Light Intensity Modulated Impedance Spectroscopy (LIMIS) in All-Solid-State Solar Cells at Open Circuit

Potentiostatic impedance spectroscopy (IS) is a well stablished characterization technique for elucidating the electric resistivity and capacitive features of materials and devices. In the case of solar cells, by applying a small voltage perturbation the current signal is recorded and the recombination processes and defect distributions are among the typical outcomes in IS studies. In this work a photo-impedance approach, named light intensity modulated impedance spectroscopy (LIMIS), is first tested in all-solid-state photovoltaic cells by recording the individual photocurrent (IMPS) and photovoltage (IMVS) responsivity signals due to a small light perturbation at open-circuit (OC), and combining them: LIMIS=IMVS/IMPS. The experimental LIMIS spectra from silicon, organic, and perovskite solar cells are presented and compared with IS. An analysis of the equivalent circuit numerical models for total resistive and capacitive features is discussed. Our theoretical findings show a correction to the lifetimes evaluations by obtaining the total differential resistances and capacitances combining IS and LIMIS measurements. This correction addresses the discrepancies among different techniques, as shown with transient photovoltage. The experimental differences between IS and LIMIS (i) proves the unviability of the superposition principle, (ii) suggest a bias-dependent photo-current correction to the empirical Shockley equation of the steady-state current at different illumination intensities around OC and (iii) are proposed as a potential figure of merit for characterizing performance and stability of solar cells. In addition, new features are reported for the low-frequency capacitance of perovskite solar cells, measured by IS and LIMIS.


Introduction
Standard potentiostatic impedance spectroscopic (IS) is a well-known and stablished technique for the characterization of the resistive, capacitive and inductive features of materials and solar cells. 1,2 In photovoltaic devices, one of the most common characterization routines is to probe the open-circuit (OC) condition under an steadystate illumination intensity and by applying a small voltage perturbation at different light intensities the IS spectra are measured and analyzed. In this way the recombination resistance , chemical capacitance and characteristic lifetimes are typically accessed.
With an alternative approach, the photo-sensitive samples have been earlier separately characterized by means of the intensity modulated photocurrent spectroscopy (IMPS) [3][4][5][6][7][8][9][10][11][12][13][14] and the intensity modulated photovoltage spectroscopy (IMVS). 9,10,[15][16][17] Particularly, the IMPS has been recently gaining attention in the field of perovskite solar cells (PSCs), mainly exploring the short-circuit (SC) condition. 13 proposed an analytical model which shows the difference between LIMIS and IS to be proportional to the surface recombination velocity. 23 In this article we further analyze this concept at OC and first present an experimental analysis of LIMIS silicon, 24 organic and perovskite solar cells. 25 Table S1 in order to facilitate the reading.

Potentiostatic impedance spectroscopy (IS) in solar cells
We may first consider a generic sample at steady-state voltage ̅ where a current density ̅ ( ̅ ) is flowing. In a first approximation, every sample can be assumed as a resistor-capacitor Voigt element with a characteristic time response constant = , as in Figure 1a. Then a small potentiostatic perturbation ̃( ) = |̃|exp[ ] can be applied in alternating current (ac) mode, being the time, the angular frequency and the imaginary unit. The total voltage would be Upon perturbation, the current may evolve as where ̅ may be the steady-state current ̅ ( ̅ ) and the phasor-related part ̃ may inform on the differential resistive and capacitive features of the sample. A typical sinusoidal ̃( ) small perturbation is illustrated in Figure 1b, to which the current may be phase shifted, as in Figure 1c. Then we can write ̃= |̃|exp[− ] and the impedance can now be introduced as The -dependence on frequency = /2 creates an impedance spectrum, which is the study subject of the impedance spectroscopy (IS). Most typically presented as ( ) = ′ ( ) + ′′( ), the Nyquist plot representation is illustrated in Figure 1d.
There the characteristic semicircle from a linear couple with single is shown.
The real part ′ carries the information on the differential resistance, and since → 0 when → 0 thus → ′ and the total differential resistance can be taken as the radius of the semicircle. On the other hand, the imaginary part ′′ informs on the capacitive features. Note that the − ′′ maximum ( = /4 in Figure 1d) belongs to the characteristic angular frequency where is the elementary electric charge, the thermal energy, the saturation current, the ideality factor, and ℎ is the bias-independent photo-generated current, typically taken as the short-circuit current . Equation ( where Ψ = Ψ is the photo-current responsivity at short-circuit that depends on the incident light spectrum, the absorption coefficient and the geometry of the absorbing materials, and is the incident light intensity in units of power density. Experimentally, under ac potentiostatic perturbation (IS measurement) the current signal results from evaluating (4.a) in (1), sampling the narrow region around the steady state condition. Figure 2a illustrates the particular case where OC is tested at constant illumination intensity.

Light intensity modulated impedance spectroscopy (LIMIS) in solar cells
Alternatively to the IS approach, in the case of photo-sensitive samples (see Figure   1a which included a photo-current source) the perturbation can be done by a light source. Then, a small perturbation ̃( ) = |̃|exp[ ] can be added to the given dc incident light power density ̅ , as in Figure 1e. The total incident light intensity in units of power density would be Upon this perturbation, both current and voltage signals can be recorded. At a given ̅ , the current would be phase shifted, as in Figure 1f, and similarly to (2) ̃= Hence a current responsivity transfer function can be defined as Likewise, at OC ( = 0) the photovoltage signal may be composed by the dc open circuit voltage ̅ and the phasor related part as Then the photo-voltage signal may have a phase shift (see Figure 1g) and taking ̃= |̃|exp[ ] thus a voltage responsivity transfer function is defined as Equations (6) and (8) Advantageously, the experimental spectra from the photo-impedance of (9) do not need voltage/current sources in the EC-based numerical simulations, resulting a simpler and less ambiguous task. Also the spectroscopic representation of the resistive (see Figure 1h) and capacitive features can be obtained too, which allows future development of analogue light intensity modulated thermal admittance spectroscopy (LIMTAS). And furthermore, a LIMIS direct comparison with IS spectra may straightforwardly inform on generation/recombination features in solar cells. Importantly, in the core of our focus is to provide a first approach to the difference between LIMIS and IS, its meaning and possible use. Accordingly, herein we define a normalized figure of merit called photo-impedance difference as where and Ψ come after (3) and (9), respectively. Note that Δ Ψ is zero when Ψ = and positive (negative) when the photo-impedance from LIMIS is larger (lower) than that from IS.

Experimental LIMIS and IS spectra
A proper analysis between IS and LIMIS at OC requires to set the same steady-state dc illumination intensity. Subsequently IMVS can be measured directly at OC and for IS and IMPS the forward bias corresponding to the same should be applied so the is cancelled. For IMPS and IMVS, the exact set of sampled frequencies is an obvious requirement. Other external parameters like temperature, humidity (when reactivity issues) or even the wire connections should be controlled to be the same during the three measurements, so the characterized state is nearly the same.
The IS, IMPS and IMVS measurements were carried out with the Zahner Zennium Pro/PP211 impedance setup using its LSW-2 white LED light source. In all cases the sample holder included N2 atmosphere.
Notably, ensuring the requirement of linear small perturbation is of upmost importance, mainly when measuring IMPS and IMVS to obtain LIMIS. In the case of IS, typically ̃< delivers accurate results, and for IMPS and IMPV keeping the ac perturbation below 10% of dc light intensity (̃< ̅ /10) provides a good empirical reference too. However, the latter rule can be not good enough in some cases, particularly for low dc illuminations approaching the ac experimental setup limit. Therefore, we use a significance parameter as described by Schiller and Kaus 29 and automatically implemented in the Zahner setup. The significance parameter goes from 0 to 1 and informs of "perfect linearity" if it equals unity. In practice, optimal results should be abode 0.98 and those below 0.95 should be discarded.
Five representative samples were experimentally characterized as summarized in Section S1.1: a silicon solar cell SiSC, an organic solar cell (OrgSC) and three perovskite solar cells (PSC1,2,3). The respective schemed structures, − curves, external quantum efficiency (EQE) spectra and 500 hours degradation tests (for the PSCs) are in Figure S1. The performance parameters are in Table S2.
The silicon device constitutes the first reference due to the simplicity and robustness of its working principles. Its characterization is presented in Section S1.2: first the IMPS and IMVS spectra at OC under different dc illumination intensities in Figure   S2, and then LIMIS and IS spectra for the SiSC are shown in Figure S3. The current and voltage responsivities in Figure S2 illustrate the expected arc-like shapes in the Nyquist representation. In Figure 3a the low frequency limits from Ψ and Ψ are plotted. From IMVS the relation Ψ = −1 −1 with ≈ 1.3 is in agreement with theoretical predictions 16,23 and the photocurrent-photovoltage trend in Figure   S4a. On the other hand, from IMPS the light-intensity independency of Ψ in almost all the range of measurement seems to fade only as approaches the built-in voltage , illustrated in the Mott-Schottky plot of Figure S4b.
By applying the LIMIS definition (9) the photo-impedance spectra can be compared with the standard IS spectra, as in Figure S3 and Figure 3b. A right-shifted Nyquist plot is apparent, reporting Δ Ψ > 0 in the measured range. For the sake of clarity, this series-resistance-like right-shift in the real part of the LIMIS impedance is going to be referred in the next as s ′.  Figure   S2) and (b) representative impedance Nyquist plot (see Figure S3). Lines in (a) belong to fitting to ∝ −1 and ∝ 0 , and in (b) refers to the EC model discussed in Section 2.3 and Figure 7b.
The organic device, with structure ITO/ZnO/PM6:Y6/MoOx/Ag, was characterized as presented in Section S1.3 including the device fabrication description. Similarly, Figure S5 shows the IMPS and IMVS spectra and Figure S6 the comparison between IS and LIMIS.  Figure S4c. Differently, from the IMPS Ψ behaves more like Ψ ∝ −1/3 in a low and medium range for the measured illumination intensities. Moreover, Figure   4b illustrate one of the Nyquist plots showing the similar arcs of the two techniques, also with the right-shifting trend for the LIMIS spectrum. More interestingly here it is that the apparent series resistance s ′ shows a negative arc in the Nyquist representation (empty dots in Figure 4b). This is an important feature whose understanding, while beyond the scope of this paper, should be attended in future works. The IMPS and IMVS spectra for PSC1 are presented in Figure S8, evidencing already a more complex response including two arcs in the Nyquist plots. Regarding the low frequency limits of the voltage and current responsivities from PSC1, in Figure 5a, the IMVS similarly gives Ψ = −1 −1 with now ≈ 1.5 following theory 16,23 and agreeing previous reports on ideality factors from mixed cation PSCs. 30,31 Distinctly, the IMPS reports a situation somehow in the middle between constant Ψ at lower light intensities and Ψ ∝ −1/3 at higher illuminations. The latter resembles the behavior of the OrgSC, probably related with the intrinsic absorber nature of both.
Applying LIMIS definition (9) allows to compare it with the IS spectra, as in Figure   S9. PSC1 brings a new feature to the impedance spectra by reporting a clear three RC constants, i.e., three arcs in the Nyquist plots and three steps in the capacitance Bode plots. Importantly, as illustrated in Figure 5b, the high frequency region of the spectra ( >1kHz) from LIMIS delivers negative values in the Nyquist plot (empty dots) and a consequent negative capacitance in the Bode plots of Figure S9. Hence, the expected high frequency arcs (plateau) of the LIMIS impedance (capacitance) spectra from PSCs are not so and instead suggest a higher complexity in terms of EC elements. The high frequency region from IS reproduces earlier described features. 32,33 On the other hand, at low frequencies ( <1kHz) LIMIS seems to reproduce very well the IS spectra, in both the impedance Nyquist plot (2 arcs in Figure 5b) and the capacitance Bode plot (2 steps in Figure 5c). respectively. This is displayed in Figure 6 for the set of studied devices. The general trend shows first a decrease as light intensity is augmented until a few tens of mW·cm -2 . In this range a rough approximation would say that Ψ ∝ (1 + −2 ). Towards 1 sun illumination intensity, the photo-impedance from LIMIS seems to exceed the impedance from IS as light intensity grows. In this latter range we could speculate that . Interestingly, in the region between the two regimes, some negative values are reported, indicating that the photo-resistance from LIMIS is lower that the total resistance from IS. This only occurs for the OrgSC, PSC2 and PSC3.
These are actually the devices with more performance issues: the OrgSC presents "S" shape above OC and the PSC2 and PSC3, besides the lower PCE, and , showed lower stability too (see Figure S1). These correlations are also a matter of further analyses, but these preliminary observations suggest that the higher Δ Ψ ′ the best, and that negative values of Δ Ψ ′ indicate performance and/or degradation issues in solar cells. Figure 6. Δ Ψ ′ as a figure of merit for checking performance and*or degradation issues: normalized real difference between T ′ and , from LIMIS and IS respectively, as a function of illumination intensity for the different studied devices. Only the cells with low performance or degradation issues show negative Δ Ψ ′ .

The differential approach to resistance and capacitance: correcting lifetimes
The derivative of the scalar current ( , ) is measured in different directions ̂1, ̂2 and ̂3 by IS, IMPS and IMVS, respectively. Thus, we can calculate them by using the concept of directional derivative and the directions in the OC surface ( = 0). For IS the derivative is found in the direction of ̃, so ̂1 = (1,0) and for IMPS, in the direction of ̃, so ̂2 = (0,1). These are the well-known partial derivatives in the axes directions. However, IMVS is not a partial derivative of ( , ) but . Hence we may redefine it as the directional derivative of ( , ) in the direction ̂3 contained in the interception between the OC surface and ( , ) (see Figure 2c).
Consequently, we can now express the transfer functions of IS (3), IMPS (6), IMVS (8) and thus LIMIS (9) as derivatives at ̅ = ̅ , respectively as Purposely, we are here interested in two main physical quantities: the total differential resistance unit area and the total differential capacitance per unit area where is the charge density. Equations (12) and (13) are total differentials that can be approached to the partial derivatives from the potentiostatic IS following (11.a) as where ( ) is that of (3) and ∝ / at each frequency. Definition (14) is the full form of (12) and (13)  We can also apply (11.b-d) to the empirical approximation of the Shockley equation (4) resulting the analogue dc parameters where Ψ = / has the same meaning as in (4.b). Note that in the assumption of bias-independent Ψ , and in agreement with Now, similarly to (14) for IS, the IMPS and IMVS respectively explore partial derivatives as in (11.b,c). This is illustrated in Figure 2c and left side of Figure 2d.
Subsequently, since (4) is not light independent, the definition (12) can be better approached as From (16) Accordingly, a better estimation of the differential capacitance in photosensitive samples would be Similarly, from (17), note that the predominant term will be the larger of the capacitances and Ψ , from IS and LIMIS respectively. Specifically, if ~Ψ then results twice that typically estimated from IS. As in the case of resistance, light charges the capacitor in addition to how the bias does it, hence it makes sense that some extra charge is stored. Accordingly, it is of crucial importance to evaluate the degree of overestimation (underestimation) of the differential resistance (capacitance) by only considering IS measurements.
Importantly, if the superposition rule (14) is valid, then = Ψ makes = /2 and = 2 . Accordingly, the LIMIS measurements would not modify the corresponding characteristic response times = = . For instance, this would be the case where the characteristic lifetimes from IS spectra coincide with some other techniques like TPV, as earlier reported. 35 However, as showed in the previous section, we found ~Ψ which may deliver a corrected lifetime including all the carrier contributions due to bias and light dependencies. Note that this result does not conflicts the reciprocity theorem of charge collection, 34  to dark recombination currents due to the injection of carriers.

Numeric approach: the equivalent circuits
After introducing LIMIS in Section 1.2, the total differential resistance and capacitance from photosensitive samples was corrected in Section 2.2, resulting as in equations (16) and (17). Differently to the that suggested by the derivatives of the dc empirical Shockley equation (15), our recent analytical analysis 23 suggested that the impedances from IS and LIMIS should differ. Accordingly, the accurate estimation of R and may include the measurement of LIMIS. In particular, the incorporation of both concepts can be represented in an equivalent circuit (EC) as in Figure 7a, where the impedance from IS and the photo-impedance Ψ from LIMIS are connected in parallel among them, excluding the non-photosensitive contributions from the ohmic series resistances . Also in Figure 7a the simplest EC including a couple of Voigt elements in parallel is illustrated, in agreement with differential definitions (16) and (17).
IS and LIMIS are measured separately, and in the next section the experimental measurement will be presented and discussed. Figure 7b,  Equivalent circuits for (a) the concept of total differential contributions from IS and LIMIS to resistance and capacitance and (b, c) the used equivalent circuits during the numerical simulation of IS and LIMIS spectra. Rseries is a series resistor, RIS and RΨ are the total resistances measured by IS and LIMIS, RHf and RLf are high and low frequencies resistors, CIS and CΨ are the total resistances measured by IS and LIMIS, and CHf and CLf are high and low frequencies capacitors, respectively.
For the silicon solar cell, the IS and LIMIS spectra were numerically simulated to the EC model of Figure 7b as presented with solid lines in Figure S3 and Figure 3b. The total C-coupled resistances = + are shown in Figure 8a as a function of the . follows an exponential law like (15.a) with ≈ 1.2, being ℎ approximately a 20% larger for LIMIS than that from IS, i.e., wider arcs as in Figure   3b. Accordingly, from (16): ≈ 0.6 • under illumination.
In addition, the right-shifting s ′ (see Figure 3b) also follows an exponential decrease as (15.a), but with ≈ 2, as in Figure 8a. This is an extra impedance contribution, different than that of the ohmic (nearly constant in Figure 8a) which may be detailed studied in the future. Here it is important to note that the high frequency part of the LIMIS spectra is particularly difficult to fit due to the lower linearity of the signal, as expressed in the significance spectra of Figure S3.
The capacitance bode plots are shown in Figure S3 with the respective simulations to in Figure 8b showing an exponential increase possibly due to diffusion capacitance. 32,37 In this case the from LIMIs is nearly half of that from IS, so from (17): With the information of and , the characteristic times = can be accessed, as presented in Figure 8c. Note that from (16) and (17) the total time response is actually a 90% of that calculated for IS. Figure 8c also present the transient photovoltage (TPV) lifetimes which nearly coincides with IS and LIMIS a lower light intensities (below ~3 mW·cm -2 ). The TPV measurements were performed with a self-made setup (see Section S1.7 for details) in order to contrast the results from the characteristic time constants. As the light intensity is augmented, the TPV signal does not decay exponentially anymore (see Figure S17) and the IS and LIMIS provide a better estimation of characteristic lifetimes. In addition, The LIMIS seems to inform on faster characteristic times ( ), possibly related with charge extraction processes, i.e., not as slow as the recombination lifetime. For the organic solar cell, Figure 9a shows , s ′ and as a function of for the IS and LIMIS spectra, as well as the numerical simulations to the EC model of Capacitance spectra are also displayed in Figure S6, and the total capacitance of the OrgSC from the fittings is presented in Figure 9b, which is basically , significantly higher and exponentially increasing in comparison with the constant geometrical capacitance = . In this case LIMIS presents a 64% higher capacitance with respect to LIMIS, so from (16): ≈ 2.64 • . Accordingly, the actual total characteristic times may be 1.35 times bigger than they are from IS, which is nearly for LIMIS. This result approaches the lifetimes from TPV below ~10 mW·cm -2 (see decays in Figure S10) and the characteristic times from IS and LIMIS, as presented in Figure 4c. In that figure it is also evident how LIMIS and IS characteristic times are similar for the OrgSC, unlike the SiCS. The and from PSC1 IS spectra is summarized in Figure 10a. More interestingly are the capacitive features and the resulting time constants. In The characteristic response times are summarized in Figure 10c. The high frequency times follow the resistance trend and even approximately agree with the TPV lifetimes (see decays in Figure S14). The low frequency times behave slightly constant and decreasing, 1 and 1 respectively, suggesting an eventual convergence around miliseconds. Two more perovskite solar cells PSC2 and PSC3 were analyzed as summarized in Sections S1.5 and S1.6, respectively, with nearly similar trends to PSC1. Nevertheless, regarding the low frequency capacitance, by eliminating the PMMA/PCBM cover towards the SnO2 in PSC2, we obtain almost totally saturated = 2 and discrepancies between LIMIS and IS (see Figure S13b) Figure S16b). The detailed analysis of these features, while only reported here, should be attended in future studies.

Bias-dependent photocurrent correction to the empirical Shockley equation around open circuit
From the previous section it was stated how the LIMIS spectra, despite resembling the IS shapes, are not the same as the IS spectra. This result from the spectroscopic ac characterization is also in agreement with the dc response in Section S1.8. In Figure   S18 the experimental J-V curves from three of the studied samples (SiSC, OrgSC and PSC1) are presented as a function of the illumination intensity, forming current threedimensional (3D) surfaces. The corresponding short-circuit currents are displayed in Figure S19 confirming the well-known relation ∝ at = 0 as (4.b).
From the experimental data in Figure S19 we can numerically find the pair ( , ) for the current roots (OC) and calculate the numerical derivatives for IS, IMPS and IMVS as (11). The code for that calculus is in Table S2 and the results are shown in Interestingly, different trends ∝ − are found depending on the sample and the illumination intensity range. Figure 11. Numerically calculated dc resistances at OC using the differential definitions of IS and LIMIS on experimental − curves at different light intensities. The experimental data is plotted in Figure S21 and the calculation code in Table S2.
The above experimental observations contradict the formulation of the well-known empirical Shockley equation (4) when applying the LIMIS definition (9) as in (15).
Note first that we already showed Ψ decreasing with light intensity at OC (see Figure   4a, Figure 5a, Figure S11d and Figure S14d), but at SC it still agrees with (4), as in Figure S19. Accordingly, a correction in (4.a) may be introduced justifying to apply the IMPS differential definition (11.b) at OC to obtain a decrease of Ψ as increase.
Note that (21) explains the three main experimental observations. First, the decrease trends of Ψ and Ψ as is augmented at OC in (21b,c) agree with the low frequency limits of IMPS and IMVS spectra, respectively. Second, from the parentheses in (21a,d) we see that Ψ, ≥ , as the experimental evidence discussed in the previous section. And third, by comparing IS and LIMIS dc resistances we realize that they converge only when Ψ ≫ , so both parentheses in (21a,d) equal unity, and as increases, as suggested more evidently by the SiSC behavior in Figure 6 and Moreover, it has been shown how the total differential resistances and capacitances are reduced and augmented, respectively with respect to IS, illustrating the photoconductivity increase under illumination for the solar cells. This effect corrects the evaluation of the lifetimes, which is a factor to consider in the typical differences when evaluating that parameter by different techniques, like TPV.

Conflicts of interest
There are no conflicts to declare.                Figure  S13) from IS and LIMIS experimental data (dots in Figure S13): (a) capacitance, (b) resistance and (c) characteristic times from LIMIS and IS (EC model in Figure 7c) and TPV (mono-exponential decay model). The lines in (b) are the fittings to the R_dc behavior with m as indicated with arrows. The lines in (a) are the fittings to = 1 exp[ / ] with = 3.7, and 1 = 5.0 × 10 −11 F • cm −2 and 1 = 8.2 × 10 −11 F • cm −2 for IS and LIMIS, respectively. (c) Low frequency limits of the voltage and current responsivities for different light intensities (see Figure S12) where the solid lines belong to fitting to Ψ V ∝ P in −1 and Ψ J ∝ P in 0 and the dashed one to Ψ J ∝ P in −1/3

S1.7. Transient photovoltage (TPV) measurements
Setup description: A Cree XP-E LED is used for white light bias. Driving the LED current with a Keithey 2400 and measuring the light intensity with a highly linear photodiode (Vishay BPW21R) allows to reproducibly adjust the light intensity with an error below 0.5% over a range of 10 -5 to 1 suns. A small perturbation is induced with a 405 nm laser diode driven by a function generator from Agilent. The intensity of the short (50 ns) laser pulse is adjusted to keep the voltage perturbation below 10 mV, typically at 5 mV. After the pulse, the voltage decays back to its steady state value in a single exponential decay. [1] The characteristic decay time is determined from a linear fit to a logarithmic plot of the voltage transient and returns the small perturbation charge carrier lifetime.

Figure S19.
Experimental short-circuit current density as a function of the illumination intensity from the experimental data in Figure S18 of the studied samples, as indicated.
The arrow points the slope of the allometric fittings: ∝ 1 .

S.2.1. Numerical deduction of the dc resistances at OC for IS and
LIMIS from the experimental J-V curves at different light intensities.  Figure S18.