Quantized conductance coincides with state instability and excess noise in tantalum oxide memristors

Tantalum oxide memristors can switch continuously from a low-conductance semiconducting to a high-conductance metallic state. At the boundary between these two regimes are quantized conductance states, which indicate the formation of a point contact within the oxide characterized by multistable conductance fluctuations and enlarged electronic noise. Here, we observe diverse conductance-dependent noise spectra, including a transition from 1/f2 (activated transport) to 1/f (flicker noise) as a function of the frequency f, and a large peak in the noise amplitude at the conductance quantum GQ=2e2/h, in contrast to suppressed noise at the conductance quantum observed in other systems. We model the stochastic behaviour near the point contact regime using Molecular Dynamics–Langevin simulations and understand the observed frequency-dependent noise behaviour in terms of thermally activated atomic-scale fluctuations that make and break a quantum conductance channel. These results provide insights into switching mechanisms and guidance to device operating ranges for different applications.

T aO x -based metal-oxide-metal devices exhibit voltageinduced resistance switching that represents a physical realization of the memristor electronic circuit element model [1][2][3] . They are interesting for device and circuit applications [4][5][6][7][8][9][10][11][12] because of their high ON-to-OFF switching endurance 13,14 , sub-nanosecond switching times 15,16 , and picoJoule switching energies 17 . At the same time, spectromicroscopic studies 18 , transmission electron microscope analysis 19 and temperature-dependent transport measurements 20 have shown that the resistance switching mechanism in the TaO x system is significantly different from that in TiO 2 -based memristors [21][22][23][24] . In particular, the primary state variable that determines the electronic transport mechanism in TaO x memristors has been identified to be the oxygen content in a TaO x channel running through the film 19,20 . Other state variables, such as the internal temperature and the diameter of the conduction channel 25 , may also influence the resistance. At low oxygen content (B20% O dissolved in Ta), the channel displays metallic conductivity, whereas at higher oxygen content (larger than 50%, a suboxide, or reduced oxide) the channel can display a range of conduction behaviours, including activated semiconducting transport, barrier tunneling and hopping, all characteristic of a Fermi glass 26 , which is reminiscent of the behaviour described by the Mooij rule for disordered transition metals 27 .
An intermediate regime of electronic transport is characterized by quantized conduction, typically observed in mechanical break junctions [28][29][30] but also in metal-insulator-metal resistive switches [31][32][33][34] . Here we focus primarily on this regime, when the resistance of the memristor is in the 3 to 15 kO range (or in the range of a few conductance quanta G Q ¼ 2e 2 /h ¼ 77.5 mS ¼ (12.9 kO) À 1 ). We observe excess noise at conductance quanta for different samples and temperatures, several orders of magnitude above the baseline. The origin of excess noise is explained in terms of atomic instabilities at the contact. These results provide insight into switching mechanisms in memristors, as well as define regions for device operation for different applications.

Current voltage characteristics near quantum point contacts.
Our devices were sputter-deposited TaO x memristors with a bottom (grounded) Pt electrode and a top Ta electrode. We measured current controlled quasi-static current-voltage (I-V) sweeps, conductance as a function of time for constant applied voltage, and frequency-dependent noise spectra (see details in Methods, Supplementary Fig. 1 and Supplementary Note 1). The I-V characteristics in Fig. 1a,b for SET and RESET operations display conspicuous telegraph-type dynamical fluctuations when the conductance is lower or equal to G Q (see also Supplementary Note 2). Thus, the device states near G Q are highly unstable, with the conductance hopping back and forth among several different and apparently discrete states as the current is ramped. The dynamical and stochastic nature of switching in this regime is shown in Fig. 1c, which displays the conductance versus time of a device held at a constant voltage of 50 mV. The conductance exhibited jumps at apparently random time intervals to more conductive states at essentially integer multiples of G Q .
In Fig. 2 we illustrate how these fluctuations affect resistance switching by cycling a device between two different states over 100,000 times. The average conductance oG4 showed a significantly enhanced variance when we attempted to reset the device to a nominal value of G Q .
Noise spectra. Figure 3 shows that the current noise spectral density in a device can be a few orders of magnitude larger than the background values for conductance values close to small integer numbers times G Q . The detailed conductance and frequency dependence of the electrical noise in a TaO x memristor is exhibited in Fig. 3. The upper panel shows the noise normalized by the square of the current at 1 kHz for room temperature and 174 K. S is the noise power spectral density in units of WHz À 1 , while S/I 2 ¼ S R gives the resistance noise spectral density in units of OHz À 1 . The lower panel shows a 2D chart of the noise amplitude plotted as a function of device conductance and frequency for room temperature data. In both plots, there is a strong peak in the noise at G Q , which rises three orders of magnitude above the baseline, and several minor peaks that appear primarily at integer values of G Q . These general features were observed for similarly prepared samples, but the peak intensities and the presence or absence of a peak at particular integer multiples of G Q (nG Q ) varied from sample to sample (see Supplementary Fig. 2). Figure 4 shows the measured frequency-dependent noise spectra over a wide range of conductance states. The frequency (f) dependence of the noise was 1/f for the high-conductance states and 1/f 2 for the low-conductance states. However, in the quantized conductance regime, the noise amplitude was significantly above the background trend for G ¼ nG Q (n ¼ 1 and 2). At G ¼ G Q , the noise displayed a 1/f 2 dependence at high frequencies but flattened out at lower frequencies.
There is a vast literature on quantized conductance in Quantum Point Contacts (QPC) in mechanical break junctions (see, for example, refs 28-30 and references therein) that show similar behaviour to Fig. 1c  experiment for noise in QPCs was performed in 2D electron gases in GaAs/AlGaAs heterojunctions 39 . For this pristine system, at integer numbers of conductance quanta, noise was actually suppressed. In a later study on clean Au mechanical break junctions 40 , a similar behaviour was observed. More recently, shot noise was evaluated in mechanical break junctions with molecules in the channel, and noise was also suppressed 41 . These findings contrast with the enhanced noise and its temperature independence reported here. We attribute the enlarged noise to thermally activated atom motion in the conductance channel of the memristive system. Resistive switches are excellent controlled environments to investigate QPCs and the corresponding noise behaviour, since the lifetime of a particular resistance state can be sufficiently long for a detailed transport analysis. The conductance state instability and enlarged noise at G ¼ nG Q values can be explained in terms of a thermally fluctuating point contact in the oxide film of the memristor. In Fig. 4, the noise frequency dependence at nG Q exhibits a Lorentzian behaviour L f ; t ð Þ / t= 1 þ ot ð Þ 2 Â Ã , which is a signature of a single fluctuator, that is, potentially a single-atom jumping inside the point contact with a particular frequency cut-off (typically several hundred Hz, refs 42,43). This fluctuator alternates among different electronic eigenstates with essentially the same conductance, since the noise peaks are relatively narrow and nearly centred at values of nG Q . Qualitatively, when a conductance channel first forms in the oxide that just bridges the electrodes of the memristor, a hot spot forms at the narrowest part of the channel, which can be an atomic-scale point contact through which most of the current in the device flows. This produces high local Joule heating, which in turn causes atomic fluctuations and severe electronic noise because of the disruption of the conductance channel. The noise peaks at nG Q with n ¼ 2, 3 and so on. may be caused by multiple point contacts in parallel or a single fluctuating contact that supports degenerate electron eigenstates 29 .
The existence of a peak in the noise at a discrete conductance can be explained by a simple local bond model of a random network of resistances r m ¼ {r,N} (ref. 44, and see Supplementary Note 3 and Supplementary Figs 3 and 4). When the concentration of the conducting bonds decreases to the critical value for disconnecting a conducting channel between the electrodes, the electronic noise diverges because the fluctuations of the conductance are determined by a small number of bonds at the point contact 45,46 , In a physical system, this divergence will be smoothed out by tunneling through the gap in the channel 46,47 and by shunting around the gap through the surrounding matrix material (for example, the nearly stoichiometric oxide).
If there are several parallel conducting paths through the device, then the fluctuations of local conductances are uncorrelated and their contributions to the total conductance are averaged. We do not know exactly the microstructure of the growing embryonic conducting channels, but our observations are quite general to any geometry that includes point contacts. It is frequently assumed that they make a dendritic (or a stalagmite/stalactite) structure. In the case of the formation of a few parallel QPCs, the amount of local heating at each would obviously be lower than the single QPC case and may lead to a series of weaker noise peaks, since the amount of local heating in an ideal case for constant current would fall off with n 2 , n being the number of parallel channels.

Discussion
We compare our theoretical expectations with the observed behaviour of the normalized noise spectra S/I 2 for resistance states around G Q shown in Fig. 4. We observe three distinct regimes: a metallic, a semiconducting or insulating one and an intermediate regime characterized by conductances of nG Q . The metallic state with R ¼ 1.28 kO exhibits 1/f noise, while the semiconducting state shows 1/f 2 behaviour at high frequencies. In the 23 kO sample (close to 0.5 G Q ), we observe the crossover from 1/f to 1/f 2 reproduced by our Molecular Dynamics-Langevin (MD-L) model (see Supplementary Fig. 5), and more resistive samples follow the 1/f 2 behaviour. The samples in the quantized conductance regime around 12 kO exhibit the largest noise power compared with samples outside of this regime, exhibiting a Lorentzian distribution with oG4 ¼ G Q . For oG4 ¼ 2G Q , (R ¼ 6.4 kO), we observe a significantly lower power but also a Lorentzian behaviour. These results are in accord with discrete atomic fluctuators at the point contact, which is the predominant source of electronic noise level for the entire device.
Noise measurements can be a very sensitive probe of the internal dynamics of atomic-scale fluctuations in memristors and related devices. The observed multistable current-voltage characteristics and severe electronic noise in TaO x memristors for quantized conductance states can be explained by a simple model of atomic thermal fluctuations that disrupt electronic eigenstates in a point contact of a conducting channel, and should be a general phenomenon in resistance switching based on atomic or ionic migration. Thus, although the possibility of having discrete conductance states in a device appears attractive, the inherent instability of the states can present a challenge for non-volatile memory applications 9,48 near point-contact regime. In previous work we showed that one can operate in quiet windows in between conductance quanta by controlling the write current 20 or using a feedback write circuit 49 . Nevertheless, there are applications where noise can be used as a resource. One example is stochastic resonance phenomena in signal processing, wherein the addition of white or 1/f noise can enhance the response of a nonlinear system to subthreshold signals [50][51][52] . This phenomenon suggests an important role for noise in information processing. In the brain, stochasticity in synaptic inputs can help in cognitive processes such as decision making and learning 53,54 . Another application is the use of noise for the realization of physical sources of random number generators 55 . Traditionally embodied in optical schemes (see, for example: http://www.idquantique.com) to harness quantum properties of light as the phenomenon generating randomness, small footprint memristor devices can present a competitive advantage in integrated circuits. Correlation tests are needed to further demonstrate the quality of the random data for cryptography and other applications such as stochastic computing 44 . These examples illustrate that beyond nonvolatility, there are novel applications for memristors when it comes to exploring microscopic phenomena governing their behaviour, with noise as an asset.
Methods TaO x devices. The TaO x films of the devices were sputtered from a tantalum oxide target (nominal composition to be Ta 2 O 5 ) with an Ar gas pressure of about 3 mtorr. The device substrate was a commercial 200 nm SiO 2 /Si(001) wafer. The disc device stack consisted of (from bottom to up) 1 nm Ti blanket adhesion layer, 100-400 nm Pt blanket bottom electrode, 10 to 18 nm TaO x blanket layer, and 100-400 nm Ta disc (50 to 200 mm diameter) top electrodes. Metallic Pt and Ti layers were deposited by electron-beam evaporation at ambient temperature and the metallic Ta contacts were DC sputtered at ambient temperature through shadow masks.
Electrical measurements. Quasi-static sweeps. Current-voltage (I-V) curves and resistance measurements were performed in a standard four-point probe configuration using an Agilent B1500A parameter analyzer equipped with 4 source measure units (SMUs). The measured contact resistance of the top and bottom electrodes was less than 60 Ohm. A linear sweep mode with constant V or I ramp rates was used for V-driven and I-driven I-V curves, respectively. Fixed gains (ranges) were used for V (I) readings to avoid any disturbance due to gain (range) adjustment. For the setup with short integration time and autogain, the measurement interval time varied between a few and tens of milliseconds. The I ramp rate set by the sweep range, the SMU integration time, the SMU delay time, and the number of data points was on the order of 10 À 2 to 10 À 5 A s À 1 . The V ramp rate used was 0.1-10 V s À 1 .