Light Capacitances in Silicon and Perovskite Solar Cells

1 Light Capacitances in Silicon and Perovskite Solar Cells Osbel Almora a, b, c and Germà Garcia-Belmonte a, * a Institute of Advanced Materials (INAM), Universitat Jaume I, 12006 Castelló, Spain b Institute of Materials for Electronics and Energy Technology (i-MEET), Friedrich-AlexanderUniversität Erlangen-Nürnberg, 91058 Erlangen, Germany c Erlangen Graduate School in Advanced Optical Technologies (SAOT), Friedrich-AlexanderUniversität Erlangen-Nürnberg, 91052 Erlangen, Germany Abstract 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 2 explanation directions are proposed here for advancing in the understanding of this challenging photovoltaic technology. Keywords: Capacitance, light, solar cells, silicon, perovskites, dielectric relaxation. Corresponding author: g[email protected] 3 1. Introduction In recent years, perovskite solar cells (PSCs), incorporating hybrid metal halide perovskite compounds as methylammonium lead iodide (MAPbI 3 ) (Kim et al., 2012; Lee et al., 2012; Yang et al., 2015) and other combinations (Saliba et al., 2016), 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 (Almora et al., 2016). 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 shortand 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 4 early noted that even in the dark, PSCs shows relatively large capacitances (~50 µ F 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; Li et al., 2016; 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. 5 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. Here we survey on the established theories about common capacitive features under illumination observed in current photovoltaic technologies. Especially important is the discussion on the conditions of observation of the chemical capacitance. As subsequently explained, the capacitive response of silicon solar cells (SiSCs) is easily connected to the charge carrier DOS occupancy and, consequently, to that predicted by the chemical capacitance framework. After that, we comment on the new phenomena and state of the art research regarding the trending field of perovskite solar cells (PSCs), in terms of its unique capacitive response. We outline here their distinctive features with respect to the 6 well-established SiSC technology and point out apparent similarities and differences concerning the observation of light capacitances of chemical origin. 2. 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 εε τ σ = (1) 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 ττ > mobile carriers can exist long enough to neutralize net charge and suppress space-charge regions. Depending on whether die rec ττ > (lifetime semiconductor regime), or die rec ττ < (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 7 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, rec die τ τ << because enhanced conductivity, so as to shrink the depletion zone built in the vicinity of the contacts at operation voltages close to the maximum power point. This leaves extended neutral regions within the semiconductor bulk in which transport of minority carrier should be controlled by diffusion (Burgelman and Grasso, 2004) while the high conductivity of majority carriers assures rapid shielding. Fig. 1 illustrates these operating regimes in semiconductor devices. Also in terms of characteristic lengths, the relaxation semiconductor regime ( rec die τ > τ ) can in turn be expressed as µ λ L D > , which indicates longer Debye lengths die τλ µ D D = than diffusion lengths L µ , being D µ the carrier diffusion coefficient. 8 Fig. 1. Scheme on semiconductor operating regimes regarding the relationship between characteristic semiconductor times. Existence of quasi-neutral regions inside the semiconductor bulk is then a necessary condition to unambiguously measure capacitances of chemical origin. The archetypical photovoltaic device showing chemical capacitances is silicon p-n solar cell. Wide absorbing electro-neutral regions of width L storage carriers under illumination and forward voltages. This effect is maximized in open-circuit when no dc current is allowed to flow. Lightor voltage-induced occupancy changes of conduction band electrons and valence band holes imply the splitting of quasi-Fermi levels (electron and hole) that, in the simplest case of large enough conductivities, is related to the output voltage as FpFn EEqV − = . (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) 2 2 exp 2 2 Fn Fp D A D A i B E E N N N N n n k T −   − −   = + +         (3a) 2 2 exp 2 2 Fn Fp A D A D i B E E N N N N p n k T −   − −   = + +         (3b) Here, D N and A N accounts for the ionized donor and acceptor impurity concentrations, i n is the intrinsic carrier density, and Tk 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 2 n Fn dn C q L dE µ = (4a) 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 2 p Fp dp C q L dE µ = − (4b) 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 16 short-circuit for the cell in (a) and a commercial mono crystalline SiSC. Adapted with permission from ref. (Almora et al., 2017a), Copyright (2017) Elsevier. 4. 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 lightand 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 CH 3 NH 3 PbI 3 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 17 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 Tk B 2/1 in 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 )/exp( BDA, TkEEnN FpFni − << , and therefore equation (5) approximates to 2 exp 2 2 Fn Fp i B B E E q L C n k T k T µ −   =     (6) However, it is certainly not the case for perovskite solar cells, as next explained. 18 Fig. 5. (a) Capacitance spectra as a function of oc V for a CH 3 NH 3 PbI 3 -based perovskite solar cell. (b) Low-Frequency capacitance for several devices based on CH 3 NH 3 PbI 3 and a variety of interlayers (2D perovskite thin capping) also as a function of oc V . Adapted with permission from ref. (Almora et al., 2018b), Copyright (2018) Elsevier. (c) Capacitance directly proportional to illumination intensity in short-circuit condition for several perovskite solar cells. Adapted with permission from ref. (Zarazua et al., 2016b), Copyright (2016) American Chemical Society. 19 Generally in CH 3 NH 3 PbI 3 -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 )2/exp( B TkVC ∝ , with exactly the exponential slope Tk B 2/1 , as shown in Fig. 5b. The capacitance usually attains values as high as -2 100 mF cm C≈ ⋅ for illumination intensity approaching 1 sun with 1.0 V oc V ≈ . For a typical perovskite absorbing layer of L = 300 nm, one can infer from equation (6) that 12 -3 10 cm i n≈ . But this large intrinsic carrier density is in conflict with the value of 5 6 -3 10 -10 cm i n≈ inferred for the parabolic band approximation )2/exp( BVC TkENNn gi −= , with effective conduction C N and valence V N band carrier densities of order 10 18 cm -3 (Giorgi et al., 2013; Menéndez-Proupin et al., 2014) and band-gap 1.6 eV g E ≈ (Bisquert et al., 2016; LopezVaro et al., 2018). Several works have confirmed the low effective DOS of the conduction band in perovskites by Burstein-Moss shift (Manser and Kamat, 2014). It is then inferred that the low effective DOS values make it difficult to assign the low-frequency capacitive mechanism to the storage of excess photo-generated carriers occurring in the perovskite semiconductor bulk. In fact, by using equation (6) and 5 6 -3 10 -10 cm i n≈ , one would expect chemical capacitances of only of the order of 6 -2 10 F cm C µ − ≈ for 1.0 V oc V ≈ (Bisquert et al., 2016). 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 LC g / 0 εε = . 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 20 drastically different to that occurring in SiSCs (see Fig. 5c) and is considered a distinctive feature of PSCs. 5. 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) should be generalized in order to include extra influences on the effective DOS occupancies and charge distributions, e.g. mobile ions, even with a change of underlying mechanism. For instance, the low-frequency light capacitance features have been recently connected to the formation of electronic accumulation zones near the outer contacts (Zarazua et al., 2016a). That model restricts the electronic space charge to thinner interfacial regions able to accumulate large amount of majority carriers, giving rise to the observed capacitance features. However, the issue of the commonly observed characteristic time scale (~1 s) remains unsolved. Such a slow time constant would indicate that mobile ions participate in the kinetics of electronic accumulation, although a comprehensive model integrating ion dynamics and electronic mechanisms still lacks. A few studies have pointed in this direction. The coupling between ion movement and surface polarization is specifically proposed, signaling that carriers show a retarded dynamics due to the slow relaxation of the accompanying ionic charge (Ravishankar et al., 2017). In fact, recent experiments probing switched photoluminescence in MAPbI 3 with symmetrical electrodes explain the time dependence of the electrical current by ion-induced modification of the doping profile, as shown in Fig. 6 (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 21 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). FIG. 6 Observation of a dark advancing front in a light-soaked perovskite. (a-f) Time-dependent PL images of a perovskite film CH 3 NH 3 PbI 3–x Cl x under an external electric field (~2 × 10 4 V m –1 ). The “+ ” and ‘–‘ signs indicate the polarity of the electrodes. The excitation intensity is ~35 mW cm –1 with a wavelength of 440 nm, and the exposure time per image is 200 ms. The channel length is ~150 µm. z(t) represents the PL quenched areas. The scale bar represents 100 μm. The color bar is a gray value with arbitrary units, indicating the PL intensity. (g, h) Electrical current I and 1/I 2 , respectively, monitored as a function of time during the measurement of experiment (a). Reproduced under Creative Commons license (http://creativecommons.org/licenses/by/4.0/) from ref. (Li et al., 2018), Copyright (2018) Nature Publishing. 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) that the coupling effect has an influence at low frequencies, as carrier recombination current becomes phase-delayed with respect to the voltage perturbation being responsible for observations of photoinduced capacitance. In this view, low-frequency capacitance is caused by out-of-phase 22 carrier recombination instead of charge accumulation currents. Sophisticated equivalent circuits (ionically gated transistor interface) have been proposed (Moia et al., 2019) accounting for the ionic-electronic interplay governing charge injection and recombination. Similar arguments appear very recently but now disregarding ionic effects and highlighting delayed electronic injection currents as the sole cause of the low-frequency capacitive phenomena (Ebadi et al., 2019). These very recent studies show the still open character of this issue: whether large capacitive effects are explained exclusively from ionic or electronic mechanisms, or more likely an interplay between ionic kinetics and electronic charge accumulation/recombination is occurring (Jacobs et al., 2018; Moia et al., 2019; Ravishankar et al., 2017). 6. 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 perovskites 35 25 − ≈ ε (Almora et al., 2017b), and also the large carrier mobility values reported for both electrons and holes in the range of 2 -1 -1 1 100 cm V s µ ≈ − (Herz, 2017), small carrier density are enough in order to fulfill the relaxation semiconductor regime. Typical dielectric relaxation time values are plotted in Fig. 7 assuming µ σ qn = as a function of the carrier concentration n . As expected, long die τ is inferred as doping level approaches intrinsic values. In order to elucidate if rec die τ > τ (relaxation semiconductor regime), an estimation of the recombination time is needed. Typical recombination times for perovskite materials can be calculated by recalling the long diffusion lengths µ L reported in the literature exceeding 1 μ m (Stranks et al., 2013). From the known relation between recombination time and diffusion length rec τ µµ DL = and assuming the Einstein expression for the diffusion coefficient µ µ TqkD B = , 23 one can readily calculate the carrier density for which dierec ττ = that results in 13 -3 4 10 cmn≈ × , independently of the carrier mobility. This last carrier concentration entails that doping levels of order 13 -3 10 cm< easily result in satisfying recdie τ >τ , a regime for which the chemical capacitance does not operate. This low value for the carrier density would signal the rather intrinsic (low doping) character of the perovskite compounds used as absorbing semiconductors in photovoltaics. This low value for the carrier density is in conflict with the doping levels usually encountered for solution-processed polycrystalline perovskite layers used in solar cells (~10 16 -10 17 cm -3 ) (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. FIG. 7 Calculation of the dielectric relaxation time as a function of the carrier density for different mobility values in the range of 2 -1 -1 1 100 cm V s µ ≈ − for 30≈ ε , with an indication of the carrier density delimiting semiconductor regimes. 24 7. 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 lownor 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 CC ≈ 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. ORCID Osbel Almora: https://orcid.org/0000-0002-2523-0203 Germà Garcia-Belmonte: https://orcid.org/0000-0002-0172-6175 Author contributions O.A. and G.G.B. jointly wrote the manuscript. Acknowledgments We acknowledge funding from MINECO of Spain under Projects MAT2016-76892-C31-R. O.A. acknowledges the financial support from the VDI/VD Innovation + Technik GmbH (Project-title: PV-ZUM) and the SAOT funded by the German Research Foundation (DFG) in the framework of the German excellence initiative. Notes The authors declare no competing financial interest. 25 References Almora, O., Aranda, C., Garcia-Belmonte, G., 2018a. Do capacitance measurements reveal lightinduced bulk dielectric changes in photovoltaic perovskites? J. Phys. Chem. C 122(25), 13450-13454. https://doi.org/10.1021/acs.jpcc.7b11703. 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