Transport in partially equilibrated inhomogeneous quantum wires

We study transport properties of weakly interacting one-dimensional electron systems including on an equal footing thermal equilibration due to three-particle collisions and the effects of large-scale inhomogeneities. We show that equilibration in an inhomogeneous quantum wire is characterized by the competition of interaction processes which reduce the electrons total momentum and such which change the number of right- and left-moving electrons. We find that the combined effect of interactions and inhomogeneities can dramatically increase the resistance of the wire. In addition, we find that the interactions strongly affect the thermoelectric properties of inhomogeneous wires and calculate their thermal conductance, thermopower, and Peltier coefficient.

Figure 1

(Color online) Inhomogeneous quantum wire connected adiabatically to two-dimensional leads and biased by the small voltage ${\mu }_l-{\mu }_r=eV$ and/or temperature difference $T_l-T_r=\Delta T$. We assume that the spatial scale $b$ of the inhomogeneities is large compared to the Fermi wavelength, and thus electrons do not experience backscattering from inhomogeneities. In this case only three-particle equilibration processes may interrupt the flow of right-moving electrons and convert some right movers into left movers. Wavy lines represent heat exchange ${\dot{Q}}^R$ between right movers and left movers.

Figure 2

(Color online) (a) Dominant three-particle collision which changes the number of right-moving electrons. (b) Equilibration mechanism, multistep backscattering of right mover into the left mover. At low temperatures each step $\delta p$ in momentum space is on the order of $\sim T/v_F$. (c) Energy-conserving two-particle scattering process that violates conservation of momentum. This process is possible due to the presence of inhomogeneities.

Figure 3

(Color online) Enhanced equilibration due to three-particle collisions in the wire segment of length ${\ell }_T$ where the inhomogeneity potential is maximal. A larger value of $e^{-\mu \left(x\right)/T}$ near $x\sim x_0$ favors stronger electron backscattering in accordance with Eq. (35). In contrast, momentum-nonconserving two-particle collisions occur throughout the wire.