radiation Dam-

age and Defects in Semiconductors, Conf. Ser. No. 16(Institute of Physics, London, 1973), p. 255.

33A. Zunger, J. Chem. Phys. {to be published).+R. S. Mulliken, J. Chem. Phys. 23, 1833 (1955}.35R. Rein, H. Fukuda, H. Win, G. A. Clark, and F. E.

Harris, in Quantum Aspects of Heterocyclic Compoundsin Chemistry and Biochemistry (Israel Academy of Sci-ence, City 1969), p. 86.L. M. Roth and A. J. Bennett, Proceedings of the TenthInternational Conference on the Physics of Semiconduc-tors, Cambridge, Mass. edited by S. P. Keller, J. C.Hensel, and F Stern, USAEC, Oak Ridge, 1970, p. 619.

3~J. M. Andre, J. Chem. Phys. 50, 1536 (1969).D. L. Beveridge and I. Jano, J. Chem. . Phys. 56, 4744(1972).W. E. Rudge, Phys. Rev. 181, 1024, 1033 (1969).D. Brust, Solid State Commun. 9, 481 (1971).D. J. Stukel, R. N. Euwema, and T. C. Collins, Phys.Rev. 179, 740 (1969).G. Blyholder and C. A. Coulson, Theor. Chim. Acta10, 316 (1968).T. L. Gilbert, in Sigma M. O. Theory, edited by O.Sinanoglu and K. B. Wiberg (Yale U, P. , New Haven,1970), p. 244.

44R. Hoffman, J. Chem. Phys. 40, 2474 (1964).4~L. C. Cusachs and J. W. Reynolds, J. Chem. Phys.

43, S160 (1965).J. Zupan, Phys. Rev. B 6, 2477 (1972).R. Vilanov, C. R. Acad. Sci. B 272 1066 (1971).

4 K. Hamrin, G. Johansson, U. Gelius, C. Nordling, and

K. Siegbahan, Phys. Sci. 1, 277 (1970).M. J. Cazaux, C. R. Acad. Sci. B 270, 700 (1970).

50S. Larach arid R. E. Shrader, Phys. Rev. 104, 68

2390 A. ZUNGE R AND A. KAT ZIR

(1956).~ &. A. Formichev, Sov. Phys. —Solid State 13, 754

(1971).M. J. Rand and J. F. Roberts, J. Electrochem. Soc.115, 423 (1968).

5 Janaf International Thermochemical Tables, Los AlamosReport No. LA-2368., 1966 (unpublished).

@R. S. Pease, Acta Crystallogr. 5, 536 (1952).~~J. Goubean, in Boron Nitrogen Chemistry, Advances

in Chemical Series, No. 42 (American Chemical Socie-ty, New York, 1964), p. 87.

56A. H. Silver and P. J. Bray, J. Chem. Phys. 32, 288(1960).A. Pullman, in Sigma M. O. Theory, edited by O. Sin-anoglu and K. B. Wiberg (Yale U. P. , New Haven, 1970),

p. 280.5 A. W. Moore and L. S. Singer, J. Phys. Chem. Solids

33, 343 (1972).S. Nagakura and J. Tanaku, J. Chem. Phys. 22, 236(1954); 23, 141 (1954).J. A. Sussman, Proc. Phys. Soc. 79, 758 (1962).G. Giessner-Prettre and A. Pullman, Theor. Chim.Acta 25, 83 (1972).W. J. Hehre, H. F. Stewart, and J. A. Pople, J. Chem.Phys. 51, 2657 (1969).

63S. Subrenic, H. Weinstein, and H. Paunz, Chem. Phys.Lett. 20, 419 (1973).H. M. Stevens, J. Chem. Phys. 52, 1397 (1970).

5S. F. Boys, Proc. R. Soc. Lond. A 200, 542 (1950).