How photoluminescence can predict the efficiency of solar cells

The efficiency of solar cells depends on the photocurrent, on the open circuit voltage and on the fill factor, which in turn depends on the diode factor. We review how photoluminescence (PL) measurements on the absorber, without finishing the solar cell, reveal the maximum open circuit voltage and the best diode factor, that can be reached in the finished device. We discuss two different ways to extract the quasi Fermi level splitting (QFLS) from absolute calibrated PL spectra, measured at a one sun excitation. The absorption spectrum of the solar cell absorber can be extracted from the PL spectra and allows the reliable determination of tail states. Tail states are responsible for radiative and non-radiative losses in the QFLS. In the ideal case the open circuit voltage is given by the QFLS. However, recombination at the interface can reduce the open circuit voltage severely. We discuss various electronic structures at the interface that lead to a reduction of the open circuit voltage. The excitation dependence of the PL allows the determination of the diode factor of the absorber alone, which would be 1 in the ideal case. Metastable defects can increase the diode factor, even for recombination in the neutral zone and for low excitation. This effect decreases the fill factor of the solar cell. 1. Solar cells and photoluminescence Thin film solar cells represent the electricity source with the lowest greenhouse gas emissions [1]. Two technologies have reached confirmed efficiencies in the lab above 23% [2–4]: Cu(InGa)Se2 and halide perovskites, with CdTe closely behind with 22.1% efficiency [2]. Thin film solar cells are complex structures, consisting of many layers and their interfaces. For the solar cell to work all materials and interfaces need to possess the appropriate electronic structure. During the development of solar cells or in industrial production, it is desirable to know already the quality of the absorber alone. Photoluminescence (PL) measurements of the absorber can provide information about the open circuit voltage [5] and the fill factor, which the absorber is able to produce when made into a solar cell. The open circuit voltage of a solar cell with ideal contacts and with ideal transport properties is given by the quasi Fermi level splitting (QFLS) of the absorber under one sun illumination [6, 7]. The fill factor of a solar cell depends critically on the diode factor of the device. We discuss in the following how PL can determine the diode factor of the absorber. It should be pointed out that these PL measurements are done at room temperature, not a cryogenic temperatures. We want to describe the absorber as close as possible to standard test conditions of the solar cell. Furthermore, room temperature PL measurements are more easily implemented in an industrial environment. Most of our examples come from chalcopyrite solar cells, but the principles are generally applicable to any solar cell technology.


Solar cells and photoluminescence
Thin film solar cells represent the electricity source with the lowest greenhouse gas emissions [1]. Two technologies have reached confirmed efficiencies in the lab above 23% [2][3][4]: Cu(InGa)Se 2 and halide perovskites, with CdTe closely behind with 22.1% efficiency [2]. Thin film solar cells are complex structures, consisting of many layers and their interfaces. For the solar cell to work all materials and interfaces need to possess the appropriate electronic structure. During the development of solar cells or in industrial production, it is desirable to know already the quality of the absorber alone. Photoluminescence (PL) measurements of the absorber can provide information about the open circuit voltage [5] and the fill factor, which the absorber is able to produce when made into a solar cell. The open circuit voltage of a solar cell with ideal contacts and with ideal transport properties is given by the quasi Fermi level splitting (QFLS) of the absorber under one sun illumination [6,7]. The fill factor of a solar cell depends critically on the diode factor of the device. We discuss in the following how PL can determine the diode factor of the absorber. It should be pointed out that these PL measurements are done at room temperature, not a cryogenic temperatures. We want to describe the absorber as close as possible to standard test conditions of the solar cell. Furthermore, room temperature PL measurements are more easily implemented in an industrial environment. Most of our examples come from chalcopyrite solar cells, but the principles are generally applicable to any solar cell technology.

QFLS
The QFLS is a measure for the open circuit voltage, because the latter is given by the difference between the quasi Fermi level of the conduction band at the electron contact and the quasi Fermi level of the valence band at the hole contact. Without gradients in the quasi Fermi levels at open circuit, the QFLS translates directly into the open circuit voltage as V OC = ∆E F /q (symbols are explained in the annex). However, in real solar cells, the contacts are not ideal and gradients in the quasi Fermi levels can form (see section 4), reducing the open circuit voltage 3 . Thus, the QFLS gives the maximum open circuit voltage an absorber is capable of.
In equilibrium any semiconductor emits black body radiation. Since a semiconductor is a grey body with a certain absorption spectrum, we need to multiply the black body emission with the absorptance of the semiconductor film or wafer [9,10]. Thus, the photon emission flux is given by: where we use the Boltzmann approximation instead of the Bose term in Planck's law for black body radiation. This is allowed [11] since we are interested in the band edge luminescence of semiconductors which are used in solar cells, i.e. have a band gap and therefore emission energies much larger than k B T. Two things are important to point out: (a) Φ and F, below in equation (2), describe the (external) photon flux, not the (internal) recombination rate, (b) A is the absorptance, i.e. the ratio of absorbed photons to incident photons, not the absorption coefficient, i.e. it depends also on the thickness of the semiconductor film or wafer [9]. This emission flux is caused by the recombination of thermally excited carriers. Thus, the spectrally integrated emission flux is given by the concentration of electrons and holes: Under excitation by the sun or by a laser, the carrier concentrations increase, which is described by the QFLS: Combining equations (1)-(3) gives the photon flux under excitation, assuming a constant QFLS throughout the absorber: For a detailed derivation, that applies also to the energy range where the Boltzmann approximation is not valid, see [9]. Equation (4) is also known as Generalised Planck's law (in Boltzmann approximation). A fit of the PL spectrum to equation (4) can be used to determine the QFLS. The usual method [12,13] assumes that the absorptance is 1 at energies high enough above the bandgap. Then a logarithmic plot of Φ/E 2 results in a straight line at energies well above the bandgap, i.e. well above the PL maximum energy, with the slope proportional to the inverse temperature at which the measurement was taken and the ordinate intercept given by the QFLS (see e.g. supplementary information of [14]). An example is given in figure 1. If the fitted temperature does not agree with the measurement temperature care needs to be taken, since a wrong temperature results also in a wrong QFLS from the fit [15]. Quite often, increased temperatures are reported in the literature, even above 100 • C. However, it is not very likely that the temperature increases considerably above room temperature during a measurement at one sun equivalent illumination, in particular if the measurement is done at a mm sized spot on a thin film absorber with a conductive back contact. Whether the ratio of non-radiative to radiative recombination is low or high has negligible influence on the temperature, since the highest radiative efficiency in the best solar cells is only 20% (see figure 2), i.e. even in the best case, non-radiative recombination is already responsible for 80% of the recombination. We check the temperature increase with the help of an infrared thermometer camera, that allows to check if the surface temperature in the laser spot is higher than the surrounding. We only see small temperature differences, usually below 5 K, and only if the illumination intensity is equivalent to several suns. The likely reason to observe a seemingly increased temperature is that the assumption of A = 1 is not valid. In some cases it is possible to obtain an acceptable fit, fixing the temperature parameter to the measurement temperature; then the intercept will result in the correct QFLS. If fixing the temperature does not result in a reasonable fit, the assumption of A = 1 can no longer be maintained. The absorptance above the band gap has to be determined by absorption-reflection or ellipsometry measurements and a fit to equation (4) using the   (7), represented by the dashed line. VOC loss of record solar cells is added. experimentally determined absorptance spectrum will reveal the correct QFLS [15]. Alternatively, the QFLS can be determined not by the fit but by the external radiative efficiency (ERE), also known as PL quantum efficiency or PL quantum yield, as discussed in the following.
Consider first an ideal solar cell at open circuit. The ideal solar cell is governed by the detailed balance between excitation and radiative recombination, as described by Shockley and Queisser [6], resulting in a QFLS of ∆E F = qV SQ OC . Combining equations (2) and (3) gives the emission flux due to radiative recombination which is balanced by the excitation flux from the sun and the thermal radiation form the environment where it is understood that G sun is the photon flux from the sun, integrated for energies above the bandgap of the absorber. Obviously, in a PL experiment the photons are not supplied by the sun but by a laser tuned to the photon flux that is contained in the sun's spectrum above the band gap of the absorber. The thermal photon flux F eq on the right hand side is essential for the derivation of the current-voltage characteristics of a solar cell [6], but can be neglected for the consideration at open circuit. In a real solar cell non-radiative recombination is the dominant recombination mechanism, which reduces the number of photogenerated carriers, and therefore the QFLS and V OC . Furthermore, the flux of emitted photons is reduced by F nr . Thus, the balance is between [6]: The second line is valid since ERE is defined as F PL /G sun . Solving equation (6) for the QFLS gives [6] ∆E F = qV SQ OC + k B T ln ERE.
To extract the QFLS from ERE, the value V SQ OC is needed. It can be determined by solving equation (5) for V SQ OC . This requires information on the band gap, since G sun depends on the band gap of the absorber, for tabulated values see e.g. [16]. The band gap can be determined from absorption-reflection or ellipsometry measurements and a Tauc plot [17,18] or from the inflection point of the quantum efficiency spectrum of the solar cell [19]. Since the Shockley-Queisser model assumes complete absorption above the band gap, equation (5) is valid independent of a direct or indirect band gap. The V SQ OC values determined from equation (5) changes, whether a 6000 K black body spectrum is used [6] or an AM1.5 spectrum, because this changes the value of G sun , and whether a model with or without ideal back reflector is applied, because this changes the value of F eq [16]. It is important to note that it is the ERE that enters into equation (7), because in the Shockley-Queisser limit the fluxes of photons (not the internal recombination rates) have to be considered [6,7]. Essentially, the photons that do not leave the absorber will be re-absorbed and then mostly lost to non-radiative recombination [20].
A comparison between QFLS extracted from a fit with fixed temperature to the high energy slope of the PL spectrum and the ERE, both obtained from the absolute calibrated PL measurement, is given in figure 2. We plot the QFLS loss with respect to V SQ OC , calculated for an AM1.5 spectrum and without back reflector, using the PL maximum energy as the band gap. We compare a number of chalcopyrite solar cells in different stages of development. The state of the art Cu(InGa)Se 2 solar cells were prepared by EMPA, the Swiss research institute on Materials Science and Technology, and ZSW, the Zentrum für Sonnenenergie-und Wasserstoff-Forschung Baden-Württemberg, and have been published before [21,22]. Cu-rich and Cu-poor Cu(InGa)Se 2 solar cells are described in [14] and the sulphide CuInS 2 solar cells in [23]. We added literature data for Si and GaAs record cells. Here we use V OC instead of QFLS. For these cells no ERE data is available. Therefore we used the reciprocity principle [24] to estimate ERE from the published external quantum efficiency (EQE) data [25]. The data for the Si cell are taken from [26] and for the GaAs cell from [27]. The data in figure 2 confirms that the correlation of equation (7) is observed, implying that all these solar cells can be described by detailed balance plus non-radiative recombination. This has also been shown in the literature, see e.g. [7,24,28]. The loss in state of the art Cu(In,Ga)Se 2 cells and the GaAs record cell is somewhat lower than the line. This is due to the fact that these optimised cells are closer to a perfect back reflector and have therefore a higher V OC . An important point to notice is that the ERE, even in the best solar cells is below 30%. Thus, the dominating recombination is still non-radiative-in general not due to bad material quality, but due to imperfect light management [29]. Experimentally, QFLS can be determined by both methods: from a fit to the spectrum, provided the fitted temperature is not wrong, or from EQE with the help of equation (7).
Obviously, to perform these analyses we need the photon flux, i.e. we need to calibrate the PL measurements for photon counting. This absolute calibration is achieved by two steps. (a) Spectral calibration: the spectral throughput of all optical components between the sample and the detector is measured with the help of a calibrated lamp with a known spectrum. A white spectralon reflector is placed at the exact position of the sample to be measured, the lamp is reflected into the detection system and the spectrum measured. The ratio between the calibrated lamp spectrum and the spectral counts of the detector gives the spectral correction function. (b) Intensity calibration is performed at a (reduced) laser power equivalent to the (expected) PL intensity by first placing a power meter at the sample position and measuring the laser power. Then a camera is placed in the same position to determine the laser profile and, thus, the intensity in the centre, where the PL is measured. Finally, the spectralon reflector is placed again at the sample position and the laser reflected through the whole optical system to the detector, which allows to determine the factor between (spectrally corrected) counts and intensity or number of photons.
QFLS depends on both quasi Fermi levels. The minority quasi Fermi level depends on the concentration of photo excitated carriers, i.e. on excitation and recombination. The majority quasi Fermi level depends on the doping level in low excitation and on the density of photo excited carriers in high excitation. For most doped absorbers, one sun illumination is still low excitation. In this case the behaviour of the quasi Fermi levels can be approximated in the following way: only the minority carrier quasi Fermi level is shifted by the excitation and the majority carrier quasi Fermi level is given by the equilibrium Fermi level. The cases beyond this approximation are discussed in section 6. For the simulation of an example see [30]. Comparing two samples, a higher QFLS can mean (a) reduced recombination activity and thus a minority carrier quasi Fermi level closer to the band or (b) a higher doping level and thus a majority quasi Fermi level closer to the band [13,31]. To separate the two effects, time resolved PL measurements are needed, which in principle allow the determination of the minority carrier lifetime. The influence of varying experimental conditions and the potential effects that could distort the result can be found in the literature [32][33][34]. If the reduction of recombination centres has a considerable effect on the QFLS, when comparing two samples, then the minority carrier lifetime will be increased. The effect on QFLS can be approximated by assuming that the carriers have a reasonably good mobility and spread evenly over the thickness of the absorber during the steady state QFLS measurement, an assumption already made when deriving equation (4). Then the minority carrier density is given by n min = G · τ /d. Comparing two (otherwise similar) samples the expected change in QFLS due to a change in minority carrier lifetime is then given by If the main effect leading to the QFLS improvement is an increase in doping level, the minority carrier lifetime remains unchanged [31]. The doping level can be cross-checked by Hall measurements (for methods how to deal with low mobility samples, see [35,36]) or by capacitance-voltage measurements on finished devices [37,38]. In most cases, both effects will play a role and the improvement in QFLS will exceed the one expected from an increase in lifetime, as given in equation (8).

Tail states
To determine the QFLS from the ERE (equation (7)), the band gap of the absorber is needed. The determination of the band gap can be severely hampered by tail states [39]. In an ideal semiconductor the density of states in the band gap is zero. However, in real semiconductors, and in particular in polycrystalline thin films, band tails occur, which are characterised by a density of states decaying exponentially into the gap. The decay constant is called the Urbach energy [40][41][42][43]. Essentially, higher Urbach energies mean there are more tail states. Exponential tail states have been first described in metal halides by Urbach, who also discovered the temperature dependence of the decay energy [44], which is now known as Urbach energy. The temperature dependence of the Urbach energy can be attributed to the vibrational disorder caused by phonons, while static disorder adds an additional constant contribution [41,45]. The static disorder can be due to structural deviations from the ideal crystal, e.g. grain boundaries, due to variations in the composition, and thus the band gap [46] or due to electrostatic fluctuations [47,48]. These bandgap or electrostatic fluctuations are characterised by low temperature PL spectra [49][50][51][52][53], which show a characteristic temperature dependence and spectral shape: the spectrum is asymmetrically broadened towards lower energies because non-local transitions allow recombination at lower energies than locally vertical ones. The peak maximum shifts first towards lower energies as the temperature is increased, because carriers become mobile and find the lowest energy states. With further temperature increase more thermal carriers are available, which flattens the electrostatic potential differences or fills the low energy states, so that the PL peak shifts towards higher energies with increasing temperature [54].
However, it is the density of states at room temperature that is relevant for the solar cell performance and is characterised by Urbach energy. The Urbach energy is determined from the exponential behaviour of the absorption coefficient below the band gap. For an accurate determination of the Urbach energy, the measurement of very low absorption coefficients is necessary to describe the states deep in the gap [55,56]. Since in general it is easier to detect single emitted photons than to detect missing photons, PL is particularly suited to measure ultra-low absorption coefficients [57]. The emission flux of an excited semiconductor contains the absorptance spectrum of the sample (equation (4)). With the knowledge of the QFLS the spectrum of the absorptance can be extracted from the emission spectrum. Since the emission spectrum extends over energies lower than the band gap up to about 100 meV above the band gap, the extracted absorption spectrum also covers this range. The lower energy limit is generally given by the detector. Using an optical model for the sample, the absorption coefficient can be extracted [15]. The original method [57] was developed for wafers, we have expanded the method to be used for thin films [15]. The method has been confirmed by comparison with the more classical method of photothermal deflection spectroscopy (PDS), which has been particularly developed to measure low absorption coefficients [58]. In fact, PL allows to measure absorption coefficients lower than PDS by more than two orders of magnitude (see figure 3) [15]. For a reliable determination of Urbach energies it is essential to measure down to very low absorption coefficients, deep in the band gap, otherwise a possible additional variation in the band gap can distort the result [47].
Several methods exist to determine the bandgap or the effective bandgap in the case of tail states [7,19]. In an ideal (direct) semiconductor the maximum of the PL emission is given by the band gap: E G + k B T/2 (see e.g. [59]). In the presence of tail states, the excited carriers will thermalise not only down to the edges of the extended band states but further into the tail states [60]. Thus, the maximum of the emission spectrum will also shift to lower energies, more so for less steep tails, as observed in a wide range of materials [7,39]. Since the detailed balance in the Shockley-Queisser limit is determined by the radiative recombination, the QFLS is limited in the best case by the energy of the emitted photons. Therefore, to determine the qV SQ OC in equation (7) we use the maximum of the PL emission spectrum as the band gap energy [39]. This shift of the PL maximum represents an additional radiative loss of QFLS and thus V OC due to tail states. However, the additional density of states also broadens the absorption and with it the luminescence spectrum towards lower energies, thereby increasing F eq (equation (2)). Because of equation (5) this will reduce the maximum open circuit voltage even in the radiative case [7,60]. We label this reduced open circuit voltage V rad OC . This loss is significant, even for rather low Urbach energies (lower than k B T): the loss increases by 5 meV for each increase of 1 meV in the Urbach energy [60]. An equally high additional loss is due to the non-radiative Shockley-Read-Hall (SRH) recombination through the tail states [60]. Thus, the V OC loss due to tail states is significant: leading at least to an additional V OC loss of 10 mV for every additional meV in Urbach energy.

Interface recombination
So far, we have treated V OC and QFLS almost as synonyms. QFLS and V OC are reduced from the radiative limit qV SQ OC by non-radiative recombination processes, reducing ERE and the minority carrier lifetime from its ideal radiative value. These recombination processes take place in the bulk of the absorber and at the interfaces [5,61]. In general, good solar cells are dominated by bulk recombination [62,63]. However, it should be noted, that the interface is always a location of defects and recombination centres. Consequently, as the quality of the absorber is improved towards the radiative limit, eventually for the best bulk, interface recombination will be the limiting factor again. On identical absorbers the quality of the interface can be inferred from QFLS and lifetime measurements [61,63]: with stronger interface recombination the effective minority carrier lifetime is reduced and consequently the QFLS is reduced (equation (8)). However, the influence of interface recombination on solar cell performance can go much beyond the reduction of QFLS. Interface recombination can cause a gradient in one (or both) of the quasi Fermi levels, typically in the minority carrier quasi Fermi level. Since the (open circuit) voltage is given by the difference between the In state of the art solar cells with optimised interfaces this loss is small, less than 10 mV [22,63]. Whether interface recombination is the dominating loss mechanism at V OC can also be studied by temperature dependent current-voltage measurements. The activation energy of the dominating recombination rate in a solar cells is given by the activation energy E a of the reverse saturation current and can be determined from the temperature dependence of the open circuit voltage by extrapolating to 0 K: Equation (10) is obtained by solving the diode equation under illumination for the open circuit voltage (see e.g. [62]). If the dominating recombination occurs in the bulk of the absorber, either in the neutral zone or in the space charge region, the activation energy is given by the band gap. Interface recombination can lead to a lower activation energy of the dominant recombination process. This reduction can be caused by several effects (figure 5): (i) unsuitable band alignment, causing an interface band gap smaller than the bulk band gap (ii) Fermi level pinning at the interface (iii) a defect layer within the absorber just below the surface [65]. Case (i) and (ii) are well established causes of dominating interface recombination and are discussed in detail in textbooks (e.g. [62,66]). In the case of unsuitable band alignment (i) ( figure 5(a)) one of the band edges of the contact material is energetically situated within the band gap of the absorber. This is labelled as cliff. The interface band gap is then reduced, as it is defined by the closest energetic distance between a valence band and a conduction band at the interface. This energetic distance is the relevant energy, because recombination across the interface is possible, i.e. electrons of one material recombine with holes from the other one [62]. This smaller interface bandgap leads to a reduced activation energy of the saturation current-and consequently to increased recombination-because recombination will occur between the lowest energy electrons and the highest energy holes on either side of the interface. If the band alignment is suitable, i.e. the band edges of the contact material lie outside the band gap of the absorber (type I), a high density of interface states can lead to Fermi level pinning (ii) ( figure 5(b)). In this case the activation energy is expected to be given by the energetic distance between the (equilibrium) Fermi level and the (bulk) majority band edge at the interface (i.e. the valence band edge for a p-type absorber) [67]. Recent device simulations [65] show, that this is a useful approximation. However, in several cases an activation energy of the saturation current lower than the band gap has been observed, that is not due to unfavourable band alignment, nor does any hint exist from e.g. photoemission experiments, that Fermi level pinning occurs [65,[68][69][70]. We recently showed [65] that a highly defective layer with a thickness of a few 10 nm just below the surface (iii) (figure 5(c)) will also lead to a significant gradient in the minority carrier quasi Fermi level, i.e. to a large interface V OC loss (equation (9)). In particular, the current voltage characteristics show the most important indicator of interface recombination: an activation energy of the saturation current smaller than the band gap. Thus, it would be more correct to say that this observation indicates recombination near or at the interface.
Chalcopyrite solar cells are particularly well suited to study the influence of interface recombination, because in general they show bulk recombination, when the absorber is grown under Cu-poor conditions, and interface recombination when it is grown under Cu-rich conditions [68,69,[71][72][73]. In particular, Cu-rich absorbers show interface recombination even with a buffer material that ensures suitable band alignment [69,74]. In devices with suitable buffers the activation energy of the saturation current equals the band gap in Cu-poor devices and is lower than the band gap in Cu-rich devices. Furthermore, in Cu-poor devices with optimised contact layers the interface V OC loss ∆V IF OC is small, less than 10 mV [22]. In Cu-poor devices with non-optimised contacts we find ∆V IF OC of up to 60 mV [14], although the current voltage characteristics give no hint on interface recombination. In this case, additional gradients of the quasi Fermi levels exist within the contact layers, which is a hint that the transport properties of the contact layers need to be improved. But ∆V IF OC is considerably larger in Cu-rich devices, values around 150 mV are typical in selenides [14,65] and above 300 mV in sulphides [72]. Such large values of ∆V IF OC (>100 mV) have always been found in conjunction with an activation energy of the saturation current smaller than the band gap. This is why we label the loss defined in equation (9) interface V OC loss. Large values indicate interface recombination, either because of unfavourable band alignment, Fermi level pinning or a defective layer just below the surface. For a given band gap V SQ OC represents the maximum open circuit voltage for idealised absorption assumptions and radiative recombination as the only recombination mechanism. If the material contains considerable tail states the energy of the radiative recombination E PL is reduced with respect to the band gap. This shift of the radiative recombination leads already to a loss of the open circuit voltage, even in the case of purely radiative recombination, and leads to V rad OC . Non-radiative recombination due to SRH recombination through deep defects or through tail states reduces the QFLS ∆E F below the value for only radiative recombination. This includes non-radiative recombination at the interface. However, interface recombination leads to an additional gradient of the minority quasi Fermi level towards the interface and therefore to an open circuit voltage that's even lower than the (bulk) QFLS.

Summary of Voc losses
PL spectroscopy provides a lot of this information: the amount of tail states allows to calculate V rad OC . Absolute calibrated PL spectra allow to extract the QFLS. Finally, the difference between V OC and ∆E F /q gives the extend of interface losses.

Higher diode factor losses due to metastable defects
In addition to elucidating the various sources of voltage losses, PL spectroscopy can also help to identify fill factor losses. The fill factor of a solar cell depends on the parasitic resistances (see e.g. [75]). To optimise them is a matter of device design. Here we concentrate on the fundamental influences that originate from the absorber material itself. The fill factor depends also critically on the diode factor of the cell (n d in equation (12) below, see e.g. [62]), which in turn depends on the main recombination mechanism [62]. Ideally the diode factor is 1. Higher diode factors will lead to lower fill factors and consequently to lower efficiencies of the solar cell. In devices that are dominated by recombination in the quasi-neutral zone the diode factor can be determined from PL measurements, since PL is measured on the absorber alone, therefore no space charge region exists [64]: experimentally a power law is found between the generation flux and the PL flux (e.g. [76] and references therein): Examples are given in figure 7, which shows a log plot of the PL flux as a function of the generation flux, resulting in a straight line over four orders of magnitude with an exponent of 1.2-1.4. On the other hand the current-voltage characteristics of the solar cell depends on the diode factor n d . Neglecting parasitic resistances it can be written: Because the generation flux G is proportional to the short circuit current of a solar cell and because the luminescence flux is given by equation (5) and because with good contacts the open circuit voltage is given by the QFLS, the exponent k and the diode factor n d of the current-voltage characteristics are the same [64,77]. Therefore, the exponent k is labelled the optical diode factor [64]. k = n d is true in devices that are dominated by recombination in the quasi-neutral zone. Devices that are dominated by recombination in the space charge region or at the interface will have a diode factor n d ⩾ k [64]. Thus, the optical diode factor k represents the minimum diode factor an absorber can achieve, similar to the QFLS representing the maximum open circuit voltage an absorber is capable of, if the formation of the solar cell and the contacts do not introduce further losses.
As seen in figure 2, the main recombination mechanism in solar cell absorbers is non-radiative, i.e. SRH recombination. The non-radiative recombination rate is given by (see e.g. [78]): Figure 7. PL flux as a function of generation flux, demonstrating the power law dependence over four orders of magnitude for two different polycrystalline Cu(In,Ga)Se2 samples. Reproduced from [30]. CC BY 4.0. Table 1. Radiative and non-radiative recombination rates. For clarity we assume a p-type semiconductor. Furthermore we assume that the dominating recombination is non-radiative, i.e. g = rtot ≈ rSRH. It should be noted that an undoped semiconductor is in the high excitation situation for any measureable excitation.
Low excitation High excitation n = ∆n, p = p0 p = n = ∆n Recombination via deep recombination centre No change to the above We consider excited semiconductors, i.e. np ≫ n 2 i . For a recombination centre deep in the gap and the semiconductor in the excited state n * , p * « n, p, whereas for a tail state and a non-degenerate semiconductor n * or p * » n, p, depending on whether we consider a conduction band or a valence band tail. Therefore in equation (13) several of the terms in the sum of the denominator can be neglected. Furthermore, in the case of low excitation by definition n min « n maj , whereas in the case of high excitation: n = p. With these simplifications we can distinguish the four cases in table 1. Furthermore the generation rate is equal to the total recombination rate, which is dominated by the non-radiative recombination: g ≈ r SRH . It should be noted that these considerations neglect the potential interaction with additional defects, which capture carriers from the bands, which then can recombine (radiatively) from the defect. In that case all the individual capture and recombination rates have to be considered simultaneously [79]. However, at room temperature the carriers are reexcited into the band and SRH recombination will give the correct recombination rates, as given in table 1.
The optical diode factor is defined by equation (11) and can therefore be written as: From table 1 it is obvious, that with SRH recombination through deep defects or through tail states, the optical diode factor in low excitation should be 1 in either case. Only in the case of high excitation and deep defect recombination, is the optical diode factor expected to be 2. This can be understood by the fact, that in low excitation only the minority carrier Fermi level moves closer to the band with increasing excitation, therefore the first term in equation (14) is 1 and the second term is 0, leading to k = 1. In the high excitation case, both Fermi levels move, thus both terms in equation (14) are 1, leading to k = 2. Interestingly, in the case of high excitation but recombination through tail states both Fermi levels move, but not according to ∆n ∼ g, but according to ∆n = √ g, thus both terms in equation (14) are 1 /2, leading to an optical diode factor of 1. Furthermore, at high excitation radiative recombination will become dominant. Based on detailed balance arguments [79], both terms in equation (14) become 1 /2, and the optical diode factor k = 1 at high enough excitation. For a simulation of the optical diode factor for the various cases see [30]. Here, we ignore the effect of Auger recombination, which could become important at high excitation [5]. Experimentally, however, in the low excitation case, where k = 1 is expected, we observe considerably larger optical diode factors, typically ranging from 1.2 to 1.3 [64]. This effect can be explained by metastable defects, which change their character from donor to acceptor upon electron capture, i.e. upon excitation [30]. Such metastable defects are well known in chalcopyrite materials [80][81][82], but also in other semiconductors (see references in [30]). In chalcopyrites one of the main metastable effects can be explained by a donor effect, that turns into an acceptor upon electron capture, e.g. due to illumination [80][81][82]. Consequently the effective p-doping increases upon illumination, and the increase depends on the available minority carrier concentration. Thus, the majority quasi Fermi level shifts, but not in direct dependence of the generation, but in dependence of the minority carrier concentration, leading to an optical diode factor larger than 1. This has been confirmed by numerical simulations, as well as by experimental measurements of the net doping level after illumination [30]. Therefore metastable defects have a direct influence on the efficiency of solar cells via the diode factor. PL provides the optical diode factor and thus the best diode factor that can be achieved in a solar cell made from the corresponding absorber.

Summary
We summarise various ways how PL can be used to establish the maximum open circuit voltage and the lowest diode factor, i.e. highest fill factor, of a solar cell made from the same absorber. Absolute calibrated PL provides photon flux spectra, which can be fitted to extract the QFLS. The spectrally integrated flux gives the ERE, which can also be used to determine the QFLS via equation (7). Once the QFLS is known, the PL spectrum allows the determination of the absorption spectrum, down to very low absorption coefficients, which is the ideal tool to measure tails states in the band gap. These tail states reduce QFLS and V OC due to an increase of radiative recombination at lower energies and due to non-radiative recombination via the tail states. This loss is significant: every meV increase in the Urbach energy reduces QFLS, and thus the V OC , by at least 10 meV. In non-optimised devices often the observed V OC is much smaller than ∆E F /q by several 100 mV. This additional loss indicates interface recombination, caused by non-ideal electronic structure at the interface: (a) the band alignment at the absorber-contact interface forms a cliff, (b) the interface has a very high density of defects states, which pin the Fermi level, (c) the surface of the absorber consists of a highly defective layer. Besides these limitations for the open circuit voltage, PL can also make predictions concerning the fill factor of the solar cell. An increased diode factor decreases the fill factor. The diode factor of the absorber alone can be measured by intensity dependent PL. The exponent of the power law is the optical diode factor. Metastable defects can cause an increase of the diode factor, even for recombination in the neutral zone and under low excitation.

Data availability statement
The data that support the findings of this study are openly available at the following URL/DOI: 10.5281/ zenodo.5145859 [83].