experimental verification of reciprocity relations in quantum thermoelectric...
TRANSCRIPT
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.