U.S. patent application number 11/876311 was filed with the patent office on 2008-02-28 for characteristic evaluation apparatus for insulated gate type transistors.
This patent application is currently assigned to Renesas Technology Corp.. Invention is credited to Kenji YAMAGUCHI.
Application Number | 20080048707 11/876311 |
Document ID | / |
Family ID | 17040475 |
Filed Date | 2008-02-28 |
United States Patent
Application |
20080048707 |
Kind Code |
A1 |
YAMAGUCHI; Kenji |
February 28, 2008 |
CHARACTERISTIC EVALUATION APPARATUS FOR INSULATED GATE TYPE
TRANSISTORS
Abstract
The accuracy of effective channel width extraction in drain
current method is improved. There are prepared a transistor with a
wide channel width serving as a reference, and a transistor with a
narrow channel width that becomes a candidate for extraction (step
ST1.1). From the characteristic curve of a plane formed by mask
channel width and source-drain conductance, there is extracted a
virtual point at which the change of source-drain conductance is
estimated to be approximately zero even if the gate overdrive is
finely changed. Then, the value of function F is calculated which
is defined by the difference between the change of the conductance
at the coordinate of the virtual point and the product obtained by
multiplying the conductance per unit width by the change of the
mask channel width (step ST1.6). From a shift amount (.delta.)
which minimizes the standard deviation of the function F to be
obtained (step ST1.7), the true threshold voltage of the transistor
with the narrow channel width is determined (step ST1.10).
Inventors: |
YAMAGUCHI; Kenji; (Tokyo,
JP) |
Correspondence
Address: |
OBLON, SPIVAK, MCCLELLAND MAIER & NEUSTADT, P.C.
1940 DUKE STREET
ALEXANDRIA
VA
22314
US
|
Assignee: |
Renesas Technology Corp.
Tokyo
JP
|
Family ID: |
17040475 |
Appl. No.: |
11/876311 |
Filed: |
October 22, 2007 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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10702455 |
Nov 7, 2003 |
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11876311 |
Oct 22, 2007 |
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10093933 |
Mar 11, 2002 |
6727724 |
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10702455 |
Nov 7, 2003 |
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09714148 |
Nov 17, 2000 |
6373274 |
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10093933 |
Mar 11, 2002 |
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09249139 |
Feb 12, 1999 |
6169415 |
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09714148 |
Nov 17, 2000 |
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Current U.S.
Class: |
324/762.09 ;
257/E21.531 |
Current CPC
Class: |
G01R 31/2837 20130101;
G01R 31/2621 20130101; H01L 22/14 20130101 |
Class at
Publication: |
324/769 |
International
Class: |
G01R 31/26 20060101
G01R031/26 |
Foreign Application Data
Date |
Code |
Application Number |
Aug 25, 1998 |
JP |
10-239148 |
Claims
1. A characteristic evaluation method for insulated gate type
transistors, comprising: a) preparing at least two insulated gate
type transistors including first and second insulated gate type
transistors that differ from each other only in mask channel width;
b) extracting a threshold voltage of said first transistor that has
a mask channel width larger than that of said second transistor,
estimating a threshold voltage of said second transistor, and
employing a value of the estimated threshold voltage as a first
estimated value; c) when a difference between a gate voltage of
said first transistor and said extracted threshold voltage of said
first transistor is defined as a first gate overdrive, a difference
between a gate voltage of said second transistors and said first
estimated value is defined as a second gate overdrive, and an X-Y
plane is assumed whose X-axis is said mask channel width and Y-axis
is source-drain conductance, (i) extracting a virtual point at
which a change of Y coordinate value is estimated to be
approximately zero even if said first and second gate overdrives
are finely changed, from a characteristic curve exhibiting a
relationship between said mask channel widths of said first and
second transistors and said source-drain conductance, (ii) defining
values of an X coordinate and a Y coordinate at said virtual point
as second and third estimated values, respectively, and (iii)
extracting a slope of said characteristic curve at said virtual
point and employing a value of the extracted slope as a fourth
estimated value; d) repeating said step c) while varying said first
estimated value; e) after said steps c) and d), (i) finding, from
said second to fourth estimated values, optimum second to fourth
estimated values with which the change of said third estimated
value is equal to a product of the change of said second estimated
value and said fourth estimated value, in reply to fine changes of
said first and second gate overdrives, (ii) determining an optimum
first estimated value that corresponds to said optimum second to
fourth estimated values, and (iii) determining a true threshold
voltage of said second transistor based an said optimum first
estimated value; and f) determining a difference between said mask
channel width and an effective channel width, based an said true
threshold voltage.
2. The method of claim 1, wherein in said step e), said
characteristic curve is approximated by using a first straight line
in said X-Y plane, said first straight line passing through a first
point that is given to said first transistor when said first gate
overdrive has a first value and a second point that is given to
said second transistor when said second gate overdrive has said
first value.
3. The method of claim 2, wherein in said step e), said optimum
second to fourth estimated values are determined from a relational
expression: F .function. ( .delta. , V gtWi ) = dW ** .function. (
.delta. , V gtWi ) + f .function. ( .delta. , V gtWi ) f 1
.function. ( .delta. , V gtWi ) dW ** 1 .function. ( .delta. , V
gtWi ) - DW * .function. ( .delta. , V gtWi ) ##EQU37## where
.delta. is a difference between the first estimated value and the
threshold voltage of said first transistor; V.sub.gtWi is said
first gate overdrive; dW** is a value of an X intercept that is
obtained by extrapolating said characteristic curve; f is said
slope of said characteristic curve at said virtual point; DW* is an
X coordinate value at said virtual point; and a prime is a
first-order differentiation of V.sub.gtWi.
4. The method of claim 2, wherein in said step e), said optimum
second to fourth estimated values are determined from a relational
expression: F .function. ( .delta. , V gtWi ) = f 2 .function. (
.delta. , V gtWi ) f 1 .function. ( .delta. , V gtWi ) dW ** 1
.function. ( .delta. , V gtWi ) - G m * .function. ( .delta. , V
gtWi ) ##EQU38## where .delta. is a difference between the first
estimated value and the threshold voltage of said first transistor;
V.sub.gtWi is said first gate overdrive; dW** is a value of an X
intercept that is obtained by extrapolating said characteristic
curve; f is said slope of said characteristic curve at said virtual
point; G.sub.m* is a Y coordinate value at said virtual point; and
a prime is the first-order differentiation of V.sub.gtWi.
5. The method of claim 2, wherein in said step e), said optimum
second to fourth estimated values are determined from a relational
expression: F .function. ( .delta. , V gtWi ) = G m ** ( .delta. ,
V gtWi ) - f .function. ( .delta. , V gtWi ) f 1 .function. (
.delta. , V gtWi ) G m ** 1 .function. ( .delta. , V gtWi ) - G m *
.function. ( .delta. , V gtWi ) ##EQU39## where .delta. is a
difference between the first estimated value and the threshold
voltage of said first transistor; V.sub.gtWi is said first gate
overdrive; G.sub.m** is a value of a Y intercept that is obtained
by extrapolating said characteristic curve; f is said slope of said
characteristic curve at said virtual point; G.sub.m* is a Y
coordinate value at said virtual point; and a prime is the
first-order differentiation of V.sub.gtWi.
6. The method of claim 2, wherein in said step e), said optimum
second to fourth estimated values are determined from a relational
expression: F .function. ( .delta. , V gtWi ) = G m ** 1 .function.
( .delta. , V gtWi ) f 1 .function. ( .delta. , V gtWi ) + DW * (
.delta. , V gtWi ) ##EQU40## where .delta. is a difference between
the first estimated value and the threshold voltage of said first
transistor; V.sub.gtWi is said first gate overdrive; G.sub.m** is a
value of a Y intercept that is obtained by extrapolating said
characteristic curve; f is said slope of said characteristic curve
at said virtual point; DW* is an X coordinate value at said virtual
point; and a prime is the first-order differentiation of
V.sub.gtWi.
7-19. (canceled)
Description
BACKGROUND OF THE INVENTION
[0001] 1. Field of the Invention
[0002] The present invention relates to a characteristic evaluation
method for insulated gate type transistors which extracts their
effective channel widths, a characteristic evaluation apparatus for
insulated gate type transistors, a method of manufacturing
insulated gate type transistors by using the above characteristic
evaluation method, and a computer readable storing medium storing a
characteristic evaluation program.
[0003] 2. Description of the Background Art
[0004] An electrically effective channel width, i.e., an effective
channel width W.sub.eff, can be determined from the drain currents
of two or more insulated gate type transistors having the same
channel length and a different channel width. This method is
generally called "drain current method." The drain current method
can directly determine the difference between an effective channel
width W.sub.eff and a mask channel width W.sub.m, namely, a channel
narrowing DW (=W.sub.m-W.sub.eff).
[0005] As a drain current method, a wide variety of methods have
been proposed heretofore. They are described, for example, in "A
New Method to Electrically determine Effective MOSFET Channel
Width" by Y. R. Ma and K L Wang, IEEE Trans. Elect. Dev., ED-29, p.
1825, 1982; "A New Method to Determine the MOSFET Effective Channel
Width" by N. D. Arora, L A. Blair and L M. Richardson, IEEE Trans.
Elect. Dev., ED-37(3), p. 811, 1990; "A Method to Extract
Gate-Bias-Dependent MOSFETs Effective Channel Width" by Y. T. Chia
and G. J. Hu, IEEE Trans. Elect. Dev., ED-38(2), p. 424, 1991; and
"A Direct Method to Extract Effective Geometries and Series
Resistances of MOS Transistors" by P. R. Karlsson and K O. Jeppson,
Proc. IEEE ICMTS, vol. 7, p. 184, 1994.
[0006] Of various drain current methods, Chia method is commonly
often used. Thus, Chia method will be briefly described here. The
total source-drain resistance R is given by the sum of a channel
resistance R.sub.ch and an external resistance R.sub.sd. Now,
supposing the following Equation 1 as the equation to express drain
current. I ds = .beta. 0 ( V gs - V th - V ds * 2 ) V ds * 1 +
.theta. .times. .times. 1 ( V gs * - V th ) + .theta. .times.
.times. 2 ( V gs * - V th ) 2 ( Eq . .times. 1 ) ##EQU1## where
.beta..sub.0, V.sub.ds* and V.sub.gs* are given by the following
Equations 2, 3 and 4, respectively, and .theta.1 and .theta.2 are
the invariables. .beta. 0 = .mu. 0 .times. C ox .times. W eff L eff
( Eq . .times. 2 ) ##EQU2## where .mu..sub.0 is a carrier mobility,
L.sub.eff is an effective channel length, W.sub.eff is an effective
channel width, and C.sub.ox is a gate insulating film capacity. V
ds * = V ds - I ds R sd ( Eq . .times. 3 ) V gs * = V gs - I ds R
sd 2 ( Eq . .times. 4 ) ##EQU3##
[0007] Neglecting the term of .theta.2, Equation 5 is obtained from
Equations 1, 3 and 4. Supposing an external resistance R.sub.sd is
inversely proportional to an effective channel width W.sub.eff, a
channel narrowing DW can be extracted through the following
procedure. I ds = .beta. 0 ( V gs - V th - V ds 2 ) V ds 1 + (
.theta. .times. .times. 1 + .beta. 0 R sd ) ( V gs - V th ) ( Eq .
.times. 5 ) ##EQU4## where the difference between a gate voltage
and a threshold voltage, (V.sub.gs-V.sub.th), is defined as a gate
overdrive V.sub.gt.
[0008] Step 1: Against a certain gate overdrive V.sub.gt,
I.sub.ds-W.sub.m characteristic is plotted in an X-Y plane whose
X-axis is mask channel W.sub.m and Y-axis is drain current
I.sub.ds, and a linear fitting is made. At that time, the
intersection with the X-axis in the X-Y plane which is obtained by
extrapolating each straight line is the channel narrowing DW
(V.sub.gt) in the gate overdrive V.sub.gt (see FIG. 1).
[0009] Step 2: By repeating step 1 while changing the gate
overdrive V.sub.gt, it can be seen how the channel narrowing DW
(V.sub.gt) depends on the gate overdrive V.sub.gt (see FIG. 1).
[0010] Prior art characteristic evaluation method for insulated
type transistors is constructed as described. In Chia method, for
example, it is necessary to know the threshold voltage of a
transistor for use in extraction. The threshold voltage of a
transistor is found by, for example, extrapolation from the
characteristic between gate voltage and source-drain current, as
shown in FIG. 2. Therefore, the error due to the uncertainty of a
threshold voltage is further pronounced with reducing transistor
size.
SUMMARY OF THE INVENTION
[0011] According to a first aspect of the present invention, a
characteristic evaluation apparatus for insulated gate type
transistors in which at least two insulated gate type transistors
that differ from each other only in mask channel width are used for
evaluation and the characteristic of a first insulated gate type
transistor having a wide mask channel width serves as a reference,
to evaluate the characteristic of a second insulated gate type
transistor having a narrow mask channel width. This apparatus
comprises: a threshold voltage estimation means that extracts the
threshold voltage of the first transistor, estimates the threshold
voltage of the second transistor, and employs a value as estimated,
as a first estimated value; an extraction means in which (i) a
difference between a gate voltage of the first transistor and the
extracted threshold voltage of the first transistor is defined as a
first gate overdrive, and a difference between a gate voltage of
the second transistors and the first estimated value is defined as
a second gate overdrive, (ii) in an X-Y plane whose X-axis is the
mask channel width and Y-axis is source-drain conductance, a
virtual point at which a change of Y coordinate value is estimated
to be approximately zero when the first and second gate overdrives
are finely changed, is extracted from a characteristic curve
exhibiting a relationship between the mask channel widths of the
first and second transistors and the source-drain conductance,
(iii) values of the X coordinate and Y coordinate at the virtual
point are defined as second and third estimated values,
respectively, and (iv) a slope of the characteristic curve at the
virtual point is extracted and a value of the slope is employed as
a fourth estimated value; a threshold voltage determination means
in which (i) from the second to fourth estimated values, optimum
second to fourth estimated values are found with which the change
of the third estimated value is equal to the product of the change
of the second estimated value and the fourth estimated value, in
reply to fine changes of the first and second gate overdrives, (ii)
an optimum first estimated value is determined which corresponds to
the optimum second to fourth estimated values, and (iii) a true
threshold voltage of the second transistor is determined based on
the optimum first estimated value; and a channel narrowing
determination means that determines a difference between the mask
channel width and an effective channel width, based on the true
threshold voltage.
[0012] According to a second aspect, the characteristic evaluation
apparatus of the first aspect is characterized in that the
extraction means approximates the characteristic curve by using a
first straight line in the X-Y plane, the first straight line
passing through a first point that is given to the first transistor
when the first gate overdrive has a first value and a second point
that is given to the second transistor when the second gate
overdrive has the first value.
[0013] According to a third aspect, the characteristic evaluation
apparatus of the second aspect is characterized in that the
threshold voltage determination means determines the optimum second
to fourth estimated values from a relational expression: F
.function. ( .delta. , V gtWi ) = dW ** .function. ( .delta. , V
gtWi ) + f .function. ( .delta. , V gtWi ) f ' .function. ( .delta.
, V gtWi ) dW ** ' .function. ( .delta. , V gtWi ) - DW *
.function. ( .delta. , V gtWi ) ##EQU5## where .delta. is a
difference between an estimated value of the threshold voltage of
the second transistor, i.e., a first estimated value, and the
threshold voltage of the first transistor; V.sub.gtWi is the first
gate overdrive; dW** is a value of an X intercept that is obtained
by extrapolating the characteristic curve; f is the slope of the
characteristic curve at the virtual point; DW* is an X coordinate
value at the virtual point; and a prime is the first-order
differentiation of V.sub.gtWi.
[0014] According to a fourth aspect, the characteristic evaluation
apparatus of the second aspect is characterized in that the
threshold voltage determination means determines the optimum second
to fourth estimated values from a relational expression: F
.function. ( .delta. , V gtWi ) = f 2 .function. ( .delta. , V gtWi
) f ' .function. ( .delta. , V gtWi ) dW ** ' .function. ( .delta.
, V gtWi ) - G m * .function. ( .delta. , V gtWi ) ##EQU6## where
.delta. is a difference between an estimated value of the threshold
voltage of the second transistor, i.e., a first estimated value,
and the threshold voltage of the first transistor; V.sub.gtWi is
the first gate overdrive; dW** is a value of an X intercept that is
obtained by extrapolating the characteristic curve; f is the slope
of the characteristic curve at the virtual point; G.sub.m* is a Y
coordinate value at the virtual point; and a prime is the
first-order differentiation of V.sub.gtWi.
[0015] According to a fifth aspect, the characteristic evaluation
apparatus of the second aspect is characterized in that the
threshold voltage determination means determines the optimum second
to fourth estimated values from a relational expression: F
.function. ( .delta. , V gtWi ) = G m ** .function. ( .delta. , V
gtWi ) - f .function. ( .delta. , V gtWi ) f ' .function. ( .delta.
, V gtWi ) G m ** ' .function. ( .delta. , V gtWi ) - G m *
.function. ( .delta. , V gtWi ) ##EQU7## where .delta. is a
difference between an estimated value of the threshold voltage of
the second transistor, i.e., a first estimated value, and the
threshold voltage of the first transistor; V.sub.gtWi is the first
gate overdrive; G.sub.m** is a value of a Y intercept that is
obtained by extrapolating the characteristic curve; f is the slope
of the characteristic curve at the virtual point; G.sub.m* is a Y
coordinate value at the virtual point; and a prime is the
first-order differentiation of V.sub.gtWi.
[0016] According to a sixth aspect, the characteristic evaluation
apparatus of the second aspect is characterized in that the
threshold voltage determination means determines the optimum second
to fourth estimated values from a relational expression: F .times.
( .delta. , V gtWi ) = G m ** ' .function. ( .delta. , V gtWi ) f '
.function. ( .delta. , V gtWi ) + DW * .function. ( .delta. , V
gtWi ) ##EQU8## where .delta. is a difference between an estimated
value of the threshold voltage of the second transistor, i.e., a
first estimated value, and the threshold voltage of the first
transistor; V.sub.gtWi is the first gate overdrive; G.sub.m** is a
value of a Y intercept that is obtained by extrapolating the
characteristic curve; f is the slope of the characteristic curve at
the virtual point; DW* is an X coordinate value at the virtual
point; and a prime is the first-order differentiation of
V.sub.gtWi.
[0017] According to a seventh aspect, a characteristic evaluation
apparatus for insulated gate type transistors in which at least two
insulated gate type transistors that differ from each other only in
mask channel width are used for evaluation and the characteristic
of a first insulated gate type transistor having a wide mask
channel width serves as a reference, to evaluate the characteristic
of a second insulated gate type transistor having a narrow mask
channel width. This apparatus comprises: a threshold voltage
estimation means that extracts the threshold voltage of the first
transistor, estimates the threshold voltage of the second
transistor, and employs a value as estimated, as a first estimated
value; an extraction means in which (i) a difference between a gate
voltage of the first transistor and the threshold voltage of the
first transistor is defined as a first gate overdrive, and a
difference between a gate voltage of the second transistor and the
first estimated value is defined as a second gate overdrive, (ii)
in an X-Y plane whose X-axis is the mask channel width and Y-axis
is source-drain conductance, a virtual point at which a change in Y
coordinate value is estimated to be approximately zero when the
first and second gate overdrives are finely changed from a first
characteristic curve exhibiting a relationship between the mask
channel widths of the first and second transistors and the
source-drain conductance, and (iii) a value of the X coordinate at
the virtual point is employed as a second estimated value,
alternatively, as a value of the X intercept of the first
characteristic curve; a threshold voltage determination means in
which (i) from the second estimated value, an optimum first
estimated value is found with which a second characteristic curve
exhibiting a relationship between the second gate overdrive and the
second estimated value in an X-Y plane whose X-axis is the second
gate overdrive and Y-axis is a value related to the second
estimated value, has a predetermined shape within a predetermined
range of the second gate overdrive, and (ii) the optimum first
estimated value is determined as a true threshold voltage of the
second transistor; and a channel narrowing determination means that
determines a difference between the mask channel width and an
effective channel width, based on the true threshold voltage.
[0018] According to an eighth aspect, the characteristic evaluation
apparatus of the seventh aspect is characterized in that the
extraction means further employs a value of the X intercept of the
first characteristic curve as a third estimated value; and the
threshold voltage determination means employs a value that is
obtained by reducing the second estimated value from twice the
third estimated value, as the value related to the second estimated
value.
[0019] According to a ninth aspect, the characteristic evaluation
apparatus of the eighth aspect is characterized in that the
threshold voltage determination means employs the first estimated
value with which a value that is obtained by reducing the second
estimated value from twice the third estimated value is best
converged on a fixed value in the predetermined range, as the
optimum first estimated value.
[0020] According to a tenth aspect, the characteristic evaluation
apparatus of the first aspect is characterized in that the channel
narrowing determination means determines a difference between the
mask channel width and an effective channel width, from a value
that is obtained by reducing the second estimated value from twice
the third estimated value when the gate overdrive is in the
vicinity of 0 V.
[0021] According to an eleventh aspect, a characteristic evaluation
apparatus for insulated gate type transistors in which at least two
insulated gate type transistors that differ from each other only in
mask channel width are used for evaluation and the characteristic
of a first insulated gate type transistor having a wide mask
channel width serves as a reference, to evaluate the characteristic
of a second insulated gate type transistor having a narrow mask
channel width. This apparatus comprises: a threshold voltage
estimation means that extracts a threshold voltage of the first
transistor, estimates the threshold voltage of the second
transistor, and employs a value as estimated, as a first estimated
value; an extraction means in which (i) a difference between a gate
voltage of the first transistor and the extracted threshold voltage
of the first transistor is defined as a first gate overdrive, and a
difference between a gate voltage of the second transistor and the
first estimated value is defined as a second gate overdrive, (ii)
under the condition that the first and second gate overdrives are
the same in an X-Y plane whose X-axis is the mask channel width and
Y-axis is source-drain resistance, a virtual point at which a
change in Y coordinate value is estimated to be approximately zero
even if the first and second gate overdrives are finely changed, is
extracted from points on a straight line passing through a first
point whose X coordinate is the mask channel width of the first
transistor and Y coordinate is the source-drain resistance of the
second transistor, and a second point whose X coordinate is the
mask channel width of the second transistor and Y coordinate is the
source-drain resistance of the first transistor, (iii) values of
the X coordinate and Y coordinate at the virtual points are defined
as second and third estimated values, respectively, and (iv) a
slope of the straight line at the virtual points is extracted and a
value of the slope is employed as a fourth estimated value; a
threshold voltage. determination means that determines a true
threshold voltage of the second transistor by using the first to
fourth estimated values; and a channel narrowing determination
means that determines a difference between the mask channel width
and an effective channel width, based on the true threshold
voltage.
[0022] According to a twelfth aspect, in the characteristic
evaluation apparatus of the eleventh aspect the threshold voltage
determination means is characterized in: (i) finding, from the
second to fourth estimated values, optimum second to fourth
estimated values with which a change of the third estimated value
is equal to the product of a change of the second estimated value
and the fourth estimated value, in reply to fine changes of the
first and second gate overdrives, (ii) determining an optimum first
estimated value that corresponds to the optimum second to fourth
estimated values, and (iii) determining the true threshold voltage
of the second transistor, based on the optimum first estimated
value.
[0023] According to a thirteenth aspect, the characteristic
evaluation apparatus of the twelfth aspect is characterized in that
the threshold voltage determination means determines the optimum
second to fourth estimated values from a relational expression: F
.function. ( .delta. , V gtWi ) = h 2 .function. ( .delta. , V gtWi
) h ' .function. ( .delta. , V gtWi ) dW ** ' .function. ( .delta.
, V gtWi ) - R # .function. ( .delta. , V gtWi ) ##EQU9## where
.delta. is a difference between an estimated value of the threshold
voltage of the second transistor, i.e., a first estimated value,
and the threshold voltage of the first transistor; V.sub.gtWi is
the first gate overdrive; dW** is a value of an X intercept that is
obtained by extrapolating the straight line; h is the slope of the
straight line; R.sup.# is a Y coordinate value at the virtual
point; and a prime is the first-order differentiation of
V.sub.gtWi.
[0024] According to a fourteenth aspect, the characteristic
evaluation apparatus of the twelfth aspect is characterized in that
the threshold voltage determination means determines the optimum
second to fourth estimated values from a relational expression: F
.function. ( .delta. , V gtWi ) = R ** .function. ( .delta. , V
gtWi ) - h .function. ( .delta. , V gtWi ) h ' .function. ( .delta.
, V gtWi ) R ** ' .function. ( .delta. , V gtWi ) - R # .function.
( .delta. , V gtWi ) ##EQU10## where .delta. is a difference
between an estimated value of the threshold voltage of the second
transistor, i.e., a first estimated value, and the threshold
voltage of the first transistor; V.sub.gtWi is the first gate
overdrive; R** is a value of a Y intercept that is obtained by
extrapolating the straight line; h is the slope of the straight
line; R.sup.# is a Y coordinate value at the virtual point; and a
prime is the first-order differentiation of V.sub.gtWi.
[0025] According to a fifteenth aspect, the characteristic
evaluation apparatus of the twelfth aspect is characterized in that
the threshold voltage determination means determines the optimum
second to fourth estimated values from a relational expression: F
.function. ( .delta. , V gtWi ) = R ** ' .function. ( .delta. , V
gtWi ) h ' .function. ( .delta. , V gtWi ) + DW .times. #
.function. ( .delta. , V gtWi ) ##EQU11## where .delta. is a
difference between an estimated value of the threshold voltage of
the second transistor, i.e., a first estimated value, and the
threshold voltage of the first transistor; V.sub.gtWi is the first
gate overdrive; R** is a value of a Y intercept that is obtained by
extrapolating the straight line; h is the slope of the straight
line; DW.sub.# is an X coordinate value at the virtual point; and a
prime is the first-order differentiation of V.sub.gtWi.
[0026] According to a sixteenth aspect, the characteristic
evaluation apparatus of the twelfth aspect is characterized in that
the threshold voltage determination means determines the optimum
second to fourth estimated values from a relational expression: F
.function. ( .delta. , V gtWi ) = dW ** .function. ( .delta. , V
gtWi ) + h .function. ( .delta. , V gtWi ) h ' .function. ( .delta.
, V gtWi ) dW ** ' .function. ( .delta. , V gtWi ) - DW .times. #
.function. ( .delta. , V gtWi ) ##EQU12## where .delta. is a
difference between an estimated value of the threshold voltage of
the second transistor, i.e., a first estimated value, and the
threshold voltage of the first transistor; V.sub.gtWi is the first
gate overdrive; dW** is a value of an X intercept that is obtained
by extrapolating the straight line; h is the slope of the straight
line; DW.sup.# is an X coordinate value at the virtual point; and a
prime is a first-order differentiation of V.sub.gtWi.
[0027] According to a seventeenth aspect, in the characteristic
evaluation apparatus of the eleventh aspect the threshold voltage
determination means is characterized in (i) finding, in an X-Y
plane whose X-axis is the second gate overdrive and Y-axis is the
second estimated value, the optimum first estimated value with
which a characteristic curve exhibiting the relationship between
the second gate overdrive and the second estimated value has a
predetermined shape in a predetermined range of the second gate
overdrive, and (ii) determining the true threshold voltage of the
second transistor, based on the optimum first estimated value.
[0028] According to an eighteenth aspect, the characteristic
evaluation apparatus of the seventeenth aspect is characterized in
that the threshold voltage determination means estimates, from the
characteristic curve in plural, an optimum characteristic curve
with which the second estimated value is best converged on a fixed
value in the predetermined range.
[0029] According to a nineteenth aspect, the characteristic
evaluation apparatus of the eleventh aspect is characterized in
that the channel narrowing determination means determines a
difference between the mask channel width and an effective channel
width, from the second estimated value when the gate overdrive is
in the vicinity of 0 V.
[0030] The characteristic evaluation apparatus of the first or
twelfth aspect allows accurate extraction of the threshold voltage
of the second insulated gate type transistor, irrespective of the
range of the second gate overdrive, thereby improving the accuracy
of effective channel width extraction.
[0031] The characteristic evaluation apparatus of the eleventh
aspect facilitates to determine the value of channel narrowing when
the first and second gate overdrives are in the vicinity of zero
because the stationary point of the second estimated value is
present in the vicinity of zero.
[0032] The characteristic evaluation apparatus of the second aspect
facilitates the slope extraction between virtual points because a
characteristic curve is approximated to a straight line. This
allows to find a virtual point as the intersection of straight
lines, and the slope at an intersection as the slope of a straight
line.
[0033] The characteristic evaluation apparatus of the third,
fourth, fifth, sixth, thirteenth, fourteenth, fifteenth or
sixteenth aspect requires no differentiation of the gate overdrive
at a virtual point, thereby reducing errors.
[0034] The characteristic evaluation apparatus of the seventh,
eighth or seventeenth aspect facilitates to determine true
threshold voltages because the second characteristic curves that
are obtained for the true threshold voltage on a graph may
approximately coincide, irrespective of mask channel width.
[0035] The characteristic evaluation apparatus of the ninth or
eighteenth aspect facilitates programming for appropriate results
by detecting an optimum characteristic curve exhibiting the best
convergence on a fixed value.
[0036] The characteristic evaluation apparatus of the tenth or
nineteenth aspect facilitates channel narrowing determination
because the channel narrowing at the gate overdrive of 0 V is
determined by using a value that is obtained by reducing the second
estimated value from twice the third estimated value,
alternatively, because the second estimated value has a stationary
point when the gate overdrive is in the vicinity of 0 V.
[0037] To solve the above problem, it is an object of the present
invention to obtain a characteristic evaluation apparatus for
insulating gate type transistors which performs evaluation of
insulated gate type transistors by using a characteristic
evaluation method for insulated gate type transistors which reduces
the error due to the uncertainty of a threshold voltage to permit
channel narrowing extraction of high accuracy.
[0038] It is another object of the present invention to obtain a
computer readable storing medium that stores a characteristic
evaluation program.
[0039] It is another object of the present invention to obtain a
manufacturing method by which insulated gate type transistors
having excellent characteristics can be manufactured easily by
using the above characteristic evaluation method.
[0040] These and other objects, features, aspects and advantages of
the present invention will become more apparent from the following
detailed description of the present invention when taken in
conjunction with the accompanying drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
[0041] FIG. 1 is a graph for explaining an effective channel length
extraction by Chia method;
[0042] FIG. 2 is a graph for explaining threshold voltage
extraction;
[0043] FIG. 3 is a graph for explaining a virtual point, G.sub.m
intercept and W.sub.m intercept in G.sub.m method;
[0044] FIG. 4 is a flowchart giving an example of the procedure of
a characteristic evaluation method for insulated gate type
transistors according to a first preferred embodiment of the
present invention;
[0045] FIG. 5 is a graph for explaining a true shift amount
determination according to the first preferred embodiment;
[0046] FIG. 6 is a graph for explaining the relationship between
channel narrowing and W.sub.m coordinate at a virtual point;
[0047] FIG. 7 is a diagram for explaining a higher-order
narrowing;
[0048] FIG. 8 is a block diagram giving an example of the
construction of a characteristic evaluation apparatus for insulated
gate type transistors according to the first preferred
embodiment;
[0049] FIG. 9 is a conceptual diagram showing the concept in which
a calculation section in FIG. 8 is implemented by a computer;
[0050] FIG. 10 is a flowchart showing the manufacturing steps for
insulated gate type transistors which employs the characteristic
evaluation method of the first preferred embodiment;
[0051] FIG. 11 is a graph showing the relationship between mask
channel length and effective channel length in manufacturing an
insulated gate type transistor;
[0052] FIG. 12 is a graph showing the relationship between
effective channel length and threshold voltage in manufacturing an
insulated gate type transistor;
[0053] FIG. 13 is a graph for explaining the outline of a second
preferred embodiment of the present invention;
[0054] FIG. 14 is a graph showing the relationship between W.sub.m
coordinate at a virtual point and threshold voltage error;
[0055] FIG. 15 is a graph for explaining the relationship between
W.sub.m intercept and threshold voltage error;
[0056] FIG. 16 is a graph for explaining the relationship between a
value that is obtained by reducing the value of W.sub.m coordinate
at a virtual point from twice the value of W.sub.m intercept, and
threshold voltage error;
[0057] FIG. 17 is a flowchart giving an example of the procedure of
a characteristic evaluation method for insulated gate type
transistors according to the second preferred embodiment;
[0058] FIG. 18 is a block diagram giving an example of the
construction of a characteristic evaluation apparatus for insulated
gate type transistors according to the second preferred
embodiment;
[0059] FIG. 19 is a graph for explaining a virtual point, R
intercept and W.sub.m intercept in Rm method;
[0060] FIG. 20 is a flowchart giving an example of the procedure of
a characteristic evaluation method for insulated gate type
transistors according to a third preferred embodiment;
[0061] FIG. 21 is a graph for explaining a true shift amount
determination according to the third preferred embodiment;
[0062] FIG. 22 is a graph for explaining the relationship between
channel narrowing and W.sub.m coordinate at a virtual point;
[0063] FIG. 23 is a block diagram giving an example of the
construction of a characteristic evaluation apparatus for insulated
gate type transistors according to the third preferred
embodiment;
[0064] FIG. 24 is a graph for explaining the outline of a fourth
preferred embodiment;
[0065] FIG. 25 is a graph showing the relationship between W.sub.m
coordinate at a virtual point and threshold voltage error;
[0066] FIG. 26 is a flowchart giving an example of the procedure of
a characteristic evaluation method for insulated gate type
transistors according to the fourth preferred embodiment;
[0067] FIG. 27 is a block diagram giving an example of the
construction of a is characteristic evaluation apparatus for
insulated gate type transistors according to the fourth preferred
embodiment;
[0068] FIG. 28 is a graph for explaining the difference between the
channel narrowing obtained by prior art characteristic evaluation
method and the channel narrowing obtained by the characteristic
evaluation method of the first or third preferred embodiment;
and
[0069] FIG. 29 is a graph showing the relationship between gate
overdrive area set for calculation in the characteristic evaluation
method of the first or third preferred embodiment, and channel
narrowing.
DESCRIPTION OF THE PREFERRED EMBODIMENTS
First Preferred Embodiment
[0070] A characteristic evaluation method for insulated gate type
transistors according to a first preferred embodiment will be
described hereafter. In this method, a channel narrowing DW is
extracted by using the drain current in the linear areas of a
plurality of transistors, each having the same mask channel length
L.sub.m and a different mask channel width W.sub.m.
[0071] The above characteristic evaluation method will be roughed
out. Firstly there are prepared at least two MOS transistors, each
having the same channel length L.sub.m and a different mask channel
width W.sub.m. In the following description, the number of MOS
transistors is limited to two. Of the two MOS transistors, one
having a wide mask channel width W.sub.m is referred to as a wide
transistor or first insulated gate type transistor, and the other
having a narrow mask channel width W.sub.m is referred to as a
narrow transistor or second insulated gate type transistor.
Subscript Wi in symbols stands for being concerned with the wide
transistor, and subscript Na stands for being concerned with the
narrow transistor. In the prior art method that is described by
referring to FIG. 2, the threshold voltages V.sub.thWi, V.sub.thNa
of the wide transistor and narrow transistor, respectively, are
extrapolated from I.sub.ds-V.sub.gs characteristic or the like. The
threshold voltage V.sub.thNa of the second insulated gate type
transistor thus obtained is a first estimated value. By changing
the threshold voltage V.sub.thNa of the narrow transistor (the
first estimated value) with the threshold voltage V.sub.thWi of the
wide transistor fixed, a coordinate (DW*, G.sub.m*) at a virtual
point at which the change in source-drain conductance is estimated
to be approximately zero even if a gate overdrive V.sub.gt is
finely changed against each of the changed threshold voltage
V.sub.thNa, is extracted from, for example, the intersection
coordinates of a plurality of characteristic curves having a
different gate overdrive V.sub.gt. In this case, the gate overdrive
V.sub.gt of the wide transistor is a first gate overdrive, and the
gate overdrive V.sub.gt of the narrow transistor is a second gate
overdrive. The coordinate DW*, coordinate G.sub.m* and slope f at
the virtual point are second, third and fourth estimated values,
respectively.
[0072] Then, by using the threshold voltages V.sub.thWi and
V.sub.thNa, the coordinate (DW*, G.sub.m*) at the virtual point is
extracted from the relationship between conductance G.sub.m and
mask channel width W.sub.m. Examples of this method is, as shown in
FIG. 3 in prior art, one in which two characteristic curves
(straight lines) representing the characteristic G.sub.m-W.sub.m
are drawn in a graph whose X-axis is mask channel width W.sub.m and
Y-axis is source-drain conductance G.sub.m, and the intersection of
the two straight lines is found to extract a virtual point. In FIG.
3, the straight line expressing the gate overdrive V.sub.gt is a
first straight line, the point that satisfies the mask channel
width W.sub.m=W.sub.mWi on the first straight line is a first
point, and the point that satisfies the mask channel width
W.sub.m=W.sub.mNa on the first straight line is a second point.
However, the estimation of the coordinate at a virtual point is not
limited to the above. Instead of a straight line passing through
two points, a curve to be determined by three or more points may be
used. Alternatively, a point in the vicinity of an intersection may
be used instead of the intersection. From among the values of a
coordinate (DW*, G.sub.m*) which express the extracted virtual
point, there is determined the value with which the change in the
value G.sub.m of the Y component of the coordinate expressing a
virtual point is estimated to be equal to the product of the change
of the value DW* of the X component of the virtual point and the
channel resistance value f per unit width.
[0073] Extraction of an effective channel width W.sub.eff in MOS
transistors will be described in detail by referring to FIG. 4.
[0074] Firstly, the I.sub.ds-V.sub.gs characteristics of two
transistors Wi and Na, each having the same mask channel length
L.sub.m and a different mask channel width W.sub.m, are measured
(step ST1.1).
[0075] From the obtained I.sub.ds-V.sub.gs characteristics, the
threshold voltages V.sub.thWi of a wide transistor and V.sub.thNa
of a narrow transistor are extracted by using extrapolation method
or the like (step ST1.2). Then, the difference
(V.sub.thNa-V.sub.thWi) between the threshold voltages V.sub.thWi
and V.sub.thNa is found. Hereafter, the difference
(V.sub.thNa-V.sub.thWi) thus found is defined as
.delta..sub.guess.
[0076] The lower and upper limits of an area in which the value
.delta. to be set as a threshold voltage difference is changed are
determined as .delta..sub.inf=.delta..sub.guess-K, and
.delta..sub.sup=.delta..sub.guess+K, respectively (step ST1.3).
Here, let K be 0.2 V, and .delta.=.delta..sub.inf is set as an
initial value.
[0077] Then, it is determined whether the value .delta. to be
calculated is present between .delta..sub.inf and .delta..sub.sup
(step ST1.4). That is, it is determined whether
.delta..sub.inf.ltoreq..delta..ltoreq..delta..sub.sup.
[0078] When the value .delta. is present between .delta..sub.inf
and .delta..sub.sup the threshold voltage V.sub.thWi of the wide
transistor is fixed to the value that has been extracted in step
ST1.2, and the threshold voltage V.sub.thNa of the narrow
transistor is supposed to be the sum of the threshold voltage
V.sub.thWi of the wide transistor and the .delta. (step ST1.5).
[0079] On the basis of the threshold voltage V.sub.thWi and
V.sub.thWi+.delta. in step ST1.5, a gate overdrive V.sub.gt is
measured. For about 20 points in a certain area .OMEGA., e.g., in
the range of the gate overdrive V.sub.gt satisfying 0.3
V.ltoreq.V.sub.gt.ltoreq.1.3 V, there are found the rate of change
DW*'(.delta.-V.sub.gtn) in the value of W.sub.m coordinate at a
virtual point, the rate of change G.sub.m*'(.delta., V.sub.gtn) in
the value of G.sub.m coordinate at a virtual point, and the
conductance f (.delta., V.sub.gtn) per unit width. From the values
thus found, the value of function F(.delta., V.sub.gtn) expressed
by Equation 6 is found. F(.delta., V.sub.gtn)=G.sub.m*'-fDW*'
[0080] where n=1, 2, . . . 20.
[0081] Next, the standard deviation of function F, .sigma.
[F(.delta.)], is calculated in the area .OMEGA. (step ST1.7). By
substituting .delta.+Q for .delta., the value of a shift amount
.delta. is changed to return to step ST1.4 (step ST1.8). Let the
value of Q be 0.01, for example.
[0082] When it is determined
.delta..sub.inf.ltoreq..delta..ltoreq..delta..sub.sup in step
ST1.4, steps ST1.5 to ST1.8 are repeated. On the other hand, when
it is not determined
.delta..sub.inf.ltoreq..delta..ltoreq..delta..sub.sup in step
ST1.4, it goes to step ST1.9 and find .delta.=.delta..sub.0, with
which the standard deviation .sigma. [F(.delta.)] becomes a
minimum. At that time, the true threshold voltage V.sub.thNa of the
narrow transistor is given by the sum of the threshold voltage
V.sub.thWi of the wide transistor and the .delta..sub.0 that has
been determined in step ST1.9.
[0083] Using the true threshold voltage V.sub.thWi+.delta..sub.0 of
the narrow transistor that has been determined in step ST1.9, the
gate overdrive V.sub.gt of the narrow transistor is measured to
find the value DW*(V.sub.gt) of W.sub.m coordinate at a virtual
point (step ST1.10). The threshold voltage V.sub.thWi of the wide
transistor at that time is based on the value that has been found
in step ST1.2, as in step ST1.5.
[0084] Let the channel narrowing DW.sub.Na of the narrow transistor
be DW.sub.Na(V.sub.gt)=dW**(V.sub.gt), where dW** is an optimum
second estimated value (step ST1.11). At the same time an effective
channel width W.sub.effNa is given by the following Equation 7.
Hereat, G.sub.m* that is obtained by using a gate overdrive
V.sub.gt providing a channel narrowing DW is an optimum third
estimated value. Further, an optimum fourth estimated value is the
conductance f of the channel per unit width which is obtained by
using a gate overdrive V.sub.gt providing a channel narrowing DW.
WeffNa(V.sub.gt)=W.sub.mNa-DW.sub.Na(V.sub.gt) (Eq. 7)
[0085] Although in step ST1.11, the channel narrowing DW.sub.Na is
determined from dW**, the channel narrowing DW(V.sub.gt) when a
gate overdrive V.sub.gt is in the vicinity of zero may be
determined as a value (2dW**-DW*), which is given from W.sub.m
coordinate at an intersection and W.sub.m intercept. In this case,
when the gate overdrive V.sub.gt is in the vicinity of zero, the
change of (2dW** -DW*) against the change of gate overdrive
V.sub.gt is extremely small, thus making it easy to determine a
channel narrowing DW.sub.Na.
[0086] Description will be now given of a concrete procedure to
determine a channel narrowing DW and the like, from the standard
deviation of the function F shown in Equation 6. In a
characteristic evaluation method for insulated gate type
transistors according to the first preferred embodiment, to reduce
the uncertainty of threshold voltage extrapolation and, in
particular, the error due to the uncertainty of threshold voltage
extrapolation for transistors having a narrow channel width, the
relationship of Equation 8 which is, for example, established
between the value DW* of W.sub.m coordinate at a virtual point and
the value dW**, is noted to apply a variation method. Here, dW** is
the value of X intercept that is obtained by extrapolating a
G.sub.m-W.sub.m characteristic curve (straight line) which is
plotted between source-drain conductance G.sub.m and mask channel
width W.sub.m, by using G.sub.m to enter a Y-axis and W.sub.m to
enter an X-axis. Hereinafter, dW** may be taken to represent the
value of W.sub.m intercept. dW ** + f f ' .times. dW ** ' - DW * =
0 ( Eq . .times. 8 ) ##EQU13##
[0087] Supposing that the threshold voltage difference between the
narrow transistor and the wide transistor is a shift amount
.delta., the value DW* of W.sub.m coordinate at a virtual point,
the value dW** of W.sub.m intercept and its rate of change dW**',
as well as the channel conductance f per unit width and its rate of
change f', are found from
G.sub.mNa(V.sub.gtWi+.delta.-V.sub.thNa+V.sub.thWi) and
G.sub.mWi(V.sub.gtWi). When a shift amount .delta. is equal to the
true threshold voltage difference .delta..sub.0 between the narrow
transistor and the wide transistor, Equation 8 is satisfied. At
that time, dW** gives a channel narrowing DW. Therefore, a channel
narrowing DW can be extracted through the following procedure.
[0088] Firstly, with respect to a certain shift amount .delta., the
value DW* of W.sub.m coordinate at a virtual point, the value dW**
of W.sub.m intercept, and the channel conductance f per unit width
are given by Equations 9 to 11. DW * = W mNa - ri W mWi ( 1 - ri )
( Eq . .times. 9 ) dW ** = W mNa - rai W mWi ( 1 - rai ) ( Eq .
.times. 10 ) f .function. ( V gtWi , .delta. ) = .times. G mWi
.times. .times. ( V gtWi ) - G mNa ( V thWi + .delta. - V gtNa + V
thWi ) W mWi - W mNa ( Eq . .times. 11 ) ##EQU14##
[0089] In Equations 9 to 11, parameters ri and rai are defined by
the following Equations 12 and 13, respectively, and V.sub.gtWi
denotes a gate overdrive on the basis of the threshold voltage
V.sub.thWi of a wide transistor having a wide mask channel width
W.sub.mWi. ri .function. ( V gtwi , .delta. ) .ident. G mNa '
.function. ( V gtWi + .delta. - V thNa + V thWi ) G mWi '
.function. ( V gtWi ) ( Eq . .times. 12 ) rai .function. ( V gtwi ,
.delta. ) .ident. G mNa .function. ( V gtwi + .delta. - V thNa + V
thWi ) G mWi .function. ( V gtWi ) ( Eq . .times. 13 )
##EQU15##
[0090] The value DW* of W.sub.m coordinate at a virtual point, the
value dW** of W.sub.m intercept and its rate of change dW**', as
well as the channel conductance f per unit width and its rate of
change f', are found by changing a shift amount .delta..
[0091] The function F in Equation 6 can be modified to redefine as
the following Equation 14, making it easy to find the function F.
When a shift amount .delta. is equal to a threshold voltage
difference .delta..sub.0 between the narrow transistor and the wide
transistor, the function F defined in Equation 14 becomes zero,
irrespective of a gate overdrive V.sub.gtWi. Then, a shift amount
.delta. with which the standard deviation of function F in an area
of gate overdrive V.sub.gtWi becomes a minimum, is determined as a
true threshold voltage difference .delta..sub.0. F .function. ( V
gtWi , .delta. ) = dW ** .function. ( V gtWi , .delta. ) + f
.function. ( V gtWi , .delta. ) f ' .function. ( V gtWi , .delta. )
dW ** ' .function. ( V gtWi , .delta. ) - DW * .function. ( V gtWi
, .delta. ) ( Eq . .times. 14 ) ##EQU16##
[0092] FIG. 5 is a graph showing one example of the relationship
between the standard is deviation of function F and shift amount
.delta.. In this graph, a minimum value is obtained when the shift
amount .delta. is -0.06 V, thus let a true threshold voltage
difference .delta..sub.0 be -0.06 V.
[0093] The value of a channel narrowing DW is determined by using
the value of the above threshold voltage difference .delta..sub.0.
For instance, it may be determined in the same manner as in step
ST1.11 of FIG. 4. Alternatively, the average of values obtained
when the gate overdrive V.sub.gt is in the vicinity of zero, among
the values DW*(.delta..sub.0, V.sub.gt) of W.sub.m coordinate at a
virtual point, may be taken as the value of a channel narrowing DW.
FIG. 6 gives an example of the results when the characteristic
evaluation method of MOS transistors according to the first
preferred embodiment (hereinafter referred to as Gm method) is
applied to a process.
[0094] Instead of Equation 14 that is used in the first preferred
embodiment, any one of Equations 15 to 17 may be used to find
function F. F .function. ( .delta. , V gtWi ) = f 2 .function. (
.delta. , V gtWi ) f ' .function. ( .delta. , V gtWi ) dW **
.function. ( .delta. , V gtWi ) - G m * .function. ( .delta. , V
gtWi ) ( Eq . .times. 15 ) F .function. ( .delta. , V gtWi ) = G m
** .function. ( .delta. , V gtWi ) - f .function. ( .delta. , V
gtWi ) f ' .function. ( .delta. , V gtWi ) G m ** ' .function. (
.delta. , V gtWi ) - G m * .function. ( .delta. , V gtWi ) ( Eq .
.times. 16 ) F .function. ( .delta. , V gtWi ) = G m ** '
.function. ( .delta. , V gtWi ) f ' .function. ( .delta. , V gtWi )
+ DW * .function. ( .delta. , V gtWi ) ( Eq . .times. 17 )
##EQU17##
[0095] In Equations 16 and 17, G.sub.m** is the value of a R
intercept that is obtained by extrapolating G.sub.m-W.sub.m
characteristic. Thus, by using mask channel width W.sub.m to enter
the X-axis and source-drain conductance G.sub.m to enter the
Y-axis, without using any coordinate at a virtual point, a
G.sub.m-W.sub.m characteristic curve (straight line) is
extrapolated to obtain the value G.sub.m** of a Y intercept and the
value dW** of a Y intercept which are found as X=0 and Y=0,
respectively. The use of the value G.sub.m** or dW** requires no
differentiation of the coordinate (DW*, R*) at a virtual point. The
accuracy is unchanged by using any one of Equations 14 to 17.
Equation 15 and 16, however, call for calculation of G.sub.m**.
Hence, Equation 14 or 17 is preferred.
[0096] Although in the first preferred embodiment, a shift amount
.delta. is determined by a value with which the standard deviation
of function F becomes a minimum, it can be determined by a value
with which the average value of functions F approaches zero, or the
minimum value of the sum of squares(.SIGMA.F.sup.2) of function F.
The above alternatives, however, might have the errors due to the
offset of the value of function F, which are caused by calculation
errors, unlike the value with which the standard deviation becomes
a minimum.
[0097] Moreover, the first preferred embodiment employs
G.sub.mNa'/G.sub.mWi' in Equation 12, for example, to improve
calculation accuracy in finding the value DW* of W.sub.m coordinate
at a virtual point. On the other hand, if easy process is desired,
a higher accuracy calculation than prior art is attainable by using
.delta.G.sub.mNa/.delta.G.sub.mWi, instead of
G.sub.mNa'/G.sub.mWi'. High-accuracy channel narrowing DW
extraction is also attainable by high-accuracy calculation of the
change in the source-drain conductance G.sub.m of a wide transistor
or narrow transistor, by means of a higher-order approximate
expression. For instance, the slope of a curve at y.sub.0 among
points that are equally spaced with a width s, as shown in FIG. 7,
can be given by a higher-order approximate expression in the
following Equation 18. y 0 ' = 1 12 h .times. ( y - 2 - 8 y - 1 + 8
y 1 - y 2 ) ( Eq . .times. 18 ) ##EQU18##
[0098] The use of the characteristic evaluation method for
insulated gate type transistors in the first preferred embodiment
permits evaluation at higher accuracy than prior art. As a result,
improvement of accuracy owing to use of G.sub.mNa'/G.sub.mWi' is
satisfactorily reflected on evaluation results than prior art.
[0099] In the calculation of G.sub.mNa'/G.sub.mWi' by G.sub.m
method, to reduce errors, resistance R is sometimes used instead of
conductance G.sub.m, as shown in Equation 19. The reason for using
the differentiation of the logarithm of resistance R is to reduce
the error due to a great change in resistance R when V.sub.gt is
brought near zero. G mNa ' G mWi ' = ( 1 / R Na ) ' ( 1 / R Wi ) '
= R Na ' R Wi ' R Wi 2 R Na 2 = ( .times. .times. n .times. .times.
R Na ) ' ( .times. .times. n .times. .times. R Wi ) ' R Wi R Na (
Eq . .times. 19 ) ##EQU19##
[0100] Description will be now given of a characteristic evaluation
apparatus for insulated gate type transistors according to the
first preferred embodiment, by referring to FIG. 8. A
characteristic evaluation apparatus for insulated gate type
transistors 1 is connected to a measuring device 3 for measuring an
object under test 2. Examples of the object under test 2 are
integrated circuits in which a wide transistor and a narrow
transistor are formed. Such an integrated circuit after being
extracted from the lot for which all manufacturing steps have been
terminated, is set to the measuring device 3 to make measurement
therefor. The measuring device 3 is controlled by a control section
4 of the characteristic evaluation apparatus 1. An input section 5
provides the control section 4 with control information. The input
section 5 is composed of a keyboard, a mouse and the like.
Measurement data obtained in the measuring device 3 is inputted to
a calculation section 6 together with the control information,
through the control section 4. The calculation section 6 extracts
an effective channel width W.sub.eff, based on the data to be
inputted from the input section 5. An output section 7 outputs the
extracted effective channel width W.sub.eff and the control
information used in the middle of extraction. Such control
information is provided from the control section 4 or calculation
section 5.
[0101] The calculation section 6 is composed of a threshold voltage
and virtual shift amount determination section 11 that determines
threshold voltages V.sub.thWi, V.sub.thNa, and virtual shift amount
.delta.; an extraction section 12 that extracts an intersection
coordinate (DW*, G.sub.m*) as the coordinate at a virtual point,
and the slope f of a characteristic curve at the intersection
coordinate; a true shift amount determination section 13 for
determining a true shift amount .delta..sub.0, and a channel
narrowing determination section 14 for determining a channel
narrowing DW (or an effective channel width W.sub.eff). Although in
this embodiment, an intersection coordinate is used as the
coordinate at a virtual point at which the change of source-drain
conductance G.sub.m is supposed to be approximately zero even if
the gate overdrive V.sub.gt is finely changed in a W.sub.m-G.sub.m
characteristic curve. The intersection coordinate may be found by
other than the method of finding an intersection, alternatively,
other point may be used as the coordinate at a virtual point, as
previously discussed. For executing calculation in the calculation
section 6, the value of a variable K for determining the upper
limit .delta..sub.sup and lower limit .delta..sub.inf in the range
of changing a shift amount .delta., the range of area .OMEGA. in
which a gate overdrive V.sub.gt is measured, and the quantity of
change Q of a virtual shift amount .delta., are inputted to the
threshold voltage and virtual shift amount determination section 11
from the input section 5. The measurement data of source-drain
current Ids and gate-source voltage V.sub.gt are provided to the
threshold voltage and virtual shift amount determination section
11, from the control section 4. The determination section 11
receives the above data, and then provides the extraction section
12 with the threshold voltage V.sub.thWi of a wide transistor and a
virtual shift amount .delta. that indicates the difference between
this threshold voltage V.sub.thWi and the threshold voltage
V.sub.thNa of a narrow transistor. In the extraction section 12,
with respect to each shift amount .delta., the rate of change
dDW*/dV.sub.gt and that of dG.sub.m/dV.sub.gt for an intersection
coordinate (DW*, G.sub.m*) in an area .OMEGA., and the slope f of a
characteristic curve are extracted by using the value of the mask
channel width W.sub.m provided from the input section 5, as well as
the source-drain current I.sub.ds and the measurement data of
gate-source voltage V.sub.gt. From the rate of change
dDW*/dV.sub.gt of W.sub.m coordinate of the intersection, the rate
of change dG.sub.m/dV.sub.gt of R coordinate of the intersection,
and the slope f of the characteristic curve which have been
extracted in the extraction section 12, the true shift amount
determination section 13 determines a virtual shift amount
.delta..sub.0 with which the standard deviation of the function F
expressed in Equation 6 becomes a minimum in an area .OMEGA.. Upon
determination of a virtual shift amount .delta..sub.0, the
extraction section 12 outputs the virtual shift amount
.delta..sub.0 and the value DW* of W.sub.m coordinate of the
corresponding intersection or the value dW** of W.sub.m intercept,
to a channel narrowing determination section 14. In the section 14,
a channel narrowing DW is determined from the value dW** of W.sub.m
intercept or the value DW* of W.sub.m coordinate at a virtual
point, and the calculation expressed in Equation 7 is carried out
to determine an effective channel width W.sub.eff. The output
section 7 outputs the channel narrowing DW and the effective
channel width W.sub.eff determined in the channel narrowing
determination section 14, the intersection coordinate (DW*,
G.sub.m*) and the slope of a characteristic curve at the
intersection coordinate extracted in the extraction section 12, and
the true shift amount .delta..sub.0 determined in the true shift
amount determination section 13.
[0102] With the above construction, it is possible to obtain a
characteristic evaluation apparatus for insulated gate type
transistors which extracts an effective channel width W.sub.eff at
a higher accuracy than prior art.
[0103] Referring to FIG. 9, the characteristic evaluation for
insulated gate type transistors as described in the first preferred
embodiment can be realized by making a computer to read an
evaluation program 30 for evaluating insulated gate type
transistors from a recording medium storing the program 30, in
accordance with the procedure in FIG. 4 as described in the first
preferred embodiment. By executing the evaluation program 30, a
measurement data 33 containing data related to an effective channel
width W.sub.eff can be extracted on the basis of a measurement data
31 provided from a measuring device 3 and a control information 32
from an input section 5 in FIG. 8, as described in the first
preferred embodiment.
[0104] Description will be now given of a method of manufacturing
an insulated gate type transistor according to the first preferred
embodiment, by referring to FIG. 10. Firstly, a target narrow
transistor and a reference wide transistor are prepared (step
ST50). Then, the electrical characteristics of both transistors are
measured (step ST51). In this step, the I.sub.ds-V.sub.gs
characteristic, off leak current I.sub.off and drain current
I.sub.dmax of each transistor are measured. The off leak current
I.sub.off is the current that flows between source and drain when,
for example, V.sub.ds=VDD and V.sub.gs=V.sub.bs=0 V, where VDD is
power supply voltage.
[0105] By the characteristic evaluation method for insulated gate
type transistors as described in the first preferred embodiment,
the threshold voltage V.sub.thNa and effective channel width
W.sub.effNa of the narrow transistor are extracted from
I.sub.ds-V.sub.gs characteristic or the like. Then, it is
determined whether the threshold voltage V.sub.thNa, effective
channel width W.sub.effNa, current I.sub.dmax, and current
I.sub.off of the narrow transistor satisfy a specification (step
ST53). If not, it returns to step ST50 to perform another
preparation of transistors by using a new mask.
[0106] Thus, the characteristic evaluation method for insulated
gate type transistors according to the first preferred embodiment
produces the following effects. Firstly, since the threshold
voltage is determined accurately from a known mask channel width
and electrical characteristics, the time required for manufacturing
is reduced, compared to the case where the section of an insulated
gate type transistor is observed with an electron microscope or the
like. Secondly, in response to a gate overdrive V.sub.gt, the range
of an effective channel width W.sub.eff in the desired mask channel
width W.sub.m is found accurately (see FIG. 11). Thirdly, the
variable range of the threshold voltage V.sub.th that corresponds
to the variable range of an effective channel width W.sub.eff is
found accurately at the same time (see FIG. 12), thus facilitating
the quality control of the threshold voltage V.sub.th in
manufacturing steps.
Second Preferred Embodiment
[0107] Description will be given of the outline of a characteristic
evaluation method for insulated gate type transistors according to
a second preferred embodiment, by referring to FIG. 13. FIG. 13 is
a graph showing the relationship between the value of (2dW**-DW*)
and gate overdrive V.sub.gt which are obtained by the
characteristic evaluation method for insulated gate type
transistors according to the second preferred embodiment.
Specifically, this graph shows the change in the value of
(2dW**-DW*) when a true threshold voltage is used for three narrow
transistors which differ one another in mask channel width
W.sub.mNa. Note that the mask channel width W.sub.mWi of a wide
transistor which serves as a reference in extracting the values
dW** and DW* of W.sub.m coordinate of these narrow transistors, is
set to the same value. A comparison of FIG. 13 with FIGS. 14 to 16
indicates that when used a true threshold voltage, the change in
the value of (2.about.dW**-DW*) against the gate overdrive V.sub.gt
is approximately the same, irrespective of the mask channel widths
W.sub.mNa of the narrow transistors. Therefore, the true threshold
voltage of a narrow transistor can be extracted by finding out one
which coincides with the characteristic curve of this graph when
the value of a gate overdrive V.sub.gt is, for example, in the
range of 0.3-1.2 V. In the second preferred embodiment, first and
second insulated gate type transistors, first and second gate
overdrives, and first and second estimated values, are also defined
as in the first preferred embodiment.
[0108] Description will be now given of an example of a
characteristic evaluation method for insulated gate type
transistors according to the second preferred embodiment. In this
method, the characteristic curve of FIG. 13 is extracted from
characteristic curves that change variously depending on the
estimated value of the threshold voltage V.sub.thNa of a narrow
transistor, namely, a first estimated value, by making use of the
fact that the standard deviations of the characteristic curves are
small in the range of 0.2-0.6 V, for example. Since in this method
the true threshold voltage of a narrow transistor is determined by
utilizing the dependence of (2dW**--DW*) on a gate overdrive
V.sub.gt, it is determined in a procedure similar to that of the
first preferred embodiment.
[0109] One example of the extraction procedure of an effective
channel width W.sub.eff in the second preferred embodiment is given
in FIG. 17. The extraction procedure of the second preferred
embodiment is different from that of the first preferred embodiment
in steps ST1.12, ST1.13 and ST1.14 to ST1.16 in FIG. 17, which
correspond to steps ST1.6, ST1.7 and ST1.9 to ST1.11 in FIG. 4,
respectively.
[0110] In step ST1.12, the value of 2dW**-DW* against, for example,
about 20 different gate overdrives V.sub.gtn are found by using the
values of W.sub.m coordinate and W.sub.m intercept. In step ST1.13,
there are calculated the average value <2dW**-DW*> and
standard deviation .sigma. [2dW**-DW*] of a value that is obtained
by reducing the value DW* of W.sub.m coordinate at a virtual point
from twice of the value dW** of W.sub.m intercept for a shift
amount .delta..
[0111] When it is judged that in step ST1.13, the calculation of a
shift amount .delta. in a predetermined range of .delta..sub.inf to
.delta..sub.sup is terminated (step ST1.4), a true shift amount
.delta..sub.0 that gives a channel narrowing DW is estimated in
step ST1.14. The true shift amount .delta..sub.0 is a shift amount
.delta..sub.0 with which a standard deviation .sigma. [2dW**-DW*]
becomes a minimum. This means that the choice of a characteristic
curve whose values are best converged on a fixed value. In step
ST1.15, a channel narrowing DW is given by, for example, the
average of the values DW* of W.sub.m coordinate at a virtual point
for a shift amount .delta..sub.0. In step ST1.16, an effective
channel width W.sub.eff is determined from the difference between a
mask channel width W.sub.m and the channel narrowing DW.
[0112] Referring to FIG. 18, a characteristic evaluation apparatus
for insulated gate type transistors according to the second
preferred embodiment will be described. A characteristic evaluation
apparatus for insulated gate type transistors 1A shown in FIG. 18
is connected to a measuring device 3 for measuring an object under
test 2, like the characteristic evaluation apparatus 1 of the first
preferred embodiment as shown in FIG. 8. In the construction of the
characteristic evaluation apparatus 1A, the same reference numerals
have been retained for similar parts which have the same functions
as in the apparatus 1 of FIG. 8. That is, the characteristic
evaluation apparatus 1A has the same structure as the apparatus 1,
except for an extraction section 12A, a true shift amount
determination section 13A and a channel narrowing determination
section 14A in a calculation section 6A. The extraction section 12A
finds (2dW**-DW*) by changing a gate overdrive V.sub.gt in an area
.OMEGA.. In the true shift amount determination section 13A, a
value with which the standard deviation .sigma. [2dW**-DW*] becomes
a minimum, is found from the value DW* of W.sub.m coordinate of an
intersection and the value dW* of W.sub.m intercept in the area
.OMEGA., to determine a true shift amount .delta..sub.0. The
extraction section 12A outputs the true shift amount .delta..sub.0
and the value DW* of W.sub.m coordinate of the corresponding
intersection or the value dW* of W.sub.m intercept, to the channel
narrowing determination section 14A. The section 14A determines a
channel narrowing DW from the average of (2dW**-DW*) when the gate
overdrive V.sub.gt is in the vicinity of 0 V, e.g., in the range of
0.2.ltoreq.V.sub.gt.ltoreq.0.6, in an area .OMEGA. for a true shift
amount .delta..sub.0, alternatively, from the value dW** of W.sub.m
intercept. In the second preferred embodiment, a value with which
the standard deviation .sigma. [2dW**-DW*] of the value (2dW**-DW*)
becomes a minimum, or a value with which the standard deviation
.sigma. [dW**] of the value dW** of W.sub.m intercept becomes a
minimum, is determined as a channel narrowing DW. Its determination
method is, however, not limited to the above, and the threshold
voltage V.sub.thNa of a narrow transistor may be determined by
selecting a characteristic curve in which the value dW** of W.sub.m
intercept or the value of (2dW**-DW*) is best converged on a fixed
value when a gate overdrive V.sub.gt is within a predetermined
range.
[0113] A method of manufacturing an insulted gate type transistor
according to the second preferred embodiment can be implemented by
employing, in step ST52 shown in FIG. 10, the evaluation method of
the second preferred embodiment in place of that of the first
preferred embodiment. This results in the same effects as in the
case where the evaluation method of the first preferred embodiment
is applied to a manufacturing method.
[0114] Referring again to FIG. 9, the characteristic evaluation for
insulated gate type transistors as described in the second
preferred embodiment is attainable by making a computer to read an
evaluation program 30 for evaluating insulated gate type
transistors from a recording medium storing the program 30, in
accordance with the procedure in FIG. 17 as described in the second
preferred embodiment.
[0115] In the channel narrowing DW extraction according to the
first or second preferred embodiment, when the mask cannel width
W.sub.mNa of a narrow transistor is significantly smaller than the
mask cannel width W.sub.mWi of a wide transistor (i.e.,
W.sub.mNa<<W.sub.mWi), the difference between the mask
channel width W.sub.mwi and a gate finished width W.sub.gWi hardly
affects on determination of the value DW* of W.sub.m coordinate at
a virtual point, thereby determines the channel narrowing DW of the
narrow transistor at high accuracy. For instance, to evaluate
device or circuit performance on the level of not more than 1.0
.mu.m in pattern width, it is required to extract the channel
narrowing DW of each transistor. For such an extraction, there are
used two transistors, i.e., a narrow transistor and a wide
transistor serving as a reference. In this case, the difference
between a gate finished width W.sub.g and a mask channel width
W.sub.m depends on the transistor, causing an error. Thus,
description will be now given of such an error. The value dW** of
W.sub.m coordinate at a virtual point when a mask channel width
W.sub.m is used is given by Equation 20. dW ** .function. ( V gt )
= ( W mNa - G mNa G mWi W mWi ) ( 1 - G mNa G mWi ) - 1 ( Eq .
.times. 20 ) ##EQU20##
[0116] If W.sub.g intercept in a plane formed by gate finished
width and source-drain conductance (i.e., W.sub.g-G.sub.m plane),
is represented by dW.sub.g**, Equation 21 is obtained. dW g **
.function. ( V gt ) = ( W gNa - G mNa G mWi W gWi ) ( 1 - G mNa G
mWi ) - 1 ( Eq . .times. 21 ) ##EQU21##
[0117] If the difference between a gate finished width W.sub.g and
a mask channel width W.sub.m is represented by .delta.W, the
difference between the gate finished width W.sub.gWi and mask
channel width W.sub.mWi of a wide transistor, and the difference
between the gate finished width W.sub.gNa and mask channel width
W.sub.nNa of a narrow transistor are represented by .delta.W.sub.wi
and .DELTA.W.sub.Na, respectively, thus the relationships of
Equations 22 and 23 are established. From Equations 20 to 23, the
difference between the coordinate value dW** of W.sub.m intercept
and the coordinate value DW.sub.g* of W.sub.g intercept is
expressed by Equation 24, where .DELTA.W is defined in Equation 25.
W gWi = W mWi + .DELTA. .times. .times. W Wi ( Eq . .times. 22 ) W
gNa = W mNa + .DELTA. .times. .times. W Na ( Eq . .times. 23 ) dW
** - dW g ** = .times. - .DELTA. .times. .times. W Na + G mNa G mWi
( 1 - G mNa G mWi ) - 1 .DELTA. .times. .times. W .apprxeq. .times.
- .DELTA. .times. .times. W Na + G mNa G mWi .DELTA. .times.
.times. W .apprxeq. .times. - .DELTA. .times. .times. W Na + W
effNa W effWi .DELTA. .times. .times. W ( Eq . .times. 24 ) .DELTA.
.times. .times. W .ident. .DELTA. .times. .times. W Wi - .DELTA.
.times. .times. W Na ( Eq . .times. 25 ) ##EQU22##
[0118] Equations 23 and 24 show that the effective channel width
W.sub.eff of a narrow transistor is extracted when the relationship
W.sub.mNa<<W.sub.mWi is established. In Equation 24, the
second term of the last expression indicates an error. If a
relative error is represented by r, Equation 26 is obtained. Then,
let be W.sub.gWi.apprxeq.W.sub.mWi, Equation 26 is modified into
Equation 27. W effNa W effWi .DELTA. .times. .times. W < r W
effNa ( Eq . .times. 26 ) W mWi > .DELTA. .times. .times. W r (
Eq . .times. 27 ) ##EQU23##
[0119] Equation 27 imposes limitations upon the size of a wide
transistor. For instance, when .DELTA.W=0.1 .mu.m and r=0.02, the
mask channel width W.sub.mWi of a wide transistor is required to be
greater than 5 .mu.m, in order to accurately extract the effective
channel width of a narrow transistor.
[0120] Also, in the case where a channel narrowing DW is determined
from (2dW**-DW*), it is desirable to determine a mask channel width
W.sub.mWi in a similar manner.
Third Preferred Embodiment
[0121] A characteristic evaluation method for insulated gate type
transistors according to a third preferred embodiment will be
described hereafter. In this method, a channel narrowing DW is
extracted by using the drain currents of linear areas in two
insulated gate type transistors that have the same mask channel
length L.sub.m and a different mask channel width W.sub.m.
[0122] The above characteristic evaluation method in the third
preferred embodiment will be roughed out. As in the first preferred
embodiment, there are firstly prepared two MOS transistors, each
having the same channel length L.sub.m and a different mask channel
width W.sub.m. Then, the threshold voltage V.sub.thWi of a wide
transistor and the threshold voltage V.sub.thNa of a narrow
transistor are extrapolated from I.sub.ds-V.sub.gs characteristic
or the like. The threshold voltage V.sub.thNa thus extracted is a
first estimated value. Under the conditions that the gate overdrive
V.sub.gt of the wide transistor, i.e., a first gate overdrive, is
equal to the gate overdrive V.sub.gt of the narrow transistor,
i.e., a second gate overdrive, a virtual point as described later
is extracted in an X-Y plane whose X-axis is mask channel width
W.sub.m and Y-axis is source-drain resistance R. This virtual point
is not present as an actual measuring point, but is a virtual point
on a straight line that passes through a first point whose
X-coordinate is the mask channel width W.sub.nWi of the wide
transistor and Y-coordinate is the source-drain resistance R.sub.Na
of the narrow transistor, and a second point whose X-coordinate is
the mask channel width W.sub.mNa of the narrow transistor and
Y-coordinate is the source-drain resistance R.sub.Wi of the wide
transistor. Such a virtual point has the characteristic feature
that the change in source-drain resistance is approximately zero
even when the first and second gate overdrives are finely changed.
Therefore, as shown in FIG. 19, this virtual point is found as the
intersection of two straight lines exhibiting the difference of
.delta. V.sub.gt between the first and second gate overdrives. The
X-coordinate (W.sub.m coordinate) and Y-coordinate of the above
intersection are represented by DW.sup.# and R.sup.#, respectively.
Note that the straight lines in the third preferred embodiment
contain curves that can be approximated to a straight line. In the
event that a virtual point is located slightly apart from the
straight lines, a point in the vicinity of an intersection may be
used.
[0123] The relationship of Equation 28 is established between the
intersection coordinate (R.sup.#, DW.sup.#) and the slope h of a
straight line in FIG. 19. In Equation 28, a prime indicates the
first-order differentiation of V.sub.gt. R.sup.#'=hDW.sup.#' (Eq.
28)
[0124] The values of DW.sup.#, (.delta., V.sub.gtWi),
R.sup.#'(.delta., V.sub.gWi), and h(.delta., V.sub.gtWi) are found
from the source-drain resistance of a narrow transistor
R.sub.Na(V.sub.gtWi+.delta.-V.sub.thNa+V.sub.thWi) and the
source-drain resistance of a wide transistor R.sub.Wi(V.sub.gtWi).
Hereat, .delta. is a shift amount to be changed in calculating the
difference between two true threshold voltages V.sub.thWi,
V.sub.thNa. When a shift amount .delta. is equal to the threshold
voltage difference between the wide and narrow transistors
(V.sub.thNa-V.sub.thWi), the relationship of Equation 28 is
established. Accordingly, the function F that is defined in
Equation 29 is zero, irrespective of the gate overdrive V.sub.gt.
F(.delta.,V.sub.gtWi)=R.sup.#'(.delta.,V.sub.gtWi)-h(.delta.,V.sub.gtWi)D-
W.sup.#'(.delta., V.sub.gtWi) (Eq. 29)
[0125] A shift amount .delta. is changed to determine the value of
a true shift amount .delta..sub.0 with which the standard deviation
of function F is a minimum in a certain area of a gate overdrive
V.sub.gt. Using the true shift amount .delta..sub.0, the value dW**
of X intercept is found by, for example, extrapolating straight
lines as shown in FIG. 19. From the obtained dW**, a channel
narrowing DW is determined. An effective channel width W.sub.eff is
a value that is obtained by reducing a channel narrowing DW from a
mask channel width W.sub.m.
[0126] Referring to FIG. 20, extraction of the effective channel
width W.sub.eff of an MOS transistor will be described in
detail.
[0127] FIG. 20 shows the steps in a characteristic evaluation
method for insulated gate type transistors according to the third
preferred embodiment. The above steps are the same as those in FIG.
4 in the first preferred embodiment which are designated by the
same reference numeral, except for step ST1.20. In step ST1.20, the
function F shown in Equation 29 is calculated. In step ST1.9, by
using a calculation result obtained in step ST1.20, a true shift
amount .delta..sub.0 is determined from a shift amount .delta. with
which the standard deviation calculated in step ST1.7 is a minimum.
Steps ST1.10 and ST1.11 in which from the above true shift amount,
a channel narrowing DW and an effective channel width W.sub.eff are
determined .delta..sub.0, respectively, are the same as in the
characteristic evaluation method of the first preferred embodiment
as shown in FIG. 4.
[0128] Although a channel narrowing DW.sub.Na is determined from
dW** in step ST1.11, a channel narrowing DW(V.sub.gt) that is
obtained when the gate overdrive V.sub.gt is in the vicinity of
zero may be determined as the value DW.sup.# of W.sub.m coordinate
at an intersection.
[0129] Description will be now given of a concrete procedure to
determine a channel narrowing DW and the like, from the standard
deviation of the function F shown in Equation 29. In the
characteristic evaluation method of the third preferred embodiment,
to reduce the uncertainty of threshold voltage extrapolation and,
in particular, the error due to the uncertainty of the threshold
voltage extrapolation of a transistor having a narrow channel
width, the relationship of Equation 30 which is, for example,
established between the value DW* of W.sub.m coordinate at a
virtual point and the value dW** of W.sub.m intercept, is noted to
apply a variation method. Hereat, since the differentiation of an
intersection coordinate (R.sup.#, DW.sup.#) in Equation 29 may
increase the error of calculated values, Equation 30 is used in
place of Equation 29. F .function. ( .delta. , V gtWi ) = dW **
.function. ( .delta. , V gtWi ) + h .function. ( .delta. , V gtWi )
h ' .function. ( .delta. , V gtWi ) dW ** ' .function. ( .delta. ,
V gtWi ) - DW # .function. ( .delta. , V gtWi ) ( Eq . .times. 30 )
##EQU24##
[0130] Firstly, to a certain shift amount .delta.,
DW.sup.#(V.sub.gtWi, .delta.) and dW**(V.sub.gtWi, .delta.) are
given by Equations 31 and 32, respectively, where rri and rai are
defined in Equations 33 and 34, respectively, and the slope h of a
straight line is given by Equation 35. DW # = W mNa - rri W mWi 1 -
rri ( Eq . .times. 31 ) dW ** = W mNa - rai W mWi 1 - rai ( Eq .
.times. 32 ) rri .function. ( V gtWi , .delta. ) = R Wi '
.function. ( V gtWi ) R Na ' .function. ( V gtWi + .delta. - V thNa
+ V thWi ) ( Eq . .times. 33 ) rai .function. ( V gtWi , .delta. )
= R Wi .function. ( V gtWi ) R Na .function. ( V gtWi + .delta. - V
thNa + V thWi ) ( Eq . .times. 34 ) h .function. ( V gtWi , .delta.
) = R Na .function. ( V gtWi + .delta. - V thNa + V thWi ) - R Wi
.function. ( V gtWi ) W mWi - W mNa ( Eq . .times. 35 )
##EQU25##
[0131] A shift amount .delta. is changed to find the value DW.sup.#
of W.sub.m coordinate, the value dW** of W.sub.m intercept and its
rate of change dW**', as well as the resistance R per unit width
and its rate of change R'.
[0132] When a shift amount .delta. is equal to the threshold
voltage difference .delta..sub.0 between narrow and wide
transistors, the function F defined in Equation 30 is zero,
irrespective of the gate overdrive V.sub.gtWi. Thus, let the value
of a shift amount d with which the standard deviation of the
function F becomes a minimum in an area of a gate overdrive
V.sub.gtWi be a true shift amount .delta..sub.0 (see FIG. 21).
[0133] Then, let the value of dW** (V.sub.gt, .delta..sub.0) of
W.sub.m intercept which is obtained by using a true shift amount
.delta..sub.0, be a channel narrowing DW(V.sub.gt).
[0134] Although in the third preferred embodiment a true shift
amount .delta..sub.0 is determined from the condition under which
the standard deviation of the function F is a minimum, it may be
determined from the condition under which the sum of values that
are obtained by squaring each of the functions F to be found for
discrete gate overdrives V.sub.gt, becomes a minimum. When
calculating gate overdrive V.sub.gt for about 20 points, the sum Z
can be expressed by Equation 36. Z = n = 1 20 .times. F 2
.function. ( V gtn ) ( Eq . .times. 36 ) ##EQU26##
[0135] Instead of Equation 30, any one of Equations 37 to 39 may be
used to find function F. F .function. ( .delta. , V gtWi ) = h 2
.function. ( .delta. , V gtWi ) h ' .function. ( .delta. , V gtWi )
dW ** ' .function. ( .delta. , V gtWi ) - R # .function. ( .delta.
, V gtWi ) ( Eq . .times. 37 ) F .function. ( .delta. , V gtWi ) =
R ** .function. ( .delta. , V gtWi ) - h .function. ( .delta. , V
gtWi ) h ' .function. ( .delta. , V gtWi ) R ** ' .function. (
.delta. , V gtWi ) - R # .function. ( .delta. , V gtWi ) ( Eq .
.times. 38 ) F .function. ( .delta. , V gtWi ) = R ** ' .function.
( .delta. , V gtWi ) h ' .function. ( .delta. , V gtWi ) + DW #
.function. ( .delta. , V gtWi ) ( Eq . .times. 39 ) ##EQU27##
[0136] In Equations 38 and 39, R** is the value of a source-drain
resistance R when the value of a mask channel width W.sub.m is set
to be zero in R-W.sub.m characteristic. Using mask channel width
W.sub.m to enter an X-axis and source-drain resistance R to enter a
Y-axis, a R-W.sub.m characteristic curve (straight line) is
extrapolated to find the value R** of a Y intercept and the value
dW** of an X intercept which are obtained as X=0 and Y=0,
respectively. The use of the value R** or dW** facilitates
calculation. Although the accuracy remains unchanged with any one
of Equations 30, and 37 to 39, it is necessary to calculate R**
when using Equation 38 or 39. Thus, Equation 30 or 37 is
preferred.
[0137] Although in the third preferred embodiment the value dW** of
W.sub.m intercept obtained when a true shift amount .delta..sub.0
is used is employed as the value of a channel narrowing DW, the
value of a channel narrowing DW obtained when a gate overdrive
V.sub.gt is in the vicinity of zero may be given by the average of
the values DW.sup.# in the neighborhood where the value DW.sup.# of
W.sub.m coordinate of an intersection has a minimum value (see FIG.
22). Since DW.sup.# has a stationary point when the gate overdrive
V.sub.gt has a value in the vicinity of zero, it is possible to
determine the value of a channel narrowing DW at higher accuracy
than the case of using the value dW* of W.sub.m intercept.
[0138] Referring to FIG. 23, a characteristic evaluation apparatus
for insulated gate type transistors according to the third
preferred embodiment can be constructed by partially modifying the
calculation section 6 of the characteristic evaluation apparatus 1
of the first preferred embodiment in FIG. 8. Specifically, the
parts to be modified are an extraction section 12B that extracts an
intersection coordinate (DW.sup.#, R.sup.#) as the coordinate of a
virtual point, the value dW** of W.sub.m intercept, the value R**
of R intercept, and the slope h of a straight line in the
intersection coordinate; a true shift amount determination section
13B that determines a true shift amount .delta..sub.0 from the
values extracted in the extraction section 12B; and a channel
narrowing determination section 14B that determines a channel
narrowing by using a value giving a true shift amount .delta..sub.0
which is selected from among the values extracted in the extraction
section 12B. Other components of the calculation section 6B in FIG.
23 are the same as those in the first preferred embodiment. The
extraction section 12B further extracts the rate of change
dDW.sup.#/dV.sub.gt, dR.sup.#/dV.sub.gt of an intersection
coordinate (DW.sup.#, R.sup.#) and the slope h of a characteristic
curve in an area .OMEGA. with respect to each shift amount .delta.,
by using the value of a mask channel width W.sub.m provided from an
input section 5, the measurement data of source-drain current
I.sub.ds and gate-source voltage V.sub.gt that are provided from a
control section 4. The true shift amount determination section 13B
determines a virtual shift amount .delta..sub.0 with which the
standard deviation of the function F shown in Equation 29 becomes a
minimum for the area .OMEGA., by using the rate of change
dDW.sup.#/dV.sub.gt of the W.sub.m coordinate of an intersection,
the rate of change dR.sup.#/dV.sub.gt of R coordinate of the
intersection, and the slope h of the characteristic curve which
have been extracted in the extraction 12B. Upon determination of a
true shift amount .delta..sub.0, the extraction section 12B outputs
the true shift amount .delta..sub.0 or the value DW.sup.# of
W.sub.m coordinate of the corresponding intersection and value dW**
of W.sub.m intercept, to a channel narrowing determination section
14B. The section 14B determines a channel narrowing DW from the
value dW** of W.sub.m intercept or the value DW.sup.# of W.sub.m
coordinate in a virtual point, and performs the calculation shown
in Equation 7, to determine an effective channel width
W.sub.eff.
[0139] Referring again to FIG. 9, the characteristic evaluation for
insulated gate type transistors as described in the third preferred
embodiment is attainable by making a computer to read an evaluation
program 30 for evaluating insulated gate type transistors from a
recording medium storing the program 30, in accordance with the
procedure in FIG. 20 as described in the third preferred
embodiment.
[0140] A method of manufacturing an insulted gate type transistor
according to the third preferred embodiment can be implemented by
employing, in step ST52 shown in FIG. 10, the evaluation method of
the third preferred embodiment in place of that of the first
preferred embodiment. This results in the same effects as in the
case where the evaluation method of the first preferred embodiment
is applied to a manufacturing method.
Fourth Preferred Embodiment
[0141] A characteristic evaluation method for insulated gate type
transistors according to a fourth preferred embodiment will be
outlined by referring to FIG. 24. FIG. 24 is a graph showing the
relationship between DW.sup.# and gate overdrive V.sub.gt that are
found by the characteristic evaluation method for insulated gate
type transistors according to the fourth preferred embodiment. This
graph shows the change in the value DW.sup.# of W.sub.m coordinate
of an intersection when a true threshold voltage is used for three
narrow transistors having a different mask channel width W.sub.mNa.
Note that the mask channel width W.sub.mWi of a wide transistor
that serves as a reference in extracting the value DW.sup.# of
W.sub.m coordinate for these transistors, is set to the same
value.
[0142] As shown by comparison of FIG. 24 with FIG. 25, if the value
of a shift amount .delta. derives from a shift amount
.delta..sub.0, the shape of a V.sub.gt-DW.sup.# characteristic
curve changes, whereas even if the value of a mask channel width
W.sub.mNa changes somewhat, the shape of a V.sub.gt-DW.sup.#
characteristic curve remains unchanged. Hence, as to other
transistor having a different mask channel width W.sub.m, it is
also possible to extract the true threshold voltage of a narrow
transistor by finding out one characteristic curve which coincides
with that in this graph when the gate overdrive V.sub.gt ranges
from 0.3 to 1.2 V, for example. In the fourth preferred embodiment,
first and second gate overdrives and first to fourth estimated
values are defined as in the third preferred embodiment.
[0143] One example of the characteristic evaluation method for
insulated gate type transistors according to the fourth preferred
embodiment will be described by referring to FIG. 26. In the method
shown in FIG. 26, from characteristic curves that change variously
depending on the estimated value of a threshold voltage V.sub.thNa,
i.e., a first estimated value, the characteristic curve in FIG. 24
is extracted based on the fact that the standard deviation of the
curve is small in the range of 0.2 to 0.6 V, for example. Since in
the evaluation method of the fourth preferred embodiment, the true
threshold voltage .delta..sub.0 of a narrow transistor is
determined by utilizing the dependence of the value DW.sup.# of
W.sub.m coordinate on a gate overdrive V.sub.gt, the true threshold
voltage .delta..sub.0 is determined in a manner similar to that in
the third preferred embodiment.
[0144] The procedure of extracting an effective channel width
W.sub.eff in the characteristic evaluation method of the fourth
preferred embodiment is the same as that of the third preferred
embodiment, except for steps ST1.30 to ST1.34 in FIG. 26, which
correspond to steps ST1.20, ST1.7 and ST1.9 to ST1.11 in FIG. 20,
respectively.
[0145] In the loop composed of steps ST1.4 to ST1.8, at step ST1.30
the value DW.sup.# of W.sub.m coordinate at a virtual point is
found. That is, the values DW.sup.# of about twenty different gate
overdrives V.sub.gtn for each shift amount .delta. are found. At
step ST1.31, the average of the twenty DW.sup.# values of
DW.sup.#(.delta., V.sub.gt) to DW.sup.#(.delta., V.sub.gtn), and
the standard deviation .sigma. [DW.sup.#] are calculated.
[0146] After repeat calculation for each shift amount .delta.
(steps ST1.14 to ST1.8) is terminated, at step ST1.32, a shift
amount .delta..sub.0 for giving a channel narrowing DW is
estimated, with which the standard deviation .sigma. becomes a
minimum. At step ST1.33, the channel narrowing DW is given by the
average of the values DW.sup.# of W.sub.m coordinates at a virtual
point when a shift amount is .delta..sub.0. At step ST1.34, an
effective channel width W.sub.eff is determined by the difference
between a mask channel width and the channel narrowing DW.
[0147] Referring to FIG. 27, description will be now given of a
characteristic evaluation apparatus for insulated gate type
transistors according to the fourth preferred embodiment. A
characteristic evaluation apparatus for insulated gate type
transistors 1C shown in FIG. 27 is connected to a measuring device
3 for measuring an object under test 2, like the characteristic
evaluation apparatus 1B of the third preferred embodiment as shown
in FIG. 23. In the construction of the characteristic evaluation
apparatus 1C, the same reference numerals have been retained for
similar parts which have the same functions as in the apparatus 1B
of FIG. 23. That is, the characteristic evaluation apparatus 1C has
the same structure as the apparatus 1B, except for an extraction
section 12C, a true shift amount determination section 13C and a
channel narrowing determination section 14A in a calculation
section 6C.
[0148] The extraction section 12C of the characteristic evaluation
apparatus 1C finds an intersection coordinate (DW.sup.#, R.sup.#)
by changing a gate overdrive V.sub.gt in an area .OMEGA.. The true
shift amount determination section 13C finds a standard deviation
.sigma. [DW.sup.#] from the value of the intersection coordinate
(DW.sup.#, R.sup.#) in the area .OMEGA., to determine a true shift
amount .delta..sub.0. The extraction section 12C outputs the true
shift amount .delta..sub.0 and the value DW.sup.# of W.sub.m
coordinate at the corresponding intersection or the value dW** of
W.sub.m intercept, to the channel narrowing determination section
14C. The channel narrowing section 14C determines a channel
narrowing DW from the average of the values DW.sup.# of W.sub.m
coordinates at virtual points within the area .OMEGA. for the true
shift amount .delta..sub.0, e.g., in the range of
0.2.ltoreq.V.sub.gt.ltoreq.0.6. Alternatively, the section 14C
determines the value dW** of W.sub.m intercept related to the true
shift amount .delta..sub.0, as a channel narrowing DW.
[0149] Referring again to FIG. 9, the characteristic evaluation for
insulated gate type transistors as described in the fourth
preferred embodiment is attainable by making a computer to read an
evaluation program 30 for evaluating insulated gate type
transistors from a recording medium storing the program 30, in
accordance with the procedure in FIG. 20 as described in the fourth
preferred embodiment.
[0150] A method of manufacturing an insulted gate type transistor
according to the fourth preferred embodiment can be implemented by
employing, in step ST52 shown in FIG. 10, the evaluation method of
the fourth preferred embodiment in place of that of the first
preferred embodiment. This results in the same effects as in the
case where the evaluation method of the first preferred embodiment
is applied to a manufacturing method.
[0151] Although in the fourth preferred embodiment, a channel
narrowing DW is determined so as to minimize the standard deviation
.sigma. [DW.sup.#] of the value DW.sup.# of W.sub.m coordinate at
an intersection or the standard deviation .sigma. [dW.sup.#] of the
value dW** of W.sub.m intercept, its determination method is not
limited to the above. For instance, the threshold voltage
V.sub.thNa of a narrow transistor may be determined by selecting a
characteristic curve in which the value DW** of W.sub.m coordinate
at an intersection is best converged on a fixed value when the gate
overdrive V.sub.gt is within a predetermined range.
[0152] When the mask channel width W.sub.mNa of a narrow transistor
is sufficiently smaller than the mask channel width W.sub.mWi of a
wide transistor (W.sub.mNa<<W.sub.mWi), Equation 31 is
approximated as shown in Equation 40. Accordingly, a channel
narrowing DW may be determined so that the standard deviation of
the value of Equation 40 is a minimum.
DW.sup.#.apprxeq.W.sub.mNa-rriW.sub.mWi (Eq. 40)
[0153] Alternatively, in Equation 40 a channel narrowing DW may be
determined under the condition that the standard deviation of a
variable rri is a minimum, because mask channel widths W.sub.mWi
and W.sub.mNa are both constants.
[0154] Alternatively, since the condition that the standard
deviation of the variation rri is a minimum is approximately equal
to that the standard deviation of its inverse number rri.sup.-1 is
a minimum, a channel narrowing DW may be determined from the
standard deviation of the inverse number rri.sup.-1.
[0155] In the channel narrowing DW extraction according to the
third or fourth preferred embodiment, when the mask cannel width
W.sub.mNa of a narrow transistor is significantly smaller than the
mask channel width W.sub.mWi of a wide transistor (i.e.,
W.sub.mNa<<W.sub.mWi), the difference between the mask
channel width W.sub.mwi and a gate finished width W.sub.gWi hardly
affects on determination of the value DW* of W.sub.m coordinate at
a virtual point, thereby determines the channel narrowing DW of the
narrow transistor at high accuracy. For instance, to evaluate
device or circuit performance on the level of not more than 1.0
.mu.m in pattern width, it is required to extract the channel
narrowing DW of each transistor. For such an extraction, there are
used two transistors, i.e., a narrow transistor and a wide
transistor serving as a reference. In this case, the difference
between a gate finished width W.sub.g and a mask channel width
W.sub.m depends on the transistor, causing errors. Thus,
description will be now given of such errors. The value DW.sup.# of
W.sub.m coordinate at a virtual point when a mask channel width
W.sub.m is used is given by Equation 41. DW # = ( W mNa - R Wi ' R
Na ' W mWi ) ( 1 - R Wi ' R Na ' ) ( Eq . .times. 41 )
##EQU28##
[0156] If W.sub.g coordinate of an intersection in a plane formed
by gate finished width and source-drain conductance (i.e., a
W.sub.g-R plane) is represented by DW.sub.g.sup.#, the following
Equation 42 is obtained. DW g # = ( W gNa - R Wi ' R Na ' W gWi ) (
1 - R Wi ' R Na ' ) - 1 ( Eq . .times. 42 ) ##EQU29##
[0157] If the difference between a gate finished width W.sub.g and
a mask channel width W.sub.m is represented by .DELTA.W, the
difference between the gate finished width W.sub.gWi and mask
channel width W.sub.mWi of a wide transistor, and the difference
between the gate finished width W.sub.gNa and a mask channel width
W.sub.mNa of a narrow transistor, are represented by
.DELTA.W.sub.Wi and .DELTA.W.sub.Na, respectively. Therefore, the
relationships of Equations 43 and 44 are established. Then, from
Equations 41 to 44, the difference between the value DW** of
W.sub.m coordinate and the value DW.sub.g* of W.sub.g coordinate at
an intersection is expressed by Equation 45, where .DELTA.W is
defined in Equation 46. W gWi = W mWi + .DELTA. .times. .times. W
Wi ( Eq . .times. 43 ) W gNa = W mNa + .DELTA. .times. .times. W Na
( Eq . .times. 44 ) DW ** - DW g ** = .times. - .DELTA. .times.
.times. W Na + R Wi ' R Na ' ( 1 - R Wi ' R Na ' ) - 1 .DELTA.
.times. .times. W .apprxeq. .times. - .DELTA. .times. .times. W Na
+ G Wi ' R Na ' .DELTA. .times. .times. W .apprxeq. .times. -
.DELTA. .times. .times. W Na + W effNa W effWi .DELTA. .times.
.times. W ( Eq . .times. 45 ) .DELTA. .times. .times. W .ident.
.DELTA. .times. .times. W Wi - .DELTA. .times. .times. W Na ( Eq .
.times. 46 ) ##EQU30##
[0158] Equations 43 and 44 show that the effective channel width
W.sub.eff of a narrow transistor is extracted when the relationship
W.sub.mNa<<W.sub.mWi is established. In Equation 45, the
second term of the last expression indicates an error. If a
relative error is represented by r, it results in Equation 26.
Therefore, again in the third and fourth preferred embodiments, to
make a relative error smaller than the desired value, the same
limitations are imposed upon the mask channel width W.sub.gWi of a
wide transistor, as in the first and second preferred
embodiments.
[0159] Consider now the influence of unequal channel lengths due to
the irregularity of finished polygate. Source-drain resistance
R.sub.tot is defined in Equation 47, where g is a channel sheet
resistance. R = L eff W eff g + R sd ( Eq . .times. 47 )
##EQU31##
[0160] Let the difference in the channel length L between a narrow
transistor and a wide transistor be .DELTA.L
(=L.sub.effNa-L.sub.effWi), Equations 47 can be modified into
Equation 48. R = L effWi W effNa ( 1 - .DELTA. .times. .times. L L
effWi ) g + R sd ( Eq . .times. 48 ) ##EQU32##
[0161] Supposing a sheet resistance g is independent of an
effective channel length L.sub.eff, Equation 48 shows that an
effective channel length L.sub.effNa appears to be increased by a
factor of (1-L/L.sub.effWi). Now, expressing a relative error by r,
an error .DELTA.r is expressed by Equation 49. W effNa .DELTA.
.times. .times. L L effWi < r W effNa ( Eq . .times. 49 )
##EQU33##
[0162] Supposing that an effective channel length L.sub.effWi is
approximately equal to a mask channel length L.sub.mWi, Equation 49
can be modified into Equation 50. L mWi > .DELTA. .times.
.times. L r ( Eq . .times. 50 ) ##EQU34##
[0163] Equation 50 imposes limitations upon the mask channel length
L.sub.mWi of a wide transistor to be used in extraction. For
instance, when .DELTA.L=0.1 .mu.m and r=0.02, the mask channel
width W.sub.mWi of a wide transistor is required to be greater than
5 .mu.m, in order to accurately extract the effective channel width
of a narrow transistor.
[0164] Description will be now given of the case where the
characteristic evaluation method for insulated gate type
transistors according to the first preferred embodiment
(hereinafter referred to as Gm method) or that of the third
preferred embodiment (referred to as Rm method) is applied to a MOS
transistor having a mask channel width W.sub.m of 0.36 .mu.m and a
mask channel length L.sub.m of 20.4 .mu.m. FIG. 28 gives a
comparison among the channel narrowing DW (obtained by Gm method),
DW (by Rm method), and DW (by Chia method). Both Gm and Rm methods
provide nearly the same result. Since it is generally difficult to
accurately determine a threshold voltage V.sub.th, Gm method and Rm
method ensure more accurate channel narrowing DW than Chia
method.
[0165] Then, it is checked how the value dW* of W.sub.m intercept
and the values DW*, DW.sup.# of W coordinate at an intersection
depend on the gate overdrive V.sub.gt in the vicinity of zero. Now,
expanding the channel narrowing DW, the slope h of a straight line
and the inverse number g (g=1/f) of the slope f of the straight
line, to the power of a gate overdrive V.sub.gt, Equations 51 to 53
are obtained where DWG1, DWG2, and A to D are an arbitrary
constant. DW = .delta. .times. .times. W - DWG .times. .times. 1 V
gt - DWG .times. .times. 2 V gt 2 + O .function. ( V gt 3 ) ( Eq .
.times. 51 ) h = A V gt + B + O .function. ( V gt ) ( Eq . .times.
52 ) g = C V gt + D + O .function. ( V gt ) ( Eq . .times. 53 )
##EQU35##
[0166] In this case, dW**, DW* and DW.sup.# are expanded as
follows. dW ** .apprxeq. .delta. .times. .times. W - DWG .times.
.times. 1 DWG .times. .times. 2 V gt 2 + O .function. ( V gt 3 ) (
Eq . .times. 54 ) DW * .apprxeq. .delta. .times. .times. W - 2 DWG
.times. .times. 1 V gt - ( 3 DWG .times. .times. 2 + D C DWG
.times. .times. 1 ) .times. .times. V gt 2 + O .function. ( V gt 3
) ( Eq . .times. 55 ) dW # .apprxeq. .delta. .times. .times. W + (
DWG .times. .times. 2 + B A DWG .times. .times. 1 ) V gt 2 + O
.function. ( V gt 3 ) ( Eq . .times. 56 ) ##EQU36##
[0167] Equations 54 to 56 indicate the following matters. When
dW**, DW* and DW.sup.# are brought to near zero, they all converge
on .epsilon. W. When DWG1 and DWG2 are both positive numbers, DW*
decreases rapidly than dW** as the gate overdrive V.sub.gt
increases. DW.sup.# has a stationary point at V.sub.gt=0, and
increases by the square of V.sub.gt as the gate overdrive V.sub.gt
increases. These indicate that the results given in FIGS. 6 and 22
are correct.
[0168] Also, the presence of the stationary point at V.sub.gt=0
suggests the possibility that .delta.W is determined so that
DW.sup.# is constant when V.sub.gt is in the vicinity of zero. This
is the case where "shift and ratio method" is applied to the
extraction of a channel narrowing DW (this method is described, for
example, in "A New "Shift and Ratio" Method for MOSFT Channel
Length Extraction," IEEE Elect. Dev. Lett., EDL-13(5), p. 267,
1992, by Y. Taur et al.). This method actually gives proper values,
however, its extraction result depends greatly on the area of a
gate overdrive V.sub.gt for calculation (see FIG. 29). On the other
hand, both Rm method and Gm method are independent of the area of a
gate overdrive V.sub.gt for calculation, and also can give nearly
the same result.
[0169] While the invention has been shown and described in detail,
the foregoing description is in all aspects illustrative and not
restrictive. It is therefore understood that numerous modifications
and variations can be devised without departing from the scope of
the invention.
* * * * *