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Theoretical approaches for nanoscale thermoelectric phenomena
Giuliano BenentiCenter for Nonlinear and Complex Systems,
Univ. Insubria, Como, Italy INFN, Milano, Italy
Quantum computation and information is a rapidly developing interdisciplinary field. It is not easy to understand its fundamental concepts and central results without facing numerous technical details. This book provides the reader with a useful guide. In particular, the initial chapters offer a simple and self-contained introduction; no previous knowledge of quantum mechanics or classical computation is required.
Various important aspects of quan-tum computation and information are covered in depth, starting from the foun-dations (the basic concepts of computational complexity, energy, entropy, and information, quantum superposition and entanglement, elementary quantum gates, the main quan-tum algorithms, quantum teleportation, and
Giuliano Benenti Giulio Casati Davide Rossini Giuliano Strini
Principles of Quantum Computation and Information
A Comprehensive Textbook
Benenti Casati Rossini Strini
Principles of Quantum
Computation and Inform
ationA
Comprehensive Textbook
World Scientific
“Thorough introductions to classical computation and irreversibility, and a primer of quantum theory, lead into the heart of this impressive and substantial book. All the topics – quantum algorithms, quantum error correction, adiabatic quantum computing and decoherence are just a few – are explained carefully and in detail. Particularly attractive are the connections between the conceptual structures and mathematical formalisms, and the different experimental protocols for bringing them to practice. A more wide-ranging, comprehensive, and definitive text is hard to imagine.”
–— Sir Michael Berry, University of Bristol, UK
“This second edition of the textbook is a timely and very comprehensive update in a rapidly developing field, both in theory as well as in the experimental implementation of quantum information processing. The book provides a solid introduction into the field, a deeper insight in the formal description of quantum information as well as a well laid-out overview on several platforms for quantum simulation and quantum computation. All in all, a well-written and commendable textbook, which will prove very valuable both for the novices and the scholars in the fields of quantum computation and information.”
–— Rainer Blatt, Universität Innsbruck and IQOQI Innsbruck, Austria
“The book by Benenti, Casati, Rossini and Strini is an excellent introduction to the fascinating field of quantum information, of great benefit for scientists entering the field and a very useful reference for people already working in it. The second edition of the book is considerably extended with new chapters, as the one on many-body systems, and necessary updates, most notably on the physical implementations. ”
–— Rosario Fazio, The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy
World Scientificwww.worldscientific.com10909 hc
ISBN 978-981-3237-22-3
S A T O R
A R E P O
T E N E T
O P E R A
R O T A S
quantum cryptography) up to advanced topics (like entanglement measures, quan-tum discord, quantum noise, quantum channels, quantum error correction, quan-tum simulators, and tensor networks).
It can be used as a broad range textbook for a course in quantum information and computation, both for upper-level undergraduate s t u d e n t s a n d f o r g r a d u a t e students. I t contains a large number of solved exercises, which
are an essential complement to the text, as they will help the student to become familiar with the subject. The book may also be useful as general education for readers who want to know the fundamental principles of quantum information and computation.
Outline
1) Thermoelectricity in the quantum coherent regime (scattering theory, Landauer formula, energy filtering)
2) Thermoelectricity in the Coulomb blockade regime (quantum dot model, kinetic equations)
3) Aspects of thermoelectricity in strongly interacting systems (phase transitions, power-efficiency trade-off, power-efficiency-fluctuations trade-off)
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; efficiency depends on geometry and size
[see G. B., G. Casati, K. Saito, R. S. Whitney, Phys. Rep. 694, 1 (2017)]
Traditional thermocouple
Quantum thermocouple
Noninteracting systems, Energy filtering,
Landauer scattering theory
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)]
Energy filters in a thermocouple geometry
What about phonons?
Necessary both: (i) to reduce phonon transport; (ii) to have an efficient working fluid (optimize the electron dynamics)
Reducing thermal conductance
[Blanc, Rajabpour, Volz, Fournier, Bourgeois, APL 103, 043109 (2013)]
Scattering theory
Scattering region connected to N terminals (reservoirs)
Describes elastic scattering (including the effect of a disorder potential), but not electron-electron interactions beyond Hartree approximation and electron-phonon interactions
Transmission matrixProbability for an electron with energy E to go from (transverse) mode m of reservoir j to mode n of reservoir i:
scattering matrix elements
transmission matrix elements
probabilitiesFrom conservation of current and condition of zero current at zero bias:
From time reversal symmetry of the scatterer Hamiltonian:
Landauer approach
Electrical current into the scatterer from reservoir i:
Fermi function
Energy current into the scatterer from reservoir i:
heat carried by an electron leaving reservoir i
Heat current:
Kirkhoff’s law of current conservation for (steady state) electrical and energy currents:
Heat current not conserved:
Heat dissipated in the reservoirs: entropy production rate
Heat (not energy) current gauge invariant
Two-terminal (thermoelectric) power production
The upper bound to efficiency is given by the Carnot efficiency (expected only at zero power; intuitively, finite currents entail dissipation):
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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Scattering theory for two reservoirs
First law of thermodynamics:
Conserved currents:
Heat currents:
Thermoelectric efficiency (power production)
Charge current
Jq.� =1h
� �
��dE(E � µ�)�(E)[fL(E)� fR(E)]
Heat current from reservoirs:
Efficiency:
Jh,↵<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
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
Example: single-level quantum dot
Dot’s scattering matrix:
The Green’s function is for a non-Hermitian effective Hamiltonian taking into account coupling to the dots
operator coupling the single-level dot to reservoirs:
Short intermezzo: 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 efficiency at maximum power (Curzon-Ahlborn efficiency) is independent of the heat conductances:
[Yvon, 1955; Chambadal, 1957; Novikov, 1958; Curzon and Ahlborn, Am. J. Phys. 43, 22 (1975)]
Within linear response:
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)]
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
Linear response for coupled (particle and heat) flows
Stochastic 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
Seebeck and Peltier coefficients
Seebeck and Peltier coefficients are related a Onsager reciprocal relation (when time symmetry is not broken, we simply have )
Interpretation of the Peltier coefficient
entropy transported by the electron flow
Entropy current:
each electron carries an entropy of advective term in thermal transport (reversible)open-circuit term in thermal transport (by electrons and phonons, irreversible)
Entropy production/ heat dissipation rate
Joule heating
heat lost by thermal resistance
disappears for time-reversal symmetric systems
To minimize dissipation large G and
small K are needed
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 length
Exhaust 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:
(TL ⇡ TR ⇡ T )<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Thermoelectric figure of merit
ZT � L2eh
detL=
GS2
KT
ZT diverging implies that the Onsager matrix is ill-conditioned, that is, the condition number diverges:
In such case the system is singular (tight coupling limit):
(the ratio Jh/Je is independent of the applied voltage and temperature gradients)
Jh � Je
Conditions for Carnot efficiency
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
⇒
Maximum refrigeration efficiency
Coefficient of performance (COP)
�(r) =Jh
P
ZT is the figure of merit also for refrigeration
Cooling power (heat extracted from the cold reservoir)
(can be >1)
ZT is an intrinsic material property?For mesoscopic systems size-dependence for G,K,S can be expectedIn the diffusive transport regime Ohm’s and Fourier’s scaling laws hold:
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
Crossover from endoreversible to exoreversible regimes
Thermoelectric generator: internal dissipation (Joule heating, thermal conductance) and external dissipation (dissipative thermal coupling to reservoirs)
[Apertet, Ouerdane, Goupil, Lecoeur, PRE 85,
031116 (2012)]
Linear response and Landauer formalism
The Onsager coefficients are obtained from the linear response expansion of the charge and thermal currents
Lee = e2TI0, Leh = Lhe = eTI1, Lhh = TI2
Wiedemann-Franz law
Phenomenological law: the ratio of the thermal to the electrical conductivity is directly proportional to the temperature, with a universal proportionality factor.
Lorenz number
Sommerfeld expansionThe Wiedemann-Franz law can be derived for low-temperature non-interacting systems both within kinetic theory or Landauer approachIn both cases it is substantiated by Sommerfeld expansion. Within Landauer approach we consider
We assume smooth transmission functions τ(E) in the neighborhood of E=µ:
Jq.� =1h
� �
��dE(E � µ�)�(E)[fL(E)� fR(E)]
To leading order in kBT/EF with
Neglected I12/I0 with respect to I2, which in turn implies LeeLhh>>(Leh)2 and
G = e2I0 �e2
h�(µ), K =
1T
�I2 �
I21
I0
�� �2k2
BT
3h�(µ)
Wiedemann-Franz law:
K
G� �2
3
�kB
e
�2
T
Wiedemann-Franz law and thermoelectric efficiency
ZT =GS2
KT =
S2
L
Wiedemann-Franz law derived under the condition LeeLhh>>(Leh)2 and therefore
Wiedemann-Franz law violated in - low-dimensional interacting systems that exhibit non-Fermi liquid behavior - (smll) systems where transmission can show significant energy dependence
(Violation of) Wiedemann-Franz law in small systems
Consider a (basic) model of a molecular wire coupled to electrodes:
Transmission:Green’s function:Level broadening functions: Self-energies:
Wide band limit: level widths energy independent:
Green’s function obtained by inverting
Take
Transmission:
Mott’s formula for thermopowerFor non-interacting electrons (thermopower vanishes when there is particle-hole symmetry)
S =1
eT
I1
I0=
1eT
���� dE(E � µ)�(E)
�� �f
�E
�
���� dE�(E)
�� �f
�E
�
Consider smooth transmissions
Electron and holes contribute with opposite signs: we want sharp, asymmetric transmission functions to have large S (ex: resonances, Anderson QPT, see Imry and Amir, 2010), violation of WF, possibly large ZT.
Metal-insulator 3D Anderson transition
x conductivi ty cr i t ical exponent
[G.B., H. Ouerdane, C. Goupil, arXiv:1602.06590; Comptes Rendus Physique, in press]
Energy filtering
No dispersion with delta-energy filtering: ZT diverges
For good thermoelectric we desire violation of WF law such that:
Thermoelectricity in the Coulomb blockade regime,
Kinetic equations. Quantum dot model
Multilevel interacting quantum dot
Discrete energy levels: ideal to implement energy filtering
[Erdmann, Mazza, Bosisio, G.B., Fazio, Taddei PRB 95, 245432 (2017)]
Study the effects of Coulomb interaction between electrons
Sequential (single-electron) tunnelling regime
Weak coupling to the reservoirs: thermal energy , level spacing and charging energy much larger than the coupling energy between the QD and the reservoirs: charge quantized
tunneling rate from level p to reservoir 𝛼
single-electron levels of the QCcapacitancenumber of electrons in the dot
electrostatic (Coulomb) interaction
Electrostatic energysingle-electron charging energy
Energy conservation
Configuration determined by occupation numbers
Non-equilibrium probability
Energy conservation for tunnelling into or from reservoirs:
Kinetic equationsOne kinetic equation for each configuration:
Stationary solution:
Steady-state currentsCharge current:
Heat current:
Energy current:
Quantum limitEnergy spacing and charging energy much bigger than
Analytical results for equidistant levels:
(energy filtering)
power factor
Coulomb interaction may enhance the thermoelectric performance of a QD
Compare interacting and non-interacting two-terminal QD with the same energy spacing
T h e r m a l c o n d u c t a n c e suppressed by Coulomb interaction: ZT is greatly increased. For a single level K=0 (charge and heat current proportional). For at least two levels Coulomb blockade prevents a second electron to enter when one is already there (electrostatic energy to be paid).
Strongly interacting systems, Electronic Phase transitions, Power-efficiency trade-off,
Power-efficiency-fluctuations trade-off, Carnot efficiency at finite power?
Generality of Onsager reciprocal relations
Short intermezzo: a reason why interactions might be interesting for thermoelectricity
thermal conductance at zero voltage
If the ratio K’/K diverges, then the Carnot efficiency is achieved
Thermodynamic properties of the working fluid
coupled equations:
Setting dN=0 in the coupled equations:
Thermodynamic cycle
maximum efficiency (over d𝜇 at fixed dT):
thermodynamic figure of merit:
Analogy with a classical gas
heat capacity at constant p or V
Power-efficiency trade-off: Is it possible to overcome the non-interacting bound?
Noninteracting systems: 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
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, PRL 121, 080602 (2018)]
Delta-energy filtering mechanism?
A mechanism for achieving Carnot different from delta-energy filtering is needed
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)
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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Non-interacting bound
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
Results can be extended to cooling
linear response
numerical data
Applications for cold atoms?
Power-efficiency trade-off at the verge of phase transitions
For heat engines described as Markov processes:
[N. Shiraishi, K. Saito, H. Tasaki, PRL 117, 190601 (2016)]
For a working substance at a critical point:
[M. Campisi, R. Fazio, Nature Comm. 7, 11895 (2016); see also Allahverdyan et al., PRL 111, 050601 (2013)]
Results compatible only with diverging amplitude A when approaching the Carnot efficiency
Power-efficiency-fluctuations trade-offFor classical Markovian dynamics on a finite set of states and overdamped Langevin dynamics, trade-off between power, efficiency, and constancy for steady-state engines:
[P. Pietzonka, U. Seifert, PRL 120, 190602 (2018)]
Bound violated in quantum mechanics, e.g. for resonant tunnelling transport (noninteracting system), but not close to Carnot efficiency. The problem for interacting systems is open.
[J. Liu. D. Segal, PRE 99, 062141 (2019)]
Carnot efficiency at finite power with broken time-reversal symmetry?
B applied magnetic field or any parameter breaking time-reversibility
such as the Coriolis force, etc.
�µ = µL � µR
�T = TL � TR
(we assume TL > TR, µL < µR)
��
�
Je = Lee(B)Fe + Leh(B)Fh
Jh = Lhe(B)Fe + Lhh(B)Fh
Constraints from thermodynamics
ONSAGER-CASIMIR RELATIONS:
POSITIVITY OF THE ENTROPY PRODUCTION:
G(B) = G(�B)K(B) = K(�B)
Both maximum efficiency and efficiency at maximum power depend on two parameters
Pmax
[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
Output power at maximum efficiency
When time-reversibility is broken, within linear response it is not forbidden from the second law 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 could in principle find small enough forces such that the linear theory holds.
Reversible part of the currents
The reversible part of the currents does not contribute to entropy production
Possibility of dissipationless transport?
[K.. Brandner, K. Saito, U. Seifert, PRL 110, 070603 (2013)]
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).
Physical model of probe reservoirsLarge but finite “reservoirs”
Voltage and temperature probe (m im ick ing e -e i ne las t i c scattering)
Voltage probe (mimicking e-ph inelastic scattering)
Non-interacting three-terminal model
tJ = (JeL, JhL, JeP , JhP )
�
k
Je,k = 0,�
k
Ju,k = 0 (Jh,k = Ju,k � (µ/e)Je,k)
tF =�
�µ
eT,�T
T 2,�µP
eT,�TP
T 2
�tFJ =4�
i=1
JiFi
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:
J = LF
�J�
J�
�=
�L�� L��
L�� L��
� �F�
F�
�
F� = �L���1L��F�
Two-terminal Onsager matrix for partially coherent transport
Reduction to 2x2 Onsager matrix when the third terminal is a probe terminal mimicking inelastic scattering
J� = L�F�, L� � L�� � L��L���1L��.
�J1
J2
�=
�L�
11 L�12
L�21 L�
22
� �F1
F2
�
First-principle exact calculation within the Landauer-Büttiker approach
Bilinear Hamiltonian
Charge and heat current from the left terminal
Onsager coefficients from linear response expansion of the currents
Transmission probabilities:
Illustrative three-dot example
Asymmetric structure, e.g..
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)]
Asymmetric power generation and refrigeration
When a magnetic field is added, the efficiencies of power generation and refrigeration are no longer equal:
To linear order in the applied magnetic field:
A small magnetic field improves either power generation or refrigeration, and vice versa if we reverse the direction of the field
The large-field enhancement of efficiencies is model-dependent, but the small-field asymmetry is generic
Transmission windows model
[V.. Balachandran, G. B., G. Casati, PRB 87, 165419 (2013);
se also M. Horvat, T. Prosen, G. B., G. Casati, PRE 86, 052102 (2012)]
�
i
�ij(E) =�
j
�ij(E) = 1 The Curzon-Ahlborn limit can be overcome (within linear response)
Saturation of bounds from 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 =43Pmax
Bounds for multi-terminal thermoelectricity
[Brandner and Seifert, NJP 15, 105003 (2013); PRE 91, 012121 (2015)]
Numerical evidence that the power vanishes when the Carnot efficiency is approached
Bounds with electron-phonon scattering
Efficiency bounded by the non-negativity of the entropy production of the original three-terminal junction. However, the efficiency at maximum power can be enhanced
[Yamamoto, Entin-Wohlman, Aharony, Hatano; PRB 94, 121402(R) (2015)]
Onsager-Casimir relations
Onsager reciprocal relations reflect at the macroscopic level the time-reversal symmetry of the microscopic dynamics, invariant under the transformation:
With an applied magnetic field one instead obtains Onsager-Casimir relations:
but in principle one could violate the Onsager symmetry:
Onsager relations and thermodynamic constraints on heat-to-work conversion
For thermoelectricity:
and in principle one could have the Carnot efficiency at finite power:
Onsager relations with broken time-reversal symmetry
Onsager relations under an applied magnetic field remain valid:
1) for noninteracting systems
2) if the magnetic field is constant
What about for a generic, spatially dependent magnetic field?
[Bonella, Ciccotti, Rondoni, EPL 108, 60004 (2014)]
Symmetry without magnetic field inversion
Analytical result for Landau gauge:
Equations of motion invariant under:
Numerical results
generic 2D case:
generic 3D case:Theoretical argument: divide the system into small volumes Time-reversal trajectories without reversing the field for
No-go theorem for finite power at the Carnot efficiency on purely thermodynamic grounds?
According to Nico Van Kampen Onsager derived his reciprocal relations in a “stroke of genius”
Onsager reciprocal relations much more general than expected so far.
Some open problems
Investigate strongly-interacting systems close to electronic phase transitions
In nonlinear regimes restrictions due to Onsager reciprocity relations might be overcome; useful for thermoelectricity?
Further investigate/optimize time-dependent driving
Power-efficiency-fluctuation trade-off for non-Markovian quantum dynamics?