Solar cell as self-oscillating heat engine

0.21 MB·English

by  Robert Alicki

Solar cell as a self-oscillating heat engine
Robert Alicki1,∗ David Gelbwaser-Klimovsky2,† and Krzysztof Szczygielski1‡
1Institute of Theoretical Physics and Astrophysics,
University of Gdan´sk, 80-952 Gdan´sk, Poland and
2 Department of Chemistry and Chemical Biology,
Harvard University, Cambridge, MA 02138, USA
5
Solar cells are engines converting energy supplied by the photon flux into work. All known
1
types of macroscopic engines and turbines are also self-oscillating systems which yield a periodic
0
motion at the expense of a usually non-periodic source of energy. The very definition of work in
2
the formalism of quantum open systems suggests the hypothesis that the oscillating “piston” is a

v necessary ingredient of thework extraction process. This aspect of solar cell operation is absent in
o theexistingdescriptionsandthemaingoalofthispaperistoshowthatplasmaoscillations provide
N thephysical implementation of a piston.

7
2
INTRODUCTION

]
h
c Thestandardmodelofworkextractionisbasedonanenginecomposedofaworkingmedium,apistonandtwoheat
e baths that are at equilibrium with different temperatures. Its importance stems from its success to set an universal
m
bound to any work extraction process, the Carnot bound, which shows, in agreement with the Kelvin’s formulation
- of the second law of thermodynamics, two different temperatures are needed for work extraction. Besides, there is a
t
a complementarypictureofanengineasaself-oscillatingsystem“focusing on their ability to convert energy inputted at
t
s one frequency (usually zero) into work outputted at another, well-defined frequency” [1]. This picture seems to apply
. to all types of turbines and motors [1, 2] and one can expect that it is equally correct for engines powered by a flux
t
a
of photons.
m
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,
d
n ii) subsequent separation of the photo-generated charge carriers in the junction.
o As noticed in [4] the often used explanation of the second process as caused by the emerging electric field in p-n
c junctioncannotbe correct. Chargeseparationis supposedtoproduceaDC currentwhich,onthe otherhand,cannot
[
be driven in a closed circuit by a purely electrical potential difference. A standard thermodynamical explanation of

2 electric current (work) generation in photovoltaic and thermoelectric heat devices is the following :
v Electrons gain energy in a form of heat current J from the hot bath, then flow against potential difference Φ
H
1
producing useful power P = J Φ, where J is an electric current. A part of the heat described by the heat current
0 E E
J is dumped to the cold bath.
7 C
0 The laws of thermodynamics put the following constraints
0
. J Φ=J J (1)
1 E H C
−
0
5
1 JH JC
0. (2)
: T − T ≤
v H C
i
X Addingkineticequationsdescribingtheprocessesofelectron-holecreation,thermalizationtotheambienttemperature
r andrecombinationoneobtainsthecorrectformulasfortheopencircuitvoltageandvoltage-currentrelation. However,
a
this picture does not explain the mechanism of persistent steady work extraction. Similarly, for a steam engine the
net pressure due to the temperature difference obviously provides the net force acting on the piston but to explain
the permanent periodic action of this engine we have to understand the details of operation of a piston linked to a
flywheel and valves.
This is exactly the place where the mechanism of self-oscillation supported by the external constant energy flow
enters into the game. In the following we propose a model in which plasma oscillations play a role of the periodic
motionofa“piston”andshowthat,indeed,underrealisticassumptionsapositivepowerissuppliedtothisessentially
classical macroscopic oscillator. Subsequently, the collective charge oscillations at THz frequencies are rectified by a
p-n junction diode (“valve”) to the output DC current.
ThemathematicalformalismisbasedonthequantumMarkovianmasterequationsforslowlydrivenopenquantum
systems studied in [5] (compare [6],[7] for the fast driving case), and consistent with thermodynamics.
2
MODEL OF QUANTUM ENGINE
We consider a model of heat engine which consists of a “working medium” called simply a system, two baths at
different temperatures, and a “work reservoir” called often a “piston” which is a system supplying to or extracting
work from the working medium. Because work, in contrast to heat, is an ordered and deterministic form of energy
we expect that a pistonshould be a macroscopicsystemoperating in the semi-classicalregime. Therefore,within the
reasonable approximationcan be replaced by external deterministic driving [8].
For the readers convenience we briefly present the formalism of Quantum Master Equations (QME) based on the
Daviesweakcouplinglimit[11],the Lindblad-Gorini-Kossakowski-Sudarshangeneratorsandtheirextensiontoslowly
varyingexternaldriving. Thenthe thermodynamicalconsequencesarediscussedandaspecialgenericclassofmodels
with 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 weakly coupled to a bath . The total Hamiltonian is a sum of three
S B
terms:
i)a”bare”Hamiltonianofthesystem-H0replacedinthefinalformulasbyH -a“physicalrenormalizedHamiltonian”
containing the lowest order corrections,
ii) Hamiltonian of the bath - H ,
B
iii) a system-bath (elementary) coupling:
H =A F , F =0. (3)
int B
⊗ h i
where A and F are observables of the system and the bath, respectively. F denotes the average with respect to
B
h i
the 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)= e−iωt F(t)F dt 0, (4)
B
h i ≥
Z−∞
b) Fourier components of the coupling operator
A(t)=U†(t)AU(t)= A(ω)eiωt, U(t)=e−iHt/~. (5)
X{ω}
Introducing system Hamiltonian spectral decomposition and Bohr frequencies
H = ǫ k k, ω = (ǫ ǫ )/~ (6)
k k l
| ih { } { − }
k
X
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 standard
derivation yields the QME in the Schr¨odinger picture
dρ(t) i
= [H,ρ(t)]+ ρ(t) (8)
dt −~ L
where
1
ρ= G(ω) [A(ω)ρ,A†(ω)]+[A(ω),ρA†(ω)] (9)
L 2
X{ω} (cid:0) (cid:1)
For general interactions H = A F
int α α⊗ α
1
ρ=P G (ω) [A (ω)ρ,A†(ω)]+[A (ω),ρA†(ω)] (10)
L 2 αβ α β α β
Xα,β X{ω} (cid:0) (cid:1)
where [G (ω)] is a positively defined relaxation matrix.
αβ
3
Properties of QME
The QME’s obtained in the weak coupling limit possess the following properties [12], [13]:
1) possesses the Lindblad-Gorini-Kossakowski-Sudarshan structure what implies that the solution of QME is a
L
completely positive, trace preserving, one-parameter semigroup.
2) Hamiltonian part commutes with the dissipative one, i.e.
ρ(t)= (t)etLρ(0)=etL (t)ρ(0), (t)ρ U(t)ρU†(t). (11)
U U U ≡
3) Diagonal (in H - basis) and off-diagonal density matrix elements evolve independently.
4) The stationary state ρ¯satisfying [H,ρ¯]=0 and ρ¯=0, always exists for finite-dimensional Hilbert spaces.
L
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/kBTG(ω), (12)
−
b) the Gibbs state is stationary
H
ρeq =Z−1exp , ρeq =0, (13)
−k T L
B
n o
c) the following quadratic form
X, ∗Y =Tr ρeqX† ∗Y (14)
eq
h L i L
is negatively defined [12],[13] , where ∗ is the Heisenberg(cid:0)picture gen(cid:1)erator given by
L
1
∗X = G (ω) A†[X,A (ω)]+ [A†,X]A (ω) . (15)
L 2 αβ β α β α
Xα,β X{ω} (cid:0) (cid:0) (cid:1)
It means that the Heisenberg picture generator ∗ can be treated as a hermitian, negatively defined operator acting
L
on the space of (complex) observables equipped with the scalar product X,Y Tr ρeqX†Y .
eq
h i ≡
(cid:0) (cid:1)
Entropy balance and the Laws of Thermodynamics
WeidentifythephysicalentropywiththevonNeumannentropyofthereduceddensitymatrixS(ρ)= k Tr(ρlnρ)
B
−
and use also the relative entropy S(ρσ)=k Tr(ρlnρ ρlnσ).
B
| −
For the solution ρ(t) of MME, and the stationary state ρ¯, (S(t) S(ρ(t)))
≡
d d
S(t)=κ(t) k Tr ρ(t)lnρ¯ (16)
B
dt − dt
(cid:0) (cid:1)
where κ(t)= dS(ρ(t)ρ¯)= k Tr [ ρ(t)][lnρ(t) lnρ¯] 0 is interpreted as an entropy production. Positivity of
−dt | − B L − ≥
theentropyproductionfollowsfromthefactthatforanycompletelypositiveandtrace-preservingmapΛ,S(ΛρΛσ)
(cid:0) (cid:1) | ≤
S(ρσ) [14]. For many independent heat baths one obtains the Second Law in the following form
|
dS J
k
0, (17)
dt − T ≥
k
k
X
where J is a heat current flowing from the k-th bath.
k
The case of a slow piston
In order to define work we introduce the time-dependent Hamiltonian H(t) = H + V(t) with slowly varying
0
and typically periodic perturbation V(t) which gives a semi-classical description of a piston. We combine now the
weak coupling assumption concerning the interaction of the system with several baths with a kind of adiabatic
4
approximation concerning the time-dependent driving. The former condition means, practically, that the relaxation
ratesaremuchsmallerthanthe correspondingBohrfrequencies. The lateris validforthe casewhenthe time scaleof
driving is muchslowerthan the time scale determined by the relevantBohrfrequencies. This is a similar situationto
standard adiabatic theorem in quantum mechanics and implies that in the derivation of Master equation we can put
the temporal values of Bohr frequencies ω(t) and transition operators A (ω(t)) satisfying (7) with H replaced
α
{ } { }
by H(t).
Under the conditions of above one obtains the following form of QME
d i
ρ(t)= [H(t),ρ(t)]+ (t)ρ(t), (18)
dt −~ Lj
j
X
where (t) is a LGKS generator obtained by a weak coupling to the j-th bath and for a fixed H(t). The properties
j
L
of generators (t) imply the Zero-th Law of Thermodynamics (β =1/k T )
j j B j
L
e−βjH(t)
(t)ρeq(t)=0 , ρeq(t)= . (19)
Lj j j Tre−βjH(t)
Using the definitions [5]: W-work provided by , Q - heat absorbed by , E - internal energy of
S S S
E(t)=Tr ρ(t)H(t) (20)
(cid:0) (cid:1)
d dH(t)
W(t)= Tr ρ(t) , (21)
dt − dt
(cid:0) (cid:1)
d dρ(t)
Q(t)=Tr H(t) (22)
dt dt
(cid:0) (cid:1) d
= Tr H(t) (t)ρ(t) Q (t), (23)
j j
L ≡ dt
j j
X (cid:0) (cid:1) X
where Q is a heat absorbed by from , one obtains the First Law of Thermodynamics
j j
S B
d d d
E(t)= Q(t) W(t). (24)
dt dt − dt
The Second Law of Thermodynamics follows again from Spohn innequality
d 1 d
S(t) Q (t)= σ (t) 0 (25)
j j
dt − T dt ≥
j
j j
X X
where σ (t) is an entropy production caused by and given by
j j
B
σ (t)=k Tr (t)ρ(t)[lnρ(t) lnρeq(t)] 0. (26)
j B Lj − j ≥
(cid:0) (cid:1)
Weak, diagonal and periodic driving
WeconsideragenericcaseofoscillatingweakdrivingV(t)whichinthelowestorderapproximationcanbereplaced
by the diagonal (in the basis of H ) operator
0
V(t)=gMsinΩt, [H ,M]=0, (27)
0
with the small coupling constant g <<1. In this case all Hamiltonians H(t) commute.
The unitary part of the dynamics U(t) governedby H(t) commutes with H and M. One can write (t)= [ξ(t)]
0
L L
where [ξ] is computed with the system Hamiltonian H +ξM and ξ(t)=gsinΩt. Again for different ξ’s the super-
0
L
operators [ξ] commute with the Hamiltonian part i[H(t), ] and one can use their lowest order expansion with
L − ·
respect to ξ
[ξ]= [0]+ξ ′[0]+ (ξ2), (28)
L L L O
5
In the next step we use the lowest order expression for the dissipative part of the super-propagator
t
Λ (t)=exp [ξ(s)]ds (29)
D
L
Z0
(cid:8) t (cid:9)
etL[0]+g (sinΩs)e(t−s)L[0] ′[0]esL[0]ds.
≃ L
Z0
Applying now the definition of work (21), using the commutation properties of the generator [ξ], and the fact that
L
[U(t),M]=0, one obtains the formula for the stationary average power output
1 t0 dH(t)
P¯ lim Tr ρ(t) dt (30)
≡−t0→∞t0 Z0 dt
1 t0(cid:0) (cid:1)
= gΩ lim Tr MΛ (t)ρ(0) cosΩtdt.
D
− t0→∞t0 Z0 (cid:16) (cid:17)
Inserting the lowest order expression (29) into (30) and using the fact that all super-operators commute one can
compute the super-operator-valuedintegrallike a usualone. Then, we apply the obviousassumption that the unper-
turbed dynamics etL[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
0
→∞
1 Ω2
P¯ = g2Tr M ′[0]ρ¯[0] . (31)
2 Ω2+( [0])2L
(cid:16) L (cid:17)
[ξ] also possesses the stationary state ρ¯[ξ], i.e. [ξ]ρ¯[ξ]=0 and hence we can use the identity
L L
′[ξ]ρ¯[ξ]= [ξ]ρ¯′[ξ] (32)
L −L
where “prime” denotes the derivative with respect to ξ. Then, replacing the Schroedinger picture generator [0] by
L
the Heisenberg picture one ∗[0] we can transform the formula (31) into
L
1 Ω2
P¯ = g2Tr ρ¯′[0] ∗[0]M . (33)
−2 Ω2+( ∗[0])2L
(cid:16) L (cid:17)
For the case when the decay rate of M is much lower than the modulation frequency we can neglect ( ∗[0])2 in (33)
L
to obtain the simplified expression
1
P¯ = g2Tr ρ¯′[0] ∗[0]M . (34)
−2 L
(cid:16) (cid:17)
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
Theobtainedlowestorderformulasforpower(33),(34)arestillconsistentwiththermodynamics. Namely,assuming
that 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,
0
2) the Heisenberg picture generator ∗[0] is a negatively defined operator on the Hilbert space equipped with the
L
scalar product X,Y Tr ρ¯[0]X†Y (compare with (14), (15)).
eq
h i ≡
Using the first property one can(cid:0)compute(cid:1)ρ¯′[0] and rewrite the formula (33) as
g2 Ω2
P¯ = Tr ρ¯[0]M ∗[0]M (35)
eq 2k T Ω2+( ∗[0])2L
B
(cid:16) L (cid:17)
g2 Ω2
= M, ∗[0]M 0 (36)
2k Th Ω2+( ∗[0])2L ieq ≤
B
L
which is obviously negative as well as its simplified version (34). Therefore, as expected, one cannot extract power
from a single heat bath.
6
Feed-back mechanism
For a reservoir composed of two equilibrium ones at different temperatures a positive output power P¯ > 0 can
be obtained and in this case the mechanism of self-oscillations works. Treating the external perturbation as caused
by a coupling of the system to a macroscopic oscillator we see that the positive power is supplied to the oscillator
increasing the amplitude of its oscillation (positive feed-back) up to the moment when the net energy supply from
the reservoir is compensated by the load attached to the oscillator.
MODEL OF SEMICONDUCTOR SOLAR CELL
Asolarcellis anengine whichproducesworkfromheatexchangedwith a non-equilibriumbath. The bathconsists
of the photonic non-equilibrium reservoircharacterizedby the local state population n[ω] and the basically phononic
heat bath at the temperature T of the device. The typical semiconductor solar cell consists of a moderately doped
p-type absorber,onbothsides ofwhichahighlydopedlayerisformed,n-type onthe topside andp-type onthe back
side, 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 (k) and E (k) . We
v c
assume a direct band structure with vertical optical transitions which preserve quasi-momentum (see fig. 1).
intraband conduction
thermalization band
photon
recombination
~ω
k photon ~ω0
gap
~ω
electron - hole k′
creation
valence
intraband band
thermalization
k
FIG.1. Schematicpictureofleadingprocessesinvolvingelectrons,photonsandphononsinasemiconductorwithadirectband.
The basic irreversible processes are the following:
i) fast intraband thermalization processes mediated by phonons and described by .
th
L
ii) optical transitions between the valence and conduction band, which create or annihilate electron-hole pairs,
described by ,
em
L
iii) non-radiative electron-hole recombination which is neglected in our idealized model.
7
Plasma oscillations and current rectification
The fundamental question in the presented approach to work generation in solar cell is the origin of periodic
oscillations which can be seen as classical. The frequency Ω is assumed in our derivations to be much smaller than
the frequency ω , but much larger than the recombinationrate in order to justify (34). The only phenomenon which
0
satisfies all these requirements is plasma oscillation visible for p-n junctions. Their appearance is due to the fact
that a p-n junction creates an interface between regions of different electron concentrations which can oscillate in
space producing collective macroscopic electric field oscillations. In several experiments such oscillations have been
observed[9,10],withtypicalfrequenciesΩ/2π 1THz,muchlowerthanω 1400THzcorrespondingtotheenergy
0
≃ ≃
gap 1eV. On the other hand Ω is much higher than the recombination rate 104s−1 what justifies the transition
∼ ∼
from (33) to (34). In the final step of cell operation the THz plasma oscillations must be converted into a direct
ħΩ
FIG.2. Schematic picture of rectification of plasma oscillations. Plasmaoscillations causedbythecollectivemotionof
free carriers represented by thedimensionless amplitude ξ(t) in theeq.(38). For small ξ plasma self-oscillation is described by
thequantumharmonicoscillatorcoherentlypumpedbythefeed-backmechanism. Duetotheasymmetricdiode-typepotential
at theedges, charge oscillations are convertedinto a DC current.
current. A qualitative picture of this mechanism is shown on Fig. 2. The plasmonic degree of freedom is represented
by the quantum levels in the asymmetric potential which is harmonic for lower energies. Asymmetry is due to the
p-n junction which defines an “easy” direction for the carrier flow (to the left). The work supplied to the oscillator
drives the unidirectional electric current.
Hamiltonians, Master equations and stationary states
Theelectronsinasemiconductoroccupyingtheconductionandvalencebandsaredescribedbytheannihilationand
creation operators ck, c†k and vk, vk†, respectively, subject to canonical anticommutation relations. The unperturbed
Hamiltonian reads
H0 = Ec(k)c†kck+Ev(k)vk†vk . (37)
k
X(cid:16) (cid:17)
In the p-n junction a non-homogeneous free carrier distribution created in a self-consistent build-in potential can be
perturbedproducingcollectiveplasmaoscillationswiththefrequencyΩ. Theassociatedtime-dependentperturbation
added to the electronic Hamiltonian (37) has a mean-field form (N - number of atoms in the sample)
1
ξ(t)M =ξ(t) Eg c†kck+vkvk† , (38)
√N
k
X(cid:0) (cid:1)
where ξ is a small dimensionless parameter describing the magnitude of deformation, E is the relevant energy scale,
g
and c†kck, vkvk† are number operators of free electrons and holes, respectively.
8
To apply the formulas derived in the previous sections we notice first that the driving perturbation (38) depends
onlyonthetotalnumbersofbothtypesofcarriersandhencedoesnotinterferewithintrabandtransitions. Therefore,
the relevant Bohr frequency is associated with the gap E yielding the time scale 10−15s, much faster than the
g
∼
modulation period 10−12s. This justifies the adiabatic approximation. The weak coupling assumption is for sure
∼
satisfied for very slow radiation recombination processes.
Amongthebasicirreversibleprocessestheintrabandthermalrelaxationisthefastest(thermalizationtime 10−12s)
∼
andtherefore,thestationarystateoftheelectronicsystemswiththetotalHamiltonianH +ξM is,withinareasonable
0
approximation, a product of grand canonical ensembles for electrons in conduction and valence band with the same
temperature T of the device and different electro-chemicalpotentials µ and µ , respectively. The associateddensity
c v
matrix has form
1 1 ξE
ρ¯[ξ]= exp Ec(k)+ g µc c†kck
Z[ξ] (−kBT k (cid:20)(cid:18) √N − (cid:19)
X
ξE
Ev(k) g µv vkvk† . (39)
− − √N −
(cid:18) (cid:19) (cid:21)(cid:27)
The electro-chemical potentials are determined by the numbers of carriers and hence by doping and radiative and
non-radiative processes of electron-hole creation and recombination.
Because the intraband thermalization to the ambient temperature T does not change the number of free electrons
and holes, i.e. ∗ M =0, the generator ∗ does not enter the formula for power (34). Here, one can doubt whether
Lth Lth
forsuchfastrelaxationtheweakcouplingconditionandhencethevalidityoftheMarkovianapproximationleadingto
holds. However, intraband relaxation does not contribute to work generation but only determines the structure
th
L
ofthe stationarystate. Theformofthis state expressedinterms ofFermi-Diracdistributions isgenerallyacceptedin
theliterature[4]andtheaccuracyoftheMarkovianapproximationforthethermalizationprocessisnotveryrelevant.
Finally,thecontributionwhichremainsintheeq. (34)describesthequasi-momentumpreserving(vertical)transitions
and reads
1
em[0]ρ= γrec(k) [ckvk†,ρvkc†k]+[ckvk†ρ,vkc†k]
L 2
k
Xn (cid:0) (cid:1)
+γex(k) [c†kvk,ρvk†ck]+[c†kvkρ,vk†ck] , (40)
(cid:0) (cid:1)o
1 1
γrec(k)= 1+n(ωk) ,γex(k)= n(ωk) (41)
τ τ
se se
(cid:2) (cid:3)
where τse is the spontaneous emission time, ~ωk =Ec(k) Ev(k), and n(ω) denotes a number of photons occupying
−
a 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 use
the properties of the quasi-free (fermionic Gaussian) stationary state (39) which allow to reduce the averages of
even products of annihilation and creation fermionic operators into sums of products of the Fermi-Dirac distribution
functions
1 1
f (k)= , f (k)= , (42)
c eβ(Ec(k)−µc)+1 v eβ(Ev(k)−µv)+1
with β =1/k T.
B
The leading order contribution to power possesses a following form
g2E2N
P¯ = g car γ (k) 1 f (k) f (k) (43)
ex c v
k T N −
B k
X(cid:16) (cid:2) (cid:3)
γ (k) 1 f (k) f (k) ,
rec v c
− −
(cid:2) (cid:3) (cid:17)
9
where Ncar = k c†kck 0+ vkvk† 0 is the total number of free charge carriers.
h i h i
One can introduce the local temperature of light T[ω] defined by
P (cid:2) (cid:3)
n(ω)
e−~ω/kBT[ω] = . (44)
1+n(ω)
For the incident sunlight on Earth a rough approximation holds
λ
n (ω)= , (45)
sun e~ω/kBTs 1
−
whereT 6000KisthetemperatureoftheSunsurfaceandλ=[R /R ]2 2 10−5isthegeometricalfactor(R
s sun 0 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 =~ω 1eV the effective temperature of sunlight T[ω ] 1000K.
g 0 0
Because the product of population numbers fo≃r free carriers given by 1 f (k) f (k) is essentia≃lly concentrated
v c
on the interval ωk [ω0,ω0+ (kBT/~)], and T 300K T[ω0] 1000−K ~ω0/kB 12000K, the expression
∈ O ≃ ≪ ≃ (cid:2) ≪ (cid:3) ≃
(43) can be approximated by
P¯ = g2Eg2NcarF¯ exp 1 1 T ~ω eΦ 1 (46)
0
k T τ × k T − T[ω ] − −
B se B 0
(cid:16) n (cid:16)h i (cid:17)o (cid:17)
where F¯ = N1 k 1+n(ωk) 1−fv(k) fc(k)>0 is independent of the size of a cell, and the voltage Φ is identified
with the difference of electro-chemical potentials, i.e. eΦ µ µ .
c v
P (cid:2) (cid:3)(cid:2) (cid:3) ≡ −
The condition for work generation by the solar cell reads
T
eΦ<eΦ =η E , η =1 (47)
0 C g C
− T[ω ]
0
what implies that Φ is an open-circuit voltage of the cell for the idealized case [15].
0
The presence of the Carnot factor η suggests also the interpretation of the eq. (47) in terms of thermodynamical
C
efficiency. Indeed,theincidentphotonofthefrequencyω >ω producesanexcitationoftheenergyclosetoE inthe
0 g
process of electron-hole creation followed by the fast thermalization of an electron to the bottom of the conduction
band, and a hole to the top of the valence one. Then, a part of energy E is transformed into useful work, equal at
g
most eΦ per single electron flowing in the external circuit. The maximal efficiency η under the conditions that
0 max
each photon with the energy higher than the gap produces an electron-hole pair and non-radiative recombination
processesareneglected,isgivenbythe productη =η η ,whereη istheso-calledultimate efficiency computed
max u C u
·
under the assumptions:
a) “... photons with energy greater than E produce precisely the same effect as photons of energy E , while photons
g g
of lower energy will produce no effect” [17],
b) the whole E is transformed into work.
g
Understandardilluminationconditionstheultimateefficiencyofasolarcellcanreach44%andtheCarnotfactoris
about 70%what yields η 31%- the Shockley’s detailed balance limit [17]. Actually, photons are absorbedalong
max
≃
their path in the absorber and n(ω ) decays exponentially with the penetration distance. Taking a more realistic
0
average value T¯ =(T[ω ]+T)/2 650K one obtains η 24%, much closer to the real performance of standard
0 max
≃ ≃
GaAs solar cells.
Conclusions
The presented model based on the idea of self-oscillations explains the dynamical origin of work generation in
photovoltaiccellswhichis notpresentinthestandard“static”picture. Themainnew ingredientisthe roleofplasma
oscillation as a “piston” which transforms the steady heat input from the photon flux into periodic motion. This
model provides a bridge between the theory of driven quantum open systems applied to heat engines and the theory
of 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 device
with the amplitude square proportionalto the power output. Such oscillations produce a weak THz radiationwhich,
in principle, could be detected.
10
The similar ideas can be applied to other types of heat engines with “hidden self-oscillations”. It seems that
thermoelectric devices basedeither onbimetallic orsemiconductor p-njunctions canbe describedby the verysimilar
models. Plasma oscillation remains a piston and sunlight is replaced by the hot bath. For organic photovoltaic
systems,protonpumpsorphotosynthesisthereexistsquitestrongevidenceoftheimportantroleofcoherentmolecular
oscillations playedin the energy and chargetransfer (see e.g. [18]). It is plausible that those oscillations canplay the
role of a piston in the work extraction mechanism as well.
Acknowledgments
R.A. andK.S. aresupportedby the Foundation forPolishScience TEAMproject co-financedby the EU European
Regional Development Fund. D.G-K is supported by CONACYT.
∗ e-mail: fi[email protected]
† e-mail: [email protected]
‡ e-mail: fi[email protected]
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