Inductive and Capacitive Hysteresis of Halide Perovskite Solar Cells and Memristors Under Illumination

The current–voltage curves of memristors exhibit significant hysteresis effects of use for information storage and computing. Here, we provide a comparison of different devices based on MAPbI₃ perovskite with different contact configurations, from a 15% efficient solar cell to a pure memristor that lacks directional photocurrent. Current–voltage curves and impedance spectroscopy give insights into the different types of hysteresis, photocapacitance, and inductance present in halide perovskites. It is shown that both halide perovskite memristors and solar cells show a large inverted hysteresis effect at the forward bias that is related to the presence of a chemical inductor component in the equivalent circuit. Based on the results, we classify the observed response according to recombination current in devices with selective contacts, to voltage-activated single-carrier device conduction in devices with symmetric contacts. These findings serve to gain an understanding of the mechanism of memristor currents in mixed ionic-electronic conductors such as halide perovskites. We establish the link in the electrical response between solar cells and memristors.

Introduction

Metal halide perovskite (MHP) is a promising photovoltaic technology that has produced very high efficiencies with solution-processed methodologies (Kim et al., 2020). MHPs can be described using an ABX₃ structure, where A = monovalent cations (i.e., methyl ammonium—MA), B = divalent cations (i.e., Pb²⁺), and X = halide anions (i.e., I⁻), with MAPbI₃ (MAPI) being the most studied configuration. In addition to remarkable electrooptical semiconductor properties, MHP semiconductors show mixed ionic-electronic conduction by ionic defect displacement (Azpiroz et al., 2015; Lopez-Varo et al., 2018; Senocrate and Maier, 2019; Zhang et al., 2020). This causes intrinsic memory effects (hysteresis) in the current–voltage ($I-V$) characteristics (Almora et al., 2016; Tress et al., 2016; Rong et al., 2017; Wu et al., 2018; Alvarez et al., 2020; Bisquert et al., 2021; Bisquert and Guerrero, 2022a) at both short and long time scales, forming an effective memristor. A memristor is a two-terminal device whose resistance depends on the history of current and voltage applied to the device. Memristors enable the storage of information by metastable modification of the device conductivity, (Pershin and Di Ventra, 2011; John et al., 2018a; Rahimi Azghadi et al., 2020; Fang et al., 2021; Kang et al., 2021; Kwak et al., 2021) and are the main candidates to realize artificial synapses for neuromorphic computation algorithms (Mehonic and Kenyon, 2016; Rahimi Azghadi et al., 2020; John et al., 2021a; Bou and Bisquert, 2021; Gogoi et al., 2021; Gong et al., 2021; Kang et al., 2021; Kwak et al., 2021; Christensen et al., 2022). Arranged in a crossbar format with a differential configuration, memristors allow synaptic propagation to be realized facilely via Kirchhoff’s Current Law and Ohm’s Law, enabling efficient in-memory computing (John et al., 2021b).

While conventional memristors are activated by purely electrical stimuli, photo-stimulation of memristive responses provides new opportunities for a variety of applications such as light-gated and electro-photo-sensitive memristors for opto-neuromorphic and arithmetic computing, (Maier et al., 2016; Tan et al., 2017; John et al., 2018b; Emboras et al., 2020; John et al., 2020; Ma et al., 2021) integrated photonic neural networks, (Stark et al., 2020; Shastri et al., 2021), and preprocessing of image data in artificial vision before the transfer to the computing unit (Yang et al., 2020; Gong et al., 2021; Kim et al., 2021). Direct processing of visual information with such optoelectronic memristors may enable simpler architectures, mitigating the need for additional electro-optical converters for signal transduction and communication. MHPs are attractive materials for developing such novel electrooptical neuromorphic platforms because of their excellent optical absorption coefficient, charge transport, bandgap tunability extending across the UV, visible, and IR spectra, and high photoluminescence (PL) quantum yield, and high extinction coefficients. Combining these outstanding electronic properties with their intrinsic ionic conduction, MHPs would allow easy implementation of electrooptical synaptic elements for information storage and computing.

MHP memristors emerge from solar cells, but their behavior in the current-voltage regime is very different. In solar cells, the inverted hysteresis is a property preferably minimized (Tress et al., 2016; Yang et al., 2017; Wu et al., 2018), while for memristors, it is amplified to permanent and reversible changes of the conductivity (Berruet et al., 2022). These characteristics have not been systematically studied yet, entailing significant attention.

Here, we aim to provide a clear understanding on the multiple features of the very complex $I-V$ curves of different devices and obtain fundamental insight about the ionic-electronic processes that cause the characteristics, in combination with insights from Impedance Spectroscopy (IS) measurements. We build different devices based on MAPI, the most simple and well-understood MHP formulation with low activation energy for ion migration that makes it suitable for memristor applications. First, we reproduce the standard solar cell (SC) configuration (FTO/c-TiO₂/m-TiO₂/MAPI/spiro-OMeTAD/Au). Two different MHP memristor devices are chosen for comparison as model systems that simplify the outer interfaces by removing the spiro-OMeTAD layer. The effect of the charge carrier selectivity of the perovskite/contact interface is studied by comparison with devices that contain PEDOT (FTO/PEDOT:PSS/MAPI/Au) and TiO₂ (FTO/c-TiO₂/mp-TiO₂/MAPI/Au). Measurements are completed in the dark and under illumination to show how the photovoltaic effects influence their operational properties. The present work establishes the link in the electrical response between solar cells and memristors. We unravel the poorly understood connection between the different regimes observed in the $I-V$ curve and the effect of light, first of its kind to the best of our knowledge.

Hysteresis of Current–Voltage

Since the hysteresis of the current–voltage (Almora et al., 2016; Tress et al., 2016; Rong et al., 2017; Wu et al., 2018; Alvarez et al., 2020; Bisquert et al., 2021; Bisquert and Guerrero, 2022a) is a central characteristic of the memristive devices, we summarize in Figure 2 on the basic idea of capacitive and inductive hysteresis as described recently (Bisquert et al., 2021; Bisquert and Guerrero, 2022a). The type of hysteresis and IS response is highly connected with the presence of capacitive and inductive currents. The hysteresis is denoted “regular” as observed in solar cells when the current of the photovoltaic quadrant is more positive in the forward scan, Figure 1B. Alternatively, hysteresis is “inverted” as observed often in memristors when the positive current of the forward scan is lower than the positive current of the backward scan, Figure 1D (Li et al., 2022; Shen et al., 2017). Regular hysteresis has previously been correlated with the presence of capacitive behavior in the IS response while the inverted hysteresis indicates the presence of chemical inductors in the response. The chemical inductor (Bisquert and Guerrero, 2022a) is a general denomination for a class of dynamical phenomena often based on a chemical reaction that produces a formal inductive response in impedance and transients without an underlying electromagnetic effect. The chemical inductor observed in halide perovskite solar cells is related to the slow ion-controlled electronic phenomena that occur at the interfaces of the devices. More details will be given in the following experiments.

FIGURE 1

FIGURE 1. Schematic response of impedance spectroscopy, current at constant sweep rate $V=st$, and hysteresis, for capacitive and inductive elements. The left panel shows the current of a constant capacitor (A), a diode with the capacitive response (B), and the impedance response (C). In the right panel, the signs and directions of the currents are schematically described for a system that is inductive at low frequency. in terms of the current-voltage (D), diode (E) and impedance (F) response.

We note that the hysteresis effects have been already observed in emerging solar cells, where ion motion is significant (Contreras et al., 2016; Elbohy et al., 2019). For example, the dye-sensitized solar cell contains a liquid electrolyte, and the strong capacitive hysteresis in $I-V$ curves is clearly observed in Figure 2, where the effect of scan rate, temperature, and use of different solvents modify the hysteresis response. Similarly, the crossing effects in the $I-V$ curves of Figure 3 show a transition between capacitive and inductive hysteresis that strongly depends on the mobile ionic species.

FIGURE 2

FIGURE 2. Normalized (reference voltage: −0.1 V) current–voltage curves of DSCs measured in the reverse scan (line) and forward scan (dash line) under a light intensity of 10 mW cm⁻² using a green LED. The influence of hysteresis-determining factors is specifically shown: (A) scan rate, (B) temperature, and (C) electrolyte solvent. Reproduced from (Contreras et al., 2016).

FIGURE 3

FIGURE 3. The forward and reverse current–voltage characteristics for DSCs fabricated with an electrolyte containing (A) NaI at 1V/s scan rate, (B) KI at 1V/s scan rate, (C) CsI at 1V/s scan rate, (D) NaI at 0.5V/s scan rate, (E) KI at 0.5V/s scan rate, (F) CsI at 0.5V/s scan rate. Reproduced with permission from (Elbohy et al., 2019).

It has been reported that the resistive switching mechanism can be modified by using different top contacts (Han et al., 2019). Here, we focus on a low reactivity Au contact which is widely used in solar cells and the switching is controlled by the interface. In halide perovskite solar cells, migrating ions that reach the external contacts generate a capacitive current as the contacts compensate for their charge providing electroneutrality in the form of a capacitor (Figure 4). In MHP, this capacitance also depends on the light intensity with a rising light-induced capacitance, as previously reported (Juarez-Perez et al., 2014; Guerrero et al., 2021). It is known that halides vacancies (V_I⁺), cations A (i.e., methyl ammonium), and metal collectors all can migrate at different rates under an external voltage bias. Halides vacancies are the fastest migrating species in MHP. When we apply a positive voltage at the FTO electrode (Figure 4A), halide vacancies will go toward gold and effectively create I⁻ excess concentration at the FTO/TiO₂ interface. The opposite is true if we apply a positive voltage at the gold contact, Figure 4B. The capacitive and inductive effects of these ions at the interfaces can be monitored by IS (Guerrero et al., 2021; Berruet et al., 2022; Taukeer Khan et al., 2022) as discussed in the latter sections.

FIGURE 4

FIGURE 4. Diagrams that show the TiO₂ and gold electrode polarization and direction of ion migration under a negative (A) and positive (B) bias in the dark, respectively, in MHP.

Results

The connection between light stimulation and production of photovoltage and photocurrent is well understood for a typical solar cell, Figure 5A. (Bisquert et al., 2004; Bisquert, 2020) Photogeneration in the semiconductor produces separated electron and hole carriers that diffuse to the metal collectors and the selectivity of contacts causes a directional photocurrent. When the voltage is increased in the forward direction, recombination is enhanced, and the photocurrent decreases until it vanishes at the open-circuit voltage ($V_{oc}$) point. In the reverse direction, there is a depletion of carriers which leaves only a small current, hence the solar cell shows a diode or rectification property. (Bisquert, 2020) However, a strong reverse bias may produce a breakdown effect with raising current in the opposite direction (Bowring et al., 2018).

FIGURE 5

FIGURE 5. Schematic of photogeneration of an electron-hole pair under illumination for (A) FTO/c-TiO₂/m-TiO₂/MAPI/spiro-OMeTAD/Au solar cell (B) optoelectronic FTO/PEDOT:PSS/MAPI/Au memristor device, (C) optoelectronic FTO/c-TiO₂/mp-TiO₂/MAPI/Au memristor device.

Figure 6 shows the device configuration of the SC device and the electro-optical response. Our PV performance of ∼15% is well aligned with reported values for the chosen formulation (Figure 6A). Hysteresis is typically observed in the current density-voltage ($j-V$) curve for this configuration with separation of the forward and backward scans even for measurements conducted in dark (Figure 6B). The hysteresis behavior is complex with marked differences as a function of the scan rates, presence of crossing points of the curves at about +0.6 V, and change in the sign of the current far away from 0 V (spikes in the semilog curves) (Li et al., 2022).

FIGURE 6

FIGURE 6. (A) Current–voltage characteristic of the FTO/c-TiO₂/m-TiO₂/MAPI/spiro-OMeTAD/Au solar cell was measured at 100 mW/cm² using a halogen-based solar simulator calibrated with a silicon photodiode. The inset table shows the characteristic parameters. Here, the photocurrent is drawn positive. (B) Current−voltage characteristics were measured under a dark condition for 5 different scan rates. Arrows indicate the sweep direction.

Figure 7 shows photovoltaic performance of the solar cell device under a monochromatic blue light at different illumination intensities, with negative current for photovoltaic effect. As expected, the photogenerated current is enhanced with light intensity thanks to the excellent carrier generation and extraction. Very interestingly, in the complete SC there is a crossing in the forward and reverse scans at $V_{oc}$ conditions with a change in tendency from regular ($V<V_{oc}$) to inverted ($V>V_{oc}$) hysteresis. In addition, the capacitive hysteresis also increases with illumination level in terms of the separation of the forward and reverse scan on the y-axis (Figure 7B). The capacitance takes values in the order of 0.2–0.9 mF/cm².

FIGURE 7

FIGURE 7. (A) Current density–voltage characteristic of an FTO/c-TiO₂/m-TiO₂/MAPI/spiro-OMeTAD/Au solar cell measured from 0 to 100 mW/cm² using a blue light source (470 nm) and 0.1 cm² active area. A positive current flow from FTO to the Au electrode. (B) A magnified view of the scale. Lines represent the forward scan, while dots represent the reverse scan. (C) Low-frequency capacitance enhancement with illumination intensity for an FTO/c-TiO₂/m-TiO₂/MAPI/spiro-OMeTAD/Au solar cell.

Figure 8 shows the $j-V$ characteristic of the solar cell device at 50 mW/cm² blue light illumination. Below $V_{oc}$, the capacitive behavior shows regular hysteresis and above $V_{oc}$ the inductive behavior is manifested in the $j-V$ as inverted hysteresis as in Figure 7A. Figures 8B,C confirm this statement, showing two capacitive arcs in the complex plane plot. Only at high dc voltages, beyond $V_{oc}$, it is possible to observe an inductive behavior (Bisquert and Guerrero, 2022a; Berruet et al., 2022), with low frequencies crossing to the fourth quadrant of the complex plane (Figure 8B inset). We note that the inductive loop is not well developed as in the next experiments clearly indicating that spiro-OMeTAD is affecting the impedance response.

FIGURE 8

FIGURE 8. (A) Current density−voltage characteristic measured at 50 mW/cm² blue light illumination for an FTO/c-TiO₂/m-TiO₂/MAPI/spiro-OMeTAD/Au solar cell. Arrows indicate sweep direction. (B–C) Corresponding complex plane plot representation of the impedance spectra at various dc voltages, measured from 1MHz to 0.1 Hz. Inset corresponds to higher dc voltages applied.

The model systems which do not contain spiro-OMeTAD have been fully characterized with two types of selective contacts, PEDOT:PSS (selective to holes), and TiO₂ (selective to electrons). The dark $I-V$ characteristics of the two memristor devices at different voltage sweep rates are shown in Figure 9. Memristor performance of the device containing Spiro-OMETAD is well aligned with previously reported results (Berruet et al., 2022). Both show inverted hysteresis loops when the voltage is positive at the Au contact similar to the SC configuration above ∼0.5 V in Figure 7A. The onset voltage for transition to the higher conductance state is larger for higher sweep rates, as explained in a recent work (Berruet et al., 2022) that will be summarized in the Model section. The loop is steeper for the PEDOT memristor, due to the fact that the compact/mesoporous TiO₂ is a better selective contact for electrons than PEDOT, which reduces the hysteresis effect in the TiO₂ memristor case. We note that some of the most recent reported memristors show performances higher than those reported here. However, top performance devices contain buffer layers (i.e., PMMA) at both perovskite/contact interfaces and reactive contacts (i.e., Ag) that complicate the device configuration and interpretation of operational mechanism. Here, we work with a simplified configuration to provide an understanding of the operational mechanism of memristors and devices that have been fabricated without a buffer layer in the perovskite/top contact interface.

FIGURE 9

FIGURE 9. Current−voltage characteristics of the (A) FTO/PEDOT:PSS/MAPI/Au and (B) FTO/c-TiO₂/m-TiO₂/MAPI/Au memristor measured under dark conditions for 5 different scan rates.

It is also important to highlight that the response in the dark is only inductive at positive voltages, there is no crossing for both model systems as compared to the SC for measurements in the dark. This result clearly points to the capacitive currents due to the migrating I⁻ ions with Spiro-OMeTAD in the dark of the SC configuration. Otherwise, only PEDOT memristors show inverted hysteresis when the voltage is negative. For TiO₂ memristors, the capacitive hysteresis is found in this region, as shown in solar cell devices in Figure 7B. This difference in PEDOT memristor performance is due to irreversible reactions taking place between ions and TiO₂ at this interface. For example, for the interface TiO₂/MAPI it has been reported that Ti−I−Pb bonds may form, as identified by Raman Spectroscopy, which can easily accommodate excess ionic charge when a negative bias is applied (Carrillo et al., 2016). This configuration could lead to neutral interfaces that would be in agreement with the low capacitance measured in the current work for the memristor containing TiO₂. Comparing these characterizations with the one previously discussed in Figure 6B, it is possible to affirm that the TiO₂ device shows closer performance to an MHP solar cell than the PEDOT device in dark conditions.

To investigate the effect of light on the memristor devices, we performed the current-voltage characterization at different illumination levels and the results are shown in Figure 10. In principle, the looping curves are like those in the dark in Figure 9. But there are also significant differences. The PEDOT device shows a symmetric opening of the curves $V=0$ for increasing light intensity, Figure 10B. This is the effect of the light-induced capacitance, previously mentioned, that is well known in halide perovskites (Juarez-Perez et al., 2014). Indeed, the measured capacitance increases with light intensity as indicated in Figure 10C. For the TiO₂ device, the photoinduced capacitive current is also observed, but it is clearly displaced to negative values of the current since the TiO₂ is selective to electrons and it is able to extract the photogenerated current. In consequence, the TiO₂ device shows diode-like characteristics and a positive photovoltage at all light intensity values. The difference between the two devices is explained by the mechanistic diagrams in Figure 5. The PEDOT device has a negligible effect on contact selectivity, hence capacitive current prevails in the photocurrent. For TiO₂ devices, the metal oxide contact acts as a selective contact that extracts preferentially electrons, hence the photogeneration gives a dominant-negative photocurrent for both signs of the voltage sweep direction. Thus, this performance is similar to the one developed by a solar cell, shown in Figure 7A.

FIGURE 10

FIGURE 10. (A) Current density–voltage characteristic of an FTO/PEDOT:PSS/MAPI/Au memristor measured from 0 to 100 mW/cm² using a blue light source (470 nm) and 0.1 cm² active area. (B) A magnified view of the scale. (C) Low-frequency capacitance enhancement with illumination intensity for an FTO/PEDOT:PSS/MAPI/Au memristor. (D) The current density–voltage characteristic of an FTO/c-TiO₂/m-TiO₂/MAPI/Au memristor was measured at different illumination intensities. Panel. (E) corresponds to a magnified view of the scale. Lines represent the forward scan and dots represent the reverse scan. (F) Corresponding low-frequency capacitance variation with illumination intensity.

For further analysis, we focus on a particular illumination, Figure 11. For PEDOT devices, it is observed that there are two loops on right and left of Figure 11A, where the current in forward and backward makes crossing pathways. As mentioned earlier in Figure 3 this property indicates a change from regular to inverted hysteresis caused by the onset of dominant inductive property in the impedance spectra (Bisquert and Guerrero, 2022a; Berruet et al., 2022). The presence of the chemical inductor (the loop in the fourth quadrant of the complex plane) is in fact obtained in the measurements of impedance spectroscopy in Figures 11B,C. While this change from capacitive to inductive behavior is shown in Figure 11E at high positive voltages, this is not obtained when negative voltages are studied in TiO₂ memristors (Figure 9B), corresponding with the loops in Figure 8D. This trend is similar to the one shown by the solar cell device, Figures 8B,C, with inductive properties significant only at high dc voltages.

FIGURE 11

FIGURE 11. (A) Current density−voltage characteristic measured at 50 mW/cm² blue light illumination for an FTO/PEDOT:PSS/MAPI/Au memristor. Arrows indicate sweep direction. (B–C) Corresponding complex plane plot representation of the impedance spectra at various dc voltages, measured from 1MHz to 0.1 Hz. Inset corresponds to higher dc voltages applied. (D) Current−voltage characteristic measured at 50 mW/cm² blue light illumination for an FTO/c-TiO₂/m-TiO₂/MAPI/Au memristor. (E–F) Corresponding complex plane plot representation of the impedance spectra at various dc voltages.

Model

In order to explain the experimental observations in a unified framework, we present a dynamical model (Berruet et al., 2022) that includes several capacitive mechanisms and a chemical inductor feature. It is formed by the system of equations

$I_{tot}=I_{ph}+i_{c0}f+\frac{u}{R_b}++C_m\frac{du}{dt}+Q_m\frac{df}{dt},(1)$

${\tau }_k\frac{df}{dt}=\left(1-f\right)-e^{-\frac{u-V_T}{V_m}}f.(2)$

The model has three independent variables: $I_{tot}$, $u$, $f$. The first two are the external current and voltage and $f$ is an occupation function ($0\le f\le 1$). The variable $f$ indicates the state of the mechanism that controls the large injection current in the activated state, with the onset potential $V_T$. $V_m$ is an ideality factor with a dimension of voltage. $i_{c0}$ is the saturation value of the injection current. Our model provides an alternative pathway to those approaches based on drift-diffusion simulations. (Li et al. 2022; Shen et al., 2017) This simple two-dimensional model already describes simultaneously the current-voltage curves and impedance characteristics. Furthermore, the model enables full analysis of time transient techniques. (Bisquert and Guerrero, 2022b).

Eq. 1 describes the components of $I_{tot}$: the photocurrent $I_{ph}$, the injection current $i_{c0}f$, a small ohmic current of constant resistance $R_b$, a capacitive charging of the interfaces with capacitance $C_m$, and a voltage-dependent capacitance correlated to $f$ with a maximum surface charging $Q_m$.

If the time derivative in Eq. 2 is suppressed, we obtain the following steady-state solution:

$f_{ss}\left(u\right)=\frac{1}{1+e^{-\left(u-V_T\right)/V_m}}.(3)$

Eq. 2 is a delay equation that states that the formation of the conduction state at a certain voltage, $f_{ss}\left(u\right)$, lags behind the changes of applied voltage with a characteristic time ${\tau }_k$. This slowing down of the response gives rise to the chemical inductor in the equivalent circuit.

The steady-state current is

$I_{app}=I_{ph}+\frac{u_{app}}{R_b}+i_{c0}{f}_{ss}\left(u\right).(4)$

In the solar cell a saturation regime $f_{ss}\approx 1$ may not be observed. Then the current in the injection regime at large voltage has the exponential form

$I_{app}\approx I_{ph}+i_{rec0}{e}^{{\mathbf{u}}_{\mathbf{app}}/V_m},(5)$

where $i_{rec0}$ is a recombination current parameter. The impedance model is obtained by small perturbation of Eqs. (1, 2). It has the expression (Berruet et al., 2022).

$Z\left(s\right)={\left[C_{m}s+R_b^{-1}+\frac{1}{R_a+L_a^{k}s}+\frac{1}{R_2+\frac{1}{C_{2}s}}\right]}^{-1},(6)$

$R_a^{-1}=\frac{d{i}_c}{du}=\frac{i_{c0}}{V_m}{f}_{ss}\left(1-f_{ss}\right),(7)$

$L_a^d={\tau }_d{R}_a,(8)$

$R_2=\frac{L_a^k{i}_{c0}}{Q_m},(9)$

$C_2=\frac{Q_m}{R_a{i}_{c0}}.(10)$

The equivalent circuit is shown in Figure 12.

FIGURE 12

FIGURE 12. Equivalent circuit model. The impedance model is obtained by a small perturbation of Eqs. (1, 2). It has the expression (Berruet et al., 2022).

Based on this model we can form several regimes of responses observed in the experimental evidence of the preceding Section.

In Figure 13 we show the effect of a constant capacitor $C_m$ that opens the steady-state $I-V$ in capacitive hysteresis as in Figures 1, 7B. Figure 14 shows the effect of voltage-activated capacitor $C_2$ that becomes active at $V_{oc}$. It causes capacitive spikes close to $V_{oc}$ as in Figures 2, 3A.

FIGURE 13

FIGURE 13. Calculation of a memristor/solar cell model. (A) Current at forward and backward scan at different rates as indicated. The grey line is the total equilibrium dc current (B) Impedance spectrum at $u=0.8$. $R_b$ = 1; $C_m=100$, $i_{c0}$ = 100, $I_{ph}=-10$, $V_T=1$, $V_m$ = 0.05, $\alpha =0$, $\left[{\tau }_m,{\tau }_d,{\tau }_k,Q_m\right]$.

FIGURE 14

FIGURE 14. Calculation of a memristor/solar cell model. (A) Current at forward and backward scan at different rates as indicated. The grey line is the total equilibrium dc current (B) Impedance spectrum $u=0.8$. $R_b$ = 1; $C_m=0.001$, $i_{c0}$ = 100, $I_{ph}=-10$, $V_T=1$, $V_m$ = 0.05, $\alpha =0$, $\left[{\tau }_m,{\tau }_d,{\tau }_k,Q_m\right]$.

Figure 15A shows the inductive hysteresis loop that is observed at a large injection current in the measuring devices. To provide a more complete description of the inductive hysteresis we extend the model as follows:

$I_{tot}=I_{ph}+\frac{u}{R_b}+i_c++C_m\frac{du}{dt}+Q_m\frac{df}{dt},(11)$

${\tau }_d\frac{d{i}_c}{dt}=i_{c0}f-i_c,(12)$

${\tau }_k\frac{df}{dt}=e^{\alpha \frac{u-V_T}{V_m}}\left(1-f\right)-e^{\left(\alpha -1\right)\frac{u-V_T}{V_m}}f.(13)$

FIGURE 15

FIGURE 15. Calculation of a memristor/solar cell model. (A) Current at forward and backward scan at different rates as indicated. The grey line is the total equilibrium dc current (B) Impedance spectrum at $u=0.95$. $R_b$ = 1; $C_m=0.001$, $i_{c0}$ = 100, $I_{ph}=-10$, $V_T=1$, $V_m$ = 0.05, $\alpha =0$, $\left[{\tau }_m,{\tau }_d,{\tau }_k,Q_m\right]$. (C) Current at forward and backward scan at different rates as indicated. $R_b$ = 1; $C_m=0.001$, $i_{c0}$ = 30, $I_{ph}=-10$, $V_T=1$, $V_m$ = 0.05, $\alpha =0.5$, $\left[{\tau }_m,{\tau }_d,{\tau }_k,Q_m\right]$.

This more general three-dimensional model has four independent variables: $I_{tot}, u$, $f, i_c$. Here the injection current $i_c$ is a variable in addition $f$. In steady-state, they satisfy the relationship

$i_c=i_{c0}{f}_{ss}.(14)$

As described before (Berruet et al., 2022), Eq. 12 represents a diffusion or migration time of ions that introduces a delay $i_c$ with respect to the external perturbation, by the characteristic time constant ${\tau }_d$. This temporal parameter indicates the time necessary to establish the configuration of high $f$ that produces the large electronic current $i_{c0}$. Eq. 13 is an extension of (2) with a Tafel coefficient $\alpha$. It is observed in Figure 15C that the expanded model provides different shapes in which the go and return currents make parallel tracks.

Finally, in Figure 16 the capacitive hysteresis is combined with an inductive hysteresis at high voltage. There is a crossing point from capacitive to inductive hysteresis in Figure 16, as observed in Figure 7A. In the impedance spectra, the low-frequency capacitive arc is transformed into an inductive feature in the fourth quadrant of the complex plane.

FIGURE 16

FIGURE 16. Calculation of a memristor/solar cell model. (A) Current at forward and backward scan at different rates as indicated. The grey line is the total equilibrium dc current (B) Impedance spectrum at $u=0.88$ and (C) $u=0.95$. $R_b$ = 1; $C_m=10$, $i_{c0}$ = 100, $I_{ph}=-5$, $V_T=1$, $V_m$ = 0.05, $\alpha =0$, $\left[{\tau }_m,{\tau }_d,{\tau }_k,Q_m\right]$.

Discussion

The previous analysis shows that a voltage-activated current with capacitive and inductive kinetic properties provides a satisfactory explanation of the current-voltage hysteresis and impedance properties of the different devices studied. We obtained a good description of the experimental results. Models with very few variables describe a large span of observed phenomena. The question is to give a definitive interpretation to the transformation equations in terms of physical mechanisms. One recent example is the combination of bidirectional photocurrent and hysteresis that has been discussed by modification of the contact layer. (Li et al. 2022).

We considered three types of devices: a solar cell with excellent selective contacts, a PEDOT memristor with both contacts selective to holes, and a hybrid TiO₂ device with partially selective contacts. In Figure 5 we showed elementary models to explain the mechanism under light conditions close to zero applied voltage that leads to the generation of a photocurrent. In the experimental figures, we have observed the transition to a large current when the Au electrode is positively biased. This large current and the associated memory effect is the key operational characteristic of a memristor. In addition, we showed that the presence of regular/capacitive hysteresis under illumination is connected with the surface ionic polarization by the high capacitance measured by IS and the large capacitive current that scales with the light intensity.

Alternatively, for measurements in the dark, large inductive currents have been observed in all types of devices measured. We can discuss the origin of the large current rise in the dark injection regime based on Figure 17.

FIGURE 17

FIGURE 17. Scheme of electrical current under strong injection regime. (A) In perfect selective contacts the current is completely due to recombination. (B) For similar contacts the device becomes a single carrier in which the current goes through from one contact to the other.

In the case of a solar cell with very good selective contacts, the large current is due to recombination as shown in Figure 17A. This current regime is usually not analyzed in solar cells as the regime beyond $V_{oc}$ is uninteresting compared to the extraction regime below $V_{oc}$. However, we have shown in the previous sections that the exponential current is well described by Eq. 5 as a recombination current that may correspond to electron-hole recombination via trap states as often reported for Light Emitting Diodes. Numerous analyses have discussed the effective ideality factor in relation to the recombination mechanisms. (Wetzelaer et al., 2015; Tress et al., 2018; Caprioglio et al., 2019).

For the operation of memristors, the injection domain is an important feature of the device operation. It is therefore essential to obtain a physical description of the variable $f$ that provides the exponential rise of the injection current at a certain voltage. Based on the model discussion, we have several options:

1) The memristor device injects electrons and holes by partially selective contacts as in Figure 17A, producing a recombination current and $f$ representing the ion-activated recombination sites as suggested in solar cells (Pockett and Carnie, 2017; Shen et al., 2017).

But another possibility is that there is no contact selectivity and the current is due to just one carrier going through the sample as in Figure 17B, i.e., electron-only devices. (Li et al., 2022) In the current work both PEDOT:PSS and Au are contacted more selective to holes than electrons due to their energy levels and have often been used for this purpose in solar cells and LEDs research. Therefore, it is likely that if the second possibility is responsible for the inductive currents and the device would be hole-only. The question is to determine the physical origin of the activation function $f$ in this case. The activation mechanisms must predict an exponential rise and a saturation effect. There are several options outlined in Figure 18:

2) The activation of surface conduction sites by reaction of the incoming ions, Figure 18A. Here is the interpretation of the function $f$ in Eqs. 1–2 is direct: It is the fraction of surface-activated sites. The chemistry of the perovskite/contact perovskite is very complex and it is possible that the presence of halides at the interface and oxidizing/reducing conditions may lead to the formation/rupture of electrically conductive sites. A representative example is the case of the interface MAPI/Au where gold halides (i.e., AuI₃) may form reversibly during polarization at positive bias and create an interface with reduced resistance to extract the charge (Pospisil et al., 2019). A negative bias, metallic Au would be recovered with increased charge transfer resistances. A similar, effect has been reported for devices containing MAPI/Ag (Solanki et al., 2020).

3) The decrease of an electronic surface barrier between the perovskite layer and the contacts, Figure 18B (Yang et al., 2013). This is caused by changes in doping ions around the interface, in a Schottky barrier diode mechanism.

4) A filamentary conductive pathway through the perovskite layer, is the standard mechanism of high current in memristors (Fang et al., 2021).

FIGURE 18

FIGURE 18. Schematics of effects of incoming ions. (A) Formation of conductive sites by the local reaction. (B) Reduction of an electrostatic barrier by increased doping.

The present experimental results and modeling do not distinguish between these options, or between models in Figure 17 and Figure 18, and additional discriminatory experimental work is needed.

Conclusion

The operation of solar cells and memristors depends crucially on the structure of contacts. We found different phenomena like directional photocurrent, photocapacitance at low bias voltage, and alternation of capacitive and inductive hysteresis domains can be explained by a model in which the rising current is activated by voltage. In the case of a solar cell, this current is well explained by the ionic-dependent recombination model. But for the memristor, different options appear since the devices are less affected by photogeneration and may become single carrier dominated. We conclude that solar cells and memristors can have very different underlying mechanisms, but more investigation is needed for the attribution of the rising current and the memory effect.

Experimental

Device Fabrication

All materials and solvents were used as received. FTO glass (Pilkington TEC 15), PEDOT:PSS (Heraeus CLEVIOS™ P VP AI 4083), titanium diisopropoxide bis(acetylacetonate) (Merk, 75% solution in 2-propanol), absolute ethanol (Sigma Aldrich, anhydrous 99.8%), acetylacetone (Sigma Aldrich, 99%), TiO₂ paste (Dyesol, DSL 18NR-T),CH₃NH₃I (MAI, Greatcellsolar), PbI₂ (TCI, 99.99%), DMF (Sigma Aldrich, anhydrous 99.8%), DMSO (Sigma Aldrich, anhydrous 99.9%), chlorobenzene (Sigma Aldrich, 99.8%), spiro-OMeTAD (Sigma Aldrich, anhydrous 99%), LITBSF (Sigma Aldrich, anhydrous 99.95%), Acetonitrile (Sigma Aldrich, anhydrous 99.9%), 4-tert-butylpyridine (TBP) (Sigma Aldrich, 99.8%). The precursor solution for the compact layer of TiO₂ (c-TiO₂) was prepared by mixing 0.4 ml of acetylacetone and 0.6 ml of titanium diisopropoxide bis(acetylacetonate) in 9 ml of absolute ethanol. In order to prepare mesoporous TiO₂ layer precursor, 150 mg of TiO₂ paste is required for each milliliter of ethanol. The MAPbI₃ precursor solution is prepared by preparation of DMF solutions (50 wt%) containing MAI and PbI₂ (1:1 mol%) and MAI, PbI_2, and DMSO (1:1:1 mol%) as reported previously. Briefly, MAI (235 mg) and PbI₂ (681.5 mg) were mixed in DMF (1 ml) and DMSO (95 μL). Spiro-OMeTAD solution requires 72.3 mg of spiro-OMeTAD for each milliliter of chlorobenzene. Then, 17.5 µL of LITBSF solution (520 mg of Li^* salt per milliliter of acetonitrile) and 28.8 µL of TBP must be added.

Devices were prepared following previously reported methods. (Aranda et al., 2017) All devices were prepared starting from FTO glass substrates. Etching was carried out with zinc powder and HCl solution (2 M). Afterward, samples were brushed, cleaned with Hellmanex solution, and rinsed with Milli-Q water. For a complete cleaning, the substrates were sonicated in acetone for 15 min and this step was repeated with a mixed 50:50 ethanol-isopropanol solution. Finally, substrates were dried with nitrogen and treated in a UV−O₃ chamber for 15 min. For PEDOT-based memristors, PEDOT:PSS solution was filtered using PTFE 0.45 μm syringe filter and spin-coated on the substrate at 3000 rpm for 30 s. This was followed by annealing at 120°C for 10 min. Alternatively, for TiO₂-based devices, a compact TiO₂ layer was deposited at 450°C by aerosol spray pyrolysis. The solution was sprayed using pure oxygen as a carrying gas and annealing was carried out at 450°C for 30 min. The mesoporous TiO₂ layer was spin-coated at 2000 rpm for 10 s and annealed following several temperature steps. Devices were transferred to a nitrogen-filled glovebox, samples are heated at 100°C for 5 min to avoid residual humidity on their surface. The perovskite precursor solution was spin-coated at 4000 rpm for 50 s, using chlorobenzene as an antisolvent. Afterward, the substrate was annealed at 100°C for 10 min. For solar cell devices, the doped spiro-OMeTAD solution was spin-coated at 4000 rpm for 30 s. At last, Au electrodes were thermally evaporated to define an active area of 0.25 cm² for measurements in the dark.

Device Characterization

Dark $I-V$ characteristics were recorded with a 10 mV step at different scan rates (from 1 V/s to 50 mV/s) using an Autolab potentiostat. The instrument was driven by the NOVA 2.1.5 software and 3 cycles were recorded from each scan rate starting in a forward direction from 0 V to 1.5 V, followed by a reverse direction from 1.5 V to −1.5 V, and finally recovered to 0 V. For light characterizations, experimental measurements were carried out using a Metrohm AutoLab optical bench with a blue LED (470 nm) as a light source. The distance between the sample and LED was adjusted to obtain 100 mW/cm² at a certain LED current and a complete light calibration was made using a Si photodiode. The LED was driven using an Autolab LED driver through the Autolab potentiostat. $I-V$ characteristics at different illumination intensities were measured from 10 to 100 mW/cm² using the same procedure followed for dark experiments. During light measurement, a mask was used to ensure a 0.1 cm² illumination area.

For IS measurements Autolab was configured to apply sinusoidal signals with a 10 mV amplitude from 1 MHz to 0.1 Hz under several illumination intensities and dc voltages, ranging from low to high illuminations and from 0 V to upper voltages. Chronoamperometry measurements during 20 s were carried out before and after IS.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author Contributions

LD, AR, RS, and AG made the measurements, AB and RJ were involved in conceptualization, BR, MK, AG, and JB planned the work, and AG and JB wrote the manuscript.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The handling editor MF declared a shared affiliation with the author(s) RE and MK at the time of review.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Acknowledgments

The authors thank Generalitat Valenciana for the project PROMETEO/2020/028. A.B. acknowledges FPI studentship funding from Ministerio de Ciencia e Innovación of Spain (BES-2017–080351). R.A.J. acknowledges the support from the ETH Zurich Postdoctoral Fellowship scheme.

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