Scaling Theory for Unipolar Resistance Switching

We investigate a reversible percolation system showing unipolar resistance switching in which percolating paths are created and broken alternately by the application of an electric bias. Owing to the dynamical changes in the percolating paths, different from those in classical percolating paths, a detailed understanding of the structure is demanding and challenging. Here, we develop a scaling theory that can explain the transport properties of these conducting paths; the theory is based on the fractal geometry of a percolating cluster. This theory predicts that two scaling behaviors emerge, depending on the topologies of the conducting paths. We confirm these theoretical predictions experimentally by observing material-independent universal scaling behaviors in unipolar resistance switching.

Figure 1

(a) Schematic of an electrode-oxide-electrode structure. (b) Initial configuration of an insulating medium with a few on-state bonds as defects. (c) Snapshot of percolating (highly conducting) paths created via the dielectric breakdown process. (d) The $\mathrm{\text{off}}\to \mathrm{\text{on}}$ switching by a dielectric breakdown, which occurs when the applied voltage exceeds some threshold voltage $V_c$. (e) The $\mathrm{\text{on}}\to \mathrm{\text{off}}$ switching by the thermal fuse effect, which occurs when the temperature of a bond exceeds some threshold temperature $T_c$ because of Joule heating. (f) Snapshot of disconnected paths ruptured by the thermal fuse effect.

Figure 2

(a) Branching tree structure of conducting paths generated by a dielectric breakdown; the paths extend from the top to the bottom. At the cross section at depth $t$, conducting spots are distributed in a region with lateral size $s_t$. (b) Slice structure of an insulating medium with thickness $\Delta t$. There are $n_t$ conducting branch segments in slice $t$. (c) Total current $I_{\mathrm{tot}}$ distributed among all branch segments in a slice.

Figure 3

(a) Electric current flowing through conducting spots (branch segments) in each slice causes Joule heating, which gives rise to a temperature gradient according to Eq. (5). When the temperature of the hottest (central) part of the slice reaches $T_c$ [see Fig. 1e], rupturing occurs. (b) In simulations, we used an $L\times L$ lattice whose four boundaries were in contact with a thermal bath at temperature $T_b=300\mathrm{K}$. $q(\overrightarrow{r})=q_0$ (constant) inside the dashed (red) box with sides of length $s$, and $q(\overrightarrow{r})=0$ elsewhere. (c) FEM simulation result. We found $q_0\sim s^{-\beta }$, with $\beta \approx 1.65\pm 0.01$; the exponent is independent of $T_c$ (400 K, 700 K, and 1000 K).

Figure 4

(a), (b), and (c) show scaling relations for $B_x$ vs $R_x$, $I_x$ vs $R_x$, and $I_x$ vs $B_x$, respectively. In each panel, ${\mathrm{NiO}}_w$, ${\mathrm{SrTiO}}_x$, ${\mathrm{FeO}}_y$, and ${\mathrm{TiO}}_z$ are denoted by ◆, ▪, ▴, and +, respectively. The data for ${\mathrm{NiO}}_w$ are obtained from Ref. 13. The two scaling regions are distinguished by the $x={10}^0$ line. (d) The two scaling regions originate from different bottleneck sizes $s_0$ of the filaments.