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68 CHAPTER 4. ELECTRONIC STRUCTURE Figure 4.6: Electronic energy dispersion calculated within the tight binding approach for nearest-neighbour hopping, with a single orbital per atom in a simple cubic lattice (E = E 0 + 2t(cos(k x a) + cos(k y a) + cos(k z a))).
4.4. TIGHT BINDING: LINEAR COMBINATION OF ATOMIC ORBITALS 69 is a nearest neighbour to atom 0), we obtain E(k) = E 0 + 2t(cos(k x a) + cos(k y a) + cos(k z a)). This dispersion is plotted in Fig. 4.6. There is a major problem with producing Bloch states from only a single orbital per site, though: this only gives a single energy band. To understand many materials, and in particular to understand semiconductors, we must extend the method so that it can produce several bands. This requires that we build the Bloch states from several orbitals per site. To see how this would work, let us first combine two orbitals per site, namely |a n 〉 and |b n 〉, to form a hybridised local orbital, and then we combine these hybridised orbitals to form a Bloch state: |ψ〉 = ∑ e ikRn (α k |a n 〉 + β k |b n 〉) (4.30) n Note that the hybridisation coefficients α k and β k can in general depend on the wavevector k. Applying our usual prescription – enter |ψ〉 into the Schrödinger eqn. Ĥ |ψ〉 = E |ψ〉, left multiply by basis states 〈a 0 | and 〈b 0 | and turn the resulting set of simultaneous equations into an eigenvector problem – gives: ( ) ( ) Ea (k) − E V k αk = 0 (4.31) E b (k) − E β k V ∗ k where E a = ∑ n 〈a eikRn 0 | Ĥ |a n〉 is the dispersion of a band, which would be formed exclusively from the atomic orbitals |a n 〉, and E b = ∑ n eikRn 〈b 0 | Ĥ |b n〉 is the dispersion of a band, which would be formed exclusively from the atomic orbitals atomic orbitals |b n 〉. The off-diagonal matrix element V k is given by V k = ∑ R n e ikRn 〈a 0 | Ĥ |b n〉 It can be shown with a bit of relabelling that V ∗ k = ∑ R n e ikRn 〈b 0 | Ĥ |a n〉 The eigenvalue problem (4.31) gives rise to two possible energies at each wavevector – we have formed two bands from the two orbitals. The effect of the off-diagonal elements in (4.31) is to hybridise the two bands which would form from each set of atomic orbitals exclusively where they become nearly degenerate. One could have approached the calculation from another angle by first forming two Bloch states from combining either the local orbitals |a n 〉 (|ψ a 〉 = ∑ n eikRn |a n 〉) or the local orbitals |b n 〉 (|ψ b 〉 = ∑ n eikRn |b n 〉). Then, we combine the Bloch states: |ψ〉 = α k |ψ a 〉 + β k |ψ b 〉 If we substitute in our expressions for |ψ a 〉 and |ψ a 〉, then we obtain exactly the same form for |ψ〉 as before, in (4.30). So whether you consider the bandstructure as arising from hopping between hybridised, molecular orbitals, or attribute it to hybridisation between bands which arise from atomic orbitals makes no difference to the formalism.
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