Electron Emission of Metals in Electric Fields III. The Transition from Thermionic to Cold Emission

The theory of Schottky emission is extended to include in the current those electrons which tunnel through the top of the potential barrier at the metal surface when rather strong electric fields are used. It is found that these tunnelling electrons contribute to the periodic deviations from the Schottky line, increasing the amplitude of the deviations. This increase leads to a better agreement with experiment, especially for large fields. This agreement requires that the Nottingham reflection coefficient be very small for fields greater than ${10}^4$ volts per cm. Expressions are developed for the electron current emitted by a metal for various field intensities ranging from the small fields of thermionic work to the large fields used in cold emission. Results are obtained to indicate the temperature and field dependence of the electron current for all fields and temperatures of interest. Of particular interest is the expression developed for the current in the "transition region," i.e., $T\sim 500-1200$ K and $F\sim {10}^7-{10}^8$ volts per cm. The modifications which must be made in the theory to take account of the polycrystalline nature of the emitting surface have no effect on the periodic deviations from the Schottky line, and very little effect on the fluctuations observed in field photo-currents.