Utilization of Temperature-Sweeping Capacitive Techniques to Evaluate Band-Gap Defect Densities in Photovoltaic Perovskites

Capacitive techniques, routinely used for solar cell parameter extraction, probe the voltage-modulation of the depletion layer capacitance isothermally as well as under varying temperature. Also defect states within the semiconductor band-gap respond to such stimuli. Although extensively used, capacitive methods have found difficulties when applied to elucidate bulk defect bands in photovoltaic perovskites. This is so because perovskite solar cells (PSCs) actually exhibit some intriguing capacitive features hardly connected to electronic defect dynamics. The commonly reported excess capacitance observed at low frequencies is originated by outer interface mechanisms and has a direct repercussion on the evaluation of band-gap defect levels. Starting by updating previous observations on Mott-Schottky (MS) analysis in PSCs, it is discussed how the thermal admittance spectroscopy (TAS) and the deep level transient spectroscopy (DLTS) characterization techniques present spectra with overlapping or even “fake” peaks caused by the mobile ion-related, interfacial excess capacitance. These capacitive techniques, when used uncritically, may be misleading and produce wrong outcomes.

2][3] These techniques allow extracting relevant information about defect levels influencing solar cell operation and can be used to understand and optimize devices.1] Accordingly, and in relation with our previous observations on Mott-Schottky (MS) analysis in PSCs, 12 we survey on the reliability of using temperature-sweeping techniques like the thermal admittance spectroscopy (TAS) and the deep level transient spectroscopy (DLTS) for characterizing trap levels in PSCs.
The standard capacitive technique for determining defect densities and spatial distributions within the semiconductor absorbers is the MS analysis. 13Its basis lies on the assumption that a depletion layer is formed in the vicinity of the contact between doped semiconductors and/or metals.A depletion capacitance 0 / dl Cw   (being  the dielectric constant and 0  the vacuum permittivity) is associated with the modulation of the depletion layer width w by an ac perturbation V at a given dc bias voltage V (ac and dc stand for alternate and direct currents, respectively), in such a way that in the one-sided abrupt p-n junction case it is approximated as Here q is the elementary charge, bi V corresponds to the built-in voltage and N accounts for the concentration of fixed ionized defects in the space charge region which defines the conductivity of the layer.
Here, assuming no degeneracy, V N is the effective density of states in the valence band, th v is the holes thermal velocity, T is the temperature and the product is the so called attempt-to-escape frequency (see it arrow-pointed in Figure 2a for the spectrum at

K T 
).The changes in occupation of the trap levels are Within the IS approach, the TAS aims at exploring the density-of-states (DOS) ) (E g corresponding to defect levels through the shift of the so-called demarcation energy 16   where  is the angular frequency of the electrical stimulus.    ) or higher temperatures, the steady state has been achieved so as to keep the defect occupancy change in-phase with the ac modulation, and thus without energy loss.Accordingly, the admittance technique can be viewed as a true energy spectroscopy, producing a traprelated step C  in the capacitance spectrum, as apparently occurs in Figure 2a.
Electronic DOS can be easily determined from the capacitance spectrum Equation ( 4) assumes homogeneous trap distribution within the semiconductor bulk and takes L as the absorber layer thickness.Importantly for perovskite solar cells, near full depletion is often attained at zero-bias because of the relatively low doping levels or almost intrinsic character, 15,19 thus wL  and there would be a limited or null validity of Equation ( 4), and even Equation (1).Furthermore, different defect distributions and band profiles have been discussed in original analyses. 18e total trap density can be approached by several means and complex formalisms, [20][21][22] however by integration in Equation (4) over frequency (energy) one readily infers a proportionality with the excess capacitance Later from the   The Arrhenius analysis of Equation ( 2) is also used in the DLTS technique 23 but differently to TAS, here the sample is bias pulsed and the resulting capacitance transients at each T are measured at a fixed frequency, typically above MHz-range.In absence of traps the capacitance signal may follow the voltage pulse shape.However, em  peaks at a corresponding T creating the spectroscopic representation.Moreover, the trap density is given as [20][21][22][23] where 2   depending on the measurement conditions (e.g.bias).1][22] Accordingly, summing up (i) MS F cm -2 measured at 100 mHz, much larger than g C typically in the order of 50 nF cm -2 ( 400 nm L  , 28   ) 24 towards high frequencies.The accumulation of mobile ions near the electrode contact has been proposed as suitable explanation through the formation of structures of double-layer kind. 257][28] Importantly, while the presence of shunt artifacts at very low frequencies probably affects s C , the presumably ionic character of this process has been widely tackled with several optoelectronic techniques in relation with the so-called hysteresis of the current-voltage curves in PSCs. 15,29 ndeed, s C lies behind of capacitive transient currents that affect steady-state operation, thus complicating solar cell characterization and reliability. 30Also stability issues have been connected to the occurrence of ionic accumulation/reactivity at the outer interfaces, which are usually visible through featured capacitive responses at low-frequencies. 31e experimental distinction among different capacitive mechanisms in such a way that is then necessary in order to thoroughly apply the MS approach. 12tt-Schottky analysis.By examining Figure 1, one can observe two different cases of MS analysis application in PSCs.In the first example (Figure 1a), a dependence of the kind V C  2 is distinguishable from quasi-equilibrium to forward bias, which obeys Equation (1) and allows extracting bi V from the intercept of the linear response and N from the slope.At reverse bias full depletion makes the capacitance collapse to g C and at higher forward bias the exponential s C dominates.Note that the linear behavior occurring at least from zero bias towards forward bias is a necessary but not sufficient condition to validate the MS plot. 12,15,32 Tere are not too many reports showing evident dl C behavior from appropriate MS analysis in PSCs, [33][34] and Figure 1a corresponding to ref. 12 is illustrative of the typical comportment of devices with CH3NH3PbI3-xClx as absorber.6] Therefore, the extraction of much lower defect densities becomes masked by additional capacitive mechanisms. 12, 15CH3NH3PbI3 illustrating different capacitive regimens.Adapted from ref. 12 with permission from American Institute of Physics.(c) MS plots for (▲) Formamidinium-and (•) CH3NH3PbI3-based PSCs when different pre-bias durations are applied.Adapted from ref. 37 , Copyright 2018 American Chemical Society.
In the case of Figure 1b the MS analysis is unpractical. 12Here  N 10 17 cm -3 and dl C cannot be unambiguously separated from the s C that in fact dominates the capacitive response contribution at typical MS measuring frequencies (1-10 kHz).
Furthermore, it has been shown very recently that a transition between these two limiting cases is even observable through MS analysis for formamidinium-based PSCs. 37Since mobile defects coexist with fixed impurities, application of pre-bias before capacitance measurement may alter the ionic distribution and the corresponding MS curve, as shown in Figure 1c.However, the effect does not persist and the solar cell relaxes to the initial situation in which the MS analysis is not applicable. 12portantly, from the above arguments every N value extracted from MS analyses in PSCs should be disregarded without the proper checking of the overlapping of dl C with the ion-related s C , whose influence gets stronger toward lower frequencies. 15However, while it is obviously not encouraged to present bad practices reporting bi V and N from MS plots like Figure 1b as accurate values in PSCs, [38][39][40][41][42][43][44][45] what could be useful is to perform qualitative relative comparisons checking changes in the apparent bi V . 46In this case it should be noticed that the exponential increase of capacitance at forward bias (see Figure 1a,b right axes) is related with the approaching and exceeding of the flat-band potential and the consequent going to the high injection current regime.
Thermal Admittance Spectroscopy.][49][50][51][52][53][54] The usual capacitance spectra shape of PSCs, as in Figure 2a, present two main features occurring at high and low frequencies, respectively.The plateau at intermediate frequencies ( 10 RC that states the dielectric capacitance step (highlighted in Figure 2b), and not to any defect density.Also interfacial layers would have similar influence, 55 but these are well-known effects in thin film solar cells. .It should be stressed that these strongly T -dependent apparent DOS values are hardly connected to the response of any defect levels.The usual observation of abrupt absorption band tails [58][59] and the results from Hall effect measurements, 30,60 suggest that photovoltaic perovskites are slightly doped semiconductors (like in Figure 1b) or at most (like in Figure 1a).
Disagreeing, capacitance values integrating the low frequency peak after Equation (4) produce total defect densities up to ~10 20 cm -3 at room temperature, which seems a completely unphysical value.Also the significantly low values extracted for ). 3,[56][57]61Last but not least, we remark that the apparent DOS peaks do not collapse in a unique curve in Figure 2b,c  occupancy of bulk electronic DOS would entail clearly in contradiction with previous observations.Indeed, thickness-independent capacitance observed in the dark is consistent with the electrode polarization at perovskite/contact layer interfaces cause by mobile ions accumulation as widely admitted. 24Therefore, we can conclude that, at least of PSCs comprising oxide selective contacts, the large low-frequency capacitance dominates the electrical response, masking possible capacitive contributions produced by electronic transitions.As known, the use of fullerene derivatives as electron extracting layers produces a reduction in the low-frequency capacitance contribution. 63 lowering s C , its masking effect becomes less pronounced so as to reveal true defect bands within the band-gap.This is suggested by the analysis in Figure 3 that compares room-temperature capacitance spectra of PSCs with different structures comprising TiO2 and fullerene layers.While the low-frequency capacitance dominates the response for oxide-based contacts, the solar cell comprising fullerene electron extracting layers exhibits a well-defined peak around 10 4 Hz, which might be related to bulk defect responses., Copyright 2016 American Chemical Society, and from ref. 28 , Copyright 2018 Elsevier.
Deep Level Transient Spectroscopy.5][66][67][68][69] The general spectra patterns include peaks around and above 300 K in a broad range of rate windows (Hz-kHz) (see Figure 4b).Here once again the masking effect of s C excess capacitance should be noticed, as highlighted in the scheme of Figure 4a with thinner red lines.The slow evolving capacitance could generate its own peaks or shift trap-related ones.The actual appearance of these phenomena is illustrated in Figure 4c for samples with symmetric contacts; i.e. without rectifying behavior.There transient capacitance from perovskite pellets is showed following typical Gouy-Chapman ionic theory as    defined. 73From the latter, a light-modulated thermal admittance spectroscopy (LM-TAS) could be implemented by applying Equation ( 4 defect levels.These problematic may require a series of systematic studies in order to make reliable deep trap level characterizations in PSCs, particularly for those devices that exhibit less pronounced excess capacitances.In this sense, it would be strategic to check variations in the device geometry and material architecture, as well as to combine characterizations with bias and light perturbations.

Figure 1 V
illustrates typical MS plot representations, i.e. can be taken from the slope and voltage intercept of the linear behaviors, specifically in Figure1a,c.Note that Equation (1) can be modified depending on the particularities of the junction, meaning that N , bi V and even the power of the expression (0.5 square root) are effective/rough approximate values.
veryT -sensitive, modifying w and hence dl C , which can be also modified due to the quasi-static charge equilibration in the space charge region.Thus the capacitance behavior under given perturbation conditions informs on t E , t N and  .In this sense two of the most general approaches include the use of (i) sinusoidal small perturbations in impedance spectroscopy (IS) analyses and (ii) squared bias pulses for evaluation of time transients.

C
(corresponding to the trap-related capacitance step respect to dl C , or g C in full depletion) as

E
and  in an Arrhenius plot (see inset of Figure2c), taken as

N
can be obtained sweeping temperatures: (ii) TAS looks for steps in the   C  spectra in excess of dl C via IS, and (iii) the DLTS evaluates slow   Ct transient changes over dl C after bias step pulses.In the case of perovskite-based devices, capacitive responses are formed by the contribution of different mechanisms.In addition to geometrical 0 excess capacitance s C in the low-frequency part of the frequency spectrum ( 1 Hz f  , see Figure 2a).Even in the dark, s C attains values as high as 50

2 - 10 4
Hz) is determined by g C , that reduces by effect of the series resistance series R in the high-frequency part of the spectra.The capacitance increments towards lower frequencies (0.1-10 Hz), which relates to the ion accumulation effect through s C .Since spectra in Figure 2a show two capacitance steps, it is in principle appealing to connect them to the response of defect bands activated at well-separated frequency (energy) ranges.Accordingly, the capacitance analysis of Equation (4) gives rise to the hypothetical DOS drawn in Figure 2b,c.Two main peaks occur as expected.The highfrequency/low-E  signature appears greatly T -independent, which more likely responds to the coupling series g pointed in Figure2a) produce the peculiar representation of measured negative E  values in Figure2c.This is a result of the low frequencies and small activation energy which result in a very low intercept in the Arrhenius plot (Figure2b), considering that each t E requires his own E  axis.2,61In typical semiconductors, traps exhibit dissimilar parameters to those encountered here ( Figure 2c), also hardly related to trap states.These parameters, 0 s N and 0 s T , whose physical meanings are still unclear were obtained from the integration of     / ln dC d and normalized to DOS units as Equation (4).This makes feasible for future works to compare results with other ways of obtaining DOS like thermally stimulated corrents (TSC) and optical techniques as photoluminescence decay.62

Figure 2 .
Figure 2. (a) Typical capacitance spectra at varying temperatures and corresponding hypothetical DOS as a function of (b) frequency and (c) demarcation energy of a CH3NH3PbI3-xClx-based PSC, exhibiting two capacitance steps.The DOS is based on Equation (4), assuming 1.0 V bi V  and nm 200  L .In inset: (b) the Arrhenius plot extracted from the derivative peak and (c) the trap density dependence on temperature.The E  axis in (c) corresponds to the

Figure 3 .
Figure 3.Comparison of (a) capacitance and (b) corresponding DOS spectra at room relaxation times  around seconds.

Figure 4 .FE 1 s x and 2 sx
Figure 4. (a) Scheme of the basic working principle of DLTS measurement; in thinner red lines a possible masking effect of slow evolving capacitance transients likely due to ionic related processes.(b) DLTS spectra for a Csx(MA0.17FA0.83)(1-x)Pb(I0.83Br0.17)3perovskite incorporated in mesoscopic solar cells.(c) Capacitance transients from CH3NH3PbI3 pellets for different voltage pulse amplitudes.Adapted from references 64 and 25 , respectively, with permission from American Institute of Physics.

Figure 5 .
Figure 5. (a) Schematic band energy diagram, (b) charge density profile and (c) absolute charge variation upon perturbation, e.g./ dV dt , of a p-type absorber perovskite the n- type contact (left) including a deep trap state neglecting (left panel) and including (right panel) mobile ionic effects, respectively.The patterns of left panel are reproduced with thinner lines in the right panels for comparison purposes.The orange square signals the broad region where

C
the application of the MS analysis of the depletion layer capacitance in PSCs suffers a masking effect caused by interfacial and ion-related excess capacitances ( dominates the low-frequency response, it spreads to the mid/high-frequency range and overlaps additional electronically-caused defect signals.This misleads the MS analysis and creates "fake" or overlapping peaks in the spectra from TAS and DLTS measurements ( s CC  ) when characterizing deeper the voltage perturbation is actually changing the occupation of the trap level, the capacitance does not return to the