# Relativistic space-charge-limited current for massive Dirac fermions

## TL;DR

This paper develops a relativistic SCLC theory for Dirac materials, revealing how current-voltage scaling varies with relativistic effects and providing a practical method to analyze 2D thin-film devices, aligning with experimental data.

## Contribution

It introduces a unified relativistic SCLC model for 1D and 2D Dirac materials, linking scaling laws to relativistic regimes and simplifying analysis without solving complex 2D problems.

## Key findings

- Scaling exponent varies from 2 to 3/2 between non-relativistic and ultra-relativistic limits.
- The 2D thin-film SCLC scaling can be derived from the 1D bulk model using Green's function approach.
- Experimental measurements in MoS$_2$ match the predicted voltage-scaling of 3/2 to 2.

## Abstract

A theory of relativistic space-charge-limited current (SCLC) is formulated to determine the SCLC scaling, $J\propto V^{\alpha}/L^{\beta}$, for a finite bandgap Dirac material of length $L$ biased under a voltage $V$. In a one-dimensional (1D) bulk geometry, our model allows ($\alpha$, $\beta$) to vary from (2,3) for the non-relativistic model in traditional solids to (3/2,2) for the ultra-relativistic model of massless Dirac fermions. For a two-dimensional (2D) thin-film geometry, we obtain $\alpha = \beta$ that varies between 2 and 3/2, respectively, at the non-relativistic and ultra-relativistic limits. We further provide a rigorous proof based on a Green's function approach that for uniform SCLC model described by carrier density-dependent mobility, the scaling relations of the 1D bulk model can be directly mapped into the case of 2D thin film for any contact geometries. Our simplified approach provides a convenient tool to obtain the 2D thin-film SCLC scaling relations without the need of explicitly solving the complicated 2D problems. Finally, this work clarifies the inconsistency in using the traditional SCLC models to explain the experimental measurement of 2D Dirac semiconductor. We conclude that the voltage-scaling $3/2 < \alpha < 2$ is a distinct signature of massive Dirac fermions in Dirac semiconductor and is in agreement with experimental SCLC measurement in MoS$_2$.