Advanced concepts for photovoltaic conversion: bridging the gap fom theory to real devices?

Jean-François Guillemoles and Stefan Kettemann

LECA, Ecole Nationale Superieure de Chimie de Paris, 11 rue Pierre et Marie Curie, 75231, Paris, France

Abstract The potential of advanced approaches (such as up/down conversion or intermediate band) have been evaluated so far considering idealized systems (radiative recombination limit, infinite carrier mobility, optimal absorption, lossless contacts). How these stringent constraints on material parameters can be relaxed without loosing too much of the benefits proposed by the advanced scheme mentioned above has been very little studied. It is furthermore essential to think in terms of real systems able to perform better, but keeping the unit costs within those of today's technologies, if the advantage of these conversion concepts is to be of any practical use. In this work, we look for the promises and problems of various approaches at different steps of entropy creation: photonic conversion, intermediate bands and thermal conversion aiming to (a) evaluate the impact of non idealities on the performance of the prospective systems and (b) propose practical material systems to embody those systems.

1.Introduction Solar flux is composed of a broad band radiation which makes it difficult to convert it

efficiently into electrical power in a single device (meaning a device using only one broad band light collector). Most of the serious attempts so far to achieve high conversion efficiency, and more specifically, an efficiency higher than that of a single p-n junction (the simplest broad band converter), have relied on the utilisation of several semiconductors each adapted to a specific region of the solar spectrum (this applies to tandem devices as well as to quantum well solar cells). In some cases, insertion of additional levels (isolated or forming bands) have been proposed to provide for additional absorption as an alternative to tandem cells.

Conversely, one could think of adapting the incident spectrum to an ideal converter. Indeed, the highest conversion efficiencies (close to 50%) have only been achieved with monochromatic light on high quality devices. It is therefore tempting to examine under which conditions, consistent with the 2nd principle of thermodynamics, an incident broad band spectrum can be converted into radiation of narrower band width with high efficiency, thus lending itself to efficient solar to electrical power conversion. The advantage of this approach is that it avoids additional contacts and does not produce additional recombination pathways for photogenerated carriers.

Finally, one could think of using the thermal energy generated from the relaxation of photogenerated carriers excess energy or from nonradiative recombination at different stages of the thermal equilibration: before equilibration with the lattice (hot carriers), before the cell becomes isothermal or before it equilibrates with the surrounding (thermoelectric, thermoionics, thermophotonics).

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The potential of the approaches mentionned above have been evaluated before considering idealized systems (radiative recombination limit, infinite carrier mobility, optimal absorption, lossless contacts). It has been very little studied how these stringent constraints on material parameters can be relaxed without loosing too much of the benefits proposed by the advanced scheme mentioned above.

From a practical point of view, it is furthermore essential to think in terms of real systems able to perform better, but keeping the unit costs within those of today's technologies, if the advantage of these conversion concepts is to be useful (scheme 1).

In the following, we look for the promises and problems of various approaches at different steps of entropy creation: photonic conversion, intermediate bands and thermal conversion aiming to (a) evaluate the impact of non idealities on the performance of the prospective systems and (b) propose practical material systems to embody those systems.

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Scheme 1 : Plot of efficiency versus unit cost of solar cells for various types of cells (I) the single crystal Si modules, (II) thin film modules (a-Si and CIGS) and (III) what the next generation of solar cell could be using advanced conversion concepts (from [20]). White and yellow dots were placed according to published values of manufacturers of resp. Si and Thin film modules.

2.Equivalence between conversion schemes Several different methods have been proposed to improve PV conversion beyond the single junction limit and limiting efficiencies of the corresponding processes have been computed. It is striking for the careful reader of the literature that some numbers, for instance around 60%, seem to appear quite often, independently of the improvement method. For instance this value was found for a tandem system with 3 cells, the intermediate band cell (Luque, 1997[1]) or up conversion (Wurfel, 2002 [2]). The argument can be made that all 3 systems rely actually on 3 optical absorption thresholds (gaps of the density of states) that can be represented as photodiodes connected electrically or optically in different ways, as shown in figure 1. In ideal systems, with only radiative recombination losses, the only free parameter for efficiency is the cell band gap. In all 3 cases outlined above the result of the optimisation of the efficiency of the global device is that obtained under constraints,

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either current or potential matching, with the same number of free parameters. Therefore, it is proposed that 63,7% is the maximum efficiency that can be achieved with 3 gaps and 86.8% the one with an infinite number of gaps. Considering the number of ways that have been proposed to reach conversion efficiencies beyond the Schockley limit, e.g. the use of tandem systems (Henry, 1980 [3]), quantum wells (Barnham, 1990 [4]), fluorescent mirrors (Gibart, 1995 [5]), impact ionization (Kolodinsky, 1994 [6]), impurity level systems (Corkish, 1993 [7]), intermediate bands (Kettemann, 1995 [8] and Luque, 1997 [1]), hot electrons (Ross 1982 [9]), thermoelectric enhancement (Kettemann, 2001[10]), the main question is now about their practical implementation in real devices. Up to date, tandem systems have been the only ones to demonstrate efficiencies beyond the Schockley limit [11], practically. But this is done through a considerable complexity of device design and at the price of a large sensitivity of the efficiency to the illumination spectrum. In the following, we turn to the investigation of the two other schemes.

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Figure 1 Different ways to connect solar cells (of different band gaps) either electrically or optically. In all three cases, the system is equivalent to 3 diodes of different band gaps extracting work out of three different spectral ranges. In all three cases, the limiting efficiencies are close to 63% (63,7% for tandem, 63,7% for IB and 63,2% for up conversion [1,2]). The colors indicate which part of the spectrum is absorbed (and emitted) by each cell.

3.Limits of ideal devices Of course, systems actually elaborated are not ideal and it is crucial to evaluate how real systems are sensitive to non idealities . For one thing, mobilities are not infinite and this impacts on the final efficiency. Even in the radiative limit, low mobilities will adversely affect conversion efficiencies: ultimately this might set the limit in organic based cell where radiative efficiencies can be high but mobilities are low. Since, as a rule, both radiative recombination efficiency (hence the dark current) and the short circuit current increase in the same way with light

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absorption, in the radiative limit, the efficiency is independent of the absorption coefficient (in the limit of high mobilities). Therefore it is just a matter of convenience to make cells in a very absorbing, but low mobility material or conversely with a high mobility material with a low absorptivity. Practical considerations, such as cost effectiveness, might ultimately favour a small quantity of a more disordered matter?

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Figure 2 Plot of efficiencies (AM 1,5) of best cells of a class as a function of the grain sizes (from [19] in the case of crystalline Si) for different types of solar cells : crystalline Si, micro-cristalline Si, amorphous Si, dye sensitized cells and CuInGaSe2 cells. Grain size is to be understood in an extended sense in polymer or amorphous materials (characteristic spatial extension of the electron wave coherence). The thermodynamic limit under the same illumination level is also represented for reference.

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conversion efficiency. Parameters for these cells are taken from the published efficiency tables [21].

Grain sizes are a measure of mobility in a class of material, and figure 2 shows how this impacts efficiencies of best cells of that class. Grain size is an element of cost and this picture can be compared to the previous (Scheme 1). Another assumption made in the calculation of limiting efficiencies is that devices work at the radiative limit. Although they normally do not, the best devices (those made out of single crystals) seems to approach that limit (figure 3), making it relevant to the state of the art to consider how devices working close to this limit are affected by imperfections. For instance, figure 3 suggests that best cells are not limited by non radiative recombination but probably by other factors (light management, contacts,?).

4. Intermediate band devices Intermediate band devices may present some advantages compared to tandem devices, such as a possible lower spectrum sensitivity. Ultimately they may be easier to built as integrated systems than the tandem type of devices. Having obtained a good understanding of their potential to raise the efficiency [1, 8], the main question is now to find the best way to realize them. It has been suggested [8] that the multiple quantum well solar cells [4] may have been one of the earliest version of such devices. Indeed, evidence pointing in that direction has been given recently [12], althougth no decisive improvement beyond the Schockley-Queisser limit has been proven experimentally, so far. A realization of this intermediate band concept was proposed recently [13], after it was realized that such intermediate band cells would work close to their optimum efficiencies (in the ideal limit defined above) without current flowing in the intermediate band (IB) itself if that band is half filled. Before such systems can be put to work, there are a number of problems to be solved.

4.1 The absorption coefficient issue One of the main difficulties with this concept is to ensure practically and efficiently the optimal optical transitions. As pointed out in [1], the most efficient conversion process is realized by having the optical transitions so that the thermalization is minimal, i.e. so that each photon is absorbed using the transition with the highest threshold possible. In the case shown in figure 4, left, the transition with lowest energies are to proceed according to path 1 until the energy is high enough to proceed along path 2, which should be dominant for energies upward until path 3 becomes in turn possible, and dominant. Therefore, an optimal use of energy imposes very different transition rates between the bands. The problem is that this is not so easy to realize in practice because generally absorption coefficients tend to vary as 1/E, where E is the transition energy, i.e. the natural slope is the converse of what is needed. Of course, there are still degree of freedom with the densities of states and the transition matrix elements, but having efficient absorption imposes that the optical absorption are allowed and therefore can be described in the dipolar (first order) approximation, as in e.g. [22], therefore they are not expected to

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depend much on energy for a given transition between 2 bands. The 3 matrix elements corresponding to the 3 transitions of figure 4 are not completely independent, and it seems difficult to have the cross sections in the order given in the previous paragraph, specially keeping them all as high as possible. There is yet a possibility, if the bands are suitably narrow, the condition given at the beginning of the section is automatically fullfiled: for instance, if the valence band width is equal to the difference in absorption thresholds between transitions 1 and 2, the desired transitions occur naturally. Yet this has an unpleasant implication: the solar spectrum being what it is, the width of the valence band would be on the order of 1 eV, quite narrow therefore, with likely implication for the hole mobility (see below). The other important constraint on the optical absorption is that the path through the intermediate band is efficient (otherwise the low energy photons are lost, or one needs more of a higher quality material, see the remark in section 3). It is immediately apparent that one needs in this band both a high electron concentration (for an efficient transition 1) and a high hole concentration (for an efficient transition 2). This can be achieved in 2 ways: (i) either the intermediate band is efficiently populated (with both electrons and holes) by photon absorption or (ii) the IB is naturally rich in both species and therefore behaves as a metallic band [13]. Of course it is necessary that such a metallic IB would not impact badly on the optical properties of the device (e.g. increased reflectivity). Case (i) gives an absorption cross section that depends on intensity of illumination as the square, so that most likely, high illumination levels will have to be reached for the processes to be efficient. This is a problem similar to that exposed in the up/down conversion section. Case (ii) relaxes the constraint on illumination level, but at a cost: by making the recombination kinetics much faster. Because of the micro-reversibility principle, in the radiative limit, both absorption and recombination are enhanced in the same ratio and this has no net influence on efficiency. This is not so anymore if non radiative recombination is included. In the metallic IB concept, there is a strong absorption probability of the intermediate band (figure 4, left). The problem of course is that such a metallic IB is very likely to be source of recombination in real systems: the high concentration of both electron and holes in the intermediate band, mandatory for having a good absorption, will increase the kinetics of any recombination path, and are very likely to make the system specially sensitive to non radiative recombination in pathes 1 and 2 because this will also increase strongly the recombination of carrier generated by path 3.

4.2 The mobility issue This system is actually equivalent to the one of the Impurity Photovoltaic (IPV) effect described in [7]: since no photocurrent is extracted nor transported in the IB, this band is formally equivalent to a deep state. Carrier mobility in this IB nevertheless adds an advantage over the IPV effect by relaxing a constraint on the absorption coefficient: it is not necessary any more that the optical cross sections of transitions 1 and 2 (figure 4) are equal, as in the IPV effect to warrant locally that the number of electrons promoted with transition 1 is equal to that promoted by 2 (since for efficient conversion, the number of photons absorbed by each transition should be equal). This constraint over the optical upconversion can now be realized globally, so that the 2 cross sections do not need to be equal. Moreover, in the IPV effect it seems impossible to have both this equality of cross sections and obey the condition on the optical cross sections given in section 4.1 that gives optimal use of the photons energy.

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Althought transporting current in the IB would not change the ultimate efficiency of such a system, it might be wondered whether allowing such transport would help the realization of such systems to show a sizeable effect. Two examples are shown in figure 4: if current is transported in the IB, one could have the front part of the cell for the collection of yellow photons and the back part being used for the red ones. This can be realized either with a built-in electrical field or with band edge engineering. We note that in all these schemes, there is some up-conversion at work, whereby two photons are used to produce an electron with a higher potential energy. In that case, the stringent requirements on the closeness to the radiative limit can be somewhat relaxed because the optical processes of the 2 step promotion of the valence band electrons take place in different part of the device and are not therefore as strongly coupled as in the metallic IB version. The sensitivity of this new version to non radiative recombination is actually equivalent to that of p/n junctions of same fundamental band gaps connected as in figure 1. The mobility necessary in the intermediate band to avoid resistive losses under 1 sun illumination can be estimated around 100 cm²/V/s (for a typical carrier concentration of 1017 cm-3), a value that can be achieved in miniband in multiple quantum well structures, but not currently in solids having narrow bands (d or f bands) such as the transition metal oxides or chalcogenides. The only possible embodiment at present seems therefore to reside with ordered assemblies of quantum dots. One should be aware that the requirement of a narrow band (to reduce thermalization losses) and a high mobility (to avoid serial resistance losses) are somewhat conflicting for the fundamental point of view of quantum mechanics because they vary inversely as a function of the overlap integral of the electronic wavefunctions of the centers (transition metals, Q-dots) forming the band.

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Figure 4: Various possible schemes of the band structure of intermediate band solar cells. From left to right: Intermediate metallic band (according to Cuadra et al. [13]), intermediate band solar cell with band edge engineering and with built-in field. Arrows mark allowed optical transitions, dotted lines indicate Fermi levels (equilibrium) or quasi Fermi levels under illumination or bias.

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5. Up/down conversion devices One step further in the logic of conversion initiated by the IB cell would be a complete decoupling of the optical and electrical conversions, since that coupling is source of recombinations. In that photovoltaic conversion scheme, a system of levels (atomic, molecular, or else) is used for photon interconversion (up or down), as illustrated in Figure 5. These schemes have been explored and found to give, in an ideal case a conversion efficiency of 63 % for up conversion and 40% only for down conversion (with the additional constraint that the 2 emitted photons had the same energy) [2]. The fact that the incoming spectrum is broad band is the main obstacle to high conversion efficiencies: a good GaAs cell illuminated by a monochromatic light slightly above its band gap has an energy conversion efficiency of typically 50% under 100 mW/cm² (as compared to 66% as a thermodynamic maximum). This fact alone shows that high efficiencies of conversion can indeed be demonstrated by 2 level type devices. If now a large number of photons of the broad band incoming spectrum can be converted into photons whose energy is matched to that of an efficient 2-level system, high efficiencies could potentially be obtained. Photon conversion is an energy conserving process, hence the energy of the converted photons is at most that of the incoming photons, but could be (and often is) smaller. More important is the efficiency of such a process. Because the number of photons produced may depend in this case on the number of incident photons, the process efficiency may be dependent on the illumination level. This is the weak point of upconversion: the conversion efficiency of the process grows with the square of the illumination intensity. It also means that these systems are particularly sensitive to non radiative recombination in the photon interconversion. Last but not least, self absorption is a crucial issue as soon as non ideal systems are considered, which have some non vanishing non radiative recombination).

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Figure 5: Illustration of up (left) and down (right) conversion schemes. In both cases, a solar cell (as indicated by the pair of levels together with their chemical potentials) is optically coupled to a non linear optical device which transforms the energy of the incident photons.

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5.1 The band width problem Several physical processes yield photon inter-conversion (up or down), these lie in the field of non linear optics and are described either in terms of waves mixing [23] or in terms of photon-photon scattering [24]. The first of these processes uses non linear polarizability of matter, and can achieve 100% photon conversion over broad ranges of incident energies. But this requires conditions of phase matching between incident and scattered waves [23] and a high intensity of the incident wave (i.e. large electrical fields) : this effect is of practical use only with laser sources or using surface enhanced scattering [27]. Other possible physical processes use transitions via intermediate states. In the case of upconversion, the following are possible [25]:

(i) successive absorption of photons where the intermediate state is long lived enough, a second absorption can take place taking the electron from that intermediate state to the upper state. An example of this process is found is fluorozirconate glasses doped with Ho(III).

(ii) addition of photon with energy transfer: this is one of the most efficient processes, the only one to date to demonstrate an effect in PV applications [5]. In that case, a first ionic species absorbs the incident radiation transfered to a second one (see figure 6).

(iii) Photon avalanche: in this process, a cross relaxation couples the transition between the lower and the intermediate state and that between the intermediate state and the upper state. An example can be found in LiYF4 doped with Tm(III).

The examples given, using rare earth doping, lead to up conversion in virtually all parts of the visible spectrum [26], but each ion has a relatively narrow range of absorption and emission. When the range is enlarged (e.g. by coupling with the lattice), the processes become generally less efficient. Other systems using organic dyes are also known [27]. In the case of upconversion there is an additionnal issue that the process efficiency depends strongly on the incident photon flux intensity (as I² for a 2-photon process) which makes it efficient only at high incoming intensities.

5.2 The gain issue Systems with up or down conversion suffer several problems. (i) Influence of parasitic absorption : because the photons to be converted have to travel through the converting optical medium (e.g. as in [2, 5]), they are prone to losses via parasitic absorption which limits the amount of light that can be converted. These losses are worsened by the lack of directionality of the spontaneous emission producing the desired wavelength: the converted photons will travel on average a distance much larger than the medium thickness before they can escape. Because of it, it may be desirable to have the photon converting medium and the PV medium being one. This could be done, for instance, with rare earth elements implanted in the semiconductor used for the PV conversion. (ii) Influence of radiative efficiency: because photons have to be radiatively converted before contributing to the device current, non radiative recombination impact both on output current and voltage, even for relatively small fractions. For instance, assuming a 90% radiatively efficient process for up conversion at each step (therefore quite a high value with today's state of the art) means 20% losses in current because of the photons

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lost in heat, while the voltage losses remain modest (assuming an ideal device operating with the upconverted photons and those of higher energy. The result is approximatively the same with down conversion. (iii) Irreversibilities: either due to the reorganization of the ions or molecules participating in the photonic conversion process or due to some thermalization inside them (for instance between some of the split f levels in the case of rare earth doping), not all of the incident energy can be retrieved in the emited photons. For instance, with upconversion using Ho(III) an excitation of 746 nm gives 550 nm photons [28], so that even with a radiative efficiency of 100% , the energetic efficiency is less than 70%. With down conversion, systems with high quantum efficiencies are known (up to 190%, [25, 29] ) but the energy efficiency of these systems is only 63% because one UV (202 nm) produces 2 photons of only 613 nm. Actually, systems with a lower quantum efficiency (140%), such as Pr(III) have a higher efficiency of conversion of 68%. For comparison, the direct conversion by a single p/n junction of optimal band gap for the same UV photons gets less than 20% of their energy. Self absorption is also an issue : since there is emission, there should be absorption at the same wavelength, and since there is non radiative recombination, photon recycling is limited. Such self absorption can nevertheless be limited if some relaxation of the intermediate level is allowed (such as in Stokes shifts), so that the overlap between absorption and emission is minimal. Moreover such relaxation mechanisms genarally increase the lifetime of the intermediate states and therefore increase the probability of up conversion versus recombination.

5.3 Practical realization Systems using upconversion of IR photons were demonstrated as early as 1995 [13], although the most efficient photon upconverting device of the time would show only 2,5% efficiency for upconverting the low energy photons under a concentration of 250 suns. This is the only demonstration up to date in the field of PV conversion, to the best knowledge of the authors, of the concepts discussed.

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Figure 6: Illustration of the APTE process, one of the most efficient known today for photon upconversion. The process is effective in a variety of matrices.

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We have essentially discussed above the utilization of rare earth elements for photon interconversion, but the same considerations would apply to organic dyes. They might have sometimes advantages. For instance, it has been discussed and is well known that strong incident flux is an essential factor for efficient upconversion. These in practice are not so easily realized using solar radiation, but surface enhanced scattering [27], provides sometimes locally the strong electric fields needed for the process. It is known that collections of metallic nanoparticles are able to generate such conditions, specially near their percolation threshold. This way, five orders of magnitude in enhancement of up conversion of a polymer used for ONL were demonstrated [27]. Last but not least, quantum dots particles showing high radiative quantum yield can be produced, including by potentially cost effective methods ("chimie douce") [30]. Those particles, behaving like tunable converters could replace advantageously either rare earth elements and molecular dyes[32].

6.Thermal Efficiency Enhancement Another mechanism which can be used to enhance the efficiency of solar cells is the thermoelectrical effect, the occurrence of a voltage opposing a temperature gradient in a device, which results in a current when the electrical circuit is closed. The excited charge carriers which are created in the optically active region of the semiconductor can be transported to the contacts driven by the temperature gradient between that hot region and its colder surroundings. When the excited, hot electrons are transported to the contacts before they can equilibrate with the lattice, this constitutes a realisation of the hot electron solar cell [14,15]. More realistically, the electrons will relax in the optically active region, emitting with their excess energy phonons. Thereby, the local lattice temperature is increased to TH. As a result, the thermoelectric force due to the temperature gradient in the cell enhances the output voltage as shown in Fig. 7. The efficiency of semiconductor thermoelements which turn temperature gradients into electrical current is limited by the so called figure of merit T Z = T S2 σ/ κ q2 with the thermopower S, the conductivity σ, and thermal conductivity κ [16]. The optimum efficiency, as defined by the maximal ratio of the output electrical power and the thermal heat transfer rate, is given by η=ηC ((1 +Z (T+TH)/2)1/2 -1)/(( 1 +Z (T+TH)/2)1/2 +T/TH ) where ηC=(TH-T)/ TH is the Carnot efficiency of a heat engine, where TH is the higher temperature than the temperature of the surroundings, T. Thus, it is limited by the Carnot efficiency ηC , in the limit of infinite figure of merit ZT. Presently, there is a strong research effort to find materials with improved figure of merit T Z, in order to increase the efficiency of thermoelements substantially [17]. Thin-film thermoelectric devices with high room-temperature figures of 2.4 for p-type Bi2Te3/Sb2Te3 superlattice devices have been reported in [31].

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Figure 7: The band diagram of a p-n-solar cell with a higher temperature TH in the optically active region than the temperature T of the surroundings. The quasi-Fermi levels of electrons and holes are shown as dashed lines. Their difference defines the voltage which is at the contacts, V, larger than at the p-n-interface, VH due to the themoelectrical effect.

It has recently been shown that the efficiency of single gap solar cells can indeed be increased beyond the Shockley-Queisser limit due to the temperature gradients inside the cell. However, because of the smallness of the realistic temperature gradients the efficiency enhancement was found not to exceed 1 %, even under strong concentration of sunlight [10]. It might therefore be of interest to combine solar cells with thermoelements to reach a larger overall efficiency. Figure 8 shows the limiting efficiency as function of the band gap EG in eV and the tempertaure TH.

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Figure 8: The efficiency limit of a single band gap solar cell under illumination of 5800 K black body radiation, combined with a thermoelectrical element as function of the band gap EG and the cell temperature TH.

Taking into account the temperature gradients, there is furthermore an enhancement of current and output voltage due to thermionic emission from the optically active region, if a semiconductor with a lower band gap than the emitter and basis is inserted there [10]. Recently, it has been experimentally demonstrated that photons can be thermally upgraded above the absorption threshold of the bulk band gap of a solar cell, when a heterostructure containing a quantum well with a lower band gap is electrically isolated but optically coupled to the solar cell [18]. Then, photons with energy below the bulk gap are absorbed in the quantum well creating electron-hole pairs with a thermal distribution such that by recombination photons with energy above the bulk gap can be created with a finite probability. As a result the quantum well region is lowered in temperature while the number of photons with energy above the bulk gap is increased, and thereby the total efficiency is enhanced. The challenge in making efficient use of the themoelectrical effect is probably to find material systems with an ideal ratio of electrical to thermal conductivity (thereby apporaching the Carnot conversion efficiency). The ideal limit would be vacuum as a medium: permeable to electrons (e.g. by tunnelling), but preventing heat to be transported by phonons, only radiatively, or by the electrons themselves.

7.Conclusions and perspectives Primo, although many concepts for PV conversion are in principle able to achieve efficiencies upto 86% in the ideal limit (complete absorption, radiative recombination limit, infinite mobility, lossless contacts), not all of them are equally easy to implement and they show different sensitivities to non idealities. Secundo, application of new concepts for high efficiency PV conversion imposes stringent constraints on materials and device architecture, some of them conflicting, and many a conundrum will have to be solved before very efficient devices are seen operating. Tertio, 3 approaches (intermediate band, photon conversion, thermal enhancement) have been studied in more details and practical material systems for the embodiment of these approaches were discussed, and requirements were compared to the state of the art. While a lot of effort has been devoted to obtaining of absolute limits of conversion efficiencies of various concepts, certainly much more remains to be done in terms of the sensitivity analysis of these ideas.

8.Acknowledgements The authors are grateful for the financial support of ADEME and the fellowship

granted by CNRS. Thanks also to J. Nelson for many reveiling discussions and to D. Vivien, G. Aka and B. Viana for their expert advices about NLO systems.

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