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In this paper, we study the Schottky transport in narrow-gap semiconductorand few-layer graphene in which the energy dispersions are highlynon-parabolic. We propose that the contrasting current-temperature scalingrelation of $J\propto T^2$ in the conventional Schottky interface and $J\proptoT^3$ in graphene-based Schottky interface can be reconciled under Kane's$\mathbf{k} \cdot \mathbf{p}$ non-parabolic band model for narrow-gapsemiconductor. 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. Forfew-layer graphene, it is found that $N$-layers graphene with $ABC$-stackingfollows $J\propto T^{2/N+1}$ while $ABA$-stacking follows a universal form of$J\propto T^3$ regardless of the number of layers. Intriguingly, the Richardsonconstant extracted from the Arrhenius plot using an incorrect scaling relationdisagrees with the actual value by two orders of magnitude, suggesting thatcorrect models must be used in order to extract important properties for manynovel Schottky devices.

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