Multilayer Capacitances: How Selective Contacts Affect Capacitance Measurements of Perovskite Solar Cells

Capacitance measurements as a function of voltage, frequency, and temperature are a useful tool to gain a deeper insight into the electronic properties of semiconductor devices in general and of solar cells in particular. Techniques such as capacitance-voltage, Mott-Schottky analysis, or thermal-admittance spectroscopy measurements are frequently employed in perovskite solar cells to obtain relevant parameters of the perovskite absorber. However, state-of-the-art perovskite solar cells use thin electron- and hole-transport layers to improve the contact selectivity. These contacts are often quite resistive in nature, which implies that their resistance will significantly contribute to the total device impedance and thereby also affect the overall capacitance of the device, thus partly obscuring the capacitance signal from the perovskite absorber. Based on this premise, we develop a simple multilayer model that considers the perovskite solar cell as a series connection of the geometric capacitance of each layer in parallel with their voltage-dependent resistances. Analysis of this model yields fundamental limits to the resolution of spatial doping profiles and minimum values of doping and trap densities, built-in voltages, and activation energies. We observe that most of the experimental capacitance-voltage-frequency-temperature data, calculated doping and defect densities, and activation energies reported in the literature are within the derived cutoff values, indicating that the capacitance response of the perovskite solar cell is indeed strongly affected by the capacitance of its selective contacts.

Popular Summary

Capacitance methods, such as capacitance-voltage-frequency measurements, Mott-Schottky analysis, and thermal-admittance spectroscopy measurements, are powerful tools to obtain important parameters of the solar cell, such as doping and defect densities, built-in voltages, and activation energies. However, the validity of these analyses assumes that the capacitance response originates solely from the absorber layer. Here, the authors demonstrate that this assumption is not valid for perovskite solar cells, since the thin and low-mobility selective-contact layers significantly contribute to the measured capacitance. Using a combination of drift-diffusion simulations and analytical modeling, they develop guidelines for the measurement of doping and defect densities, built-in voltages, and activation energies from these capacitance methods. These guidelines can be applied to any photovoltaic technology that incorporates low-conductivity charge-transport layers in addition to the absorber layer.

Figure 1

(a) Band diagram of the reference PSC upon which the simulations of this paper are based (see Table S1 for parameters). (b) EC model of the solar cell in the dark. $R_s$, $R_{\mathrm{HTL}}$, $R_{\mathrm{pero}}$, and $R_{\mathrm{ETL}}$ are the series resistance and voltage-dependent resistances of the individual layers, and $C_{g,\mathrm{HTL}}$, $C_{g,\mathrm{pero}}$, and $C_{g,\mathrm{ETL}}$ are the geometric capacitances of the individual layers. (c) Simulated capacitance evolution versus measurement frequency for the EC of (b), with $(C(Z))$ and without $(C(Z-R_s))$ a series resistance, for well-separated time constants of each $R||C$ element. Three plateaus in the capacitance are observed at high frequency ($C_1$), intermediate frequency ($C_2$), and low frequency ($C_3$), the analytical solutions of which are derived in Sec. A3 in the Supplemental Material [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31] and shown in (c). Corresponding EC representing the capacitance plateau is shown in the inset next to each plateau. Note that the high-frequency drop in capacitance is simply a consequence of the series resistance, as derived in Sec. A3 in the Supplemental Material [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31].

Figure 2

Experimentally measured (a) capacitance versus voltage plots, (b) Mott-Schottky plots, and (c) doping profiles for $\mathrm{ITO}/\mathrm{PTAA}/{\mathrm{CH}}_3{\mathrm{NH}}_3\mathrm{Pb}({\mathrm{I}}_{0.8}{\mathrm{Br}}_{0.2}{)}_3/\mathrm{PCBM}/\mathrm{BCP}/\mathrm{Ag}$ samples with increasing thickness of the PTAA layer, as indicated in the inset. Fits of the capacitance versus voltage to the equation $C=C_g+C_0\exp (qV/m{k}_{B}T)$ to obtain the m value are shown in Fig. S8(a) in the Supplemental Material [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. Measurement frequency is 25 469 Hz.

Figure 3

(a) Mott-Schottky plots of different PSCs and a silicon solar cell reported in the literature. Corresponding calculated doping densities are shown in the label, obtained from their corresponding doping profiles [see Fig. S8(b) in the Supplemental Material 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. (b) Simulated Mott-Schottky plots of a PSC with different acceptor doping densities of the perovskite layer, as shown in the label, using scaps. Orange and red lines are simulations of an intrinsic perovskite layer with different thicknesses. Simulated Mott-Schottky plots for an intrinsic layer and dopant density of ${10}^{15}$ cm⁻³ are very similar to each other and to the experimental plots shown in (a). Literature data are obtained from refs. [35, 36, 43, 44, 45, 46], as mentioned in the corresponding label.

Figure 4

(a) Onset voltage of the space-charge capacitance as a function of doping density for different thicknesses of the absorber layer, calculated by setting the space-charge capacitance [Eq. (S1) in the Supplemental Material [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31] ] equal to the geometric capacitance. For thin-film solar cells, dopant densities in excess of ${10}^{16}$ cm⁻³ are required to observe the space-charge capacitance at reverse bias. (b) scaps simulations of Mott-Schottky plots of a PSC (without selective contacts and 0-eV injection barrier on either side) with different doping densities of the perovskite layer, as shown in the label. Arrow indicates the shift in the onset voltage of the linear Mott-Schottky region with increasing doping density.

Figure 5

Simulated evolution of (a) capacitance, (b) corresponding Mott-Schottky plot, (c) resistances, and (d) characteristic frequencies versus applied voltage based on the multilayer model developed in Sec. 2a. $V_{\mathrm{int}}\cong V_{\mathrm{ext}}$ is assumed in these simulations, which is a good approximation for the relevant voltage range in dark measurements [see fig. S6(a) in the Supplemental Material [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31] ]. Parameters are discussed in Table S2 in the Supplemental Material [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. Green and blue dashed lines in (a) correspond to the low-voltage capacitance limit, $C_{\mathrm{LV}}$, and high-voltage capacitance limit, $C_{\mathrm{HV}}$ [Eqs. (S50) and (S52) in the Supplemental Material [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31] ]. Measurement frequency is chosen to be $f_{\exp }={10}^3$ Hz.

Figure 6

(a) Plot of reported apparent built-in voltages obtained from Mott-Schottky analysis in the literature versus corresponding open-circuit voltages. Solid line indicates the region where $V_{\mathrm{BI}}=V_{\mathrm{OC}}$. Larger built-in voltages are associated with larger open-circuit voltages. (b) Simulated Mott-Schottky plots for different open-circuit voltages shown in the label, using the multilayer model. Measurement frequency is chosen to be $f_{\exp }={10}^3$ Hz. Multilayer model predicts the trend observed in (a) for the evolution of built-in voltage versus open-circuit voltage. Literature data correspond to Refs. [36, 43, 44, 45, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67]. Variation of the built-in voltage of the multilayer model as a function of frequency is shown in Fig. S13(b) in the Supplemental Material [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31].

Figure 7

(a) Simulated spatial doping profiles (from CV simulations) of a thin-film PSC for different thicknesses of the perovskite layer, using scaps and the multilayer model developed in Sec. 2a. Arrow indicates the reduction of the minimum doping density obtained in the plateau region, versus perovskite-layer thickness. (b) Comparison of experimental (from Ref. [68]) and simulated spatial doping profiles for different PSCs. Analytical approximation of the forward-bias behavior of the doping profile, as predicted by Eq. (17) ($m=1.5$ is assumed), is also shown. Slopes of experimental data match nicely with this limit. (c) Comparison of minimum doping density resolvable from capacitance measurements and calculated minimum doping densities (lowest value of plateau region of doping profile) versus implied perovskite-layer thickness from different capacitance measurements of PSCs. Blue data points correspond to DLCP measurements in Ref. [68]. Black data points correspond to CV measurements in Refs. [35, 36, 43, 44, 45, 61, 66, 67, 69, 70], where the implied perovskite-layer thickness is calculated using Eq. (S2) in the Supplemental Material [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31] (the validity of which is assumed during a Mott-Schottky analysis) with ${\epsilon }_r=30$ from the low-bias saturation-capacitance value. Black dotted lines indicate the typical thin-film perovskite-layer thicknesses used. Area between the red and green lines sets the range of the limiting doping-density value, depending on the m value of the capacitance transition. PSCs generally show large m values, see Fig. S8(a) in the Supplemental Material [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. Star indicates the only data point (from Ref. [68]) that lies above the resolution limit and the black point filled with red indicates a device without selective contacts [70].

Figure 8

(a) Experimental dark capacitance evolution versus frequency for different applied voltages for an $\mathrm{ITO}/\mathrm{PTAA}/{\mathrm{CH}}_3{\mathrm{NH}}_3\mathrm{Pb}({\mathrm{I}}_{0.8}{\mathrm{Br}}_{0.2}{)}_3/\mathrm{PCBM}/\mathrm{BCP}/\mathrm{Ag}$ PSC. (b) is a magnified plot of (a), showing an additional capacitance step (indicated by the arrow) at frequencies between 10⁻¹ and 10⁴ Hz at low forward biases. (c) scaps simulations of the capacitance-voltage-frequency behavior of a PSC [band diagram in Fig. 1]. Mobilities of the PCBM and PTAA layers are set to $\mu ={10}^{-6}\mathrm{c}{\mathrm{m}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$ in (c). (d) Simulations of the capacitance-voltage-frequency behavior from the multilayer model. $C_{\mathrm{LV}}$ corresponds to the low-voltage capacitance plateau of Eq. (S50) in the Supplemental Material [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. Multilayer model reproduces the general capacitance features observed in experiments versus applied voltage and frequency. Steep rise in capacitance at low frequencies seen in (a) is related to the mobile ionic density in the PSC, which are not included in the simulations of (c) and (d).

Figure 9

(a) Simulated evolution of capacitance versus frequency for different temperatures using the multilayer model. D1 step generally observed for n-i-p PSCs at high frequencies is labeled. (b) Comparison of TAS data reported in the literature for PSCs with simulated TAS results using the multilayer model (dashed) for the D1 step using different built-in voltages, $V_{\mathrm{BI},\mathrm{TL}}$, for the electron-transport layer. Literature data are obtained from refs. [88, 89, 90, 91, 92, 93], as mentioned in the corresponding label. Fits of simulated activation energies to the analytical solution in Eq. (22) are shown in Fig. S14 in the Supplemental Material [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. Experimental data sets from the same literature reference are shown in light and dark colors. Electron-transport-layer mobility is reduced to $2\times {10}^{-3}\mathrm{c}{\mathrm{m}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$ for the simulations in (a) and (b), while $V_{\mathrm{BI},\mathrm{ETL}}=V_{\mathrm{BI},\mathrm{HTL}}=0.1$ V is used for (a).