Modelling Double-Gate MOSFET

In this part, we only consider double-gate MOSFET. First, Table 5.1 shows the correla-tion between every independent variables. Because of our design of experiments, we can find that there is collinearity phenomenon among some variables. Figs. 5.3, 5.4, 5.5, 5.6, 5.7, 5.8 show the scattering plots : each independent variable against standard deviation of wave packet. Obviously, we can find that there are not only linear correlations but also quadratic correlations between dependent variable and some independent variables. So, we will add square terms into the model. Because there is collinearity phenomenon, effect of some variables may be weakened so that variables are not significant. Besides, we also add interaction terms to check whether there are interaction effects or not.

All models we establish will be analyzed by stepwise method:

Independent variables : 1. Variables 1∼ 6

2. Square terms of Variables 1∼ 6

3. Second order interaction terms of Variables 1∼ 6 Model I Dependent variable : Standard deviation of wave packet

Residual analyses of model I are shown as Fig. 5.9 and Fig. 5.10. Clearly, we can know model I is bad from Fig. 5.10.

We try to transform the dependent variable by power transformation. Fig. 5.11 is Likeli-hood plot of power transformation. LikeliLikeli-hood achieves maximum when λ = −0.148. By the reason stated before, we prefer to set λ= 0. So, we try to take log of standard deviation of wave packet.

Model II Dependent variable :ln(Standard deviation of wave packet)

Figure 5.12 and Fig. 5.13 show the residual analyses of model II. Obviously, model II is not good, either.

1D quantum correction may be not enough for devices with ultra-short channel [48][49][50].Now, we reduce data and retain data whose channel length longer than 20nm only. Then we get model III and model IV.

Model III Dependent variable : Standard deviation of wave packet Model IV Dependent variable : ln(Standard deviation of wave packet)

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Residual analyses are shown from Fig. 5.14 to Fig. 5.17.

We find that results of model IV is good from plots of residual analysis.

Lg Vg Vd tsi tox N

Table 5.1: Correlation Table: pairwise correlation between every variables. There may be collinearity phenomenon if there is high correlation between two variables.

Following are ANOVA table, residual statistics, coefficient table, and formula.

Source SS DF MS F₀ R²

Regression 9.729 9 1.081 21929.330 0.999 Residual 0.015 314 4.93e − 5

Total 9.744 324

Table 5.2: ANOVA table for significance of regression in multiple regression in double-gate MOSFET. R² is almost equal to 1, so the model is good in terms of explanatory ability.

Minimum Maximum Mean Std. deviation Predicted value 1.4146 2.1109 1.6934 0.17332

Residual -0.009687 0.009424 0 0.00436

Table 5.3: Residual statistics in double-gate MOSFET: It shows points have maximum and minimum. Absolute value of residual are all less than 0.001.

Table 5.4: Coefficients table in double-gate MOSFET: the first frame is name of variable; second is the coefficient of variable; third is standard deviation of coefficient; the last frame is value of t distribution.

ln(a) = 4.297 + (1.372V g²− 3.372V g) + (0.196tox²− 0.339tox) (5.2) + (0.00018Lg² − 0.02Lg) − 0.005V d − 0.003N − 0.000018tsi

+ 0.0001Lg × tox − 0.143V g × tox,

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where unit of N is1e17/cm³.

After generating formula, we have to check whether the model is reasonable. From scat-tering plots, we can find that response variable appears monotone decreasing or increasing.

On the other hand, our model is a second order formula. If maximum or minimum occur during the interval we simulate, then the results are suspect. In our model, there are three second order terms, Vg, tox, and Lg. Maximum or minimum occur about at Vg=1.23, tox=0.86, Lg=55.55. All of them are out of range of simulation. So, the formula is rea-sonable.

Because we used stepwise method to select variables. Variable selected into the model earlier has larger effect. In table 5.4, effect of variable is decreasing according to the order of variables. We know that gate voltage, thickness of oxide, and channel length dominate the variation of standard deviation of wave packet. In fact, effects of drain voltage and thickness of bulk are insignificant. It may resulted from collinearity of variables. It doesn’t mean that drain voltage and thickness of bulk will not influence the results.

Form table 5.3, maximum of residual occurs when ln(a) = 2.1112. In the other word, maximum of residual is 0.08146 when a = 8.258. If we want to simplify the model, we may delete some variables whose effects are too small and estimate model again.

Figure 5.3: Scattering Plot : Channel length vs. Standard deviation of wave packet. It appears quadratic trend. Therefore, we will add the quadratic terms into the model.

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Figure 5.4: Scattering plot : Gate voltage vs. Standard deviation of wave packet. It appears quadratic trend. Therefore, we will add the quadratic terms into the model.

Figure 5.5: Scattering plot : Drain voltage vs. Standard deviation of wave packet. It appears quadratic trend. Therefore, we will add the quadratic terms into the model.

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Figure 5.6: Scattering plot : Thickness of bulk vs. Standard deviation of wave packet. It appears quadratic trend. Therefore, we will add the quadratic terms into the model.

Figure 5.7: Scattering plot : Thickness of oxide vs. Standard deviation of wave packet. It appears quadratic trend. Therefore, we will add the quadratic terms into the model.

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Figure 5.8: Scattering plot : Doping concentration vs. Standard deviation of wave packet. From this figure, the effect of doping concentration may be not so evident.

Figure 5.9: Normal plot of Model I. Y axis is cumulate probability of normal distribution, and X axis is cumulate probability of observed residual. The result is satisfied.

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Figure 5.10: Scattering plot : Fitted value against residual of model I. It appears nonlinear pattern. Therefore, we may conclude that the model I is not good.

Figure 5.11: Likelihood plot of power transformation. Likelihood achieve maximum when λ= −0.148. We prefer to set λ= 0.

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Figure 5.12: Normal plot of Model II. Y axis is cumulate probability of normal distribution, and X axis is cumulate probability of observed residual. The result is satisfied.

Figure 5.13: Scattering plot : Fitted value against residual of model II.

It appears nonlinear pattern. Therefore, we may conclude that the model II is not good.

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Figure 5.14: Normal plot of Model III. Y axis is cumulate probability of normal distribution, and X axis is cumulate probability of observed residual. The result is satisfied.

Figure 5.15: Scattering plot : Fitted value against residual of model III.

It appears nonlinear pattern. So the model III is not good.

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Figure 5.16: Normal plot of Model IV. Y axis is cumulate probability of normal distribution, and X axis is cumulate probability of observed residual. The result is satisfied.

Figure 5.17: Scattering plot : Fitted value against residual of model IV.

It appears flat band pattern, so the model IV is good.

Combining result of normal plot. The model IV is good.

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