Simulation Approach

The most popular model to simulate the magnitude spectrum of LER is the Gaussian and exponential autocorrelation functions with two parameters：

S

k

π

e

, where ∆ is the rms amplitude of the roughness, and Λ is the correlation length which is a fitting parameter for a particular type of LER. k is the index of discrete sampling points defined as k=i(2 / Ndx)π with dx the spacing of sampling points.

Fig. 3.2 shows a real LER power spectrum compared with the Gaussian and exponential models with the specified values of ∆ and Λ [10]. With appropriate choices of rms amplitude and correlation length combined with randomly selected phases that can make sure each rough line is unique, we can rebuild rough lines as shown in Fig. 3.3. It can be seen that the Gaussian model is smoother than the exponential one due to lack of high frequency component. Besides, unlike planar MOSFETs, multi-gate LER comes from both gate length and fin width dimension deviations. It has been confirmed that σ_{total_LER}² =σ_{fin_LER}² +σ_{gate_LER}² [34]. Fig.

As for the determination of model parameters, the value of rms amplitude can be obtained from the ITRS. Unlike rms amplitude, the choice of correlation length should be extracted from the real line edge patterns. P. Oldiges et al. [32] reported that the values of correlation length vary between 10 and 50nm from their measurements.

Based on the SEM analysis, A. Asenov et al. [10] indicated that the values of correlation length are in the range of 20-30nm. Due to lack of the experimental data, we determine the value of correlation length from the literatures mentioned above in the following simulations.

As in chapter 2, we perform Monte Carlo simulation with 150 samples to capture the stochastic behavior regarding LER. Besides, we use the same physical TCAD simulation models and current extraction criterion as in chapter 2. To relieve the simulation burden, we use continuous doping in the following simulations.

3.1.3 Results and Discussions

Before investigating the impacts of model parameters on device characteristics, we first qualitatively evaluate their influences on the LER patterns and try to gain more insights about the model. In Fig. 3.5, we set correlation length to 20nm with various rms amplitude values. It can be seen that increasing rms amplitude gives rise to larger amplitude spectrum. Moreover, from Fig. 3.5 (a) (b), we find the difference in amplitude spectrum is about four times when the rms amplitude becomes two times larger. This can be explained by (3.1). Besides, increasing the rms amplitude would

lead to rougher line as shown in Fig. 3.5 (c). In Fig. 3.6 (a), we vary the correlation length with constant rms amplitude. We can find that the magnitude spectrum with smaller correlation length would spread out to higher frequency region, which results in the rougher LER (more high frequency components) as observed in Fig. 3.6 (b).

Moreover, in Fig. 3.6 (b), we find that increasing correlation length would also lead to worse LER (σ_LER =1.23, 1.87 nmfor correlation lengths equal to 20 and 50nm).

Fig. 3.7 shows the rms amplitude dependence of σV_thfor lightly doped devices with 3 kinds of AR. It can be seen that increasing the rms amplitude, or rougher line edge, may result in severer threshold voltage variation for both gate and fin components. Besides, for devices with AR=0.5 or 1, gate LER is the dominant mechanism due to poor electrostatic integrity. With the increasing of AR, we can find that the influences of gate- and fin- LER become comparable in our FinFET (AR=2) devices. For devices designed with AR=5 as described in [34], fin LER would dominate the overall LER variation. In Fig. 3.8, we design the total width of FinFET (AR=2) devices to three times of Lg at different technology nodes and compare the importance of gate and fin LER. It can be seen that the fin LER is the main contributor to device LER variations because of the significant shrinkage of fin width at smaller devices. From above discussions, we conclude that gate and fin LER will become dominant for devices with critical dimension in channel length and fin width, respectively.

In Fig. 3.9, we investigate the impacts of correlation length on device variation.

Similar to the results in Fig. 3.7, increasing correlation length leads to higher Vth

fluctuation. In addition, we find that V_th fluctuations would saturate as correlation

variations induced by the fin LER will saturate as correlation lengths reach the gate length, 25nm, and gate LER shows different saturation levels because of the different fin widths for three various AR devices. This property is also observed in planar transistors [10]. To explain the phenomenon, we demonstrate three LER patterns associated with different correlation lengths as shown in Fig. 3.10. If we assume that the device critical dimension is equal to 50nm, and assign the corresponding LER pattern for each device as shown in the figure, it can be seen that as the correlation length decreases, the discrepancy of LER pattern for each device increases. This explains the initial increase for small correlation lengths. When the correlation length approaches the device critical dimension, there is less LER pattern dispersion and therefore the V_th variation saturates.

For heavily doped transistors, all of above characteristics are similar to lightly doped ones. In Fig. 3.11, we compare total LER (σ_{fin_LER}² +σ_{gate_LER}² ) for devices with different doping concentration and AR. It can be seen that heavily doped devices show better immunity to LER because of the superior gate control especially for the devices with small AR. Moreover, increasing doping concentration can significantly suppress LER at the price of worse RDF as mentioned in chapter 2. Fig. 3.12 illustrates the comparison of RDF and LER. It can be seen that RDF dominates in heavily doped devices. For lightly doped devices, LER is the most important contributor to device variations. Under the consideration of RDF and LER, lightly doped FinFET (AR=2) has better immunity to device intrinsic parameter fluctuations.

So far, we have discussed several properties of LER and its influences on multi-gate devices. We tackle the problem with individual assessment of RDF and

LER, that is, we assume that they are independent events. Actually, this is confirmed in planar MOSFETs [10]. However, there are several studies against this argument. It is believed that there exist coupling between RDF and LER. S. Xiong et. al. [35]

performed process simulation and found LER enhanced lateral diffusion. M. Hane et.

al. [36] considered both RDF and LER simultaneously and confirmed the existence of

coupling. To assess this problem, we make a simple examination of dopant number in the channel with different degree of LER. The determination of dopant number is similar to the full Monte Carlo simulation in section 2.2.2. Fig. 3.13 show the dopant number histogram plots with no LER, LER with rms amplitude=1.5nm and 2.5nm, respectively. It can be seen that different rough patterns have the same average but different spreading in dopant number. Rougher line edge pattern seems to be more diverse. This is because worse LER causes larger discrepancy in the channel volume which is closely related to the total dopant number. In other words, we may observe the coupling between RDF and LER in aspect of dopant number. We will further investigate the coupling between RDF and LER in the future.