Advanced Workshop on Energy Transport in Low-Dimensional Systems: Achievements and Mysteries October 2012

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1 Advanced Workshop on Energy Transport in Low-Dimensional Systems: Achievements and Mysteries October 2012 Thermal Conductance of Uniform Quantum Wires K. MATVEEV Argonne National Laboratory Argonne U.S.A.

2 Konstantin Matveev Trieste, October 18, 2012 In collaboration with A. Levchenko, T. Micklitz, J. Rech, Z. Ristivojevic

3 Motivation: Experiments with quantum wires, quantized conductance Theory: Thermal conductance of quantum wires Non-interacting wires, Wiedemann-Franz law; Classification of electron-electron scattering processes; Small corrections to thermal conductance in short wires; Strong suppression of thermal conductance in long wires.

4 From Berggren & Pepper, 2002 As a function of gate voltage conductance shows multiple steps of height

5 Top view Electron motion across the channel Gate voltage changes the number of parallel channels in the wire Each channel has conductance

6 Current: Conductance:

7 Heat current: Thermal conductance: Wiedemann-Franz law: cf.

8 Uniform quantum wire: No disorder Interactions inside the wire No interactions in the leads What is the thermal conductance?

9 Non-trivial two particle collisions are forbidden by the conservation laws Three particle collisions conserve both energy and momentum

10 Collision integral can be linearized in at

11 (a) (b) (c) fast less fast very slow Scattering rates: These scattering processes relax the electron gas toward different equilibrium states

12 Gibbs distribution: In a uniform wire momentum is conserved, and In general, all conserved quantities enter the exponent of the Gibbs distribution

13 (a) Numbers, momenta, and energies of the left- and right-moving particles are conserved separately The distribution supplied by the leads belongs to this class:

14 (b) Numbers of the left- and right-moving particles are conserved separately, in addition to the total momentum and energy The distribution supplied by the leads does not belong to this class:

15 (c) Only the total number of particles, momentum, and energy are conserved

16 In a long wire electron distribution reaches the equilibrium form corresponding to the dominant scattering process (a) (b) Equilibration due to type (a) processes is more efficient: In wires shorter than, type (b) processes can be accounted for perturbatively At the distribution function has the form The 6 parameters depend on position

17 Integro-differential equation: 6 ordinary differential equations: 4 conservation laws Easily solvable!

18 (b) The correction is due to the curvature of electron spectrum near the Fermi level. Even in long wires it remains small as The equilibration length is model-specific. For unscreened Coulomb interactions we found without spins with spins [Levchenko, Micklitz, Ristivojevic & KM, PRB 84, (2011)]

19 (c) Electron distribution: Heat current depends only on the velocity u

20 Consider the energy current of right-moving electrons The total energy current Exclude the outgoing current Substitute the known expression for the incoming currents

21 Momentum of the electron gas The total momentum is conserved, Momentum change: Energy change: Rate of energy change due to backscattering:

22 (c) In a system without net electric current the electron distribution relaxes towards the usual Fermi function Standard relaxation law: Combining with and we find

23 Old result in a short wire,. Small thermal conductance, but finite thermal conductivity in a long wire, [Micklitz, Rech & KM, PRB 81, (2010)]

24 In long wires the electrical conductance saturates Momentum conservation results in infinite conductivity But the thermal conductance is small: Strong violation of the Wiedemann-Franz law: Thermoelectric figure of merit:

25 Electron-electron scattering reduces thermal conductance of quantum wires Small reduction of thermal conductance in short wires In (exponentially) long wires the thermal conductance is strongly suppressed

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