Statistical mechanics of band states and impurity states in semiconductors

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Abstract
Thermodynamic analogies have previously been invoked to interpret the temperature- and pressure-dependent electronic levels in a semiconductor as standard Gibbs free energies. In this paper it is shown by means of statistical mechanical arguments using the pressure ensemble that standard Gibbs free energies are the quantities determining equilibrium occupations of electronic levels and hence are the thermodynamic quantities that enter detailed balance relationships. Optical band gaps are identified with the 'nophonon line' gap for a transition in which the occupation numbers for the phonons and remaining electrons are unchanged. An exact expression for no phonon optical transition energies is given. It is shown rigorously that the optical no-phonon-line energy is identical to the corresponding free-energy change for any transition between band states and also for transitions involving localised states provided: (i) the effects of electronic degeneracy changes are excluded from the free energy definition; (ii) the localised state is not associated with a local lattice mode.