Universal Scaling Laws in Schottky Heterostructures Based on Two-Dimensional Materials

We identify a new universality in the carrier transport of two-dimensional (2D) material-based Schottky heterostructures. We show that the reversed saturation current ($\mathcal{J}$) scales universally with temperature ($T$) as $\log (\mathcal{J}/T^{\beta })\propto -1/T$, with $\beta =3/2$ for lateral Schottky heterostructures and $\beta =1$ for vertical Schottky heterostructures, over a wide range of 2D systems including nonrelativistic electron gas, Rashba spintronic systems, single- and few-layer graphene, transition metal dichalcogenides, and thin films of topological solids. Such universalities originate from the strong coupling between the thermionic process and the in-plane carrier dynamics. Our model resolves some of the conflicting results from prior works and is in agreement with recent experiments. The universal scaling laws signal the breakdown of $\beta =2$ scaling in the classic diode equation widely used over the past sixty years. Our findings shall provide a simple analytical scaling for the extraction of the Schottky barrier height in 2D material-based heterostructures, thus paving the way for both a fundamental understanding of nanoscale interface physics and applied device engineering.

Figure 1

(a) Band diagram of a graphene-based Schottky heterostructure showing the thermionic transport over a Schottky barrier (${\mathrm{\Phi }}_{B0}$). (b),(c) Schematic drawing of a graphene-based (b) vertical and (c) lateral 2D/3D Schottky heterostructure.