Figure 3 shows random discrete active As distribution in the Si NW calculated by the KMC simulation. The histogram shows the normal distribution curve, and therefore, 200 seeds are large enough to represent the randomness. The average number of active As atoms in the NW is 27 with the standard deviation of 5. Out of 300 As atoms implanted into the 3-nm-wide Si region, only approximately 10% of As atoms are active in the Si NW. Most of the As atoms are in the oxide (approximately 40 atoms), at the oxide/Si interface (approximately 50), in As-vacancy (As-V) clusters (approximately 90), and As precipitates (approximately 90) (see Figure 1). As-V clusters and As precipitates are inactive and immobile. They are formed when As concentration exceeds approximately 10²⁰ cm⁻³ (for As-V clusters) and the solubility limit (for As precipitates) 14 15 . In Sentaurus, not only As-V clusters but also As-Si interstitial (I) clusters (inactive and immobile) are taken into account, but As precipitates are not. In the present study, therefore, As-Si interstitial clusters in Sentaurus are interpreted as As precipitates. The calculation results show that the As activation ratio decreases with higher As dose because inactive As species (As-V clusters and As precipitates) are more likely to be formed. In NWs with smaller widths and heights, the As activation is found to be lower because more As atoms are closer to the oxide/Si interface and hence are piled up at the interface.

Figure 4 shows the I _d-V _g characteristics at V _d = 0.5 V of 100 devices with different discrete As distributions (gray lines). In the figure, their average value 〈I _d〉 (open circles) and the I _d of a continuously doping case in the S/D extensions (solid circles) are also shown for comparison. For the continuously doping case, the S/D extensions are uniformly n-doped with a concentration of 3 × 10²⁰ cm⁻³, which corresponds to the average active As concentration in the Si NWs (see Figure 3). The I-V characteristics of devices uniformly n-doped with 2 × 10²⁰, 2.5× 10²⁰, and 3.5 × 10²⁰ cm⁻³ are also calculated, and the results show only slight differences (within 10%) compared with the 3 × 10²⁰ cm⁻³ case. Figure 5 represents the carrier density profiles and the location of active As atoms in some representative devices. Equidensity surfaces at V _d = V _g = 0.5 V (blue and green surfaces for 3 × 10²⁰ and 1 × 10²⁰ cm⁻³, respectively) and dopant positions (yellow dots) are shown. Figure 5 (a), (b), (c), and (d) correspond to the I-V characteristics of continuously doped (solid circles in Figure 4), high-current (red dashed line), medium-current (green dashed line), and low-current (blue dashed line) devices, respectively. The drain current of NW devices with random discrete As distribution is found to be reduced compared to that with uniform As distribution. This reduction is ascribed to ionized impurity scattering, which is taken into account for random As distribution, but not for uniform As distribution. The normalized average current 〈I _d〉/I ₀ (I ₀ is the drain current of the continuously doped device) is found to be approximately 0.8 and decreases with V _g, as shown in Figure 6. The standard deviation of the 100 samples is found to be σI _d ~ 0.2〈I _d〉.

In order to investigate the cause of the drain current fluctuation, we examine the correlation between I _d and the factors related to random As distributions. The factors are extracted from the random As positions, based on a simple one-dimensional model as schematically shown in Figure 7, where blue dots represent active As atoms. The factors are an effective gate length (L _g ^*), standard deviations of interatomic distances in the S/D extensions (σ _s and σ _d), their sum (σ = σ _s + σ _d), and the maximum separation between neighboring impurities in the S extension (S _s), in the D extension (S _d), and in the S/D extensions (S). The effects of the number of As dopants in the S/D extensions are also examined, with the factors of the number of active As in the S extension (N _s), in the D extension (N _d), and in the S/D extensions (N). Figure 8 represents the correlation between I _d and these factors, and Table 1 summarizes the correlation coefficients for the off-state (V _g = 0 V) and the on-state (V _g = 0.5 V) at V _d = 0.05 and 0.5 V. The correlation coefficient r is classified as follows: 0.0 < |r| < 0.2, little correlation; 0.2 < |r| < 0.4, weak correlation; 0.4 < |r| < 0.7, significant correlation; 0.7 < |r| < 0.9, strong correlation; and 0.9 < |r| < 1.0, extremely strong correlation. We highlight clear correlations in Table 1. Note that the threshold voltage is closely related to the off-current because I _d varies exponentially with V _g at the subthreshold region.

Clear correlations are shown in italics.

Significant correlations between I _d and L _g ^* are found at the off-state with V _d of both 0.05 and 0.5 V. Negative correlation means that I _d tends to decrease with increasing L _g ^*. The sum of the standard deviations of interatomic distances in the S/D extensions (σ) shows a clear correlation at the on-state with V _d = 0.05 V. Concerning the maximum separation, a clear correlation at the on-state with V _d = 0.5 V and that with V _d = 0.05 V are found with S _s and S, respectively, while little correlation with S _d is seen at any cases. These results demonstrate that the effective gate length (L _g ^*) is a main factor for the off-state, where the channel potential mainly governs the I V characteristics. We mention that the off-current becomes larger when active As atoms penetrate into the channel region, which is not taken into account in the present simulation. This increase in off-current can be explained in terms of the ion-induced barrier lowering 16 , where the potential barrier in the channel is significantly lowered by attractive donor ions, which enhances the electron injection from the source. For the on-state, random As distribution in the S extension (S _s) is an important factor at high V _d due to current injection from S, and that in the S/D extensions (σ and S) is dominant at low V _d because the back-flow current from D also contributes the current.

On the other hand, little or weak correlations between I _d and the number of As dopants are found. The weak positive correlations with N _s and N at the off-state are attributed to a tendency that a larger number of dopants lead to smaller L _g ^*. In order to further investigate the effect of the number of As, I _d-V _g characteristics of NWs implanted at a smaller dose of 2 × 10¹⁴ cm⁻² were calculated. The average number of active As atoms in this NW is 16, which averages 1.8 × 10²⁰ cm⁻³. The average and standard deviation of the on-current in this NW are almost the same as those in the 1 × 10¹⁵ cm⁻² NW. This is consistent with little or weak correlations between I _d and the number of As dopants as we mentioned above. However, a few out of 100 NW devices of 2 × 10¹⁴ cm⁻² have on-current which is only about one half its average. This is attributable to the large interatomic distances of discrete As atoms in these devices. These results indicate that the on-current fluctuation is caused by the fluctuation of interatomic distances of discrete As atoms, not by the fluctuation of the number of As. The off-current fluctuation can be reduced by a process in which dopants in the S/D extensions are likely to exist near the channel region. In contrast, the on-current fluctuation may be inherent in ultra-small NW transistors because interatomic distance is determined by random atomic movement.