  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$_2$.
