APPROXIMATE NON-EMPIRICAL TIGHT-BINDING BAND STRUCTURE CALCULATIONS WITH APPLICATIONS TO DIAMOND AND SILICON

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1 APPROXIMATE NON-EMPIRICAL TIGHT-BINDING BAND STRUCTURE CALCULATIONS WITH APPLICATIONS TO DIAMOND AND SILICON J. Linderberg, J. Chr. Poulsen To cite this version: J. Linderberg, J. Chr. Poulsen. APPROXIMATE NON-EMPIRICAL TIGHT-BINDING BAND STRUCTURE CALCULATIONS WITH APPLICATIONS TO DIAMOND AND SILICON. Journal de Physique Colloques, 1972, 33 (C3), pp.c3-123-c < /jphyscol: >. <jpa > HAL Id: jpa Submitted on 1 Jan 1972 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 JOURNAL DE PHYSIQUE Colloque C3, supplkment au no 5-6, Tome 33, Mai-Juin 1972, page C3-123 APPROXIMATE NON-ER/IPIRICAL TIGHT-BINDING BAND STRUCTURE CALCULATIONS WITH APPLICATIONS TO DIAMOND AND SILICON J. LINDERBERG and J. Chr. POULSEN Department of Chemistry, Aarhus University, DK-8000 Aarhus, Denmark Rhsumh. - On s'intkresse a l'approximation des cc liaisons fortes >> avec une evaluation non empirique des parametres figurant dans I'hamiltonien. Les termes d'kchange pour la bande de valence sont inclus de maniere auto-coherente dans l'approximation du recouvrement diffkrentiel nul. Les rksultats pour le diamant sont trks proches de ceux d'autres calculs. Abstract. - The tight-binding approximation is considered with a non-empirical evaluation of hamiltonian parameters. Valence band exchange terms are included in the zero differential overlap approximation by means of a self-consistent procedure. Results for diamond are closely similar to other calculations. Introduction. - Slater and Koster [I] presented several years ago the tight-binding method as an interpolative scheme for obtaining information on energy bands in crystals when results of accurate calculations were known at symmetry points. It was at the time considered to be unrealistic to attempt an evaluation of the parameters without recourse to such data. A few years ago it was shown that an operator relationship [2] could be used for the direct calculation of the tight-binding energy parameters and that such a scheme leads to simple calculations. The motivation for us to be interested in the tightbinding method lies in its relation to the most common methods for the calculation of properties of molecules. Such calculations as are performed in the tight-binding framework relies on prescriptions for the evaluation of matrix elements. Different schemes have been developed for different properties which is an unfortunate state of affairs. It is the purpose of the present investigation to test yet another prescription and to try later to derive a variety of different properties. Particularly we wish to calculate spectral and cohesive properties both of which are reflected in the structure of the imaginary part of the dielectric susceptibility. In the present paper we limit ourselves to a discussion of the Hartree-Fock approximation. The second section contains the definition of the hamiltonian and the methods for determinating its elements. It is followed in the third section by a description of the calculation of a self-consistent Hartree-Fock solution. The results for diamond and silicon are presented in the fourth part of the paper. A discussion of the methods and the results can be found in the fifth and final section. Hamiltonian. - Second quantization language will be used and the electron creation and annihilation t operators, as, and as, respectively, are defined with respect to an orthonormal set of valence shell atomic orbitals ( us(r)) and spinors, o = A model hamiltonian is defined as H = c as a:a as, + C Brs a:a asa + sa The so-called hopping term contains parameters P, which are defined to be zero when orbitals r and s are on the same atomic center. The electron interaction term is already at this stage simplified by the zero-differential overlap approximation so that only Coulomb integrals, with the interpretation enter the model. We will also consider the position and momentum operators for electrons, and t r = 2 (r)rs am asa P = C (- ihv)rs asa. The restriction to valence shell orbitals gives an incomplete basis and this causes difficulties with respect to the basic operator identities. It is for instance impossible to satisfy the canonical commutation relation, between the components of position and momentum operators and the number operator No,. We will, (3) (4) Article published online by EDP Sciences and available at

3 C3-124 J. LINDERBERG AND J. CHK. POULSEN in spite of this failure, attempt to satisfy the dynamical relation from the consideration that it is meaningful to accept the three representations (I), (3), and (4) as a model for the electronic system we wish to consider. The commutator of the position operator and the electron interaction terms should vanish and we are led to postulate that the two-center integrals (r),, should be discarded and that the electron repulsion integrals y,, only should depend on the atoms to which the orbitals r and s are associated. The limitation to one-center integrals in the position operator simplifies eq. (6) so that we obtain, if r belongs to the set of orbitals an atom A and s belongs to the set on atom B, and it is clear that only the components of position and momentum along the axis AB give nontrivial information. We will use eq. (7) to determine PrS. To this end we define a coordinate system such that the z-axis passes through atoms A and B with B at the origin and A at the distance R from the origin on the positive axis. Then we calculate the matrix elements from the set of atomic orbitals assuming that the orthonormal set ( u,(r) ) is generated from a standard set ( us@) ) by means of a Lowdin symmetric orthogonalization [3] and that it is admissable to represent this procedure by the expansion form and a separation occurs between a- and n-orbitals. Thus it holds as obtained earlier 121 that &,=,- ii2 as,, The remaining a-orbitals are real and give rise to a system of equations. Let us denote them as s and p where the latter signifies the po-orbital. We wish to determine the matrix where rows are labeled by orbitals on A and columns by orbitals on B. Correspondingly we have an overlap matrix It is important for the results not to discard the integrals and for the two centers and we obtain for the gradient the formulae : limited to linear terms in the overlap integrals The one-center elements (z),, are then those calculated with the functions ( vt(r) ), while it can be seen that the two-center integrals generally will be small. The gradient integrals (a/az),, are calculated partly from the formula The system of equations is then obtained from eq. (7) as, in atomic units RPss + PA bps - PB bsp = (L) 9 SS which is applicable when the orbitals ( ut(r) ) are strictly one-center orbitals. We also need one-center elements of the gradient operator since they enter through the orthogonalization process. We want to consider the particular case when each atomic site is the center for one s-orbital and three p-orbitals. Rotational symmetry around the z-axis is assumed to hold This completes the description of the determination of the hopping terms in the hamiltonian.

4 APPROXIMATE NON-EMPIRICAL TIGHT-BINDING BAND STRUCTURE CALCULATIONS C3-125 A Hartree-Fock approximation to the hamiltonian (1) leads to the Fock matrix with elements The off-diagonal elements are frs = Prs - ~ r< s aru > 7 (19) and in the present study we omit the exchange term for other than nearest neighbors. This allows us to use only one parameter y for nearest neighbor interactions since all nearest neighbor terms represent interactions between the same kind of atoms in the zincblende and diamond structure. The diagonal elements have been left as parameters until further experience is gained from the use of the hopping terms and the free parameter is the difference between elements for atomic s- and p-orbitals. The calculations in this paper depend upon two parameters for diamond and silicium, the difference between f,, and fpp and the two-center y. We expect that y should be about equal to e2/r, the Coulomb energy of an electronic charge at each center. Calculational procedure; - Overlap integrals, their derivatives with respect to distance, and atomic integrals p and ic were calculated with the use of analytical atomic orbitals given by Clemedti [4]. Hopping parameters firs were evaluated for first and second neighbors while contributions from more distant atoms were discarded. The self-consistent determination of the Fock matrix calls for the calculation of bond orders < a:, a,, >. This was done from the resolvent or Green's function to the Fock matrix defined through the equation Coulson's contour integral method [5] was used to obtain 2 < a;,a,, > = a,, + nu' 1 d~ GrX+ + iy), -a, (21) where 8, is the Fermi energy. A given Fock matrix was transformed to the basis of Bloch sums, and used to calculate the resolvent in this basis. A numerical integration procedure was used for the integral in eq. (21) and the transformation from the Bloch basis to the atomic orbital basis was performed through a sampling technique over the Brillouin zone proposed by Hazelgrove 161. These procedures and the iterations to a self-consistent Fock-matrix have been rapidly convergent. Results. - HamiItonian parameters and bond orders as obtained by the procedures described above are given in Table I for diamond and they are compared to the parameters derived by Slater as appropriate to the calculations by Herman [7]. The energy eigenvalues at symmetry points and along symmetry lines are displayed in figure 1. The general features FIG Energy eigenvalues of the Fock matrix for Diamond along symmetry lines in the Brillouin zone calculated for the selfconsistent solution obtained for y =.5 a. u. = 1 Ry. m Parameters for Diamond in Atomic Units. Notations given in the text (") Calculated with y =.342 6, fpp - f,, =.I39 5. (b) Calculated with y =.500 0, fpp - f,, =.I39 5. (? Reference [7], f,, - f,, =.5.

5 J. LINDERBERG AND J. CHR. POULSEN Parameters for Silicon in Atomic Units. Notation given in the text ("? Calculated with y =.225 0, f,, - f,, = are closely similar to not only Herman's calculations but also to the elaborate calculations by R. Keown and later by Chaney, Lin and Lafon [S]. Results for silicium are given in Table I1 and figure 2. The valence band is again in accord with other models [9] while the conduction band differs somewhat. The band gap at r agrees with the established value. Both in diamond and silicon we find an atomic charge FIG Energy eigenvalues of the Fock matrix for Silicon along symmetry lines in the Brillouin zone calculated for the selfconsistent parameters of Table 11. distribution very close to the traditional sp3 configuration. Discussion. - We conclude that a tight-binding model as the one we have defined in this paper may serve as a means of obtaining ideas about the band structure of solids beyond the purely qualitative features. Further exploration is necessary, particularly with regard to electronegativity and Coulomb interaction parameters. It is also important to extend the present results to the evaluation of dielectric and optical properties in order to test the reliability of transition moments. The cdculation of cohesive and elastic properties should be a test of the variation of the hopping parameters with distance. A principle advantage of the model discussed in this paper is the simplicity of the calculations. We anticipate that it is possible to solve the Green's function eq. (20) effectively also for large aperiodic problems, particularly at complex energy values as those required in the integral (21). The present calculations have been performed at the regional computation center at Aarhus University and required times of the order of minutes on their CDC 6400 system, but there are several procedures that can be made more efficient yet. References [I] SLATER (J. C.) and KOSTER (G. F.), Phys. Rev., 94,1498. HERMAN (F.) and CALLAWAY (J.), Phys. Rev., 1953, 89, [2] LINDERBERG (J.), Chem. Phys. Letters, 1967, 1, [3] LOWDIN (P. O.), J. Chem. Phys., 1950,18,365. SLATER (J. C.), Technical Report no 4, Massachusetts Institute of Technology. I41 CLEMENTI (E.), Tables of Atomic Functions, S~PP~. to 181 (R.), phys. R~~., 1966, 150, 568. IBM J' of and Devezopment ' CHANEY (R. C.), LIN (C. C.) and LA~N (E. E.), Phys. [5] LINDERBERG (J.), Chem. Phys. Letters, 1970, 5, 134. Rev., 1971, B 3, 459. [6] HAZELGROVE (C. B.), Math. Computation, 1961,15,323. [9] BASSANI (F.) and YOSHIMINE (M.), Phys. Rev., 1963, [7] HERMAN (F.), PhyS. Rev., 1952,88, , 20.