Laboratoire de Physique et Modélisation des Milieux Condensés Univ. Grenoble & CNRS, Grenoble, France - PDF

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Laboratoire de Physique et Modélisation des Milieux Condensés Univ. Grenoble & CNRS, Grenoble, France The best quantum thermoelectric at finite power output Robert S. Whitney Preprint arxiv: Aachen

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Laboratoire de Physique et Modélisation des Milieux Condensés Univ. Grenoble & CNRS, Grenoble, France The best quantum thermoelectric at finite power output Robert S. Whitney Preprint arxiv: Aachen Nov 2013 OVERVIEW Ioffe (1958) Inst. Semicond. Leningrad MY QUESTION: What is MAX. efficiency at GIVEN power output? 2 Watts (80-90 Volts) Quantum thermoelectric Nonlinear Landauer-Büttiker T HOT = 572 K T = 305 K COLD parasitic heat flows phonons & photons inelastic / relaxation in quantum system CENTRAL RESULTS Heat-engine efficiency: η eng = P/J Output : power= P = VI Input : heat-current = J Refrigerator efficiency coeff. of performance (COP): η fri = J/P Output : J Input : P [1] ABSOLUTE upper bound on Power Output: P P qb P qb is quantum-bound (ill-defined in classical thermodyn) [2] MAX. efficiency, η eng (P), at given power P function of P/P qb = Quantum unlike Carnot efficiency = classical ( ) Example low power : η eng (P) ηeng Carnot 1 α 1 P/Pqb + INTRODUCTION Bulk versus Quantum : for LARGE T Linearity requires temp. drop to be small on scale of thermalization HOT Scale of thermalization heat bulk thermoelectric LINEAR electric current COLD HOT heat quantum thermoelectric Scale of thermalization COLD electric current NON-LINEAR Nonlinear regime : efficiency η is meaningful, but ZT is NOT. Muralidharan-Grifoni (2012), Whitney (2013), Meier-Jacquod (2013), Michelini s poster Linear formula η/η Carnot = ZT+1 1 ZT+1+1 Dictionary (for linear people) ZT = η = η Carnot ZT = 3 η = 1 3 η Carnot ZT = 0 η = 0 METHOD: scattering theory ÝÓÒ linear response Heat current: J L = ( dǫ h ǫ T RL(ǫ) f [ ǫ evl k B T L ] f transmission function T RL (ǫ) = tr [ S (ǫ)s(ǫ) ] [ ]) ǫ evr k B T R Fermi-Dirac for ingoing particles: f [ (ǫ ev j ) / k B T j ] obeys 2nd law thermodyn, etc. Whitney PRB (2013) ÝÓÒ linear-response : Hartree-like interactions self-consistent Christen-Büttiker (1996) Self-consistent loop: S(ǫ) potential-distrib. in system ORIGIN of THERMOELECTRICITY HOT Fermi sea quantum system's transmission COLD Fermi sea I = 0 V = V stop P = IV = 0 Mahan,Sofo (1996). Humphrey,Linke (2005) Vanishing transmission width reversibility (no entropy generated) Carnot efficiency η = η carnot = 1 T R /T L... but no power ORIGIN of THERMOELECTRICITY HOT Fermi sea quantum system's transmission COLD Fermi sea I 0 V 0 P = IV = max. Efficiency at max. power vanishing transmission width; Esposito,Lindenberg,van den Broeck (2009) Curzon Ahlborn efficiency Curzon, Ahlborn (1975), Novikov (1957), Chambadal (1957) non-vanishing transmission width: Nakpathomkun, Xu, Linke (2010), Leijnse, Wegewijs, Flenberg (2010) Hershfield, Muttalib, Nartowt (2013),...Others... higher max power but lower efficiency at that power ANSWERING MY QUESTON What is MAXIMUM EFFICIENCY for ÁÎ Æ power output? OPTIMIZING EFFICIENCY for ÁÎ Æ POWER OUTPUT (n+1) variables: n slices + bias,v one constraint : power = P want minimum heat-flow J for given P Proof: changing heightτ γ of slice γ, decreasesj (increases efficiency) if 0 J ( ) ǫγ τ γ = P ev J P P τ γ V primed =d/dv TRANSCENDENTAL EQUATION For given temperatures T L (hot) & T R (cold) ǫ 0 = ev 1 T R /T L ǫ 1 = ev J P primed = d/dv transmission ǫ 0 ǫ 1 energy Energy-integrals in J and P are Fermi-functions top-hat P & J are sums of logs and dilog.-functions ln [ 1+e (ǫ evj)/kbtj ] & Li2 [ e (ǫ ev j)/k BT j ] Getǫ 1 from above transcendental eq. for givent L (hot) &T R (cold) zero power output OPTIMAL TOP-HAT WIDTH increasing power output max. power output transmission transmission transmission energy energy energy UPPER-BOUND on POWER OUTPUT for N transverse modes Refrigerator cooling power: J 1 2 J qb π2 12h N(k BT L ) 2 Heat-engine electrical power: P P qb A 0π 2 6h N ( k B T L k B T R ) 2 with A Purely ÕÙ ÒØÙÑ, i.e. irrelevant forn J qb = Pendry (1983) as limit on entropy flow = single mode fermionic analogue of black-body Large N Bjorn Sothmann s talk Jordan et al (2013) For Ioffe s Kerosene Radio set-up : J qb,p qb 10nW per transverse mode 100W needs cross-section 1cm 1cm MAX. EFFICIENCY for GIVEN POWER OUTPUT Heat-eng. : small power output, P. ( ) η = η Carnot P 1 α 1 + P qb wherep qb is upper-bound Fridge : small power output, J. ( ) η = η Carnot J 1 α 2 + J qb wherej qb is upper-bound PARASITIC PHONON/PHOTON FLOWS Black-body photons: J ph (T 4 hot T4 cold ) J ph phonon in nanostructures Heron et al ( ) P Total efficiency η el&ph = J el +J ph Max efficiency - don t care what P J el (not givenp ) No phonons/photons: max η el&ph = NARROW transmission = η el&ph η Carnot Phonons/photons dominate: P=VI max η el&ph max P = WIDE transmission Max efficiency at given P : P η el (P) η el&ph (P) = P +η el (P)J ph RELAXATION modelled as Buttiker voltage probe (1988) Sometime inelastic scattering may increase efficiency see Casati s talk & Ora s talk Ê ÄÄ À Ê ÈÊÇ Ä Å ANSWERED in 2 limits : low power and max power Over-estimate never exceed results without relaxation.... open question for intermediate powers CONCLUSIONS Max. efficiency at given power : top-hat transcendental eq. for position/width Width grows with power Results : [1] max. possible power (quantum) [2] max. possible efficiency (quantum) transmission energy Is this the BEST thermoelectric at finite power output?? Open question: relaxation at intermediate power outputs? Open question: strongly correlated systems (Kondo, Luttinger, etc)? How to make top-hat? Buttiker said top-hat = band = chain quantum-dots
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