Decoherence of coherent transport in a disordered one-dimensional wire: Phenomenological model

We model the effect of phase-breaking collisions on the coherent electron transport in a disordered one-dimensional single-channel wire. In our model the phase-breaking collisions break the wire into segments, where each segment is an independent series resistor with coherent electronic resistance and the segmentation is a stochastic process with Poisson distribution of phase-breaking scattering times. The wire resistance as a function of the wire length $L$, coherence length $L_{\phi}$, and localisation length $\xi$ is calculated and the transition from coherent to incoherent transport is traced quantitatively. In the coherent regime ($L < L_{\phi}$) the resistance fluctuates from wire to wire with a characteristic log-normal distribution of resistances, the typical resistance increases as $\exp(L/\xi)$, and the mean resistance increases as $\exp(2L/\xi)$ (or faster if disorder is strong). As $L$ exceeds $L_{\phi}$, decoherence suppresses the resistance fluctuations and narrows the resistance distribution. As a result, at $L \gg L_{\phi}$ the mean resistance increases as $\beta L-c$ and the typical resistance as $\beta L - c'$, where $\beta$ is the wire resistivity, $c$ is a constant shift due to the decoherence near the source electrode, and $c' \gg c$ is the shift related to the resistance self-averaging in a single wire. Numerical results are given for a GaAs quantum wire. It is noted that coherent transport in such wire can exhibit peculiar deviations from universal scaling owing to strong backscattering by impurities.