Model of Multifilamentary Resistive Switching for a Memristor with Hopping Conductivity

Abstract

Many applications, including learning neuromorphic computing systems, require the analogue or, at least, multilevel resistive switching (RS) of memristor devices. In metal–insulator–metal (MIM) structures it is often associated with the creation of a large number of conductive channels, or filaments, but in the majority of cases, especially for the vacancy mechanism of RS, the direct observation of many filaments is difficult or impossible. In this work, a model of the explicit filament conductivity distribution for a nanocomposite (NC) LiNbO₃-based memristor, that nonetheless may be applicable to other types of MIM memristors with hopping conductivity over dangling bonds in an amorphous insulator layer, is derived from physical principles. This model, in addition to indirect evidence from magnetic and capacitance measurements, reaffirms the multifilamentary RS mechanism by the approximation of experimental data on synaptic plasticity, i.e., a change in the conductivity of a memristor by a pair of pulses applied to it. The results of this research offer opportunities to study RS in MIM structures at the macroscopic level using model filament distributions and substantiates the possibility of successfully applying the considered NC memristors as synaptic elements of neuromorphic networks.

INTRODUCTION

Currently, there is increased interest in the study of memristive structures of the metal–insulator–metal type (MIM), demonstrating reversible resistive switching (RS) effects, in connection with the prospects of their use for creating elements of multilevel memory and arrays of synaptic memristors in the crossbar variant, necessary for building neuromorphic computing systems (NCS) aimed at solving artificial intelligence problems by hardware emulation of the principles of operation of biological neural networks [1–8].

The RS effects usually observed in MIM structures are due to the electromigration of oxygen vacancies (anions) or metal cations [9, 10]. As a result, filamentary conductive channels (filaments) are formed (or destroyed) in the oxide insulating layer, and the structure switches accordingly to the low-resistance state (LRS) or high-resistance state (HRS). The nature of the formation of filaments is largely random, which is one of the main reasons for degradation of the properties of memristors during cyclic RS [9, 10]. Another disadvantage of anion or cation memristive MIM structures is that their stable operation requires, as a rule, forming, which consists in applying a relatively high voltage to the structure, upon which filaments (bridges) are formed. However, other RS mechanisms are also possible that do not require forming. For example, those associated with the recharging of localized electronic states in the region of the Schottky barrier and/or in the bulk of the oxide [11, 12] and references there), the electrical repolarization of a ferroelectric oxide [13, 14], redox reactions in organic materials [15], the electron-drag effect [16], and, finally, with a temperature-induced metal–insulator transition in Mott materials (NbO₂, VO₂, V₂O₃), which was discovered relatively long ago, but is still the subject of discussion and research [17]. Nevertheless, the greatest interest is shown in oxide memristive structures as systems that can have a multilevel character of the RS at long retention times of resistance states, and are also quite technologically advanced in fabrication and can be easily integrated into modern microelectronic technology [1, 9].

Despite the significant accumulated experimental material, there is still no microscopic theory of reversible RS effects. The latter, in particular, is due to both difficulties in describing the interrelated nonequilibrium processes of thermal, electron, and ion transport at the nanoscale, as well as additional effects: the participation of several RS channels associated, for example, with the simultaneous manifestation of cation and anion transports [18], synergistic contributions in the RS of the electric polarization of a ferroelectric and ion transport [14], phenomena such as quantization of the structure conductivity [19, 20], etc.

In many memristor structures of the MIM type, a filament type of RS has been demonstrated, a particular variant of which is the formation of single metal bridges between electrodes, observed experimentally in [21]. Generally speaking, direct observation of the filament type of RS seems to be a big problem, since in fact the only option for such observation is electron-microscopy studies, but they are difficult in the case of low electron contrast of the structures under study. An important subtype is the multifilamentary mechanism of resistive switching of MIM memristors, when the transition between different resistance states is due to the formation of not one, but many (>10) conducting channels. A large number of filaments determines the possibility of a gradual change in the conductivity and, accordingly, the multilevel or close to analog character of RS, which is necessary in neuromorphic applications. But even the very fact of confirming and studying the multifilamentary type of RS is associated with great experimental difficulties. For some types of MIM structures, including nanocomposite samples with a dielectric layer at the bottom electrode (metal–nanocomposite–metal structures M/NC/M) studied in this work, indirect experimental evidence was obtained for the formation of precisely a large number of filaments, rather than single filaments in the process of RS [22, 23].

In this paper, along with a brief review of previously obtained indirect data in favor of multifilamentary resistive switching in M/NC/M memristor elements based on LiNbO₃ with metal granules, an original physical model of RS is proposed, which takes into account the distribution of filaments in terms of conductivity values. Such a model makes it possible to interpret the data of pulsed measurements from a variation in the conductivity of the NC memristor and to explain a number of features in the corresponding dependences in the case of a transition from HRS to LRS and vice versa. Moreover, a comparison is made using a monofilamentary model built on the same physical principles, on the basis of which a conclusion is made about the nature of the RS in the structures under study. In conclusion, considerations are given about the universality of the proposed model and the possibility of its application to the study of the microscopic mechanism of RS for other filament-type memristor elements, as well as some open questions and prospects related to the development of a new approach.

RESULTS AND ITS DISCUSSION

Experimental Evidence for a Multifilamentary RS Mechanism

In the case of metal–nanocomposite–metal structures (M/NC/M) on the basis of metal-oxide nanocrystals, the transition to the conducting state should be determined by percolation chains, i.e., the given spatial position and concentration of metal nanogranules in nanocrystals; therefore, the RS stability should be high [24]. The importance of percolation effects in the RS of M/NC/M structures was first demonstrated in [24–26]. We note that previously tangible results were achieved in M/Pt–SiO₂/M structures, in which the active layer was created by the magnetron sputtering of a composite Pt/SiO₂ target and consisted of a SiO₂ matrix with dispersed atomic nanoclusters of Pt [27]. It is shown that in this case the maximum number of RS N_max exceeds 3 × 10⁷ at a retention time of resistance states of t_r > 6 months. The mechanism responsible for RS was not established in [27]. Presumably, it is associated with the motion of oxygen vacancies and the formation of conductive filament channels formed during RS, as well as with the local heating and coarsening of Pt nanoparticles in them (from 2–3 to 3–4 nm). In [28], M/W–SiO₂/n-Si structures with a heavily doped bottom contact n-Si (ρ < 0.05 Ω cm), in which the active layer was created by implantation of SiO₂ by W ions, were studied. It is shown that such structures do not require forming. In this case, even in the presence of metal nanoparticles W in the active switching layer W–SiO₂, a sufficiently large resistance ratio in the HRS (off) and LRS (on) states R_off/R_on > 10⁴ at N_max ≈ 4 × 10² and t_r ≈ 10⁴ s is achieved [28]. We note that the plasticity of the memristor structures developed in [27, 28] has not been studied. The possible relationship between the RS structures and the effects of their percolation conduction has not been studied either.

In studies of M/NC/M structures based on (CoFeB)_x(LiNbO₃)_100–x NC [24, 25], a memristive effect was found with the ratio R_off/R_on reaching ~100 at some optimal value x = x_opt ≈ 8–15 at % below the NC percolation threshold. The effect is well reproduced when the number of RS cycles is >10⁵ and barely depends on the type of contacts [24, 25, 29]. In this case, the molding effect is barely manifested, i.e., the voltage U_RS. The switching acts from HRS to LRS during the first and subsequent switching acts are similar, in contrast to metal—oxide—metal (MOM) structures based on homogeneous oxides [2, 3, 9]. Finally, the synthesized M/NC/M structures possessed a high degree of plasticity (smooth character of variation in the resistance state in the window R_off–R_on), which made it possible to emulate the important properties of biological synapses [4, 6, 8].

The multilevel nature of RS in combination with a long retention time of resistance states is achieved in structures with a developed oxide layer at the bottom electrode, which is formed during initial growth of the nanocrystal at a fairly high partial oxygen pressure P_O2 ≈ 2.5 × 10^–5 Torr in the mode of its given flow (Fig. 1). In this case, the optimal metal content for observing RS shifts to the region of large values x = x_opt ≈ 15 at % [30].

Fig. 1.

Dark-field high-angle STEM image of the nanocomposite boundary CoFeB-LiNbO₃ with the bottom electrode Cr/Cu separated by a thin layer of LiNbO₃ ~15 nm thick (a). M/NC/M structure in the intermediate resistance state (b). The green color shows the amorphous matrix of LiNbO₃, containing CoFeB metal nanogranules (black ovals) and a nonequilibrium phase of Co and Fe atoms with a concentration of n_i ~(10²¹–10²²) cm⁻³ (grey circles). The green dotted line separates the high-resistivity layer near the bottom electrode of the structure from the predominantly stoichiometric LiNbO₃, in which there is no nonequilibrium atomic-metal phase. The gray areas surrounding the chains of granules represent a metal condensate, which arises due to the nucleation of Co and Fe atoms and oxygen vacancies during current flow through the structure.

Subsequently, however, it turned out that the existence of an oxide layer does not guarantee the observation of stable RS in M/NC/M structures. Another important NC parameter for observing RS is the presence of dispersed magnetic atoms in the insulating matrix in sufficiently large amounts (N_d ~ 10²² cm^–3); at low N_d ~ 10²¹ cm^–3 RS becomes unstable and practically ceases to be observed [31]. This fact, large limiting currents in the case of RS (up to 100 mA; inset in Fig. 2a), as well as the recent discovery of the effect of a strong increase in the capacitance (by a factor of 8) upon switching the M/NC/M structure from HRS to LRS (Fig. 2a), led us to a qualitative multifilamentary RS model [30], which is illustrated in Fig. 1b.

Fig. 2.

Dependence of the relative change in the resistance and capacitance of the M/NC/M structure during its switching between the HRS and LRS on the content of the metal phase at a frequency of 1 kHz. The inset shows 6 I–V cycles (a). The dependence of magnetization on the magnetic-field strength M(H) for M/NC/M structures with x ≈ 15 at % measured at T = 2 K before and after the RS cycle (b). The inset shows the dependence of the decrease in magnetization ΔM(H) = (M_PS – M_RS) after RS, where M_PS and M_RS are the saturation magnetizations in the pristine state and after RS, respectively.

In the pristine state, immediately after fabrication of the M/NC/M structure, the dispersed atoms are uniformly distributed in the insulating matrix of the NC. However, when a voltage is applied and a current is passed, the nucleation of dispersed atoms can occur around chains of granules that form percolation paths (Fig. 1b), and, as a consequence, the formation of conducting “metallized” chains (MC). Obviously, the manifestation of nucleation effects is due in this case to strong supersaturation of the system under consideration with the atomic phase of metals and the presence of metal nuclei (nanoparticles) in it. Similar effects were observed, for example, in memristive structures based on SiO₂ with dispersed Pt or W atoms [27, 28].

When a rather large positive voltage is applied to the upper electrode, the structure tends to transform into the LRS due to the movement of oxygen vacancies and cations to the lower electrode along MCs and a decrease in the effective gap l_eff between the MCs and the lower electrode in the amorphous dielectric layer (Fig. 1b). On the other hand, the capacitance of the structure in this situation should increase, since, in the first approximation, the capacitance value C ∝ 1/l_eff. This fact is clearly manifested when studying the correlation in the behavior of the resistance ratio R_off/R_on with a change in the ratio of the capacitance of the M/NC/M structures in the HRS (C_off) and LRS (C_on) states performed for samples with different contents of the metal phase x (Fig. 2a). It can be seen that the behavior is anticorrelated (with increasing R_off/R_on ratio increases C_on/C_off) in accordance with the model proposed in [30]. We note that in the case of the presence of one or a small number of filaments with a small total cross-sectional area compared to the electrode area, a significant change in the total capacitance would not be observed due to the small capacitance of the gaps between filaments and the lower electrode compared to the total capacitance of the structure.

Additional confirmation of the multifilamentary model is the observation of a decrease in the magnetization of the M/NC/M structure during its RS (Fig. 2b), which can be interpreted as follows. The dispersed atomic metal phase is mainly Fe²⁺ and Co²⁺ ions with a magnetic moment per ion of m_i = 5.4 and 4.8 μ_B, respectively, where μ_B is the Bohr magneton ([30] and references there). Meanwhile, the magnetic moment of Co and Fe atoms is much lower: m_m = 1.72, 2.22, and 2.33 μ_B for Co and Fe metal atoms and CoFe alloy, respectively. Therefore, one can expect a decrease in the magnetization of the M/NC/M structure after its RS if we assume, as described above, partial nucleation of the magnetic Fe²⁺ and Co²⁺ ions into metal granules during the first RS, which in this case is a variant of soft forming (without a significant change in the switching voltage U_RS). A drop in the magnetization was actually found in samples in which the contribution of the paramagnetic component to the magnetization exceeded the contribution of the ferromagnetic component or was comparable to it (Fig. 2b).

The multifilamentary nature of the RS also follows from analysis of the measurement data of the limiting currents in the samples under study. In [30] breakdown currents were measured for M/NC/M structures with different contents of the metal phase x. The presented data clearly indicate the existence of two breakdown mechanisms, one of which occurs when the content of the metal phase is less than 13.5 at %. The limiting current of irreversible RS in this case is I_break ≈ 20–30 μA. However, at high values x ≈ 13.5 at % and more, there is a sharp (almost 4 orders of magnitude) increase in I_break up to ~100 mA. The observed effect of a strong increase in the limiting current during RS with increasing x is naturally associated with the manifestation of the multifilamentary mechanism of RS. If we assume that such a current flows through single MCs, the transverse size of which slightly exceeds the diameter of the granules and is ~10 nm, as for the filaments of most memristors [9, 10, 32], then the current density would reach about 10¹¹ A/cm². This is a physically unrealizable value in comparison with the limiting current (~10⁷ A/cm²) that metal conductors can withstand without being destroyed by electromigration effects ([16] and references there). Hence it follows that, during the RS of the M/NC/M structure to the LRS, its conductivity is determined by many MCs (>10⁴).

The multifilamentary nature of RS not only ensures the stable RS of M/NC/M structures, but also a smooth change in their resistance in the window R_off–R_on, which in turn determines the possibility of their use in neuromorphic electronics [1–7]. Despite the presented experimental data, which testify in favor of the formation of a large number of filaments in the structure M/NC/M during RS, they do not reveal the microscopic nature of the transition between resistance states, especially from the HRS to LRS state, in which one can expect a rather sharp change in the type of conductivity due to the formation of filaments adjacent to the lower electrode (see below). In this case, modeling of the corresponding processes can help to establish, at least qualitatively, the monofilamentary or multifilamentary nature of RS.

Model of Monofilamentary RS in a NC Memristor

According to the qualitative picture presented in the previous section, the RS of memristive structures based on a NC is explained by the formation of a high-resistance amorphous LiNbO₃ interface layer (thickness of l_max ≈ 15–20 nm) containing multiple gaps between the edges of low-resistance filaments and the lower electrode of the M/NC/M structure. This dielectric layer determines the variation in the capacitance and resistance of the entire structure by changing its effective thickness l_eff ≤ ∼20 nm (when varying the lengths of individual filaments) by applying an appropriate electric field. It is important that the conductivity of the structure in the intermediate resistance state has a hopping character, determined by a high density of localized states near the Fermi level, which are caused by dangling bonds in the amorphous dielectric layer and oxygen vacancies. Obviously, under these conditions, inelastic resonant-tunneling effects can occur, which are described by the exponential dependence of the gap resistivity on its thickness l [33]:

$${{r}_{{\text{g}}}}\left( {l,\,\,T} \right) = {{{{\rho }}}_{{\text{d}}}}\left( T \right)\exp \frac{{2l}}{{a\left( {n + 1} \right)}},$$

(1)

where a is the radius of the localized state, and n is the number of available energy states in the resonant chain of localized defects.

At the limit of low temperatures T and the number of resonant states n = 2, the temperature dependence of the resistivity of the dielectric layer ${{{{\rho }}}_{{\text{d}}}}{\kern 1pt} \left( T \right)$ has a power character: ${{{{\rho }}}_{{\text{d}}}}{\kern 1pt} \left( T \right)$ ∝ T^–4/3 [33]. As the temperature rises, so does the value n. At n > 6 dependence ${{{{\rho }}}_{{\text{d}}}}{\kern 1pt} \left( T \right)$ gradually transforms into an activation dependence of the conductivity along isolated impurity chains [34, 35], which, in turn, transforms into bulk conductivity with a variable hop length, $\ln {{{{\rho }}}_{{\text{d}}}}{\kern 1pt} \left( T \right)$ ∝ (T₀/T)^α, with α =1/4 for the Mott law and α =1/2 in the presence of a Coulomb gap in the density of states near the Fermi level (the Efros–Shklovskii law) [33, 36, 37].

Considering a model filament in the form of a cylinder with the cross-sectional area S and using a simple approach to describe the change in the resistance of a single filament with a change in the thickness of the gap l between the filament boundary and the lower electrode of the structure [38], and also taking into account the comments made above, we obtain formulas for the conductivity $g$ of the filament and its small change in the following form [6]:

$$g\left( {l,\,\,T} \right) = \frac{S}{{{{{{\rho }}}_{{\text{f}}}}\left( {d - l} \right) + {{{{\rho }}}_{{\text{d}}}}l\exp \frac{{2l}}{{{{a}_{{{\text{eff}}}}}}}~}} \approx \frac{{{{g}_{{{\text{on}}}}}c}}{{c + x{{e}^{x}}}},$$

(2)

$$\left| {\Delta g\left( {T,x} \right)} \right| \approx \frac{{{{g}_{{{\text{on}}}}}c{{e}^{x}}\left( {x + 1} \right)}}{{{{{\left( {c + x{{e}^{x}}} \right)}}^{2}}}}\Delta x,$$

(3)

where ${{a}_{{{\text{eff}}}}} = a\left( {n + 1} \right)$, $x = 2l{\text{/}}{{a}_{{{\text{eff}}}}}$, $c = 2{{{{\rho }}}_{{\text{f}}}}d{\text{/}}\left( {{{{{\rho }}}_{{\text{d}}}}{{a}_{{{\text{eff}}}}}} \right)$, ${{g}_{{{\text{on}}}}} = S{\text{/}}{\kern 1pt} \left( {{{{{\rho }}}_{{\text{f}}}}d} \right)$, and the sign of approximate equality takes into account the fact that $l \ll d$. Here d ∼ 1 μm is the thickness of the NC layer, which approximately corresponds to the total thickness of the structure and the total length of the filament (Fig. 1a, as well as [30]). Formula (3) is obtained by differentiation (2), so here and below we will consider the approximation of only small increments.

A change in the filament length can occur under the action of one applied voltage pulse or a pair of pulses overlapping in time and, accordingly, superimposed on each other, if one of them is applied to the upper and the second to the lower electrode of the structure. Such a situation takes place, for example, in the case of using the dynamic plasticity of an STDP-type memristor to tune it as part of a pulsed neural network [4, 6]. But here we consider an experimental dependence of this type (Fig. 3) with the aim of its further approximation by model dependences.

Fig. 3.

Dependence of the amplitude of the maximum change in the conductivity of the NC memristor on its initial conductivity value. The curves were obtained using overlapping paired pulses according to the STDP plasticity recording type. The dots represent the experimental values, the straight lines are eye guides. The delay between pulses Δt > 0 corresponds to a potentiation (increase) in the conductivity, while a negative delay Δt < 0 corresponds to depression (decrease) in the conductivity of the memristor.

We note that the effect of a pair of superimposed presynaptic and postsynaptic pulses on the memristor (with a delay between them of several ms) is practically equivalent to applying a unipolar pulse to it [4]: in the case of the data in Fig. 3 with V_sp ≈ 5 V and duration Δt ≈ 1 ms during potentiation or V_sp ≈ –5 V during depression. We note also that the data presented in Fig. 3 were obtained for different initial conductivity values (at the moment of the start of the pulse pair), but in all cases near the edge of the HRS, since the high-conductivity state of this NC memristor is characterized by the value G_on = 10 mS $\gg$ G₀.

The change in the gap width between the filament and the lower electrode can be estimated taking into account the drift in the electric field of oxygen vacancies [6]: Δl ≈ –μ(V_g/l)Δt, where μ is the mobility of oxygen vacancies, and ${{V}_{{\text{g}}}} = {{V}_{{{\text{sp}}}}}{\text{/}}\left( {1 + c{\text{/}}x{{e}^{x}}} \right)$ is the voltage across the gap.

It can be assumed that the high resistance of the NC structure with one filament corresponds to a gap width of at least ${{l}_{{{\text{max}}}}}~\sim ~20$ nm (typical thickness of the oxide layer at the lower electrode of the NC structure). Under this and other necessary conditions (a ≈ 1 nm, n = 4, and g_on/g_off ~ 2000 (see below)) this gives the maximum value ${{x}_{{{\text{max}}}}} \equiv L = 2{{l}_{{{\text{max}}}}}{\text{/}}{{a}_{{{\text{eff}}}}} \cong$ 8, using Eq. (2) the value $c \approx \frac{{{{g}_{{{\text{off}}}}}}}{{{{g}_{{{\text{on}}}}}}}L{{e}^{L}} \approx$ 12, so x = 1 (corresponding to l ≈ 2.5 nm) and lower values reasonably approximate the LRS state ($g \geqslant 0.8~{{g}_{{{\text{on}}}}}$) of a monofilamentary NC structure. This reasoning allows us to consider $x \geqslant 1$ in the following estimation:

$$\begin{gathered} \left| {\Delta g\left( {x,T} \right)} \right| \approx \frac{{{{g}_{{{\text{on}}}}}c{{e}^{x}}\left( {x + 1} \right)}}{{{{{\left( {c + x{{e}^{x}}} \right)}}^{2}}}}\frac{{4{{\mu }}\Delta t}}{{a_{{{\text{eff}}}}^{2}}}\frac{{{{V}_{{{\text{sp}}}}}}}{{x\left( {1 + \frac{c}{{x{{e}^{x}}}}} \right)}} \\ = \frac{{{{g}_{{{\text{on}}}}}c{{{\left( {x{{e}^{x}}} \right)}}^{2}}}}{{{{{\left( {c + x{{e}^{x}}} \right)}}^{3}}}} \cdot \frac{1}{x}\left( {1 + \frac{1}{x}} \right)\frac{{4\mu {{V}_{{{\text{sp}}}}}\Delta t}}{{a_{{{\text{eff}}}}^{2}}} \\ \approx \Gamma \frac{g}{{{{g}_{{{\text{on}}}}}}}{{\left( {1 - \frac{g}{{{{g}_{{{\text{on}}}}}}}} \right)}^{2}}, \\ \end{gathered} $$

(4)

where Γ $= 4{{g}_{{{\text{on}}}}}{{\mu }}{{V}_{{{\text{sp}}}}}\Delta t{\text{/}}a_{{{\text{eff}}}}^{2}\left( {1{\text{/}}{{x}_{{{\text{eff}}}}}} \right)\left( {1 + 1{\text{/}}{{x}_{{{\text{eff}}}}}} \right)$ is a constant, and ${{x}_{{{\text{eff}}}}} \approx$ 5 is the characteristic value x within the range of its change (approximately from 1 to 8). The last transition in (4) is applicable due to the fact that, depending on $\Delta g\left( x \right)$ exponential factors prevail, (4) tends to 0 at x → 0 and, in general, the whole expression does not change much numerically when taking into account the variability of the factor $\frac{1}{x}\left( {1 + \frac{1}{x}} \right)$.

It is important to note in the monofilamentary model that the mobility of oxygen vacancies greatly depends on the effective temperature of the filament. To describe the drift or diffusion of ions/vacancies, the simple one-dimensional rigid point ion model of Mott and Gurney is usually used ([39, 40] and references therein):

$$\begin{gathered} {{\mu }}\left( T \right) \approx \frac{{e{{D}_{0}}}}{{kT}}\exp \left( { - \frac{{{{E}_{{\text{a}}}}}}{{kT}}} \right) \\ \cong \frac{{e{{D}_{0}}}}{{kT}}\exp \left( { - \frac{{{{E}_{{\text{a}}}}{\text{/}}k}}{{{{T}_{0}} + V_{{{\text{sp}}}}^{2}g\Delta t{\text{/}}\left( {{{m}_{{\text{f}}}}{{C}_{{\text{f}}}}} \right)}}} \right) \\ = \frac{{e{{D}_{0}}}}{{kT}}\exp \left( { - \frac{{{\theta }}}{{1 + \left( {g{\text{/}}{{g}_{{{\text{on}}}}}} \right){{\tau }}}}} \right), \\ \end{gathered} $$

(5)

where e is the electron charge, D₀ ~ 2 × 10⁻³ cm² s⁻¹ is the pre-exponential factor of the diffusion coefficient, k is the Boltzmann constant, ${{E}_{{\text{a}}}}$ ~ 0.5–1 eV is the activation energy of oxygen-vacancy migration in amorphous LiNbO₃, and T is the local temperature. Let us assume here, for a rough estimate, the Joule heating of a filament with the mass ${{m}_{{\text{f}}}}$ and effective heat capacity ${{C}_{{\text{f}}}}$ from room temperature ${{T}_{0}}$ to some $T$ without heat dissipation during application to the structure of a pulse with amplitude ${{V}_{{{\text{sp}}}}}$. The pre-exponential factor with the inverse temperature will be assumed to be a constant compared to the exponential dependence on T. The dimensionless activation energy ${{\theta }} = {{E}_{{\text{a}}}}{\text{/}}{\kern 1pt} \left( {k{{T}_{0}}} \right)$ has a value of 20 to 40 for ${{E}_{{\text{a}}}}$ ~ 0.5–1 eV, and the dimensionless temperature to which a highly conductive filament can be heated is ${{\tau }} = V_{{{\text{sp}}}}^{2}{{g}_{{{\text{on}}}}}\Delta t{\text{/}}({{m}_{{\text{f}}}}{{C}_{{\text{f}}}}{{T}_{0}})$ = $\Delta {{T}_{{\max }}}{\text{/}}{{T}_{0}}$, in order of magnitude ∼1 in the case of local heating up to ∼300°C in accordance with typical estimates [39, 40].

We note that in ${{{{\rho }}}_{{\text{d}}}}{\kern 1pt} \left( T \right)$ there is another power-law or weaker exponential dependence on the temperature, as mentioned above. However, taking it into account only slightly changes the position of the maximum on the ∆γ(γ) curve (see below) and does not fundamentally affect the conclusions of this analysis, so we will not take this contribution into account in the qualitative analysis.

Finally, combining (4) and (5), we obtain the conductivity (for convenience, in a dimensionless form) of a monofilamentary NC structure:

$$\left| {\Delta \gamma \left( \gamma \right)} \right| \approx {{\Gamma }_{0}}\gamma {{\left( {1 - \gamma } \right)}^{2}}\exp \left( { - \frac{\theta }{{1 + \gamma \tau }}} \right),$$

(6)

where ${{\gamma }} \equiv g{\text{/}}{{g}_{{{\text{on}}}}}$, ${{\Gamma }_{0}} = 4\frac{{e{{D}_{0}}}}{{k{{T}_{0}}}}\frac{{{{V}_{{{\text{sp}}}}}\Delta t}}{{a_{{{\text{eff}}}}^{2}}}\left( {1{\text{/}}{{x}_{{{\text{eff}}}}}} \right)\left( {1 + 1{\text{/}}{{x}_{{{\text{eff}}}}}} \right)$. It can be seen that (6) has a linear dependence on γ at the very beginning of the curve, i.e., at a conductivity close to the HRS. Further, curve (6) has a maximum approximately at γ ≈ 0.7–0.9, with reasonable values of the parameters included in the formula (Fig. 4). Moreover, $\Delta {{\gamma }}\left( {{\gamma }} \right)~$ tends to 0 as γ approaches 0 or 1, defining the “soft bounds” of the resistance window in the monofilamentary NC memristor model.

Fig. 4.

Dependence (5) Δγ(γ) for the monofilamentary RS model in a NC memristor. The inset on a logarithmic scale shows the initial section of the dependence (solid curve) together with the experimental values in Fig. 3 (points), rearranged in relative conductivity coordinates, taking into account that the maximum conductivity of the NC structure G_max = 10 mS.

Analysis of the inset to Fig. 4 shows that the monofilamentary model cannot satisfactorily explain the experimental changes in the conductivity under the action of voltage pulses applied to the structure: the initial section of the model curve at any values of the fitting parameters changes too sharply compared to the experimental dependences. Moreover, at the approximation parameters that most closely approximate the experimental data near the HRS (inset in Fig. 4), Δγ(γ) has unrealistically large values of conductivity variation at most points in the range of the initial conductivity γ (Fig. 4, especially at the maximum of the curve), at which the approximation of small increments adopted in the consideration ceases to work.

Multifilamentary RS Model in a NC Memristor

Let us consider a RS model in a NC memristor with a certain distribution of filaments along the length or, which is the same, according to the size of the gap between the boundary of the highly conductive part of the filament and the lower electrode of the memristor structure. The gap value uniquely determines the total conductivity of filamentary channels, so we can alternatively consider the distribution of filaments by their conductivity using the probability density function (PDF) η(γ). The conductivity of the entire NC structure (in dimensionless form, where 1 corresponds to the state of low resistance LRS, and HRS for convenience of notation and without loss of generality is denoted by zero value γ) is then defined as

$$G \cong \mathop \smallint \limits_0^1 {{\gamma }}\left( {{\gamma }} \right)d{{\gamma }}.$$

(7)

When a voltage pulse is applied to the structure, the filaments change their length and hence their conductivity to varying degrees according to Eq. (6). Therefore, the original distribution η(γ) is replaced by some new distribution $\eta {\kern 1pt} ^*{\kern 1pt} \left( \gamma \right)$, then the variation in the conductivity of the entire NC structure is given by the expression

$$\Delta G = \mathop \smallint \limits_0^1 \gamma \left( {\eta {\kern 1pt} ^*{\kern 1pt} \left( \gamma \right) - \eta \left( \gamma \right)} \right)d\gamma .$$

(8)

We note that when there is a change in $\Delta {{\gamma }}$, part of the filaments $\eta \left( {\gamma - \Delta \gamma } \right)d\left( {\gamma - \Delta \gamma } \right)$ with the conductivity ${{\gamma }} - \Delta {{\gamma }}$ determines the contribution $\eta {\kern 1pt} ^*{\kern 1pt} \left( \gamma \right)d\gamma$ to a new distribution of filaments, i.e., $\eta \left( {\gamma - \Delta \gamma } \right)d\left( {\gamma - \Delta \gamma } \right)$ = $\eta {\kern 1pt} ^*{\kern 1pt} \left( \gamma \right)d\gamma$, hence

$$\eta {\kern 1pt} ^*{\kern 1pt} \left( \gamma \right) = \eta \left( {\gamma - \Delta \gamma } \right)\left( {1 - \Delta \gamma {\kern 1pt} '\left( \gamma \right)} \right).$$

(9)

Substituting the found expression in (8), we find:

$$\begin{gathered} \Delta G = \mathop \smallint \limits_0^1 \gamma \left[ {\eta \left( {\gamma - \Delta \gamma } \right)\left( {1 - \Delta \gamma {\kern 1pt} '} \right) - \eta \left( \gamma \right)} \right]d\gamma \\ \approx \mathop \smallint \limits_0^1 \gamma \left[ {\left( {\eta - \eta {\kern 1pt} '{\kern 1pt} \Delta \gamma } \right)\left( {1 - \Delta \gamma {\kern 1pt} '} \right) - \eta } \right]d\gamma \\ = - \mathop \smallint \limits_0^1 \gamma \left( {\eta \Delta \gamma } \right){\kern 1pt} '{\kern 1pt} d\gamma + \mathop \smallint \limits_0^1 \gamma \Delta \gamma \Delta \gamma {\kern 1pt} '{\kern 1pt} \eta {\kern 1pt} '{\kern 1pt} d\gamma \\ = \mathop \smallint \limits_0^1 \eta \Delta \gamma d\gamma - \mathop \smallint \limits_0^1 \eta \left( {\gamma \Delta \gamma \Delta \gamma {\kern 1pt} '} \right){\kern 1pt} '{\kern 1pt} d\gamma \\ ~ \approx \mathop \smallint \limits_0^1 \eta \left( \gamma \right)\Delta \gamma \left( \gamma \right)d\gamma . \\ \end{gathered} $$

(10)

When deriving (10), the first transition with approximate equality was carried out by expanding $\eta \left( {\gamma - \Delta \gamma } \right)$ in a Taylor series to a term linear in Δγ, and the second approximate transition is carried out taking into account the smallness of the integral term quadratic in Δγ (respectively, in Δγ ' also, since dimensionless Δγ and Δγ ' are of one order of smallness according to (6)). In addition, integration in parts and the fact that $\Delta \gamma \left( \gamma \right)$ is equal to 0 on both limits of integration were used in the second to last transition.

Thus, the change in the conductivity of the entire NC structure in the case of small changes in the conductivity of each filament separately can be calculated as the average change in the conductivity of the filaments of the structure.

The above considerations are rather general, and the result in terms of ∆G greatly depends on: (1) the specific form of an equation of type (6) of the monofilamentary model for changing conductivity ∆γ(γ), on the one hand, and (2) a specific PDF for channel conductivity in the multifilamentary RS model of a memristive structure, on the other hand. The true distribution of filaments, both in length and in conductivity, is most likely of a complex nature and is barely accessible for an exhaustive experimental study. Moreover, the true distribution must obey the found transient condition (9). Finding such a distribution, even in a model form, seems to be a difficult task. At the same time, for a rough estimate, we can try to approximate it with some simple and reasonable distribution from a physical point of view, for example, an exponential distribution of gaps over thickness:

$${{\rho }}\left( x \right) = x_{0}^{{ - 1}}\exp \left( { - \frac{{L - x}}{{{{x}_{0}}}}} \right),$$

(11)

where x₀ is the characteristic dimensionless-gap thickness, which determines the conductivity of the NC structure: the smaller x₀, the higher the resistance, and vice versa; $L = 2{{l}_{{{\text{max}}}}}{\text{/}}{{a}_{{{\text{eff}}}}}$ (Section 2). We note that formally the carrier x in the distribution (11) has a range of values (–∞, L], while real x cannot take negative values. At the same time, as we will see below, this formal restriction is not an obstacle for the purposes of evaluative consideration. Indeed, in accordance with the developed approach, it is necessary to switch from the distribution of filaments by the gap thickness to the distribution by their conductivity. For this x in (11) must be expressed in terms of γ. The use of Eq. (2) for these purposes is inconvenient due to its transcendence and, accordingly, unsolvability in elementary functions. Instead, we use the equation without the pre-exponential factor x in the denominator:

$${{\gamma }}\left( x \right) = \frac{c}{{c + x{{e}^{x}}}} \approx \frac{c}{{c + {{e}^{x}}}},$$

(12)

which is approximately valid for large x when the exponential factor prevails. At the same time, if dependence (11) is used as the basic distribution, approximation (12) becomes justified for all values x, since it determines the change in conductivity from the minimum value $\left( { \sim {\kern 1pt} c{\text{/}}{\kern 1pt} \left( {c + {{e}^{L}}} \right)} \right)$ to the maximum γ = 1 at, formally, x = –∞. Applying (12), on the one hand, and using the condition to change the variable in the distribution ${{\rho }}\left( {x\left( {{\gamma }} \right)} \right)dx = \left( {{\gamma }} \right)d{{\gamma }}$, we get

$$\eta \left( \gamma \right) = x_{0}^{{ - 1}}{{\left( {\frac{c}{{{{e}^{L}}}}} \right)}^{{1/{{x}_{0}}}}}{{\gamma }^{{ - \frac{1}{{{{x}_{0}}}} - 1}}}{{\left( {1 - \gamma } \right)}^{{\frac{1}{{{{x}_{0}}}} + 1}}}.$$

(13)

Distribution (13) is key for the purposes of the study. It has a fairly simple form, although it is not integrable in elementary functions; it is defined in the range from ${{{{\gamma }}}_{{{\text{min}}}}} = c{\text{/}}{\kern 1pt} \left( {c + {{e}^{L}}} \right)$ to ${{{{\gamma }}}_{{{\text{max}}}}} =$ 1, i.e., does not contain physically inconsistent conductivity values γ (as opposed to basic ${{\rho }}\left( x \right)$ in relation to x) and therefore can be used to obtain average values using formulas (2) and (6) of the monofilamentary model. In a certain sense, the model distribution (13) has become independent of the source ${{\rho }}\left( x \right)$, as if guessing its appearance initially, without using the underlying distribution.

Once again, we emphasize the formality of the values x in the basic distribution (11): the true thickness of the gap between the edge of the filament and the lower electrode of the structure cannot take negative values. Nevertheless, a distribution of type (11) can easily be renormalized to the range of values x ∈ [0, L] and then use the exact expression in (12) with the term $x{{e}^{x}}$ in the denominator. In this case, the maximum conductivity value γ = 1 is achieved, as it should be, with a physically consistent x = 0, not formal $x = - {\kern 1pt} \infty$. Unfortunately, for the considered type of hopping conduction with nonresonant tunneling, such a conclusion can only be drawn numerically due to the transcendental nature of Eq. (12) with the exact equality with respect to x. Qualitatively, this will lead to sharper changes in the conductivity near γ ≈ 1 with a transition to the limit $\Delta {{\gamma }} \to \infty$ at $x \to$ 0, which can already be seen from Eq. (4), for which the empirically determined minimum value is also noted x ≈ 1. Sharper jumps in the conductivity of the subgroup of filaments with γ ≈ 1 can provide a better fit of the fitting curves to the experimental data in Fig. 6 (see below), especially at the inflection point, which should become sharper in accordance with the nature of the empirical dependences (Fig. 1). At the same time, the purpose of this study was to obtain a relatively simple analytical (rather than tabular) form for the approximate distribution of filaments in terms of conductivity, with the help of which it is easy to analyze both the qualitative nature of the considered distribution and directly calculate the expected values of any variables of interest. Work on the derivation and application of a more accurate distribution model in tabular form is planned to be carried out in the future.

Fig. 5.

Distribution density (13) η(γ) over the conductivity of filaments in the multifilamentary model of the RS of a NC memristor. The inset on a linear scale with a break along the y axis shows the lower values of the distribution: noticeable subpopulations of filaments are found mainly near γ = 0 and significantly smaller subpopulations appear around γ = 1 for x₀ > 1 (a). The model conductivity of the NC structure depending on the parameter x₀ of the distribution (b). Conductivity saturates with increasing x₀. The inset shows the initial section of the same dependence.

Fig. 6.

Model dependences of the variation in conductivity (6) under the action of a voltage pulse applied to the memristive NC structure, averaged using the multifilamentary model distribution (13), on the initial value of the memristor conductivity, disregarding (dashed curves) and taking into account the diffusion of oxygen vacancies (solid curves: red is diffusion and drift are co-directed, black are oppositely directed). Dots are experimental data. The inset demonstrates the same dependence, but on a linear scale and in almost the entire range of conductivity.

The distribution η(γ), as well as the basic dependence (11), is determined by the characteristic parameter x₀, which can take any positive value: the larger its value, the closer the distribution η(γ) to the LRS state (Fig. 5a). Thus, x₀ sets the conductivity value of the entire structure of the NC memristor. So, with growth of x₀ from a value less than 1 to a value near 2, in the distribution η(γ) small subpopulations of filaments appear with conductivity values near γ = 1 (inset to Fig. 5a), which mainly determine the overall conductivity of the structure. This can be seen from the increase in the growth rate of the overall conductivity, starting from x ≈ 2 (inset in Fig. 5b). This correlates with the traditional assumption within the framework of the multifilamentary approach that only a small fraction of highly conductive filaments out of their total number makes the largest contribution to the conductivity of the entire structure as a whole. We note that the normalization of the model distribution (13) to 1 follows from its derivation from the normalized basic distribution (11) and is confirmed by numerical integration. The apparent inconsistency with the normalization, based on the analysis of Fig. 5a, where the curves for higher values x₀ are visually located higher in the entire range of conductivity changes, due to the fact that the lower x₀, the greater the part of the filaments with a conductivity close to 0. This is not visible in Fig. 5a due to the very sharp nature of the decline in the distribution density η(γ) near 0.

Using distribution (13), any macroscopic mean values of the quantities can be calculated. In particular, the average value of the filament conductivity, which determines the conductivity of the memristive NC structure as a whole, is shown in Fig. 5b. Expected saturation of the conductivity is observed (that is, its approach to the maximum value G_on) with increasing distribution parameter x₀.

The dotted gray curves in Fig. 6 show the approximation of the experimental data on the change in the total conductivity of the structure by the multifilamentary model dependence (10) when fitting the following model parameters: g_on/g_off for a single filament, activation energy of oxygen-vacancy migration θ, the characteristic temperature of Joule heating of the highly conductive filament τ, as well as the model parameter x_eff. All other parameters of the model were fixed according to their values indicated above, obtained from simple physical considerations or from tabular data. Taking into account the simplicity and approximate nature of the model, quite satisfactory agreement is observed, especially in comparison with the result of approximation by the monofilamentary model dependence (inset in Fig. 4). Moreover, $\Delta G{\text{/}}{{G}_{{{\text{on}}}}}$ turns out to be small over the entire range of relative conductivity $G{\text{/}}{{G}_{{{\text{on}}}}}$, which indicates the applicability of the developed model of small increments in the case under consideration.

Best fit is observed with the following approximation parameters: θ = 27 (that is, the activation energy of the motion of vacancies E_a ≈ 0.7 eV, which is in full agreement with the above range of assumed values from 0.5 to 1 eV); τ = 0.44, which corresponds to the characteristic heating temperature of the filament up to ∼160°С (it is necessary to emphasize the conditionality of such a parameter as the temperature of a nanoscale filament; therefore, the parameter under consideration should be treated only as some effective value of the model); x_eff = 2.1; and finally g_on/g_off = 2000. The value of the ratio of the limiting resistances for an individual filament turns out to be 1–2 orders of magnitude higher than for the memristor structure as a whole, which can be interpreted as the fact that only a small part of the area of the lower contact of the memristor element in the LRS is occupied by highly conductive filaments: the fill factor is K_c ∼ 20–100/2000 ≈ 1–5% area if the ratio R_off/R_on ≈ 20–100 for the structure as a whole. This looks like a plausible result and agrees in order of magnitude with the estimate K_c ≈ 1% in terms of capacitance measurements of the NC structure (Section “Experimental Evidence for a Multifilamentary RS Mechanism” and [30]) and in this work is obtained directly from the correspondence of empirical data on the change in conductivity and calculated dependences obtained using a model multifilament distribution.

Even more accurate agreement with the experiment can be obtained if the goal is to take into account, at least in a rough approximation, along with drift in an electric field, the diffusion of oxygen vacancies at the edge of the filament, where there is a concentration gradient V_O. Let us assume for simplicity that this gradient is constant regardless of the conductivity (and, accordingly, the length) of the filament. Further, simplifying the real situation, we estimate the diffusion displacement of the filament edge as ${{{v}}_{D}}\Delta t$, where ${{{v}}_{D}} = - D\nabla {{n}_{V}}{\text{/}}{{n}_{V}}$ is the diffusion rate against the concentration gradient $\nabla {{n}_{V}}$, and ${{n}_{V}}$ is the concentration of oxygen vacancies on the side of the highly conductive part of the filament. Of course, in the presence of a gradient, constant displacement of the filament edge occurs, but it is negligible in the absence of Joule heating due to the exponential dependence of the diffusion coefficient D on the temperature noted in (5); therefore, it is taken into account only in the time interval $\Delta t$, during which a voltage pulse is applied to the structure.

Under these assumptions, the diffusion term is added to expression (3) as a component Δx of the kind $\pm \frac{{2D}}{{{{a}_{{{\text{eff}}}}}}}\left| {\frac{{\nabla {{n}_{V}}}}{{{{n}_{V}}}}} \right|$, where the sign is chosen depending on the potentiation (“+” sign) or depression (“–” sign) of the conductivity of the structure by the applied voltage pulse. Expression (6) is rewritten taking diffusion into account as follows:

$$\begin{gathered} \left| {\Delta \gamma \left( \gamma \right)} \right| \approx {{\Gamma }_{0}}\gamma \left( {1 - \gamma } \right)\exp \left( { - \frac{\theta }{{1 + \gamma \tau }}} \right) \\ \times \;\left[ {\left( {1 - \gamma } \right) \pm {{x}_{{\operatorname{eff} }}}\frac{{k{{T}_{0}}}}{{e{{V}_{{{\text{sp}}}}}}}\frac{{{{a}_{{\operatorname{eff} }}}}}{2}\frac{{\nabla {{n}_{V}}}}{{{{n}_{V}}}}} \right]. \\ \end{gathered} $$

(14)

The term with the sign “$\pm$” on the right side of formula (14) reflects in the simplest version the effect of the diffusion of oxygen vacancies on the change in the conductivity of a single filament, and we will consider it an additional constant fitting parameter of the model.

The solid curves in Fig. 6 demonstrate the solution of the approximation problem taking into account the fixed diffusion term in (14), which was chosen to be positive for potentiation (diffusion is codirectional with drift, Δt > 0 in Fig. 3, red curves in Fig. 6) and negative for depression of the conductivity of the memristor (diffusion is opposite to drift, Δt < 0 in Fig. 3, black solid curves in Fig. 6). It is obvious that there is a better agreement between theory and experiment, which is observed at the value of the parameter ${{x}_{{{\text{eff}}}}}\frac{{k{{T}_{0}}}}{{e{{V}_{{{\text{sp}}}}}}}\frac{{{{a}_{{{\text{eff}}}}}}}{2}\frac{{\nabla {{n}_{V}}}}{{{{n}_{V}}}}$ ≈ 0.05, corresponding to the value ${{\left( {\frac{{\nabla {{n}_{V}}}}{{{{n}_{V}}}}} \right)}^{{ - 1}}} \approx {{a}_{{{\text{eff}}}}}$/9 ≈ 0.5 nm. This means that the diffusion blurring of the filament edge is estimated at about 5 Å, which seems to be a physically plausible value and indirectly confirms the overall consistency of the developed model. It is important that all other parameters of the model have the same values as in the case of approximation without diffusion (dashed curves in Fig. 6).

Thus, the set of the presented model data and their comparison with the experimental dependences of the variation in the conductivity of a NC memristor under the action of applied voltage pulses testifies in favor of the multifilamentary nature of RS. The distribution of filaments over conductivities for the HRS structure is most likely described by the vast majority of filaments in the high-resistance state (with a large gap between the boundary of the highly conductive part and the lower electrode) with a small (surface filling factor of the electrode K_c = 0–0.05%) subpopulation of filaments in the low-resistance state. When there are practically no highly conductive channels, the total conductivity of the structure is small and slowly changes with increasing parameter x₀ of the distribution (Fig. 5b, inset), which determines its degree of bias towards the LRS (Fig. 5a). For some parameter value x₀ ∼1.5 the proportion of LRS filaments begins to be g_off/g_on ∼1/2000 of the area of the minimum of the electrodes of the memristor structure, due to which the conductivity of highly conductive filaments becomes comparable with the conductivity of the entire low-conductivity part of the structure, and its total resistance with increasing x₀ starts to drop rapidly (Fig. 5b, inset). It is reasonable to assume that the inflection point on both the experimental and theoretical dependences of the conductivity variation in Figs. 3, 5, and 6 (approximately at G/G_on = 0.03, which corresponds to x₀ ≈ 2) is due to the effect of a sharp increase in the conductivity of the NC structure as a result of the formation of a sufficient number of highly conductive filaments that shunt the conductivity of the rest of the structure. Further, the conductivity of the NC memristor increases almost linearly with an increase in parameter x₀ up to values of ∼20 due to the formation of more and more new filaments in the LRS state, after which saturation of the total conductivity is observed and its asymptotic exit to the value G_on when filling K_c = (G_on/G_off)/(g_on/g_off) ≈ 20–100/2000 = 1–5% of the area of the smallest contact of the NC structure with filaments in a highly conductive state.

To confirm this interpretation, let us estimate the filling factor of the NC structure in the LRS based on the following simple considerations. At the inflection point on the curves of conductivity variations (G/G_on ≈ 0.03) $G \propto {{g}_{{{\text{off}}}}}\left( {S - {{S}_{{\text{c}}}}} \right) + {{g}_{{{\text{on}}}}}{{S}_{{\text{c}}}} \approx {{g}_{{{\text{off}}}}}S\left( {1 + \frac{{{{g}_{{{\text{on}}}}}}}{{{{g}_{{{\text{off}}}}}}}\frac{{{{S}_{{\text{c}}}}}}{S}} \right)$ ≅ $2{{g}_{{{\text{off}}}}}S$, where S is the area of the smallest memristor electrode, and S_c $\ll$ S is the electrode area occupied by highly conductive filaments at a given conductivity value. The last jump to 2g_offS is valid for the reason that at the inflection point, according to our assumption, the resistance of highly conductive filaments is comparable in value with the resistance of the rest of the active layer of the memristive structure. In the case of a memristor, the LRS can be written similarly: ${{G}_{{{\text{on}}}}} \propto ~{{g}_{{{\text{off}}}}}S\left( {1 + \frac{{{{g}_{{{\text{on}}}}}}}{{{{g}_{{{\text{off}}}}}}}\frac{{{{S}_{{{\text{c}}{\text{,on}}}}}}}{S}} \right)$, where S_c,on is the filling area for the structure in the low-resistance state. Then $G{\text{/}}{{G}_{{{\text{on}}}}} = 2{\text{/}}\left( {1 + \frac{{{{g}_{{{\text{on}}}}}}}{{{{g}_{{{\text{off}}}}}}}\frac{{{{S}_{{{\text{c}}{\text{,on}}}}}}}{S}} \right)~$, whence we obtain the required estimate ${{K}_{{{\text{c}}{\text{,on}}}}}\, \equiv \,\frac{{{{S}_{{{\text{c}}{\text{,on}}}}}}}{S}\, = \,\frac{{{{g}_{{{\text{off}}}}}}}{{{{g}_{{{\text{on}}}}}}}\left( {2\frac{{{{G}_{{{\text{on}}}}}}}{G}\, - \,1} \right)$ ≈ 0.03, which is in excellent agreement with the range of the fill factor found above for the NC structure in the LRS from 1 to 5% and corresponds to the realistic value of the ratio R_off /R_on ≈ 2000 × 0.03 = 60 for the NC structure as a whole. Consequently, at the inflection point of the curves on a logarithmic scale in Fig. 3, that is, near G ≈ 0.03G_on, the total conductivity of highly conductive filaments indeed becomes comparable with the conductivity of the rest of the structure, and our model concept of the multifilamentary RS mechanism receives additional confirmation.

CONCLUSIONS

For the first time, an explicit form of the model distribution of filaments over the conductivity values was proposed, and with its help, realistic estimates of the change in the total conductivity of a NC memristor structure under the action of voltage pulses were obtained. Peculiarities in the experimental dependences of memristor plasticity (the inflection point at the transition from LRS to HRS, the total amplitude of the observed conductivity variations, the difference between the depression and potentiation curves) are explained using actual physical processes and realistic values of the model parameters. The entire set of data testifies in favor of multifilamentary RS in NC memristors based on LiNbO₃, which indicates the fundamental and practical possibility of a gradual change in their resistance states and, therefore, prospects for application in NCS.

The developed RS model is valid only for small changes in conductivity. In the case of many memristive MIM structures described in publications, the gap width, which determines the total resistance, is rather small (on the order of a few nanometers or less). Then, very slight displacements of the filament edges, even on the scale of several interatomic distances, can significantly affect the overall conductivity of the structure, and our approach becomes inapplicable, although the presented qualitative consideration of the multifilamentary RS most likely remains valid in general terms.

In the derivation, as a key approximation, we used the assumption of the hopping character of the structure conductivity with inelastic resonant tunneling along point defects in the dielectric layer (with a high density of localized states at the Fermi level ~(10²¹–10²²) eV^—1 cm^–3). This assumption may well be valid for a fairly wide class of amorphous dielectrics used to create memristive structures of the MIM type. Thus, for these structures, it is possible to apply the proposed theoretical model approach to establish both the very nature of the RS and the features of the multifilamentary RS from a microscopic point of view. We hope that this will contribute to further improvement of both the microarchitecture itself and the technology for creating memristor structures for neuromorphic and other applications.