Second Order Meanfield Approximation for calculating Dynamics in Au-Nanoparticle Networks

Abstract

Exploiting physical processes for fast and energy-efficient computation bears great potential in the advancement of modern hardware components. This paper explores non-linear charge tunneling in nanoparticle networks, controlled by external voltages. The dynamics are described by a master equation, which describes the development of a distribution function over the set of charge occupation numbers. The driving force behind this evolution are charge tunneling events among nanoparticles and their associated rates. In this paper, we introduce two meanfield approximations to this master equation. By parametrization of the distribution function using its first- and second-order statistical moments, and a subsequent projection of the dynamics onto the resulting moment manifold, one can deterministically calculate expected charges and currents. Unlike a kinetic Monte Carlo approach, which extracts samples from the distribution function, this meanfield approach avoids any random elements. A comparison of results between the meanfield approximation and an already available kinetic Monte Carlo simulation demonstrates great accuracy. Our analysis also reveals that transitioning from a first-order to a second-order approximation significantly enhances the accuracy. Furthermore, we demonstrate the applicability of our approach to time-dependent simulations, using eulerian time-integration schemes.

Figures (8)

• Figure 1: A $4\times4$ nanoparticle network with four electrodes $U_0$, $U_1$, $U_2$, and $U_3$. The electrodes are attached at the corners. The tunnel junctions are depicted as capacitors and resistors in parallel.
• Figure 2: Different approximations to the full Gaussian distribution with $\langle n \rangle = 4.43$ and $(\Delta n )^2$ = 0.6 are shown. Beyond the distributions used in MF2 and QMF2, we have also included the distribution resulting from maximization of the Shannon entropy, also restricted to a four-dimensional phase space, denoted restricted Gaussian distribution.
• Figure 3: A circuit diagram of the single-electron-transistor. To the left and right, the two connected electrodes (source and drain) and the gate, realized by the capacitively coupled silicon substrate, are shown. Tunnel junctions are indicated as capacitors and resistors in parallel.
• Figure 4: For a non-zero gate voltage of $0.05\,V$, the first order method has problems replicating the mean occupation number and its variance. The second order methods still accurately reproduce the mean and variance of the master equation. The shaded areas represent the interval of $\pm\Delta n$ predicted by the methods compared to the master equation. It can be seen that for low input voltages, no charge resides on the island, a non-linear effect called Coulomb blockade (see mensing2023kinetic).
• Figure 5: For a random voltage configuration, the mean occupation numbers are plotted from left to right for the KMC, first order MF1 and second order QFM2 method. A continuous gradient can be seen. Results of all methods lie in qualitative agreement. $E_1 = -0.10\,V$, $E_2 = -0.23\,V$, $E_3 = 0.07\,V$, $E_4 = 0.05\,V$ are set and the gate voltage is $G = 0.13\,V$.