## 1 Introduction  
Semiconductor devices have been basically progressed with the shrinking of MOSFETs (metal-oxide-semiconductor field-effect-transistors), which are used as the key component in them. The shrinkage has been performed following the rule of the constant electric field in MOSFETs, which decreases signal voltage [1, 2]. In addition, power consumption of electronic devices has skyrocketed because the amount of digital data generated is growing at a rate faster than Moor's law [3, 4]. The reduction of power consumption strongly requires decreasing supply voltage of MOSFETs because power consumption (P) is proportional to the square of the supply voltage (V\({}_{\mathrm{dd}}\)) as follows [5]. \[\mathrm{P}=\mathrm{C}_{\mathrm{L}}\mathrm{V}_{\mathrm{dd}}^{2}\mathrm{f} \tag{1}\]  
where \(\mathrm{C}_{\mathrm{L}}\) represents the load capacitor and \(\mathrm{f}\) represents the switching frequency of the circuit. The growth of clock frequency in the leading edge logic devices has stopped by exceeding the heat extraction capability. However, the downscaling has been continued to reduce the cost. In the other devices, the downscaling the device size has also been continued to reduce the power consumption and the other reasons. As a result, as the device size is reduced, the signal voltage of MOSFETs decreases. Memory devices' power consumption and supply voltage also have to be reduced [6, 7, 8, 9] because of the same reasons as the logic devices and reducing a leakage current. On the other hand, a decrease in signal voltage degrades the reliability of electronic circuits, including analog and digital devices. The logic (bit) error rate (LER) is given by the following equation. \[\mathrm{LER}=\mathrm{P}_{0}\int_{-\infty}^{\alpha}\mathrm{f}_{0}(\mathrm{x}) \mathrm{d}\mathrm{x}+\mathrm{P}_{1}\int_{\alpha}^{\infty}\mathrm{f}_{1}( \mathrm{x})\mathrm{d}\mathrm{x} \tag{2}\]  
where \(\mathrm{P}_{0}\) and \(\mathrm{P}_{1}\) represent the probabilities of signals \"0\" and \"1\", respectively, \(\alpha\) represents the identification level between \"0\" and \"1\", and \(\mathrm{f}_{0}(\mathrm{x})\) and \(\mathrm{f}_{1}(\mathrm{x})\) represent noise amplitude densities superimposed on \"0\" and \"1\", respectively. If \(\mathrm{f_{0}}\) and \(\mathrm{f_{1}}\) are Gaussian noise, the following equations apply. \[\mathrm{f_{0}(x)}=\frac{1}{\sqrt{2\pi\sigma^{2}}}\mathrm{e}^{-\frac{x^{2}}{2 \sigma^{2}}},\ \mathrm{f_{1}(x)}=\frac{1}{\sqrt{2\pi\sigma^{2}}}\mathrm{e}^{-\frac{(x-A_{ \mathrm{S}})^{2}}{2\sigma^{2}}} \tag{3}\]  
where \(\mathrm{A_{S}}\) represents the signal amplitude and \(\sigma\) represents the standard deviation of the noise. When \(\mathrm{P_{0}}=\mathrm{P_{1}}=1/2\) and \(\mathrm{\alpha=A_{S}/2}\) are assumed, the LER is given by the following equation from Equations (2) and (3). \[\mathrm{LER}=\frac{1}{2}\mathrm{erfc}\bigg{(}\frac{\mathrm{A_{S}}}{2\sqrt{2 \sigma^{2}}}\bigg{)} \tag{4}\]  
\[\mathrm{erfc(x)}=1-\mathrm{erf(x)}=1-\frac{2}{\sqrt{\pi}}\int_{0}^{x}\mathrm{ e}^{-t^{2}}\mathrm{dt}=\frac{2}{\sqrt{\pi}}\int_{x}^{\infty}\mathrm{e}^{-t^{2}} \mathrm{dt} \tag{5}\]  
When Nyquist transmission rate is the same as the signal band, the signal to noise (S/N) ratio (dB) is given by the following equation. \[\frac{\mathrm{S}}{\mathrm{N}}=\frac{\mathrm{E_{b}}}{\mathrm{N_{0}}}=20\log \bigg{(}\frac{\mathrm{A_{S}}}{\sigma}\bigg{)} \tag{6}\]  
where \(\mathrm{E_{b}}\) and \(\mathrm{N_{0}}\) represent signal and noise energy per second, respectively. From Equations (4) and (6), the LER is given by the S/N ratio as follows [10]. \[\mathrm{LER}=\frac{1}{2}\mathrm{erfc}\bigg{\{}\frac{1}{2\sqrt{2}}10^{\frac{ \mathrm{S}}{\mathrm{(S)}}}\bigg{\}} \tag{7}\]  
Figure 1 shows the LER as a function of S/N (dB) and A/\(\sigma\) ratios [10]. The LER decreases with an increase in S/N (\(\mathrm{A_{S}/\sigma}\)) ratio. On the other hand, to guarantee that a system does not make a mistake even once during the operation period, the LER should be reduced as shown in the following equation.