Analysis of the Electrical ReRAM Device Degradation Induced by Thermal Cross‐Talk

A switching of resistive memory cells leads to a local accumulation of Joules heat in the device. In resistive RAM (ReRAM) arrays, the heat generated in one cell spreads via common electrode metal lines to the neighboring cells and may cause their performance degradation. The performance degradation results in reduced number of switching cycles and, in extreme cases, even in a loss of a bit, caused by the rupture of the nanofilament. The authors propose a thermal analysis of the thermal cross‐talk, describe its impact on cells’ electric performance, and identify three major mechanisms for the ReRAM reliability: (i) thermal conductivity, (ii) the specific heat capacity, and (iii) geometry of the electrodes. Several ReRAM arrays are manufactured to vary thermal conductivity, specific heat and geometry of the electrodes by depositing eight different inert electrodes: Pt(50 nm)/Ti(30 nm), Ru(50 nm)/Ti(30 nm), Co(50 nm)/Ti(30 nm), Pt(50 nm/Cu(100 nm)/Ti(30 nm), Pt(50 nm)/ Cu(200 nm)/Ti(30 nm), Ru(50 nm/Cr(30 nm), Ru(50 nm)/Ti(50 nm), and Rh(50 nm)/Cr(30 nm). The experimentally found differences of the degradation of electric performance of the array cells performed under identical circumstances can be correctly predicted by the proposed thermal analysis using the material properties and geometry parameters of the electrodes.


Introduction
Traditional non-volatile memory based on floating gate MOSFET transistor has reached its scaling limits, and, hence, the development of alternative memory cells has become imperative. Resistive Random Access Memory (ReRAM) is one of the prime candidates to replace the current non-volatile technology due to its simple architecture, excellent scaling potential, low power consumption, high switching speed, large on/ temperature. Walczyk et al [15] found that the R off /R on ratio decreased from a value of 20 to ≈5 over the temperature interval of 213K-413 K. Such changes in R on and R off of ReRAM cells also cause significant degradation in the computational accuracy of ReRAM based neuromorphic computing systems. [5,6] Specifically, a hotter cell is more susceptible to generate an incorrect output. The thermal reliability of ReRAM is also a concern for embedded memories for automotive applications. [16] Chang et al [14] found two different modes of resistive switching for identical devices with the exception of a different thickness of the bottom Pt electrode. The difference in switching behavior has been explained in terms of the temperature-dependent stability of the CF. Thinner thickness of the bottom electrode makes the dissipation of Joule heat less efficient and hence the filament will be heated to a higher temperature and become more unstable at the same switching conditions. A comprehensive review of resistive switching phenomena has been given. [17] Switching of a cell repeatedly leads to an increasing deposition of Joule heat in the device. It has been shown [18,19] that the Joules heat is transported preferentially along the electrode metal lines affecting the neighboring cells disposed along the same electrode lines and causing the deterioration of their electrical properties. Moreover, it has been shown [19,20] that the thermal cross-talk can erase a programmed neighboring cell, if the nanofilament in the cell is fragile enough and sufficient amount of heat has been transferred to it from the heated cell. For some combinations of specific amount of heat transfer and the strength of the Cu filament, the bit of the cell may be erased temporarily by remote heating. The filament recovers spontaneously after a couple of tens of seconds. [19,20] The goal of this work is to provide a thermal analysis explaining the various electric degradation phenomena in terms of material properties and geometries of the electrodes of the memory cells.

Experimental Section
In the most common architecture, ReRAM cells lie at the intersection of perpendicular metal electrode lines in a crossbar array as shown in Figure 1a. The cross-section of the baseline Cu/TaO x /Pt/Ti and of the derivative ReRAM devices shown in Figure 2 was fabricated in a crossbar array on a thermally oxidized Si wafer [3] with a SiO 2 layer with 730 nm thickness. Notably, the seven different devices differed only in the construction of the inert electrode. Cu (150, 100, and 200 nm), Pt (50 nm), Rh(50 nm), Co(50 nm), Ru(50 nm), Cr (30 nm), and Ti (30 nm) layers were deposited in a Kurt Lesker e-beam PVD-250 chamber and patterned by lift-off technology with a photoresist thickness of 2 µm to make sure that the sidewalls are sloped gently. A thin Ti or Cr layer of 30 nm was used between Pt, Ru, Rh, Co, and SiO 2 to improve the adhesion of the inert electrode proper layer and also to modulate the thermal properties of the composite electrode. The thickness of TaO x was 25 nm. The thickness of the TaO x on the sloped sidewalls of Pt line was 98% of the planar thickness as confirmed by Ta 2 O 5 atomic layer cells manufactured with Ta 2 O 5 deposition by atomic layer deposition (ALD). [21] The oxygen-deficient TaO x (x ≈ 1.9) was deposited also in the PVD-250 chamber by evaporating Ta 2 O 5 pellets without O 2 injection into the evaporation chamber. The details of the sample manufacturing process were given, [21] where the role of the parameter x in TaO x (as opposed to the stoichiometric Ta 2 O 5 ) on resistive switching properties has been discussed in detail. The width of metal lines varied between 5 and 35 µm resulting in rectangular device areas of the device in the range of (5 to 35) x (5 to 35) µm 2 . The neighboring line pitch was 150 µm and the lateral dimensions of the contact pads were 100 µm × 100 µm.
To protect the manufactured memory arrays from damages by moist environment, [22] the manufactured devices were kept in a desiccator in a clean room. The memory cells used in this investigation are described in much detail. [23,24] The electrical characterization was performed at room temperature on a probe station equipped with Keithley 4200-SCS analyzer. To set the ramp rate, Keithley equipment allows one to choose the voltage step height and it automatically determines the pertinent time step. As an example, if a step height of 0.025 V is selected, the Keithley analyzer returns a time step of 80 mV, resulting in a ramp rate of 0.313 V s -1 .
When a positive voltage is applied to the Cu electrode, Cu cations are generated according to the redox reaction: and migrate under the influence of the electric field in the solid electrolyte to be electrochemically reduced on the surface of the Pt, Ru, Rh, Co cathode which acts as an effective www.advelectronicmat.de diffusion barrier for Cu atoms. As more and more Cu atoms accumulate, the copper atom protrusion grows and a metallic filament (CF) of Cu atoms as building blocks forms a conductive path between two electrodes. An abrupt onset of conductance occurs at a threshold voltage V set . The filament can be partially undone by applying a negative voltage to the Cu electrode and grounding the inert electrode. The abrupt rupturing of the filament occurs in the I-V characteristics at a threshold voltage, V reset . Several typical I-V characteristics for set and reset operations are shown in Figure 1b. Before a measurement is taken, the two grounded needles were placed on the cell contacts for at least 20 s to make sure that the cell capacitor is fully discharged. Then, for the SET operation, the voltage of the Cu electrode starting at 0 V was ramped at a ramp rate (rr) ranging from 0.01 V s -1 to 2.50 V s -1 . During the SET operation, a compliance current (I cc ) of 5 µA to 1 mA were imposed without an off-chip resistance lest the device be damaged. This work is interested in cells that have been set prior to the heating of a neighboring (heated) cell into a LRS state under various levels of compliance current, I cc and that are connected by a thermal path to the heated device as shown in Figure 3.
Since this paper is concerned with thermal properties of the electrodes, and particularly of the inert electrode, Table 1 summarizes the important material properties of the materials involved in manufacturing of the devices shown in Figure 2.
Since several inert electrodes are composite electrodes and in order to expedite the thermal analysis given below, the effective thermal conductivity for the composite inter electrodes was calculated. The effective thermal conductivity was calculated as a weighted average of the respective layer thicknesses and is shown in Table 2.
It is known that the imposition of I cc during the set operation controls the resistance of the LRS state, R on , via the relation: [2] = R K/I on cc n (2) where K extracted from experimental data for the Cu/TaO x /Pt devices yields K ≈ 0.29 V and n ≈ 1. It has been shown [2] that the constant K for n = 1 represents the minimal voltage under which the device can be set. A typical R on on I cc dependence is shown for Cu/TaO x /Pt/Ti devices in Figure 4. This was confirmed following the procedure outlined [2] by applying a constant voltage to a reset cell and measuring the current as a function of time. For all voltages in excess of 0.287 V, the cell can be set to an ON-state, whereas for voltages smaller than 0.287 V, the cell remains in the OFF-state, even when the voltage is applied for a long period of time. The dependence of R on on I cc allows one to form stable but weak, i.e., highly resistive (R on = 50 kΩ) Cu filaments at I cc = 5 µA, and strong, low resistive (R on = 500 Ω) Cu The filaments at I cc = 0.2 mA. For compliance current values below 5 µA, the filament did not form at all or became volatile and ruptures unaided and spontaneously. On the other hand, robust low-resistance Cu filaments formed at I cc > 0.25 mA were prone to become inoperable, i.e., meaning that the cell has been permanently damaged (shorted).
The retention properties of all devices has been found to be satisfactory. Figure 5 shows the retention of Ru and Pt devices  www.advelectronicmat.de (a) for the on-state characterized by R on , and (b) for the off-state characterized by R off . It can be seen from Figure 5 that the level of noise for Ru devices is higher than for Pt devices for both bit states, characterized by R on and R off . While R off fluctuations of Ru devices are irrelevant as the outlier lie in the tens of GΩ range, the outliers for R on are more of a concern, as its variation can be as high as 75%. Such large fluctuations could pose, inter alia, a problem for the multibit storage applications and for the stability of the sense amplifiers when reading the cell. For comparison the variations of R on and R off in Pt devices are less than 10% as discussed in more detail. [26] As described in more detail, [8,19] a frequent switching of a cell led to a deposition of Joules heat in that cell that can spread to neighboring cells disposed along one of the electrodes shared with the heated cell. Even those cells that do not share any of the two electrodes with the heated cell can be affected by the thermal cross-talk provided the intermediate cells are set in the ON-state whereby the Cu filament provides a thermal conduit for the heat transport. This phenomenon allowed remote heating of neighboring cells to a various degree dependent on the amount of heat dissipated in the heated cell, on the selected shared electrode and the distance of the cell from the heated cell, see Figure 3.
Several thermal models have been put forward in the literature to estimate the temperature distribution in ReRAM cells. G. Zhou et al [6] discussed how localized Joule heat affect the strength of the CF. Mickel et al [26] proposed a geometrically equivalent circuit to calculate the heat transport through the conductive filament, yielding temperature distribution of the memory cell as a function of the spatial dimensions. The critical temperature responsible for the rupturing of a filament was calculated to be ≈1225 °K. In another work, Mickel et al [27] proposed a set of constitutive equations describing the evolution of the heat transport. Based on this approach, the temperature responsible for the rupturing of the filament may be even as high as ≈1500 °K. Sun et al [28] also found that the peak temperature of the filament is somewhere between 600 °C and 900 °C. Most of the models assumed a thermal boundary condition at the mesoscopic electrodes to be the room temperature. These results clearly refuted this assumption. The electrode lines were heating up considerably and enabled the thermal cross-talk between the cells of the array. Karpov et al [29] discussed a thermal model for a metallic filament with the temperature boundary condition other than room temperature that is adequate to the thermal crosstalk situation. Fangohr et al [30] investigated Joule heat in nanowires with various constriction geometries and found that the temperature at the constriction is as high as 1336 °K. Uenuma et al. [31] observed by infrared detection the position of the CF from Joule heating at the surface of the electrode on the CF and estimated the temperature to be larger than 900 °C. Sato et al. [32] found that the temperature in the reset process reaches the same temperature level in each reset and estimated the temperature of filaments with radii 10 nm -40 nm to be between 1000 °C and 1200 °C, depending on the resistivity of the filament. Thus, there is a wide consensus that the local peak temperature of the filament can be very high and this temperature is bound to increase further when the cell is switched on and off sequentially with high switching frequency. This work presents an experimental evidence that a strongly heated cells may lead to local melting of the Cu electrode at 1085 °C.

Thermal Cell-to-Cell Cross Talk Effects in ReRAM Arrays
Switching of a cell repeatedly leads to an accumulation of depositions of Joules heat in the device. The heating of the cell was accomplished by applying N set-reset cycles. It is noted that during a set cycle, the heat dissipation is very small and negligible compared with the reset cycle. During the set operation before V set is reached the current is below 1 nA, and when threshold voltage is reached the current is limited in our experiments by I cc to less than 100 µA, whereas the current during the reset operation is in the mA range. Therefore, the heating occurs predominantly during the reset operation and can be represented by heat pulses of a pulse duration of several seconds (t res ) with a pulse period of 40-60 s, the time needed to operate the probe station in different bias conditions. This means that during the reset operations when the cell is heated up, there follows a prolonged cool-off period of ≈40 s. Thus, the duty cycle of the heating is smaller than ≈0.05. During the off pulse time, the cell and the system will cool off due to Newtonian losses, but not completely. The fraction of the heat retained in the cell is only a fraction of the heat deposited; this fraction is described by the factor f loss . Therefore, several cycles are needed to accumulate enough heat in the device to reach a certain temperature. It has been recently shown [18,19] that the Joules heat is transported along the shared electrode metal lines, which affects the neighboring cells and causes the deterioration of their electrical properties. [19] The heat generated in the heated memory cell can be controlled by the voltage ramp rate, rr, applied during the reset operation and by the number, N, of consecutive set-reset cycles, and by the R on resistance of the Cu CF. During the electrical characterization of the memory cells, it has been noticed that the electrical switching of a fresh cell has been degraded when a direct neighbor of this cell, has been heated by repeated SET-RESET cycles. The degradation manifests itself by the onset of volatility after the device has been cycled only a few times. This degradation of electrical properties disappeared when the cell was tested again after 5 min or a longer cool-off time period. We assume that the thermally induced degradation subsides after times of the order of 2-5 min. Tests between various Pt and Cu line proved that no sneak paths [33] are present.
In order to quantify the thermal cross talk, a metric of so called "marginal" memory cell has been introduced. [18,19] By a cell being "marginal" we mean following cell's behavior after being subject to a specific SET condition: We set a device at an I cc of only 10 µA and a voltage ramp rate rr = 1.1 V s -1 . When the cell is set at such conditions, the cell becomes volatile after a limited number of sequential SET-RESET switching cycles. One manifestation of volatility are spontaneous RESET events. The marginality of the device is demonstrated further when the device is set at slightly lower and slightly higher I cc . When applying I cc < 7 µA, the device cannot be set permanently in a LRS state. When the same device is set at higher compliance currents of 8-10 µA, the LRS state is stable although only for a very small number of sequential switching cycles, typically 11-14, after which the cell becomes volatile, eventually. A test has been applied to ≈100 cells set at I cc = 10 µA and a maximum of consecutive switching cycles has been determined. We obtain a mean of M x = 12.7 cycles and standard deviation σ = 1.3. The cells set at I cc = 10 µA display a rather tight distribution of R on values with the mean value of 27.7 kΩ with a standard deviation, σ = 13.2 kΩ.
A low rr = 0.1 V s -1 has been applied during the RESET operation to allow for a long heating time. For a low rr, the current lingers for a long time and hence low rr maximizes heat dissipation. In [34] it has been shown that V reset decreases with decreasing ramp rate, rr, as V reset ∼ rr 3 When the maximum number of switching cycles (M x ) is reached, the device is driven to a highly unstable performance. On the other hand, a device set at I cc = 40-100 µA can be switched repeatedly more than hundreds of times. Thus, a "marginal" device demarcates the boundary between stable and unstable (volatile) behavior of the cell. Hence, a "marginal" device set at I cc = 10 µA is our "canary in the coal mine" and the maximum number of switching cycles, M x , is our metric to quantify the degree of the thermal cross talk. We find that such a marginal device when heated remotely by another cell will have a maximum number of switching cycles smaller than M x = 13, depending on the amount of heating experienced from the heated cell. Thus, the measure M x defines the onset of the cell's volatility in response to the local temperature of the cell. Thus, the remote heat superposes on the heat generated locally by the switching of the cell itself. In our experimental setup, the time between stop of the heating of the heated cell and characterization of the

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probed cell is ≈50s -the time required to replace the needles on the probe station.
The metric for degradation is established customarily by relating the M x for the remotely heated cell to M x of an unheated cell as: The mean of M x for an unheated cell has been found to be 13 SET-RESET cycles, as mentioned before. The cell degradation results for cells disposed along the Pt and Cu electrodes are shown in Table 3 based on 125 devices measured. It can be seen that for the nearest neighbors the degradation is about D = 67%. In contrast, the degradation of the 4 th device located on the Pt electrode is much smaller, D = 13%. The results will be explained below by the heat transfer over a distance of n × 150 µm where n = 1,2,3,4, corresponding to the 1 st , 2 nd , 3 rd , 4 th neighboring cell as shown in Figure 3. The degradation of electrical performance of the neighboring cells due to thermal cross-talk is shown in Table 3.
As discussed extensively in [19] Table 3, it is seen that the degradation of the neighboring cells along the Cu electrode is much more pronounced than of the neighboring cells along the Pt electrode. The finding that the degradation along the Pt electrode is smaller than along the Cu electrode was, at first, surprising [19] as one would assume that the Cu filament has a broad base with the Pt electrode while a small contact area at the tip of a truncated cone with the Cu electrode. The increased degradation along the Cu line led to the revision of the shape of the filament from a come to more an hourglass shape [19] which has large contact area with both electrodes. The reason for the larger degradation of the devices along the Cu electrode is the much higher rate of heat transfer along the Cu electrode than along the Pt electrode. The rate of heat transfer is proportional to the product of thermal conductivity and the cross-section area of the electrode. The thermal conductivity of Cu is almost 8 times larger than of Pt/Ti see Table 2 and the cross-section of the Cu electrode is 2.5 times larger than that of the Pt/Ti electrode. Therefore the rate of heat transfer along the Cu electrode is more than 20 times larger than along the Pt electrode. Consequently, the neighboring cells along the Cu electrode are heated up to higher temperatures than along the Pt electrode.

Thermal Analysis, Predictions of ReRAM cells' Degradation and Verification
The heating of the cell A can be controlled by a number of sequential switching (set-reset) cycles, N, the level of I cc , and by the voltage ramp rate, rr, during the reset cycles. To maximize the heating, a high I cc leading to a robust low resistance Cu CF, and/or by a low rr = 0.1 V s -1 may be chosen for the reset operation providing a long heating time prior to reaching the threshold voltage V reset . For a low rr, the current lingers for a long time and hence low rr maximizes heating of the cell. For a cell to which a constant voltage ramp rate is applied, the Joules heat Q JH can be calculated by Equation (4) [34,35] where relation between R on and I cc (see Equation (2)) has been used to obtain the final expression.
Here the reset time, t res , at an applied voltage ramp rate rr is given by t res = V res /rr. [34,35] Depending on the chosen values for I cc and rr, Q JH can vary from 3 to 60 µJ.
Q JH is the energy that went into the memory array and is solely determined by external parameters, the applied voltage ramp rate rr, and the experimental values for R on and V reset . The primary locus of the heat generation is the nanofilament characterized by the on-resistance R on . Since the nanofilament is totally encapsulated in the memory cell, the question arises where the generated heat is being stored. The heat transferred to a material increases the material's temperature according to where c s is specific heat of the material, m is the mass of the material, and ΔT is the temperature difference, with a reference temperature of the room temperature of the device. In evaluating Equation (5), one has to take into account the temperature dependence of the specific heat capacity whichfor an exact solution -would require numerical solution. [37] The heat capacity of metals varies with temperature according to the Debye theory and displays strong dependence with up to slightly below the Debye temperature, T D , of the material. [36] Around and above T D , the specific heat capacity of metals increase only modestly with increasing temperature. For example, the specific heat capacity for Cu and Pt increases from T = 300 °K T = 1350 °K by 16%. Since T D (Pt) = 240 °K and T D (Cu) = 340 °K, and the temperature interval of operation of our ReRAM devices is 300 °K-1350 °K, a constant specific heat capacity appears to be an acceptable 1 st order approximation. Because the heat capacity is slightly underestimated in our estimates at high temperatures, our ΔT values at high temperatures are slightly overestimated by the same percentage.
Among other things, Equation (5) indicates that the increase of the temperature difference ΔT depends on the shape of the electrode. In order to evaluate Equation (5) for a given heat generated, one needs to calculate the total mass of the electrode Table 3. The average maximum number of switching cycles, M x (ave) and degradation, D, of the neighbor cells of the heated device for the array with inert electrode of Pt(50 nm) array for a 10 mm wide line. [19] #neighbor which is given by the product of mass density times the total volume of the electrode (width × length × thickness) including the volume of the connected contact pads. Clearly, the thicker or wider the electrode is, the larger its mass, and the smaller the temperature increase, ΔT. Hence the geometry, i.e., the volume of the electrode, determine the amount of the temperature increase ΔT. Let us assume for a moment that the heat Q JH generated in the filament is stored entirely in the Cu nanofilament. One may ask then, what the temperature difference would be in order to accommodate a heat of 10 µJ in a Cu NF. In order to answer it, one has to calculate the volume of the Cu NF, estimate its mass density, and its specific heat. Assuming a truncated cone shape of nanofilament with a radius of 8 nm at the bottom, and 5 nm top radius, and 25 nm height, given by the thickness of the TaO x , the volume comes to 4.6 × 10 -18 cm 3 . [37] This volume and geometry lead to a R on resistance of ≈1000 Ω, if one assumes the resistivity of the Cu NF of 5 × 10 -4 Ωcm, ≈20 times higher than that the resistivity of bulk Cu. Indeed, such R on value is observed experimentally at I cc = 10 µA. Given that the mass density of Cu is 9.0 g cm -3 and that of Ta 2 O 5 8.2 g cm -3 , then the mass density of the filament has to lie somewhere in-between, say at 8.5 g cm -3 , the total mass of Cu NF comes to 4 × 10 -17 g. The specific heat capacity of Cu NF must also lie between the heat capacity of Cu of 0.385 J (g -1 °K -1 ) and that of Ta 2 O 5 of ≈750 J (g -1 °K -1 ). Assuming the extreme case of 750 J (g -1 °K -1 ) and using Equation (5), one can calculate the temperature difference ΔT needed to store such energy within the nanofilament alone. The temperature difference relative to the room temperature comes to at least 3 × 10 8 °K, which is ten times higher than the sun's core temperature of 2.7 × 10 7 °K. Assuming the other extreme of the specific heat capacity being c s (Cu-CF) = 0.385 J (g -1 °K -1 ), the temperature would be at 3 × 10 11 °K even higher. Because this is impossible, it is clear that this kind of energy cannot be stored in the filament because of its exceedingly tiny volume. The heat instead has to be stored at least in the two (Cu and Pt) electrodes, by the dominant heat conduction transfer, and in the filament whose volume contribution can be neglected. In addition, since, the electrodes have a very large surface to volume ratio, a large part of energy delivered to the electrodes will be dissipated due to Newtonian losses, including radiation and buoyancy-driven convection. [38] Thus, one may assume that over one heating period during the reset operation, only a fraction of Q JH will really be stored in the electrode material, or Q st = Q JH × f loss , where f loss denotes a fraction of the heat retained in the material after all heat losses to the ambient. The coefficient f loss can be related to the coefficient of heat transfer, r, to the ambient, f loss ≈ exp (-rt). [39] When a memory cell is switched back and forth in quick succession N times, the heat, Q Nst , stored in the electrodes is given in Equation (6) Q Where f diss , is a factor smaller than unity describing the dissipation time between the switching cycles and accounting also for the cool-off period during the set operation when no current is flowing or very small current limited by I cc and by the heat loss when transported between the heated and probed neighboring cells. m in Equation (6) is now the total mass of the two electrodes connected by the filament.
For an evaluation of Equation (6), we possess an important experimental reference. Under certain switching conditions for medium-to-large R on and small rr, chosen deliberately in order to maximize the heating, when a cell is switched on and off repeatedly, we observe melting of the Cu electrode over the heated cell extending even to the contact pads as shown in Figure 6. In Figure 6a, the melting of the top Cu electrode over the ReRAM cell is shown. Because of the high thermal diffusivity, the high temperature spreads even to the contact pad as shown in Figure 6b and distorts their square shape visibly while the contact pads for the Pt electrodes remain intact. In this context, we observe that the Cu electrode line runs entirely over a thin 25 nm TaO x but the Cu pads are located on a 730 nm SiO 2 field oxide. In addition, underneath the TaO x layer, there is an array of Pt electrode lines which appears to indicate that the heat transfer from the Cu electrode through the thin TaO x with the Pt lines underneath is more efficient in heat transport than through the thick SiO 2 field oxide. Hence, after several heating cycles the temperature of the pad is higher than the temperature of the electrode line.

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This particular melting of Cu electrode has been observed for 25 switching cycles, each cycle being characterized by Q JH = 40 µJ, according to the values used for rr, and experimental values for R on and for V reset . Since the melting of Cu occurs at 1085 °C for Cu layers thicker than 100 nm, [40] we can calculate from Equation (6) the product f loss × f diss using the specific heat capacities for Cu, c s = 0.385 K (g -1 °K -1 ), and for Pt c s = 0.13 J (g -1 °K -1 ), and obtain f loss × f diss = 0.063.
With this information, one can now go back and calculate what is the average temperature increase in the CF-electrodes system after one single switching cycle used for the experiments shown in Table 3 for which one reset cycle has Q JH = 10 µJ. Thus, we can repeat the calculation using Equation (6), but now taking as a mass, the entire mass of the Cu and Pt electrodes including the contact pads of square shape with 100 µm on the side, and using the respective specific heat capacities for Cu: c s = 0.385 K J -1 (g -1 °K -1 ) and for Pt: c s = 0.13 J (g -1 °K -1 ) and calculate the ΔT. Using Q JH = 10 µJ, f loss = 0.1, we obtain from the dimensions of our ReRAM array ΔT = 27 °C, assuming that the heat is uniformly distributed. Naturally, the temperature, at least initially, when the heated cell is only being heated, is not uniform, but as stated before, because of the high thermal diffusivity, reaches such state within a short time period. As the heat spreads from the heated cell to other cells the maximum temperature of the neighboring cell will be lower.
One can reliably estimate f diss from the solution of 1-D heat equation for an isolated metal rod: [41] Where T is the temperature, k is the thermal diffusivity, p is the heat provided, c s is the specific heat capacity, and ρ is the mass density. Equation (7) for the metallic rod with one end being heated has been solved explicitly. [41] It is noted that the thermal diffusivities of Cu and Pt are substantially higher than for TaO x of k = 0.02 cm 2 s -1 , (see Table 1).
The characteristic time scale for the Equation (7) is given by t ts = L 2 /k. If one takes as a typical distance L the space between two electrodes, i.e., 150 µm and the thermal diffusivity k of Cu of 1.11cm 2 s -1 , one obtains t ts = 200 µs. Since the pulse period of our set-reset cycles is 60 s, which is 3 × 10 5 times larger than t ts , one may evaluate Equation (7) under steady state condition. In the steady state condition the temperature is a linear function of the space coordinate. Of course, during the period of 60 s, the electrode is cooling off due to the Newtonian losses and the fraction of the heat remaining in the electrode is estimated by the factor f loss as described before. Thus, assuming that the electrode is heated by a cell at one end, the temperature along the electrode varies linearly.
From the equation L k t = × tr , one can estimate the time, t tr , required for a temperature front to move over the length L = 850 µm, from the heated cell to the far-off pad of the electrode. For the Cu electrode, this time is smaller than 10 ms, and larger for the Pt electrode, due to its five times smaller thermal diffusivity (Table 1). From this, it is clear that for the neighboring cells being characterized after the heating of the heated cell has been stopped for 40-60 s, the temperature distribution may well be represented by a steady state, which according to the solution of Equation (7) for a perfectly insulated metal rod is linear function of temperature as a function the longitudinal spatial coordinate, x, of the electrode.
Since in steady state the temperature is a linear function of location, assuming that the first cell on the electrode has been heated and has reached temperature T c while the far end pad of the electrode is at T o < T c , then the temperature drop from cell to cell is given by (T c -T o ) × 150/(5 × 150+100) = (T c -T o ) × 0.18, where 150 denotes 150 µm distance between two cells and 5 × 150 + 100 = 850 µm is the distance of the heated cell from the far end contact pad. In the initial stage of heating one can assume that T o is still at 27 °C.
Counting the heat loss traveling to further neighboring cells a good estimate for f diss is 0.6-0.7. Using the experimental finding f loss × f diss = 0.063 the latter result allows one to estimate f loss as 0.11-0.09 which justifies our use of f loss = 0.1 above. Thus the critical temperature of the heated cell drops to lower values as the heat is being transferred along the common electrode to the neighboring cells.
The critical temperature itself, T c , at which one cell becomes unstable, depends on the robustness of the Cu CF and, hence, it increases with the strength of the filament. For a fragile filament with R on of several tens of kΩ, the critical cell temperature will be as low 350 °C. The temperature of 350 °C can be estimated from number of cycles leading to a rupture of a given Cu filament. From our heating experiments of a probed cell with a strong Cu filament of R on = 5kΩ set at I cc = 100 µA, 25 cycles of heating of the heated cell were needed to observe the onset of the melting of the Cu electrode pads. This allows one to estimate ΔT for a single cycle ΔT = (1085 °C-27 °C)/25 = 42 °C. Subsequently, we have repeated the experiment with the same heated cell but the probed cell was set at much weaker filament of R on = 20kΩ set at I cc = 20 µA, and 7-8 cycles of heating were sufficient to rupture the cell. Since one cycle raises the temperature by 42 °C, then 42 °C × 8 + 25 °C = 361 °C. Thus a critical temperature to rupture the weaker Cu filament is ≈350 °C.
But a strong Cu filament, with low resistance of several hundred Ohm, will exhibit a critical temperature close to melting temperature of Cu, i.e., 1085 °C, as demonstrated in the experiment shown in Figure 6. In Figure 7, we plot the critical temperature for a weak filament for all 5 cells disposed along the Pt line, with the first cell being the heated cell and the remaining the neighbor cells being remotely heated. The heated cell is heated with 13 maximum heating switching cycles to reach T c = 350 °C and the neighboring cells will exhibit lower tem- , for the 1 st , 2 nd , 3 rd , and 4 th neighbor, respectively as discussed above.
In Figure 7, for every neighbor cell, the temperature difference of the cell to T c is also indicated. According to Sato et al., [32] the temperature in the reset process reaches the same temperature in each reset, thus it is bound to reach the same T c , at which further switching is not possible. If one divides the critical temperature T c by the temperature difference of ΔT = 27 °C, the temperature increase due to one switching cycle, one can calculate M x of cycles that a remotely heated neighboring cell may switch. These numbers are also indicated in Figure 7; for example for the 1 st neighboring cell one obtains 130/27 = 4.81, indicating that the cell can be switched for 4-5 more cycles, at most. If one compares the predicted number of additional www.advelectronicmat.de switching cycles with the corresponding experimental numbers M x (ave) in Table 3, one finds a remarkable agreement.
One can test the predictive power of the above methodology by manufacturing ReRAM arrays with enhanced thermal properties of the electrodes. We have accomplished this by replacing the inert electrode of Pt(50 nm) by Rh(50 nm), and by the composite electrodes Pt(50 nm)/Cu(100 nm) and Pt(50 nm)/ Cu(200 nm). The relevant material properties of the materials used are summarized in Table 1. Two important properties of the bottom inert electrode are being changed, the thermal conductivity is enhanced by a factor of 4, the product m × c s increases in case of the added 200 nm Cu layer by a factor of 1.95, and in case of the added 100 nm Cu layer by a factor of 1.47. Repeating now the same electric characterization experiments of the degraded cells using the same parameters for Q JH , we can calculate the ΔT for a single heating cycle. For the Pt(50 nm) array, we obtained ΔT = 26.5 °C and for Pt(50 nm)/ Cu(100 nm), we can predict a ΔT = 18.0 °C, and for Pt(50 nm)/ Cu(200 nm) a ΔT = 13.5 °C. Clearly with the enhanced capacity of the bottom electrode to store heat, the number of heating cycles to reach the same T c , or to render the cell inoperable, is bound to increase. The maximum number of switching cycles will increase even more, on account of the much higher thermal conductivity of the bottom electrode which allows a much faster spread of the heat pulse along the electrode.
The thermal conductivity of the composite inert electrode Pt(50 nm)/Cu(200 nm) increased more than fivefold leading to a fivefold increase of thermal diffusivity, the f diss should be smaller than for the Pt(50 nm) by a factor at least 3-4 times. This would indicate that for the new array for the same switching parameters, the maximum number of switching steps should be 6 to 8 times larger, 78-104 maximum switching cycles. For the case of an array with Pt(50 nm)/Cu(100 nm), the corresponding quantities are ΔT = 18.0 °C and the maximum switching cycles of ≈26. The data of maximum switching cycles for the two Pt/Cu inert electrodes and for the Rh electrode is shown in Table 4. The analogous electric characterization experiments performed on Pt(50 nm)/Cu(100 nm) and Pt(50 nm)/Cu(200 nm) arrays in the same as the experiments on the Pt(50 nm) array show that the maximum number of cycles to reach the state of cell's inoperability increases as shown in Table 5 and Table 6, confirming very accurately the predictions based on the previous thermal analysis, as shown in Table 5 and Table 6 for the Pt(50 nm)/Cu(100 nm) and Pt(50 nm)/Cu(200 nm), respectively.
From Table 5 and 6, it is clearly seen that with higher specific heat capacity and higher thermal conductivity of the bottom inert electrode, the electric performance degradation decreased drastically. In case of the Pt(50 nm)/Cu(200 nm) inert electrode, no degradation at all of the cells disposed along the Pt electrode is observed, and, in case of the devices along the Cu electrode, only the first neighbor suffers a low degradation of 11%. Clearly the higher thermal conductivity and specific heat capacity lead to much smaller local temperature than for the Pt(50 nm)/ Ti(30 nm) ReRAM array. In case of Pt(50 nm)/Cu(200 nm), this kind of experimental characterization performed for this work is rather tedious because the cells that fail for a Pt(50 nm) electrode only after 13 times, but for Pt(50 nm)/Cu(200 nm) switch now for 72-90 times, requiring a lot of student's time.
The proposed thermal analysis can also be extended to the dependence of cell degradation on the width of the line of the same ReRAM array. With the increase of the width, the volume of the electrode increases proportionally and thus, according to Equation (5) the temperature difference ΔT decreases. Our experiments so far have been using 10 µm Cu and Pt lines as a standard. We have repeated the analysis for the Pt(50 nm) ReRAM array for a 35 µm wide Pt line. Clearly, the mass of the Pt electrode increased by a factor of 3.5 and the total product m × c s increased by a factor of 1.59 about the same as the product m × c s for the Pt(50 nm) 10 µm electrode. However, the thermal conductivity remained the same, because the Pt material did not change. The electrical characterization of the 35 µm wide Pt line of the Pt(50 nm) array, shown in Table 7, shows that the degradation of the neighboring cells is smaller than for the 10 µm Pt line of the Pt(50 nm) arrays but somewhat worse than for the Pt(50 nm)/Cu(100 nm) ReRAM array. Table 7 confirms further the validity of the thermal analysis and the accuracy of its predictions. In addition, in an earlier   Ti and Cr layers of 20 nm serving as glue layers. In our thermal analysis, we find that the m × c s for Rh and Pt electrode are similar, with the product for Rh being only slightly higher than that of Pt (see Table 2). But the thermal conductivity of Rh is a factor 2 larger than for Pt, so we expect the temperature of the neighboring cells lower in case of Rh electrode than for the Pt electrode. The experimentally found enhancement factor of 26/15 = 1.73 lies in the reasonable range of the estimates of the thermal analysis proposed here. The case of the Rh(50 nm) electrode is, in a sense, an opposite to the case of a wider Pt (35 µm) electrode. In case of a wider Pt line the product m × c s increased by a factor of 3.5, but the thermal conductivity was the same compared with Pt line of 10 µm width. In case of Rh 10 µm electrode, the product m × c s is the same, but the thermal conductivity increased by a factor of 2. Furthermore, in Figure 2, two identical devices Cu/TaO x / Ru have been fitted with different glue layers Ti and Cr. The main reason why Ti has been replaced with Cr is Cr's almost fivefold higher thermal conductivity 94 W mK -1 than for Ti of 18 W mK -1 . One finds that the electric performance of nominally identical devices depends sensitively on the nature of the glue layer. We find that the V reset value for Ru/Cr is 0.4 V larger (V reset(Ru/Cr) = -3.8 V) than for the Ru/Ti device (V reset(Ru/Ti) = -3.4 V). Such result was, in fact, expected based on our analysis. The device Ru/Cr experiences much higher heat removal rate from the Cu CF during the reset operation than the Ru/Ti device. In case of Ru/Ti, the Joules heat generated in the filament cannot be as easily dissipated and lingers therefore for some time around the hot spot in the filament. Thus, lower thermal conductivity of the inert electrode requires less Joules heating than an inert electrode with high thermal conductivity as the heat accumulates in the nanofilament in a shorter time than in the case of high thermal conductivity of the inert electrode, and hence lower V reset voltage. It is also instructive to compare V reset values of the other devices. We find V reset for Co/Ti and Pt/Ti device, to be V reset (Pt/Ti) = -0.9 V, V reset (Co/Ti) = -1.0 V or essentially the same, which correlates again very well with the same thermal conductivities of the two metals of 69 W Km -1 . Finally there is large difference between the V reset for Ru devices and Pt and Co devices of ≈2.5 V, which is in perfect agreement with the much higher thermal conductivity of Ru compared with Pt and Co devices.
It has been thus amply demonstrated that three mechanisms: (i) the thermal conductivity, (ii) the material's capacity per degree K to absorb heat or m × c s , and (iii) the geometry of the electrodes, contribute in almost equal ways to the effects of the thermal cross-talk on the device performance. Once calibrated to a base line array, our methodology can quantitatively predict the improvement or worsening of the degradation effect in terms of material properties and geometry parameters.
Finally, we wish to comment on the cycling ability of various devices. Although initially Cu/TaO x /Pt and Cu/TaO x /Ru devices behave similarly, albeit with different V form , V set , and V reset threshold voltages, the main difference between the two devices is the cycling ability which in case of Ru devices is very limited. [25] At most stable switching conditions the Pt device could undergo upward of hundred switching cycles, while the Ru device's cycling ability was limited to 13 cycles. The limited cycling ability was found to be caused by the degraded inertness properties of the Ru electrode. Similarly, it has been shown that Cu/TaO x /Ti and Cu/TaO x /Ta devices displayed very limited switching capability compared with Cu/TaO x /Pt devices. [42] The limited cycling ability was attributed to reduced stopping power of the counter-electrode (dissolution of cations in the counterelectrode), to reduced cation nucleation rate at the counter-electrode, and to reactivity of Ti and Ta inert electrode metals with the dielectric of the switching layer.

Conclusions
In conclusion, we have presented ample experimental evidence of how material properties and geometries of the ReRAM electrodes impact the intensity of thermal cross-talk and how the thermal cross-talk, in turn, impacts the various manifestations of electrical performance degradation of the resistive memory cells. A comprehensive thermal analysis of the thermal crosstalk has been provided which quantifies the amount of device degradation in terms materials properties such as thermal conductivity, specific heat capacity, thermal diffusivity, and mass density of the electrode materials. Since the degradation effects can be described in terms of basic material properties, our  academic memory arrays which conform with commercial arrays only in the dimensions of the layer thickness, but are much larger in lateral dimensions, still provide an excellent baseline to extract reliably the underlying physics valid also for the high density commercial ReRAM arrays. The Joule heat density in commercial cutting edge memory arrays is much more severe, about two orders of magnitude higher than in our memory arrays. [19] Hence, the presented thermal analysis may serve as reliable guide in material selection in an effort to mitigate ReRAM degradation and reliability issues caused by thermal-cross-talk. The question as to what may be the optimum choices for the inert electrode is still not satisfactorily resolved. In terms of thermal cross-talk, low electrical conductivity material leads to lower V reset values -which is clearly an advantage -and distant neighboring cells are little affected by the slow heat transfer along the line. On the other hand, the heat lingers longer in the cell and in the electrode with low thermal conductivity. In case of an electrode with high thermal conductivity, the heat is dissipated faster, but now even distant neighboring cells are affected by the heat transfer. The optimum design of ReRAM memory array must take into consideration how the memory is being operated.