# Dynamic Single-Electron Transistor Modeling for High-Frequency Capacitance Characterization
## 2 Method of Modeling  
### Proposed Model  
The pictorial and symbolic schematics of the SETs are shown in Figure 1a,b, respectively. In the proposed model of Figure 1c, the gate current \(I_{\text{g}}\) and the drain current \(I_{\text{d}}\) is calculated based on the time-dependent master equation [29]. The \(I_{\text{g}}\) represents the contribution of the gate terminal (input) capacitance \(C_{\text{gg}}\) and gate-related transcapacitances \(C_{\text{gd}}\) and \(C_{\text{gs}}\). The contribution of other capacitances is also included in \(I_{\text{d}}\) and \(I_{\text{s}}=-(I_{\text{d}}\) + \(I_{\text{g}})\). Although three-terminal configuration is considered at this time for simplicity, it is also possible to extend the model to four-terminal configuration [13] with the second gate (backgate), which is useful in some circuit applications [4; 11] and the analysis of the background noise effect [30] that has a large impact on the SET-based circuit integrity. When the master equation is solved, it is assumed that number of electrons (\(n\)) in the Coulomb island can be in four states, i.e., \(n=-1\), \(0\), \(1\) or \(2\). The \(I_{\text{d}}\) is given by  
\[I_{\text{d}}=e{\sum}P_{\text{n}}\left\{\Gamma_{\text{d}}(n-1,n)-\Gamma_{\text{ d}}(n+1,n)\right\} \tag{1}\]  
Here, \(P_{\text{n}}\) represents the probability that the number of electrons in the island is \(n\) whereas \(\Gamma_{\text{d}}\) (\(n-1\), \(n\)) and \(\Gamma_{\text{d}}\) (\(n+1\), \(n\)) are the tunneling rates of electrons from \(n\) to \(n-1\) and \(n\) to \(n+1\) at the drain junction, respectively. Figure 1: (a) SET pictorial schematic, (b) symbolic schematic, and (c) model schematic with capacitance components inside the SET. To calculate the \(I_{\text{g}}\), potential of the Coulomb island \(V_{\text{isl}}\) needs to be determined first by  
\[V_{\text{isl}}=\frac{C_{\text{g}}V_{\text{g}}+C_{\text{s}}V_{\text{s}}+C_{\text {d}}V_{\text{d}}+Q_{\text{isl}}}{C_{\Sigma}} \tag{2}\]  
Here, \(C_{\text{g}}\), \(C_{\text{s}}\), and \(C_{\text{d}}\) are the gate, source and drain capacitances, respectively, and \(C_{\text{$\sum$}}=C_{\text{g}}+C_{\text{s}}+C_{\text{d}}\). Average charge \(Q_{\text{isl}}\) in the Coulomb island is given by  
\[Q_{\text{isl}}=-e\sum nP_{\text{n}} \tag{3}\]  
By substituting (3) in (2), \(V_{\text{isl}}\) can be rewritten as  
\[V_{\text{isl}}=\frac{C_{\text{g}}V_{\text{g}}+C_{\text{s}}V_{\text{s}}+C_{ \text{d}}V_{\text{d}}-e\sum nP_{\text{n}}}{C_{\text{$\sum$}}} \tag{4}\]  
Here, \(V_{\text{g}}\), \(V_{\text{s}}\) and \(V_{\text{d}}\) are gate, source and drain voltages, respectively. The \(I_{\text{g}}\) flowing from the gate terminal to the Coulomb island via gate capacitor can be expressed as the change in the charge \(Q_{\text{g}}\) in gate capacitor with respect to time. \[ \begin{array}{ll}I_{\text{g}}&=\frac{dQ_{\text{g}}}{dt}=\frac{d}{dt}\left( V_{\text{g}}-V_{\text{isl}}\right)C_{\text{g}}\\ &=\frac{dt}{dt}\left\{V_{\text{g}}-\left(\frac{C_{\text{g}}V_{\text{g}}+C_{ \text{d}}V_{\text{d}}+C_{\text{s}}V_{\text{s}}+e\sum nP_{\text{n}}}{C_{\Sigma }}\right)\right\}C_{\text{g}}\\ &=\frac{dt}{dt}\left\{\frac{C_{\text{g}}}{C_{\Sigma}}\left(V_{\text{g}}C_{ \Sigma}-C_{\text{g}}V_{\text{g}}-C_{\text{d}}V_{\text{d}}-C_{\text{s}}V_{ \text{s}}+e\sum nP_{\text{n}}\right)\right\}\end{array}  \tag{5}\]  
The SPICE circuit simulator [27] is used as a solver for the simultaneous differential equation [31] of the master equation to obtain the \(P_{\text{n}}\) and \(\Gamma_{\text{d}}\) in (1) and (5). In the following sections, the simulation results will be shown based on the parameters summarized in Table 1. These parameters are set rather arbitrarily to realize the cutoff frequency of 1 GHz where experimental verification is relatively easy and are also viable considering that the subattofarad capacitances have already been realized [1; 2; 3], and the tunneling resistance could be varied in a wide range by changing the insulator thickness.