Yield Analysis and Performance Optimization Using FastBlaze and SPAYN

Introduction

In previous Simulation Standard articles (Nov 97 & Nov 98) FastBlaze has been presented as a new, highly efficient approach to simulating advanced HEMTs and MESFETs.

Conventional device simulators often suffer from slow execution times, leading to a trade off between mesh density and physical model complexity against CPU run time and convergence. This requires engineers to compromise accuracy to achieve a reasonable throughput.

By focusing solely upon FET simulation, FastBlaze has been able to greatly optimize the physical solution procedure, enabling the use of the most sophisticated physical models while maintaining a fast execution (typically less than 1 minute for a full set of DC ID-VDS curves) This speed enables both complex and extensive experimental designs to be completed on a reasonable time scale, permitting experiments that would be prohibitively expensive for more traditional device simulators.


Figure 1. Scatter plot showing VDbrk vs. gmpk from Monte Carlo experiment.

 

 

Experimental Design

In this exercise we illustrate the power of FastBlaze, when coupled to SPAYN by implementing a simple but large experimental design.

We took a relatively complicated double-recessed, double delta-doped pHEMT structure (Figure 2.) and randomly varied 9 of the physical input parameters (see Table 1).


Table 1. Input and response variables for the experiment.

 

 


Figure 2. Double recessed, double -doped pHEMT structure.

 

There are two common distribution choices for Monte Carlo experimental design, a uniform random distribution suits device optimization, as the input parameter plane is more efficiently covered, however, by using a Gaussian profile, which is more representative of natural variations in the device, we can also perform yield analysis directly upon the experimental output.

 

Device Optimization

When the results from the Monte Carlo experiment are investigated in SPAYN the user can identify "good" devices by visually inspecting the scatter plots. In Figure 1 a device has been chosen with both high gmpk and breakdown. SPAYN then displays each of the input variables for that simulation in a pop-up box.

A simple yet effective method of optimization is then to center the input values on this "good" device and re-run the whole experiment. Figure 3 displays the combined scatter plots for both experiments, showing a marked improvement in both gmpk and VDbrk.


Figure 3. Scatter plot showing VDbrk vs. gmpk for both
the nominal and optimized device structures.

 

Figure 4 shows histograms of both gmpk and VDbrk for the nominal and optimized cases. The qualitative improvement in the parameters is immediately observable, with the mean gmpk increasing from 410 to 490 mS/mm and VDbrk from 24 to 28 V. Further there is no significant change in the standard deviation of either distribution.



Figure 4. Distribution analysis. Here the mean value of both VDbrk and gmpk have been increased in the optimized structure with no significant change in the standard deviations.

 

 

Yield Analysis

Since both experiments were performed using Gaussian distributions the output distribution functions can be evaluated directly to obtain the yield values.

In our example we choose an arbitrary "fail" point for the breakdown voltage and integrate both the nominal and optimized distributions to calculate the expected yield from a wafer. Choosing a minimum of 20 Volts breakdown the yield from the nominal structure is 92.2% where as the optimized is greater than 99%. If the minimum is shifted to 22 Volts, the nominal structure's yield drops to 80% where as the optimized is still greater than 98%.

 

SPAYN Parameter Analysis

The previous examples demonstrate a straight forward optimization technique, however, the inter-dependencies of each input parameter have not been analyzed. SPAYN provides the tools for investigating these relationships via the correlation matrix.

SPAYN can also be used to generate an analytical "black box" model of this data set through regression analysis. This is more computationally efficient than FastBlaze however is strictly limited to this structure.

For more information on the techniques described below please refer to the SPAYN User's Manual.

 

Correlation

SPAYN can be used to investigate the inter-parameter dependencies through the correlation matrix.An abridged matrix showing the most significant variables is presented in Table 2.


Table 2. A bridged correlation matrix for the combined nominal and optimized experimental data.

This is useful when determining which parameters will have a larger influence in the regression models. In this case we can see that the gate length, second recess length and the total recess depth are highly correlated with VDbrk (|r| > 0.5).

 

Regression

Regression analysis can be performed within SPAYN to generate response surfaces for the target parameters. These regression equations can then be used to predict new values for the response variables far more efficiently than by using any physical simulator.

Using VDbrk as the response variable and the 9 predictor parameters from Table 1, an analysis of variance (ANOVA) was performed to identify the most suitable model. If a more complete ANOVA analysis was required, an engineer might also add or remove individual parameters.

There are several model selection criterion available, the most commonly used being p-value and adjusted R2.

From the ANOVA table (table 3) all of the p-values up to model 3. are highly significant, hence our regression model should include all linear (x), interaction (xy) and quadratic terms (x). Adding extra parameters to the regression equation (models 4,5 and 6) produce non-significant p-values, indicating that it is not worth including them in the model.


Table 3. Analysis of variance (ANOVA) for VDbrk.

 

As an alternative we can also use the adjusted R value as the selection criterion. In this case model 4 should be used i.e. all linear (x), interaction (xy), quadratic (x2) and interaction (xyz) terms.

The computational demands for evaluating a regression model are minimal when compared to a full physical simulation, hence we chose model 4 from the ANOVA table.

After fitting the regression equation a residual analysis is performed to check the model. This is accomplished by plotting the residuals against the estimated values and also individual predictor variables (see Figure 5)


Figure 5. Residual analysis of model 4.

 

A visual inspection of these plots should not reveal any discernible trends. If any patterns were apparent this would indicate a poor model and further analysis would be required.

Finally the response surface was then used in place of FastBlaze to re-generate the original Monte Carlo data. Note: 1000 samples were simulated in under 1 second on a Sun ULTRA 10 workstation. VDbrk was then compared between the two data sets, the first generated by FastBlaze and the second via the SPAYN regression model. A deviation of less than 5% was typically observed, illustrating the validity of this approach.

 

Conclusions

By looking at the input parameters from the simulated "good" devices we can draw some conclusions about general design criteria for this type of device. First we have obvious changes, decreasing the gate length and increasing the second recess length to increase gm and breakdown respectively. The recess fraction, the ratio of 1st and total recess depths should be fixed for this structure at 0.8. Finally, the doping density of the 1st delta should be lowered to increase breakdown, whilst raising the density of the 2nd delta to maintain gm.

The SPAYN statistical analysis was used to confirm the device optimization and corresponding improvement in yield. Further, the correlation matrix revealed which input parameters are most significant.

An analysis of variance was performed to select an appropriate regression model which was checked with a residual analysis. Figure 5 shows no discernible trends indicating a good model. Finally the regression equation was used to predict new values for the response variables with good accuracy validating the model selection.