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Next: The rescaling functions Up: The tight binding model Previous: The band energy model

The bond energy model

The model shown in the preceding section enables the calculation of the energy bands of a system of well defined configuration. Thus, the parameters suitable for the diamond lattice, for instance, cannot be used for calculations in graphite. To describe systems with a wide variety of coordinations with the same tight binding parameters, a total energy scheme has to be employed, which accounts for interactions other than the single-electron one's, with an explicit dependency on the interatomic distances.

The tight binding model has been developed on the basis of two major approximations. The first to be considered is the adiabatic approximation [95], which is based on the fact that electrons move typically $\sim 10^2 - 10^3$ faster than the ions. The latter can thus be considered in their ground state at any moment for a particular instantaneous ionic configuration, and the electronic and ionic degrees of freedom can therefore be separated. The second approximation consists in reducing the N-body problem to a one-electron scheme, where each electron moves independently of the others, and experiences an effective interaction due to the other electrons and to the ions. Within these approximations, the one-particle electronic part of the total Hamiltonian can be written in the form

\begin{displaymath}\hat{H} = \hat{T}_e + \hat{U}_{ee} + \hat{U}_{ei},
\end{displaymath} (7.11)

where $\hat{T}_e$ is the kinetic energy operator of the electrons, $\hat{U}_{ee}$ and $\hat{U}_{ei}$ are the electron-electron and electron-ion interactions respectively. Following the notation of Horsfield et. al. [96], the single-particle Schrödinger equation is

\begin{displaymath}\hat{H} \vert n\rangle = \epsilon^{(n)} \vert n \rangle,
\end{displaymath} (7.12)

where $\vert n\rangle$ is a single particle (doubly occupied) eigenfunction, and $\epsilon^{(n)}$ is the corresponding eigenvalue. It has to be mentioned that the k dependency of $\vert n\rangle$ and $\epsilon^{(n)}$ does not appears explicitly in the notation for clarity (see equation (7.7)). We shall return to this point below. The eigenfunctions are expanded in an atomiclike (Löwdin) orbitals set

\begin{displaymath}\vert n\rangle = \sum_{i \alpha} C_{i\alpha}^{(n)} \vert i\alpha\rangle
\end{displaymath} (7.13)

where i is a site index and $\alpha$ an orbital index. It has to be noted that the basis used to expand the wave functions may be non-orthogonal. However, in the present work, orthogonal basis functions are used. The influence of this choice on the results will be discussed further.

Taking into account the orthonormality of the eigenstates, the eigenvalues and eigenstates of the Hamiltonian are therefore found by solving the matrix equation

\begin{displaymath}\sum_{j \beta} H_{i \alpha,j \beta}C_{j \beta}^{(n)}
= \epsilon^{(n)} C_{i\alpha}^{(n)},
\end{displaymath} (7.14)


\begin{displaymath}H_{i \alpha,j \beta} = \langle i\alpha \vert \hat{H} \vert j\beta \rangle
\end{displaymath} (7.15)

are the matrix elements and

\begin{displaymath}\sum_{i\alpha} C_{i\alpha}^{(n)}C_{i\alpha}^{(m)}
\equiv \sum...
... i\alpha \rangle \langle i\alpha\vert m\rangle
\end{displaymath} (7.16)

The off-diagonal matrix elements $H_{i \alpha,j \beta} = \langle
i\alpha \vert \hat{H} \vert j\beta \rangle$, for $i \alpha \neq j \beta$, are called hopping integrals, and the on-site elements $H_{i
\alpha,i \alpha}$ are the atomic orbital energies. In the tight binding approach, these hopping integrals and the on-site matrix elements are constants to be fitted on the basis of the following approximations (besides the two-center approximation introduced above):

(i) Only atomic orbitals whose energy is close to that of the energy bands we are interested in, are used [90]. This is the minimal basis set approximation. Thus, for instance, only the 2s(one orbital) and 2p (three orbitals: px,py, and pz) orbitals are considered in the case of diamond and 3s and 3p orbitals for silicon, to describe the occupied (valance) bands. For these two materials there are 16 possible hopping integrals. However, it can be shown [97] that only hopping integrals between orbitals with the same angular momentum about the bond axis, are non-vanishing. There remain therefore just four nonzero hopping integrals, labeled ($ss\sigma$), ($sp\sigma$), ($pp\sigma$), and ($pp\pi$). $\sigma$stands for orbitals with 0 angular momentum about the bond axis and $\pi$ for orbitals with angular momentum $\pm$ 1. The dependence of these hopping integral in the distance between the atoms will be considered further.

(ii) We consider only hopping integrals between two atoms separated by a distance shorter than a suitable cutoff. Obviously, to reduce the number of parameters to be fitted, a cutoff which includes the nearest neighbors is appropriate. However, the orthogonalized functions (Löwdin) extend further than those (non-orthogonal) from which they are derived, because the orthogonalization procedure involves orbitals from nearby atoms. Thus, interactions extending beyond first nearest neighbors have to be taken into account when an orthogonal basis is used.

Considering the approximations above, the off-diagonal elements of the Hamiltonian matrix $H_{i \alpha,j \beta} = \langle
i\alpha \vert \hat{H} \vert j\beta \rangle$ (for $i \alpha \neq j \beta$) are fitted to electronic band structure of the equilibrium crystal phase, as calculated by more accurate first-principle models [98]. Sets of hopping integrals can thus be obtained for each crystalline structure considered.

On the basis of the Harris-Foulkes formulation of the density functional theory [99,100], Sutton et. al. [101] shown that the cohesive energy of the crystal can then be written in the form

\begin{displaymath}U_{\rm coh} = U_{\rm band} + U_{\rm rep} - U_{\rm atoms} =
... -
\sum_{i\alpha} N_{i\alpha}^{\rm atom} \varepsilon_{i\alpha}
\end{displaymath} (7.17)

where $U_{\rm band}$ is the band energy, $U_{\rm rep}$ is the repulsive potential, given as a sum of pair potentials, and $U_{\rm
atoms}$ is the total energy of the free atoms. $\epsilon^{(n)}$ are the eigenvalues obtained from the diagonalization of the Hamiltonian matrix. Within the adiabatic approximation, the electrons are assumed to be in their ground state, so that all the states below the Fermi level are occupied, and the summation that appears in the band energy is made over these occupied k states. $U_{\rm rep}$ accounts for the ion-ion repulsion, for the double counting of electron-electron interactions that appears in the band energy, for the repulsion of overlapping orbitals due to Pauli's principle and for the exchange-correlation energy related to the N-body electronic interaction. $N_{i\alpha}^{\rm atom}$ is the occupancy of an atomic state in the free atom, and $\varepsilon_{i\alpha} =
H_{i\alpha,i\alpha}$ is the on-site Hamiltonian matrix element.

The form of the repulsive energy $U_{\rm rep}$ proposed by Xu et. al. [102] and used in the present research is

\begin{displaymath}U_{\rm rep} = \sum_i f \left ( \sum_j \phi(r_{ij}) \right ),
\end{displaymath} (7.18)

where f is a functional expressed as a 4th-order polynomial, $\phi(r)$ is a pairwise potential between atom i and atom j, and described below, and rij is the interatomic distance between the atoms.

next up previous contents
Next: The rescaling functions Up: The tight binding model Previous: The band energy model
David Saada