Relativistic space-charge-limited current for massive Dirac fermions

A theory of relativistic space-charge-limited current (SCLC) is formulated to determine the SCLC scaling, J alpha V-alpha / L-beta, for a finite band-gap Dirac material of length L biased under a voltage V. In one-dimensional (1D) bulk geometry, our model allows (alpha, beta) to vary from (2,3) for the nonrelativistic model in traditional solids to (3/2,2) for the ultrarelativistic model of massless Dirac fermions. For 2D thin-film geometry we obtain alpha = beta, which varies between 2 and 3/2, respectively, at the nonrelativistic and ultrarelativistic limits. We further provide rigorous proof based on a Green's-function approach that for a 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 a 2D Dirac semiconductor. We conclude that the voltage scaling 3/2 < alpha < 2 is a distinct signature of massive Dirac fermions in a Dirac semiconductor and is in agreement with experimental SCLC measurements in MoS2.