Examination of Two-Band $E(k)$ Relations for Band-to-Band Tunneling

Four different two-band $E(k)$ relations for band-to-band tunneling in the literature are examined. Three of them are continuous functions, from Franz, Kane, and Flietner, respectively. One is piecewise, consisting of two linear lines from the band edges in a $k^{2}$ versus $E$ plot. It is shown that while all three continuous models converge to the same function when the effective masses of the two bands are equal, they give different $k^{2}$ or mass-barrier products when the masses are unequal. In particular, Flietner’s model can result in a mass-barrier even higher than the piecewise linear model when the masses are highly mismatched. In addition to the four published models, a fifth, generalized power- $n~E(k)$ model is introduced, which covers both Franz’s model and the piecewise linear model as two special cases.