Coherent “Metallic” Resistance and Medium Localization in a Disordered One-Dimensional Insulator

It is believed that a disordered one-dimensional (1D) wire with coherent electronic conduction is an insulator with the mean resistance $⟨\rho ⟩\simeq e^{2L/\xi }$ and resistance dispersion ${\Delta }_{\rho }\simeq e^{L/\xi }$, where $L$ is the wire length and $\xi$ is the electron localization length. Here we show that this 1D insulator undergoes at full coherence the crossover to a 1D “metal,” caused by thermal smearing and resonant tunneling. As a result, ${\Delta }_{\rho }$ is smaller than unity and tends to be $L/\xi$ independent, while $⟨\rho ⟩$ grows with $L/\xi$ first nearly linearly and then polynomially, manifesting the so-called medium localization.

Figure 1

Top panel: Mean resistance $⟨\rho ⟩$ and typical resistance ${\rho }_t$ vs normalized length $L/\xi$ for various normalized temperatures $T/T_{\xi }$. The dashed line is the metallic dependence $\rho =L/\xi$. Middle panel: $⟨\rho ⟩$ and ${\rho }_t$ vs $T/T_{\xi }$ for various $L/\xi$. Bottom panel: Dispersion of the resistance (left) and conductance (right) vs $L/\xi$ for various $T/T_{\xi }$. The inset shows the comparison with the estimate $0.5\sqrt{T_{\xi }/T}$ (dashed line) derived in the text.

Figure 2

(a) Transmission probability $\mathcal{T}(\epsilon )$ versus energy $\epsilon$ for one random configuration of disorder and distribution $P(\mathcal{T})$ averaged over many configurations. Energy axis is in units ${\epsilon }_F$, where ${\epsilon }_F=35\mathrm{m}\mathrm{e}\mathrm{V}$. The length parameters are $L/\xi =8$ and $L=80\mu \mathrm{m}$, other parameters are the the same as quoted for Fig. 1. (b) The $\mathcal{T}(\epsilon )$ dependence from (a) in linear scale. (c) Schematic generalization of (a),(b) (see the text).

Figure 3

Mean resistance and typical resistance versus $L/\xi$ for various $T/T_{\xi }$. The dashed line is the metallic resistance $\rho =L/\xi$. Inset to the left panel: The $T=12.5T_{\xi }$ curve is shown in more detail to stress the sublinear concave shape. The open circles show the limit $⟨\rho ⟩=⟨G{⟩}^{-1}-1$, where $⟨G⟩$ is given by Eq. (12) with $P(\mathcal{T})$ generated numerically. Inset to the right panel: The typical resistance simulated up to $L/\xi =200$, but only for $T/T_{\xi }\le 0.125$ owing to huge computational time.

Figure 4

(a) Open circles are the two-terminal resistance data of Ref. 14, measured at $T=4.2\mathrm{K}$ in a series of GaAs quantum wires with a single occupied channel. In these wires $\xi \simeq 1.5\mu \mathrm{m}$ and ${\epsilon }_F\simeq 3.5\mathrm{m}\mathrm{e}\mathrm{V}$, as the quantized ballistic conductance is observed for $L\lesssim 1.5\mu \mathrm{m}$ and the lowest quantized step is centered roughly at the mentioned ${\epsilon }_F$ value [14]. For these parameters $T_{\xi }\simeq 2\mathrm{K}$. The full line shows the simulated mean two-terminal resistance, $⟨1/G⟩$. In our simulation the value $\xi \simeq 1.5\mu \mathrm{m}$ can be realized through many different combinations of parameters $N_I$ and $R_I(k_F)$, but the results remain the same provided the disorder is weak. So we do not need to consider sample-dependent details. (b) Open circles are the experimental data from inset (a) reduced by the contact resistance (the two-terminal resistance at $L\to 0$). To prove the superlinear rise with $L$ clearly, the experimental data are compared with the linear fit $\rho =L/\xi$. The best fit is obtained for $\xi =0.6\mu \mathrm{m}$ (dashed line). However, it overestimates all experimental data for $L\le 2\mu \mathrm{m}$ and the value $\xi =0.6\mu \mathrm{m}$ is in contradiction with the ballistic conductance observed for $L\lesssim 1.5\mu \mathrm{m}$ [14]. For $\xi =1.5\mu \mathrm{m}$ (full line) the fit is indeed good for the ballistic wire lengths, but for $L>2\mu \mathrm{m}$ the experimental data grow superlinearly. Main panel: The calculated mean resistance $⟨\rho ⟩$ (full circles), standard deviation from $⟨\rho ⟩$ (vertical bars), and resistance distribution $P(\rho )$ are shown. Open circles are the same as those in inset (b).