As distribution by KMC simulation
Figure 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 3nmwide 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 Asvacancy (AsV) clusters (approximately 90), and As precipitates (approximately 90) (see Figure ). AsV clusters and As precipitates are inactive and immobile. They are formed when As concentration exceeds approximately 10
^{20} cm
^{−3} (for AsV clusters) and the solubility limit (for As precipitates) [
14,
15]. In Sentaurus, not only AsV clusters but also AsSi interstitial (I) clusters (inactive and immobile) are taken into account, but As precipitates are not. In the present study, therefore, AsSi 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 (AsV 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.
NEGF simulation
Figure 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 ndoped with a concentration of 3
×
10
^{20} cm
^{−3}, which corresponds to the average active As concentration in the Si NWs (see Figure ). The
I
V characteristics of devices uniformly ndoped with 2
×
10
^{20}, 2.5× 10
^{20}, and 3.5
×
10
^{20} cm
^{−3} are also calculated, and the results show only slight differences (within 10%) compared with the 3
×
10
^{20} cm
^{−3} case. Figure 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
×
10
^{20} and 1
×
10
^{20} cm
^{−3}, respectively) and dopant positions (yellow dots) are shown. Figure , (b), (c), and (d) correspond to the
I
V characteristics of continuously doped (solid circles in Figure ), highcurrent (red dashed line), mediumcurrent (green dashed line), and lowcurrent (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 . The standard deviation of the 100 samples is found to be
σI_{d}~
0.2
I_{d}.
Drain current fluctuation
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 onedimensional model as schematically shown in Figure , 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 represents the correlation between
I_{d} and these factors, and Table summarizes the correlation coefficients for the offstate (
V_{g}=
0 V) and the onstate (
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 . Note that the threshold voltage is closely related to the offcurrent because
I_{d} varies exponentially with
V_{g} at the subthreshold region.
 Table 1Summary of correlation factors of drain current 
Significant correlations between
I_{d} and
L_{g}^{*} are found at the offstate 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 onstate with
V_{d}=
0.05 V. Concerning the maximum separation, a clear correlation at the onstate 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 offstate, where the channel potential mainly governs the
IV characteristics. We mention that the offcurrent becomes larger when active As atoms penetrate into the channel region, which is not taken into account in the present simulation. This increase in offcurrent can be explained in terms of the ioninduced 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 onstate, 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 backflow 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 offstate 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
×
10
^{14} cm
^{−2} were calculated. The average number of active As atoms in this NW is 16, which averages 1.8
×
10
^{20} cm
^{−3}. The average and standard deviation of the oncurrent in this NW are almost the same as those in the 1
×
10
^{15} 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
×
10
^{14} cm
^{−2} have oncurrent 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 oncurrent fluctuation is caused by the fluctuation of interatomic distances of discrete As atoms, not by the fluctuation of the number of As. The offcurrent 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 oncurrent fluctuation may be inherent in ultrasmall NW transistors because interatomic distance is determined by random atomic movement.