A Memristive Oscillator

Memristors, or more generally “memristive systems,” are nonlinear electric components that change their resistance depending on a set of state variables. Meanwhile, negative differential resistance (NDR) is an uncommon electrical property that occurs in a few nonlinear electric components. Such a nonlinear property is now widely used in amplifiers and oscillators. It finds that one of the bilayer‐type nickel‐dithiolene complexes shows nonlinear transport with the NDR property and self‐oscillates under a bias current by simply adding a capacitor in parallel. Surprisingly, the Lissajous figure of this molecular material shows a “pinched hysteresis loop” typical of a memristor, and an inductive reactance emerges when a bias current or voltage is applied to the sample. Herein, a new type of oscillator that uses the NDR and hidden inductive properties of a memristive system is reported. The findings suggest that a memristive system can mimic other passive elements, such as an inductor, and can oscillate itself without having them in the circuit. The oscillation mechanism is straightforward and applicable to a wide variety of memristive materials, including resistors with large negative temperature coefficients, discharge tubes, and even nerve cells in the living body.


Introduction
Memristors are considered the final piece of the theoretical quartet of fundamental electrical components after the resistor, capacitor, and inductor. [1]It behaves like a resistor but the resistance is dependent on the history of the electric current passing through it.1b] After the proposal of the memristor concept, the physical realization of the memristor took several decades until the first candidate was found in a resistive random-access memory (ReRAM). [2]emristors are considered versatile materials that can be used in next-generation low energy-consumption memories, analog computing, and neuromorphic computing.The molecular oscillator that gradually reveals its memristor properties is the bilayer-type nickel-dithiolene complex (Et-4BrT)[Ni(dmit) 2 ] 2 , where Et-4BrT is ethyl-4-bromothiazolium and dmit is 1,3-dithiole-2-thione-4,5dithiolate.The Ni(dmit) 2 molecules form two crystallographically independent layers with different molecular arrangements inside the unit cell (inset of Figure 1a). [3]Owing to its characteristic molecular arrangement, layer A is considered the (Mott insulating) magnetic layer, and layer B is the conducting layer.Basic physical properties can be found in the literature. [3,4]This molecular material shows a semiconductor-to-insulator transition ≈T SI = 60 K (Figure 1a), and a ferromagnetic transition at T C = 1 K.Although the insulating ground state of layer B has not yet been confirmed, the origin of the ferromagnetic property in layer A is considered to be due to the Nagaoka-Penn ferromagnetism. [4]

Negative Differential Resistance and Self-Oscillations
To investigate the origin of the semiconductor-insulator transition, we studied the current-voltage (I-V) characteristics of (Et-4BrT)[Ni(dmit) 2 ] 2 .The I-V characteristics show nonlinear transport below T SI , and a remarkable negative differential resistance (NDR) property is observed below 35 K (Figure 1b).The appearance of the NDR property was due to the large resistance drop caused by the applied current.Because of the differences in the sample resistance and contact resistance, the threshold current/voltage and NDR slope differ for each sample, as shown in Figure 1b,c.Similar NDR properties have been previously observed in several molecular materials, such as TTF-CA, mixed stacked charge transfer salts, organic conductors, and MXand MMX-chain complexes. [5]5b] The analysis can be found in the Supporting Information.Therefore, the observed NDR is not only due to the heating effect of the sample but also due to carrier decondensation or multiplication by the applied current.
The NDR is widely used in "negative resistance oscillators" such as the Esaki and Gunn diode oscillators. [6]In general, a diode with NDR property is connected across inductor L and capacitor C to create a resonating LCR circuit.NDR cancels the positive resistance of the resonator circuit, creating a lossless resonator in which spontaneous oscillations occur (Figure 1d).Another notable self-oscillation mechanism that uses NDR is the "relaxation oscillator." [6,7]In this case, the circuit contains an NDR device that serves as a resistance-switching device and a capacitor in parallel; sawtooth waves are generally observed in the latter.See the Supplementary Information for the mechanisms and related equations.
Figure 2a shows the I bias dependence of the self-oscillation using (Et-4BrT)[Ni(dmit) 2 ] 2 at 36 K. Here, only a capacitor of C = 33 F is added in parallel to the sample, and I bias is applied to the circuit (Figure 1e).As shown in Figure 2a, the voltage across the sample V pq starts to self-oscillate above I bias = 100 A, where the sample enters into the NDR region (cf. Figure 1c).Sawtoothwave oscillations are observed.The operating frequency proportionally increases with I bias (blue solid circle in Figure 2b), and then the oscillation is damped above I bias = 850 A.Although the sample is still in the NDR range with I bias = 850 A, as shown in Figure 1c, the oscillation damps above the frequency of f c ≈2 Hz (Figure 2b).
Because the circuit in Figure 1e contains only a capacitor, and a sawtooth wave is generated, the relaxation oscillator-type mechanism appears to play a role in the oscillation.However, the operating frequency and damping phenomena are difficult to explain using such a mechanism (see Supplementary Information for discussion).In contrast, the damping can occur in "negative resistance oscillators" when the oscillation's damping factor  = (r − R)/2L of Equation S4 (Supporting Information) becomes negative.When the load resistance R load is inserted into the circuit (Figure 2c inset), a smooth transition from the self-oscillation ( ≥ 0) to damping ( < 0) regime is observed (Figure 2c) because the total resistance R increases with R load .The damped oscillations ( < 0) were reproduced well using Equation S4 (Supporting Information) (dotted curve in Figure 2c), and several sets of  and angular frequency  were obtained from these oscillations.The relation between  2 and  2 in Figure 2d is in accordance with the resonance condition for a negative resistance oscillator,  2 = 1 LC −  2 .Thus, the damping effect favors a negative resistance-type oscillation, although L is missing in the circuit.The interception of  2 in Figure 2d corresponds to  2  = 0 = 1∕LC , and a huge inductance of ≈L = 1062 H is expected.

Evidence of a "Coil-Less" Inductance
Our oscillation results suggest the presence of a large L in Figure 1e, and that the NDR and inductance may both originate from the sample.For this reason, we performed impedance spectroscopy of (Et-4BrT)[Ni(dmit) 2 ] 2 under a bias voltage V bias (not under I bias owing to instrument capability).Figure 3a shows the frequency dependence of the phase shift between the applied ac voltage and the detected ac current.The positive and negative phase angles suggest the existence of inductive and capacitive reactances in the sample, respectively. [8]An inductive phase angle was observed when V bias was finite, and the positive phase angle increased with an increase in V bias .A capacitive phase angle was observed above ≈1 kHz.
The Cole-Cole plots of the impedance (Figure 3b) show a clearer picture of the inductive and capacitive components, which are half-circles on the positive and negative sides of the imaginary impedance, respectively (see also Figure 3c). [8]he inductive half-circle shifts to the left and increases as V bias increases.The observed Cole-Cole plots can be fitted with the equivalent circuit shown in Figure 3d.The fitting curves are shown as thick curves in Figure 3b.The obtained fitting parameters R s1 , R s2 , and L s for each bias voltage are shown in Figure 3e.C s has no V bias dependence, and its value is always ≈50 nF.We assumed that this capacitive reactance was an extrinsic effect of the electrodes.The left end of the inductive half-circle is the resistance for zero frequency, which corresponds to the so-called "slope resistance" R s1 , and the right end of the inductive halfcircle is the high-frequency resistance, which corresponds to R s1 +R s2 , known as the "static resistance."The former and latter are represented by triangles and squares in Figure 3b,c,e.Our impedance spectroscopy analysis clearly shows the appearance of a large inductance L s by applying V bias (Figure 3e).Moreover, consistent with the I-V characteristics, both the slope resistance R s1 and static resistance R s1 +R s2 decrease with an increase in V bias (Figure 3e).V bias could not be applied above 4 V because the sample was destroyed owing to the lack of a current limit in our instrument.Owing to this instrumental limitation, the slope resistance (triangular symbols, R s1 ) did not reach a negative value, as shown in Figure 3b,e.In other words, the NDR region could not be reached within a V bias of 4 V.However, it is evident that the inductive half-circle should partially enter the negative side of the resistance (schematically shown as a red curve in Figure 3c) by applying V bias or I bias because the I-V curve of (Et-4BrT)[Ni(dmit) 2 ] 2 shows an NDR property under I bias (i.e., R s1 <0 for NDR).In this case, there is a limited frequency range in which both the NDR (R s1 <0) and finite inductance (L s ≠0) hold (thick red curve in Figure 3c), and the limit of this condition is the cut-off frequency.Hence, our sample indeed has an inductive property under I bias or V bias , but the oscillation is damped because the negative slope resistance becomes positive above the cut-off frequency (f c ≈2 Hz for sample B).

Pinched Hysteresis Loop
To investigate the origin of the induction, we measured the ac I-V characteristics, namely, the Lissajous figures of (Et-4BrT)[Ni(dmit) 2 ] 2 .We applied a sinusoidal ac current to the sample and monitored the voltage across the sample.Figure 4a shows the Lissajous curves for the selected frequencies.The Lissajous curves for frequencies up to 10 Hz show a "pinched hysteresis loop," where the curve passes through the origin, and hysteresis loops are formed in the first and third quadrants.The hysteresis loop is accompanied by an NDR for low frequencies such as 0.2 and 0.7 Hz, but the NDR feature disappears for frequencies above 2 Hz (Figure 4a), which is consistent with f c .Subsequently, the area of the hysteresis loop shrinks at higher frequencies and becomes a straight line at 100 Hz.This frequency dependence of the pinched hysteresis loop is a typical property of memristors (or more generally, memristive systems). [1]1b] In contrast to a linear resistor, memristive systems have a dynamic relationship between current and voltage, which results in a pinched hysteresis loop.As mentioned earlier, the resistance of our sample depends on the temperature and current, V = R(T, I) • I, from the analysis of the I-V curve.As explained in the Supporting Information, the time evolution of the body temperature of the sample is a function of the temperature and current, dT dt = f (T, I).1b] Finally, we show how the inductance appears under I bias as a result of the memristive-pinched hysteresis loop.The I bias dependence of the Lissajous curve is shown in Figure 4b.The applied ac current (red curve) and the resulting voltage (blue curve) as a function of time for I bias = 0 and 150 A are shown in Figure 4c,d, respectively.Starting from the pinched hysteresis loop, a sin-gle clockwise loop is formed as I bias increases (Figure 4b).For I bias = 0 A, a small deviation between the ac voltage and current is noticed; however, the phase remains the same at I(t) = V(t) = 0 (Figure 4c).On the other hand, a clear phase shift is observed for I bias = 150 A (Figure 4d), in which the voltage precedes the alternating current.Because of the pinched hysteresis loop, the memristive system can mimic the inductive component by forming a loop with a certain amount of I bias or V bias .
The appearance of inductive reactance in a dynamic resistor was predicted by Mauro in 1961. [9]Although the memristor concept did not exist at that time, by applying the same equations used in a memristive system, Mauro explicitly provided analytical evidence that a circuit containing thermonegative element  T = R∕R T < 0 will have an NDR property and a spontaneous inductance should emerge (see Supporting Information). [9]The inductance obtained by Mauro can be rewritten as L = 2C p | T | I −2 in the NDR region, where C p is the heat capacity of the sample.Next, the oscillating frequency is expected to be linear with the , which is consistent with our result shown in Figure 2b.The temperature coefficient,  T , of our memristive sample is ≈−3.3 between 30 and 40 K, which is three orders of magnitude greater than that of semiconductors such as Si and Ge. [10]The heat capacity of sample B is estimated to be C p = 2.3 × 10 −4 J K −1 at 30 K. [4] Next, a linear frequency slope of f = (2346.4)•I bias is expected from the estimated C p , | T |= 3.3, and C = 33 F (red line in Figure 2b).The observed frequency is in agreement with the prediction by Mauro. [9]Using the same parameters, the inductance at I bias = 300 A is expected to be ≈L = 1548 H, which is in the same order as the one obtained from the analysis of Figure 2d (L = 1062 H).

Conclusion and the Universality of Memristive Oscillations
It has been stated that (Et-4BrT)[Ni(dmit) 2 ] 2 exhibits both inductive and NDR properties because of its memristive nature (i.e., pinched hysteresis loop).Such unique and hybrid properties lead to the self-oscillation of the sample by simply adding a single capacitor to complement the resonating LCR circuit.We call this new type of oscillator a memristive oscillator.The cut-off frequency of the memristive oscillator arises from the nature of the memristive system, where the NDR slope inside the hysteresis loop disappears as the operating frequency increases.The existence of a cut-off frequency is a good index for the memristive oscillator, in addition to the existence of NDR and large inductance in the material.
The memristive properties of our sample essentially originate from i) the semiconductor-to-insulator transition and ii) the decondensation mechanism of the insulating state by the current, which leads to a dynamic resistance sensitive to T and I.The inductive reactance of our sample, which originates from the hysteresis loop, might have arisen as a result of inertia, the thermal inertia of thermonegative elements, or the inertia of carrier motion after decondensation.A recent theoretical study by Peronaci et al., suggests that Mott insulators can be thought of as a memristive system where the state variable is the density of doublon excitations, and can oscillate with the addition of a capacitor in parallel. [11]According to this study, a carrier avalanche multiplication occurs in the presence of a strong electric field, and NDR should be observed.Meanwhile, retarded current under the oscillating voltage owing to the time evolution of doublon density is predicted.Although the insulating ground state of layer B needs to be clarified, layer A of (Et-4BrT)[Ni(dmit) 2 ] 2 is expected to be in a Mott insulating state from previous studies, and the observation of NDR and the emergence of inductive reactance might originate from the same mechanism (i.e., carrier avalanche multiplication of the Mott insulating state). [3,4]e believe that memristive oscillation is a universal phenomenon, and is not restricted to our sample or memristive systems with the state variable x = T.As mentioned above, many molecular materials exhibit NDR properties, and some of them even oscillate. [5]The NDR property and self-oscillation with the addition of a capacitor have also been reported for transition-metal dichalcogenide and ReRAMs. [12]It is possible that some of these materials are memristive.Materials with metal-to-insulator transitions are also good candidates for memristive oscillators.Thermistors (resistors with negative temperature coefficients) that are also memristive systems with x = T can be used as oscillators. [13]Thermistors are reported to have NDR, inductive properties, and a cut-off frequency when used as oscillators. [9,13,14]The evidence of memristive oscillation for the thermistor is provided in the Supporting Information (Figure S5).
1b] The Lissajous curve of the Neon lamp has a pinched hysteresis loop with the NDR (Figure S6a, Supporting Information). [15]Although the authors could not determine its inductance, the hysteresis loop of the Lissajous curve is clockwise, and the emergence of inductance upon I bias is expected.Neon lamps oscillate with the same circuit configuration, as shown in Figure 1e (known as the Pearson-Anson oscillator), and a cut-off frequency of the oscillation exists.For these reasons, the Pearson-Anson oscillator can be regarded as a memristive oscillator.
A living body may have already adopted such an oscillatory mechanism.1b,16] The axon membrane of a giant squid shows both capacitive and inductive reactances and its resistance seems to decrease with the removal of Ca 2+ ions, which may lead to NDR. [14,17] This suggests that the axon membrane can act as a memristive oscillator without additional capacitors.In fact, axons in a low Ca 2+ concentration environment seem to give rise to the self-oscillations of impulses or repetitive bursts. [18]Consequently, auto-repetitive electrical impulses of the cardiac pacemaker or brainwaves in the body may be the result of memristive oscillation.
Our study revealed that the memristive oscillation originates from the hybrid electric properties (NDR and inductance) of the memristors.This suggests that memristors hold the potential to streamline the number of electrical components within oscillatory circuits.While the cut-off frequency of the memristive system with x = T is presently too low for practical applications, an ideal memristor like ReRAM may exhibit a higher cut-off frequency, making it suitable for applications in communication systems and neuromorphic computing.Another advantage of the hybrid electrical properties within memristors is their inductive behavior without the need for a traditional coil.Typically, coils tend to be relatively large compared to other electrical components, and we believe the memristor can contribute to the miniaturization of inductors and communication systems as well.

Experimental Section
Single crystals of (Et-4BrT)[Ni(dmit) 2 ] 2 were obtained through galvanostatic electrolysis of a mixture of ( n Bu 4 N)[Ni(dmit) 2 ] (4.5 mg) and (Et-4BrT)BF 4 (45 mg) as the supporting electrolyte in acetoneacetonitrile (3:1, v/v; 20 mL) at 20 °C under argon. [3,4]Three single crystal samples from the same batch were chosen in this study: sample A (0.5 × 0.35 × 0.1 mm 3 ), sample B (0.33 × 0.25 × 0.16 mm 3 ), and sample C (0.5 × 0.4 × 0.1 mm 3 ).The transport and the I-V characteristics measurements of sample A were performed using the conventional two-terminal method.See Supporting Information for more detailed electrical configuration.The resistance was measured using a resistance bridge (Lakeshore LS 380), and the I-V characteristics were measured using a source meter (Keithley 2450).Oscillation measurements of sample B were performed by adding capacitances in parallel, and a bias current I bias was applied (Keythley 6221) to the circuit.The oscillating voltage across the sample was fed into an oscilloscope.The ac I-V characteristics (Lissajous curve) of sample B were measured by applying an ac current (Keythley 6221) to the sample and the load resistance (6 kΩ) in series.The ac current (x-axis of the Lissajous curve) was converted from the load resistance voltage, and the ac voltage (y-axis of the Lissajous curve) was obtained from the sample voltage.Impedance spectroscopy measurements of sample C were performed using a combination of a frequency response analyzer (Solartron Analytical 1252A) and a dielectric interface (Solartron Analytical 1296).The Cole-Cole plots were analyzed using the ZView software (Ametek Scien-tific Instruments).In all measurements, the single-crystal sample was fed into the cryostat and cooled at a rate of 2 K min −1 .The instruments and additional passive components (if required) were connected to the cryostat (Figure S7b, Supporting Information).

Figure 1 .
Figure 1.Transport properties and circuit diagrams.a) Typical transport property of (Et-4BrT)[Ni(dmit) 2 ] 2 measured using the two-terminal method.The inset shows the crystal and molecular structures of (Et-4BrT)[Ni(dmit) 2 ] 2 .b) Current-controlled I-V characteristics of (Et-4BrT)[Ni(dmit) 2 ] 2 (sample A) from 9 to 200 K.The inset is the enlarged view of the I-V curves for temperatures higher than 28 K. c) The I-V characteristics of sample B. The NDR slope differs for each sample.d) Circuit diagram of the negative resistance oscillator.NDR (r) cancels the positive circuit resistance (R).e) Circuit diagram of our oscillation experiment.

Figure 2 .
Figure 2. Oscillation measurements of (Et-4BrT)[Ni(dmit) 2 ] 2 .a) I bias dependence of the oscillation at 36 K (Sample B).C = 33 F is used in Figure 1e, and the voltage across the sample V pq was monitored.b) I bias dependence of the operating frequency (blue symbols) obtained from the saw-tooth wave in (a).Damped oscillations at I bias >850 A are excluded from the data.The red line represents the expected linear line for memristive oscillation.c) Load resistance R load dependence of the oscillations for 35 K and C = 33 F (Sample B).The inset is the circuit diagram including R load .The dotted dark blue curve for R load = 4 kΩ is the typical fitting of the oscillation in the damping regime ( < 0).d)  2 −  2 diagram, where the values are obtained from the fitting of the damping waves ( < 0) in (c) using Equation S4 (Supporting Information).

Figure 3 .
Figure 3. Impedance spectroscopy of (Et-4BrT)[Ni(dmit) 2 ] 2 .a) V bias dependence of the phase angle from the impedance spectroscopy measurements at 30 K (Sample C). b) Cole-Cole plots of the same impedance spectrum with several selected values of V bias (colored plots), and the fitting curves (thick colored curves).c) Schematic drawing of the behavior of the Cole-Cole plots with the increase in bias current or voltage.The zero-frequency resistance is the slope resistance R s1 (triangle symbol), and the high-frequency resistance is the static resistance R s1 + R s2 (square symbol).A cut-off frequency appears when the slope resistance R s1 is negative.d) Equivalent circuit of our memristive sample.e) V bias of L s , R s1 , and R s1 +R s2 , obtained from the fitting curves in (b).

Figure 4 .
Figure 4. Lissajous curves of (Et-4BrT)[Ni(dmit) 2 ] 2 .a) Lissajous curves of (Et-4BrT)[Ni(dmit) 2 ] 2 (Sample B) at 35 K for several selected frequencies.A pinched hysteresis loop that is typical of memristive systems is observed.b) I bias dependence of the Lissajous curves for the same sample at 35 K. c,d) time domain map of the Lissajous curve for I bias = 0 and 150 A, respectively.The applied ac current and the resulting voltage as a function of time are presented as red and blue curves, respectively.