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Thermodynamics and reciprocity of solar energy conversion
Physica E 14 (2002) 71 – 77
www.elsevier.com/locate/physe
Thermodynamics and reciprocity of solar energy conversion T. Markvarta; ∗ , P.T. Landsbergb a School
b Faculty
of Engineering Sciences, University of Southampton, Higheld, Southampton SO17 1BJ, UK of Mathematical Studies, University of Southampton, Higheld, Southampton SO17 1BJ, UK
Abstract A model is developed which describes the useful conversion of radiation in di(erent systems: solar cells, photochemical and photobiological systems (as in photosynthesis). The connection with irreversible thermodynamics is emphasised for cases when the di(erences in temperature and potential are small. Some conjectures are made as to the extension of this work to systems with larger -uxes of heat and photons. ? 2002 Elsevier Science B.V. All rights reserved. PACS: 84.60.J; 92.20.L; 05.70.L Keywords: Photovoltaics; Photosynthesis; Irreversibility
1. Introduction The fundamental limits on solar cell e7ciency have been aired many times since the modern solar cell has seen the light of day. In their seminal paper on the e7ciency of p-n junction solar cells, Shockley and Queisser [1] obtained the
application to photosynthesis (see, for example Refs. [12–14]). This paper considers the thermodynamic constraints on the photovoltaic or photochemical solar energy conversion, with two principal aims. First, solar energy conversion is considered in a rather general setting, which embraces both the photochemical and photovoltaic mechanisms. This similarity has been noted before on a number of occasions (see, for example Ref. [5]) but a clear demonstration that both these phenomena can be described by a modi
1386-9477/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 2 ) 0 0 3 5 2 - 1
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T. Markvart, P.T. Landsberg / Physica E 14 (2002) 71 – 77
expressed in simple mathematical terms and extended to other systems such as the photosynthetic energy conversion in purple bacteria. The second aim of this paper is to examine the conversion mechanism as an irreversible process, and apply the appropriate formalism (see, for example Ref. [17]). Since, to our knowledge, there is at present no satisfactory description of solar energy conversion as an explicitly irreversible process, a plausible generalisation of Onsager’s relations is proposed which is consistent with the currently available data. Similar relations have been known since the beginning of solid-state electronics [18] but their full impact on solar cell operation has only recently been recognised [19 –23]. 2. A simple model for solar energy conversion Processes that convert light into other forms of energy are dealt with in electronics, chemistry or physics, and barriers between these disciplines usually emphasise di(erences rather than similarities. Those of these processes which are of a quantum nature (principally the photovoltaic and photochemical conversion mechanisms) generally consist of the absorption of a photon of light by a molecule or semiconductor structure, promoting an electron to an excited state, followed by charge separation—by a semiconductor (or metal=semiconductor junction) in the solar cell or between two other molecules (donor D and acceptor A) in a photochemical reaction. The latter reaction—which can be represented as -
P*
D++A
P
D+A
(1)
can also be used as a simple model to illustrate the applicability of the solar cell equation. This simple model consists of a light absorber where the absorption of a photon of light promotes an electron from the ground state P to an excited state P∗ (Fig. 1). The total rate of excitation g of the absorber is a combination of two processes, illumination (inducing transitions with rate gopt ) and thermal excitation, with rate go . The lifetime of the excited state with respect to transitions to the ground state will be denoted
P*
µA
∆µ A (Electron reservoir)
µD D (hole reservoir)
hν P
Fig. 1. Schematic diagram of a photochemical energy converter.
by . The conversion process is modelled by adding two other components: an electron reservoir (which accepts an electron from the excited state) and a hole reservoir (which accepts a ‘hole’ from the ground state or, in other words, donates an electron to the ground state). In steady state, the net excitation rate is the di(erence between the generation rate G and recombination rate R. If the population of the excited state p is small and the two level system always remains electrically neutral, we can write 1 G − R = (gopt + go ) − p po p −1 ; = gopt − po
(2)
where we used the fact that the rate of thermal generation in thermal equilibrium, go , is equal to the rate of recombination po =. The chemical potential of electrons in these reservoirs will be denoted by e and h . In photochemistry, these correspond to the chemical potentials of the reduced and oxidised species in the reaction (1). One can then write Eq. (2) in terms of the di(erence K = e − h , K po e kB T − 1 ; G − R = gopt − (3) where p=po of Eq. (2) has now been replaced by ee =kB T =eh =kB T . Under ideal circumstances with no other losses present, the overall rate I of charge separation (i.e., also of energy extraction) is equal to the net rate of
T. Markvart, P.T. Landsberg / Physica E 14 (2002) 71 – 77
73
excitation (3). In more general situations, one can still write [13,14] I = I‘ − Io (eK=kB T − 1);
(4)
where I‘ and Io are constant parameters which express the forward and reverse rates of the chemical reaction (see also Fig. 4 below). In particular, the forward rate I‘ is a product of the quantum yield and the excitation rate gopt : I‘ = gopt :
hν
µe e∆ψ µh
(5)
For a solar cell, of course, Eq. (4) becomes the familiar Shockley equation, with I‘ and Io playing the role of the light-generated and dark saturation current, respectively. The chemical potential di(erence K becomes the product eK of the light-induced electrostatic potential di(erence K across the junction (i.e., the measured voltage) and the electron charge e. Note that in both cases, the argument of the exponential is the free energy per electron produced by the chemical reaction or photovoltaic process. As shown in Fig. 1, the photochemical process can thus be represented with the aid of the two-level ‘quantum converter’ P which ‘pumps’ electrons from a low-energy donor D to a high-energy acceptor A. As a result, the level of the acceptor reservoir rises above the equilibrium level , but the self-energies of the electrons and holes (represented by the standard redox potentials of the couples D+ =D and A=A− or, in the graphical representation, by the positions of the two ‘buckets’) remain the same. This picture can be compared with the operation of the solar cell (Fig. 2) where the conversion centre (a semiconductor junction) creates electrical energy by raising the energy of the electron reservoir rather than by
Electrical load Fig. 2. Schematic diagram of the operation of a solar cell.
centration of the high-energy chemical species. Thus, the bacterial photosynthetic system provides, in some sense, a bridge between, and a generalisation of, the photochemical and photovoltaic conversion mechanisms (see Eq. (6) below). The photosynthetic reaction centre of purple bacteria consists of a protein complex which binds cofactors of the active branch of the electron transport chain: the primary electron donor (a bacteriochlorophyll dimer), a bacteriochlorophyll molecule (denoted by B), bacteriopheophytin H and a quinone Q (Fig. 3; see, for example, [24,25]). The quinone QB of the inactive branch forms the
74
T. Markvart, P.T. Landsberg / Physica E 14 (2002) 71 – 77
Antenna
P
*
+
P B
+
-
ν hν
Exciton transport
+
P Q
-
Reaction rate
P H
Photochemical reaction
MPP
Solar cell
P Electrochemical potential
work Fig. 3. Electron transport in bacterial photosynthesis. K is the light-induced electrostatic potential di(erence across the junction.
One can show [15,16] that the rate of the photosynthetic charge separation also follows an equation of the form (4) but the exponent becomes the di(erence of electrochemical potentials of the electron on the two sides of the photosynthetic membrane. Thus, the three special cases that we have considered all comply with the conversion Eq. (4), but the electrochemical potential K contains the following di(erent contributions: eK (solar cell) pq kB T ln (photochemistry) K = po qo p q (photosynthesis); eK + kB T ln po qo (6) where p and q are the probabilities of electron and hole occupation of the acceptor and donor states. Although the operating points of the photovoltaic and photochemical converters all lie on the characteristic (4), there will usually be some di(erences in how this operating point is determined (Fig. 4). In a solar cell, for example, the load may be a resistor which can be represented by a straight line in the I –V diagram. In the photochemical case or photosynthesis, the ‘load curve’ is more likely to be an exponential function of the chemical potential, re-ecting the linear dependence of the chemical rates on the concentrations of reactants. The dependence of the operating point on the electric
Fig. 4. The characteristic (4) (full line). The typical load curve for a solar cell is shown by the dashed line. The dotted curves are load curves for di(erent photochemical reactions or for photosynthesis with di(erent values of the electric
point and extract the maximum energy from sunlight without resorting to electronic devices. In thermodynamic terms, the photochemical= photovoltaic conversion process shown in Figs. 1 and 2 can be thought of as a heat engine which absorbs heat in the form of photons from a high-temperature source (the sun) and emits heat at near ambient temperatures. This heat -ux pumps electrons from the low-energy reservoir h to the high-energy reservoir e , producing work or high-energy chemical compounds. In general, this is a thermodynamically irreversible process. An exception is the operation at or very near the open circuit where, under ideal conditions, the solar cell operation can be described by an analogue of the Carnot cycle [4,7] with turnover rate equal to the optical lifetime . At other points of the characteristic (4), solar cell operation is accompanied by entropy generation. For example, at short circuit, a photon with energy equal to the energy gap generates a separated electron and hole whose combined free energy is zero. To describe this process, one must employ the language of irreversible thermodynamics.
3. Irreversible thermodynamics of solar energy conversion A schematic diagram of the energy converter is shown in Fig. 5. Photons, i.e. heat, are absorbed from
T. Markvart, P.T. Landsberg / Physica E 14 (2002) 71 – 77
Ts
T
I qin
I
g
1/ τ
Iqout
I
75
where KT 1 1 1 = = − : K T Ts T T Ts
µe
µh
Fig. 5. The solar energy converter as a heat engine (see text for details).
a reservoir held at temperature Ts , corresponding to heat -ow Iqin . This is the net photon -ux absorbed by the converter, i.e., including incident photons less those emitted by the converter which originate from electron-hole pairs not in thermal equilibrium at temperature T . These pairs may be those created by light before charge separation or the neutral pairs produced by a di(erence of the electrochemical potentials h and e . Ts denotes any temperature higher than T and will later be equated to the temperature of solar radiation. Heat is emitted at ambient temperature T , producing heat -ow Iqout . The heat -ux from the high to low temperature reservoir drives a particle current I , pumping particles from the hole reservoir at electrochemical potential h to the electron reservoir at a higher potential e . Both particle reservoirs are at the ambient temperature T . The entropy production rate due to irreversible operation of the converter will be denoted by . Let us assume, for a moment, that the di(erences K = e − h ;
(7)
KT = Ts − T
(8)
are in some sense small. One can then try to understand the coupling between the -ows in terms of equations of the irreversible thermodynamics (see, for example Ref. [26]), and write the particle and heat -ows as linear combinations of the ‘forces’ (7) and (8): K 1 − I = Lpp + Lpq K ; (9) T T K 1 Iqin = Lqp + Lqq K ; (10) T T
Eqs. (9) and (10) can be justi
(11)
Eq. (11) shows that K(1=T ) and K=T are the correct ‘forces’ in the Onsager terminology which drive the conversion process. Iqin and −I are the associated -ows, corroborating Eqs. (9) and (10). The Onsager reciprocity relation [27] then states that Lpq = Lqp :
(12)
So far, this formalism appears to bear little relevance to the operation of solar cell or the photosynthetic apparatus that was discussed earlier. We shall show presently that this may be due simply to the fact that, for any useful solar energy converter, the di(erences (7) and (8) are not small. Indeed, solar radiation can be approximated by black body radiation at temperature near 6000 K, far above the usual ambient temperature of 300 K. Similarly, the voltage achieved by the solar cell is usually in the region of 0:5 V, giving energy of 0:5 eV. This is very much in excess of the ambient thermal energy of 25 meV. Can the current relations (9) and (10) be generalised to encompass these large di(erence of potential or temperature? Although, to our knowledge, a general theory of this type does not exist, we shall provide a recipe which appears to be consistent with the known data. Furthermore, the formalism which arises is similar to the linear irreversible thermodynamics of chemical reactions where small deviations from equilibrium can be extended to large values by using the law of mass action rather than the minimum of the
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T. Markvart, P.T. Landsberg / Physica E 14 (2002) 71 – 77
entropy production (see, for example, the extension of the equations of linear irreversible thermodynamics in a recent treatment [28]). We thus propose the following replacements: K K → kB exp −1 ; (13) T kB T
1 kB h 1 1 K → exp −1 ; − T h kB T Ts
(14)
where h is the energy di(erence of the two levels of the converter and, as before, T , Ts are the ambient and solar temperatures. Before writing down the resulting equations, the exponential in Eq. (14) will be related to the absorption and emission parameters of the converter by using detailed balance between these two processes (e.g. based on the well-known relationship between Einstein’s A and B coe7cients which are discussed in a similar manner to the present treatment (for example, in Ref. [8], p. 86, or Ref. [16]). In the case of non-degenerate statistics, this relation can be written in terms of the total excitation rate g = gopt + go (as de
h : (15) g = exp − kB Ts Eq. (14) can now be written using the equilibrium population of the upper two levels of the converter po ,
h 1 1 exp −1 − kB T Ts gopt g − po = : (16) = po po Using Eqs. (13), (14) and (16) then gives K kB − 1 − Lpq gopt ; (17) I = −kB Lpp exp kB T hpo K kB − 1 + Lqq Iqin = kB Lqp exp gopt : (18) kB T hpo The
generated (I‘ ) and dark saturation currents (Io ) as I‘ = −
kB Lpq gopt ; hpo
Io = kB Lpp :
(19) (20)
Furthermore, Eq. (5) indicates that the coe7cient of Lpq in Eq. (19) (which is negative) is related to the quantum yield or collection probability by Lpq = −
hpo
: kB
(21)
Using now the Onsager reciprocity relation (12), the coe7cient in front of the square brackets of the
: (22) kB Lqp = −h We now note that the
exp kB T Equating (23) and (24) now gives K p − po −1 = exp po kB T
(25)
in agreement with the reciprocity relations obtained previously in [20 –23] for the carrier transport in solar cells. 4. Conclusions We have shown that a generalised form of equations can be used to describe a range of solar energy conversion processes, including the photovoltaic and photochemical conversion as well as the photosynthetic energy conversion in purple bacteria. From a
T. Markvart, P.T. Landsberg / Physica E 14 (2002) 71 – 77
thermodynamic point of view, a surprising result has been found, namely that the case of photosynthesis covers a wider range of conditions than does the photochemistry or the solar cell (see Eq. (6)). Particularly interesting is the possibility this creates for the photosynthetic organism to control the operating point and optimise the energy produced by photosynthetic reaction, as shown in Fig. 4. Irreversible thermodynamics has been shown to provide a new insight into the conversion process. The Onsager relations between the driving forces and -ows can be generalised to yield the Shockley solar cell equation and the reciprocity relations. Rau and Brendel [23], in particular, already discussed the relationship between these two aspects of the reciprocity relation for a small applied bias. Acknowledgements PTL is indebted to the NATO linkage grant PST.CLG.975 758 for support. References [1] W. Shockley, H.J. Queisser, J. Appl. Phys. 32 (1961) 510. [2] H.A. MUuser, Z. Physik 148 (1957) 380. [3] P. Baruch, C. Picard, R.M. Swanson, Proceedings of the Third European Photovoltaic Solar Energy Conference, Cannes, 1980. [4] P. Baruch, J.E. Parrott, J. Phys. D 23 (1990) 739. [5] P. WUurfel, J. Phys. C 15 (1982) 3967.
77
[6] A. deVos, H. Pauwels, Appl. Phys. 25 (1981) 119. [7] P.T. Landsberg, T. Markvart, Solid-State Electron. 42 (1998) 657; P.T. Landsberg, V. Badescu, J. Phys. D 33 (2000) 3004. [8] P.T. Landsberg, V. Badescu, in: S. Sieniutycz, A. de Vos (Eds.), Thermodynamics of Energy Conversion and Transport, Springer, New York, 2000, p. 72. [9] P.T. Landsberg, in: S. Sieniutycz, P. Salamon (Eds.), Advances in Thermodynamics, Vol. 3, Taylor and Francis, Bristol, 1991. [10] P.T. Landsberg, V. Badescu, Second World Conference on Photovoltaic Energy Conversion, Vienna, 1998, p. 62. [11] A. Luque, A. Marti, Phys. Rev. B 55 (1997) 6994. [12] R.T. Ross, M. Calvin, Biophys. J. 7 (1967) 595. [13] R.S. Knox, in: J. Barber (Ed.), Primary Processes of Photosynthesis, Elsevier, Amsterdam, 1977, p. 55. [14] M.D. Archer, J.R. Bolton, J. Phys. Chem. 94 (1990) 8028. [15] T. Markvart, P.T. Landsberg, Twelfth Workshop on Quantum Solar Energy Conversion, Selva Gardena, 2000, unpublished. [16] T. Markvart, Prog. Quantum Electron. 24 (2000) 107. [17] J. Jou, J. Casas-Vazques, G. Lebon, Extended Irreversible Thermodynamics, 2nd Edition, Springer, Berlin, 1996. [18] W. Shockley, M. Sparks, G.K. Teal, Phys. Rev. 83 (1951) 151. [19] K. Misiakos, F.A. Lindholm, Appl. Phys. Lett. 58 (1985) 4743. [20] C. Donolato, IEEE Trans. Electron Dev. ED-37 (1990) 1165. [21] T. Markvart, IEEE Trans. Electron Dev. ED-43 (1996) 1034. [22] M.A. Green, J. Appl. Phys. 81 (1997) 269. [23] U. Rau, R. Brendel, J. Appl. Phys. 84 (1998) 6412. [24] G.R. Fleming, R. van Grondelle, Phys. Today, 48, 1994. [25] V. Sundstrom, Prog. Quantum Electron. 24 (2000) 187. [26] O. Kedem, S.R. Caplan, Trans. Faraday Soc. 61 (1965) 1897. [27] L. Onsager, Phys. Rev. 37 (1931) 405. [28] L.S. Garcia-Colin, F.J. Uribe, J. Non-Equilibrium Thermodyn. 16 (1991) 89.
www.elsevier.com/locate/physe
Thermodynamics and reciprocity of solar energy conversion T. Markvarta; ∗ , P.T. Landsbergb a School
b Faculty
of Engineering Sciences, University of Southampton, Higheld, Southampton SO17 1BJ, UK of Mathematical Studies, University of Southampton, Higheld, Southampton SO17 1BJ, UK
Abstract A model is developed which describes the useful conversion of radiation in di(erent systems: solar cells, photochemical and photobiological systems (as in photosynthesis). The connection with irreversible thermodynamics is emphasised for cases when the di(erences in temperature and potential are small. Some conjectures are made as to the extension of this work to systems with larger -uxes of heat and photons. ? 2002 Elsevier Science B.V. All rights reserved. PACS: 84.60.J; 92.20.L; 05.70.L Keywords: Photovoltaics; Photosynthesis; Irreversibility
1. Introduction The fundamental limits on solar cell e7ciency have been aired many times since the modern solar cell has seen the light of day. In their seminal paper on the e7ciency of p-n junction solar cells, Shockley and Queisser [1] obtained the
application to photosynthesis (see, for example Refs. [12–14]). This paper considers the thermodynamic constraints on the photovoltaic or photochemical solar energy conversion, with two principal aims. First, solar energy conversion is considered in a rather general setting, which embraces both the photochemical and photovoltaic mechanisms. This similarity has been noted before on a number of occasions (see, for example Ref. [5]) but a clear demonstration that both these phenomena can be described by a modi
1386-9477/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 2 ) 0 0 3 5 2 - 1
72
T. Markvart, P.T. Landsberg / Physica E 14 (2002) 71 – 77
expressed in simple mathematical terms and extended to other systems such as the photosynthetic energy conversion in purple bacteria. The second aim of this paper is to examine the conversion mechanism as an irreversible process, and apply the appropriate formalism (see, for example Ref. [17]). Since, to our knowledge, there is at present no satisfactory description of solar energy conversion as an explicitly irreversible process, a plausible generalisation of Onsager’s relations is proposed which is consistent with the currently available data. Similar relations have been known since the beginning of solid-state electronics [18] but their full impact on solar cell operation has only recently been recognised [19 –23]. 2. A simple model for solar energy conversion Processes that convert light into other forms of energy are dealt with in electronics, chemistry or physics, and barriers between these disciplines usually emphasise di(erences rather than similarities. Those of these processes which are of a quantum nature (principally the photovoltaic and photochemical conversion mechanisms) generally consist of the absorption of a photon of light by a molecule or semiconductor structure, promoting an electron to an excited state, followed by charge separation—by a semiconductor (or metal=semiconductor junction) in the solar cell or between two other molecules (donor D and acceptor A) in a photochemical reaction. The latter reaction—which can be represented as -
P*
D++A
P
D+A
(1)
can also be used as a simple model to illustrate the applicability of the solar cell equation. This simple model consists of a light absorber where the absorption of a photon of light promotes an electron from the ground state P to an excited state P∗ (Fig. 1). The total rate of excitation g of the absorber is a combination of two processes, illumination (inducing transitions with rate gopt ) and thermal excitation, with rate go . The lifetime of the excited state with respect to transitions to the ground state will be denoted
P*
µA
∆µ A (Electron reservoir)
µD D (hole reservoir)
hν P
Fig. 1. Schematic diagram of a photochemical energy converter.
by . The conversion process is modelled by adding two other components: an electron reservoir (which accepts an electron from the excited state) and a hole reservoir (which accepts a ‘hole’ from the ground state or, in other words, donates an electron to the ground state). In steady state, the net excitation rate is the di(erence between the generation rate G and recombination rate R. If the population of the excited state p is small and the two level system always remains electrically neutral, we can write 1 G − R = (gopt + go ) − p po p −1 ; = gopt − po
(2)
where we used the fact that the rate of thermal generation in thermal equilibrium, go , is equal to the rate of recombination po =. The chemical potential of electrons in these reservoirs will be denoted by e and h . In photochemistry, these correspond to the chemical potentials of the reduced and oxidised species in the reaction (1). One can then write Eq. (2) in terms of the di(erence K = e − h , K po e kB T − 1 ; G − R = gopt − (3) where p=po of Eq. (2) has now been replaced by ee =kB T =eh =kB T . Under ideal circumstances with no other losses present, the overall rate I of charge separation (i.e., also of energy extraction) is equal to the net rate of
T. Markvart, P.T. Landsberg / Physica E 14 (2002) 71 – 77
73
excitation (3). In more general situations, one can still write [13,14] I = I‘ − Io (eK=kB T − 1);
(4)
where I‘ and Io are constant parameters which express the forward and reverse rates of the chemical reaction (see also Fig. 4 below). In particular, the forward rate I‘ is a product of the quantum yield and the excitation rate gopt : I‘ = gopt :
hν
µe e∆ψ µh
(5)
For a solar cell, of course, Eq. (4) becomes the familiar Shockley equation, with I‘ and Io playing the role of the light-generated and dark saturation current, respectively. The chemical potential di(erence K becomes the product eK of the light-induced electrostatic potential di(erence K across the junction (i.e., the measured voltage) and the electron charge e. Note that in both cases, the argument of the exponential is the free energy per electron produced by the chemical reaction or photovoltaic process. As shown in Fig. 1, the photochemical process can thus be represented with the aid of the two-level ‘quantum converter’ P which ‘pumps’ electrons from a low-energy donor D to a high-energy acceptor A. As a result, the level of the acceptor reservoir rises above the equilibrium level , but the self-energies of the electrons and holes (represented by the standard redox potentials of the couples D+ =D and A=A− or, in the graphical representation, by the positions of the two ‘buckets’) remain the same. This picture can be compared with the operation of the solar cell (Fig. 2) where the conversion centre (a semiconductor junction) creates electrical energy by raising the energy of the electron reservoir rather than by
Electrical load Fig. 2. Schematic diagram of the operation of a solar cell.
centration of the high-energy chemical species. Thus, the bacterial photosynthetic system provides, in some sense, a bridge between, and a generalisation of, the photochemical and photovoltaic conversion mechanisms (see Eq. (6) below). The photosynthetic reaction centre of purple bacteria consists of a protein complex which binds cofactors of the active branch of the electron transport chain: the primary electron donor (a bacteriochlorophyll dimer), a bacteriochlorophyll molecule (denoted by B), bacteriopheophytin H and a quinone Q (Fig. 3; see, for example, [24,25]). The quinone QB of the inactive branch forms the
74
T. Markvart, P.T. Landsberg / Physica E 14 (2002) 71 – 77
Antenna
P
*
+
P B
+
-
ν hν
Exciton transport
+
P Q
-
Reaction rate
P H
Photochemical reaction
MPP
Solar cell
P Electrochemical potential
work Fig. 3. Electron transport in bacterial photosynthesis. K is the light-induced electrostatic potential di(erence across the junction.
One can show [15,16] that the rate of the photosynthetic charge separation also follows an equation of the form (4) but the exponent becomes the di(erence of electrochemical potentials of the electron on the two sides of the photosynthetic membrane. Thus, the three special cases that we have considered all comply with the conversion Eq. (4), but the electrochemical potential K contains the following di(erent contributions: eK (solar cell) pq kB T ln (photochemistry) K = po qo p q (photosynthesis); eK + kB T ln po qo (6) where p and q are the probabilities of electron and hole occupation of the acceptor and donor states. Although the operating points of the photovoltaic and photochemical converters all lie on the characteristic (4), there will usually be some di(erences in how this operating point is determined (Fig. 4). In a solar cell, for example, the load may be a resistor which can be represented by a straight line in the I –V diagram. In the photochemical case or photosynthesis, the ‘load curve’ is more likely to be an exponential function of the chemical potential, re-ecting the linear dependence of the chemical rates on the concentrations of reactants. The dependence of the operating point on the electric
Fig. 4. The characteristic (4) (full line). The typical load curve for a solar cell is shown by the dashed line. The dotted curves are load curves for di(erent photochemical reactions or for photosynthesis with di(erent values of the electric
point and extract the maximum energy from sunlight without resorting to electronic devices. In thermodynamic terms, the photochemical= photovoltaic conversion process shown in Figs. 1 and 2 can be thought of as a heat engine which absorbs heat in the form of photons from a high-temperature source (the sun) and emits heat at near ambient temperatures. This heat -ux pumps electrons from the low-energy reservoir h to the high-energy reservoir e , producing work or high-energy chemical compounds. In general, this is a thermodynamically irreversible process. An exception is the operation at or very near the open circuit where, under ideal conditions, the solar cell operation can be described by an analogue of the Carnot cycle [4,7] with turnover rate equal to the optical lifetime . At other points of the characteristic (4), solar cell operation is accompanied by entropy generation. For example, at short circuit, a photon with energy equal to the energy gap generates a separated electron and hole whose combined free energy is zero. To describe this process, one must employ the language of irreversible thermodynamics.
3. Irreversible thermodynamics of solar energy conversion A schematic diagram of the energy converter is shown in Fig. 5. Photons, i.e. heat, are absorbed from
T. Markvart, P.T. Landsberg / Physica E 14 (2002) 71 – 77
Ts
T
I qin
I
g
1/ τ
Iqout
I
75
where KT 1 1 1 = = − : K T Ts T T Ts
µe
µh
Fig. 5. The solar energy converter as a heat engine (see text for details).
a reservoir held at temperature Ts , corresponding to heat -ow Iqin . This is the net photon -ux absorbed by the converter, i.e., including incident photons less those emitted by the converter which originate from electron-hole pairs not in thermal equilibrium at temperature T . These pairs may be those created by light before charge separation or the neutral pairs produced by a di(erence of the electrochemical potentials h and e . Ts denotes any temperature higher than T and will later be equated to the temperature of solar radiation. Heat is emitted at ambient temperature T , producing heat -ow Iqout . The heat -ux from the high to low temperature reservoir drives a particle current I , pumping particles from the hole reservoir at electrochemical potential h to the electron reservoir at a higher potential e . Both particle reservoirs are at the ambient temperature T . The entropy production rate due to irreversible operation of the converter will be denoted by . Let us assume, for a moment, that the di(erences K = e − h ;
(7)
KT = Ts − T
(8)
are in some sense small. One can then try to understand the coupling between the -ows in terms of equations of the irreversible thermodynamics (see, for example Ref. [26]), and write the particle and heat -ows as linear combinations of the ‘forces’ (7) and (8): K 1 − I = Lpp + Lpq K ; (9) T T K 1 Iqin = Lqp + Lqq K ; (10) T T
Eqs. (9) and (10) can be justi
(11)
Eq. (11) shows that K(1=T ) and K=T are the correct ‘forces’ in the Onsager terminology which drive the conversion process. Iqin and −I are the associated -ows, corroborating Eqs. (9) and (10). The Onsager reciprocity relation [27] then states that Lpq = Lqp :
(12)
So far, this formalism appears to bear little relevance to the operation of solar cell or the photosynthetic apparatus that was discussed earlier. We shall show presently that this may be due simply to the fact that, for any useful solar energy converter, the di(erences (7) and (8) are not small. Indeed, solar radiation can be approximated by black body radiation at temperature near 6000 K, far above the usual ambient temperature of 300 K. Similarly, the voltage achieved by the solar cell is usually in the region of 0:5 V, giving energy of 0:5 eV. This is very much in excess of the ambient thermal energy of 25 meV. Can the current relations (9) and (10) be generalised to encompass these large di(erence of potential or temperature? Although, to our knowledge, a general theory of this type does not exist, we shall provide a recipe which appears to be consistent with the known data. Furthermore, the formalism which arises is similar to the linear irreversible thermodynamics of chemical reactions where small deviations from equilibrium can be extended to large values by using the law of mass action rather than the minimum of the
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T. Markvart, P.T. Landsberg / Physica E 14 (2002) 71 – 77
entropy production (see, for example, the extension of the equations of linear irreversible thermodynamics in a recent treatment [28]). We thus propose the following replacements: K K → kB exp −1 ; (13) T kB T
1 kB h 1 1 K → exp −1 ; − T h kB T Ts
(14)
where h is the energy di(erence of the two levels of the converter and, as before, T , Ts are the ambient and solar temperatures. Before writing down the resulting equations, the exponential in Eq. (14) will be related to the absorption and emission parameters of the converter by using detailed balance between these two processes (e.g. based on the well-known relationship between Einstein’s A and B coe7cients which are discussed in a similar manner to the present treatment (for example, in Ref. [8], p. 86, or Ref. [16]). In the case of non-degenerate statistics, this relation can be written in terms of the total excitation rate g = gopt + go (as de
h : (15) g = exp − kB Ts Eq. (14) can now be written using the equilibrium population of the upper two levels of the converter po ,
h 1 1 exp −1 − kB T Ts gopt g − po = : (16) = po po Using Eqs. (13), (14) and (16) then gives K kB − 1 − Lpq gopt ; (17) I = −kB Lpp exp kB T hpo K kB − 1 + Lqq Iqin = kB Lqp exp gopt : (18) kB T hpo The
generated (I‘ ) and dark saturation currents (Io ) as I‘ = −
kB Lpq gopt ; hpo
Io = kB Lpp :
(19) (20)
Furthermore, Eq. (5) indicates that the coe7cient of Lpq in Eq. (19) (which is negative) is related to the quantum yield or collection probability by Lpq = −
hpo
: kB
(21)
Using now the Onsager reciprocity relation (12), the coe7cient in front of the square brackets of the
: (22) kB Lqp = −h We now note that the
exp kB T Equating (23) and (24) now gives K p − po −1 = exp po kB T
(25)
in agreement with the reciprocity relations obtained previously in [20 –23] for the carrier transport in solar cells. 4. Conclusions We have shown that a generalised form of equations can be used to describe a range of solar energy conversion processes, including the photovoltaic and photochemical conversion as well as the photosynthetic energy conversion in purple bacteria. From a
T. Markvart, P.T. Landsberg / Physica E 14 (2002) 71 – 77
thermodynamic point of view, a surprising result has been found, namely that the case of photosynthesis covers a wider range of conditions than does the photochemistry or the solar cell (see Eq. (6)). Particularly interesting is the possibility this creates for the photosynthetic organism to control the operating point and optimise the energy produced by photosynthetic reaction, as shown in Fig. 4. Irreversible thermodynamics has been shown to provide a new insight into the conversion process. The Onsager relations between the driving forces and -ows can be generalised to yield the Shockley solar cell equation and the reciprocity relations. Rau and Brendel [23], in particular, already discussed the relationship between these two aspects of the reciprocity relation for a small applied bias. Acknowledgements PTL is indebted to the NATO linkage grant PST.CLG.975 758 for support. References [1] W. Shockley, H.J. Queisser, J. Appl. Phys. 32 (1961) 510. [2] H.A. MUuser, Z. Physik 148 (1957) 380. [3] P. Baruch, C. Picard, R.M. Swanson, Proceedings of the Third European Photovoltaic Solar Energy Conference, Cannes, 1980. [4] P. Baruch, J.E. Parrott, J. Phys. D 23 (1990) 739. [5] P. WUurfel, J. Phys. C 15 (1982) 3967.
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[6] A. deVos, H. Pauwels, Appl. Phys. 25 (1981) 119. [7] P.T. Landsberg, T. Markvart, Solid-State Electron. 42 (1998) 657; P.T. Landsberg, V. Badescu, J. Phys. D 33 (2000) 3004. [8] P.T. Landsberg, V. Badescu, in: S. Sieniutycz, A. de Vos (Eds.), Thermodynamics of Energy Conversion and Transport, Springer, New York, 2000, p. 72. [9] P.T. Landsberg, in: S. Sieniutycz, P. Salamon (Eds.), Advances in Thermodynamics, Vol. 3, Taylor and Francis, Bristol, 1991. [10] P.T. Landsberg, V. Badescu, Second World Conference on Photovoltaic Energy Conversion, Vienna, 1998, p. 62. [11] A. Luque, A. Marti, Phys. Rev. B 55 (1997) 6994. [12] R.T. Ross, M. Calvin, Biophys. J. 7 (1967) 595. [13] R.S. Knox, in: J. Barber (Ed.), Primary Processes of Photosynthesis, Elsevier, Amsterdam, 1977, p. 55. [14] M.D. Archer, J.R. Bolton, J. Phys. Chem. 94 (1990) 8028. [15] T. Markvart, P.T. Landsberg, Twelfth Workshop on Quantum Solar Energy Conversion, Selva Gardena, 2000, unpublished. [16] T. Markvart, Prog. Quantum Electron. 24 (2000) 107. [17] J. Jou, J. Casas-Vazques, G. Lebon, Extended Irreversible Thermodynamics, 2nd Edition, Springer, Berlin, 1996. [18] W. Shockley, M. Sparks, G.K. Teal, Phys. Rev. 83 (1951) 151. [19] K. Misiakos, F.A. Lindholm, Appl. Phys. Lett. 58 (1985) 4743. [20] C. Donolato, IEEE Trans. Electron Dev. ED-37 (1990) 1165. [21] T. Markvart, IEEE Trans. Electron Dev. ED-43 (1996) 1034. [22] M.A. Green, J. Appl. Phys. 81 (1997) 269. [23] U. Rau, R. Brendel, J. Appl. Phys. 84 (1998) 6412. [24] G.R. Fleming, R. van Grondelle, Phys. Today, 48, 1994. [25] V. Sundstrom, Prog. Quantum Electron. 24 (2000) 187. [26] O. Kedem, S.R. Caplan, Trans. Faraday Soc. 61 (1965) 1897. [27] L. Onsager, Phys. Rev. 37 (1931) 405. [28] L.S. Garcia-Colin, F.J. Uribe, J. Non-Equilibrium Thermodyn. 16 (1991) 89.