An elementary theory of electronic semi-conductors, and some of their possible properties

Abstract

The formal theory of electronic semi-conductors was started by A. H. Wilson. It has recently been carried further by Bronstein, and has, of course, been applied (for example to the theory of rectifying contacts) and discussed in a number of other places. There is, however, I think, still room for a more general but quite elementary discussion of a number of possible models, all of which represent possible varieties of semi-conductors; it is to be understood throughout that only electronic conduction is in question. Granted certain general quantum mechanical theorems, the elementary theory can be made both simple and exact. After presenting such a version of the theory here, it is my purpose to show how the proposed model semi-conductors can account for the positive or negative or zero Hall coefficients which are observed, and also for the large positive or negative thermoelectric power of a semi-conductor-metal thermocouple and for the sign relationships of the two effects. The possibility of the explanation of an abnormal sign for the Hall coefficient is, of course, no new thing; it was first contemplated in the work of Peierls on metals and at his suggestion taken over (but not elaborated) by Bronstein ( loc. cit .) for semi-conductors. But a satisfactory (even elementary) theory requires us to consider the conduction in a solid as a mixed phenomenon due to two almost independent families of electrons. This has not been undertaken by Bronstein or by Peierls and such a theory is given here. It may prove of importance in further study of semi-conductors, beyond the phenomena on which attention is concentrated here. The scope of this paper then is as follows. We give an elementary theory of the effective number of “conduction” electrons in model semi-conductors of various types. We take account both of the few ordinary electrons in otherwise empty levels, and of the few holes or vacancies in otherwise completely full electron levels which function as positive electrons. We work out for semi conductors, conducting partly by electrons and partly by holes, the isothermal Hall coefficient and the thermo-electric power of the thermocouple formed by the semi-conductor and an ideal metal. We thus show in detail how abnormal signs of the Hall effect and the thermo-electric power can be fitted into the theory.