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A Theoretical Perspective on Transient Photovoltage and Charge Extraction Techniques

Transient photovoltage (TPV) is a technique frequently used to determine charge carrier lifetimes in thin-film solar cells such as organic, dye sensitized and perovskite solar cells. As this lifetime is often incident light intensity dependent, its relevance to understanding the intrinsic properties of a photoactive material system as a material or device figure of merit has been questioned. To extract complete information on recombination dynamics, the TPV measurements are often performed in conjunction with charge extraction (CE) measurements, employed to determine the photo-generated charge carrier density and thereby the recombination rate constant and its order. In this communication, the underlying theory of TPV and CE is reviewed and expanded. Our theoretical findings are further solidified by numerical simulations and experiments on organic solar cells. We identify regimes of the open-circuit voltage within which accurate lifetimes and carrier densities can be determined with TPV and CE experiments. A wide range of steady-state light intensities is required in performing these experiments in order to identify their 'working dynamic range' from which the recombination kinetics in thin-film solar cells can be determined.


Introduction
Thin-film devices based on organic and perovskite semiconductors hold the potential for the use within sustainable energy production [1][2][3][4] as well as other optoelectronic applications [5][6][7][8][9] such as sensors, photodetectors, and light-emitting diodes. In order to further improve and optimize the device performance over both short and long timescales, a better understanding of loss mechanisms such as recombination of the charge carriers is necessary. [10][11][12] For example, various different techniques, both electrical and optical, have been used to investigate the charge carrier recombination in thin-film solar cells. One of the more popular opto-electrical methods used to probe the recombination dynamics in this regard is the small-perturbation transient photovoltage technique (TPV), often measured in combination with charge extraction (CE) or differential charging/transient photocurrent. [13][14][15][16] In TPV, the device under test is initially held at open-circuit under steady-state illumination. 13 To probe the recombination lifetime of the excess charge carriers, i.e. the carriers introduced into the device as a consequence of the steady-state light bias, an additional (weak) short light pulse is applied to the device and the corresponding induced voltage transient, developed over a large load resistance, is measured. The decay of this small photovoltage is then recorded from which the TPV lifetime is extracted. Ideally, this lifetime is associated with the steady-state recombination rate of charge carriers within the bulk. The small light perturbation increases the excess carrier density in the device as = oc + , giving rise to a subsequent perturbation in the recombination rate ℛ = ℛ 0 + ℛ; here, oc is the photo-generated steadystate (dc) carrier density and ℛ 0 is the associated (time-independent) recombination rate, which is balanced by an equal steady-state photo-generation rate 0 of charge carriers at open-circuit.
The decay of the voltage transient, induced as a result of the small perturbation, is approximately given by 13 leading to a first-order kinetics of Δ ( ) ∝ exp(− ⁄ ), where is the associated pseudofirst-order bulk recombination lifetime.
In general, the open-circuit voltage is related to the steady-state light intensity or photogeneration rate via oc ∝ ( id ⁄ ) ln( 0 ), where id is the diode ideality factor and ⁄ is the thermal voltage. [17][18][19] Since the open circuit voltage is logarithmically dependent on the steady-state (dc) photo-generated carrier density, the recombination lifetime from TPV measurements is parametrically dependent on the oc , taking the form 13,19 = 0 exp (− ) (2) where is the slope parameter that ideally depends only on the order of the steady-state recombination dynamics within the bulk, while 0 is a voltage-independent prefactor. For example, in the case of second-order bulk recombination ℛ = 2 , the carrier recombination dynamics after the perturbation pulse is given by For δ ≪ oc the bulk recombination lifetime is thus where is the important (and unknown) associated slope parameter, and ,0 is a prefactor. 19 Measuring CE in conjunction with TPV allows for the recombination rate ∼ CE ⁄ to be mapped as a function of carrier density, providing the sought-after information concerning the dominating recombination mechanism.
In accordance with Eq. (1), one expects the lifetime extracted from the TPV experiment to reflect the recombination of photo-induced carriers. However, this interpretation has been questioned by Street who suggested that the extracted TPV lifetimes in organic solar cells is set by the device capacitance and given by 20,21 where dark is the dark steady-state current density and is the capacitance of the active layer.
This "lifetime" can therefore be interpreted as an internal discharging time of the diode itself rather than reflecting the actual recombination dynamics of photo-induced carriers. Similar where 0 is the dark saturation current density of the device. The capacitive contribution, corresponding to the second term on the right-hand side of Eq. (7) and dominating at lower voltages, is equivalent to the internal diode discharging lifetime suggested by Street.
In the same work, Kiermasch and co-workers also found the charge density (Eq. (5)), as extracted from CE, to be strongly influenced by capacitive effects, in particular at low voltages, leading to unrealistically large values for the slope parameter at these intensities. 24 As a consequence, using the charge carrier density obtained from CE in combination with the lifetimes extracted from TPV to probe the underlying recombination reaction order might often result in recombination orders > 2, which do not necessarily reflect the recombination of steady-state photo-induced carriers within the bulk. These type of higher recombination orders have usually been explained in terms of trap-assisted recombination via localized tail states, both in organic [25][26][27] and perovskite 28 solar cells. However, it has also been pointed out that higher recombination orders might be artefacts caused by non-uniform carrier profiles in thin devices. 19,29 Given this background, and the recent insights of Kiermasch

Theory
A schematic picture of the electrical set up used for transient measurements such as TPV and CE is shown in Figure 1. In general, for a planar current flow across the (non-ideal) diode or solar cell device, the transient current density is given by 30,31 is the average conduction current density in the active layer and the second term on the right-hand side is the corresponding average displacement current density, with ( ) being the voltage across the active layer: where ext ( ) is the external applied voltage (by a voltage source), L is the load resistance (input terminal to the oscilloscope), and s is the total series resistance of the electrodes, the external wires, and the internal resistance of the voltage source. In the set up shown in Figure   1 In conjunction with the Poisson and the charge carrier continuity equations, the average conduction current can generally be expressed as 32 as shown in the Supporting Information, with eff being an effective charge carrier density in the device, where ̅ and ̅ are the spatial averages of the total hole and electron carrier densities across the active layer, respectively, while where ( , ) is the total space charge density in the active layer. el ′ can be interpreted as an additional electrode charge induced by spatially non-uniform space charge distributions inside the active layer. Finally, D is the sum of the net recombination-generation current density and the leakage current due to parasitic shunt resistance sh (in Ωm 2 ); (13) where ( , ) is the photo-generation rate of free charge carriers, ℛ ( , ) is the carrier recombination rate in the bulk, and surf ( ) = n (0, ) + p ( , ) is the total surface recombination current of electrons at the anode ( = 0) and holes at the cathode ( = ); n p ⁄ ( , ) is the local electron/hole current density at the position . We note that, in general, (or surf ) also includes a contribution from Ohmic conduction currents (e.g. in case of a doped layer).
The above set of equations generally describes planar (one-dimensional) electrical current transients in sandwich-type devices. It should be noted that under dc conditions (∂⁄ = 0), the total current density reduces to , taking its steady-state value. In the dark, this current density can usually be approximated as which is the (non-ideal) generalized Shockley diode equation. Here, the applied voltage across the diode under dc conditions is given by = ext − D s,tot , with ,tot = L + s being the total series resistance of the external circuit (see Figure 1). Note, however, that in practice, the dark dc current is always measured without a separate load resistor so that = ext − D s in Eq. (14). In the following, the general theory [Eqs. (8) to (13)] is applied to TPV and CE relevant conditions.

Small-Perturbation Transient Photovoltage
In TPV, a large load resistance ( L ≫ s ) is connected in series with the device. This is to maintain open-circuit conditions during the perturbation light pulse; in practice, a load resistance of 1 MΩ is commonly used, although it is preferable to use much larger resistances to avoid shunting over the measurement load at low voltages. 33 From the (ideally) exponential decay of the voltage transient, the associated small perturbation TPV lifetime is determined. A schematic picture is illustrated in Figure 2a where 0 = 0 ⁄ is the geometric capacitance per unit area of the active layer. Provided that the perturbation is small, also the change in the current D will be small and where B,eff = ሾ ℛ ̅ eff ⁄ ሿ −1 is an effective carrier recombination lifetime, and is the differential resistance of the diode itself (excluding the shunt resistance), where Eq. (14) was used in the last step. Note that the shunt resistance ℎ plays an identical role to the load resistance in Eq. (18), consistent with previous experimental findings. 24,33 In the limit when the electron and hole densities are homogenous across the active layer, corresponding to sufficiently high steady-state light intensities, we obtain B,eff = Under these conditions and provided that B,eff ≫ 0 0 (and 0 ≪ , ℎ ), TPV thus embodies a real carrier lifetime which is also relevant under steady-state conditions. Note that this lifetime is only a constant in the case of true first order recombination, but the general assumption is that B is allowed to be dependent on the carrier concentration and should thus represent recombination dynamics of any order. We also note that for the case eff = ̅ = ̅, Eq. (18) becomes equivalent to the TPV lifetime expression previously proposed by Credgington et al. 34,35 , and can be simplified where is given by the capacitance of the active layer, assuming eff to be governed by the dark back-injected carriers. In this limit, TPV is limited by a composite RC-time constant of the device, being ruled by the smallest of the load resistance, the shunt resistance, and the differential resistance of the diode itself.
We note that the RC decay in the limit of low light intensities is not fully determined by the geometric capacitance 0 , but rather by a total capacitance , as given by Eq. (21). This total capacitance is given by the sum of 0 and a space charge term that monitors the change in the background charge carrier distribution (as induced by the voltage perturbation) during the voltage transient. Only when the change in the space charge distribution is negligible, will be given by the geometric capacitance 0 . However, for the case of a doped active layer, for example, the photo-induced voltage change across the active layer will induce a small change in the (doping-induced) depletion layer thickness as well, and Eq. (20) reduces in this case to the depletion layer capacitance = 0 ⁄ , as shown in the Supporting Information.

Charge Extraction
A schematic picture of the CE method is shown in Figure 2c where Δ = Δ = oc is the excess steady-state carrier density at open-circuit conditions. Note that in case of pure second-order recombination, we then expect CE = = √ 0 / , as before.

Comparison with Drift-Diffusion Simulations
To gain further insights into the physical meaning of the above theoretical analysis, we turn to transient device simulations based on a 1D drift-diffusion model. [37][38][39] The device model numerically solves Eq. (8) and (9)   At smaller light intensities (lower oc ), the TPV lifetime is governed by capacitive effects associated with back-injected dark carriers. In this regime, the TPV lifetime is very well approximated by Eq. (20) (depicted by blue short-dashed line in Figure 4a), assuming to be given by the dark dc capacitance ( Figure S1). We note that this capacitance exhibits a voltage dependence under forward bias, increasing from ≈ 1.1 0 at = 0 to ≈ 1.4 0 at = 0.5 V, as can be seen from Figure S1. The increase in , relative to 0 , can be understood in terms of back-injected dark carriers at the contacts penetrating deeper into the active layer, effectively decreasing the active layer thickness (in a geometric capacitor approximation). This capacitance is sometimes also referred to as the chemical or diffusion capacitance. [43][44][45] It should be stressed that TPV is ultimately limited at lower voltages by the RC-times of the load resistance and/or the shunt resistance of the cell. The smaller of these two will set the RCtime constant of the system and defines the upper limit for the lifetime that can be extracted by the measurement. This is demonstrated in Figure 4b in case of a 1 MΩ load resistance (assuming L ≪ ℎ ); as TPV approaches the RC-limit set by the load resistance, a saturation of the extracted lifetime TPV → L will occur at low voltages, in accordance with both Eq.

The Validity of Complete Extraction and Uniform Carrier Distributions in CE
We next turn to charge extraction. The CE carrier density, shown in Figure 5, is given by CE = CE / , with CE obtained by integrating the simulated extraction current transients and correcting for the geometric capacitance (see Eq. (22)). In Equation (25), it was assumed that (i) the recombination during the extraction process is negligibly small, and (ii) the carrier distributions are homogenous. To check the validity of these two assumptions, we have simulated CE current transients for the device considered in Figure 4a. The corresponding extracted carrier density CE is shown in Figure 5a. This is to be compared to the excess carrier density oc = √ 0 / , as expected from Eq. (25), indicated by the red line (with a slope parameter of = 2). It can be seen, for the case = 10 −4 cm 2 /Vs, that Eq. (25) overestimates the actual CE at higher oc , but underestimates it at low oc . , corresponds to the time it takes a carrier to traverse the inter-electrode distance. It can be seen that for the simulated second order dynamics, the pseudo-first order bulk lifetime B becomes smaller than tr as oc exceeds 0.8V resulting in a substantial recombination during the extraction process at larger intensities, explaining the incomplete charge extraction at these intensities. On the other hand, a much better agreement is obtained for the case with = 10 −2 cm 2 /Vs, corresponding to a hundred times shorter transit time tr (the other parameters are kept the same as before), in Figure 5a.
We can thus conclude that the underestimation of the carrier density seen at high intensities is related to the fact that higher-order (>1) recombination during the charge extraction process, starts to compete with extraction at higher intensities (see Eq. (4)). Therefore, a necessary condition to avoid recombination during the extraction pulse is that tr ≪ B . This limitation can be potentially overcome by applying a strong reverse bias extraction voltage pulse, as shown by Kniepert et al, 46 although this will also give rise to a larger amount of displacement charges and thereby a larger noise on the extracted charge evaluation.
At low light intensities (small oc ), the bimolecular recombination during the extraction process is negligible. However, as seen from Figure 6b, the assumption of uniform carrier distributions is no longer valid. Instead, the excess carrier distributions are highly non-uniform, varying exponentially with distance within the active layer, with the electron and hole profiles being mirror-symmetric to each other. This type of behaviour is generally expected for (dark) carriers, originating from the contacts. [47][48][49][50][51] Accordingly, at low light intensity, the excess carrier densities of charges injected into the undoped active layer can be approximated by for oc < bi , where bi is an effective built-in potential that accounts for the energy-level bending in the vicinity of the injecting contacts, 52 and a and c are the associated effective hole and electron densities in the dark at the anode ( = 0) and cathode ( = ) regions, respectively. Then, by substituting Eq. (26) and (27) into Eq. (24) and integrating, we find when ( bi − oc ) ≫ ⁄ . In Figure 5b, we have compared Eq. (28) with the simulated CE data and a good agreement can found when assuming a = c and bi in Eq. (28) to be fixed and independent of the light intensity. In general, however, these quantities depend on the applied voltage (the injection level) as well; subsequently, a weak intensity dependence (via oc ) of a , c and bi is to be expected. We note that these findings are consistent with previous results reported by Deledalle et al. 29

Discussion
Based on the recombination lifetime obtained from TPV, and the corresponding steady-state carrier density extracted with CE, the recombination rate and the lifetime as a function of carrier density is usually mapped. 13 The subsequent recombination rate, as derived from TPV and CE, can be expressed as where = ⁄ and 1 + is the recombination order.
In line with the above findings, however, due to the inaccuracy of at low light intensities, the TPV/CE recombination rate plotted against this extracted charge density is heavily plagued by capacitive effects. As discussed earlier this is caused by spatially separated carrier distributions and/or shunts. Only at high enough carrier densities, when ≈ and ≈ , can the recombination rate constant be calculated from the recombination/charge plot. In this high carrier density regime, provided that the recombination during the extraction process in CE can be minimized, the slope of the recombination rate on a log-log plot, directly gives the associated recombination order.
We should note that here we considered balanced electron and hole mobilities and a relatively thin junction. Imbalanced mobilities or thicker junctions (which make the transit times of electrons and holes imbalanced) result in more problems in the validity of assumption (i), as one needs to consider the lifetime of the slower carriers and their transit time. On the other hand, in thicker devices, the photo-induced excess charge carrier profiles are generally more uniform throughout the active layer. 29 In addition, since capacitive effects can be reduced by increasing the active layer thickness, thicker devices are in general better suited for TPV but also for ensuring the validity of assumption (ii) in CE. This is consistent with the conclusions of previous work by Kirchartz and Deledalle. 19,29,54

Experimental Demonstration on P3HT:PCBM Solar Cells
To demonstrate the relevance of the above theoretical findings, we turn to experimental results obtained from organic bulk heterojunction solar cells, based on P3HT:PCBM. In Figure 7a and At small , good agreement is obtained between the estimated and the experimental in this case. Conversely, at high enough , where we expect the recombination between bulk carriers to instead dominate the TPV response, a slope parameter of = 2 is obtained. A similar situation is true for the CE data in Figure 7b, where = 2 is obtained at high enough . The corresponding TPV/CE recombination rate, using the obtained slope = 1, is shown in Figure 7c as a function of . Accordingly, a slope of two is obtained in the relationship between total recombination rate and carrier density, i.e. ℛ ⁄ ∝ 2 , consistent with pure second-order recombination dominating at these light intensities. Furthermore, under these conditions (slope = 2), the measured data allows one to extract the second-order recombination coefficient via = (2 ) −1 , in accordance with Eq. (4); from the vs plot, depicted in Figure 7d, we find ≈ 6 × 10 −12 cm 3 s ⁄ .
The experimental results in this work suggests that the dominating recombination mechanism in P3HT:PCBM is second-order near 1 sun incident light intensities with a constant secondorder recombination coefficient, being in line with previous findings by various other methods. 46,49,[55][56][57][58] Furthermore, this also corroborates recent work by Tvingstedt and Deibel, who concluded that trap-assisted recombination via exponentially distributed tail states is not the dominating recombination mechanism in this system. 18

Conclusions
In conclusion, based on the analytical derivations and numerical device simulations on realistic organic solar cells, the relation between the bulk recombination of photo-induced charge carriers and the lifetime, as extracted from the TPV transients, has been clarified. At higher the lifetime is given by the sought-after bulk lifetime governed by bulk recombination. At lower light intensities and open-circuit voltages, the TPV lifetimes are instead limited by a composite RC-time constant which is dominated by the smallest of the load resistance of the measurement circuit, the shunt resistance (associated with parasitic leakage currents), and the internal (differential) resistance of the diode itself. For CE, the determination of the photo-generated carrier density under open-circuit conditions is more challenging (and requires uniform carrier distributions) for higher order recombination. At higher , the CE measurement can be susceptible to recombination during the charge extraction process, which might lead to the extracted carrier density underestimating the actual steady-state density prior to the pulse.
A necessary requirement for the correct determination of in this regime is that ≪ .
At lower , however, the extracted CE density is dominated by capacitive effects caused by the extraction of spatially separated (back-injected) charge carriers. After accounting for the dark carrier profile (and their displacement current effects), analytical approximations for the capacitive CE regime could be obtained. Finally, the theoretical behaviour is reproduced experimentally on organic solar cells based on P3HT:PCBM.

Associated Content
Supporting Information: Analytical derivations, details regarding the device model, and additional simulations