Current-Temperature Scaling for a Schottky Interface with Nonparabolic Energy Dispersion

In this paper, we study the Schottky transport in a narrow-gap semiconductor and few-layer graphene in which the energy dispersions are highly nonparabolic. 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 $\mathbf{k}·\mathbf{p}$ nonparabolic band model for narrow-gap semiconductors. Our model suggests a more general form of $J\propto (T^2+\gamma k_B{T}^3)$, where the nonparabolicty parameter $\gamma$ provides a smooth transition from $T^2$ to $T^3$ scaling. For few-layer graphene, we find that $N$-layer 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 2 orders of magnitude, suggesting that correct models must be used in order to extract important properties for many Schottky devices.

Figure 1

Nonparabolic energy dispersion and model of Schottky transport. (a) Kane’s nonparabolic band, (b) low-energy Dirac cone in graphene, low-energy dispersion of trilayer graphene with (c) $ABA$-stacking and (d) $ABC$-stacking orders, and (e) Schottky transport model in a metal-semiconductor interface. ${\mathrm{\Phi }}_0$ is the work function of the metal, $\chi$ is the electron affinity of the semiconductor, and the Schottky barrier is $\mathrm{\Phi }={\mathrm{\Phi }}_0-\chi$. Inset shows the coordinate system and the geometry of the Schottky interface studied in this work; (f) crystal structure of the graphene layer. Solid (dashed) circle denotes the $A$ ($B$) sublattice; (g) $ABA$-stacking order; (h) $ABC$-stacking order.

Figure 2

Arrhenius plot of the Kane-Schottky model using different scaling relation with $\gamma k_B={10}^{-3}{\mathrm{K}}^{-1}$. Kane-Schotkky, Schottky, and Dirac-Schottky scaling relations are denoted by the solid, dashed, and dotted lines, respectively. The reverse bias is set to $V=-1\mathrm{V}$ to ensure current saturation.

Figure 3

Schottky transport in few-layer graphene with $ABA$- and $ABC$-stacking orders. $J\text{−}V$ characteristics of (a) $ABA$ stacking, (b) $ABC$ stacking at $T=300\mathrm{K}$ and $\eta =1.1$ with for $N=(3,4,5,7,10)$, (c) the $N$ dependence of the normalized Schottky current, and (d) the temperature dependence of ${\overline{J}}^{(N)}/{\overline{J}}_{\text{Dirac}}$. Solid (dashed) lines denote $ABA$ ($ABC$) stacking; (e) Arrhenius plot of tetralayer graphene ($N=4$). The inset shows the $N$ dependence of ${\mathcal{A}}_{\mathrm{fit}}/{\mathcal{A}}_{ABC}^{(N)}$.