Percolating through Networks of Random Thresholds: Finite Temperature Electron Tunneling in Metal Nanocrystal Arrays

We investigate how temperature affects transport through large networks of nonlinear conductances with distributed thresholds. In monolayers of weakly coupled gold nanocrystals, quenched charge disorder produces a range of local thresholds for the onset of electron tunneling. Our measurements delineate two regimes separated by a crossover temperature $T^{*}$. Up to $T^{*}$ the nonlinear zero-temperature shape of the current-voltage curves survives, but with a threshold voltage for conduction that decreases linearly with temperature. Above $T^{*}$ the threshold vanishes and the low-bias conductance increases rapidly with temperature. We develop a model that accounts for these findings and predicts $T^{*}$.

Figure 1

(a) Evolution of $IV$ curves with temperature for an array of length $N=128$. The $IV$s are fully symmetric with respect to bias voltage reversal. Above $T^{*}$ the $IV$s no longer have a finite voltage threshold, as shown in the inset. (b) Below $T^{*}$, $IV$s at different temperatures can be collapsed by translating the voltage scale by an amount $V_{\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}}(T)$, with no alteration of the current scale. The inset shows the zero-bias conductance, $g_0=dI/dV$, as a function of temperature.

Figure 2

(a) Scaling behavior of current $I$ with excess voltage $V-V_t(T)$ above threshold for the same array as in Fig. 1 (middle curve) and two other arrays with different degrees of disorder. For each array, four different temperatures are plotted. The dotted line corresponds to $\zeta =2.27$. (b) The effective threshold as a function of temperature for several arrays of length $N$ and effective charging energy $e{V}_t(0)/\alpha N$.

Figure 3

Normalized threshold, $V_t(T)/V_t(0)$ as a function of the dimensionless temperature variable $x=k_{B}TP(0)$ for all arrays. The dashed line is the model prediction. Inset: probability density of charging energies $P(\Delta E)$. With capacitive coupling in the presence of offset charges on all nearest neighbors, $P(\Delta E)$ evolves to a more rounded shape.