1.10.6.1 Energy of a Part-Filled Band

An alternate starting point to defining potentials is tight-binding theory. As this already has localized orbitals, it gives a more intuitive path from quantum mechanics to potentials. Consider a band with a den­sity of electronic states (DOS) n(E) from which the cohesive energy becomes •Ef

(E — Eo)n(E) dE where E0 is the energy of the free atom, which to a first approximation lies at the center of the band.

For example, a rectangular d-band describing both spin states and containing N electrons, width W has n(E) = 10/ W, and Ef = W(N — 5)/10 + E0 whence (Figure 3),

This gives parabolic behavior for a range of energy — related properties across the transition metal group, such as melting point, bulk modulus, and Wigner — Seitz radius. For a single material, the cohesion is proportional to the bandwidth. Even for more com­plex band shapes, the width is the key factor in determining the energy.

The width of the band can be related to its second moment9 here: (E — Eo)2n(E) dE

E0+W/2

Eo—W/2

= 5 W2/6

To build a band in tight-binding theory, we set up a matrix of onsite and hopping integrals (Figure 4). For a simple s-band ignoring overlap, n(E)

Figure 4 Matrix of onsite and hopping integrals for a planar five-atom cluster — in tight binding this gives five eigenstates, each of which contributes one level to the ‘density of states’: five delta functions. In an infinite solid, the matrix and number of eigenstates become infinite, so the density of states becomes continuous. Of course, tricks then have to be employed to avoid diagonalizing the matrix directly.

S = (Ф,1 Vi |Ф,> h{rtJ ) = (Фі | Vi |фу>

The electron eigenenergies come from diagonalizing this matrix (there are, of course, cleverer ways to do this than brute force). Typically, we can use them to create a density of states, n(E), which can be used to determine cohesive energy (as above).

The width of this band depends on the off-diagonal terms (in the limit of h = 0, the band is a delta function). One can proceed by fitting S and h, or move to a further level of abstraction.