Inductive and Capacitive Hysteresis of Current-Voltage Curves: Unified Structural Dynamics in Solar Energy Devices, Memristors, Ionic Transistors, and Bioelectronics

Hysteresis observed in the current-voltage curves of both electronic and ionic devices is a phenomenon where the curve's shape is altered on the basis of the measurement speed. This effect is driven by internal processes that introduce a time delay in the response to an external stimulus, leading to measurements being dependent on the history of the past disturbances. This hysteresis effect has posed challenges, particularly in solution-processed photovoltaic devices such as halide perovskite solar cells, where it significantly complicates the evaluation of performance quality. In other devices, such as memristors and organic electrochemical transistors for neuromorphic applications, hysteresis is an inherent aspect of their functionality, facilitating transitions between different conductivity states. Natural and artificial ionically conducting channels also exhibit pronounced hysteresis, a crucial component for generating action potentials in neurons. In this study, we aim to categorize various forms of hysteresis by identifying shared elements among diverse physical, chemical, and biological conducting systems. Our method involves examining hysteresis from multiple angles, using simplified models that capture essential response types. We analyze system behavior using techniques such as linear sweep voltammetry and impedance spectroscopy and transient currents resulting from small voltage steps. Our investigation reveals two primary hysteresis types based on how current responds to rapid sweep rates: capacitive hysteresis and inductive hysteresis. These terms correspond to the dominant component in the equivalent circuit, determining the transient time response. Remarkably, these concepts provide insights into vastly different systems, spanning solar cells, capacitors, transistors, electrofluidic nanopores, and protein ion channels. The consistency in electrical responses across the different cases enables the identification of the primary cause of hysteresis. We also elucidate the frequency dependence of hysteresis and the stepwise responses of solar cells, illustrating how fundamental relaxations contribute to the overall surplus or deficit of current during extensive voltage sweeps that define the current-voltage curve.

Popular Summary

Many systems exhibit a time- and measurement-dependent change in response to an applied electric field, known as current-voltage hysteresis. Such hysteresis can be a device feature, enabling the memory effects of a memristor device, or a hindrance, obscuring the analysis of perovskite-based solar cells. In this Perspective, the author considers current-voltage hysteresis across a variety of electronically and ionically conducting systems and uses simple models to propose general and widely applicable principles. The author finds that all current-voltage hysteresis can be identified as capacitive-type, inductive-type, or a combination of the two and describes how the hysteresis types are manifested in common experimental measurements. The presented archetypes offer a shared framework to understand current-voltage hysteresis across solar cells, electronic devices, memory devices, and biological systems.

Figure 1

(A) (a) I-V curves measured in forward scan (FS) and reverse scan (RS) for a perovskite solar cell using ${\mathrm{CH}}_3{\mathrm{NH}}_3{\mathrm{Pb}\mathrm{I}}_3$. The maximum power point (mpp) is indicated. The voltage settling time was 200 ms and the light intensity was AM1.5 G 1 sun (100 mW cm⁻²). (b),(c) Time-dependent photocurrent response as a function of voltage settling time in (b) forward scan and (c) reverse scan. Reproduced with permission from H.-S. Kim, N.-G. Park, J. Phys. Chem. Lett. 5, 2927–2934 (2014). Licensed under a Creative Commons Attribution (CC BY 4.0) license. (B) (a) Characteristic current-voltage curves of a hafnium oxide–based memristive device switching from the high-resistance state to the low-resistance state and back. Voltage ramps at three different speeds: gray, slow (43.6 mV s⁻¹); blue, medium (480 mV s⁻¹); orange, fast (4.8 V s⁻¹). The inset shows readout I-V characteristics of different resistance states depending on the ramp speed of the switching. A sketch of the layer stack of the memristive device is given on the right side. (b) Complex plane impedance plots after application of 2.1 V for 1 and 30 s. Reproduced with permission from R. Marquardt, F. Zahari, J. Carstensen, G. Popkirov, O. Gronenberg, G. Kolhatkar, H. Kohlstedt, M. Ziegler, Adv. Electron. Mater. 9, 2201227 (2023). Licensed under a Creative Commons Attribution (CC BY 4.0) license. (C) (a) Experimental cyclic voltammetry curves of an $\mathrm{Al}$ electrode in 0.01 M ${\mathrm{Na}}_2{\mathrm{SO}}_4$ recorded at scan rates between 100 and 2000 mV s⁻¹ and (b) the capacitive current as a function of the scan rate. Reproduced with permission from O. Gharbi, M. T. T. Tran, B. Tribollet, M. Turmine, V. Vivier, Electrochim. Acta 343, 136109 (2020). Copyright 2020, Elsevier. (D) Current-voltage relations of the egg cell membrane at four different $\mathrm{K}$ concentrations (10, 25, 50, and 100 mM) in $\mathrm{Na}$-free medium. Continuous lines, instantaneous current; broken lines, steady-state current. Reproduced with permission from S. Hagiwara, S. Miyazaki, N. P. Rosenthal, J. Gen. Physiol. 67, 621–638 (1976). Copyright 1976, Rockefeller University Press.

Figure 1

(A) (a) I-V curves measured in forward scan (FS) and reverse scan (RS) for a perovskite solar cell using ${\mathrm{CH}}_3{\mathrm{NH}}_3{\mathrm{Pb}\mathrm{I}}_3$. The maximum power point (mpp) is indicated. The voltage settling time was 200 ms and the light intensity was AM1.5 G 1 sun (100 mW cm⁻²). (b),(c) Time-dependent photocurrent response as a function of voltage settling time in (b) forward scan and (c) reverse scan. Reproduced with permission from H.-S. Kim, N.-G. Park, J. Phys. Chem. Lett. 5, 2927–2934 (2014). Licensed under a Creative Commons Attribution (CC BY 4.0) license. (B) (a) Characteristic current-voltage curves of a hafnium oxide–based memristive device switching from the high-resistance state to the low-resistance state and back. Voltage ramps at three different speeds: gray, slow (43.6 mV s⁻¹); blue, medium (480 mV s⁻¹); orange, fast (4.8 V s⁻¹). The inset shows readout I-V characteristics of different resistance states depending on the ramp speed of the switching. A sketch of the layer stack of the memristive device is given on the right side. (b) Complex plane impedance plots after application of 2.1 V for 1 and 30 s. Reproduced with permission from R. Marquardt, F. Zahari, J. Carstensen, G. Popkirov, O. Gronenberg, G. Kolhatkar, H. Kohlstedt, M. Ziegler, Adv. Electron. Mater. 9, 2201227 (2023). Licensed under a Creative Commons Attribution (CC BY 4.0) license. (C) (a) Experimental cyclic voltammetry curves of an $\mathrm{Al}$ electrode in 0.01 M ${\mathrm{Na}}_2{\mathrm{SO}}_4$ recorded at scan rates between 100 and 2000 mV s⁻¹ and (b) the capacitive current as a function of the scan rate. Reproduced with permission from O. Gharbi, M. T. T. Tran, B. Tribollet, M. Turmine, V. Vivier, Electrochim. Acta 343, 136109 (2020). Copyright 2020, Elsevier. (D) Current-voltage relations of the egg cell membrane at four different $\mathrm{K}$ concentrations (10, 25, 50, and 100 mM) in $\mathrm{Na}$-free medium. Continuous lines, instantaneous current; broken lines, steady-state current. Reproduced with permission from S. Hagiwara, S. Miyazaki, N. P. Rosenthal, J. Gen. Physiol. 67, 621–638 (1976). Copyright 1976, Rockefeller University Press.

Figure 2

(a) Equivalent circuit model. (b) Impedance spectra for $C_m=1$, $R_b=1$, $R_a=1$, and $L_a=0.1$ (arc A) $\mathrm{or}{L}_a=20$ (arc B). The red point is the dc resistance. (c),(d) Current-voltage curves at different voltage sweep rates $v_s$, $u=v_{s}t$. (c) The state variable responds fast, and the capacitive current with capacitance $C_m=1$ is added to the direct current. The steady-state current (slow) is shown as the gray line. (d) The capacitance is removed, $C_m=0$, and the variable w contributes to the current. The purple line corresponds to an infinitely fast scan. The parameters are $R_b=1$, $R_a=1$, ${\tau }_k=0.01$, and $b=1$. (e) A more general equivalent circuit model. The curves in (b), (c), (d) are given in arbitrary units to illustrate the general shapes caused by the indicated parameters.

Figure 2

(a) Equivalent circuit model. (b) Impedance spectra for $C_m=1$, $R_b=1$, $R_a=1$, and $L_a=0.1$ (arc A) $\mathrm{or}{L}_a=20$ (arc B). The red point is the dc resistance. (c),(d) Current-voltage curves at different voltage sweep rates $v_s$, $u=v_{s}t$. (c) The state variable responds fast, and the capacitive current with capacitance $C_m=1$ is added to the direct current. The steady-state current (slow) is shown as the gray line. (d) The capacitance is removed, $C_m=0$, and the variable w contributes to the current. The purple line corresponds to an infinitely fast scan. The parameters are $R_b=1$, $R_a=1$, ${\tau }_k=0.01$, and $b=1$. (e) A more general equivalent circuit model. The curves in (b), (c), (d) are given in arbitrary units to illustrate the general shapes caused by the indicated parameters.

Figure 3

(a) Time-varying voltage scan rate with a positive scan direction part (1) and a negative scan direction part (2) of the cycle. (b)–(d) Current-voltage curves. The thick line represents the stationary current and the dashed line represents the current under the varying speed indicated in (a). (c)–(e) Basic impedance spectra and the associated equivalent circuit. The arrow indicates the direction of increasing angular frequency. (b),(c) Capacitive system. (d),(e) Inductive system.

Figure 3

(a) Time-varying voltage scan rate with a positive scan direction part (1) and a negative scan direction part (2) of the cycle. (b)–(d) Current-voltage curves. The thick line represents the stationary current and the dashed line represents the current under the varying speed indicated in (a). (c)–(e) Basic impedance spectra and the associated equivalent circuit. The arrow indicates the direction of increasing angular frequency. (b),(c) Capacitive system. (d),(e) Inductive system.

Figure 4

(a) Current-voltage curve measured at frequency ${\mathrm{\Omega }}_s=10\text{Hz}$ for a multipore membrane in 100 mM KCl solution at neutral pH. The inset shows the electrochemical cell with the membrane. (b) Impedance spectra at different reverse voltages, with the corresponding Bode plot of the imaginary part of the impedance in (c). (d),(e) Impedance spectra and Bode plots at forward voltage. Adapted from P. Ramirez, J. Cervera, S. Nasir, M. Ali, W. Ensinger, S. Mafe, J. Colloid Interface Sci. 655, 876–885 (2024) with permission from Elsevier.

Figure 4

(a) Current-voltage curve measured at frequency ${\mathrm{\Omega }}_s=10\text{Hz}$ for a multipore membrane in 100 mM KCl solution at neutral pH. The inset shows the electrochemical cell with the membrane. (b) Impedance spectra at different reverse voltages, with the corresponding Bode plot of the imaginary part of the impedance in (c). (d),(e) Impedance spectra and Bode plots at forward voltage. Adapted from P. Ramirez, J. Cervera, S. Nasir, M. Ali, W. Ensinger, S. Mafe, J. Colloid Interface Sci. 655, 876–885 (2024) with permission from Elsevier.

Figure 5

(a) Operation of an organic electrochemical transistor and geometric dimensions. The green zone is the electrolyte, the gray zone is the OMIEC layer, and the yellow zones are the gate (G), source (S), and drain (D) electrodes. The anions inserted into the film are the source of electronic holes that carry the current. (b) Hysteresis in current-voltage curves due to the ion diffusion effect at different voltage sweep rates $(v_r)$ and the equilibrium curve (gray). The parameters are as follows: $L=10$, $d=0.1$, $w=1$, $X_0=100$, $u_0=0$, $V_m=0.2$, $q{\mu }_p{u}_d=1$, $R_s=0.1$, $C_s=1$, and ${\tau }_d=100$. Impedance spectra for different sets of parameters: $R_s=0.25$, $C_s=1$, and (c) $R_a=5$, $L_a=0.1$, $R_B=5$, and $C_{\mu }=10$, (d) $R_a=1$, $L_a=10$, $R_B=3$, and $C_{\mu }=100$, (e) $R_a=1$, $L_a=100$, $R_B=3$, and $C_{\mu }=10$, and (f) $R_a=1$, $L_a=3$, $R_B=1$, and $C_{\mu }=10$. The indicated vectors are the time constants $(R_a^{-1}+R_B^{-1}{)}^{-1}{C}_s$, $R_B{C}_{\mu }$, and $L_a/R_a$. (g) Equivalent circuit model. Adapted with permission from J. Bisquert, J. Phys. Chem. Lett. 14, 10951−10958 (2023). Licensed under a Creative Commons Attribution (CC BY 4.0) license.

Figure 5

(a) Operation of an organic electrochemical transistor and geometric dimensions. The green zone is the electrolyte, the gray zone is the OMIEC layer, and the yellow zones are the gate (G), source (S), and drain (D) electrodes. The anions inserted into the film are the source of electronic holes that carry the current. (b) Hysteresis in current-voltage curves due to the ion diffusion effect at different voltage sweep rates $(v_r)$ and the equilibrium curve (gray). The parameters are as follows: $L=10$, $d=0.1$, $w=1$, $X_0=100$, $u_0=0$, $V_m=0.2$, $q{\mu }_p{u}_d=1$, $R_s=0.1$, $C_s=1$, and ${\tau }_d=100$. Impedance spectra for different sets of parameters: $R_s=0.25$, $C_s=1$, and (c) $R_a=5$, $L_a=0.1$, $R_B=5$, and $C_{\mu }=10$, (d) $R_a=1$, $L_a=10$, $R_B=3$, and $C_{\mu }=100$, (e) $R_a=1$, $L_a=100$, $R_B=3$, and $C_{\mu }=10$, and (f) $R_a=1$, $L_a=3$, $R_B=1$, and $C_{\mu }=10$. The indicated vectors are the time constants $(R_a^{-1}+R_B^{-1}{)}^{-1}{C}_s$, $R_B{C}_{\mu }$, and $L_a/R_a$. (g) Equivalent circuit model. Adapted with permission from J. Bisquert, J. Phys. Chem. Lett. 14, 10951−10958 (2023). Licensed under a Creative Commons Attribution (CC BY 4.0) license.

Figure 6

(A) Characteristic I-V response on a linear scale of the FTO/PEDOT:PSS/${\mathrm{CH}}_3{\mathrm{NH}}_3{\mathrm{Pb}\mathrm{Br}}_3$/$\mathrm{Au}$ memristor device with differing upper vertex voltages of (a) 0.25 V, (b) 0.75 V, and (c), 1.25 V, with the arrows indicating the scan direction. Corresponding I-V response on a semilogarithmic scale for upper vertex voltages of (d) 0.25 V, (e) 0.75, and (f) 1.25 V, with the arrows and numbers indicating the scan direction and sequence, respectively. Voltage-dependent impedance spectral evolution measured at (g) 0 V, (h) 0.3 V, and (i) 0.6 V exhibiting a transition from a low-frequency capacitive response at low applied voltages to a low-frequency inductive response at high applied voltages. Figure courtesy of Cedric Gonzales. (B) Results of the measurement of a perovskite solar cell with structure FTO/${\mathrm{CH}}_3{\mathrm{NH}}_3{\mathrm{Pb}\mathrm{Br}}_3$/spiro-OMeTAD in the dark. (a) Evolution of hysteresis at different scan rates in the forward direction on a logartithmic vertical scale in the dark. (b) Time constants resulting from the impedance parameters as a function of cell voltage. Two time constants are indicated in the nomenclature of the present paper: $R_A{C}_A$ and $L_a/R_a$. Reproduced with permission from C. Gonzales, A. Guerrero, J. Bisquert, J. Phys. Chem. C 126, 13560–13578 (2022). Licensed under a Creative Commons Attribution (CC BY 4.0) license.

Figure 6

(A) Characteristic I-V response on a linear scale of the FTO/PEDOT:PSS/${\mathrm{CH}}_3{\mathrm{NH}}_3{\mathrm{Pb}\mathrm{Br}}_3$/$\mathrm{Au}$ memristor device with differing upper vertex voltages of (a) 0.25 V, (b) 0.75 V, and (c), 1.25 V, with the arrows indicating the scan direction. Corresponding I-V response on a semilogarithmic scale for upper vertex voltages of (d) 0.25 V, (e) 0.75, and (f) 1.25 V, with the arrows and numbers indicating the scan direction and sequence, respectively. Voltage-dependent impedance spectral evolution measured at (g) 0 V, (h) 0.3 V, and (i) 0.6 V exhibiting a transition from a low-frequency capacitive response at low applied voltages to a low-frequency inductive response at high applied voltages. Figure courtesy of Cedric Gonzales. (B) Results of the measurement of a perovskite solar cell with structure FTO/${\mathrm{CH}}_3{\mathrm{NH}}_3{\mathrm{Pb}\mathrm{Br}}_3$/spiro-OMeTAD in the dark. (a) Evolution of hysteresis at different scan rates in the forward direction on a logartithmic vertical scale in the dark. (b) Time constants resulting from the impedance parameters as a function of cell voltage. Two time constants are indicated in the nomenclature of the present paper: $R_A{C}_A$ and $L_a/R_a$. Reproduced with permission from C. Gonzales, A. Guerrero, J. Bisquert, J. Phys. Chem. C 126, 13560–13578 (2022). Licensed under a Creative Commons Attribution (CC BY 4.0) license.

Figure 7

(a) Current density−voltage characteristic measured at 50 mW/cm² blue light illumination for a FTO/PEDOT:PSS/${\mathrm{CH}}_3{\mathrm{NH}}_3{\mathrm{Pb}\mathrm{I}}_3$/$\mathrm{Au}$ memristor. Arrows indicate the sweep direction. (b),(c) Corresponding complex plane plot representation of the impedance spectra at various dc voltages, measured from 1 MHz to 0.1 Hz. The inset corresponds to higher dc voltages applied. Reproduced with permission from L. Munoz-Diaz, A. J. Rosa, A. Bou, R. S. Sanchez, B. Romero, R. A. John, M. V. Kovalenko, A. Guerrero, J. Bisquert, Front. Energy Res. 10, 914115 (2022). Licensed under a Creative Commons Attribution (CC BY 4.0) license.

Figure 7

(a) Current density−voltage characteristic measured at 50 mW/cm² blue light illumination for a FTO/PEDOT:PSS/${\mathrm{CH}}_3{\mathrm{NH}}_3{\mathrm{Pb}\mathrm{I}}_3$/$\mathrm{Au}$ memristor. Arrows indicate the sweep direction. (b),(c) Corresponding complex plane plot representation of the impedance spectra at various dc voltages, measured from 1 MHz to 0.1 Hz. The inset corresponds to higher dc voltages applied. Reproduced with permission from L. Munoz-Diaz, A. J. Rosa, A. Bou, R. S. Sanchez, B. Romero, R. A. John, M. V. Kovalenko, A. Guerrero, J. Bisquert, Front. Energy Res. 10, 914115 (2022). Licensed under a Creative Commons Attribution (CC BY 4.0) license.

Figure 8

(A) Time-dependent photocurrent response of a planar perovskite solar cell on compact ${\mathrm{Ti}\mathrm{O}}_2$ with 500-nm ${\mathrm{CH}}_3{\mathrm{NH}}_3{\mathrm{Pb}\mathrm{I}}_3$ film as the light-absorber layer and 150-nm spiro-OMeTAD as the hole transport layer under reverse and forward stepwise scans with (a) 1-s step time and (b) 0.1-s step time. (c) Current-voltage response with different scan rates. CV, cyclic voltammetry. Reproduced with permission from B. Chen, M. Yang, X. Zheng, C. Wu, W. Li, Y. Yan, J. Bisquert, G. Garcia-Belmonte, K. Zhu, S. Priya, J. Phys. Chem. Lett. 6, 4693−4700 (2015). Copyright 2015, American Chemical Society. (B) Current response of the linear model to two consecutive voltage steps, from time $t_0$, of duration $\mathrm{\Delta }t$, the first with amplitude $\mathrm{\Delta }V=0.1$ and the second with amplitude $\mathrm{\Delta }V=0.2$. The common parameters are $R_b=1$, $R_a=0.1$, $R_s=0.05$, $t_0=0.1$, $\mathrm{\Delta }t=0.05$, and ${\tau }_k=0.0005$. The different cases are (a) $C_m=0.1$, (b) $C_m=1$, and (c) $C_m=1$, and added photocurrent $I_{\text{photo}}=2$. The red lines indicate the applied voltage steps and the gray lines indicate the steady-state current at the given voltage. The orange arrows indicate the excess current at the end of the cycle with respect to the steady-state value. The curves in (B) are given in arbitrary units to illustrate the general shapes caused by the indicated parameters.

Figure 8

(A) Time-dependent photocurrent response of a planar perovskite solar cell on compact ${\mathrm{Ti}\mathrm{O}}_2$ with 500-nm ${\mathrm{CH}}_3{\mathrm{NH}}_3{\mathrm{Pb}\mathrm{I}}_3$ film as the light-absorber layer and 150-nm spiro-OMeTAD as the hole transport layer under reverse and forward stepwise scans with (a) 1-s step time and (b) 0.1-s step time. (c) Current-voltage response with different scan rates. CV, cyclic voltammetry. Reproduced with permission from B. Chen, M. Yang, X. Zheng, C. Wu, W. Li, Y. Yan, J. Bisquert, G. Garcia-Belmonte, K. Zhu, S. Priya, J. Phys. Chem. Lett. 6, 4693−4700 (2015). Copyright 2015, American Chemical Society. (B) Current response of the linear model to two consecutive voltage steps, from time $t_0$, of duration $\mathrm{\Delta }t$, the first with amplitude $\mathrm{\Delta }V=0.1$ and the second with amplitude $\mathrm{\Delta }V=0.2$. The common parameters are $R_b=1$, $R_a=0.1$, $R_s=0.05$, $t_0=0.1$, $\mathrm{\Delta }t=0.05$, and ${\tau }_k=0.0005$. The different cases are (a) $C_m=0.1$, (b) $C_m=1$, and (c) $C_m=1$, and added photocurrent $I_{\text{photo}}=2$. The red lines indicate the applied voltage steps and the gray lines indicate the steady-state current at the given voltage. The orange arrows indicate the excess current at the end of the cycle with respect to the steady-state value. The curves in (B) are given in arbitrary units to illustrate the general shapes caused by the indicated parameters.

Figure 9

Steady-state current measurements using forward and reverse stepwise voltage sweeps for two metal halide perovskite solar cells differing in the contact: (a),(b) spiro-OMeTAD and (c),(d) $\mathrm{Cu}\mathrm{I}$-based. (e) Complex plane impedance plots for the two devices, and fit to the equivalent circuit model. (f) Variation in low-frequency resistance and capacitance in both devices. Impedance measurements were performed under constant illumination. OC, open circuit. Reproduced with permission from G. A. Sepalage, S. Meyer, A. Pascoe, A. D. Scully, F. Huang, U. Bach, Y.-B. Cheng, L. Spiccia, Adv. Funct. Mater. 25, 5650–5661 (2015). Copyright 2015, Wiley.

Figure 9

Steady-state current measurements using forward and reverse stepwise voltage sweeps for two metal halide perovskite solar cells differing in the contact: (a),(b) spiro-OMeTAD and (c),(d) $\mathrm{Cu}\mathrm{I}$-based. (e) Complex plane impedance plots for the two devices, and fit to the equivalent circuit model. (f) Variation in low-frequency resistance and capacitance in both devices. Impedance measurements were performed under constant illumination. OC, open circuit. Reproduced with permission from G. A. Sepalage, S. Meyer, A. Pascoe, A. D. Scully, F. Huang, U. Bach, Y.-B. Cheng, L. Spiccia, Adv. Funct. Mater. 25, 5650–5661 (2015). Copyright 2015, Wiley.

Figure 10

Current response of the linear model to two consecutive voltage steps, from time $t_0$, of duration $\mathrm{\Delta }t$, the first with amplitude $\mathrm{\Delta }V=0.1$ and the second with amplitude $\mathrm{\Delta }V=0.2$. The common parameters are $R_b=1$, $R_a=0.1$, $R_s=0.05$, $t_0=0.5$, $\mathrm{\Delta }t=1$, and $C_m=1$. The different cases are (a) ${\tau }_k=0.3$ and (b) ${\tau }_k=3$. The red lines indicate the applied voltage steps and the gray lines indicate the steady-state current at the given voltage. The orange arrows indicate the excess current at the end of the cycle with respect to the steady-state value. The curves are given in arbitrary units to illustrate the general shapes caused by the indicated parameters.

Figure 10

Current response of the linear model to two consecutive voltage steps, from time $t_0$, of duration $\mathrm{\Delta }t$, the first with amplitude $\mathrm{\Delta }V=0.1$ and the second with amplitude $\mathrm{\Delta }V=0.2$. The common parameters are $R_b=1$, $R_a=0.1$, $R_s=0.05$, $t_0=0.5$, $\mathrm{\Delta }t=1$, and $C_m=1$. The different cases are (a) ${\tau }_k=0.3$ and (b) ${\tau }_k=3$. The red lines indicate the applied voltage steps and the gray lines indicate the steady-state current at the given voltage. The orange arrows indicate the excess current at the end of the cycle with respect to the steady-state value. The curves are given in arbitrary units to illustrate the general shapes caused by the indicated parameters.

Figure 11

(A) (a) I-V curves for a multipore membrane in 100 mM KCl solution at neutral pH, parametrically in the electrical potential scan rate, characterized by the signal frequency $f={\mathrm{\Omega }}_s/2\pi$, obtained with a voltage amplitude of 2 V. The arrows indicate the signal time evolution. Reproduced with permission from P. Ramirez, V. Gomez, J. Cervera, S. Mafe, J. Bisquert, J. Phys. Chem. Lett. 14, 10930–10934 (2023). Copyright 2023, American Chemical Society. (B) Complex plane plot of the impedance of a ${\mathrm{CH}}_3{\mathrm{NH}}_3{\mathrm{Pb}\mathrm{I}}_3$-based mesoscopic solar cell, measured at 950 mV under equivalent 1000 W m⁻² solar irradiation. The Lissajous curves (current versus voltage) corresponding to a sinusoidal perturbation (V_rms= 22 mV) under the same measuring conditions (light and dc bias) are shown at the top for the indicated frequencies. Reproduced with permission from N. Pellet, F. Giordano, M. I. Dar, G. Gregori, S. M. Zakeeruddin, J. Maier, M. Grätzel, Prog. Photovolt. Res. Appl. 25, 942–950 (2017). Copyright 2017, Wiley. (C) Simulation of halide perovskite solar cell hysteresis at different scan velocities. (a) Measurement procedure to record the current-voltage curves and band diagrams starting at open-circuit voltage (OC) and progressing to short-circuit current (SC) and back. (b) Simulated fast-hysteresis power conversion efficiency plot in forward and reverse scans and the characteristic efficiencies at slow, medium, and fast scan speeds. (c) Corresponding simulated current-voltage curves. Reproduced with permission from V. M. Le Corre, J. Diekmann, F. Peña-Camargo, J. Thiesbrummel, N. Tokmoldin, E. Gutierrez-Partida, K. P. Peters, L. Perdigón-Toro, M. H. Futscher, F. Lang, J. Warby, H. J. Snaith, D. Neher, M. Stolterfoht, Solar RRL 6, 2100772 (2022). Copyright 2022, Wiley.

Figure 11

(A) (a) I-V curves for a multipore membrane in 100 mM KCl solution at neutral pH, parametrically in the electrical potential scan rate, characterized by the signal frequency $f={\mathrm{\Omega }}_s/2\pi$, obtained with a voltage amplitude of 2 V. The arrows indicate the signal time evolution. Reproduced with permission from P. Ramirez, V. Gomez, J. Cervera, S. Mafe, J. Bisquert, J. Phys. Chem. Lett. 14, 10930–10934 (2023). Copyright 2023, American Chemical Society. (B) Complex plane plot of the impedance of a ${\mathrm{CH}}_3{\mathrm{NH}}_3{\mathrm{Pb}\mathrm{I}}_3$-based mesoscopic solar cell, measured at 950 mV under equivalent 1000 W m⁻² solar irradiation. The Lissajous curves (current versus voltage) corresponding to a sinusoidal perturbation (V_rms= 22 mV) under the same measuring conditions (light and dc bias) are shown at the top for the indicated frequencies. Reproduced with permission from N. Pellet, F. Giordano, M. I. Dar, G. Gregori, S. M. Zakeeruddin, J. Maier, M. Grätzel, Prog. Photovolt. Res. Appl. 25, 942–950 (2017). Copyright 2017, Wiley. (C) Simulation of halide perovskite solar cell hysteresis at different scan velocities. (a) Measurement procedure to record the current-voltage curves and band diagrams starting at open-circuit voltage (OC) and progressing to short-circuit current (SC) and back. (b) Simulated fast-hysteresis power conversion efficiency plot in forward and reverse scans and the characteristic efficiencies at slow, medium, and fast scan speeds. (c) Corresponding simulated current-voltage curves. Reproduced with permission from V. M. Le Corre, J. Diekmann, F. Peña-Camargo, J. Thiesbrummel, N. Tokmoldin, E. Gutierrez-Partida, K. P. Peters, L. Perdigón-Toro, M. H. Futscher, F. Lang, J. Warby, H. J. Snaith, D. Neher, M. Stolterfoht, Solar RRL 6, 2100772 (2022). Copyright 2022, Wiley.