A quantum theory of electron emission from a metal–dielectric structure in high electric fields

A generalized formula for electron emission current as a function of temperature, field, and electron work function in a metal–dielectric system has been derived with regard to the quantum nature of image forces. For free electrons, the Fermi–Dirac distribution and a quantum image potential obtained in terms of the electron polaron theory have been used. The well-known Richardson–Schottky formula (for thermionic emission) and Fowler–Nordheim formula (for field emission) have been derived in the limit of the image force classical potential. It has been shown that at high temperatures and electric fields $E\ge$ 10 MV/cm, the polaron contribution grows with field and declines with rising temperature. The decrease in emission current is associated with the increase in effective work function of electrons, which is due to the electron polaron effect. Extrapolation formulas suitable for theoretical estimation of thermionic and cold emission currents have been obtained.