We have theoretically investigated the effects of random discrete distribution of implanted and annealed arsenic (As) atoms on device characteristics of silicon nanowire (Si NW) transistors. Kinetic Monte Carlo simulation is used for generating realistic random distribution of active As atoms in Si NWs. The active As distributions obtained through the kinetic Monte Carlo simulation are introduced into the source and drain extensions of n-type gate-all-around NW transistors. The current–voltage characteristics are calculated using the non-equilibrium Green's function method. The calculated results show significant fluctuation of the drain current. We examine the correlation between the drain current fluctuation and the factors related to random As distributions. We found that the fluctuation of the number of dopants in the source and drain extensions has little effect on the on-current fluctuation. We also found that the on-current fluctuation mainly originated from the randomness of interatomic distances of As atoms and hence is inherent in ultra-small NW transistors.
Keywords:Silicon nanowires; Random discrete dopant distribution; Gate-all-around transistors; Kinetic Monte Carlo; Non-equilibrium green's function
Fluctuation due to random discrete dopant (RDD) distribution is becoming a major concern for continuously scaled down metal-oxide semiconductor field-effect transistors (MOSFETs) [1-4]. For ultra-small MOSFETs, not only random location of individual dopant atoms but also fluctuation of the number of active impurities is expected to have significant impacts on the device performance. Effects of the RDD distribution are usually analyzed with a randomly generated RDD distribution. The actual RDD distribution, however, should be correlated with the process condition and can be different from a mathematically generated one. In the present study, we investigate the effects of random discrete distribution of implanted and annealed arsenic (As) atoms in source and drain (S/D) extensions on the characteristics of n-type gate-all-around (GAA) silicon nanowire (Si NW) transistors. We investigate a GAA Si NW transistor since it is considered as a promising structure for ultimately scaled CMOS because of its excellent gate control [2,5-7]. Kinetic Monte Carlo (KMC) simulation is used for generating realistic random distribution of active As atoms in Si NWs. The current–voltage characteristics are then calculated using the non-equilibrium Green's function (NEGF) method. Our results demonstrate that the on-current fluctuation mainly originated from the randomness of the dopant location and hence is inherent in ultra-small NW transistors.
Random discrete As distribution in a Si NW is calculated using Sentaurus KMC simulator (Synopsys, Inc., Mountain View, CA, USA) [8-10]. Figure 1 shows an example of the calculated discrete As distribution in a Si NW (3 nm wide, 3 nm high, and 10 nm long) with 1-nm-thick oxide. The Si NW is implanted with As (0.5 keV, 1 × 1015 cm−2) and annealed at 1,000°C with a hold time of 0 s. Statistical variations are investigated using 200 different random seeds. The active As distributions obtained through the KMC simulation are then introduced into the S/D extensions of n-type Si NW MOSFETs, whose device structure is given in Figure 2. In the present study, we consider only an intrinsic channel, and impacts of possible penetration of dopant atoms into the channel region are not examined. To mimic metal electrodes, the S/D regions are heavily doped with Nd = 5 × 1020 cm−3 (continuously doping). We simulate 100 samples using 200 different random seeds (each sample needs two random seeds for S/D extensions). The drain current-gate voltage (IdVg) characteristics are calculated using the NEGF method with an effective mass approximation [11,12]. The discrete impurities are treated with a cloud-in-cell charge assignment scheme . Phonon scattering is not taken into account in the present calculation.
Figure 1. Discrete As distribution in a Si NW. Cross-sectional view (left) and entire view (right). Red dots show active As atoms in Si. Blue dots, As precipitates; light blue dots, As-V clusters; orange dots, As at the oxide/Si interface; and yellow dots, As in the oxide.
Figure 2. Schematic diagram of the n-type GAA Si NW MOSFET. Discrete distributions of the active As atoms are introduced into the S/D extensions. To mimic metal electrodes, the S/D regions are heavily doped with Nd = 5 × 1020 cm−3 (continuously doping). The channel region is intrinsic. We simulated 100 samples using 200 different random seeds (each sample needs two random seeds for S/D extensions).
Results and discussion
As distribution by KMC simulation
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 3. Histogram of the number of active As atoms in the Si NWs. Si NWs (3 nm wide, 3 nm high, and 10 nm long) with 1-nm-thick oxide are implanted with As (0.5 keV, 1 × 1015 cm−2) and annealed at 1,000°C with a hold time of 0 s. Two hundred different random seeds were calculated.
Figure 4 shows the Id-Vg characteristics at Vd = 0.5 V of 100 devices with different discrete As distributions (gray lines). In the figure, their average value 〈Id〉 (open circles) and the Id 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 Vd = Vg = 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 〈Id〉/I0 (I0 is the drain current of the continuously doped device) is found to be approximately 0.8 and decreases with Vg, as shown in Figure 6. The standard deviation of the 100 samples is found to be σId ~ 0.2〈Id〉.
Figure 4. Id-Vgcharacteristics of GAA Si NW transistors atVd = 0.5 V. Gray lines show the Id-Vg of 100 samples with different discrete As distributions. Open circles represent their average value 〈Id〉. The continuously doping case with Nd = 3 × 1020 cm−3 in the S/D extensions is shown by solid circles for comparison.
Figure 5. Carrier density profiles and location of active As atoms in NW devices. Equidensity surfaces (blue and green surfaces) and dopant positions (yellow dots) for (a) continuously doped, (b) high-current (red dashed line in Figure 4), (c) medium-current (green dashed line in Figure 4), and (d) low-current (blue dashed line in Figure 4) devices. Vd = Vg = 0.5 V.
Figure 6. Average and standard deviation of drain current in NW devices. Average current 〈Id〉 and standard deviation σId vs. Vg. I0 is the drain current of the continuously doped device.
Drain current fluctuation
In order to investigate the cause of the drain current fluctuation, we examine the correlation between Id 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 (Lg*), standard deviations of interatomic distances in the S/D extensions (σs and σd), their sum (σ = σs + σd), and the maximum separation between neighboring impurities in the S extension (Ss), in the D extension (Sd), 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 (Ns), in the D extension (Nd), and in the S/D extensions (N). Figure 8 represents the correlation between Id and these factors, and Table 1 summarizes the correlation coefficients for the off-state (Vg = 0 V) and the on-state (Vg = 0.5 V) at Vd = 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 Id varies exponentially with Vg at the subthreshold region.
Figure 7. One-dimensional model to analyze drain current fluctuation. Blue dots represent active As atoms. Lg*, effective gate length; σ = σs + σd, sum of the standard deviations of interatomic distances in the S/D extensions; Ss, the maximum separation between neighboring impurities in the S extension; Sd, that in the D extension; S, that in the S/D extensions. si and di are interatomic distances in the S/D extensions. The effects of the number of As dopants in the S extension (Ns), in the D extension (Nd), and in the S/D extensions (N) are also examined.
Figure 8. Correlation coefficients between drain current and factors related to random As distributions. Blue and red circles represent correlation coefficients at Vd = 0.05 and 0.5 V, respectively. The coefficient of 0 means no correlation, and those of ±1, the strongest correlation.
Table 1. Summary of correlation factors of drain current
Significant correlations between Id and Lg* are found at the off-state with Vd of both 0.05 and 0.5 V. Negative correlation means that Id tends to decrease with increasing Lg*. The sum of the standard deviations of interatomic distances in the S/D extensions (σ) shows a clear correlation at the on-state with Vd = 0.05 V. Concerning the maximum separation, a clear correlation at the on-state with Vd = 0.5 V and that with Vd = 0.05 V are found with Ss and S, respectively, while little correlation with Sd is seen at any cases. These results demonstrate that the effective gate length (Lg*) is a main factor for the off-state, where the channel potential mainly governs the IV 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 , 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 (Ss) is an important factor at high Vd due to current injection from S, and that in the S/D extensions (σ and S) is dominant at low Vd because the back-flow current from D also contributes the current.
On the other hand, little or weak correlations between Id and the number of As dopants are found. The weak positive correlations with Ns and N at the off-state are attributed to a tendency that a larger number of dopants lead to smaller Lg*. In order to further investigate the effect of the number of As, Id-Vg 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 Id 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.
We have theoretically investigated the effects of random discrete distribution of implanted and annealed As atoms in the S/D extensions on the device characteristics of n-type GAA Si NW transistors. KMC simulation is used for generating realistic random distribution of active As atoms in Si NWs, and the current–voltage characteristics are calculated using the NEGF method. The fluctuation of drain current is observed with the normalized standard deviation of approximately 0.2. The correlation between the drain current and the factors related to random As distribution is examined. The results indicate that the on-current fluctuation is not directly due to the fluctuation of the number of dopants in the S/D extensions. The on-current fluctuation may be caused by the randomness of As dopant positions in the S/D extensions and hence is inherent in ultra-small NW transistors.
GAA: gate-all-around; KMC: kinetic Monte Carlo; MOSFET: metal-oxide semiconductor field-effect transistors; NEFG: non-equilibrium Green's function; NW: nanowire; RDD: random discrete dopant; S/D: source and drain.
The authors declare that they have no competing interests.
MU carried out the KMC calculations to obtain random discrete As distributions in the S/D extensions of NW transistors and drafted the manuscript. KMI supervised the KMC simulation. GM and HM participated in the NEGF simulation of NW transistors. NM carried out the NEGF calculations and analyzed the I-V characteristics of NW transistors. All authors read and approved the final manuscript.
We acknowledge Dr. Ignacio Martin Bragado for the fruitful discussions on KMC modeling.
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