## Abstract

The growth of a filament in an oxide medium with top electrode of Ag and bottom electrode of Pt is analyzed by considering the microscopic processes. The filament, either made up of Ag metal ions or oxygen vacancies, is considered to nucleate at the interface and grow under the influence of the applied electric field and injection of the defects from the interface. The flux of current is used to calculate the joule heating and the resulting temperature change in the filament after solving the thermal diffusion equation. In addition, the concentration of defects in the filament is evaluated from the equilibrium concentration and the loss of defects from the filament using the analytical solution to the mass diffusion equation. The rise in temperature, the concentration of defects, electrical current carried, and the height and radius of the filament are determined for several increments in time. The results showed that the growth of the filament is a complex process dependent on the several interdependent parameters. The important parameters are the initial concentration of defects, the applied voltage, activation energy for atomic jumps, free energy of formation of defects and the activation energy for diffusion. The numerical results are presented for ZnO with Ag top electrode and Pt bottom electrode by considering different choices of the parameters to illustrate their relative importance.

## Introduction

Nucleation and growth of filaments are considered [1, 2] the mechanisms of transformation from high resistance to low resistance state in memristor devices. Filament formation may be either from an active metal like Ag from the metal electrode [2] or from excess oxygen vacancy concentration introduced from the medium near an inert Pt electrode [1]. In particular, stoichiometric ZnO film with Pt as the bottom electrode and Ag as the top electrode exhibited unipolar switching characteristics [3]. Several examples of films exhibiting different switching characteristics are presented in the literature [4,5,6,7]. Low-resistance metal-rich filament formation or dissolution has been observed experimentally by both transmission electron microscopy [8] and conducting atomic force microscopy [9] giving rise to the observed resistance characteristics and nonvolatile memory. The microscopic mechanisms consist of electric field-induced filament nucleation at the interface between the electrode and the insulating oxide [10] and an increase in temperature resulting from joule heating accompanied by increase in vacancy or active metal concentration, called Soret force [11], and growth of the filament toward the opposite electrode. The reverse process of dissolution is also expected when the polarity is reversed. There is also an opposing influence of diffusion of vacancies or active metal ions away from the filament in the presence of concentration gradient around the filament, called Fick force [12]. The balance of these three forces is responsible for the filament growth. Other microscopic processes have also been suggested for the increase in temperature [13], but a quantitative formulation is required in all materials. For example, Fick force is counterbalanced by Gibbs–Thomson effect through surface diffusion and surface/interface energy instability in Ag and Cu filaments in TiO_{2}, SiO_{2}, MgO_{x} or SiO_{x}N_{y} dielectric films leading to filament breaking [14]. In situ high resolution and nanoparticle dynamics simulation demonstrate that Ag atoms disperse under electrical bias and regroup spontaneously under zero bias because of interfacial energy minimization [15]. Threshold and memory switching mechanisms are observed in Ag/TiO_{2}/Pt device as a result of change in filament morphology with applied compliance current. Whereas formation of crystalline conductive state is observed at the tip of the wider filament in memory switching, spontaneous phase transformation of the crystalline state to amorphous structure in a less wide filament is responsible for threshold switching [16]. Here, we consider the common features of the filament growth in both types of filaments consisting of vacancies or metal ions under low electric field.

## Analysis

In the present work, we formulate the filament growth using analytical solution to the diffusion equation that is applicable to both thermal diffusion and mass transport. The filament configuration is schematically shown in Fig. 1. The analysis is carried out to determine the influence of different thermodynamic parameters on the filament growth in the vertical direction between the two electrodes at low electric field of below 1MV/cm. The high concentration gradient at the interface between Ag electrode and ZnO, solubility of Ag in ZnO and the large applied electric field, as high as 10^{4} V/cm is responsible for diffusion of Ag^{+} ions in ZnO. The presence of one-dimensional high diffusivity paths such as grain boundaries and dislocations is responsible initially for formation of locally concentrated region of Ag. When lattice diffusion is primarily responsible, planar film of Ag is formed very close to the Pt electrode maintained at a negative potential with respect to the Ag electrode. The Gibbs–Thomson [17] effect is considered responsible for conversion of the Ag regions or the planar film into filamentary shaped nuclei as a result of reduction of interface energy of the film. Similarly, diffusion of V_{o}^{++} defects from the medium around the positive Pt electrode is responsible for formation of filamentary nuclei at the Pt or Ag electrode when maintained at negative potential with respect to the other Pt electrode. Although the defects responsible for filament formation in ZnO are V_{o}^{++}, the filament itself is a Zn-rich region.

The concentration of either Ag^{+} ions or V_{o}^{++} defects in the filaments formed in ZnO depends on several factors such as solubility and the diffusivity. It is also important to consider the effect of enhanced electrostatic field at the tip of the filament. It has been shown [18] that the electrostatic field ahead of a planar crack tip is enhanced by a factor (d/r) ^{1/(N+1)} where r is the radial distance from the crack tip and d is distance from the tip at which the field becomes equal to the applied electric field. Also, the factor N that appears in the exponent varies from 30 to 50 [18] depending on the purity of ZnO. The Ag^{+} interstitials occupy a position closer than a lattice atom of Zn so that r could be assumed to be a/2 where a is the lattice parameter equal to 0.325 nm. If d is close to 100 a, the enhancement factor is below 1.1 which is not significant. The enhancement due to a three-dimensional filament is expected to be smaller than that due to two-dimensional crack tip. The hopping rate of ions, equal to ν_{o} exp{-(E_{a}-ZqaE/2)/kT} which is used in Eq. (1) below, is almost equal to ν_{o} exp(-E_{a} /kT) at low electric field as the effect of the field becomes negligible. This is called voltage–time dilemma in memristors [12].

The growth of the nucleus is formulated by the Butler–Volmer equation [19] or the slightly different Mott–Gurney [20, 21] ionic current equation to give the current density, J, in the filament in the form:

In Eq. (1), *I* is the current, A is the cross-sectional area, πr_{o}^{2}, of the filament of circular cylindrical shape, Zq is the net charge associated with the ion, N_{i} is the concentration of the charged ions, a is the jump distance assumed to be the shortest distance between jump sites, ν_{o} is the attempt frequency assumed to be 10^{13} s^{−1}, E_{a} is the activation energy for atomic jumps, E is the electric field acting on the filament, given by V/(L–h) where V is the applied voltage, L is the total thickness of the film and h is the filament height. We assume the filament is conducting and hence electric field is mostly confined from the filament tip to the top electrode. Further to clarify, Zq for Ag^{+} ions is 1.6 × 10^{–19} C and that for V_{o}^{++} defects is 3.2 × 10^{−19}C. The parameter, kT, is the product of Boltzmann constant, k, and temperature, T, that represents the thermal energy. Vertical growth of the filament, dh, in a time interval, dt, as a result of charge balance is given by

In Eq. (2), N_{m} is the concentration of metal ions or oxygen vacancies fully occupied. Further to clarify, N_{m} is the concentration of oxygen ions in ZnO that is same as that of Zn ions or Ag ions in ZnO and it is a constant. Ag may occupy Zn positions or interstitial positions in the ZnO lattice, but the latter is found to be more energetically favorable. Simultaneous growth of the filament in both radial and vertical directions in a time interval dt could also be written in the form:

where dr_{o} is the increment in the radial direction. In addition to Eq. (3), we can use the principle that growth in the vertical or radial direction is proportional to exposed surface area assuming flux to be same.

In order to determine the growth rate dh/dt from Eq. (2), the quantity N_{i} is needed. The concentration of V_{o}^{++} or Ag^{+} ions that represent the parameter N_{i} is determined by the balance of equilibrium concentration and the outward flux due to concentration gradients. The equilibrium concentration of vacancies or Ag ions at any temperature, T, is given by:

In Eq. (4), N_{L} is the number of lattice sites per unit volume and G_{v} is the free energy of formation of the defect, either Ag^{+} or V_{o}^{++} in ZnO. If significant portion of the ions or defects are created at the metal/oxide interface, their contribution to the concentration may be determined by solving the diffusion equation under the action of drift force due to the applied electric field which is the dominant term. Thus, evaluation of temperature of the filament becomes the important task. We consider a concentrated line source of temperature, ΔT_{o}, in the film and determine the temperature distribution as the solution to the thermal diffusion equation in cylindrical coordinates in the form:

In Eq. (5), ΔT_{o} is the initial temperature difference between the filament and the surrounding medium and ΔT is the resulting temperature difference around the filament as a function of time, t. A factor of πr_{o}^{2} is introduced to maintain dimensionality of the equation. Further, D is the thermal diffusivity in ZnO. We consider the temperature increment over an interval of time, dt, and evaluate the new temperature from the previous value. Thermal energy flux going out of the source at a radial distance of r_{o} of the filament is

In Eq. (6), K is the thermal conductivity of ZnO. The energy, E, going out of the filament of height, h, in time dt is.

E (going out of the filament)

In Eq. (7), since thermal diffusion takes place only when the temperature is above the value reached at the previous interval of time, dt, a factor of 0.5 is introduced to get the average value in the interval, dt. When the interval of time dt is small, an average value is a good representation of the temperature increment. Also, the maximum value of the exponential term as a function of time at *r*_{o}^{2} = 4Ddt is used and therefore the term 1/e is the inverse of the base of the natural logarithm (2.71). In Eq. (7), C_{v} is the volume heat capacity of ZnO equal to K/D. An additional contribution to the thermal flux arises from the contact of the filament to the electrode that can be written in the form

In Eq. (8), J_{int} is the flux across the interface, and G and ΔT are the interface thermal conductance and the temperature difference between the filament and the electrode, respectively. However, the above term will not significantly contribute since the interface area of contact is much smaller than the surface area of the filament.

The balance of thermal energy is written in the form:

where Δ*T*_{o} = (*T*_{2}−*T*_{1}), *T*_{2} is the new temperature and *T*_{1} is the previous temperature for the increment in time, dt. The first term on the left of Eq. (9) is the energy input from joule heating, and the second term is the thermal energy loss from thermal flux out of the filament. The term on the right is the net thermal energy gain in the filament giving rise to the temperature rise. The value of *T*_{2} is evaluated from Eq. (9) for the known value of *T*_{1}, I and V and dt. We can now determine the equilibrium concentration of N_{i} at the new temperature, T_{2}. If the equilibrium concentration at T_{1} is N_{i1} and that at T_{2} is N_{i2}, diffusion of V_{o}^{++} defects or Ag^{+} ions away from the filament is expected to alter the value of N_{i2}. The solution to the atomic mass diffusion equation in cylindrical coordinate system, assuming a concentrated line source at the center, is written in the form:

In Eq. (10), M is the number of defects per unit length of the source. When the filament is nucleated, the concentration is N_{vo}. This is the value before the interval of time, dt. However, the increase in temperature is responsible for an increase in concentration over the interval of time, dt. The new concentration is the sum of the initial and the additional concentration to give

Also, D_{v} is the diffusion coefficient of the oxygen vacancy defects or Ag ions, r is the radial distance and t is the time interval. The flux of point defects from the source radially is given by

Using the above expression for flux, the number of atoms per unit volume that moved out of the filament during increment in time, dt, is determined to be

The factor of one half in Eq. (13) is included to take into consideration that diffusion occurred over an interval of time, dt, with concentration changing continuously and an average value is a good representation. The new value of concentration of defects or Ag ions after diffusion over interval of time, dt, is given by:

The value of N_{i}, given by Eq. (14), is the new value of N_{vo} that will be used for the next increment in time, dt. The ratio of the number of oxygen vacancy defects or Ag ions added to the filament radially to the number in the vertical direction is proportional to the exposed surface areas of the filament and hence equal to 2 h/r_{o}. Therefore, the increment in the size of the filament in the radial direction dr_{o} = (2 h/r_{o}) dh. Thus, the new radius of the filament, r_{o} + dr_{o}, is also determined.

## Numerical analysis

The numerical analysis is performed assuming known values of the parameters for ZnO such as the number of Zn lattice sites per unit volume, *N*_{L} = 4.15 × 10^{22} cm^{−3}, jump distance, *a* = 0.325 nm, jump frequency, *ν*_{o} = 10^{13} s^{−1}, volume heat capacity, C_{v} = 2.8 J/cm^{3}K and thermal diffusion coefficient, *D* = 0.012 cm^{2}/s in thin films [22, 23], thickness of the film, *L* = 0.5 µm, the initial dimensions of the cylindrical filament or the nucleus, r_{o} = 10 nm and height h = 10 nm, and growth step time increment, dt = 0.1 ms. The influence of the other parameters on the growth of the filament is analyzed by choosing the activation energy for the jump, E_{a}, free energy for formation for the V_{o}^{++} or defect of Ag ions in ZnO, G_{v}, activation energy for diffusion of the defect, G_{m} with the diffusion coefficient given by D_{v} = 0.011 exp (-G_{m}/kT) in cm^{2}/s, the initial concentration of the defects or the Ag atoms in the filament, N_{i} = N_{vo} and the initial temperature, T = 300 K. Using these values, the flux J, current I and the new temperature given by Eq. (9) and the new concentration of defects given by Eq. (14) are evaluated after every interval of time, dt. The increment in height, dh, given by Eq. (2) is also evaluated to give the new value of height, h, and radius, r_{o}, of the filament.

The iterative procedure is used to evaluate all the parameters for any number of steps provided the temperature T does not reach a high value leading to very fast vertical growth or starts decreasing and thus prevents the filament growth. It should be mentioned that, as pointed out previously [1], the field-induced movement of the charged defects, either oxygen vacancies or Ag ions, is considered negligible as the term, Sinh(ZqaE/2kT), is assumed to be close to unity. The initial concentration of defects, N_{i}, in the filament is not known and may take different values depending on the microstructure, temperature and diffusivity. However, this factor is included in the analysis through the initial concentration in the nucleus of the filament. Table 1 and 2 list the various thermodynamic properties chosen from the literature, for V_{o}^{++} [24] and Ag^{+} in ZnO [25], so that the uncertainty in the values is taken into consideration. In particular, the activation energy for diffusion of oxygen vacancies and Ag ions is very sensitive to the presence of grain boundaries and dislocations [26]. Also, the values are chosen to emphasize the influence of each property on the filament growth.

## Results

It is seen, from Fig. 2, that filaments 3 and 6 showed a rapid increase in temperature. Of these, filament 3 exhibited higher rate as the initial concentration of vacancies injected is much higher and the applied voltage, 2 V, is also higher. Although the initial concentration is higher in filament 2, the rate of increase is lower compared to that in filaments 3 and 6 as the applied voltage is only 1 V with all the other parameters kept same. Similarly, comparison of results from filaments 4 and 6 shows that a higher applied voltage was responsible for a higher rate of increase in temperature in filament 6, whereas that in filament 4 was very low initially. Comparison of results from filaments 6 and 7 shows that, for all the same chosen parameters except the free energy of formation of a vacancy, G_{v}, the rate of increase in temperature is lower in filament 7 because the value of G_{v} is higher and thus the equilibrium vacancy concentration in the filament is lower although the diffusion of vacancies is similar because the value of G_{m} is same. Results from filament 1 are not shown because the initial vacancy concentration is below the minimum (2 × 10^{20} cm^{−3}) required for any change in temperature even when the applied voltage is increased by a factor of 5.

Filament 4 is an interesting situation wherein the minimum vacancy concentration is present and the temperature increased slowly. However, the diffusion of vacancies away from the filament dominated when the temperature increased and the net vacancy concentration in the filament decreased with increment in time, as shown in Fig. 3, with the net result that the temperature started to decrease in the later stages. Comparison of results from filaments 4 and 8 shows that for all the same parameters, increasing the activation energy for diffusion of vacancies, G_{m}, results in the vacancy concentration in the filament 8 to increase in the later stages and thus the temperature continues to increase. Comparison of results from filament 5 with that from filament 8 indicates that a much higher initial concentration of vacancies increases the temperature earlier in filament 5 even with the presence of higher activation energy for the jumps, E_{a}. However, the exponential term associated with activation energy for jumps of defects reduces the flux, J in Eq. (1), even with higher initial vacancy concentration and results in lower rate of increase in temperature. Comparison of results from filaments 3 and 5 illustrates this effect of lower value of E_{a} and a higher applied voltage in filament 3 although with smaller initial concentration of vacancies N_{vo}.

Thus, a higher initial vacancy concentration, N_{vo}, lower activation energy for jumps E_{a}, higher activation energy for vacancy diffusion, G_{m}, from the filament and lower free energy of formation of vacancies, G_{v}, gives rise to faster and earlier increase in temperature. Higher voltage also helps to improve the rate of increase in the temperature. However, the relative effect of the different parameters on the filament growth is more complex as described above.

The change in the vacancy concentration for each filament configuration is shown in Fig. 3. The increase in temperature of the filament has two effects. First, the equilibrium concentration increases, however, the second effect is the diffusion of vacancies out of the filament. The combined effect is a complex variation in concentration of vacancies. In filament configurations 2, 3, 6 and 7, initially the temperature increase is not significant enough to increase the equilibrium vacancy concentration, but diffusion out of the filament dominates. Soon the temperature increase is responsible for a significant increase in vacancy concentration that dominates the diffusion from the filament. In filament 6, the increase in temperature in the final stages is very rapid to enable a second minimum. The increase in temperature in filaments 5 and 8 is not large enough, as a result, the net concentration is dominated by the diffusion out of the filament so that the value decreased. It is important to note that although vacancy concentration decreases with increase in temperature, the filament growth continues provided the increase in the exponential term in Eq. (1), given by exp(-E_{a}/kT), is larger than the decrease vacancy concentration.

As pointed out earlier, the effect of external field through the term, Sinh(ZqaE/2kT), that is assumed to be unity is negligible and it may be influencing the growth differently. The increase in temperature in filament 4 is not significant and the diffusion of vacancies is responsible for a net decrease in the vacancy concentration. Comparison of results from filaments 4 and 6 shows the influence of applied voltage to give rise to a faster increase in temperature so that vacancy concentration started to increase after reaching a minimum in filament 6, whereas the temperature did not increase significantly in filament 4 and the vacancy concentration decreased continuously.

The current carried through the device in the presence of the filament with oxygen vacancies is shown in Fig. 4. The current carried is a net result of change in temperature and vacancy concentration. The change in temperature has a more significant effect on the current as it appears in the exponential term, exp (-Ea/kT). Thus, although the vacancy concentration decreases, current carried by the filament continued to increase.

The growth or the increase in the height, h, of the filament, shown in Fig. 5, illustrates the increase starts in early stages in filaments 2, 3, 6 and 7 as a result of thermal energy from increase in temperature. The growth is delayed in filaments 5 and 8, however, it is stable. Growth initially take place in filament 4 as the temperature increases. However, growth is not sustained subsequently because increase in temperature is not enough and vacancy concentration decreases due to diffusion from the filament. Comparison of results from filaments 4 and 8 clearly shows that when the activation energy for diffusion of vacancies, G_{m}, is increased from 0.5 to 0.6 eV, vacancy diffusion from the filament is reduced and temperature continues to increase enabling the sustained filament growth in filament 8 only.

Results from filament 1, presented in Fig. 6, show that a minimum concentration of Ag ions at 10^{20} cm^{−3} is required before the temperature starts to increase. However, after an initial increase, the temperature remains constant even upon increase in applied voltage up to 5 V, as shown in filament 4. The lower activation energy for diffusion of Ag ions is responsible for Ag diffusion from the filament with the result that temperature does not increase. Comparison of results from filaments 4 and 5 shows an increase in temperature continuously as the Ag diffusion from the filament 5 is reduced as a result of increase in the activation energy for diffusion, G_{m}, from 0.45 to 0.71 eV with the net effect that the concentration of Ag ions increases after reaching a minimum.

Also, comparison of results from filaments 4 and 8 illustrates that an increase in the initial concentration to 10^{21} cm^{−3} is responsible for rapid increase in temperature.

Another choice to increase the temperature of the filament is to reduce the free energy of formation of Ag ions, G_{V}, to 0. 26 eV and increase the voltage V to 2 V as in filament 9. The results from filament 9 illustrate that initially the temperature increased less rapidly, but it took several steps to finally increase rapidly. The reason, as will be seen in the concentration of Ag ions presented in Fig. 7, is that in every increment in time, dt, the temperature increases, and diffusion of Ag ions from the filament also increases with the net result that the concentration of Ag ions oscillates before it finally increases rapidly. If only the free energy of formation, G_{V}, is reduced to 0.26 eV with voltage kept at 1 V, as in filament 10, the temperature increases continuously but slower than in filament 9 initially and more rapidly subsequently compared to the results from filament 9. These different results arise from the competing effects of increasing the concentration of Ag ions and the diffusion of Ag ions from the filament due to increase in temperature. However, it is important to note that the temperature increase is much more rapid in filament 9 as the effect of higher applied voltage takes over after several increments of time. The effect of lower free energy of formation of Ag defect is also seen by comparison of results from filaments 5 and 6 with faster increase in temperature in filament 6 although the activation energy for diffusion is slightly lower. Higher initial concentration of Ag ions in filaments 7 and 8 gives rise to faster increase in temperature although the lower activation energy for diffusion in filament 8 is responsible for reduced rate of increase in temperature.

The changes in the concentration of Ag ions in the filament are shown in Fig. 7 with increment in time. The Ag ion concentration decreased continuously as a result of diffusion from the filaments 1, 2, 3 and 4 and higher free energy of formation of Ag ions, G_{v}. Higher applied voltage was responsible for higher temperature and higher diffusion so that Ag concentration decreased faster in filament 4. However, Ag concentration in filament 10 increased after reaching a minimum with increase in temperature as a result of lower free energy of formation, G_{v}. On the other hand, the Ag ion concentration in filament 9 oscillated even with lower free energy of formation as a result of higher voltage and higher temperature resulting in diffusion of Ag ions from the filament.

The waviness of Ag ion concentration may be reduced with smaller intervals of time that are used in the numerical simulation. Comparison with the results from filament 6 shows that higher activation energy, G_{m}, of 0.71 eV for diffusion increases the concentration of Ag ions after reaching a minimum. Furthermore, when the free energy of formation is reduced to 0.26 eV with activation energy G_{m} at 0.71 eV, the Ag ion concentration increased continuously with no noticeable dip in filament 6. The higher initial concentration of Ag ions in filament 8 with lower activation energy for diffusion, G_{m}, lead to significant diffusion of Ag ions and the concentration reached a minimum and started to increase as a result of higher temperature. Higher initial concentration and higher activation energy, G_{m}, are both responsible in filament 7 to increase the temperature and Ag ion concentration with a smaller minimum.

The results shown in Fig. 8 illustrate that although the current increases initially, the value decreases as the temperature and the Ag ion concentration decrease in filaments 1, 2, 3 and 4. Thus, increase in applied voltage is insufficient to increase the temperature. In comparison, the increase after a minimum in concentration of Ag ions as a result of decrease in the energy of formation is responsible for continuous increase temperature and current carried by the device in filament 10. All other filaments showed the expected current variation.

The change in the height of the Ag filament is shown in Fig. 9. Although there is an increase in the temperature initially, the height has not changed in filaments 1 to 4 for reasons mentioned previously. Comparison of results from filaments 1 and 4 shows that the filament height is lower for filament 4 even with higher voltage as the Ag concentration is lower at higher temperature. The increase in Ag concentration in filaments 9 and 10 is responsible for faster growth of the filament after several increments of time. The higher growth rate of filament 9 compared to that of filament 10 in the final stages is mainly due to the higher applied voltage. The oscillating Ag concentration is responsible for stepwise growth in filament 9 in the initial stages. Higher Ag concentration due to lower free energy of formation, G_{V}, in filament 6 compared to that in filament 5 is also seen to give rise to faster growth rate. Comparison of results from filaments 7 and 8, both with higher initial Ag concentration, shows that lower activation energy for diffusion, G_{m}, is responsible for slower growth in filament 8.

The change in the dimensions of the filament, h and r_{o}, is shown in Fig. 10. Similar variations are observed for all filaments formed either by oxygen vacancies or Ag ions. The value of r_{o} is found to be slightly higher than that of h in the initial stages of growth indicating that radial growth is favorable. However, as the tempearture and oxygen vacancy or Ag ion concentration increase, the increase in the value of r_{o} becomes much faster than that of h illustrating that the filament will grow radially. The possibility of filament bifurcation is not considered in the present calculations.

## Discussion

The above numerical simulation using the analytical solution of the thermal diffusion and the atomic mass diffusion equations shows that the filament growth is a complex phenomenon dependent on several interrelated properties. The increment in time, dt = 0.1 ms, is chosen to get a meaningful result within a few increments. Smaller increments of time will smoothen the graphs, however, it is expected that the results will be close to that presented. In the present analysis, both types of filaments are modeled using the same microscopic processes. The values of several thermodynamic parameters are chosen from results published.

The first important parameter that controls the growth of either type of filament is the initial concentration of the defects either oxygen vacancies or Ag ions, introduced into the nucleus that is stable. As mentioned previously, the influence of the electric field could be important in the formation of the nucleus. If the Pt electrode interface is the source of vacancies or the metal electrode is the source of the Ag ions, diffusion due to concentration gradient and the drift flux term *J* = CMF, where *C* is the concentration at the interface, *M* = D/kT is the mobility and *F* = ZqE is the force, will be responsible for movement of defects and formation of the filament at the electrode of opposite polarity. Grain boundaries and dislocations are important high diffusivity paths. Therefore, the initial concentration is assumed to be above the minimum required for an increase in temperature. Higher electric field is expected at the tip of the filament that is responsible for an increase in the initial concentration before the growth starts. The equilibrium concentration of defects at singularities [18, 27] takes the general form *C* = N_{L} exp(ZqaE/kT) with the singularity in E represented in the form (d/r)^{1/(N+1)} where N is a constant between 30 and 50 and *r* is the radial distance from the spherical tip of the filament. However, the enhancement in the electric field at the tip is found to be insignificant.

The second important parameter is the temperature of the filament. In the present analysis the initial temperature is maintained constant at 300 K. The increase in temperature due to joule heating is found to depend on the flux of charged defects, given by Eq. (1), that in turn depends on the concentration of charged defects and the probability of atomic level jumps which are exponentially dependent terms on the free energy of formation of defects, G_{v}, and the activation energy for atomic level jumps, E_{a}, respectively. In addition, thermal diffusion takes place simultaneously reducing the temperature. The thermal diffusion is modeled by considering the filament as a concentrated line source and the thermal energy flux from the filament is determined. The balance of joule heating and thermal flux from the filament are used to calculate the net temperature increase in the filament. As expected, higher initial concentration of defects or lower value of G_{v} and lower activation energy for jumps, E_{a} are found to be responsible for larger increase in temperature. It is important to appreciate that one of the terms can be a dominant factor and be responsible for increase in temperature. A rapid increase in temperature is responsible for higher concentration of defects and diffusion of defects from the filament.

The third important parameter is the diffusion of defects from the filament that has been modeled by treating the filament as a line source and solving the mass diffusion equation in cylindrical polar coordinates. The flux of defects from the filament and the number of defects diffused out from the filament are evaluated to get the net concentration of defects in the filament. The activation energy for diffusion of defects and the temperature of the filament are found to be important. If the activation energy, G_{m}, is higher, there is less diffusion and consequently the concentration in the filament remains higher leading to higher temperature and growth rate. The microstructure and the defect structure of the medium are very important as the diffusional processes take place with lower activation energy. The microstructure in the dielectric material is also important as gain boundaries and defects are responsible for charge accumulation. The line and surface defects act as sources of defects.

The fourth important factor is the applied voltage. Filament growth is initially slow and starts to increase when the temperature reaches higher value and the successful jump frequency *ν*_{o}exp(-E_{a}/kT) increases more than the decrease in defect concentration. Finally, when the temperature reached sufficiently high values, both the above terms are favorable and growth becomes rapid. The term due to electric field in the Eq. (1) for flux, given by, Sinh(ZqaE/2kT), is assumed to be equal to unity and thus did not give rise to a direct effect on the filament growth. However, joule heating given by IVdt is responsible for increase in temperature and the filament growth. Thus, higher applied voltage is responsible for higher electric field as well as increased joule heating, thus, increasing temperature more rapidly. The present analysis is applicable to the formation of the filament between electrodes at low electric field.

## Availability of data

Available upon request from the author.

## Code availability

Available from the author.

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Jagannadham, K. Microscopic mechanisms of filament growth in memristor.
*Appl. Phys. A* **127, **229 (2021). https://doi.org/10.1007/s00339-021-04367-2

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### Keywords

- Memristor device
- Filament growth
- ZnO
- Modeling
- Active electrode