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Threshold voltage model for narrow width MOSFETs
Solid-State Electronics Vol. 36, No. 9, pp. 1251-1260, Printed in Great Britain
1993
0038-I 101/93 $6.00 + 0.00 Pergamon Press Ltd
THRESHOLD VOLTAGE MODEL FOR NARROW WIDTH MOSFETs SUNG-HUN OH’, AHMAD USMAN’, JULIAN J. SANCHEZ~
Arizona
for Solid State Electronics AZ 85287 and *Intel Chandler, AZ 85224, U.S.A. (Received
12 November
THOMASA. of Electrical Engineering, W. Chandler Road,
1992; in revised form 5 February
1993)
Abstract-An analytical threshold voltage model for narrow width MOSFETs is presented. The model is based on a Fourier solution of the two-dimensional Poisson’s equation for the fully recessed oxide isolation scheme. Realistic boundary conditions are used, which make this formulation more accurate than charge conservation models and applicable to different oxide isolation schemes. The present model provides an analytical closed-form expression for the threshold voltage as a function of design and process parameters such as device width, oxide isolation type, ion implantation, and substrate bias. The model requires only one fitting parameter whose sensitivity to process and design parameters is investigated. The results obtained for the fully recessed oxide isolation case are compared with numerical data, and good agreement is obtained for device widths down to 0.25 pm for uniformly doped substrates and 0.4 urn for nonuniformly doped substrates. Surface potentials and electric fields are also generated from the solutions predicted by this model. The same Fourier solution technique can be used to model different isolation schemes by changing the fitting parameter. This is demonstrated with the LOCOS and fully recessed isolation schemes. The speed and accuracy of this model make it extremely useful for CAD applications.
1. INTRODUCTION
desire to reduce the device size to increase packing density and enhance performance and reliability has led to several innovative isolation schemes. One isolation scheme able to provide increased packing density is the fully recessed oxide isolation. As the individual device dimensions are scaled down, the sensitivity of various device parameters with geometry increases, and the second order effects become more significant. Of the device parameters modified by device size reduction, the threshold voltage is the most significant. An accurate prediction of the threshold voltage is required since it is used to determine the circuit noise margins, gain, mobility, effective channel length and width. In a conventional MOSFET, local oxidation of silicon (LOCOS) process is used to isolate and define the width of the device. The increase in threshold voltage is the result of the equipotential contour spreading apart as the thin gate oxide transitions into thick field oxide. In addition, the field channel implant which penetrates into the thin gate oxide region during processing also contributes to an increase in threshold voltage as the width of the device decreases. By contrast, in a fully recessed structure, as shown in Fig. 1, the electrostatic potential increases near the edges of the channel due to the concentration of field lines which exist at the comers and the side walls. This results in the threshold voltage decreasing with decreasing device width. This phenomena is referred to as the inverse narrow width effect[l]. The
SSE MieC
Several techniques have been used to model the threshold voltage for small geometry MOSFETs. The first approach is based on the charge conservation principle. In this technique, when the channel length becomes comparable to the source and drain depletion width, the charge sharing underneath the gate with the source and drain depletion regions becomes important. For narrow width devices, charge conservation models attempt to determine the amount of fringe charge induced at the edge of the isolation regions by some simple geometrical approximation. Jeppson[2], Akers[3], Merkel[4], Hong and Chang[S], and DeMassa and Chien[6] have developed models based on this approach. The advantages of these models lie in their simplicity and ability to analytically display trends. However, these models have difficulties in fitting experimental data over the range of process parameters typically employed in fabrication. The second technique used to determine the threshold voltage is numerical modeling which is based on finite difference or finite element methods. These models solve the continuity, transport and Poisson’s equations iteratively. Though these models are far more accurate, they are not useful for CAD simulations because they consume too much computer time, making them very expensive. Kroell and Ackermann[7j and Noble and Cottrell[I?] have developed models based on this technique. Another approach based on the Fourier solution of Poisson’s equation has been used by several authors. This approach provides a good balance between
1251
1252
SUNG-HUNOH et al.
(a)
(b) Fig. 1. Fully recessed isolation scheme and surface potential.
accuracy and simulation time and has been successfully used to model short channel MOSFETs. A two
dimensional ana’:vt.ical solution of Poisson’s equation is obtained with approximate boundary conditions. Unlike numerical modeling, these solutions are less computation intensive and the accuracy depends on the degree of approximation used in the boundary conditions. Ratnakumar and Meindlp] derived a short channel model based on an analytical solution of the two dimensional Poisson’s equation in the depletion region under the gate of a MOSFET. They reduced Poisson’s equation to an equivalent Laplace equation by use of a trial function. Their model includes the effect of back bias and nonuniform doping, but it tends to overestimate the short channel effects because they assumed a constant surface potential over the length of the channel with discontinuities at the source and drain junctions. Also, infinite source and
drain junction depths were assumed, which further overestimates the short channel effect. Poole and Kwong[lO] improved this model by dropping the constant surface potential assumption and using Gauss’s law at the interface. However, they assumed an infinitely deep junction, which again results in overestimation of short channel effects. Pfiester et al.[l 1] improved the original model proposed by Poole and Kwong by considering the effect of finite source-drain junction depths. However, the boundary conditions on the sourc+drain junctions were assumed to be rectangular which again resulted in an overestimation of the short channel effect. Kendall and Boothroyd[ 121improved the model of Nester et al.[l l] by including the effects of curved junctions. Their model also includes ion implantation, the effect of finite junction depth, and does not require fitting parameters. The model is in good agreement with numerical data, down to half micron
Threshold voltage model for narrow width MOSFETs
channel lengths. However, it diverges significantly for high substrate bias. All of these Fourier models have been developed for short channel devices but have not been extended to narrow width devices. Extension of these models to include narrow width effects has been hampered by the lack of boundary conditions along the width direction, since these boundary conditions are floating rather than fixed. The approach used in the present work is similar to the one used by Kendall and Boothroyd[lZ]. The boundary conditions along the width direction are extracted by plotting the potential along the sidewall under various process and bias conditions using the numerical program PISCES[ 131.From the analysis of this data, an equation for the boundary conditions is generated by interpolation. Once the boundary conditions are established along the sidewalls, the Poisson’s equation is transformed into a homogeneous partial differential equation through use of trail functions. The resulting Laplace equation is then solved by separation of variables. In this way, narrow width effects are incorporated into the solution for the potential. 2.
MODEL
FORMULATION
A threshold voltage model based on an approximate two dimensional analytical solution of Poisson’s equation for a fully recessed MOSFET is presented. The width cross section of the fully recessed isolation scheme is shown in Fig. l(a), where the thick oxide grown into the substrate with 90” recessed angle defines the width of the MOSFET. Note that the origin is in the middle (z = 0) and the device width is W,
1253
a2u(Y),~N -K
4
-g-
(3)
*-
and
J2T(Y) 4 -=~,Nty)=f(y) aY*
yields the Laplace equation as:
(5) The trial functions U(y) and T(y) are obtained later in this paper by solving eqn (3) and (4) under the appropriate boundary conditions. 2.2. Boundary conditions and trial factions The boundary condition at the oxide-silicon interface (y = 0) is given by applying Gauss’s law at the gate, which yields: ~(O,z)-R.T..~(O,z)=V,,,
(6)
where T,, is the gate oxide thickness,
and VGM=
vG-
vFB~
also, V, is the gate voltage, 65, is the permittivity of the oxide and VFBis the flatband voltage. The depletion width is not constant along the width of the device. However, we can still define a boundary condition at the minimum depletion width edge. The electric field at the depletion edge can be defined by assuming the vertical hole current flow to be zero. Hence: kT 1 ap(y) E(d)=--4 p(d) ay
4’
(7)
where k is Boltzmann‘s constant, T is the temperature and p is the hole concentration. Also the electric field at d in terms of potential is given by:
f$(d, z) =
-E(d),
where E(d) = 0 for uniform doping.
1254
et al.
SUNGHUNOH
The boundary conditions along the sidewalls are derived by interpolating the potential profile extracted from the simulation program PISCES. From the analysis of the PISCES data, it was found that the sidewall potential is not a strong function of the field oxide thickness, but is a function of both the doping and the gate oxide thickness. These effects must be incorporated into any solution describing the potential along the sidewall. Another constraint imposed on the boundary conditions is that they must be of such a form that analytical solutions are possible. This requirement is simply imposed to reduce the complexity of the solution by providing analytically integrable functions for the series coefficients. For the fully recessed isolation scheme, the potential along the sidewall as a function of distance is shown in Fig. 2. It was observed that the best fits were obtained when the potential profile along the sidewalls is fitted using a quadratic equation with two fitting parameters and is given by:
-aw ay
=0
and U(d) = c,
- Kdsy
u(y)=;y2-Kdy+;d2+c.
The trial function T(y) associated with the nonuniform profile is obtained by solving eqn (4) with the following boundary conditions: YY) ay
= -E(d)
x
y=d
--
(9)
where Z’(Y) is the trial function associated with the nonuniform doping and is zero for the uiniform case, K is given by qN,/c, , A and B are the fitting parameters and &r represents the potential at the comers shown in Fig. l(b). To satisfy the boundary condition of eqn (8), the coefficient B of eqn (9) must equal A. Thus, at f W/2:
kT
1
T(d)
=
7
- 2d) + 9s + T(Y) - T(0).
(10)
The trial function U(y) associated with the background doping (uniform profile) is found by solving equation (3) with the following boundary conditions: POTENTIAL ALONG THE SIDEWALLS I I I I I I I
0.7 0.6
-
Potential at sidewall
------
Potentialatcenter
dN(Y) aY
(14)
=d
kT
N.4 log fi % log NA + N(d)’ p(d) q
(15)
where quasi-neutrality is assumed and T(d) = 0 for uniform doping. Integrating eqn (4) twice with the above boundary conditions yields: T(Y) = G(Y) - YNd)
+ E(d)1 f T(d) - G(d) + dP(d)
0.8~
(13)
and
+ 6X + T(Y) - T(O),
4(Y, I!IW/Z) = YY(Y
(12)
where the parameter c is some constant determined in eqn (28). Integrating eqn (3) twice using the above boundary conditions yields:
q N*+N(d)
f$(Y, + W/2) = YY2
(‘1)
y=d
+
EWI, (16)
where F(y) represents the zero-order moment of the distribution and is given by eqn (4) as: F(Y) =
For a Gaussian profile, F(y) is obtained as: F(Y) =
_I
!
where Nr represents the peak concentration dopant distribution and is given by:
of the
Np =
where D is the dose, RP is the range of the distribution and AR,, is the straggle. F(d) and F(0) are obtained by evaluating F(Y) at d and 0, respectively. The function G(y) represents the first-order moment of the dopant distribution and is given by:
o
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 DEE"CH [clml
Fig. 2. Potential along the sidewalls (fully recessed isolation scheme).
Assuming a Gaussian profile for the dopant distribution, G(y) becomes:
Threshold voltage model for narrow width MOSFETs
1255
Combining eqn (24) with the boundary condition (21) gives:
XgLf(~)
kCv) = JJcos(k.v)
(18) J2i(z)
The comer potential & is derived by matching the semiconductor electric field at the gate, obtained from eqn (6), with the electric field at the sidewalls at y = 0 and z = f W/2, obtained from eqn (lo), yielding:
(19)
1
= VoM-y-c
+ R . T,,[F(O) -F(d) - E(d) - Kd] - T(O),
(27)
where:
To solve for y. in eqn (27), c is chosen such that the right hand side of eqn (27) vanishes: c=V,,-T(O)-? + R . T,, [F(O) - F(d) - E(d) - KdJ.
(28)
Since the right hand side of eqn (27) vanishes, so must the left hand side: I-R*T,$tan(Y.)=O or
$(O,z)+T(O)+U(O)-R-7',,
[Jl(y,z)+TOt)+Ll(yllh-,=
(26)
15,= k,d.
Notice that the potential &r can be larger or smaller than the center potential 2&, depending on the fitting parameter A. Hence the potential q&r for any isolation scheme can be correctly obtained by choosing A. Substituting eqn (2) into eqns (6), (8) and (10) generates boundary conditions for Laplace equation as follows: At the oxide-silicon interface
va4. (20) Using eqns (23), (25) and (26) yields:
At the depletion edge:
$
l-Rt,,x~tan(Y~)
+F(d) + E(d) -F(O)].
x$
~nWsW,vll
and substituting eqn (26) into eqn (23) and then substituting the result into eqn (20) yields:
-Yexp[-(y.$$]].
4 sT= VGM-R.Tox(Kd.4
+
~0s k,(y - d)
(d, z) = 0.
J. x sinh(k,z) cos(Y*)
At z = &-W/2:
+&T- WY) -
m% (22)
where T(0) = 0 for uniform doping. 2.3. Total solution Using the boundary conditions at the interface, eqn (20), and at the depletion edge, eqn (21); the Laplace equation is solved by the technique of sepatration of variables. This method assumes that the Laplace solution can be written as the product of two functions as follows: $(v, z) = k(y)i(z),
(23)
where k(y) is a function of y only and i(z) is a function of z only. Substituting eqn (23) into eqn (5) and solving for k(y) and i(z) yields: hb) = J, sin(k,y) + Jz cos(k,y)
(24)
i(z) = Js sinh(k,z) + J, cosh(k,z),
(25)
, COdY”) 11
+ - J:, cosh(k,z)
where J,, = J2J,, and JL = J2J,. The coefficients J. and J; are determined using the boundary conditions given by eqn (22) with the orthogonality of the Fourier series. From eqns (2) and (30), the total solution for the Poisson’s equation is now given by:
d(~.z)=~~{cosk.(~ -d)[&sWkz) II
+
--&osh(k.z) n
II
+ T(y) + V(Y).
(31)
2.4. Threshold voltage Once the Laplace equation is solved and the coefficients Jn and J’. determined, all the terms in eqn (31) are known and the potential can be evaluated. From orthogonality J, = 0 and the potential as a function of y and z is obtained as:
and
where J, , Jz, J3, J, and k. are arbitrary constants.
(30)
Q(~, z)
-so [--&
cos k,,ti - d)coWnz) 1 + U(Y) + Tti),
(32)
SUNG-HUN OH et al.
1256 -HOLD
VOLTAGE VS WIDTH
equals the majority carrier concentration at the depletion edge. For uniform doping, the onset of inversion occurs when N,(d) equals NA and the surface potential reduces to 2& (twice the Fermi potential of the substrate). To determine the threshold voltage Vm,, eqn (32) is used to calculate the minimum surface potential i.e.
‘C N~~lx’O’6cm-‘,$l=“~A.
acdVBs=-2.W
>________
d@ 0) = -
FULLY
------
LOCOS
RECESSED
A-0.5
where U(0) is obtained y = 0 and is given by:
A-1.5
: 0
5 J; + U(0) + T(O),
(38)
“=ZO
by evaluating
eqn (13) at
U(O)=$d’+c.
1
2
3 4 5 6 DEVICE WIDTH [FITI]
Fig. 3. Threshold voltage as a function of device width for different isolation schemes.
Eqn (38) can be arranged for Vo, as:
f’o, = 4 (0, 0) - R . To, x [F(O)-W)-E(d)-W
where:
- BNW, (3%
where
R.T,;y,siny, d
1>
(33)
where a,=l+-.
sin 2y,
R
1
T ynsin Y, ox. d
.
2Yfl To evaluate the threshold voltage VTNWrthe following three conditions are imposed on #~(y, z): g
At threshold:
and d(O, 0) is given by eqn (36) eqn (39) becomes, after simplifying:
(0, W,) = 0
9M Wm)= T(d)
(40)
(35)
and (36) where ni is the intrinsic carrier concentration, NA is the substrate doping, N,(d) is the depletion edge doping, and V,, is the source to bulk voltage. For a Gaussian distribution N,(d) is given by:
THRESHOLD VOLTAGE VS WIDTH ,111
-4
l
N,(d) = NA+ N,exp[
-(sy].
(37)
The first condition determines the minimum surface potential position W,,, in the channel. By symmetry W, = 0 and this condition need not be extracted. The second condition is used to extract the minimum depletion depth d. The third condition is used to define the threshold voltage after Doucet and Van DeWiele[l4]. Their definition of inversion more closely relates to the commonly used techniques to extract threshold voltage experimentally. For nonuniform doping, the onset of inversion is defined when the minority carrier concentration at the surface
MINIMOS
N =5.5~10’~&~ A ~,,$50A
A=10.71-15.222W+7.785W2-1.2633&
-=
0.6 0
1
2
3
4 5 6 DEVICE WIDTH [~ml
Fig. 4. Threshold voltage as a function of substrate bias for LOCOS isolation scheme.
MOSFETs
POTENTIAL
1257
PROFILE
W=lp
0.6
A=O.8
N A = lx 10’6an -’
z
Tm =200A
VBS=ov
8 2E
OS 0.4
&
0.3 0.2
Fig. 5. 3-D plot of potential.
where C,, is the oxide layer capacitance per unit area and is equal to c~JT.,~. Equation (41) can be written as: VTNW =
VT,
-
BNW 9
(42)
where vTL=
410qyy VFB+kT
and represents the threshold voltage of a nonuniformly doped long channel device[lS]. The physical significance of eqn (42) is now immediately evident. Since V,, is the threshold voltage for a long channel device and BNWis the correction term to account for narrow width effects, eqn (42) states that the threshold voltage of a narrow width MOSFET is equal to the long channel threshold voltage minus a correction factor BNw. For a wide device BNw approaches zero and the model predicts the correct long channel threshold voltage. Note that pNwcan be. negative or positive depending on whether the structure is fully recessed or isoplanar in nature.
To determine VTNw, an initial guess is made for d to estimate yn using eqn (29). Equation (29) is then solved together with eqn (35) to determine a new value for d. This process is repeated until the difference between successive values of d is small and the solution has converged. Only the first four terms of the series in eqn (32) are needed for the required degree of accuracy. These equations are solved using a multi-dimensional damped Newton’s method and the threshold voltage V,, is obtained from eqn (41). 3. RESULTS
3.1. Fitting parameters Once the Fourier solution is obtained, the fitting parameter A must be determined. The sensitivity of A to the various process parameters such as oxide thickness, substrate doping and the design parameters such as device width and back bias, must also be investigated. The proper choice of A allows modeling of the inverse narrow width (fully recessed isolation) and the narrow width (LOCOS process) effects (Fig. 3).
SUNG-HUN OH et al.
1258 FIlTINC l.l
PARAMETER
A VS WIDTH
TRRRSHOLD 0.5
7 NA=l~lO”cm’~
1 .
E
0.4
g
0.3 0.2
9
0.1
4
0.9
f! 5:
0.8
s i3 *
0.7
8
Ls &f
0.6 .’
6
0.5 0
0.25
I
I
I
I
I
0.5
0.75
1
1.25
1.5
DEVICE
WIDTH
Hence, this model is quite general in nature, and can be used to model different isolation schemes using the correct choice of the fitting parameter. In Fig. 4, the threshold voltage shifts obtained from the three dimensional numerical program MINIMOS 5.0 [16] for the LOCOS process is shown. Threshold voltage was obtained by increasing the gate voltage until the minority carrier concentration at the surface equals the majority carrier concentration at the depletion edge. To extract the fitting parameters on various process parameters for the fully recessed isolation structure, several threshold voltages were obtained from PISCES simulations. Substrate doping concentrations from 1 x 10’5cm-3 to 1 x 10i7cm-3 and gate oxide thicknesses of 100400 8, were investigated. For the fully recessed isolation schemes, the range of values for A was limited such that the surface potential rises at the corners of the sidewalls and the threshold voltage matches with the PISCES results. A typical plot of the potential as a function of y and z is shown in Fig. 5. Notice that the potential at the THRESHOLD VOLTAGE VS WIDTH I 1 1 I 1 0
PISCES
T,,x=4wn
o
c
-0.1
VOLTAGE
I
I
1
2
I
VS WIDTH f
I
$
-0.2 0
[pm]
Fig. 6. Fitting parameter A as a function of device width.
2.5
8
t
3 4 5 DEVICE WIDTH
6 [Km]
Fig. 8. Threshold voltage as a function of device width (for different oxide thicknesses).
corners is higher than the potential in the middle. Both of these results are in agreement with previous theory. Plots of A vs doping, gate oxide thickness and device width were generated indicating that A was not sensitive to changes in oxide thickness or doping concentration concentrations for less than 1 x 10i6cm-‘. Figure 6 shows the variation of parameter A as a function of device width. The variation of A seems to have a quadratic relationship with respect to device width. However, the best fit using a linear equation was extracted for simplicity. For a given device width and uniform doping case, the following linear expressions may be used to calculate A: A = 0.32 + 0.593 W < 1
for
NA < 1 x lOi cme3
< 1 for
NA > 1 x 10’6cm-3
and A =0.3796+1.067W
(44) where W is the device width in microns.
0.6 _
THRESHOLD VOLTAGE VS DOSE I I I I I
veS=Ov N =lxlo”em0
0 0
1
2
3 4 5 6 DEVICE WIDTH [Km]
Fig. 7. Threshold voltage as a function of device width (for different substrate dopings).
0
I
I
I
2
4
6
I
I
8
10
I2
DOSE [ 10 ’ ‘cm- *I
Fig. 9. Threshold voltage as a function of dose.
Threshold voltage model for narrow width MOSFETs THRESHOLD VOLTAGE VS ENERGY
4. CONCLUSIONS
O.* -
.
0
20
40
60
PISCES
80
100
:
120
ENERGY [KeV] Fig. 10. Threshold voltage as a function of energy.
For nonuniform doping, it was observed that A is also not a very strong function of the implant dose D or the energy E. For a given device width and nonuniform doping, the value of A can be calculated from the following equation: A =0.1088+0.905W~1 3x
10’1cm-2
for x 10’2cm-2
and 30 keV d E < 100 keV.
1259
(45)
The error in threshold voltage that is introduced by neglecting the weak dependence of A on dose and energy is less than 20 mV. 3.2. Threshold voltage Once the fitting parameters are evaluated, the threshold voltage is calculated using eqn (42). In Figs 7 and 8, the threshold voltage as a function of device width is presented for 3 values of oxide thickness. Eqn (44) was used to determine the fitting parameter A. As predicted by the simple charge sharing theory, the present theory also indicates that the inverse narrow width effect is more pronounced for higher doping and thicker gate oxide. The difference between the threshold voltage calculated by the present model and PISCES is at worst case 20mV, and can be further reduced by adjusting eqn (44). Some deviation from the numerical data is due to the weak dependence of A on substrate doping and oxide thickness, which was not included. Finally, for the nonuniform case, the plots of threshold voltage vs dose and energy are given in Figs 9 and 10. Eqn (45) was used to calculate the fitting parameter A. Note that all the trends agree with previous theory. For comparison, numerical data from PISCES is also shown on these same plots. The slight deviation from PISCES data can be attributed to the weak dependence of A on dose and energy, which was not included.
A threshold voltage model for the inverse narrow width effects of a fully recessed MOSFET is presented in this paper. The model is based on a Fourier solution of the two dimensional Poisson’s equation. The accuracy of this approach depends on the accuracy of the boundary conditions used in solving Poisson’s equation. Realistic boundary conditions are used, which not only provide insight into the physics of the device, but also make the model more accurate than the charge conservation models. The Fourier mode1 is significantly faster than numerical models and furthermore its accuracy approaches that obtainable from the numerical models. Some of the important features of this model are listed as follows: 1. The model provides a good balance between
accuracy and speed. 2. Realistic boundary conditions are incorporated in the general solution. 3. The narrow width effect is described as the superposition of the long channel threshold voltage and a correction term. 4. The model is able to accomodate device widths down to 0.25 and 0.4pm for uniform and nonuniform doping, respectively. 5. Substrate bias and ion implantation, both important for VLSI design are incorporated. 6. The approach adopted makes this model more general in nature and it can be extended to different isolation schemes by only changing the fitting parameter. 7. Correct trends for surface potential and electric fields are also generated. 8. Short channel solutions developed using the Fourier or Green Function’s methods can be extended to full three dimensional solutions using the proposed boundary conditions for any isolation scheme. The sixth and last feature are of extreme importance because several other modern isolation schemes can be modeled for small geometry devices resulting in a useful CAD simulation model for VSLI applications.
REFERENCES
I. N. Shigyo, M. Konaka and R. L. M. Dang, Eiecfron Lerr. 18, 274 (1982). 2. K. 0. Jeppson, Electron Lert. 11, 297 (1975). 3. L. A. Akers, Electron Lett. 17, 49 (1981). 4. G. Merkel, Solid-St. Electron. 23, 1027 (1980). 5. K. M. Hong and Y. C. Chang, J. appl. Phys. 61, 2387 (1987). 6. T. A. DeMassa and H. S. Chien, Solid-State Electron. 29, 409 (1986). 7. K. E. Kroell and G. K. Ackermann, Solid-Sk Electron. 19, 77 (1976). 8. W. P. Noble and P. E. Cottrell, IEDM, p. 582 (1976). 9. K. N. Ratnakumar and J. D. Meindl. IEEE J. Solid St. Circuits SC-17, 937 (1982).
1260
SONG-HUNOH et al.
10. D. R. Poole and D. L. Kwong, IEEE Electron Device Lert. EDLll, 443 (1984). 11. J. R. Pfiester, J. D. Shott, and J. D. Meindl, IEEE Trans. Electron Devices ED-32, 333 (1985). 12. J. D. Kendall and A. R. Boothroyd, IEEE Elecfron Deuice Left. EDL-7, 401 (1986).
13. PISCES II-B User’s Manual, Stanford University. 14. G. Doueet and F. Van DeWiele, Solid-St. Electron. 16, 417 (1973). 15. R. R. Troutman, IEEE Trans. Electron Devices ED-24, 182 (1977). 16. MINIMOS 5 User’s Guide, March (1990).
1993
0038-I 101/93 $6.00 + 0.00 Pergamon Press Ltd
THRESHOLD VOLTAGE MODEL FOR NARROW WIDTH MOSFETs SUNG-HUN OH’, AHMAD USMAN’, JULIAN J. SANCHEZ~
Arizona
for Solid State Electronics AZ 85287 and *Intel Chandler, AZ 85224, U.S.A. (Received
12 November
THOMASA. of Electrical Engineering, W. Chandler Road,
1992; in revised form 5 February
1993)
Abstract-An analytical threshold voltage model for narrow width MOSFETs is presented. The model is based on a Fourier solution of the two-dimensional Poisson’s equation for the fully recessed oxide isolation scheme. Realistic boundary conditions are used, which make this formulation more accurate than charge conservation models and applicable to different oxide isolation schemes. The present model provides an analytical closed-form expression for the threshold voltage as a function of design and process parameters such as device width, oxide isolation type, ion implantation, and substrate bias. The model requires only one fitting parameter whose sensitivity to process and design parameters is investigated. The results obtained for the fully recessed oxide isolation case are compared with numerical data, and good agreement is obtained for device widths down to 0.25 pm for uniformly doped substrates and 0.4 urn for nonuniformly doped substrates. Surface potentials and electric fields are also generated from the solutions predicted by this model. The same Fourier solution technique can be used to model different isolation schemes by changing the fitting parameter. This is demonstrated with the LOCOS and fully recessed isolation schemes. The speed and accuracy of this model make it extremely useful for CAD applications.
1. INTRODUCTION
desire to reduce the device size to increase packing density and enhance performance and reliability has led to several innovative isolation schemes. One isolation scheme able to provide increased packing density is the fully recessed oxide isolation. As the individual device dimensions are scaled down, the sensitivity of various device parameters with geometry increases, and the second order effects become more significant. Of the device parameters modified by device size reduction, the threshold voltage is the most significant. An accurate prediction of the threshold voltage is required since it is used to determine the circuit noise margins, gain, mobility, effective channel length and width. In a conventional MOSFET, local oxidation of silicon (LOCOS) process is used to isolate and define the width of the device. The increase in threshold voltage is the result of the equipotential contour spreading apart as the thin gate oxide transitions into thick field oxide. In addition, the field channel implant which penetrates into the thin gate oxide region during processing also contributes to an increase in threshold voltage as the width of the device decreases. By contrast, in a fully recessed structure, as shown in Fig. 1, the electrostatic potential increases near the edges of the channel due to the concentration of field lines which exist at the comers and the side walls. This results in the threshold voltage decreasing with decreasing device width. This phenomena is referred to as the inverse narrow width effect[l]. The
SSE MieC
Several techniques have been used to model the threshold voltage for small geometry MOSFETs. The first approach is based on the charge conservation principle. In this technique, when the channel length becomes comparable to the source and drain depletion width, the charge sharing underneath the gate with the source and drain depletion regions becomes important. For narrow width devices, charge conservation models attempt to determine the amount of fringe charge induced at the edge of the isolation regions by some simple geometrical approximation. Jeppson[2], Akers[3], Merkel[4], Hong and Chang[S], and DeMassa and Chien[6] have developed models based on this approach. The advantages of these models lie in their simplicity and ability to analytically display trends. However, these models have difficulties in fitting experimental data over the range of process parameters typically employed in fabrication. The second technique used to determine the threshold voltage is numerical modeling which is based on finite difference or finite element methods. These models solve the continuity, transport and Poisson’s equations iteratively. Though these models are far more accurate, they are not useful for CAD simulations because they consume too much computer time, making them very expensive. Kroell and Ackermann[7j and Noble and Cottrell[I?] have developed models based on this technique. Another approach based on the Fourier solution of Poisson’s equation has been used by several authors. This approach provides a good balance between
1251
1252
SUNG-HUNOH et al.
(a)
(b) Fig. 1. Fully recessed isolation scheme and surface potential.
accuracy and simulation time and has been successfully used to model short channel MOSFETs. A two
dimensional ana’:vt.ical solution of Poisson’s equation is obtained with approximate boundary conditions. Unlike numerical modeling, these solutions are less computation intensive and the accuracy depends on the degree of approximation used in the boundary conditions. Ratnakumar and Meindlp] derived a short channel model based on an analytical solution of the two dimensional Poisson’s equation in the depletion region under the gate of a MOSFET. They reduced Poisson’s equation to an equivalent Laplace equation by use of a trial function. Their model includes the effect of back bias and nonuniform doping, but it tends to overestimate the short channel effects because they assumed a constant surface potential over the length of the channel with discontinuities at the source and drain junctions. Also, infinite source and
drain junction depths were assumed, which further overestimates the short channel effect. Poole and Kwong[lO] improved this model by dropping the constant surface potential assumption and using Gauss’s law at the interface. However, they assumed an infinitely deep junction, which again results in overestimation of short channel effects. Pfiester et al.[l 1] improved the original model proposed by Poole and Kwong by considering the effect of finite source-drain junction depths. However, the boundary conditions on the sourc+drain junctions were assumed to be rectangular which again resulted in an overestimation of the short channel effect. Kendall and Boothroyd[ 121improved the model of Nester et al.[l l] by including the effects of curved junctions. Their model also includes ion implantation, the effect of finite junction depth, and does not require fitting parameters. The model is in good agreement with numerical data, down to half micron
Threshold voltage model for narrow width MOSFETs
channel lengths. However, it diverges significantly for high substrate bias. All of these Fourier models have been developed for short channel devices but have not been extended to narrow width devices. Extension of these models to include narrow width effects has been hampered by the lack of boundary conditions along the width direction, since these boundary conditions are floating rather than fixed. The approach used in the present work is similar to the one used by Kendall and Boothroyd[lZ]. The boundary conditions along the width direction are extracted by plotting the potential along the sidewall under various process and bias conditions using the numerical program PISCES[ 131.From the analysis of this data, an equation for the boundary conditions is generated by interpolation. Once the boundary conditions are established along the sidewalls, the Poisson’s equation is transformed into a homogeneous partial differential equation through use of trail functions. The resulting Laplace equation is then solved by separation of variables. In this way, narrow width effects are incorporated into the solution for the potential. 2.
MODEL
FORMULATION
A threshold voltage model based on an approximate two dimensional analytical solution of Poisson’s equation for a fully recessed MOSFET is presented. The width cross section of the fully recessed isolation scheme is shown in Fig. l(a), where the thick oxide grown into the substrate with 90” recessed angle defines the width of the MOSFET. Note that the origin is in the middle (z = 0) and the device width is W,
1253
a2u(Y),~N -K
4
-g-
(3)
*-
and
J2T(Y) 4 -=~,Nty)=f(y) aY*
yields the Laplace equation as:
(5) The trial functions U(y) and T(y) are obtained later in this paper by solving eqn (3) and (4) under the appropriate boundary conditions. 2.2. Boundary conditions and trial factions The boundary condition at the oxide-silicon interface (y = 0) is given by applying Gauss’s law at the gate, which yields: ~(O,z)-R.T..~(O,z)=V,,,
(6)
where T,, is the gate oxide thickness,
and VGM=
vG-
vFB~
also, V, is the gate voltage, 65, is the permittivity of the oxide and VFBis the flatband voltage. The depletion width is not constant along the width of the device. However, we can still define a boundary condition at the minimum depletion width edge. The electric field at the depletion edge can be defined by assuming the vertical hole current flow to be zero. Hence: kT 1 ap(y) E(d)=--4 p(d) ay
4’
(7)
where k is Boltzmann‘s constant, T is the temperature and p is the hole concentration. Also the electric field at d in terms of potential is given by:
f$(d, z) =
-E(d),
where E(d) = 0 for uniform doping.
1254
et al.
SUNGHUNOH
The boundary conditions along the sidewalls are derived by interpolating the potential profile extracted from the simulation program PISCES. From the analysis of the PISCES data, it was found that the sidewall potential is not a strong function of the field oxide thickness, but is a function of both the doping and the gate oxide thickness. These effects must be incorporated into any solution describing the potential along the sidewall. Another constraint imposed on the boundary conditions is that they must be of such a form that analytical solutions are possible. This requirement is simply imposed to reduce the complexity of the solution by providing analytically integrable functions for the series coefficients. For the fully recessed isolation scheme, the potential along the sidewall as a function of distance is shown in Fig. 2. It was observed that the best fits were obtained when the potential profile along the sidewalls is fitted using a quadratic equation with two fitting parameters and is given by:
-aw ay
=0
and U(d) = c,
- Kdsy
u(y)=;y2-Kdy+;d2+c.
The trial function T(y) associated with the nonuniform profile is obtained by solving eqn (4) with the following boundary conditions: YY) ay
= -E(d)
x
y=d
--
(9)
where Z’(Y) is the trial function associated with the nonuniform doping and is zero for the uiniform case, K is given by qN,/c, , A and B are the fitting parameters and &r represents the potential at the comers shown in Fig. l(b). To satisfy the boundary condition of eqn (8), the coefficient B of eqn (9) must equal A. Thus, at f W/2:
kT
1
T(d)
=
7
- 2d) + 9s + T(Y) - T(0).
(10)
The trial function U(y) associated with the background doping (uniform profile) is found by solving equation (3) with the following boundary conditions: POTENTIAL ALONG THE SIDEWALLS I I I I I I I
0.7 0.6
-
Potential at sidewall
------
Potentialatcenter
dN(Y) aY
(14)
=d
kT
N.4 log fi % log NA + N(d)’ p(d) q
(15)
where quasi-neutrality is assumed and T(d) = 0 for uniform doping. Integrating eqn (4) twice with the above boundary conditions yields: T(Y) = G(Y) - YNd)
+ E(d)1 f T(d) - G(d) + dP(d)
0.8~
(13)
and
+ 6X + T(Y) - T(O),
4(Y, I!IW/Z) = YY(Y
(12)
where the parameter c is some constant determined in eqn (28). Integrating eqn (3) twice using the above boundary conditions yields:
q N*+N(d)
f$(Y, + W/2) = YY2
(‘1)
y=d
+
EWI, (16)
where F(y) represents the zero-order moment of the distribution and is given by eqn (4) as: F(Y) =
For a Gaussian profile, F(y) is obtained as: F(Y) =
_I
!
where Nr represents the peak concentration dopant distribution and is given by:
of the
Np =
where D is the dose, RP is the range of the distribution and AR,, is the straggle. F(d) and F(0) are obtained by evaluating F(Y) at d and 0, respectively. The function G(y) represents the first-order moment of the dopant distribution and is given by:
o
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 DEE"CH [clml
Fig. 2. Potential along the sidewalls (fully recessed isolation scheme).
Assuming a Gaussian profile for the dopant distribution, G(y) becomes:
Threshold voltage model for narrow width MOSFETs
1255
Combining eqn (24) with the boundary condition (21) gives:
XgLf(~)
kCv) = JJcos(k.v)
(18) J2i(z)
The comer potential & is derived by matching the semiconductor electric field at the gate, obtained from eqn (6), with the electric field at the sidewalls at y = 0 and z = f W/2, obtained from eqn (lo), yielding:
(19)
1
= VoM-y-c
+ R . T,,[F(O) -F(d) - E(d) - Kd] - T(O),
(27)
where:
To solve for y. in eqn (27), c is chosen such that the right hand side of eqn (27) vanishes: c=V,,-T(O)-? + R . T,, [F(O) - F(d) - E(d) - KdJ.
(28)
Since the right hand side of eqn (27) vanishes, so must the left hand side: I-R*T,$tan(Y.)=O or
$(O,z)+T(O)+U(O)-R-7',,
[Jl(y,z)+TOt)+Ll(yllh-,=
(26)
15,= k,d.
Notice that the potential &r can be larger or smaller than the center potential 2&, depending on the fitting parameter A. Hence the potential q&r for any isolation scheme can be correctly obtained by choosing A. Substituting eqn (2) into eqns (6), (8) and (10) generates boundary conditions for Laplace equation as follows: At the oxide-silicon interface
va4. (20) Using eqns (23), (25) and (26) yields:
At the depletion edge:
$
l-Rt,,x~tan(Y~)
+F(d) + E(d) -F(O)].
x$
~nWsW,vll
and substituting eqn (26) into eqn (23) and then substituting the result into eqn (20) yields:
-Yexp[-(y.$$]].
4 sT= VGM-R.Tox(Kd.4
+
~0s k,(y - d)
(d, z) = 0.
J. x sinh(k,z) cos(Y*)
At z = &-W/2:
+&T- WY) -
m% (22)
where T(0) = 0 for uniform doping. 2.3. Total solution Using the boundary conditions at the interface, eqn (20), and at the depletion edge, eqn (21); the Laplace equation is solved by the technique of sepatration of variables. This method assumes that the Laplace solution can be written as the product of two functions as follows: $(v, z) = k(y)i(z),
(23)
where k(y) is a function of y only and i(z) is a function of z only. Substituting eqn (23) into eqn (5) and solving for k(y) and i(z) yields: hb) = J, sin(k,y) + Jz cos(k,y)
(24)
i(z) = Js sinh(k,z) + J, cosh(k,z),
(25)
, COdY”) 11
+ - J:, cosh(k,z)
where J,, = J2J,, and JL = J2J,. The coefficients J. and J; are determined using the boundary conditions given by eqn (22) with the orthogonality of the Fourier series. From eqns (2) and (30), the total solution for the Poisson’s equation is now given by:
d(~.z)=~~{cosk.(~ -d)[&sWkz) II
+
--&osh(k.z) n
II
+ T(y) + V(Y).
(31)
2.4. Threshold voltage Once the Laplace equation is solved and the coefficients Jn and J’. determined, all the terms in eqn (31) are known and the potential can be evaluated. From orthogonality J, = 0 and the potential as a function of y and z is obtained as:
and
where J, , Jz, J3, J, and k. are arbitrary constants.
(30)
Q(~, z)
-so [--&
cos k,,ti - d)coWnz) 1 + U(Y) + Tti),
(32)
SUNG-HUN OH et al.
1256 -HOLD
VOLTAGE VS WIDTH
equals the majority carrier concentration at the depletion edge. For uniform doping, the onset of inversion occurs when N,(d) equals NA and the surface potential reduces to 2& (twice the Fermi potential of the substrate). To determine the threshold voltage Vm,, eqn (32) is used to calculate the minimum surface potential i.e.
‘C N~~lx’O’6cm-‘,$l=“~A.
acdVBs=-2.W
>________
d@ 0) = -
FULLY
------
LOCOS
RECESSED
A-0.5
where U(0) is obtained y = 0 and is given by:
A-1.5
: 0
5 J; + U(0) + T(O),
(38)
“=ZO
by evaluating
eqn (13) at
U(O)=$d’+c.
1
2
3 4 5 6 DEVICE WIDTH [FITI]
Fig. 3. Threshold voltage as a function of device width for different isolation schemes.
Eqn (38) can be arranged for Vo, as:
f’o, = 4 (0, 0) - R . To, x [F(O)-W)-E(d)-W
where:
- BNW, (3%
where
R.T,;y,siny, d
1>
(33)
where a,=l+-.
sin 2y,
R
1
T ynsin Y, ox. d
.
2Yfl To evaluate the threshold voltage VTNWrthe following three conditions are imposed on #~(y, z): g
At threshold:
and d(O, 0) is given by eqn (36) eqn (39) becomes, after simplifying:
(0, W,) = 0
9M Wm)= T(d)
(40)
(35)
and (36) where ni is the intrinsic carrier concentration, NA is the substrate doping, N,(d) is the depletion edge doping, and V,, is the source to bulk voltage. For a Gaussian distribution N,(d) is given by:
THRESHOLD VOLTAGE VS WIDTH ,111
-4
l
N,(d) = NA+ N,exp[
-(sy].
(37)
The first condition determines the minimum surface potential position W,,, in the channel. By symmetry W, = 0 and this condition need not be extracted. The second condition is used to extract the minimum depletion depth d. The third condition is used to define the threshold voltage after Doucet and Van DeWiele[l4]. Their definition of inversion more closely relates to the commonly used techniques to extract threshold voltage experimentally. For nonuniform doping, the onset of inversion is defined when the minority carrier concentration at the surface
MINIMOS
N =5.5~10’~&~ A ~,,$50A
A=10.71-15.222W+7.785W2-1.2633&
-=
0.6 0
1
2
3
4 5 6 DEVICE WIDTH [~ml
Fig. 4. Threshold voltage as a function of substrate bias for LOCOS isolation scheme.
MOSFETs
POTENTIAL
1257
PROFILE
W=lp
0.6
A=O.8
N A = lx 10’6an -’
z
Tm =200A
VBS=ov
8 2E
OS 0.4
&
0.3 0.2
Fig. 5. 3-D plot of potential.
where C,, is the oxide layer capacitance per unit area and is equal to c~JT.,~. Equation (41) can be written as: VTNW =
VT,
-
BNW 9
(42)
where vTL=
410qyy VFB+kT
and represents the threshold voltage of a nonuniformly doped long channel device[lS]. The physical significance of eqn (42) is now immediately evident. Since V,, is the threshold voltage for a long channel device and BNWis the correction term to account for narrow width effects, eqn (42) states that the threshold voltage of a narrow width MOSFET is equal to the long channel threshold voltage minus a correction factor BNw. For a wide device BNw approaches zero and the model predicts the correct long channel threshold voltage. Note that pNwcan be. negative or positive depending on whether the structure is fully recessed or isoplanar in nature.
To determine VTNw, an initial guess is made for d to estimate yn using eqn (29). Equation (29) is then solved together with eqn (35) to determine a new value for d. This process is repeated until the difference between successive values of d is small and the solution has converged. Only the first four terms of the series in eqn (32) are needed for the required degree of accuracy. These equations are solved using a multi-dimensional damped Newton’s method and the threshold voltage V,, is obtained from eqn (41). 3. RESULTS
3.1. Fitting parameters Once the Fourier solution is obtained, the fitting parameter A must be determined. The sensitivity of A to the various process parameters such as oxide thickness, substrate doping and the design parameters such as device width and back bias, must also be investigated. The proper choice of A allows modeling of the inverse narrow width (fully recessed isolation) and the narrow width (LOCOS process) effects (Fig. 3).
SUNG-HUN OH et al.
1258 FIlTINC l.l
PARAMETER
A VS WIDTH
TRRRSHOLD 0.5
7 NA=l~lO”cm’~
1 .
E
0.4
g
0.3 0.2
9
0.1
4
0.9
f! 5:
0.8
s i3 *
0.7
8
Ls &f
0.6 .’
6
0.5 0
0.25
I
I
I
I
I
0.5
0.75
1
1.25
1.5
DEVICE
WIDTH
Hence, this model is quite general in nature, and can be used to model different isolation schemes using the correct choice of the fitting parameter. In Fig. 4, the threshold voltage shifts obtained from the three dimensional numerical program MINIMOS 5.0 [16] for the LOCOS process is shown. Threshold voltage was obtained by increasing the gate voltage until the minority carrier concentration at the surface equals the majority carrier concentration at the depletion edge. To extract the fitting parameters on various process parameters for the fully recessed isolation structure, several threshold voltages were obtained from PISCES simulations. Substrate doping concentrations from 1 x 10’5cm-3 to 1 x 10i7cm-3 and gate oxide thicknesses of 100400 8, were investigated. For the fully recessed isolation schemes, the range of values for A was limited such that the surface potential rises at the corners of the sidewalls and the threshold voltage matches with the PISCES results. A typical plot of the potential as a function of y and z is shown in Fig. 5. Notice that the potential at the THRESHOLD VOLTAGE VS WIDTH I 1 1 I 1 0
PISCES
T,,x=4wn
o
c
-0.1
VOLTAGE
I
I
1
2
I
VS WIDTH f
I
$
-0.2 0
[pm]
Fig. 6. Fitting parameter A as a function of device width.
2.5
8
t
3 4 5 DEVICE WIDTH
6 [Km]
Fig. 8. Threshold voltage as a function of device width (for different oxide thicknesses).
corners is higher than the potential in the middle. Both of these results are in agreement with previous theory. Plots of A vs doping, gate oxide thickness and device width were generated indicating that A was not sensitive to changes in oxide thickness or doping concentration concentrations for less than 1 x 10i6cm-‘. Figure 6 shows the variation of parameter A as a function of device width. The variation of A seems to have a quadratic relationship with respect to device width. However, the best fit using a linear equation was extracted for simplicity. For a given device width and uniform doping case, the following linear expressions may be used to calculate A: A = 0.32 + 0.593 W < 1
for
NA < 1 x lOi cme3
< 1 for
NA > 1 x 10’6cm-3
and A =0.3796+1.067W
(44) where W is the device width in microns.
0.6 _
THRESHOLD VOLTAGE VS DOSE I I I I I
veS=Ov N =lxlo”em0
0 0
1
2
3 4 5 6 DEVICE WIDTH [Km]
Fig. 7. Threshold voltage as a function of device width (for different substrate dopings).
0
I
I
I
2
4
6
I
I
8
10
I2
DOSE [ 10 ’ ‘cm- *I
Fig. 9. Threshold voltage as a function of dose.
Threshold voltage model for narrow width MOSFETs THRESHOLD VOLTAGE VS ENERGY
4. CONCLUSIONS
O.* -
.
0
20
40
60
PISCES
80
100
:
120
ENERGY [KeV] Fig. 10. Threshold voltage as a function of energy.
For nonuniform doping, it was observed that A is also not a very strong function of the implant dose D or the energy E. For a given device width and nonuniform doping, the value of A can be calculated from the following equation: A =0.1088+0.905W~1 3x
10’1cm-2
for x 10’2cm-2
and 30 keV d E < 100 keV.
1259
(45)
The error in threshold voltage that is introduced by neglecting the weak dependence of A on dose and energy is less than 20 mV. 3.2. Threshold voltage Once the fitting parameters are evaluated, the threshold voltage is calculated using eqn (42). In Figs 7 and 8, the threshold voltage as a function of device width is presented for 3 values of oxide thickness. Eqn (44) was used to determine the fitting parameter A. As predicted by the simple charge sharing theory, the present theory also indicates that the inverse narrow width effect is more pronounced for higher doping and thicker gate oxide. The difference between the threshold voltage calculated by the present model and PISCES is at worst case 20mV, and can be further reduced by adjusting eqn (44). Some deviation from the numerical data is due to the weak dependence of A on substrate doping and oxide thickness, which was not included. Finally, for the nonuniform case, the plots of threshold voltage vs dose and energy are given in Figs 9 and 10. Eqn (45) was used to calculate the fitting parameter A. Note that all the trends agree with previous theory. For comparison, numerical data from PISCES is also shown on these same plots. The slight deviation from PISCES data can be attributed to the weak dependence of A on dose and energy, which was not included.
A threshold voltage model for the inverse narrow width effects of a fully recessed MOSFET is presented in this paper. The model is based on a Fourier solution of the two dimensional Poisson’s equation. The accuracy of this approach depends on the accuracy of the boundary conditions used in solving Poisson’s equation. Realistic boundary conditions are used, which not only provide insight into the physics of the device, but also make the model more accurate than the charge conservation models. The Fourier mode1 is significantly faster than numerical models and furthermore its accuracy approaches that obtainable from the numerical models. Some of the important features of this model are listed as follows: 1. The model provides a good balance between
accuracy and speed. 2. Realistic boundary conditions are incorporated in the general solution. 3. The narrow width effect is described as the superposition of the long channel threshold voltage and a correction term. 4. The model is able to accomodate device widths down to 0.25 and 0.4pm for uniform and nonuniform doping, respectively. 5. Substrate bias and ion implantation, both important for VLSI design are incorporated. 6. The approach adopted makes this model more general in nature and it can be extended to different isolation schemes by only changing the fitting parameter. 7. Correct trends for surface potential and electric fields are also generated. 8. Short channel solutions developed using the Fourier or Green Function’s methods can be extended to full three dimensional solutions using the proposed boundary conditions for any isolation scheme. The sixth and last feature are of extreme importance because several other modern isolation schemes can be modeled for small geometry devices resulting in a useful CAD simulation model for VSLI applications.
REFERENCES
I. N. Shigyo, M. Konaka and R. L. M. Dang, Eiecfron Lerr. 18, 274 (1982). 2. K. 0. Jeppson, Electron Lert. 11, 297 (1975). 3. L. A. Akers, Electron Lett. 17, 49 (1981). 4. G. Merkel, Solid-St. Electron. 23, 1027 (1980). 5. K. M. Hong and Y. C. Chang, J. appl. Phys. 61, 2387 (1987). 6. T. A. DeMassa and H. S. Chien, Solid-State Electron. 29, 409 (1986). 7. K. E. Kroell and G. K. Ackermann, Solid-Sk Electron. 19, 77 (1976). 8. W. P. Noble and P. E. Cottrell, IEDM, p. 582 (1976). 9. K. N. Ratnakumar and J. D. Meindl. IEEE J. Solid St. Circuits SC-17, 937 (1982).
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SONG-HUNOH et al.
10. D. R. Poole and D. L. Kwong, IEEE Electron Device Lert. EDLll, 443 (1984). 11. J. R. Pfiester, J. D. Shott, and J. D. Meindl, IEEE Trans. Electron Devices ED-32, 333 (1985). 12. J. D. Kendall and A. R. Boothroyd, IEEE Elecfron Deuice Left. EDL-7, 401 (1986).
13. PISCES II-B User’s Manual, Stanford University. 14. G. Doueet and F. Van DeWiele, Solid-St. Electron. 16, 417 (1973). 15. R. R. Troutman, IEEE Trans. Electron Devices ED-24, 182 (1977). 16. MINIMOS 5 User’s Guide, March (1990).