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

Potential barrier profiles for large applied fields and/or high temperature are developed for the study of field and thermionic emission electron sour ces intended for radio frequency power tube applications. The numerical imp lementation provides a fast and flexible method to obtain the barriers whic h govern current density, and yet allows for complications such as nanoprot rusions, adsorbates, "internal'' field emission, the sputtering of low work function emission sites, and so on. The model consists of (i) a modified f orm 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 triangula r representation of the barrier with a single adjustable parameter which en ables both the solution of Schrodinger's equation in terms of Airy function s and thus an exact evaluation of the electron density near the barrier, an d (iv) a numerical integration of Poisson's equation to evaluate the dipole potential and positive background boundary. An iterative calculation is pe rformed such that the barrier used in the solution of Schrodinger's equatio n becomes equivalent to the barrier predicted from the exchange-correlation and dipole potentials. As a test of the method, evaluations of the work fu nction of various metals are made. A good correspondence is found between t he potential profiles and an "analytic'' image charge potential (which cont ains modifications to the standard image charge model). Modifications to th e Richardson-Laue-Dushman and Fowler Nordheim equations, so as to obtain cu rrent density estimates, are described. The (only) adjustable parameter use d to correlate theory and experimental work functions is the magnitude of t he ionic core "radius,'' which is often close to the actual radius of the m etal ions in the test cases considered. The temperature and field dependenc e of the work function, which is dependent upon electron penetration of the barrier and its effect on the dipole potential, are investigated. The meth od is suggested to be suitable for the analysis of more complex potential b arrier profiles that are encountered in actual (realistic) thermionic and f ield emission electron sources. The limitations of the model are discussed and methods to circumvent them are proposed. [S0021-8979(99)07005-X].