Conducting-insulating transition in adiabatic memristive networks

The development of neuromorphic systems based on memristive elements—resistors with memory—requires a fundamental understanding of their collective dynamics when organized in networks. Here, we study an experimentally inspired model of two-dimensional disordered memristive networks subject to a slowly ramped voltage and show that they undergo a discontinuous transition in the conductivity for sufficiently high values of memory, as quantified by the memristive ON-OFF ratio. We investigate the consequences of this transition for the memristive current-voltage characteristics both through simulation and theory, and demonstrate the role of current-voltage duality in relating forward and reverse switching processes. Our work sheds considerable light on the statistical properties of memristive networks that are presently studied both for unconventional computing and as models of neural networks.

Figure 1

The network conductance for several values of $G_{\mathrm{on}}$ are plotted against the applied voltage for both (a) $G_{\mathrm{off}}\to G_{\mathrm{on}}$ and (b) $G_{\mathrm{on}}\to G_{\mathrm{off}}$ processes. The network conductance $G$ has been scaled to vary from 0 to 1 and the range of the applied voltage shortened to focus on the point of transition. The insets show a typical network biased by a voltage $V_{+}$ just following the transition where the formation of a (a) conducting backbone and (b) crack may be observed.

Figure 2

Simulated hysteresis curves of memristive networks for $G_{\mathrm{on}}=$ (a) 4, (b) 100, and (c) 1000. The discrete jump in conductance becomes evident in the $I-V$ curves for large values of $G_{\mathrm{on}}$ The asymmetry of the curves is the result of element thresholds depending on the current, and thus the transition occurring at a factor of $1/G_{\mathrm{on}}$ lower voltage than the corresponding forward transition. The insets show analytically reached $I-V$ curves, which demonstrate similar asymmetry and the emergence of a jump in the current. The reverse branch has been rescaled and plotted in the second row for $G_{\mathrm{on}}=$ (d) 4, (e) 100, and (f) 1000. For large $G_{\mathrm{on}}$, the transition appears as a noisy area near to the $y$ axis. Here the jumps in conductivity at a fixed voltage give rise to sharp decreases in the current that are opposed by the subsequent increase in voltage.

Figure 3

The top row shows hysteresis curves switching from $G_{\mathrm{on}}=$ (a) 4, (b) 100, and (c) 1000 to $G_{\mathrm{off}}=1$ in a current controlled network. For a current-controlled network switching from $G_{\mathrm{on}}\to G_{\mathrm{off}}$ the transition gives a discrete jump, as in the voltage-controlled $G_{\mathrm{off}}\to G_{\mathrm{on}}$ case. After the mapping in Eq. (2), the reverse branch of the hysteresis curves (dashed) have been plotted over the forward branch for $G_{\mathrm{on}}=$ (d) 4, (e) 100, and (f) 1000 demonstrating the duality between the two processes. The insets show analytical $I-V$ curves resulting from a cluster approximation, showing identical behavior to the voltage-controlled forward switching case.

Figure 4

$h\left(f\right)=\sqrt{\frac{G\left(f\right)}{\langle g\rangle \left(f\right)}}$ is plotted for several values of $G_{\mathrm{on}}$. In the inset, $\langle g\rangle \left(f\right)$ and $G\left(f\right)$ are plotted for $G_{\mathrm{on}}=100$. Note that when the average conductance increases more quickly than the network conductance as for small $f, h\left(f\right)$ is decreasing and vice versa for large $f$.

Figure 5

The left-hand side (blue, dark gray) and right-hand side (red, light gray) of Eq. (6) are plotted for several values of the applied voltage. For low values of $G_{\mathrm{on}}$ their intersection gives a solution that is a smooth function of the voltage [(a) $G_{\mathrm{on}}=10$]. A transition develops for intermediate values that can appear continuous [(b) $G_{\mathrm{on}}=30$]. For sufficiently large values of $G_{\mathrm{on}}$, a transition occurs where the solution jumps discontinuously [(c) $G_{\mathrm{on}}=60$]. The transition voltage is highlighted in (b) and (c).

Figure 6

Solving the self consistency equations (6) and (8) for a uniform distribution leads to the hysteresis curves above. Here they are plotted for $G_{\mathrm{on}}=$ (a) 10, (b) 30, (c) 50. For small values of $G_{\mathrm{on}}/G_{\mathrm{off}}$, the networks are smooth, but as the ratio is increased a discrete jump emerges in the forward direction. A similar jump near the end of the reverse branch is only barely discernible.

Figure 7

The left-hand side (blue, dark gray) and right-hand side (red, light gray) of Eq. (12) are plotted for several values of the applied voltage for the distribution $\text{Uniform}(0,1)$. Here they are plotted for $G_{\mathrm{on}}=$ (a) 1.5, (b) 2, (c) 10. For a collection of memristors in series, the transition is the completion of the conducting backbone, with all elements in the $G_{\mathrm{on}}$ state.

Figure 8

The systems and subunits considered in the cluster approximations. As a model of the conducting backbone embedded in the network we consider (a) a series chain of memristors in parallel subject to a slowly ramped voltage $V+$ composed of subunits of (b) pairs of memristors in parallel subject to a slowly ramped current. As a model of the crack, we consider (c) a parallel strip of memristors in series subject to a slowly ramped current consisting of subunits of (d) pairs of memristors in series subject to a slowly ramped voltage.

Figure 9

Avalanche sizes binned just above the transition for randomly diluted networks ($p=0.6,G_{\mathrm{on}}=100$) of size $N_{x/y}=32,64,128$ (1000 realizations) and $N_{x/y}=256$ (100 realizations). As the network size increases, the avalanche size distribution approaches the asymptotic form $P\left(s\right)\sim s^{-3/2}$ given by the mean-field theory, subject to a finite-size cutoff.