Thermionic emission is the term for current due to the thermal excitation of electrons to conductive energy states. Thermionic emission is the conduction mechanism in vacuum tubes (also known as thermionic valves), cathode ray tubes, and Schottky diodes.
The energy distribution of electrons in a solid is given by the Fermi-Dirac distribution, which is an exponential function of temperature. Thermionic emission results from those electrons that have enough equilibrium energy to cross potential energy barriers. For example, vacuum tubes and cathode ray tubes use heated metal cathodes in which some electrons have enough energy to leave the cathode, after which they are accelerated to an anode by an electric field. This creates convection current--current due to moving charges in air (or in these cases vacuum).
The current from all thermionic emission sources is given by the Richardson equation:
I = CT2e-qΦ/kT,
where C is a constant that depends on the application, T is temperature, Φ is the potential energy barrier that must be surmounted, q is the electron charge, and k is Boltzmann's constant.