Laboratoire de Physique et Modelisation ´ des Milieux Condenses ´ Univ. Grenoble & CNRS, Grenoble, France
The best quantum thermoelectric at finite power output
Robert S. Whitney Preprint arXiv:1306.0826 Aachen — Nov 2013
OVERVIEW Ioffe (1958) Inst. Semicond. Leningrad
2 Watts (80-90 Volts)
MY QUESTION: What is MAX. efficiency at GIVEN power output?
♣ Quantum thermoelectric ♣ Nonlinear Landauer-Buttiker ¨
THOT = 572 K TCOLD = 305 K www.neazoi.com/technology/thermocouple.htm
♦ parasitic heat flows — phonons & photons ♦ inelastic / relaxation in quantum system
CENTRAL RESULTS Heat-engine efficiency: ηeng = P/J Output : power = P = V I Input : heat-current = J
Refrigerator efficiency ≡ coeff. of performance (COP): ηfri = J/P Output : J Input : P
[1] ABSOLUTE upper bound on Power Output: P ≤ Pqb ♣ Pqb is quantum-bound (ill-defined in classical thermodyn) [2] MAX. efficiency, ηeng (P ), at given power P ♣ function of P/Pqb
⇐= Quantum unlike Carnot efficiency = classical
Carnot Example low power : ηeng (P ) ≤ ηeng 1 − α1
p P/Pqb + · · ·
INTRODUCTION
Bulk versus Quantum : for LARGE ∆T Linearity requires temp. drop to be small on scale of thermalization Scal e therm of aliza tio
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NON-LINEAR
Nonlinear regime : efficiency η is meaningful, but ZT is NOT. Muralidharan-Grifoni (2012),
Whitney (2013), Meier-Jacquod (2013),
Michelini’s poster
Dictionary (for linear people) Linear formula√
η/ηCarnot =
√ZT +1−1 ZT +1+1
ZT = ∞ ⇐⇒ η = ηCarnot ZT = 3 ⇐⇒ η = 13 ηCarnot ZT = 0 ⇐⇒ η = 0
METHOD: scattering theory beyond linear response Heat current:
JL =
Z
∞
−∞
dǫ ǫ TRL (ǫ) h
ǫ−eVL ǫ−eVR f −f kB TL kB TR
♦ transmission function TRL (ǫ) = tr S † (ǫ) S(ǫ) ♦ Fermi-Dirac for ingoing particles: f (ǫ − eVj ) kB Tj ♣ obeys 2nd law thermodyn, etc.
beyond
Whitney PRB (2013)
linear-response : Hartree-like interactions – self-consistent Christen-Buttiker ¨ (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 = Vstop P = IV = 0
Mahan,Sofo (1996). Humphrey,Linke (2005)
Vanishing transmission width ⇒ reversibility (no entropy generated) ⇒ Carnot efficiency η = ηcarnot = 1 − TR /TL ... but no power
ORIGIN of THERMOELECTRICITY HOT Fermi sea
quantum system's transmission
COLD Fermi sea
I = 6 0 V = 6 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 GIVEN power output?
OPTIMIZING EFFICIENCY for GIVEN POWER OUTPUT (n + 1) variables: n slices + bias, V one constraint : power = P ♣ want minimum heat-flow J for given P
Proof: changing height τγ of slice γ , decreases J ∂P ǫγ J′ ∂J × = − if 0 > ∂τγ P eV P′ ∂τγ V
(increases efficiency)
primed = d/dV
For given temperatures TL ǫ0 =
eV 1 − TR /TL
(hot)
& TR
ǫ1 = eV
(cold) ′
J P′
primed = d/dV
transmission
TRANSCENDENTAL EQUATION
ǫ0
ǫ1 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 )/kB Tj & Li2 −e−(ǫ−eVj )/kB Tj Get ǫ1 from above transcendental eq. for given TL (hot) & TR (cold)
OPTIMAL TOP-HAT WIDTH
transmission
max. power output
transmission
increasing power output
energy
energy
transmission
zero power output
energy
UPPER-BOUND on POWER OUTPUT for N transverse modes Refrigerator cooling power: J ≤
1 π2 Jqb ≡ N (kB TL )2 2 12h
Heat-engine electrical power: 2 A0 π 2 P ≤ Pqb ≡ N kB TL − kB TR 6h
with
A0 ≃0.192
• Purely quantum, i.e. irrelevant for N → ∞
• Jqb = 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 : Jqb , Pqb ∼ 10nW per transverse mode ⇒ 100W needs cross-section 1cm×1cm
MAX. EFFICIENCY for GIVEN POWER OUTPUT Heat-eng. : small power output, P . ! s P + ··· η = η Carnot 1 − α1 Pqb where Pqb is upper-bound
Fridge : small power output, J . ! s J Carnot η=η 1 − α2 + ··· Jqb where Jqb is upper-bound
PARASITIC PHONON/PHOTON FLOWS 4 4 Black-body photons: Jph ∝ (Thot − Tcold )
phonon in nanostructures Heron et al (2009-11)
Total efficiency ηel&ph =
Jph Jel
P Jel + Jph
Max efficiency - don’t care what P
P=VI Max efficiency at given P :
(not given P ) • No phonons/photons: max ηel&ph = NARROW transmission =⇒ ηel&ph → ηCarnot • Phonons/photons dominate: max ηel&ph ⇔ max P = WIDE transmission
ηel&ph (P ) =
P ηel (P ) P + ηel (P )Jph
RELAXATION modelled as Buttiker “voltage probe” (1988) Sometime – inelastic scattering may increase efficiency see Casati’s talk & Ora’s talk
REALLY HARD PROBLEM !!
ANSWERED in 2 limits : low power and max power Over-estimate never exceed results without relaxation. ... open question for intermediate powers
♣ Max. efficiency at given power : “top-hat” transcendental eq. for position/width
⇒ Width grows with power ♣ Results : [1] max. possible power
transmission
CONCLUSIONS
energy
(quantum)
[2] max. possible efficiency (quantum)
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”
