17 Tight Binding Method for Electronic Structure Calculation

5.4.1 Introduction to tight binding method

Similar to nearly free electron method for calculating band structure, the tight-binding model (TBM) belongs in the class of methods based on independent electrons approximation. TBM model is the most popular method for realistic band structure calculation, that takes explicitly into account the presence of the periodic lattice potential. In contrast to the nearly free electron model, TBM describes the electronic states starting from the isolated atomic orbitals. TBM can produce excellent quantitative results for bands derived from localized atomic orbitals. It is most appropriate when electrons move

Figure 5.24 The evolution of the energy spectrum of Carbon from an atom (a), to a molecule (b), to a solid (c).

through the crystal slowly, as it is the case in carbon based nanotructures, like carbon nanotube and graphene in which the electrons are in some sense tightly bound to the atom and only hop because staying put on a simple atom costs high energy.

TBM uses atomic orbitals as basis wave functions. The energy spectrum gradually evolves as we go from atom to an assembly of atoms to form the solid. Let us consider carbon as example. In a free atom electrons moves in a potential well, as shown in Figure 5.24. The atomic spectrum consists of a series of discrete energy levels, which are denoted by 1s, 2s, 2p, etc. The carbon atom contains six electrons, two of which occupy the 1s shell and two are in 2s shell which are completely full, and the rest two electrons in the 2p shell. If the two carbon atoms are assembled to form the molecule C2, the potential “seen” by electrons is now the double well shown in Figure 5.24. Due to a coupling between atoms, each of the atomic levels (1s, 2s, 2p) has split into two closely spaced levels which are 1s, 2s, 2p, etc., molecular energy levels, composed of two sublevels. The amount of splitting depends strongly on the internuclear distance of the two atoms in the molecule. If the distance between the two nuclei reduces, the perturbation becomes stronger leading to larger the splitting which also depends on the atomic orbital. The splitting of the 2p level is larger than that of the 2s level, which is larger still than that of the 1s level. This is due to poorer overlap 1s wavefunction compared to that for 2s and 2p wavefunctions. It follows that, generally speaking, the higher the energy, the greater the splitting incurred, as shown in Figure 5.25.

The above considerations may be generalized to a N no of carbon atoms. The carbon solid, such as graphite, diamond, and nanotructure such as graphene and carbon nanotube, may then be viewed as the limiting case in which the number of atoms has become very large, resulting in a large carbon molecule. Each atomic levels is split into N closely spaced sublevels, where N is the number of atoms in the solid. As N is very large (~ 1023) the sublevels are extremely close spaced, so that discreteness of the energy levels are blurred resulting continuum band of energy known as energy band. Thus the

Figure 5.25 Schematic representation of the formation of tight-binding bands as the spacing between atoms is reduced.

1s, 2s, 2p levels give rise  to the 1s, 2s, and 2p  energy bands, respectively  as shown in Figure 5.24. The regions separating these bands are energy gaps which is a  regions of forbidden energy.

Generally the higher the band the greater its width due to strong perturbation, which is the cause of the level broadening. On the contrary, the low energy states correspond to tightly bound orbitals, which are affected marginally by the perturbation resulting narrower width of the band.

5.4.2 Tight Binding Formalism

In this formalism, the wave function can be written as a linear combination of fixed energy independent orbitals, where each orbital is associated with a specific atom in the molecule or crystals. Here, one assumes a form for the Hamiltonian and overlap matrix elements without specifying anything about the orbitals except their symmetry. Generally speaking, if N atoms are included in the tight binding model, the electronic wave functions can be expressed a linear combination of Bloch functions.

Here Ci,j and Φj(k, r) are expansion coefficients and Bloch functions which can be expressed as,

Here R is the position of the atom and denotes the atomic wave function in state j. Because of the translational symmetry of the unit cells in the direction of the lattice vectors, ai, (i=1, 2, 3), any wave function of the lattice, , should satisfy Bloch’s theorem

where      ⃗ is a translational operation along the lattice vector ai and k is the wave vector.

where we use the periodic boundary condition for the M= N-1/3 unit vectors in each ai direction,

consistent with the boundary condition imposed on the translation vector TMai=, 1. From this boundary condition, the phase factor appearing in Eq. satisfies exp{ikMai} = 1, from which the wave number k is related by the integer p,

The eigen energy, Ej(k) of the jth band, is given by

where H is the Hamiltonian of the solid. After making substitution for , we obtain the following equation:

where, Hjj’ = <Φj|H|Φj’ > is transfer integral matrix element and Sjj’ = <Φj|Φj’> is overlap integral matrix element.

Minimizing the energies Ei with respect to the expansion coefficients Ci,j leads to the following equation,

The band energies Ei can be determined from the generalized eigenvalue equation by solving the secular equation

where the number of solutions is dependent on the number of orbitals per unit cell.

5.4.3 Procedure for obtaining the energy dispersion

In TBM, the one-electron energy eigenvalues Ei(k) are obtained by solving the secular equation Eq. (2.14). The Ei(k) is a periodic function in the reciprocal lattice, which can be described within the

first Brillouin zone. In higher dimension, it is difficult to show the energy dispersion relations over the whole range of k values, so Ei(k) is generally plotted along the high symmetry directions in the Brillouin zone. Using TBM, the bandstructure is calculated by following steps,

1.   First identify the unit cell and the unit vectors, ai.

2. Then identify the coordinates of the atoms in the unit cell.

3.  Select the n atomic orbitals which will constitute basis set for calculation.

4.   Specify the Brilouin zone and the reciprocal lattice vectors, bi and identify the high symmetry directions in the Brillouin zone, and k points along the high symmetry axes.

5.  For the selected k points, calculate the transfer and the overlap matrix element, Hij and Sij

6.  For the selected k points, solve the secular equation and obtain the eigenvalues Ei(k)(i= 1,..,n)and the coefficients Cij(k).

Tight-binding calculations are not self-consistent calculations in which the occupation of an electron in an energy band would be determined self-consistently.

Figure 5.26 shows the band structure of Si and diamond calculated by tight binding method as described above.

Figure 5.26 Tight binding band structure of Si (top) and Diamond (bottom).