# Application of Single-Electron Transistor to Biomolecule and Ion Sensors
## 3 Single-Electron Transistor with Multiple Islands  
In the fabricated SET sensors, Si dots were used as Coulomb islands, and the tunnel barriers were formed by the constriction of Si nanowire. The necessary condition for observing single-electron effects, including Coulomb oscillations, is that the charging energy of a Coulomb island should be larger than the energy of operation temperature:  
\[e^{2}/(2C_{\Sigma})\,\gg\,k_{B}T \tag{5}\]  
Consequently, for the room-temperature operation of Si SETs with a single Coulomb island, the island size should be reduced to below 10 nm. In fact, several reports on Si SETs have indicated that the dimension of the Coulomb island should be around 5 nm or below for the room temperature operations [34; 35; 36], although the precise size to achieve the room temperature SET operation depends on the individual device structure. However, it seems difficult to fabricate such Si SETs with the extremely small Coulomb island reproducibly, even by using current available fabrication techniques. One way to reduce the difficulty is to utilize serially connected islands instead of a single island in the channel region [37]. In such a serially connected multiple-island system, the effective junction capacitance \(C_{\mathit{eff}}\) is smaller than that in a single-island system because the plural junction capacitances are connected in series. In the case of infinite number of serially distributed junction capacitances, for example, the capacitance \(C_{h}\) seen from the edge of a half infinite array is smaller than a junction capacitance \(C\) as can be understood from the following relation:  
\[C_{h}{}^{-1}=C^{-1}+(C_{0}+C_{h})^{-1} \tag{6}\]  
where \(C_{0}\) is a parasitic capacitance including gate and substrate capacitances [38]. The smaller \(C_{\mathit{eff}}\) increases the charging energy \(e^{2}/(2C_{\Sigma})\) and leads to the increase in the operation temperature. Another reason for the advantage of serially connected islands to the room-temperature SET operation is that it increases the probability of the existence of an extremely small island with an extremely small total capacitance.  
With serially connected multiple Coulomb islands, a reliable SET operation is also possible. Generally, in a Coulomb blockade regime, a cotunneling process gives rise to leakage current [39]. The temperature dependence of the valley current of Coulomb oscillations was found to be well described by the inelastic cotunneling theory at high temperatures in the low \(V_{d}\) region [40]. Since the inelastic cotunneling current can be suppressed by multiplexing Coulomb islands, SETs with multiple islands are promising for reliable operation [41]. Furthermore, it was theoretically found that periodic Coulomb oscillations appear with the period determined by the average capacitance in the multiple-island system under the small deviation of gate capacitances [42].  
Owing to the above advantages in the multiple-island system, the room-temperature operation of an exclusive-or (XOR) circuit [43] and the high-temperature operation of an exclusive-not-or (EXNOR) circuit [44] have been achieved. Therefore, for the SET sensors in this study, serially connected multiple islands were fabricated in the channel region.