Pulse length and amplitude dependent resistive switching mechanisms in Pt-Pr_0.67Ca_0.33MnO₃-Pt sandwich structures

Our devices consist of Pr_0.67Ca_0.33MnO₃ (PCMO) films sandwiched between Pt electrodes. The films are successively deposited by the ion beam sputtering technique on 10 × 10 mm MgO single crystal substrates. First, about 450 nm of Pt as the bottom electrode, followed by 300 nm of PCMO, are deposited at 1023 K under 1.4 · 10⁻⁴ mbar O₂ and 1.0 · 10⁻⁴ mbar Xe (sputter gas) background pressure. All PCMO films reveal the orthorhombic crystal structure with dominating [001] orientation perpendicular to the substrate, but with different degrees of twining and different volume fractions (<10%) of (112) misorientations.

In order to confine the top electrode, a photo resist lacquer (Allresist, U-4040) with small holes is used as an insulator on the extended PCMO film. The holes with diameters of a few μm are created by a photolithography technique and provide the top electrode contacts on the PCMO after the final Pt layer is deposited on top of the stack (see schematic in figure 1(b)). The top electrode Pt layer is deposited at room temperature without oxygen partial pressure. A low deposition temperature is necessary because deposition at high temperatures would lead to a decomposition of the organic lacquer which remains on the sample. The lacquer also provides an insulating interlayer for the supply lines leading to the top electrodes on the film. Five different devices were used for this contribution. The devices are fabricated using the same procedure and showed only insignificant deviations in the relevant properties. The different top electrode diameters lead to different overall resistance. Therefore abbreviations with the electrode diameter will be used: A, C (both 5 μm) and B, D, E (4 μm).

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In the following, the excitation pulse is characterized by the resistance R, the current density J, the voltage drop U and the pulse duration t_p. Scaling of the resistance with respect to device cross-section and thickness [3] implies a rather homogeneous current transport across the device. We therefore calculate J from the current and the full device cross section for comparison of differently sized samples. Nevertheless, the actual current-carrying cross section might be smaller i.e. J should be interpreted as an overall current density. Underlying device diameters are mentioned in the figure captions.

Figure 1(a) shows a typical U-J and R-J dependence, respectively. For small currents, the resistance is current independent. At higher current densities, a nonlinear regime evolves which is characteristic for polaronic transport. Nonvolatile resistive switching commonly occurs beyond J ≥ 10⁸ A m⁻². The excitation pulse can be either applied in voltage-controlled or current-controlled modes, respectively. Using a J-controlled pulse mode avoids a current runaway due to nonlinear R(U) behavior and thus does not require compliance limits. However, the occurrence of PS and NS switching is independent of the measurement mode. Therefore current and voltage amplitude are in general treated as pulse amplitude. The remanent resistance changes are probed after the excitation pulse at small probe pulses U_probe in the ohmic regime. The resistances R⁺ and R⁻ correspond to the probe resistance after positive and negative excitation pulses. The switching amplitude is defined as ΔR/R⁻ = (R⁺−R⁻)/R⁻.

Two different measurements setups are used for the pulsed measurement of J-U curves and remanent resistance changes: cross plane electric transport properties at voltage or current pulses of duration t_p = 1 ms (probe and excitation pulse) are obtained by using a Keithley 2430 current/voltage source. U_probe was fixed at 0.08 V and the delay time between subsequent probe and excitation pulses was about 200 ms. For electric excitation on time scales down to 100 ns, a pulse generator (Agilent 81101A) was used (figure 1(b)) in combination with a Keithley 2000 multimeter. The generator and multimeter were electrically separated from the device by a relay box leading to a longer delay time of about 500 ms between excitation and probe of the remanent resistance. The multimeter measured the remanent resistance at U_probe = 0.2 V for about 300 ms. The applied pulse trains used for both measurement setups are sketched in the insets of figures 2(a) and (b).

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For the experiments involving the pulse generator, a calibration procedure is required to gain actual voltage drop U. Since the pulse generator has an internal resistance of 50 Ohm, it is configured such that the sample resistance is equal to the value of the internal resistance. Consequently, when setting the generator e.g. to U_app = 1.5 V, a voltage of 2 U_app = 3 V will be applied to the series circuit of sample and internal resistance which leads to a sample voltage of 1.5 V for a 50 Ohm sample. Since the sample resistance is voltage dependent, we have used an oscilloscope to measure the actual voltage drop for various U_app (figure 1(c)). This calibration between U and U_app was also applied to another sample (see caption of figure 7(a)).

Figure 2 demonstrates the remarkable similarity of the influences of the pulse current density and the pulse duration on the nonvolatile resistance changes. The actual development of the two resistance states R⁺ and R⁻after the application of a positive/negative pulse for a fixed t_p = 1 ms and increasing J is plotted in figure 2(a). Figure 2(c) shows the corresponding switching amplitude ΔR/R⁻ = (R⁺−R⁻)/R⁻. At small current densities no difference between R⁺ and R⁻ is visible. A positive switching amplitude (R⁺ > R⁻) evolves above J ≈ 10⁹ A m⁻², i.e. the HR state corresponds to R⁺. The amplitude increases up to a maximum at about J ≈ 2.5 · 10⁹ A m⁻² and then decreases with increasing current density. At ≈ 3.5 · 10⁹ A m⁻² the switching amplitude vanishes, i.e. the resistance states R⁺ and R⁻ are equal. Larger currents now exhibit R⁺ < R⁻ and therefore negative switching amplitude (the HR state corresponds to R⁻) evolves, where the switching polarity is reversed. Note the good reversibility of the process: while successively reducing the pulse current density from its maximum value, the R⁺ and R⁻ branches are very close to the R⁺ and R⁻ branches for increasing J. Furthermore, at low currents, the initial resistance value of the cycle is approached, where R⁺ ≈ R⁻. Only the first cycling of the device (starting from virgin resistance state) causes some irreversible changes in R⁺ and R⁻ (supplement S6). The initial resistance is typically about 3–4 times higher than the virgin resistance ([3], supplement S3).

It is worthwhile to note that, after a first initial sequence, further sequences with the same maximum pulse length or maximum current (voltage) are well reproducible. The switching amplitude ΔR is voltage/current dependent, but nearly independent of the actual R⁺ and R⁻ (see supplement S6), most likely due to the strong nonlinearity of device resistance in the switching regime. Therefore the amplitude between R⁺ and R⁻ is only current dependent i.e. R⁻ decreases and R⁺ increases when J is decreased after J_max. The final resistance after each cycle is the sum of all previous switching processes and is, remarkably, almost constant for switching sequences which are symmetric with respect to the increase and decrease of applied pulse amplitude/length.

The qualitative same R⁺/R⁻ characteristics can be observed for alternating pulses with variable pulse length at fixed stimulation amplitude U_app (figure 2(b)). For very small pulse lengths t_p ≤ 10⁻⁶ s no switching is observed. With increasing t_p positive switching also occurs (ΔR/R⁻ (t_p) > 0, see figure 2(d)) until a maximum is reached. For longer pulses, ΔR/R⁻ (t_p) is reduced again and shows a sign reversal at about t_p = 5 · 10⁻² s.

Figure 3(a) gives an overview of the combined effect of pulse length and pulse amplitude (here expressed by U_app) on ΔR/R⁻. Since the negative switching has much larger switching amplitudes, the modulus was taken. Note that the curve at U_app = 1.5 V (green line) is the same as in figure 2(d). The polarity change from PS to NS is now represented by a dip in |ΔR/R⁻| at a characteristic pulse length t_p which shifts to longer pulse lengths for decreasing U_app. For U_app = 1.0 V, the PS to NS transition is shifted beyond the studied maximal pulse length of t_p = 1 s. Therefore, no negative switching is observed. Also the onset of the positive switching shifts to increased pulse lengths with decreasing U_app. The shortest visible time scale for switching of t_p ≈ 10⁻⁶ s is given by the time constant of the RC circuit of the measurement setup and is thus not an intrinsic time scale of the device (see also figure 4).

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Figure 3(b) shows the temperature dependence of |ΔR/R⁻|. A strong increase in the PS amplitude with increasing resistance is observed at lower temperatures, where the thermally activated polaronic transport (R(T) ∝ T exp(E/kT)) in PCMO leads to a strong increase in the resistance (see supplement S1). Remarkably, the switching amplitude ΔR/R⁻ follows an almost universal curve at positive switching, i.e. the resistance difference ΔR(t_p) = R⁺−R⁻ scales with initial R(T). In contrast to the PS regime, both the onset as well as the magnitude of the NS switching amplitude is strongly temperature dependent: the transition to negative switching is successively shifted to longer pulse lengths and |ΔR/R⁻| drops with decreasing temperature. Below T = 240 K, the transition is even shifted to t_p > 1 s outside the accessible measurement range of our study.

In addition to the study of the remanent resistance change after the application of the stimulation pulse, we have conducted experiments to analyze the resistance evolution during a stimulation pulse. For such experiments based on the pulse generator, the nonlinear resistance behavior of the device in combination with the internal 50 Ohm resistance (in series) of the pulse generator has to be taken into account. It influences the time evolution of the applied voltage which is actually applied to a sample as a function of time and U_app: the temporal evolution of the resistance, current density and voltage during a 1 ms pulse applied by the pulse generator are demonstrated in figure 4 for a typical sample. Figure 4(a) shows the time evolution of the dynamic device resistance during an excitation pulse of 1 ms with different values of U_app. For one of the pulses (U_app = +2 V), the corresponding voltage and current densities are plotted in figure 4(b).

At the beginning of the pulse (t < 10⁻⁸ s), the sample exhibits a greater resistance (ohmic resistance of this device is about 600 Ω) than the internal 50 Ohm resistance of the pulse generator and experiences almost the full voltage amplitude of 2 × U_app (not visible in figure 4(b) due to limited measurement range). Since the device resistance is high, the current is low. In the regime 10⁻⁸ s ≤ t ≤ 10⁻⁷ s the device resistance has dropped well below the ohmic regime, but no voltage dependence is visible for the different applied U_app. Between 10⁻⁷ s ≤ t ≤ 10⁻⁶ s this dependence becomes visible in our measurement setup. However, the resistance drop of the sample is overlaid by more or less reproducible oscillations on a time scale of the RC circuit (extended contacts on the device, bonds, cable, etc) for all times below 10⁻⁶ s. For t > 10⁻⁶ s, the device resistance is reduced to a value comparable to the internal 50 Ohm, leading to a decrease in the actual U with increasing current density J. A well-measurable dynamic resistance is therefore established above 10⁻⁶ s. This time scale is independent of the applied U_app. Note that this transient regime for t < 10⁻⁶ s might suppress initial resistive switching at shorter pulse lengths (see also figure 2).

The J-U characteristic of devices commonly reveals a small asymmetry with respect to the current direction. It is therefore worthwhile to note that the slightly higher resistance at positive excitation pulses (figure 4(a)) is not related to the formation of the HRS in the PS regime. In general, small remanent resistance changes have only negligible influence on the dynamic, nonlinear polaron resistance, especially at higher voltages. The measureable difference between the HRS and LRS decreases with increasing U_probe, as the dynamic resistance curves for both states approach each other at higher voltages (see [3]). Therefore small switching effects occurring in the PS regime are invisible in the dynamic resistance. However, the slight increase of the resistance during the pulse with –U_app = 2.0 (red curve) indicates a crossover to negative switching (for further support, see supplemental material S2). This implies that during a 1 ms pulse the initial positive switching is followed by the negative switching in time. The observed switching amplitude seems to be a result of the overlay of an initial fast but saturating positive switching and a slower but more pronounced negative switching process. As a consequence it is important to note that the switching amplitude in figure 2(d) can be qualitatively interpreted as the actual development of the remanent resistance during a single, long pulse since the switching evolution during long pulse lengths contains the characteristics of shorter ones.

The time and space dependent temperature evolution due to Joule heating can dramatically influence the underlying mechanisms of the resistance change. The influence is therefore studied numerically [20] by finite element modeling using COMSOL Multiphysics^®. We partition the total device into different areas, where the partial differential equation of heat conduction is solved in the time domain. These areas are (with thickness): the MgO substrate (1 mm), the bottom (600 nm) and top electrode (500 nm) made of Pt, the insulating layer (PMMA), the passive area of the PCMO layer below the PMMA resin and the electrically contacted area of the PCMO layer (300 nm) between the top and the bottom electrode. In good approximation to the experimental geometry, the device is axisymmetric (see figure 5(a)). The heat equation is therefore treated in cylinder coordinates:

$$\begin{eqnarray*}&&\frac{\partial T}{\partial {{r}^{2}}}+\frac{1}{r}\frac{\partial T}{\partial r}+\frac{1}{{{r}^{2}}}\frac{{{\partial }^{2}}T}{\partial {{\theta }^{2}}}+\frac{{{\partial }^{2}}T}{\partial {{z}^{2}}}-\frac{1}{D}\frac{\partial T}{\partial t}+\frac{p}{\kappa \;}=0\end{eqnarray*}$$

including the thermal diffusivity D = $\kappa$/(ρ c_p) and the Joule heating volume power density p = R(T) J² A/d. The values used for the local thermal conductivity $\;\kappa ,$ specific heat c_p and densitiy ρ are given in table 1. These three values are given at room temperature and approximated as temperature independent.

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Due to the dominating resistance contribution, the Joule heating (p-term) is exclusively induced in the electrically contacted PCMO layer between the top and the bottom electrodes, which represents the diameter (3 μm) of the device. The heat generated depends on the temperature dependent electric resistance R(T). This results in a drop in the Joule heating power during temperature increase. We apply a fit function R(T) = 77.86/T^0.5 exp(1272.26/T) Ω to the experimentally measured data points (between 140 and 300 K) and extrapolate R(T) up to temperature T = 2000 K, where no experimental values are accessible (see figure S1(a) in supplement).

All outer boundaries are adiabatic (the top electrode on the surrounding PMMA is extended to 10 mm) except the substrate mount. Here, we consider an ideal thermal contact between the 1 mm thick substrate and the mount so that the ambient temperature is given as an isothermal boundary condition. Since PCMO is a poor thermal conductor and exhibits a relatively high electric resistance, a tremendous temperature increase is expected in short intervals. As a consequence both the time discretization and the spatial mesh discretization were chosen to be small. We apply time steps down to t_s = 1 · 10⁻¹³ s and a finite element mesh with an element size of d_m ≈ 2.5 · 10⁻⁸ m in the domain of the PCMO layer. The additional resistance drop due to the applied voltage is disregarded in this work. Thus we consider exclusively R(T) instead of R(T, U).

Figures 5(b)–(f) show the spatial distribution of the temperature evolution at J = 10¹⁰ A m⁻² at different time steps. Clearly, the temperature increase at the bottom electrode is much smaller compared to the top electrode and also compared to the film center. In the stationary state close to a 1 s pulse duration, the highest temperatures evolve at the PCMO interface with the top electrode. This effect might become even more pronounced if a series resistance of the PCMO-top electrode interface was taken into account.

The simulated temperatures can be compared with the experimental observation of the melting of the top electrode material by performing switching experiments in a SEM (see figure S1(b) in supplementary material). We conclude from this calibration that the absolute temperatures are somewhat overestimated in the finite element calculations. This is due to disregarding the resistance drop by the applied voltage and the slight decrease in κ(T), reducing the actual generated power and increasing the heat conduction for a given current density. We observe that the timescale of about 10⁻⁶ s for the cross over between the rapid temperature increase and a moderate further increase due to the warming up of the bottom electrode and the substrate does not strongly depend on the current density (figure 5(g)). In the following, we define the regime of slowly increasing temperature with T ≈ (¾-1) T_end as the quasi-stationary state. We therefore expect that this timescale is not significantly affected by the approximations used. It is important to note that Joule heating at pulse amplitudes and time scales relevant for NS lead to temperatures up to a few hundred K in the device.

Another important result of the simulations is that the temperature increase in the stationary state is proportional to the heating power density P per unit area, i.e. ΔT = q U J = q P. This result is used for the simulations of the microscopic processes involved in PS and NS. We obtain q = 1.0 · 10⁻⁷ Km² W⁻¹ from experiments (see supplement figure S1(b)) which is below the q_sim ≈ 1.6 · 10⁻⁷ Km² W⁻¹ obtained from the simulations after approaching the stationary state. It is plausible to assume a general calibration T ≈ 0.6 T_sim also qualitatively valid during the temporal temperature evolution as can be seen in figure 5(g).

In order to identify the interfaces which are involved in the resistance change, a third Pt-electrode ('center') was prepared as a reference close to the top and bottom electrodes by conventional lithography and focussed ion beam etching (figures 6(a) and (b)). The freestanding top electrode was contacted by a conductive micro-tip in an advanced SEM system, as described in more detail in [3]. After applying the excitation pulse between top and bottom electrodes, the remanent resistances were subsequently measured in two-point geometry (Keithley 2430, 1 ms pulses) for all three contact combinations by means of a relay box: between top and center, between bottom and center and between bottom and top electrodes (figure 6(a), pulse setup) with delay times of about 500 ms between all pulses.

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In order to validate the self-consistency of the three measured resistance values, e.g. that center and top includes the contributions of the involved interfaces and a PCMO bulk part, we introduce a series equivalent circuit (figure 6(c)). The two different interfaces, together with an associated bulk contribution, are represented by three different resistance contributions (R_TOP, R_CENTER and R_BULK) which can be calculated from the experimentally measured resistances (table 2). The model was confirmed by shunting two of the three electrodes and comparing experimental and calculated resistances (columns 6 and 7). The contribution of the top electrode is much higher than that of the bottom electrode. This implies a voltage drop mainly at the top electrode.

Figure 6(d) shows the remanent resistances of all electrode combinations during a voltage cycle. The R_top-bottom exhibits the typical NS mode (HRS for negative voltages). A similar resistance change (note the comparable amplitude) is observed only at R_top-center. R_{bottom-center} remains essentially unchanged. Therefore the resistance change in NS takes place at the top electrode interface. It is out of the frame of this contribution to discuss the rather small resistance change in PS. However, we found strong evidence that PS also takes place at the top electrode (see supplement S7). This seems plausible, since the electric field, as well as Joule heating, is much more prominent at the top electrode than at the bottom electrode.

Field enhanced migration models have been frequently applied to simulate remanent switching effects [e.g. 28, 29]. Here we use a similar approach in order to show that the main features of the contour plot in figure 7(a) can be qualitatively simulated (figure 7(b)), if only a single interface and the migration of one species is considered. The model is based on the following assumptions:

• (i)  
  The device consists of three volumes: Pt electrode, PCMO interface region and PCMO bulk.
• (ii)  
  The interfacial region of PCMO to the top electrode represents a switchable 'blocking layer' [2], i.e. a serial resistance model can be applied. The resistance change during switching only takes place in the PCMO interface area due to changes of the mean vacancy concentration C_int and the overall device resistance depends linearly on C_int.
• (iii)  
  Applying a negative bias to the top electrode causes a vacancy flux J_NS = C_bulk · v_bulk from the bulk into the interface volume, which represents the NS mechanism. At the same time, the PS mechanism is represented by a vacancy flux J_PS = C_int · v_int out of the interface volume, due to the reduction of PtO_x to Pt. ΔJ = J_PS−J_NS is the net vacancy flux to the interface volume and v_int and v_bulk are the vacancy drift velocities for the interface and bulk regions, respectively.
• (iv)  
  The top region of the PCMO film reveals most probably an oxygen deficit due to ion sputter deposition. In our model we only change the distribution and not the number of vacancies, i.e. no thermal vacancy formation during the pulses is considered. The latter might give rise to nonreversible resistance changes in the real device.
• (v)  
  We restrict the simulation to timescales above 10⁻⁶ s, where the dynamic resistance during pulse excitation is almost constant (figure 4). In accordance with the thermal simulation the constant resistance implies stationary (Joule-heating induced) temperature distribution in the device.
• (vi)  
  The concentration change depends linearly on the pulse duration (stationary flux: ΔC_int = ΔJ · t_p). Since the metal TE interface region can only absorb limited amounts of oxygen, the experimentally observed (figure 3(a)) maximum resistance change of 10% was used as a saturation limit for PS.

For the drift velocities of PS and NS one can use a rather general model:

$$\begin{eqnarray}&&v\left( {{U}_{{\rm hop}}}\left( U \right),T\left( U \right) \right)={{V}_{0}}\cdot {{e}^{-\frac{E}{k\cdot T}}}\cdot \;{\rm sinh} \left( \frac{Q\cdot {{U}_{{\rm hop}}}}{2\cdot k\cdot T} \right)\end{eqnarray} \tag{ 1 }$$

where E is the activation barrier of the rate-limiting step and Q is the vacancy charge, i.e. Q = 2 · e. U_hop is the voltage drop between adjacent sites which can be calculated from U_hop = F · d_hop, where F is the local electric field and d_hop is the hopping distance for interface and bulk, respectively.

The resistance change is given by

$$\begin{eqnarray}\begin{array}{lllllllllllllll} \Delta R\left( {{C}_{\operatorname{int}}} \right) & = & \frac{\partial R}{\partial {{C}_{\operatorname{int}}}}\cdot \Delta {{C}_{\operatorname{int}}} \\ {} & = & \frac{\partial R}{\partial {{C}_{\operatorname{int}}}}\cdot \left( {{C}_{\operatorname{int}}}\cdot {{v}_{\operatorname{int}}}-{{C}_{{\rm bulk}}}\cdot {{v}_{{\rm bulk}}} \right)\cdot {{t}_{{\rm p}}} \\ \end{array}\;\end{eqnarray} \tag{ 2 }$$

For small variations in the vacancy concentrations we assume that the observed switching regime is determined by the difference between the drift velocities in the different regions. Combining the different pre-factors results in:

$$\begin{eqnarray}\begin{array}{lllllllllllllll} \frac{\Delta R}{R} & = & {{\left( \frac{\Delta R}{R} \right)}_{{\rm PS}}}-{{\left( \frac{\Delta R}{R} \right)}_{{\rm NS}}} \\ {} & \approx & \left( {{Y}_{{\rm PS}}}\cdot {{e}^{-\frac{{{E}_{{\rm PS}}}}{k\cdot T}}}\cdot {\rm sinh} \left( \frac{Q\cdot {{U}_{{\rm hop}-{\rm PS}}}}{2\cdot k\cdot T} \right) \right. \\ {} & {} & \left. -{{Y}_{{\rm NS}}}\cdot {{e}^{-\frac{{{E}_{{\rm NS}}}}{k\cdot T}}}\cdot \;{\rm sinh} \left( \frac{Q\cdot {{U}_{{\rm hop}-{\rm NS}}}}{2\cdot k\cdot T} \right) \right)\cdot {{t}_{{\rm P}}} \\ \end{array}\;\end{eqnarray} \tag{ 3 }$$

In the saturation limit of PS the first contribution has to be replaced by the maximum PS switching ratio of about 0.1 (figure 3(a)). In order to calculate the unknown pre-factors (Y_PS and Y_NS) and the activation energies (E_PS and E_NS) we have adjusted the simulation to the experimental resistance changes as demonstrated in figure 7(c). We have assumed that the ΔR/R = 0.04 contour line (black data points) is caused only by PS leading to (ΔR/R)_PS = 0.04 and therefore Y_PS and E_PS. The crossover line between PS and NS (red data points) with ΔR/R = 0 corresponds to a saturated PS contribution of 0.1 which is compensated by NS, i.e. this line can be treated as a pure NS contour line with (ΔR/R)_NS = 0.1 for estimating Y_NS and E_NS.

For the adjustment Joule heating has to been taken into account. The stationary temperature T during pulse excitation was calculated by T = 300 K + 10⁻⁷ K m² W⁻¹ · P (see section 3.3). Using the experimental voltage dependence of the device resistance R(U) = (180.9−170 · (U−1.15)^0.5) Ω the temperature during a voltage pulse can be approximated by T(U) = (305 + 380 · (U−0.46)^2.7) K.

The hopping voltage has to be estimated for both bulk and interface by an estimate of the corresponding local electric field. As pointed out in the beginning of the discussion, comparing the device resistance with the expected PCMO film (bulk) contribution suggests that most of the applied voltage ( ≈ 0.94 U) drops on the interface leaving only a minor fraction (0.06 U) for the bulk. In the suggest model, the NS is governed by bulk diffusion of oxygen vacancies. The electric field across the bulk (thickness 300 nm) is rather small (of the order of F_bulk ≈ 0.06 · U /300 V nm⁻¹). For U = 1V and d_hop = a₀ ≈ 0.4 nm the value eU_hop-NS ≈ 0.08 meV is negligibly small compared to the activation barrier. Best agreement with the experimental data was obtained by the fit in figure 7(c) with E_NS = 0.9 ± 0.2 eV.

The estimation of the activation energy in PS is not obvious because the suggested model for PS involves both diffusion of vacancies at the interfacial area of PCMO and the oxidation step at the Pt top electrode. If the latter represents the rate-limiting step, the relevant hopping distance might be of the order of Pt-O bonding length (d_hop ≈ 0.2 nm). Furthermore, a realistic estimate of the electric field in the interfacial area is required. The three point measurement does not reveal rectifying behavior due to Schottky-like space charge layers at the top electrode. Therefore, we do not expect a pronounced contact resistance at the PtO_x/PCMO interface. For a coarse estimate we assume that the thickness of the interfacial PCMO area is of the order of 10 nm and d_hop ≈ 0.2 nm. Therefore the electric field in the interfacial area is about F_int ≈ 0.94 U/10 V nm⁻¹ and eU_hop-PS ≈ 19 meV (for U = 1 V). For such small hopping voltages an activation barrier of E_PS = 0.7 ± 0.2 eV gives best agreement for PS in figure 7(c).

For the PS we assume that the HRS is formed at positive bias at the top electrode by electrochemical oxidation of metallic Pt to PtO_x at the Pt-PCMO interface. The standard formation enthalpies of PtO₂ and Pt₃O₄ by reacting solid Pt with gaseous O₂ are H_F = 0.83 eV and 1.7 eV, respectively [30, 31]. However, the formation energy of PtO_x at a solid interface of Pt to an oxide may be different to the H_F values obtained from gas phase reactions. The observed activation barrier of 0.7 eV is therefore comparable to the formation enthalpy of PtO_x.

We do not consider the formation of thermal vacancies during the pulse excitation, i.e., the resistance change (equation (2)) is governed by the redistribution of existing vacancies and the activation energy should be comparable to the migration barrier. Therefore, a comparison of the obtained activation energy for NS from simulation and direct measurements of the vacancy diffusion coefficient in PCMO would be useful. However, to the best of our knowledge, no literature values have been published, and even for La_1−xSr_xMnO₃ only oxygen tracer and oxygen self-diffusion data has been published. Typical activation energies are of the order of 3.0 eV for LaMnO₃ and La_0.9MnO₃ [32, 33] and 3.5 eV for La_0.5Sr_0.5MnO₃ [34]. In addition, the barrier seems to depend on the oxygen content, e.g., the activation energy in oxidation and reduction is different in La_1−xSr_xMnO₃ [35].

In order to obtain a realistic estimate of the activation energy for oxygen vacancy diffusion, an estimation of the vacancy formation enthalpy ΔH^F is required; i.e. E_vac = E_self−ΔH^F. Reported values range from ΔH^F = 3.16$\$eV for La_0.3Sr_0.7MnO₃ [36] to ΔH^F = 2.24 eV for LaCoO₃ [37]. The relatively large value given by [36] would give rise to unreasonably small values for the activation energy for vacancy migration below 0.3 eV. Since the value was determined by a new and less established electro-chemical approach, we find the value given by Ishigaki for LaCoO₃ more reliable, which results in a vacancy migration barrier of the order of 1 eV. This is in accordance with our simulation that implies an activation barrier for the migration of preformed vacancies of 0.9 eV.

It is important to note that Joule heating gives rise to a simple relation between the slope of the contour plots and the activation energy of the involved activation energies. Using equations (1) and (2) with the power dependence of Joule heating, one can easily show that the inverse slope is (see supplement S4)

$$\begin{eqnarray}&&\frac{{\rm d}\;{\rm ln} \left( {{t}_{{\rm p}}} \right)}{{\rm d}U}\approx \frac{-e}{2\cdot k\cdot {{T}_{0}}}\cdot \left( 1.3\;E-\frac{{{U}_{{\rm hop}}}}{U} \right)\end{eqnarray} \tag{ 4 }$$

T₀ corresponds to room temperature (300 K) and E in [eV]. For a negligible contribution of the U_hop term (as predicted) one would expect a slope dominated by the activation energy. Indeed, experimentally (figure 7(a)) we observe that the inverse slopes for small voltages (U < 1.3 V) in the PS and the NS regime are not very different and are in the order of:

$$\begin{eqnarray}&&\frac{{\rm d}\;{\rm ln} \left( {{t}_{{\rm p}}} \right)}{{\rm d}U}\approx \frac{-e}{2\cdot k\cdot {{T}_{0}}}\end{eqnarray} \tag{ 5 }$$

This rather general result confirms that the underlying mechanism must be of a thermally activated nature. The suggested model is plausible, but without direct experimental proof the results would also be compatible with alternative explanations e.g. mechanisms with opposing diffusion polarities such as coexisting anion and cation diffusion. Since the cation diffusion coefficient in manganites are two orders of magnitude smaller than oxygen diffusion coefficients [38], we have not considered this possibility in detail.

The simulation (figure 7(d)) depicts the essential features of the experimental contour plot (figure 7(a)). At small voltages/short pulse duration the switching amplitude increases exponentially due to the PS contribution. With increasing voltage/pulse duration the PS contribution saturates and the switching amplitude remains almost constant until pronounced contribution from NS causes a strong decrease and the crossover to negative switching amplitudes. However, the simulation does not reflect the strong increase in the slope at higher voltages. The discrepancy between simulation and experiment is most probably related to the uncertainness of T due to the increased Joule heating, the lack of controlled stimulation in the transient regime in our measurement setup (pulse lengths <10⁻⁶ s, see figure 4) and the limited applicability of equation (2) at higher voltages.