Exchange-correlation, dipole, and image charge potentials for electron sources: Temperature and field variation of the barrier height

Abstract

Potential barrier profiles for large applied fields and/or high temperature are developed for the study of field and thermionic emission electron sources intended for radio frequency power tube applications. The numerical implementation provides a fast and flexible method to obtain the barriers which govern current density, and yet allows for complications such as nanoprotrusions, adsorbates, “internal” field emission, the sputtering of low work function emission sites, and so on. The model consists of (i) a modified form of the Wigner Lattice expansion of the electron ground state energy to evaluate the exchange and correlation potential, (ii) a simplified form of the ionic core potential to correct the “Jellium” model, (iii) a triangular representation of the barrier with a single adjustable parameter which enables both the solution of Schrödinger’s equation in terms of Airy functions and thus an exact evaluation of the electron density near the barrier, and (iv) a numerical integration of Poisson’s equation to evaluate the dipole potential and positive background boundary. An iterative calculation is performed such that the barrier used in the solution of Schrödinger’s equation becomes equivalent to the barrier predicted from the exchange-correlation and dipole potentials. As a test of the method, evaluations of the work function of various metals are made. A good correspondence is found between the potential profiles and an “analytic” image charge potential (which contains modifications to the standard image charge model). Modifications to the Richardson–Laue–Dushman and Fowler Nordheim equations, so as to obtain current density estimates, are described. The (only) adjustable parameter used to correlate theory and experimental work functions is the magnitude of the ionic core “radius,” which is often close to the actual radius of the metal ions in the test cases considered. The temperature and field dependence of the work function, which is dependent upon electron penetration of the barrier and its effect on the dipole potential, are investigated. The method is suggested to be suitable for the analysis of more complex potential barrier profiles that are encountered in actual (realistic) thermionic and field emission electron sources. The limitations of the model are discussed and methods to circumvent them are proposed.