Resistive switching in metallic Ag₂S memristors due to a local overheating induced phase transition

Abstract

Resistive switchings in nanometer-scale metallic junctions formed between an inert metallic tip and an Ag film covered by a thin Ag₂S layer are investigated as a function of temperature at different biasing conditions. The observed switching threshold voltages along with the ON and OFF state resistances are quantitatively understood by taking the local overheating of the junction volume and the resulting structural phase transition of the Ag₂S matrix into account. Our results demonstrate that the essential characteristics of the resistive switching in Ag₂S based nanojunctions can be routinely optimized by suitable sample preparation and biasing schemes.

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Self-assembled nanostructures with tunable electrical properties have been intensively studied as potential candidates to replace the present day CMOS devices. Conducting nanofilaments formed by reversible solid state electrochemical reactions between metallic electrodes have been demonstrated to be suitable for logical and non-volatile resistance switching random access memory (ReRAM)^1–15 operations at dimensions which are inaccessible in present day's silicon based technology.¹⁶ The resistance of such a two-terminal device, called memristor,¹⁷ is modified at higher bias voltages (V > V_th), while readout is performed at V ≪ V_th without altering the stored information.

Oxidation, activated ionic transport and reduction were identified as the leading mechanisms of nanofilament formation and rupture in solid state electrolytes^{4,7,15,18–35} giving rise to resistive changes in the tunneling regime between insulating R_OFF ≫ 10 MΩ and R_ON ≈ h/2e² = 12.9 kΩ resistance values, the latter is often considered as the hallmark of atomic sized contacts. Such devices, however, could only be operated at reduced switching speeds due to their fundamental RC limitations. Intensive studies of the metallic regime with electrical conduction taking place via 10–15 nm wide filaments were started recently in metal oxide based memristive systems where a Joule heating assisted activated ion and/or oxygen vacancy migration have been found to play a key role in the observed resistive switchings.^36–38

Our previous experiments^39,40 based on the ionic conductor Ag₂S have demonstrated reproducible, nanosecond timescale resistive switchings and robustness against low voltage read-out operations at room temperature exhibiting R_OFF/R_ON ratios of 2–10. Resistive switching is attributed to metallic conduction channels with changing diameter in the range of 2–5 nm while they are exposed to current densities as high as 10¹⁰ A cm⁻².

Here we report on extensive measurements and a thorough statistical analysis performed in order to explore the effect of the environment's temperature, the series resistance of the biasing circuit and the amplitude of the biasing signal on the technologically most relevant parameters, the ON and OFF switching threshold voltages, R_ON and R_OFF. We discuss the role of a highly non-equilibrium mechanism characteristic to the metallic regime. We demonstrate that our data are in quantitative agreement with a model taking pronounced power dissipation released at the nanojunction's length scale into account, giving rise to a local overheating of the junction up to the superionic phase transition of the Ag₂S layer even at cryogenic ambient temperatures. This accelerates the field driven, thermally activated ionic migration processes as well as electrode reactions. The observed resistive switchings are attributed to an enhanced filament reconfiguration accompanying the structural phase transition of the Ag₂S matrix.

During sample fabrication an 80 nm thick Ag layer was deposited onto a Si substrate followed by a 5-minute long sulfurisation resulting in a 30 nm thick stoichiometric Ag₂S layer on top of the Ag electrode.⁴¹ Nanometer-scale junctions were created between the Ag₂S surface and a mechanically sharpened PtIr tip in STM geometry. The V_drive triangular voltage output of a low output impedance data acquisition card was acting on the memristive junction and on a variable series resistor R_S as shown in the inset of Fig. 1(b). The device's current was monitored by a current amplifier operating at a typical gain of 10⁴. The V_bias voltage drop on the junction was determined numerically as V_bias = V_drive − I·R_S where the 50 Ω input impedance of the current amplifier was also accounted for in R_S. By convention, a positive bias corresponds to a positive voltage applied to the Ag layer with respect to the PtIr electrode. The measurements were performed in the variable temperature insert of a standard ⁴He cryostat between 4.2 K and room temperature. Approximately 10³ stable current–voltage (I–V) traces were acquired at each temperature at various R_S values ranging from 50 to 400 Ω.

Fig. 1 (a) Representative I–V traces of metallic PtIr–Ag₂S–Ag junctions recorded at various temperatures from T = 4.2 K to room temperature. The direction of the loops indicated by the arrows was identical for all curves. The traces are vertically shifted for clarity. (b) The OFF to ON switching threshold voltage V_th as a function of temperature. The dots and the error bars correspond to the average and standard deviation as evaluated for ∼10³ traces at each temperature, respectively. The inset shows the electrical circuit diagram of the biasing setup.

Typical I–V traces recorded within 300 ms (corresponding to a typical, low bias sweep rate of 5–10 V s⁻¹) at selected temperatures are illustrated in Fig. 1(a). At low bias voltages the initial OFF state resistance R_OFF is measured. Increasing V_bias to a temperature dependent switching threshold V_th = 180–320 mV a reconfiguration takes place resulting in a drop of the junction's resistance from R_OFF ≈ 10³ Ω to R_ON ≈ 10² Ω. Decreasing V_bias preserves R_ON until a negative threshold of 80–250 mV is reached where the onset of the transition from the ON to the initial OFF state occurs. The direction of the resulting hysteresis loops was identical for all curves and is indicated by the arrows in Fig. 1(a). The monotonic decrease in V_th with increasing temperature is emphasized by the statistical analysis of the ∼10⁴ stable I–V traces (approximately 2000 at each temperature) as shown in Fig. 1(b).

The influence of R_S on R_ON and R_OFF is shown in Fig. 2(a) and (b), respectively. The initial OFF state resistances are determined to a great extent by the controlled approach procedure of the STM tip and thus are not expected to exhibit any dependence on the biasing scheme. Consequently, a systematic R_S dependence of R_OFF was not observed, either, as demonstrated by the uniformly scattered red dots in Fig. 2(b). On the other hand, R_ON is seen to be proportional to R_S as indicated by the blue squares in Fig. 2(a). The latter behavior is attributed to the decreasing voltage drop on the junction as its resistance is decreasing toward R_S upon an OFF to ON transition resulting in a rate limitation for a further resistance change as discussed in ref. 40. This is also in agreement with the observed dependence of R_OFF and R_ON on the bias voltage amplitude as displayed in Fig. 2(c) for a representative single junction measured at room temperature, i.e. R_ON saturates at the scale of R_S as the driving amplitude is increased.

Fig. 2 Average and standard deviation of the ON (a) and OFF (b) state resistances R_ON and R_OFF acquired at various temperatures at 4.2 K < T < 300 K and at V⁰_drive = 0.6 V as a function of the series resistance R_S. Note the different scales on (a) and (b). (c) R_ON/R_S (blue squares) and R_OFF/R_S (red dots) as a function of the bias voltage amplitude measured in a single junction at T = 300 K and R_S = 50 Ω.

Next we investigate the self-heating effect characteristic to nanometer-scale metallic junctions exposed to finite bias voltages and show that our data acquired over a broad temperature range from T = 4.2 K to room temperature are consistent with a phase change model exploiting extensive power dissipation released in the junction's volume.

Power dissipation in point contacts is concentrated in a small volume around the contact center, leading to an elevated temperature in the junction area.⁴² Assuming that only electrons, subject to a diffusive motion, carry the heat and the L_i inelastic diffusion length exceeds the d contact diameter, i.e. l_e < d < L_i with l_e being the elastic mean free path, the junction's local temperature is given by

where T is the equilibrium temperature of the leads⁴³ as it was demonstrated experimentally in ferromagnetic contacts by the observation of a finite bias induced ferromagnetic phase transition.⁴⁴ A detailed derivation of eqn (1) along with the discussion of the so-called thermal (d > L_i) and ballistic (d < l_e) regimes are given in the Appendix. Inserting our typical junction diameters³⁹ of 2–5 nm, L_i ∼10 nm and the observed switching threshold of a few 100 mV into eqn (1) reveals that the T_J = T_C = 451 K bulk structural phase transition temperature of Ag₂S is easily reached even at an ambient temperature of T = 4.2 K. While the low temperature acanthite phase of Ag₂S is a semiconductor with very low conductivity, its high temperature argentite phase is a superionic conductor exhibiting a highly enhanced ionic mobility.⁴⁵ It has also been demonstrated^8,22 that the bulk T_C is considerably reduced in the presence of an applied electric field facilitating the initial growth of the metallic Ag nanofilaments in high OFF state resistance devices. The detailed microscopic picture of the electric field driven ionic motion is a subject of on-going molecular dynamical simulations. However, the above numerical estimation implies that a Joule heating assisted local superionic phase transition of the Ag₂S solid electrolyte may play a major role in the observed resistive switchings. Based on a recent study on the dynamics of nanoscale metallic inclusions in dielectrics³⁵ we anticipate a continuous, bulk filament growth in our Ag₂S system characterized by high oxidation and reduction rates as well as by a high ion mobility.¹ It is to be added that another relevant, highly non-equilibrium contributions, e.g. surface atom diffusion and electron wind forces, may also arise from the strong, inhomogeneous electric field and high current density, respectively.

Based on the above considerations we interpret the observed switching cycle along with the schematic illustrations displayed in Fig. 3 as follows. We point out that in a steady state the junction's temperature is determined by the P = I·V_bias power dissipation resulting in the emergence of isothermal regions along the I·V_bias = constant lines in the I–V plane of the device as indicated by the dashed lines in Fig. 3(b). Note that the steady state is classified here by a time scale which is shorter than that of the resistance change at the actual bias voltage⁴⁰ but long enough for the establishment of a stable temperature profile around the contact volume. We emphasize that the I·V_bias = constant equation of the isothermal lines directly follows from eqn (1) by assuming an orifice-shaped, diffusive nanojunction where G = 1/R = σd with σ being the electrical conductivity⁴⁶ and thus T_J² − T² is proportional to G·V_bias² = I·V_bias. We believe that a narrowest cross-section dominating the electronic transport properties necessarily exists along the metallic filament and its shape can be well approximated by an orifice-like profile. However, a nanowire-like, uniformly elongated constriction of the length l can also be taken into account here. In the latter case G = σd²/l applies, resulting in isothermal regions situated along the steeper, I·V_bias³ = constant lines. In reality, these two scenarios can be considered as limiting cases for the isothermal lines corresponding to arbitrary filament geometries. The exact form of these lines, however, does not play a role in our argumentation.

Fig. 3 (a) Schematic illustration of the recorded voltage signals. V_drive(t) (black) is the triangular voltage acting on R_S and the junction. The V_bias voltage drop is indicated by red (blue) in the OFF (ON) state. The four V_bias values characterizing the resistive switching cycle are labeled accordingly. (b) Schematic illustration of the switching cycle at the I–V plane starting from the OFF state at zero bias in the framework of the thermally assisted phase change model. Along the dashed hyperbolae the I·V_bias dissipated power along with the T_J local temperature of the junction are constant. Electrical conductance via the metallic filament in the OFF (c) and ON (e) states as well as during the OFF to ON (d) and ON to OFF (f) transitions are labeled on the I–V characteristics. (d) & (f) An enhanced filament growth/shrinkage is expected to take place when the local temperature of the biased junction reaches the T_C = 451 K critical temperature of the superionic phase transition of the Ag₂S matrix in either field direction.

Compared to the predictions of eqn (1) the actual junction temperature may be reduced due to the fact that the filament is embedded in a semiconducting Ag₂S matrix exhibiting finite heat conduction. However, based on its poor electrical conductivity we do not expect a considerable cooling power on the junction mediated by the acanthite phase Ag₂S in our non-planar arrangement.

While ionic motions do occur within the filament volume also at T_J < T_C leading to a slow reconfiguration of the conducting channel and thus to a small, systematic positive deviation from a linear I–V characteristics in the initial OFF state at a negatively charged inert electrode [Fig. 3(c)], the onset of an enhanced filament growth takes place when at V_bias = V₁ = V_th the local temperature around the contact reaches T_C and the surrounding Ag₂S matrix enters its superionic phase [Fig. 3(d)]. Due to the rapid filament growth taking place in the superionic argentite surrounding in the presence of strong electric fields during the OFF to ON switching, V_bias can be decreasing while V_drive is still increasing. This is not only the consequence of the increasing fraction of V_drive dropping on the series resistance at higher current levels but is also in agreement with eqn (1), i.e. at larger d the T_J ≥ T_C condition can be fulfilled at lower V_bias. At decreasing V_bias during the transition the characteristic time scale of the resistance change is exponentially diverging as reported in ref. 40. Consequently, a “final” ON state resistance is never strictly reached. However, any further resistance change would require either an unreasonably long time or unreasonably high V_drive, as will be discussed below. Therefore the OFF to ON transition at a given voltage sweep rate is naturally terminated as the junction's resistance becomes comparable to R_S, in agreement with the data shown in Fig. 2(a).

At oppositely charged electrodes the direction of ionic motion is reversed and acts toward the (partial) destruction of the conducting filament [Fig. 3(e)]. In this field direction the T_J = T_C condition is fulfilled at a lower, V₃ < V₁ bias in accordance with eqn (1) as d_ON > d_OFF. Independently from the relationship of V₁ and V₂, V₃ < V₂ also holds because both points on the linear I–V trace belong to the same ON state exhibiting an identical resistance while they are subjected to different driving voltages of V_drive(2) > |V_drive(3)|, as illustrated in Fig. 3(a) and (b). The V₄ > V₁ relationship relies on similar considerations. These arguments make it evident that the hysteresis loops are necessarily asymmetric in V_bias exhibiting a longer and smoother transition from the ON to the OFF state rather than vice versa, which is in full agreement with the representative traces shown in Fig. 1(a).

Another important consequence of the V₃ < V₁ condition is T_J(2) > T_J(1), i.e. the resistive switching is not an isothermal process. This implies that during a decreasing positive bias further resistance change is expected to take place in the ON state as long as T_J > T_C. The competing time scales of this inherent resistance change and that of the driving signal are demonstrated in Fig. 4. A single device was identically biased to its OFF to ON transition within 2.5 ms by a triangular signal with an amplitude of V⁰_drive = 0.5 V. This was followed by different voltage sweeps at rates of −200 V s⁻¹ (blue trace) and −2.5 V s⁻¹ (green trace). In the latter case the on-going resistance change at higher biases corresponding to T_J > T_C resulted in a further, well visible decrease in R_ON and a rounded upper corner of the hysteresis loop. In contrast, the fast sweep rate in the former experiment outperformed the reduced rate of the resistance change and yielded to an apparent linear I–V trace and a sharp corner. The actual shape of the transition along with the achievable R_OFF/R_ON ratios are thus also largely determined by the frequency of the driving signal.⁴⁰

Fig. 4 I–V traces of a single junction recorded at different voltage sweep rates after the OFF to ON transition. The inset illustrates the different driving schemes with −200 V s⁻¹ (blue) and −2.5 V s⁻¹ (green) sweep rates. V⁰_drive = 0.5 V, R_OFF = 380 Ω, R_ON = 75 Ω (blue) and 45 Ω (green), R_S = 50 Ω.

At V_bias = V₃ the Ag₂S matrix becomes superionic again, fueling an accelerated filament destruction [Fig. 3(f)]. However, in contrast to the one responsible for the OFF to ON switching, this process is inherently self-limiting as at a decreasing d the magnitude of V_bias necessary to keep the T_J = T_C condition satisfied is also increasing. Therefore, in order to achieve higher R_OFF values one has to supply an increased V_bias. The unambiguous identification of the terminating mechanism of the ON to OFF switching is not as straightforward as in the opposite case. However, the firm experimental finding that upon increasing the magnitude of the negative bias R_OFF always takes its initial value indicates that the Ag₂S matrix exposed to the strong electric field in the vicinity of the filament has a limited capability to incorporate the dissolved Ag⁺ ions and this saturation concentration is close to the stoichiometric value which is present at the beginning of the switching cycle. It is also to be noted that in the Ag rich environment the argentite phase can survive even at room temperature.²² Such residues may also contribute to the filament structure refining the simple picture of pure Ag channels depicted in Fig. 3(c)–(f).

The relevance of the above scenario is further supported by re-plotting the V₁(T) = V_th(T) data of Fig. 1(b) along with the V₃(T) ON to OFF switching threshold voltages against eqn (1) in a linearized fashion with T_J = T_C = 451 K, as shown in Fig. 5(a). While the thermal limit with L_i/d = 1 clearly remains unchallenged at the onset of the OFF to ON switching, even the low temperature enhancement in L_i, which presumably occurs due to suppressed phonon scattering, falls below an upper limit of L_i/d = 6. On the other hand, the ON to OFF switching takes place close to the thermal limit signalling that the relevant transport length scales, L_i, d and l_e fall close to each other within a few nanometers.

Fig. 5 (a) V₁² = V_th² (red dots) and V₃² (blue squares) as a function of T². The dots and the error bars correspond to the average and standard deviation as evaluated for ∼10³ traces at each temperature, respectively. The dashed lines are calculated by eqn (1) with T_J = T_C = 451 K and L_i/d = 6 (upper line) and 1 (lower line), the latter representing the thermal limit. (b) Logarithmic plot of V₁/V₃versus R_OFF/R_ON with a linear fit to the data.

Fig. 5(b) shows the relationship of the ON and OFF switching threshold voltages to the corresponding R_OFF and R_ON values. As pointed out in the discussion of eqn (1), the requirement that the onsets of both transitions must occur as T_J approaches the same T_C sets the conditions of (V₁/V₃)² = (R_OFF/R_ON)^α with α = 1 and α = 1/2 for the two limiting junction geometries of an orifice and a uniform nanowire, respectively. Arbitrary junction geometries are manifested in the intermediate values of α. The satisfactory linear fit to the corresponding logarithmic plot displaying data acquired at all investigated temperatures and R_S values reveals α = 0.61 ± 0.06 and thus provides a strong quantitative evidence to our picture.

In the intermediate regime situated on the verge of ballistic and diffusive conductance the d_OFF and d_ON average filament diameters are estimated based on the interpolating formula⁴⁷ based on the Boltzmann equation for arbitrary ratio of d/l_e

where Γ(l_e/d) is a numerically determined monotonic function with Γ(0) = 1 and Γ(∞) = 0.694. Note that the first term is exactly the Sharvin resistance by placing σ = l_e²n/ħk_F Drude conductivity into the formula, thus it is actually independent of l_e. The junction diameters are calculated by taking λ_F = 0.4 nm, being the Fermi wavelength in bulk Ag, into account. This approximation results in the temperature independent values of d_OFF = 1.8 ± 0.7 nm and d_ON = 4.8 ± 1.2 nm confirming the range of validity of eqn (1), i.e. the l_e ≤ d < L_i assumption with l_e = 1.8 nm being the elastic mean free path in the argentite phase of Ag₂S.⁴⁸ These values are also in agreement with our recent study on the conducting channel distribution carried out by low temperature point contact Andreev reflection spectroscopy.³⁹

In conclusion, we studied the resistive switching characteristics to metallic Ag₂S nanojunctions. We propose that the dominating driving force of the resistance change in the highly non-equilibrium metallic regime is self-heating assisted electric field driven ionic transport giving rise to an intensive expansion/shrinkage of the metallic conduction channel as the bias dependent local temperature of the nanojunction approaches the superionic phase transition temperature of the surrounding Ag₂S solid electrolyte. Based on the statistical analysis of ∼10⁴I–V traces acquired at different biasing conditions over a wide temperature range we found that the technologically most relevant parameters, the ON and OFF switching threshold voltages as well as R_ON and R_OFF are determined by the controllable means of sample preparation, the series resistance of the biasing circuit and the ambient temperature, in quantitative agreement with the proposed model. In spite of the self-assembling nature of filament formation/destruction, our results demonstrate the merits of engineering and large scale production of nanometer-scale Ag₂S based memory cells exhibiting uniform behavior at technologically optimal device parameters.

Appendix

In this appendix power dissipation and local overheating in nanometer-scale junctions are discussed. In biased point contacts the major part of the potential drops within a characteristic distance of d from the contact center thus the power dissipation is also concentrated in a small volume resulting in elevated junction temperatures under an applied voltage. Assuming that the heat is conducted just by the electrons, the equations of heat conduction can be solved for an arbitrary point contact geometry.⁴⁹ These calculations are valid in the so-called thermal regime, where the d contact diameter is larger than the L_i inelastic diffusive length. In this limit the T_J temperature of the contact center is determined by the bias voltage aswhere T is the temperature of the electrodes and is the Lorentz number. The power is dissipated inside the junction within a characteristic volume of ∼d³. As the resistance of a diffusive contact is R = 1/σd with σ being the electrical conductivity,⁴⁶ the dissipated power scales with the contact diameter as P = V_bias²σd. According to eqn (3) these two factors give a contact temperature which is independent of d.

In smaller nanojunctions with d < L_i the potential also drops within a distance of d from the contact center.⁴⁶ However, in this regime the electrons must travel a distance of L_i to dissipate the energy gained from the potential drop. Therefore the power P = V²σd is dissipated in a volume of ∼L_i³, regardless of the contact diameter. As the magnitude of power dissipation scales with d whereas its corresponding volume is constant, T_J is expected to decrease with decreasing contact size. These considerations can be quantified by the following calculation.

We consider an orifice-like contact and assume that the power dissipation takes place within a sphere with radius b around the contact whereas the power density is constant in this volume. The continuity equation for the thermal current density is given as

The equation for the heat conduction is . If the heat is conducted solely by the electrons, the Wiedemann–Franz relation between the κ heat conductivity and the σ electrical conductivity is applied as thus can be written as . Assuming that in the steady state the temperature depends only on the r distance from the contact center the continuity equation can be integrated as

with the boundary condition of T(r) → T at r → ∞.

In a voltage biased diffusive contact and R = 1/σd, so the temperature of the junction at r = 0 is

In the thermal regime the power is dissipated in a characteristic distance of d from the contact center, thus b = d should be substituted into eqn (7) in agreement, apart from a small deviation in the numerical coefficient, with eqn (3).

As this simple calculation successfully reproduces the known results in the thermal limit, it can also be employed for smaller contacts with d < L_i. In this case b = L_i is to be inserted into eqn (7) with the appropriate coefficient adopted from eqn (3) as

Eqn (8) shows that T_J is reduced in smaller contacts and the magnitude of this reduction is determined by d/L_i.

In even smaller contacts with d < l_e the electrons perform a ballistic motion and the resistance is determined by the Sharvin formula, R = 16l_e/3πd²σ thereby the temperature of the junction is written as

It must be noted that the validity of these calculations in the d < l_e regime is limited. As the electrons can only thermalize within a characteristic distance of L_i, the equations of heat conduction are not strictly applicable at shorter length scales. However, while the major part of the temperature increase takes place at a longer length-scale of r > L_i, the above formulae still provide a good estimation for the contact temperature within an error of ∼33%. These calculations are also overestimating T_J when not only the distant electrodes are cooled but also the whole contact surface, for instance, in measurements performed under ambient conditions.

Acknowledgements

This work was supported by the Hungarian Research Funds OTKA K105735, K112918 and by the European Union 7th Framework Programme (Grant No. 293797). The technical assistance of A. Magyarkuti is acknowledged.

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