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Self-Consistent LCAO Calculation of the Electronic Properties of Graphite

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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 forma- tion" "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 de- scription of systems exhibiting some sort of The matrix elements of the Bloch function are heteropolarity, such as binary
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