Should A Good Organic Solar Cell Be A Good Organic LED ?

We revisit the argument that link the efficiency of a solar cell to its reverse operation as a LED, in the case where the material is organic. In organic cells, exciton transport is an intermediate process between sunlight absorption and the generation of electric current. We show that quenching exciton radiation can be beneficial to cell efficiency, without contradicting the general rule prevailing for semiconductor cells. Our treatment allows us to discuss both bulk heterojunction and planar junctions.


Introduction
"A good solar cell should be a good LED" -This simple rule rapidly became a central design principle for solar cells after it was first enacted by Owen D. Miller, Eli Yablonovitch, and Sarah R. Kurtz in 2012 [1]. However, it only strictly holds for semi-conductor cells. In this communication, we 12 revisit the derivation of this statement and explain how to apply it to organic solar cells. What makes 13 organic cells distinct from inorganic ones is the limiting role played by excitons. In organic cells, the 14 absorption of a photon first gives rise to a long-lived exciton. It is only at the interface between the 15 donor and acceptor materials that the local electric gradient can break the exciton into a free hole and a 16 free electron. Counter-intuitively, we show here that if the exciton emissivity can be reduced, as in [2], 17 while leaving electron emission unaffected, then one may improve the open-circuit voltage V oc .

Classical theory
The famous statement by Miller, Yablonovitch and Kurtz directly comes from Shockley and Queisser's theory of solar cells. In its simplest form, this theory expresses the conservation of the number of particles : the current of electron-hole pairs generated by photon absorption, I s , is eventually converted either into a particle current in the external circuit I(V), where V is the electric potential, or into a recombination current. We write the latter as I R,e (V) + I NR,e (V), where "R" and "NR" stand for "radiative" and "non radiative", respectively, and the subscript "e" is there to express that the processes in question involve electrons. In particular, I R,e (V) is a current of photons re-emitted by the cell through electrons-hole recombination. The detailed balance reads I s = I R,e (V) + I NR,e (V) + I(V). (1) With good approximation, the recombination terms follow the classical expression of an ideal diode [3][4][5], so that one may rewrite the previous equation as where q is the absolute electric charge and β = 1/k B T. From this, one can compute the open-circuit voltage V oc : Suppose now that the solar cell operates in reverse, with electric current being injected into it. Then the cell becomes a Light Emitting Diode and the injected electrons produce photons with an LED quantum 33 efficiency Q LED defined as With this efficiency in mind, we can express the V oc in terms of its ideal value, V rad oc as [6] If Q LED = 1, recombination is only radiative and the second term vanishes, yielding V oc = V o r c ad .
Otherwise (Q LED < 1), V oc is reduced by non-radiative recombination. From this follows the famous statement at the centre of this communication.

Organic solar cells
In organic solar cells, the absorbed photons first produce excitons, which must migrate to the donoracceptor interface before any usable electric current can be produced. Denoting the exciton current at the interface by I χ , Eqs. (1) and (2) = ln I s I R,e (0) + I NR,e (0) = qβV rad oc + ln Q LED + ln (IQE) .
The last expression differs from (8) by the last term which is the logarithm of the ratio between the generated free carriers and the absorbed photons, i.e. the internal quantum efficiency (IQE), if 45 a perfect charge collection efficiency is assumed at short-circuit condition, which is commonly the 46 case [10][11][12]. Hence, if IQE is less than unity, V oc is further decreased due to the incomplete transport 47 and dissociation of generated excitons. From equation (12), we can make two important observations. Firstly, bulk and planar heterojunction are characterized by distinct realization of Eq. (12). On the one hand, bulk-heterojunctions make IQE as close to 1 as possible, but can have poor Q LED due to dead-ends in the electron path toward the electrodes. On the other hand, with planar heterojunctions, interferences can be optimized in order to maximise Q LED . However IQE decreases rapidly as soon as the thickness of the active layer exceeds the diffusion length of excitons, which severely limits photon absorption. Fortunately, photonic management can significantly increase the diffusion length for exciton with large fluorescence quantum yield, allowing one to guarantee a good IQE [2].
Secondly, as excitons diffuse in direction to a dissociation interface, some of them disintegrate, either radiatively or non-radiatively, making I χ < I s . Explicitly, where I R,χ and I NR,χ are the radiative and non-radiative exciton recombination current, respectively. 58 Here, it is important to note that I R,χ is quite distinct from I R,e (V), since the two quantities relate to 59 distinct particles. Crucially, since excitons are electrically neutral, their radiation is independent of the 60 voltage V. For, the same reason, their transport within the cell, contrary to electron, is insensitive to 61 the local electric field. Using (13), we can rewrite the IQE as As I s is far larger than the total exciton recombination current, the logarithm of the IQE in (12) can be expanded and we finally get Hence, if one manages to decrease I R,χ , while keeping I NR,χ as small as possible, one can improve 65 the V oc .
As it turns out, it is possible, in principle to reduce drastically I R,χ through interference 66 management, without affecting I s . Indeed, organic cells are ultra-thin compared to optical wavelengths 67 and, in such confined environment, exciton fluorescence can be tuned by properly adjusting the cell 68 geometry [13].
That one can reduce I R,χ while maintaining I s may seem slightly counter-intuitive because one usually assumes that the emissivity of the cell is equal to its absorptivity. Consequently, decreasing 71 the emitted current should decrease the absorbed photons current. However, as far as exciton is 72 concerned, absorption takes place at energies larger than the bandgap while exciton emission occurs 73 only at sub-bandgap energies. Secondly, and more importantly, absorption involves sun photons 74 that enter the cell at nearly-normal incidence, whereas radiation by excitons occurs in all directions. 75 In such a strongly confined environment as an organic solar cell, photon emission becomes highly 76 anisotropic. As a result, there exists an asymetry between absorption of sun photons and radiative 77 decay of excitons, see Figure 1. Thermodynamically speaking, this is a non-classical effect.
In conclusion, while it remains true that a good organic solar cell must be a good LED, it is only desirable to have good LED luminescence from electron-hole recombination. Meanwhile, exciton being electrically neutral, their fluorescence should be minimized so as to maximize IQE.
Author Contributions: All authors have contributed equally to all aspects of the work, including manuscript 82 preparation. All authors have read and agreed to the published version of the manuscript.

Abbreviations
The following abbreviations are used in this manuscript: LED Light Emitting Diode IQE Internal Quantum Efficiency