Physical origins of current and temperature controlled negative differential resistances in NbO2

Negative differential resistance behavior in oxide memristors, especially those using NbO2, is gaining renewed interest because of its potential utility in neuromorphic computing. However, there has been a decade-long controversy over whether the negative differential resistance is caused by a relatively low-temperature non-linear transport mechanism or a high-temperature Mott transition. Resolving this issue will enable consistent and robust predictive modeling of this phenomenon for different applications. Here we examine NbO2 memristors that exhibit both a current-controlled and a temperature-controlled negative differential resistance. Through thermal and chemical spectromicroscopy and numerical simulations, we confirm that the former is caused by a ~400 K non-linear-transport-driven instability and the latter is caused by the ~1000 K Mott metal-insulator transition, for which the thermal conductance counter-intuitively decreases in the metallic state relative to the insulating state.

(a) X-ray map of a device operated using current-sweep (i.e.: a voltage source with a very large series resistance (R S )), which was much greater than the magnitude of NDR, R NDR (200-400 Ω). (b) X-ray map of a device operated with voltage source and a series resistance of ~3 kΩ (R S >R NDR ). X-ray map of a device operated with a voltage source and a series resistance (from the contact electrodes) of about 300 Ω (R S <R NDR ). These maps show that as R S is reduced below R NDR , there is a localized conduction channel that forms and causes an irreversible change to the NbO 2 . 1000 Ω (at low temperatures), which changes resistance value abruptly to 500 Ω at a critical temperature of 1070 K. This yields a current-voltage plot with a range of current values for which there is no solution of the resulting curve (solid blue line with no data-markers). This leaves the system unstable in this current range, leading the system to utilize another state parameter to achieve stability. This is often some material property (e.g.: R th ) or formation of localized conduction channels, wherein the crosssectional area of the conduction channel becomes the state parameter. (b) An identical simulation as in (a), except that there is an increase in R th at the critical temperature, as used in the rest of the manuscript. (d) In-operando x-ray spectromicroscopy on a fresh device showing that the there was a crystal structure change (Peierls transition) accompanying a Mott transition (shifts in the lowest conduction bands) upon driving a current higher than that required for causing NDR-2.

Supplementary Note 1: Additional details on temperature and electrical measurements
Quasi-static current-voltage behavior was measured using an Agilent B1500 parameter analyzer and a Cascade probe station. The parameter analyzer was controlled through a General Purpose Interface Bus (GPIB) using software programs written in Igor. The voltages measured across the device were measured over 1-10 ms at every data-point, which was sufficiently long to allow for attainment of steady-state.
Thermoreflectance measurements were performed using a Microsanj NT220 thermoreflectance analyzer, the details of which (including construction, operating principles, calibration procedures, etc.) are available elsewhere. 2 The typical error we expected from the measurements were ~5% (of measured temperature) dependent on measurement parameters including time-of integration, number of averaging cycles, noise, etc. To account for other sources of errors like drift in sample behavior, noise/distortions in the drift-correcting piezoelectric stages, etc., we report a total error of 10% in Fig. 2c. Thermoreflectance measurements involved repeated cycling of the devices through many cycles of high and low current levels (through time-multiplexing) using pulsed current signals. The wait-time ('τ' in Fig. 2a) was set to at least 1 ms for acquisition of steady-state data reported in Fig. 2. Attainment of steady-state within 1 ms was verified by collecting dynamic temperature data by varying τ (Fig. 2d). In all the experimental measurements, a study of NDR-2 was performed using current levels sufficient to exceed NDR-2, because it is impossible to achieve a stable access of NDR-2 using any combination of electrical parameters, since NDR-2 is a manifestation of a temperature-controlled instability.

Supplementary Note 2: Temperature gradients within the device structure
Since the temperature maps show a fairly uniform temperature distribution within the crosspoint area and no significant temperature gradients outside the crosspoint area, it is fair to assume that heat transfer in the lateral direction is negligible with regard to the measurements. Along the direction perpendicular to the membrane/device surface, we consider the following heat transfer equation: where T NbO is the temperature of the niobium oxide layer; T S is the temperature at the surface of the material stack; k f is the effective thermal conductivity of the layers contacting the niobium oxide layer, with a total thickness of t f ; h air is the heat transfer coefficient of air that also accounts for convection (100 Wm -2 K -1 ); σ s is the Stefan's constant (5.6×10 -8 Wm -2 K -4 ); the ambient temperature of the bath, T amb , was set to 300 K. Supplementary Equation 1 accounts for heat transfer through the solid layers in contact with the niobium oxide, through air (including convective cooling) and by radiation (assuming a perfect blackbody). We used extreme-case values for k f of 1 Wm -1 K -1 , and for t f of 200 nm (considering the 150 nm of freely suspended silicon nitride membrane beneath the device and the other layers in the stack). By solving Supplementary Equation 1, we found that the difference between T NbO and T S for all values of T NbO between 300.1 K and 1500 K was <0.1 K. This shows that the temperature gradient across the thickness of the silicon nitride membrane (150 nm) was sufficiently insignificant, thereby allowing us to ignore such temperature gradients across the thickness of the electrodes (<20 nm).

Supplementary Note 3: Estimating the R th jump from the electrical data
Using the static current-voltage sweep, it is possible to estimate the magnitude of the abrupt change in R th upon MIT. Consider the MIT events M1 (insulator state (Ins) at the onset of insulator-to-metal transition) and M2 (metallic state (Met) at the onset of metal-to-insulator transition) occurring during increase and decrease of current, respectively, as marked in Supplementary Fig. 8. Assuming that the MIT occurs at a constant temperature, 3 we can represent T MIT as: where the superscripts (Ins and Met) denote the quantities measured at the specified MIT events M1 or M2, respectively. Thus, it is possible to calculate the ratio of change in R th from the currents and voltages measured at M1 and M2. From this, we estimated the ratio in Supplementary Equation 3 to be 0.77 from the current-voltage curve, while that obtained from direct temperature mapping was 0.74. This good agreement between the two estimates obtained from different experiments supports the larger conclusion of an increase in R th in the metallic state of NbO 2 .

Supplementary Note 4: In-operando x-ray characterization technique
One of the greatest challenges in physically understanding memristor operation has been the extremely localized, low signal, atomic-scale material changes associated with large changes in the cell resistance, the result of which is uncertainty and controversy on the details of the operating mechanisms. 4 Experiments are usually performed using destructive techniques like cross sectional electron microscopy, non-standard device construction, and amplifying the material changes by using stronger operating conditions. 5,6 To overcome these limitations, we developed an ultra-sensitive measurement technique to probe the electronic, structural, and chemical properties during regular operation of a cell. 4,7 In order to achieve the necessary spatial and spectral resolution, we employed a synchrotron-based scanning transmission x-ray microscope (STXM) with spatial resolution of <30 nm and spectral resolution of ~70 meV. 8 Prototype devices for this experiment were fabricated atop 200 nm thick freely suspended Si 3 N 4 membranes to enable x-ray transmission. 9 Further, in order to overcome the signal-limiting issues of spatial drift, background absorption changes, stochasticity in cell operation and drift in cell behavior, we constructed an in-operando time-multiplexed experimental setup ( Supplementary Fig. 12). We incorporated an adaptive cell-switching technique that utilized feedback-enabled resistance switching together with a verification read, while synchronously gating the detector signal (from the asynchronous synchrotron x-ray pulses) into two different counters corresponding to physical measurements only when the target resistance states were successfully achieved. The result was an integration of signal at every spatial and spectral position over many verified switching events, which averages stochastic processes during cell switching. More importantly, the time-multiplexing provided a reduction in effects due to spatial drift of the sample and other background changes by over 5 orders of magnitude, detailed elsewhere. 4 The adaptive system also corrected for temporal drifts in cell switching behavior, which is another major source of distortion in such measurements.

Supplementary Note 5: Comparison of electrical, temperature and x-ray measurements to VO2
In Fig. 2c the abrupt jump in temperature predicted by the model in Fig. 2e is not observed, mainly because the size of the effect is within the error bars on the experimental data. Since NbO 2 undergoes a Mott + Peierls transition at ~1000 K, which is challenging to perform accurate thermal measurements on, we show here a similar effect in VO 2 , a Mott insulator that undergoes a Mott + Peierls transition at ~340 K, which is easier to measure/calibrate. In Supplementary Fig. 13, we show a set of previously published data that reproduces the current-driven 'four-sided' hysteretic NDR-2 measured in VO 2 ( Supplementary   Fig. 13b), along with local temperature measurement ( Supplementary Fig. 13c) and in-operando x-ray spectromicroscopy ( Supplementary Fig. 13d). 3,9 The local temperature is 340 K at the threshold current, which causes the sharp NDR-2 and is the precise Mott transition temperature of VO 2 . Having established that we are indeed studying the Mott transition-driven NDR of VO 2 , we also see that the temperature undergoes abrupt hysteretic jumps accompanying the Mott transition and NDR-2, consistent with our MIT model. This also shows that there are abrupt jumps in R th for VO 2 . Further, using x-ray spectromicroscopy, we showed that there is a Peierls transition that accompanies NDR-2, the Mott transition and the abrupt change 9 in R th of VO 2 . This data details the crystal phases that are associated with the different R th . Similar x-ray absorption data has been provided for NbO 2 as well (Fig. 3).
However, VO 2 does not exhibit NDR-1 because the MIT occurs at a temperature lower than the nonlinear conductance thermal runaway regime of NbO 2 . Another oxide, TiO 2 , exhibits NDR-1 but not NDR-2, because it does not have a Mott transition. 10 Thus, NbO 2 is seen to be unusual if not unique in its nonlinear electronic behavior.

Supplementary Note 6: Statistical analysis of the OD distributions
In Fig. 3, the mean of the OD inside the crosspoint area (M ≈ 1.62×10 -3 in Fig. 3d and M ≈ 2.47×10 -3 in Fig. 3f) contained a relatively negligible error, M err , estimated from the sample standard deviation as M err = /√ ≈ 1.622×10 -7 for N>3500 representing many measurement cycles. The standard deviations (S ≈ 0.73×10 -3 ) were similar for all the distributions reported in Fig. 3 and was a function of the dwell-time of the focused x-rays within each pixel of the map.