Thermionic Emission, Field Emission, and the Transition Region

Although the theories of thermionic and field emission of electrons from metals are very well understood, the two types of emission have usually been studied separately by first specifying the range of temperature and field and then constructing the appropriate expression for the current. In this paper the emission is treated from a unified point of view in order to establish the ranges of temperature and field for the two types of emission and to investigate the current in the region intermediate between thermionic and field emission. A general expression for the emitted current as a function of field, temperature, and work function is set up in the form of a definite integral. Each type of emission is then associated with a technique for approximating the integral and with a characteristic dependence on the three parameters. An approximation for low fields and high temperatures leads to an extension of the Richardson-Schottky formula for thermionic emission. The values of temperature and field for which it applies are established by considering the validity of the approximation. An analogous treatment of the integral, for high fields and low temperatures, gives an extension of the Fowler-Nordheim formula for field emission, and establishes the region of temperature and field in which it applies. Also another approximate method for evaluating the integral is given which leads to a new type of dependence of the emitted current on temperature and field and which applies in a narrow region of temperature and field intermediate between the field and thermionic emission regions.