Experimental verification ofreciprocity relations in quantum

thermoelectric transport

J. Matthews, F. Battista, D. Sanchez, P. Samuelsson, H. Linke

PRB 90, 165428 (2014)

Workshop on Quantum Thermoelectrics, Marseille, November 2014

Outline

Onsager symmetry relations, Seebeck and Peltier.Additional symmetries, microreversibility.Symmetry breaking, mechanisms.

Symmetries in thermoelectric transport

Four terminal ballistic anti-dot geometry.Electrical conductance matrix, symmetry properties.Thermoelectric reciprocity relations. Quantitative analysis of symmetry properties.Symmetry suppression at large heating.

Experiment, method and results

Open questions

Origin of asymmetry and symmetry suppression

Thermoelectric transport

- electrical and thermal conductance

Transport coefficients

- Seebeck coefficient or thermopower

- Peltier coefficient

Charge and heat current flow Linear response

Onsagers magnetic field symmetries OnsagerPR ’31

𝑀 −𝐵 = −𝐿(𝐵)𝜃

Multiterminal system

1

Voltage and thermal bias

Butcher, JPCM ’90

2

3

4

Mesoscopic quantum transport

Transport relations Linear response

4x4 sub-matrices

Scatteringsub-matrix

Electrical conductance matrix elements Büttiker, PRL ’86

Transmission coefficient

Microscopic reversibility, Schrödinger equation

In line with Onsagers relations

Benoit et al, PRL ’86

𝛼 ≠ 𝛽

Thermal conductance matrix elements

Thermoelectric transport coefficients

Symmetry relation

For weak energy dependence on scale .

Wiedemann-Franz law

Symmetry relation

Not predicted by Onsager

Butcher, JPCM ’90, Jacquod et al, PRB ’12

Following Onsager

Symmetry breaking

Pure dephasing Voltage probe model, energy conserving

Symmetry relation survives.

Inelastic scattering

No ”quantum symmetry”

Voltage probe Serra , Sanchez PRB ’11, Saito et al, PRB ’11

Energy dependent scattering broken Wiedemann-Franz law

Additionalcondition

Can the symmetry be observed in experiment?

Thermopower symmetry

Thermopower, magnetic field symmetries

No multi-terminal experiment!

Godijn et al, PRL ’99 Two terminal chaotic cavity

Experimental setupFour-terminal ballistic anti-dot geometry. Matthews et al, PRB ’14.

System properties

2DEG in InP/GaInAs Independent heating at

all four terminals. Current bias and voltage

measurements at all terminals

Background temperatureq=240mK

Measurement approach1. Electrical bias

Drive a current 𝐼𝛼 𝑡 = −𝐼𝛽 𝑡 = 𝐼 cos𝜔𝑡, with𝜔

2𝜋= 37𝐻𝑧,

between terminals a and b.

Extract Fourier components of induced voltage ∆𝑉𝛼 𝑡 =

𝑛∆𝑉𝛼(𝑛)cos 𝑛𝜔𝑡 at terminals.

In linear response, only ∆𝑉𝛼(1)

is non-zero. Determine electrical conductance matrix elements 𝐺𝛼𝛽.

2. Thermal bias

Drive a heating current 𝐼𝐻 𝑡 = 𝐼𝐻 cos𝜔𝑡 through the heating wire at

terminal a terminal temperature ∆𝜃𝛼 𝑡 = 𝑛 ∆𝜃𝛼(𝑛)cos 𝑛𝜔𝑡.

Extract Fourier components ∆𝑉𝛼(𝑛)

of induced voltage at terminals.

∆𝑉𝛼(2)

dominates (Joule heating) From thermal voltages and 𝐺𝛼𝛽, determine thermoelectric coeff. 𝐿𝛼𝛽.

Electrical biasCurrent bias and voltage measurements at all terminalsFull conductance matrix

Properties

Open conductor, >

Large degree of symmetry, , at B=0 ≈ T

but not perfect…

Resistance reciprocity relations

Multi-terminal resistance as a function of magnetic field

Büttiker, PRL ’86

Representative traces

Origin of deviations from perfect symmetry is unclear(magnetic impurities?)

Thermal biasAll terminal potentials are left floating no current flow

Terminal g is heated, other terminals are assumed to stay cold

Sweeping magnetic field . We find

= + d with and assumemagnetic field independent .

d ≪

extracted

We can test the predicted symmetry

Magnetic field traces

Pair of L-coefficients (arb. units).

Symmetry predicted

Symmetry not predicted

Symmetries are clearly present but with noticeable deviations

Origin of deviations unclear (meas. problem, inelastic scattering, unjustified model assumptions,…?)

Quantification, degree of symmetry

The degree of symmetry is quantified with the Pearson, or r, coefficient

where the renormalized L-coefficients are defined as ( … is averageover B-field)

−1 ≤ ≤ 1

Set of traces

Symmetry breakdown

Increasing the thermal bias, the symmetries tend to be suppressed

Possible explanations: Non-linear thermal transport regime. Increased inelastic scattering. Unwanted heating of cold terminals.

Sanchez, Lopez, PRL 13, Meair, Jacquod JPCM ’13

Summary

Thermoelectric symmetry properties in mesoscopicconductors.

Experiment on four-terminal ballistic anti-dot.Independent heating of all terminals.Strong support for thermoelectric reciprocity relations.Deviations from perfect symmetry.Symmetry suppression with increased heating voltage.