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

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?

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)]

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)

Finite-time thermodynamics I: endoreversible cyclic engines

Dissipation is due to finite thermal conductances between heat reservoirs and the ideal heat engine

Output power:

Optimize power with respect to

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)]

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)]

Fokker-Planck equation:

is the mobility Gaussian distribution

Exactly solvable model

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:

Low-dissipation enginesThe entropy production vanishes in the limit of infinite-time cycles:

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)]

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)]

Thermoelectric efficiency (power production)

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

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)]

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:

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)]

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)]

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

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)]

Boxcar transmission in topological insulators

[Chang et al., Nanolett., 14, 3779 (2014)]

Graphene nanoribbons with heavy adatoms and nanopores

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

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:

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

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

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)]

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)]

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:

Thermoelectric figure of merit

ZT � L2eh

detL=

GS2

KT

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

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:

�max

�(�max)Pmax

Efficiency versus power

⇒

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

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,

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)

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

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)

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

Non-decaying correlation functions

Anomalous thermal transport

Carnot efficiency at the thermodynamic limit

ZT =�S2

kT

(R. Luo, G. B., G. Casati, J. Wang, arXiv:1710.08823)

Delta-energy filtering mechanism?

A mechanism for achieving Carnot different from delta-energy filtering is needed

Power vs. efficiency

Validity of linear response

The agreement with linear response improves with N

Non-interacting classical bound (but quantum mechanics needed)

charge current

heat current

Maxwell-Boltzmann distribution

(in 1D)de Broglie thermal

wave length

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

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

Overcoming the (2D) non-interacting bound

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