from thermal rectifiers to thermoelectric...
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From thermal rectifiers to thermoelectric devices
Giuliano BenentiCenter for Nonlinear and Complex Systems,
Univ. Insubria, Como, ItalyINFN, Milano, Italy
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OUTLINE
Can we derive the Fourier law of heat conduction from dynamical equations of motion, without any a priori statistical assumptions? And in quantum mechanics?
Can we control the heat current? Towards thermal diodes and thermal transistors
Coupled heat and charge transport: mechanisms for optimizing the figure of merit of thermoelectric efficiency
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Fourier Heat Conduction Law (1808)“Théorie de la Propagation de la Chaleur dans les Solides”
�T
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1808 - J.J. Fourier: study of the earth thermal gradient
19 century: Clausius, Maxwell, Boltzmann, kinetic theory of gas, Boltzmann transport equation
1914 - P. Debye: κ~Cvl, conjectured the role of nonlinearity to ensure finite transport coefficients
1936 - R. Peierls: reconsidered Debye's conjecture
1953 - E. Fermi, J. Pasta and S.Ulam: (FPU) numerical experiment to verify Debye's conjecture
An old problem, and a long history
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“It seems there is no problem in modern physics for which there are on record as many false starts, and as many theories which overlook some essential feature, as in the problem of the thermal conductivity of (electrically) nonconducting crystals.”
R. E. Peierls (1961), Theoretical Physics in the Twentieth Century.
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QUESTION: Can one derive the Fourier law of heat conduction from dynamical equations of motion without any statistical assumption?
REMARK:(Normal) heat flow obeys a simple diffusion equation which can be regarded as a continuous limit of a discrete random walk Randomness should be an essential ingredient of thermal conductivity
deterministically random systems are tacitly required by the transport theory
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Methods: nonequilibrium simulations
(Maxwellian heat baths)
S. Lepri, et al, Phys. Rep. 377, 1 (2003); A. Dhar, Adv. Phys. 57, 457 (2008)
Heat bath TH Heat bath TL Heat flux J
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Free electron gases at the reservoirs with Maxwellian distribution of velocities
Heat flux J = limt��
1t
�
i
�Ei
Internal temperature Ti = �v2i �
Ding-a-ling model
chaos for �2/E � 1
(m = kB = 1)
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(G. Casati, J. Ford, F. Vivaldi, W.M. Visscher, PRL 52, 1861 (1984))
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Methods: equilibrium simulations
Green-Kubo formula��
normal heat conduction
anomalous heat conduction
S. Lepri, et al, Phys. Rep. 377, 1 (2003); A. Dhar, Adv. Phys. 57, 457 (2008)
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Momentum-conserving systemsSlow decay of correlation functions, diverging transport coefficients (Alder and Wainwright, PRA 1, 18 (1970))
FPU revisited: chaos is not sufficient to obtain Fourier law(Lepri, Livi, Politi, EPL 43, 271 (1998))
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3D
2D
1D
(normal heat conduction)
(anomalous heat conduction)
(anomalous heat conduction)
For momentum-conserving systems
From hydrodynamic theory:
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Fourier law in quantum mechanicsQuantum chaos ensures diffusive heat transport and decay of ``dynamical’’ (energy-energy) correlation functions.
Exponential sensitivity to errors absent in quantum mechanics but not necessary to obtain the Fourier law
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Can we control the heat current?
Towards thermal diodes and thermal transistors
Thermal rectification: everyday’s experience when there is thermal convection (transfer of matter, e.g.: heating a fluid from below or from the top surface)
Thermal recifiers much less intuitive in solid-state devices, but not forbidden by thermodynamics
Let us focus on electrical insulators (heat carried by lattice vibrations: phonons)
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Rectification factor
Ratio between reverse and forward heat flow
Why can these flows be different?
Assuming the Fourier law:
reversing boundary conditions we can change temperature and local thermal conductivity distributions
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Two ingredients are needed:
- Temperature-dependent thermal conductivity- breaking of the inversion symmetry of the device in the direction of the flow
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Microscopic model
Morse on-site potential: Nonlinearity needed to have temperature-dependent phonon bands
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Linearized model (around the equilibrium position):
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phonon-band mismatch:
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Effective (temperature-dependent) harmonic model:
(M. Terraneo, M. Peyrard, G. Casati, PRL 88, 094302 (2002))21
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With continuos variation of the vibrational properties versus space (it amounts to stacking an infinity of interfaces):
(for a pedagogical review, see G. B., G. Casati, C. Mejia-Monasterio, M. Peyrard, in Lecture Notes in Physics, in press)
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A different model: carbon nanotube bundles
Interplay between Kapitza resistance and thermal rectification. R up to 1.2 (work in progress)
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An historical note: Volta and the discovery of thermoelectricity
1794-1795: letters from Volta to Vassali. “I immersed for some half-minute the end of such (iron) arc into boiling water and, without letting it to cool down, returned to experiments with two glasses of cold water. And it was then that the frog in water started contracting...”
(see Anatychuk et al, “On the discovery of thermoelectricity by A.Volta”)
Thermoelectric transport
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Abram Ioffe (1950s): Doped semiconductors have large thermoelectric effect
The initial excitement about semiconductors in the 1950s was due to their promise, not in electronics but in refrigeration. The goal was to build environmental benign solid state home refrigerators and power generators
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Thermoelectric (Peltier) refrigerators have poor efficiency compared to compressor-based refrigerators
Niche applications: space missions, medical applications, laboratory equipments, air conditioning in submarines (reliability and quiet operation more important than cost)
car’s seats cooler/heater Use vehicle waste heat to improve fuel economy
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Mildred Dresselhaus et al. (Adv. Materials, 2007): “a newly emerging field of low-dimensional thermoelectricity, enabled by material nanoscience and nanotechnology... Thermoelectric phenomena are expected to play an increasingly important role in meeting the energy challenge for the future...”
Small scale thermoelectricity could be relevant for cooling directly on chip, by purely electronic means. Nanoscale heat management is crucial to reduce the energy cost in many applications of microelectronics.
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Coupled 1D particle and heat transport
Stochastic baths: ideal gases at fixed temperature
and chemical potential
Onsager relation:
Positivity of entropy production:
X1 = ��µ
X2 = ��� = �T/T 2
� = 1/T
�µ = µL � µR
�� = �L � �R
�T = TL � TR
(we assume TL > TR, µL < µR)
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Onsager and transport coefficients
G =�
J�
�µ/e
�
�T=0
� G =e2
TL��
� =�
Jq
�T
�
J�=0
� � =1
T 2
detLL��
Note that the positivity of entropy production implies that the (isothermal) electric conductance G>0 and the thermal conductance Ξ>0
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Maximum efficiency
Find the maximum of η over X1, for fixed X2 (i.e., over the applied voltage ΔV for fixed temperature difference ΔT)
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Thermoelectric figure of merit
ZT =L2
q�
detL=
GS2
�T
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Efficiency at maximum power
Find the maximum of ω over X1, for fixed X2 (over the applied voltage ΔV for fixed ΔT)
Output power � = �TX1(L��X1 + L�qX2)
X1 = � L�q
2L��X2
Efficiency at maximum power
�(�max) =�C
2ZT
ZT + 2� �CA �
�C
2
ηCA Curzon-Ahlborn upper bound
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�max
�(�max)
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Reducing thermal conductivity
[Blanc, Rajabpour, Volz, Fournier, Bourgeois, APL 103, 043109 (2013)]
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ZT diverges iff the Onsager matrix is ill-conditioned, that is, the condition number:
diverges
In such case the system is singular (tight-coupling limit):
(the ratio Jq/Jρ is independent of the applied voltage and temperature gradients)
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Charge current
Non-interacting systems, Landauer-Büttiker formalism
Heat current from reservoir α
Jq.� =1h
� �
��dE(E � µ�)�(E)[fL(E)� fR(E)]
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If transmission is possible only inside a tiny energy window around E=E✶ then
Thermoelectric efficiency
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Carnot efficiency
[Mahan and Sofo, PNAS 93, 7436 (1996); Humphrey et al., PRL 89, 116801 (2002)]
Energy filtering mechanism
Carnot efficiency obtained in the limit of reversible transport (zero entropy production) and zero output power
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Is energy-filtering necessary to get Carnot efficiency?
No, for interacting systems with momentum conservation
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Short intermezzo: a reason why interactions might be interesting for thermoelectricity
Thermodynamic cycle
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Analogy with a classical gas
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Interacting systems, Green-Kubo formulaThe Green-Kubo formula expresses linear response transport coefficients in terms of dynamic correlation functions of the corresponding current operators, cal- culated at thermodynamic equilibrium
Non-zero generalized Drude weights signature of ballistic transport
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Conservation laws and thermoelectric efficiencySuzuki’s formula (which generalizes Mazur’s inequality) for finite-size Drude weights
Qn relevant (i.e., non-orthogonal to charge and thermal currents), mutually orthogonal conserved quantities
Assuming commutativity of the two limits,
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Momentum-conserving systems
Consider systems with a single relevant constant of motion, notably momentum conservation
Ballistic contribution to det(L) vanishes as
k � detLL��
� ��, � < 1
� � L�� � �
S � L�q
L��� �0
ZT =�S2
�T � �1�� �� when ���
(G.B., G. Casati, J. Wang, PRL 110, 070604 (2013))
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For systems with more than a single relevant constant of motion, for instance for integrable systems, due to the Schwarz inequality
Equality arises only in the exceptional case when the two vectors are parallel; in general
detL � L2, � � �, ZT � �0
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Example: 1D interacting classical gasConsider a one dimensional gas of hard-point elastically colliding particles with unequal masses: m, M
injection rates
ZT depends on the system size
(integrable model)ZT = 1 (at µ = 0)
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Non-decaying correlation functions
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D��(�) =TN2
2�(mN1 + MN2),
Duu(�) =9T 3N2
8�(mN1 + MN2),
D�u(�) =3T 2N2
4�(mN1 + MN2).
m + M
Finite-size Drude weights: analytical results vs. numerics
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Ballistic behavior of Onsager coefficients
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Anomalous thermal transport
ZT =�S2
kT
(G.B., G. Casati, J. Wang, PRL 110, 070604 (2013))
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Energy-filtering mechanism?At a given position x compute:
Density of particles crossing x from left
Density of particles crossing x from right
“transmission function”
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There is no sign of narrowing of D(E) with increasing the system size L
A mechanism for increasing ZT different from energy filtering is needed
(K. Saito, G.B., G. Casati, Chem. Phys. 375, 508 (2010))
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And when time-reversal is broken?
B applied magnetic field or any parameter breaking time-reversibility
such as the Coriolis force, etc.
X1 = ��µ
X2 = ��� = �T/T 2
� = 1/T
�µ = µL � µR
�� = �L � �R
�T = TL � TR
(we assume TL > TR, µL < µR)
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Constraints from thermodynamics
ONSAGER-CASIMIR RELATIONS:
POSITIVITY OF THE ENTROPY PRODUCTION:
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Both maximum efficiency and efficiency at maximum power depend on two parameters
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(G.B., K. Saito, G. Casati, PRL 106, 230602 (2011) )
The CA limit can be overcome within linear response
When |x| is large the figure of merit y required to get Carnot efficiency becomes small
Carnot efficiency could be obtained far from the tight-coupling condition
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OUTPUT POWER AT MAXIMUM EFFICIENCY
When time-reversibility is broken, within linear response is it possible to have simultaneously Carnot
efficiency and non-zero power.Terms of higher order in the entropy production, beyond linear response, will generally be non-zero. However, irrespective how close we are to the Carnot efficiency, we can find small enough forces such that the linear theory holds.
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Reversible part of the currents
J revi � Lij � Lji
2Xj , i = �, q
J irri � LiiXi +
Lij + Lji
2Xj
S = J�X1 + JqX2 = J irr� X1 + J irr
q X2
The reversible part of the currents do not contribute to entropy production
Possibility of dissipationless transport?
(K. Brandner, K. Saito, U. Seifert, PRL 110, 070603 (2013))
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How to obtain asymmetry in the Seebeck coefficient?
For non-interacting systems, due to the symmetry properties of the scattering matrix
This symmetry does not apply when electron-phonon and electron-electron interactions are taken into account
Let us consider the case of partially coherent transport, with ine las t ic processes s imula ted by “conceptual probes” (Buttiker, 1988).
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Non-interacting three-terminal model
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Three-terminal Onsager matrixEquation connecting fluxes and thermodynamic forces:Equation connecting fluxes and thermodynamic forces:
In block-matrix form:
Zero-particle and heat current condition through the probe terminal:
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Two-terminal Onsager matrix for partially coherent transport
Reduction to 2x2 Onsager matrix when the third terminal is a probe terminal mimicking phase-breaking.
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Illustrative three-dot example
Asymmetric structure, e.g..
First-principle exact calculation within the Landauer-Büttiker multi-terminal approach
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Asymmetric Seebeck coefficient
(K. Saito, G. B., G. Casati, T. Prosen, PRB 84, 201306(R) (2011) )(see also D. Sánchez, L. Serra, PRB 84, 201307(R) (2011) )
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Bounds beyond the second law of thermodynamicsfrom the unitarity of S-matrix
Bounds obtained for non-interacting 3-terminal transport (K. Brandner, K. Saito, U. Seifert, PRL 110, 070603 (2013))
�(⇥max) =47
�C at x =43
(Numerics: V. Balachandran, G. B., G. Casati, PRB 87, 165419 (2013));
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Magnetic thermal switch (n-terminal setup)
Set voltages (for fixed thermal affinities) to obtain conditions on the currents from a subset K of the n terminals:
J(B) = J(r)(B) + J(i)(B)
(R. Bosisio, S. Valentini, F. Mazza, G.B., V. Giovannetti, R. Fazio, F. Taddei, PRB 91, 205420 (2015))
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Heat current multiplier:
Heat path selector:
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Fully reversible heat:
Heat current swap:
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Example: interferometer model
Fully reversible heat: Heat path selector:
(R. Bosisio, S. Valentini, F. Mazza, G.B., V. Giovannetti, R. Fazio, F. Taddei, PRB 91, 205420 (2015))
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Switch also applicable to phononic currents
(O. Entin-Wolman, A. Aharony, PRB 85, 085401 (2012))
Due to electron-phonon coupling the thermal current from the bosonic terminal has a reversible component
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Multi-terminal thermoelectricityPossibility to exploit additional terminals to decouple charge and heat flows and improve thermoelectric efficiency
The third terminal is not necessarily a probe (F. Mazza, R. Bosisio, G. B., V. Giovannetti, R. Fazio, F. Taddei, New J. Phys. 16, 085001 (2014))
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Heat-charge separation
(F. Mazza, S. Valentini, R. Bosisio, G.B., R. Fazio, V. Giovannetti, F. Taddei,PRB 91, 245435 (2015))
Improved thermoelectric performances (in the low-temperature Sommerfeld regime)
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Open problems
To combine thermal rectification with thermoelectric power generation or cooling
Investigate strongly-interacting systems close to electronic phase transitions
In nonlinear regimes restrictions due to Onsager reciprocity relations might be overcome
Investigate time-dependent driving
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