# Application of Single-Electron Transistor to Biomolecule and Ion Sensors
## 2 Mechanism of High Sensitivity in Biomolecule Detection Based on a Single-Electron Transistor
The dependence of memory characteristics on the number and distribution of the floating dot were systematically analyzed experimentally and numerically for dots-on-nanowire memories with multiple Si nanoscale floating dots [27; 28]. From the numerical results in the case where the width of the channel is wider than the size of the floating dot (Figure 3) [28], the potential in the non-existing region of a floating dot was lower than that in the existing region (Figure 4). This indicates the existence of a percolation path for electrons to pass through the channel when the width of the channel is wider than the size of the dot. SET sensors should have extremely narrow Si nanowire barriers for the tunnel current to flow at room temperature. The FET sensors used for the comparison with the SET also should have such a narrow nanowire channel region that the tunnel current flows through it, making the percolation effect of carriers well suppressed in this nanowire region. Because the Coulomb oscillations is modulated and weighted by the current flowing through the nanowire barrier regions, the \(g_{m}\) at the corresponding \(V_{g}\) increases with increasing \(V_{g}\) for the SET sensors (Figure 5). Therefore, the \(g_{m}\) in the SET sensor can be larger than that in the FET sensor, whose channel consists of the nanowire barrier region, by increasing \(V_{g}\). Hence, the sensitivity in the SET sensor can be higher than that in the FET sensor with the nanowire channel by increasing \(V_{g}\). Figure 4: Calculated potential profile perpendicular to the current direction for dots-on-nanowire memories with single and double Si floating dots. Schematic distribution of the single dot is shown in Figure 3. Reproduced with permission from [28], Copyright 2009, AIP Publishing LLC. Figure 3: Schematic view of dot distribution in a dots-on-nanowire memory with one Si nanoscale floating dot whose width is smaller than that of the channel. The finding can also be confirmed by the theoretical analysis [22]. The equivalent circuit used for the analysis is shown in the inset of Figure 6. Here, the number of Coulomb islands is assumed to be one, which corresponds to the case where only the smallest island among the multiple islands fabricated acts as a Coulomb island. The \(I_{d}\)-\(V_{g}\) characteristics for the SET were calculated using the analytical formula of \(I_{d}\)-\(V_{g}\) characteristics for the Coulomb oscillations in the classical regime, weighted by the \(V_{g}\) dependence of current flowing through the nanowire barrier regions. The analytical formula used for Coulomb oscillations is based on the orthodox theory as follows [29; 30; 31]:  
\[I_{n}=\frac{e}{2R_{\Sigma}\mathbb{C}_{\Sigma}}\frac{\left(\widetilde{V}_{g,n}^ {2}-\widetilde{V}_{d}^{2}\right)\sinh\left(\widetilde{V}_{d}/\widetilde{T} \right)}{\widetilde{V}_{g,n}\sinh\left(\widetilde{V}_{g,n}/\widetilde{T} \right)-\widetilde{V}_{d}\sinh\left(\widetilde{V}_{d}/\widetilde{T}\right)} \tag{1}\]  
where  
\[\widetilde{V}_{g,n}=\frac{2C_{g}V_{g}}{e}-\frac{\left(C_{g}+C_{sub}+C_{s}-C_{d }\right)V_{d}}{e}-1-2n,\ \widetilde{V}_{d}=\frac{C_{\Sigma}V_{d}}{e},\ \widetilde{T}=\frac{2k_{B}TC_{\Sigma}}{e^{2}},\text{and}\ R_{ \Sigma}=R_{T}+R_{T} \tag{2}\]  
where \(k_{B}\), \(e\), \(T\), \(V_{d}\), \(C_{g}\), \(C_{s}\), \(C_{d}\), \(C_{sub}\), \(C_{\Sigma}\), and \(n\) are the Boltzmann constant, elementary charge, operation temperature, drain voltage, gate capacitance, junction capacitances at source and drain, capacitance between the island and the substrate, total capacitance, and electron number in the Coulomb island. Here, \(T\) was set to room temperature. The values of \(C_{g}\), \(C_{s}\), and \(C_{\Sigma}\) were obtained from the measurement results of Coulomb oscillations for the SET biosensor in buffer solution [22]. \(R_{\Sigma}\) is the total resistance of the barrier regions, and \(R_{\Sigma}=2R_{T}\), where \(R_{T}\) is the resistance of a tunnel junction.