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Power-efficiency trade-off in thermoelectricity:
From scattering theory to interacting systems
Giuliano BenentiCenter for Nonlinear and Complex Systems,
Univ. Insubria, Como, Italy INFN, Milano, Italy
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General motivationCarnot efficiency can be obtained only for infinitely slow heat engines, so the extracted power vanishesWhat is the maximum allowed efficiency at a given power output?
For steady-state (thermoelectric) quantum systems modeled by the Landauer-Büttiker scattering theory a (rather restrictive) upper bound exists
[Whitney, PRL 112, 130601 (2014); PRB 91, 115425 (2015)]
Is it possible to overcome this bound for interacting systems, thus allowing a better power-efficiency trade-off?
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Finite time thermodynamics
In an ideal Carnot engine conversion processes are quasi-static and the extracted power reduces to zero.
How much the efficiency deteriorates when heat to work conversion takes place in a finite time?
Finite time thermodynamics: finite-time steady-state conversion processes or thermodynamic cycles; the efficiency at the maximum output power is an important concept
[Andresen, Angew. Chem. Int. Ed. 50, 2690 (2011)]
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Cyclic thermal machines
The upper bound to efficiency is given by the Carnot efficiency:
Carnot efficiency obtained for quasi-static transformation (zero extracted power)
The ideal Carnot engine is a reversible machine, since there is no dissipation (no entropy production)
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Finite-time thermodynamics I: endoreversible cyclic engines
Dissipation is due to finite thermal conductances between heat reservoirs and the ideal heat engine
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Output power:
Optimize power with respect to
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The efficient at maximum power (Curzon-Ahlborn efficiency) is independent of the heat conductances:
Within linear response:
[Yvon, 1955; Chambadal, 1957; Novikov, 1958; Curzon and Ahlborn, Am. J. Phys. 43, 22 (1975)]
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Finite-time thermodynamics II: exoreversible cyclic engines
Irreversibility only arises due to internal dissipative processes
Time-dependent probability density
Stochastic thermodynamics
Time-dependent trapping potential
[Seifert, Rep. Prog. Phys. 75, 126001 (2012)]
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Fokker-Planck equation:
is the mobility Gaussian distribution
Exactly solvable model
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Schmiedl-Seifert efficiency at maximum power:
related to the ratio of entropy production during the hot and cold isothermal steps of the cycle
for the symmetric case
[Schmiedl and Seifert, EPL 81, 20003 (2008)]
Within linear response:
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Low-dissipation enginesThe entropy production vanishes in the limit of infinite-time cycles:
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The CA limit is recovered for symmetric dissipation:
dots: efficiencies of various thermal power plants
[Esposito, Kawai, Lindenberg, Van den Broeck, PRL 105,
150603 (2010)]
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Steady-state (thermoelectric) power production
The upper bound to efficiency is given by the Carnot efficiency:
SLeft (L) reservoir
Right (R) reservoir
T ,L L T ,R R
P = [(µR � µL)/e]Je<latexit sha1_base64="AVVkidoGiZKGjobmFXl/ZhZ6+wk=">AAAB/nicbVDLSsNAFJ3UV62vqODGTbAIdWFNRFAXQtGNiIsqxhbSECbT23bo5MHMRCixC3/FjQsVt36HO//GSZuFth64l8M59zJ3jh8zKqRpfmuFmdm5+YXiYmlpeWV1TV/fuBdRwgnYJGIRb/pYAKMh2JJKBs2YAw58Bg2/f5H5jQfggkbhnRzE4Aa4G9IOJVgqydO36mdOpRUk3u1+1q/3DsC98sDTy2bVHMGYJlZOyihH3dO/Wu2IJAGEkjAshGOZsXRTzCUlDIalViIgxqSPu+AoGuIAhJuO7h8au0ppG52IqwqlMVJ/b6Q4EGIQ+GoywLInJr1M/M9zEtk5cVMaxomEkIwf6iTMkJGRhWG0KQci2UARTDhVtxqkhzkmUkVWUiFYk1+eJvZh9bRq3hyVa+d5GkW0jXZQBVnoGNXQJaojGxH0iJ7RK3rTnrQX7V37GI8WtHxnE/2B9vkDXKKUlw==</latexit><latexit sha1_base64="AVVkidoGiZKGjobmFXl/ZhZ6+wk=">AAAB/nicbVDLSsNAFJ3UV62vqODGTbAIdWFNRFAXQtGNiIsqxhbSECbT23bo5MHMRCixC3/FjQsVt36HO//GSZuFth64l8M59zJ3jh8zKqRpfmuFmdm5+YXiYmlpeWV1TV/fuBdRwgnYJGIRb/pYAKMh2JJKBs2YAw58Bg2/f5H5jQfggkbhnRzE4Aa4G9IOJVgqydO36mdOpRUk3u1+1q/3DsC98sDTy2bVHMGYJlZOyihH3dO/Wu2IJAGEkjAshGOZsXRTzCUlDIalViIgxqSPu+AoGuIAhJuO7h8au0ppG52IqwqlMVJ/b6Q4EGIQ+GoywLInJr1M/M9zEtk5cVMaxomEkIwf6iTMkJGRhWG0KQci2UARTDhVtxqkhzkmUkVWUiFYk1+eJvZh9bRq3hyVa+d5GkW0jXZQBVnoGNXQJaojGxH0iJ7RK3rTnrQX7V37GI8WtHxnE/2B9vkDXKKUlw==</latexit><latexit sha1_base64="AVVkidoGiZKGjobmFXl/ZhZ6+wk=">AAAB/nicbVDLSsNAFJ3UV62vqODGTbAIdWFNRFAXQtGNiIsqxhbSECbT23bo5MHMRCixC3/FjQsVt36HO//GSZuFth64l8M59zJ3jh8zKqRpfmuFmdm5+YXiYmlpeWV1TV/fuBdRwgnYJGIRb/pYAKMh2JJKBs2YAw58Bg2/f5H5jQfggkbhnRzE4Aa4G9IOJVgqydO36mdOpRUk3u1+1q/3DsC98sDTy2bVHMGYJlZOyihH3dO/Wu2IJAGEkjAshGOZsXRTzCUlDIalViIgxqSPu+AoGuIAhJuO7h8au0ppG52IqwqlMVJ/b6Q4EGIQ+GoywLInJr1M/M9zEtk5cVMaxomEkIwf6iTMkJGRhWG0KQci2UARTDhVtxqkhzkmUkVWUiFYk1+eJvZh9bRq3hyVa+d5GkW0jXZQBVnoGNXQJaojGxH0iJ7RK3rTnrQX7V37GI8WtHxnE/2B9vkDXKKUlw==</latexit>
(TL > TR, µL < µR)<latexit sha1_base64="Kb2JlA5x3IdX0EPP99KZNN4Xg9M=">AAAB/3icbVDLSsNAFJ3UV62vqAsXboJFqFBKIoIKIkU3LrqopbGFJoTJdNIOnUzCzEQooRt/xY0LFbf+hjv/xkmbhbYeuJfDOfcyc48fUyKkaX5rhaXlldW14nppY3Nre0ff3XsQUcIRtlFEI971ocCUMGxLIinuxhzD0Ke4449uM7/ziLkgEWvLcYzdEA4YCQiCUkmeflBpe43rtteqOlUnTLzGVdZbJ55eNmvmFMYisXJSBjmanv7l9COUhJhJRKEQPcuMpZtCLgmieFJyEoFjiEZwgHuKMhhi4abTAybGsVL6RhBxVUwaU/X3RgpDIcahryZDKIdi3svE/7xeIoMLNyUsTiRmaPZQkFBDRkaWhtEnHCNJx4pAxIn6q4GGkEMkVWYlFYI1f/IisU9rlzXz/qxcv8nTKIJDcAQqwALnoA7uQBPYAIEJeAav4E170l60d+1jNlrQ8p198Afa5w+yEpS/</latexit><latexit 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⌘C = 1� TR
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Charge current
Landauer formalism for thermoelectricity
Heat current from reservoirs:
Jq.� =1h
� �
��dE(E � µ�)�(E)[fL(E)� fR(E)]
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Thermoelectric efficiency (power production)
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Carnot efficiency
[Mahan and Sofo, PNAS 93, 7436 (1996); Humphrey et al., PRL 89, 116801 (2002)]
Delta-energy filtering and Carnot efficiency
Carnot efficiency obtained in the limit of reversible transport (zero entropy production) and zero output power
If transmission is possible only inside a tiny energy window around E=E✶ then
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Heat-to-work conversion through energy filtering
Flow of heat from hot to cold but no flow of charge
[see G. B., G. Casati, K. Saito, R. S. Whitney, Phys. Rep. 694, 1 (2017)]
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Bekenstein-Pendry bound
There is an purely quantum upper bound on the heat current through a single transverse mode
[Bekenstein, PRL 46, 923 (1981); Pendry, JPA 16, 2161 (1983) ]
For a reservoir coupled to another reservoir at T=0 through a -mode constriction which lets particle flow at all energies:
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Maximum power of a heat engine
Since the heat flow must be less than the Bekenstein-Pendry bound and the efficiency smaller than Carnot efficiency also the output power must be bounded
Within scattering theory:
[Whitney, PRL 112, 130601 (2014); PRB 91, 115425 (2015)]
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Efficiency optimization (at a given power)Find the transmission function that optimizes the heat-engine efficiency for a given output power
[Whitney, PRL 112, 130601 (2014); PRB 91, 115425 (2015)]
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Trade-off between power and efficiency
Effic
ienc
y
Carnot efficiency
Maximum possible power, P max
gen
forbidden1
2
power generated, Pgen
Result from (nonlinear) scattering theory
[Whitney, PRL 112, 130601 (2014); PRB 91, 115425 (2015)]
increase voltage
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Power-efficiency trade-off including phonons
Power output, P
Effi
cien
cy
0
strong phonons
no phonons
weak phonons
[see Whitney, PRB 91, 115425 (2015)]
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Boxcar transmission in topological insulators
[Chang et al., Nanolett., 14, 3779 (2014)]
Graphene nanoribbons with heavy adatoms and nanopores
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Is it possible to overcome the non-interacting bound?
For P/Pmax<<1,
Bound not favorable for power-efficiency trade-off; due to the fact that delta-energy filtering is the only mechanism to achieve Carnot for noninteracting systems
For interacting systems it is possible to achieve Carnot without delta-energy filtering
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Linear response for coupled (particle and heat) flowsStochastic baths: ideal gases at fixed temperature
and electrochemical potential
�µ = µL � µR
�T = TL � TR
(we assume TL > TR, µL < µR)
Onsager relation (for time-reversal symmetric systems):
Positivity of entropy production:
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Onsager and transport coefficients
Note that the positivity of entropy production implies that the (isothermal) electric conductance G>0 and the thermal conductance K>0
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Local equilibrium
Under the assumption of local equilibrium we can write phenomenological equations with ∇T and ∇µ rather than ΔT and Δµ
In this case we connect Onsager coefficients to electric and thermal conductivity rather than to conductances
charge and heat current densities
� =�
je
�V
�
�T=0
, � =�
jh
�T
�
je=0
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Traditional versus quantum thermoelectrics
Relaxation length (tens of n a n o m e t e r s a t r o o m temperature) of the order of the mean free path; inelastic s c a t t e r i n g ( p h o n o n s ) thermalizes the electrons
Structures smaller than the relaxation length (many microns at low temperature); quantum interference effects; Boltzmann transport theory cannot be applied
[see G. B., G. Casati, K. Saito, R. S. Whitney, Phys. Rep. 694, 1 (2017)]
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Linear response?
(exhaust gases)
(room temperature)
Linear response for small temperature and electrochemical potential differences (compared to the average temperature) on the scale of the relaxation lengthExhaust pipe: temperature drop over a mm scale: temperature drop of 0.003 K on the relaxation length scale (of 10 nm)
[Vining, Nat. Mater. 8, 83 (2009)]
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Maximum efficiency
Find the maximum of η over , for fixed (i.e., over the applied voltage ΔV for fixed temperature difference ΔT)
Within linear response and for steady-state heat to work conversion:
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Thermoelectric figure of merit
ZT � L2eh
detL=
GS2
KT
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Efficiency at maximum power
Find the maximum of P over , for fixed (over the applied voltage ΔV for fixed ΔT)
Output power
Maximum output power
Power factor
Pmax =T
4L2
eh
LeeF2
h =14
S2G(�T )2
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Efficiency at maximum power
�(�max) =�C
2ZT
ZT + 2� �CA �
�C
2
ηCA Curzon-Ahlborn upper bound
P quadratic function of , with maximum at half of the stopping force:
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�max
�(�max)Pmax
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Efficiency versus power
⇒
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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
Qm 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 vanishes since
ZT =�S2
�T � �1�� �� when ���
(G.B., G. Casati, J. Wang, PRL 110, 070604 (2013))
DeeDhh �D2eh = 0
(� < 1)
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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
DeeDhh �D2eh = ||xe||2||xh||2 � �xe,xh� � 0
xi = (xi1, ..., xiM ) =1
2�
��JiQ1��
�Q21�
, ...,�JiQM ��
�Q2M �
�
�xe,xh� =M�
k=1
xekxhk
� �2
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Example: 1D interacting classical gas
Consider a one dimensional gas of elastically colliding particles with unequal masses: m, M
ZT depends on the system size
(integrable model)ZT = 1 (at µ = 0)
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Quantum mechanics needed: Relation between density and electrochemical potential
Maxwell-Bolzmann distribution of
velocities
Reservoirs modeled as ideal (1D) gases
injection rates
grand partition function
density
de Broglie thermal wave length
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Non-decaying correlation functions
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Anomalous thermal transport
Carnot efficiency at the thermodynamic limit
ZT =�S2
kT
(R. Luo, G. B., G. Casati, J. Wang, arXiv:1710.08823)
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Delta-energy filtering mechanism?
A mechanism for achieving Carnot different from delta-energy filtering is needed
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Power vs. efficiency
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Validity of linear response
The agreement with linear response improves with N
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Non-interacting classical bound (but quantum mechanics needed)
charge current
heat current
Maxwell-Boltzmann distribution
(in 1D)de Broglie thermal
wave length
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0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
PêPmax
hêhC
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Overcoming the non-interacting bound
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Multiparticle collision dynamics (Kapral model) in 2D
S t r e a m i n g s t e p : f r e e propagation during a time τ
Collision step: random rotations of the velocities of the particles in cells of linear size a with respect to the center of mass velocity:
Momentum is conserved
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Overcoming the (2D) non-interacting bound
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ConclusionsThanks to interactions, for a given power it is possible to overcome the bound of efficiency which applies for classical non-interacting systemsNon-integrable momentum-conserving systems exhibit a power-efficiency trade-off which is optimal within linear response
Our results are based on the fact that such systems can achieve the Carnot efficiency at the thermodynamic limit without delta-energy filteringExtension of our results to purely quantum systems?
Results can be extended to cooling