Time dependent, self-consistent simulations of field emission from silicon using the Wigner distribution function

Abstract

Field emission current calculations from semiconductors typically rely on the assumptions that the emitted current is negligible and that the band bending at the surface may be calculated via Poisson’s equation with a Fermi–Dirac momentum distribution of electrons [the zero emitted current approximation (ZECA)]. This approach cannot take into account complications due to quantum confinement near the surface, and problems associated with scattering for large applied field. We have developed and applied a time-dependent, self-consistently calculated Wigner distribution function (WDF) approach for dealing with the semiconductor field emission problem, and have applied it to the case of silicon for the cases of 300 and 900 K. In particular, effects of self-consistency and scattering on the density and potential profiles were examined, comparing the results to the ZECA approach to show when quantum and scattering effects become significant. At low fields, the WDF approach may be coupled with the transmission coefficient approach to yield reasonable current estimates. At high fields, it is shown that current saturation effects set in, in which the current no longer exponentially rises with applied field. Finally, the time dependence of the current and particle densities after a sudden shift in the applied field are examined.