Linear temperature dependence of conductivity in Si two-dimensional electrons near the apparent metal-to-insulator transition

In a high mobility two-dimensional electron system in Si, near the critical density, $n_c=0.32\times10^{11}$cm$^{-2}$, of the apparent metal-to-insulator transition, the conductivity displays a linear temperature ($T$) dependence around the Fermi temperature. When $\sigma_0$, the extrapolated T=0 conductivity from the linear T-dependence, is plotted as a function of density, two regimes with different $\sigma_0(n)$ relations are seen, suggestive of two different phases. Interestingly, a sharp transition between these two regimes coincides with $n_c$, and $\sigma_0$ of the transition is $\sim$ $e^2/h$, the quantum conductance, per square. Toward T=0, the data deviate from linear $\sigma(T)$ relation and we discuss the possible percolation type of transition in our Si sample.