Self-Consistent LCAO Calculation of the Electronic Properties of Graphite

PHY SINAI RKVIK% 8 VOLUME 17, %UMBER 2 15 JAGUAR Y 1978

I Self-consistent LCAO calculation of thf: electronic properties of graphite. I. The regular graphite lattice Alex Zunger* Department of Theoretical Physics and Applied Mathematics, Soreq Nuclear Research Centre, Yavne, Israel (Received 11 August 1975) The electronic properties of the regular graphite lattice are investigated within self-consistent LCAO (linear combination of atomic'orbitals) scheme based as a modified extended-Huckel approximation. The band structure and interband transition energies agree favorably with previous first-principles calculations. Good agreement with experimental data on the density of valence states, energetic position of the lowest conduction states, equilibrium unit-cell parameters, cohesive energy and vibration force constants, is obtained. The McClure band parameters that were previously adjusted to 'obtain agreement with Fermi-surface data and the electronic specific heat, are reasonably reproduced. The charge distribution and bonding characteristics of the covalent 'graphite structure, are discussed. The same calculation scheme is used in part II of this article (following paper) to discuss properties associated with point defects in graphite. The correlation between the electronic. properties of the regular and point-defect-containing lattice is studied.

I. INTRODUCTION Brillouin zone (BZ) and more than a single band are i:mportant in determining the defect levels. \ In view of the accumulation of rather detailed Truncated-crystal calculations" "solving for the experimental data on the electronic properties of eigenvalue problem of a relatively large cluster both regular and point-defect containing semi- of atoms with a defect placed at its center, have conducting covalent solids, theoretical interest yielded a wealth of information regarding the has developed in methods that are capable of charge distribution around the defect and the re- treating these related phenomena by a unified ap- laxation mechanisms; however, owing to the proach. The electronic properties of point defects presence of a substantial number of dangling bonds in covalent solids have been treated either by on the cluster's surface, significant unrealistic methods using the perfect-lattice periodic states charge inhomogeneity is generated in the cluster. as zero-order eigenvectors' ' or by methods that Except for clusters with special symmetries, "one have utilized the local atomic orbitals surrounding cannot establish a correlation between the cluster's the defect site to construct the "defect mole- energy eigenvalues and those of the perfect in- cule. "' ' The first approach, usually implemented finite solid. '4 either by using various forms of resolvent meth- Recently, a new model has been suggested" " ods' ' or by direct expansion techniques, ' does not for treating point defects in covalent solids. This provide a natural way of introducing self-consis- model is based on the representation of the one- tency in the description of the charge rearrange- electron energies of the defect by the eigenvalue. ment accompanying the defect formation, and in spectrum of a small Periodic cluster of atoms many cases requires the knowledge of the perfect- treated in the I.CAO (linear combination of atomic crystal eigenstates in the Wannier representation. orbitals) representation. Undesired surface Lattice distortions around the defect site are effects are completely eliminated and a simple usually not amenable to treatment by these meth- correspondence between the cluster levels and ods owing to the large perturbation associated that of the infinite solid is established. Lattice with such a rearrangement. The second approach, relaxations can be conveniently treated and cou- using the defects. local-environment representa- pling of the defect states with the bulk crystal tion, does not provide a correlative scheme be- bands at a discrete subset of points in the BZ is tween the eigenvalue spectrum associated with the automatically introduced. In the present paper defect and the perfect-solid band structure, and is we treat the electronic structure of the perfect deficient both in the description of the coupling lattice and in part II" (following paper) we treat of the defect states with the band states and in the vacancy problem in graphite using this "small- treating the relaxation of the lattice. Effective- periodic -cluster" (SPC) approach. We extend mass theories'" on the other hand, are not suit- our previous treatment of point defects in hexa- able for description of deep defect levels associa- gonal systems"" by introducing recently devel- ted with a deep short-range potential, since in oped theorems ' for the mean value points in the these systems, a rather subst8, ntial part of the BZ in conjunction with the small-periodic-cluster

626 SELF-CONSISTENT LCAO CALCULATION OF THE. . .I.. . 627

approach, enabling a detailed study of self-con- homopolar crystals containing defects. " There sistency effects and emphasizing the density-of- is some strong evidence" " that for systems with states aspects of the problem. Both the localized relatively homogeneous charge distribution (e.g, and the extended crystal states in the presence homopolar molecules and solids or systems with of the vacancy are examined. small electronegativity difference between the The problem of describing the electronic states constituent atoms}, the EXH method provides a of perfect lattice and radiation-damage induced reasonable approximation to the Hartree-Pock- defects in graphite, is one of great importance, Roothaan method. both from the fundamental and practical point of The details of the calculations involved in EXH view. In the past years, a large amount of ex- band structure are described in Sec. II and in Sec. perimental data concerning the electronic struc-, III we discuss the results of optical transitions, ture of both the perfect and the defect structure positions of high-symmetry points with respect has been accumulated. The perfect lattice has to the Fermi energy, electron density of states, been studied by optical absorption, ' reflect- imaginary part of the dielectric function, valence- ance, ""electron energy loss, "'"thermoreflect- charge distribution, unit-cell parameters, force ance, "soft-x-ray emission, "photoelectron spec- constants, and various band parameters related to troscopy (ESCA),""secondary-electron emis- Fermi-surface data. Comparisons with both sion (SEE),"'"and photoemission. " The d'etails available experimental data and with other theo- of the Fermi surface have been analyzed by the retical approaches are discussed. In the second de Haas —Van Alphen effect, "'"cyclotron reso- article (part II), we use the self-consistent EXH nance, "'"and magnetoreflectance. " Electronic method in a small-periodic-cluster approach to heat capacity has been measured and analyzed calculate various properties related to the vacancy 42~' by several authors. The crystal force con- in graphite, i.e., energy of localized states, den- stants have been deduced""' from neutron-scat- sity of states in the presence of the vacancy, tering data" by fitting the calculated phonon spec- Jahn-Teller distortions, energy of vacancy for- tra to experiment while the lattice sublimation mation, and migration and charge redistribution eriergy has been measured by thermochemical due to the defect. A completely self-consistent techniques. "" The properties associated with approach is used. Relations with previous ap- point vacancies in graphite have also received proaches and with available experimental data are experimental attention. Experiments on electron discussed in detail. paramagnetic resonance of damaged-graphite" ~' lattice-parameter change due to defect formation" "and displacement energy by electron im- II. METHOD OF CALCULATION pact" "have been performed. Optical absorp- The Bloch function constructed from tion of irradiated graphite, "diffusion, thermal- 4~(k, r) atomic orbital = situated at the site annealing, and stored-energy-release experi- X,(p, 1, ... , o} n(n = in the unit cell is given ments measuring the energy of vacancy formation 1, ... , h) by and migration"~' have been performed. The =N ' ' — — electron dynamics in the presence of defects has 4 (k r) Pe'"'"&g (r R d ) been examined by Hall-effect experiments, "mag- '"'" " netoresistivity, and electrical resistivity. where n = 1, ... ,N numbers the unit cells and d Theoretical studies on the electronic properties is the position vector of site & in the unit cell. of graphite have been performed both within the Crystal eigenfunctions belonging to the jth band m -electron " approximation" and within the all- (j = 1, ... , vh) are given by valence electron model. """ The electronic structure of the point vacancy in the same system has been treated by the nearest-neighbor "defect molecule" approach. ' In this paper (part I) we treat the electronic The expansion coefficients C~,.(k) and the band band structure of graphite within the LCAO tight- eigenvalues e,.(k) are determined from the solu- binding scheme using a self-consistent modifica- tion of the variational equations: tion to the extended-Huckel (EXH)" approximation to the interaction integrals. It has been previously ' 8' 3~ [E,',(k) -S,~(k)e,.(k)]C„(k)=0. indicated that a charge-self -consistent ga=I P scheme is essential for maintaining realistic description of systems exhibiting some sort of The matrix elements of the Bloch function are heteropolarity, such as binary crystals" "or given in the atomic basis set by AI, E X ZUWGKR

Eug(k) eik (Ru-Rp) all occupied bands, Q . If no charge transfer g occurs between the various sublattices, this equals the valence electronic of the x(X, (r —R, —d ) ~E~X,(r —R„—1(()) charge free atom. (e) The net atomic charge q, is given by -=pe" E"(O,n), th.e difference between Q and the core charge Z These quantities are defined in Appendix A. We have used the Lowdin definition of atomic eik (Ru-RP3 S up(k) g charges rather than the Mulliken" or the Ros and Schmit" definitions since the latter methods in- x (X„(r—R, —d )~X,(r —R —d, )) volve a somewhat artificial division of the overlap charge between nonequivalent atoms. This — eik R„Su~ l((0 n) complication does not arise when orthogonal Low- P din functions are used. In systems exhibiting a where R, is the position vector of the origin unit nonhomogeneous charge distribution (such as heteronuclear solids and molecules or homo- cell and E,", ,(P0, )nand S ~P(0, n) are the matrix elements of the Hartree-Pock operator F and the nuclear systems with point defects), the. Lowdin unity operator, respectively. definition has been shown to better reproduce The secular equations [Eq. (3)] are solved by accurately ca).culated molecular multipoles~ " 97 applying the Lowdin orthogonal transformation' and energies. to the basis Bloch functions followed by a direct. The overlap matrix elements S,„(0,n) are cal- diagonalization of the orthogonal eigenvalue equa- culated directly from a valence basis set (p, = 2s, tions. The resulting wave functions are subjected 2p„, 2p„and 2p, ) of Slater orbitals using stan- to a Lowdin charge analysis. " The elements dard Slater exponents (1.625 for carbon). The of the charge matrix are computed by replacing the energy matrix elements are calculated from a Brillouin zone integration by a numerical quadra. - self -consistent extended-Huckel approach, simi- ture" over 180 inequivalent k points in the —,', sec- lar to the approximation previously suggested for tion 'of the two-dimensional hexagonal Brillouin molecules" '" zone. The population analysis is used to compute E',~(n, m) = ,'G, „[E"„,'(q-„-)+E',P(q,)]S,'.(n, m), (5) the following quantities: (a) the charge contribu- t" '0' tion the p-basis orbital to the state The where is the Wolfsberg —Helmholz constant by ~ kj). dispersion curves of this orbital charge q (kj) is The diagonal E,"„(q ) element is given by: calculated for all occupied bands, through the — ' Brillouin zone. (b) A band-by-band integration of E..(q.)=E:(0)-q.&.'.. .. Z q y... (6) l0/nn q~(k, j) yields the contribution of the jth band to the p-orbital charge R ~, (c) The contribution E~(0) is the li-orbital energy of the neutral atom of the jth band to the total electronic charge on at site & and is obtained from atomic ionization site &, D,. is computed by summing R„over the potentials determined spectroscopically'""" and basis orbitals p, . The quantity D, for the defect is the change in the p, -orbital energy of level j will be used as a measure to the localiza- atom & due to deviation from charge neutrality tion of the defect wave function on given sites &. at this site. p„,z is the two-center spherically (d) The total electronic charge on site (k due to averaged Coulomb repulsion integral:

-R, -d, — -R„-d. B(r. -R(- 8» y„., „=&X..(r, -R„-d.)X.,(r. ) X..(r, )X. g2

/ calculated from the formula listed by Roothaan. '0' Hartree-Pock calculations or from experiment, We use a spherically averaged Coulomb repulsion consists of a carbon 2s term E;,(0) and three rather than calculate the individual orbital repul- degenerate carbon 2p terms E» (0) =E; (0) sjLon integrals since this averaging'" has only little =E» (0). The values usually taken for these effect on the band pattern. The repulsion inte- quantities in quantum-chemical '"calculations on gral y„,,z behaves asymptotically as an electro- carbon containing molecules" are E;,(0) static point charge potential" — ' and = —19.44 eV and = —11.40 eV. In molecular ~ R„—R, p~ E»(0) is screened at shorter range. systems having axial symmetry (e.g. , acetylene, The diagonal neutral atom orbital energies ethylene) or a reflection plane (e.g., benzene), E,(0) appearing in Eq. (6) and taken from atomic the threefold degeneracy of the Zp orbitals is SELF-CONSISTENT LCAO CALCULATION OF THK. . .I.. . 629 lifted due to the different crystal-field screening simple physical meaning-an electron in orbital in the various axial directions. Penetration effects p, on atom c. has a binding energy E~(0) corre- of an atomic orbital on a given center into the sponding to a neutral system, and experiences screened field of other nuclei were shown to both a potential due to accumulation of charge on lower the energy of the planar terms F» (0) this site and a screened Madelung-type electro- = E» (0) relatively to the out-of-plane term E» (0) static potential due to the field on other atoms. in aromatic molecules having their o manifold In a neutral system (q = 0), the diagonal element in the x-y plane. ' ' Since a first-principles way simply reduces to the atomic ionization potential of introducing this effect requires a detailed cal- E~(0), as used in the conventional non-self-con- culation of all penetration and two-electron inte- sistent extended-Huekel method, ' The inclusion grals, it has been previously suggested'" that of the two first terms corresponds to the iterative such an "atoms-in-molecules" effect could be extended Huckel approximation" '" while the introduced phenomenologically by splitting the inclusion of all three terms corresponds to the in-plane and out-of =plane orbital energies arti- modified iterative extended Huckel in which ficially. Calculations on several organic mole- screened Madelung effects are introduced. ""'" cules showed that a splitting of about -0.3 To test the adequacy of the employed atomic ~ E»(0) ~ yields satisfactory agreement with experiment parameters for graphite, the one-electron ener- regarding the ordering of molecular levels, ioni- gies of the closely related benzene molecule were zation potentials, and o-m splitting. In a previous computed with the same parameters. A carbon- study on graphite" such a procedure has indeed carbon bond length of 1.40 A and a carbon-hydro- yielded satisfactory results. We consequently gen bond length of 1.10 A were used. The con- split the F»(0) element symmetrically around the ventional" value for the hydrogen 1s,orbital ener- commonly employed degenerate value of —11.40 gy E,",(0) = —13.6 eV was used. The energies e,. eV taking: of the ten valence molecular levels (for which experimental photoelectron data exist) were com- E' (0)=E; (0)= —13.11 eV, pared both with experimental data"'"' & ',."' and () with the results of an extended-basis-set Hartree- E;p (0) = —9.69 eV, Fock calculation, '" performed with the same nuclear geometry. The standard deviation and the commonly used 2s value: E;,(0) = —19.44 0 1/2 eV. The Wolfgang-Helrnholtz constants are taken (~cele empt) 2 expt)2 i ~i (~i as'". t=l i =1

2.0, p, = p=p„. was found to be 0.093 for the. present extended- G Huckel calculation and 0.150 for the extended basis 1.75, otherwise. set Hartree-Pock calculation. The one-center term &"„„,[Eq. (6)] can be The calculations were. performed in the following obtained from empirical interpolation schemes way: one guesses the elements of the charge ma- relating the spectroscopically observed atomic trix ( isolated atom values are used as initial ionization potential with the net atomic and computes the elements ' ' E~(q) guess) diagonal energy charge, and, " from Hartree-Pock calcula- [Eq. (6)] up to a maximal interaction range R,. tions'" or ean be deduced from the m-electron These are used together with the computed overlap transitions in benzene. "' All these estimates matrix to calculate all the energy elements [Eqs. yield values around 11.6 + 1.1 eV. Inclusion of (4) and (5)] and to solve the eigenvalue problem ionic configurations and correlation effects into [Eq. (3)] at a selected grid in It space. The eigen- the Hartree-Fock estimate yields a value of vectors are used to calculate a refined charge 10.53 eV,"' while fitting various calculated prop- matrix and orbital charges [Eqs. (Al)-(A10)], and erties of carbon-containing molecules to experi- these in turn are employed for the calculation of ment, using diagonal matrix elements of the form the energy elements [Eqs. (5) and (6)]. The pro- E,(q) = E„(0)+&„q yieMs a value of 11 eV.'" This cess is repeated to obtain a constancy in the value is sufficiently close to'those yielded by other charge-matrix elements and band energies be- approximations and it would be consequently used tween successive iterations of less than 10 'e and in this work. Calculations on the charged vacancy. 10 eV, respectively. Damping of the iteration in graphite (Sec. II) indicated that orbital ener- cycle'" is used ta facilitate the convergence. The gies and atomic charges were rather insensitive iteration cycle converges readily for the case of (-7% and -4% for energies and charges, respec- a perfect lattice of graphite owing to the lack of tively) to changes in &„' „,between 10 and 12 eV. interatomic charge transfer in the system. Thus, Expression (6) for the diagonal element has a in this system the simple non, -iterative extended- ALEX ZUNGER

Huckel method yields good results, as previously ined. 180 inequivalent k points in the» irreduc- demonstrated. " However, when lattice imperfec- ible zone were sufficient to obtain converged re- tions are present (see II),"iterations towards sults for the charge density elements (error less self-consistency substantially change the final than 10 '%) and total energy (10 %) in a Simpson charges and the energy eigenvalue spectrum. In integration scheme. " Using the special k points the convergence limit one computes the total calculated by Cunningham" for the irreducible energy per atom: hexagonal BZ by applying the mean-value algorithm of Chadi and Cohen, "it was found that the dk n,-e,- k . (10) six special points CCo B Z j=]. = (k„,k,) (9, 2/9 v 3), (—,', 4/9 W3), ('g, 4/9~3, Numerical first and second derivatives of E,/N ' with respect to the totally symmetric stretching (-', , 2/9~3 (— 4/9W3)(~9', 2/9v 3) mode are performed to yield equilibrium carbon- + —' —' with respective weights of ', —, were carbon distance@ and force constant, respectively. , already sufficient to obtain an accuracy of 0.2% The minimization procedure used to obtain the for the total energy. This remarkable y, ccuracy is unit cell parameters at static equilibrium is the further improved when one employs an artifi. cially standard steepest-descent method in which the large unit cell instead of the primitive cell, thus displacement 5g in the vector required to ap- $ reducing the dispersion of the energy bands. In proach equilibrium is iteratively determined by calculations involving a large unit cell (20-50 6)= -AVIE„,($), (ii) atoms instead of two) associated with the defect "only the six special points are used. where is a vector whose components are the problem, $ The charge density [Eq. (12)] in the primitive unit structure parameters to be optimized and A is a cell calculations is computed throughout this work numerical scaling factor. using the 180 k points. The lattice sums in direct The final eigenvectors are used to the compute space [Eq. (4)] are found to converge for ~ 22 A. electronic charge density in the valence bands: 8, Since deviations from charge neutrality associated a with the defect structure occur only in the vicinity V(r)= dk dT, ' ' ' d~~ g of the defect site, a discrete summation over 22 A is sufficient to include even all the electrostatic x O,*. (k, r, .. r„)@,(k, r, ... , r„). . , contributions [Eqs. (6) and (7)] and Fourier-transform convergence procedures'" are not required. Using the computed band structure, the elec- This can be expressed by the atomic basis orbitals tronic density of states given by X„and their expansion coefficients C„",. [Eqs. (1) and The Slater atomic orbitals are then ex- (2)]. (14) panded in a series of Gaussians using the 6-G expansion of Hehre et a/. "' The charge density is is calculated. 180 k points in the ~» irreducible thus expressed in terms of the coefficients C~&(k) zone are used to generate 10' points by means of and products of Gaussians centered on various Lagrangian interpolation. Energy channels of sites. These multicenter products can be expres- 0.02 eV are used. The joint density of states sed as a sum of Gaussians centered on single intermediate sites. ' ' lt has been previously shown (e) = — dk5(e —c,(k) + e„(k)), in molecular calculations'" that the electrostatic J„, Poisson potential V„„generated by a charge density expressed as a sum of Gaussians, can be where e,(k) and E„(k) denote conduction and valence analytically calculated directly from the Poisson energy bands, respectively, are similarly cal- equation: culated. Near the discontinuities in J„,(c) the derivatives of the valence and conduction bands V'V„„(r)= -4mp(r). (i3) are analytically computed in order to determine the transition at critical This is Following this procedure, we calculate the elec- energy points. done the algorithm developed Delhale and tronic contribution to the electrostatic potential by by Andre. '" The k derivative is given direct dif- I „„(r).This is used for obtaining the perturba- by ferentiation of Eq. (3): tive potential exerted by the defect (see Part II). The convergence of the integrals over the BZ d~,. - dE(k) [Eqs. (10) and (12)] as a function of the k grid used (k), (16) dk and that of the sums in dir'ect space [Eq. (4)] as a function of the interaction range R„was exam- where SELF-CON SISTKAT LCAO CALCULATIO'Ã OF THK. ..I.. .

iteration cycle due to charge rearrangement in all the oceuPied 0+@ bands in the Brillouin zone. This is a distinct improvement over both non-self- consistent all-valence electron methods" "and &-electron models. ""

III. RESULTS

A. Band-structure and optical properties

In the two-dimensional graphite structure there FIG. 1. Two-dimensional Brillouin zone of graphite are two atoms per unit cell. The primitive trans- with the notation for high symmetry points of Lomer lation vectors are a, = a(1,0) and a, = a(&, v 3/2) (Ref. 74). where the unit cell parameter a, expressed by the carbon-carbon bond length is v with 0 R, 3R, R, = 1.42 A.'" The reciprocal lattice vectors are ) — j R+0 ~)yak'Rg b, = (2m /a)(1, —1/0 3) and b', ~ (2w/a)(0, 2w /W3). dk Q The first BZ is a hexagon and is shown in Fig. 1 Critical points of interband transition are deter- along with the notation of high symmetry points mined from of Lomer. " The symmetry properties of the hexagonal layer structure and the classification of &ge,(k) —e„(k)]= 0. electronic states, have been discussed by Lomer'4 In the approximation of constant transition moment and by Bassani and Parravicini. " ' for the m and. o bands, the imaginary part of the Figure 2 shows the results for the band struc- dielectric function, e,(0, ~) is taken to be propor- ture along the I' -P —Q lines of the present cal- tional to J'„,(E)/&'. culation together with the results of Painter and It should be indicated that although in the two- Ellis" and that of Bassani and Parravicini. " dimensional graphite structure (with reflection Figure 3 shows the band structure obtained by symmetry in the basal plane) the electronic states the self-consistent INDO (intermediate neglect of can be rigorously divided into m and 0 states, in differential overlap) method, similar to the com- our self-consistent calculation scheme, no such 'plete neglect of differential overlap (CNDO) meth- separation is invoked. Thus, for instance, Ham- od previously used 80, 8x The results of the self- iltonian matrix elements coupling w Bloch states consistent extended-Huckel calculations of the are allowed to modify through the self-consistency present work [Fig. 2(b)j correspond to the con-

Q2„

' p+ 5

Q2„ / p+ Q+ / 2 / / / / / -Q2 Q' Q29 9 Q 29 P~ ( p + -- 1~9 r Q2U rr Q2„

p+ Q29 + + I Q29 Q29 -20 p+ I p+

Q, 9 QI -30 P Q-

FIG. 2. Band structure of the two-dimensional graphite along the P-J'-Q directions. {a) Painter and Ellis, Ref. 77; {b) present work, extended Huc'kel; {c) Bassani and Parravicini, Ref. 76. Full lines constitute the 0 bands, the broken lines the r bands. Fermi level is at P3 ~ 632 ALEX ZU5GER

+l5 Corbato" for energies lower than about 10 eV above the Fermi level E~. Points at higher energy, in the conduction band (i.e., Q;„and Q;„) are +X / ~5U X substantially overestimated the present cal- / by / culation. In applications of the EXH band structure / Q / 2g we will restrict ourselves to energies well below E~+ 10 eV. The pseudopotential calculation" yields reason- — able results for the main m gap at Q,, Q~ and the main o gap at I", —I",„but underestimates strongly other transition energies. The order of transi- )5 tions seems to be incorrectly revealed by this 0) calculation. The INDO band structure (Fig. 3) agrees well with the previously published results 0) of Messmer et al."and Kapsomenos" using the ~ -Z5 similar CNDO approximation but disagrees mark- edly with experimental data (overestimation of the + Q)„ low lying interband transition energies by a factor

Qig of 2-3 and of the width of the valence bands by a factor of 3-5). In previous applications of the CNDO method to band-structure calculations of various carbon- containing organic polymers (see Ref. 123) and references therein), no detailed comparison with experimental optical data was made. Our above -55 conclusions on the inadequacy of the CNDO approximation to graphite and our similar experi- FIG. 3. 1&TOO band structure of the two-dimensional ence on boron nitride, "suggests that this might graphite along the P-I -Q directions (present work). also be a poor approximation for carbon polymers. Figure 4 shows the imaginary part of the. dielectric function perpendicular to the c axis, &»(0, ~) vergence limit of both direct and reciprocal space as calculated from EXH band structure in the con- sums, discussed in Sec. II. stant oscillator strength approximation. This is In Table I we have compared the theoretically compared with the experimentally determined calculated interband transition energies with the function using ref lectivity" and electron energy experimental reflectance, " ' energy loss, "photo- loss" data. Only the overall features of the cal- emission, "and secondary-electron-emission" " culated &»(0, &u) function could be compared with spectra. Complete assignment of the experimental experiments since the details of the energy de- spectra is still lacking. The lowest g —g * transi- pendence of the oscillator strength is not included tion between the saddle points Q,„and Q~ states appears to be consistently determined by all workers at around 4.5-4.8 eV, while the lowest o —v* transition (I"~- I",„) has been attributed by Bassani and Parravicini" to the weak shoulder 60— observed in the reflectance spectra" at 6 eV and by Painter and Ellis" to the main reflectance 3. 40— I &Zu &29 peak at 14.5 eV. It was argued however by the Al former authors that an artificial increase of the 4J r=r- 20 2LI 29 calculated o —o ~ gap from 6 to about 10 eV wouM 1 yield better fit with experiment for the calculated a pe '&o) e,(0, near the Q~ -Q;„ transition. Lateq, '0 l0 l5 20 SEE""and photoemission" experiments estab- Energy {eV) lished the onset of the o.—o* transitions at 11.5 FIG. 4. Experimental and calculated (in arbitrary ~0.1 eV. units) &»(0, Q. ( ) Calculation; (-.—.-) experimental The results of the present study are in good data from reflectance spectra of Taft and Philipp (Ref. agreement both with experiment and with the first- 23); (-—--) experimental data from electron energy principle calculations of Painter and Ellis" and loss of Tosatti and Bassani (Ref. 28). SELF-CON SIS'f ENT LCAO CALCULATION OF THE. . .I. ~ ~ 633

TABLE I. Comparison between experimental and calculated interband transitions for graphite. Energies given in ev. E J.c and F. ~~ c denote polarization with respect, to the c axis.

Expt. EXH Painter and Bassani and Tsukada ~ Transition Results present work Ellis Parravicini Corbato et al. " Haeringen '

\ ~ Q~, @2~ (E &c) 4.5, 4.8, 4.7 4 5i 4.6 "4 85' I" I'Su (E ~c) ll. 50 + 0.1 " 12.1 10.7 3r llf @~u ~E~c~ 15.0+ 0.5" 14.0 15.3 4 9 14.8 &20 "16' 17.2 13.0 22.4 18.7 8.9 15.0 ' 15.8 13.0 14.2 14.5 +2" dipole 10.4 9.0 4.3 forbidden

~Discre'te variational method, Hef. 77. I Beflectivity, Hef. 24. "Empirical tight binding, Bef. 76. "Heflectivity, Hef. 25. ' First-principles LCAO, Bef. 75. ' Thermoreflectance, Hef. 29. ~ ~-orthogonalzzed-plane-wave, Bef. 73. ' Adjusted value. 'Empirical pseudopotential, Bef. 79. "Secondary electron emission, Hef. 35, 36. Beflectivity, Ref. 23. ~ Reflectivity, Bef. 26.

in the calculation. The agreement below the m band and subsequently relaxed due to electron- plasma frequency (~~, =—7 eV)""'"is satisfactory. electron and electron-phonon scattering into lower. At higher energies the calculated e„(0,e) shows minima in the energy bands, the energetic posi- a small shoulder at around 17—18 eV which is ab- tions observed in SEE experiments provide lower sent in the reflectance spectra and seems to ap- bounds. Again, it is observed that the present pear at about 15-16 eV in the electron energy loss EXH calculation are in good agreement with ex- data. Owing to the occurrence of the Q', „state at periment for the lowest zone-center conduction- relatively high energy in our calculation, the band levels but in rather poor agreement with Q~ - Q,'„transition is not contributing much to the high-energy zone-edge antibonding levels. calculated e»(0, ~) around 15-20 eV in contrast with the results of Bassani and Parravicini. " B. Density of states ln Table II we compare the energetic position of some high symmetry points in the conduction band, The partial density of states of graphite is relative to the theoretical Fermi level E„(located shown in Fig. 5 together with the notation for at the P, point) with the experimental determina- special points. Saddle-paint singularities occur at tion by SEE.." With this technique, only energy the Q point while the P point creates local minima levels located above the vacuum level (E~+ 4), in the density of states. Since a detailed density where the work function 4 for graphite is 4.7 eV,'" of states has not been published previously, only are determined (e.g. , higher than the I",„level). the gross features of the present calculation can Since electrons are initially excited into levels in be compared with previous results. The charac- high density-of-state regions in the conduction teristic widths of the valence bands and the amount

TABLE II. Energies of conduction states at high symmetry points relative to the calculated Fermi energy, 'compared with SEE data. Energies in eV.

Expt. EXH Painter and State results present work Ellis" Corbato d

7.7 +0.5 8.0 7.5 3.0 11.0 12.2 +0.5 12.3. 10.5 9.0 12.8 8.7 + 0.5 19.1 8.0 8.2' 9.6 ~Reference 35. 'Reference 76. "Reference 77. Reference 75. ALEX ZUNGEH,

new x-ray photoemission data of McFeely" yield m. —bands a value of 24 eV in reasonable agreement with the 60— Q2„ present calculation. The energetic overlap be- 0. tween valence and 7t bands has also been specu- latively established from various experiments. Oscillator-strength sum rules for m and 0 transi- g 0 tions indicate a rough overlap of 1-3 eV, '" O while analysis of soft x-ray data"" suggests an cr- bands 60— overlap of about 1.6 eV. The valence O. -m over- O lap observed in si.mple aromatic molecules is of the order of 1 eV "' Several x-ray photoelectron spectra" "and emission" spectra of graphite, have been re- 0 -50 -24 -IS -12 ported. Although direct comparison between x-ray Energy {eV) photoelectron and emission spectra with the calculated profile is complicated the need to FIG. 5, Partial density of states of graphite. by weight the latter by the energy dependent transition probabilities, '" the peak positions and the of o. -g band overlap at the zone center are given general features are rather unaffected. In Table in Table III. Coulson's" analysis of the soft x- IV we have compared the various features ob- ray spectroscopy data of Chalkin" suggests a tained in the combined experimental x-ray photo- width of 5.0+0.5 eV for the g-valence subband, electron" and emission"" spectra with the theo- in agreement with the present calculation. The retical assignment based on our calculated density experimental determination of the total valence of states. The exact position of our calculated bandwidth is rather controversial. Early K-emis- peaks and their parentage was determined by the sion results of Chalkin" suggest a value of about procedure outlined in Eq. (16). The energies of 15-17 eV while ESCA results of Thomas et al." the calculated values are given relative to the and of Hamrin et al."suggest values of 31+2 and theoretical Fermi energy as yielded by each au- 30 eV, respectively. The soft x-ray emission thor. The results of the present work are in very spectra arise mainly from the 2P part of the valence good agreement with experiment and in moderate band (due to the 2P-1s selection rule for K emis- agreement with the results of Painter and Ellis, " sion), while the ESCA spectra gives more weight while the calculated results of Bassani and Parra- to the 2s part (a cross-section ratio o„/a» =13 vicini" reveal systematically lower binding ener- has been suggested). '" The value obtained by gies. Owing to very pronounced cross section Chalkin is thus a lower bound to which the 2s part effects the occurrence of the relative minima at of the spectrum (with contributions from the vici- -10 and at -14 eV in our density of states near nity of the I';g point) should be added. The large the I", and P', points, respectively, cannot be width of the observed ESCA bands was attribu- verified by direct comparison with experiment ted"'" to the presence of surface impurities such and it is not sure whether it is an artifact of the as oxygen whose traces are directly detected in calculation model, or not. The maximum inten- the spectra. Recent x-ray emission spectra of sity in the experimental photoelectron spectra is pure graphite obtained by Muller et al."indicate observed at the peak assigned to the Q,'„point and a width of 22 eV for the valence band while the at the flat top of the I",~ -Q;~ peak, while in the

TABLE III. Partial valence bandwidths (W, and W~), total valence bandwidth (W~,&), and

0-m. band overlaps (Dg —, r) as obtained in various calculations. Energy given in

EXH Painter and Bassani and Property present work Ellis e Parravic ini Corbato'

w 5.6 7 35 8.0 5.0 9.8 Wa 17.0 14.5 16" 11.3 16.6 w~ot 21.2 19.3, a20 7~ 13.8 18.5 1.4 2.7, '3 0~ 1.3 6.0

~Reference 77. Reference 75. b Reference 76. Reference 35, extended basis set. SELF-CONSISTENT LCAO CALCULATION OF THE. . .I. . .

TABLE IV. Characteristic features of the valence density of states in graphite. Energies measured relative to the Fermi level, iri eV.

Theoretical Energy Painter Exper imental assignment (present and Bassani and feature Energy (Present work). work) Ellis Par ravicini

&-band peak 4c Q2„saddle point 2.5 Top of 0 4cl z+ 45 - valence band 0-band peak 8+1d Q&, saddle point 7.2 7 ' 5 5.5

Sharp peak 13.8, 13.0 9+&„saddle point 13.1 14.2 11~ 3

Local minima 12-13.8 deep near I'+3 13~ 6 14.0 12.5

SmaH peak 17' 16.2 15 ~ 5 13.3 Flat region 17-20 Q+ ~+ 17—21 -13-15 -12-14

~ Bottom of 21 ~ 2 19.3, 20.7 13.8 valence band

'Reference 77. 'X-ray photoemission, Ref. 33. Reference 76. X-ray emission spectra, Ref. 32. K-emission spectra, Ref. 30. 5 Reference 35. X-ray photoemission, Ref. 34.

density of states, the Q'" peak is more intense. ca]culated72 from the band structure (half of the Examination of the calculated unit cell charges v —v" splitting at Q) tobe 2.25 eV. Bothy, and ' at the Q,'„point [(q", (k,j)+q", (k,j)] (see Appendix y, " are calculated by numerically fitting a A) reveals 24/o 2s character and 76% 2p character. separately performed three-dimensional EXH The Q' state, on the other hand, around which a band structure along the II —I' -H line to the high density of states is accumulated, is a pure analytical formula of E(kH. , y„y, " ') given by ' 2P state. Using the estimated cross section ratio Wallace. " This yields effective values of y, " for 2s and 2p atomic contributions to the spectral =2.53 eV and y, =0.32 eV. These values are ob- intensity (o„/o»= 13),'" one obtains that the Q;„ tained from an all-valence electron calculation state should indeed reveal higher intensity than the using five in-plane neighboring atomic shells and Q'" peak, in agreement with experiment. Our considering the interaction of a given plan with its '4 assignment agrees with that of McFeely et al. nearest neighbor planes. In g tight-binding calcu- except for the 13.8 eV peak. These authors spec- lations retaining nearest neighbors only, a single ulate that this peak originates from a Aigk density value is obtained for y, . McClure" obtained values ' of states near the I", point. Our calculation re- of y, " = 2.8 + 0.1 eV and y, = 0.27 eV from a fit veals a relatively sharp increase in the band ener- of a simplified z band structure with only nearest- gy near P,', and consequently a minima rather than neighbor interactions to the observed high-tem- a maxima in the density of states. perature diamagnetic susceptibility and the de Haas-van Alphen effect. Cyclotron resonance = C.- McClure band parameters data fitted by Nozieres" yield (yor" ')'/y, 25 eV, compared with McClure's value of 29 eV and our The present band-structure calculation can be value of 20 eV. Magnetoref lection data of Dressel- used to obtain theoretical estimates of the McClure haus and Mavroides4' give y, = 0.395 eV. Long- y, and y, band parameters"' "previously used in wavelength and uv optical experiments have yielded simple 7t approximations to fit various Fermi sur- y, = 0.14 eV" and y, = 0.40 eV,""while the ultra- face and optical data. "" Since the optical be- violet optical spectra of the lowest Q-saddle point havior at low energies is determined by the band transition suggest y' = 2.25 —2.4 eV."''" structure at the vic'inity of the Q point, while The electronic contribution to the heat capacity Fermi surface data are determined by the behavior of graphite has been calculated by Bowman and at the II -P line in the three-dimensional zone Krumhansl" using the g -band-structure param- (perpendicular to the P point at the zone corner), eters of Wallace. " Using the y, and y, effective we have determined the corresponding two values parameters determined in the present study (aver- ' for y,. The optical parameter y',"is directly age values of y', "and y, " were employed) and ALE X ZUNGER

l00% 2s occurring due to s —p overlap, comes from the two highest 0. bands. The band orbital charge itself can be decomposed into contributions from various parts of the Bril- louin zone [q„(k,j) in Eq. (A6)]. This is shown in 50% Fig. 6 where the dispersion of the 2s —2p, character of the three valence 0 bands along the P —I' —Q directions is revealed. The o., band changes from a pure 2s Bloch state at the zone center to about 2 and —, 2s character at the P and Q points, re- I00%2p spectively. The 0, band reveals an opposite behavior, being a pure 2P, Bloch state at the zone center and gaining some 2s character at the P and Q. The third o, band is a pure 2p, Bloch state FIG. 6. 2s character of the three lowest valence 0 throughout the P —I' —Q directions. The band-by- bands along P-I -Q directions. band orbital charge representation throughout the BZ shown here, should be viewed as the solid-state analog of the molecular charge decomposition assuming a value of 0.02 eV for the shift of the "~~ Fermi level relative to the center of gravity of used in quantum chemistry. A two-dimensional real-space representation of the charge density the 7] bands at II,""we obtain C„„=0.43 && 10~ cal mol ' K ' compared with the experimental val- contributed by each band at a given point in the ' BZ is easily obtained when the 2s —2p mixing co- ues ' . of 0.6&& 10 '.. Although the agreement obtained is rather favor- efficients are given as in Fig. 6. "bond order" able, it should be noted that this merely reflects The concept introduced by Coulson the fact that the overall level distribution around for describing r bonding effects can be identified in our calculation scheme ',~(0, which de- the Fermi energy in our model is correct, and by P, n) notes the charge-density'coefficient the thus, it probably does not constitute a sensitive relating n wave-function amplitudes of the two translation- test to the adequacy of the theoretical Fermi surface. ally inequivalent atoms & and p that are separated by n unit cells. Calculating these values from our = D. Charge distribution and bonding eigenvectors, we obtain P (0, 0) 0.525, P'&"~(0, 2)= —0.183, and P'~'"(0 3)= —0.052. The We next turn to consider the charge distribution value at P (0, 0) can be compared with the value in the valence band of graphite. A carbon atom in of 0.662 obtained for the benzene molecule, '" in- its "reactive" state has an electronic configuration dicating a 52.5/o double-bond character in the form- 2s"2p„"2p,'~2p,", or using the notation appropri- er case [P;;;,(0, 0) = 1 corresponds to a 100% double ate for a crystal field with reflection symmetry bond]. The negative values for the bond order of 0„ in the xy plane: 2s"2p,"2p", . This corre- distant separated atoms, indicate "antibonding" sponds to the usually assumed sp' hybridization nodal character. in planar carbon systems. The total orbital valence charges calculated from the band structure E. Equilibrium unit-cell parameters and bond energy using the four occupied valence bands and 180 in- equivalentkpoints in the irreducible BZ [P~~(n) in The unit-cell parameters of graphite under static 2s'08762P,""'42/', 'ooo' Eq. (A6)] comes out tobe .equilibrium are calculated by minimizing the total which corresponds to a total electronic charge crystal energy per unit cell, as a function of the [Q in Eq. (A9)] of 4.0000e with o hybridization of unit cell parameter a. This minimization pro- sp'" which is close to the commonly accepted cedure, carried out using the steepest-descent ideal sp' hybridization. When the total valence technique [Eq. (11)] yields a„=2.482 A corre- orbital charges are broken into separate contri- sponding to a carbon-carbon bond length of 1.435 butions from the individual bands [R, in Eq. (AV)] A, which is in only moderate agreement with the it is observed that the lowest valence band (de- experimental value of 1.42 A (experimental unit noted o, ) contributes most of the 2s charge having cell parameter a= 2.46 A).'" The cohesive energy the configuration (2s""'2po'""2po. o),, while the is obtained from the total energy per atom at next two o bands (o, and v, ) give rise to the con- equilibrium, after subtracting from it the sum of (2so' ""2p' m and figuration 0' ""2p")g, 0'pp 03. The band free-atom orbital energies the promotion has the electronic configuration (2so2Po2P,"oo),. energy E, gained by the atom when it returns to Thus, most of the 0 covalent bonding in the solid its ground (2s'2p') 'P electronic configuration. SELF-CONSISTENT LCAO CALCULATION OF THE. . .I.. ~ 637

The latter energy has been calculated by Goldfarb The carbon-carbon symmetric stretching force and Jaffe'" to be 6.93 eV and by Jordan and Long- constarit was computed from differentiation of the uet-Higgins' to be 7.61 eV. These yield cohesive total energy curve at the minimum with respect energies E, of 8.73 and 8.05 eV, respectively. It to the carbon-carbon distance. This yields is interesting to observe that using the two special k = 5.6 && 10' dyn/cm. This can be compared with k points for the hexagonal two-dimensional BZ the corresponding values for the benzene mole- calculated by Cunningham, "instead of the 180 k cule k= 6.71 && 10' dyn/cm" and the naphthalene points in the irreducible zone, we obtained an molecule k = 4.789 x 10' dyn/cm. " The latter error of only 1.1% in the cohesive energy relative values were used in calculations for the phonon to the dense mesh results, while the six-point spectrum of graphite"*" yielding very good agree- representation already yields an error of only ment with the measured heat-capacity data and 0.2%. No direct experimental determinations of phonon dispersion curves. " the cohesive energy of graphite are available. The sublimation energy of graphite at 298 K was F. Behavior under pressure determined to be E,= 7.4 eV.~ " Adding to this zero-point energy (c'alculated from the phonon The effect of external hydrostatic pressure on density of states of Young and Koppel" to be 0.18 the band structure of the two-dimensional graphite eV) one obtains a value of 7.58 eV for the static was investigated by repeating the band calculation energy. Since an average of 2 bonds are involved for various values of the unit cell parameter a. in the sublimation process, ' the experimental Figure 7(a) shows the variation of some high-sym- bond energy per atom is , (E,+ E„,—)= 9.7-10.3 eV metry energy levels with the carbon-carbon dis- or Es = 4.85-5.15 eV per bond (using E „,= 6.93 tance R,= (I/v 3)a and Fig. 7(b) exhibits a similar eV'" and =7.61 '" respectively). This plot of the total and m valence bapdwidths (W„, E„, eV, * compares well with the calculated value of gE, and W„respectively) and the onset of ~ -z and +E, )=10.44 eV or Es=5.22 eV per bond. Inde- 'a -o band to band transitions (Q,„-Q~ and I"~ pendent estimates can be obtained from surface I';„, respective'iy). It is interesting to note that energies. If we denote by I/W3dR, the number of upon compression, the bonding 1",„level, repre- bonds unit area involved in creation of two c senting the lower edge of the m valence subband, per 0 surfaces, where d= 3.3 A is the interlayer dis- is stabilized while the upper edge of the a subband, tance, the surface energy o, is given by 2Es/ the I'3~ antibonding state increases in energy there- &3dR,. Using. the measured o, value of 4800 by increasing the a -71 overlap. For lower den- erg/cm', '" one obtains a bond energy of 4.9 eV sities (R, = 1.50 A) these bands exchange their which is in reasonable agreement with the present- order and the a -7t overlap vanishes. The a —7l ly calculated value and with that published previously. " 0 30 The contribution of the m valence band to the . (b) cohesive energy was calculated by integrating the m band the of the energy over occupied part zone. 25 This yields a stabilization of 2.28 eV coming from this band above. Early r electron calculation on graphite using only nearest-neighbor interaction -IO 20 ~ " revealed a binding energy of 1.08y, where y is ) the resonance integral. Comparison with our re- Q) g -I5 l5 P sults of a full band-structure calculation yields B,n CD LLI LLI effective value of y= 2.09 eV. This can be com- p+ I pared with various empirical estimates previously -20 10 suggested, . such as y=, 3.0 eV,"and y= 2.5 eV.'" Most of the binding in the crystal results from the tT bands. The greatest contributions to the co- —25 hesive energy from these bands were calculated to arise from the vicinity of the BZ points P and —30 0 At these points the stabilization of the energy 'i Q. I.IO l.30 I.50 I I 0 l30 I.50 bands relative to the atomic energy levels (field stabilization) is high and s —p hybridization is I' FIG. 7. Dependence of some electron energies at very pronounced (Fig. 6). At the point s -p high-symmetry points (a) and the bandwidth, the onsets hybridization is low, resulting in a low crystal- of 0 (Y* and & 7t* transitions (b) on the interatomic field effect. distance. 638 ALEX ZU1VGER band splitting (Er- —Er+ ) is a measure of the from neutron diffraction). 2Q 3g departure from degeneracy between the in-plane The behavior of the band structure for various 2p„and 2p, orbitals (forming the 1"~ level) and values of the unit-cell parameters as well as the the out-of-plane 2p, orbitals (forming the I ~ contribution of various bands and points in the level), these orbitals being degenerate in spheri- Brillouin zone to the orbital charges, are also cally symmetric free carbon state. In layered ' discussed. The overall success of the extended- hexagonal crystals having atoms of different elec- Huckel method in accounting for the wealth of tronegativity in the unit cell (e.g. , hexagonal boron experimental data on regular graphite, seems to nitride") this o -)( overlap is small due to the justify its use also for further investigation on larger stability of the l"~ level, while in small point defect problems in graphite (following arti- aromatic molecules it is of the order of 1-2 cle) &9 eV."'"' The ionization potential of graphite (-Ep ) is shown to be reduced upon compression, APPENDIX A: ORBITAL AND ATOMIC CHARGE reaching for a compression of 22% about 70% of DEFINITIONS its value in under zero pressure. equilibrium We give here briefly the working definitions of Similarly, the onsets of the a-o. ~ and p -g* orbital and atomic charges used in this work. To transitions are greatly increased compression by avoid arbitrary partitioning of two-center charge of the lattice, the latter transition exhibiting a terms to equal parts, we transform our eigen- lower sensitivity due to the relatively low value value problem of the 2p, —2p, overlap. Experiments on the optical properties of graphite under high pressure or E(k) C(k) = $(k) C(k) & (k) (Al high temperature could be useful for elucidating in the overlapping basis set [g'(k), $(k), and C(k) the interesting features of o -7t band interchange are Fock, overlap, and eigenvector coefficients and other effects introduced by the variation of matrices, respectively] to an orthogonal eigen- the lattice constants. value problem

IV. SUMMARY I(k) C(k) = &(k) C(k) (A2)

The extended-Huckel approximation to the LCAO using the Lowdin transformation matrix elements was used to calculate the elec- =$ ~~ ~( ~ ~( tronic properties of the two-dimensional graphite. I(k) (k)P(k)$ (k) C(k)=$ (k) C(k). The band structure is shown to agree favorably (AS) with previously published first-principles calcula- The wave functions . are tions for the valence bands and the low-energy crystal C, (k, r) [E(l. (2)] part of the conduction bands. The calculated prop- given now in the orthogonalized Bloch representa- erties compared with experimental data, are: tion 4~(k, r) by: (a) Interband transition energies (compared with . = (A4) optical data). (b) Energy of high symmetry points +,(k, r) Q Q C;,(k)4;(k, r). in the valence bands (compared with ESCA data). The elements of the crystalline charge matrix are (c) Energy of high symmetry points in the conduc- given by: tion bands (compared with SEE data). (d) Dielec- tric e»(0, &) function. (e) Density of states (com- ~ cc pared with the general features of the soft x-ray P''(u, m) = — dk Q a,.c'„,.(k) cs,.(k)e'"' emission and ESCA spectra). (f) Effective values BZ j=1 for the McClure y, and y, band parameters (used (A6) to fit experimental Fermi surface data and optical where ~ is the BZ volume, cr„, is the number of data through the Slonzewski and Weiss model). occupied bands in the ground state, and the inte- (g) The parameter yielding the electronic contri- gration is defined nn the occupied part of the BZ. bution to the specific heat in the Bowman and The. charge contributed by the pth basis function Krumhansl model. (h) E(equilibrium unit cell pa- on band at point k in the BZ, (k, is related rameters of graphite (compared with x-ray data). j q~ j ) to the diagonal element of the charge matrix by: (i) Cohesive energy (compared with thermochemical estimates) .. (j) Symmetric vibration stretching c force constants (compared with the value used in P„(n)= — dk q"(k,j). (A6) lattice-dynamical calculations on graphite to ob- BZ tain good fit with the experimental phonon heat The contribution of the jth band to the p, -orbital capacity and with the phonon dispersion curves charge is given by: -17 SELF-CONSISTENT, LCAO CALCULATION OF THE. ..I.. .

The total electronic charge on site & due to all R„~= — dkq„(k, j) (A&) BZ occupied bands, in the ground state, is given by: and the contribution of the jth band to the total electronic charge, on site &, in the ground state, (A9) is given by and is equal to the atomic valence charge (4e) if no charge transfer takes The net atomic (A8) place. charge on site & is given simply by the unscreened The BZ integration in Eqs. (A5}-(A8) is replaced charge: by a numerical quadrature over 180 k points in the (A10) —,', segment of the BZ. The sum of D,. over the sites & that are nearest neighbors to a vacant where Z is the core charge (+4 for carbon} on site (where j denotes here the defect band), is si.te +. For neutral isolated atoms and for an taken as a measure of the localization of the defect atom in a homonuclear solid where all sites are charge density (paper II). crystallographically equivalent, q„ is zero.

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