MOSFET MODELING FOR VLSI SIMULATION - Theory and Practice Episode 16 doc

In Table 12.2, the numbers in the bracket show standard deviation times the amount of the shift in the parameter value from their respective mean typical values.. Mean typical, minimum s

Idrive is a function of the model parameters and device W and L, the optimization was carried out using W = 3 pm and L = 1 pm It should be pointed out that the parameters were obtained using two steps of optimi- zation In the first step, parameters pertaining to the threshold voltage model were optimized These parameters were then fixed and other

parameters for Idrive were then optimized

In Table 12.2, the numbers in the bracket show standard deviation times the amount of the shift in the parameter value from their respective mean

(typical) values Note that the change in the parameter value is less than 30

for the so called four independent parameters AL, AW, Cox and Vfb This is

understandable because overall change for Idrive is the effect of other parameters too

The optimized DC parameter values shown in Table 12.2 are then used to

calculate Idrive bounds These bounds are shown as dotted line in Figure 12.2 Clearly, the optimization technique results in a more realistic WCF

compared to the principal factor method

The histrogram of Idrive, for W J L , = 12.5/1 n-channel devices, based on data collected from 3 different lots (117 die locations from different wafers) for a typical 1 pm CMOS technology is shown in Figure 12.3 The vertical lines designated as TS and TF are the slow and fast bounds, respectively, generated by the principal factor method, while the corresponding bounds generated by the optimization method discussed above are designated as

0 s and OF, respectively Note from this figure that the bounds generated

Table 12.2 Mean (typical), minimum (slow) and maximum (fast) valuesjor some important

n-channel parameters obtained using the optimization method

by the principal factor method are 8-1 1 % higher compared to the optimization method

Although optimization method generates realistic WCF, however, it needs large amount of statistically meaningful data that is not always available Same is the case with the Factor rotation method To a first approximation, WCF could be generated using principal factor method In the latter approach, it is more appropriate to replace Vfb as independent parameter

by N , , the bulk concentration

[3] P Cox, P Yang, S S Mahant-Shetti, and P Chatterjee, ‘Statistical modeling for

efficient parametric yield estimation of MOS VLSI circuits’, IEEE Trans Electron

References 579 [6] T K Yu, S M Kang, I N Hajj, and T N Trick, ‘Statistical performance modeling and parametric yield estimation of MOS VLSI’, IEEE Trans Computer Aided Design, [7] P Tuohy, A Gribben, A J Walton, and J M Robertson, ‘Realistic worst-case parameters for circuit simulation’, IEEE Proc., 134, Pt I, pp 137-140 (1987) [8] S Inohira, T Shinmi, M Nagata, T Toyabe, and K Iida, ‘A statistical model including

parameter matching for analog integrated circuits simulation,’ IEEE Trans Computer Aided Design, CAD-4, pp 621-628 (1985)

[9] M Pelgrom, A Duinmaijer, and A Welbers, ‘Matching properties of MOS transistor’, IEEE J Solid-state Circuits, 24, pp 1433-1439 (1989)

[lo] C Michael and M Ismail, ‘Statistical modeling of device mismatch for analog MOS integrated circuits’, IEEE J Solid-state Circuits, 27, pp 154-166 (1992)

[11] L A Glasser and D W Doubberpuhl, The Design and Analysis of VLSI Circuits,

Addison-Wesley Publishing Co., Reading, MA, 1985

[12] W Maly and A J Strojwas, ‘Statistical simulation of the IC manufacturing process’, IEEE Trans on Computer Aided Design, CAD-1, pp 120-131 (1982)

[I31 S R Nassif, A J Strojwas, and S W Director, ‘FABRICHSII: A statistical based IC

fabrication process simulator,’ IEEE Trans on Computer Aided Design, CAD-3,

pp 40-46 (1984) Also see Report (Feb 1990) on FABRICS 11: ‘A statistical simulator

of the IC manufacturing process’, Department of Electrical Engineering, Carnegie- Mellon University, Pittsburgh, PA, 15213

[14] S R Nassif, A J Strojwas, and S W Director, ‘A methodology for worst-case analysis

of integrated circuits’, IEEE Trans on Computer Aided Design, CAD-5, pp 104-1 13 (1 986)

CAD-6, pp 1013-1022 (1987)

[lS] D G Rees, ‘Foundations ofStatistics’, Chapman and Hall, New York, 1987 [16] R E Walpole and R H Myers, Probability and Statisticsfor Engineers and Scientists,

McGraw Hill, New York, 1976

[17] C W Helstrom, Probability and Stochastic Processes f o r Engineers, Macmillan Publishing Company, New York, 1984

[18] N D Arora and L M Richardson, ‘MOSFET modeling for circuit simulation’ in

Advanced M O S Device Physics (N G Einspruch and G Gildenblat, eds.), VLSI Electronics: Microstructure Science, Vol 18, pp 236-276, Academic Press Inc., New York, 1989

[19] V Bernett and T Lewis, ‘Outliers in Statistical Data, John Wiley & Sons, New York,

Appendix

Appendix A Important Properties of Silicon,

Silicon Dioxide and Silicon Nitride at 300 K

Magnitude prefix Multiple factor Symbol

Appendix E Methods of Calculating 4s from the Implicit

Rearranging Eq (6.30) for q5s yields

Assume 4: is an initial guess of 4s then the next value of the estimate of 4s

is given by the Schroder series expression [1]

Y” (3y”)Z - y’y”’ K 3

where only the first 5 terms in the series are shown and taken into account

The prime on y denotes the order of the derivative of the function f(4,)

is significant in weak inversion

A good initial guess for the surface potential is suggested [2]

where

and 4ss is 4s in weak inversion given by Eq (6.90) That is,

The semi-empirical Eq (E.3) is such that in strong inversion #: M v,b + V,

and in weak inversion 4: M 4ss; therefore, it follows the general behavior

Appendix F 583

of the surface potential 4s The absolute value sign in Eq (4) is to prevent the argument of the logarithm from becoming negative in weak inversion With the initial guess given by Eq (E.3), an accurate estimation of C#Is is obtained in all the regions of device operation using Eq (E.2) Only one or two iterations are normally required

Other non-iterative approaches for calculating 4s, such as storing values

of 4s in a 2-D array [3], or approximating the potential using cubic spline

functions [4], have also been proposed

An approximate solution of Eq (E.l) in different regimes of device operation has also been suggested Since in strong inversion, defined as Vgb > V g b h ,

the logarithm term varies very little, an approximate expression for 4s is given by

4,(strong inversion) % 24f + V,, + 6Vt (E.6)

A better estimate (within 1% of the exact solution) for 4, in strong inversion

is obtained by substituting (E.3) in the right hand side of Eq (E.1) For weak inversion region, defined as v g b < Vgbm, 4s is given by Eq (E.5) A better

estimate can be found by substituting C#Iss in Eq (E.1) However, for moderate inversion, no simple relationship exist

S Yu, A F Franz, and T G Mihran, ‘A physical parametric transistor model for

CMOS circuit simulation’, IEEE Trans Computer-Aided Design, CAD-7, pp 1038-1052 (1988)

H J Park, P K KO, and C Hu, ‘A charge sheet capacitance model of short channel MOSFET’s for SPICE, IEEE Trans Computer-Aided Design, CAD-10, pp 376-389 (1991)

Appendix F Charge Based MOSFET Intrinsic

Capacitances

In this appendix, the expressions for the intrinsic capacitances for large and wide device will be presented These capacitances are based on the charge equations given in section 7.3 and the definition of the capacitance equation

(7.40) In order to write the equations in a tractable form we first define

some auxiliary functions

Note that the sum of all the four capacitances CGS, CGB, C G D and C G G is

zero The bulk capacitances based on Eq (7.60) can be written as

f4( 1 - 2 4 + 2h, h4 12h2

Duth + 0.5( - h4 + 1 2 ) + + Clh4(f1 + o.sf,)]

Saturation Region Differentiating Eq (7.61~) with respect to V,, V,, V, and

Vg Yields the corresponding capacitances C,,, C,,, C,, and C,,, respectively,

which are shown below:

Again, differentiating (7.61d) with respect to V,, V,, V, and Vg we get the

corresponding capacitances C,,, C,,, C,, and C,,, respectively, which are given by the following equations

CBB=Coxt - d

3a

[ vth

The transcapacitances corresponding to the drain charge will be

which were defined earlier

Suppose that the threshold voltage, v t h , of a MOSFET is measured at different temperatures T Let us define temperature as the variable x , and

v t h as the variable y Clearly x (temperature) is an independent variable

and y (observed Vth) is the dependent variable Suppose there are m

measured data points of y versus x That is, there will be m number of data points ( y l , xl), ( y 2 , x2), , (ym,xm), where yi is a measured value of Vt,, at ith temperature xi The best fit line that relates y to x is called the regression

line, represented by the equation

where yi is the predicted value of y at temperature x i , obtained using

Eq (G.1) The constants a and b are called regression coefficients Note that

since Eq (G.l) is the equation of a straight line, the parameters a and b

are the intercept and slope, respectively, of the straight line

In order to find the best fitting regresion passing through the cluster

of data points, we use the so called least squares criterion that results in the

smallest sum of squared deviations of the data points from the line Stated

mathematically, we determine the slope b and y-intercept a of (G.l) such that

is minimum, where yi is the observed value and yI is the predicted value

The slope b of the best-fitting line, based on the least squares criterion, can

be shown to be [l]

and

((3.3)

where the C (summation) is over all measurement points from i = 1 to m

Note that there is always an error ei associated with any measurement x i Generally this measurement error ei is unknown When this is the case, the

least square formulation described above is recommended If, however, one

knows that ei has a variance o;, some other estimate procedure might be better

Reference

[l] N R Draper and H Smith, Applied Regression Analysis, 2nd ed., John Wiley & Sons,

New York, 1981

If a variable x is observed repeatedly in an experiment, the m observed values

xl, x2 '"x, constitute a sample of size m from which the characteristics

of x can be estimated

Mean The mean value of a variable x, denoted by 2, is defined as the sum

of the observed values divided by the number of values Thus, for the mean

X of x we have

Since m represents a subset of the full set of observations, called the

population, that might have been observed, 2 is called the sample mean in

order to distinguish it from the so called population mean, which is generally

denoted by p in statistical theory

Variance A measure of spread of the value of x from its mean 2 is provided

by the sample variance It is the average of the sum of the squared deviation

from the mean and is ordinarily represented by s2 Thus,

Note the denominator used in calculating sample variance is '(m - 1)' not 'm' For computational purposes it is more appropriate to use the following equation

Appendix H 589 due to its better round-off error properties Equation (H.3) can be shown

to be the same as Eq (H.2) The variance of a population is generally denoted

by o2 and is expressed as

m Standard Deviation The square root of the sample variance is called the sample standard deviation denoted by s It is a measure of the absolute variability in a data set and is a most common measure of disperson used

Note that CV is a unitless number

Gaussian or Normal Distribution A quantitative measure of the frequency

of occurrences of one or more specific values of a random variable' x is called the probability P r ( x ) of x The relationship between the possible values of x and the corresponding probabilities is called the probability distribution of x The probability distribution associated with a continuous random variable x is specified in terms of the probability density function

(PDF) f ( x ) The PDF f ( x ) has the following properties:

0 Probability must be positive, therefore, PDF must be greater than zero, that is,

f (x) 2 0

A random variable x could be discrete or continuous It is called continuous if it can

take any numerical value in a given range The model parameter pi ( i = 1,2."n) is an example of a continuous random variable

0 The probability that x has a value in the interval ( a I X < b) is given

by the area under the curve f ( x ) between a and 6 , that is,

where p is the mean value of x and is a quantitative measure of the location

of the center of the curve f ( x ) and o is the standard deviation and is a

measure of the dispersion about the mean value Note that the symbol p

and 0 used here are population mean and population standard deviation

in order to distinguish them from the sample mean 2, and sample standard

deviation s defined earlier The constant 1/,,/27~ has been chosen to ensure

that the area under this curve, obtained by integrating the density function

f(x), is equal to unity, or probability is 1.0 Figure H.l shows different curves

obtained using Eq (H.7) with mean p = 3 and standard deviation 0 = 0.5 and 2 Note that the curve is symmetric about its mean p , which locates the peak of the bell (see Figure H.l) The interval running one 0 in each direction from p has a probability of 0.683, the interval from p - 2 0 to

p + 20 has a probability of 0.954, and the interval from p - 30 to p + 30

has a probability of 0.997 In other words,

Pr(p - 0 5 X i p + a) = 0.683,

Pr(p - 2a 5 X i p + 2 0 ) = 0.954,

Pr(p - 30 i X I p + 30) = 0.997

The curve never reaches zero for any value of x , but because the tail areas

outside ( p - 30, p + 30) are very small, we usually terminate the curve at

these points

By convention, the probability associated with a particular variable is

usually expressed as a percentage statement rather than as a decimal

Appendix H 59 1

X

Fig H l Plot of Normal or Gaussian distribution function f ( x ) for two different values

of standard deviation u = 0.5 and 2 The mean value of x is 3

probability Suppose the process mean of the Vth is 0.5V and that the standard deviation is 0.03 V, then from the equation above the following facts would emerge

0 68.3% of the v,,, will lie within 0.5 f 0.03 V (vfh f 0)

95.4% of the V,, will lie within 0.5 f 0.06 V (Vfh k 20)

0 99.7% of the V,,, will lie within 0.5 L- 0.09 V (v,,, f 30)

Integration of PDF Gives the Probability The probability that the random

variable lies in the range (- CQ, x) is given by

D ( x ) = [" f ( x ) d x

J - 0 0

The function D(x) is called the distribution function

Standard Normal Distribution If we define a variable z such that

then the normal distribution (H.7) becomes

(H.lO)

The new transformed variable z has a mean of zero and standard deviation

of 1 This particular normal distribution function f ( z ) is called the standard normal distribution The probability or the area to the left of the curve for

a specified value of z has been tabulated and is usually given in most of

the statistical textbooks

Covariance The extent of the relationship between the two variables defined in the same sample space can be determined from their scatter plot2

It is often difficult to judge the dependence quantitatively from such plots, except in cases when the relationship between the two parameters is very

strong One measure of the degree of the linear association between the two parameters, which is often used, is the covariance Just as variance measures

the spread of values of a variable around its mean, the Covariance C,,

measures the joint distribution of the variables x and y around their mean

It is the sum of the deviations of the paired x i and y i values of the variables

x and y from their respective means Thus,

A variable can be converted into standarized or unitless form using the

relationship given in Eq (H.9), so that the new transformed variable z has

a mean of zero and standard deviation of 1 The covariance between the standarized variables can be used to estimate the degree of interrelation between the variables, in a manner not influenced by measurement units,

and is called the correlation ~ o e f i c i e n t ~ When based on a sample of data

(from population), the correlation coefficient is denoted by r and in turn

is an estimate of the population correlation coefficient denoted by p The

correlation coffiicient r between the two normalized parameters z , and z ,

A scatter plot is simply a plot of data points between two variables x and y A normalized scatter plot for po (low field mobility) versus 0 (mobility degradation parameter) is shown

in Figure H.3 (p 597) Note that in this Figure dotted line (ellipse) is not part of the

scatter plot

If the variables are reasonably commensurable, the covariance form has greater statistical appeal