Two-dimensional hopping conductivity in a $\delta$-doped $\mathrm{GaAs}/{\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ heterostructure

We present zero magnetic-field resistivity measurements of the variable-range hopping (VRH) in a two-dimensional electron gas created at a δ-doped $\mathrm{GaAs}/{\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ heterostructure. When either the temperature or carrier density are reduced the longitudinal resistivity shows a crossover from two-dimensional Mott VRH, $\rho \mathrm{}\left(T\right)={\rho }_M\exp {(T}_M{/T)}^{1/3}$ to Efros-Shklovskii (ES) Coulomb-gap behavior $\rho \mathrm{}\left(T\right)={\rho }_{\mathrm{ES}}\exp {(T}_{\mathrm{ES}}{/T)}^{1/2}.$ In both regimes, near the crossover, the data collapse onto universal curves with a temperature independent prefactor ${\rho }_M\approx {\mathrm{}(1/2)(h/e}^2)$ or ${\rho }_{\mathrm{ES}}\approx {h/e}^2.$ The latter result is in close agreement with measurements of Si metal-oxide-semiconductor field-effect transistors. We discuss the possible implications of our data on the theory of phonon-assisted VRH.