Noise Tailoring in Memristive Filaments

In this study, the possibilities of noise tailoring in filamentary resistive switching memory devices are investigated. To this end, the resistance and frequency scaling of the low-frequency 1/f-type noise properties are studied in representative mainstream material systems. It is shown that the overall noise floor is tailorable by the proper material choice, as demonstrated by the order-of-magnitude smaller noise levels in Ta2O5 and Nb2O5 transition-metal oxide memristors compared to Ag-based devices. Furthermore, the variation of the resistance states allows orders-of-magnitude tuning of the relative noise level in all of these material systems. This behavior is analyzed in the framework of a point-contact noise model highlighting the possibility for the disorder-induced suppression of the noise contribution arising from remote fluctuators. These findings promote the design of multipurpose resistive switching units, which can simultaneously serve as analog-tunable memory elements and tunable noise sources in probabilistic computing machines.


■ INTRODUCTION
In traditional electrical engineering, noise is considered as an issue, which is to be suppressed to the lowest possible level. 1,2 Accordingly, the introduction of new components is usually preceded by lengthy material optimization steps to decrease the low-frequency, 1/f-type noise generated by material imperfections. 3,4 The emergence of novel neuromorphic computing architectures, 5,6 however, brings a paradigm change in noise engineering, demonstrating that tailored noise can be harvested as a useful computing resource in probabilistic computing schemes. As specific examples, stochastic magnetic tunnel junctions were utilized to solve integer factorization in a probabilistic bit computing architecture, 7 whereas a Hopfield neural network of resistive switching memory (RRAM) units was applied to solve nondeterministic polynomial-time (NP)hard max-cut problems. 8 Noise tuning served as a key ingredient in the operation in both approaches.
Resistive switching memories, or memristors, 9,10 have already demonstrated their pioneering role in the development of information technologies, including energy-efficient, fast, and compact applications in mass data storage, 11 in-memory computing, 12 or the hardware implementation of artificial neural networks (ANNs). 13 −16 In the latter case, the nonvolatile and analog-tunable RRAM resistance states serve as the neural weights of ANNs and dense crossbar RRAM architectures perform massively parallel operations, such as single time-step vector-matrix multiplication. Similar crossbar architectures served as the hardware accelerator in the above RRAM-based combinatorial optimization machines, 8 where the simulated annealing protocol was realized by the amplification/suppression of the intrinsic RRAM noise using an external hysteretic threshold circuitry.
To support further potential applications in noise engineering, here, we analyze how the intrinsic noise of RRAM devices can be tailored. More specifically, (i) we study the influence of material choice on the base noise level; (ii) we investigate how the relative noise level scales with the analog-tunable resistance states of RRAMs; and (iii) we deliver a fundamental understanding of the resistance scaling of the noise by model considerations. To this end, we analyze the resistance and frequency dependence of the intrinsic noise in Ta 2 O 5 -and Nb 2 O 5 -based resistive switching devices and compare these results to the markedly different noise levels observed in Agbased resistive switching filaments. The above transition-metal oxide memristors represent well-established resistive switching systems with robust switching characteristics, 17,18 multilevel programming, 19,20 and ultrafast switching. 20,21 Ta 2 O 5 is an especially widely studied compound with great potential for near-future neuromorphic computing applications. 6,22 Controversially, the noise analysis of these systems is very limited, 23−26 and the detailed investigation and understanding of the noise's resistance scaling is lacking. Our reference systems, the Ag-based filamentary devices, serve as another fundamentally important platform in the RRAM technology. 27−33 Our related noise analysis 34 highlighted a universal resistance scaling behavior, revealing that the resistance fluctuations are dominated by the internal fluctuations of the Ag nanowires, whereas the embedding environment does not have an important influence on the noise.
The 1/f-type noise in nanoscale devices may originate from either atomic fluctuations or charge trapping/detrapping effects. In our previous study, we have demonstrated that our Nb 2 O 5 scanning tunneling microscope (STM) pointcontact devices preserve the metallic conduction through electronically transparent, unbroken filaments down to the level of single-atom diameters. 35 In this unbroken filamentary regime, charge-trap states in the embedding insulating matrix are efficiently screened by the metallic filament; therefore, we rather consider atomic fluctuations inside or at the surface of the filament as the dominant noise source. In the transitionmetal oxide systems, we consider oxygen vacancies as the major fluctuators. In our further analysis, we solely treat this metallic unbroken filamentary regime, which is related to the resistance regime below the inverse conductance quantum, G 0 −1 ≈ 12.9 kΩ. This regime is favorable in RRAM crossbars utilized to implement vector-matrix multiplication operations. 6,14,15 Previous noise studies have indicated that the noise level may depend of the resistance state in selected RRAM systems. 34,36−41 Prior studies mostly described the noise's resistance scaling in the metallic regime by a simple geometrical model relying on a cylinder (or prism) geometry with a single two-state fluctuator at the filament surface. 36,39−41 Here, we analyze the noise data in terms of our recently proposed model, which considers the scattering on dynamical defects (or two-level systems, TLSs) in a metallic point-contact geometry, also taking into account the crossover between the diffusive and ballistic transport regimes. 34 This model is able to describe an ensemble of fluctuators, which also accounts for the dominance of the fluctuators located nearby the narrowest part of the filament, and the suppressed contribution of more remote fluctuators. The fitting of the noise data with this model uncovers the material specificity of the two key parameters, the TLS density and the electron mean free path. Our analysis brings a counterintuitive conclusion demonstrating that the order-of-magnitude noise suppression of the transition-metal oxide resistive switching units compared to the Ag-based systems is primarily related to the enhanced level of disorder in the former material systems. This disorder enhancement yields an increased suppression of those "remote" fluctuators' noise contribution, which are located outside the narrowest region of the filaments.

■ RESULTS AND DISCUSSION
To study the noise characteristics of Ta 2 O 5 and Nb 2 O 5 memristive systems, we have applied two approaches: (i) we have established resistive switching junctions in both material systems by touching thin-film structures at various lateral positions with the PtIr tip of a custom-designed STM (see the insets in Figure 1a,b). With this approach, we could collect noise data on a statistical ensemble of independent junctions with various resistances. (ii) We have investigated the variation of the noise with the device resistance in Al/Nb 2 O 5 /Pt crosspoint RRAM structures (see the inset in Figure 1c), where the resistance states were tuned by voltage pulses. The growth of the oxide layer was performed by either anodic oxidation (STM devices) or reactive sputtering (crosspoint devices). Both oxide growth protocols were optimized to achieve a pentoxide stoichiometry, which was confirmed by Xray photoelectron spectroscopy (XPS) analysis. The peaks in the XPS spectra characteristic of the Ta 2 O 5 and Nb 2 O 5 compositions are demonstrated in Figure 1d,e, respectively. More details on the preparation and electroforming of the devices are provided in the Experimental Section. Figure 1a−c exemplifies the resistive switching current−voltage (I(V)) characteristics of the Ta 2 O 5 (dark brown) and Nb 2 O 5 (light brown) STM point-contact devices and Nb 2 O 5 crosspoint devices (gray). We observed stable switching cycles, which are illustrated by the reproducibility of ≈100 consecutive I(V) curves on each panel.
A reliable noise study builds on the detailed understanding and careful separation of the simultaneous noise contributions present in the measurement setup and a corresponding data processing. Therefore, we first describe the scheme of our analysis shown in Figure 2. The measurement circuit ( Figure  2c) includes a driving unit, the memristor device, a current amplifier, and an R S serial resistor. The latter terminates the switching at the low-resistance state once the R M memristor resistance becomes comparable to R S . The V bias voltage drop on the memristor junction is calculated as V bias = V drive − I × R S . More details on the measurement instrumentation are provided in the Experimental Section.   Figure 2b. The voltage cycle leaves the junction in its high-resistance state (HRS). Next, a lower-amplitude voltage staircase is applied, where we evaluate the mean current (red curve in Figure 2b) and the power spectral density (PSD) of the current noise ( Figure 2d) for each voltage step (see the Experimental Section for more details of the PSD calculation). Then, another higher-amplitude I(V) measurement is performed (green curves in Figure 2a,b), preparing the junction in the low-resistance state (LRS) for the subsequent noise measurement (blue curves in Figure 2a,b,e). The sequence is completed by performing a final I(V) measurement. To grant the mechanical stability of the STM pointcontact devices, only those data are accepted for later analysis, where these three I(V) curves are reproducible also agreeing with the low-bias I(V) branches calculated from the voltage staircase data. In the case of the crosspoint devices, such a validation of each resistance state is not necessary. Instead, we measure the I(V) curve at the beginning, and later positive and negative voltage pulses are used for the fine analog tuning of the resistance states. The noise is measured in each state with the same protocol using identical voltage staircase signals.
We wish to study the resistance scaling of the steady state S G (f) = (ΔG) f 2 /Δf conductance noise PSD of the memristor junctions, where the (ΔG) 2 mean-square conductance deviation is evaluated within a small Δf bandwidth around the central frequency f. 1 According to Ohm's law, this conductance noise PSD converts to current noise PSD as i.e., the V 2 voltage scaling of the current noise is a benchmark that the steady-state conductance noise is studied, and voltage-induced fluctuations and/or nonlinear features are excluded from the analysis. To evaluate the ΔI/I relative current fluctuation, we remove the S 0 zero-bias noise floor and integrate the current noise PSD in the frequency interval between 100 Hz and 50 kHz as Note that in the case of steady-state noise measured in the linear part of the I(V) curve, ΔI/I = ΔG/G = ΔR/R holds, and these relative current/ conductance/resistance fluctuations are independent of the driving amplitude. Figure 2f demonstrates that the ΔI/I relative current fluctuations calculated from the spectra in Figure 2d,e are indeed voltage-independent within the scattering of the data, confirming that the steady-state conductance noise is measured in both states. For further analysis, we have set two criteria on the data: (i) the integrated noise PSD should be at least 1 order of magnitude larger than the S 0 noise floor integrated for the same band and (ii) nonlinear features should be excluded, i.e., we use a voltage interval, where the variance of the R M = V/I resistance and the ΔI/I relative current fluctuation are, respectively, less than 5 and 20% compared to the mean value for every device. Accordingly, we have used the 75−115 mV voltage interval for further analysis as a safe margin (see the gray interval in Figure  2f), where the negligibility of the background noise and nonlinear features is satisfied for all devices.
We have found that the frequency dependence of the noise spectra of Nb 2 O 5 -and Ta 2 O 5 -based memristors is often qualitatively different from the typical spectra acquired in Agbased memristors. 34 In such cases, the log S I vs log f spectrum cannot be fitted with a single line, rather a bump is superimposed on an ∼1/f γ background, as exemplified in Figure 2g. This behavior can be modeled by a spatial distribution of the fluctuators, as illustrated in the inset of  conductance noise with a temporal correlation function described by a single time constant, C G (Δt) = ⟨ΔG(t) × ΔG(t + Δt)⟩ = ⟨ΔG 2 ⟩ × exp(−Δt/τ), yielding a Lorentzian noise spectrum in the frequency space: S G (f) = (4⟨ΔG 2 ⟩ × τ)/ (1 + (2πf) 2 τ 2 ) denoted by the orange line in Figure 2g. In the case of the remote fluctuators, the superposition of such Lorentzians characterized by different time constants yield a 1/f-type envelope. The latter is well described by the ∼1/f γ spectrum, where γ is usually close to unity, as represented by the green line in Figure 2g. Relying on this observation, we have evaluated all PSD spectra following two procedures: (i) we have calculated the numerical integral of the noise yielding the actual (ΔI/I) numint relative current fluctuation of the junction irrespective of the spectrum shape; (ii) we have fitted each spectrum with the superposition of a Lorentzian spectrum and a 1/f γ spectrum (see the yellow fitting curve in Figure 2g). Based on the latter method, the relative current fluctuations arising from the nearby fluctuator, (ΔI/I) Lor , and the remote fluctuators, (ΔI/I) 1/f , can be decomposed by separately integrating the individual Lorentzian fitting function and the ∼1/f γ term within the same 100 Hz to 50 kHz frequency band.
In Figure 3, we analyze the resistance scaling of the noise including its specificity to (i) the nature of the state (HRS or LRS), (ii) the preparation method (STM devices relying on anodic oxidation or crosspoint devices fabricated by reactive sputtering), and (iii) the material system (Nb 2 O 5 , Ta 2 O 5 , Ag 2 S). Furthermore, we investigate the relative contribution of the nearby fluctuators as well as the γ frequency scaling exponent of the remote fluctuators. Finally, to extract the relevant parameters for practical noise engineering, we employ our recently proposed point-contact noise model. 34 Figure 3a demonstrates the clear resistance scaling of (ΔI/I) numint evaluated in the HRSs (dark red) and LRSs (dark blue) of several Nb 2 O 5 STM devices as well as of a Nb 2 O 5 crosspoint device (gray). In the former case (STM devices), the data represent several (>20) independent pointcontact junctions with various LRS and HRS resistances, whereas in the latter case, the multilevel programming of a particular crosspoint device was achieved by voltage pulses with typical amplitudes of ±(2−3) V (see the Experimental Section for the pulsing scheme). This analysis demonstrates that all of the blue, red, and gray data points follow the same resistance scaling tendency, implying that the noise is not specific to the preparation method, neither to the HRS or LRS nature of the state, but it solely depends on the device resistance in a certain material system. This conclusion agrees with our previous noise measurements on Ag-based filaments, where Ag 2 S and AgI embedded as well as stand-alone Ag nanowires exhibited the same universal resistance scaling.
The black data points in Figure 3c−e show the resistance scaling of (ΔI/I) numint for Nb 2 O 5 , Ta 2 O 5 , and Ag 2 S STM pointcontact devices, respectively. To better resolve the average resistance dependencies, a statistical ensemble of noise data acquired on independent devices is grouped into resistance bins, which are equally spaced along the logarithmic resistance axis. The data points and error bars represent the mean values and standard deviations for the various noise measurements corresponding to a resistance bin. Note that the Ag 2 S noise data represent our previous measurements acquired by a similar STM point-contact arrangement. 34 Here, these data are reevaluated according to the protocol used for the analysis of the Nb 2 O 5 and Ta 2 O 5 noise data in panels (c) and (d), i.e., we have evaluated (ΔI/I) numint in the same frequency band, applied the resistance binning, and performed the decomposition of the (ΔI/I) Lor and (ΔI/I) 1/f contributions. This comparison shows that the noise increases with increasing resistance for all material systems; however, the overall noise level is markedly lower in the transition-metal oxide memristors than in the Ag 2 S devices.
The (ΔI/I) Lor noise contribution of the nearby fluctuators is shown by the yellow dots in Figure 3c−e. This analysis also highlights a clear difference between the transition-metal oxide systems and Ag 2 S: in the former case, the noise of the nearby fluctuators is comparable to the total noise, whereas in the latter, (ΔI/I) Lor is more than an order of magnitude smaller than (ΔI/I) numint . This is also demonstrated by the green lines showing the relative noise contribution of the nearby fluctuators compared to the total noise, (ΔI Lor ) 2 /((ΔI Lor ) 2 + (ΔI 1/f ) 2 ), demonstrating an ≈30% noise contribution of the The black (yellow) squares represent the resistance scaling of (ΔI/I) numint ((ΔI/I) Lor ). For each material system, the data rely on >20 independent junctions and 5 repeated I(V) and noise measurement cycles on each junction, and from each noise measurement cycle, four different bias steps are used for the analysis (see the gray interval in Figure 2f). The best-fitting theoretical curves describe the diffusive (red) to ballistic (blue) crossover (see text). The fitting curves are calculated with the numerical values of I Ω = 767.29 34 and the k F = 1.2 × 10 10 m −1 coincident Fermi wavenumbers of Ag and Nb. 46 Relying on the similar band structures of Nb and Ta, we use the same value for Ta as well. The green curves and the corresponding green axis on the right show the relative noise contribution of the nearby fluctuator as deduced from the Lorentzian fitting component (see text). (f−h) Frequency scaling exponents as deduced from the 1/f γ fitting component. nearby fluctuators for Nb 2 O 5 and Ta 2 O 5 , and a significantly smaller value for Ag 2 S. We note that the statistically underpinned significance of nearby fluctuators in transitionmetal oxide memristors agrees with previously reported 1/f 2type noise spectra of Ta 2 O 5 RRAM devices 25 representing the high-frequency limit of a dominant Lorentzian spectrum.
The γ frequency scaling exponents are close to unity for all of the three systems, as shown in Figure 3f−h. Note that in the Ag 2 S system, the noise spectrum is highly dominated by a 1/ftype dependency, i.e., the fitting provides more precise values for γ. In contrast, the significant Lorentzian contribution in the Ta 2 O 5 and Nb 2 O 5 noise spectra also yields a consequently higher error in the fitted γ values.
In our previous work, we have quantitatively analyzed the resistance scaling of the noise in terms of our model taking electron scattering on dynamical defects (TLSs) into account. 34 In particular, the model considers the conducting filament as a point-contact device with a realistic geometry, where the ΔG conductance noise contribution of a dynamical defect scales with the probability that an electron returns to the junction after scattering on the TLS. 47 This return probability strongly depends on the relation of the d diameter of the junction and the l mean free path, i.e., the average distance at which the electrons loose their momentum due to scattering on lattice defects, impurities, or at the filament surface. 46 In the ballistic regime, where the l electron mean free path is larger than the d diameter (see Figure 3b (top), where the dark gray/ white circles represent a TLS and the arrows illustrate possible electron trajectories), this return probability scales with the square of the solid angle at which the junction is seen from the TLS position. However, if d becomes larger than l (see Figure  3b (bottom)), the diffusive motion of the electrons reduces the return probability by a factor of (2l/d) 2 compared to the ballistic regime. 47 Relying on these geometrical coefficients and assuming a constant ρ TLS TLS density as well as the validity of the Maxwell (Sharvin) conductance formulas in the diffusive (ballistic) regimes, we estimate the resistance scaling of the relative current fluctuations as 34 where G 0 = 2e 2 /h is the universal conductance quantum, k F is the Fermi wavenumber, I Ω is a constant resulting from a solid angle integral, C = ΔG/G 0 sets the average amplitude of the conductance noise resulting from a TLS located at the center of the point contact, and l TLS = ρ TLS −1/3 is the average spacing of neighbor TLSs. In the diffusive (ballistic) limit, ΔI/I scales with the 3/2 (1/4) power of the memristor resistance, respectively. Note that this model is derived for an orifice-like geometry, but it also well approximates a more realistic (e.g., hyperboloid) point-contact geometry. 47 This model has two fitting parameters, l and C/l TLS 3/2 . Using a C = 0.5 estimate, 34,48 we can extract the two relevant length scales, l and l TLS , from the fitting. Note that the fitting is performed on the binned data in panels (c−e); however, the raw data (see the red and blue dots for the Nb 2 O 5 STM point-contact junctions in panel (a)) and the binned data provide practically the same fitting results. This model fits well to our Ag 2 S memristor noise data, 34 yielding l Ag 2 S = 1.02 ± 0.02 nm and l TLS,Ag 2 S = 1.93 ± 0.03 nm. The best-fitting curve composed of the diffusive and ballistic branches according to eqs 1 and 2 is shown by the red and blue dashed lines in Figure 3e. In Figure 3c,d, we fit the Nb 2 O 5 and Ta 2 O 5 noise data with the same model (red and blue dotted lines), yielding l Nb 2 O 5 = 0.484 ± 0.036 nm, l TLS,Nb 2 O 5 = 3.27 ± 0.33 nm, l Ta 2 O 5 = 0.380 ± 0.021 nm, and l TLS,Ta 2 O 5 = 3.71 ± 0.29 nm. As a reference, the fitting curve of the Ag 2 S noise data is reproduced on both panels as a red/blue dashed line. The reduced noise level in the transition-metal oxide memristive systems compared to the Ag 2 S data is the most pronounced in the diffusive regime, where Nb 2 O 5 (Ta 2 O 5 ) memristors exhibit a factor of ≈14 (≈31) reduction of ΔI/I with respect to Ag 2 S devices, according to the offset of the diffusive fitting lines. This significant noise reduction has a twofold origin: the decrease of the electron mean free path and the decrease of the TLS density.
The above analysis allows us to draw clear conclusions on the aspects of noise engineering in the metallic regime of filamentary resistive switching memories: (i) The diffusive regime is ideal for noise tailoring, as the ΔI/I ∼R M 3/2 resistance scaling is steep enough to customize the noise level within a sufficiently wide range by the analog tuning of the resistance states. As the ballistic regime is reached, a less steep resistance scaling is achieved, which hinders further noise tailoring. It is also noted that the mean free path is mainly extracted from the resistance threshold, where the noise evolution deviates from the diffusive scaling (≈1.5 kΩ, ≈2.5 kΩ, and ≈400 Ω for Nb 2 O 5 , Ta 2 O 5 , and Ag 2 S, respectively). In the transition-metal oxide devices, this boundary is close to the inverse conductance quantum, G 0 −1 ≈ 12.9 kΩ, i.e., the ballistic regime spans less than an order of magnitude along the resistance axis. The even larger resistances are clearly outside the validity range of our point-contact model; however, other studies considering fluctuations in broken filaments reported saturated noise in the R M > G 0 −1 regime. 36,38−41 Based on all of these, we propose the ≲2 kΩ (≲400 Ω) resistance range for the transition-metal-oxide (Ag 2 S) systems as an optimal working range for noise tuning. At larger resistances (i.e., in the ballistic regime and in the regime of broken filaments), the saturated character of the relative noise levels is detrimental for efficient noise tailoring. Furthermore, in close vicinity to the inverse conductance quantum, the truly atomic-scale diameter of the filament introduces an extreme sensitivity to the precise atomic position of a nearby fluctuator. 25 (ii) It is evident that the device noise can be reduced by decreasing the density of dynamical defects generating the noise, which was also demonstrated by a recent study relying on training-induced noise reduction. 49 Our analysis, however, highlights a further, less obvious noise reduction strategy via the suppression of the remote fluctuators' noise contribution by decreasing the electron mean free path. If the degree of static disorder can be increased without introducing further nearby fluctuators, the suppression of the remote fluctuators' noise delivers a significant overall noise reduction. Moreover, as the noise scales with a higher (5/2) power of l than l TLS (3/2 power), a factor of 2 reduction in l yields twice as large reduction in ΔI/I, as a factor of 2 increase in l TLS considering a constant memristor resistance. Alternatively, one can study the scaling of the relative current fluctuation as a function of the filament size by converting the memristor resistances to the d filament diameter according to the Maxwell (Sharvin) formula in the diffusive (ballistic) regime. 34 This conversion yields , again demonstrating that the enhancement of static disorder yields noise reduction in the diffusive regime.
(iii) In the transition-metal oxide memristors, the noise contribution of remote fluctuators is so much suppressed that a single nearby fluctuator gives a major contribution to the total noise, which is a great advantage if noise reduction is targeted. However, the noise contribution of a single fluctuator is sensitive to the actual atomic position of the fluctuator within the narrowest filament region, which may change upon the resistive switching cycles. If noise tailoring is considered, it may be beneficial to eliminate this sensitivity by choosing material systems (like Ag-based memristors) with a higher overall noise level and a smaller relative contribution of the nearby fluctuators. (iv) Finally, we consider the material aspects behind the significant noise suppression in transition-metal oxide filaments compared to Ag filaments. In the former case, the host metal is highly decorated with oxygen impurities. Furthermore, electron transport is dominated by d-orbitals, making the conductance sensitive to the details of the actual bond structure. Both aspects significantly decrease the mean free path of the electrons compared to Ag filaments, which are considered as pure metallic wires, where the highly delocalized s electrons are less sensitive to the actual atomic landscape. While the mean free path relies on all scattering processes including the static disorder of the filament, the noise solely originates from dynamical defects, where atoms are fluctuating between metastable positions. We consider unstable oxygen vacancies (Ag atoms) driven by temperature-activated Langevin dynamics 50 as the major noise sources in these transition-metal-oxide (silver) filaments. Interestingly, our analysis highlights that the enhanced level of static disorder in the Nb 2 O 5 and Ta 2 O 5 systems is not accompanied by an enhanced dynamical defect density. On the contrary, l TLS is further increased compared to the Ag 2 S system, i.e., these transition-metal oxide filaments are more stable against internal fluctuations than the Ag filaments.

■ CONCLUSIONS
In summary, we have performed a comparative study on the resistance scaling of the low-frequency noise in mainstream RRAM systems. We demonstrated that the noise characteristics of Ta 2 O 5 and Nb 2 O 5 memristors are well described by our point-contact model recently developed for the noise analysis of Ag-based filamentary resistive switches. In the diffusive transport regime, the relative noise levels of all of these systems exhibit a universal and sufficiently steep resistance scaling power for efficient noise tailoring. However, we found markedly different overall noise levels in the silverbased and the transition-metal-oxide-based systems. Our analysis yields a counterintuitive explanation for this difference highlighting the role of disorder-induced noise suppression in Ta 52 Electroforming and Resistance Tuning. In the Nb 2 O 5 crosspoint devices, the electroforming was performed by applying 500 μs long voltage pulses with gradually increasing amplitude on the memristor device and a 100 kΩ serial resistor. After each step, the pulse amplitude was increased by 1 mV. Between subsequent pulses, a lower-amplitude I(V) curve was measured to read out the low-bias resistance. The forming process was stopped once the low-bias device resistance became smaller than 100 kΩ. This occurred typically after pulse amplitudes of 2−3 V. After the electroforming, the serial resistance was reduced to 1 kΩ or 100 Ω depending on the targeted LRS resistance, and the device was operated in the regime of unbroken metallic filaments. The multilevel programming was achieved by similar positive or negative writing pulses with somewhat smaller amplitudes (1−2 V) and similar readout I(V) curves.
In the STM point-contact devices, mechanically sharpened PtIr tips were applied with a 250 μm wire diameter. A micron-scale tip-radius is estimated, with possible nanometer-scale protrusions. The forming is performed by the gentle mechanical approach of the thin film by the STM tip. This approach is stopped once the junction resistance becomes smaller than 100 kΩ. Afterward, a dedicated electroforming procedure is not required, and the junction is ready to study resistive switching I(V) traces. 20 Note that the very similar noise characteristics of STM point contact and crosspoint Nb 2 O 5 devices (see Figure 3a) imply that the fundamentally different forming steps of these two types of devices do not influence the noise characteristics.
Details of the Noise Measurement System. The measurement circuit (Figure 2c) supplies the V drive driving voltage using an Agilent 33220A arbitrary waveform generator. An additional RC filter on the output is utilized to further reduce the noise of the voltage source. The current through the memristor (I) is measured by a Femto DLPCA-200 current amplifier. The output of the amplifier is digitized with 22-bit resolution and 500 kS/s sampling rate with an NI PCI-5922 data acquisition card applying an antialiasing filter at the Nyquist frequency. At each constant voltage level, typically N = 100 000 data points are acquired. To avoid transient features, the first and last parts of this dataset are dropped and the remaining points are partitioned to nine segments with N = 2 13 points in each segment. For each segment, the noise spectrum is calculated by the fast Fourier transform (FFT) algorithm. Note that the spectral density of the current noise (or the noise PSD) is evaluated as where Δt is the time delay between the neighbor acquisition events and ΔI(t) is the deviation of the current from its mean value. The noise spectra calculated for the nine segments are averaged.