3.1 - Introduction

This exercise is an introduction to experimental design and model fitting using the RS1 optimisation tool. It begins by defining the problem that is to be solved. The process of experimental design with the RS1 software is then demonstrated as a worked example. Sample results are provided that can then be analysed.

At the start of the exercise, all commands to be entered are shown. As the exercise progresses and you become more familiar with the software, less help is given and you are expected to work out how to do some things by yourself. Demonstrators will be available if you need assistance during the exercise.

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3.2 - Objectives

The objective of this exercise is to show how experimental design techniques can be used to analyse the manufacturability of a particular fabrication process prior to its implementation.

The idea can also be extended to optimise the performance of a given set of device characteristics whilst minimising sensitivity to process variations. This involves investigating many complex process parameter interactions and trade-offs.

Although a series of physical experiments can be performed, this can be time-consuming and expensive. Careful use of semiconductor process and device simulations can provide the results for any particular set of input conditions, at an acceptable cost and timescale.

The effects of some process parameters can be easily ascertained, however many more subtle interactive effects may be missed by simply looking at one input variable at a time. Testing every possible combination of variables would involve an extremely large number of experiments or simulations, and is therefore not practical.

It is preferable to use an established experimental design technique to reduce the number of simulations required, but which will still allow the necessary effects and interactions to be observed. The use of Response Surface Methodology (RSM) allows this reduced set of results to be analysed to show the overall effect of varying all of the input parameters and thus find their optimum values that comply with the tolerances set on the device performance.

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3.3 - The Problem

The problem chosen to illustrate this technique is that of optimising certain performance aspects of an n-channel LDD (lightly doped drain) device in terms of a particular set of process parameters.

Figure 1:LDD transistor diagram

The chosen process/design control variables are listed in table 1.

CONTROL FACTORS	SYMBOL
Polysilicon gate sidewall slope	S
LDD implant dose	D
LDD implant energy	E

Table 1 Process variables or control factors

In altering these variables or FACTORS, it is desired to observe the following effects on the RESPONSES listed in table 2.

RESPONSES	SYMBOL
Ensure the punchthrough voltage >10V	VPT
Minimize the peak electric field	EPK
Maximise the transconductance	GM
Minimise the series resistance	RS

Table 2 Response factors

The range of variation that is of interest for each of the above FACTORS is quoted in table 3 and will be used to choose the actual simulation settings.

CONTROL FACTOR	RANGE
Sidewall Slope	70 to 110°
LDD Implant Dose	1E13 to 1E15atomscm-2
LDD Implant Energy	30 to 90 keV

Table 3 Ranges of process variables.

SOLUTION TECHNIQUE

In order to solve the above problem, there are four steps which need to be performed. These are :-

Design an experiment using RS/1 Discover.
Perform these experiments and return results to RS/1 database.
Create individual response surfaces using RS/1 Explore.
Define composite response formulae to optimise device performance.

Part 2 involves performing all of the necessary simulations (or experiments), which can be performed using a selection of TMA's process, device and parameter extraction software. This is an extremely computationally intensive operation and not within the scope of this problem. Therefore, the results from these simulations will be supplied to enable the remainder of the solution to be attained. Parts (1), (3) & (4) are all performed using RS/1, for which the basic operating procedure is presented below. Example screen dumps of each stage are also presented with this documentation.

N.B. Menu options which appear as highlighted, are available by simply pressing 'RETURN', or they can be selected by their name or number. The '>' next to the options indicate which options the software considers to probably be the most appropriate at the stage of the experiment.

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3.4 - Experiment Design

There are five main steps within RS1 which must be performed to decide the experiments:

MAKE the experiment
DEFINE the factors
DEFINE the specifications
DEFINE the responses
DEFINE the model

These are detailed below.

To start RS/1, use the following command in the terminal window:

rs1ws <return>

Note: do not run this as a background task.

3.4.1 - Make an Experiment

When the RS/1 window appears, select: Make -> RS/Discover Objects -> Experiment from the drop down menu.

Enter the experiment name and title at the prompts (you can use either upper or lower case throughout the program):

New EXPERIMENT name: LDD_OPT Title of experiment: LDD_OPT Use technical choices from and old experiment? [No] <return>




Screen Shot 1: Blank experiment


3.4.2 - Define the Factors


The next step in the process is to define the control factors and the range over which they will be varied.  The various aspects of the factors that must be defined are detailed below.


Factor Names:
 The names of the factors must
begin with a letter and can have only letters, numbers and underscores.
Factor names cannot have blank spaces.
Abbreviation: 
The factors are identified by their abbreviation,D, E or S, in this case.

Factor Role: 
In most situations CONTROLLED
will be selected. Choose UNCONTROLLED for a factor that will be measured
and later used in the analysis, but which cannot be explicitly set. Enter
HELP at the prompt or see the RS/Discover Reference Manual for more information
on the factor roles available.


Factor Settings:  
For quantitative factors,
the range is entered in the form of min to max, or a list of at least three
specific values to look at. For qualitative factors, enter the names of
the levels separated by commas. For example, 0 to 100; or 10, 50, 100;
or red, blue, green. Enter ranges as 'low value' TO 'high value'(see table
4 for the specific values for this exercise


Factor Scale:
 Orthogonal is the most common
choice. In a mixture experiment, choose 0 for formulation factors and orthogonal
for process factors.


Units of Measurement: 
This records the units
of measurements, simply enter a small string with no spaces as given in
table
4.


Ease of Varying this Factor: 
If it is very
difficult to change the levels of a factor and it is reasonable to sacrifice
complete randomisation of the order of runs, enter HARD. Enter MODERATE
for the second most difficult to change factor. Since these two options
restrict the randomisation of runs, EASY should be chosen for all
three cases.


Factor Precision: 
This affects the number
of decimals in the printout. It does not affect the number of significant
figures in the calculations. Specify a number of decimal places to match
the accuracy of the measurements, for example, use .001 to measure
the to three decimal places.


Descend through the menus DEFINE, FACTORS, and ADD
then input the data below for each factor in turn. The commands for the first factor, SLOPE, are 
shown below, and the factor summary table (table 4) gives the values for the other factors.
EXPER> [DEFINE] <return>


EXPER.DEFINE> [FACTORS] <return>


EXPER.DEFINE.FACTORS> [ADD] <return>

New factor name: slope <return>


Factor abbreviation: [S] S <return>


Factor role: [CONTROLLED] <return>


Factor settings: 70 to 110 <return>


Factor settings indicate that factor type is QUANTITATIVE


Factor Scale: [ORTHOGONAL] <return>


Units of measurement: deg <return>


Ease of varying this factor: [EASY] <return>


Factor precision: [0.01] <return>

CONTROL FACTORS	ABBREVIATION	ROLE	SETTINGS	SCALE	UNITS	EASE	PRECISION
Slope	S	Controlled	70 to 110	Orthogonal	Degrees	Easy	0.01
Dose	D	Controlled	1E13 to 1E15	Orthogonal	Atomscm-2	Easy	0.01
Energy	E	Controlled	30 to 90	Orthogonal	keV	Easy	0.01

Table 4 Summary of inputs for factor definitions

Once all three factors have been entered, hit <return> when asked for a new factor name to end this section, and type up to move up the menus, until the menu shown in screen shot 6 is seen.

Screen Shot 6: Defined control factors

For more detail as to how the data is input for this problem see screen dumps 2, 3, 4, 5, and 6.

3.4.3 - Define the Specifications

This part of the experimental design defines the specifications. The various options are described and a summary table, table 5 shows all the values to be entered.

Objective:

There are three objectives to choose from. Choose SCREENING for a two level design with many factors. Choose RSM (response surface methodology) for a three or four level design with relatively few factors. Choose TAGUCHI for an experiment with an inner and outer array, that is, one that has noise factors.

Max. Blocksize:

This option is not available since there are no blocking factors because operating conditions aren't expected to change during the experiment. If changing conditions would affect the experimental results, you would enter the maximum number of runs that can be performed with homogenous conditions.

Model Type:

RS/Discover will suggest a model based on the objective (eg QUADRATIC, CUBIC).

Design Type:

RS/Discover will suggest a design. To choose another design, enter its name. RS/Discover will not accept a design that is inappropriate for the stated objective and model type. Enter HELP or see the reference manual for information on the different designs available.

Centerpoints/block:

Use this to change the number of centerpoints in the design. For simulation this need never be greater than one.

Number of units:

Consider increasing it if costs and operating conditions allow. Otherwise, accept the default (14)

Run Order:

Choose STANDARD for simulation data. You could choose EASE if one of the factors is defined as hard or moderate to vary.

Move down the menus DEFINE and SPECIFICATIONS then input the data below. For more details on the input of data see screen dump 7.

OBJECTIVE	MAX BLOCK SIZE	MODEL TYPE	DESIGN TYPE	CENTRE PTS	NO OF UNITS	RUN ORDER
RSM	n/a	Quadratic	CCF	0	14	Standard

Table 5 Summary of inputs for specification definitions

Use the /EXIT command to get back to the menu.

3.4.4 - Define the Responses

This section defines the responses and their ranges. As with the previous sections, the options are discussed and the values to be entered are given in the summary table

New response Name:

The names of responses must begin with a letter and can only have letters, numbers and underscores. Response names cannot have blank spaces.

Units of Measurement:

These can be letters, numbers, and symbols, or the field can be left blank.

Response Range:

Specify the range of values possible for the response. Enter it in the form min to max, for example, 50 to 150.

Response Precision:

This affects the number of decimals in the printout. It does not affect the number of significant figures in the calculations done by the computer. Specify a number of decimal places to match the accuracy of the measurements.

The responses for this exercise are listed in table 6. However all input details can be found in the screen dump below. The input process for entering details of responses such as names, abbreviations, ranges, precision and units can be obtained from screen dumps 8, 9, 10, 11.

Enter ranges as 0 TO next multiple of 10,100,1000 etc. N.B. Use keV, not eV, etc as units to keep numbers sensibly sized. After entering all the responses, press RETURN at the New response name prompt to display the following screen:

Screen Shot 10: Response input details

RESPONSE NAME	ABBREV	UNITS	RANGE
PUNCHTHROUGH_VO	VPT	Volts	0 to 30
PEAK_FIELD	EPF	kVm-1	0 to 1000
TRANSCONDUCTANC	GM	Siemens	1 to 30
SERIES_RESISTAN	R_SD	Ohms	1 to 2000

Table 6 Summary of inputs for reponse definitions

3.4.5 - Generate Worksheet

Select NEXT.AUTOGENERATE as shown in screen dump 12.

At this stage the experiments have been defined. These should now be carried out and the results fed back to RS1. The experiments can be viewed by issuing the command:

!display ldd_opt@worksheet <return>

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3.5 - Enter Simulation Results

Ideally the results would be entered by taking the ENTER option of the EXPER menu and entering the values. As this part is too time-consuming for this course, the results have been supplied. To see the experiment matrix which has been defined by RS1, and the results of these experiments, exit this experiment by typing EXIT. Now load in the results as a new experiment as follows:

Once you have returned to the RS/1 prompt (#), click on the return to browser button above the terminal window.
Download the file: LDD_OPT_ARX.ARX into your own filespace.
Choose DEARCHIVE from the FILE Menu.
Dearchive the LDD_OPT_ARX.ARX file into an experiment called LDD_OPT1.

The experiments and their results can be seen in table 4.

Dose	Energy	Slope	VPT	EPEAK	Gm	Rs
1×10^13	30	70	19.8	280	5.2	1420
1×10^15	30	70	16.3	370	8	1020
5.05×10^14	60	70	16.6	290	7.2	1090
1×10^13	90	70	14.5	250	6.6	1150
1×10^15	90	70	12.45	350	8.5	940
5.05×10^14	30	90	16.7	420	7.7	620
1×10^13	60	90	14.9	320	7.7	720
1×10^15	60	90	11.2	470	9.8	480
5.05×10^14	90	90	10.25	380	9.1	590
1×10^13	30	110	14.8	560	8.8	425
1×10^15	30	110	11.3	770	11.6	245
5.05×10^14	60	110	10.6	620	10.8	305
1×10^13	90	110	8.5	530	10.2	405
1×10^15	90	110	6.45	730	12.1	210

Table 4 Simulation data for RS/1

Screen dumps 13, 14 show the completed worksheet containing the results that have been read in from the file.

Screen Shot 13: After dearchiving results.

Screen Shot 14: Dearchived results.

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3.6 - Fit Models to Experimental Data

Having completed the experiments, we now want to fit models to the results. Return to the command mode RS/1 and edit the experiment LDD_OPT1, by typing EDIT LDD_OPT1. Take ANALYSE option from EXPER me nu, then FIT MODEL option as shown in screen dump 15 and 16. Now these models can be examined and refined. For each of the four responses in the experiment use the following pr ocedure as illustrated in the associated screen dumps.

SELECT the RESPONSE/MODEL, if you are asked for a model, enter DESIGN, when asked for a response, enter either VPT, GM, EPK or R_SD.
REFINE MODEL, as shown in screen dump 17.
Select STEP, as shown in screen dump 18.
Type AUTO, this removes unnecessary terms from the response surface polynomial. Mulreg will go through the process of deleting non-significant terms one at a time. At the end, Mulreg will display the coefficients table of the refin ed model, this is shown in screen dump 19.
Type UP to go back up the menu tree, see screen dump 20.
Choose CHECK FIT, screen dump 21.
CHECK Y Transform, this tests to see if a transformation on the Y variable (the response) will yield a more accurate model, it should resemble screen dump 22.
CHECK Variance, this tests if a weight to stabilize variance will yield a more accurate model. This should look similar to screen dump 23.
CHECK Robust, this tests if a robust instead of a least squares fit will improve the fit of the model, as shown in screen dump 24.
Select NEXT to return to the FIT MODELS menu.
ACCEPT MODEL When asked to temporary save design, accept the default design_copy, when asked for a model name enter LDD_OPT_EPK, LDD_OPT_GM, LDD_OPT_R_SD or LDD_OPT_VPT as shown in screen dump 24b shown below. When asked which model to accept, repeat the above model name.
Repeat for all four responses. When finished choose accept models and check your screen with screen dump 24b.

Screen Shot 24b: The four new fitted models.

Now simulation data can be used to provide further information or the response can be displayed graphically as required. This is achieved from the INTERPRET and VISUALISE menu options from the very top level menu, it can also be reached from the analys e menu. Screen dumps 25 and 26 show the process. Initially, just the individual response data will be made available for plotting.

From the top level menu, select INTERPRET.VISUALIZE
Select RESPONSE and add each of the variables (VPT, EPK, GM and R_SD).
Select CONTOUR/3D.GRAPHTYPE and ensure that this is set to CONTOURPLOT.
Use INTERPRET.VISUALIZE.CONTOUR/3D.MAKE to make a graph, ready for specifications. Use MODIFY to define the graph variables. Fix the slope, and use the dose and energy as the variables. Step through the responses as the surf ace variables. Make all responses PLOT as opposed to HIDE.
Select PRODUCE graph and agree to DISPLAY. (An example plot is given in screen dump 27 below).

Screen Shot 27: Sample Plot of EPK, GM, R_SD & VPT varying against energy and dose.

The next task is to define a composite response formula which will enable a suitable process design point to be chosen that meets all the device specifications listed earlier.

Go to INTERPRET.DEFINE FORMULA menu options to enter a suitable formula which links the four responses to the design criterion. The formula is entered as a text expression and takes the form:

formula_name = mathematical expression involving RESPONSE names

N.B. Only the right side of the equation should be entered

An example of a composite response is:

COMP_RES = meanVPT/VPT + EPK/meanEPK + meanGM/GM + R_SD/meanR_SD

Mean values can be calculated manually and entered into the above expression. However, RS/1 can be used to calculate and store in a file the mean values for the formula by typing !CALL $MEASURE(COLS 4 to 7 OF LDD_OPT1@WORKSHEET,"TEMP"). This creates a table called TEMP which can be displayed using !DISPLAY TEMP. Another command for displaying the mean value for a single response is !MEASURE COLS 4 OF LDD_OPT1@WORKSHEET.

Another formula that could be used for each term is:

-1 <= (y - MEDIAN) / (0.5 * RANGE) <= +1

Are there any other formulae that you could use for this?

Once the mean values of EPK, VPT, GM and R_SD are known, select ADD from the INTERPRET.DEFINE FORMULA menu. This will ask you for a name, enter COMP_RES. It will then ask you if want to use the text editor to enter the expression, type NO and enter the expression.

Screen dump 28 shows the procedure for entering the formula for COMP_RES which is given above. Composite responses allow the importance of various parameters to be accounted for so that the region where the re is least sensitivity to control factor settings can be selected. You can also weight the importance of the different parameters in the light of engineering knowledge to help select the best process combination.

Once the composite response equation has been defined it can be displayed as a contour plot. To do this, another graph specification must be made which plots this composite response surface as well as three of the responses. Screen dump 29 shows the input details, although you should set all responses to PLOT, rather than HIDE. When plotted, this will enable a suitable operating point to be chosen.

Return to the graph specification (INTERPRET.VISUALISE.CONTOR/3D.MODIFY) and select SURFACES. Type the name of your expression (it's COMP_RES) and confirm that you wish to add it. Press return to accept the defaul t contour levels. Produce the graphs and confirm it looks like screen dump 30.

Screen Shot 30: Final graph showing composite response curve.

The INTERPRET.OPTIMIZE option can now be used to help select the best operation conditions as illustrated in screen dump 31. You need to select to MINIMIZE the COMP_RES function an d then PERFORM the optimization.

What is the optimum value of the COMP_RES funtion?

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3.7 - Summary

In this exercise you have learned how to design an experiment using the RS1 software. Sample results have been provided and you used these to learn how to fit expermental models to the results.

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