Thermal conductance and energy relaxation rate in quantum wires due to three-particle collisions

We study the problem of energy relaxation in a one-dimensional electron system. The leading thermalization mechanism is due to three-particle collisions. We show that for the case of spinless electrons in a single channel quantum wire the corresponding collision integral can be transformed into an exactly solvable problem. The latter is known as the Schrödinger equation for a quantum particle moving in a Pöschl-Teller potential. The spectrum for the resulting eigenvalue problem allows for bound-state solutions, which can be identified with the zero modes of the collision integral, and a continuum of propagating modes, which are separated by a gap from the bound states. The inverse gap gives the time scale at which counterpropagating electrons thermalize.We study the problem of energy relaxation in a one-dimensional electron system. The leading thermalization mechanism is due to three-particle collisions. We show that for the case of spinless electrons in a single channel quantum wire the corresponding collision integral can be transformed into an exactly solvable problem. The latter is known as the Schrödinger equation for a quantum particle moving in a Pöschl-Teller potential. The spectrum for the resulting eigenvalue problem allows for bound-state solutions, which can be identified with the zero modes of the collision integral, and a continuum of propagating modes, which are separated by a gap from the bound states. The inverse gap gives the time scale at which counterpropagating electrons thermalize.

arXiv:1005.0154Physical Review LettersMay 11, 201133 citations

Three-particle collisions in quantum wires: Corrections to thermopower and conductance

Anders Mathias Lunde, Karsten Flensberg, L. I. Glazman

We consider the effect of electron-electron interaction on the electron transport through a finite length single-mode quantum wire with reflectionless contacts. The two-particle scattering events cannot alter the electric current and therefore we study the effect of three-particle collisions. Within the Boltzmann equation framework, we calculate corrections to the thermopower and conductance to the leading order in the interaction and in the length of wire $L$ . We check explicitly that the three-particle collision rate is identically zero in the case of several integrable interaction potentials. In the general (nonintegrable) case, we find a positive contribution to the thermopower to leading order in $L$ . The processes giving rise to the correction involve electron states deep in the Fermi sea. Therefore, the correction follows an activation law with the characteristic energy of the order of the Fermi energy for the electrons in the wire.We consider the effect of electron-electron interaction on the electron transport through a finite length single-mode quantum wire with reflectionless contacts. The two-particle scattering events cannot alter the electric current and therefore we study the effect of three-particle collisions. Within the Boltzmann equation framework, we calculate corrections to the thermopower and conductance to the leading order in the interaction and in the length of wire $L$. We check explicitly that the three-particle collision rate is identically zero in the case of several integrable interaction potentials. In the general (nonintegrable) case, we find a positive contribution to the thermopower to leading order in $L$. The processes giving rise to the correction involve electron states deep in the Fermi sea. Therefore, the correction follows an activation law with the characteristic energy of the order of the Fermi energy for the electrons in the wire.

arXiv:cond-mat/0703042Physical Review BJun 15, 200774 citations

Energy Relaxation and Thermalization of Hot Electrons in Quantum Wires

Torsten Karzig, L. I. Glazman, Felix von Oppen

We develop a theory of energy relaxation and thermalization of hot carriers in clean quantum wires. Our theory is based on a controlled perturbative approach for large excitation energies and emphasizes the important roles of the electron spin and finite temperature. Unlike in higher dimensions, relaxation in one-dimensional electron liquids requires three-body collisions and is much faster for particles than holes which relax at nonzero temperatures only. Moreover, comoving carriers thermalize more rapidly than counterpropagating carriers. Our results are quantitatively consistent with a recent experiment.We develop a theory of energy relaxation and thermalization of hot carriers in clean quantum wires. Our theory is based on a controlled perturbative approach for large excitation energies and emphasizes the important roles of the electron spin and finite temperature. Unlike in higher dimensions, relaxation in one-dimensional electron liquids requires three-body collisions and is much faster for particles than holes which relax at nonzero temperatures only. Moreover, comoving carriers thermalize more rapidly than counterpropagating carriers. Our results are quantitatively consistent with a recent experiment.

arXiv:1007.1152Physical Review LettersNov 24, 201058 citations

Interaction-induced corrections to conductance and thermopower in quantum wires

Alex Levchenko, Zoran Ristivojević, Tobias Micklitz

We study transport properties of weakly interacting spinless electrons in one-dimensional single-channel quantum wires. The effects of interaction manifest as three-particle collisions due to the severe constraints imposed by the conservation laws on the two-body processes. We focus on short wires where the effects of equilibration on the distribution function can be neglected and the collision integral can be treated in perturbation theory. We find that interaction-induced corrections to conductance and thermopower rely on the scattering processes that change the number of right- and left-moving electrons. The latter requires transition at the bottom of the band which is exponentially suppressed at low temperatures. Our theory is based on the scattering approach that is beyond the Luttinger-liquid limit. We emphasize the crucial role of the exchange terms in the three-particle scattering amplitude that was not discussed in previous studies.We study transport properties of weakly interacting spinless electrons in one-dimensional single-channel quantum wires. The effects of interaction manifest as three-particle collisions due to the severe constraints imposed by the conservation laws on the two-body processes. We focus on short wires where the effects of equilibration on the distribution function can be neglected and the collision integral can be treated in perturbation theory. We find that interaction-induced corrections to conductance and thermopower rely on the scattering processes that change the number of right- and left-moving electrons. The latter requires transition at the bottom of the band which is exponentially suppressed at low temperatures. Our theory is based on the scattering approach that is beyond the Luttinger-liquid limit. We emphasize the crucial role of the exchange terms in the three-particle scattering amplitude that was not discussed in previous studies.

arXiv:1012.5539Physical Review BJan 19, 201116 citations

Coulomb blockade of tunneling into a quasi-one-dimensional wire

К. А. Матвеев, L. I. Glazman

Tunneling into the correlated electron system in a quantum wire is considered. For the first time, the problem is solved for an arbitrary number of transverse electron modes in the wire. The tunnel density of states has power-law asymptotics at low energies, \ensuremath{\nu}(\ensuremath{\epsilon})\ensuremath{\propto}${\mathrm{\ensuremath{\epsilon}}}^{\mathrm{\ensuremath{\alpha}}}$. A sample formula expressing \ensuremath{\alpha} in terms of velocities of soundlike excitations (plasmons) propagating along the wire is obtained and the eigenvalue equation for these velocities is found. The general result for \ensuremath{\alpha} allows one to extrapolate correctly between the known cases of a one-mode wire and a wide three-dimensional wire.Tunneling into the correlated electron system in a quantum wire is considered. For the first time, the problem is solved for an arbitrary number of transverse electron modes in the wire. The tunnel density of states has power-law asymptotics at low energies, \ensuremath{\nu}(\ensuremath{\epsilon})\ensuremath{\propto}${\mathrm{\ensuremath{\epsilon}}}^{\mathrm{\ensuremath{\alpha}}}$. A sample formula expressing \ensuremath{\alpha} in terms of velocities of soundlike excitations (plasmons) propagating along the wire is obtained and the eigenvalue equation for these velocities is found. The general result for \ensuremath{\alpha} allows one to extrapolate correctly between the known cases of a one-mode wire and a wide three-dimensional wire.

Physical Review LettersFeb 15, 1993170 citations

Infrared catastrophe in a two-quasiparticle collision integral

Olga V. Dimitrova, V. E. Kravtsov

Relaxation of a non-equilibrium state in a disordered metal with a spin-dependent electron energy distribution is considered. The collision integral due to the electron-electron interaction is computed within the approximation of a two-quasiparticle scattering. We show that the spin-flip scattering processes with a small energy transfer may lead to the divergence of the collision integral for a quasi one-dimensional wire. This divergence is present only for a spin-dependent electron energy distribution which corresponds to the total electron spin magnetization M=0 and only for non-zero interaction in the triplet channel. In this case a non-perturbative treatment of the electron-electron interaction is needed to provide an effective infrared cut-off.Relaxation of a non-equilibrium state in a disordered metal with a spin-dependent electron energy distribution is considered. The collision integral due to the electron-electron interaction is computed within the approximation of a two-quasiparticle scattering. We show that the spin-flip scattering processes with a small energy transfer may lead to the divergence of the collision integral for a quasi one-dimensional wire. This divergence is present only for a spin-dependent electron energy distribution which corresponds to the total electron spin magnetization M=0 and only for non-zero interaction in the triplet channel. In this case a non-perturbative treatment of the electron-electron interaction is needed to provide an effective infrared cut-off.

arXiv:0710.3222Journal of Experimental and Theoretical Physics LettersFeb 1, 20084 citations

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. We check explicitly that the three-particle collision rate is identically zero in the case of several integrable interaction potentials. In the general (nonintegrable) case, we find a positive contribution to the thermopower to leading order in

L

. The processes giving rise to the correction involve electron states deep in the Fermi sea. Therefore, the correction follows an activation law with the characteristic energy of the order of the Fermi energy for the electrons in the wire.