Coulomb drag between quantum wires with different electron densities

We study the way the backscattering electron-electron interaction generates Coulomb drag between quantum wires with different densities. At low temperature $T$ the system can undergo a commensurate-incommensurate transition as the potential difference $\mid W\mid$ between the two wires passes a critical value $\Delta$, and this transition is reflected in a marked change in the dependence of drag resistivity on $W$ and $T$. At high temperature a density difference between the wires suppresses Coulomb drag induced by backscattering, and we use the Tomonaga-Luttinger model to study this suppression in detail.

Figure 1

$N_{r,K}$ for interaction parameter $K=1$, 0.75, 0.5, and 0.25.

Figure 2

Drag in arbitrary units as a function of $\omega \propto I$ in units of $\mid Q\mid =u\mid \delta k_F\mid$ for interaction parameters $K=0.75$ and 0.25 at temperatures $T=0.25Q$ (smooth curves) and $T⪡Q$. For large currents $(\mid \omega \mid ⪢\mid Q\mid )$ the drag goes as ${\omega }^{4K-2}$. At low temperatures the drag shows a ${\mid \omega -Q\mid }^{2K-1}\Theta ({\omega }^2-Q^2)$ singularity near the threshold $\mid \omega \mid \sim \mid Q\mid$, which is smeared over a regime where $\mid \mid \omega \mid -\mid Q\mid \mid \sim T$.

Figure 3

Normalized drag resistivity ${\tilde{\rho }}_D$ [cf. Eq. (9)] as function of $Q∕T=u\delta k_F∕T$ for the interaction parameter $K=1$, 0.75, 0.5, and 0.25. respectively.

Figure 4

The inverse drag resistivity of a pair of wires at low temperature, $T⪡\Delta$, as a function of the potential difference $W$ (schematic). For $\mid W\mid$ less than the energy gap $\Delta$ the charges order in a zigzag formation (commensurate phase) and the drag is exponentially large, ${\rho }_D\sim a\exp (\Delta -\mid W\mid )∕T$ (I). For $W$ above the gap, $\mid W\mid -\Delta ⪢T$, the drag resistance decreases as ${(\mid W\mid -\Delta )}^{-1∕2}$ with increasing $W$ (II). At large $W⪢\Delta$ the drag resistance decays linearly with $W$ (III).