By Conway B.E.(ed.), et al.
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That the electronic density profile is broader than the positive-charge (ionic) background profile. This, of course, is the origin of the surface dipole potential, which is an important part of the work function. 58 This yields, in addition to an electron density profile, the energy of the ion-electron system. The energy as a function of ionic configuration, say, £(R), becomes the potential energy which determines the arrangement of the ions. In principle, each ionic configuration is weighted by a Boltzmann factor, exp[—E(R)/kT].
The difference in energies of the two functions is the band gap. 41 The result is that the energy as a function of k follows the parabola E = h2k2/2m except near k = rrnr/d, where there are distortions such that the energy for k just above rmr/d differs by a finite amount (the band gap) from the energy for k just below mir/d. The origin of allowed bands and band gaps may also be understood in terms of the electronic energy levels of neutral atoms which are allowed to come together so that their charge distributions overlap (tight-binding approximation).
The value of zw will be determined to make the system electrically neutral overall. The wave functions for electrons at an infinite barrier are products of imaginary exponentials (free-particle wave functions) in the x- and y-directions, and sine functions vanishing at zw in the z-direction. The electron density «_(z) is obtained by adding together the squares of the electronic wave functions, starting with zero energy and 24 Jerry Goodisman including enough to give the density of Eq. (11) for z -» -oo (bulk).