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Physical review. B, Condensed matter

Interaction enhanced thermopower in a Luttinger liquid

Ilya V. Krive, Eduard N. Bogachek, Andrew G. Scherbakov, Uzi Landman

Published Feb 12, 2001• 14 citations
Publisher

Abstract

The thermopower ${S}_{L}(T)$ of a finite Luttinger liquid (LL) wire containing an impurity and connected to leads of noninteracting electrons is calculated. It is shown that at low temperatures ${k}_{B}T\ensuremath{\ll}{\ensuremath{\Delta}}_{L}\ensuremath{\simeq}\ensuremath{\Elzxh}{\mathbf{v}}_{F}/gL$ (L is the length of the wire, ${\mathbf{v}}_{F}$ is the Fermi velocity of the electrons, and g is the correlation parameter of the LL), ${S}_{L}(T)$ is described by a Mott-type formula. However, at higher temperatures, that is, when ${\ensuremath{\Delta}}_{L}<~{k}_{B}T\ensuremath{\ll}{\ensuremath{\varepsilon}}_{F},$ the expression for ${S}_{L}(T)$ contains an interaction-dependent factor. For strong interelectron interaction, the enhancement of the thermopower is large and it is much more pronounced for spinless electrons.

DOI
10.1103/physrevb.63.113101

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The thermopower of a Luttinger liquid (LL), originating from the energy dispersion of electrons at the Fermi level and/or from the backscattering of electrons by impurities, is analytically evaluated. It is shown that in both cases the thermopower is described by a corresponding Fermi-liquid formula renormalized by an interaction-dependent factor. For an infinite LL the renormalization coefficients decrease with an increase of the electron-electron interaction. In a realistic situation, when a LL wire is connected to leads of noninteracting electrons, the dispersion-induced thermopower in the limit of strong repulsive interaction is strongly suppressed, ${S}_{W}^{(d)}\ensuremath{\sim}{g}^{2}{S}_{0}^{(d)}\ensuremath{\ll}{S}_{0}^{(d)}$ (here S0(d){S}_{0}^{(d)}S0(d)​ is the corresponding Fermi-liquid value for the thermopower and $g\ensuremath{\ll}1$ is the LL correlation parameter), while the impurity-induced thermopower ${S}_{W}^{(i)}\ensuremath{\sim}{S}_{0}^{(i)}/g$ is enhanced by the interelectron interaction.
We theoretically investigate transport in a spin-incoherent one-dimensional electron system, which may be realized in quantum wires at low-electron density and finite temperature. Both the pure and disordered cases are considered, both in finite wires and in the thermodynamic limit. The effect of Fermi-liquid leads attached to the finite-length system is also addressed. In the infinite system, we find a phase diagram identical to that obtained for a spinless Luttinger liquid, provided we make the identification g=2gcg=2{g}_{c}g=2gc​, where ggg is the interaction parameter in a spinless Luttinger liquid and gc{g}_{c}gc​ is the interaction parameter of the charge sector of a Luttinger-liquid theory for electrons with spin. For a finite-length wire attached to Fermi-liquid leads, the transport depends on the details of the disorder in the wire. A simple picture for the crossover from the spin-incoherent regime to the spin-coherent regime as the temperature is varied is also discussed, as well as some physical implications.
Interacting Tomonaga-Lüttinger liquid with impurity for interaction constant K=1/2K = 1/2K=1/2: Thermopower investigation, entropy variation and heat capacity densities associated with thermoelectric particle transport
We investigate thermoelectric and thermodynamic properties of a Tomonaga-Luttinger liquid with interaction parameter K=1/2K=1/2K=1/2 in the presence of a localized impurity. Using bosonization and refermionization, we derive an exact expression for the thermopower in the nonlinear regime and obtain the corresponding Seebeck coefficient in linear response. At low temperature, the Seebeck coefficient follows a Mott-like relation governed by the energy dependence of the transmission coefficient and is found to be closely related to the entropy per particle. This connection allows us to derive using Kelvin formula the thermopower contribution to the entropy density variation and the charge-carriers contribution to the heat capacity density. Both quantities exhibit Mott-like behavior in the low-temperature regime. We apply our results to a fractional quantum Hall quantum point contact and to a one-channel quantum conductor coupled to an Ohmic environment. Our results demonstrate the close connection between thermoelectric transport and thermodynamic properties in interacting one-dimensional quantum systems.
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The inhomogeneous Tomonaga-Luttinger liquid model describing an interacting quantum wire adiabatically coupled to noninteracting leads is analyzed in the presence of a weak impurity within the wire. Due to strong electronic correlations in the wire, the effects of impurity backscattering, finite bias, finite temperature, and finite length lead to characteristic nonmonotonic parameter dependencies of the average current. We discuss oscillations of the nonlinear current voltage characteristics that arise due to reflections of plasmon modes at the impurity and quasi-Andreev reflections at the contacts, and show how these oscillations are washed out by decoherence at finite temperature. Furthermore, the finite-frequency current noise is investigated in detail. We find that the effective charge extracted in the shot noise regime in the weak backscattering limit decisively depends on the noise frequency $\ensuremath{\omega}$ relative to vF∕gL{v}_{F}∕gLvF​∕gL, where vF{v}_{F}vF​ is the Fermi velocity, ggg the Tomonaga-Luttinger interaction parameter, and LLL the length of the wire. The interplay of finite bias, finite temperature, and finite length yields rich structure in the noise spectrum which crucially depends on the electron-electron interaction. In particular, the excess noise, defined as the change of the noise due to the applied voltage, can become negative and is nonvanishing even for noise frequencies larger than the applied voltage, which are signatures of correlation effects.
Thermal transport through one-dimensional systems of strongly interacting electrons is studied in two cases: (i) a finite Luttinger liquid (LL) wire containing a strong impurity, and (ii) a LL ring connected by quantum wires to noninteracting electron reservoirs. It is shown that, for (i), a temperature-induced resonant transport takes place under certain conditions. In case (ii) the heat conductance under resonance conditions does not depend on the electron-electron interaction and the system thermal conductance coincides with the maximal Fermi-liquid thermal value.