Memristor devices have been investigated for their properties of resistive modulation that can be used in data storage and brain-like computation elements as artificial synapses and neurons. Memristors are characterized by an onset of high current values under applied voltage that produces a transition to a low resistance state or successively to different stable states of increasing conductivity that implement synaptic weights. Here, we develop a nonlinear model to explain the variation with time of the voltage and the resistance and compare it to experimental results on ionic–electronic halide perovskite memristors. We find separate experimental signatures of the capacitive discharge and inductive current increase. We show that the capacitor produces an increase step of the resistance due to the influence of the series resistance. In contrast, the inductor feature associated with inverted hysteresis causes a decrease of the resistance, as observed experimentally. The chemical inductor feature dominates the potentiation effect in which the conductivity increases with the voltage stimulus. Our results enable a quantitative characterization of highly nonlinear electronic devices using a combination of techniques such as time transient decays and impedance spectroscopy.
The continuous development of artificial intelligence powered by neural networks, which can receive physical inputs from a wide range of sensors, has paved the way for the creation of various applications capable of performing numerous automated tasks. For instance, autonomous vehicles utilize various sensors, such as cameras, radars, and lidars, to perceive their environment and make decisions accordingly. In the healthcare industry, wearable devices can track vital signs and provide alerts in the case of abnormal readings. Similarly, in the manufacturing sector, machines equipped with sensors can detect anomalies in the production process and take corrective measures automatically.
However, there is a concern that the expansion of artificial intelligence for improved efficiency and productivity across various industries is accompanied by a significant amount of energy consumption. Accordingly, new computational systems that mimic the natural brain are being developed.1 Well suited for perception, cognition, and motor tasks, these methods are adaptive, fault-tolerant, and scalable and process information using energy-efficient, asynchronous, and event-driven techniques. One candidate for the development of brain-like neuromorphic systems is the memristor element that can be formed in several material platforms for the functions of synapses and neurons.2–9 Memristors are highly nonlinear systems that show resistive switching under voltage cycling.10,11 One type of system shows an abrupt jump to a low resistance state where an ohmic response is observed at high voltages.12,13 These are useful as volatile memories and only show two states of current: the highly resistive state during the sweep in the forward direction of the I–V curve and the low resistive state in the backward direction. On the other hand, to use a memristor as a synapse, it is necessary to program it with distinct non-volatile resistive states that can support spike timing-dependent plasticity, as shown in Fig. 1.10,14–17
By construction, memristors show intense hysteresis and memory effects that have been the topic of current investigations especially for the multistate memristors for synaptical functions.17,18 One central property of a memristor is the temporal evolution of the resistance that is described in Fig. 2 for a halide perovskite CsPbBr3 nanocrystalline film device.9 The experimental results show the dominant effect of a chemical inductor19,20 in the temporal response of the memristor. In this paper, we apply a recently developed nonlinear memristor model21–23 for a quantitative description and interpretation of the current transients and the peaks of memristor resistance under voltage pulses.
II. THE MEMRISTOR MODEL
Thus, ic varies from 0 at low voltage to a saturation value ic0 at high voltage, with the activation potential VT and an ideality factor Vm with dimension of voltage. The physical mechanisms governing the change of fss are the filamentary conductive pathway6,25 or the decrease of a surface barrier between the perovskite layer and the contacts.26,27
III. AC IMPEDANCE PROPERTIES
The model of Eq. (7) is the recently described impedance of a chemical inductor.19 It is characteristically observed in halide perovskite devices in the high voltage domain.28,29 The equivalent circuit is shown in Fig. 3(b), and the impedance parameters are represented in Fig. 5(a). These elements are changing by orders of magnitude under the application of voltage corresponding to the highly nonlinear nature of the system.
In the following, we explore the interplay of capacitance and inductance in synaptic memristors based on the nonlinear neuron-style model developed in perovskite solar cells, Eqs. (1)–(3).20,32 Capacitive transients have been already described for semiconductor LEDs,33 and the potentiation of synapses was explained based on the same model.34
The connection between the impedance features and hysteresis properties under voltage scan has been elucidated in halide perovskites32,35 and related areas.36 In capacitive hysteresis, the current increases at a higher scan rate. However, when the low frequency impedance is dominated by a chemical inductor, the current decreases at a higher scan rate and the inverted hysteresis pattern of Fig. 1(b), which is a frequent characteristic of memristors, is obtained. Figure 4(b) shows the result of continuously cycling the applied voltage in the model of Eqs. (1)–(3) at a constant scan rate bs. The current–voltage curves are characterized by inverted hysteresis due to the effect of the inductor.32,37
IV. TIME TRANSIENT RESPONSE
The current response of a perovskite memristor to a voltage step pulse to the value Vapp can be understood from the equivalent circuit. The best way for explaining this response is to separate it into different moments, as indicated in Fig. 6(a).
The very first moment corresponds to the high frequency part of the impedance spectrum. At this point, the parallel capacitance is shorted, and therefore, the parallel resistances Rb and Ra have no current going through them. The total resistance of the system is reduced only to the series resistance Rs. Accordingly, the memristor will initially take the current value corresponding to the gray line, ΔI = Vapp/Rs, point A, which corresponds to the minimum value of impedance, as in the spectrum of Fig. 5(d).
Immediately after, the capacitance gets charged. Then, the current starts decreasing by the time constant τs, until the value of the ac resistance is maximum, point B. This can be clearly seen through the impedance spectrum: at intermediate frequencies, the spectrum crosses the real axis at the maximum value of the resistance. If the time constant of the R–L branch τd is large enough, the inductor will take a long time to charge and will initially behave as an open circuit; therefore, the total resistance of the system at this intermediate step will be close to Rs + Rb (green line), i.e., all the current goes through Rb. The current response will follow its minimum value, point B; see Fig. 6(b).
When time reaches the order τd, the inductor starts passing current. The essential property of the inductor is that once the capacitor has already been fully charged, the current still increases with time, as shown in Fig. 6(a).20 These effects are well known in solid-state electronics.38–41 When it is completely activated, the current reaches the magenta line, point C. Here, the inductor behaves as a short circuit and the total ac resistance of the system decreases from its maximum point. Looking at the impedance spectrum, at lower frequencies, the impedance decreases from the maximum resistance to a lower value through the arc of the fourth quadrant, Fig. 5(d). At zero frequency, the impedance gets the Rdc value, which is the series resistance Rs plus the resulting resistance of parallel Rb and Ra, Eq. (10).
Finally, we observe the inversion of the current when the voltage is set to 0. If the internal voltage at the moment of switch off is uf, the voltage in the resistance must be −uf = ItotRs. Hence, the negative current increment is the same as the initial step, point D.
To sum up, we have described transient patterns that are well observed in the potentiation of perovskite synapses.5 In applying a step, we have a maximum at the very first moment, when the total resistance is only the series resistance. Then, the current decreases to its minimum, when the resistance maximizes due to the slow response of the inductor. Finally, the inductor permits the flow of current and the total resistance reduces. At this point, the current gradually increases to its steady-state value. A negative current is observed upon disconnecting the voltage.33 These patterns consisting of slow charging features due to a chemical inductor property can be observed in the transient behavior of different types of devices, as in photoswitching experiments in light detectors and photoelectrochemical cells.42,43 In Fig. 7, the feature is shown for halide perovskite photodetectors.42 It is observed that the current rises after the initial spike in cases (b) and (c), while (a) and (d) show a different type of response.
We have described here a single charge–discharge cycle. In the potentiation of synapses over multiple steps, the system remembers the voltage of the previous cycle5,42,44–49 and the current increases gradually, as shown in Fig. 1(c). We conclude that the chemical inductor is an essential feature for the gradual rise of the synaptic conductivity.18,24 These properties have been recently explained elsewhere.34
V. RESISTANCE TRANSIENTS
We conclude that in the capacitive transient, the resistance increases. Note that Rf > Rdc with respect to the small perturbation value Rf indicated by the magenta line, Eq. (10).
When the applied voltage pulse is larger in Figs. 8(c) and 8(d), the inductor is stimulated. The small parallel resistance Ra is activated, and the total resistance decreases in the inductive domain, as shown in Fig. 8(d). Overall, there is a peak shape in the resistance vs time: the rising capacitive part and the decreasing inductive part. These features are, indeed, reported in Fig. 2 9 and in the supplementary material of Ref. 5.
The previous analysis in Figs. 8(b)–8(d) shows that the resistance measured as instantaneous V/I is, in general, different from the small ac perturbation resistance, that is, the low frequency value of the impedance, obtained as Rdc = Z(ω = 0), Eq. (10). The use of V/I is convenient for transient measurement, as it is well defined at any moment. In contrast to this, ac impedance gives more general information, but it needs to be measured at stationary points. This is not feasible, in general, as the memristor has an internal dynamics of the state variable x that cannot be controlled externally. Therefore, it is not warranted that one can measure the full ac impedance up to very low frequency unless the I–V curve can be stabilized, as explained elsewhere.50
The calculation of the current and resistance response is shown in Fig. 9 for the same decays as in Fig. 8. In Fig. 9(a), we observe that the charging of the capacitor by the voltage step activates Ra and produces the stepping down of the total dc resistance toward the equilibrium value Rdc. When the applied voltage pulse is larger, the inductor is stimulated. As a consequence, in Fig. 9(b), after the capacitive decrease of the resistance, another feature appears in which it increases and then decreases with time. Therefore, it must be remarked that the ac measurements should be made once the system is stabilized, and the impedance value is constant with time.
VI. GENERAL SIGNIFICANCE
Here, f is a conductivity function, and the changes of the “slow” state variable x are controlled by a driven adaptation function g(u, x) with the characteristic time τk. The above system of equations corresponds to the general structure of a voltage-controlled memristor,51 where impedance and stability properties have been recently described.19,23,52,53 For any combinations of configurational functions , the equivalent circuit in Fig. 3(b) is obtained in every case. Hence, the transient properties discussed in Sec. IV can be obtained in very different circumstances.
However, there are different possibilities according to the time constant and specific form of configurational functions. The proposed model is applicable in a wide range of systems, providing accurate fittings where a saturation current (ic0) is observed at high voltages, such as in Fig. 4. These systems gradually increase their current with the applied voltage with a representative non-ohmic response. The actual saturation current values depend on the conditioning steps applied to the sample (i.e., voltammetry cycles), and multiple current states may be reached that are useful for analog computing systems.5,9,18,24,54 This saturation current is not to be confused with the compliance current that is typically programmed during the measurement to avoid the damage of the sample.
As mentioned before, other resistive switching systems instead present a sudden jump to a low resistance state with ohmic characteristic at high voltages.12,13 They are useful as volatile memories and only show two states of current, the highly resistive state during the sweep in the forward direction of the I–V curve and the low resistive state in the backward direction. The second type of systems will not be fitted adequately with the proposed model.
It has been previously observed that the electrical response of halide perovskite memristors shows different stages according to the stationary applied potential. It evolves from a totally capacitive response at low applied voltage to a dominant inductive feature at the onset of the low resistance state. The latter feature causes the typical inverted hysteresis feature of the memristor.21,37,50 Here, we showed a nonlinear neuron-style memristor model that enables a simultaneous analysis of small signal ac impedance and the transient behavior after a voltage pulse. The current in the capacitive domain is a decay after an initial peak. Correspondingly, the total resistance increases to the resistance value characteristic of the stationary state of the memristor. However, when the voltage pulse is higher, it activates the inductive feature. Accordingly, after the initial capacitive increase, the current rises again and the resistance traces a peak. The chemical inductor causes an increase of the current with time that explains the positive potentiation feature in which a high conductivity is achieved in successive voltage cycles. These observations provide a new tool for the quantitative interpretation of memristive synaptic dynamics by the combination of frequency and time domain electrical measurements.
This study forms part of the Advanced Materials program and was supported by MCIN with funding from the European Union Next Generation EU (Grant No. PRTR-C17. I1) and Generalitat Valenciana. This work received funding from the Universidad Rey Juan Carlos, Project No. M2993.
Conflict of Interest
The authors have no conflicts to disclose.
Juan Bisquert: Conceptualization (lead); Formal analysis (equal); Writing – original draft (lead). Agustín Bou: Formal analysis (equal). Antonio Guerrero: Formal analysis (equal). Enrique Hernández-Balaguera: Formal analysis (equal).
Data sharing is not applicable to this article as no new data were created or analyzed in this study.