Download Effects Of The Inversion Layer Centroid On MOSFET Behavior

IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 44, NO. 11, NOVEMBER 1997
1915
Effects of the Inversion Layer
Centroid on MOSFET Behavior
Juan A. López-Villanueva, Pedro Cartujo-Casinello,
Jesus Banqueri, Member, IEEE, F. Gámiz, and Salvador Rodrı́guez
Abstract— The effects of the average inversion-layer
penetration, which are termed the inversion-layer centroid, on
the inversion-charge density and the gate-to-channel capacitance
have been analyzed. The quantum model has been used, and a
variety of data have been obtained by self-consistently solving
the Poisson and Schrödinger equations. An empirical expression
for the centroid position that is valid for a wide range of
electrical and technological variables has been obtained and has
been applied to accurately model the inversion-layer density
and capacitance.
I. INTRODUCTION
Q
UANTIZATION of the transverse electron motion in the
inversion layer of a metal-oxide-semiconductor (MOS)
transistor has been widely studied in the last 30 years.
An excellent review can be found in [1]. A remarkable consequence for MOSFET behavior is the fact that electron density
does not reach its maximum at the oxide-semiconductor interface but instead inside the semiconductor, while it almost
vanishes right at the interface. At first, the main interest
on quantum effects focused on the two-dimensional (2-D)
properties of the electron gas, which were apparent mainly at
low temperatures since multisubband occupation is produced
at 300 K. Nevertheless, the extension of the electron density
inside the semiconductor and its effects on capacitance were
noted early on by Pals, even at 300 K [2]. Two consequences
of this fact that are important for device behavior are that
1) the electric potential value at the interface is greater than
that predicted by the classical model and 2) when the gate
voltage is varied, the main modification of electron charge at
the semiconductor is produced away from the interface, thus
leading to reduced capacitance and transconductance [3].
However, although quantum effects were known early on,
they were not seriously considered by the modeling community until very recently, as classical models worked reasonably
well with previous transistor generations. In those transistors,
gate oxides were relatively thick, and therefore, most of the
gate voltage drop was produced at the oxide, making the
error in the interface potential to be relatively unimportant.
Furthermore, the average distance of the electron density from
the interface was small compared with the oxide thickness.
The situation has obviously changed in modern transistor
Manuscript received November 13, 1996; revised April 2, 1997. The review
of this paper was arranged by Editor D. Verret. This work was supported by
the CICYT of the Spanish Government under Contract TIC95-0511.
The authors are with the Departamento de Electrónica, Facultad de Ciencias,
18071 Granada, Spain (e-mail: [email protected]).
Publisher Item Identifier S 0018-9383(97)07730-7.
generations, in which oxide thickness is comparable with the
extension of the inversion layer. Inversion-layer penetration
is therefore important and must be incorporated in accurate
models.
Quantum effects have been included in different ways. Some
examples follow.
1) The classical electron density used in classical driftdiffusion simulators has been corrected in order to force
it to vanish at the oxide-semiconductor interface, and
suitable boundary conditions are derived [4].
2) As energy levels below the ground subband minimum
are forbidden for the electrons, an effective bandgap
widening has been proposed to modify the intrinsic
carrier density used in drift diffusion simulators near
the semiconductor-oxide interface [5].
3) The oxide thickness is modified by adding a constant
value representing the average inversion-layer penetration [6], [7]. Flatband and threshold voltages are also
modified [6].
4) The oxide capacitance is corrected by adding a term
representing the difference between the classical and
quantum average penetrations, which is modeled as a
function of the inversion and depletion charges [8], [9].
Nevertheless, despite increasing interest in modeling the effects of average inversion-layer penetration, there are several
aspects that require further clarification, such as the correction to be applied, whether this correction is the same
in current–voltage characteristics and in capacitance–voltage
characteristics, and what the dependence of the average inversion layer on the surface potential and the doping level is,
among other parameters.
Some of the above questions are addressed here. To do
so, a self-consistent solution of the Schrödinger and Poisson
equations is obtained in a variety of conditions, and a detailed
examination of the average penetration of the inversion layer
into the semiconductor, which is termed the inversion-layer
centroid, is performed. Some useful expressions are provided
throughout the sections below.
II. THE INVERSION-LAYER CENTROID
To obtain the inversion-layer centroid, an accurate solution of the Poisson and Schrödinger equations is obtained
within the Hartree approximation. After the first variational
calculation by Stern and Howard [10], these equations were
self-consistently solved in the pioneering work by Stern (and
the unpublished one by Howard) [11], [12], where the most
0018–9383/97$10.00  1997 IEEE
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IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 44, NO. 11, NOVEMBER 1997
important features of quantization were shown. Several other
authors have added some improvements [13], and some of
the authors of this paper developed a procedure that provided
a good convergence that starts from the classical solution,
even in the accumulation case at room temperature [14]. This
procedure has been used as the basis for a Monte Carlo
simulator of low-field transport in a quantized inversion layer,
which has allowed us to accurately predict the effect of
charged impurities on electron mobility [15]–[17]. A complete
simulator including inversion-layer quantization has also been
reported by Fischetti and Laux [18]. The procedure detailed
in [14] has been used here to obtain the electric potential,
the inversion and depletion charges, the energy of the lowest
subband minima, and the inversion-layer centroid as a function
of the applied gate voltage for different doping levels and
profiles and different oxide thicknesses. We chose the number
of subbands considered in the calculation so that the addition
of a new subband would not change the results, even in weak
inversion. This is not a serious problem in our calculation as all
the higher subbands are included in a three-dimensional (3-D)
gas. Oxide thickness has been observed to have no significant
influence on the results provided below.
The electron density in the inversion layer as a function
of the distance from the silicon-SiO interface is shown in
Fig. 1. The total electron concentration is shown as a solid
line, while the electron concentrations contained in the three
lowest subbands are plotted in dashed lines. Long dashed
lines correspond to the two ellipsoids perpendicular to the
interface in a transistor with a (100)-oriented substrate. Short
dashed lines correspond to the four parallel ellipsoids. As can
be observed, the total electron density has a shape similar
to the electron density contained in the ground subband,
although multisubband occupation causes the total electron
density to widen, and the centroid of the total density is
therefore different from the ground subband centroid. This
similarity in shapes is the reason for the relatively good
behavior of the variational expression, which was obtained
by Stern and Howard by considering only occupation of the
ground subband [10], even in the multisubband occupation
case at room temperature, as reported by several authors [9],
[19]. According to this approximation, the centroid position
depends on the depletion and inversion charges as
(1)
and
are the modulus of the depletion and
where
inversion-charge densities per unit area, respectively, and
is a fitting parameter.
We have tested this dependence in our results and have
compared it with other expressions. The most general one, due
to multisubband occupation, is to assume a dependence on the
effective electric field, as defined by Sabnis and Clemens [20],
which is the average transverse electric field in the inversion
layer, regardless of the shape of the electron density. This
transverse effective electric field is given by
(2)
where
is the silicon permittivity.
Fig. 1. Total electron concentration as a function of the distance from the
silicon-SiO2 interface (solid line) and electron concentrations contained in the
three lowest subbands (dashed lines). Long dashed lines correspond to the two
ellipsoids perpendicular to the interface in a transistor with a (100)-oriented
substrate. Short dashed lines correspond to the four parallel ellipsoids. Doping
concentration: 5 1 1017 cm03 . Electron density per unit area: 1012 cm02 .
Therefore, we can also test the suitability of the following
empirical expression
(3)
The result of this comparison for the best
and
values
is shown in Fig. 2, in which the inversion-layer centroid is
plotted versus the electron density per unit area. Numerical
results are shown in solid lines, and the best fit obtained
with (1) and (3) in the text is shown in long- and mediumdashed lines, respectively. The substrate-doping concentration
is
cm in Fig. 2(a), and 3 10
cm in Fig. 2(b).
Approximation 3 behaves better for the lowest doping case
[Fig. 2(a)], whereas for the highest doping case [Fig. 2(b)], (1)
and (3) behave comparably, providing a small absolute error.
Equation (1) may provide a slightly better agreement with
the numerical results in this case, which can be interpreted
as being a consequence of the greater confinement in the
ground subband. Both the
and
values depend on the
doping concentration. Therefore, to obtain a general expression
applicable across a wide range of doping levels, we propose a
more general empirical approximation given by
(4)
MV/cm
and are fitting parameters. We have obtained the
where
values of these parameters that produce the best fit for doping
cm to
levels ranging from
cm , obtaining very good agreement with the numerical
results, with a maximum error of about 1%. The resulting
and values show a linear dependence with the doping
level in a semilogarithmic scale and can be expressed as
cm
cm
nm
(5)
When (5) is used in (4), the maximum error is increased
to about 3% in the worst cases, which is still fairly small. In
obtaining this approximation, values for the surface potential
within the moderate and strong inversion regions, as defined
by Tsividis [21], have been used. These regions have been
LÓPEZ-VILLANUEVA et al.: EFFECTS OF THE INVERSION LAYER CENTROID
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also tested the following expression.
MV
cm
(a)
(b)
Fig. 2. Inversion-layer centroid versus the electron density per unit area.
Numerical results are shown in solid lines. The best fit obtained with (1), (3),
and (6) in the text are shown in long-dashed lines, medium-dashed lines, and
short-dashed lines, respectively. The substrate-doping concentration is (a) 10
17 cm03 and (b) 3 11018 cm03 .
chosen since the inversion-layer centroid produces noticeable
is not negligible with respect to
effects when
the surface potential , as explained in the next section. At
the limit between the moderate and weak inversion regions,
is lower than 0.5 mV in all the cases considered
here. Therefore, although the average penetration of the inversion layer rapidly increases in weak inversion when the
surface potential is reduced, its effect is negligible due to the
exponential decrease of the inversion charge.
We also show in the next section that the inversion-layer
(where
centroid effects are noticeable when
is the oxide permittivity and
is the physical oxide
thickness) is not negligible versus one. These effects are
thus important mainly for thin oxides. However, according to
scaling rules, thin oxides must be used in transistors with high
bulk doping levels so that an empirical expression specific for
high doping concentrations is also interesting. One advantage
of such an expression is that the dependence of parameter
on the doping concentration in (4) can be obviated, thus
making it applicable to a variety of doping profiles. As the
values vary from 0.38 to 0.27 in the range
cm
to
cm , the value
used in (3) is a good choice in this range. Furthermore, we
values obtained for different
have analyzed the different
doping concentrations of the silicon bulk and have observed
can also produce
that a dependence on approximately
a reasonable good agreement in this range. Therefore, we have
(6)
has been introduced here to express
The silicon permittivity
the charges in electric field units (in megavolts per centimeter).
is a reference length acting as a fitting parameter. The
curves obtained for the best
value are also shown in Fig. 2
in short dashed lines. As seen in this figure, (6) provides
a slightly better fit than (3). However, the great advantage
of (6) is that it can be used for different doping levels and
profiles and even for different bulk biases. In the range of
doping concentrations considered now, we have observed that
the constant value
nm provides good agreement
with numerical results (a higher value should be considered for
lower doping levels, and a different value should in fact be
chosen). Such a good agreement is shown in Fig. 3, where the
inversion-layer centroid is represented versus electron density
per unit area for different doping concentrations [Fig. 3(a)]
and for different bulk biases [Fig. 3(b)]. Numerical results
are shown in solid lines, whereas results obtained with (6)
in the text, with
nm, are shown in dashed lines.
The substrate-doping concentration in Fig. 3(a) ranges from
10 cm (curve 1) to
cm (curve 5), with the
substrate biased at 0 V, whereas in Fig. 3(b), ,the substratedoping concentration is kept fixed at 10 cm , and the
substrate bias is changed from
V to
V (from top to bottom). The maximum error is about 0.1 nm,
which is comparable to the error of the numerical procedure
itself as image effects and exchange-correlation effects have
not been considered [22]. The validity of (6), with
nm, has also been tested for nonuniform doping profiles.
Fig. 4(a) shows the inversion-layer centroid versus the electron
density per unit area for the two doping profiles depicted in
Fig. 4(b). Numerical results are plotted in solid lines, and
results obtained with (6), with
nm, are shown
in dashed lines. Curve 1 corresponds to a 10-nm low-doped
epitaxial layer grown on a highly doped substrate plotted in
short-dashed lines in Fig. 4(b), and Curve 2 corresponds to
a Gaussian profile that peaked at 100 nm from the interface,
which is depicted in long-dashed lines in the same figure. Good
agreement can also be noted here. The strong influence of the
doping level was already remarked on by Pals [2] and has been
recently underscored by Arora et al. [8], who commented on
the inaccuracy of simple models that propose a constant value
independently of the technological variables.
III. EFFECT ON THE INVERSION CHARGE DENSITY
We have analyzed the effects of the centroid on the inversion
charge as they have a direct effect on the I–V characteristics.
The influence of the inversion-layer centroid on the total band
bending in the semiconductor has been recognized since the
pioneering papers on the subject [2], [12]. If is the coordinate perpendicular to the interface, multiplying the Poisson
equation by , and integrating it, we obtain
(7)
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IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 44, NO. 11, NOVEMBER 1997
(a)
(a)
(b)
Fig. 3. Inversion-layer centroid versus electron density per unit area. Solid
lines: numerical results. Dashed lines: results obtained with (6) in the text,
1:2 nm. (a) Substrate-doping concentration is (1) 1017 cm03 ,
with zI 0
(2) 2 1 1017 cm03 , (3) 5 1 1017 cm03 , (4) 1018 cm03 , and (5) 3 1 1018
cm03 . The substrate is biased at 0 V. (b) Substrate-doping concentration is
1017 cm03 , and the substrate bias is VSB = 0; 1; 2; 3; and 5 V, respectively,
from top to bottom.
=
where
modulus of the electron charge;
doping profile;
depletion layer width.
If
is the 3-D electron density in the inversion layer, the
average penetration depth is defined as
(8)
[
has been assumed to vanish at
and higher distances
from the silicon-oxide interface].
Equation (7) can also be obtained by assuming that the
inversion layer is a sheet of charge with infinitesimal width
(a delta function) placed at
from the silicon-oxide interface
and with charge density per unit area
[23]. The first term
of the right-hand member can be defined as a depletion-layer
band bending, which is increased by the second term in order
to obtain the total surface potential. Therefore, we can define
(9)
Although (7) and (8) do not depend on the
shape, the
effect of the term containing the inversion-layer centroid is
more important with the quantum model than with the classical
(b)
Fig. 4. (a) Inversion-layer centroid versus the electron density per unit area
for the two doping profiles shown in figure. (b) Solid lines: Numerical results.
Dashed lines: Results obtained with (6) in the text, with zI 0 = 1.2 nm. Curve
1: for profile depicted in short-dashed lines. Curve 2: for profile depicted in
long-dashed lines.
model as the value of the centroid provided by the former
is greater. This is the reason for the higher surface potential
value obtained with the quantum model. Both
and
are represented in Fig. 5 as a function of the applied gate
voltage for two extreme values of the doping concentration,
cm and
cm (an
-poly has been used
as gate metal).
is shown in solid lines and
in shortdashed lines. The value
, where
is the distance
between the Fermi level and the intrinsic level in the silicon
,
bulk, is shown with solid squares, and the value
defined by Tsividis as the limit between the moderate and
strong inversion regions [21], is shown with horizontal dashed
lines. It is apparent that
is used instead of
to define the
threshold voltage in standard MOSFET models and that
is almost constant in the strong inversion region, whereas
strongly diverges. Furthermore,
must be used to calculate
the depletion charge. With a constant doping profile, it is well
known that
(10)
is the Boltzmann constant and
is the absolute
where
temperature.
The correction
within the root symbol accounts for
the effect of the majority carrier tail in the depletion layer
edge. An interesting treatment of the corrections necessary
for a nonconstant doping profile was provided by Baccarani
LÓPEZ-VILLANUEVA et al.: EFFECTS OF THE INVERSION LAYER CENTROID
Fig. 5. Surface potential versus the gate voltage for two different doping-concentration values: (1) 17 cm03 and (2) 1 18 cm03 . Solid lines:
F points.
s . Short-dashed lines: dep . Solid squares show the s
1 kB T=q . (tox
Horizontal long-dashed lines correspond to F
nm).
10
3 10
2 +6
=2
= 10
et al. [24]. Equation (10) has been observed to produce a
good agreement with our numerical results for constant doping
profiles, with an error lower than 1%, whereas the maximum
error increases to about 4% if the correction
is not
included.
can be calculated
By using the definition given in (9),
according to the well-known expression
(11)
is the gate voltage, and
is the flatband voltage,
where
which includes the difference between the work functions in
the metal and the semiconductor and the effect of the fixed
charge within the oxide.
is the oxide capacitance per unit
area, which is defined as
(12)
Equation (11) is exact regardless of the shape of the charge
distributions, provided the interface-state density and poly
depletion are negligible, as is obtained by simply applying
Gauss’s law and imposing the continuity of the electric field
times the permittivity at the oxide silicon interface. To obtain
an expression for the inversion-layer charge, (9) and (11) can
be used, leading to
(13)
with
(14)
The denominator of (14) can be termed the “electrical oxide
thickness,” according to Arora et al. [8]. Equation (14) can also
be interpreted as a series combination of
and a “centroid
. Although this same correction of
capacitance” given by
the oxide thickness has been proposed by several other authors
[6], [8], [9], it is worthwhile to emphasize the following points:
1) Equation (14) is not an empirical correction, as it has
been derived directly from well-known equations. This
correction is a consequence of the increase in the surface
potential with the inversion charge and should not be
1919
used together with the actual surface potential
but
with
.
2)
appearing in (14) is not the difference between the
quantum and classical centroids but the actual centroid
position obtained according to the chosen model (either
quantum or classical). Quantum effects have not explicitly been used in (7)–(14). Nevertheless, it is expected
that the quantum model provides a more realistic value
for the centroid.
3) Introducing the centroid correction leads to using
instead of
in the definition of threshold voltage.
can also be expressed according to (10) if an effective
constant doping concentration is used.
is divided by
instead of by
. The body4)
effect coefficient is therefore not affected by the centroid
correction.
5) As (7) and (11) have been obtained with no approximations, (13) applies even in weak inversion, provided
that accurate expressions for
and
are used in
this regime.
The electron concentration per unit area
as a
function of
is shown in Fig. 6 for the two extreme doping
values 10 cm and 3 10 cm on a semilogarithmic
scale [Fig. 6(a)] and on a linear scale [Fig. 6(b)]. Data obtained
with the numerical procedure detailed in the previous section
are shown with a solid line, whereas data obtained using (13)
with the numerical values for
and
are shown in solid
circles. These data are compared with those obtained with a
simple strong-inversion model in which
is assumed to
be approximately equal to
.
is
obtained with (10), and
is calculated by using (6), (13), and
(14), with an iterative procedure. The result is shown in Fig. 6
with solid squares, which also produces a reasonably good
agreement with numerical results except in weak inversion.
The shift observed in weak inversion to higher gate voltages
is mainly due to the constant value used for
, which
overestimates the surface potential in this region. The result
for the highest doping case can be improved if a correction
greater than
is used, according to the results in
can be approximated by a straight line in strong
Fig. 5.
inversion, as shown in Fig. 6(b), but the value deduced for
has been observed to be very sensitive to the range of the
curve used to apply the least square routine, thus leading to
inaccurate results.
IV. EFFECT
ON THE
GATE CAPACITANCE
The correction given by (14) is not directly justified in
the gate capacitance expression as it has been obtained only
for the
relationship. Nevertheless, it has been
used empirically even for the capacitance [8], [9], [19]. In
fact, Baccarani and Wordeman showed that if the centroid
capacitance is compared with the actual inversion-layer capacitance, the two curves cross each other and do not merge even
asymptotically (see Fig. 2 in [3]). Therefore, if a deviation
capacitance
is searched as
(15)
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IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 44, NO. 11, NOVEMBER 1997
(a)
Fig. 7. Inversion-layer capacitance versus gate voltage. Dots: Numerical
results. Long-dashed lines: the “centroid capacitance” defined in text.
Short-dashed lines: the “corrected centroid capacitance” defined in (19)
in text.
(b)
Fig. 6. Electron concentration per unit area in the inversion layer versus
the gate voltage for the two extreme doping values: (1) 1017 cm03 and (2)
3 1 1018 cm03 . Solid lines: Data obtained with the numerical procedure. Solid
circles: Results obtained by using (13) with the numerical values for zI and
dep . Solid squares: Results obtained with a simple strong-inversion model
but with centroid correction. Results are shown in (a) logarithmic scale and
(b) linear scale. tox = 10 nm.
with
where the superindex “si” stands for “strong inversion.”
,
which is computed numerically according to (16), is compared
with
and the “centroid capacitance”
in Fig. 7 as
a function of the gate voltage. Numerical results are shown
in dots, long-dashed lines show the “centroid capacitance,”
and the “corrected centroid capacitance,” which is defined in
(19), is plotted in short-dashed lines. The crossing between
and
observed by Baccarani and Wordeman is
also produced with the quantum-model data, and the better
asymptotic behavior given by
is apparent. Hartstein and
Albert also proposed an inverse-capacitance term proportional
but multiplied by a numerical factor [22], which is
to
simply 2/3 with our approximation. [Instead of the factor 2/3,
a factor
would have been obtained if the more general
(4) had been used.] By using this asymptotic behavior, we can
define a deviation capacitance that does not change its sign.
given by
(20)
(16)
then
is negative for gate-voltage values greater than that of
the cross point. To obtain a suitable asymptotic expression, we
used the following reasoning. Regarding (9), in the very strong
inversion limit, any increase in
is produced by the increase
, while
, and thus
, remain almost constant. This
in
is the well-known effect of the depletion-layer screening by
the inversion layer. Therefore, the derivative
(17)
must vanish at this limit.
Here, we can use the empirical (6), which is obtained by
fitting the numerical data produced by the quantum model, and
consider
as a parameter (which is not affected by the
variations). The result is
(18)
In the case
, the derivative of (17) approaches
unity, and (18) provides
(19)
is basically due to the nonnegThe physical origin of
ligible influence of the depletion charges. By using (18) and
(20), we can obtain
(21)
Although (21) cannot be easily used for modeling, it aids in
. This deviation capacitance is partly
the interpretation of
due to the nonnegligible
versus
but mainly due to the
nonvanishing derivative [which is defined in (17)], which is
a measure of the change in the amount of the space charges
in the depletion layer due to variations in the gate voltage.
This derivative should decrease as the screening effect of
increases. Therefore, following Takagi and Toriumi, we can
assay a dependence
. These authors proposed this
defined above and claimed to have obtained
dependence for
good agreement without the change in sign predicted. This may
be due to the fact that they are actually using the difference
between the centroids obtained with the quantum and classical
models for instead of the centroid itself. This difference may
be close to
. We have also obtained a good agreement
to the empirical expression
(22)
LÓPEZ-VILLANUEVA et al.: EFFECTS OF THE INVERSION LAYER CENTROID
Fig. 8. Gate-to-channel capacitance versus gate voltage. Solid line: Numerical results. Symbols: Adjustment with (24). Doping concentration: 1017 cm03
(solid circles) and 3 1 1018 cm03 (solid squares). tox = 10 nm.
with being a fitting factor depending on the doping concentration, ranging from 0.774 for
cm to 0.391 for
cm . The error in (22) is acceptably low
except for very high
values, where it is not very important
because
dominates. We have compared the numerical
gate-to-channel capacitance as defined in the split-capacitance
method
(23)
to the value provided by the approximation:
(24)
where the approximation given in (6) is used for , and the
correction term for the depletion-layer capacitance given by
Sodini et al. [25] has been neglected because it contributes to
less than 1% for
higher than 10 cm
Results for this
comparison are shown in Fig. 8, where the gate-to-channel
capacitance is plotted versus gate voltage. Numerical results
are plotted as a solid line, and the adjustment with (24) is
shown in symbols (solid circles for a doping concentration
of
cm , and solid squares for 3
10 cm ). Good
agreement of the results from (24) with the numerical results
is apparent for the two extreme doping concentrations.
V. CONCLUSIONS
The increase in the average penetration of the inversion
layer, termed the inversion-layer centroid, is one of the main
quantum effects on MOSFET behavior. This averaged penetration can be described as a function of the depletion and
inversion charges by a compact empirical expression, which
is valid across a wide range of electrical and technological
variables.
One important effect of the centroid on the static characteristics is the strong increase of the surface potential with
the gate voltage, even in the strong inversion regime. If a
“depletion-layer surface potential,” which is less dependent
on the gate voltage, is defined, the centroid effect leads to a
modification of the oxide thickness, resulting in an effective
electrical oxide thickness that is dependent on the gate voltage itself. This effective electrical oxide thickness is not an
1921
empirical correction as it has been derived directly from wellknown equations. Introducing the centroid correction leads to
using the “depletion layer surface potential” in the definition
of threshold voltage as well. The body-effect coefficient is
therefore not affected by the centroid correction.
Concerning the inversion-layer capacitance, the “centroid
capacitance,” which was defined in the text, is confirmed not to
be a correct asymptotic value in the very strong-inversion limit.
Nevertheless, it is a suitable asymptotic value if it is multiplied
by a constant derived from the empirical expression for the
centroid. Good agreement with the numerical results for the
gate-to-channel capacitance is obtained with this expression.
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1922
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Juan A. López-Villanueva (M’90), for photograph and biography, see p. 846
of the May 1997 issue of this TRANSACTIONS.
Pedro Cartujo-Casinello was born in 1969. He
received the B.S. degree in physics in 1995 from
the University of Granada, Spain. He is currently
pursuing the Ph.D. degree at the Department of
Electronics, University of Granada.
His current research interest is focused on the
study of MOSFET modeling for analog circuits.
Jesus Banqueri (M’95) was born in 1965. He graduated with a degree in physics in 1988 and received
the Ph.D. degree in 1994 from the University of
Granada, Spain, with a thesis on the experimental
study of electron mobility in MOS transistors.
Since 1990, he has been working on MOS device physics, including characterization, simulation,
and modeling of MOSFET’s. His current research
interests include modeling of MOSFET’s for analog
applications, and in particular, modeling of the
mobility and determination of the series resistance
and the effective channel length in submicron MOSFET’s. He is an Associate
Professor at the University of Granada, Spain.
F. Gámiz, for photograph and biography, see p. 846 of the May 1997 issue
of this TRANSACTIONS.
Salvador Rodrı́guez was born in 1971. He graduated with a degree in physics in 1994 and in
electronic engineering in 1997, both at the University of Granada, Spain. He is currently pursuing
his Ph.D. degree at the University of Granada,
Spain, where he is a Teaching Assistant. His current
research interest is the study of hole confinement in
heterostructures.