Solar cell as self-oscillating heat engine
Robert Alicki, David Gelbwaser-Klimovsky, Krzysztof Szczygielski
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov Solar cell as a self-oscillating heat engine
Robert Alicki , ∗ David Gelbwaser-Klimovsky , † and Krzysztof Szczygielski ‡ Institute of Theoretical Physics and Astrophysics,University of Gda´nsk, 80-952 Gda´nsk, Poland and Department of Chemistry and Chemical Biology,Harvard University, Cambridge, MA 02138, USA
Solar cells are engines converting energy supplied by the photon flux into work. All knowntypes of macroscopic engines and turbines are also self-oscillating systems which yield a periodicmotion at the expense of a usually non-periodic source of energy. The very definition of work inthe formalism of quantum open systems suggests the hypothesis that the oscillating “piston” is anecessary ingredient of the work extraction process. This aspect of solar cell operation is absent inthe existing descriptions and the main goal of this paper is to show that plasma oscillations providethe physical implementation of a piston.
INTRODUCTION
The standard model of work extraction is based on an engine composed of a working medium, a piston and two heatbaths that are at equilibrium with different temperatures. Its importance stems from its success to set an universalbound to any work extraction process, the Carnot bound, which shows, in agreement with the Kelvin’s formulationof the second law of thermodynamics, two different temperatures are needed for work extraction. Besides, there is acomplementary picture of an engine as a self-oscillating system “focusing on their ability to convert energy inputted atone frequency (usually zero) into work outputted at another, well-defined frequency” [1]. This picture seems to applyto all types of turbines and motors [1, 2] and one can expect that it is equally correct for engines powered by a fluxof photons.The standard description of the photovoltaic cell involves the following processes [3, 4]:i) generation of the charge carriers due to the absorption of photons,ii) subsequent separation of the photo-generated charge carriers in the junction.As noticed in [4] the often used explanation of the second process as caused by the emerging electric field in p-njunction cannot be correct. Charge separation is supposed to produce a DC current which, on the other hand, cannotbe driven in a closed circuit by a purely electrical potential difference. A standard thermodynamical explanation ofelectric current (work) generation in photovoltaic and thermoelectric heat devices is the following :Electrons gain energy in a form of heat current J H from the hot bath, then flow against potential difference Φproducing useful power P = J E Φ, where J E is an electric current. A part of the heat described by the heat current J C is dumped to the cold bath.The laws of thermodynamics put the following constraints J E Φ = J H − J C (1) J H T H − J C T C ≤ . (2)Adding kinetic equations describing the processes of electron-hole creation, thermalization to the ambient temperatureand recombination one obtains the correct formulas for the open circuit voltage and voltage-current relation. However,this picture does not explain the mechanism of persistent steady work extraction. Similarly, for a steam engine thenet pressure due to the temperature difference obviously provides the net force acting on the piston but to explainthe permanent periodic action of this engine we have to understand the details of operation of a piston linked to aflywheel and valves.This is exactly the place where the mechanism of self-oscillation supported by the external constant energy flowenters into the game. In the following we propose a model in which plasma oscillations play a role of the periodicmotion of a “piston” and show that, indeed, under realistic assumptions a positive power is supplied to this essentiallyclassical macroscopic oscillator. Subsequently, the collective charge oscillations at THz frequencies are rectified by ap-n junction diode (“valve”) to the output DC current.The mathematical formalism is based on the quantum Markovian master equations for slowly driven open quantumsystems studied in [5] (compare [6],[7] for the fast driving case), and consistent with thermodynamics. MODEL OF QUANTUM ENGINE
We consider a model of heat engine which consists of a “working medium” called simply a system, two baths atdifferent temperatures, and a “work reservoir” called often a “piston” which is a system supplying to or extractingwork from the working medium. Because work, in contrast to heat, is an ordered and deterministic form of energywe expect that a piston should be a macroscopic system operating in the semi-classical regime. Therefore, within thereasonable approximation can be replaced by external deterministic driving [8].For the readers convenience we briefly present the formalism of Quantum Master Equations (QME) based on theDavies weak coupling limit [11], the Lindblad-Gorini-Kossakowski-Sudarshan generators and their extension to slowlyvarying external driving. Then the thermodynamical consequences are discussed and a special generic class of modelswith diagonal, weak driving , applicable to the theory of solar cells, is presented.
Master equation for open system with constant Hamiltonian .The total system consisting of a system S weakly coupled to a bath B . The total Hamiltonian is a sum of threeterms:i) a ”bare” Hamiltonian of the system - H replaced in the final formulas by H - a“ physical renormalized Hamiltonian”containing the lowest order corrections,ii) Hamiltonian of the bath - H B ,iii) a system-bath (elementary) coupling: H int = A ⊗ F , h F i B = 0 . (3)where A and F are observables of the system and the bath, respectively. h F i B denotes the average with respect tothe stationary state of the bath.Two main ingredients enter the QME derived using Davies weak coupling limit procedure [11]:a) the spectral density of the bath G ( ω ) = Z + ∞−∞ e − iωt h F ( t ) F i B dt ≥ , (4)b) Fourier components of the coupling operator A ( t ) = U † ( t ) AU ( t ) = X { ω } A ( ω ) e iωt , U ( t ) = e − iHt/ ~ . (5)Introducing system Hamiltonian spectral decomposition and Bohr frequencies H = X k ǫ k | k ih k, { ω } = { ( ǫ k − ǫ l ) / ~ } (6)one obtains the relations [ H, A ( ω )] = − ~ ωA ( ω ) , A ( − ω ) = A † ( ω ) . (7)where A ( ω ) called transition operators or Lindblad operators correspond to energy exchange of ~ ω . The standardderivation yields the QME in the Schr¨odinger picture dρ ( t ) dt = − i ~ [ H, ρ ( t )] + L ρ ( t ) (8)where L ρ = 12 X { ω } G ( ω ) (cid:0) [ A ( ω ) ρ, A † ( ω )] + [ A ( ω ) , ρA † ( ω )] (cid:1) (9)For general interactions H int = P α A α ⊗ F α L ρ = 12 X α,β X { ω } G αβ ( ω ) (cid:0) [ A α ( ω ) ρ, A † β ( ω )] + [ A α ( ω ) , ρA † β ( ω )] (cid:1) (10)where [ G αβ ( ω )] is a positively defined relaxation matrix . Properties of QME
The QME’s obtained in the weak coupling limit possess the following properties [12], [13]:1) L possesses the Lindblad-Gorini-Kossakowski-Sudarshan structure what implies that the solution of QME is a completely positive, trace preserving, one-parameter semigroup .2) Hamiltonian part commutes with the dissipative one, i.e. ρ ( t ) = U ( t ) e t L ρ (0) = e t L U ( t ) ρ (0) , U ( t ) ρ ≡ U ( t ) ρU † ( t ) . (11)3) Diagonal (in H - basis) and off-diagonal density matrix elements evolve independently.4) The stationary state ¯ ρ satisfying [ H, ¯ ρ ] = 0 and L ¯ ρ = 0, always exists for finite-dimensional Hilbert spaces.5) For a single heat bath (i.e. a reservoir in the thermal equilibrium state):a) spectral density satisfies the KMS condition G ( − ω ) = e − ~ ω/k B T G ( ω ) , (12)b) the Gibbs state is stationary ρ eq = Z − exp n − Hk B T o , L ρ eq = 0 , (13)c) the following quadratic form h X, L ∗ Y i eq = Tr (cid:0) ρ eq X † L ∗ Y (cid:1) (14)is negatively defined [12],[13] , where L ∗ is the Heisenberg picture generator given by L ∗ X = 12 X α,β X { ω } G αβ ( ω ) (cid:0) A † β [ X, A α ( ω )] + (cid:0) [ A † β , X ] A α ( ω ) (cid:1) . (15)It means that the Heisenberg picture generator L ∗ can be treated as a hermitian, negatively defined operator actingon the space of (complex) observables equipped with the scalar product h X, Y i eq ≡ Tr (cid:0) ρ eq X † Y (cid:1) . Entropy balance and the Laws of Thermodynamics
We identify the physical entropy with the von Neumann entropy of the reduced density matrix S ( ρ ) = − k B Tr( ρ ln ρ )and use also the relative entropy S ( ρ | σ ) = k B Tr( ρ ln ρ − ρ ln σ ).For the solution ρ ( t ) of MME, and the stationary state ¯ ρ , ( S ( t ) ≡ S ( ρ ( t ))) ddt S ( t ) = κ ( t ) − k B ddt Tr (cid:0) ρ ( t ) ln ¯ ρ (cid:1) (16)where κ ( t ) = − ddt S ( ρ ( t ) | ¯ ρ ) = − k B Tr (cid:0) [ L ρ ( t )][ln ρ ( t ) − ln ¯ ρ ] (cid:1) ≥ entropy production . Positivity ofthe entropy production follows from the fact that for any completely positive and trace-preserving map Λ, S (Λ ρ | Λ σ ) ≤ S ( ρ | σ ) [14]. For many independent heat baths one obtains the Second Law in the following form dSdt − X k J k T k ≥ , (17)where J k is a heat current flowing from the k -th bath. The case of a slow piston
In order to define work we introduce the time-dependent Hamiltonian H ( t ) = H + V ( t ) with slowly varyingand typically periodic perturbation V ( t ) which gives a semi-classical description of a piston. We combine now theweak coupling assumption concerning the interaction of the system with several baths with a kind of adiabaticapproximation concerning the time-dependent driving. The former condition means, practically, that the relaxationrates are much smaller than the corresponding Bohr frequencies. The later is valid for the case when the time scale ofdriving is much slower than the time scale determined by the relevant Bohr frequencies. This is a similar situation tostandard adiabatic theorem in quantum mechanics and implies that in the derivation of Master equation we can putthe temporal values of Bohr frequencies { ω ( t ) } and transition operators { A α ( ω ( t )) } satisfying (7) with H replacedby H ( t ).Under the conditions of above one obtains the following form of QME ddt ρ ( t ) = − i ~ [ H ( t ) , ρ ( t )] + X j L j ( t ) ρ ( t ) , (18)where L j ( t ) is a LGKS generator obtained by a weak coupling to the j -th bath and for a fixed H ( t ). The propertiesof generators L j ( t ) imply the Zero-th Law of Thermodynamics ( β j = 1 /k B T j ) L j ( t ) ρ eqj ( t ) = 0 , ρ eqj ( t ) = e − β j H ( t ) Tr e − β j H ( t ) . (19)Using the definitions [5]: W -work provided by S , Q - heat absorbed by S , E - internal energy of S E ( t ) = Tr (cid:0) ρ ( t ) H ( t ) (cid:1) (20) ddt W ( t ) = − Tr (cid:0) ρ ( t ) dH ( t ) dt (cid:1) , (21) ddt Q ( t ) = Tr (cid:0) dρ ( t ) dt H ( t ) (cid:1) (22)= X j Tr (cid:0) H ( t ) L j ( t ) ρ ( t ) (cid:1) ≡ X j ddt Q j ( t ) , (23)where Q j is a heat absorbed by S from B j , one obtains the First Law of Thermodynamics ddt E ( t ) = ddt Q ( t ) − ddt W ( t ) . (24)The Second Law of Thermodynamics follows again from Spohn innequality ddt S ( t ) − X j T j ddt Q j ( t ) = X j σ j ( t ) ≥ σ j ( t ) is an entropy production caused by B j and given by σ j ( t ) = k B Tr (cid:0) L j ( t ) ρ ( t )[ln ρ ( t ) − ln ρ eqj ( t )] (cid:1) ≥ . (26) Weak, diagonal and periodic driving
We consider a generic case of oscillating weak driving V ( t ) which in the lowest order approximation can be replacedby the diagonal (in the basis of H ) operator V ( t ) = gM sin Ω t, [ H , M ] = 0 , (27)with the small coupling constant g <<
1. In this case all Hamiltonians H ( t ) commute.The unitary part of the dynamics U ( t ) governed by H ( t ) commutes with H and M . One can write L ( t ) = L [ ξ ( t )]where L [ ξ ] is computed with the system Hamiltonian H + ξM and ξ ( t ) = g sin Ω t . Again for different ξ ’s the super-operators L [ ξ ] commute with the Hamiltonian part − i [ H ( t ) , · ] and one can use their lowest order expansion withrespect to ξ L [ ξ ] = L [0] + ξ L ′ [0] + O ( ξ ) , (28)In the next step we use the lowest order expression for the dissipative part of the super-propagatorΛ D ( t ) = exp (cid:8)Z t L [ ξ ( s )] ds (cid:9) (29) ≃ e t L [0] + g Z t (sin Ω s ) e ( t − s ) L [0] L ′ [0] e s L [0] ds. Applying now the definition of work (21), using the commutation properties of the generator L [ ξ ], and the fact that[ U ( t ) , M ] = 0, one obtains the formula for the stationary average power output¯ P ≡ − lim t →∞ t Z t Tr (cid:0) ρ ( t ) dH ( t ) dt (cid:1) dt (30)= − g Ω lim t →∞ t Z t Tr (cid:16) M Λ D ( t ) ρ (0) (cid:17) cos Ω t dt. Inserting the lowest order expression (29) into (30) and using the fact that all super-operators commute one cancompute the super-operator-valued integral like a usual one. Then, we apply the obvious assumption that the unper-turbed dynamics e t L [0] drives, asymptotically, any initial state ρ (0) to the stationary state denoted by ¯ ρ [0]. Finally,the limit t → ∞ in (30) can be performed leading to the second order approximation for the average output power¯ P = 12 g Tr (cid:16) M Ω Ω + ( L [0]) L ′ [0]¯ ρ [0] (cid:17) . (31) L [ ξ ] also possesses the stationary state ¯ ρ [ ξ ], i.e. L [ ξ ]¯ ρ [ ξ ] = 0 and hence we can use the identity L ′ [ ξ ]¯ ρ [ ξ ] = −L [ ξ ]¯ ρ ′ [ ξ ] (32)where “prime” denotes the derivative with respect to ξ . Then, replacing the Schroedinger picture generator L [0] bythe Heisenberg picture one L ∗ [0] we can transform the formula (31) into¯ P = − g Tr (cid:16) ¯ ρ ′ [0] Ω Ω + ( L ∗ [0]) L ∗ [0] M (cid:17) . (33)For the case when the decay rate of M is much lower than the modulation frequency we can neglect ( L ∗ [0]) in (33)to obtain the simplified expression ¯ P = − g Tr (cid:16) ¯ ρ ′ [0] L ∗ [0] M (cid:17) . (34)The compact formula of above will be used to derive the specific expression for the solar cell power. No output power from a single heat bath
The obtained lowest order formulas for power (33), (34) are still consistent with thermodynamics. Namely, assumingthat the reservoir is a thermal equilibrium bath at the temperature T we have the following properties:1) ¯ ρ [ ξ ] is the Gibbs state with respect to the Hamiltonian H + ξM ,2) the Heisenberg picture generator L ∗ [0] is a negatively defined operator on the Hilbert space equipped with thescalar product h X, Y i eq ≡ Tr (cid:0) ¯ ρ [0] X † Y (cid:1) (compare with (14), (15)).Using the first property one can compute ¯ ρ ′ [0] and rewrite the formula (33) as¯ P eq = g k B T Tr (cid:16) ¯ ρ [0] M Ω Ω + ( L ∗ [0]) L ∗ [0] M (cid:17) (35)= g k B T h M, Ω Ω + ( L ∗ [0]) L ∗ [0] M i eq ≤ Feed-back mechanism
For a reservoir composed of two equilibrium ones at different temperatures a positive output power ¯
P >
MODEL OF SEMICONDUCTOR SOLAR CELL
A solar cell is an engine which produces work from heat exchanged with a non-equilibrium bath. The bath consistsof the photonic non-equilibrium reservoir characterized by the local state population n [ ω ] and the basically phononicheat bath at the temperature T of the device. The typical semiconductor solar cell consists of a moderately dopedp-type absorber, on both sides of which a highly doped layer is formed, n-type on the top side and p-type on the backside, respectively. The electronic states in the valence band and in the conduction band are labeled by the index k which corresponds to the quasi-momentum ~ k (spin can be easily added) with the energies E v ( k ) and E c ( k ) . Weassume a direct band structure with vertical optical transitions which preserve quasi-momentum (see fig. 1). electron - holecreation conductionbandvalencebandintrabandthermalization k gap ~ ω photon ~ ω k recombinationphoton ~ ω k ′ intrabandthermalization FIG. 1. Schematic picture of leading processes involving electrons, photons and phonons in a semiconductor with a direct band.
The basic irreversible processes are the following:i) fast intraband thermalization processes mediated by phonons and described by L th .ii) optical transitions between the valence and conduction band, which create or annihilate electron-hole pairs,described by L em ,iii) non-radiative electron-hole recombination which is neglected in our idealized model. Plasma oscillations and current rectification
The fundamental question in the presented approach to work generation in solar cell is the origin of periodicoscillations which can be seen as classical. The frequency Ω is assumed in our derivations to be much smaller thanthe frequency ω , but much larger than the recombination rate in order to justify (34). The only phenomenon whichsatisfies all these requirements is plasma oscillation visible for p-n junctions. Their appearance is due to the factthat a p-n junction creates an interface between regions of different electron concentrations which can oscillate inspace producing collective macroscopic electric field oscillations. In several experiments such oscillations have beenobserved [9, 10], with typical frequencies Ω / π ≃ T Hz , much lower than ω ≃ T Hz corresponding to the energygap ∼ eV . On the other hand Ω is much higher than the recombination rate ∼ s − what justifies the transitionfrom (33) to (34). In the final step of cell operation the THz plasma oscillations must be converted into a direct ħ Ω FIG. 2.
Schematic picture of rectification of plasma oscillations.
Plasma oscillations caused by the collective motion offree carriers represented by the dimensionless amplitude ξ ( t ) in the eq.(38). For small ξ plasma self-oscillation is described bythe quantum harmonic oscillator coherently pumped by the feed-back mechanism. Due to the asymmetric diode-type potentialat the edges, charge oscillations are converted into a DC current. current. A qualitative picture of this mechanism is shown on Fig. 2. The plasmonic degree of freedom is representedby the quantum levels in the asymmetric potential which is harmonic for lower energies. Asymmetry is due to thep-n junction which defines an “easy” direction for the carrier flow (to the left). The work supplied to the oscillatordrives the unidirectional electric current. Hamiltonians, Master equations and stationary states
The electrons in a semiconductor occupying the conduction and valence bands are described by the annihilation andcreation operators c k , c † k and v k , v † k , respectively, subject to canonical anticommutation relations. The unperturbedHamiltonian reads H = X k (cid:16) E c ( k ) c † k c k + E v ( k ) v † k v k (cid:17) . (37)In the p-n junction a non-homogeneous free carrier distribution created in a self-consistent build-in potential can beperturbed producing collective plasma oscillations with the frequency Ω. The associated time-dependent perturbationadded to the electronic Hamiltonian (37) has a mean-field form ( N - number of atoms in the sample) ξ ( t ) M = ξ ( t ) 1 √ N E g X k (cid:0) c † k c k + v k v † k (cid:1) , (38)where ξ is a small dimensionless parameter describing the magnitude of deformation, E g is the relevant energy scale,and c † k c k , v k v † k are number operators of free electrons and holes, respectively.To apply the formulas derived in the previous sections we notice first that the driving perturbation (38) dependsonly on the total numbers of both types of carriers and hence does not interfere with intraband transitions. Therefore,the relevant Bohr frequency is associated with the gap E g yielding the time scale ∼ − s , much faster than themodulation period ∼ − s. This justifies the adiabatic approximation. The weak coupling assumption is for suresatisfied for very slow radiation recombination processes.Among the basic irreversible processes the intraband thermal relaxation is the fastest (thermalization time ∼ − s )and therefore, the stationary state of the electronic systems with the total Hamiltonian H + ξM is, within a reasonableapproximation, a product of grand canonical ensembles for electrons in conduction and valence band with the sametemperature T of the device and different electro-chemical potentials µ c and µ v , respectively. The associated densitymatrix has form ¯ ρ [ ξ ] = 1 Z [ ξ ] exp ( − k B T X k (cid:20)(cid:18) E c ( k ) + ξE g √ N − µ c (cid:19) c † k c k − (cid:18) E v ( k ) − ξE g √ N − µ v (cid:19) v k v † k (cid:21)(cid:27) . (39)The electro-chemical potentials are determined by the numbers of carriers and hence by doping and radiative andnon-radiative processes of electron-hole creation and recombination.Because the intraband thermalization to the ambient temperature T does not change the number of free electronsand holes, i.e. L ∗ th M = 0, the generator L ∗ th does not enter the formula for power (34). Here, one can doubt whetherfor such fast relaxation the weak coupling condition and hence the validity of the Markovian approximation leading to L th holds. However, intraband relaxation does not contribute to work generation but only determines the structureof the stationary state. The form of this state expressed in terms of Fermi-Dirac distributions is generally accepted inthe literature [4] and the accuracy of the Markovian approximation for the thermalization process is not very relevant.Finally, the contribution which remains in the eq. (34) describes the quasi-momentum preserving (vertical) transitionsand reads L em [0] ρ = 12 X k n γ rec ( k ) (cid:0) [ c k v † k , ρ v k c † k ] + [ c k v † k ρ, v k c † k ] (cid:1) + γ ex ( k ) (cid:0) [ c † k v k , ρ v † k c k ] + [ c † k v k ρ, v † k c k ] (cid:1)o , (40) γ rec ( k ) = 1 τ se (cid:2) n ( ω k ) (cid:3) , γ ex ( k ) = 1 τ se n ( ω k ) (41)where τ se is the spontaneous emission time, ~ ω k = E c ( k ) − E v ( k ), and n ( ω ) denotes a number of photons occupyinga state with the frequency ω . Power and efficiency
One can insert all elements computed in the previous section into the expression for power (34). Then we usethe properties of the quasi-free (fermionic Gaussian) stationary state (39) which allow to reduce the averages ofeven products of annihilation and creation fermionic operators into sums of products of the Fermi-Dirac distributionfunctions f c ( k ) = 1 e β ( E c ( k ) − µ c ) + 1 , f v ( k ) = 1 e β ( E v ( k ) − µ v ) + 1 , (42)with β = 1 /k B T .The leading order contribution to power possesses a following form¯ P = g E g k B T N car N X k (cid:16) γ ex ( k ) (cid:2) − f c ( k ) (cid:3) f v ( k ) (43) − γ rec ( k ) (cid:2) − f v ( k ) (cid:3) f c ( k ) (cid:17) , where N car = P k (cid:2) h c † k c k i + h v k v † k i (cid:3) is the total number of free charge carriers.One can introduce the local temperature of light T [ ω ] defined by e − ~ ω/k B T [ ω ] = n ( ω )1 + n ( ω ) . (44)For the incident sunlight on Earth a rough approximation holds n sun ( ω ) = λe ~ ω/k B T s − , (45)where T s ≃ K is the temperature of the Sun surface and λ = [ R sun /R ] ≃ × − is the geometrical factor ( R sun - Sun radius) which takes into account the photon density reduction at large distance from the source. In particular,for the typical value of the energy gap E g = ~ ω ≃ eV the effective temperature of sunlight T [ ω ] ≃ K .Because the product of population numbers for free carriers given by (cid:2) − f v ( k ) (cid:3) f c ( k ) is essentially concentratedon the interval ω k ∈ [ ω , ω + O ( k B T / ~ )], and T ≃ K ≪ T [ ω ] ≃ K ≪ ~ ω /k B ≃ K , the expression(43) can be approximated by¯ P = g E g N car ¯ Fk B T τ se × (cid:16) exp n k B T (cid:16)h − TT [ ω ] i ~ ω − e Φ (cid:17)o − (cid:17) (46)where ¯ F = N P k (cid:2) n ( ω k ) (cid:3)(cid:2) − f v ( k ) (cid:3) f c ( k ) > voltage Φ is identifiedwith the difference of electro-chemical potentials, i.e. e Φ ≡ µ c − µ v .The condition for work generation by the solar cell reads e Φ < e Φ = η C E g , η C = 1 − TT [ ω ] (47)what implies that Φ is an open-circuit voltage of the cell for the idealized case [15].The presence of the Carnot factor η C suggests also the interpretation of the eq. (47) in terms of thermodynamicalefficiency. Indeed, the incident photon of the frequency ω > ω produces an excitation of the energy close to E g in theprocess of electron-hole creation followed by the fast thermalization of an electron to the bottom of the conductionband, and a hole to the top of the valence one. Then, a part of energy E g is transformed into useful work, equal atmost e Φ per single electron flowing in the external circuit. The maximal efficiency η max under the conditions thateach photon with the energy higher than the gap produces an electron-hole pair and non-radiative recombinationprocesses are neglected, is given by the product η max = η u · η C , where η u is the so-called ultimate efficiency computedunder the assumptions:a) “... photons with energy greater than E g produce precisely the same effect as photons of energy E g , while photonsof lower energy will produce no effect” [17],b) the whole E g is transformed into work.Under standard illumination conditions the ultimate efficiency of a solar cell can reach 44% and the Carnot factor isabout 70% what yields η max ≃
31% - the Shockley’s detailed balance limit [17]. Actually, photons are absorbed alongtheir path in the absorber and n ( ω ) decays exponentially with the penetration distance. Taking a more realisticaverage value ¯ T = ( T [ ω ] + T ) / ≃ K one obtains η max ≃ GaAs solar cells.
Conclusions
The presented model based on the idea of self-oscillations explains the dynamical origin of work generation inphotovoltaic cells which is not present in the standard “static” picture. The main new ingredient is the role of plasmaoscillation as a “piston” which transforms the steady heat input from the photon flux into periodic motion. Thismodel provides a bridge between the theory of driven quantum open systems applied to heat engines and the theoryof photovoltaic devices. The formulas (44) and (47) explain in a simple way the meaning of the “light temperature”,the Carnot bound, and the linear relation between the open circuit voltage and the device temperature.The experimental verification of this model should provide the evidence of THz plasma oscillation in the devicewith the amplitude square proportional to the power output. Such oscillations produce a weak THz radiation which,in principle, could be detected.0The similar ideas can be applied to other types of heat engines with “hidden self-oscillations”. It seems thatthermoelectric devices based either on bimetallic or semiconductor p-n junctions can be described by the very similarmodels. Plasma oscillation remains a piston and sunlight is replaced by the hot bath. For organic photovoltaicsystems, proton pumps or photosynthesis there exists quite strong evidence of the important role of coherent molecularoscillations played in the energy and charge transfer (see e.g. [18]). It is plausible that those oscillations can play therole of a piston in the work extraction mechanism as well.
Acknowledgments
R.A. and K.S. are supported by the Foundation for Polish Science TEAM project co-financed by the EU EuropeanRegional Development Fund. D.G-K is supported by CONACYT. ∗ e-mail: fi[email protected] † e-mail: [email protected] ‡ e-mail: fi[email protected][1] A. Jenkins, Self-oscillation , Physics Reports , 167-222 (2013)[2] A. A. Andronov, A. A. Vitt and S.E. Khakin,
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