Axon-like active signal transmission | Nature

EOC nomenclature

EOC, by definition, refers to semi-stability. However, since previous reports have used the phrase ‘edge of chaos’ to describe destabilized manifestations of the underlying EOC, such as chaotic dynamics⁴³, here we refer to the distinct semi-stable behaviour exhibited by our devices as semi-stable EOC.

Material growth and device fabrication

LaCoO₃ blanket films were grown on a LaAlO₃ substrate through pulsed laser deposition using a Nd:YAG (neodymium-doped yttrium aluminum garnet) laser to ablate a stoichiometric LaCoO₃ ceramic target, at a wavelength l = 266 nm, operated at 5 Hz. The oxygen partial pressure of the chamber was 100 mtorr and the substrate temperature was maintained at 650 °C during growth. The as-grown films were then annealed in air at 500 °C for 1 h to reduce oxygen deficiencies. Although Fig. 2b refers to the state of LaCoO₃ at 300 K as low spin, since the spin crossover in LaCoO₃ begins well below room temperature, the state at 300 K—relative to that at 600 K—is a lower-spin state.

An MA6 mask aligner and an MLA150 maskless aligner were used to fabricate the patterned electrodes on blanket LaCoO₃ films. The AZ5214E reversible photoresist was used in its image reversal process. After baking at 110 °C, the resist received 50 mJ cm^–2 exposure and, after exposure was baked at 115 °C. Then, it received a flood exposure of 900 mJ cm^–2 and was developed in AZ400K-1:4 for 1 min. Then, 100 nm of Pt electrodes (with an adhesive Cr or Ti layer of less than 5 nm) were sputter deposited followed by lift-off in Remover PG at room temperature assisted by mechanical agitation. The electrodes were also deposited by e-beam evaporation in some of the samples, with similar results. The electrode gap (8–12 µm) in the specific structure shown in Fig. 4a and the metal thickness were chosen for the ease of fabrication using optical lithography, and the lengths of the transmission line (varying from 0.2 to 4 mm) were chosen to represent typical chip-layout dimensions.

Electrical measurements

Setup

The devices were tested using FormFactor ACP40-GSG-250 radio-frequency probes connected to the d.c. + a.c. source current through subminiature version A connectors and mini coaxial cables totalling a length of less than 1.2 m. Electrical phase-shift measurements were directly performed using an a.c. voltage supplied and measured by an Agilent 33220A 20 MHz function generator and an Agilent InfiniiVision MSO7054A 500 MHz oscilloscope, with d.c. power supplied by a Keysight B2911A source measure unit (SMU), with the spot measure time manually set to greater than 300 ms (to fully approach the steady state) and auto-ranging turned off (to avoid range-switching artefacts).

Repeatability of I–V behaviour

Quasistatic current-controlled I–V sweeps were repeated for a variety of device dimensions under identical conditions, using identical replicates, and with more than six consecutive I–V sweep cycles (Supplementary Fig. 5). Although there are quantitative differences from the first cycle to the second and subsequent cycles, all of them are qualitatively similar. There was typically negligible quantitative variation between cycles after the first cycle. For example, the range in voltage at 2 mA varies by 20 mV (0.1%) for the 3 µm device (Supplementary Fig. 5a), and by 260 mV (0.7%) for the duplicate 10 µm device (Supplementary Fig. 5f). The consistency and repeatability of these measurements verifies that the observed NDR was robust and probably not due to artefacts such as noise or gradual device damage. The NDR-onset current bias was identified using Savitzky–Golay smoothing and differentiation (Supplementary Fig. 6). For transmission-line structures, the cycle-to-cycle variation was higher, probably due to the inhomogeneous nucleation of phase transition throughout the much larger active volume.

Power supply and test circuit

Setup

Because LaCoO₃ devices manifest current-controlled NDR, measuring frequency-dependent phase shifts as a function of current bias required coupling tunable d.c. current and a.c. current sources. It was also necessary for the two to be reactively decoupled so that the reactance of the d.c. source would be screened from the a.c. source. We achieved this setup by coupling a Keysight SMU in the current-source mode (the same one used for I–V sweeps) to an Agilent function generator using a diode and 200 and 5.1 kΩ resistors (Supplementary Fig. 7a). We found that the diode effectively screened the SMU without introducing artefacts so long as the diode was fully forward biased, for currents greater than 0.1 mA.

The 200 kΩ resistor simultaneously performed two crucial roles. First, it converted the signal from the function generator from 5 V to less than 25 µA, so that the signal was small with respect to the d.c. bias. Second, it generated a sufficient non-reactive load such that the voltage from the function generator and the current through the circuit and device were in phase with an error of 0.01 radians, so that the function generator voltage is a valid substitute for the circuit current with respect to computing phase shifts. This ability to take the supplied a.c. voltage as essentially in phase with the a.c. current through the circuit as well as the LaCoO₃ was experimentally necessary, because the current through the circuit was too small and noisy to reliably trigger the oscilloscope and varied compared with the d.c. current. By contrast, the function generator always produced a reliable noise-free 5 V (a.c.) signal. The 5.1 kΩ resistor was used to dampen oscillations during I–V sweeps and was retained during subsequent measurements for consistency. This test circuit was used for all the measurements of the two-terminal test structures, including during operando thermal mapping.

For the transmission-line measurements, larger voltages were required; therefore, the 200 kΩ resistor had to be removed. Without the resistor, the voltage from the function generator was too large to fully forward bias the diode. Therefore, the diode was replaced with a capacitor to provide a d.c. block between the function generator and the d.c. source (Supplementary Fig. 7b). The 5.1 kΩ resistor was similarly replaced with a 1.1 kΩ resistor to reduce the total d.c. voltage sourced from the SMU from around 50 to about 30 V (as the transmission lines are more conductive structures and therefore operate at higher d.c. currents).

A common measurement pitfall

Reactive phase shifts are routinely measured using black-box impedance analysers. However, we measured the dynamic properties of our nonlinear devices by directly analysing their response to a.c. signals from a function generator recorded on an oscilloscope, which we have found to be more reliable for three reasons. First, we could directly observe that the dynamic signals were small enough to minimize distortion and power lost to higher harmonics. Second, a direct signal-versus-time measurement can be modelled with standard physics-based simulators. Third, we could account for all the parasitic reactive components in the measurement system in our models. A black-box impedance analyser used for such measurements often leads to highly erroneous results. For example, a measurement of \(\Delta \phi \approx \frac{{\rm{\pi }}}{2}\) at 100 Hz when biased at the cusp of NDR (Fig. 3c; raw data are shown in Supplementary Fig. 8) would be interpreted as a pure inductor with an unreasonable magnitude of several millihenry. Similarly, \(\frac{{\rm{\pi }}}{2} < | \Delta \phi | \le {\rm{\pi }}\) at 100 Hz for bias currents of 3–4 mA would be interpreted as the presence of an actively powered circuit like an operational amplifier. These anomalies also confounded Hodgkin and Huxley in their initial investigations of the neuron¹⁶. Thus, it is important to carefully understand and analyse phase shifts in nonlinear components.

Signal processing

Sinusoidal amplitudes and phases

Amplitudes A and absolute phases ϕ (with respect to cosine) of the sinusoidal signals were manually computed from raw oscilloscope time series s(t) by taking their inner product with a cosine and sine at the base frequency f, and normalizing by the number of complete cycles N as

To illustrate the uncertainties in the phase contour plot (Fig. 3c), Supplementary Fig. 9 displays the cross-sections of Fig. 3c.

Electrical uncertainty quantification

For sinusoidal phases and amplitude, uncertainty was estimated with leave-one-out cross-validation. That is, temporarily removing one of the N cycles (N > 15) from the dataset, repeating the amplitude–phase calculation for the remaining N – 1 cycles and then taking the standard deviation of the distribution. Uncertainties in gain ratios and phase differences were propagated from amplitudes and phases in the standard way. Typical phase uncertainties were found to be of the order of 0.01 radians—too small to be visible in most plots. Similarly, uncertainties in the gain were also too small to be visible in most plots.

PSDs

PSDs were computed using the fast Fourier transform implementation of MATLAB R2021B, and a Hann windowing function was used to eliminate spectral leakage before normalizing by the Nyquist frequency. Welch averaging with three sections and 30% overlap was used to trade some spectral resolution of the periodograms for improved statistical variation in the fast Fourier transform amplitudes. The frequency resolution after Welch averaging was better than 1.5 Hz.

The data in Fig. 3d were normalized by the 100 Hz frequency component to decouple the amplification from the expected Johnson–Nyquist scaling. The PSD of the Johnson–Nyquist noise has a known scaling with temperature T and local resistance dR as

Here k_B is the Boltzmann constant. Thus, normalizing the periodograms by the PSD at one fixed frequency component eliminates the Johnson–Nyquist scaling, as for a periodogram at a given bias, all the components have the same T and dR values. When comparing the data before and after normalization (Supplementary Fig. 10), the most important features of the data are preserved. In particular, the local maxima in the noise at 60 Hz and their harmonics for current biases near the NDR onset at roughly 3 mA are in both raw and normalized data and therefore not an artefact of normalization. By contrast, the raw data exhibit a rapid decrease in absolute magnitude as the bias approaches 3 mA from either direction. This is a direct effect of the expected Johnson–Nyquist scaling, since the differential resistance also rapidly decreases near the NDR onset. The rapid decrease in the noise baseline obscures the shape of the amplification peak. Thus, normalizing by the PSD at 100 Hz, as we have done in the main text, clarifies the features of the amplification peak without introducing artefacts.

The cross-sections of the PSDs (Supplementary Fig. 11a,b) at a fixed bias current display the detailed features of the periodograms. The most obvious feature is the largest peak at 100 Hz corresponding to the input sinusoid, but the data also clearly show a strong peak at 60 Hz and its harmonics (along with a peak at 1 kHz), which is only present in the data after the NDR onset at roughly 3 mA. The cross-sections at a constant frequency (Supplementary Fig. 11c) show that the noise amplification of about 20 dB exists over a large bandwidth.

Transmission-line measurements

Shape of the I–V curve

The quasistatic I–V curves corresponding to both test structures and transmission lines (Supplementary Figs. 5, 12, 14 and 15) exhibit an abrupt jump around (either before or after) the onset of NDR. This behaviour is probably associated with the abrupt localization of spatial thermal gradients, driven by energy minimization principles, which was first theorized in 1936 (ref. ⁴⁶) and formalized later⁴⁷. Although this effect deserves a dedicated study, owing to the increasing nonlinearities in current nanoscale electronic components, in our experiments, we verified that the abrupt jump had no effect on our observations. Specifically, our measurements show that EOC and amplification are associated with the onset of NDR and not the abrupt features in the I–V curves. The transmission lines had an end-to-end resistance of 6.6 kΩ mm^−1 (measured at 0.1 V) and the 1 mm line had a signal-to-ground resistance of 1 kΩ.

Insertion delays

To eliminate potential insertion delays, we characterized the delays induced by the setup by placing two probes on the same pad of a transmission line, which produced delays smaller than any delays measured across different pads on the same transmission line (delays were less than 0.003 × 2π radians).

Thermal-mapping measurements

Setup

Operando thermal maps were acquired with an FLIR SC6700 infrared camera and a120 Hz acquisition rate. Raw data as the infrared signal were calibrated to temperature (Supplementary Fig. 21) using independent isothermal data obtained with an Instec HCP421V-MP+ temperature stage, which was independently verified to 1 K accuracy with a Pt metallic line thermometer. We note that the calibration data extended to 400 K, greater than any of the experimental data (measured to a maximum of 370 K); the calibration provided only interpolation, not extrapolation. Composite thermal images were obtained as pixel-wise averages over >30 frames from the thermal video. Gaussian fits to thermal profiles were computed by a least-squares analysis.

Reproducibility of the thermal measurements

The thermal-mapping experiment was attempted at several PDR biases, where—in contrast to the results shown in Fig. 5—only net heating would be expected. However, the observed temperature change was not statistically different from zero for the different cases (with and without a small signal in addition to the bias). This issue is an instrumental limitation: because the baseline temperature in the PDR is substantially lower, the signal to the IR camera is exponentially smaller.

Net device cooling during super-quadrature phase shifts (Fig. 5) is a substantial claim. Hence, we verified it by duplicating the result at a second independent NDR bias (I_bias = 4.1 mA) (Supplementary Fig. 22). In contrast to Fig. 5 (at I_bias = 3 mA, where the maximum frequency at which relative cooling could be observed was about 2 kHz), for I_bias = 4.1 mA, this frequency was more than 10 kHz. This observation indicates a notable increase in the bandwidth on slightly changing the bias.

Thermal uncertainty quantification

Uncertainties in thermal data were estimated as the standard deviation of more than 30 frames from the FLIR thermal video and then propagated using the temperature-calibration power laws. Uncertainties in the Gaussian-fit peak temperatures were estimated by bootstrapping. For each measured point (x, T) with measured uncertainty δT in the cross-section, a normal distribution \({\mathscr{N}}(T,\delta T)\) was generated, a sample was selected at random from the distribution and finally the new sampled data were fit with Gaussians, and the Gaussian fits were recorded. The sampling process was repeated for N = 5,000 times, and the uncertainty in the peak of the Gaussian fit was taken as the standard deviation of the 5,000 bootstrap simulations.