Theory of Field Emission from Semiconductors

The field emission current density $j_c$, originating from the conduction band, is derived for an arbitrary degeneracy (i.e., Fermi energy) at the surface. The theory allows for a difference between the effective and free electron masses; detailed results being worked out for spherical energy surfaces. Simple formulas for $j_c$ are presented which involve correction factors that are slowly, varying functions of the temperature, field $F$, and Fermi energy, and have been computed numerically; $j_c$ is approximately proportional to the emission probability of an electron either at the Fermi level or at the bottom of the conduction band for positive or negative Fermi energies, respectively. Strong deviations from linearity, of a $\ln j_c$ versus ($\frac{1}{F}$) plot, require that the Fermi energy at the surface depend markedly on $F$. The emission current for the intermediate, or $T-F$, range is also considered. The field emission current density $j_v$ originating from the valence band is also discussed. As an example, numerical results are given for germanium. For this case, $j_v$ exceeds $j_c$ at room temperature, except when the surface is strongly degenerate $n$ type. The theory is qualitatively consistent with Allen's experimental results for a clean germanium surface.