Below the Urbach Edge: Solar Cell Loss Analysis Based on Full External Quantum Efficiency Spectra

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Below the Urbach Edge: Solar Cell Loss Analysis Based on Full External Quantum Efficiency Spectra

Item Type Article

Authors Holovský, Jakub;Ridzoňová, Katarína;Peter Amalathas,

Amalraj;Conrad, Brianna;Sharma, Rupendra Kumar;Chin, Xin Yu;Bastola, Ebin;Bhandari, Khagendra;Ellingson, Randy J.;De Wolf, Stefaan

Citation Holovský, J., Ridzoňová, K., Peter Amalathas, A., Conrad, B., Sharma, R. K., Chin, X. Y., Bastola, E., Bhandari, K., Ellingson, R. J., & De Wolf, S. (2023). Below the Urbach Edge: Solar Cell Loss Analysis Based on Full External Quantum Efficiency

Spectra. ACS Energy Letters, 3221–3227. https://doi.org/10.1021/

acsenergylett.3c00951 Eprint version Publisher's Version/PDF

DOI 10.1021/acsenergylett.3c00951

Publisher American Chemical Society (ACS)

Journal ACS Energy Letters

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Below the Urbach Edge: Solar Cell Loss Analysis Based on Full External Quantum Efficiency Spectra

Jakub Holovsky , ́ * Katarína Ridzon ̌ ová, Amalraj Peter Amalathas, Brianna Conrad,

Rupendra Kumar Sharma, Xin Yu Chin, Ebin Bastola, Khagendra Bhandari, Randy J. Ellingson, and Stefaan De Wolf

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^sı Supporting Information

ABSTRACT: We suggest a new solar cell loss analysis using the external quantum efficiency (EQE) measured with sufficiently high sensitivity to also account for defects. Unlike common radiative-limit methods, where the impact of deep defects is ignored by exponential extrapolation of the Urbach absorption edge, our loss analysis considers the full EQE including states below the Urbach edge and uses corrections for band-filling and light-trapping. We validate this new metric on a whole range of photovoltaic materials and verify its accuracy by electrical simulations. Any deviations between the new metric and experimental open-circuit voltage are due to the presence of spatially localized defects and are explained as violations of the assumption of flat quasi-Fermi levels through the device.

I

n recent years the so-called radiative limit has become popular to evaluate the maximally attainable open-circuit voltage, VOC,rad, of a solar cell based on the assumption that carrier recombination occurs only through electroluminescence.¹ The excess electroluminescence, compared to its equilibrium value Φ_EL0, is described as Φ_EL (V) = Φ_EL0[exp(qV/kT) − 1], where V represents applied voltage, kis the Boltzmann constant,Tis the absolute temperature, and qis the elementary charge. For an illuminated solar cell in open circuit, as no external current flows, one may assume that the emitted photon flux,Φ_EL(VOC), equals the flux of the absorbed photons.²From this, one obtains

V kT

q

EQE E

E ln 1

d

OC rad d

AM G EL ,

1.5 0

lm ooo nooo

|} ooo

~ooo

= + ·

(1) where EQEis the external quantum efficiency (EQE) of the solar cell and Φ_AM1.5G is the photon flux of the reference AM1.5G solar spectrum. The integrations are over the whole solar and luminescence spectra. Despite its elegance, this approach to calculateVOC,radhas not been used much. Rather, the reciprocity relation between absorptance and photon emission^3,4is mostly used (seeeq 2), whereΦ_EL0is replaced by blackbody radiation spectrum multiplied by EQEUrbach, which represents theEQEobtained when replacing the realistic absorption edge by the Urbach exponential tail.⁵ See the

example inFigure S1. The reciprocity is valid when the Urbach energy is smaller than kT (26 meV at room temperature),³ which is the case for most high-quality semiconductors.⁶Then the expression for V_OC,radfor Urbach energy smaller than kT takes the form of eq 2, which, as stated, is frequently employed.⁷⁻⁹

V kT

q

EQE E

ln 1 d

OC rad d

Urbach AM G Urbach bb K ,

1.5 ,300

lm ooo nooo

|} ooo

~ooo

+ ·

· ₍₂₎

In this, Φ_bb,300Kis the room-temperature blackbody radiation, calculated as 2πhc²/[λ⁴E²(e^E/kT− 1)], where h is the Planck constant,cthe speed of light,λthe photon wavelength, andE the photon energy. Also note that for a sufficiently high carrier mobility, the absorptance can be taken to be identical to the EQE.¹⁰The novelty ofeq 2compared to traditional Shockley−

Queisser, radiative-limit calculations¹ is that it takes into account non-abrupt absorption edges of the band-gap by

Received: May 12, 2023 Accepted: June 23, 2023

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considering the Urbach exponential tail. Eq 2 also eliminates the challenge of absolute scaling of Φ_EL0 or EQEUrbach. Moreover, the Urbach tail extrapolation provides spectral data starting at zero photon energy, which is required for integration. Since we know that deep defects critically limit device performance, we aim to analyze how to account for their contributions, which inherently means accounting also for the case when the Urbach energy is larger than kT. A recent approach focusing on the role of mid-gap states in bulk heterojunction devices has been proposed, using also high- sensitivity EQE with alternative parametrization by Gaussians attributed to charge-transfer states and mid-gap states.^11,12 This approach is attributing diode ideality factors 1 and 2 to the two respective EQE contributions. Here we follow another approach, the recently introduced band-filling (BF) concept, in the radiative-limit calculations¹³ to address the problem of reciprocity for Urbach energy larger thankT. We show that the ideality factor >1 can also be reproduced by the BF concept.

This concept means that once quasi-Fermi levels are split due to photoexcitation, the deep states energetically situated between these levels do not effectively take part in the recombination process. This statement is not in conflict with the fact that the states between the quasi-Fermi levels still absorb, because absorption and emission are not anymore in the reciprocity for deep defect states or states in tails that are decaying slower thankT. This is in line with the very basis of the Shockley−Read−Hall (SRH) model, where only empty trap states can trap an electron and vice versa for holes.¹⁴ Mathematically, the final SRH equations allow all kinds of trends,¹⁵while the BF concept can reproduce only those where the carrier lifetime increases with quasi-Fermi level splitting,¹⁶ corresponding to the behavior of a solar cell with the ideality factor≥1. We should also note here that the trends given by Auger recombination cannot be reproduced by the BF concept. Applying the BF correction requires knowledge of the effective masses of electrons and holes, which may be difficult. However, by assuming equal effective masses, a simplified correction to the absorption coefficient (or absorptanceAifαγd≪1) can be written as follows:¹³

A E qV A E qV

kT

( ) tanh

BF OC 4

i OC

kjjj y

{zzz

> (3)

The BF correction shown ineq 3limits the integration range foreq 2to photon energies above qVOC, giving more realistic values ofV_OC,rad,BF(the BF subscript indicates BF correction) but also more correct theoretical electroluminescence spectra.¹³

Following this, we define here a new metric calculated from the full EQE and the experimentally measuredV_OCthat allows a quick loss analysis of a device. The drawback of the application ofeq 3is that it requires knowledge ofV_OCof the cell before calculating VOC,rad,BF. But from this point of view, our approach is similar to the so-called external radiative efficiency introduced by Green,¹⁷ without ignoring the problems of EQE truncation and fulfilling reciprocity due to absorption edge steepness, however. The new metric, denoted as V_OC,EQE, is defined by eq 4, using the full EQE without extrapolation, with BF correction, and by applying several other corrections that will follow:

V kT

q

EQE E

ln 1

d

OC EQE d

BF AM G BF bb K ,

1.5 ,300

l mooo nooo

| }ooo

~ooo

+ ·

· ₍₄₎

The symbolEQE_BFhere represents theEQEcorrected byeq 3, and this makes the difference compared to eq 2. In the following text, we will discuss several corrections, starting with the effect of light-trapping and absorber thickness. For the operation of the solar cell, the question of absorber thickness is critical, and this is the same for the validity of reciprocity assumption. Becauseeq 4is only logarithmically sensitive, with sufficient accuracyEQEcan be approximated byeq 5based on the Beer−Lambert law, modified to illustrate the light-trapping path enhancementγ,

EQE

d

1 e

1

d d

d 1

1

l moooo

noooo ₍₅₎

where α is the absorption coefficient and d the thickness, which means from now on always the physical absorber thickness. The possible effect of the back reflector should be treated separately and is not automatically included inγ. Then, the assumption of reciprocity between absortion and emission critically depends on the limit of eq 5. In the solar cell’s spectral operating range, theEQEis saturated (αγd≫1) and the thickness has only a minor effect (this is the case for the integral of the product withΦ_AM1.5G); in the spectral range of luminescence, the sensitivity to thickness is still linear (αγd≪ 1) and thickness has a fundamental effect (this is the case of the integral of the product withΦ_bb,300K). To the best of our knowledge, this has not been accounted for in previous analyses ofV_OC,rad. To illustrate this effect, we appliedeq 2for different thicknesses of crystalline silicon (c-Si) and hydrogenated amorphous silicon (a-Si:H) solar cells. The results are shown inFigure 1and demonstrate that theVOC,radis thickness

dependent according toy=a−kT/qln(x). Only because this dependence usually (e.g., for c-Si) corresponds well to the reduction of excess carrier density and reduction of quasi- Fermi level splitting, there was not much discrepancy observed between predicted and measured voltages.

While changing the light-trapping path enhancement γ or changing the physical thicknessdhas optically the same effect according to eq 5, it will not have the same effect on the device’s electrical performance, as the light-trapping path enhancement does not change the volume density of defects.

Figure 1. Simulation of the effect of varying thickness on the VOC,radfor crystalline silicon (c-Si) and hydrogenated amorphous silicon (a-Si:H). Inset: trend in obtained values ofVOC,radfitted by

−kTln(d).

ACS Energy Letters http://pubs.acs.org/journal/aelccp Letter

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Consequently, a discrepancy appears. To correct for this light- trapping effect, we use here an already reported correction¹⁸of light extraction (LE) efficiency, η_LE, to account for the difference between external and internal luminescence. In the original equation, the corrected value V_OC,rad,LEwas penalized compared to VOC,rad that was calculated from internal luminescence. In eq 6 we use the same expression but take the external luminescence of a smooth slab as the reference.

Since for the smooth slab of refractive index n the external luminescence is reduced by a factorη_LE= 1/2n²relative to the internal luminescence,¹⁹we renormalize by the factor 2n². In this way, eq 4 remains valid for the case of a smooth slab becauseeq 1calculates withextractedlight.

V V kT

q ln(2n )

OC rad LE, , OC rad, LE

= + 2

(6) For the limit of perfect light extraction η_LE = 1, we obtain a voltage boostkT/qln(2n²) compared to the case without light extraction enhancement.²⁰ So far, we just took and renormalized the existing correction to light extraction.

However, for isotropic absorptance, the improvement in light outcoupling is identical to the improvement in light incoupling,²¹ and for the perfectly scattering layer�in the low absorption limit�the light absorptance is enhanced also by γ = 2n².²² Note that the same result can be obtained through etendue expansion,²¹as the etendue inside the layer is 2n² times higher than outside (factor 2 comes from the fact that the light inside occupies a full solid angle, versus only one hemisphere for the light outside). This means we know the two extreme cases when γ = 1 and η_LE = 1/2n² for smooth slabs and γ= 2n² andη_LE= 1 for perfect light scattering. In both cases,γ= 2n²η_LEholds. The last step is to assume that this also holds between the two extreme cases. Based on that assumption, we correct for light-trapping (labeled LT) by replacing 2n²η_LEineq 5byγand obtainingeq 7:

V V kT

q ln( )

OC EQE

LT OC EQE

, , +

(7) For smooth layers, this again does not change anything because in the case of smooth layersγ= 1. We performed ray- tracing simulations²³of the absorptance in an a-Si:H layer for different surface roughness to show that, with certain accuracy, γ can be taken as a spectrally independent constant, as we obtained fairly similar spectra as for smooth layers of various thicknesses, seeFigure S3.

Note that application of light-trapping does not itself increase the voltage of the device as it might seem fromeq 7 because, as demonstrated by Figure 1, light-trapping will effectively increaseEQEin the low-energy part and therefore at the same rate reduce VOC,EQE. A net positive effect can be achieved if the light-trapping enhancement γ is combined, ideally with γ times reduced thickness. Then EQE should remain similar (according toFigure S3), giving similarV_OC,EQE, while the second term in eq 7will yield a positive effect and increase V_OC,EQE. Therefore, in the case when two solar cells have the same EQE, the one that has the higher γ(and thus correspondingly lower thickness) should give the higher voltage. On the other hand, if these two cells have the same γ but different thicknesses (they must then have different absorption coefficients), our analysis gives identical VOC,EQE. This is clearer from the point of view of recombination of active tail states because, as will be shown later, it is not the

volume density but the total number of tail states in the layer that matters. Therefore, instead of the defect-related absorption coefficient, it is the defect-relatedEQEthat matters, which, in the low absorption range, is a product of the absorption coefficient times thickness. It also means that knowingγandEQEis sufficient to evaluate the total number of tail states or to evaluate V_OC,EQE.

We have assumed that the states contributing toEQE are homogeneously distributed volume states. For such homogeneously distributed defects, the recombination rate is evenly distributed and we can assume a pair off latquasi-Fermi levels, as required by the reciprocity law⁴(Figure 2A). But in the case

ofspatially localizeddefects�such as “cracks” in microcrystalline silicon (μc-Si); grain boundaries (GB) in multicrystalline Si, CdTe, or CIGS; localized charges; and segregated phases in mixed-halide perovskites�quasi-Fermi levels are deformed (Figure 2B). This equally applies to interfaces which typically contain large populations of defects. These spatially localized defects represent local sinks for excess carriers due to a local quasi-Fermi level bending (toward mid-gap), expressing local depletion of free charge carriers; these charges can funnel over some distance to the localized recombination centers. On the other hand, the overall optical absorption of these spatially localized low-dimensional defects remains insignificant compared with absorption by the overall volume defects. The visibility in the EQE spectrum is even more reduced due to poor carrier collection from these defects. It is well known that, although “cracks” inμc-Si²⁴⁻²⁶might be effectively invisible in the EQE, they still reduce the VOC considerably. The proportionality between radiative recombination and optical absorption is lost in such cases.

In our case, the high sensitivity EQE was measured by Fourier-transform photocurrent spectroscopy (FTPS), which enabled a fast and sensitive measurement of photocurrents as a function of the incident photon wavelength. The frequency dependence and relatively high modulation frequency used during FTPS measurements mean that FTPS is not strictly equivalent to EQEand does not match the absolute scale of EQE.²⁷These, however, are not issues for the application ofeq Figure 2. Sketch illustrating that spatially localized defects present on surfaces or grain boundaries (GB) may lead to deformed quasi- Fermi levels, which have implications for the validity of reciprocity relations.

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4, where the sensitivity is logarithmic and the equation is insensitive to any scaling constant.

The shapes of theΦ_AM1.5GandΦ_bb,300Kspectra imply that the main contribution to the integral in the denominator ineq 4 comes from the region of the EQEwhere its slope is exactly opposite to the slope of the Φ_bb,300K, with the latter being dominated by an exp(−E/kT) term. Plotting the exponential line exp(E/kT) tangential toEQEBFis used to search for the most important spectral region (Figure 3). The first step to

assess the role of homogeneously distributed volume states in the device should be to check if the touching point is in the region of the deep defect absorption (indicated by circles).

Note that this is not the case for c-Si and CdTe and pure iodide perovskite material in the fresh state (MAPI fresh) where also the BF correction does not have any effect.

In most cases whenE₀<kT, the most important integration region is just below the absorption edge. However, in some cases, the deep-defect-related absorptance may be sufficiently

intense to dominate the integral ineq 4. This is the case for a- Si:H or an unoptimized mixed-halide Cs0.17FA0.83Pb(I0.6Br0.4)3

perovskite. In this case, as well as in the case whenE₀>kT, the BF effect becomes especially important for obtaining physically relevant values ofV_OC,EQE. InFigure 4a, we plot as a function of VOCthe values of our metric,VOC,EQE, including light-trapping correction with and without BF correction. We see that the values are greater than the experimentalVOCvalues only after including the BF correction. The corresponding experimental EQEcurves are shown inFigure S7.

A comparison of VOC,EQE values with and without light- trapping correction can be found inFigure 4b for a-Si:H and c- Si. The BF correction was, this time, included in all cases. For a-Si:H, the calculated optical limit ofV_OC,EQEis larger than the experimentally measuredVOConly after correction to the light- trapping, assuming γ = 10.²⁸ The corresponding EQEcurves for different thicknesses are shown inFigure S2. Regarding c-Si technologies, the absorption coefficient is the same except perhaps for very slight differences in the Urbach energy, but the differences were negligible (Figure S4). Since the absorber thicknesses are similar (the absorber was assumed to be 250 μm in all cases), we would expect very similar theoretical values of the radiative limit for all c-Si technologies (refer also to discussion of Figure 1). Again, this is true only after correction to the light-trapping. From the lowest to the highest VOC, the data points in the graph represent multicrystalline Si (multi-Si), monocrystalline Al back surface field (mono-Al- BSF), monocrystalline interdigitated back contact (mono- IBC), and monocrystalline silicon heterojunction solar cell (SHJ). The low-efficiency technologies haveγ≈ 4, while the high-efficiency technologies exhibit γ ≈ 10. For the determination of these values, refer to Figure S5. Because here the experimental voltages have negligible effects on BF correction, we could, only in this case, use values representative for the state of the art of each technology rather than actual measured values. Light-trapping path enhancement in other thin-film technologies was assumed to be the same as for a- Si:H (γ ≈ 10) except for μc-Si:H, where γ ≈ 3.5 is also observed, as evaluated fromFigure S6.

To find possible demonstrations of the effect of spatially localized defects, we plot inFigure 4c theVOC,EQEforμc-Si:H, c-Si, and phase-segregated perovskite without the BF correction (the reason is that due to BF correction VOC,EQE

may reproduce a trend coming purely from the trend inVOC; refer to Supporting Information ∂V_OC,EQE/∂V_OC described Figure 3. Normalized EQE spectra obtained by FTPS after

applying the BF correction. Dashed lines represent the functions y∝exp(x/kT),kT= 25.85 meV that are tangential toEQE. Plotted are only the cases when the touching point is in the defect absorption region (indicated by a circle), showing the strong role of deep defects in theVOC,EQE.

Figure 4. a) Band-filling correction necessary to obtain physically meaningful values (VOC,EQE>VOC) for Cs0.17FA0.83Pb(I0.6Br0.4)3and a-Si:H.

b) Light-trapping correction necessary to obtain values (VOC,EQE>VOC) or values reflecting actual technological advancement for a-Si:H and c-Si, respectively. The dashed line represents the case whereVOC,EQE=VOC. c) Data points demonstrating that for c-Si,μc-Si, and differently phase-segregated mixed-halide perovskites, the trend betweenVOC,EQEandVOCis lost, likely due to spatial localization.

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later). We see thatV_OC,EQEwithout the BF correction shows no trend, meaning that EQE itself does not contain information on any difference inV_OCand the horizontal data point spread is due to nonradiative recombination centers that are invisible to EQE. In the case ofμc-Si these are cracks, in the case of c-Si these are surfaces, and in the case of mixed-halide perovskites these are spatially localized segregated regions with low band- gap. The corresponding experimental EQE curves are in presented inFigures S4, S6, and S8.

In parallel to the new metric, we can evaluate the recombination by electrical simulations. Using simple differ- ential equations, assuming nonradiative Shockley−Read−Hall recombination, it follows that theV_OCshould again scale with the logarithm of the total number of recombination centers.

More advanced simulations using the software AFORS-HET²⁹ were also done.³⁰The variation was only in the concentration of homogeneously distributed defects. InFigure 5, simulations

together with experimental data on a set of a-Si:H cells of different thicknesses and different levels of defects are shown as a function of product of defect density times thickness. To link the defect absorptance with the volume density we used a scaling constant 1 ppm·cm between density and absorption coefficient.³⁰Becausex-axis is the product of thickness times defect density, it makes the picture somewhat more complex.

Here, we wish to focus only on the effect of defect variations.

The simulated data points show first a plateau until a certain threshold, followed further by a trend−2kT/qln(x). The same trends are observed by the experiment as well as the results from the new metric. Interestingly, the factor 2 cannot be produced directly by the previous equations where only kT/

q ln(x) is applied. It is produced indirectly by the BF correction through the implicit dependence ofeq 4onVOC, as explained later.

The overall comparison between experimental VOC and V_OC,EQEfor all materials is shown inFigure 6. Note that some of the data points do not represent state-of-the art devices or material quality (CdTe, CIGS, and mixed-halide perovskites).

ExperimentalVOCvalues for CdTe (FTPS spectra of CdTe are in Figure S9) and CIGS are rather low, and mixed-halide perovskites do not exceed the best voltage of MAPI. On the

other hand, the device prepared from pure bromide represents the state of the art in terms of VOC; similarly, a-Si:H devices give rather highV_OC. The following loss analysis will reveal the limiting factors. We assume that all homogeneously distributed bulk defects are visible in theEQEspectrum, and as a result, the performance according to our metricVOC,EQE(green arrow in the inset of Figure 6) is determined only by the bulk absorber quality and cell optics. The experimental values of V_OC, conversely, are also limited by localized defects and losses due to structures (contacts) that are invisible inEQE(orange arrow in the inset of Figure 6). Reducing these losses should increase the experimental VOC, moving the data point horizontally to the right, but only up to the red line for whichVOC=VOC,EQE, the case when all the device deficiencies are reflected in the EQE spectrum. At this point localized defects and contacts are no longer the limiting factor, and the optimization should be focused on bulk defects and optics, to move the point both vertically and horizontally (in the inset of Figure 6, both the green arrows rise).

The previous reasoning was based on the fact that there were two independentparameters,V_OC,EQEandV_OC(the green arrows in the inset of Figure 6), while only one was additionally limited by localized defects and contacts. Now we also take into account the implicit dependence ofVOC,EQE

(andV_OCalso) onV_OCstemming from the BF correction (thin black arrows in the inset of Figure 6). The strength of this dependence can be assessed by considering two extreme ceases:VOC= 0 andVOC=qEG, whereEGis the band-gap. We can easily figure out that the corresponding value of V_OC,EQE fulfils 0 < VOC,EQE < qEG, and this already indicates that the value of ∂V_OC,EQE/∂V_OC should be in the range from 0 to 1.

This can be supported not only empirically (Figure S10) but also theoretically. We assume the shape of defect absorptance

asEQEdef∝e^E/E^U,def, where theEU,defis analogous to the Urbach

energy for the defect region. Note that the integral in the denominator of eq 4 is sensitive mainly to EQEdef if the criterion of the tangent lines is fulfilled (Figure 3). Then it can be mathematically derived (refer to the Supporting Informa- Figure 5. Experimentally obtained and calculatedVOCas a function

of the logarithm of the total amount of defects for different absorber thicknesses, for the case of a-Si:H. The full lines show logarithmic trends; the dotted line is a guide for eyes.

Figure 6. Data points demonstrating that the new metric focused on optical material evaluation gives values higher than electrically measuredVOC. The thick red line represents the case when device performance is limited by bulk optical properties,VOC,EQE=VOC.

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tion) that∂V_OC,EQE/∂V_OC≅1−kT/E_U,def, which falls into the range from 0 to 1. So, the implicit dependence can be considerable, but its strength at any time can be evaluated from the slope of the defect density tailEU,def. This also means that even for the device that is less bulk defects- and optics-limited, there still might be some coupled vertical and horizontal movement of the data points because the improvement of localized defects and contacts may lead to simultaneous improvement in V_OC,EQE. Moreover, this also has a con- sequence for the dependence of VOC,EQE on EQEdef which is investigated in Figure 5. It can be shown using the rules for derivatives of composite functions (the complete derivation is given in the Supporting Information) that

V kT

E kT

q EQE 2 EQE

OC EQE

U def

def def ,

,

i kjjjj jj

y {zzzz

zz ₍₈₎

where δEQE_def/EQEdefis the relative change of homogeneous defect absorptance in the material. This coupling finally explains the factor 2 in the dependence of VOC,EQE on the number of defects for a-Si:H (Figure 5).

To conclude, with the use of recent advancements in understanding the thermodynamic limits of solar cells, namely, the implementation of band-filling, we attempted to include sub-band-gap states to define a new metric,V_OC,EQE, that allows for more advanced and accurate loss analysis of solar cells. This metric assumes that under (non-equilibrium) conditions of illumination in the case of flat quasi-Fermi levels, the reciprocal mechanism to optical absorptance is any recombination and not only radiative recombination. Inputs for this analysis include high-sensitivity EQE measurements, such as those based on Fourier-transform photocurrent spectroscopy (FTPS), together with experimentalV_OCand the light-trapping path enhancement factor. The analysis determines how much the performance is limited by the detailed shape of theEQE spectrum, i.e., by homogeneously distributed bulk defects and optical management, and how much it is limited by other deficiencies such as spatially localized defects including unpassivated interfaces and contacts (causing quasi-Fermi level bending). Due to the band-filling effect, the two limiting factors might be coupled, depending on the slope of defect absorptance tail, giving an explanation of the observed factor 2kT in the defect density dependence obtained from simulations and experiments (Figure 5). For the loss analysis, the first is the criterion of the touching point of the exponential line exp(E/kT) tangential to theEQEcurve. If this point is in the region of defect absorptance, then the optimization of bulk defects and optics may improve performance. The second is the plot ofV_OC,EQEvsV_OCand the vicinity of data points to the red line inFigure 6. The closer to the red line, the more the bulk defects and optics are limiting factors and the higher is the chance that optimization of bulk defects and optics improves the observed V_OC. The last criterion is the data point shift during an optimization step, ultimately confirming the type of actual limiting factor.

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ASSOCIATED CONTENT

*^sı Supporting Information

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsenergylett.3c00951.

FTPS spectra of the cells, Φ_AM1.5G and Φ_bb,300K, additionalEQEspectra of most of the devices, graphical

analysis of light-trapping in c-Si solar cells, and analytical derivation of dVOC,EQE/dVOCand derivation of dVOC,EQE/ dEQE (PDF)

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AUTHOR INFORMATION Corresponding Author

Jakub Holovský−Faculty of Electrical Engineering, Centre for Advanced Photovoltaics, Czech Technical University in Prague, Prague 166 27, Czechia; Institute of Physics, Academy of Sciences of the Czech Republic, Prague 162 00, Czechia; orcid.org/0000-0002-4222-6070;

Email:[email protected] Authors

Katarína Ridzoňová−Institute of Physics, Academy of Sciences of the Czech Republic, Prague 162 00, Czechia;

Faculty of Mathematics and Physics, Charles University, Prague 121 16, Czechia

Amalraj Peter Amalathas−Faculty of Electrical Engineering, Centre for Advanced Photovoltaics, Czech Technical University in Prague, Prague 166 27, Czechia; Department of Physics, Faculty of Science, University of Jaffna, Jaffna 40000, Sri Lanka; orcid.org/0000-0002-9252-1975

Brianna Conrad−Faculty of Electrical Engineering, Centre for Advanced Photovoltaics, Czech Technical University in Prague, Prague 166 27, Czechia

Rupendra Kumar Sharma−Faculty of Electrical Engineering, Centre for Advanced Photovoltaics, Czech Technical University in Prague, Prague 166 27, Czechia Xin Yu Chin−Institute of Microengineering (IMT),

Photovoltaic and Thin-Film Electronics Laboratory, École Polytechnique Fédérale de Lausanne (EPFL), Neuchâtel 2002, Switzerland

Ebin Bastola −Wright Center for Photovoltaics Innovation and Commercialization, Department of Physics and Astronomy, The University of Toledo, Toledo, Ohio 43606, United States; orcid.org/0000-0002-4194-3385 Khagendra Bhandari− Wright Center for Photovoltaics

Innovation and Commercialization, Department of Physics and Astronomy, The University of Toledo, Toledo, Ohio 43606, United States; orcid.org/0000-0002-4386-6301 Randy J. Ellingson− Wright Center for Photovoltaics

Innovation and Commercialization, Department of Physics and Astronomy, The University of Toledo, Toledo, Ohio 43606, United States; orcid.org/0000-0001-9520-6586 Stefaan De Wolf− KAUST Solar Center (KSC), King

Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia; orcid.org/0000- 0003-1619-9061

Complete contact information is available at:

https://pubs.acs.org/10.1021/acsenergylett.3c00951

Author Contributions

J.H. provided the concept, all analytical calculations, and manuscript writing. J.H. and K.R. performed the high- sensitivity EQE measurements. A.P.A. and X.Y.C. prepared perovskite cells. B.C. and R.K.S. provided semiconductor simulations. E.B., K.B., and R.J.E. prepared CdTe cells. S.D.W.

provided crystalline Si cells and contributed to editing the manuscript.

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Notes

The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of Air Force Research Laboratory or the U.S. Government.

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

This work was supported by Czech Ministry of Education, Youth and Sports grant no. CZ. 02.1.01/0.0/0.0/15_003/

0000464 − “Centre of Advanced Photovoltaics” and Czech Science Foundation grant number no. 23-06285S. E.B., K.B., and R.J.E. acknowledge support from the Air Force Research Laboratory under agreement number FA9453-18-2-0037. We acknowledge here also a fruitful discussion with Thomas Kirchartz.

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