Inductive and capacitive hysteresis of current-voltage curves: unified structural dynamics in solar energy devices, memristors, ionic transistors, and bioelectronics

PRX ENERGY 3, 011001 (2024) Perspective Inductive and Capacitive Hysteresis of Current-Voltage Curves: Unified Structural Dynamics in Solar Energy Devices, Memristors, Ionic Transistors, and Bioelectronics Juan Bisquert * Institute of Advanced Materials (INAM), Universitat Jaume I, 12006 Castelló, Spain (Received 25 August 2023; revised 29 November 2023; published 8 January 2024) 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. DOI: 10.1103/PRXEnergy.3.011001 I. INTRODUCTION Hysteresis in current-voltage curves is a phenomenon observed in various electronic, ionic, and molecular devices, significantly influencing their operational traits [1–9]. This effect leads to the current-voltage curve’s behavior being contingent on the kinetic properties of the measurement process. This manifestation arises from a delay in the current response relative to voltage changes, thereby altering the steady-state current-voltage relationship during time-varying perturbations [10]. *[email protected] Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. The main approach to hysteresis is obtained in the voltage sweep technique with scan velocity vsused to measure the current-voltage curve. The voltage uvaries with time as u=vst.(1) The measurement is performed in a cycle of forward and reverse directions, or vice versa, so as to return to the starting state, and it may be repeated many times to check the stability of the response. This method is applied in many kinds of devices to obtain the current-voltage curve at a low sweep velocity and to provide the dynamic response by measuring at increasing sweep rates. In general, hysteresis indicates that those physical phenomena that produce the current take some time to respond to the changes of applied voltage; hence, the current response depends on the history of the perturbation. Therefore one can apply, for example, a sinusoidal voltage covering the range (−u1,u1), i.e., u=u1sin(st), instead 2768-5608/24/3(1)/011001(23) 011001-1 Published by the American Physical Society JUAN BISQUERT PRX ENERGY 3, 011001 (2024) of Eq. (1), and the hysteresis will be manifest as well. These measurements extend over a wide voltage window in highly nonlinear systems and typically the response is difficult to interpret in terms of the internal processes. The specifical physical mechanisms of hysteresis are analyzed in different research fields [1–7,9]. However, the related studies are not able to reveal the general principles of hysteresis that occurs across a wide variety of conducting systems, from photovoltaics to biology. Recent findings obtained in a variety of systems that shows intense hysteresis, such as in memristors and halide perovskite solar cells [11–16], enable us to provide a general classification of the main hysteresis types, using very simple models. As the phenomena that lie behind the hysteresis respond in different types of measurement of a time-dependent perturbation, this gives us an opportunity to probe the system in different ways to correlate the responses and obtain insight into the dominant phenomena. This is the approach we use in this work, in contrast to elaborating a mechanistic model for each separate physical system. The advantage of this approach is that it combines the measurements of different experimental techniques, as the voltage scan over a wide window, impedance spectroscopy on a set of stationary states, the transient response to a set of step perturbations, and the frequency dependence of the current-voltage curves. By merging these methods on the basis of simple models that contain just a few parameters, one can obtain much insight into the dynamical properties of a system and one can predict and study important types of behavior. Of course, these methods can be also regarded as a first step towards elaborated molecular or transport models that implement particular mechanisms, sometimes using large parameter sets [12,17–21]. These more complex models must finally obey the general rules that will be outlined here. We start by reviewing some specific systems of interest to see what general properties we can infer from observations of hysteresis. Since very different systems spanning biological and artificial ion channels, perovskite solar cells, memristors, and electrochemical transistors show properties that appear closely related, there must be an underlying structure to hysteresis effects. We establish such structure by analyzing the underlying equivalent circuit in the frequency domain [10,22]. We show that this approach provides a strong tool for the analysis of rather involved hysteresis features in a variety of experimental systems. II. SOME EXAMPLES OF HYSTERESIS RESPONSE Hysteresis has been a prominent phenomenon in lead halide perovskite solar cells, and its significance has been widely recognized [1–3,22–26]. In photovoltaic devices, hysteresis as shown in Fig. 1(A) is a major problem that complicates the achievement of a stationary currentvoltage curve, which is essential to determine the power conversion efficiency (PCE). This drawback makes it necessary to apply time-consuming advanced measurement protocols such as maximum power point tracking [30– 35]. However, in other devices, such as memristors, as shown in Fig. 1(B) [27], the hysteresis loop is an intrinsic phenomenon providing a functionality that needs to be optimized. Indeed, the defining property of memristors [36–43] is the resistive switching from a low-resistance state to a high-resistance state in a set process that occurs at some threshold voltage. The low-resistance state is maintained in the return part of the cycle, and a reset process in the negative voltage side recovers the initial high resistance. Also, in field-effect transistors and in ionic transistors for memories, synapsis, and neurons [44–48], the hysteresis is a main functional property. Capacitors and electrochemical supercapacitors are central devices for electronics and energy storage. While they do not conduct direct current, they show hysteresis under voltage cycling, as presented in Fig. 1(C), which also shows a general characteristic of hysteresis in currentvoltage curves: the effect becomes amplified when the voltage scan velocity increases. The previous three systems have in common the property that the electrical current is electronic, although it is strongly influenced by ionic phenomena. Halide perovskites are mixed ionic-electronic conductors, where the slow hysteresis response is attributed to ionic reorganization in the sample [49–52]. In memristors, the resistive transition is associated with the buildup of an ionic filament [53,54]. The capacitive response in Fig. 1(C) is due to the ionic-electronic double layer of a metal plate in contact with a solution [28]. Hysteresis can be found also in many systems that have exclusively ionic conduction, such as ionically conducting glasses [4] and electrokinetic transport in nanopipettes [5]. Ionic current rectification is a frequently observed occurrence in both protein ion channels that form naturally and synthetic nanopores [55]. These phenomena are amply studied because transport through biological channels and pores plays a central role in many physiological processes in living organisms [56]. Inwardly rectifying potassium (KIR) channels in cell membranes control the passive and active electrical properties of cells, and they also link cellular metabolic state and membrane excitability in vivo. These channels operate by intracellular divalent cations and other molecules blocking the asymmetric open channel pores [57]. Figure 1(D) shows the hysteresis in the property of inward rectification of the KIR channel at different potassium concentrations [29]. The steady-state measurement produces a much larger current than the fast measurement. This is another way to look at hysteresis: to compare the current between a very slow measurement and a very fast one. The hysteresis of ion channels in the 011001-2 INDUCTIVE AND CAPACITIVE HYSTERESIS . . . PRX ENERGY 3, 011001 (2024) (A) (B) (C) (D) (a) (a) (b) (b) (b) (c) (a) FIG. 1. (A) (a) I-Vcurves measured in forward scan (FS) and reverse scan (RS) for a perovskite solar cell using CH3NH3PbI3.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-Vcharacteristics 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 Al electrode in 0.01 MNa2SO4recorded at scan rates between 100 and 2000 mV s−1and (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 K concentrations (10, 25, 50, and 100 mM) in 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. 011001-3 JUAN BISQUERT PRX ENERGY 3, 011001 (2024) cell membrane is a central property of neuron functionality. In the model of Hodgkin and Huxley for the dynamics of action potentials in neurons [58,59], the concerted delay of the sodium and potassium channels that conduct opposite currents is responsible for the production of the action potentials by which neurons transmit information. Artificial solid-state nanopores and nanochannels can be regarded as mimics of protein ion channels and have also been developed for sensing applications. The underlying physical cause of rectification arises from the asymmetry in the geometry of the nanopore and/or the distribution of surface charges and their polarities along the sidewalls of the nanopore [55]. Conical nanopores are observed to display hysteresis by decreasing current at higher scan rates [5,60–62]. III. DISTINCTION BETWEEN CAPACITIVE AND INDUCTIVE RESPONSE We briefly analyze the physical nature of hysteresis. In the examples in Fig. 1we can observe two different kinds of hysteresis behavior with respect to the velocity of the external perturbation. In the capacitor corresponding to Fig. 1(C) the current increases with the scan rate. The square shape becomes progressively larger. However, in the ionic channel corresponding to Fig. 1(D) the fastest scan rates produce a decreasing slope and lower total current, as found also in conical nanopores [5,60–62]. As we are concerned with the current-voltage curve, the ac impedance provides a natural framework to analyze the response. In general, a physical model of any kind of device is highly nonlinear. However, operating at a stationary point of the current-voltage curve, one can perform a small signal measurement at angular frequency ωof the voltage-current ratio, and this is called the “impedance” Z. Then an arbitrary complex physical system becomes a linear model that can be represented in terms of resistances, capacitors, and inductors [63–65]. This method also serves to analyze stability and other significant properties of the nonlinear system [66–69]. In halide perovskite solar cells, the connection between the impedance components and the hysteresis behavior is understood [10,25, 51,70,71], with important application for stabilization of the current-voltage curves and determining the stationary performance [72]. One needs to keep in mind that the elements of the equivalent circuit depend strongly on the stationary voltage, so nonlinear properties, as the hysteresis behavior, may change along the measured current-voltage curve. This feature is commented on in Sec. VII. The constitutive equation of a constant capacitor of capacitance Cis the charge Qto voltage Vrelation Q=CV.(2) The current under voltage sweep (1) is given by I=dQ dt =Cvs.(3) Thus, the forward current is positive and the reverse current is negative; see Fig. 1(C)(a). In each case the current is proportional to the scan rate. This fact is well known and it is widely used for the characterization of electrochemical capacitors, as shown in Fig. 1(C)(b) [28]. The impedance of the capacitor is obtained by the Laplace transform of Eqs. (2) and (3), and the result is Z=1 iωC.(4) Now consider an inductor with inductance L, which is described by the equation V=LdI dt ,(5) with the correspondent impedance Z=iωL.(6) On the basis of Eq. (4) we can define a generalized complex capacitance C∗as follows: C∗=1 iωZ.(7) We obtain the capacitance of the inductor as [73] C∗=− 1 ω2L.(8) Therefore, the inductor element is associated with a “negative capacitance” [74,75]. We observe that a capacitor and an inductor provide very different impedance responses, due to their intrinsically different time delays. Next we analyze the significance of these properties for the interpretation of hysteresis. IV. LINEAR MODEL WITH MEMORY TO EXPLAIN THE BASIC TYPES OF HYSTERESIS As mentioned, ionic-electronic devices may require very complex models, providing intricate hysteresis properties. However, there is an intrinsic structure to many models that we can analyze on the basis of a simple model with two equations that link the current, Itot, the voltage, u,andan additional internal state variable, w. To explore these basic 011001-4 INDUCTIVE AND CAPACITIVE HYSTERESIS . . . PRX ENERGY 3, 011001 (2024) properties of hysteresis, we consider the linear equations Itot =u Rb +w+Cm du dt ,(9) τk dw dt =u Rw −w b, (10) where Cmis a capacitance, and Rband Rware constant resistances. The total current in Eq. (9) has three components: an instantaneous current u/Rb, a capacitive current, and a slow variable current wthat is determined by the relaxation equation (10), with constants τkand b. We calculate the impedance spectroscopy response of this model for a small sinusoidal perturbation of angular frequency ω. Since Eqs. (9) and (10) form a linear system, the Laplace transform for the variable s=iωis ˆ Itot =ˆu Rb +w+Cmsˆu, (11) τksˆw=ˆu Rw −ˆw b. (12) Here the circumflex accent indicates a small perturbation of the variable y. We obtain the impedance as follows: Z(s)=ˆu ˆ Itot =Cms+Rb −1+1 Ra+Las−1 , (13) where the resistance and the inductor are defined as Ra=Rw b, (14) La=Rwτk, (15) and the inductor characteristic time is given by τL=La Ra =bτk. (16) The equivalent circuit that represents Eq. (13) is shown in Fig. 2(a). This model indicates the three parallel branches mentioned before: the capacitive charging, the direct conduction mode Rb, and the slow inductive branch (Ra,La). The capacitive element has been included in Eq. (9),as most devices do have an intrinsic capacitance. On the other hand, the inductive element is due to the delay equation (10) for the slow variable. This is not a general property of devices, but it is more common that one may suspect, especially when ionic-electronic mixed effects are present. Since the inductor is not based on electromagnetism, it is generally described as a chemical inductor [73,76]. When the inductor effect is negligible, the impedance is formed by a positive RbCmarc; see Fig. 2(b),arcA. If, however, the inductor parameter is large, the impedance traces a loop in the fourth quadrant that represents a “negative capacitance” feature, as shown by arc Bin Fig. 2(b). Figure 2(e) shows a more general equivalent circuit [77] with an RACAline that produces a double-arc feature at low frequencies that will be discussed later. This feature, corresponding to surface polarization of ion diffusion that is blocked at the contacts, was originally observed in perovskite solar cells by light-modulated techniques [78]and by impedance methods [79,80]. In the following we argue that the combination of the standard RbCmindicated in Fig. 2(a) with the delay mode represented by the RaLabranch explains the existence of both types of hysteresis mentioned earlier in current-voltage curves of electronic devices as capacitors and memristors, and the two types will be classified as capacitive and inductive hysteresis [11–16,81,82]. The stationary current-voltage characteristic of the model is given by Idc =1 Rb +1 Rau(17) and the current will be modified when the voltage scan rate vsis finite. We now restrict the model to the time dependence in Eq. (1). We first discuss the case in which the slow current wresponds without delay, to focus on the capacitive current; hence, w=u/Ra, and the total current is given by Itot =u Rb +u Ra +Ic, (18) where Ic=Cmvs. (19) As shown in Fig. 2(c), under a forward scan, a positive capacitive current is added to the stationary current. The capacitive current is proportional to the scan rate vs,as already stated in Eq. (3). We now discuss the second component of the timedependent current, the slow variable wthat responds to the applied voltage with characteristic time τk.FromEq.(10), we obtain for a constant sweep rate τkvs dw du =u Rw −w b. (20) This equation needs to be integrated and inserted into Eq. (9) to obtain the current. For the initial condition w(0)=0, the solution is Itot(u)=u Rb +u Ra +vsτL Ra (e−u/(vsτL)−1). (21) 011001-5 JUAN BISQUERT PRX ENERGY 3, 011001 (2024) (a) (b) (e) (c) (d) FIG. 2. (a) Equivalent circuit model. (b) Impedance spectra for Cm=1, Rb=1, Ra=1, and La=0.1 (arc A) or La=20 (arc B). The red point is the dc resistance. (c),(d) Current-voltage curves at different voltage sweep rates vs,u=vst. (c) The state variable responds fast, and the capacitive current with capacitance Cm=1 is added to the direct current. The steady-state current (slow) is shown as the gray line. (d) The capacitance is removed, Cm=0, and the variable wcontributes to the current. The purple line corresponds to an infinitely fast scan. The parameters are Rb=1, Ra=1, τk=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. The inductive current IL=Itot −Idc is negative in the forward scan and positive in the reverse scan, as shown in Fig. 2(d), corresponding to the negative capacitance of Eq. (8).Ifvsis small, the full current in Eq. (17) is activated, indicated by the gray line in Fig. 2(d). However, if vsτLu, the inductive current through Ravanishes in Eq. (21) and the remaining fast component Ifast =u Rb (22) is indicated by the purple line in Fig. 2(d). This representation provides an excellent account of the properties of the KIR channel observed in Fig. 1(D). V. GENERAL CHARACTERISTICS OF INDUCTIVE AND CAPACITIVE HYSTERESIS To obtain a general system that displays both capacitive and inductive hysteresis, we can write the dynamical equations with the same general structure of Eqs. (9) and (10), 011001-6 INDUCTIVE AND CAPACITIVE HYSTERESIS . . . PRX ENERGY 3, 011001 (2024) but with arbitrary nonlinear functions for conductivity, f, and time delay of the internal variable, g, corresponding to the specific physical situation: Itot =Cm du dt +f(u,w), (23) τk dw dt =g(u,w). (24) The delay equation (24) is very typical of broad kinds of model, such as the Hodgkin-Huxley model in neurobiology [58,59], the related models for protein ion channels with memory effects [83,84], and the generic model of a memristor with an internal state variable [36,40]. It was reported in very old measurements that ion channels in neurons contain a large inductor component [85–87]. In the field of halide perovskite solar cells, this model represents the essential mechanism for the electronic recombination controlled by ionic motion that has been suggested in many papers to explain the negative capacitance observed in the device [15,21,79,88,89]. To determine the equivalent circuit, we obtain the small perturbation expansion: ˆ Itot =Cmsˆu+fuˆu+fwˆw, (25) τksˆw=guˆu+gwˆw, (26) where the subindex indicates a partial derivative, fu= ∂f/∂uand so forth. Equations (25) and (26) reproduce Eqs. (11) and (12) and the impedance model is the same as Eq. (13).The equivalent circuit elements are given by [69,73,77] Rb=1 fu , (27) Ra=− gw fwgu , (28) La=τk fwgu . (29) These elements may all be functions of the voltage according to the properties of the functions fand gproducing evolutions of the hysteresis properties that are discussed below. The inductor time constant is given by τL=−τk gw . (30) Consider the special role of the constant bin Eq. (10) and more generally gwin Eq. (26) for the system’s stable response. If b<0(gw>0), then the inductor time constant τLis negative, which produces an exponential growth in Eq. (24). Then the system is unstable [90]. Hence, we require that gw<0. We can now generalize the results of Fig. 2for a nonlinear system of the type of Eqs. (23) and (24). We apply a voltage schedule with varying sweep rates, as indicated in Fig. 3(a). In the first part we have vs>0 and in the second part we have vs<0, so as to return to the initial voltage. Similarly, the voltage can be cycled with frequency s. (a) (b) (c) (d)(e) FIG. 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. 011001-7 JUAN BISQUERT PRX ENERGY 3, 011001 (2024) We assume first a system that is dominated by the capacitive transient current; see Fig. 3(b). In the first part of the cycle it is Ic>0, while in the second part it is Ic<0. Therefore, the current will describe a clockwise loop as indicated in Fig. 3(b). We term this loop a “capacitive hysteresis”, and the associated impedance spectrum is a positive arc; see Fig. 3(c). If, on the other hand, the system is dominated by the slow current component, then we have IL<0inthefirst part and IL>0 in the second part. The hysteresis loop is counterclockwise, and we term it “inductive hysteresis”; see Fig. 3(d). The correspondent impedance spectrum is a negative arc as indicated in Fig. 3(e). From Fig. 3we conclude that for a capacitive hysteresis the forward current is larger than the stationary curve, and the current in the reverse direction is lower. However, for solar cell devices, the representation is often reversed. This is because the total current is given by Itot =Iphoto −Irec, (31) where Iphoto is the photocurrent, taken to be positive, and Irec is the recombination current associated with Eq. (23), which contains the hysteresis effects. Hence, for the forward scan the capacitive hysteresis current appears lower than the reverse scan current, as shown in Fig. 1(A) [22], and the effect of capacitive and inductive hysteresis in the solar cell current is opposite that in Fig. 3. In the field of halide perovskite solar cells, the capacitive hysteresis is termed “regular” and the inductive hysteresis is termed “inverted” [12,18,26,91]. A major problem for determining the PCE in perovskite solar cells with hysteresis, mentioned long ago [1], is that the performance is different in the forward and return cycles, consequently necessitating equilibration methods by successive slow scans [34,72]. The PCE is a product of current and voltage at the maximum power point. We can see in Fig. 1(A) that the capacitive hysteresis produces a larger PCE in the reverse scan than in the forward cycle [22], while for inductive hysteresis the converse is true: the PCE is larger in the forward direction [10,12]. These properties are independent of the sign convention adopted to plot the current. In the literature there are different ways to quantify the capacitive and inductive hysteresis. A hysteresis index is based on the integral of the hysteresis loop [8,92]: A=V2 V1 IdV. (32) This or equivalent integrals (such as the hysteresis enrichment charge [5,62]) provide a positive area for inductive hysteresis and a negative area for a capacitive hysteresis, or vice versa, according to the sign convention for the current. VI. INTERPRETATION OF SYSTEMS WITH CAPACITIVE AND INDUCTIVE HYSTERESIS Starting from the reference linear system formed by Eqs. (9) and (10) we pass to the more general model of Eqs. (23) and (24) that produces a general connection between the dominant elements of impedance spectroscopy and the type of hysteresis loop that will be obtained [14,16]. Many materials and systems may require still additional features with respect to Eqs. (23) and (24), depending on the morphology, the number of internal processes, and their evolution with the applied voltage [11, 93]. Nevertheless, the basic distinction of capacitive and inductive hysteresis observation provides a useful diagnostic technique, as the equivalent circuit obtained from the measurement of small-perturbation impedance spectroscopy can tell us the dominant hysteresis type that is expected in the large perturbation of linear sweep voltammetry, according to the frequency or velocity of the measurement [13,15,22]. Furthermore, the observation of the prevalent type of hysteresis can produce a guideline for the basic structural equations needed in a specific model system. To further determine the significance of these questions, in the following we provide a set of examples where the connection between hysteresis and impedance illustrates the kinetic and physical elements of the system. First we return to the hafnium oxide–based memristor corresponding to Fig. 1(B) and we note that the large inductive (“inverted”) hysteresis loop characteristic of the set process of the memristor, in which the resistance switches to a high-conductance state [94], reveals the appearance of the associated inductor in the complex plane plot of the impedance, Fig. 1(B)(b), as predicted by the general model. Equations (23) and (24) are the standard constitutive equations of a memristor [36,40,95], and we can remark that memristors, in the voltage range of the set process, will display a large inverted hysteresis loop in the set cycle, where the current increases at a certain threshold of voltage or current [12–14]. The associated chemical inductor appears naturally in the impedance response, as observed in Fig. 1(B)(b). A similar connection of inverted hysteresis and the inductive loop has been reported for halide perovskite memristors and solar cells [14,16,96], and these devices are discussed in Sec. VII. To further analyze the change of hysteresis type, we consider the properties of conical ionic channels in an electrolyte solution, shown in Fig. 4(a), that show rectifying properties due to the electrical interaction between the functionalized charges on the conical pore surface and the nanoconfined ionic solution [55,98]. Nanopore memristors have been studied for their strong hysteresis properties [5,61,62], and inductive impedance has been reported [60], but the correlation between the type of hysteresis and the inductive element was not established. 011001-8 INDUCTIVE AND CAPACITIVE HYSTERESIS . . . PRX ENERGY 3, 011001 (2024) (a) (b) (c) (d) (e) FIG. 4. (a) Current-voltage curve measured at frequency s=10 Hz for a multipore membrane in 100 mMKCl 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. We show in Fig. 4results obtained recently by Ramirez et al.[97]. On the positive voltage side of the currentvoltage curve in Fig. 4(a), the memristive pores show a strong inverted hysteresis, accompanied by the notorious inductive impedance loop in Fig. 4(d). The forward and reverse scans of the current-voltage curve cross at the origin, and it follows that the type of hysteresis is changed. On the negative side, the hysteresis is capacitive, and the impedance shows purely capacitive spectra; see Fig. 4(b). Remarkably, it has been shown that the side of inductive hysteresis of the pore response can be transposed from a positive potential to a negative potential by modulation of the electrolyte composition [8], opening interesting applications for nanofluidic neuromorphic circuits. 011001-9 JUAN BISQUERT PRX ENERGY 3, 011001 (2024) capacitive hysteresis in terms of elementary relaxations, we now apply a similar method to the linear model of Eqs. (9) and (10), represented as the equivalent circuit in Fig. 2(a). When a voltage step is imposed on the conducting system, it is important to consider the presence of the series resistance, as indicated in Fig. 2(a). Then the applied voltage Vapp is divided between the active zone voltage, u,and the series resistance, as follows: Vapp =RsItot +u. (42) We apply two consecutive voltage steps V,2Vof duration tand measure the current. The results of the calculation are shown in Fig. 8(B) for a system in which the inductive process is very fast and can be ignored, so that the time response is dominated by the capacitance and resistors. In each diagram the gray line is the stationary current that can be expected at the given applied voltage, i.e., Idc =Vapp Rtot , (43) where Rtot =Rs+1 Rb +1 Ra−1 . (44) We observe that when a voltage step is applied at t0in Fig. 8(B)(a), there is a capacitive peak. This is because in the initial instant the capacitor is discharged and all the (a) (b) (c) (d) (e)(f) FIG. 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) CuI-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. 011001-16 INDUCTIVE AND CAPACITIVE HYSTERESIS . . . PRX ENERGY 3, 011001 (2024) voltage goes to the series resistance [121], so the initial peak height is V/Rs. Thereafter the capacitor charges with a characteristic time τC=RsCm, until the equilibrium current Idc is reached. The next increase Vraises the current again. In Fig. 8(B)(b) the capacitance is 10 times larger, so the charging time is 10 times longer. The charging cannot be completed in each step, and a difference Iis accumulated at the end point. This is the positive capacitive hysteresis current. We remarked regarding Eq. (31) that the representation of the photocurrent in a solar cell is inverted with respect to the dark current. For Fig. 8(B)(c), we apply Eq. (31) and flip the capacitive peaks, so that Ibecomes negative. The resulting pattern explains well the current response to voltage steps in a forward scan in Figs. 1(A)(b) and 8(A)(b). We can deduce from the model that in a faster scan, the steps will be shorter in time, and the accumulated deficit of current will be larger. This is the property of capacitive hysteresis that is observed in Fig. 8(A)(c). For Fig. 9the step procedure is applied to two perovskite devices that differ in the hole extraction contact [120]. In the first cell [Figs. 9(a) and 9(b)] the relaxation is not completed in each step and the sample shows significant hysteresis. In the second case [Figs. 9(c) and 9)(d)] the charging time τCof the capacitor is much shorter, so the hysteresis is reduced. This was achieved by decrease of both the low-frequency resistance and the capacitance, as shown in Fig. 9(f). We now turn our attention to the influence of the inductor in the step response of the current, shown in Fig. 10. The transient begins with a capacitive peak, as in the former case, but now additional parallel resistance Rais activated when the inductor responds in time of order τL [121]. Hence, the current increases with time until the equilibrium current is reached; see Fig. 10(a). In Fig. 10(b) the inductor time is longer, and the final current cannot be achieved. This produces a total negative deficit, which explains the property of negative hysteresis in terms of the elementary step behavior. A full classification of the timedomain response when several inductive and capacitive elements act in concert was presented in Ref. [111]. In summary, capacitive and inductive processes provide opposite types of time transient response. This is closely related to the different frequency-domain response of both elements indicated in Figs. 3(c) and 3(e).Ina capacitive process the impedance decreases at increasing frequency. Correspondingly, the current of capacitive charging decreases with time. However, the impedance of the inductor decreases at lower frequency. Consequently, the current of the inductive process increases with time, as shown in Fig. 10. The inductive property is an essential component of the phenomenon of synapse potentiation that is necessary for neuromorphic computation elements [8,112,121–123]. IX. THE FREQUENCY DEPENDENCE OF HYSTERESIS In this section we comment on the manifestation of hysteresis as a function of the frequency sof measurement of current-voltage curves [114]. This is shown in Fig. 11(A) for the porous nanochannels of Fig. 4[8]. The voltage is scanned by a sinusoidal wave. At low frequencies the full stationary current is observed, but at higher frequencies at positive voltage the inductive hysteresis becomes significant and the current remains lower due to removal of the slow-response component. We consider an important result shown in Fig. 11(B) [72], which illustrates the correspondence of the hysteresis effect and the frequency of measurement, according to the (a) (b) FIG. 10. Current response of the linear model to two consecutive voltage steps, from time t0, of duration t, the first with amplitude V=0.1 and the second with amplitude V=0.2. The common parameters are Rb=1, Ra=0.1, Rs=0.05, t0=0.5, t=1, and Cm=1. The different cases are (a) τk=0.3 and (b) τk=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. 011001-17 JUAN BISQUERT PRX ENERGY 3, 011001 (2024) (A) (C) (B) (a) (b) (c) FIG. 11. (A) (a) I-Vcurves for a multipore membrane in 100 mMKCl solution at neutral pH, parametrically in the electrical potential scan rate, characterized by the signal frequency f=s/2π, 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 CH3NH3PbI3based mesoscopic solar cell, measured at 950 mV under equivalent 1000 W m−2solar irradiation. The Lissajous curves (current versus voltage) corresponding to a sinusoidal perturbation (Vrms=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 fasthysteresis 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. type of impedance response that prevails at the frequency chosen. By changing the cycling of the current-voltage curve (with a small amplitude of 22 mV), we change the point of measurement in the complex plane plot of the impedance. It is observed that when the imaginary part of the impedance becomes large, corresponding to a capacitive response at this frequency, the hysteresis increases, as shown in the Lissajous curves. In Fig. 11(B) the two components introducing time lag in the respective arcs (one at low and another one at high frequency) are capacitive, corresponding to the model of Figs. 2(e) and 5(c). The low-frequency inductor response is not presented in Fig. 11(B), although it could exist, but the impedance is not reported up to very low frequency. If we measure up to such a low frequency, and find the inductor component, we can infer that the inductive feature of the hysteresis could appear, causing the Lissajous figure to spin in the opposite direction, as shown in Fig. 3. 011001-18 INDUCTIVE AND CAPACITIVE HYSTERESIS . . . PRX ENERGY 3, 011001 (2024) Figure 11(C) shows a study of the hysteresis traits of drift-diffusion simulated halide perovskite operation, which corresponds well to the experimental observations in the same work [113]. The solar cell is stabilized at opencircuit voltage and then measured in reverse and forward cycles, as shown in Fig. 11(C)(a). The PCE is larger in the reverse scan, so this model contains only capacitive hysteresis, as discussed in Sec. V, and the impedance picture must be the same as in Fig. 11(B). The trend of hysteresis as a function of frequency shown Fig. 11(C)(b) is the same as that in Fig. 11(B): the system surmounts a capacitive arc centered at a scan rate of 10 V s−1. At a high frequency of the scan, the system becomes a pure resistance, and the total PCE increases slightly. This is because the low-frequency arc in Fig. 11(B) has been removed, and it means that the low-frequency arc is deleterious for the performance, because it is a recombination loss at the contacts, caused by the ionic distribution, that is avoided in the fast measurement. In the study in Ref. [114], the efficiency and hysteresis increase at increasing frequency as in Fig. 11(C), but then start to decrease at the highest frequencies. This means that the scan is affecting the high-frequency arc that holds the PCE of the solar cell. However, the previous considerations should be regarded as preliminary intuitive explanations. In the voltage scan the system changes over a wide voltage range that covers a large variation of the impedance parameters [63,124], and a detailed quantitative transformation between the time domain and the frequency domain becomes necessary [11]. We remark that one can obtain a transition of the kind of hysteresis (capacitive to inductive or vice versa) in two different ways: by changing the voltage modifying the dominant low-frequency component, as shown in Fig. 6, or by operating at a fixed voltage, by changing the frequency of the measurement of the current-voltage curve, as in Fig. 11. X. CONCLUSION Our comprehensive exploration has unveiled the intricate nature of hysteresis across a spectrum of electronic, ionic, and mixed ionic-electronic devices. Through a multidimensional approach, we have delineated two fundamental hysteresis archetypes—capacitive and inductive—rooted in distinctive transient responses. These archetypes, revealed through diverse analytical methods, provide a unifying framework to decipher hysteresis in systems as disparate as solar cells, capacitors, transistors, and ion channels. Furthermore, the use of complementary methods facilitates the prediction and classification of more concrete physical and molecular models that account for the complicated evolution of the current with time in highly heterogeneous and nonlinear systems. Our analysis provides a criterion for making these advanced physical models, such as those elaborated by drift-diffusion equations and polarization assumptions. From the knowledge of basic structural conditions for equations that produce capacitive and inductive hysteresis and their combinations, we have a yardstick to measure advanced physical models that contain specific physical effects. We can first determine the evolution of hysteresis in an experimental system using the frequency analysis and general descriptive models, and can then develop concrete specific explanations that satisfy the overall behavior. We emphasize that the fact that a model providing some type of description of hysteresis is not enough to validate such a model. Many equivalent models can be made, provided that they satisfy the basic structural conditions that give the right evolution of the equivalent circuit. A more stringent test of models and theories is needed, at least considering a variety of experimental response methods. By elucidating the underlying mechanisms driving hysteresis, we increase our understanding of its pervasive presence in both natural and engineered contexts, contributing to the broader understanding of dynamic responses in complex systems. ACKNOWLEDGMENTS This work was funded by the European Research Council via Advanced Grant No. 101097688 (PeroSpiker). I am grateful for discussions with Agustín Bou, Antonio Guerrero, Cedric Gonzales, Enrique Hernández-Balaguera, and Patricio Ramirez. [1]H.J.Snaith,A.Abate,J.M.Ball,G.E.Eperon,T. Leijtens, N. K. Noel, S. D. Stranks, J. T.-W. Wang, K. 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