Variable-range hopping transport in modulation-doped n-channel Si/Si_1-xGe_x quantum well structures

We have measured the temperature- and field-dependent longitudinal conductivity, σ_xx, in the Landau level tails of two modulation-doped Si/Si_1-xGe_x two-dimensional electron gas samples at temperatures below 1 K. The temperature dependence of σ_xx at the minima of the Shubnikov-de Haas oscillations obeyed a relation of the form σ_xx^min (T) ∝ (1/T) exp[-(T₀T) ^1/2 ], in agreement with published models of variable-range hopping between localized states by Ono and by Polyakov and Shklovskii. However, the value and magnetic field dependence of the characteristic temperature, T₀, cannot be explained quantitatively on Ono's model, which is based on Gaussian localization of the electron wavefunction on a scale given by the magnetic length. Polyakov and Shklovskii used exponential wavefunctions to derive an alternative expression for the characteristic temperature, and to model the conductivity in the vicinity of the peaks between adjacent quantum Hall plateaux. Our results have been analysed according to this theory, and show good agreement: the magnetic field dependence of the corresponding characteristic temperature, T₁ (v), obeys the power law relation. T₁ ∝ (Δv)^γ, as expected from theory, while the experimental value of γ, 0.90 ± 0.07, agreed with that determined from a half-width analysis of the σ_xx peaks; however, this value differs from the theoretically predicted figure of approx. 2.3.