Local equilibrium Under the assumption of local equilibrium we can write phenomenological equations with ∇ T and ∇ µ rather than Δ T and Δ µ charge and heat current densities In this case we connect Onsager coefficients to electric and thermal conductivity rather than to conductances � j e � j h � � , σ = κ = � V � T � T =0 j e =0
II. Landauer formalism (scattering theory)
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 matrix Probability 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 probabilities Conservation of current and condition of zero current at zero bias from: 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. The generated power equals heat and so is also gauge invariant.
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sha1_base64="Kb2JlA5x3IdX0EP9KZN4Xg9M=">AB/3icbVDLSsNAFJ3UV62vqAsXboJFqFBKIoIKIkU3LrqopbGFJoTJdNIOnUzCzEQoRt/xY0LFbf+hjv/xkmbhbYeuJfDOfcyc48fUyKkaX5rhaXldW14npY3Nre0f3XsQUcIRtlFEI971ocCUMGxLIinuxhzD0Ke49uM7/ziLkgEWvLcYzdEA4YCQiCUkmeflBpe43rteqOlUnTLzGVdZbJ5eNmvmFMYisXJSBjmanv7l9COUhJhJRKEQPcuMpZtCLgmieFJyEoFjiEZwgHuKMhi4abTAybGsVL6RhBxVUwaU/X3RgpDIcahryZDKIdi3svE/7xeIoMLNyUsTiRmaPZQkFBDRkaWhtEnHCNJx4pAxIn6q4GkEMkVWYlFYI1f/IisU9rlzXz/qxcv8nTKIJDcAQqwALnoA7uQBPYAIEJeAav4E170l60d+1jNlrQ8p198Afa5w+yEpS/</latexit> <latexit sha1_base64="Kb2JlA5x3IdX0EP9KZN4Xg9M=">AB/3icbVDLSsNAFJ3UV62vqAsXboJFqFBKIoIKIkU3LrqopbGFJoTJdNIOnUzCzEQoRt/xY0LFbf+hjv/xkmbhbYeuJfDOfcyc48fUyKkaX5rhaXldW14npY3Nre0f3XsQUcIRtlFEI971ocCUMGxLIinuxhzD0Ke49uM7/ziLkgEWvLcYzdEA4YCQiCUkmeflBpe43rteqOlUnTLzGVdZbJ5eNmvmFMYisXJSBjmanv7l9COUhJhJRKEQPcuMpZtCLgmieFJyEoFjiEZwgHuKMhi4abTAybGsVL6RhBxVUwaU/X3RgpDIcahryZDKIdi3svE/7xeIoMLNyUsTiRmaPZQkFBDRkaWhtEnHCNJx4pAxIn6q4GkEMkVWYlFYI1f/IisU9rlzXz/qxcv8nTKIJDcAQqwALnoA7uQBPYAIEJeAav4E170l60d+1jNlrQ8p198Afa5w+yEpS/</latexit> Two-terminal (thermoelectric) power production Right ( R ) Left ( L ) reservoir reservoir S T , T , L L R R P = [( µ R − µ L ) /e ] J e ( T L > T R , µ L < µ R ) The upper bound to efficiency is given by the Carnot efficiency (expected only at zero power; intuitively, finite currents entail dissipation): η C = 1 − T R T L
Scattering theory for two reservoirs Conserved currents: Heat currents: First law of thermodynamics:
Second law for scattering theory For two terminals: monotonically decaying function implies For arbitrary number of terminals proof by Nenciu (2007): The second law implies that when the reservoirs are at the same temperature the system cannot generate electrical power Joule heating:
Scattering theory & Nernst’s unattainability principle Dynamical formulation of the third law of thermodynamics: it is impossible to reach absolute zero temperature in finite time Extracting heat from reservoir i with rate J i , we change its temperature: Heat capacity of a free-electron reservoir: Nernst principle satisfied (in a weak form)
How to obtain the best steady-state heat to work conversion?
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)]
<latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> Thermoelectric efficiency (power production) in the Landauer approach Charge current Heat current from reservoirs: � ∞ J q. α = 1 J h, α dE ( E − µ α ) τ ( E )[ f L ( E ) − f R ( E )] h −∞ Efficiency:
Delta-energy filtering and Carnot efficiency If transmission is possible only inside a tiny energy window around E=E ✶ then Carnot efficiency Carnot efficiency obtained in the limit of reversible transport (zero entropy production) and zero output power [Mahan and Sofo, PNAS 93, 7436 (1996); Humphrey et al., PRL 89, 116801 (2002)]
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:
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 Carnot efficiency f o r b i d d e 1 n Efficiency Maximum 2 possible power, P max gen increase voltage power generated, P gen Result from (nonlinear) scattering theory [Whitney, PRL 112 , 130601 (2014); PRB 91 , 115425 (2015)]
Power-efficiency trade-off including phonons no phonons Efficiency weak phonons strong phonons 0 Power output, P [see Whitney, PRB 91 , 115425 (2015)]
Boxcar transmission in topological insulators Graphene nanoribbons with heavy adatoms and nanopores [Chang et al., Nanolett., 14, 3779 (2014)]
Linear response and Landauer formalism The Onsager coefficients are obtained from the linear response expansion of the charge and thermal currents L ee = e 2 TI 0 , L eh = L he = eTI 1 , L hh = TI 2
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 expansion The Wiedemann-Franz law can be derived for low- temperature non-interacting systems both within kinetic theory or Landauer approach In both cases it is substantiated by Sommerfeld expansion. Within Landauer approach we consider � ∞ J q. α = 1 dE ( E − µ α ) τ ( E )[ f L ( E ) − f R ( E )] h −∞ We assume smooth transmission functions τ (E) in the neighborhood of E=µ:
To leading order in k B T/E F with G = e 2 I 0 ≈ e 2 I 2 − I 2 ≈ π 2 k 2 K = 1 � � B T 1 h τ ( µ ) , τ ( µ ) 3 h T I 0 Neglected I 12 /I 0 with respect to I 2 , which in turn implies L ee L hh >>(L eh ) 2 and Wiedemann-Franz law: � 2 G ≈ π 2 � k B K T e 3
Wiedemann-Franz law and thermoelectric efficiency ZT = GS 2 T = S 2 K L Wiedemann-Franz law derived under the condition L ee L hh >>(L eh ) 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: Take Transmission: Green’s function obtained by inverting
Mott’s formula for thermopower For non-interacting electrons (thermopower vanishes when there is particle-hole symmetry) � � � ∞ − ∂ f −∞ dE ( E − µ ) τ ( E ) S = 1 = 1 I 1 ∂ E � � � ∞ eT I 0 eT − ∂ f −∞ dE τ ( E ) ∂ 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 conductivity critical exponent [G.B., H. Ouerdane, C. Goupil, Comptes Rendus Physique 17 , 1072 (2016)]
Energy filtering For good thermoelectric we desire violation of WF law such that: No dispersion with delta-energy filtering: ZT diverges
III. Rate equations
Conditions for a rate equation Consider systems weakly coupled to the environment. Non-Markovian effects are neglected as well as the generation of coherences (in the system’s energy eigenbasis). Interaction effects may be included.
Rate equations Transition rate from state a to state b (due to the coupling to reservoir i): The probability to find the system in state |b ⟩ at time t obeys a rate equation:
Local detailed balance The rates can be derived from microscopic Hamiltonian via Fermi golden rule, or be considered phenomenological constants. They obey the local detailed balance principle: change of entropy in reservoir i when it induces a system’s transition from a to b. Clausius relation:
⇒ (Steady-state) currents Probability current at time t for the transition from a to b induced by reservoir i: Steady-state solution of the rate equations: Steady-state probability currents: Kirkhoff’s law:
Particle and energy currents By taking the steady-state probability currents, we obtain the steady-state charge and heat currents: There is no net flow of particle and energy into the system at equilibrium:
Equilibrium (dynamical definition) Connect the system to reservoirs at the same temperature and electrochemical potential A state at equilibrium must obey the detailed balance: From the local detailed balance: We then derive the equilibrium state
Output power and the first law of thermodynamics The power generated at reservoir i depends on the reference electrochemical potential, but the overall power is gauge-invariant: This is a consequence of Kirkoff’s law and of the relation: We obtain the first law of thermodynamics: the rate of heat absorption equals the rate of work production:
Second law of thermodynamics Change of entropy in reservoir i: System in general not in a thermal state: use Shannon entropy rather than Clausius definition: Total entropy production rate: Use local detailed balance to prove that
Example: single-level quantum dot Neglect electron spin; charging energy too high to have doble occupancy Rate equations for the dot’s dynamics:
Particle and energy currents Particle currents: Set Heat currents: Entropy production at steady state:
Power and efficiency Generated power: Efficiency of a heat engine: Coefficient of performance for a refrigerator:
⇒ ⇒ Steady-state solution Use Using local detailed balance:
⇒ <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> Carnot efficiency Reversible engine if the entropy production rate vanishes: This leads to Carnot efficiency: The price to pay is that the power is zero: E 0 − E 1 − ( N 0 − N 1 ) µ L = − ✏ 1 T L T L = E 0 − E 1 − ( N 0 − N 1 ) µ R = −− ✏ 1 + µ = − ✏ 1 + ✏ 1 (1 − T R /T L ) = − ✏ 1 T R T R T R T L
Multilevel interacting quantum dot Discrete energy levels: ideal to implement energy filtering Study the effects of Coulomb interaction between electrons [Erdmann, Mazza, Bosisio, G.B., Fazio, Taddei PRB 95 , 245432 (2017)]
Sequential (single-electron) tunnelling regime single-electron levels of the QC capacitance number of electrons in the dot electrostatic (Coulomb) interaction tunneling rate from level p to reservoir 𝛽 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 Electrostatic energy single-electron charging energy
Energy conservation Configuration determined by occupation numbers Non-equilibrium probability Energy conservation for tunnelling into or from reservoirs:
Kinetic (rate) equations One kinetic (rate) equation for each configuration: Stationary solution:
Steady-state currents Charge current: Energy current: Heat current:
Quantum limit Energy spacing and charging energy much bigger than Analytical results for equidistant levels: power factor (energy filtering)
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).
IV. Thermodynamic bounds on heat-to-work conversion
Can interactions improve the power-efficiency trade-off? What is the role of a magnetic field? Is it possible to have Carnot at finite power? What is the role played by fluctuations? Thermodynamic uncertainty 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 d 𝜈 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/P max <<1, [Whitney, PRL 112 , 130601 (2014); PRB 91 , 115425 (2015)] 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 formula The 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 efficiency Suzuki’s formula (which generalizes Mazur’s inequality) for finite-size Drude weights Q m 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 D ee D hh − D 2 eh = 0 ZT = σ S 2 T ∝ Λ 1 − α → ∞ when Λ → ∞ κ ( α < 1) (G.B., G. Casati, J. Wang, PRL 110, 070604 (2013))
For systems with more than a single relevant constant of motion, for instance for integrable systems, due to the Schwarz inequality eh = || x e || 2 || x h || 2 � � x e , x h � � 0 D ee D hh � D 2 � � x i = ( x i 1 , ..., x iM ) = 1 � J i Q 1 � , ..., � J i Q M � � � 2 Λ � Q 2 � Q 2 1 � M � M � � x e , x h � = x ek x hk k =1 Equality arises only in the exceptional case when the two vectors are parallel; in general det L ∝ L 2 , κ ∝ Λ , ZT ∝ Λ 0 ∝ Λ 2
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