Quantum Transport and Thermodynamics Giuliano Benenti Center for Nonlinear and Complex Systems, Univ. Insubria, Como, Italy INFN, Milano, Italy Outline 1) Basic thermodynamics of nonequilibrium states (linear response, Onsager relations, efficiency of thermal machines, finite-time thermodynamics) 2) Landauer formalism (scattering theory) (energy filtering, scattering theory and the laws of thermodynamics) 3) Rate equations (local detailed balance, examples of thermal machines) 4) Thermodynamic bounds on heat-to-work conversion (power-efficiency trade-off, thermodynamic uncertainty relations and power-efficiency-fluctuations trade-off) What is quantum in energy conversion? Ex: Traditional versus quantum thermoelectrics Structures smaller than the Relaxation length (tens of relaxation length (many nanometers at room microns at low temperature); temperature) of the order of quantum interference effects; the mean free path; inelastic Boltzmann transport theory s c a t t e r i n g ( p h o n o n s ) cannot be applied; efficiency thermalizes the electrons depends on geometry and size [see G. B., G. Casati, K. Saito, R. S. Whitney, Phys. Rep. 694, 1 (2017)] Traditional thermocouple Quantum thermocouple Ex: Cooling by heating A third (hot) reservoir can help cooling [Pekola & Hekking, PRL 98, 210604 (2007); Mari & Eisert, PRL 108, 120602 (2012), …] [Cleuren, Rutten, Van den Broeck, PRL 108, 120603 (2012)] I. Basic thermodynamics of nonequilibrium states The Nobel Prize in Chemistry 1968: From the award ceremony speech “Professor Lars Onsager has been awarded this year’s Nobel Prize for Chemistry (1968) for the discovery of the reciprocal relations, named after him, and basic to irreversible thermodynamics… Onsager’s reciprocal relations can be described as a universal natural law…It can be said that Onsager’s reciprocal relations represent a further law making possible a thermodynamic study of irreversible processes…It represents one of the great advances in science during this century. According to Nico Van Kampen Onsager derived his reciprocal relations in a “stroke of genius” Irreversible thermodynamic Irreversible thermodynamics based on the postulates of equilibrium thermostatics plus the postulate of time- reversal symmetry of physical laws (if time t is replaced by -t and simultaneously applied magnetic field B by -B) The thermodynamic theory of irreversible processes is based on the Onsager Reciprocity Theorem Refs.: Thermodynamic forces and fluxes Irreversible processes are driven by thermodynamic forces (or generalized forces or affinities) Xi Fk Fluxes Ji characterize the response of the system to the applied forces Entropy production rate given by the sum of the products of each flux with its associated thermodynamic force S = S(U, V, N1,N2, ...)=S(E0,E1,E2, ...) dS ∂S dEk = = XkkJJk k dt ∂Ek dt k k F k Linear response Purely resistive systems: fluxes at a given instant depend only on the thermodynamic forces at that instant (memory effects not considered) J = L + L + ... i ijFj ijkFjFk j j,k Fluxes vanish as thermodynamic forces vanish Linear (and purely resistive) processes: J = L i ijFj j Lij Onsager coefficients (first-order kinetic coefficients) depend on intensive quantities (T,P,µ,...) Phenomenological linear Ohm’s, Fourier’s, Fick’s laws Onsager reciprocal relations Relationship of Onsager theorem to time-reversal symmetry of physical laws Consider delayed correlation moments of fluctuations (without applied magnetic fields) δE (t) E (t) E , δE =0, j ≡ j − j j δE (t)δE (t + τ) = δE (t)δE (t τ) = δE (t + τ)δE (t) j k j k − j k δEk(t + τ) δEk(t) δEj(t + τ) δEj(t) lim δEj(t) − = lim − δEk(t) τ 0 τ τ 0 τ → → δE δE˙ = δE˙ δE j k j k Assume that fluctuations decay is governed by the same linear dynamical laws as are macroscopic processes δE˙ = L δ k kl Fl l L δE δ = L δ δE kl j Fl jl Fl k <latexit sha1_base64="(null)">(null)</latexit> j l Xl Assume that the fluctuation of each thermodynamic force is associated only with the fluctuation of the corresponding extensive variable δE δ = k δ h<latexit sha1_base64="(null)">(null)</latexit> j Fli − B jl Onsager relations: L = L <latexit sha1_base64="(null)">(null)</latexit> jk kj 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: Linear response for coupled (particle and heat) flows Stochastic baths: ideal gases at fixed temperature and electrochemical potential ∆µ = µ µ L − R Onsager-Casimir relations: ∆T = T T L − R (we assume TL >TR,µL <µR) Onsager reciprocal relations for time-reversal symmetric systems: Positivity of entropy production Entropy production rate given by the sum of the products of each flux with its associated thermodynamic force S = S(U, V, N1,N2, ...)=S(E0,E1,E2, ...) dS ∂S dEk = = XkkJJk k dt ∂Ek dt k k F k For thermoelectricity: Linear response: Positivity of entropy production rate: 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 by a Onsager-Casimir reciprocal relation (when time symmetry is not broken, we simply have ) Interpretation of the Peltier coefficient Entropy current: entropy transported by the electron flow 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 To minimize disappears for time-reversal dissipation large G and symmetric systems small K are needed Linear response? (exhaust gases) (room temperature) [Vining, Nat. Mater. 8, 83 (2009)] 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) Maximum efficiency Within linear response and for steady-state heat to work conversion: Find the maximum of η over , for fixed (i.e., over the applied voltage ΔV for fixed temperature difference ΔT) (T T T ) <latexit sha1_base64="(null)">(null)</latexit> L ⇡ R ⇡ Thermoelectric figure of merit L2 GS2 ZT eh = T ≡ det L K Conditions for Carnot efficiency 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): J J h ∝ e (the ratio Jh/Je is independent of the applied voltage and temperature gradients) Efficiency at maximum power Output power Find the maximum of P over , for fixed (over the applied voltage ΔV for fixed ΔT) Maximum output power 2 T Leh 2 1 2 2 Pmax = h = S G(∆T ) 4 Lee F 4 Power factor P quadratic function of , with maximum at half of the stopping force: Efficiency at maximum power η ZT η η(ω )= C η C max 2 ZT +2 ≤ CA ≡ 2 ηCA Curzon-Ahlborn upper bound ηmax η(Pωmax) Efficiency versus power ⇒ Maximum refrigeration efficiency Cooling power (heat extracted from the cold reservoir) Coefficient of performance (COP) J η(r) = h P (can be >1) ZT is the figure of merit also for refrigeration ZT is an intrinsic material property? For mesoscopic systems size-dependence for G,K,S can be expected In 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 charge and heat current densities In this case we connect Onsager coefficients to electric and thermal conductivity rather than to conductances j j σ = e , κ = h V T =0 T j =0 ∇ ∇ ∇ e 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. Two-terminal (thermoelectric) power production Left (L) Right (R) reservoir S reservoir TL , L TR , R P =[(µ µ )/e]J <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>