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Limiting efficiency of coupled thermal and photovoltaic converters
Solar Energy Materials & Solar Cells 58 (1999) 147}165
Limiting e$ciency of coupled thermal and photovoltaic converters Antonio Luque*, Antonio MartmH Instituto de Energn& a Solar, Universidad Polite& cnica de Madrid, Ciudad Universitaria, 28040 Madrid, Spain Received 27 May 1998; accepted 20 October 1998
Abstract This paper presents a general energetic and entropic analysis of ideal photovoltaic and solar-thermal converters. Its purpose is to determine the e$ciency limit when both types of converters operate together (hybrid converters). It has been found that, while in practical cases hybrid converters may give very high e$ciency (61.7% vs. 40.0% of the solar thermal at 500 K and 40.7% of the photovoltaic at 300 K), in the limiting case of a system formed by an in"nite number of band gaps, the e$ciency of hybrid converters, 86.8%, is strictly equivalent to photovoltaic or solar thermal converters. Conversely, hybrid systems operating with one gap give an e$ciency of 86.7%, very close to the previous one and higher than the top e$ciency achievable with a single temperature solar thermal, 85.4%. 1999 Elsevier Science B.V. All rights reserved. Keywords: Photovoltaic; Thermal; Limiting e$ciency; Stack
Notation A c *e $ E E e e e
area speed of light quasi-fermi level split rate of energy associated rate of energy associated rate of energy associated rate of energy associated energy of a photon
with with with with
the the the the
escaping radiation entering radiation monochromatic escaping radiation entering monochromatic radiation
* Corresponding author. 0927-0248/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 4 8 ( 9 8 ) 0 0 1 9 9 - 8
148
e $ e $ e % e
e + H h g k k k k k N N n n Q Q q S S S p ¹ ¹ ¹ < < = = = w X X z
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quasi-fermi level of electrons quasi-fermi level of holes semiconductor bandgap lower limit of the range of energies of interest upper limit of the range of energies of interest eH tendue or Lagrange invariant of the bundle of luminous rays reaching or leaving the cell Planck's constant e$ciency of the converter Boltzman's constant chemical potential associated with the escaping radiation chemical potential associated with the escaping radiation under open-circuit conditions chemical potential associated with the entering radiation chemical potential of the thermal equivalent emitted radiation rate of escaping photons rate of photons absorbed from the entering radiation rate of monochromatic escaping photons rate of monochromatic photons absorbed from the entering radiation rate of heat delivered to the ambient rate of heat delivered by the cell electron electric charge rate of entropy associated with the escaping radiation rate of entropy associated with the entering radiation rate of irreversible entropy generation Stefan}Boltzman's constant temperature of the ambient temperature of the cell temperature of the sun cell voltage open-circuit voltage rate of work (power) produced at the photovoltaic converter rate of work (power) produced at the thermal engine total rate of work produced at the hybrid system total rate of work produced at the hybrid system when operating with monochromatic radiation grand potential #ow associated with the entering radiation grand potential #ow associated with the escaping radiation normalised variable de"ned as (e-k)/k¹
1. Introduction A solar energy converter is a device that receives solar radiation and produces useful power. In compliance with the laws of thermodynamics, the converter must
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supply some heat to the ambient and, in addition, it must emit some radiant energy. The possibility of emitting some radiant energy can also be understood as a consequence of the need that if there is a path for rays to enter into the receiver, by time reversal, the same path has to allow the rays to escape (in the absence of external "elds violating the time micro-reversibility). This is a general situation for luminous photons with the standard terrestrial energies where photovoltaic conversion takes place. Presumably, in less ideal cases, additional paths for escaping rays also exist, linking the receiver not with the source but with the darker parts of the surroundings. The nature of this escaping radiation de"nes the nature of the solar converter. If it is free radiation at a certain temperature } the temperature of the receiver, ¹ } then the converter is solar-thermal. If it is matter-coupled (luminescent) radiation characterised by a non-zero chemical potential at the ambient temperature, then the converter is photovoltaic. The solar cell used as a receiver in photovoltaic converters is the device producing this luminescent radiation. Hybrid systems, which are dealt within this paper, are also possible. For them, the receiver is a solar cell operating at a temperature ¹ above the ambient that produces work in combination with a thermal engine operated between the cell, ¹ , and the ambient, ¹ , temperatures. For such a hybrid system, the escaping radiation will be a matter-coupled radiation at a temperature above the ambient and in some cases will be characterised by a zero chemical potential at some wavelengths and by a non-zero chemical potential at others. The hybrid systems which combine photovoltaic conversion and di!erent uses of the heat rejected have sometimes been proposed [1]. This paper does not deal with such systems. They provide two di!erent goods, electricity (non-entropic useful work) and heat (with entropy) whose economic values are di!erent, the latter varying with the temperature at which the heat is rejected. In this paper we will assume that the heat rejected by the solar cells is used to drive a thermal engine that produces mechanical work which can be theoretically converted into electricity without losses. Then, the practical question addressed here is if this approach will allow for a better e$ciency conversion of solar energy into useful work. The question is relevant because if we raise the cell temperature to obtain a better engine e$ciency then the e$ciency of the cell decreases. The global balance is uncertain. This question has been partially answered by De Vos [2] and his reply is a$rmative. However, he does not refer to a general case but to a speci"c one because he only deals with what we call low pass hybrid converters, discussed later in this paper. In fact, we shall prove that hybrid systems, at their best con"guration, cannot exceed the e$ciency of photovoltaic systems. They are strictly equivalent. This point is important because it means that hybrid converters cannot reach the Landsberg's e$ciency [3] either, which is the e$ciency upper bound of any solar converter that obeys the time micro-reversibility. That is in spite of the fact that hybrid systems may ful"l one of the two thermodynamic conditions (but not both simultaneously) necessary to achieve this e$ciency: zero entropy generation rate or thermal equivalent to the ambient temperature of the emitted radiation. It is worth mentioning that none of the known ideal devices is able to attain the Landsberg's e$ciency. Therefore, it is unknown if there is a theoretical reason that prevents solar converters from achieving it.
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Our paper deals with ideal converters, which means that the production of irreversible entropy is avoided as much as possible and is isotropically illuminated by the sun by means of an ideal concentrator. In Section 2 we will describe the theory of the hybrid converter which shows the contributions from the photovoltaic and thermal components to the total power delivered by the system and to the irreversible entropy generation rate. Maximum e$ciencies and entropy generation rates will be computed in Section 3 where results are also discussed. Section 4 will emphasise the topics concerning the limiting e$ciency of the hybrid converters and Section 5 will summarise the conclusions of this work. 2. Theoretical analysis In this section we present the theoretical expression of the useful power and of the irreversible generation of entropy in the photovoltaic cell, the thermal engine and the complete hybrid converter. The results obtained are summarised in Table 1 for the readers' convenience. 2.1. The photovoltaic cell A sketch of a hybrid converter is shown in Fig. 1. The inner box represents the photovoltaic component (or solar cell) whose operation will be reviewed in the following paragraphs. The cell absorbs radiant energy from the solar photons at a rate E and emits it at a rate E . It also produces electric (useful) work at a rate = and delivers heat to the crystal network at a rate Q . The "rst law of thermodynamics is then written for this device as E "E #= #Q and the second law as
(1)
Q S #S "S # , ¹
(2)
Table 1 Summaries of the output power, =, contributions to the hybrid converter and the entropy generation rate, S , from the photovoltaic converter and the thermal engine =
S
Photovoltaic
q<(N !N )
1 1 k q< X X E ! #N 1 ! # ! 1 ¹ ¹ ¹ ¹ ¹ ¹
Thermal
¹ (E !q
Total hybrid converter
¹ ¹ (E !E ) 1! #q<(N !N ) ¹ ¹
0 1 1 k q< X X E ! #N 1 ! # ! 1 ¹ ¹ ¹ ¹ ¹ ¹
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Fig. 1. Sketch of the hybrid converter showing the di!erent energy and heat #uxes involved as well as the power delivered.
In this equation, S's are rates of entropy, associated with the entering radiation (subindex e) and with the escaping radiation (subindex c). The entropy supplied to the cell crystal network } here acting as a cold reservoir } is Q /¹ , ¹ being the cell temperature. S is a positive term added to express that the entropy of the right-hand member of the equation, being the e!ect, is higher than the entropy supplied by the source radiation. S can be referred to as the rate of irreversible entropy generation. The expression of the thermodynamic variables used in these formulas is presented in Table 2 [4}6]. They are a function of the temperature ¹, of the chemical potential k, and of the lower and upper limits of the energetic range (e and e , respectively) for
+ which photons are absorbed. In a solar cell this range extends usually from the semiconductor band gap (e "e ) to in"nity (e "R).
% + Eqs. (1) and (2) can be combined into [7] (E !¹ S )"(E !¹ S )#= #¹ S (3) this being the rather fundamental equation that gives the rate of irreversible entropy generation provided = is known. In this respect, the Shockley}Queisser (SQ) model [8] } which presents an ideal solar cell that prevents the production of avoidable entropy}states that the ideal solar cell is made up of a semiconductor with a conduction band and a valence band separated by a band gap e . In this cell, the photons % from the source (sun) are absorbed at a rate N , generating electron}hole pairs, one per absorbed photon. Electrons and holes may recombine again, but only by emission of luminescent photons that may be reabsorbed (photon recycling [9]) or escape from the cell. Non-radiative recombination } a process producing irreversible entropy } is absent and therefore, the resulting recombination equals the rate of escaping photons N . In addition, the SQ cell has two ideal contacts so that only holes go through the positive contact and only electrons through the negative one. By balancing of
152
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Table 2 Some thermodynamic formulae 2H E(¹, k, e , e )"
+ hc
C+
e de
"
C exp
e!k !1 k¹
C+ e(¹, k, e) de
C
C+ 2H C+ e de N(¹, k, e , e )" " n(¹, k, e) de
+ hc e!k C exp C !1 k¹
C+ k!e 2H C+ ln 1!exp e de" u(¹, k, e) de X(¹, k, e , e )"k¹
+ k¹ hc C C E!kN!X S" ¹
H E(¹, 0, 0, R)" p¹ p 4H S(¹, 0, 0, R)" p¹ 3p Note. dH"dydzdqdr#dzdxdrdp#dxdydpdq (p, q, r are optical cosines) is the element of eH tendue or Lagrange invariant of the bundle of luminous rays reaching or leaving the receiver. For instance, if the photons are escaping isotropically from an area A, then H"pA. The eH tendue is invariant when the rays proceed through any optical system. r is the Stefan}Boltzman constant.
particles, the extracted current is q times the di!erence between generation and recombination rates, that is, I/q"N !N (q<).
(4)
SQ also consider that the semiconductor presents in"nite mobility ("nite mobility also produces irreversible entropy) so that the quasi-Fermi levels are constant throughout the whole cell. However, as electrons and holes are outside the thermal equilibrium, their quasi-Fermi levels are di!erent, with values e and e , and the cell $ $ voltage is q<"e !e "*e . In addition, these authors consider that the emitted $ $ $ radiation is in thermal equilibrium with the electron and hole gases (for further discussion on this topic see Ref. [10]) so that the photons are emitted at room temperature with a chemical potential k"*e "q<. Obviously, the electric power $ delivered is = "q<(N !N ).
(5)
As mentioned before, the generation of irreversible entropy can then be calculated by introducing this power into Eq. (3). Together with the expression of the entropy in
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Table 2 we obtain, after some mathematical handling,
1 1 k q< X X ! S "E ! #N # ! , (6) ¹ ¹ ¹ ¹ ¹ ¹ where ¹ is the sun temperature and k , usually zero, is the sun or source chemical potential. The X's (see Table 2) are grand potential #ows. 2.2. The thermal engine The thermal engine operates between the hot reservoir, made up of the heated crystal network of the solar cell at the temperature ¹ , and the environment as cold reservoir at the temperature ¹ . The "rst law of thermodynamics applied to the thermal engine is now expressed by Q "Q #= . (7) The heat Q can be obtained from Eq. (1) and (5): Q "(E !q
¹ = "[(E !q
(9)
2.3. The hybrid converter The reversible-engine power output } which can be converted, ideally without losses, into electric power } added to the solar cell output leads to the total useful work of the hybrid system:
¹ ¹ ="= #= "(E !E ) 1! #q<(N !N ) . ¹ ¹
(10)
The application of the "rst and second laws of thermodynamics to the hybrid system, that is, to the outer box of Fig. 1, leads to the set of equations E "E #=#Q
(11)
Q S #S "S # . ¹
(12)
and
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Elimination of Q and = between these three equations leads to the same expression given in Eq. (6). This is a consequence of the zero production of entropy in the reversible engine.
3. Discussion Two cases will be considered for discussion. In the "rst one, referred to as case a or low pass hybrid converter, the receiver only absorbs photons with energy above the band gap of the solar cell, leaving the rest of the photons available for further use, for example, in a second solar cell located beneath. The second case, case b or opaque hybrid converter, occurs when the photons below the semiconductor band gap are also absorbed in a perfect sub-band absorber located beneath the cell. This absorber is at the cell temperature ¹ and the heat produced is converted reversibly into work in the thermal engine. The power and entropy generation rates produced in both cases are calculated and discussed in this section. 3.1. Power The solar source will be taken as a blackbody radiator at ¹ "6000 K, its chemical potential being k "0. The ambient temperature will be taken as ¹ "300 K. These values are assumed for the purpose of comparing the results with other published works. However, slightly di!erent values would be more accurate for the characterisation of the sun and the Earth heat reservoirs. For case a, using the nomenclature in Table 2, E "E(¹ , 0, e ,R). The photons % with energy below e are of no use in the system under consideration, although they % may have some use in another energy converter. In case b, E "E(¹ , 0, 0,R) because here the low energy photons also contribute to the heat input. In both cases N is limited to the photons capable of producing electron-hole pairs, that is N "N(¹ , 0, e ,R). % Looking at the radiation emitted by the solar cell, in case a we "nd that E "E(¹ , q<, e ,R). In case b we must consider two energy regions: one for photons with % energies higher than the solar cell band gap e (emitted from the solar cell) and the % other for photons with energies between 0 and e (emitted from the sub-bandgap % black absorber). Although photons in both regions are at a temperature ¹ , photons emitted with energies above the band gap are characterised by a chemical potential k "q<, while photons emitted with energies below the band gap are characterised by k "0. Therefore, when adding the contribution from both regions, we "nd that E "E(¹ , q<, e ,R)#E(¹ , 0, 0, e ). % % Isotropic illumination on the cells is assumed which may be achieved with an ideal concentrator (such as that in Table 2, with H"pA). It is important to know that such ideal concentrators are not trivial but that they exist for speci"c shapes of the absorber. For instance, for a disc shape the concentrator must have a variable index of refraction [11]. The well- known compound parabolic concentrator [5], often considered as ideal, is only so in a simplifying two-dimensional case. Hence, the power
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Fig. 2. E$ciency as a function of the band gap energy for di!erent conditions of study as labelled and pointed with the arrows. Plots describe the maximum e$ciency for the hybrid converter (dashed line) and cells operated at short circuit voltage and open circuit voltage. When referring to case `ba, the photons below the bandgap are used for the energy conversion while when referring to case `aa, they are not. Maximum e$ciency for the pure photovoltaic case is also plotted for reference (`photovoltaica).
output depends on the cell temperature ¹ and voltage q<"k and, in the general case, it also depends on the semiconductor bandgap e . % If the cell is kept in short circuit, the emitted radiation, with k"q<"0, is free radiation. In this case, the converter is purely solar-thermal formed by a black-body absorber in case b or with an energy absorption threshold at the semiconductor band gap in case a. We present in Fig. 2 the e$ciency vs. bandgap curve, obtained by optimising it with respect to the temperature ¹ . The e$ciency g is de"ned here as the ratio of the useful power output to the incoming solar power in the whole spectrum, that is, to E(¹ , 0, 0,R). Notice that for case b, the e$ciency is independent of e . In % this case (and for case a with e "0) the e$ciency is given by the well known formula % [12,13]
¹ ¹ g" 1! 1! , (13) ¹ ¹ obtained in this paper from Eq. (10) by using the energy expression of Table 2. In this formula, the "rst term accounts for the fraction of energy absorbed and not radiated, while the second term is the Carnot e$ciency of the ideal engine. The highest e$ciency is a trade-o!: for very high absorber temperature the Carnot e$ciency is higher but the losses by radiation also increase. An e$ciency maximum of 85.4% occurs in this case for a temperature of 2544 K. As it is to be expected, in case a where the absorption of photons is incomplete, the e$ciency decreases with the band gap. In Fig. 3 we also present the temperature at which the optimum e$ciency is obtained.
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Fig. 3. Temperature as a function of the band gap energy, at which the maximum e$ciency for the di!erent cases of study (which are labelled) is obtained.
Another con"guration of interest is when the cell is in open circuit. In this case no electric contact is made to the absorber or no cell structure actually exists, but only a semiconductor absorber, provided it keeps the ideal conditions of not presenting non-radiative recombination and keeping the carrier mobility in"nite. In this con"guration, as no current is extracted, the output power comes from the thermal engine only. According to Eq. (4), N "N . This causes the absorber to emit strong radiation, well above the one corresponding to its temperature, which will bring about a strong decrease in the e$ciency as the band gap increases. This strong radiation is caused by a chemical potential k, for the radiation with photon energies above e , equal to the quasi-Fermi levels splitting (q< ) in the semiconducting % absorber. For the rest of the spectrum the radiation emitted is zero in case a or purely thermal (k"0) in case b. Again we represent in Fig. 2 the e$ciency optimised with respect to the temperature for this con"guration and, in Fig. 3, the optimum operation temperature. In addition, we plot in Fig. 4 the open-circuit voltage for the optimal temperature as a function of the band gap energy. For the purpose of comparison we have also represented in Fig. 2 the well-known value of the e$ciency of an ideal photovoltaic converter (a solar cell) at the ambient temperature, optimised with respect to the voltage [14]. It can be observed that the e$ciency limit of the photovoltaic converters is much smaller than the one for the solar-thermal converters (q<"0). The voltages at which these e$ciencies occur are presented in Fig. 4. The use of a semiconductor absorber is rather interesting because it is a good example of non-ful"lment of the law of Kirchho!. According to this law, as stated in a classical book of thermodynamics [15], `the ratio of the absorbing and the emitting
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Fig. 4. Normalised voltage as a function of the band gap energy, at which the maximum e$ciency for the di!erent cases of study (which are labelled) is obtained. Notice the cases at which the voltage is negative.
power of a body is independent of the nature of the body, of the chosen point and of the direction, and only dependent on the temperature of the body and on the wavelengtha. This is not accomplished with the semiconductor. In fact, the semiconductor, with a back absorber, absorbs the totality of the incident radiation but emits di!erently (the ratio of the absorbing and the emitting power depends then on the nature of the bodies). The reason for this is the lack of thermodynamic equilibrium in the receiver, which causes the splitting of the quasi-Fermi levels and suggests that, in solar energy, more care should be taken with the common application of the Kirchho!'s law because, with the solar photons as an energy source, luminescent excitation may exist. This observation is not restricted to semiconductors. It has to be said that such subtleties were in the mind of the classic author [15] when he wrote `we are going to study...the purely thermal radiation or incandescent radiation. We do not study the luminescent radiation...a. The question that motivated our paper refers to the maximum e$ciency that can be obtained in the hybrid system if we optimise the e$ciency simultaneously with respect to the cell temperature and to the voltage. This maximum e$ciency is represented vs. the band gap also in Fig. 2 for case a and b. What we observe is that it exceeds, although slightly, the e$ciency of the hybrid system when it is in short circuit, or in other words, the e$ciency of the purely solar-thermal converter. The maximum maximorum is of 86.7% for a band gap of 0.10 eV instead of the 85.4% of the ideal solar-thermal case. Thus, it can be said that the hybrid systems have an e$ciency advantage, greater for photovoltaic converters and less for the solar thermal converters. This statement is valid both for case a } as already obtained by de Vos [2] } and for case b. However we shall see that this is not the general case.
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How to achieve this higher e$ciency is perhaps more interesting. For this purpose, we present the temperature and the voltage for the maxima in Figs. 3 and 4, respectively. The observation of the voltage curve tells us that the highest e$ciencies (obtained at low band-gaps) are obtained for negative voltages. This means that some of the power produced by the thermal engine is absorbed by the solar cell and used to reduce the chemical potential of the emitted radiation. In this way, the radiant energy sent back to the sun is reduced and the temperature of the converter can be increased (see Fig. 4), resulting in a higher Carnot e$ciency of the engine. This is interesting because it has been shown that the e$ciency upper bound of a solar converter (Landsberg's e$ciency) is achieved when the radiated energy is equivalent, in the way described by Luque and MartmH (LM) [7], to a thermal radiation at the ambient temperature. We wonder if, by shaping the emitted radiation with some reverse bias, using the power produced with the thermal engine, this equivalent radiation may be reproduced. This point will be dealt with in detail in Section 4. Finally, in Figs. 5 and 6, we present the e$ciency vs. bandgap for hybrid converters at di!erent temperatures for the cases a and b, respectively. For the sake of comparison, we have included in these "gures the maximum e$ciency curves (appearing at variable temperatures) also drawn in Fig. 2. The purely photovoltaic case is also shown as the curve at the ambient temperature. These curves tell us that hybrid converters may be of practical interest, in particular for low absorber temperatures, if the solar cell and the thermal engine can be kept relatively free of losses. For instance, for the ideal devices at 500 K, the purely thermal converter gives an e$ciency of 40%, according to Eq. (13), while the hybrid system of case b, with an ideal cell of 1.10 eV, gives an e$ciency of 62.7%. The best cold solar cell, of 1.11 eV, gives the 40.7%. 3.2. Entropy Eq. (6) represents the rate of entropy generation: it has to be non-negative. It can be considered as a function of two variables: q< and ¹ . The derivatives with respect to them are *S N 1 *X N !N , "! ! " ¹ ¹ *(q<) ¹ *(q<) *S E X 1 *X q
(14)
In the determination of these derivatives we have used the basic thermodynamic relationships [4] stating that the grand potential #ow (X) derivatives with respect to the chemical potential (q<) and with respect to the temperature, with the sign reversed, are respectively the particle (in this case with e'e ) and the entropy #ows. % The entropy expression in Table 2 has also been used. Looking at these formulae we see that, for constant temperature, an extremum appears for open circuit conditions. For direct (positive) current in the cell (<)< )
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Fig. 5. Maximum e$ciency as a function of the band gap energy and for constant temperature of operation, for the hybrid converter } case a in which photons below the band gap are not used for conversion into work. For reference of the reader, the maximum e$ciency, when the temperature of operation is also optimised, is also plotted as a dashed line. Short-dashed lines plot the maximum e$ciency for the hybrid-converter in which photons below the band gap are used for energetic conversion into work (hybrid converter } case b).
Fig. 6. Maximum e$ciency, as a function of the band gap energy for constant temperature of operation, for the hybrid converter- case b in which photons below the band gap are used for conversion into work. For reference of the reader, the maximum e$ciency, when the temperature of operation is also optimised, is also plotted as a dashed line. Also for reference, short-dashed lines plot the maximum e$ciency for the hybridconverter in which photons below the band gap are used for energetic conversion into work (hybrid converter } case a).
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the derivative is negative and for reverse current (<*< ) the derivative is positive. Therefore, the extremum is a minimum. It must be noticed that this is also true even for the case that the source is at a lower temperature than the cell (< (0). The extremum along the line of minima given by k"q< (or N "N ) is obtained by also setting to zero the derivative with respect to ¹ while keeping the open circuit condition (N "N ). This occurs when the E "E . Then, the two equations in Eq. (14) are ful"lled for ¹ "¹ and <"0 but we still do not know if it is minimum minimorum. However,when <"0, E is an increasing function of ¹ so that S decreases for temperatures below ¹ and increases for temperatures above it. Thus, a minimum also occurs at the mentioned point in the direction of the line of minima. It is therefore a minimum minimorum. At this point it is easy to see, looking at Eq. (6), that the entropy is just zero. Therefore, for any other combination of < and ¹ the entropy is higher and therefore positive. The preceding discussion, valid for cases a and b, is illustrated for case a in Fig. 7, representing the entropy vs. temperature with the condition of <"< for three di!erent band gaps. For the condition of zero entropy production (¹ "6000, q<"0), the reversibility is con"rmed by reversing the time, so that the entering and escaping bundles of rays are interchanged while zero power is delivered or received. In this way we have proven the thermodynamic coherence of the models utilised, in the sense that an increase in entropy always takes place. We have also proven that for a given combination of cell temperature and voltage (¹ "¹ , <"0), the conversion process is reversible. Consequently, we may assert that the hybrid converter we have modelled is ideal in the sense we mentioned that it can produce the least possible entropy. In fact, it is zero for certain values of the couple (¹ , q<).
Fig. 7. Normalised irreversible entropy rate that is produced in the hybrid converter in open circuit as a function of the temperature for several illustrative band gap energies for the case a. Normalising factor, S , 1 is 4/3p¹. 1
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4. E7ciency limit of hybrid converters The upper limit of the e$ciency of any solar converter subject to the time microreversibility was obtained by Landsberg and Tonge [3] for solar-thermal converters as
4¹ 1 ¹ g"1! # 3 ¹ 3¹
(15)
(93.3% for 6000}300 K) and generalised by LM [7] for emitted radiation of any kind. These authors emphasised that, in order to achieve this limit, two conditions were necessary: (i) entropy generation should be zero and (ii), the emitted radiation should be thermal equivalent to a radiation temperature ¹ . This equivalence means that the emitted radiation is characterised by a chemical potential k and a temperature ¹ satisfying
¹ ¹ k "e 1! #k "0 V ¹ ¹
(16)
for all the energies e of the photons emitted. However, it is not known how to make a device ful"lling these conditions. We have observed that high e$ciencies may be obtained by injecting the power of a thermal engine into the solar cell. In this way, the reverse bias (q<"k, negative) of the cell reduces the radiated energy and this allows for a higher temperature of the thermal engine and thus, for a higher e$ciency. This reverse bias is also obtained with the preceding Eq. (16), for ¹ '¹ , when k "0. However, we observe that, to achieve the Landsberg s limiting e$ciency, this reverse biasing or k should be a function of the photon energy. Therefore, if a solution exists to our problem it has to be within the framework of a stack of converters, each one set to operate at a monochromatic radiation of photons of energy e. In this stack, each one of these monochromatic cells would be made of a semiconductor of band gap e, covered at the front by an ideal low-pass "lter that transmits the photons with energy below e#de and at the back, by another ideal low-pass "lter of energy cut-o! at e. This cell has to be illuminated by an ideal concentrator that sends all the luminescent photons emitted back to the sun. The front low-pass "lter is actually the back "lter of the preceding cell in the stack. The aim of the back low-pass "lter is to assure that the luminescent photons emitted backwards are not actually able to escape and therefore this loss of energy is prevented. An ideal mirror can do the same but it would prevent the sunlight from entering deeper in the stack. It should be mentioned, however, that ArauH jo and MartmH [16] proved that for a photovoltaic stack, the use of the preceding "lters was irrelevant as the number of gaps approaches in"nity. Indeed, in the hybrid converter, every cell in the stack will supply its heat to a di!erent Carnot engine.
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By using the nomenclature in Table 2 we can write the output power per unity of band gap (d=/de"w) as a particularisation of Eq. (10):
¹ k ¹ e!k k¹ w"(e !e ) 1! # (e !e ) "(e !e ) 1! , (17) ¹ ¹ e k¹ e where we have used that e " en. If we substitute in this equation the chemical potential obtained from Eq. (16) by k "0 we realise that the output power is zero, that is, we expend all the power produced by the thermal engine to keep the emitted radiation equivalent to a thermal one at the ambient temperature. In the less common case of operating the cell at a below-ambient temperature, the chemical potential to produce a radiation equivalent to a thermal one at the ambient temperature is positive and the cell is thus generating power, but all this power is to be spent on keeping the cell cool, with the Carnot engine operating as a refrigerator. Again the output power is zero. It can be seen we have been able to ful"l one of the conditions (condition ii in the way de"ned by LM) for achieving the Landsberg s e$ciency. However, in the preceding case the entropy generation is not zero, so the other condition has not been ful"lled. For monochromatic cells, the entropy generation is zero [7] only if the cell is in open circuit, that is for n "n (or e "e ). This means that the arguments of these variables are equal:
e e!k ¹ Nk "e 1! . " (18) k¹ k¹ ¹ In this case k is not zero, but k "e(1!¹ /¹ ), and the total power produced is V V again zero, now because the energy received from the sun is full balanced by the one emitted back by the cell. Obviously, the maximum power production corresponds in this case to a di!erent condition } not ful"lling any one of the conditions for the Landsberg's e$ciency } characterised by certain values of ¹ and k. However, for its calculation it is worth, noting Table 2 and the Eq. (17), that w is a function of ¹ and k only through the variable z de"ned as e!k z" . (19) k¹ In other words, the power output is the same for all the couples of values (k, ¹ ) situated in any straight line obtained by "xing z to a constant value (Fig. 8). This means that for any hybrid converter operating at an energy e, with a chemical potential k, and at a receiver temperature ¹ we can "nd a purely photovoltaic one, operating at ambient temperature ¹ , giving the same power, with a chemical potential k "e(1!¹ /¹ )#k¹ /¹ or a purely solar thermal one (k"0) operat ing with a temperature at the receiver ¹ "¹ e/(e!k). That is, no theoretical advantage exists, when using stacked structures, for the hybrid system as compared to the purely photovoltaic one or to the purely solar thermal one.
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Fig. 8. Illustrative representation of the factor z"(e!k)/k¹ for di!erent constant values of z, also showing the lines corresponding to the solar thermal converter and to the pure photovoltaic converter.
Fig. 9. E$ciency as a function of the energy for a monochromatic converter (connected line). The optimum value of z"(e!k)/k¹ that gives the result is also plotted (dashed line).
We represent the power vs. e optimised with respect to z and normalised with respect to the input energy density #ux e(¹ , 0, e) in Fig. 9. The quotient of the integration of the optimal w throughout all the e divided by the input power form the sun, E(¹ , 0, 0,R) gives the highest e$ciency conversion of an ideal stacked system with an in"nite number of elements which results in 86.8% (6000-300 K) [17}19]. Novel photovoltaic devices with higher-than-one quantum e$ciency, have recently been analysed [20,21,7]. Their e$ciency is limited to 85.9%. (Originally the "gure was
164
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calculated for ¹ "5760 K, giving 85.4% and has been recalculated by us for ¹ "6000 K) and therefore it does not give better results. 1 Therefore, the stack analysed in this section can give more e$ciency than any known solar converter. The fact that all the monochromatic cells with the same value of z give the same e$ciency adds a degree of freedom to the design of the stack. For instance, it can be a stack of solar cells at ambient temperature, each operating with a di!erent voltage. It may also be a stack of pure solar-thermal converters, each one at a di!erent temperature, with a di!erent engine. But we can, for instance, put all the cells at the same temperature (di!erent from the ambient), and optimise their operating voltages so that only one thermal engine is needed. Other combinations implying arbitrary links between voltage and temperature may equally lead to converters reaching the e$ciency maximum maximorum.
5. Conclusions We have presented here a comprehensive analysis of the photovoltaic and the solar-thermal converters including hybrid converters where both e!ects are used together. In this analysis, such devices are de"ned by the chemical potential and the temperature of the radiation they emit. The discussion of the operation of these devices has been supported also by a study of the entropy production. Under certain conditions the system is reversible, (produces zero entropy) and this proves that our analysis is related to ideal systems producing the least possible entropy. The study of the hybrid converters has also been extended to the case of an in"nite stack of monochromatic converters, for which the temperature and the chemical potential may vary continuously as a function of the photon energy. Obviously the latter are those with more degrees of freedom and allow for the most optimum design. For this case, it has been proven that hybrid converters can give the same e$ciency as photovoltaic or solar thermal devices working alone. To some extent, they are strictly equivalent, giving a top e$ciency of 86.8% for a source temperature of 6000 K and an ambient temperature of 300 K. However, if we restrict our analysis to devices operating with a single cell, acting as the absorber of the hybrid system, then the hybrid converters give an e$ciency of 86.7%, very close to the previous one and higher than the top e$ciency achievable with a single temperature solar thermal of 85.4% and much above the one achievable with a single solar cell at room temperature of 40.7%. Letting impractical situations aside (due to excessive temperature), in many practical situations, hybrid converters give more e$ciency than solar-thermal or photovoltaic ones. Furthermore, the values for ideal devices are such (at 500 K, 61.7% for the hybrid system vs. 40.0% for the solar-thermal or 40.7% for the photovoltaic one at 300 K) that it is very probable that practical hybrid converters can attain more e$ciency than practical solar-thermal or photovoltaic converters alone. Another important question is whether the Landsberg's e$ciency } the e$ciency upper bound of any solar converter (obeying the time micro-reversibility) of 93.3% } can be achieved with some type of hybrid converter. The answer is negative. Hybrid
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converters may separately ful"ll the two conditions for achieving the Landsberg's e$ciency: they may generate zero entropy or they may emit a radiation equivalent to a room temperature thermal radiation, but both conditions cannot be achieved simultaneously. In fact, in both cases the energy produced is zero. Therefore, we continue to ignore how to reach the Landsberg's e$ciency or if this e$ciency may even be reached, even in ideal situations. Finally, the case of the semiconductor receiver (not necessarily a solar cell) is interesting as it constitutes a good example of the failure of the Kirchho!'s law. This is due to the lack of thermodynamic equilibrium in the receiver, which causes the splitting of the quasi-Fermi levels and suggests that, in solar energy, the common application of Kirchho!'s law should be done more carefully because, with the solar photons as an energy source, luminescent excitation may exist. This observation is not restricted to semiconductors.
References [1] J. Loferski, C. Case, D. Doodlesack, B. Roesler, R. Dobbins, T. Russell, J. Beall, J. Kirkorian, Proc. 16 Photovoltaic Specialists Conf., IEEE, New York, 1982, p. 188. [2] A. de Vos, Endoreversible Thermodynamics of Solar Energy, Oxford University Press, New York, 1992. [3] P.T. Landsberg, G. Tonge, J. Appl. Phys. 51 (1980) R1. [4] E.D. Landau, E.M. Lifchithz, Physique Statistique, MIR, Moscow, 1967. [5] W.T. Welford, R. Winston, The Optics of Non-imaging Concentrators, Academic press, New York, 1978. [6] J.C. Min ano, in: A. Luque (Ed.), Solar Cells and Optics for Photovoltaic Concentration, Ch. 12, Adam Hilger, Bristol, 1989. [7] A. Luque, A. MartmH , Phys. Rev. B 55 (1997) 6994. [8] W. Shockley, H.J. Queisser, J. Appl. Phys. 32 (1961) 510. [9] A. MartmH , J.L. Balenzategui, J. Appl. Phys. 82 (8) (1997) 4067. [10] A. Luque, A. MartmH , Phys. Rev. Lett. 78 (1997) 5014. [11] J.C. Min ano, Appl. Opt. 22 (1983) 2751. [12] M. Castan s, Revista Geo"Hsica 35 (1976) 227. [13] A. de Vos, H. Pauwels, Appl. Phys. 25 (1981) 119. [14] G.L. ArauH jo, A. MartmH , Sol Energy Mater. Sol Cells 33 (1994) 213. [15] G. Bruat, Cours de Thermodynamique, Masson, Paris, 1933, pp. 303 and 312. [16] G.L. ArauH jo, A. MartmH , Proc. 22nd Photovoltaic Specialists Conf., IEEE, New York, 1991, p. 290. [17] H. Pauwels, A. De Vos, Solid State Electron. 24 (9) (1981) 835. [18] De. Vos, H. Pauwels, J. Appl. Phys. 25 (1981) 119. [19] A. MartmH , G.L. ArauH jo, Sol. Energy Mater. Sol. Cells. 43 (1996) 203. [20] J.H. Werner, S. Kodolinski, H.J. Queisser, Phys. Rev. Lett. 72 (1994) 3851. [21] J.H. Werner, R Brendel, K.J. Queisser. Proc. 1st World Conf. Photovoltaic Energy Conversion, IEEE, New York, 1994, p. 1742.
Limiting e$ciency of coupled thermal and photovoltaic converters Antonio Luque*, Antonio MartmH Instituto de Energn& a Solar, Universidad Polite& cnica de Madrid, Ciudad Universitaria, 28040 Madrid, Spain Received 27 May 1998; accepted 20 October 1998
Abstract This paper presents a general energetic and entropic analysis of ideal photovoltaic and solar-thermal converters. Its purpose is to determine the e$ciency limit when both types of converters operate together (hybrid converters). It has been found that, while in practical cases hybrid converters may give very high e$ciency (61.7% vs. 40.0% of the solar thermal at 500 K and 40.7% of the photovoltaic at 300 K), in the limiting case of a system formed by an in"nite number of band gaps, the e$ciency of hybrid converters, 86.8%, is strictly equivalent to photovoltaic or solar thermal converters. Conversely, hybrid systems operating with one gap give an e$ciency of 86.7%, very close to the previous one and higher than the top e$ciency achievable with a single temperature solar thermal, 85.4%. 1999 Elsevier Science B.V. All rights reserved. Keywords: Photovoltaic; Thermal; Limiting e$ciency; Stack
Notation A c *e $ E E e e e
area speed of light quasi-fermi level split rate of energy associated rate of energy associated rate of energy associated rate of energy associated energy of a photon
with with with with
the the the the
escaping radiation entering radiation monochromatic escaping radiation entering monochromatic radiation
* Corresponding author. 0927-0248/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 4 8 ( 9 8 ) 0 0 1 9 9 - 8
148
e $ e $ e % e
e + H h g k k k k k N N n n Q Q q S S S p ¹ ¹ ¹ < < = = = w X X z
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quasi-fermi level of electrons quasi-fermi level of holes semiconductor bandgap lower limit of the range of energies of interest upper limit of the range of energies of interest eH tendue or Lagrange invariant of the bundle of luminous rays reaching or leaving the cell Planck's constant e$ciency of the converter Boltzman's constant chemical potential associated with the escaping radiation chemical potential associated with the escaping radiation under open-circuit conditions chemical potential associated with the entering radiation chemical potential of the thermal equivalent emitted radiation rate of escaping photons rate of photons absorbed from the entering radiation rate of monochromatic escaping photons rate of monochromatic photons absorbed from the entering radiation rate of heat delivered to the ambient rate of heat delivered by the cell electron electric charge rate of entropy associated with the escaping radiation rate of entropy associated with the entering radiation rate of irreversible entropy generation Stefan}Boltzman's constant temperature of the ambient temperature of the cell temperature of the sun cell voltage open-circuit voltage rate of work (power) produced at the photovoltaic converter rate of work (power) produced at the thermal engine total rate of work produced at the hybrid system total rate of work produced at the hybrid system when operating with monochromatic radiation grand potential #ow associated with the entering radiation grand potential #ow associated with the escaping radiation normalised variable de"ned as (e-k)/k¹
1. Introduction A solar energy converter is a device that receives solar radiation and produces useful power. In compliance with the laws of thermodynamics, the converter must
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supply some heat to the ambient and, in addition, it must emit some radiant energy. The possibility of emitting some radiant energy can also be understood as a consequence of the need that if there is a path for rays to enter into the receiver, by time reversal, the same path has to allow the rays to escape (in the absence of external "elds violating the time micro-reversibility). This is a general situation for luminous photons with the standard terrestrial energies where photovoltaic conversion takes place. Presumably, in less ideal cases, additional paths for escaping rays also exist, linking the receiver not with the source but with the darker parts of the surroundings. The nature of this escaping radiation de"nes the nature of the solar converter. If it is free radiation at a certain temperature } the temperature of the receiver, ¹ } then the converter is solar-thermal. If it is matter-coupled (luminescent) radiation characterised by a non-zero chemical potential at the ambient temperature, then the converter is photovoltaic. The solar cell used as a receiver in photovoltaic converters is the device producing this luminescent radiation. Hybrid systems, which are dealt within this paper, are also possible. For them, the receiver is a solar cell operating at a temperature ¹ above the ambient that produces work in combination with a thermal engine operated between the cell, ¹ , and the ambient, ¹ , temperatures. For such a hybrid system, the escaping radiation will be a matter-coupled radiation at a temperature above the ambient and in some cases will be characterised by a zero chemical potential at some wavelengths and by a non-zero chemical potential at others. The hybrid systems which combine photovoltaic conversion and di!erent uses of the heat rejected have sometimes been proposed [1]. This paper does not deal with such systems. They provide two di!erent goods, electricity (non-entropic useful work) and heat (with entropy) whose economic values are di!erent, the latter varying with the temperature at which the heat is rejected. In this paper we will assume that the heat rejected by the solar cells is used to drive a thermal engine that produces mechanical work which can be theoretically converted into electricity without losses. Then, the practical question addressed here is if this approach will allow for a better e$ciency conversion of solar energy into useful work. The question is relevant because if we raise the cell temperature to obtain a better engine e$ciency then the e$ciency of the cell decreases. The global balance is uncertain. This question has been partially answered by De Vos [2] and his reply is a$rmative. However, he does not refer to a general case but to a speci"c one because he only deals with what we call low pass hybrid converters, discussed later in this paper. In fact, we shall prove that hybrid systems, at their best con"guration, cannot exceed the e$ciency of photovoltaic systems. They are strictly equivalent. This point is important because it means that hybrid converters cannot reach the Landsberg's e$ciency [3] either, which is the e$ciency upper bound of any solar converter that obeys the time micro-reversibility. That is in spite of the fact that hybrid systems may ful"l one of the two thermodynamic conditions (but not both simultaneously) necessary to achieve this e$ciency: zero entropy generation rate or thermal equivalent to the ambient temperature of the emitted radiation. It is worth mentioning that none of the known ideal devices is able to attain the Landsberg's e$ciency. Therefore, it is unknown if there is a theoretical reason that prevents solar converters from achieving it.
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Our paper deals with ideal converters, which means that the production of irreversible entropy is avoided as much as possible and is isotropically illuminated by the sun by means of an ideal concentrator. In Section 2 we will describe the theory of the hybrid converter which shows the contributions from the photovoltaic and thermal components to the total power delivered by the system and to the irreversible entropy generation rate. Maximum e$ciencies and entropy generation rates will be computed in Section 3 where results are also discussed. Section 4 will emphasise the topics concerning the limiting e$ciency of the hybrid converters and Section 5 will summarise the conclusions of this work. 2. Theoretical analysis In this section we present the theoretical expression of the useful power and of the irreversible generation of entropy in the photovoltaic cell, the thermal engine and the complete hybrid converter. The results obtained are summarised in Table 1 for the readers' convenience. 2.1. The photovoltaic cell A sketch of a hybrid converter is shown in Fig. 1. The inner box represents the photovoltaic component (or solar cell) whose operation will be reviewed in the following paragraphs. The cell absorbs radiant energy from the solar photons at a rate E and emits it at a rate E . It also produces electric (useful) work at a rate = and delivers heat to the crystal network at a rate Q . The "rst law of thermodynamics is then written for this device as E "E #= #Q and the second law as
(1)
Q S #S "S # , ¹
(2)
Table 1 Summaries of the output power, =, contributions to the hybrid converter and the entropy generation rate, S , from the photovoltaic converter and the thermal engine =
S
Photovoltaic
q<(N !N )
1 1 k q< X X E ! #N 1 ! # ! 1 ¹ ¹ ¹ ¹ ¹ ¹
Thermal
¹ (E !q
Total hybrid converter
¹ ¹ (E !E ) 1! #q<(N !N ) ¹ ¹
0 1 1 k q< X X E ! #N 1 ! # ! 1 ¹ ¹ ¹ ¹ ¹ ¹
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Fig. 1. Sketch of the hybrid converter showing the di!erent energy and heat #uxes involved as well as the power delivered.
In this equation, S's are rates of entropy, associated with the entering radiation (subindex e) and with the escaping radiation (subindex c). The entropy supplied to the cell crystal network } here acting as a cold reservoir } is Q /¹ , ¹ being the cell temperature. S is a positive term added to express that the entropy of the right-hand member of the equation, being the e!ect, is higher than the entropy supplied by the source radiation. S can be referred to as the rate of irreversible entropy generation. The expression of the thermodynamic variables used in these formulas is presented in Table 2 [4}6]. They are a function of the temperature ¹, of the chemical potential k, and of the lower and upper limits of the energetic range (e and e , respectively) for
+ which photons are absorbed. In a solar cell this range extends usually from the semiconductor band gap (e "e ) to in"nity (e "R).
% + Eqs. (1) and (2) can be combined into [7] (E !¹ S )"(E !¹ S )#= #¹ S (3) this being the rather fundamental equation that gives the rate of irreversible entropy generation provided = is known. In this respect, the Shockley}Queisser (SQ) model [8] } which presents an ideal solar cell that prevents the production of avoidable entropy}states that the ideal solar cell is made up of a semiconductor with a conduction band and a valence band separated by a band gap e . In this cell, the photons % from the source (sun) are absorbed at a rate N , generating electron}hole pairs, one per absorbed photon. Electrons and holes may recombine again, but only by emission of luminescent photons that may be reabsorbed (photon recycling [9]) or escape from the cell. Non-radiative recombination } a process producing irreversible entropy } is absent and therefore, the resulting recombination equals the rate of escaping photons N . In addition, the SQ cell has two ideal contacts so that only holes go through the positive contact and only electrons through the negative one. By balancing of
152
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Table 2 Some thermodynamic formulae 2H E(¹, k, e , e )"
+ hc
C+
e de
"
C exp
e!k !1 k¹
C+ e(¹, k, e) de
C
C+ 2H C+ e de N(¹, k, e , e )" " n(¹, k, e) de
+ hc e!k C exp C !1 k¹
C+ k!e 2H C+ ln 1!exp e de" u(¹, k, e) de X(¹, k, e , e )"k¹
+ k¹ hc C C E!kN!X S" ¹
H E(¹, 0, 0, R)" p¹ p 4H S(¹, 0, 0, R)" p¹ 3p Note. dH"dydzdqdr#dzdxdrdp#dxdydpdq (p, q, r are optical cosines) is the element of eH tendue or Lagrange invariant of the bundle of luminous rays reaching or leaving the receiver. For instance, if the photons are escaping isotropically from an area A, then H"pA. The eH tendue is invariant when the rays proceed through any optical system. r is the Stefan}Boltzman constant.
particles, the extracted current is q times the di!erence between generation and recombination rates, that is, I/q"N !N (q<).
(4)
SQ also consider that the semiconductor presents in"nite mobility ("nite mobility also produces irreversible entropy) so that the quasi-Fermi levels are constant throughout the whole cell. However, as electrons and holes are outside the thermal equilibrium, their quasi-Fermi levels are di!erent, with values e and e , and the cell $ $ voltage is q<"e !e "*e . In addition, these authors consider that the emitted $ $ $ radiation is in thermal equilibrium with the electron and hole gases (for further discussion on this topic see Ref. [10]) so that the photons are emitted at room temperature with a chemical potential k"*e "q<. Obviously, the electric power $ delivered is = "q<(N !N ).
(5)
As mentioned before, the generation of irreversible entropy can then be calculated by introducing this power into Eq. (3). Together with the expression of the entropy in
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153
Table 2 we obtain, after some mathematical handling,
1 1 k q< X X ! S "E ! #N # ! , (6) ¹ ¹ ¹ ¹ ¹ ¹ where ¹ is the sun temperature and k , usually zero, is the sun or source chemical potential. The X's (see Table 2) are grand potential #ows. 2.2. The thermal engine The thermal engine operates between the hot reservoir, made up of the heated crystal network of the solar cell at the temperature ¹ , and the environment as cold reservoir at the temperature ¹ . The "rst law of thermodynamics applied to the thermal engine is now expressed by Q "Q #= . (7) The heat Q can be obtained from Eq. (1) and (5): Q "(E !q
¹ = "[(E !q
(9)
2.3. The hybrid converter The reversible-engine power output } which can be converted, ideally without losses, into electric power } added to the solar cell output leads to the total useful work of the hybrid system:
¹ ¹ ="= #= "(E !E ) 1! #q<(N !N ) . ¹ ¹
(10)
The application of the "rst and second laws of thermodynamics to the hybrid system, that is, to the outer box of Fig. 1, leads to the set of equations E "E #=#Q
(11)
Q S #S "S # . ¹
(12)
and
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Elimination of Q and = between these three equations leads to the same expression given in Eq. (6). This is a consequence of the zero production of entropy in the reversible engine.
3. Discussion Two cases will be considered for discussion. In the "rst one, referred to as case a or low pass hybrid converter, the receiver only absorbs photons with energy above the band gap of the solar cell, leaving the rest of the photons available for further use, for example, in a second solar cell located beneath. The second case, case b or opaque hybrid converter, occurs when the photons below the semiconductor band gap are also absorbed in a perfect sub-band absorber located beneath the cell. This absorber is at the cell temperature ¹ and the heat produced is converted reversibly into work in the thermal engine. The power and entropy generation rates produced in both cases are calculated and discussed in this section. 3.1. Power The solar source will be taken as a blackbody radiator at ¹ "6000 K, its chemical potential being k "0. The ambient temperature will be taken as ¹ "300 K. These values are assumed for the purpose of comparing the results with other published works. However, slightly di!erent values would be more accurate for the characterisation of the sun and the Earth heat reservoirs. For case a, using the nomenclature in Table 2, E "E(¹ , 0, e ,R). The photons % with energy below e are of no use in the system under consideration, although they % may have some use in another energy converter. In case b, E "E(¹ , 0, 0,R) because here the low energy photons also contribute to the heat input. In both cases N is limited to the photons capable of producing electron-hole pairs, that is N "N(¹ , 0, e ,R). % Looking at the radiation emitted by the solar cell, in case a we "nd that E "E(¹ , q<, e ,R). In case b we must consider two energy regions: one for photons with % energies higher than the solar cell band gap e (emitted from the solar cell) and the % other for photons with energies between 0 and e (emitted from the sub-bandgap % black absorber). Although photons in both regions are at a temperature ¹ , photons emitted with energies above the band gap are characterised by a chemical potential k "q<, while photons emitted with energies below the band gap are characterised by k "0. Therefore, when adding the contribution from both regions, we "nd that E "E(¹ , q<, e ,R)#E(¹ , 0, 0, e ). % % Isotropic illumination on the cells is assumed which may be achieved with an ideal concentrator (such as that in Table 2, with H"pA). It is important to know that such ideal concentrators are not trivial but that they exist for speci"c shapes of the absorber. For instance, for a disc shape the concentrator must have a variable index of refraction [11]. The well- known compound parabolic concentrator [5], often considered as ideal, is only so in a simplifying two-dimensional case. Hence, the power
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Fig. 2. E$ciency as a function of the band gap energy for di!erent conditions of study as labelled and pointed with the arrows. Plots describe the maximum e$ciency for the hybrid converter (dashed line) and cells operated at short circuit voltage and open circuit voltage. When referring to case `ba, the photons below the bandgap are used for the energy conversion while when referring to case `aa, they are not. Maximum e$ciency for the pure photovoltaic case is also plotted for reference (`photovoltaica).
output depends on the cell temperature ¹ and voltage q<"k and, in the general case, it also depends on the semiconductor bandgap e . % If the cell is kept in short circuit, the emitted radiation, with k"q<"0, is free radiation. In this case, the converter is purely solar-thermal formed by a black-body absorber in case b or with an energy absorption threshold at the semiconductor band gap in case a. We present in Fig. 2 the e$ciency vs. bandgap curve, obtained by optimising it with respect to the temperature ¹ . The e$ciency g is de"ned here as the ratio of the useful power output to the incoming solar power in the whole spectrum, that is, to E(¹ , 0, 0,R). Notice that for case b, the e$ciency is independent of e . In % this case (and for case a with e "0) the e$ciency is given by the well known formula % [12,13]
¹ ¹ g" 1! 1! , (13) ¹ ¹ obtained in this paper from Eq. (10) by using the energy expression of Table 2. In this formula, the "rst term accounts for the fraction of energy absorbed and not radiated, while the second term is the Carnot e$ciency of the ideal engine. The highest e$ciency is a trade-o!: for very high absorber temperature the Carnot e$ciency is higher but the losses by radiation also increase. An e$ciency maximum of 85.4% occurs in this case for a temperature of 2544 K. As it is to be expected, in case a where the absorption of photons is incomplete, the e$ciency decreases with the band gap. In Fig. 3 we also present the temperature at which the optimum e$ciency is obtained.
156
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Fig. 3. Temperature as a function of the band gap energy, at which the maximum e$ciency for the di!erent cases of study (which are labelled) is obtained.
Another con"guration of interest is when the cell is in open circuit. In this case no electric contact is made to the absorber or no cell structure actually exists, but only a semiconductor absorber, provided it keeps the ideal conditions of not presenting non-radiative recombination and keeping the carrier mobility in"nite. In this con"guration, as no current is extracted, the output power comes from the thermal engine only. According to Eq. (4), N "N . This causes the absorber to emit strong radiation, well above the one corresponding to its temperature, which will bring about a strong decrease in the e$ciency as the band gap increases. This strong radiation is caused by a chemical potential k, for the radiation with photon energies above e , equal to the quasi-Fermi levels splitting (q< ) in the semiconducting % absorber. For the rest of the spectrum the radiation emitted is zero in case a or purely thermal (k"0) in case b. Again we represent in Fig. 2 the e$ciency optimised with respect to the temperature for this con"guration and, in Fig. 3, the optimum operation temperature. In addition, we plot in Fig. 4 the open-circuit voltage for the optimal temperature as a function of the band gap energy. For the purpose of comparison we have also represented in Fig. 2 the well-known value of the e$ciency of an ideal photovoltaic converter (a solar cell) at the ambient temperature, optimised with respect to the voltage [14]. It can be observed that the e$ciency limit of the photovoltaic converters is much smaller than the one for the solar-thermal converters (q<"0). The voltages at which these e$ciencies occur are presented in Fig. 4. The use of a semiconductor absorber is rather interesting because it is a good example of non-ful"lment of the law of Kirchho!. According to this law, as stated in a classical book of thermodynamics [15], `the ratio of the absorbing and the emitting
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Fig. 4. Normalised voltage as a function of the band gap energy, at which the maximum e$ciency for the di!erent cases of study (which are labelled) is obtained. Notice the cases at which the voltage is negative.
power of a body is independent of the nature of the body, of the chosen point and of the direction, and only dependent on the temperature of the body and on the wavelengtha. This is not accomplished with the semiconductor. In fact, the semiconductor, with a back absorber, absorbs the totality of the incident radiation but emits di!erently (the ratio of the absorbing and the emitting power depends then on the nature of the bodies). The reason for this is the lack of thermodynamic equilibrium in the receiver, which causes the splitting of the quasi-Fermi levels and suggests that, in solar energy, more care should be taken with the common application of the Kirchho!'s law because, with the solar photons as an energy source, luminescent excitation may exist. This observation is not restricted to semiconductors. It has to be said that such subtleties were in the mind of the classic author [15] when he wrote `we are going to study...the purely thermal radiation or incandescent radiation. We do not study the luminescent radiation...a. The question that motivated our paper refers to the maximum e$ciency that can be obtained in the hybrid system if we optimise the e$ciency simultaneously with respect to the cell temperature and to the voltage. This maximum e$ciency is represented vs. the band gap also in Fig. 2 for case a and b. What we observe is that it exceeds, although slightly, the e$ciency of the hybrid system when it is in short circuit, or in other words, the e$ciency of the purely solar-thermal converter. The maximum maximorum is of 86.7% for a band gap of 0.10 eV instead of the 85.4% of the ideal solar-thermal case. Thus, it can be said that the hybrid systems have an e$ciency advantage, greater for photovoltaic converters and less for the solar thermal converters. This statement is valid both for case a } as already obtained by de Vos [2] } and for case b. However we shall see that this is not the general case.
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How to achieve this higher e$ciency is perhaps more interesting. For this purpose, we present the temperature and the voltage for the maxima in Figs. 3 and 4, respectively. The observation of the voltage curve tells us that the highest e$ciencies (obtained at low band-gaps) are obtained for negative voltages. This means that some of the power produced by the thermal engine is absorbed by the solar cell and used to reduce the chemical potential of the emitted radiation. In this way, the radiant energy sent back to the sun is reduced and the temperature of the converter can be increased (see Fig. 4), resulting in a higher Carnot e$ciency of the engine. This is interesting because it has been shown that the e$ciency upper bound of a solar converter (Landsberg's e$ciency) is achieved when the radiated energy is equivalent, in the way described by Luque and MartmH (LM) [7], to a thermal radiation at the ambient temperature. We wonder if, by shaping the emitted radiation with some reverse bias, using the power produced with the thermal engine, this equivalent radiation may be reproduced. This point will be dealt with in detail in Section 4. Finally, in Figs. 5 and 6, we present the e$ciency vs. bandgap for hybrid converters at di!erent temperatures for the cases a and b, respectively. For the sake of comparison, we have included in these "gures the maximum e$ciency curves (appearing at variable temperatures) also drawn in Fig. 2. The purely photovoltaic case is also shown as the curve at the ambient temperature. These curves tell us that hybrid converters may be of practical interest, in particular for low absorber temperatures, if the solar cell and the thermal engine can be kept relatively free of losses. For instance, for the ideal devices at 500 K, the purely thermal converter gives an e$ciency of 40%, according to Eq. (13), while the hybrid system of case b, with an ideal cell of 1.10 eV, gives an e$ciency of 62.7%. The best cold solar cell, of 1.11 eV, gives the 40.7%. 3.2. Entropy Eq. (6) represents the rate of entropy generation: it has to be non-negative. It can be considered as a function of two variables: q< and ¹ . The derivatives with respect to them are *S N 1 *X N !N , "! ! " ¹ ¹ *(q<) ¹ *(q<) *S E X 1 *X q
(14)
In the determination of these derivatives we have used the basic thermodynamic relationships [4] stating that the grand potential #ow (X) derivatives with respect to the chemical potential (q<) and with respect to the temperature, with the sign reversed, are respectively the particle (in this case with e'e ) and the entropy #ows. % The entropy expression in Table 2 has also been used. Looking at these formulae we see that, for constant temperature, an extremum appears for open circuit conditions. For direct (positive) current in the cell (<)< )
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Fig. 5. Maximum e$ciency as a function of the band gap energy and for constant temperature of operation, for the hybrid converter } case a in which photons below the band gap are not used for conversion into work. For reference of the reader, the maximum e$ciency, when the temperature of operation is also optimised, is also plotted as a dashed line. Short-dashed lines plot the maximum e$ciency for the hybrid-converter in which photons below the band gap are used for energetic conversion into work (hybrid converter } case b).
Fig. 6. Maximum e$ciency, as a function of the band gap energy for constant temperature of operation, for the hybrid converter- case b in which photons below the band gap are used for conversion into work. For reference of the reader, the maximum e$ciency, when the temperature of operation is also optimised, is also plotted as a dashed line. Also for reference, short-dashed lines plot the maximum e$ciency for the hybridconverter in which photons below the band gap are used for energetic conversion into work (hybrid converter } case a).
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the derivative is negative and for reverse current (<*< ) the derivative is positive. Therefore, the extremum is a minimum. It must be noticed that this is also true even for the case that the source is at a lower temperature than the cell (< (0). The extremum along the line of minima given by k"q< (or N "N ) is obtained by also setting to zero the derivative with respect to ¹ while keeping the open circuit condition (N "N ). This occurs when the E "E . Then, the two equations in Eq. (14) are ful"lled for ¹ "¹ and <"0 but we still do not know if it is minimum minimorum. However,when <"0, E is an increasing function of ¹ so that S decreases for temperatures below ¹ and increases for temperatures above it. Thus, a minimum also occurs at the mentioned point in the direction of the line of minima. It is therefore a minimum minimorum. At this point it is easy to see, looking at Eq. (6), that the entropy is just zero. Therefore, for any other combination of < and ¹ the entropy is higher and therefore positive. The preceding discussion, valid for cases a and b, is illustrated for case a in Fig. 7, representing the entropy vs. temperature with the condition of <"< for three di!erent band gaps. For the condition of zero entropy production (¹ "6000, q<"0), the reversibility is con"rmed by reversing the time, so that the entering and escaping bundles of rays are interchanged while zero power is delivered or received. In this way we have proven the thermodynamic coherence of the models utilised, in the sense that an increase in entropy always takes place. We have also proven that for a given combination of cell temperature and voltage (¹ "¹ , <"0), the conversion process is reversible. Consequently, we may assert that the hybrid converter we have modelled is ideal in the sense we mentioned that it can produce the least possible entropy. In fact, it is zero for certain values of the couple (¹ , q<).
Fig. 7. Normalised irreversible entropy rate that is produced in the hybrid converter in open circuit as a function of the temperature for several illustrative band gap energies for the case a. Normalising factor, S , 1 is 4/3p¹. 1
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4. E7ciency limit of hybrid converters The upper limit of the e$ciency of any solar converter subject to the time microreversibility was obtained by Landsberg and Tonge [3] for solar-thermal converters as
4¹ 1 ¹ g"1! # 3 ¹ 3¹
(15)
(93.3% for 6000}300 K) and generalised by LM [7] for emitted radiation of any kind. These authors emphasised that, in order to achieve this limit, two conditions were necessary: (i) entropy generation should be zero and (ii), the emitted radiation should be thermal equivalent to a radiation temperature ¹ . This equivalence means that the emitted radiation is characterised by a chemical potential k and a temperature ¹ satisfying
¹ ¹ k "e 1! #k "0 V ¹ ¹
(16)
for all the energies e of the photons emitted. However, it is not known how to make a device ful"lling these conditions. We have observed that high e$ciencies may be obtained by injecting the power of a thermal engine into the solar cell. In this way, the reverse bias (q<"k, negative) of the cell reduces the radiated energy and this allows for a higher temperature of the thermal engine and thus, for a higher e$ciency. This reverse bias is also obtained with the preceding Eq. (16), for ¹ '¹ , when k "0. However, we observe that, to achieve the Landsberg s limiting e$ciency, this reverse biasing or k should be a function of the photon energy. Therefore, if a solution exists to our problem it has to be within the framework of a stack of converters, each one set to operate at a monochromatic radiation of photons of energy e. In this stack, each one of these monochromatic cells would be made of a semiconductor of band gap e, covered at the front by an ideal low-pass "lter that transmits the photons with energy below e#de and at the back, by another ideal low-pass "lter of energy cut-o! at e. This cell has to be illuminated by an ideal concentrator that sends all the luminescent photons emitted back to the sun. The front low-pass "lter is actually the back "lter of the preceding cell in the stack. The aim of the back low-pass "lter is to assure that the luminescent photons emitted backwards are not actually able to escape and therefore this loss of energy is prevented. An ideal mirror can do the same but it would prevent the sunlight from entering deeper in the stack. It should be mentioned, however, that ArauH jo and MartmH [16] proved that for a photovoltaic stack, the use of the preceding "lters was irrelevant as the number of gaps approaches in"nity. Indeed, in the hybrid converter, every cell in the stack will supply its heat to a di!erent Carnot engine.
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By using the nomenclature in Table 2 we can write the output power per unity of band gap (d=/de"w) as a particularisation of Eq. (10):
¹ k ¹ e!k k¹ w"(e !e ) 1! # (e !e ) "(e !e ) 1! , (17) ¹ ¹ e k¹ e where we have used that e " en. If we substitute in this equation the chemical potential obtained from Eq. (16) by k "0 we realise that the output power is zero, that is, we expend all the power produced by the thermal engine to keep the emitted radiation equivalent to a thermal one at the ambient temperature. In the less common case of operating the cell at a below-ambient temperature, the chemical potential to produce a radiation equivalent to a thermal one at the ambient temperature is positive and the cell is thus generating power, but all this power is to be spent on keeping the cell cool, with the Carnot engine operating as a refrigerator. Again the output power is zero. It can be seen we have been able to ful"l one of the conditions (condition ii in the way de"ned by LM) for achieving the Landsberg s e$ciency. However, in the preceding case the entropy generation is not zero, so the other condition has not been ful"lled. For monochromatic cells, the entropy generation is zero [7] only if the cell is in open circuit, that is for n "n (or e "e ). This means that the arguments of these variables are equal:
e e!k ¹ Nk "e 1! . " (18) k¹ k¹ ¹ In this case k is not zero, but k "e(1!¹ /¹ ), and the total power produced is V V again zero, now because the energy received from the sun is full balanced by the one emitted back by the cell. Obviously, the maximum power production corresponds in this case to a di!erent condition } not ful"lling any one of the conditions for the Landsberg's e$ciency } characterised by certain values of ¹ and k. However, for its calculation it is worth, noting Table 2 and the Eq. (17), that w is a function of ¹ and k only through the variable z de"ned as e!k z" . (19) k¹ In other words, the power output is the same for all the couples of values (k, ¹ ) situated in any straight line obtained by "xing z to a constant value (Fig. 8). This means that for any hybrid converter operating at an energy e, with a chemical potential k, and at a receiver temperature ¹ we can "nd a purely photovoltaic one, operating at ambient temperature ¹ , giving the same power, with a chemical potential k "e(1!¹ /¹ )#k¹ /¹ or a purely solar thermal one (k"0) operat ing with a temperature at the receiver ¹ "¹ e/(e!k). That is, no theoretical advantage exists, when using stacked structures, for the hybrid system as compared to the purely photovoltaic one or to the purely solar thermal one.
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Fig. 8. Illustrative representation of the factor z"(e!k)/k¹ for di!erent constant values of z, also showing the lines corresponding to the solar thermal converter and to the pure photovoltaic converter.
Fig. 9. E$ciency as a function of the energy for a monochromatic converter (connected line). The optimum value of z"(e!k)/k¹ that gives the result is also plotted (dashed line).
We represent the power vs. e optimised with respect to z and normalised with respect to the input energy density #ux e(¹ , 0, e) in Fig. 9. The quotient of the integration of the optimal w throughout all the e divided by the input power form the sun, E(¹ , 0, 0,R) gives the highest e$ciency conversion of an ideal stacked system with an in"nite number of elements which results in 86.8% (6000-300 K) [17}19]. Novel photovoltaic devices with higher-than-one quantum e$ciency, have recently been analysed [20,21,7]. Their e$ciency is limited to 85.9%. (Originally the "gure was
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calculated for ¹ "5760 K, giving 85.4% and has been recalculated by us for ¹ "6000 K) and therefore it does not give better results. 1 Therefore, the stack analysed in this section can give more e$ciency than any known solar converter. The fact that all the monochromatic cells with the same value of z give the same e$ciency adds a degree of freedom to the design of the stack. For instance, it can be a stack of solar cells at ambient temperature, each operating with a di!erent voltage. It may also be a stack of pure solar-thermal converters, each one at a di!erent temperature, with a di!erent engine. But we can, for instance, put all the cells at the same temperature (di!erent from the ambient), and optimise their operating voltages so that only one thermal engine is needed. Other combinations implying arbitrary links between voltage and temperature may equally lead to converters reaching the e$ciency maximum maximorum.
5. Conclusions We have presented here a comprehensive analysis of the photovoltaic and the solar-thermal converters including hybrid converters where both e!ects are used together. In this analysis, such devices are de"ned by the chemical potential and the temperature of the radiation they emit. The discussion of the operation of these devices has been supported also by a study of the entropy production. Under certain conditions the system is reversible, (produces zero entropy) and this proves that our analysis is related to ideal systems producing the least possible entropy. The study of the hybrid converters has also been extended to the case of an in"nite stack of monochromatic converters, for which the temperature and the chemical potential may vary continuously as a function of the photon energy. Obviously the latter are those with more degrees of freedom and allow for the most optimum design. For this case, it has been proven that hybrid converters can give the same e$ciency as photovoltaic or solar thermal devices working alone. To some extent, they are strictly equivalent, giving a top e$ciency of 86.8% for a source temperature of 6000 K and an ambient temperature of 300 K. However, if we restrict our analysis to devices operating with a single cell, acting as the absorber of the hybrid system, then the hybrid converters give an e$ciency of 86.7%, very close to the previous one and higher than the top e$ciency achievable with a single temperature solar thermal of 85.4% and much above the one achievable with a single solar cell at room temperature of 40.7%. Letting impractical situations aside (due to excessive temperature), in many practical situations, hybrid converters give more e$ciency than solar-thermal or photovoltaic ones. Furthermore, the values for ideal devices are such (at 500 K, 61.7% for the hybrid system vs. 40.0% for the solar-thermal or 40.7% for the photovoltaic one at 300 K) that it is very probable that practical hybrid converters can attain more e$ciency than practical solar-thermal or photovoltaic converters alone. Another important question is whether the Landsberg's e$ciency } the e$ciency upper bound of any solar converter (obeying the time micro-reversibility) of 93.3% } can be achieved with some type of hybrid converter. The answer is negative. Hybrid
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converters may separately ful"ll the two conditions for achieving the Landsberg's e$ciency: they may generate zero entropy or they may emit a radiation equivalent to a room temperature thermal radiation, but both conditions cannot be achieved simultaneously. In fact, in both cases the energy produced is zero. Therefore, we continue to ignore how to reach the Landsberg's e$ciency or if this e$ciency may even be reached, even in ideal situations. Finally, the case of the semiconductor receiver (not necessarily a solar cell) is interesting as it constitutes a good example of the failure of the Kirchho!'s law. This is due to the lack of thermodynamic equilibrium in the receiver, which causes the splitting of the quasi-Fermi levels and suggests that, in solar energy, the common application of Kirchho!'s law should be done more carefully because, with the solar photons as an energy source, luminescent excitation may exist. This observation is not restricted to semiconductors.
References [1] J. Loferski, C. Case, D. Doodlesack, B. Roesler, R. Dobbins, T. Russell, J. Beall, J. Kirkorian, Proc. 16 Photovoltaic Specialists Conf., IEEE, New York, 1982, p. 188. [2] A. de Vos, Endoreversible Thermodynamics of Solar Energy, Oxford University Press, New York, 1992. [3] P.T. Landsberg, G. Tonge, J. Appl. Phys. 51 (1980) R1. [4] E.D. Landau, E.M. Lifchithz, Physique Statistique, MIR, Moscow, 1967. [5] W.T. Welford, R. Winston, The Optics of Non-imaging Concentrators, Academic press, New York, 1978. [6] J.C. Min ano, in: A. Luque (Ed.), Solar Cells and Optics for Photovoltaic Concentration, Ch. 12, Adam Hilger, Bristol, 1989. [7] A. Luque, A. MartmH , Phys. Rev. B 55 (1997) 6994. [8] W. Shockley, H.J. Queisser, J. Appl. Phys. 32 (1961) 510. [9] A. MartmH , J.L. Balenzategui, J. Appl. Phys. 82 (8) (1997) 4067. [10] A. Luque, A. MartmH , Phys. Rev. Lett. 78 (1997) 5014. [11] J.C. Min ano, Appl. Opt. 22 (1983) 2751. [12] M. Castan s, Revista Geo"Hsica 35 (1976) 227. [13] A. de Vos, H. Pauwels, Appl. Phys. 25 (1981) 119. [14] G.L. ArauH jo, A. MartmH , Sol Energy Mater. Sol Cells 33 (1994) 213. [15] G. Bruat, Cours de Thermodynamique, Masson, Paris, 1933, pp. 303 and 312. [16] G.L. ArauH jo, A. MartmH , Proc. 22nd Photovoltaic Specialists Conf., IEEE, New York, 1991, p. 290. [17] H. Pauwels, A. De Vos, Solid State Electron. 24 (9) (1981) 835. [18] De. Vos, H. Pauwels, J. Appl. Phys. 25 (1981) 119. [19] A. MartmH , G.L. ArauH jo, Sol. Energy Mater. Sol. Cells. 43 (1996) 203. [20] J.H. Werner, S. Kodolinski, H.J. Queisser, Phys. Rev. Lett. 72 (1994) 3851. [21] J.H. Werner, R Brendel, K.J. Queisser. Proc. 1st World Conf. Photovoltaic Energy Conversion, IEEE, New York, 1994, p. 1742.