Stimulated Ionic Telegraph Noise in Filamentary Memristive Devices

Random telegraph noise is a widely investigated phenomenon affecting the reliability of the reading operation of the class of memristive devices whose operation relies on formation and dissolution of conductive filaments. The trap and the release of electrons into and from defects surrounding the filament produce current fluctuations at low read voltages. In this work, telegraphic resistance variations are intentionally stimulated through pulse trains in HfO2-based memristive devices. The stimulated noise results from the re-arrangement of ionic defects constituting the filament responsible for the switching. Therefore, the stimulated noise has an ionic origin in contrast to the electronic nature of conventional telegraph noise. The stimulated noise is interpreted as raising from a dynamic equilibrium establishing from the tendencies of ionic drift and diffusion acting on the edges of conductive filament. We present a model that accounts for the observed increase of noise amplitude with the average device resistance. This work provides the demonstration and the physical foundation for the intentional stimulation of ionic telegraph noise which, on one hand, affects the programming operations performed with trains of identical pulses, as for neuromorphic computing, and on the other hand, it can open opportunities for applications relying on stochastic processes in nanoscaled devices.

STN amplitude as a function of the average device resistance through a phenomenological filamentary model, in which ionic re-arrangements localized at the edges of the CF produce the observed current fluctuations.
It must be mentioned that, while conventional RTN influences the reading operation of the memristive devices, the described STN affects their programming operation when performed through trains of identical pulses. Such programming scheme is intensively investigated for applications in hardware neural networks. As a matter of fact, in a large number of publications [10][11][12][13][14][15][16][17][18] also by other authors, the reported resistance evolution as a function of the number of identical pulses shows a large superimposed noise which is not contextually discussed and may be ascribed to STN. On the other side, we envision that the intentional activation of random resistance variations can be exploited for those applications relying on stochastic phenomena of nanoscaled devices, like random number generation [19][20][21][22] , physical unclonable function [23][24][25] , stochastic or chaos computing or emulation of stochastic cognitive neural processes [26][27][28][29][30][31] .

Results
Evidence of Stimulated Telegraph Noise. In filamentary memristive devices, the electric conduction through CFs with nanometric effective cross-sections is affected by the occasional trapping and release of charges into and from surrounding defects [2][3][4][5][6][7] . This phenomenon is particularly relevant for high resistance values in which the conductive path is partially interrupted or very thin 32 . In this manuscript, we demonstrate that telegraphic current fluctuations can be generated by ionic re-arrangement in HfO 2 -based memristive devices stimulated by voltage pulses, as described in the Methods Section. Figure 1 reports a representative example of stimulated telegraph noise (STN) measured on a HfO 2 -based device programmed in a state with average resistance of about 13 kΩ. Concerning the data reported in Fig. 1a, 0.7 V pulses are delivered to the device every 1s (grey thin line in Fig. 1a right axis) and the current is read in between at a low voltage of 0.1 V (green thick line in Fig. 1a left axis). The current displays clear jumps in correspondence of the pulse delivery; in comparison the resistance value in between the pulses is more stable. The pulse parameters (pulse voltage and pulse time width) are chosen so that the fluctuations produced by the pulse trains average to zero and no appreciable switching is produced at the end of the measurement. Additional STN traces can be found in Supplementary Figs S1 and S2. Figure 1b  www.nature.com/scientificreports www.nature.com/scientificreports/ RTN. Figure 1b demonstrates also that the reading noise of our experimental apparatus is much smaller than the intrinsic device noise, in both cases of conventional RTN and STN. In summary, Fig. 1 demonstrates that voltage pulses can drive zero-average telegraphic current fluctuations. The resistance fluctuations stimulated by the high voltage pulses in Fig. 1a are stable for at least 1 s, which can only be ascribed to an ionic phenomenon. Indeed, according to related literature results, conventional RTN fluctuations, due to charge trapping and release, persist on average for few ms for the same reading voltages as for the present experiments 7,33,34 .
In the following, we refer to the average resistance value (i.e. the average in time of the signal affected by STN in Fig. 1a or equivalently the value in Fig. 1b with no pulse stimulation) as the unperturbed or average resistance value. Conversely, the resistance values that deviates from the average because of a stimulating voltage pulse are called perturbed resistance values. The noise amplitude is evaluated as the standard deviation of the measured resistance during pulse stimulation.
Tuning noise with pulse parameters. The pulse stimulation allows the noise characterization for different stimulus strengths (voltage and time width of the pulses). The immediate effect of the delivery of trains of identical pulses to an RRAM device is the change of its average resistance. Figure 2a,b reports the resistance dynamics of a device initialized in a low resistance state (resistance value of about 0.7-0.9 kΩ) and subjected to trains of identical pulses with various voltages from 0.3 to 0.9 V. Voltage polarity is chosen so as to increase the resistance (RESET operation). The resistance is read once after each pulse at 0.1 V. At the beginning of the pulse stimulation, the device resistance gradually drifts in an analogue fashion from low to high values, as evidenced in Fig. 2a with logarithmic x axis. In this regime, noise and non-volatile switching are superimposed. After a certain number of pulses, the resistance saturates around an almost constant average value which is perturbed by STN (evident in Fig. 2b with linear x axis). Therefore, STN is produced by pulses whose voltage is high enough also to promote the defect migration responsible for the neat gradual resistance change itself.
Resistance dynamics driven by trains of identical pulses as those shown in Fig. 2a,b have been widely reported in the recent literature papers and most of them show a superimposed significant noise in the RESET transitions, as well [10][11][12][13][14][15][16][17][18] . Since these works are mainly aimed at demonstrating gradual or analogue transitions of RRAMs devices, the noise is rarely discussed. Figure 2b shows an apparent dependence of the noise amplitude on the pulse voltage (from pulse voltages of 0.3 V to 0.9 V for the same time width). However, the increase of the pulse voltage produces an increase of the average saturation resistance value. The unperturbed resistance value and the noise amplitude at the saturation are collected for several pulse voltages and time widths. Figure 2c reports the resistance window, defined as the ratio between the final saturation resistance and the initial value, as a function of the voltage and the time width of the stimulating pulses. The noise amplitude is evaluated as the standard deviation of the resistance values recorded over the last 100 pulses, where the random fluctuations average to zero. In this manner, the phenomenology of the switching is completely isolated from that of noise, which cannot be easily done with other characterization methods based on quasi static voltage ramps 6,9 . The map of the noise amplitude is reported in Fig. 2d and correlates very well with the trend of the resistance window, indicating that the average resistance value determines the amplitude of the STN. Furthermore, the maps of Fig. 2c,d can be used to identify the kinetics of the switching process and of the processes causing the noise, respectively. In particular, as a guide for the eye, white dashed lines are superimposed to the maps of Fig. 2c,d as an indication of the threshold voltage (vs time width) for the non-volatile switching and noise to occur, respectively. The matching of the both kinetics indicates that the origin of both phenomena must be ascribed to the same process of migration of ionic species.
Additional data for the resistance dynamics, resistance window and noise maps can be found in Supplementary  Fig. S3. As shown in Supplementary Figs S4 and S5, the evolution of the STN in the SET process is less evident because high voltages bring the device to low resistance values. Therefore, voltages in principle high enough to www.nature.com/scientificreports www.nature.com/scientificreports/ stimulate noise are actually applied to low resistance device conditions where the noise is level is intrinsically low. Generally speaking, we can expect that the higher the average saturation resistance value in which a device can settle, the higher the STN amplitude. As a matter of principle, large and evident STN can be obtained also during SET operations if the saturation resistance value is sufficiently high. In addition, Supplementary Fig. S6 shows representative resistance evolutions driven by SET and RESET pulses as starting from similar initial resistance values. The higher STN amplitude shown by the RESET evolutions in comparison to those of the SET indicates that the occurrence of STN does not depend on the previous device programming; rather it is a further confirmation that the parameter governing the STN amplitude is the average resistance value.
Noise stimulation at the dynamic equilibrium between drift and diffusion. In the previous paragraphs, we demonstrate the STN phenomena and its dependence on the resistance value of the device. We show that STN can be isolated from the non-volatile resistance switching in the saturation tails of the analogue resistance dynamics produced by trains of pulses and we provide further evidence that noise and non-volatile switching are caused by the same processes.
Filament formation and dissolution processes responsible for the non-volatile resistive switching are ascribed to drift (driven by electric field) and diffusion (driven by concentration gradient) of defects or ionic species that introduce electronic levels in between the forbidden gap of the insulating material. The description of the noise in the present manuscript derives from a previous modelling approach applied to HfO 2 -based device by the same authors 3 and from other models reported in the literature 35,36 . We model the low resistance state of the device through a vertical cylindrical CF with radius of 3 nm connecting top and bottom electrodes. During a RESET process, an insulating gap is opened within the CF, increasing the device resistance. The electric conduction within the switching material is described through a defect density, n D , and a conductivity, σ 35,36 , that depends on the defect concentration: where k B is the Boltzmann's constant, T is the temperature, E a is the activation energy for conduction and σ 0 is a conductivity pre-factor. E a is modelled according to the following equation 35,36 : www.nature.com/scientificreports www.nature.com/scientificreports/ where E a,0 = 0.01 eV, n D,crit = 5.7 ⋅ 10 3 m −1 is the critical defect density value above which the conduction becomes metallic. It is worth noticing that the CF evolves only in the vertical direction so that n D can be considered as the defect density per unit length (along the vertical coordinate). The pre-factor of the conductivity per unit length σ 0 is proportional to the defect density and varies linearly from 0.93 Ω −1 m −1 nm 2 for n D = 0 to 9 ⋅ 10 6 Ω −1 m −1 nm 2 35,36 . In summary, a CF shorting the two device electrodes is described as a high concentration of defects with cylindrical symmetry. We model the opening of a gap as a decrease of the defect density in the middle of the CF. First, we model the resistance as a function of the filament/gap structure. The different resistance states, in which the HfO 2 -based device stack can settle, result from a vertical conductive filament (CF) characterized by an interrupting gap with different length. Indeed, Fig. 3a shows the n D profile along the CF for different gap lengths. The highest resistance state is associated to the largest (deepest) insulating gap within the CF (yellow line), while in the lowest resistance state the gap extension is almost negligible (blue line). For a constant reading current flowing through the device, the electric field along the CF axis is peaked on the centre of the gap and the larger the gap, the higher the peak, as shown in Fig. 3b. As a result, the voltage drop between the two terminals of the devices (along the CF) is higher, the larger the gap length, as shown in Fig. 3c. The resistance shown in Fig. 3d is the unperturbed value when voltages and currents are too low to produce any defect re-arrangement and it follows the trend reported in Fig. 3d as a function of the gap length.
In agreement with the picture of drift and diffusion of defects, we suppose that zero-average STN is the result of a dynamic equilibrium between the physical processes driving the resistance change itself 37,38 . According to literature models, drift and diffusion combine to modify the concentration of defects, n D , inside the material according to the following equation: where D is the diffusion constant, μ the defect mobility and F the electric field. We assume that in a condition of dynamic equilibrium the diffusive term (first term on the right-hand side of equation (3)) balances the drift term (second term on the right-hand side of equation (3)). It is well known that the diffusion describes an entropic mechanism that tends to smooth out the concentration gradients. The diffusion is mainly important at the CF/gap interface, where the concentration gradients are the maximum. At equilibrium, the drift must counteract the   www.nature.com/scientificreports www.nature.com/scientificreports/ smoothing action of diffusion. Therefore, we can model the effect of a pulse that produces the deviation of the device resistance from the average value as a change of the slope of the n D profile at the CF/gap interface around the average (unperturbed) value. In particular, diffusion tends to reduce the slope of the n D profile (reduce the gradient) and drift produces the opposite effect so that on average no net resistance change occurs. Figure 4a sketches the CF configuration in two representative cases of sharp and smooth profiles, which, in a pictorial view, can be associated to close packed and slightly dispersed defect arrangements. Orange balls represent electrically active defects. In case of a sharp profile of the CF edge, diffusion is predominant (Fig. 4a top), while in the case of smooth CF edge drift tends to re-establish a sharp CF edge (Fig. 4a bottom). A representative slight slope change is shown in Fig. 4b for a particular value of the gap length (=0.7 nm). The n D profile, zoomed around one of the CF/gap interfaces, is plotted for the unperturbed (straight line) and for the perturbed (dashed line) cases. The slope change can be tuned to fit the experimental STN amplitude as a function of the unperturbed resistance value. The fit is shown in Fig. 4c and the fitting parameters are reported in Table 1. The good agreement justifies the model assumptions and indicates that the STN can be described through a phenomenological model that grasps a dynamic equilibrium of drift and diffusion tendencies acting on the CF/gap interface. Additional data on different sample align to the same experimental trend shown in Fig. 4c, as can be appreciated in Supplementary Fig. S7.

Discussion
In the present work, we demonstrate that ionic random telegraphic fluctuations of the resistance of filamentary memristive devices can be intentionally stimulated by pulses with suitable voltage levels. When a device in the low resistance state is stimulated by trains of identical RESET pulses, its resistance initially drifts in an analogue fashion towards high values. After a certain number of pulses, the resistance variation saturates and only random zero-average noise can be appreciated. In the saturation region, therefore, we can distinguish random resistance variations from the switching of the average resistance. The stimulated noise is telegraphic in the sense that voltage pulses trigger jumps in the resistance values that are afterward maintained at low read voltage until a new stimulus is provided. Noise is the result of voltage pulse stimulation because it does not occur in absence of pulses. The long retention times of the perturbed resistance values (measured up to 1 s) are only compatible with an ionic phenomenon, because any electronic process of trapping and release would result in a retention time for the perturbed state of few ms 7,33,34 . Furthermore, we analyse the kinetics of the processes activating the STN by varying pulse width and pulse voltage over a wide range of values. The qualitative agreement between the kinetics of processes producing non-volatile switching and STN indicates that both phenomena are driven by the same processes of drift and diffusion of defects. In agreement with all the experimental observations, we describe the STN as a dynamic equilibrium of the drift and diffusion tendencies at the CF/gap interfaces. Indeed, as described by Marchewka et al. 37 , during the RESET process, drift and diffusion approach an equilibrium configuration. In the present paper, we harness the dynamic nature of such equilibrium situation to generate telegraphic noise. On the contrary, the SET process of filamentary devices is typically self-accelerated due to thermal runaway 17,39,40 . This fact prevents the establishment of an equilibrium, which could be an additional factor hindering the identification of STN during SET transitions. On the other hand, it could be expected that, if the SET process is self-limiting, i.e. does not require external current limitation preventing device breakdown as in the present case, then the low resistance state is reached as an equilibrium between drift and diffusion. Therefore, as a matter of principle, in such case, telegraph noise can be stimulated during SET, provided the saturation resistance is sufficiently high.
The description of the switching in terms of gap and CF is purely phenomenological and descriptive. Indeed, a CF should be considered a region where defects reach such a high density that the localized states that they introduce in the forbidden energy band gap tend to interact and delocalize promoting electronic conduction in the formerly insulating material. Conversely, the insulating gap within the CF forms as a depletion (or de-activation) 41 of defects in a region that was previously densely rich of them 3,42-45 . The proposed model intends to describe the noise within the same descriptive framework of CF and gap picture used to model the resistance switching and correctly describes the highlighted features of STN.
The present work, therefore, stands out from the previous literature in that random telegraphic variations are characterized in an un-explored voltage range and are ascribed to re-arrangement of defects constituting the CF responsible for the conduction in the oxide of memristive devices. As a matter of fact, usually conventional RTN is measured at low voltages where only electronic trapping and de-trapping processes can take place [3][4][5]33,[46][47][48][49][50][51][52] . Only few works try to extend the characterization of RTN to quasi-static voltage ranges in which significant ionic contribution is present, but the simultaneous occurrence of noise and deterministic switching events makes their discrimination complex 6,9 . Conventional electronic RTN has been the subject of wide research in the field of resistance switching devices, because it is an important source of noise affecting the read-out margin of RRAMs memory arrays 8 . We demonstrate here that ionic telegraphic variations appear also in the programming operation when it is operated through trains of identical pulses. Such programming scheme is specifically suitable for the expanding field of hardware neural networks in which a plastic response of memristive devices to repeated stimuli is desired. In particular, STN affects the ultimate resolution, i.e. number of distinguishable resistance levels, and the accuracy of a programmed resistance state of a filamentary memristive device. On the other hand, in most of the literature reports dealing with analogue plastic response of memristive devices, the switching dynamics is often associated with a large superimposed noise [10][11][12][13][14][15][16][17][18] , which is rarely discussed and which can be ascribed to STN.
On the other side, the intentional activation of random zero-average noise with tuneable amplitude as a function of pulse voltage and time width can open new opportunities for applications exploiting the inherent variability of memristive devices, such as random number generation, physical unclonable functions, stochastic learning systems and chaos and stochastic computing.