  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.
