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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
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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
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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
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Multiterminal system
1
Voltage and thermal bias
Butcher, JPCM ’90
2
3
4
Mesoscopic quantum transport
Transport relations Linear response
4x4 sub-matrices
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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
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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
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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?
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Thermopower symmetry
Thermopower, magnetic field symmetries
No multi-terminal experiment!
Godijn et al, PRL ’99 Two terminal chaotic cavity
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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
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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.
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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…
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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?)
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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
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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,…?)
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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
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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
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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.