Reaching optimal efficiencies using nano-sized photo-electric devices Bart Cleuren in collab. with Bob Rutten and Massimiliano Esposito July 20, 2009 UCSD bart.cleuren@uhasselt.be Bart Cleuren Reaching optimal efficiencies using nano-sized photo-electric devices
Introduction: Carnot efficiency Sadi Carnot η c = 1 − T c (1796-1832) T h T c Q c = Q h Q h T h T h T c T c W = (1- )Q h T h fundamental result, universal upper limit no energy losses solar cells: η c ≈ 95% reversible operation (entropy production = 0) → isothermal parts are infinitely slowly work → power = 0 = cycle time → ∞ Bart Cleuren Reaching optimal efficiencies using nano-sized photo-electric devices
Introduction: Solar Cells in reality: lower efficiency η ≈ 24% reasons: energy losses / dissipation heat generation due to electron/hole relaxation within band thermal recombination processes non-zero power output / irreversible operation in practice: operation at maximum power output Bart Cleuren Reaching optimal efficiencies using nano-sized photo-electric devices
Introduction: efficiency at maximum power F.L. Curzon and B. Ahlborn, Am. J. Phys. 43, 1974 � T c η ca = 1 − T h remarks: (cfr. previous talk) not an upper limit ↔ Carnot (see further) eff. @ max. power: highest for strongly coupled systems universality for strongly coupled systems in the linear term: η = η c 2 + O ( η 2 c ) and sometimes also in the quadratic term Bart Cleuren Reaching optimal efficiencies using nano-sized photo-electric devices
Introduction: efficiency at maximum power F.L. Curzon and B. Ahlborn, Am. J. Phys. 43, 1974 � T c η ca = 1 − T h remarks: (cfr. previous talk) not an upper limit ↔ Carnot (see further) eff. @ max. power: highest for strongly coupled systems universality for strongly coupled systems in the linear term: η = η c 2 + O ( η 2 c ) and sometimes also in the quadratic term topic of this talk: efficiency at maximum power of a nano-sized solar cell Bart Cleuren Reaching optimal efficiencies using nano-sized photo-electric devices
Nano solar cell E r E l nano structure with 2 energy levels (no band structure!) Bart Cleuren Reaching optimal efficiencies using nano-sized photo-electric devices
Nano solar cell T T μ r E r μ l E l nano structure with 2 energy levels (no band structure!) contacts: two electron reservoirs at the same (ambient) temperature but with different chemical potential µ r = µ l + qV Bart Cleuren Reaching optimal efficiencies using nano-sized photo-electric devices
Nano solar cell T s T T μ r E r μ l E l nano structure with 2 energy levels (no band structure!) contacts: two electron reservoirs at the same (ambient) temperature but with different chemical potential µ r = µ l + qV solar excitation/recombination of electrons Bart Cleuren Reaching optimal efficiencies using nano-sized photo-electric devices
Nano solar cell ”a minimal model for solar energy conversion” T s T T μ r E r μ l E l nano structure with 2 energy levels (no band structure!) contacts: two electron reservoirs at the same (ambient) temperature but with different chemical potential µ r = µ l + qV solar excitation/recombination of electrons thermal (non-radiative) excitation/recombination of electrons Bart Cleuren Reaching optimal efficiencies using nano-sized photo-electric devices
Nano solar cell: electron dynamics flow of electrons: stochastic description (master equation for driven open quantum systems) 0 l r E l E r E r E l E r E l coupling constants with the reservoirs: Γ l , Γ r , Γ nr and Γ s Bart Cleuren Reaching optimal efficiencies using nano-sized photo-electric devices
Nano solar cell: electron dynamics flow of electrons: stochastic description (master equation for driven open quantum systems) 0 l r E l E r E r E l E r E l p i ∝ e − β ( E i − µ ) in equilibrium: grand-canonical distribution Bart Cleuren Reaching optimal efficiencies using nano-sized photo-electric devices
Nano solar cell: electron dynamics stationary electron (particle) current: J = k l 0 p 0 − k 0 l p l T T μ r E r J μ l E l Bart Cleuren Reaching optimal efficiencies using nano-sized photo-electric devices
Nano solar cell: electron dynamics stationary electron (particle) current: J = k l 0 p 0 − k 0 l p l T T μ r E r J s J nr μ l E l two contributions: J = J s + J nr with: J s → pumping of sun, ∝ Γ s J nr → non-radiative excitation/recombination, ∝ Γ nr Bart Cleuren Reaching optimal efficiencies using nano-sized photo-electric devices
Nano solar cell: thermodynamics heat flows due to excitation/recombination: ˙ Q s = ( E r − E l ) J s Q l T, μ l ˙ ε r Q nr = ( E r − E l ) J nr Q s T s Q r T, μ r ε l T heat flows from contacts: Q nr ˙ P Q l = ( E l − µ l ) J ˙ Q r = ( E r − µ r ) J work source power: conservation of energy P = ( µ r − µ l ) J = ( qJ ) V Bart Cleuren Reaching optimal efficiencies using nano-sized photo-electric devices
Nano solar cell: thermodynamics heat flows due to excitation/recombination: ˙ Q s = ( E r − E l ) J s Q l T, μ l ˙ ε r Q nr = ( E r − E l ) J nr Q s T s Q r T, μ r ε l T heat flows from contacts: Q nr ˙ P Q l = ( E l − µ l ) J ˙ Q r = ( E r − µ r ) J work source power: conservation of energy P = ( µ r − µ l ) J = ( qJ ) V efficiency: = ( µ r − µ l ) J η = P ˙ ( E r − E l ) J s Q s Bart Cleuren Reaching optimal efficiencies using nano-sized photo-electric devices
Nano solar cell: efficiency at maximum power setting Γ nr = 0 only solar excitation/recombination J = J s each absorbed photon pumps one electron ! 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 [ Γ l = Γ r = Γ s = Γ and Γ nr = α Γ ] Bart Cleuren Reaching optimal efficiencies using nano-sized photo-electric devices
Nano solar cell: efficiency at maximum power setting Γ nr = 0 only solar excitation/recombination J = J s when Γ nr � = 0 → decrease of efficiency due to dissipation 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 [ Γ l = Γ r = Γ s = Γ and Γ nr = α Γ ] Bart Cleuren Reaching optimal efficiencies using nano-sized photo-electric devices
Nano solar cell: efficiency at maximum power entropy production: Q l T, μ l ε r ˙ Q l + ˙ ˙ Q r + ˙ Q s Q s Q nr T s Q r ˙ T, μ r S i = − − ε l T s T T Q nr P Since P = ˙ Q s + ˙ Q l + ˙ Q r + ˙ Q nr work source combination gives the familiar expression: S i = ˙ ˙ Q s F U + JF N with thermodynamic forces: F U = 1 T − 1 F N = µ l − µ r � � = − qV ; T s T T Bart Cleuren Reaching optimal efficiencies using nano-sized photo-electric devices
Nano solar cell: efficiency at maximum power linear regime (small thermodynamic forces): ˙ Q s ≈ L UU F U + L UN F N L ij = Onsager coefficients J ≈ L NU F U + L NN F N L NU = L UN : cross coupling @ max. power: κ 2 � 1 − 3 � η = η c ˙ S i = F 2 4 κ 2 U L UU 2 − κ 2 2 with: efficiency is maximal when κ 2 = 1 → STRONG COUPLING and determinant of Onsager matrix = 0 Bart Cleuren Reaching optimal efficiencies using nano-sized photo-electric devices
Nano solar cell: efficiency at maximum power setting Γ l = Γ r = Γ s = Γ and Γ nr = α Γ gives: 1.0 f( � ) 0.8 η = η c 0.6 2 f ( α ) 0.4 0.2 0.0 0 1 2 3 4 5 6 � → without strong coupling: fast decrease of efficiency Bart Cleuren Reaching optimal efficiencies using nano-sized photo-electric devices
Nano solar cell: efficiency at maximum power Second order expansion (still strong coupling): η = η c 2 + 0 . 09288 η 2 c + . . . No universality! Bart Cleuren Reaching optimal efficiencies using nano-sized photo-electric devices
Nano solar cell: efficiency at maximum power Second order expansion (still strong coupling): η = η c 2 + 0 . 09288 η 2 c + . . . No universality! No collapse of forces and fluxes at second order! J = L F + ( M UU F 2 U + M UN F U F N + M NN F 2 N ) + . . . with F = ( E r − E l ) F U + F N Bart Cleuren Reaching optimal efficiencies using nano-sized photo-electric devices
Nano solar cell: efficiency at maximum power Second order expansion (still strong coupling): η = η c 2 + 0 . 09288 η 2 c + . . . No universality! No collapse of forces and fluxes at second order! J = L F + ( M UU F 2 U + M UN F U F N + M NN F 2 N ) + . . . with F = ( E r − E l ) F U + F N But: M UU − E g M UN + E 2 g M NN = 0 as a consequence of the fluctuation theorem Bart Cleuren Reaching optimal efficiencies using nano-sized photo-electric devices
Nano solar cell: efficiency at maximum power Solar energy converter: Thermoelectric converter: T s T l T r T T μ r μ r E r μ l μ l E l Bart Cleuren Reaching optimal efficiencies using nano-sized photo-electric devices
Nano solar cell: efficiency at maximum power Solar energy converter: Thermoelectric converter: T l T F U T r F U T T μ r μ r E r μ l F N μ l E l F N 3 reservoirs 2 reservoirs mixed statistics (Fermi - Bose) Fermi statistics Bart Cleuren Reaching optimal efficiencies using nano-sized photo-electric devices
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