Current-temperature scaling for a Schottky interface with non-parabolic energy dispersion

In this paper, we study the Schottky transport in narrow-gap semiconductor and few-layer graphene in which the energy dispersions are highly non-parabolic. We propose that the contrasting current-temperature scaling relation of $J\propto T^2$ in the conventional Schottky interface and $J\propto T^3$ in graphene-based Schottky interface can be reconciled under Kane's $\mathbf{k} \cdot \mathbf{p}$ non-parabolic band model for narrow-gap semiconductor. Our new model suggests a more general form of $J\propto \left(T^2 + \gamma k_BT^3 \right)$, where the non-parabolicty parameter, $\gamma$, provides a smooth transition from $T^2$ to $T^3$ scaling. For few-layer graphene, it is found that $N$-layers graphene with $ABC$-stacking follows $J\propto T^{2/N+1}$ while $ABA$-stacking follows a universal form of $J\propto T^3$ regardless of the number of layers. Intriguingly, the Richardson constant extracted from the Arrhenius plot using an incorrect scaling relation disagrees with the actual value by two orders of magnitude, suggesting that correct models must be used in order to extract important properties for many novel Schottky devices.