Light intensity dependence of the photocurrent in organic photovoltaic devices

present a combined theoretical and experimental study of intensity-dependent photocurrent (IPC), a tool for understanding solar and indoor device fundamentals, to identify different photovoltaic device performance-limiting photocurrent loss mechanisms based on their unique signatures in IPC. The competition between recombination and extraction of carriers deﬁnes the charge collection efﬁciency and, therefore, the overall performance of organic photovoltaic devices, including solar cells and photodetectors. In this work, we describe different compo-nents of the steady-state light intensity-dependent photocurrent (IPC) and charge collection efﬁciency under operational conditions. Further, we demonstrate how different loss mechanisms can be identiﬁed based on their unique signatures in the IPC. In particular, we show how IPC can be used to distinguish ﬁrst-order, trap-assis-ted recombination from other ﬁrst-order photocurrent loss mechanisms, which dominate at the low-intensity characteristic of indoor light-harvesting applications. The theoretical framework is presented and veriﬁed by a one-dimensional drift-diffusion device model. Finally, the extended IPC methodology is validated on organic thin-ﬁlm photovoltaic devices. We conclude that the relatively straightforward measurement of IPC over a large dynamic range can be a powerful tool for understanding solar and indoor device fundamentals. recombination. The results are very general and apply to a wide variety of photovoltaic systems, including indoor photovoltaics, concentrator solar cells, and photodetectors. between injected carriers near the


INTRODUCTION
Photovoltaic devices based on organic semiconductors, including solar cells, indoor photovoltaic cells, and photodetectors, hold great promise for sustainable energy and light-harvesting technologies. [1][2][3][4] However, these systems generally suffer from large non-geminate recombination of charge carriers, limiting the collection of photogenerated charge carriers and, ultimately, the performance of organic photovoltaic (OPV) devices and the dynamic range of photodetectors. To improve device performance and overcome this loss mechanism, it is vital to better understand the competition between recombination and extraction of photogenerated charge carriers. 5,6 The effect of non-geminate recombination is strongly dependent on the illumination light intensity. Considering that indoor light photovoltaic cells and photodetectors operate under vastly different light intensity regimes compared with outdoor solar cells, a comprehensive understanding of the intensity dependence of charge collection (over a very broad range of intensities) is needed to chart the full potential of OPV-based technologies.
In general, the light absorption in the active layer (or junction) of an OPV device results in formation of strongly bound electron-hole pairs, so-called excitons. [7][8][9] Separation of the excitons into free charge carriers is generally a multi-step process. To facilitate this process, an active layer based on a donor (D)-acceptor (A) blend, a bulk heterojunction (BHJ), is typically used. In this structure, excitons generated in D or A phase can diffuse to a D-A interface, undergo charge transfer (CT), and ultimately dissociate into free electrons and holes in the A and D phases, respectively. 7 The electrons and holes are then transported within these phases and extracted at the electrodes. The corresponding photocurrent density J ph at a given excitation wavelength l and light intensity I L is given by J ph = EQE 3 ql hc I L ; (Equation 1) where EQE is the photovoltaic external quantum efficiency, h the Planck constant, c the speed of light, and q the elementary charge. The EQE can be further summarized as EQE = h abs 3 h CGY 3 h col , where h abs is the absorptance, h CGY is the quantum efficiency for photogenerated excitons to be converted into free charge carriers, and h col is the charge collection efficiency.
The charge collection efficiency describes the probability that photogenerated charge carriers in the bulk can be extracted at the contacts. Apart from being extracted, on their way to the electrodes, free charge carriers may encounter each other and recombine via a non-geminate process. Because of the low charge carrier mobilities in organic semiconductors, non-geminate recombination is a particularly important loss mechanism in organic solar cells and photodetectors. 6,10 Apart from limiting the short-circuit current density (J SC ), recombination is also responsible for loss in the fill factor and open-circuit voltage (V OC ), ultimately limiting the power conversion efficiency as well. 6,7,11 In contrast to h abs and h CGY , h col generally depends on I L . This manifests as a J ph ðI L ÞfI a L dependence, where aðI L Þ is the slope of log J ph versus log I L and depends on the competition between charge extraction and the dominant charge carrier recombination kinetics within the prevailing intensity regime. [12][13][14][15] Importantly, a is not a constant and varies as a function of I L . Understanding the complex dependence of a on I L is one of the aims of the work reported here.
The fact that different recombination mechanisms scale differently with I L and carrier density has been utilized to understand recombination processes in solar cells by using I L -dependent V OC [16][17][18] and photocurrent (I L -dependent photocurrent [IPC]) measurements as well as transient photovoltage (TPV) and charge extraction (CE) techniques. 15,19 However, it has been pointed out that TPV and CE suffer from capacitive effects associated with non-uniform charge carrier distributions, limiting the range of reliable recombination order estimation. [20][21][22] Intensity-dependent V OC measurements, on the other hand, are very sensitive to parasitic leakage currents at low light intensities and surface recombination of charge carries at the electrodes at high light intensities. 16,23 Finally, IPC measurements offer the possibility to investigate the photocurrent losses as a function of the incident I L under different operational conditions. [24][25][26][27][28] However, because incomplete dissociation of excitons and intermediary CT states is first order, it has been extremely challenging to distinguish first-order, trap-assisted recombination from the former based on IPC measurements because both scale linearly with carrier density and I L .
Given this background, it is evident that methods to reliably quantify and distinguish different recombination mechanisms under operational conditions, especially those that are of first order, are currently lacking. In this work, we clarify the influence of different first-and higher-order photocurrent loss mechanisms on the device photocurrent and charge collection efficiency, supported by drift-diffusion simulations. We demonstrate how IPC, when performed sensitively over a broad range of light intensities, can be used to unambiguously differentiate trap-assisted recombination from other first-order losses. This sets IPC apart from other techniques commonly used to study recombination losses in organic solar cells and opens a new window to investigate the photocurrent loss induced by first-order, trap-assisted recombination. The results are very general and apply to a wide variety of photovoltaic systems, including indoor photovoltaics, concentrator solar cells, and photodetectors.

Theoretical background
We investigate the steady-state photocurrent of a typical OPV diode device based on a BHJ active layer sandwiched between a hole-extracting anode contact and an electron-extracting cathode contact. The analysis is restricted to photocurrents near short-circuit conditions, corresponding to applied voltages V close to zero, where the dark current is negligible. The BHJ active layer is treated as an effective semiconductor where the electron transport takes place in the A, and the hole transport occurs in the D. The anode and cathode contacts are assumed to be situated at x = 0 and x = d, respectively, where d is the thickness of the active layer.
Under these conditions, the current density is given by the sum of the electron and hole current densities in the active layer, J ph = J n ðxÞ + J p ðxÞ, where Here, the terms À vJ n =vx and vJ p =vx represent the CE rates for electrons and holes (note that À vJ n =vx = vJ p =vx).
From Equation 1, the charge collection efficiency can then be expressed as is the spatial average of the photogeneration rate of free charge carriers in the active layer. For perfect charge collection, h col = 1, we obtain J ph = qGd. In this limit, the photocurrent is only limited by first-order losses related to absorption and the charge carrier generation process. In real devices, however, there are likely always charge collection losses induced by recombination. In the general case, h col depends on the charge carrier generation rate, reflected by an intensity dependence of the EQE; this may be expressed as h col fI S L ; where S = a À 1. Only when the photocurrent density is linear with I L (a = 1) will the corresponding h col and, hence, EQE be independent of I L .
To clarify the effect of recombination on the photocurrent and the charge collection efficiency, we use a one-dimensional drift-diffusion model. 23,29 The device model numerically calculates the current density based on the prevailing charge transport properties, space charge effects, and recombination within the effective semiconductor layer. For the simulations, a device having an active layer thickness of 100 nm with a relative permittivity ε of 3.5 and an electrical D-A energy level gap E g of 1.3 eV, is considered. We assume selective contacts, J n ð0Þ = J p ðdÞ = 0, with injection barriers of 0.1 eV for majority carriers. Finally, the charge carrier photogeneration rate is assumed to be uniform throughout the active layer, with an equivalent I L of 1 sun corresponding to G = 1:1310 22 cm À3 s À1 .

The competition between CE and bimolecular recombination
We first consider the case of balanced current transport with different degrees of bimolecular recombination in the active layer. This process follows second-order recombination kinetics having a charge carrier recombination rate given by where b is the bimolecular recombination coefficient. The recombination coefficient is typically expressed relative to the Langevin recombination constant (b L ) as b = zb L , where z is the reduction factor relative to the Langevin rate, and b L hqðm n + m p Þ= εε 0 , where ε 0 is the vacuum permittivity. Figure 1A shows the simulated (solid lines) J SC as a function of I L for different recombination constants b = zb L . The corresponding h col versus intensity is depicted in Figure 1B. The dashed black lines in Figures 1A and 1B represent the limiting cases for perfect charge collection (h col = 1), with slope parameters of a = 1 and S = 0, respectively. As shown in Figure 1A ( Figure 1B), the photocurrent (charge collection efficiency) is linear (constant) at low intensities, whereas at higher intensities, J ph (h col ) starts to deviate from linearity and instead follows a power of 1/2 (À1/2) dependence. The onset of the deviation depends on b and shifts to higher intensity with decreasing z.
At high enough photogeneration rates, the bimolecular recombination rate starts dominating over the CE rate in Equation 3. Under these conditions, and that m n n = m p p; hence, n = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m p G=m n b q . Accordingly, the photocurrent J ph = 2J n is obtained as with m eff = 2 ffiffiffiffiffiffiffiffiffiffi ffi m n m p p . Equation 7 represents an upper limit for the photocurrent set by second-order recombination. The associated charge collection efficiency in this limit approaches h col = L eff =d, where is an effective charge collection distance that depends on the generation rate via the effective bimolecular recombination lifetime t b = 1= ffiffiffiffiffiffi ffi bG p .
As evident from Figure 1, the analytical limits (dotted lines) show good agreement with the numerical drift-diffusion simulations (solid lines) at high photogeneration rates when bimolecular recombination dominates, corresponding to L eff = d ( 1. This also explains the observed slope parameters of a = 1=2 and S = À 1= 2, as seen at high light intensities in Figures 1A and 1B, respectively, in accordance with Equation 7.
At low light intensities, when L eff =d [ 1, second-order recombination is negligible, and we expect J SC fG. Under these conditions, the collection efficiency is constant and close to unity because most of the photogenerated carriers are extracted. The corresponding charge collection loss, resulting in a fixed h col < 1, can be attributed to effective first-order recombination between injected dark background carriers near the contacts and photogenerated carriers. 30,31 Another source of first-order recombination loss that is present in the case of non-selective contacts is surface recombination; i.e., the extraction of the wrong carrier type at the contacts. [32][33][34][35][36] These first-order losses are generally small at short circuit but may become important for thin active layers and at high injection/voltage levels.
The parameter L eff =d determines the onset of the transition from the linear (a = 1) to the nonlinear (a = 1=2) photocurrent versus intensity regime, where second-order recombination dominates. The associated onset charge carrier generation rate, corresponding to the generation rate at which L eff =d = 1, can be approxi- m n m p ½m n + m p z under short-circuit conditions. Hence, the onset of second-order recombination predominantly depends on the ratio between the smaller charge carrier mobility and the (Langevin) reduction factor z. 37 In other words, to avoid significant second-order recombination losses of the photocurrent, the mobilities and z must be such that L eff [ d for a given operating I L (G).
The effect of an external series resistance Apart from second-order recombination within the device, the photocurrent at high light intensities will eventually also be limited by the series resistance of the external circuit. For an externally applied voltage V , the actual applied voltage drop across the device is given by V dev = V À JR S , where JR S represents the resistive voltage loss across the rest of the circuit (e.g., electrodes, external wires, etc.) having a total specific series resistance R S in units of Ucm 2 . After accounting for this resistive voltage loss, the high-intensity photocurrent limit in the case of bimolecular recombination (Equation 7) is modified as Although the effect of the R S is negligible at small intensities, it becomes prevalent at high intensities because the resistive voltage loss scales with the current.
In the limit of high intensities (large currents), the current density approaches Equation 10 gives the ultimate upper limit of the photocurrent density as set by the external R S . This result is independent of the predominant recombination mechanism within the device. In the absence of an external applied voltage (corresponding to a short-circuit condition), the maximum voltage drop across a device under illumi- The effect of external series resistive losses is depicted in Figures 2A and 2B, where the I L -dependent J SC and h col , respectively, are simulated for different R S . Ideally, the V OC scales as V OC f À kT=q Á lnðI L Þ with the I L when bimolecular recombination dominates (kT=q is the thermal voltage). However, in reality, V OC often saturates at high intensity. Ultimately, the short-circuit current tails off under such conditions (i.e., a/0), with h col approaching an inverse I L dependence, manifested by a slope parameter of S = À 1. The dotted lines in Figure 2B representing the theoretical limits based on Equation 10 are in good agreement with the simulations.
The influence of space charge effects The above considerations assume nearly balanced electron and hole transport. However, this assumption generally breaks down for conditions when space charge effects dominate inside the active layer. Such conditions may occur when a significant amount of space charge is present in the active layer; e.g., caused by doping or imbalanced mobilities. 12 , and the thin, black solid line is a guide to the eye with a slope of S = À 1 corresponding to theory explained above.

OPEN ACCESS
an electric field redistribution inside the active layer, resulting in formation of a space charge region (where the electric field is concentrated) and a neutral region (where the electric field is small). Subsequently, the charge collection is efficient only within the space charge region (of width w sc ), whereas the collection from the neutral region is limited to a diffusion length L s from this region. 43 For w sc + L s < d, the photocurrent density can then be approximated as The corresponding charge collection efficiency is given by h col = ðw sc + L s Þ=d.
In case of doping-induced space charge, the space charge region is composed of ionized dopants. Under these conditions, w sc and h col are generally expected to be independent of I L , suggesting a = 1. A slight enhancement in h col is expected for low-mobility materials at higher intensities, when the density of photogenerated charge carriers become comparable with the dopant density. 43 The effect of unintentional doping is mainly relevant for thicker active layers (when d > w sc ) but typically negligible in OPVs with thin active layers. 44 In the case of imbalanced mobilities, the space charge region is induced by a buildup of photogenerated carriers with the lower mobility at high enough light intensities. 12,45 Specifically, in the case of space charge induced by imbalanced mobilities, it can be shown that w sc = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where m s is the mobility of the slower charge carrier. 39 Here, N sc is the mobilityinduced space charge density (inside the space charge region) given by N sc z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi εε 0 G=ðqm s Þ p , and V 0 is the built-in potential across the space charge region. Hence, for high enough intensities (and low m s ) when w sc ( d, the photocurrent is nonlinear and takes the form J ph fG 3=4 , following an a = 3=4 power dependence of G. If the charge collection from the neutral region is negligible, then the charge collection efficiency simplifies to suggesting an S = À 1=4 power dependence between h col and the I L . Figures 3A and 3B show the simulated J SC and the charge collection efficiency plotted as a function of I L for the case of an increasing mobility imbalance. The mobility ratio varies from balanced mobilities (m p = m n = 1) to highly imbalanced mobilities of m p = m n = 10 À 3 , assuming z = b=b L = 0:1 to be fixed in all cases. As shown in Figures 3A and 3B, the onset of the nonlinear regime where a = 3= 4 depends on the degree of imbalanced mobility (or, rather, the mobility of the slower charge carrier) and shifts to lower intensity the more imbalanced the charge carriers. Dotted lines in Figure 3B are estimated limits in accordance with the analytical expression in Equation 12 for the corresponding mobility ratios. The mobilityinduced space charge effect (Equation 12) is competing against the bimolecular recombination-dominated regime (Equation 8) at high intensities because of the strong bimolecular recombination (b=b L = 0:1). At smaller b=b L , however, the space charge effect becomes more pronounced ( Figure S1).

The effect of trap-assisted recombination
In the case where a significant amount of trap states is present in the bulk of the D-A blend, the recombination of free charge carriers via those sub-gap states becomes dominant at low intensities. This type of non-geminate recombination is referred to as trap-assisted recombination, which is commonly described in terms of ll Shockley-Read-Hall (SRH) statistics: a free electron (or hole) is first captured by a trap; the trapped electron (hole) can then recombine with a free hole (or electron). In the case of trap states within the D-A gap, situated at a trap depth D t below the electron transport level, the corresponding trap-assisted recombination rate of free electrons and holes is given by where t n and t p are the SRH lifetimes for electrons and holes, respectively, and Here, N c and N v denote the effective densities of transport states for electrons in the A and holes in the D, respectively; k is the Boltzmann constant; and T is the temperature.
We first consider the case of mid-gap trap states. In this case, n 1 and p 1 are small. In the low-intensity limit, the carrier densities near the anode and cathode are dominated by dark background holes and electrons, respectively. 35 Thus, we expect p [ n and, thus, R = n=t n , for photogenerated electrons within the hole-dominated (anode-side) region of the active layer. Then, assuming the driving force for CE of photogenerated carriers to be drift-dominated vE Fn =vxzq½V À V bi =d, where V bi is the builtin voltage, Equations 2 and 3 can be readily solved for J n ðxÞ. 46 Concomitantly, we find J n ðxÞ = ÀqGL n ½1 À expð À x =L n Þ, where L n = m n t n jV bi À V j=d is the electron drift length in the hole-dominated region. A similar treatment can be applied for photogenerated holes within the electron-dominated (cathode-side) region of the active layer, where n [ p and R = p=t p ; hence, the hole current density is obtained as J p ðxÞ = ÀqGL p ½1 À expð½x À d =L p Þ, with L p = m p t p jV bi À V j=d being the hole drift length in the electron-dominated region.
Finally, demanding that nðd Ã Þ=t n = pðd Ã Þ=t p , where x = d Ã is the point separating the hole-and electron-dominated regions, an approximation for the photocurrent, J ph = J n ðd Ã Þ + J p ðd Ã Þ, can be obtained. 46 Subsequently, when the recombination of charge carriers is dominated by SRH recombination via mid-gap trap states, the corresponding charge collection efficiency at low light intensities is given by in accordance with the Hecht equation. 47 Noting that h col is independent of the I L (i.e., S = 0), the photocurrent density is expected to be linear with intensity I L in this regime; i.e., a = 1. Figures 4A-4D show the simulated J SC and charge collection efficiency as a function of I L for the case of A-type trap states at varying trap depths D t and SRH lifetimes, assuming t p = t n . The photocurrent is indeed found to follow a linear intensity dependence, J SC fG (a = 1), at low intensities, seen as a constant h col plateau. In the case of mid-gap traps, h col is well approximated by Equation 15 at the lowest intensities (indicated by the star symbol in Figure 4C). However, in the case of shallower trap states, an increased h col is obtained at low intensities (Figure 4), resulting in deviation from Equation 15. This deviation can be attributed to n 1 no longer being negligible in the active layer at low intensities.
The effect of a decreasing D t (i.e., increasing n 1 ), can be understood by considering photogenerated electrons in the hole-dominated region of the active layer, noting

OPEN ACCESS
that the dark hole density is of the form pðxÞzN expðq½V À V bi x =½kTdÞ, 15 where N is an effective hole density at the anode contact. In the low-intensity limit, the SRH recombination rate via non-mid-gap traps is first order (R = n=t n ) as long as p [ n 1 . However, because of the exponential decay of pðxÞ, at larger distances from the anode, eventually p ( n 1 . At these distances, Equation 13 instead simplifies to R = np=ðt n n 1 Þ, becoming effectively second order, and is thus expected to be negligibly small at low intensities. Assuming t p zt n and pðx 0 Þ = n 1 , the width x 0 of the hole-dominated region, where first-order SRH recombination prevails, can be obtained as Here, b = kT lnðN c =NÞ represents the energy level bending from accumulated dark holes at the anode contact (assuming N v = N c ). An analogous situation is expected for the electron-dominated region at the cathode.
For non-mid-gap traps, at low intensities, Equation 13 may subsequently be approximated as Rzn=t n for 0 < x < x 0 (hole-dominated region) and Rzp=t p for d À x 0 < x < d (electron-dominated region), whereas Rz0 otherwise (recombination-free zone). Hence, the charge collection efficiency in the low-intensity limit of a device dominated by SRH recombination via non-mid-gap traps can be approximated as ) for x 0 < d=2. Accordingly, the effect of decreasing the trap depth is to reduce the width x 0 of the hole and electron-dominated regions, resulting in a wider recombination-free zone within the active layer and, thus, an enhanced h col . Indeed, a good agreement between Equation 17 and the simulated h col is seen for non-midgap traps at low intensities (indicated by circle-shaped symbols in Figures 4C and  4D).
At higher intensities, however, h col ( Figure 4) is seen to undergo a transition into a moderate-intensity regime with h col saturating to an approximately constant value (az 1) that is independent of the trap depth. For mid-gap traps, this transition results in a slightly increased h col , whereas for shallower traps, h col is generally decreased relative to the low-intensity limits. This transition can be attributed to the increased density of photogenerated carriers eventually starting to dominate over dark injected carriers inside the active layer. As the photogenerated electron and hole densities become comparable with each other inside the active layer in this case (so that n $ p), Equation 13 instead takes the form R $ n=ðt p + t n Þ. This effectively results in an increased average lifetime for photogenerated charge carriers inside the active layer and, hence, an increased h col . 26,48 For shallower, non-mid-gap traps, the transition to the moderate intensity regime occurs when the photogenerated carrier density in the active layer exceeds n 1 . Under these conditions, a light-induced trap-filling occurs in the bulk (traps start to behave as deep traps), and the trap-assisted recombination in the active layer is no longer negligible, becoming first order (R $ n). This will ultimately result in a reduced charge collection efficiency, appearing as a second h col plateau in the EQE with a lower magnitude (compared with the first one) at moderate intensities. The intensity at which the transition between the low-intensity and the moderate-intensity regime occurs depends on the trap depth, as seen in Figure 4. This unique feature allows SRH ll OPEN ACCESS recombination to be distinguished from other first-order losses. It should be noted, however, that SRH recombination between photogenerated charge carriers (at moderate intensities) is also strongly influenced by space charge effects induced by trapped charges. 25 Finally, we stress that this analysis is limited to SRH recombination via traps within the gap, far away from the conduction and valence level edges. In case of exponentially distributed tail states below the edges, 49,50 however, a different scenario is generally expected. For conditions when trap-assisted recombination via such tail states is the dominant recombination mechanism, it has been suggested that the slope of the photocurrent decreases with I L , being strongly dependent on the characteristic energy E U of the trap distribution. 27 However, if space charge from the trapped charge carriers is considerable, 51,52 then we expect the charge collection to be limited by a trap-induced space charge region. This effect is also expected to become more prominent at higher intensities and thicker active layers. 27,51 Under such conditions, the charge collection efficiency is of the form (Note S1; Figure S2) for exponential tail states with l > 1, where l = EU kT . For exponentially distributed tail states, we thus expect S to vary between S = À 1=4 and S = 0 in this case, depending on E U . The corresponding intensity dependence of the photocurrent is described by a = 1 À 1=ð2 + 2lÞ. This dependence has also been suggested previously by Hartnagel and Kirchartz. 27 Other recombination orders and effects The above considerations may be qualitatively extended to the case with a general, but fixed, charge carrier recombination order d for photogenerated charge carriers. For pzn, the corresponding recombination rate is of the form with the associated recombination coefficient given by B. Provided that d > 1, and assuming balanced transport (m n = m p ) and recombination with injected dark carriers to be negligible, the recombination rate term in Equation 3 will be negligible at small intensities (small n) and J ph = À qGd (i.e., all carriers are extracted). On the other hand, at high intensities, when the recombination term dominates, we have RzG. In accordance with Equation 3, J n in this limit is again expected to be uniform in the active layer. Subsequently, from Equation 19, we find n = ðG=BÞ 1=d and, thus, J ph fG 1=d , corresponding to slopes of a = 1=d and S = ð1 À dÞ=d. For example, for the case of third-order recombination kinetics (d = 3), expected in conjunction with Auger recombination, we obtain a = 1=3 and S = À 2= 3.
In these analyses we have made several simplifications and assumptions. Most notably, we have assumed effects from the absorption profile and optical interference to be negligible. Such effects generally give rise to non-uniform charge generation profiles within the active layer. 53,54 In thick junctions in particular, where the majority of charge carriers are generated at the transparent contact, space charge effects may become significant. 55 For thin active layers (which is the topic of this work), however, the influence on the photocurrent is expected to remain negligible. At very high light intensities, thermal effects may also start to play a role. These effects are generally associated with a turnover of the V OC with increasing intensity 56 but could also influence the photocurrent. We have also assumed a linear dependence between the carrier generation rate and the I L (  57 Under normal operating intensity conditions, however, these effects are generally negligible in BHJ organic solar cells.

Experimental demonstration
To validate and exemplify the theoretical framework, we next conducted experimental intensity dependent photocurrent (IPC) measurements on OPV cells. The steady-state IPC measurements were performed at an excitation wavelength of 520 nm, and no bias voltage was applied to the devices (short-circuit conditions). From the raw IPC data, we calculated the corresponding EQE versus I L according to indole-2,10-diyl)bis(methanylylidene))bis(5,6-dif luoro-3-oxo-2,3-dihydro-1H-indene-2,1-diylidene))dimalononitrile (Y6) device was selected as a high-efficiency, state-of-the-art, non-fullerene acceptor organic solar cell; the corresponding short-circuit current densities and normalized EQEs are shown in Figures 5C-5F, respectively. Guided by the theoretical considerations above, different photocurrent loss mechanisms can be identified and will be briefly discussed in what follows.
At low incident light intensities, below equivalent light intensities of I L z 10 À 4 sun, the PM6:ITIC, PBDB-T:EH-IDTBR, and PM6:Y6 devices exhibit a linear photocurrent regime (i.e., a = 1, indicated by black solid lines in Figures 5A, 5C, and 5E) that directly translates into constant EQE plateaus ( Figures 5B, 5D, and 5F). At moderate intensities, within the range 10 À 3 < I L < 10 À 1 sun, a second linear photocurrent regime can be observed in all three devices. These photocurrent regimes translate into slightly down-shifted, second EQE plateaus indicating the presence of charge carrier recombination via non-mid-gap trap states in the D-A active layer. The corresponding relative loss in the EQE induced by trap-filling in the D-A bulk at moderate intensities amounts to 3%, 5%, and 10% in the PM6:Y6, PBDB-T:EH-IDTBR, and PM6:ITIC devices, respectively (indicated by the gray shaded areas in Figure 5).
At intensities above 1 sun, the photocurrent of the PM6:ITIC device eventually starts to deviate from linearity exhibiting a photocurrent (EQE) versus intensity power dependence of a = 1=2 (S = À 1=2), consistent with bimolecular recombination between free charge carriers dominating over CE in this regime. The corresponding limits set by bimolecular recombination, as expected in accordance with Equation 7, are indicated by the red solid lines in Figures 5A and 5B. The Similarly, the photocurrent of the PM6:Y6 device was also found to eventually deviate from linearity and seemingly approach a slope of S = À 1=4 at the highest intensities. This suggests that a build-up of a mobility-induced space charge is dominating at the highest intensities and that b=b L is small (cf. Figure S1), consistent with the observed mobility imbalance of six and the reduced bimolecular recombination in PM6:Y6. 58 Because of the relatively small mobility imbalance, combined with the reduced bimolecular recombination in this system, mobility-induced space charge effects are expected to remain negligible under 1 sun conditions. 39 Finally, based on the magnitudes of the EQE plateau at low intensity, referred to as EQE 1 , the EQE plateau at moderate intensity, EQE 2 , and the transition in between them, one can estimate the trap depth D t for the non-mid-gap trap states, as we have reported in a previous publication. 25 The corresponding D t was estimated to be 0.4, 0.45, and 0.44 eV for the PBDB-T:EH-IDTBR, PM6:ITIC, and PM6:Y6 devices, respectively. Based on these trap depths, we estimate x 0 =d of 0.2, 0.25, and 0.24 for the PBDB-T:EH-IDTBR, PM6:ITIC, and PM6:Y6 devices, assuming V bi = 1 V and b = 0:2 eV. To get a rough estimate of the SRH lifetimes, we assume EQE 1 (EQE 2 ) to be given by Equation 17 (Equation 15), allowing for a relation between the ratio EQE 1 = EQE 2 and L n =d to be obtained. Here, we assume L n zL p for PM6:ITIC and L n [ L p for PBDB-T:EH-IDTBR and PM6:Y6, yielding L n =dz3:6, L n =dz1:4 and L n = dz 1:4, respectively. Subsequently, with L n =d = m n tV bi =d 2 , we find an SRH lifetime of tz 2:2 ms for PM6:ITIC, tz0:6 ms for PM6:Y6, and tz0:3 ms for PBDB-T:EH-IDTBR.
We expect the influence of EEA and R S limitations to be negligible in the intensity range chosen to probe the IPC of the three devices. EEA is generally significant at exciton densities above 10 17 cm À3 , whereas for typical exciton lifetimes around 0.5 ns, exciton densities of around 10 13 cm À3 are expected at generation rates equivalent to 1 sun. 57 Hence, EEA is expected to be negligible below intensities corresponding to 1,000 suns. Limitations from the R S , on the other hand, are negligible as long as the photocurrent is smaller than Equation 10. For our devices, we expect the external R S to be smaller than 25 U. With a pixel area of 0.04 cm 2 of our devices, we then expect resistive effects to be negligible for J SC < 1:0 A/cm 2 .

DISCUSSION
Based on the above considerations, it is evident that the intensity dependence of the photocurrent and charge collection efficiency can be used as a probe to distinguish between different dominant recombination mechanisms. It is important to note that a simple deviation from linearity (i.e., a < 1), does not automatically imply the presence of second-order recombination because nonlinear photocurrent behavior is also obtained for the case of space-charge build-up as a result of imbalanced mobilities or external R S limitations independent of the underlying dominant recombination mechanism. Table 1 summarizes the expected signatures and expected J SC and EQE power dependences in steady-state IPC measurements for different recombination and photocurrent loss mechanisms. For comparison, the corresponding EQE versus photocurrent power dependence has also been included because this dependence has been frequently reported in the literature. 38,59 Finally, it is important to emphasize that, because the associated photocurrent loss mechanisms vary with intensity, and transition regimes occur in between them, it is important to measure the J SC or, respectively, h col (respectively, EQE) over a broad range of light intensities. Failing to do so may lead to misjudgment of the predominant photocurrent loss mechanisms and the corresponding recombination orders.
To conclude, we have presented a combined theoretical and experimental study aimed at delivering an improved understanding of photocurrent and charge collection efficiency losses and their origins in OPV devices, including solar cells and photodetectors. Our results highlight that different first-and higher-order recombination loss mechanisms show unique intensity dependencies that can be used in IPC measurements when performed sensitively over a broad range of light intensities to unambiguously identify them. In particular, we have shown how IPC, compared with other commonly used measurement techniques, can be used to distinguish first-order, trap-assisted recombination from other first-order photocurrent loss mechanisms.
The results we present here are, in general, not only important for determining predominant recombination loss mechanisms needed to optimize device fabrication and ultimately enhance the efficiency of organic solar cells around 1-sun operational conditions but are also relevant for understanding other light-harvesting devices, such as photodetectors (usually working at low light intensities), indoor photovoltaic applications (working regime of $ 1=100 sun intensities) and concentrator solar cells (performing under multiple sun intensities 60 ).

EXPERIMENTAL PROCEDURES
Resource availability Lead contact Further information and requests for resources and materials should be directed to, and will be fulfilled by, the lead contact, OJS (o.j.sandberg@swansea.ac.uk).

Materials availability
This study did not generate new unique materials.

Data and code availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request. Steady-state intensity-dependent photocurrent measurements A commercial laser with variable output power was used as a light source and modulated at a low frequency (<0.1 Hz) using a function generator (Keysight Technology, 33500B series). The light perturbation allowed precise and noise-reduced measurements of the external photocurrent. Here, the choice of modulation frequency f was limited by the technical aspects of the IPC setup itself and the physical processes governing the device under test (DUT) (e.g., charge carrier transport, [de]trapping time, and RC time of the circuit). A motorized two-wheel attenuator (Standa, 10MCWA168) containing different optical density (OD) filters was used to vary the input light intensity. The combination of variable laser output power and two-wheel attenuator with a low-frequency photocurrent modulation allowed sensitive measurements of the photocurrent over I L of multiple orders of magnitude. A silicon (Si) reference photodiode (Thorlabs, SM05PD1A) was used for light power calibration and in situ intensity tracking. Two source-measure units (Keithley 2450) were used to simultaneously read the DUT photocurrent and Si photodiode current, while different bias voltages could be applied on the DUT. For the initial calibration process, an additional NIST-calibrated silicon photodiode power sensor (Thorlabs, S121C) was used.

Device fabrication
The PBDB-T:EH-IDTBR (PM6:ITIC and PM6:Y6) solar cell was fabricated with inverted (conventional) device architecture: glass/ITO/ZnO/active layer/MoO 3 /Ag (glass/ ITO/poly(3,4-ethylenedioxythiophene) polystyrene sulfonate (PEDOT:PSS)/active layer/2,9-bis[3-(dimethyloxidoamino)propyl]anthra[2,1,9-def:6,5,10-d 0 e 0 f 0 ]diisoquinoline-1,3,8,10(2H,9H)-tetrone (PDINO)/Ag). Detergent solution was used to clean the commercial ITO patterned glass electrodes for 10 min, followed by a series of ultra-sonication in distilled water, acetone, and 2-propanol, each for 10 min. Prior to oxygen plasma treatment, a nitrogen stream was used to dry the substrates. The ZnO electron transport layer was prepared by dissolving 200 mg zinc acetate dihydrate (Sigma-Aldrich) in 2 mL 2-methoxyethanol (Sigma-Aldrich) using 56 mL ethanolamine (Sigma-Aldrich) as a stabilizer. After stirring the solution overnight under ambient conditions, the ZnO solution was spin coated for 30 s at 4,000 rpm onto the ITO substrates and subsequently annealed for 1 h at 200 C to obtain a ZnO layer thickness of roughly 30 nm. PEDOT:PSS solution was first diluted with the same volume of water and then cast at 4,000 rpm on ITO substrates, followed by thermal annealing at 155 C for 15 min to form a 10-nm film. PBDB-T was purchased from Zhiyan (Nanjing, China), and EH-IDTBR was purchased from Solarmer (Beijing, China). A total concentration of 16 mg/mL in CB:DIO (99:1 [v/v]) with a D:A ratio of 1:1 was used to prepare the PBDB-T:EH-IDTBR solution. The solution was spin coated at 900 rpm and thermally annealed at 100 C for 10 min to form a film with a thickness of around 110 nm. PM6 was purchased from Solarmer (Beijing, China), and ITIC was purchased from Zhi-yan (Nanjing, China). A total concentration of 14 mg/mL in CF:DIO (99:1 [v/v]) with a D:A ratio of 1:1 was used to prepare the PM6:ITIC solution.
The solution was spin coated at 3,000 rpm and thermally annealed at 100 C for 10 min to form a film with a thickness of around 110 nm. Y6 was purchased from Solarmer (Beijing, China). Here, a total concentration of 16 mg/mL in CF:1-chloronaphthalene (99.5:0.5 [v/v]) with a D:A ratio of 1:1.2 was used to prepare the PM6:Y6 solution. The final solution was spin coated at 3,000 rpm and thermally annealed for 10 min at 100 C to form an active layer film with a thickness of around 100 nm. After spin coating of the PBDB-T:EH-IDTBR, PM6:ITIC, and PM6:Y6 active layer, 7 nm of MoO 3 was evaporated on PBDB-T:EH-IDTBR to form the hole transport layer, while 10 nm of PDINO was spin coated from solution (1.5 mg/mL PDINO at 2,000 rpm) on PM6:ITIC and PM6:Y6 to form the electron transport layer. Finally, 100 nm of Ag was ll OPEN ACCESS evaporated as the top electrode on all devices with a pixel area of 0.04 cm 2 . The base pressure in the vacuum chamber was less than 10 À6 mbar. The substrates were then finalized by sealing with a coverglass using UV light-annealed glue from Bluefix. All film thicknesses were measured with a Dektak 3ST profilometer.