How band tail recombination influences the open-circuit voltage of solar cells

The power conversion efficiency of solar cells strongly depends on the open-circuit voltage V OC which, in turn, depends on the recombination activity within the device. A possible source of detrimental charge carrier recombination is band tails. An empirical linear relationship between V OC loss and the Urbach energy of the band tails has been shown in the past. Here we discuss how band tails influence the radiative recombination and the nonradiative recombination in the bulk of the absorber. First, we show through photoluminescence that the band tails can be willfully tuned in state-of-the-art thin-film Cu (In,Ga)Se 2 (CIGSe) absorbers and solar cells on a 20% efficiency level and beyond through the incorporation of alkali atoms. In the second part, we compare our CIGSe results to published results from other solar cell technologies. This comparison reveals that CIGS solar cells follow the previously described empirical trend: an increase in the open-circuit voltage with decreasing band tails. Finally, we model the influence of tail states on the radiative and nonradiative recombination losses: Radiative recombination is increased because carriers thermalize into the tail states and nonradiative recombination of free carriers in the bands is increased because of Shockley – Read – Hall recombination through the tail states. The comparison with experimental data shows that the influence of tail states is even worse than the increase in radiative and SRH recombination predicted by our model. Our results thus suggest that band tails act as one of the main remaining voltage limitations in the majority of state-of-the-art solar cells.


| INTRODUCTION
The efficiency of a solar cell defines how much solar energy can be converted into electricity and strongly depends on the solar cell's open-circuit voltage V OC .The latter is a quantity that is dictated by the charge carrier recombination activity within the semiconducting absorber of the solar cell.Ideally, the only recombination is radiative band-to-band recombination, which is an unavoidable and necessary process to keep the detailed balance in the ideal case.In real systems, however, several additional harmful recombination channels are present and limit the V OC .An improvement in the V OC is generally achieved by reducing the harmful recombination processes.One potential cause of harmful recombination is band tails.Band tails are defined as a density of states (DOS) that extend from the bands into the band gap of a semiconductor and are caused by structural and thermal disorder. 1,2The investigation of band tails has recently caught renewed attention in a wide variety of solar cell materials such as chalcopyrites 3 , kesterites 4 , perovskites [5][6][7] , and quantum dot lead sulfides 8 .Furthermore, several studies reveal a correlation between band tails and the V OC 5,6,8 , which suggests a causal relationship.
Consequently, as a source of voltage losses in solar cells, the charge carrier recombination activity through band tails appears to play a crucial role.
To identify the exact influence of the band tails on the open-circuit voltage, a material system is required where the band tails can be willfully tuned.An example of such a system is stateof-the-art Cu (In,Ga)Se 2 (CIGSe) thin-film solar cells where the band tailing can be manipulated by the addition of an alkali-based postdeposition treatment (PDT). 9The alkali PDT is a process where alkali fluorides are deposited onto the surface of the absorber after growth.When done at elevated temperatures, the alkali atoms can diffuse into the absorber.While Rudman et al. introduced the alkali PDT already back in 2004 by using the light alkali element Na 10 , the recent increases in efficiency from 20.3% 11 in 2011 to the current record efficiency of 23.35% 12 in early 2019 are due to the usage of the heavier alkali elements such as K 13 , Rb 14 , and Cs 14,15 .Even though the alkali PDT improves all solar cell parameters, a direct influence is distinguishable for the open-circuit voltage V OC .7][18][19][20] An encompassing review of relevant literature can be found in ref 9 , where the role of band tails for the V OC loss was already pointed out.
In this contribution, we thoroughly investigate the band tails in state-of-the-art CIGSe absorbers grown in either a low-temperature (suitable for the deposition on flexible substrates) or a hightemperature co-evaporation process.A precise description of the samples is given in the experimental Section 2. As the alkali incorporation into the absorbers leads to a well-reported increase of the V OC in both low-and high-temperature absorbers 14,21,22 , we can use CIGSe as a reference system that allows us to monitor and quantify the influence of band tail recombination on the solar cell performance.
Ultimately, this allows us to identify whether a change in the band tails is responsible for the observed increase in the V OC and thus to declare if band tails merit the attention that they so prominently attract.

| Sample preparation
In total, nine different sets of CIGSe samples have been investigated in this study.This specific combination of various samples is selected to represent a broad range of preparation-specific methods such as growth conditions, substrates, and PDTs.Table 1 summarizes important selected sample properties.Absorbers that share a similar label (e.g., H1 and H2) have been grown at identical conditions but received a different PDT (or none at all).As such, the samples can be allocated into different groups: • H1 and H2 • L1, L2, and L3 • LN1 and LN2 • LS and LC The samples LS and LC are included as they represent highly efficient devices.The labeling letter "L" refers to a growth process at low temperatures, while the letter "H" designates samples that were grown at Note: Indicated are the sample label, the PDT, the presence of alkalis within the absorber, the compositional [Cu]/([Ga] + [In]) (CGI) ratio, the band gap energy E g , the open-circuit voltage V OC , and the power conversion efficiency (without antireflective coating; typically the efficiencies can be increased around 1% absolute or even more by applying an anti-reflective coating).Safe for the band gap energy, the indicated values all represent averages across several solar cells (typically 6 to 10).The determination of the band gap energy is described in detail in the supporting section S1.
high temperatures.All samples are present as CdS-covered absorbers as well as finished solar cell devices without antireflective coating.
[25][26] .The preparation process of sample LC is detailed in ref. 27 .Finally, the recorded PL signal is spectrally calibrated using the known spectrum of a commercial halogen lamp.Intensity-calibrated measurements are described in more detail in Section S2 of the supporting information and are used to determine the quasi-Fermi level splitting (QFLS) according to Planck's generalized law. 28,29QFLSs values are presented in Section S4 of the supporting information.Once the QFLS is known, Planck's generalized law can be used to determine the absorptivity. 30,31However, state-of-the-art Cu (In,Ga)Se 2 absorbers show generally interferences near and below the absorption edge. 32These interferences do not disturb the determination of the QFLS 32,33 , but they are particularly strong at low absorption and thus in the region of the tail states.We developed a reversible surface treatment to remove the interferences. 34,35With this layer, we can no longer calibrate the intensity of the incoming or the outcoming light.
However, the absorption of the layer is spectrally flat, that is, the spectral calibration is still reliable and allows to determine the spectral shape of the absorption, albeit not the absolute value.It is thus possible to determine Urbach energies from the absorption spectra as described in Section S2 of the supporting information.In this case of spectral calibration only, the incident photon flux is not equivalent to an AM1.5 spectrum.

| Current-voltage characterization
The solar cell parameters are determined from current-voltage measurements using an ABA-class solar simulator in a 4-probe configuration at 25 C and using a simulated AM1.5G spectrum with a Si cell for calibration.

| Compositional analysis
The average elemental composition of the samples is measured by using a 45-keV home-built X-ray fluorescence spectrometer.
From the recorded spectra, the Kα peaks of the Cu, In, Ga, and Se species are fitted.The measured CGI ratios are believed to deviate from the real compositional ratio by less than 0.03 in absolute.
The depth-dependent compositional measurements are carried out with a time-of-flight secondary ion mass spectrometer (ION-TOF GmbH TOF SIMS5) using the ions O 2 + for sputtering and Bi + for analysis.

| External quantum efficiency
External quantum efficiency (EQE) measurements are used as a supporting tool for the definition of the band gap energies of the samples.The band gap determination is described in detail in Section S1 of the supporting information.The EQE measurements are done at 25 C with a white light halogen lamp as light source.
The light is chopped at 270 Hz and split into its monochromatic spectral components in a LOT MSH-300 monochromator before reaching the solar cell subjected to a light bias of about 0.15 suns.
A certified Si cell and a calibrated Ge cell are used to calibrate the intensity., the extension of the DOS into the band gap follows an exponential energy dependence.In such a case, the band tails are labeled as Urbach tails. 43Figure 1a illustrates an example of the DOS reaching exponentially into the band gap in a direct semiconductor.
According to Cody et al 1 and John et al 44 , Urbach tails are caused by phonons and by static disorder, that is, by lattice vibrations and defects, that lead to deviations from the perfect crystal lattice.In the case of CIGSe, origins of disorder and hence Urbach tails are likely to be related to variations in the bond lengths 45 and potential fluctuations of which there are two kinds: band gap fluctuations and electrostatic potential fluctuations. 2,46nd gap fluctuations in CIGSe arise from inhomogeneities in the elemental composition.As the band gap energy is most strongly influenced by the gallium content 47,48 , local variations in the [Ga]/([Ga] + [In]) (GGI) ratio lead to local variations of the band gap energy. 49Band gap fluctuations that are due to this alloy disorder are unavoidable.In addition to the Ga content, variations of the copper content also play a role as nonstoichiometry can lead to strain-induced fluctuations of the band edges. 46lectrostatic potential fluctuations arise from local inhomogeneities of charged defects, which is a consequence of Cu-poor (i.e., state of the art) CIGSe being a compensated semiconductor.The randomly distributed charged defects exert a Coulomb potential onto the charge carriers in the bands, leading to fluctuations of the bands, 42,50,51 which however flatten at room temperature. 52,72In addition, charged defects are known to accumulate at grain boundaries, leading to a band bending which is another source of disorder. 52,53 Above the band gap energy E g , α follows the classical square root-like behavior (see, e.g., 54 ), while below E g , α shows the Urbach-like exponential decay that is mathematically described as α $ expð E EU Þ (see e.g. 55).The amount of subband gap absorption can be described by the characteristic Urbach energy E U : The larger E U , the more the joint DOS extends into the band gap.In the case of Urbach energies E U larger than 25 meV (i.e., k B T), as is typically the case in kesterites 4 , the QFLS is low as charge carriers populate tail states rather than the band edge.In systems where E U is lower than 25 meV, band tails are generally assumed to be benign.Here, we investigate the influence of such "weak" tail states on the open-circuit voltage of the solar cells.

| Influence of alkali atoms on band tails in Cu (In,Ga)Se 2
To investigate the influence of the alkali atoms on the band tails in CIGSe, the absorption coefficient α spectra need to be measured.
From the multiple experimental techniques that are able to determine α, we use photoluminescence spectroscopy (PL) as our main experimental technique since it allows for the detection of α over magnitude. 30,31Extracting the absorption coefficient from photoluminescence spectra recorded at room temperature involves determining the QFLS and the absorptivity from the PL spectra, by analyzing them in terms on Planck's generalized law.An exponential fit to the low-energy slope of the absorption coefficient at energies significantly lowers than the band gap energy then enables us to extract the Urbach energy E U .The detailed procedure is given in ref. 31 and in Section S2 of the supporting information.In this study, the Urbach energy is extracted from the low-energy slope of the In the case of an ideal direct semiconductor, the conduction band (CB) and valence band (VB) densities of states do not extend into the band gap (dashed lines).However, in the case of a real semiconductor, the density of states (DOS) may extend exponentially into the band gap and thus lead to states that are available for optical transitions (full lines).(b) Absorption coefficient α spectra of an ideal direct semiconductor (dashed lines) and of a real semiconductor (full lines).An exponentially decaying joint DOS leads to an absorption coefficient α that shares the same energetic dependence.Depending on how strongly the DOS extends into the band gap, the absorption coefficient shows a different decay below the band gap energy E g , which is characterized by the Urbach energy E U .The more the DOS extends into the band gap, the larger E U and the more states are available for transitions.Above E g , the absorption coefficient shows the classical square-root-like dependence on the photon energy absorption coefficient deep in the gap and not from the energy range just below the band gap, which can lead to unphysically high Urbach energies. 4,31gure 2 shows the absorption coefficient spectra of the CdScovered CIGSe absorbers L1 (with NaF and RbF-PDT-red curve), L2 (with NaF-PDT-blue curve), and L3 (without alkalis-green curve) in a semilogarithmic representation.For the fitting range specified in the graph, the exponential fits (dashed lines) yield an Urbach energy of 14.7 meV for sample L1, 15.8 meV for sample L2, and 16.6 meV for sample L3.These results thus clearly reveal a strong influence of the alkali atoms on the band tails: A reduction of the Urbach energy in the CIGSe absorber that contains Na (L2) compared to the one that does not contain any alkalis (L3).In addition, the presence of the heavier alkali atoms Rb (L1) leads to an additional decrease of the Urbach energy.In this specific example, the range over which the fit is done extends over 2 orders of magnitude in absorption coefficient which is only possible because the measurement extends over 4 orders of magnitude or more.The fitting range in Figure 2 is only one of multiple different fitting ranges that are selected for the extraction of E U .Different fitting ranges lead to slightly different E U (the exact influence is showcased in Figure S6 of the supporting information) which are used to perform a statistical analysis and identify a standard deviation.The standard deviation is then used to form the error bars in the following sections and figures.It can thus be concluded that the presence of alkali atoms within the CIGSe absorber leads to a reduction of the Urbach energy, that is, of the band tails.In ref. 9 , the same observation was made through EQE measurements.
Additionally, this reduction is stronger if a heavier alkali species such as Rb is present compared to a lighter species such as Na.It is likely that alkali accumulation at grain boundaries removes charged defects and thus band bending at the grain boundaries, as discussed in detail in ref 9 .The important point for the current study is that it is possible to manipulate the band tails in the same material.

| Influence of Cu content on band tails of Cu (In,Ga)Se2
To demonstrate that the band tails are strongly manipulated by the alkali treatments, we also study the dependence of the Urbach energy on the Cu content.As mentioned in Section 3, one origin of band tails can be electrostatic potential fluctuations that are present in state-of-the-art CIGSe absorbers.From literature reports, it is known that the magnitude of the electrostatic potential fluctuations decreases with increasing [Cu]/([Ga] + [In]) (CGI) ratio. 42,56A recent related study, where a multitude of CIGSe solar cells with different alkali PDT processes were investigated, reveals a reduction of the band tails with increasing CGI ratio from 0.8 to 1. 22 Figure 3 shows the extracted Urbach energy E U from photoluminescence in dependence of the CGI ratio for all the samples listed in Table 1.Generally, a slight expected trend is distinguishable as the band tails are reduced in samples that exhibit a higher CGI ratio.However, this trend is only clear for the samples grown at low temperatures (L1, L2, L3, LN1, LN2, and LS) and the LC sample.The samples grown at high temperatures, H1 and H2, do not follow this trend as their Urbach energies are considerably lower yet their CGI ratio is not higher compared to the other samples.Considering only the samples that were prepared under identical conditions but received different PDTs, that is, the samples that share the same label (i.e., L1-3, LN1-2, and H1-2), Figure 3 offers a very important conclusion: The presence of alkali atoms in the absorbers influences the band tails more than the CGI ratio.
F I G U R E 2 Absorption coefficient α spectra in semilogarithmic representation of the CdScovered graded absorbers L1 (red curve), L2 (blue curve), and L3 (green curve).The dashed lines represent the fit applied in the indicated exemplary fitting range (0.98 to 1.05 eV) that yields the Urbach energies E U as displayed in the bottom-right corner of the graph.The values of E U depend only weakly on the exact fitting range.Also shown are the band gap energies (dotted lines) as indicated in Table 1 of Section 2. The determination of the band gap energies is explained in Section S1 of the supporting information

| Correlation between band tails and voltage losses
The previous sections have shown that the presence of alkali atoms is an effective driver for the reduction of band tails in CIGSe absorbers and solar cells.Additionally, it is known from multiple published studies that the alkali atoms also lead to an increase of the optoelectronic quality of CIGSe solar cells through an increase in the open-circuit voltage V OC . 14,22,33These findings hint at a correlation between the V OC and the Urbach energy E U that has experimentally already been demonstrated in other materials. 5,6,8Figure 4 shows the open-circuit voltage deficit with respect to the Shockley-Queisser limit V SQ OC À V OC of the samples investigated in this study in dependence of their extracted Urbach energies.Compared to the aforementioned studies 5,6,8 , where the main figure of merit is the open-circuit voltage deficit E g À qV OC , the quantity V SQ OC À V OC is more accurate when representing materials with significantly different band gap energies.
In this case, for matters of reference and comparison, some selected literature data for GaAs, c-Si, CH 3 NH 3 PbI 3 , and CdTe are included in Figure 4 (triangle symbols).Additionally, different data sets are plotted for the reference technologies to show the possible variability of the selected parameters in literature.The inclusion of GaAs is motivated by the fact that it represents the single-junction solar cell technology with the highest efficiency 57 and can thus be used as a reference point.For the GaAs data point labeled (1), we selected the highest V OC (from the record device, i.e., 1.1272 V 57 ) and the lowest E U (6.9 meV 36 from a single crystal) while the GaAs point labeled (2) represents a slightly inferior combination (taken from ref. 5 and references therein).In both cases, to compute V SQ OC , the band gap energy of 1.42 eV was used. 58The inclusion of c-Si in Figure 4 is insightful because it shows that CIGSe solar cells can achieve a similar open-circuit voltage deficit while still exhibiting a lower power conversion efficiency.The c-Si data point labeled (3) represents the V OC of the current champion device (0.738 V 57 ) with the lowest published Urbach energy from a different crystal (9.6 meV 37 ).The c-Si data point labeled (4) is taken from ref. 5 and references therein and shows that the location of the c-Si data point on the graph depends strongly on the various published Urbach energies.The value of 1.12 eV was used for the band gap of the c-Si data points. 58The two perovskite data points labeled (5) are taken from the recent study by Ledinsky et al 6 while the data point ( 6) is taken from the older study by De Wolf et al 5 Finally, for the CdTe data point (7) we took the V OC from the record device. 57The band gap energy was extracted via linear extrapolation from the EQE curve that was given in ref. 59 .The CdTe Urbach energy was extracted from the study by Rakshani. 38The inclusion of literature data is generally problematic as, in general, Urbach energy data are not reported for devices fabricated for photovoltaic purposes.Consequently, for the literature data we use V OC loss and Urbach energy values from different samples, which will be at least partly responsible for some of the outliers to the general trend in (3) and the alkali-free L3 sample strongly deviate from this trend and show a V OC loss higher than the one given by the trend with Urbach F I G U R E 3 Urbach energy E U as extracted from photoluminescence as a function of the absorber CGI ratio.The samples grown in a hightemperature process are displayed as squares while the samples grown in a low-temperature process are represented by circles.The sample LC is displayed by an open-diamond symbol.The color code alludes to the presence of alkalis in the samples and is explained in the bottom-left corner.The vertical error bars are extracted from a statistical analysis as described for Figure 2 energies.The deviation of the c-Si data point (3) from the trend might be explained by the fact that we chose the lowest Urbach energy ever measured in c-Si for this data point.The c-Si data point (4) shows that even an indirect semiconductor can follow the empirical trend.
The trend observed in Figure 4 is also observed with QFLSs instead of open-circuit voltages (see Figure S7 and ref. 9 ) with a very similar slope.The QFLS is determined from the PL spectra measured on the absorber before junction formation, that is, without the formation of a significant space-charge region.We have already previously demonstrated that the difference between QFLS and V OC is very small (below 10 mV) in state-of-the-art Cu(In,Ga)Se 2 devices. 33This observation indicates that the recombination mechanisms leading to the V OC loss occur in the quasi-neutral region of the absorber, not in the space-charge region.
Band tails can contribute to voltage losses through both radiative and nonradiative recombination.First, we calculate the loss due to additional radiative recombination.As shown in Figure 1, band tails are states within the band gap that participate in absorption events.
Consequently, the number of photons that can be absorbed is larger in a semiconductor with band tails than in an identical semiconductor without band tails.However, following the principle of detailed balance, states that lead to absorption processes also lead to emission processes in thermal equilibrium. 61 where α 0 and E 0 are material-specific parameters.For reasons of simplicity, we set E 0 ¼ E g .Additionally, to ensure that the absorptivity equals unity above E g , as assumed in the Shockley-Queisser model, we set α 0 to 10 5 cm À1 .The latter is a strong simplification as the absorption coefficient follows a square root behavior at energies above the band gap energy. 55It should be noted that Equation 1 is a useful approximation for the absorption behavior of tail states to calculate voltage losses only in the case of Urbach energies smaller than k B T. For higher Urbach energies, the fact that αðEÞ changes with the occupation of states, that is, with the position of the quasi-Fermi Open-circuit voltage deficit with respect to the SQ limit V SQ OC À V OC in dependence of the Urbach energy E U of the samples investigated in this study.The square symbols represent the CIGSe samples grown at high temperatures, the circles the samples grown at low temperatures and the open-diamond symbol represents the LC sample.The color code alludes to the presence of alkali atoms and is explained in the top-left corner.The V OC values were measured on the finished solar cell devices while E U was extracted from the absorbers.Figure S7 of Section S4 in the supporting information shows the influence of E U on the quasi-Fermi level splitting deficit and reveals a similar trend in the absorbers.The triangle symbols represent selected literature data for GaAs (1, 2), c-Si (3, 4), CH 3 NH 3 PbI 3 (5, 6), and CdTe (7) described in the text.Also shown are estimated voltage losses in case recombination processes are occurring via band tails through a radiative channel (dashed line), a nonradiative channel (dotted line), or a combination of both (continuous line), all offset to the GaAs data point (1) as it serves as reference.The gray dashed line is a guide to the eye and shows the empirical trend levels, needs to be taken into account. 63,64From the absorption coefficient, the absorptivity aðEÞ can be determined through the simplified where we assume the absorber thickness d to be 2.5 μm.Using the whole absorber thickness here is a simplification since the emission is not from the whole absorber but just from the notch region around the band gap minimum. 33The error is negligible with respect to the other simplifications made here.Compared to the Shockley-Queisser model where a step function for aðEÞ is assumed, Expression 2 is slightly larger than 0 below E g and slightly below unity above it.Equation 2 is inserted into the well-known expressions for the short-circuit current density j SQU SC and saturation current density The quantity Φ sun ðEÞ represents the photon flux density of the AM1.5 spectrum, while the quantity Φ 0 ðE, TÞ describes the thermal black body photon flux density, where we use a temperature T of 296 K.
The current densities are determined by numerical integration.Finally, both short-circuit and saturation current densities are inserted into the open-circuit voltage expression of the SQ model The exact details, parameters, and restrictions of the SQU model are described in Section S5 of the supporting information.Since we want to compare with real world open-circuit voltages, we calculate the decrease of V SQU OC with increasing Urbach energy using the GaAs data (1) in Figure 4 as the reference point.According to these calculations, represented by the black dashed line in Figure 4, an increase of the Urbach energy from 10 to 15 meV leads to an increased voltage loss of approximately 24 mV which is not negligible but also not significant enough to explain the observed losses.Similar calculations in literature show very similar voltage losses for the Urbach energy region under investigation. 8,67 addition to radiative recombination processes, states within the band gap also participate in nonradiative recombination processes of the Shockley-Read-Hall (SRH) type. 68,69Contrary to the radiative recombination process where at least one of the states is a carrier trapped in a tail state, the SRH recombination process considers the recombination of free carriers where the initial and final states are an extended band state.While the probability of SRH recombination processes increases the more the states are situated between the demarcation levels (i.e., close to mid gap) 70 , nonradiative recombination processes can also occur through states close to the band edges. 58To estimate the voltage losses in a p-type semiconductor that are resulting from nonradiative band tail recombination, we equated the band tails with a defect density NðE T Þ that has a peak defect density N 0 at the conduction band minimum E C and decays exponentially into the band gap according to with E T being the energy of the tail state.For the DOS given in Equation 6, the nonradiative recombination rate of free carriers in the bands can be calculated through the SRH recombination rate 68,69 with n and p the free carrier concentrations of electrons and holes, respectively, and n Á ¼ N C exp½ ETÀEC kBT and p ¼ N V exp½ EVÀET kBT the auxiliary charge carrier densities.The lifetimes τ n=p are expressed as τ n=p ¼ ½σ n=p v th NðE T Þ À1 where v th is the thermal velocity and σ n=p is the recombination cross section for electrons and holes, respectively.By assuming the recombination cross-section to be equal for both electrons and holes, the SRH recombination rate can be written as follows For tail states with Urbach energies smaller than k B T, the auxiliary charge carrier densities are always considerably higher than n or p.
Thus, we end up with the expression (the detailed arguments are given in Section S6 of the supporting information) with the quantity E R being the decay energy of the SRH recombination rate that is a consequence of the decaying nature of the recombination activity of tail states with decay energy E U according to It is obvious from Equation 10 that the approximations we make here work only for Urbach energies smaller than k B T. The other parameters needed to calculate absolute values to R SRH are not known.Therefore, we calculate only the increase of the open-circuit loss ΔΔV nrad OC as the Urbach energy increases.As explained in detail in @Section S6 of the supporting information, the increase ΔΔV nrad OC in open-circuit voltage loss following an increase of the Urbach energy E U in the case of nonradiative tail recombination is mostly dictated by the decay energy where E A R constitutes the reference decay energy.Choosing, for example, as reference value an Urbach energy of 10 meV, the reference SRH recombination rate decay energy E A R would equate to 16.4 meV according to Equation 10.Increasing the Urbach energy to 15 meV would then lead to an SRH recombination rate decay energy E B R of 36.4 meV.Such an increase in the Urbach energy finally leads to an increase in the nonradiative voltage loss ΔΔV nrad OC of approximately 20 mV according to Equation 11.For a broader Urbach energy range, the nonradiative voltage losses are represented by the dotted black line in Figure 4 as well as in Figure S10 in the supporting information.
In Section S7 of the supporting information, we present the results from SCAPS simulations where near-identical results are obtained.
Finally, it is important to mention that the above calculations are only based on the presence of a conduction band tail (cf.Equation 6).However, since electrons are the minority charge carriers in a p-type semiconductor, the inclusion of an additional valence band tail does not influence the calculated results.From the dotted black line in Figure 4, we see that the voltage losses due to nonradiative recombination are similar in magnitude to the radiative losses and can, thus, also not explain the observed trend.
The losses due to radiative recombination through tail states and due to nonradiative recombination via tail states are of the same magnitude.This might be surprising in two ways.Firstly, for inorganic semiconductors the radiative loss can be usually neglected with respect to the nonradiative loss. 66However, the tail states will increase radiative recombination significantly, since states at lower energies contribute more due to the exponential increase of the black body photon flux density Φ 0 that enters into Equation 4. Our model covers the tail states deep down into the band gap.And the modeled nonradiative loss is due to SRH recombination through states near the band edges, which do not contribute as much as deep states.Thus, secondly, it may seem surprising that states so close to the edges lead at all to a nonradiative recombination loss.Das et al. recently investigated the recombination activity of defects as function of their energy in the band gap and found that only defects near midgap contribute significantly to nonradiative recombination. 71However, assuming a defect density of 10 15 cm À3 , they find a nonradiative lifetime for defects near the band edges of about 100 ms, for a range of different semiconductors.The DOS in the band tails is certainly much higher, in the order of the effective DOS in the band.With about 4 orders of magnitude higher DOS, the nonradiative lifetime, only due to tail states, is in the same range as the radiative lifetime.Thus, a comparable voltage loss due nonradiative recombination via the tail states and due to radiative recombination through tail states is to be expected.
Since both radiative and nonradiative recombination processes occur simultaneously, the continuous black line in Figure 4 visualizes the total expected voltage loss due to band tail recombination, which is obtained by summing up the radiative and nonradiative voltage losses.The comparison between the measured data and the calculated losses indicates that band tails lead to about a factor of 2 more V OC loss than expected from radiative and SRH recombination processes and appear to contribute to the losses via an additional (nonradiative) channel.One possible nature of this additional channel could be nonlocal recombination processes through tails that involve tunneling of electrons and holes, for example, in the band bending around charged extended defects.The additional recombination has to be a mechanism that does not depend on the details of the material, as we observe a general correlation between band tails and voltage losses across a broad spectrum of technologies in Figure 4 and in previous studies. 5,6,8The fact that voltage losses correlate so well to the Urbach energy across a wide variety of technologies strongly hints at a causal relationship: Recombination through tail states appears to limit the open-circuit voltage in state-of-the-art photovoltaic devices.This effect is worse than expected from the influence of tail states on radiative and SRH recombination.
[Correction added on 11 April 2022, after first online publication: citation of references 53 'Abou-Ras D, et al.' to 71 'Das B, et al.' have been corrected in this version] The Urbach tails represent a joint DOS within the band gap that are available for optical transitions.With the absorption coefficient α being proportional to the joint DOS 2 , the optical absorption edge replicates the characteristic exponential decay of the Urbach tails within the band gap. Figure 1b illustrates the absorption coefficient behavior in function of the photon energies involved.

Figure 4 .
Figure 4.The advantage of the new experimental data added in this study is that Urbach energies and V OC losses have been determined on the same samples.Recently, a new study by Liu et al has revealed a record V OC for a CH 3 NH 3 PbI 3 solar cell. 60The inclusion of this data point in Figure 4 is discussed in Section S8 of the supporting information.The data in Figure 4 exhibit a reduction of the voltage losses with decreasing Urbach energies.Using GaAs as a reference point, an empiric trend (gray dashed line) is observable that suggests that the voltage losses are caused by the Urbach tails.Only the c-Si data point Thus, the presence of sub band gap states leads to an increase in the radiative equilibrium recombination which, in turn, leads to a reduction of the open-circuit voltage.Charge carriers thermalize into the tail states and lose energy in the process.Thus, these recombination processes are transitions where at least one of the states (initial and/or final) is a carrier trapped in a tail state.To estimate the voltage losses through radiative band tail recombination, we modify the Shockley-Queisser model62 to allow for absorption and emission from Urbach tails (labeled in the following Shockley-Queisser-Urbach [SQU] model).This is done by defining the following absorption coefficient55

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CONCLUSION Since band tails constitute a DOS that extend into the band gap, they induce additional radiative and nonradiative recombination channels that contribute to losses in the open-circuit voltage V OC of solar cell devices.The correlation between voltage losses and Urbach energy has recently been investigated for various solar cell technologies.Empirically, a linear trend between V OC loss and the Urbach energy has been demonstrated in the literature.Alkali treatments on state-of-the-art thin-film Cu(In,Ga)Se 2 absorbers with efficiencies of 20% and above allow to manipulate the amount of tailing, as measured by the Urbach energy through photoluminescence and photocurrent spectroscopy.We determined the open-circuit voltage loss and the Urbach energy in the same sample.The relation of the voltage deficit with respect to the theoretical Shockley-Queisser limit in dependence of the Urbach energy for various solar cell technologies confirms the previously observed empirical trend: a linear reduction of the voltage losses in devices that show weaker band tails.We implemented two theoretical models that account for the voltage losses through both radiative and SRH nonradiative band tail recombination: Radiative recombination is increased because the carriers that thermalize into the tail states can recombine radiatively; nonradiative recombination removes carriers from the extended band states via SRH recombination through the tail states.Through the models, we demonstrate that, in fact, an increase in Urbach energy of only 1 meV leads to an increase in voltage loss as high as 10 mV.The modeled contributions of radiative and nonradiative recombination to the additional voltage loss are about 5 mV each.However, the measured voltage losses turn out to be more severe by a factor of 2 compared to the theoretically predicted ones.Consequently, an additional (nonradiative) loss mechanism must be active in these devices that go beyond straightforward SRH recombination.Given the generality of the empirical trend, we suggest that band tails are also responsible for this additional voltage loss, possibly through nonlocal recombination processes involving the tunneling of electrons and holes.Thus, we conclude that the recombination through tail states is the main cause of the voltage loss in these state-of-the art devices.This loss is accounted for to 25% by radiative recombination, to 25% by SRH recombination, and to 50% by an additional nonradiative mechanism that is not understood yet.It appears that band tails are the main driver for open-circuit voltage losses in the majority of stateof-the-art solar cell technologies.

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Selected properties of the samples that feature in this study