Avoiding Ionic Interference in Computing the Ideality Factor for Perovskite Solar Cells and an Analytical Theory of Their Impedance-Spectroscopy Response

Impedance spectroscopy (IS) is a straightforward experimental technique that is commonly used to obtain information about the physical and chemical characteristics of photovoltaic devices. However, the nonstandard physical behavior of perovskite solar cells (PSCs), which are heavily influenced by the motion of mobile ion vacancies, has hindered efforts to obtain a consistent theory of PSC impedance. This work rectifies this omission by deriving a simple analytic model of the impedance response of a PSC from the underlying drift-diffusion model of charge-carrier dynamics and ion-vacancy motion, via an intermediate model that shows extremely good agreement with the drift-diffusion model in the relevant parameter regimes. Excellent agreement is demonstrated between the analytic impedance model and the much more complex drift-diffusion model for applied biases (including both open circuit and the maximum power point at 0.1 and 1 Sun) close to the cell’s built-in voltage $V_{\mathrm{bi}}$. Both models show good qualitative agreement to experimental IS data in the literature and predict many of the observed anomalous features found in impedance measurements on PSCs. The analytic model provides a practical and useful tool with which to interpret PSC impedance data and extract physical parameters from IS experiments. We define a physical parameter, $n_{\mathrm{el}}$ (the electronic ideality factor), that is of particular significance to PSC physics, since, in contrast to the apparent ideality factor, the value of $n_{\mathrm{el}}$ can be used to identify the dominant source of recombination in the cell independent of its ionic properties.

Figure 1

(a) Nyquist, (b),(d) frequency plots with key impedance parameters labeled for a typical PSC impedance spectra exhibiting two features. (c) An $RC$-$RC$ equivalent circuit used to extract resistances and capacitances. The labeled frequencies are related to the angular frequencies via $f=\omega /2\pi$. The low-frequency semicircle can lie below the axis on the Nyquist plot, resulting in negative LF resistance and capacitance values (see, for example, Refs. [25, 26]).

Figure 2

(a) Diagram illustrating the potential drops $V_1$-$V_4$ across a perovskite solar cell at steady state. Inset shows the charge contained within the perovskite space-charge layers at steady state. (b) Diagram illustrating the potential across a PSC with a nonzero uniform bulk electric field after a rapid reduction to the applied voltage.

Figure 3

Comparison between solutions obtained for the electron and hole densities from the full drift-diffusion model [Eqs. (1)–(15) of Ref. [41] ] and the Boltzmann approximation. In both cases $V_{\mathrm{dc}}=V_{\mathrm{OC}}$. Left: carrier density at equally spaced intervals over a low-frequency period (1 mHz). Right: the equivalent but over an intermediate and high-frequency period (25 kHz). The parameters used are detailed in Table 1, under 0.1-Sun equivalent illumination and with recombination at the ETL/perovskite interface ($R_l$) from Table 2.

Figure 4

Simulated impedance spectra for a PSC with recombination at the ETL/perovskite interface ($R_l$), under 0.1-Sun equivalent illumination and with different anion vacancy densities $N_0$. Spectra for three dc voltages are shown, including at open circuit, $V_{\mathrm{dc}}=V_{\mathrm{OC}}$ [i.e., 0.93 V in (a) and 0.91 V in (b)]. The impedance perturbation amplitude is 10 mV, recombination parameters are listed in Table 2, while other cell parameters are listed in Table 1. Frequency plots for (a),(b) are presented in Figs. 5 and S4 within the Supplemental Material [64], respectively.

Figure 5

Real ($R$) and imaginary ($X$) components of impedance versus frequency for the spectra presented in Fig. 4.

Figure 6

Simulated impedance spectra at open circuit with different recombination mechanisms. $R_n$, electron-limited bulk SRH ($V_{\mathrm{OC}}$=0.94 V); $R_p$, hole-limited bulk SRH ($V_{\mathrm{OC}}$=0.92 V); $R_b$, bimolecular bulk recombination ($V_{\mathrm{OC}}$=0.95 V); $R_r$, perovskite and HTL interfacial ($V_{\mathrm{OC}}$=0.95 V). Cell and recombination parameters as listed in Tables 1 and 2, respectively.

Figure 7

Real ($R$) and imaginary ($X$) components of impedance versus frequency for the spectra presented in Fig. 6.

Figure 8

Electronic ideality factor, $n_{\mathrm{el}}$, and apparent ideality factor, $n_{\mathrm{ap}}$, calculated at different open-circuit voltages (corresponding to different illumination intensities) from impedance spectra obtained analytically and numerically. Calculated from the same spectra as used for Fig. S13(f) within the Supplemental Material [64]. Parameters used to calculate the numerical and analytic spectra are those from Table 1 for a cell with hole-limited interfacial recombination ($R_l$).

Figure 9

Electronic ideality factor and the apparent ideality factor calculated from impedance spectra at open circuit for the five recombination types considered in this work and cell parameters from Table 1. The rightmost entry is for multiple recombination mechanisms, as displayed in Fig. 12.

Figure 10

Illustration of the potential across a PSC during high-frequency (above approximately $100$ Hz) impedance measurements. The ion distribution is unable to adjust over the short timescale of a period and so the potential drops across the space-charge layers $V_{1-4}$ remain in their steady-state configuration. Therefore, at high frequencies the applied voltage results in an approximately uniform oscillating electric field $E(t)$ across the perovskite.

Figure 11

Illustration of the effect of an oscillating potential difference on the distribution of the electric potential across a PSC during impedance measurements at ultralow frequencies (below approximately $10$ mHz). The slow modulation of applied voltage allows ionic charge to fill and deplete the perovskite space-charge layers in phase with the applied potential. The charging and discharging of space-charge layers effectively screens the bulk electric field.

Figure 12

Simulated impedance spectra for a PSC with four types of recombination under 0.1-Sun equivalent illumination. Spectra at three dc voltages are shown, including $V_{\mathrm{OC}}=0.88$ V. Specifically, the recombination taking place is bimolecular ($R_b$) and hole-limited SRH ($R_p$) in the bulk and at both the ETL/perovskite ($R_l$) and perovskite/HTL ($R_r$) interfaces. Recombination parameters for each type are specified in Table 2 and the additional parameters are given in Table 1. Figure S9 within the Supplemental Material [64] presents the corresponding frequency plot for this Nyquist diagram.