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Journal of Statistical Mechanics Theory and Experiment

Coulomb drag and tunnelling between quantum wires

P. Schlottmann

Published Dec 4, 2006• 2 citations
Publisher

Abstract

Charge carriers moving in a quantum wire may induce, via the Coulomb interaction and momentum conservation, a drag current in another quantum wire located nearby. In addition, tunnelling of carriers between the two parallel wires is considered. For spinless carriers the problem can be mapped onto the one-dimensional Hubbard model. The exact solution of the Hubbard chain and conformal field theory are used to calculate the critical exponents for the temperature dependence of the drag current between the two wires. The exponents are non-universal and depend on the interaction strength, the band filling and the tunnelling amplitude. For a repulsive interaction the drag current is opposite to the driving current, while for an attractive interaction they are parallel.

DOI
10.1088/1742-5468/2006/12/p12003

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Charge carriers moving in a quantum wire may induce, via the Coulomb interaction and momentum conservation, a drag current in another quantum wire located nearby. In addition, tunnelling of carriers between the two parallel wires is considered. For spinless carriers the problem can be mapped onto the one-dimensional Hubbard model. The exact solution of the Hubbard chain and conformal field theory are used to calculate the critical exponents for the temperature dependence of the drag current between the two wires. The exponents are non-universal and depend on the interaction strength, the band filling and the tunnelling amplitude. For a repulsive interaction the drag current is opposite to the driving current, while for an attractive interaction they are parallel.
The exact solution of the one-dimensional Hubbard model, the supersymmetric $t\ensuremath{-}J$ model, and the gas of fermions interacting via a $\ensuremath{\delta}$-function potential is used to calculate the drag current between two parallel quantum wires. The critical exponent for the drag current at low temperatures is obtained by means of the mesoscopic energy spectrum and conformal field theory. For a repulsive interaction between the carriers the drag current is opposite to the driving current, while for attractive potentials the two currents are parallel.
We calculate the critical exponent of the temperature dependence of the drag current between two parallel quantum wires. The two wires have carriers with equal mass but they may have different electron density. The electrons interact via a $\ensuremath{\delta}$-function spin-exchange potential, which is attractive or repulsive depending on whether the interacting particles are in a spin-singlet or spin-triplet state. This interaction leads to the formation of spin-singlet or spin-triplet bound states of the Cooper type (preformed hard core bosons that do not condensate). Depending on the parameters, the drag current can be parallel or opposite to the driving current. The critical exponent of the drag current is calculated using the Bethe Ansatz solution of the model and conformal field theory.
The exact solution of the one-dimensional Calogero–Sutherland models with spin-exchange and sinh− 2(r) and r − 2 interactions, and the supersymmetric t–J-model with long-range exchange interaction is used to calculate the drag current between two parallel quantum wires. The critical exponent for the drag current at low temperatures is obtained by means of the mesoscopic energy spectrum and conformal field theory. Since the interaction in the above models is repulsive, the drag current is opposite to the driving current.
We observed negative Coulomb drag for parallel coupled quantum wires, in which electrons flow in the opposite directions between the wires. This only occurred under the conditions of strong correlation in the wires, that is, low density, high magnetic field, and low temperature, and cannot be addressed by a standard theory of momentum transfer. We propose a Coulomb drag model in which formation of a Wigner crystal state in the drag wire and a particle-like state in the drive wire is taken into account.
The spin Coulomb drag is the decay of the spin current in a metal as a consequence of the Coulomb interaction between up- and down-spin carriers. This interaction reduces the spin current but does not change the spin-polarization. We calculate the critical exponents of the resistivity for up- and down-spin electrons and the transresistivity for the spin-polarized Hubbard chain with nonmagnetic impurities within the Kubo formalism using the Bethe ansatz solution and conformal invariance. Due to the Luttinger liquid properties the temperature dependence of the transport correlation functions follow power laws of TTT with nonuniversal exponents.
We study Coulomb drag in a pair of parallel one-dimensional electron systems within the framework of the Tomonaga-Luttinger model. We find that Coulomb coupling has a much stronger effect on one-dimensional wires than on two-dimensional layers: At zero temperature the transresistivity diverges, due to the formation of locked charge density waves. At temperature well above a crossover temperature T∗{T}^{*}T∗ the transresistivity follows a power law $\ensuremath{\rho}\ensuremath{\propto}{T}^{x},$ where the interaction-strength dependent exponent x is determined by the Luttinger liquid parameter ${K}_{c\ensuremath{-}}$ of the relative charge mode. At temperature below T∗{T}^{*}T∗ relative charge displacements are enabled by solitonic excitations, reflected by an exponential temperature dependence. The crossover temperature T∗{T}^{*}T∗ depends sensitively on the wire width, interwire distance, Fermi wavelength and the effective Bohr radius. For wire distances $\overline{d}\ensuremath{\gg}{k}_{F}^{\ensuremath{-}1}$ it is exponentially suppressed with ${T}^{*}{/E}_{F}\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}\overline{d}{k}_{F}/(1\ensuremath{-}{K}_{c\ensuremath{-}})].$ The behavior changes drastically if each of the two wires develop spin gaps. In this case we find that the transresistivity vanishes at zero temperature. We discuss our results in view of possible experimental realizations in $\mathrm{GaAs}\ensuremath{-}{\mathrm{Al}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{As}$ semiconductor structures.