6

Modeling of Statistical Manufacturing

Sensitivity and of Process Control and
Metrology Requirements for a 0.18-km
NMOSFET
Peter M. Zeitzoff
International SEMATECH, Austin, Texas

I. INTRODUCTION

Random statistical variations during the IC manufacturing process cause corresponding

variations in device electrical characteristics. These latter variations can result in substan-
tial reductions in yield and performance, particularly as IC technology is scaled into the
deep submicron regime. In an effort to understand and deal with this problem, simulation-
based modeling of the manufacturing sensitivity of a representative 0:18-mm NMOSFET
was carried out, and the results were used to analyze the process control and metrology
requirements.
In the sensitivity analysis, eight key device electrical characteristics (the threshold
voltage, source/drain leakage current, drive current, etc.; these eight characteristics are
also called the responses) were modeled as second-order polynomial functions of nine key
structural and doping parameters (gate oxide thickness, gate length, source/drain exten-
sion junction depth, etc.; these nine parameters are also called the input parameters). The
polynomial models were embedded into a special Monte Carlo code that was used to do
statistical simulations. In these simulations, the mean values and the statistical variations
of the nine structural and doping parameters were the inputs, and the resulting probability
density functions (pdf's) of the eight key device electrical characteristics, as well as their
mean values and statistical variations, were the outputs. Through numerous Monte Carlo
statistical simulations with a variety of values for the input parameter variations, the level
of process control of the structural and doping parameters required to meet speci®ed
targets for the device electrical characteristics was analyzed, including the tradeoffs. In
addition, the metrology requirements to properly support the establishment and mainte-
nance of the desired level of process control were analyzed.

Copyright © 2001 Marcel Dekker, Inc.

 II. DESIGN OF AN OPTIMAL 0.18-lm NMOSFET

Figure 1 illustrates the overall structure and many of the important structural parameters
for the 0.18-mm NMOSFET (1). The eight key electrical parameters listed and de®ned in
Table 1 were chosen to characterize the optimal device's electrical performance. In the
table, DIBL means ``drain induced barrier lowering,'' which is a short-channel effect. The
primary goal was to design a device that showed maximum drive current (at least 450 mA/
mm) while satisfying the targets in the table for peak off-state leakage, DIBL, peak sub-
strate current (to ensure hot-carrier reliability), etc. The optimal device was meant to be
broadly representative of industry trends, although this is a relatively low-power transistor
due to the 50-pA/mm limit on the leakage current. Due to the short gate length of the
device, it was necessary to include a boron ``halo'' implant as part of the device structure,
in order to obtain an optimal combination of turnoff and drive current performance for
the device. The effectiveness of the halo implant in suppressing short-channel effects as
well as maintaining hot-carrier reliability has been previously reported (2,3). This implant,
along with a boron VT adjust channel implant, was found to improve both the VT rolloff
with decreasing channel length and the device reliability while maintaining acceptable Idsat
vs. Ileak characteristics of the device. Because of the 1.8-V power supply
(Vdd ˆ 1:8 V nominal) assumed for this technology, consideration was given to ensuring
hot-carrier reliability. This was done through the use of a device in which shallow source-
drain (S/D) extensions were doped with a peak concentration of 4  1019 cm 3 and were
self-aligned to the edge of the oxide grown on the polysilicon gate (15 nm from the
polysilicon edge). The deep S/D regions were self-aligned to the spacer oxide edge and
had a junction depth of 150 nm, which was held constant throughout the analysis.
The process simulators, TSUPREM-3 (4) (one-dimensional) and TSUPREM-4 (5)
(two-dimensional), were used to generate the doping pro®les for the various regions of the
device. Due to the uncertain accuracy of two-dimensional diffusion models for arsenic-
implanted junctions with short thermal cycles, the one-dimensional vertical pro®le of both
the shallow and deep S/D junctions was simulated using TSUPREM-3. For each junction,
the two-dimensional pro®le was then generated by extending the vertical pro®le laterally
using a complementary error function with a characteristic length corresponding to 65%
of the vertical junction depth. Conversely, the two-dimensional halo implant pro®le was
directly simulated using TSUPREM-4. The VT adjust implant vertical pro®le was simu-

Figure 1 Schematic cross section of the 0:18-mm NMOSFET structure. The nominal values of the
structure parameters and the maximum variations that were used in the sensitivity analysis are listed
in Table 2. The polysilicon reoxidation thickness, tre-ox , was ®xed at 15 nm for all simulations.

Copyright © 2001 Marcel Dekker, Inc.

 Table 1 Key Device Electrical Characteristics and Target Values for the Optimal Device.

Electrical characteristic Target value

Threshold voltage (from extrapolated linear I-V, @ Vd ˆ 0:05 V), VT (V)  0:5
Drive current (@ Vg ˆ Vd ˆ Vdd ), Idsat (mA=mm of device width)  450
Peak off-state leakage current (@ Vd ˆ 2 V, Vg ˆ 0, T ˆ 300K), Ileak  50
(pA/mm of device width)
DIBL (Vt @ ‰Vd ˆ 0:05 VŠ Vt @ ‰Vd ˆ Vdd Š),
VT (mV)  100
Peak substrate current (@ Vd ˆ Vdd ), Isub (nA/mm of device width) < 200
Subthreshold swing (@ Vd ˆ 0:05 V), S (mV/decade of Id )  90
Peak transconductance (@ Vd ˆ 2:0 V); gsm (mS/mm of device width)  300
Peak transconductance (@ Vd ˆ 0:05 V); glm (mS/mm of device width)  30

Vdd ˆ 1:8 V (nominal).

lated using TSUPREM-3, and was then extended laterally without change over the entire
device structure. A composite pro®le containing all the foregoing individual pro®les was
generated and imported to the device simulator, UT-MiniMOS (6) (where UT stands for
University of Texas at Austin and UT-MiniMOS is a version of the MiniMOS device
simulator with modi®cations from UT). UT-MiniMOS was chosen to simulate the device's
electrical characteristics because it has both the UT hydrodynamic (HD) transport model
based on nonparabolic energy bands and the UT models for substrate current (7), quan-
tum mechanical effects (8±10), and mobility in the inversion layer (11±13). Also, UT-
MiniMOS has adaptive gridding capability, and this capability was used to adapt the
grid to the potential gradient and the carrier concentration gradients during the
simulations.
The optimal device structure was determined by examining a large number of simu-
lated devices with different halo peak depths and doses. For each value of the halo peak
depth and dose, the boron VT adjust implant (also called the channel implant) dose was
adjusted to satisfy the requirement that the maximum off-state leakage current is 50 pA/
mm at room temperature (see Table 1). A number of simulations were performed to
examine the ranges of variation. The result of these simulations was the selection of a
boron halo implant dose of 1:5  1013 cm 2 with a peak doping pro®le depth of 80 nm and
a boron channel implant dose of 5:65  1012 cm 2 in order to obtain maximum drive
current while meeting all the other targets in Table 1.
In Table 2 the nine key structural and doping parameters for the optimal device are
de®ned, and the optimal value for each is listed. For Lg , Tox , Tsp , Xj sh† , Nsh , and Rs , the
optimal values were selected from technology and scaling considerations, and the values
chosen are broadly representative of industry trends. For Nch , Nhalo , and d, the optimal
values were determined from simulations aimed at de®ning an optimal device structure,
as explained earlier.

III. DETERMINATION OF SECOND-ORDER MODEL EQUATIONS

A primary aim of this analysis was to obtain a set of complete, second-order empirical
model equations relating variations in the structural and doping (input) parameters of the
0:18-mm NMOSFET to the resulting variations in the key device electrical characteristics
listed in Table 1. (This technique is also known as response surface methodology (14).) In

Copyright © 2001 Marcel Dekker, Inc.

 Table 2 Nominal Value and Maximum Variation for Key Structural and Doping (Input)
Parameters

Optimal, nominal Maximum

i Input parameter value variation

1 Gate length, Lg (mm) 0.18 15%

2 Gate oxide thickness, Tox (nm) 4.50 10%
3 Spacer oxide width, Tsp (nm) 82.50 15%
4 Shallow-junction doping pro®le depth, 50 10%
Xj sh† (nm), @ ND ˆ 4:36  1018 cm 3
(where ND ˆ NA for the nominal device)
5 Peak shallow-junction doping, Nsh (cm 3 ) 4  1019 10%
6 Channel dose, Nch (cm 2 ) 5:65  1012 10%
7 Halo dose, Nhalo (cm 2 ) 1:5  1013 10%
8 Halo peak depth, d (nm) 80 10%
9 Series resistance (external), Rs ( -mm) 400 15%

this analysis, the nominal or design center device was identical to the optimal NMOSFET
from the previous section, and the variations were with respect to this nominal device.
Hence, the ``optimal'' values of the input parameters in Table 2 are also the ``nominal''
values for this analysis. Also listed in Table 2 are the maximum variation limits for each
input parameter. Since the model equations are accurate only for variations less than or
equal to these maximum limits, these limits were intentionally chosen to be large to give a
wide range of validity to the model equations. However, typical IC manufacturing lines
have manufacturing statistical variations considerably less than the maximum variations
listed in the table. In the next section, Monte Carlo simulations employing the model
equations were used to explore the impact of smaller, more realistic variations.
A three-level Box±Behnken design (15) was performed in order to obtain the
responses of the output parameters to the input parameters. Besides the centerpoint,
where all factors were maintained at their nominal values, the other data points were
obtained by taking three factors at a time and developing a 23 factorial design for
them, with all other factors maintained at their nominal values. The advantage of this
design was that fewer simulations were required to obtain a quadratic equation as com-
pared to other designs. A total of 97 simulations (96 variations plus the one nominal
device simulation) was required for this analysis for the case of nine input factors. One
drawback of this design, however, is that all of the runs must be performed prior to
obtaining any equation, and it is not amenable to two-stage analyses. Hence, there is
no indication of the level of factor in¯uence until the entire experiment has been
conducted.
In contrast to the nine input parameters that were varied (see Table 2), several device
parameters, such as the deep S/D junction pro®le and its peak doping, were held constant
throughout all of the simulations. In addition, a background substrate doping of
5  1015 cm 3 and an interface charge of 3  1010 cm 2 were uniformly applied. The
eight key device electrical characteristics listed in Table 1 were the response variables.
After the completion of the 97 simulations, two sets of model equations were generated
for each response, one in terms of the actual values of the input parameters, and the other
in terms of their normalized values. The normalized values were calculated using the
following equation:

Copyright © 2001 Marcel Dekker, Inc.

 xi x i †

xi ˆ 2 ; i ˆ 1; 2; . . . ; 9 1†
di
where x i is the nominal value of the ith input parameter, xi is the actual value of the ith
input parameter for any given run,
xi is the normalized value of the ith input parameter,
and di is the magnitude of the difference between the two extreme values for xi used in the
sensitivity analysis. Note that, from the de®nition in Eq. (1), it is clear that 1 
xi  1,
and
xi ˆ 1 corresponds to xi equal to its minimum possible value,
xi ˆ 0 corresponds
to xi equal to the nominal value, and
xi ˆ 1 corresponds to xi equal to its maximum
possible value. Furthermore, for any input parameter, di is twice the maximum variation
listed in Table 2 (for example, i ˆ 1 for gate length, then
x1 ˆ
Lg and
d1 ˆ 2  0:15  0:18 mm ˆ 0:054 mm). The equations that use the
xi 's for their variables
will be called normalized model equations. In the remainder of this chapter, only the
normalized equations will be considered.
After each of the 97 simulations was performed, the eight electrical responses
were extracted for each device and then entered into a design matrix. Analysis of
variance (ANOVA) methods were used to estimate the coef®cients of the second-
order model equations and to test for the signi®cance of each term in the model
(16). For the model as a whole, information such as the coef®cient of determination,
R2 , and coef®cient of variation was generated using the data from the ANOVA. A
diagnosis of the model was performed by using normal probability plots to evaluate
the normality of the residuals. A transformation of the response was made if necessary.
The ANOVA was performed again if a transformation was made, and comparisons of
the normal probability plots and coef®cients of determination were made to decide
whether to use the transformed data or not. In addition, plots of Cook's distance vs.
run order and Outlier-t vs. run order were generated in order to check for the occur-
rence of any extraneous data (outliers). The corresponding model was then used to
generate a set of reduced equations, i.e., equations including only those terms with at
least a 95% level of signi®cance (terms with a lower signi®cance value were discarded
from the ANOVA). As mentioned before, for input parameters outside of the range of
values used in this analysis, the model equations are not guaranteed to be accurate,
and they are expected to become less and less accurate as the values move further and
further outside the range.
The ®nal resulting normalized model equations for the eight key electrical responses
are listed in the appendix at the end of this chapter. As mentioned earlier, all of the input
parameters take on a value between 1 and ‡1, as determined by Eq. (1) . The advantages
of the normalized equations are:

Because the input variables are dimensionless, the coef®cients are independent of
units.
The relative importance of any term is determined solely by the relative magnitude of
the coef®cient of that term. For example, in the model equation for the satura-
tion drive current, Idsat is most sensitive to the normalized gate length (
Lg ),
followed by the oxide thickness (
Tox ), the shallow junction depth (
Xsh ), the
spacer oxide width (
Tsp ), and the channel dose,
Nch . Also, this attribute of
the normalized equations simpli®es the generation of reduced equations by
dropping less signi®cant terms.
For all the normalized parameters, the mean value is zero and, as will be explained
later, the maximum value of the standard deviation is 1/3.

Copyright © 2001 Marcel Dekker, Inc.

 In some cases, the output responses were transformed into a form that yielded a
more reliable model equation. Typically, if the residual analysis of the data suggests that,
contrary to assumption, the standard deviation (or the variance) is a function of the mean,
then there may be a convenient data transformation, Y ˆ f y†, that has a constant var-
iance. If the variance is a function of the mean, the dependence exhibited by the variance
typically has either a logarithmic or a power dependence, and an appropriate variance-
stabilizing transformation can be performed on the data. Once the response is trans-
formed, the model calculations and coef®cients are all in terms of the transformed
response, and the resulting empirical equation is also in terms of the transformation. In
this study, both a logarithmic and an inverse square root dependence were used in trans-
forming three of the output responses so that a better model equation ®t was obtained (see
the appendix).

IV. MONTE CARLO±BASED DETERMINATION OF PROCESS CONTROL

REQUIREMENTS

Monte Carlo simulations were utilized for several purposes.

1. For a given set of input parameter statistical variations, to determine the
probability density function (pdf) and the resulting mean value and statistical
variation for each of the device electrical characteristics
2. To determine the impact on the statistical variations of the device electrical
parameters of changing the input parameter statistical variations by arbitrary
amounts
3. To analyze whether the device electrical targets (for example, the maximum
leakage current speci®cation) are reasonable
4. To ®nd an optimal set of reductions in the input parameter variations to meet
device electrical targets
5. To analyze process control and metrology requirements
The normalized model equations obtained in the previous section were utilized in
Monte Carlo simulations to extract the pdf of each response for a speci®ed set of statistical
variations of the nine normalized structural and doping parameters. A special Monte
Carlo code was written that treats each of these parameters as a random input variable
in the normalized model equations. This is a good approximation, since any correlations
among these parameters are weak and second order. Each of the random input variables
comes from a normal distribution with a speci®ed mean and standard deviation, siN ,
where the ``N'' designates that this is for a normalized variable. (Note that, due to the
de®nition of the normalized parameters, the mean value for all of them is zero.
Furthermore, the maximum allowable value of all the standard deviations is set by the
``maximum variations'' in Table 2 and corresponds to 3siN ˆ 1, or siN ˆ 1=3, for all of
the input variables.) For each trial and for each of the nine normalized input parameters,
the Monte Carlo code randomly selected a value by using a random number generator that
returns a series of nonuniform deviates chosen randomly from a normal distribution that
correctly predicts the speci®ed standard deviation. For each trial, the set of randomly
generated values for the nine input parameters was used as inputs to the second-order
normalized model equations, which were evaluated to calculate the values of each of the
eight electrical characteristics. A large number of such trials (typically 5000±10,000) was

Copyright © 2001 Marcel Dekker, Inc.

 Table 3 Monte Carlo Results with Maximum Input Parameter Statistical Variations (3siN ˆ 1
for all i)

Monte Carlo
Response (key device Target value for simulated value for
electrical characteristics) Critical parameter critical parameter critical parameter

VT (mV), extrapolated 3s variation < 50±60 97.1

VT (mV), DIBL Maximum value < 100 121.8
Idsat (mA=mm) Minimum value > 450 430.2
Ileak (pA/mm) Maximum value < 50 95.5
Isub (nA/mm) Maximum value < 200 55.6
S (mV/decade) Maximum value  90 86.8
gsm (mS/mm) Minimum value > 300 397.8
glm (mS/mm) Minimum value > 30 48.0

Note: The bold font indicates critical parameter values which do not meet their respective target.

run to generate the pdf's of the characteristics. The pdf's were then analyzed to obtain the
mean and standard deviation of each of the device electrical characteristics.
In Table 3, the de®nition of critical parameter and the target value for this parameter
are listed for each of the responses. The target values are meant to be broadly representa-
tive of industry trends. The critical parameter is either the 3s statistical variation or the
maximum or minimum value of the response, where the maximum value is calculated as
the mean value ‡ ‰3s statistical variation], while the minimum value is calculated as the
mean value ‰3s statistical variation]. In this set of Monte Carlo simulations, all the input
parameter variations, the siN 's were set to the maximum value of 1/3, corresponding to the
``Maximum variations'' in Table 2. Figure 2 shows a typical pdf, for the substrate current,
Isub . The mean value and the standard deviation, s, are listed at the top. The crosses
indicate the Monte Carlo±simulated pdf, while the solid curve is a ®tted Normal prob-
ability distribution with the same mean and s. The pdf is clearly not a Normal distribu-
tion, although the input parameter statistical distributions are Normal. The non-Normal,
skewed pdf for Isub (and the other responses) is due to the nonlinear nature of the model
equations (17), and the amount of skew and the departure from the Normal distribution
vary considerably from response to response. The Monte Carlo simulation results are
listed in Table 3, where the mean value and s from the Monte Carlo simulations were
used to calculate the ``simulated value'' in the last column. The targets for the ®rst four
responses in the table (VT ,
VT due to DIBL, Idsat , and Ileak ) were not met, but the targets
for the last four parameters in the table (Isub , S, gsm , and glm ) were met. To bracket the
problem, and to determine whether the targets for the ®rst four responses are realistic, all
the input parameter variations were reduced in two stages, ®rst to a set of more ``realistic''
values and second to a set of ``aggressive'' values. These sets are listed in Table 4; they
re¯ect the judgment of several SEMATECH experts (18). In the table, both the statistical
variations of the normalized input parameters, 3siN , and the corresponding statistical
variations in percentage terms of the input parameters are listed. A Monte Carlo simula-
tion was performed for each of these sets of variations, and the simulation results are listed
in Table 5. The targets for the third and fourth responses, Idsat and Ileak , were satis®ed with
the ``realistic'' input variations, but the targets for VT and
VT were satis®ed only with
the ``aggressive'' input variations. The conclusion is that the targets for all the responses
can probably be met but that it will be especially dif®cult to meet them for VT and
VT
(DIBL).

Copyright © 2001 Marcel Dekker, Inc.

 Figure 2 Probability density function (pdf) of substrate current (Isub ), from Monte Carlo simula-
tion. The input parameter statistical variations are maximum: 3siN ˆ 1 for all i.

Table 4 Sets of Normalized Input Parameter Statistical Variations, 3siN a

Maximum Realistic Aggressive

Input statistical statistical statistical
parameter variation variation variation

Lg ˆ
x1 1 (15%† 2/3 (10%† 1/2 (7:5%†

Tox ˆ
x2 1 (10%† 1/2 (5%† 1/4 (2:5%†

Tsp ˆ
x3 1 (15%† 2/3 (10%† 1/3 (5%†

Xsh ˆ
x4 1 (10%† 1/2 (5%† 1/4 (2:5%†

Nsh ˆ
x5 1 (10%† 1 (10%† 3/4 (7:5%†

d ˆ
x6 1 (10%† 1/2 (5%† 1/4 (2:5%†

Nhalo ˆ
x7 1 (10%† 3/4 (7:5%† 1/2 (5%†

Nch ˆ
x8 1 (10%† 1 (10%† 3/4 (7:5%†

Rs ˆ
x9 1 (15%† 2/3 (10%† 1/2 (7:5%†
a
For each 3siN , the corresponding statistical variation as a percentage of the mean value of the non-normalized
input parameter is in parentheses.

Copyright © 2001 Marcel Dekker, Inc.

 Table 5 Monte Carlo Results for Different Levels of Input Parameter Statistical Variations

Monte Carlo simulated value of critical parameter

Responses (key device Critical Target Maximum input Realistic input Aggressive input
electrical parameters) parameter value parameter variation parameter variation parameter variation
VT (mV), extrapolated 3s variation < 50±60 97.1 66.1 46:0

VT (mV), DIBL Maximum value < 100 121.8 104.8 95.8
Idsat (mA/mm) Minimum value > 450 430.2 457.8 482.6
Ileak (pA/mm) Maximum value < 50 95.5 30.3 15.3
Isub (nA/mm) Maximum value < 200 55.6 49.4 46.2
S (mV/decade) Maximum value  90 86.8 85.7 85.1
gsm (mS/mm) Minimum value Maximize 397.8 427.8 442.0
glm (mS/mm) Minimum value Maximize 48.0 52.1 54.1
Note: The bold font indicates critical parameter values that do not meet their respective targets.

Copyright © 2001 Marcel Dekker, Inc.

 An important point is that there is a difference between the nominal values for the
eight key device electrical characteristics and the mean values for these characteristics from
the Monte Carlo statistical simulations. This is undesirable, since the nominal is the
optimal device center calculated in the design optimization section. Ideally, the difference
between the mean and the nominal should be zero, and in any case it should be minimized.
From an analysis of the model equations it can be shown that this difference is dependent
on the size of the second-order terms. The form of the equations is
X X
X
2
yi ˆ Ai ‡ Bij
xj ‡ Cijk
xj
xk ‡ Dij
xj 2†
j j;k;j6ˆk j

where yi is one of the eight key electrical device characteristics and Ai , Bij , Cijk , and Dij are
coef®cients. For the optimal value of yi from the design optimization (denoted by yi;opt ),
all the
xj 's are zero, since yi;opt corresponds to all input parameters at their nominal
values and hence all
xj 's set to zero. Then
yi;opt ˆ Ai 3†
However, for the mean value of yi , denoted by hyi i:
X X X
hyi i ˆ Ai ‡ Bij h
xj i ‡ Cijk
xj
xk † ‡ Dij
xj †2 4†
j j;k;j6ˆk j

However, for each
xj the probability distribution is the Normal distribution centered
about zero. Because this distribution is symmetric about its center, h
xj i ˆ
h
xj
xk †i ˆ 0, since
xj and
xj
xk † are odd functions. On the other hand,
xj †2
is an even function; hence h
xj †2 i 6ˆ 0 19†. In fact, s2jN ˆ h
xj †2 i h
xj i†2 , and since
2 2
h
xj i ˆ 0, h
xj † i ˆ sjN . Hence,
X
hyi i yi;opt ˆ Dij s2jN 5†
j

Clearly, the nonlinear, second-order relationship between the responses and the input
parameters causes a shift in the mean value of the responses from their optimal values.
Using Eqs. (3) and (5):
P 2
hyi i yi;opt j Dij sjN
ˆ 6†
yi;opt Ai
The right-hand side of Eq. (6) can be used as a metric to evaluate the expected
relative difference between the mean value and the optimal value for any of the responses.
This metric, call it the expected shift of the mean, can be directly evaluated from the
normalized model equation before a Monte Carlo simulation is run to determine hyi i.
After such a simulation is run and hyi i is determined, the expected shift of the mean can be
compared to the ``actual shift of the mean,'' hyi i yi;opt †=yi;opt . These calculations were
done for the case where all the normalized input parameter statistical variations are
maximum, 3siN ˆ 1. The results are listed in Table 6 for all the responses. For all the
responses except the leakage current, the absolute value of the actual shift of the mean is
small, at less than 5% in all cases and less than 1% in most, and the expected and actual
shifts are quite close to each other. Even for leakage current, the actual shift of the mean is
a tolerable 11%, but the expected and actual shifts are relatively far apart.
Next, Monte Carlo simulation was used to meet the targets for the output para-
meters with an optimal set of reductions in the input parameter statistical variations. Each
input parameter statistical variation was reduced in steps, as listed in Table 7. (Note that,

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 Table 6 From Monte Carlo Simulations, the Mean, Standard Deviation (s), and Actual Shift
of the Mean; From the Normalized Model Equations, the Expected Shift of the Mean

Actual shift of Expected shift

mean of mean
i yi ˆ response Ai ˆ yi;opt Mean ˆ hyi i s hyi i yi;opt †=yi;opt 1=9†j Dij =Ai †

1 Vt (mV) 454.9 452.4 32.4 0:00550 0:00630

2
VT , DIBL (mV) 72.3 74.1 15.9 0.02490 0.02751
3 log Isat ) (uA/um) 2.721 2.7243 0.03 0.00121 0.00135
4 log Ileak ) (pA/um) 0.385 0.428 0.517 0.11169 0:08139
5 Isub (nA/um) 35.4 36.9 6.3 0.04237 0.04112
6 S 1=2 (mV/dec.) 1=2 0.10916 0.10906 0:00092 0:00080
7 gsm (mS/mm) 470.6 467.4 23.2 0:00680 0:00633
8 gsm (mS/mm) 58.63 59.2 3.7 0.00972 0.01006

for each input parameter, the maximum variation is the same as that used in the previous
section [see Table 2] and earlier in this section [see Table 4], and the minimum is half or less
than half of the maximum.) The straightforward approach is to run a series of Monte
Carlo simulations covering the entire range of possible combinations for the input para-
meter variations. However, the number of simulations is 46,656 for each response (see
Table 7), an unreasonably high number. In order to reduce the number of simulations to a
more manageable total, the following procedure was used. For each response, the normal-
ized model equation was examined to select those input parameters that are either missing
from the equation or included only in terms with small coef®cients. Since, as noted pre-
viously, these inputs are unimportant in in¯uencing the response, the variation was held
®xed at its maximum value for each of these selected parameters. As shown in Table 8,
following this procedure, two parameters were selected for each response, and hence the
number of Monte Carlo simulations was reduced to a more manageable 3888 or 5184.
Since each Monte Carlo simulation took about 3 seconds to run on a Hewlett-Packard
workstation, the total simulation time was about 3±4 hours for each response. (Table 8
does not include listings for Isub , S, gml , or gsm , since those responses are within speci®ca-
tion for the maximum values for the variation of all the input parameters, as shown in
Table 3.) The outputs from the Monte Carlo simulations were imported to a spreadsheet
program for analysis and display. By utilizing the spreadsheet capabilities, the input

Table 7 Steps in Input Parameter Statistical Variation, 3siN a

Maximum Minimum No. of No. of

Parameters variation variation Step size steps combinations

Lg ,
Xsh ,
Nsh ,
d, 1 (10%) 1/2 (5%) 1/4 (2.5%) 3 36 ˆ 729

Nhalo ,
Nch

Lg ,
Tsp ,
Rs 1 (15%) 0.3 (4.5%) 0.233 (3.5%) 4 43 ˆ 64
ÐÐÐÐ
Total number of 64  729 ˆ 46,656
combinations
a
For each 3siN , the corresponding statistical variation as a percentage of the mean value of the non-normalized
input parameter is in parentheses.

Copyright © 2001 Marcel Dekker, Inc.

 Table 8 Number of Steps in Input Parameter Statistical Variation

No. of
Response
Lg
Tox
Tsp
Xsh
Nsh
d
Nhalo
Nch
Rs combinations

VT 4 3 4 3 3 1 3 3 1 3888

VT 4 3 4 3 3 3 3 1 1 3888
Idsat 4 3 4 3 1 1 3 3 4 5184
Ileak 4 3 4 3 3 1 3 3 1 3888

Note: The bold font indicates that the number of steps has been reduced from the number in Table 7.

variations were then iteratively reduced from their maximum values to meet the targets for
the responses.
As already discussed, it was most dif®cult to meet the targets for VT and for
VT
due to DIBL. Hence, these two were dealt with ®rst. From the size of the coef®cients in the
normalized model equation for VT , the terms containing
Lg ,
Tox ,
Xsh , and
Nch are
the most signi®cant. Thus, the statistical variations of only these parameters were reduced
to meet the VT target, while the variations of the other input parameters were held at their
maximum values. Contour plots of constant 3s variation in VT were determined using the
spreadsheet program. The results are shown in Figure 3, where the statistical variations of
Tox and Xsh were ®xed at their realistic values of 5% each (corresponding to 3siN ˆ 1=2),
and the statistical variations of Lg and Nch were varied. Along Contour 1, the 3s variation
in VT is 50 mV, and the variations of both Lg and Nch are less than 7.5%. Since these
variations are quite aggressive (see Table 4), the 50-mV target will be dif®cult to meet.
Along Contour 2, the 3s variation in VT is 60 mV. This target is realistic because the
variations of both Lg and Nch on the contour are achievable, particularly in the vicinity of
the point where the variations are about 9.5% for Lg and 7.5% for Nch (see Table 4).
Figure 4 also shows contours of constant 3s variation in VT ; the only difference from
Figure 3 is that the statistical variation of Xsh is 7.5%, not 5% as in Figure 3. The 60-mV

Figure 3 Contours of constant 3s variation in VT (mV).

Copyright © 2001 Marcel Dekker, Inc.

 Figure 4 Contours of constant 3s variation in VT (mV), with Xsh variation of 7.5%.

contour here, labeled Contour 3, is shifted signi®cantly to the left from the 60-mV contour
in Figure 3 and hence is much more dif®cult to achieve. For the case where the variation of
Tox is 7.5% while that of Xsh is 5%, the 60-mV contour is shifted even further to the left
than Contour 3. The contour plots can be utilized to understand quantitatively the impact
of the statistical variations of the key input parameters and how they can be traded off to
reach a speci®c target for VT variation. Looking particularly at Contour 2 in Figure 3, and
utilizing ``realistic'' values of the variations as much as possible (see Table 4), an optimal
choice for the variations is 5% for Tox and Xsh , 7.5% for Nch , and 9.5% for Lg .
Next, the requirements to meet the target for
VT due to DIBL were explored.
From the size of the coef®cients in the normalized model equation for
VT , the terms
containing
Lg ,
Xsh ,
Tsp , and Tox are the most signi®cant. Thus, the variations of only
these parameters were reduced to meet the
VT target, while the variations of the other
input parameters were held at their maximum values. Figure 5 shows contours of constant

Figure 5 Contours of constant maximum
VT (DIBL) (mV).

Copyright © 2001 Marcel Dekker, Inc.

 maximum value of
VT , where the variations of both Tox and Xsh were held at 5%, as in
Figure 3. Along Contour 4, the value is 100 mV, the target value. The variations of Lg and
Tsp on this contour are realizable, particularly in the vicinity of the point where the
variations are about 9% for Lg and 8% for Tsp . Utilizing the same reasoning as discussed
earlier for meeting the VT target, this point is about optimal.
Finally, the requirements to meet the targets for Idsat and Ileak were explored. From
the size of the coef®cients in the normalized model equations, the terms containing
Lg ,

Tox , and
Xsh are the most signi®cant for Idsat , while the terms containing
Lg ,
Xsh ,

Nch , and
Tox are most signi®cant for Ileak . Figure 6 shows the contours of constant
minimum value of Idsat , with the variation of Tox held at 5% as in Figures 2, 3, and 4.
Along Contour 5, the minimum Idsat is 450 mA/mm, the target value. Similarly, Figure 7
shows the contours of constant maximum value of Ileak , with the variations of all input
parameters except Xsh and Lg held at their maximum variations. Along Contour 6, the
maximum Ileak is 50 pA/mm, the target value. The input parameter variations along both
Contours 5 and 6 are signi®cantly larger than those required to meet the targets for VT
and
VT due to DIBL (see Contour 2 in Figure 3 and Contour 4 in Figure 5). Hence, if
the VT and
VT targets are met, then the targets for Ileak and Idsat are also automatically
met.
Tying all the foregoing results and discussion together, only ®ve input parameters,
Lg ; Tox , Xsh , Nch , and Tsp , need to be tightly controlled (i.e., the statistical variation of
each of them must be notably less than the ``Maximum statistical variation'' in Table 4) to
meet the targets in Table 3 for the device electrical characteristics. The other four input
parameters, d, Nhalo , Nsh , and Rs , can be relatively loosely controlled (i.e., the statistical
variation of each of them can be equal to or possibly larger than the maximum variation in
Table 4), and the targets will still be met. An optimal set of choices that satis®es all the
output response targets is: statistical variation of Lg  9%, statistical variation of
Tox  5%, statistical variation of Xsh  5%, statistical variation of Nch  7:5%, and sta-
tistical variation of Tsp  8%. Note that this optimal set is challenging, since the statistical
variation values range from ``realistic'' to ``aggressive'' according to the classi®cation in
Table 4. In particular, the 9% requirement on gate length control will be pushing the
limits, since, according to the 1999 International Technology Roadmap for Semiconductors

Figure 6 Contours of constant minimum Idsat (mA/mm).

Copyright © 2001 Marcel Dekker, Inc.

 Figure 7 Contours of constant maximum Ileak . Maximum statistical variation for all input para-
meters except Lg and Xsh .

(20), the gate CD (critical dimension) control is 10%. This approach can also be used to
determine tradeoffs. If, for example, the process variation of Tsp can only be controlled to
12%, it is evident from Contour 4 of Figure 5 that the control of Lg would have to be
tightened so that its process variation is 8% or less.
For process control purposes, let UL be the upper speci®cation limit for the process,
let LL be the lower speci®cation limit, and let MEAN be the mean value for the structural
and doping parameters. A very important quantity is the ``process capability,'' Cp . For the
ith input parameter, Cp ˆ UL LL†=6si , where si is the standard deviation of the ith
(non-normalized) input parameter. The goal is to control the process variations and the
resulting si so that Cp  1. For Cp much less than 1, a non-negligible percentage of the
product is rejected (i.e., noticeable yield loss) because of input parameter values outside
the process limits, as illustrated schematically in Figure 8. For Cp  1, the statistical
distribution of the input parameter values is largely contained just within the process
limits, so the cost and dif®culty of process control are minimized, but very little product
is rejected because of input parameter values outside the process limits. Finally, for Cp
much larger than 1, the actual statistical distribution of the input parameter is much
narrower than the process limits, and hence very little product is rejected, but the cost
and dif®culty of process control are greater than for the optimal case, where Cp  1. In
practice, because of nonidealities in IC manufacturing lines and the dif®culty of setting
very precise process limits, the target Cp is typically somewhat larger than 1, with 1.3 being
a reasonable rule of thumb (21,22). In practical utilization of the previous Monte Carlo
results, especially the optimal set of variations, it makes sense to set
UL ˆ MEAN ‡ optimal 3si variation† and LL ˆ MEAN optimal 3si variation) for
all of the key structural and doping parameters. Using these formulas, the values of LL
and UL are listed in Table 9 for all the input parameters.
Meeting the process control requirements for the statistical variation of the ®ve key
input parameters is dependent on the control at the process module level. For example, to
meet the channel implant dose (Nch ) requirement, the channel implant dose and energy as
well as the thickness of any screen oxide must be well controlled. As another example, to

Copyright © 2001 Marcel Dekker, Inc.

 Figure 8 Impact of Cp variation.

meet the Tox requirement, the gas ¯ows, temperature, and time at temperature for a
furnace process must be well controlled. Through empirical data or simulations, the
level of control of the process modules necessary to meet the requirements on the input
parameter statistical variations can be determined. Of course the tighter the requirements
on the input parameter variations, the tighter the required level of control of the process
modules.

V. METROLOGY REQUIREMENTS

The metrology requirements are driven by the process control requirements, i.e., the UL
and LL in Table 9 for the input parameters. In-line metrology is used routinely in the IC
fabrication line to monitor these parameters, to ensure that they stay between the LL and
UL, or to raise an alarm if they drift out of speci®cation. For in-line metrology on a well-
established, well-characterized line, the most important characteristic is the measurement
precision, P, where P measures the repeatability of the measurement. For a measurement
of a given parameter with a particular piece of measurement equipment (for example, a
measurement of Tox using a particular ellipsometer), P is determined by making repeated
measurements of Tox on the same wafer at the same point. P is de®ned to be 6sMETROL ,
where sMETROL is the standard deviation of the set of repeated measurements (23). A key
parameter is the ratio of measurement precision, P, to the process tolerance, T, where
T ˆ UL LL. Then P=T ˆ 6sMETROL = UL LL). It is important that P=T  1, or
measurement errors will reduce the apparent Cp (or, equivalently, the apparent process
standard deviation will increase). This point is illustrated in Figure 9, where a simpli®ed
process control chart for one of the key doping and structural parameters, such as the gate
oxide thickness or gate length, is shown. The position of each X indicates the value of an
in-line measurement (or the mean value of multiple measurements on one or several wafers
from a lot) if there were no metrology errors. The cross-hatched area around each X
indicates the uncertainty introduced because of the random errors from the metrology,
i.e., the in¯uence of the nonzero sMETROL . In the following, let sPROC be the statistical
standard deviation due to random process variations, as discussed in the previous sections.
In Case 1 and Case 2 in the ®gure, Cp  1, so 6sPROC  UL LL). In Case 1, P=T  1,

Copyright © 2001 Marcel Dekker, Inc.

 Table 9 Nominal Values and Optimal Statistical Variations for Input Parameters

Maximum 3si Maximum 3si

variation (optimal variation (optimal
Input parameter MEAN case) [%] case) [units] UL MEAN ‡ 3si ) LL (MEAN 3si )

Gate length, Lg (mm) 0:18 mm 9% 0:0162 mm 0:196 mm 0:164 mm

Gate oxide thickness, Tox (nm) 4.50 nm 5% 0:225 nm 4.73 nm 4.28 nm
Spacer oxide width, Tsp (nm) 82.5 nm 8% 6:6 nm 89.1 nm 75.9 nm
Shallow-junction doping pro®le depth, 50 nm 5% 2:5 nm 52.5 nm 47.5 nm
Xj sh† (nm)
Peak shallow-junction doping, 4  1019 cm 3
10% 4  1018 cm 3
4:4  1019 cm 3
3:6  1019 cm 3

Nsh cm 3 †
Channel dose, Nch cm 2 † 5:65  1012 cm 2 7:5% 4:24  1011 cm 2 6:07  1012 cm 2
5:23  1012 cm 2

Halo dose, Nhalo cm 2 † 1:5  1013 cm 2 10% 1:5  1012 cm 2 1:65  1013 cm 2
1:35  1013 cm 2

Halo peak depth, d (nm) 80 nm 10% 8 nm 88 nm 72 nm

Series resistance (external), Rs -mm† 400 -mm 15% 60 -mm 460 -mm 340 -mm

Copyright © 2001 Marcel Dekker, Inc.

 Figure 9 Simpli®ed process control charts showing the impact of different P=T ratios.

so 6sMETROL  UL LL†  6sPROC . Under those circumstances, as shown in the ®g-

ure, for most of the measurements the error due to sMETROL does not impact whether a
given measurement is within the process speci®cation limits. For Case 2, however, where
P=T ˆ 0:7 and hence 6sMETROL ˆ 0:7 UL LL†  6sPROC , the errors due to sMETROL
are becoming comparable to (UL LL). As shown in Case 2, the metrology error can
cause the measured value to lie outside the process speci®cation limits even though the
actual parameter value lies inside these limits. If sMETROL cannot be reduced, then in order
to ensure that the process parameter stays within the process speci®cation limits, sPROC
must be reduced, as shown in Case 3. Depending on the amount of reduction required, this
can be costly and dif®cult, since it requires more stringent process control.
These considerations can be quantitatively evaluated as follows (24). Since sMETROL
and sPROC are standard deviations due to independent randomly varying processes,
the total standard deviation, sTOTAL ˆ s2METROL ‡ s2PROC †1=2 . Letting
CP;PROC ˆ UL LLg=6sPROC and CP;TOTAL ˆ UL LL=6sTOTAL , CP;TOTAL is the
apparent Cp and sTOTAL is the apparent standard deviation. From these equations and
the de®nition of P=T, for given values of P=T and CP;PROC ,

   ! 1=2
CP;TOTAL   P 2
ˆ 1 ‡ CP;PROC 7†
CP;PROC T

Since (UL LL) is ®xed by speci®cation, then CP;TOTAL =CP;PROC ˆ sPROC =sTOTAL . A
plot of CP;TOTAL =CP;PROC versus CP;PROC , with P=T as a parameter varying from 0.1 to
1.0, is shown in Figure 10. This plot illustrates the impact of the measurement variation as

Copyright © 2001 Marcel Dekker, Inc.

 Figure 10 Impact of P=T on Cp and s, where CP;PROC is an independent variable.

characterized by the parameter P=T. For a given value of CP;PROC (and hence of sPROC ),
CP;TOTAL decreases (and hence sTOTAL increases) rapidly with P=T; and since the goal is to
maximize Cp and minimize s, the increase of P=T imposes a signi®cant penalty.
An alternate way to evaluate the impact of P=T variation is shown in Figure 11,
where CP;TOTAL and P=T are the independent variables and CP;TOTAL =CP;PROC is plotted
as in Figure 10, but versus the independent variable, CP;TOTAL . The parameter P=T varies
from 0.1 to 0.8. As in Figure 10, since (UL LL) is ®xed by speci®cation, then
CP;PROC =CP;TOTAL ˆ sPROC =sTOTAL . Using the de®nitions of P=T, CP;TOTAL , CP;PROC ,
sPROC , and sTOTAL , the equation is
   !1=2
CP;PROC sPROC   P 2
ˆ ˆ 1 CP;TOTAL 8†
CP;TOTAL sTOTAL T

In general, for each key doping or structural parameter in an IC technology, there is a

speci®ed target for Cp , and since (UL LL) is also speci®ed, there is a corresponding
target for si . Ideally, through process control techniques, sPROC would be regulated to
match the si target, and hence CP;PROC would equal the Cp target. However, because of
the random variations in the measurements, the total standard deviation is sTOTAL , not
sPROC . As a result, sTOTAL is matched to the si target and CP;TOTAL equals the Cp target.
The horizontal axis in Figure 11 then represents the Cp target as well as CP;TOTAL . The
curves in the ®gure show, for any given Cp target, the percentage reduction from the si
target that is required in sPROC , as illustrated schematically in Case 3 of Figure 9. For
example, for Cp target ˆ 1:3 and P=T ˆ 0:1, sPROC must be reduced only about 1% from
the s target. For P=T ˆ 0:2, the reduction is about 3%; for P=T ˆ 0:3, the reduction is
about 7%; and the reduction is about 38% for P=T ˆ 0:6. Clearly, for small P=T, up to

Copyright © 2001 Marcel Dekker, Inc.

 Figure 11 Impact of P=T on Cp and s, where CP;TOTAL is an independent variable.

perhaps 0.3 for this case, the reduction is relatively small and tolerable, but for large P=T,
the required reduction is intolerably large. Note that the curves for P=T  0:4 go to zero at
some value of CP;TOTAL ˆ Cp0 , where Cp0 < 2:5, the maximum value plotted. This occurs
for P=T†Cp0 ˆ 1 [see Eq. (8)], which corresponds to sMETROL ˆ sTOTAL , and hence
sPROC ˆ 0, which is impossible to achieve. In this case, the entire budget for random
variation is absorbed by the metrology variation, leaving none available for the random
process variations.
Tying the foregoing considerations together with the process speci®cation limits (UL
and LL) in Table 9, the required metrology precision as re¯ected by the sMETROL values
for P=T ˆ 0:1 and for P=T ˆ 0:3 are listed in Table 10. Note that in most cases, the
required precision is quite high. For example, for Tox , with P=T ˆ 0:1, the sMETROL
value of 0.0075 nm is less than 0.1 AÊ; even for P=T ˆ 0:3, the 0.023-nm value is just
over 0.2 AÊ. Similarly, for P=T ˆ 0:1, the sMETROL value for Lg is just over 5 AÊ and for
xj sh† , the sMETROL value is less than 1 AÊ.
The in-line metrology is limited by practical issues. Ideally, the in-line measure-
ments are rapid, nondestructive, direct, and done on product wafers. However, in a
number of cases, test wafers are used, and sometimes the measurement is destructive.
Of the ®ve key input parameters requiring ``tight control'' (i.e., 3sPROC < 10%, as
discussed in the previous section and listed in Table 9), four parametersÐthe gate length
(Lg ), the gate oxide thickness (Tox ), the spacer oxide width (Tsp ), and the channel dose
(Nch )Ðcan be routinely and directly measured on product wafers. Lg and Tsp are mea-
sured optically or via SEM after etch, Tox is measured via ellipsometry, and Nch is
measured via the thermal wave re¯ectance (25) technique. Alternatively, Nch is some-
times monitored on test wafers using secondary ion mass spectrosopy (SIMS) or four-
point sheet resistance measurements, and Tox is sometimes measured on test wafers. The

Copyright © 2001 Marcel Dekker, Inc.

 Table 10 sMETROL values for P=T ˆ 0:1 and P=T ˆ 0:3

sMETROL sMETROL
Input parameter Mean UL LL (P=T ˆ 0:1) (P=T ˆ 0:3)

Gate length, Lg (mm) 0:18 mm 0:0324 mm 0:00054 mm 0:0016 mm

Gate oxide thickness, 4.50 nm 0.45 nm 0.0075 nm 0.023 nm
Tox nm†
Spacer oxide width, 82.5 nm 13.2 nm 0.22 nm 0.66 nm
Tsp (nm)
Shallow-junction doping 50 nm 5 nm 0.083 nm 0.25 nm
pro®le depth,
Xj sh† (nm)
Peak shallow-junction 4  1019 cm 3
8  1018 cm 3
1:33  1017 cm 3
4  1017 cm 3

doping, Nsh (cm 3 )

Channel dose, 5:65  1012 cm 2
8:48  1011 cm_ 2 1:41  1010 cm 2
4:24  1010 cm 2

Nch cm 2 †
Halo dose, 1:5  1013 cm 2
3  1012 cm 2
5  1010 cm 2
1:5  1011 cm 2

Nhalo cm 2 †
Halo peak depth, 80 nm 16 nm 0.27 nm 0.8 nm
d (nm)
Series resistance 400 -mm 120 -mm 2 -mm 6 -mm
(external), Rs -mm†

other parameter requiring tight control, the S=D extension (``shallow'') junction depth
(Xsh ), cannot currently be routinely measured on product wafers. Consequently, Xsh is
typically monitored on test wafers via SIMS. However, in the future, the Boxer-Cross
technique (26) shows promise of becoming practical for in-line measurements on product
wafers. For the four input parameters requiring ``looser control'' (i.e., 3sPROC  10%, as
discussed in the previous section and listed in Table 9), none are routinely measured on
product wafers. For Nhalo and the halo peak depth (d), given the loose control needed
and the typically tight control on the halo implant dose and energy, monitoring test
wafers via SIMS is generally adequate. The peak shallow-junction doping (Nsh ) can be
monitored using the same SIMS that is used to measure Xsh . Finally, at the end of the
wafer fabrication process, routine electrical monitoring of test transistors on product
wafers is used to measure Rs . Typically, for each lot in an IC manufacturing line, the in-
line measurements are made on a reasonable sample size, and good statistics for the
standard deviation (sTOTAL ) and direct veri®cation of meeting process control limits are
obtained.
As mentioned earlier, for in-line metrology used for process control in well-estab-
lished and well-characterized IC fabrication processes, the key requirement is precision
(i.e., repeatability), since there is a well-established, optimal baseline, and the main
requirement is to ensure that the process does not drift unacceptably far from the
optimal. However, for establishment of new or strongly modi®ed process modules or
process ¯ows, it is necessary to measure the values of key parameters, such as Tox and
Lg , to a relatively high degree of absolute accuracy in order to understand and estab-
lish the optimal baseline. For example, in establishing the 0.18-mm NMOSFET process,
it is important that Tox be accurately measured so that the real Tox of the process can
be set close to 4.5 nm. (Note that this metrology with high absolute accuracy is often

Copyright © 2001 Marcel Dekker, Inc.

 different from the in-line metrology. For example, transmission electron microscopy
(TEM) measurements and/or electrical C-V measurements with corrections for quan-
tum effects and polysilicon depletion can be used to measure Tox on test wafers.)
However, once an optimal process is well established, the in-line metrology can indicate
that the nominal Tox is 4.8 nm as long as that corresponds to a real nominal Tox of 4.5
nm and as long as the in-line metrology has adequate precision. In other words, for in-
line metrology the precision is critical, but adjustments can be made for consistent
measurement bias.

VI. CONCLUSIONS

A 0.18-mm NMOSFET device was designed and optimized to satisfy a speci®ed set of
electrical characteristics. This optimized device was the nominal design center for a simu-
lated sensitivity analysis in which normalized second-order polynomial model equations
were embedded within a special Monte Carlo code. Monte Carlo simulations with the code
were used to correlate the random statistical variations in key electrical device character-
istics to the random variations in the key structural and doping parameters. Using these
simulations, process control tradeoffs among the different structural and doping para-
meters were explored, and the level of process control required to meet speci®ed statistical
targets for the device electrical characteristics was analyzed. It turns out that meeting these
targets requires tight control of ®ve key structural and doping parameters: the gate length,
the gate oxide thickness, the shallow source/drain extension junction depth, the channel
dose, and the spacer width. Making process control tradeoffs based on estimates of
industry capability, an optimal set of 3s statistical variations was chosen for the ®ve
parameters: 9%, 5%, 5%, 7.5%, and 8%, respectively. If the estimates of industry cap-
ability were different, the tradeoffs would be changed and hence the optimal set of varia-
tions would be changed.
The optimal set of parameter statistical variations drives the in-line metrology. The
key requirement is that the metrology precision for any measured parameter be no more
than 10±30% of the optimal statistical variation for that parameter. Also, the impact of
not meeting the precision requirements was quantitatively analyzed, and it was found that
the more the metrology precision departs from the requirements, the tighter the process
control must be.

ACKNOWLEDGMENTS

I am grateful to a number of people from International SEMATECH who were helpful in

preparing this chapter. Jack Prins and Paul Tobias provided valuable inputs on the sta-
tistical modeling aspects of this work. Larry Larson provided important input on the
process control techniques and capability of the industry, and Alain Diebold provided
valuable input on metrology. My former colleague William E. Moore (now at AMD, Inc.)
was very helpful in running the Monte Carlo simulations.
I am grateful to Professor Al F. Tasch of the University of Texas at Austin and to his
former students, S. Are®n Khan, Darryl Angelo, and Scott Hareland. They were instru-
mental in developing the device optimization and the Monte Carlo code under a contract
with International SEMATECH.

Copyright © 2001 Marcel Dekker, Inc.

 APPENDIX: NORMALIZED MODEL EQUATIONS

In the following equations, each of the eight output responses is given as a function of the
normalized set of input factors. Each of these input factors, for example,
Lg , will have a
range of values between 1 and ‡1. Referring to Table 2, a 1 value for
Lg would
correspond to a gate length, Lg , of 0:18 mm 15% ˆ 0:153 mm, while a ‡1 value for
Lg
would indicate an Lg ˆ 0:18 mm ‡ 15% ˆ 0:207 mm. For the nominal case, each normal-
ized input variable would have a value of 0.

VT mV† ˆ 454:9 ‡ 56:1
Lg ‡ 52:4
Tox ‡ 41:4
Nch 33:6
Xsh ‡ 9:9
Nsh
‡ 9:7
Tsp ‡ 7:1
Nhalo 1:6
d 19:9
L2g 2
5:9
Tox 2
3:8
Tsp
‡ 3:8
d 2 ‡ 21:9
Lg 
Xsh 6:9
Lg 
Tsp 6:2
Lg 
Nsh

VT mV† ˆ 72:3 39:4
Lg ‡ 17:8
Xsh 12:7
Tsp ‡ 8:4
Tox 5:9
Nhalo
4:7
Nch 3:0
Nsh 1:5
d ‡ 12:4
L2g ‡ 7:8
Tsp
2 2
2:3
Nsh
11:9
Lg 
Xsh ‡ 5:9
Lg 
Tsp 5:3
Lg 
Tox ‡ 4:4
Tsp

d ‡ 2:4
Lg 
Nhalo ‡ 2:3
Tsp 
Nhalo

Note: In the next two equations, a logarithmic transformation was used to achieve better
normality and ®t.

Idsat log‰mA=mmŠ† ˆ 2:721 0:060
Lg 0:052
Tox ‡ 0:028
Xsh 0:019
Tsp
0:016
Nch 0:008
Nhalo 0:007
Rs 0:005
Nsh
‡ 0:015
L2g ‡ 0:009
Tox
2 2
‡ 0:009
Tsp ‡ 0:013
Xsh

Nch 0:011
Tsp 
Nch 0:008
Lg 
Tox
0:006
Lg 
Xsh

Ileak log‰pA=mmŠ† ˆ 0:385 1:189
Lg ‡ 0:571
Xsh 0:508
Nch 0:417
Tox
0:241
Tsp 0:144
Nsh 0:127
Nhalo ‡ 0:011
d
0:424
L2g ‡ 0:104
Tsp
2
0:080
d 2 2
0:063
Nsh
2
‡ 0:055
Tox 0:449
Lg 
Xsh ‡ 0:156
Lg 
Tsp
‡ 0:112
Lg 
Nsh 0:088
Lg 
Tox

Isub nA=mm† ˆ 35:4 17:6
Lg 1:9
Tsp 1:7
Nch 1:7
Rs ‡ 1:5
Nsh
1:5 ;
Tox ‡ 1:2
Xsh ‡ 6:2
L2g ‡ 2:0
Tox
2 2
‡ 1:7
Tsp
2 2
‡ 1:7
Xsh ‡ 1:5
Nch ‡ 3:2
Lg 
Nch
Note: In the following equation, an inverse square root transformation was used to achieve
better normality and ®t.

Copyright © 2001 Marcel Dekker, Inc.

 1=2 1=2
S mV=decade† ˆ 0:10916 0:00137
Tox 0:00085
Lg 0:00050
Nch
0:00018
Nhalo 0:00016
Tsp ‡ 0:00016
Xsh
2
0:00030
Xsh 0:00027
L2g 2
0:00022
Nhalo
‡ 0:00047
Lg 
Xsh 0:00028
Lg 
Tsp

gsm mS=mm† ˆ 470:6 50:4
Tox 41:7
Lg ‡ 14:4
Xsh 6:0
Nhalo ‡ 2:9
Nch
2 2
14:3
Xsh 12:5
Nhalo 20:2
Tox 
Nch

glm mS=mm† ˆ 58:63 9:44
Lg 3:93
Tox ‡ 2:74
Xsh 1:69
Tsp 1:53
Rs
0:85
Nch 0:47
Nsh ‡ 0:43
d 0:34
Nhalo ‡ 1:68
L2g
2 2 2 2
‡ 1:36
Tox ‡ 0:94
Nhalo ‡ 0:69
Nch ‡ 0:64
Tsp 1:33
Lg

Xsh ‡ 0:78
Lg 
Tsp

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Copyright © 2001 Marcel Dekker, Inc.

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Copyright © 2001 Marcel Dekker, Inc.