Light Capacitances in Silicon and Perovskite Solar Cells

The framework on which the physics of silicon solar cells (SiSCs) is based robustly predicts dependences of capacitance on light intensity and voltage, even when most recent innovations are considered as the incorporation of transition metal oxide/Si heterojunctions. However, perovskite solar cells (PSCs) challenge most of the established paradigms, claiming for rethinking of known theories and devising novel models. Here we tackle this scenario by probing and comparing light-induced capacitance responses yielded by these two major exponents in the field of photovoltaic research. SiSCs light capacitances can be easily interpreted in the framework of the so-called chemical capacitance. Current approaches addressing the intriguing low-frequency capacitive features of PSCs are outlined and compared. Here, apparent similarities and differences between both photovoltaic technologies are highlighted, concerning the observation of light capacitances of chemical origin. It is concluded that, contrary to that occurring in SiSCs, bulk electronic chemical capacitances are not straightforwardly observed in PSCs. As capacitive features exhibited by PSCs are believed to be critically connected to performance degradation and device instability, future research and


Introduction
In recent years, perovskite solar cells (PSCs), incorporating hybrid metal halide perovskite compounds as methylammonium lead iodide (MAPbI3) (Kim et al., 2012;Lee et al., 2012;Yang et al., 2015) and other combinations , have achieved power conversion efficiencies above 24% (Green et al., 2019), comparable to those exhibited by silicon solar cells (SiSCs). The impressive optoelectronic performance of PSCs stems from the reported high absorption coefficient, long charge diffusion length and high carrier mobility, reduced recombination rate, low exciton binding energy, and tunable direct band gap (Green et al., 2014;Oga et al., 2014;Park, 2015;Wehrenfennig et al., 2014;Zhao and Zhu, 2016). Despite their superior optoelectronic properties, there remains a set of unresolved issues that block commercialization of PSCs, mainly related to their chemical stability and photovoltaic parameter degradation/variation. Among other operating alterations, there is extensive literature concerning the hysteretic effects on the current density-voltage (J-V) characteristics (Jeon et al., 2014;Snaith et al., 2014;Unger et al., 2014). Hysteresis has a detrimental influence on the photovoltaic operation, device reliability and stability so as to advance in its alleviation. This technological drawback was suggested to be connected to their uncommon capacitive and photo-capacitive response .
Light capacitance in PSCs was reported to exhibit an intriguing behavior in the lowfrequency part of the capacitive spectrum (Juarez-Perez et al., 2014). A huge increment of capacitance is commonly encountered at 0.1-10 Hz, which results proportional to the light intensity, both in short-and open-circuit conditions (Juarez-Perez et al., 2014;Zarazua et al., 2016a). Values as high as 0.1 F cm -2 can be readily achieved at standard 1 sun illumination, a feature hardly observed in other kind of solar cell technologies. It was early noted that even in the dark, PSCs shows relatively large capacitances (~50 µF cm -2 ) at low frequencies as compared with geometrical capacitance, irrespective of the perovskite bulk thickness (Almora et al., 2015). This fact indicated that interfaces between the absorbing semiconductor and contacting layers would be behind these uncommon capacitive experimental trends (Bergmann et al., 2016;Carrillo et al., 2016;Kim et al., 2017). Simultaneously, it was reported a hysteretic effect of the J-V curve under operation. It has been ascribed to several mechanisms: intrinsic ferroelectricity of the perovskite compounds (Wei et al., 2014), delayed trap filling upon illumination (Shao et al., 2014), ionic charge accumulation at the outer interfaces (Almora et al., 2019b;Meloni et al., 2016). The connection between most significant hysteresis trends and the capacitive features has been firmly established (Chen et al., 2015). It is also known that photovoltaic perovskites possess a non-negligible ionic conductivity (Azpiroz et al., 2015;Meloni et al., 2016). This fact has suggested the participation of mobile ionic species in the explanation of either the achieved level (Kim et al., 2018) or the slow kinetics (Bag et al., 2015) of the observed hysteresis and photo-capacitive effect. Also the formation of an electronic space-charge accumulation zone at the interfaces between the perovskite and contacting layers has been proposed (Zarazua et al., 2016a). In any case, the capacitive issue does not have a widely accepted and shared explanation. But it deserves further attention as light capacitance exhibited by PSCs is connected to performance degradation and device instability. Therefore, a deep analysis of the capacitive and photocapacitive response in PSCs is needed and here it is compared with the capacitance behavior of a well-known technology such as the silicon solar cell.
Standard Physics establishes two types of well-known capacitive mechanisms. On the one hand, semiconductors possess a given polarizability, which establishes the amount of surface charge per unit area that can be stored by effect of the applied voltage. This charge is characterized by the dielectric geometrical capacitance ( g C ), and the depletion layer capacitance dl C if present (Almora et al., 2019a), and is often easily accessed by basic impedance spectroscopy techniques (IS) (Lopez-Varo et al., 2018). In addition, for some semiconductors it is also possible to storage charge and energy directly in the bulk region between the metallic plates/electrodes. This charge accumulation can be characterized by the concept of chemical capacitance ( µ C ) (Bisquert, 2003). It establishes the effect of Fermi level displacements on the occupancy change in the density-of-states (DOS) of electronic charge carriers. For a small perturbation regime, and assuming that the output voltage easily connects to the internal Fermi level, the response accounting for charge storage is essentially capacitive in nature (Bisquert, 2014). This is particularly observed when effective electrical fields are shielded, either because a large density of majorities or compensating mobile charges.

Chemical capacitance measuring conditions
The first necessary condition to be fulfilled by a given semiconductor material in order to assure the observation of chemical capacitance features relates to the competition between dielectric relaxation time and carrier recombination time. Long relaxation times, of the order of or exceeding recombination times, entails carriers slowly achieve steady-state conditions. As chemical capacitance is measured by small perturbation of a given steady state, it is necessary that the carriers in the semiconductor bulk dielectrically relax before measurement. This also relates the presence or absence of extended quasi-neutral regions within the active layer bulk.
The dielectric relaxation time can be expressed in terms of the material permittivity 0 εε (being ε the dielectric constant, and 0 ε the vacuum permittivity), and its conductivity σ as 0 die The occurrence of quasi-neutral regions relies on the property that carrier lifetime (recombination time, rec τ ) is much larger than the dielectric relaxation time die τ . If die rec τ τ > mobile carriers can exist long enough to neutralize net charge and suppress space-charge regions.
Depending on whether die rec τ τ > (lifetime semiconductor regime), or die rec τ τ < (relaxation semiconductor regime) the physics governing the device operation changes drastically (Fonash, 2010). In the relaxation semiconductor regime, electro-neutrality is not a justifiable assumption.
Therefore, space charge of dielectric relaxation-dependent decay is allowed to occur (van Roosbroeck and Casey, 1972). Low-conductivity amorphous p-i-n SiSCs develop photogenerated hole space charge regions near the p contact that concentrate the voltage drop (Schiff, 2003). In the extreme case currents should be space charge limited as occurring in organic lightemitting diodes based on low-mobility polymers or molecules (Bozano et al., 1999).
A device will be then functioning in lifetime or relaxation regime depending on the materials properties and operating conditions. For doped enough semiconductors,   (2) 9 being q the elementary charge. The general expressions for electron n and hole p concentrations derive from Boltzmann statistics, electro-neutrality conditions and the massaction law, is written as (Brendel, 2005) Here, D N and A N accounts for the ionized donor and acceptor impurity concentrations, i n is the intrinsic carrier density, and T k B represents the thermal energy. Excess carrier concentration produced by light irradiation or bias voltage can be accessed by exploring chemical capacitive effects. The chemical capacitance (Bisquert, 2014), also known as diffusion capacitance (Sze and Ng, 2007), informs on the occupancy of conduction band bulk electrons (Bisquert et al., 2004) as where the capacitance is given per unit area. Since at forward bias or under usual illumination levels the device can work in high-injection conditions, the occupancy change of valence band bulk holes gives rise to an additional capacitance per unit area as Accordingly, from equation (4) and taking into account equation (3) (Brendel, 2005), which relates the splitting of the quasi-Fermi levels and the carrier concentrations at a given temperature, one can take directly the partial derivatives and the capacitance per unit area can be expressed as (Mora-Seró et al., 2008) 2 2 2 2 exp 2 exp 2 And similarly, in the case of bulk holes Here lengths n L and p L account as corrections on the effective quasi-neutral region widths, in relation to the bulk thickness L , due to asymmetric partial derivatives effects or deviations from equation 3.
It is important to note that equation (5) Therefore, a transition between two regimes in the exponential of the open-circuit voltage (Fermi level splitting) of equation (2). It is important to stress that equation (5) will be readily of application in open-circuit conditions, since this maximizes the charge storage in the absorbing material by suppressing dc current flow and transport gradients.
We also note that the introduced expressions assume large carrier mobility, homogeneous carrier density profiles and perfectly selective contacts (Mora-Sero et al., 2009).
The experimental observation of the chemical capacitance is also restricted by resistive processes occurring in series with respect to it. As equation (5) refers to the total frequency-independent chemical capacitances of the device, the practical measurement of this capacitances by IS depends on the coupled resistance ( R ), which may shift the spectra toward lower or higher frequencies, depending on the measuring response frequency ( ) In other words, the larger the resistivity, the lower are the frequencies at which chemical capacitances could be measured, as illustrated in Fig. 2.

Chemical capacitance in Si solar cells
For SiSCs, the relationships of equation (5)   Well-separated electron and hole chemical capacitances are not always observable, and one of the chemical capacitances dominate (Almora et al., 2017a;Mora-Sero et al., 2009). Only when the chemical capacitive effect is measured in a sufficiently wide frequency window, the unambiguous extraction of the capacitance parameters is feasible. It is also noted that the chemical capacitance response is masked by the capacitive contribution of the contact, as observed in Fig. 3 for low voltages . The space-charge depletion zone in the vicinity of the contact produces an additional capacitance in excess of the chemical capacitance that is therefore invisible.
In other cases is even possible observing the regime change in the exponential slope from (Almora et al., 2017a), as shown in Fig. 4a for a Si/V2O5 heterojunction device.
Furthermore, Fig. 4b presents the spectra corresponding to a Si/MoO3 heterojunction solar cell in the analogue situation to Fig. 4a. There, the characteristic capacitance plateaus around 1 kHz in the capacitance spectra exhibit growing values by more than two orders of magnitude under light irradiation, as expected from equation (5) in the corresponding oc V range. The effect of the diminishing series resistance is also evidenced as a reduction in capacitance at higher frequencies.
Interestingly, another consequence of equation (5) is that in short-circuit condition ( no increment in the capacitance is expected. This is nicely confirmed in the experiment, as shown in Fig. 4c for the capacitance of both a Si/V2O5 device and a commercial Si homojunction solar cell at different illumination intensities. Only a slight increase of a few nF cm -2 appears, a typical the geometrical capacitance thermal effect. All these evidences obviously agree with the recombination character of doped silicon as used in solar cell technologies. A simple calculation considering equation (1), and doping densities approaching 10 15 cm -3 for monocrystalline Si, yields relaxation times approximately equal to 7 ≈ die τ ns, which is orders of magnitude shorter than typical recombination times in the range of µs.

The unexpected capacitive phenomena of perovskite solar cells
Perovskite solar cells most commonly exhibit two capacitive responses as a function of the measuring frequency (Fig. 5a). A transition occurs between geometrical capacitances dominating at high frequencies to a light-and voltage-dependent low-frequency increment. As already noted, capacitance increments following an exponential law with voltage have been typically understood in terms of chemical capacitances. Therefore, it is in principle appealing trying to correlate the light-induced increase of the low-frequency (0.1-1 Hz) capacitance usually observed for perovskite-based solar cells (Fig. 5a) with the occurrence of chemical capacitance features. Note that the exponential increase of capacitance does not take place above 100 Hz as it is the case of SiSCs (see the example of Fig. 4b). Some experimental data show a slightly enhanced capacitance in PSCs for the intermediate frequency plateau of the capacitance spectra. This is also illustrated in Fig. 5a in which a capacitive plateau appears at 3 4 10 10 Hz − that increases with light. However, the increment hardly attains one order of magnitude, certainly small for the exponential dependences of equation (5), and several measurement factors could be considered as contributing to it (Almora et al., 2018a).
The light-induced exponentially-growing low-frequency capacitance ( Lf C ) at open-circuit is illustrated in Fig. 5b for a set of PSCs comprising 3D perovskite layers based on CH3NH3PbI3 and a variety of interlayers (2D perovskite thin capping). By examining Fig. 5b, it also noted that the ground capacitance 0 Lf C from 0 to 0.4 V is not g C , but a different surface capacitance possibly related with electrode polarization (irrespective of the bulk thickness), as introduced in previous papers (Almora et al., 2015).
At a first glance, the slope equaling Fig. 5b does not result conflictive with equation (5). Certainly, for low doped semiconductors and in high-injection (high enough illumination) it is derived that , and therefore equation (5) approximates to 2 exp 2 2 However, it is certainly not the case for perovskite solar cells, as next explained. Generally in CH3NH3PbI3-based PSCs, the low-frequency capacitance has been found (Almora et al., 2018b;Contreras-Bernal et al., 2017;Zarazua et al., 2016b;Zarazua et al., 2017) to follow , with exactly the exponential slope  . A value certainly small in comparison to the measured low-frequency capacitances, and slightly larger than that attained by the geometrical capacitance that depends on the dielectric properties of the perovskite as A second reason reinforcing this conclusion is the very low frequencies (0.1-1 Hz) of observation, hardly connected to a pure electronic bulk process (Almora et al., 2019a).
In addition, equation (6) also fails to predict the linear increase of the low-frequency capacitance (in short-circuit conditions) with the light intensity or photocurrent, as shown in Fig. 5c for several PSCs including different material absorbers and absorber thicknesses. This behavior is drastically different to that occurring in SiSCs (see Fig. 5c) and is considered a distinctive feature of PSCs.

Ionic-electronic coupling
Provided the physical inconsistency of attributing a bulk-related chemical capacitance origin to the low-frequency capacitive response, we can infer that (i) chemical capacitance are not observed at all in PSCs or (ii) equation (5) (Li et al., 2018). Here, the drift of vacancies modulate the local electron and hole concentration and consequently the steady-state electronic properties. Other proposals consider a rather exotic light-enhanced ionic conductivity by several orders of magnitude, affecting the overall electrical response of PSCs under illumination (Kim et al., 2018), while the capacitance is explained by stoichiometric (ionic) polarization effects (Gregori et al., 2016). Previous works (Pockett et al., 2017) also suggested that the recombination mechanism in PSCs depends on the ionic environment, in such a way that ionic vacancies move to increase the recombination resistance on the ms to s timescale. It has been argued (Jacobs et al., 2018)

Perovskite semiconductor regime
The previous discussion leads us to consider if perovskites used as solar absorbing materials may be considered as relaxation semiconductors or not. Given the dielectric constant of  (Shao et al., 2014). Thus, suggesting that the chemical capacitance in PSCs is not observed because of the low effective density-of-states of the conduction band, as aforementioned, even satisfying the recombination regime.

Concluding remarks
In summary, contrary to that reported for silicon solar cells capacitances extracted in perovskite solar cells can hardly be interpreted in terms of the established formulation of chemical capacitances. Neither the low-nor high-frequency capacitive responses show the expected exponential trends and values. Here we propose two different explanations: (i) the low effective density-of-states of the conduction band that implies chemical capacitances presumably masked by the geometrical capacitance values as g C C ≈ µ . (ii) The fact that perovskite compounds can be considered as relaxation semiconductors ( rec die τ > τ ) with long times for carriers to slowly achieve steady-state conditions. For this last explanation to be satisfied, perovskites should be rather intrinsic or slightly doped semiconductors. (iii) It cannot be completely discarded that mobile ions influence the electronically originated capacitance of so as to situate its occurrence in the low-frequency part of the spectra.