The Design of Experiment (DoE) Analysis

The statistical design of experiments (DoE) is an efficient procedure for planning experiments so that the data obtained can be analyzed to yield valid and objective conclusions. In an experiment, one or more process variables (factors) are changed in order to observe the effect that the changes have on one or more output variables (responses). It is widely used in multidisciplinary design to create approximations of the output. This method is much more efficient to run and gives a functional relationship between design factors (input, x) and responses (output, y). An experimental design formally represents a sequence of experiments to be performed and expressed in terms of factors or design variables set at specified levels. It is represented mathematically by a matrix X where the rows represent the experimental runs and the columns denote the particular setting for each factor for each run [12].

The procedure begins with determining the objectives of an experiment and selecting the process factors for the study. The statistical theory underlying DoE generally begins with the concept of process models.

The best way is to begin with a process model of the box type with several discrete or continuous input factors that can be controlled, and one or more measured output responses as can be seen in Figure 5.3. Experimental designs are used to derive an approximation model linking the outputs and inputs, which generally contain first and second order terms. The experiment often has to account for a number of uncontrolled factors that maybe discrete such as different machine or operators, or continuous such as temperature or humidity.

131 Figure 5.3: A box type example of a process model for DoE [13].

The empirical models which fit to the experimental data take either linear or quadratic forms where a linear model with two factors (x1 and x2) is as shown below:

(5.1)

Where y is response for given levels of the main effects (x1 and x2) and ε is the experimental

error. The x1x2 term is included to account for a possible interaction effect between x1 and x2.

The constant β0 is the response of y when both main effects are zero.

A quadratic form is a second order model which adds two more terms to the linear model namely, and to build a model as shown below:

(5.2)

This model is typically used in response surface DoE with suspected curvature.

Before the type of design of experiment is selected, the design objectives need to be determined. There are four types of design objectives that are mainly used in DoE, namely, comparative, screening, response surface method and regression model objectives. The four design objectives are summarised in Table 5.1 with their functions respectively.

Controlled inputs (factors) Discrete factors Outputs (responses) PROCESS Continuous factors

132 Table 5.1: Types of design of experiments with their functions.

Designs Functions

Comparative

 Choose between alternatives with narrow scope, suitable for initial comparison

 Choose between alternatives with broad scope, suitable for confirmary comparison

Screening

 To identify which factors or effects are important  2 to 4 factors = full factorial

 >3 factors, starts with as small design as possible

 Trying to extract the most important factors from a large list of initial factors (fractional factorial design)

Response surface modelling

 To achieve one or more of the following objectives: - Hit a target

- Maximize or minimize a response

- Reduce variation by locating a region where the process is easier to manage

- Make a process robust

Regression modelling  To estimate a precise model , quantifying the dependence of response variables on process inputs

The comparative objective is used when it needs to be decided whether one important factor among the other factors under investigation is significant or whether or not there is a significant change in the response for different levels of that factor. The screening objective is used when it is required to select or screen out important main effects from the many less important ones. The response surface method objective is used to estimate the interaction and quadratic effects of factors to be investigated. The process involves finding optimal process settings, troubleshooting process problems and making a process more robust against external and non-controllable influences. The final design objective is the regression model objective when it is required to model a response as a mathematical function of a few continuous factors and it is used as a guideline to build a good model parameter.

For each design objective, except the regression model objective, several DoE methods can be used depending upon the number of factors as shown in Table 5.2. Each of the DoE methods is briefly described below.

133 Table 5.2: Summary of design methods for each type of design objectives.

Number of factors Comparative objective

Screening objective Response surface method objective 1 1-factor completely randomized design - - 2 – 4 Randomized block design Full or fractional factorial Central composite design

5 or more Randomized block design Fractional factorial or Plackett-Burman Screen first to reduce number of factors

For a single factor, the comparative objective is usually used which incorporates 1-factor completely randomized design. For two to four factors, all three objectives can be applied, the randomized block design is used for comparative objective; the full or fractional factorial design for screening purposes and the central composite design (CCD) to fulfil response surface method objective. For five or more factors, the same design method is used for comparative objective, whilst fractional factorial or Plackett-Burman design is used for screening. However for response surface method objective, it is required to screen first in order to reduce the number of factors before proceeding with CCD. For the research work in this thesis, the response surface method objective is selected with the Plackett-Burman and

CCD designs incorporated into the design of experiments. 5.2.2.3.1. 1-factor completely randomized design

This design compares the values of a response variable based on the different levels of the main factors. For completely randomized design, the levels of the main factors are rando mly assigned to the experimental units.

5.2.2.3.2. Randomized block design

This design involves one factor or variable that is of primary interest but there are also several secondary factors that may affect the measured result but are not of primary interest. An important technique called blocking can be used to reduce or eliminate the contribution to experimental error caused by these factors. Basically, this design involves blocking factors followed by randomization. Blocking is used to remove the effects of a few of the most important secondary variables whilst randomization is then used to reduce the contaminating effects of the remaining secondary variables.

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5.2.2.3.3. Full factorial design

The most basic experimental design is full factorial design. The most common designs for the full factorial are the 2k, which are used to evaluate main effects and interactions, and 3k designs for evaluating main and quadratic effects and interactions, for k is number of factors, which is equal to 2 and 3 levels respectively. The most common experimental design is the 2- level design because it is ideal for screening design, being simple and economical.

The two-level design uses +1 and -1 notation to denote the high and the low levels respectively for each factor. The use of +1 and -1 for the factor settings is called data coding. This aids in the interpretation of the coefficient fits to any experimental model. The centre point for this design is zero.

5.2.2.3.4. Fractional factorial design

This design is where not all factor level combinations are considered and the designer can choose which combinations are to be excluded. Thus, only an adequately chosen fraction of the treatment combinations is required for the complete factorial experiment is selected to be run. Even if the number of factors in a design is small, the 2k runs specified for a full factorial can quickly become large. In order to counter this problem, only a fraction of the runs specified by the full factorial design is used. Properly chosen fractional factorial designs for two-level experiments have the desirable properties of being both balanced and orthogonal. A design is said to be balanced when each factor has the same number of levels. The fractional factorial design only focuses on fractions of two-level designs because the two- level fractional designs are the most used in engineering.

5.2.2.3.5. Plackett-Burman design

This design is a two level fractional factorial design used for screening experiments, where only a few specifically chosen runs are performed to investigate just the main effects, assuming all interactions are negligible when compared with few important main effects. The Plackett-Burman designs can be performed efficiently for 25 runs or more. These designs accept up to 47 factors, which sometimes can be narrowed down to 10 factors or less.

In the research work Plackett-Burman design was carried out on 31 device and interconnect parameters associated with the low-swing driver schemes described in Chapter 3. Through

135 Plackett-Burman screening the main effects or factors has been reduced to 12 as listed in Table 5.3. The device parameters include the threshold voltage, Vtho, gate-oxide thickness,

tox and other parameters such as carrier mobility, µo and effective gate length, Leff, whilst

the interconnect parameters consists of resistivity, ρ, interconnect dimensions (width, w; spacing, s; thickness, t; interlayer dielectric height, h) and the inter-level metal insulator permittivity, εk. Environmental factors such as Vdd and temperature are also included in the

variability analysis. As the main parameters have been identified through the Plackett- Burman design, the next step will involve an implementation of these parameters on the RSM design.

Table 5.3: The main parameters of variability. Device parameters Interconnect parameters

Vth εk tox ρ Leff s µo w Vdd t Temp h

5.2.2.3.6. Central composite design (CCD)

CCD is a two level factorial or fractional factorial design, augmented by centre points and

axial points [13]. The centre points are where all values of the factors are in mid-range whilst the axial points are positioned at mean, ±α for each factor, where α is the variation in the factor , which gives the estimation of the curvature of the response surface. For k input factors, CCD requires (2k + k + 1) experimental runs to build a second order model of the

output parameters. The desirable features of this design are their orthogonality, where there is minimal variance of the regression coefficients, and rotability, which means equal precision of estimation in all directions.

The experimental design consists of devising a set of experiments in which the range of input parameters can be altered systematically between three levels (-1, 0, +1) which represent (-

3σ, 0, +3σ) variations respectively. The circuit output of interest is measured and calculated

at each of the design points to build mathematical models of the output. For the 12 input parameters used for this analysis, DoE (CCD) technique is computed by using Minitab,

136 indicating 154 experiments is required, where the experiment points for each parameter is recorded in Appendix II.

5.2.2.4. Work Flow for Variability Analysis implementing DoE method