Theory of Semiconductor-To-Metal Transitions

Abstract

This paper discusses a general model for a semiconductor-to-metal transition, in which the energy gap between the valence and conduction bands decreases linearly with the number of electrons excited across the gap. It is shown that this model results in a rapid disappearance of the forbidden gap with rising temperature according to either a first-order or a second-order phase transition, depending on the magnitude of the relative change in gap with the number of excited carriers. Two possible physical models are treated in detail. In one the energy gap results from the splitting of the first Brillouin zone by an antiferromagnetic exchange interaction, and in the other it results from a crystalline-structure distortion to lower symmetry. The latter model is considered in detail in terms of the pairing of ions in a one-dimensional crystal. With these models, using plausible values of the parameters, the explicit relationship between energy gap and freecarrier concentration is estimated. The thermodynamic theory is worked out for the limiting cases of band width large and small compared to the zero-temperature gap. In the narrow-band limit it is found that the parameters of the model are such as to give a second-order transition for the antiferromagnetic case and a first-order transition in the crystalline-distortion model. Using these models, the transition temperature can be evaluated explicitly in terms of the zero-temperature gap. A number of results relating experimentally measurable quantities such as the pressure coefficient of the transition temperature and the energy gap can be derived.