Distribution function for impurity states in semiconductors.

For a monovalent donor impurity in a semiconductor, the number of electrons that can be bound to an impurity site is either zero or one. The one bound electron can have either direction of spin. For the discussion of the occupancy of such bound states, one does not apply the usual Fermi-Dirac statistics. A new derivation of the electron distribution function is presented in terms of creation and annihilation operators and the appropriate projection operators for the case of no interaction with the phonons. With the use of double-time temperature-dependent Green's Functions, the electron and phonon distribution functions are derived when there is interaction between the bound electron and phonons. Under certain circumstances, one can speak of a quasi-particle spectrum and the distribution functions have the same form as the interaction-free case but with renormalized energies, which are temperature dependent. The temperature dependence of the distribution functions is then two-fold; one, the usual, statistical, dependence, the other due to the temperature dependence of the energies themselves. The latter quantity requires detailed knowledge of the wave-function, the interaction potential, and energy spectrum of the donor impurity. An application is made to phosphorus donors in silicon. The shifts in the energy levels are found to be small. The agreement with experiment is not completely satisfactory.Science, Faculty ofPhysics and Astronomy, Department ofGraduat