Benchmarking Stochasticity behind Reproducibility: Denoising Strategies in Ta2O5 Memristors (2025)

Steady-State Noise Map and Device-to-Device Noise Variation

In previous studies, it was demonstrated that noise maps, i.e., the steady-state ΔG/G relative device noise values plotted as a function of the G device conductance, are useful device fingerprints. (20−32) The relevant transport mechanisms and key sources of the fluctuations are reflected by dedicated ΔG/G vs G dependencies, and a trend change in the ΔG/G vs G plot indicates a change in the transport mechanism. Figure 1a presents such a noise map based on the steady-state noise analysis of nine different Ta₂O₅ memristive devices. For each device, a broad range of different conductance states were programmed to map the conductance dependence of the noise characteristics as well as the state-to-state and device-to-device noise variation (see the circles in Figure 1a demonstrating measurements taken at 148 different conductance states, with the color shades representing the various devices). The noise characteristics are expected to exhibit a mostly conductance-independent ΔG/G vs G relation in the broken filamentary regimes, while a strongly conductance-dependent ΔG/G vs G relation is anticipated in the nonbroken filamentary regimes (see our review paper in ref (20)). In the case of atomic-sized filaments, the crossover is expected close to the G₀ = 77.48 μS = 2e²/h quantum conductance. (20) These core expectations are clearly confirmed by the observed tendencies in the ΔG/G vs G plot in Figure 1a (see the top axis in the units of 2e²/h): indeed, at G < 2e²/h, a mostly saturated relative noise amplitude is observed, while at G > 2e²/h, the relative noise amplitude exhibits a steep decrease with increasing conductance. In the latter case, the best fitting of the data is close to the ΔG/G ∼ G^–3 trend, which can be explained by the effect of a single fluctuator in point-contact-like junction geometry, while the crossover at G ≈ 2e²/h in the noise map underpins the truly atomic-sized filamentary nature of the active region. See Figure S3 for a better visibility of the ΔG/G vs G trends as well as the fitting and discussion of the two corresponding transport regions.

The noise map of the Ta₂O₅ memristive devices (Figure 1a) also exhibits a remarkable, order-of-magnitude, device-to-device, and state-to-state noise variation for measurements sharing similar conductance values. To decide whether the observed noise levels are small or large, it is worth comparing the noise data with the outstanding ΔG_ref = 2 μS reference conductance resolution achieved in ref (15) over a similar conductance range. As the ΔG noise levels are normalized to the conductance, ΔG_ref should also be normalized to G, which is a line on the log–log plot (see the thick black dashed line in Figure 1a). Compared to this reference conductance resolution, the shaded areas illustrate other possible conductance resolutions with a factor of 2 (1 bit) resolution difference between the different shades. Note that, on the left axis, ΔG represents the standard deviation of the conductance, but to achieve the ΔG_ref conductance resolution, rather the peak-to-peak noise should be below the resolution limit. Therefore, as a safe margin, the relative noise should satisfy ΔG/G < ΔG_ref/8G, i.e., to have a 3-bit better standard deviation of the noise as the envisioned reference resolution (see the black solid line). Exemplifying the same condition in the time domain, Figure 1b illustrates the temporal noise traces of the blue and red data points in Figure 1a, facilitating (blue) or conflicting (dark red) the reference conductance resolution.

The ΔG_ref/8G line in Figure 1a clearly crosses the noise data, with a significant portion of the data points being below/above the reference line. This observation prompts the question whether a memristive device with ΔG > ΔG_ref/8 is inherently too noisy for the reference resolution or whether a dedicated protocol could yield a significant noise reduction, keeping the conductance state practically unchanged. The study of non-steady-state noise will help to answer this question.

Prior to investigating noise reduction possibilities, we further analyze the shape of ΔG/G vs G tendencies in Figure 1a. Clearly, the transition between the nonbroken and broken filamentary regimes is the most critical, where the largest portion of the data points violates the condition for the reference resolution. This is understandable because the atomic-scale active region is very sensitive to any nearby fluctuations. At much lower conductances (G ≪ 2e²/h), the weak (mostly constant) ΔG/G vs G dependence of the broken filamentary regime helps to satisfy the reference resolution condition. Similarly, at G ≫ 2e²/h, the steep decay of ΔG/G with the widening of the filament also helps to keep the noise below the reference. All of these observations underpin the importance of plotting noise maps like Figure 1a because the satisfiability of the reference resolution condition strongly depends on the chosen conductance range.

Full-Cycle Nonlinear Noise Spectroscopy

So far, steady-state fluctuations have been investigated, which can be considered as a baseline for the read-out noise: regardless of the accuracy of the instrumentation used, the resolution of the conductance read-out cannot be better than steady-state noise. We then move beyond steady-state noise measurements, showing that noise benchmarking is possible all along the entire switching cycle. The scheme of our nonlinear noise spectroscopy measurements is demonstrated in Figure 2. As a key requirement, we wanted to match nonlinear noise and nonlinear I(V) data. However, the so-called time–voltage dilemma is well-known for memristive systems; (1,35−38) i.e., the shape of the I(V) curve and especially the switching threshold strongly depend on the speed of the measurement. Accordingly, comparative I(V) and noise measurements should rely on voltage sweeps sharing the same amplitude and period time, or even better, the I(V) and noise data should be extracted from the same measurement.

From the current response to the stepwise voltage drive, one can plot a traditional I(V) curve at the discrete voltage levels (black circles in Figure 2e), which clearly shows bipolar switching between a high-conductance state (HCS) and a low-conductance state (LCS). This discretized I(V) curve perfectly matches the previously acquired conventional I(V) curve (gray line in the background), which was measured by a continuous triangular voltage sweep with the same overall amplitudes and period as the discretized I(V) curve. Note that both I(V) curves are plotted as a function of the V_bias = V_drive – IR_series voltage drop on the memristor; i.e., the voltage drop on the applied R_series = 110 Ω series resistor is subtracted from the drive voltage. From exactly the same discretized measurement, one can also plot the ΔI/I vs V_bias full-cycle noise curve (Figure 2f).

The representative full-cycle noise measurement in Figure 2 exhibits numerous remarkable features. First, the HCS displays a high degree of linearity up to the switching threshold voltage; i.e., the linearity condition of steady-state noise measurements is clearly satisfied. In the steady state, however, the ΔG standard deviation of the G = I/V conductance should be voltage-independent. This is strongly violated by Figure 2c1–c4, where the high-bias measurements (Figure 2c2,c3) exhibit much larger noise than the neighboring low-bias measurements (Figure 2c1,c4) (see the same measurements with a vertical scale highlighting the relative fluctuations in Figure S4). This means that in Figure 2c2,c3 the applied voltage excites a high level of fluctuations compared to the low-bias measurements even though the voltage remains below the switching threshold and I(V) is highly linear. The same feature is also demonstrated by the corresponding colored points in Figure 2f. Furthermore, in the low-bias measurements (Figure 2c1,c4), the noise spectrum (Figure 2d1,d4) is dominantly Lorentzian-type, which is characteristic of a single dominant fluctuator with a specific fluctuation time constant. (20) As a sharp contrast, the high-bias noise measurements (Figures 2d2,d3) display 1/f-type spectra, which is characteristic of a large number of relevant fluctuators with a broad distribution of fluctuation times. (20) This indicates the voltage-induced activation of a large number of fluctuators. See Section 2 in the Supporting Information for more details on the decomposition of the spectra to Lorentzian and 1/f-type contributions.

Similar features are also found along repeated switching cycles, as demonstrated in Figure 3a,b. Here the gray curves in the background display 10 subsequent I(V) curves (a) and full-cycle noise curves (b). The I(V) curves exhibit a remarkable cycle-to-cycle reproducibility, which contrasts the noise measurements, where a huge cycle-to-cycle variation is experienced (see Section 4 in the Supporting Information for a more detailed analysis and figure). The colored circles represent the average I(V) curve and average full-cycle noise curve for the 10 cycles. These average curves unambiguously show multiple characteristic regimes: (i) a steady-state regime at low voltages with voltage-independent relative noise levels, (ii) a non-steady-state regime at slightly increased voltages still in the linear, with a nonswitching conduction regime of the current–voltage characteristics exhibiting an order of magnitude increase of relative noise, and (iii) a switching regime where the resistive transition occurs. The color coding of the corresponding data points for set/reset transitions is (i) red/blue, (ii) dark red/dark blue, and (iii) purple. The relative noise increase in the non-steady-state regimes (dark red and dark blue) is attributed to voltage-induced activation of ionic motions around the active region, but this is not yet a resistive switching, only a fluctuation with mostly unchanged mean conductance. In this sense, the non-steady-state region is considered to be a precursor regime: the increasing noise forecasts the proximity of the switching.

To put the full-cycle noise data into perspective, Figure 3c plots the average relative noise data (red, dark-red, blue, dark-blue, and purple curve segments in Figure 3b) on top of the steady-state noise map reproduced from Figure 1a (gray circles). Compared to the strong, order-of-magnitude device-to-device variation of the gray steady-state noise data, the steady-state regime of the average voltage-dependent noise curve (red and blue circles) exhibits a smaller variation in both the relative noise and conductance. The conductance of the LCS corresponds to a more oxygen-saturated conducting filament with a transport deviating from metallic conduction and more prone to instabilities, which explains the broader variation in conductance. By definition, the non-steady-state regime is the voltage range, where the conductance is mostly unchanged, but the noise significantly deviates from the steady-state noise. Accordingly, the corresponding dark-red and dark-blue non-steady-state noise data are positioned above the steady-state noise data in Figure 3c. Finally, the noise data in the switching region (purple) do not grow above the device-to-device variation of the steady-state noise data, but the purple points are positioned around the largest possible steady-state noise values at the given G = I/V_bias conductance of the actual point on the switching curve.

Cycle-to-Cycle Noise Variation

In addition to the voltage-induced excitation of non-steady-state fluctuations, the example measurements in Figure 2 display an additional remarkable feature. At the beginning of the full cycle (Figure 2c1), the G_avg = 408.3 μS conductance is accompanied by ΔG = 0.13 μS conductance noise. At the end of the full cycle (Figure 2c4), the conductance returns to a very similar value (≈3% conductance change); meanwhile, the noise reduces by more than a factor of 4 (see ΔG = 0.03 μS in Figure 2c4), and a comparison of the light-blue and light-cyan points is given in Figure 2f. This means that a single switching cycle remarkably manipulates the dominant fluctuators and related noise performance.

The related cycle-to-cycle noise variation is even better displayed in Figure 3d, where the blue and red points demonstrate the steady-state (low-bias) noise values measured between the subsequent switching cycles of Figure 3a,b compared with the device-to-device and state-to-state variation of the steady-state noise values (gray circles reproduced from Figure 1a). Even though the repeated switching cycles with seemingly reproducible I(V) curves are intended to restore the same device states in each cycle, it is clear that the cycle-to-cycle variation of the steady-state noise values spans the same wide range as the device-to-device and state-to-state variation.

The previous observations mean that the seemingly reproducible switching process yields the reconfiguration of the fluctuators along the switching; i.e., a completed switching cycle yields a conductance very similar to that of the previous cycle, while the fluctuations of the active region completely change. This can cause even an order-of-magnitude decrease of the steady-state noise from one cycle to the other but a similarly large increase as well, which can be interpreted by switching OFF or ON a highly dominant fluctuator along a resistive switching cycle. This cycle-to-cycle noise variation does not allow a deterministic denoising strategy, but by trial-and-error, along ∼10 switching cycles, one can find a device state for which the steady-state noise is close to the bottom end of noise levels’ device-to-device variation; i.e., the noise of the actual state is close to the smallest possible noise value for the given device pool. Accordingly, the such-adjusted low-noise state is applicable as a high-resolution synaptic weight in a NN.

Subthreshold Denoising

In the following, we analyze the possibility of subthreshold denoising on the Ta₂O₅ cross-point devices, following the voltage-cycling strategies illustrated at the bottom of Figure 3. The bottom strategy (switching cycling) illustrates just the above-described scheme, where the steady-state noise of a device can be manipulated throughout repeated switching cycles. Alternatively, one can ramp up and down the applied voltage, always staying below the switching threshold, and study how the steady-state noise is manipulated by such subthreshold cycling measurements. By a definition of the steady state, no major variation of the noise properties is expected if a steady-state subthreshold cycling is performed, i.e., if the applied voltage cycle does not exceed the steady-state region. A non-steady-state subthreshold cycling may already yield considerable noise variation, offering the possibility for subthreshold denoising. These schemes are discussed in the following by demonstrating a noise benchmarking protocol that provides comprehensive insight into the noise properties along the above cycling strategies.

We analyzed the variation of the steady-state noise with the subthreshold measurement protocol. First, a long read-out noise measurement is executed at V_read = −100 mV. Next, the voltage is ramped up toward the negative polarity in a stepwise fashion to a V_max level that is still below the switching threshold. Afterward, the voltage is ramped down to zero in a stepwise fashion. Finally, another noise measurement is performed at V_read, as illustrated through the corresponding driving signal in Figure 4f. The results of such measurements are summarized in Figure 4a–e. The horizontal axis shows the number of executed cycling periods, while Figure 4a presents the V_max voltage reached along the given cycle. First, a V_max value of −100 mV is applied. Each cycling amplitude is repeated 10 times to observe the cycle-to-cycle variation at a given voltage amplitude, and then V_max is increased by −50 mV toward the negative polarity. Negative voltages were chosen based on the findings evident in Figure 3b, which shows that the subthreshold non-steady-state noise increase is most pronounced under negative polarity. This schematic is repeated for 130 cycles in total, reaching V_max = −700 mV. The latter value relies on our experience with a large number of devices. It represents a voltage level where the switching threshold is not yet reached; i.e., the steady-state conductance is mostly unchanged, but the steady-state noise regime is well exceeded. Eventually, only the part exhibiting stable conductance is included in the analysis. Practically, this means that cyclings leading to a >10% change in the conductance are excluded. This was the case for the measurements of Figure 4b,d, where the cyclings with maximal amplitudes above V_max = −650 mV and −550 mV resulted in an enhanced change of the conductance.

Such cycling periods are presented on three different conductance states (Figure 4b–d), well representing the entire conductance region of the usual measurements. The experiments yield two useful quantities for further evaluation: (i) the steady-state relative noise level after a certain cycling period is evaluated along the V_read read-out plateau (read-out noise demonstrated by colored circles with dots in the middle in Figure 4b–d); (ii) the relative noise is also evaluated at each voltage plateau of the cycling, and the highest voltage (V_max) is used in this analysis, as demonstrated by the gold triangles in Figure 4b–d. As a reference, the conductance is also evaluated after each cycle at V_read, as demonstrated by the colored squares in Figure 4e using the same colors for the various conductance states as those in the noise data in Figure 4b–d. The initial steady-state relative noise and conductance values are indicated by solid horizontal colored lines to emphasize any relative change.

We emphasize that the conductance (Figure 4e) is extremely stable along all of these subthreshold voltage cycles. The noise variation measured at V_max (triangles in Figure 4f) is not necessarily monotonic, but a general increasing trend is identified in accordance with the non-steady-state noise region in Figure 3. This is attributed to the tendency that a non-steady-state voltage is likely to excite further fluctuators compared to the steady state, but sometimes it is also possible that the applied voltage pushes a certain fluctuator to a state where it stops fluctuating or, alternatively, it modifies its fluctuation frequency, which also alters the integrated current fluctuation. Section 5 in the Supporting Information presents a detailed example of such voltage-induced tuning of a dominant fluctuator that can lead to an initial decrease of noise before the increasing tendency of the non-steady state is observed.

The read-out noise (colored circles in Figure 4g) is mostly constant until V_max = −450 mV or V_max = −500 mV voltage amplitudes, corresponding to device states with 5.7G₀ and 2.9G₀ conductances, which is apparent in panels b and c of Figure 4, respectively. At even higher V_max, the read-out noise values do not show a general increasing tendency, but rather a stochastic variation between the subsequent subthreshold cycles is observed, such that the actual noise values can even be significantly smaller than the initial read-out noise. The results in Figure 4d are obtained at a lower conductance of 0.8G₀ (green), where the device is less stable than at the above-mentioned higher conductances. Around the quantum conductance unit, the smallest rearrangements in the conductive filament strongly influence the overall conductance and the steady-state read-out noise. Such instabilities hinder the unambiguous identification of the steady-state and non-steady-state regimes. Nevertheless, the results do suggest an onset of the non-steady state around −400/–450 mV with a considerably increased maximal amplitude noise relative to the initial values and, more apparently, an increased cycle-to-cycle variation of the read-out noise.

The results of Figure 4b–d can also be summarized in comparison to the reference figure on the device-to-device steady-state noise variation (gray circles in Figure 4g). In the same figure, the vertical extent of the light-blue, light-red, and light-green shaded areas encloses the intervals, where the relative read-out noise values scatter for the initial 10 cycles in Figure 4b–d, i.e., for the measurements with V_max = −100 mV. These noise values span a significantly narrower noise interval than the device-to-device variation represented by the gray circles. This means that the low voltage cycling keeps the device’s noise mostly stable. As a sharp contrast, the blue, red, and green circles in Figure 4g exhibit the read-out noise values during the final 10 cycles in Figure 4b–d, where the V_max voltage amplitudes reach the maximal values in the non-steady-state regime. These noise values span a noise interval similar to that of the device-to-device variation at the same conductance. All of these observations demonstrate that an initial high noise of a memristive device (like the red point and red curve in Figure 1a,b) is not an immutable property, and not even complete switching cycles are needed to tune the noise. With subthreshold cycling, the full available noise range can be traversed with a few voltage sweeps and a lower noise state can be set.

Device Fabrication

The detailed noise spectroscopy measurements and evaluation are performed on Ta/Ta₂O₅/Pt cross-point devices. (40) The sample is fabricated on top of a SiO₂ substrate where a Ti adhesive layer of 10 nm and a Pt bottom layer of 40 nm thickness are deposited by electron-beam evaporation. Subsequently, the 5-nm-thick Ta₂O₅ layer is sputtered by reactive high-power-impulse magnetron sputtering. Finally, the 65-nm-thick Ta top layer and an additional Pt capping layer are sputtered on top of the Ta₂O₅ layer.

Noise Measurement Setup

The Ta/Ta₂O₅/Pt cross-point devices are investigated in a Faraday cage and with shielded cables in order to eliminate external instrumental noise appearing in the measurements. The circuit diagram of the noise measurement setup is depicted in Figure 5. An Agilent 33220A arbitrary waveform generator serves as a voltage source with a simple low-pass RC filter on the output. The latter decreases the output noise of the generator and can be tuned in accordance with the time scale of the actual measurement. The device current is amplified by an Femto DLPCA-200 current amplifier and recorded by a National Instruments PXI-5922 digitizer that offers a 6 MHz aliasing-free bandwidth. The R_series series resistor has a current limiting role in the low-resistance states of the devices.

With this method, the noise spectra are evaluated within the 100 Hz to 50 kHz frequency range, which we use as a standard for our measurements in order to ensure comparability of noise characteristics recorded for different parameters, e.g., different memristor resistances. The upper limit is fundamentally set by the limited bandwidth of the current amplifier in the high-gain (10⁷ V/A) mode, which is used at high resistances. On the other hand, the 100 Hz bottom limit allows a full-cycle noise measurement with ≈80 subsequent voltage plateaus per cycle such that the overall measurement time of the full cycle remains reasonable (≈40 s).