The Tight-Binding Approximation

The application of the method to bulk systems is most easily introduced by first considering a lattice of atoms with overlapping s orbitals, 3, and corresponding free atomic energy levels, Es. Generalizing the LCAO method 

The phase factor automatically guarantees that ( ) satisfies Bloch s theorem, eqn (5.30), since 

Making the usual assumption that the crystalline potential, V, is given by the sum of overlapping atomic potentials, v, we have 

The second contribution on the right-hand side is the shift in the on site energy due to the neighbouring atomic potentials. In the spirit of our earlier treatment of diatomic molecules we will neglect this crystal field term. It does not fundamentally alter the band structure of either transition metals or semiconductors. The band structure, E(k), can, therefore, be written within the approximation as 

The band structure for a simple cubic lattice may now be quickly found. Assuming that the bond integrals couple only to the six first nearest neighbours with position vectors, R, equal to ( a,0,0), (0, a,0), and (0,0, a), eqn (7.8) gives 

Suppose then that we start with a set of atomic wavefunctions 

Our first task is to construct states which can be used as the basis for expansion of the crystal wavefunctions. These states must obey Bloch s theorem, and we call them Xkii(r)  

In the above equation we only need to consider matrix elements of states with the same k index, because 

At this point we introduce an important approximation in the spirit of the TEA, we take the overlap matrix elements in Eq. (4.7) to be non-zero only for the same orbitals on the same atom, i.e. only for m = j = i,R = 0, which is expressed by the relation 

This is referred to as an orthogonal basis , since any overlap between different orbitals on the same atom or orbitals on different atoms is taken to be zero.  

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Whereas the tight-binding approximation works well for certain types of solid, for other s. items it is often more useful to consider the valence electrons as free particles whose motion is modulated by the presence of the lattice. Our starting point here is the Schrodinger equation for a free particle in a one-dimensional, infinitely large box ... 

TOTAL ENERGY CALCULATIONS IN THE TIGHT-BINDING APPROXIMATION... 

We will limit ourselves here to transition metals. It is well known that in these metals, the cohesive properties are largely dominated by the valence d electrons, and consequently, sp electrons can be neglected save for the elements with an almost empty or filled d valence shelP. Since the valence d atomic orbitals are rather localized, the d electronic states in the solid are well described in the tight-binding approximation. In this approximation, the cohesive energy of a bulk crystal is usually written as ... 

Here, u is the displacement of the /ith molecule from its equilibrium position and M the reduced mass of each molecular site. Second, the electron is described within the frame of the tight-binding approximation, where it is assumed that the effect of the potential at a given site of the one-dimensional chain is limited to its nearest neighbors. In that case, the energy dispersion of the electron is given by... 

Within the tight binding approximation, it imphes a decrease in electron locahzation energy ... 

Fig. 3 (a) Crystal structure of (DMET)2FeBr4. The dotted and dashed lines denote the intermo-lecular anion—anion and donor-anion contacts, respectively, (b) Fermi surfaces obtained for a donor layer around z = 1/2 using the tight-binding approximation. The solid arrow represents the nesting vector Q (a b )/2... 

The Hiickel approximation (3.136) is equivalent to neglect of fi and p" (the tight-binding approximation), leading to the simpler Hiickel-type matrix h(HMO) ... 

These moments can be calculated using the tight binding approximation. Introducing a complete set of atomic orbitals za) satisfying the equations... 

The simplest model of a solid is a linear chain of N atoms, with one end of the chain corresponding to the surface. If an atomic orbital r> (r = 1,..., N) is associated with the rth atom, then, in the tight-binding approximation, the matrix elements of the Hamiltonian for the solid, can... 

We begin by considering a one-dimensional model in which the crystal is represented by a straight chain of similar atoms and a foreign atom is in interaction with one end of the chain. This is the simplest model of the chemisorption process which may be expected to yield useful results (9). If the normal electronic structure of the chain consists of just one band, this one-dimensional model is easily treated in the tight-binding approximation. 

The strict generalization of the one-dimensional model treated in Sec. III,A leads to a crystal with its surface completely covered by adsorbed atoms. For this system, the tight-binding approximation gives a difference equation and boundary conditions which can be solved directly. The results show an important new feature. For a simple cubic lattice, the energy levels are given by the usual equation... 

Let the chain atoms be numbered by an index m(—N m N), and let the two foreign atoms X and n be in interaction with the chain at positions - -n and —n. The wave functions for the system are either even or odd in the center of symmetry at the chain atom number 0. It is sufficient, therefore, to consider the range w 0. In the tight-binding approximation, we now have Equation (4) for all m n with the boundary conditions... 

To establish a quantitative relation between F and G for the entire tip and the entire sample, we have to consider all the states in the tip and the sample. A rigorous treatment is complicated. The following treatment is based on the approximate additivity of atomic force and tunneling conductance with respect to the atoms of the tip. In other words, the force between the entire tip and the sample can be approximated as the sum of the force between the individual atoms in the tip and the entire sample, so does tunneling conductivity. Because the tip is made of transition metals, for example, W, Pt, and Ir, the tight-binding approximation, and consequently, additivity, are reasonable assumptions. Under this approximation, the total force is... 

In order to understand the concept, it is customary to present a sort of paradox arising from the solution of the Hamiltonian (11) in the tight-binding approximation (Eq. (12)). 

In the case of narrow bands - and this will be the case of hybridized 5 f bands when 5 f electrons are itinerant - an approximate treatment has to be done. Kubo and Obata have studied the case of transition metals in the tight binding approximation. The narrow band susceptibility is the sum of 4 terms... 

In Eq. (3.106), A/2 is the period of the lattice potential, Xq is the x-coordinate of the bottom of the harmonic potential and f/ determines the tightness of the transverse confinement of a particle in the lattice. Using the tight-binding approximation, the condensate order parameter is [80, 81]... 

Consider a simple cubic lattice of p valent ajtoms which form nearest neighbour bonds only. Show that the bandstructure E(k, 0,0) is given within the tight binding approximation by... 

Methods of calculating energy hands 2.2.1 The tight-binding approximation s-bands... 

A natural extension of the tight-binding approximation is to write... 

Subsequent cellular methods, on which there is an enormous literature, will not be described here. We shall, however, need to introduce- certain ideas, particularly that of the pseudopotential. We begin by introducing the concept of the muffin-tin potential due to Ziman (1964a). This is illustrated in Fig. 1.9. The tight-binding approximation is appropriate for states with energies below the muffin-tin zero ( bound bands in Ziman s notation). If the energy is above the... 

For a simple cubic lattice and a half-filled zone, we may set in the tight-binding approximation... 

This compares the width for a given value of V0 with the bandwidth for the crystalline state. The factor 1.75 comes, as in (40), from comparing N(E) in midband for a simple cubic, using the tight-binding approximation. 

Much of this book is concerned with the properties of narrow bands to which the tight-binding approximation is appropriate. In this case, if the band is half full or nearly so, the short-range repulsion between the electrons may have very important effects on the properties of the electrons in the bands, producing magnetic moments and non-conducting properties. These are a major theme of this book. At this point we introduce the Hubbard intra-atomic energy ... 

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