Neuromorphic behavior in percolating nanoparticle films

We show that the complex connectivity of percolating networks of nanoparticles provides a natural solid-state system in which bottom-up assembly provides a route to realization of neuromorphic behavior. Below the percolation threshold the networks comprise groups of particles separated by tunnel gaps; an applied voltage causes atomic scale wires to form in the gaps, and we show that the avalanche of switching events that occurs is similar to potentiation in biological neural systems. We characterize the level of potentiation in the percolating system as a function of the surface coverage of nanoparticles and other experimentally relevant variables, and compare our results with those from biological systems. The complex percolating structure and the electric field driven switching mechanism provide several potential advantages in comparison to previously reported solid-state neuromorphic systems.

Figure 1

Map of $200\times 200$ system of overlapping disks (gray) with locations of switching events shown using black dots. The electrodes (not shown) are at the left and right edges of the system. $V_{\max }=1\mathrm{V}$ and $p=0.55$ (left), $p=0.65$ (right). At higher coverages the larger size of the connected groups [33] means that there are fewer tunnel or switching junctions between groups and a much higher conductance prior to the switching process (by up to 8 orders of magnitude); it also means that there are fewer locations at which switching can occur.

Figure 2

Current flowing between the electrodes as a function of time (which corresponds to the number of voltage steps $N_V)$ for $P_{\uparrow }=10\%$ and (a) $p=0.55$, (b) $p=0.65$. (c) Corresponding voltage ramp with $V_{\max }=1\text{V}$ and $V_{\mathrm{step}}=0.025$. The increase in current as a function of applied voltage and in subsequent voltage cycles is due to the insertion of high conductances (representing atomic wires) in tunnel gaps between particles when the electric field in the gap exceeds a defined threshold $E_{\mathrm{th}}$.

Figure 3

As for Fig. 2 but with $P_{\uparrow }=1\%$. (a) $p=0.55$, (b) $p=0.65$, (c) corresponding voltage ramp. The data in (b), as in many other figures below, terminates when the conductance converges to within 1%.

Figure 4

Number of tunnel junctions replaced $\left(N_R\right)$ and system conductance $\left(G\right)$ as a function of time (number of voltage steps $N_V)$ for $p=0.55$ and $P_{\uparrow }=1$, 10, and 80% (black, red, and blue curves, respectively, and from right to left). (a) and (d) $V_{\max }=0.5\text{V}$, (b) and (e) $V_{\max }=1\text{V}$, (c) and (f) $V_{\max }=5\text{V}$. Colored dots mark the points at which the last tunneling conductor on the primary conduction path between the contacts is replaced by an Ohmic conductor $G_{\mathrm{Ohmic}}$.

Figure 5

Data from Figs. 4 and 4 (i.e., $p=0.55,V_{\max }=5\text{V})$ replotted on a linear horizontal scale so as to allow the behavior at small $N_V$ to be seen more clearly and to emphasize the nearly exponential initial increase in $G$. Colored dots mark the points at which the last tunneling conductor on the primary conduction path between the contacts is replaced by an Ohmic conductor $G_{\mathrm{Ohmic}}$.

Figure 6

As for Fig. 4 but for $p=0.65$.

Figure 7

Direct comparison of conductance as a function of time $\left(N_V\right)$ for a range of parameter values: $p=0.55$ and 0.65, $V_{\max }=1$ V (dashed lines) and 5 V (solid lines), and $P_{\uparrow }=1\%$ and 40%. (From right to left) $p=0.55,P_{\uparrow }=1\%$ (red); $p=0.55,P_{\uparrow }=40\%$ (blue); $p=0.65,P_{\uparrow }=1\%$ (green); $p=0.65,P_{\uparrow }=40\%$ (black).

Figure 8

(a) $G$ as a function of $N_V$ for $P_{\uparrow }=100\%,V_{\max }=5\mathrm{V}$. (From bottom to top) $p=0.55$, 0.57, 0.60, 0.62, 0.65. (b) The same data plotted as function of $N_V-N_V^C$ in order to show the power law variation of the conductance discussed in the text. The dashed line indicates a slope of 0.5.

Figure 9

$G$ versus $N_V$ when both creation (with probability $P_{\uparrow }=1\%)$ and breaking (with probability $P_{\downarrow }=100\%)$ of atomic wires is allowed in the simulation (at $E_{\mathrm{th}}=0.9$ and $I_{\mathrm{th}}=0.1$, respectively). (a) $p=0.55$, (b) $p=0.65$, (c) corresponding voltage ramp. Note the change in scales from (a) to (b).

Figure 10

Same as for Fig. 9 except with $P_{\downarrow }=1\%$. (a) $p=0.55$, (b) $p=0.60$, (c) $p=0.65$, (d) corresponding voltage ramp. Note that (a) is on a different vertical scale from (b) and (c).

Figure 11

Dependence of critical number of steps required to create an Ohmic connection across the system $N_V^C$, on coverage $p$, extracted from the data in Figs. 4 and 6. (Top to bottom) $P_{\uparrow }=1\%$, at $V_{\max }=1$ V; $P_{\uparrow }=1\%$, at $V_{\max }=2$ V; $P_{\uparrow }=1\%,V_{\max }=5$ V; $P_{\uparrow }=10\%$, at $V_{\max }=5$ V; $P_{\uparrow }=80\%$, at $V_{\max }=1$ V. Lines serve only to connect the data points.

Figure 12

Conductance change at constant voltage $(V=1$ V) for $p=0.55$ (black, right), 0.57, 0.60, 0.62, and 0.65 (red, left). Dashed lines are from ramped voltages for comparison.

Figure 13

Schematic I(V) curve for a single tunnel junction showing jump from an initial high resistance (tunneling) state to a low resistance state on formation of an atomic scale wire, and a hysteresis loop (solid lines) that results from a quasideterministic memristive behavior; see Sec. 4e. Note that while tunneling is a voltage-dependent process, for simplicity in this schematic we have assumed that even in the high resistance (tunneling) state the resistor still obeys Ohm's law at low V; different I(V) characteristics in the two states would change the shape of the curve but not the existence of a hysteresis loop. Any bias dependence will be unimportant in most applications because $G_{\mathrm{high}}$ exceeds $G_{\mathrm{low}}$ by many orders of magnitude—it is only the ability of the device to switch that is important to the observation of neuromophic behavior reported here.

Figure 14

Illustration of an avalanche of successive switching events resulting from application of a constant voltage (1 V). Locations of switching events (black dots) are shown for a $100\times 100$ subset of a $200\times 200$ system of overlapping disks (gray) with $p=0.55$ for $N_V=1$ (a), $N_V=2000$ (b), and $N_V=5500$ (c), corresponding to the conductivity data in Fig. 4.