Reference

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Publications

82-2471

CS1: A Two-Dimensional Finite

Element Charge-Sheet Model of A

Short-Channel MOS Transistor

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Electronics and

Electrical Engineering

Semiconductor Devices & Circuits Division

Washington, DC 20234

April 1982

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. DEPARTMENT OF COMMERCE

lONAL BUREAU OF STANDARDS

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LIBRARY MAY 14 1982

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NBSIR 82-2471

CS1: A TWO-DIMENSIONAL FINITE

ELEMENT CHARGE-SHEET MODEL OF A

SHORT-CHANNEL MOS TRANSISTOR

C. L Wilson and J. L Blue

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Electronics and

Electrical Engineering

Semiconductor Devices & Circuits Division

Washington, DC 20234

April 1982

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director
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Table of Contents

Page

Abstract 1

1 . Introduction 2

2. Theory 3

A. Two-Dimensional Charge-Sheet Model 3 B. Numerical Procedures 6

3. Computer Implementation 9

A. Software Modules 9

B. Physical Restrictions 11

4. Device Model Verification . 13

5. Conclusions 31

Acknowledgments ...... 31

References 32

Appendix A. Channel Current Calculation 34

Appendix B. File CS1.HLP 35

Appendix C. File CS1.EXP 40

List of Figures

Page

1 . The geometry used in the charge-sheet model 5

2. a) Two-dimensional level 2 mesh for the 1.12-ym transistor .... 14

b) Two-dimensional equipotentials for the 1.12-ym transistor ... 15

3. Convergence of the direct method for the test problem 18

4. Convergence of the multigrid method for the test problem 19

5. Relationship between the surface potential change and the Newton

step size for the test problem 20

6. a) Relationship between surface potential change (kT/q) and drain

current change (arbitrary unit, Jj-j = 1 ) for the level 1 mesh . . 21

b) Relationship between surface potential change (kT/q) and drain

current change (arbitrary unit, Jq = 1 ) for the level 2 mesh . . 22

iii
List of Figures (continued) Page

7. Comparison of the drain current calculated by the one- and twodimensional models .......... 24

8. Comparison of the subthreshold and triode region drain current

measured and calculated using the one- and two-dimensional models . 25

9. Comparison of the triode region drain current measured and calculated using the one- and two-dimensional models 27

10. Comparison of the output characteristics measured and calculated

using the two-dimensional model for the 3.20-ym transistor .... 28

11. a) Comparison of the output characteristics measured and calculated using the two-dimensional model for a 1.89-)om transistor . 29

b) Comparison of the output characteristics measured and calculated using the one-dimensional model for a 1.89-jjm transistor . 30

C-1 . Example of level 1 mesh 50

C-2. Example of level 2 mesh 51

C-3. Example of level 3 mesh 52

C-4. Example of level 1 projection 53

C-5. Example of level 2 projection 54

C-6. Example of level 3 projection 55

C-7. Example of contour plot 56

List of Tables

Page

1. Available Computer Types 12

2. Transistor Parameters 12

3. Iteration Convergence 16

iv
CS1 ; A Two-Dimensional Finite Element Charge-Sheet Model

of a Short-Channel MOS Transistor

C. L, Wilson and J. L, Blue*

National Bureau of Standards

Washington, DC 20234

Abstract

A two-dimensional charge-sheet model for short-channel MOS transistors has been developed. The unique feature of the model is that

the effect of channel inversion layer charge is included as a nonlinear integral boundary condition on the two-dimensional electrostatic field in the transistor. The average inversion layer charge

density and source-drain current are obtained directly from the

model rather than from the electron density or electron quasi-Fermi

level. The model retains all of the physical detail of more com- plex two-dimensional models such as sensitivity to source-drain

profile shape, channel profile, and oxide field shape. This allows

the model to represent the changes in drain current associated with

short-channel effects while still allowing simple comparison with

long-channel models. For long-channel transistors, the results of

this model are identical to Brews' long-channel charge-sheet model.

The accuracy of this model is verified by modeling a sequence of

transistors with channel lengths between 4,6 and 1,1 ym. In shortchannel transistors, effects previously attributed to high field

mobility are explained by simple two-dimensional electrostatics.

The simulations produced using this model have been compared to

experimental measurements on an array of n-channel MOSFETs; the

model is in good agreement for transistors with channel lengths as short as 1,1 ym. In this verification process, the model represented accurately the onset of subthreshold current, channellength-induced threshold voltage offset, and drain-field-induced

output conductance changes.

From studies of numerical accuracy, we conclude that the chargesheet model can easily simulate drain current with an accuracy

which exceeds that required for most applications. To obtain 5-

percent accuracy for drain current, a 146 element mesh is sufficient. Refinement of the 146 element mesh to a 455 element mesh

gives a current which is accurate to 0,16 percent. Average computer time for a high accuracy solution is 2.5 min on a DEC-20,'*’

The numerical solutions were obtained using general-purpose software for solving elliptic partial differential equations. Problems

* NBS Center for Applied Mathematics, Scientific Computing Division,

t Certain commercial equipment, instruments, or materials are identified in

this paper in order to adequately specify the experimental procedure. Such

identification does not imply recommendation or endorsement by the National

Bueau of Standards, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.

1
with exact solutions have been solved to test the correctness and

accuracy of the codes. Also, the physics included in this model

and the geometry of the transistor can be easily changed. The finite element method used allows refinement of oblique triangles

which is important in achieving computational efficiency. The

program is portable and has been run on a DEC-20, a VAX 11/780, a Cyber 175, and a Uni vac 1108.*

Key words; elliptic solvers; finite elements; interactive graphics; MOS transistor.

1 . Introduction

A two-dimensional formulation of a charge-sheet model of an MOS transistor

and an associated computer program is presented. This formulation differs

from the formulation of De La Moneda [1] in that it includes the effects of

source-drain channel shape and arbitrary channel doping profile shape. The

formulation differs from the formulation proposed by! Brews [2] in that strong

two-dimensional electrostatic effects are included by solving Poisson's equation using adaptive finite element techniques in the bulk semiconductor. In

this formulation, a new approach that differs from both [1] and [2] is used

to couple the one-dimensional current flow to the two-dimensional potential.

The effects of the charge in the inverted channel of the transistor is included as a nonlinear boundary condition at the interface between the semiconductor and the gate oxide. This formulation allows the two-dimensional

electrostatic field and drain current to be calculated without the inclusion

of an additional two-dimensional current continuity equation. This results

in a great improvement in computational efficiency which allows finite element model calculations to be performed on a minicomputer. This model is

accurate in all regions of inverted transistor operation and works well even

in regions of operation where the channel current produces large changes in

the electrostatic potential in the channel. The model retains the computational efficiency of single equation models [3] without imposing the computer

requirements of coupled equation models [4,5,6]. The complexity and cost of

the model lies between the long-channel one-dimensional model of Brews, which

runs well on a desk top calculator, and the coupled equation models, which

are best suited to large computers. Unlike previous single equation models

[3], the two-dimensional charge-sheet model works in all regions of inverted

transistor operation including weak inversion. Use of the finite-element

method also allows the adaptive mesh procedures [7] previously applied to

MESFETs [8] to be applied to MOSFETs.

In section two, the formulation of the two-dimensional charge-sheet model and

the integral boundary condition at the semi conductor -oxide interface are discussed. In this formulation, the total channel current is computed as a function of the two-dimensional electrostatic potential. The computation of

total channel current allows the effects of this mobile charge to be treated

as an effective surface charge and to be included in subsequent calculations

of the electrostatic potential as a nonlinear integral boundary condition.

In section two, the numerical procedures used to calculate the electrostatic

potential in two dimensions with a nonlinear boundary condition are also

discussed. The partial differential equation is transformed into a system of

2
nonlinear equations using adaptive, computer-generated, finite elements. The

nonlinear field equations are solved using a damped Newton's method. The

resultant system of linear equations is solved either by direct methods or by

a multigrid iteration.

In section three, the specific procedures used to implement the model are discussed. The structure of the computer program is discussed and the function of each of the principal modules is outlined. The installation requirements of the program are discussed as are the sources of the software components which were developed by other authors. The physical limitations of the

model are then discussed. These simplifications are responsible for the low

cost of the model but may, in some cases, lead to modeling inaccuracies.

Solutions to these physical limitations are outlined.

In section four, the modeling of an array of MOS transistors is discussed.

First, the numerical accuracy of the model is discussed. Next, the relationship between long-channel theory, as represented by the one-dimensional

charge-sheet model, and the two-dimensional model is discussed. Using the

two-dimensional model, it is possible to obtain accurate models of transistors with channel lengths less than 1.0 ym. The transistors used in this

work have a nominal channel doping density of 1 x 10^^ cm“^. This low channel doping density increases short-channel effects by increasing the Debye

length in the channel of the transistor. If, for example, transistors with

channel doping density of 1 x 10^^ cm”^ had been used, then short-channel

effects comparable to those seen in the 1.12-ym transistor would occur in a transistor with 0.35-ym channel length. In reference [9], the onset of

short-channel effects is calculated for the subthreshold region; a transistor

is classified as short channel if excess subthreshold current is present.

For the channel doping and oxide thickness used in this study, this shortchannel limit is at 5.6 ym; all of the transistors studied are short-channel

transistors from a subthreshold current criterion.

2. Theory

A. Two-Dimensional Charge-Sheet Model

The general form of the static semiconductor device equations is given by

reference [10]:

V^\j; = -p/e

P = + P • ")

(1 )

q

- n = n. e

1 (2)

q(4. - p = n. e

1

\j;)/kT

(3)

V • fy nVd) 1 =

'' n ^n^

R (4)

3
V (MpPV(|)p) = - R (5)

where ij; is the electrostatic potential; p is the net space charge density; e is the dielectric permittivity; is the net ionized impurity density; n and p are the electron and hole densities; nj^ is the intrinsic carrier

density; (|)p are the quasi-Fermi levels;

pp are the electron

and hole mobilities; and R is the net volume generation-recombination rate.

The coordinate system used in these calculations is shown in figure 1 . The

source-and-drain regions of the device are assumed to be uniform at the source or drain potential and are removed from the calculation. This improves the stability of the calculation by removing two regions in which

space charge neutrality is achieved by subtracting a large mobile carrier

density from a large doping density. This geometry was discussed previously

in reference [10]. If recombination is neglected and the model is explicitly

restricted to n-channel transistors by neglecting hole current, then eqs (1)

to (5) become:

= q(n - N^)/e

V exp (q(ip - (|)^ ) /kT ) V (|)^ = 0

( 6 )

(7)

so that the electron current density is given by:

J = qu n. exp

n ^n 1 kT V(j) ( 8 )

If eq (6) is used in both the oxide and semiconductor regions of the transistor, then an interior constraint must be imposed at the oxide-semiconductor

interface

ox 8x oesc 3x o+

Q (9)

where and are the permittivities of the gate oxide and the semiconductor, respectively; the subscripts o+,o- represent the two sides of the

interface; and Q is the total net interface surface charge density. The

inclusion of these dielectric effects would require modification of eq (1):

V • (eVi|;) = - p . (10)

An alternate approach, used here, is to separate the transistor into oxide

and semiconductor regions and use eq (9) as a matching boundary condition.

Direct solution of eqs (6) and (7) is possible, and several methods exist for

doing this [4,5,6]. In surface channel transistors, with or without implanted channels, the electron current is concentrated in a thin sheet near

the semiconductor-oxide interface. If eq (6) is solved in two dimensions to

obtain i|;(x,y), then this interface-associated charge is given by:

4
Gate ip = ipg

i’igure 1. The geometry used in the charge-sheet model.

5
00

N(y) = f V* W * J n /

1 + [e -1] —J /

dx e u J n

dy.

and the total electron current in the channel is given by:

f" j PV dx e

(11 )

I = Wqn.

1 3V.[e - 1 ] /

dy.

, 3^ dx e ^ p

where 3 = q/kT, the weighted mobility is given by y* = — f

( 12 )

n d

is the source-drain voltage, L is the channel length, and W is the channel

width. The derivation of these equations is given in Appendix A. The charge

calculated in eq (12) is used in eq (9) to combine the competing effects of

gate surface inversion and inversion charge screening caused by current. The

two-dimensional charge-sheet model is given by eqs (6), (9), and (11) where Q

in eq (9) is:

2 * ^surface "‘y' (13)

and drain current is given by eq (12).

B. Numerical Procedures

An important feature of this work is that the numerical solutions were obtained using general-purpose software for solving elliptic partial differential equations (PDEs). Although some efficiency is sacrificed, the benefits

obtained by using the general-purpose software far outweigh the loss in efficiency. Problems with exact solutions have been solved to test the correctness and accuracy of the codes and to determine optimal strategies and optimal values for parameters in the code. The physics included in the model and

the geometry of the problem can be easily changed. The numerical software

was inspired by finite-element software of Bank and Sherman [7] and retains

most of their philosophy. Currently, a general-purpose package to solve

coupled nonlinear systems of time -dependent partial differential equations is

being developed. The current report uses an intermediate version of the

package.

This version of the package solves the PDE

V • [a(x,y,u) Vu) ] = f

/ 3u 9u \ (14)
in a region bounded by straight line segments and circular arc segments. On

each segment of the boundary, u must obey one of two types of boundary conditions :

(1) g(x,y,u) = 0 (15)

(2) 3u/3n + g(x,y,u) = 0 . (16)

This is a fairly general example of a PDE arising from conservation of a flux and includes both Poisson's equation and the hole and electron continuity equations (in the multiple-PDE version).

At the heart of the package is a module which solves a linear elliptic PDE

using linear finite elements on a mesh of triangles. (Boundary "triangles"

may have one curved side.) The initial triangulation is an input to the

package; succeeding triangulations are calculated adaptively by the package.

For a given triangulation with M vertices, the calculation proceeds as follows. The M finite-element basis functions {bj^} are linear on each triangle; bjj, is 1 at vertex m and 0 at all other vertices. The set of vertices

on type 1 boundaries is denoted by D (for Dirichlet). The solution is

approximated by a sum of basis functions

M

u(x,y) = \ ^ a b (x,y) . (17) m m

m=1

The coefficients are determined by a Galerkin method [11]; the error in the

PDE is made orthogonal to each of the basis functions except those in D.

(aVu) + f

)

b

m

m~1 , 2 , . .

,

M

m not in D

(18)

The remaining conditions on the b's are that the type 1 boundary conditions

hold exactly [11].

9

= 0, m in D . 'mm mm-'

Equation (18) is integrated by parts to yield

(19)

aVu »Vbm

+ fbm )^/

2

agb

m

0 ( 20 )

where the single integral is over the type-2 parts of the boundary only.

Equations (19) and (20) comprise M nonlinear equations in M unknowns, the

coefficients

7
The M nonlinear equations are solved by an iterative process, a damped Newton's method. Let v be the Newton step and use a fraction of v as a correction to u; V has the same form as u.

v(x,y)

m

b (x,y) m

(21 )

To calculate v, we replace u by u + v in eqs (19) and (20) and linearize in

V, The linearization of eq (19) yields

am

3g(x ,y ,u ) m m m

3u 3 = 0, m in D , m

( 22 )

and the linearization of eq (20) yields

3v 3f —\l

3x 3(3u/3y) 3y /j

(23)

/

agb = r m m

2

Substituting for u and v from eqs (17) and (21) and doing the integrals nu- merically gives a set of M linear equations for the correction coefficients

{3jj,}» These linear equations are sparse; there are typically only nine

nonzero elements per row, on average, and M can be a few hundred or a few

thousand. A satisfactory way to solve these equations is directly, with

Gaussian Elimination, using the Yale Sparse Matrix Package [12].

Unless u is close to a solution of the nonlinear finite -element equations,

u + V may be worse than u. A damped Newton's method is used: replace u by

u + Xv, where X is chosen so that max |r_ (u + v) I < max |r (u)|. This

m “• m avoids divergence of the iterative process.

In practice, the initial triangulation may be too coarse to give the desired

accuracy or may need local refinement to represent the solution accurately in

a region of high curvature. After the package has converged to an approximate solution on the initial triangulation, the approximate solution may be

used to estimate the error and produce a new and finer triangulation adaptively. There are various ways to approximate the error in each triangle.

For the present work, a simple method is used. In the silicon, the PDE is

= -p/e . (24)

//I

3a aVv'Vb + —— v7u»Vb

m 3u m

+ b

-//(

/9f .. . 3f

\ 3u ^ * 3( 3u/3x)

^ 9a . _ -

3u * ®

3u / m

( a Vu • Vb + f b_J

\ m/

8
Since ij; is linear in each triangle, = 0. We estimate the relative error

in each triangle by

triangle

The refined triangulation is obtained by dividing the triangles with the

largest estimated errors, such as all triangles with estimated error larger

than 1/4 of the largest error.

For the refined triangulation, the previously calculated u is a good guess at

the new u, so that few Newton iterations should be necessary. The hardest

work is done on the initial triangulation, the one with the fewest vertices.

If there are two or more triangulations, the linear equations on the finest

triangulation may be solved by a multilevel iteration [13,7], saving both

time and storage space. However, for the present problem, sufficient accuracy could be obtained with M of only a few hundred, so that a direct solution

using the Yale Sparse Matrix Package was quite adequate.

Using this procedure, the solution of eqs (9) to (11) is started by first

obtaining a trial solution for the linearized form of eq (10), using the

solution of eq (11) obtained from the one-dimensional charge-sheet model with

eq (9) as a boundary condition. The Jacobian of g required for the solution

of eq (22) contains the Jacobian of eq (11). This Jacobian is approximated

by

= q3N(y) . (26)

Once the surface charge and approximate local field term are obtained from

the one-dimensional model and a local-field condition, the field term in eq

(9) associated with the oxide is obtained using a fast Poisson solver [14].

The trial solution is then used to obtain a solution for eq (11) by quadrature and an additional fast Poisson solution in the oxide. This iterative

procedure is repeated for each Newton step until the requested convergence is

obtained or the specified maximum iteration count is reached.

3. Computer Implementation

A. Software Modules

The distributed source of the CS1 program is divided into four files:

CS1.F0R, SYSTEM. FOR, PDESUB.FOR, and DEVLIB.FOR. These four files are ANSI

66 FORTRAN IV and can be compiled on any FORTRAN compiler which meets this

standard. The files CS1 .HLP, CS1.EXP, and CS1 .LNK are also provided.

CS1 .HLP provides a set of user instructions identical to Appendix B of this

report. CS1 .EXP contains an example set of program input and output which is

identical to Appendix C of this report. The file CS1 .LNK is a sample set of

linkage control statements which have been used to link the program on the

DEC TOPS -20 system.

9
The file CS1.F0R is the actual charge-sheet model program. This file contains the subroutines which control input and output, model initial approximations, the oxide field solution, and the calls to the subroutines which

solve the partial differential equation. The starting approximation is identical to the charge-sheet model originally developed by Brews in [2]. The

output of this long-channel model is printed at the beginning of each CS1

calculation. The oxide solution is calculated using the fast methods discussed in reference [14]; these subroutines are in the file DEVLIB.FOR. A

complete copy of the Swarztrauber and Sweet package is not included. Only

those parts of the package that are used by CS1 are included.

The CS1 module of the program contains parameters TOL and IP(6) which are

used to control the accuracy of the solution. The parameter TOL is used to

control the iteration accuracy at each multigrid level. The parameter IP(6)

is used to control the number of multigrid levels. These parameters are set

to yield very accurate solutions in the version of the program which is distributed. This level of accuracy greatly exceeds that required for most

applications, but is necessary, during installation and testing, to verify

that the program has been correctly installed. Resetting these parameters to

lower accuracy by increasing the iteration tolerance and/or decreasing the

number of multigrid levels will greatly improve run times.

The file SYSTEM. FOR contains subroutines which are used in the other three

program files but which are system dependent. The subroutines FOPEN and

FCLOSE are used to open and close input and output files. In the DEC TOPS-20

version supplied on the tape, the file FCLOSE is a dummy subroutine. The

subroutines ISTIME and IFTIME are used to start and stop clocks which are used to measure the execution times of the parts of the partial differential

solution process. If these routines are not used and dummy subroutines are substituted, ISTIME must be less than, NOT equal to, IFTIME. If these subroutines are dummy routines, then the timing information in the output

summary will be incorrect.

The subroutines PLTUTL, CURVEA, and ISCALE contain calls to vector graphics

subroutines. The calls in the supplied program can be used with the TCS

software supplied by Tektronix and can be used to drive either Tektronix

iff 4010 terminals or Hewlett-Packard 2647 terminals. Other graphics packages

can easily be substituted if the necessary initialization, absolute motion,

and absolute vector drawing primitives are available in either software or terminal firmware. If these subroutines are made into dummy subroutines,

graphics output will not be available from the program except through the

plot output file. The data written into the plot output file, file 2, are a sequence of non-negative x- and y-coordinate pairs. The first line of the

file contains the minimum and maximum values of the x- and y-coordi nates.

Each plot is terminated by a set (or sets) of negative coordinate pairs. The

subroutines R1MACH, D1MACH, and II MACH are used to set machine constants

which are used in the other program modules to set machine-dependent quantities. The original versions of the functions are discussed in reference [15] as are the procedures used to calculate these constants for a type of com- puter which is not presently included. A list of the types of computers

which have these constants stored in comment statements, in the existing

* See disclaimer on page 1

10
versions of functions, is given in table 1. To convert the program from any

of these computer systems to another, the data statements for the old

computer are converted to comments, and the C's on the data statements for

the new computer are removed

The conversion of the program from one computer type to another should not

require modification of any part of the program which is not in the SYSTEM

module. The program was converted from the DEC-20 to the VAX 11/780 in a few

minutes of editing time. This assumes that some plotting package is available and that timing subroutines are available on the target computer

system.

The module PDESUB.FOR contains the subroutines used to solve the partial

differential equations. The theory of these subroutines has been discussed

in section 2B above.

The module DEVLIB.FOR contains subroutines used by the other three modules.

These subroutines have been obtained from several sources. The error handling software, memory management software, and machine-independent constant

software were obtained from [15]. The fast elliptic solver was obtained from

[14]. In the initial approximation quadrature from [16] and root finding

routines from [17] are used. In the PDESUB module, the Yale Sparse Matrix

Package is used [12].

B. Physical Restrictions

Three physical restrictions have been imposed in the present implementation

of the CS1 program. First, the two-dimensional electric field dependence of

the mobility has not been included. Second, the detailed slopes of the

source-and-drain doping profiles have not been included in the model. Third,

the work function difference between the gate material and the bulk semiconductor is not included.

The first limitation is the most serious problem. In circuit models and in

some two-dimensional device simulation programs, the field dependence of the

mobility is empirically included using an expression of the form a/(1+b*E)

where a and b are empirically determined parameters and E is the magnitude of

the electric field. Recent work by Cooper and Nelson [18] suggests that this

model is not adequate to characterize the field dependence of the mobility.

On the other hand, the Cooper and Nelson data do not include several effects

which are known to affect the low field value of the mobility; these are the

effect of bulk doping and oxide interface quality. Since no definitive model

for the two-dimensional dependence of the mobility is available, none has

been included. Future versions of the program are expected to include such a mode 1

The details of the two-dimensional source-and-drain profiles are included in

the model only as two elliptic abrupt junctions. This allows the details of

the majority carrier quasi-Fermi level to be neglected and greatly simplifies

the program, making the charge-sheet calculation much more efficient. On the

other hand at high source-drain-voltages, this approximation can lead to

convergence problems and/or to inaccurate modeling of the potential near the

drain. In addition, this model places the burden of specifying lateral and

11
Table 1. Available Computer Types.

Type Model Speed (DEC-20=1)

Burroughs 1700

CDC

5700/6700/7700

6000/7000 3.3 (Cyber 175)

Cray 1 Data General Eclipse S/200

Harris 220

Honeywell 600/6000

IBM 360/370

Xerox Sigma 5/7/9

PDP-1

0

Ka or Ki 1.0 (Kl)

PDP-1

1

VAX 11/780 0.72

Uni vac 1100 0.57 (1108)

Table 2. Transistor Parameters.

Name Value

Channel Lengths 4.65, 3.20, 1.89, 1.12 ym*

Vertical Junction Depth 1.05 ym

Lateral Junction Diffusion o .

00 Channel Impurity Density 1.0 X 10^^ cm Source-Drain Impurity Density 1.0 X 10^0 cm Oxide Thickness 51 . 5 nm Channel Width 81 ym

* These lengths were obtained from scanning electron microscope

measurements of polysilicon length made by P. Roitman.

12
vertical junction depths on the program user. Other available twodimensional device simulation programs, such as MINIMOS [4], attempt to include simple two-dimensional process models in the device modeling programs.

These simplified models do not agree with more complex two-dimensional models

[19] or with experimental measurement of two-dimensional junction shape [20].

If the user of the program wishes to use these simplified models, then the

work of Kennedy and O'Brien can be used [21]. If more complicated models

[19] or experimental data are available, then these can be used.

The least serious of the restrictions is that the work function difference

between the gate material and the bulk semiconductor is not included. This

difference, if it is known, can be included in the gate bias.

4. Device Model Verification

This section is divided into three parts. In the first part, the numerical

accuracy of the model is evaluated. In the second part, the one- and twodimensional models are compared for a transistor similar to the transistor

studied by Brews [2]

.

In the third part, experimentally measured data are compared to the one- and two-dimensional models. The transistors used in

this experimental study were taken from an array of transistors fabricated

using an SEM-compatible transistor array of the type discussed in reference

[22]. This pattern has been incorporated into test pattern NBS-28A [22]. In

this study the polysilicon gate lengths were measured using a scanning electron microscope, and the metallurgical channel lengths were obtained using

estimates of lateral-to-vertical arsenic diffusion in reference [19]. The

relevant device simulation parameters are given in table 2. The heavily

doped source-and-drain regions are excluded from the region of calculation

because the potential in the regions is pinned to the source or drain potential plus the built-in potential, and the ratio of channel to source-drain

depletion depth is more than 100 to 1.

The numerical accuracy of two different adaptive meshing procedures is discussed. The first procedure, referred to as the direct method, uses adaptive

mesh generation to refine the mesh. After each adaptation, the resulting

linear equations are solved by direct methods. Multigrid iteration is not

used. The second method, referred to as the multigrid method, uses adaptive

mesh generation to obtain a sequence of more refined mesh levels. A multilevel iteration over these levels is then used for the solution of the linear

equations

.

The problem used as a test case for numerical accuracy studies is a 1.12-ym

n-channel transistor with 0.2 V of applied source-drain voltage and a gate

voltage 0.7 V above threshold. This transistor, under these bias conditions,

carries a drain current of 88 pA. The level 2 mesh and a contour plot of the

electrostatic potential for this calculation are shown in figures 2a and 2b.

The progress of the iteration process for this simulation, using the direct

method, is given in table 3. In this case, and all other convergent cases,

the direct and multigrid methods yield similar accuracies for the specified

number of level 1 and level 2 iterations. The direct method is more robust

and has converged in cases where the multigrid method fails.

13
14
15
12

34

5

6

78

9

10

1112

13

14

15

16

17

18

19

20

2122

23

24

25

Table 3. Iteration Convergence.

Level 1 Level 2

Newton

norm

A\{;s AJ

s Newton

norm

All; AJ

s

36.1 3.31 771 24.9 3.78 0.36

14.6 1.28 634 24.9 3.70 0.34

7.24 1.25 340 3.63 0.809 0.87

7.22 1.25 146 0.798 0.679 0.37

2.14 1 .24 59.9 0.653 0.556 0.31

1.78 1.22 23.7 0.548 0.488 0.30

1.22 1.19 8.8 0.488 0.473 0.29

1.15 1.13 3.01 0.462 0.447 0.26

1.05 1.03 1.01 0.431 0.416 0.23

0.94 0.91 0.53 0.398 0.385 0.21

0.829 0.803 0.43 0.365 0.353 0.18

0.722 0.698 0.36 0.333 0.323 0.16

0.624 0.602 0.28 0.304 0.294 0.14

0.537 0.510 0.22 0.276 0.268 0.12

0.459 0.442 0.17 0.250 0.244 0.11

0.392 0.377 0.14 0.227 0.221 0.09

0.334 0.322 0.11 0.205 0.200 0.08

0.285 0.274 0.08 0.186 0.181 0.07

0.233 0.242 0.07 0.168 0.164 0.06

0.207 0.199 0.05 0.152 0.148 0.04

0.176 0.169 0.03 0.137 0.134 0.03

0.150 0.144 0.03 0.124 0.122 0.02

0.127 0.123 0.02 0.113 0.113 0.02

0.108 0.104 0.01 0.104 0.103 0.01

0.092 0.089 0.01 0.094 0.095

0.076 0.078 0.088 0.087

16
The convergence of each iteration is evaluated by plotting the average Newton

step size (Newton norm) as a function of the iteration number. These data

are shown for the direct case in figure 3. The corresponding multigrid case

is shown in figure 4. In each case the convergence sequence is divided into

two regions. For the first few iterations, the Newton norm falls rapidly

from a very inaccurate initial value to a value of approximately 1 , in

normalized units, or approximately 0.026 V. The convergence then proceeds

linearly, at a slower uniform rate, until the convergence criterion is

reached or the maximum allowed iteration count is exceeded. As the number of

mesh points is increased, by increasing the level of the mesh, the

convergence rate decreases ; it takes more iterations to reduce the Newton

norm to some specified value. The reduced rate of convergence is the result

of the approximate Jacobian in eg (26) used in eq (22). The smallest Newton

norm is obtained with the level 2 mesh after 23 iterations. Solutions

obtained with other channel lengths and bias voltages using different numbers

of quadrature points for the solution of eq (11) and different oxide mesh

densities show, in all cases, that the optimal solution is obtained using the

level 2 mesh.

Although the error evaluation technique discussed above provides an accurate

measure of the convergence of the numerical method, two other error evaluation techniques are also useful. In one method, the maximum change in the

surface potential at the semiconductor-oxide interface after each Newton step

is compared to the Newton norm of that step. A graph of this relationship is

shown in figure 5. In the initial iterations of each convergence sequence,

the Newton norm falls rapidly, while only mimimal improvement in the surface

potential is made. As the Newton norm approaches 1.0, the values of Aij^g and the norm become identical. Again, the iteration is divided into two

parts. In the first part, the two-dimensional potential is refined, and the

surface potential is only crudely calculated. This process converges

rapidly. The second part of the process is dominated by successive refinements of and the two-dimensional potential is dominated by surface

potential boundary effects. The level 3 iteration divergence occurs because

the Jacobian in eq (26) is not sufficiently exact to allow the Newton method

to converge for the multigrid solution of the linear equations.

The most important error estimation method compares Aipg to AJ^j/ the difference in the drain current between the value at the present iteration and

the final value of the drain current. These data are plotted for the test

problem in figures 6a and 6b for both level 1 and level 2. In the level 1 case, the value of AJ^j falls slowly as Aij^g decreases from 30 to 2. As

Aipg decreases from 2 to 1 , the current convergence is very rapid. Finally,

if A\pg < 1, the current converges quadratically with Aipg.

Only the quadratic region of convergence is important for the level 2 case shown in figure 6b. The initial error in the first region is represented by

two points near A\j<g = 3, and no structure is visible. The relationship of

AJ^ and Aij;g to the Newton norm is important in that it allows the convergence of the equation solution process to be monitored by the Newton norm. For small Newton norms, less than one, the approximation AJ^ = constant

(Norm) is valid. A request for a 0.1 Newton norm (i.e., 0.1 kT/q) allows

the error in the current to be reduced to 0.16 percent and a 0.4 Newton norm produces a 1.6-percent error.

17
NEWTON

NORM

Figure 3. Convergence of the direct method for the test problem.

18
NEWTON

NORM

ITERATION

Figure 4. Convergence of the multigrid method for the test problem.

19
All/

NEWTON NORM

Figure 5. Relationship between the surface potential change and the Newton

step size for the test problem.

20
AJq

AIb

Figure 6. a) Relationship between surface potential change (kT/q) and drain

current change (arbitrary unit, = 1) for the level 1 mesh.

21
Figure 6. b) Relationship between surface potential change (kT/q) and drain

current change (arbitrary unit, = 1) for the level 2 mesh.
From this discussion of numerical accuracy, one can conclude that the accuracy of this implementation of the charge-sheet model can greatly exceed the

accuracy required for most applications. To obtain 5-percent accuracy for

drain current, a level 1 mesh using 10 Newton steps and a relative Newton

step convergence criterion of 2 percent are sufficient. Refinement of the

level 1 mesh to level 2 gives a which is more accurate than required.

Next, the difference between the one- and two-dimensional charge-sheet model

in their long-channel limit is discussed. A transistor similar to the transistor shown in figure 5 of Brews' paper [2] is considered. The substrate

doping is 10^^ cm“^, the oxide thickness is 100 nm, and for the twodimensional case the polysilicon length is 4.6 ym, with 0.3 ym of lateral

junction diffusion at each edge, and 0.5 ym of vertical junction depth. The

one-dimensional theory used here is Brews' eq (11); no simplifications have

been made. The results of these one- and two-dimensional calculations are shown in figure 7. Although the transistor is a short-channel transistor in

the sense discussed in reference [9] , no threshold voltage offset or change

in output conductance in the saturation region is calculated. The two calculations are in good agreement throughout the region shown. Similar agreement

has been obtained for a transistor with 8.6-ym polysilicon, 50-nm gate oxide,

and substrate doping of 5 x 10^*^ cm“^. In the long-channel limit, the oneand two-dimensional charge-sheet models are in good agreement. If the characteristics of the device being modeled are not dominated by short-channel

effects in the triode and saturation regions, the one-dimensional chargesheet model provides an adequate model of the transistor well below the limit

discussed in reference [9]. This is true since the effects studied in [9]

for subthreshold regions are not associated with large current changes in

other regions of device operation.

The two-dimensional charge-sheet model can be used to characterize MOS transistors in regions where both threshold voltage offset and variable output

conductance are important. In the triode region of operation, as the channel

length of the transistor is decreased, the two-dimensional electrostatic

fields associated with the source-and-drain junctions extend significantly

into the channel region and cause surface inversion, even when the gate potential is substantially below threshold. This is an electrostatic effect

and is well modeled by the two-dimensional charge-sheet model. This region

of transistor operation is shown in figure 8 for transistors with 4.65-, 3.2-,

and 1.12-ym channel lengths. For the 4.65-ym transistor, both one- and twodimensional theories are adequate to model the triode region and the high

current part of the subthreshold region shown in figure 8. For the 3.2-ym

transistor, the two-dimensional model is in good agreement with the measured

data, while the one -dimensional model is diverging from these data. For the

1.12-ym transistor, the two-dimensional model shows a threshold voltage offset of 0.060 V. This threshold voltage offset results in a 26-percent in- crease in measured current, as compared to the one-dimensional model, when

the gate voltage is 0.25 V. At some channel length below 1.0 ym, these twodimensional electrostatic effects will become so large that the channel cur- rent will no longer be confined to the surface of the transistor and the

charge-sheet model will fail. This is not the case for the short-channel

transistors modeled here.

23
COoXo

CO

CM

a

(U CO 3 (U

tji •rH

O

Em e

24

Comparison

of

the

drain

current

calculated

by

the

oneand

two-dimensional
T3d

(C

<Ud

10d

(Ue

4-1 c

<uddod

•H

(0 T3do

•H

Cn

dJ

(U to 'O -H O (1) •H T3

>4

O

d e

T3 >H d (0

(tj

do

T3 -H

V4 T3

4d I 4-) O ^ 5 d 4J

CO 'V

0)

d

x;

m

4-> I IW 0) o do do <u CO xi •H 4-1 ^4

(0 tji a d B -H O to u d

T3 • CU

CX) 4-) m

Q) -H d d d o

D^t—I

•H

do

25
As the source-drain voltage is increased, the apparent threshold voltage

offset produced by two-dimensional electrostatics is increased, along with

the size of the depletion region around the drain. This effect is shown in

figure 9 for a source-drain voltage of 0.5 V; four transistors are shown with

1.1 2-, 1.89-, 3.20- and 4.65-pm channel lengths. In the shortest channel

transistor, at 0.25-V gate voltage the measured drain current is twice the

value given by the one-dimensional theory. Agreement with the twodimensional model is within 3 percent. For the 1.89-ym transistor, the

agreement with two-dimensional theory is within 5 percent; at 0.25-V gate

voltage, the measured current is 1.5 times the current given by the onedimensional model. For the long-channel lengths, the measured data and the

two-dimensional model both agree with long-channel theory to better than 1 percent.

In the saturation region, neither of the charge-sheet models is satisfactory

for the 1.12-ym transistor. The two-dimensional model does not converge,

because field reversal effects in the channel occur along the entire channel

length. As the source-drain voltage is increased, the depletion region

around the drain expands under the channel. This expanded drain depletion

region causes current flow away from the surface channel and, at some voltage

below punch- through, causes the field in the channel to reverse. This field

reversal effect shortens the effective channel length and draws charge away

from the semiconductor-oxide surface charge sheet. The effect of this distributed charge is not adequately represented in the charge-sheet model and

leads to convergence problems in the saturation region of the transistor.

This effect decreases with increasing channel doping and would not occur

until 0.5 pm if channel doping were 1 x 10^^ cm~^*

Although some convergence problems occur due to field reversal in the

channel, it is possible to make useful comparisons between measured data and

the two-dimensional charge-sheet model for transistors with channel lengths

longer than 1.12 pm. The transition between output characteristics which

look like long-channel characteristics in the saturation region and those

which look like short-channel characteristics occurs between 3. 20- pm and

1.89-

pn channel lengths. The model results are compared for these two cases

in figures 10 and 11. In figure 11a the 1.89-pm transistor model is compared

to measured data. In figure 11b the experimental data are compared to the

long-channel theory of the one -dimensional charge-sheet model. There is no similarity between the long-channel theory and the measured data. The longchannel theory fails because the effect of source-drain field-induced channel

inversion is neglected. This channel inversion is dominated by the sourcedrain field, since the calculated threshold voltage offset for this transistor, shown in figure 8, is not large enough to explain this change in drain

current. The source-drain component of the field produces such an offset, as

is shown in figure 9. The additional source-drain-induced offset dominates

the effect shown in figure 11.

Two interesting physical effects are also observed in the calculation of the

1.89-

pm channel transistor. In the saturation region of the transistor, a significant part of the total channel is in the field reversal region. If

the charge associated with this field reversal region is removed from the

calculation of eqs (11) and (12), good agreement between the model and the

experimental data is obtained as shown in figure 11a. This observation also

26
EN2,4,6,8

27

Figure

9.

Comparison

of

the

triode

region

drain

current

measured

and

calculated

using

the

oneand

two-dimensional

models.
MEASURED

28

Figure

10.

Comparison

of

the

output

characteristics

measured

and

calculated

using

the

two-dimensional

model

for

the

3.20-iim

transistor.
Il

!

Vso |V|

Figure 11. a) Comparison of the output characteristics measured and calculated using the two-dimensional model for a 1.89-ym transistor.

29
VSD |V)

Figure 11. b) Comparison of the output characteristics measured and calculated using the one-dimensional model for a 1.89-ym transistor.

30
explains the modulation of the output conductance in figure 11a without the

inclusion of field dependent mobility. The effective decrease in conductivity occurs because the part of the channel which is not experiencing field

inversion is constrained by strong electrostatic effects originating in the

drain. In the transition region between the triode and saturation regions in

figure 11b, the long-channel theory, with the same mobility, overestimates

the drain current. Strong two-dimensional electrostatic effects at low

source-drain voltages slow the process of channel inversion and at higher

source-drain voltages enhance it, shortening the channel by field inversion.

5. Conclusions

A two-dimensional charge-sheet model of a surface-channel MOS transistor has

been developed. When this model is implemented using adaptive finite element

methods, an efficient high-accuracy transistor simulation program is possible. The simulations produced using this model have been compared to experimental measurements; the model is in good agreement for transistors with

channel lengths as short as 1.12 ym. In this verification process the model

represented accurately the onset of subthreshold current, channel -lengthinduced threshold voltage offset, and drain-field-induced output conductance

changes. These drain-field-induced output conductance changes are successfully modeled by constant channel mobility and do not require use of high

field mobility effects.

Acknowledgments

The authors wish to thank C. W. Gwyn for providing computer support for this

work, E. H. Nicollian for providing the experimental transistors, J. R.

Brews for helpful discussion of his charge-sheet model, and P. Roitman for

SEM measurements.

31
References

1. De La Moneda, F., Threshold Voltage from Numerical Solutions of the TwoDimensional MOS Transistor, IEEE Trans, Circuit Theory CT-20 , 666-673

(1973).

2. Brews, J. R. , A Charge-Sheet Model of the MOSFET, Solid-State Electronics 2^, 345-355 (1978).

3. Greenfield, J. A., and Dutton, R. W., Nonplanar VLSI Device Analysis

Using the Solution of Poissons 's Equation, IEEE Trans, Electron Devices

ED-27 , 1520-1532 (1980).

4. Selberherr, S., Schutz, A., and Potzl, H. W., MINIMOS - A TwoDimensional MOS Transistor Analyzer, IEEE Trans, Electron Devices ED-27 , 1540-1549 (1980).

5. Buturla, E. M., and Cottrell, P. E. , Two-Dimensional Finite-Element

Analysis of Semiconductor Steady-State Transport Equations, Int, Conf, on Computational Methods in Nonlinear Mechanics, Austin, Texas, 1974.

6. Fichtner, W., and Rose, D. J. , On the Numerical Solution of Nonlinear

Elliptic PDE's Arising from Semiconductor Device Modeling, Elliptic

Problem Solvers, M. Schultz, Ed. (Academic Press, New York, 1981), pp.

277-284.

7. Bank, R. E., and Sherman, A. H. , PLTMG User's Guide, CNA-152, Center for

Numerical Analysis, University of Texas at Austin, September 1979.

8. Laux, S. E., and Lomax, R. I., Numerical Investigation of Mesh Size

Convergence Rate of Finite Element Method in MESFET Simulation, SolidState Electronics 24, 485-493 (1981).

9. Brews, J. R. , Fichtner, W. , Nicollian, E. H., and Sze, S. M. , Generalized Guide for MOSFET Miniaturization, Electron Device Letters EDL-1 , 2-

4 (1980).

10. Wilson, C. L. , and Blue, J. L. , Semiconductor Device Simulation, Elliptic Problem Solvers, M. Schultz, Ed. (Academic Press, New York, 1981),

pp. 435-439.

11. Strang, G. , and Fix, G. J., An Analysis of the Finite Element Method

(Prentice-Hall, Englewood Cliffs, N.J., 1973).

12. Eisenstat, S. C., Gursky, M. C., Schultz, M. A. and Sherman, A. H.

,

Yale Sparse Matrix Package I. The Symmetric Codes, Research Report

#112, Department of Computer Science, Yale University, 1976.

13. Brandt, A., Multi-Level Adaptive Solutions to Boundary-Value Problems,

Math, Comp, 3^, 333-390 (1977).

32
14 . Swarztrauber, P. , and Sweet, R. , Efficient FORTRAN Subprograms for the

Solution of Elliptic Partial Differential Equations, NCAR Technical

Note, NCAR-TN/IA-109, 1975.

15. Fox, P. A., Hall, A. D. , and Schryer, N. L. , The PORT Mathematical Subroutine Library, ACM Trans, Mathematiaat Software 2^, 104-126 (1978).

16. Blue, J. L., Automatic Numerical Quadrature - DQUAD, Bell Labs Comput.

Sci. Tech. Rep. 25, February 1975; also. Automatic Numerical Quadrature,

Belt Syst. Teak, J, 1651-1678 (1977).

17. Brent, R. P. , Algorithms for Minimization Without Derivatives (PrenticeHall, Englewood Cliffs, N.J., 1973), pp. 188-189.

18. Cooper, J. A., Jr., and Nelson, D. F. , Measurement of High-Field Drift

Velocity of Electrons in Inversion Layers on Silicon, IEEE Electron

Device Letters edl-2 , 171-173 (1981).

19. Warner, D. D. , and Wilson, C. L., Two-Dimensional Concentration Dependent Diffusion, Bell Syst, Tech, J, 1-41 (1980).

20. Wilson, C. L., Scanning Electron Microscope Measurements on Short Channel MOS Transistors, Solid-State Electronics 23, 345-356 (1980).

21. Kennedy, D. P., and O’Brien, R. R. , Analysis of the Impurity Atom Distribution Near the Diffusion Mask for a Planar p-n Junction, IBM J,

Res, Dev, % 179-186 (1965).

22. Mitchell, M. A., and Linholm, L. w. , Semiconductor Measurement Technology: Test Patterns NBS-28 and NBS-28A: Random Fault Interconnect

Step Coverage and Other Structures, NBS Spec. Publ. 400-65 (March

1981).

33
Appendix A

Channel Current Calculation

Following the general method presented by Brews [2], a vertical average

(along the x-direction) of the electron quasi-Fermi level is calculated over

the channel. The channel charge is then given by:

N(y)

z

(A-1

)

This follows from eq (2) and the arguments presented in reference [2].

The integral form of eq (7) then yields;

V • J = 0

which can be rewritten as:

y*

d(j)^(y) I dy

J

(A-2)

(A-3)

This equation has the solution

z (A-4)

Since the total change in the quasi-Fermi level must equal the source-drain

voltage drop, for potentials relative to the source:

z = 1 o (A-5)

Substitution of these factors into (A-1) yields eq (11). The current is then

given by:

(A-6)

which reduces to eq (12).

34
1

Appendix B

File CS1 .HLP

. Interactive Execution

The program is executed by entering:

RUN CS1 .EXE

A prompt will appear for the next input; enter a file name after each prompt, followed by a carriage return (denoted <cr>)

FILENAMEKcr>

FILENAME2<cr>

FILENAMES <cr>

The first and third files must have been created prior to this

execution. They contain input data used by the program. The

second file may be created during execution. The FILENAMES

must be legal TOPS-20 file names; they are limited to eight

characters by an internal buffer in CS1.

A. FILES

1 ) Filel contains data for each point to be

calculated. The file format is shown below.

2) File 2 is an empty file for output of plotting

data if the plots are to be saved. This file name must be entered even if no plotting is to be done.

TOPS-20 will create the file if plotting data is to

be stored; other operating systems may require the file

to be created before the run begins.

3) Files contains a table of channel doping profiles.

B. Examples of FILE FORMATS.

Each point to be calculated requires 5 lines of input.

Data on a given line is free-format. Comments may appear

after all required data fields. After each point

is calculated, the program will look for additional data

until a source doping of zero is found (first line of input),

after which the program will terminate.

EXAMPLE OF Filel : Data is in units given.

Line 1

;

Source doping/cm* *3

3.0el9

Line 2: Vertical junction depth ( urn ), Lateral junction depth ( urn ), Poly-Si length(um)

note: Channel length = Poly-Si length-2*Lateral junction depths

0.55 0.84 5.00

Line 3: Drain( volts ) ,Gate(volts ) , Substrate (volts

)

note: Source voltage = 0.0.

0.15 +1.158 0.0

35
Line 4: Oxide thickness (urn) /Oxide dielectric constant/ Interface charge (cm** 2)

0.057 4.0 0.0

Line 5: Channel mobi lity( cm* *2/V-sec) /Channel width (urn)

400.0 20.0

Line 6; Line 6 may contain 0.0 for stopping program/ or may be

first line of data for next point to be calculated. Last line in

data file must be zero.

EXAMPLE of File 3; Data for doping density.

Line 1

Contains integer/ Ndop/ denoting the length of the profile table.

2Lines 2 through Ndop+1

Each contains an x (urn) and the doping density at x (1/cm**3)

0.0 3.e15

100.0 3.e15

C. OUTPUT DISPLAY

All outputs (except the drain current) are in intrinsic Debye

lengths (33um) and thermal volts (25mV)/ or units derived from

these such as, Thermal volts/Debye length.

1 . After program execution/ the following prompts appear on

the display:

ENTER 0/1/ 2/

FOR CURRENT AND NEXT POINT/ 2D PRINTING/

2D PLOTTING/ CURRENT AND STOPPING.

0 — print current. Continue to next point in data file/ if any.

1 “ print additional information about point just calculated.

2 — plot additional information about point just calculated.

3 — print current and terminate execution. Do not continue

to next data point in data file/ if any.

If response 1 or 2 is given/ further prompts will be furnished.

2. OPTIONS for printing (response 1

)

The prompt is

ENTER SUMMARY /NX /NY

SUMMARY — for a summary of the number of mesh points and triangles

created/ how much memory allocated/ and how much time

used/ enter 1. If not desired/ enter 0.

NX & NY — how many points along X and Y are to be printed in the

print matrix. If NX and NY are both negative/ a scaled integer printer plot of the potential will result.

If NX and NY are both positive/ a rectangular grid is

36
placed on the region; at each point X, Y, U (the

potential), UX, and UY (the derivatives of U in the

X and Y directions) will be printed. At grid points outside

the region, large numbers will be printed.

3. OPTIONS for plotting (response 2)

The prompt is

1:TRIANGLES, 2:C0NT0URS, 3:SURFACE, 4:PR0FILES

Enter 1, 2, 3, or 4, to plot on screen. Enter 101, 102, 103, or 104, to save plot data on File2.

Enter 0 to terminate plotting.

Each plot mode will furnish one or more further prompts. In every

screen-plotting case, the final prompt before drawing the plot is

Enter g or G to when ready to plot

This gives the user time to clear the screen of to do any work

necessary to put the terminal in plotting mode.

After each plot is drawn on the screen, the program will wait for

a carriage return, and then re-issue the above prompt. This gives

the user time to clear the screen or to do any work necessary to

change the terminal out of plotting mode.

1 ; TRIANGLE PLOTS

The prompt is

ENTER LEVEL

Enter level number (1, 2, ...) to plot the mesh of that level.

Enter 0 to terminate triangle plotting.

After the triangle plot has been drawn, the program waits

for a carriage return, and then reissues the Triangle

Plot prompt.

2: CONTOUR PLOTS

The first prompt is

1:FUNCTI0N, 2: ABS ( GRADIENT)

Enter 1 to plot the function (potential), or 2 to plot the

absolute value of its gradient (absolute value of electric

field; units are thermal volts/Debye length).

Enter 0 to terminate contour plotting.

The next prompt is

NUMBER OF CONTOURS

37
Enter the number of contour lines to be drawn. They will be

drawn equally-spaced between (and including) the minimum and

maximum values found.

After the contour plot has been drawn, the program waits

for a carriage return, and then reissues the first Contour

Plot prompt.

3: SURFACE PLOTS

The first prompt is

1:FUNCTI0N, 2: ABS ( GRADIENT)

Enter 1 to plot the function (potential), or 2 to plot the

absolute value of its gradient (absolute value of electric

field; units are thermal volts/Debye length).

Enter 0 to terminate surface plotting.

The next prompt is

ENTER LEVEL

Enter level number (1, 2, ...) to make a plot of the surface

using the mesh of the given level.

Enter 0 to return to the First Surface Plot prompt

The next prompt is

ENTER VIEWING DIRECTION—NX,NY,NZ

Enter three integers. The surface plot is a (parallel)

projection of the surface as seen from indicated direction.

Thus (0,0,1) will give a top view (like a Triangle Plot),

and (0,-l,0) will give a side view.

After the surface plot has been drawn, the program waits

for a carriage return, and then reissues the first Surface

Plot prompt.

4: PROFILE PLOTS

The first prompt is

ENTER NUMBER OF PROFILES

Enter an integer, the number of profiles to be drawn.

Enter 0 to terminate profile plotting.

The next prompt is

ENTER INDICES OF PROFILE PLANE(S), NX AND NY

Enter two integers. A profile plot is a slice through the

solution, perpendicular to the given direction. Thus (0,1)

will give a slice parallel to the X axis.

After the profile plot has been drawn, the program waits

for a carriage return, and then reissues the first Profile

Plot prompt.

38
2.

BATCH EXECUTION (for DEC-20 only)

A. Submit File

Create a submit file which contains:

1 . RUN CS1 .EXE

2. FILENAMEI

3. FILENAME2

4. FILENAMES

5. (Any other input used for the the equivalent interactive

execution)

.

B. Output will go to file CS1 .LOG which can be

examined with the editor.

3.

SUMMARY OF MODEL

A two-dimensional charge sheet model of a surface-channel

MOS transistor has been developed. When this model is implemented

using adaptive finite element methods, an efficient high-accuracy

transistor simulation program is possible. The simulations produced

using this model have been compared to experimental measurements;

the model is in good agreement for transistors with channel lengths

as short as 1.1 2um. In this verification process the model represented

accurately the onset of subthreshold current, channel-length-induced

threshold voltage offset, and drain-field output conductance

changes.

39
Sample input data (first file)

Appendix C

File CS1 .EXP

1 .E20

0.80 0.59 3.00

0.15 1.158 0.0

0.0570 4.0 O.OeOO

416.0

20.0

1 .E20

0.80 0.59 3.00

0.15 1.658 0.0

0.0570 4.0 O.OeOO

416.0

20.0

0.0

Sample input channel profile (third file)

20.0 1.E15

200.0

1.E15

Sample run using data shown above (DEC 20).

In this run all user input is labled by (user input).

@run csl .exe INPUT FILE NAME FOR INPUT DATA,

FILE NAME MUST BE <= 8 CHARACTERS.

cs1.dat (user input)

INPUT FILE NAME FOR OUTPUT DATA PLOTTING

csl .pit (user input)

INPUT FILENAME FOR DOPING PROFILE DATA

csl.dop (user input)

BETA = 3.868472E+01

PS IB = 2.871 279E-01

LB = 1 .292317E-05

COX = 6.210526E-08

BPSO = 2.101 317E+01

BPSI = 2.652973E+01

PS(0) = 6.857934E-01

NO = 9.071783E+10

PS(L) = 8.104595E-01

NL = 3.404201E+10

ID = 6.762688E-06

10 = 1.181 714E+02

I. . . = 3.564658E+01

VG = 4.479690E+01

VSD = 5.802708E+00

LCH = 5.45441 IE-02

40
4 FOR CHANNEL PRINTING

1 (user input)

enter summary, nx,ny

1 -6-11 (user input)

3 MULTI-GRID LEVELS

VERTICES 146 387

TRIANGLES 322 982

STORAGE ALLOCATION

ALLOCATED USED

MAXT 4591 1691

MAXV 1722 617

MAXVSM 2296 1153

MAXJA 9749 4348

MAXU 1099 1099

MAXJU 1099 1062

TOTAL 54686 23223

TIMING DISTRIBUTION

SECONDS PERCENT

GRID 0.226 0.118

ASMBLY 103.892 54.249

MG 84.522 44.135

ADAPT 2.870 1.499

TOTAL 191 .510 100.000

scaled values from -1.111E+01 to 2.500E+01 scale 2.500E+01

9999 9999 17 -40 -44 -44

9999 9999 12 -42 -44 -44

9999 9999 -2 -43 -44 -44

89 54 -22 -44 -44 -44

100 13 -37 -44 -44 -44

94 -1 -41 -44 -44 -44

88 5 -39 -44 -44 -44

63 38 -29 -44 -44 -44

9999 9999 -12 -44 -44 -44

9999 9999 0 -43 -44 -44

9999 9999 4 -42 -44 -44

ENTER 0 FOR CURRENT AND NEXTPOINT

1 FOR 2D PRINTING

2 FOR 2D PLOTTING

3 -FOR CURRENT AND STOPPING

4 FOR CHANNEL PRINTING

1 (user input)

enter summary, nx, ny

0 6 11 (user input)

IX Y U UX UY

1 0.00000

2 0.01839

0.00000 1.701 41 E+38 1.701 41 E+38

0.00000 1.70141E+38 1.70141E+38

1 .701 41 E+38

1 .701 41 E+38

41
34

5

6

78

9

10

1 1 12

1 3 14

1 5 16

17

18

19

20

21

22

23

24

25

26

27

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29

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31

32

33

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37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

0.03678 0.00000 9.58144E-01 -1 .24680E+03 -9.78029E+01

0.05517 0.00000 -1.041 1 1E+01 -1 .82800E+02 -8.38715E+01

0.07355 0.00000 -1 .10993E+01 -1 .13871E+01 -2.47837E+00

0.09194 0.00000 -1.11075E+01 4.58229E-01 2.03329E-01

0.00000 0.01253 1 .70141E+38 1 .70141E+38 1 .70141E+38

0.01839 0.01253 1 .70141E+38 1 .70141E+38 1 .70141E+38

0.03678 0.01253 -7.82947E-02 -1 .24951E+03 -1 .99585E+02

0.05517 0.01253 -1 .07115E+01 -8.98376E+01 -5.09973E+00

0.07355 0.01253 -1 .11026E+01 -7.23250E+00 2.03329E-01

0.09194 0.01 253 -1 .11075E+01 2.46640E-01 O.OOOOOE+00

0.00000 0.02505 1 .70141E+38 1 .70141E+38 1.701 41 E+38

0.01839 0.02505 1 .70141E+38 1 .70141E+38 1 .70141E+38

0.03678 0.02505 -3.04848E+00 -1 .04036E+03 -3.75744E+02

0.05517 0.02505 -1 .08778E+01 -5.80910E+01 -6.96095E+00

0.07355 0.02505 -1 .11 101E+01 2.81 150E-01 2.59381E-01

0.09194 0.02505 -1.11075E+01 -2.32784E-02 O.OOOOOE+00

0.00000 0.03758 1 .58477E+01 1 .79824E+03 4.59544E+03

0.01839 0.03758 9.53071E+00 -1 .01526E+03 -9.01933E+02

0.03678 0.03758 -7.12977E+00 -5.72754E+02 -2.55661E+02

0.05517 0.03758 -1 .10460E+01 -5.12203E+01 -3.48379E+01

0.07355 0.03758 -1 .11071E+01 5.61262E-03 -5.07924E-03

0.09194 0.03758 -1 .11075E+01 -1 .92022E-02 O.OOOOOE+00

0.00000 0.0501

2.20992E+01 -1 .44968E+03 1 .27753E+02

0.01839 0.05011 1 .2761 3E+00 -8.08550E+02 -3.37286E+02

0.03678 0.0501

-9.83393E+00 -2.56518E+02 -1 .56215E+02

0.05517 0.0501

-1 . 10851 E+01 -1 .14023E+01 -8.79710E+00

0.07355 0.0501

-1.11074E+01 5.61262E-03 -2.44136E-01

0.09194 0.05011 -1 . 1 1075E+01 -7.34552E-03 O.OOOOOE+00

0.00000 0.06264 2.35587E+01 -1 .60102E+03 1. 26531 E+02

0.01839 0.06264 -3.05892E-01 -8.09359E+02 9.27894E+01

0.03678 0.06264 -1 .02042E+01 -1 .25966E+02 2.96531E+01

0.05517 0.06264 -1 .1 1043E+01 -2.24213E+00 8.89146E+00

0.07355 0.06264 -1 .11076E+01 1 .89679E-01 -1 .47744E-02

0.09194 0.06264 -1 . 1 1075E+01 -7.34552E-03 O.OOOOOE+00

0.00000 0.07516 2.49961E+01 -1 .49096E+03 1 .00300E+02

0.01839 0.07516 3.18066E+00 -8.78082E+02 4.96746E+02

0.03678 0.07516 -9.27189E+00 -3.4691 5E+02 2.35055E+02

0.05517 0.07516 -1 .10720E+01 -1 .86051E+01 1 .51825E+01

0.07355 0.07516 -1 .11074E+01 1 .20671E-03 2.49914E-01

0.09194 0.07516 -1 .11075E+01 -7.34552E-03 O.OOOOOE+00

0.00000 0.08769 2.22886E+01 1 .46264E+03 -3.13579E+03

0.01839 0.08769 1 .35906E+01 -1 .14082E+03 1 .08186E+03

0.03678 0.08769 -5.55180E+00 -7.1 1942E+02 3.34245E+02

0.05517 0.08769 -1 .09763E+01 -4.73173E+01 2.89867E+01

0.07355 0.08769 -1 .1 1088E+01 1 .20671E-03 -1 .45270E-01

0.09194 0.08769 -1 .1 1075E+01 -1 .94337E-02 O.OOOOOE+00

0.00000 0.10022 1 .70141E+38 1 .70141E+38 1 .701 41 E+38

0.01839 0.10022 1 .70141E+38 1 .70141E+38 1 .701 41 E+38

0.03678 0.10022 -5.58964E-01 -1 .21393E+03 4.47622E+02

0.05517 0.10022 -1 .06938E+01 -8.02369E+01 3.40412E+01

0.07355 0.10022 -1 .11140E+01 -3.81814E+00 -9.89015E-01

0.09194 0.10022 -1 . 1 1075E+01 9.71475E-02 O.OOOOOE+00

42
55 0.00000 0.11274 1 .70141E+38 1 .70141E+38 1 .701 41 E+38

56 0.01839 0.11274 1 .70141E+38 1 .701 41 E+38 1 .701 41 E+38

57 0.03678 0.11274 3.00789E+00 -1 .44263E+03 2.27028E+02

58 0.05517 0.11274 -1 .03946E+01 -1 .50872E+02 1 .31435E+01

59 0.07355 0.11274 -1.11073E+01 -2.80341E+00 2.98414E+00

60 0.09194 0.11274 -1 .1 1075E+01 1 .12634E+00 O.OOOOOE+00

61 0.00000 0.12527 1.701 41 E+38 1 .70141E+38 1.70141E+38

62 0.01839 0.12527 1 .70141E+38 1 .701 41 E+38 1 .701 41 E+38

63 0.03678 0.12527 4.26292E+00 -1 .42063E+03 1 .20125E+02

64 0.05517 0.12527 -9.91306E+00 -2.91113E+02 1.18065E+02

65 0.07355 0.12527 -1 .10732E+01 -4.72232E-01 7.43968E-01

66 0.09194 0.12527 -1 . 1 1075E+01 -1 .97903E+00 2.9841 4E+00

ENTER 0 FOR CURRENT AND NEXTPOINT

1 FOR 2D PRINTING

2 FOR 2D PLOTTING

3 FOR CURRENT AND STOPPING

4 FOR CHANNEL PRINTING

2 (user input)

1;triangles, 2:contours, 3:surface, 4;profiles

1 (user input)

enter level

1 (user input)

enter g or G when ready to plot

g (user input)

(See figure C-1 for graphic output)

Lenter g or g when ready to plot

g (user input)

enter level

2 (user input)

enter g or G when ready to plot

g (user input)

(See figure C-2 for graphic output)

enter g or G when ready to plot

g (user input)

enter level

3 (user input)

enter g or G when ready to plot

g (user input)

(See figure C-3 for graphic output)

enter g or G when ready to plot

g (user input)

enter level

0 (user input)

l:triangles, 2:contours, 3:surface, 4;profiles

3 (user input)

1:function, 2 ;abs( gradient)

1 (user input)

enter level

1 (user input)

enter viewing direction — nx,ny,nz

3 -1 1 (user input)

enter g or G when ready to plot

g (user input)

43
(See figure C-4 for graphic output)

enter g or G when ready to plot

g (user input)

1: function, 2 cabs (gradient

)

1 (user input)

enter level

2 (user input)

enter viewing direction — nx,ny,nz

3 -1 1 (user input)

enter g or G when ready to plot

g (user input)

(See figure C-5 for graphic output)

enter g or G when ready to plot

g (user input)

Icfunction, 2:abs (gradient)

1 (user input)

enter level

3 (user input)

enter viewing direction — nx,ny,nz

3 -1 1 (user input)

enter g or G when ready to plot

g (user input)

(See figure C-6 for graphic output)

enter g or G when ready to plot

g (user input)

Icfunction, 2 cabs (gradient)

0 (user input)

Ictriangles, 2ccontours, 3csurface, 4cprofiles

2 (user input)

Icfunction, 2c abs (gradient

)

1 (user input)

number of contours

-8 (user input)

umin and umax

-10 25 (user input)

ulev

-1.000E+01 5.000E+00 O.OOOE+00 5.000E+00

1.500E+01 2.000E+01 2.500E+01

enter g or G when ready to plot

g (user input)

(See figure C-7 for graphic output)

enter g or G when ready to plot

g (user input)

Icfunction, 2c abs (gradient)

0 (user input)

Ictriangles, 2ccontours, 3c surface, 4cprofiles

0 (user input)

ENTER 0 FOR CURRENT AND NEXTPOINT

1 FOR 2D PRINTING

2 FOR 2D PLOTTING

3 FOR CURRENT AND STOPPING

4 FOR CHANNEL PRINTING

0 (user input)

1 .OOOE+01

44
normalized current 3.90387E+07 current 1.61832E-05 15-2.23085E-01

BETA = 3.868472E+01

PSIB = 2.871279E-01

LB = 1 .292317E-05

COX = 6.210526E-08

BPSO = 2.101317E+01

BPS I = 2.810985E+01

PS(0) = 7.266396E-01

NO = 2.659051E+1

1

PS(L) = 8.694479E-01

NL = 2.012935E+11

ID — 2.564667E-05

10 = U181714E+02

I. .. = 1 .351853E+02

VG = 6.413927E+01

VSD = 5.802708E+00

LCH = 5.45441 IE-02

level 1

1 dpsi -1 .421E+01 a -4.181E+06 Umax 3.953E+01

2 dpsi 1 .567E+01 a -3.541E+08 Umax 1 .835E+01

3 dpsi -1 .018E+00 a -2.516E+08 ximax 8.423E+00

4 dpsi -1 .028E+00 a -1 .643E+08 Umax 5.007E+00

5 dpsi -1 .041E+00 a -1 .297E+08 Umax 4.235E+00

6 dpsi -1 .057E+00 a -1 .233E+08 Umax 1 . 060E+00

7 dpsi -1 .057E+00 a -1 .246E+08 Umax 1 . 060E+00

8 dpsi -1 .057E+00 a -1 .230E+08 Umax 1 . 060E+00

9 dpsi -1 .014E+00 a -1 .179E+08 vimax 1 .016E+00

10 dpsi -9.497E-01 a . -1 . 1 17E+08 Umax 9.503E-01

11 dpsi -8.875E-01 a -1 .060E+08 Umax 8.911E-01

12 dpsi -7.569E-01 a -1 .014E+08 Umax 7.653E-01

13 dpsi -5.473E-01 a -9.766E+07 Umax 5.573E-01

14 dpsi -3.251E-01 a -9.461E+07 Umax 3.286E-01

15 dpsi -8.178E-04 a -9.460E+07 Umax 2.1 10E-01

level 2

1 dpsi 7.274E-02 a -9.51 1E+07 Umax 2.110E-01

2 dpsi 8.938E-01 a -9.898E+07 Umax 2.595E+00

3 dpsi -5.622E-01 a -9.256E+07 Umax 7.61 IE-01

4 dpsi -4.720E-01 a -9.026E+07 Umax 6.152E-01

5 dpsi -3.972E-01 a -8.976E+07 Umax 4.997E-01

6 dpsi -3.338E-01 a -8.998E+07 Umax 4.071E-01

7 dpsi -2.794E-01 a -9.034E+07 Umax 3.320E-01

8 dpsi -2.329E-01 a -9.056E+07 Umax 2.708E-01

9 dpsi -1 .933E-01 a -9.058E+07 Umax 2.208E-01

10 dpsi -1 .600E-01 a -9.048E+07 Umax 1 .834E-01

1

1

dpsi -1 .321E-01 a -9.034E+07 Umax 1.539E-01

12 dpsi -1 .090E-01 a -9.022E+07 Umax 1 .294E-01

13 dpsi -9. 146E-02 a -9.014E+07 Umax 1 .090E-01

14 dpsi -7.847E-02 a -9.010E+07 Umax 9. 195E-02

15 dpsi -6.725E-02 a -9.007E+07 Umax 7.772E-02

level 3

1 dpsi -2.044E-02 a -9.016E+07 Umax 7.772E-02

2 dpsi -6.264E-01 a -8.872E+07 Umax 2.303E+00

45
3 dpsi -6.057E-01 a -9.076E+07 Umax 6.299E-01

4 dpsi -5.519E-01 a -9.087E+07 Umax 5.671E-01

5 dpsi -4.956E-01 a -9.052E+07 Umax 5.175E-01

6 dpsi -4.541E-01 a -9.032E+07 Umax 4.759E-01

7 dpsi -4. 178E-01 a -9.029E+07 Umax 4.396E-01

8 dpsi -3.854E-01 a -9.032E+07 Umax 4.071E-01

9 dpsi -3.561E-01 a -9.032E+07 Umax 3.775E-01

10 dpsi -3.294E-01 a -9.029E+07 Umax 3.502E-01

1 1 dpsi -3.049E-01 a -9.023E+07 Umax 3.249E-01

12 dpsi -2.824E-01 a -9.017E+07 Umax 3.015E-01

1 3 dpsi -2.616E-01 a -9.012E+07 Umax 2.798E-01

14 dpsi -2.424E-01 a -9.007E+07 Umax 2.597E-01

15 dpsi -2.246E-01 a -9.003E+07 Umax 2.41 IE-01

ENTER 0 FOR CURRENT AND NEXTPOINT

1 FOR 2D PRINTING

2 FOR 2D PLOTTING

3 FOR CURRENT AND STOPPING

4 FOR CHANNEL PRINTING

1 (user input)

enter summary,;nx,ny

1 11 11 (user input)

3 MULTI-GRID LEVELS

VERTICES 146 403 613

TRIANGLES 322 1028 1676

STORAGE ALLOCATION

ALLOCATED USED

MAXT 4591 1681

MAXV 1722 613

MAXVSM 2296 1165

MAXJA 9749 4396

MAXU 1099 1099

MAXJU 1099 1062

TOTAL 54686 23301

TIMING DISTRIBUTION

SECONDS PERCENT

GRID 0.232 0.119

ASMBLY 105.177 53.804

MG 86.983 44.497

ADAPT 3.090 1.581

TOTAL 195.482 100.000

I X Y U UX UY

1 0.00000 0. 00000 1 .70141E+38 1 .70141E+38 1 .70141E+38

2 0.00919 0. 00000 1 .70141E+38 1 .70141E+38 1.70141E+38

3 0.01839 0. 00000 1 .70141E+38 1 .70141E+38 1 .70141E+38

4 0.02758 0. 00000' 1.57054E+01 -1 .91704E+03 -1.00523E+02

5 0.03678 0. 00000 9.65422E-01 --1 .24682E+03 -9.76190E+01

6 0.04597 0. 00000 -7.26674E+00 --6.01051E+02 -1.08911E+02

46
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0.05517 0.00000 -1 .04098E+01 -1 .831 19E+02 -8.39450E+01

0.06436 0.00000 -1 .09944E+01 -1 .14218E+01 -2.47309E+00

0.07355 0.00000 -1 .10994E+01 -1 .14218E+01 -2.47309E+00

0.08275 0.00000 -1.11 1 17E+01 4.65918E-01 2.09808E-01

0.09194 0.00000 -1 .11075E+01 4.65918E-01 2.09808E-01

0.00000 0.01253 1 .70141E+38 1 .70141E+38 1 .70141E+38

0.00919 0.01253 1 .70141E+38 1 .70141E+38 1 .70141E+38

0.01839 0.01 253 1 .70141E+38 1 .70141E+38 1 .70141E+38

0.02758 0.01253 1 .47769E+01 -2.0841 3E+03 -8.83702E+01

0.03678 0.01253 -6.45350E-02 -1 .25021E+03 -1 .98808E+02

0.04597 0.01253 -8.00539E+00 -4.72531E+02 -3.76761E+01

0.05517 0.01253 -1 .07103E+01 -9.00897E+01 -5.09873E+00

0.06436 0.01253 -1 .10358E+01 -7.26375E+00 2.09808E-01

0.07355 0.01253 -1 . 1 1026E+01 -7.26375E+00 2.09808E-01

0.08275 0.01253 -1 .1 1097E+01 2.47587E-01 O.OOOOOE+00

0.09194 0.01 253 -1.11075E+01 2.47587E-01 O.OOOOOE+00

0.00000 0.02505 1 .70141E+38 1 .70141E+38 1 .70141E+38

0.00919 0.02505 1 .70141E+38 1 .70141E+38 1 .70141E+38

0.01839 0.02505 1 .70141E+38 1 .70141E+38 1 .70141E+38

0.02758 0.02505 9.52872E+00 -1 .81628E+03 -8.72109E+02

0.03678 0.02505 -3.031 73E+00 -1 .03973E+03 -3.72893E+02

0.04597 0.02505 -9. 16492E+00 -3.181 13E+02 -8.87428E+01

0.05517 0.02505 -1 .08767E+01 -5.83974E+01 -6.99410E+00

0.06436 0.02505 -1 .10863E+01 -7.26375E+00 -6.99410E+00

0.07355 0.02505 -1 . 1 1 101E+01 2.96845E-01 2.71334E-01

0.08275 0.02505 -1 .1 1077E+01 2.47587E-01 2.71 334E-01

0.09194 0.02505 -1.11075E+01 -3.47702E-02 O.OOOOOE+00

0.00000 0.03758 1 .72286E+01 1 .17692E+03 4.87063E+03

0.00919 0.03758 1 .81529E+01 -8.54785E+02 -1 .09740E+03

0.01839 0.03758 9.68705E+00 -1.021 34E+03 -8.89039E+02

0.02758 0.03758 -8.99877E-02 -1 .1 1267E+03 -8.57369E+02

0.03678 0.03758 -7.08274E+00 -5.77684E+02 -2.55017E+02

0.04597 0.03758 -1 .04765E+01 -6.02732E+01 -3.92674E+01

0.05517 0.03758 -1 . 10444E+01 -5.21483E+01 -3.52405E+01

0.06436 0.03758 -1 .1 1096E+01 2.96845E-01 2.0371 IE-01

0.07355 0.03758 -1 .1 1071E+01 9.63477E-02 -4.61240E-02

0.08275 0.03758 -1 .11072E+01 -3.47702E-02 -4.61240E-02

0.09194 0.03758 -1 .1 1075E+01 2.24505E-03 O.OOOOOE+00

0.00000 0.0501

1

2.31712E+01 -1 .51404E+03 1 .07122E+02

0.00919 0.05011 1 .09044E+01 -1 .02262E+03 -2.42571E+02

0.01839 0.0501

1

1 .65833E+00 -8.28871E+02 -3.25365E+02

0.02758 0.0501

1

-5.55641E+00 -5.65257E+02 -3.50660E+02

0.03678 0.0501

1

-9.77549E+00 -2.66561E+02 -1 .58139E+02

0.04597 0.0501

1

-1 .09121E+01 -5.65621E+01 -1 .53389E+01

0.05517 0.0501 1 -1 .10835E+01 -1.11795E+01 -5.04539E+00

0.06436 0.0501

1

-1 . 1 1083E+01 9.63477E-02 1 .93562E-01

0.07355 0.0501

1

-1 .11074E+01 9.63477E-02 1 .93562E-01

0.08275 0.0501

1

-1.11075E+01 2.24505E-03 -7.86801E-05

0.09194 0.0501

1

-1 .11075E+01 2. 30819E-03 O.OOOOOE+00

0.00000 0.06264 2.46360E+01 -1 .6441 5E+03 1 .24823E+02

0.00919 0.06264 1 .05622E+01 -1 .31833E+03 8.54931E+01

0.01839 0.06264 1 .47207E-01 -8.31568E+02 9.46786E+01

0.02758 0.06264 -6.78125E+00 -6.32757E+02 1 .46283E+02

47
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1

112

0.03678 0.06264 -1 .01494E+01 -1 .40768E+02 2.95493E+01

0.04597 0.06264 -1 .09480E+01 -4.17099E+01 6.84505E+01

0.05517 0.06264 -1 .1 1023E+01 -1 .51335E+00 2.04004E+00

0.06436 0.06264 -1 .1 1069E+01 -5.90518E-02 -7.86801E-05

0.07355 0.06264 -1 .11074E+01 -5.90518E-02 -7.86801E-05

0.08275 0.06264 -1 .11075E+01 2.30819E-03 O.OOOOOE+00

0.09194 0.06264 -1 .1 1075E+01 2.30819E-03 O.OOOOOE+00

0.00000 0.07516 2.59830E+01 -1 .53400E+03 8.44740E+01

0.00919 0.07516 1 .32984E+01 -1 .08408E+03 4.79736E+02

0.01839 0.07516 3.56422E+00 -8.95785E+02 4.86827E+02

0.02758 0.07516 -4.30413E+00 -6.39271E+02 4.65338E+02

0.03678 0.07516 -9.20783E+00 -3.58072E+02 2.36151E+02

0.04597 0.07516 -1 .07994E+01 -8.72578E+01 2.58347E+01

0.05517 0.07516 -1 .10703E+01 -1 .83162E+01 1 .09657E+01

0.06436 0.07516 -1 .11084E+01 1 .06331E-01 -2.05005E-01

0.07355 0.07516 -1.11074E+01 1 .06331E-01 -2.05005E-01

0.08275 0.07516 -1.11075E+01 1 .44525E-03 1 .07529E-03

0.09194 0.07516 -1.11075E+01 2.30819E-03 O.OOOOOE+00

0.00000 0.08769 2.36266E+01 6.81750E+02 -3.01507E+03

0.00919 0.08769 2.32034E+01 -9.481 53E+02 1 .32428E+03

0.01 839 0.08769 1 .37377E+01 -1 .14021E+03 1.04329E+03

0.02758 0.08769 2.71884E+00 -1 .26672E+03 9.99422E+02

0.03678 0.08769 -5.52942E+00 -7.16435E+02 3.34261E+02

0.04597 0.08769 -9.981 66E+00 -1 .50930E+02 7.62547E+01

0.05517 0.08769 -1 .09754E+01 -4.75933E+01 2.90905E+01

0.06436 0.08769 -1 .10790E+01 -3.83210E+00 4.80330E+00

0.07355 0.08769 -1 .11088E+01 1 .06331E-01 -1 .04312E-01

0.08275 0.08769 -1 . 1 1080E+01 8.51 575E-02 -1 .04312E-01

0.09194 0.08769 -1 .11075E+01 1 .44525E-03 O.OOOOOE+00

0.00000 0.10022 1 .70141E+38 1 .701 41 E+38 1 .701 41 E+38

0.00919 0.10022 1 .70141E+38 1 .701 41 E+38 1 .70141E+38

0.01839 0.10022 1 .70141E+38 1 .701 41 E+38 1 .701 41 E+38

0.02758 0.10022 1 .37814E+01 -1 .98749E+03 9.59081E+02

0.03678 0.10022 -4.89539E-01 -1 .22193E+03 4.45889E+02

0.04597 0.10022 -8.08566E+00 -4.40321E+02 1 .25887E+02

0.05517 0.10022 -1 .06945E+01 -8.01578E+01 3.3541 1E+01

0.06436 0.10022 -1 .10436E+01 -1 .38424E+01 8.60929E+00

0.07355 0.10022 -1.11140E+01 -3.8321 OE+00 -1 .01022E+00

0.08275 0.10022 -1 .11 105E+01 1.13642E+00 -1 .01022E+00

0.09194 0.10022 -1.11075E+01 8.51575E-02 O.OOOOOE+00

0.00000 0.11274 1.701 41 E+38 1 .701 41 E+38 1 .70141E+38

0.00919 0. 1 1 274 1 .701 41 E+38 1 .701 41 E+38 1 .701 41 E+38

0.01839 0.11274 1 .70141E+38 1 .701 41 E+38 1.701 41 E+38

0.02758 0.11274 1 .97528E+01 -2.29747E+03 1 .03905E+02

0.03678 0.11274 3.05657E+00 -1 .43381E+03 2.32941E+02

0.04597 0.11274 -6.61 363E+00 -6.41455E+02 4.29263E+01

0.05517 0. 1 1 274 -1 .03770E+01 -1 .55745E+02 1.32586E+01

0.06436 0.11274 -1 .10155E+01 -1 .44971E+01 -8.19146E+00

0.07355 0.11274 -1 .11073E+01 -2.85756E+00 2.99369E+00

0.08275 0.11274 -1.111 79E+01 1 . 1 3642E+00 O.OOOOOE+00

0.09194 0.11274 -1 .11075E+01 1 . 13642E+00 O.OOOOOE+00

0.00000 0. 1 2527 1 .701 41 E+38 1 .701 41 E+38 1 .701 41 E+38

0.00919 0.12527 1 .701 41 E+38 1 .70141E+38 1 .70141E+38

48
113 0.01839 0.12527 1 .70141E+38 1 .70141E+38 1 .70141E+38

114 0.02758 0.12527 2.07662E+01 -2.12272E+03 1 .01953E+02

115 0.03678 0.12527 4.18054E+00 -1 .44197E+03 9.42864E+01

116 0.04597 0.12527 -5.69846E+00 -7.66748E+02 1 .21720E+02

117 0.05517 0.12527 -9.95308E+00 -2.76592E+02 1 .04345E+02

118 0.06436 0.12527 -1 .09852E+01 -4.05405E+01 4.23887E+00

119 0.07355 0.12527 -1 . 10732E+01 -4.31471E-01 6.62310E-01

120 0.08275 0.12527 -1 . 10893E+01 -1 .97889E+00 2.99369E+00

121 0.09194 0.12527 -1 .11075E+01 -1 .97889E+00 2.99369E+00

ENTER 0 FOR CURRENT AND NEXTPOINT

1 FOR 2D PRINTING

2 FOR 2D PLOTTING

3 FOR CURRENT AND STOPPING

4 FOR CHANNEL PRINTING

0 (user input)

normalized current 9.00330E+07 current 3.73224E-05 1 5-2. 2461 4E-01

CPU time 29:18.05 Elapsed time 2:23:34.40

49
Figure C-1. , Example of level 1 mesh.

50
Figure C-2. Example of level 2 mesh.

51
Figure C-3. Example of level 3 mesh.

52
Figure C-4. Example of level 1 projection.

53
Figure C-5. Example of level 2 projection.

54
Figure C-6. Example of level 3 projection.

55
Figure C-7. Example of contour plot.

56
NBS"114A (REV. 2-6C) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBSIR 82-2471

2. Performing Organ. Report No. 3. Publication Date

April 1982

4.

TITLE AND SUBTITLE

CSl: A Two-Dimensional Finite Element Charge-Sheet Model of a Short-Channel MOS

Transistor

5.

AUTHOR(S)

C. L. Wilson and J. L. Blue

6.

PERFORMING ORGANIZATION (If joint or other than NBS, see instructions) 7 . Contract/Grant No. NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE 8 . Type of Report & Period Covered

WASHINGTON, D.C. 20234

9.

SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

10. SUPPLEMENTARY NOTES

^21 Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bi bliography or literature survey, mention it here) A two-dimensional charge-sheet model for short-channel MOS transistors has been developed. The unique

feature of the model Is that the effect of channel Inversion layer charge Is Included as a nonlinear Integral boundary condition on the two-dimensional electrostatic field In the transistor. The average Inversion layer charge density and source-drain current are obtained directly from the model rather than from

the electron density or electron quasI-FermI level. The model retains all of the physical detail of more

complex two-dimensional models such as sensitivity to source-drain profile shape, channel profile, and

oxide field shape. This allows the model to represent the changes In drain current associated with shortchannel effects while still allowing simple comparison with long-channel models. For long-channel transistors, the results of this model are Identical to Brews' long-channel charge-sheet model. The accuracy

of this model Is verified by modeling a sequence of transistors with channel lengths between 4.6 and 1.1 pm. In short-channel transistors, effects previously attributed to high field mobility are explained by

simple two-dimensional electrostatics.

The simulations produced using this model have been compared to experimental measurements on an array of

n-channel MOSFETs; the model Is In good agreement for transistors with channel lengths as short as 1.1 pn.

In this verification process, the model represented accurately the onset of subthreshold current, channellength-lnduced threshold voltage offset, and draln-f leld-Induced output conductance changes.

l'

i!

i

From studies of numerical accuracy, we conclude that the charge-sheet model can easily simulate drain

current with an accuracy which exceeds that required for most applications. To obtain 5-percent accuracy

for drain current, a 146 element mesh Is sufficient. Refinement of the 146 element mesh to a 455 element

mesh gives a current which Is accurate to 0,16 percent. Average computer time for a high accuracy solution Is 2,5 min on a DEC-20,

The numerical solutions were obtained using general-purpose software for solving elliptic partial differential equations. Problems with exact solutions have been solved to test the correctness and accuracy of

the codes. Also, the physics Included In this model and the geometry of the transistor can be easily

changed. The finite element method used allows refinement of oblique triangles which Is Important In achieving computational efficiency. The program Is portable and has been run on a DEC-20, a VAX 11/780, a Cyber 175, and a Univac 1108,

12.

KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Elliptic solvers; finite elements; Interactive graphics; MOS transistor.

13. AVAILABILITY 14. NO. OF

Unlimited

PRINTED PAGES

1 1 For Official Distribution, Do Not Release to NTIS

Ijn Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

61

15. Price

Order From National Technical Information Service (NTIS), Springfield, VA. 22161

to oo

USCOMM-DC 6043-P80
01. Summary date

Yr. Mo. Day

1 1 2 1 5

04. Software date

Yr. Mo. Day

L_2_

FEDERAL INFORMATION PROCESSING STANDARD SOFTWARE SUMMARY

02. Summary prepared by (Name and Phone)

C. L. Wilson (301) 921-3541

05. Software title CSl: A Two-Dimensional Finite Element ChargeSheet Model of a Short-Channel MOS Transistor

06. Short title

03. Summary action New Replacement Deletion

Previous HDInternal Software ID

07. Internal Software ID CSl

08. Software type

Automated Data

I I System ^ Computer Program

I [ Subroutine/Module

09. Processing mode ^ Interactive g] Batch

) I Combination

10. General

Application area

Computer Systems

I I Support/Utility • Q Scientific/Engineering

[~| Bibliographic/Textual

Management/

[ I Business

Process Control Other

Specific MOS Transistor

Modeling

11.

Submitting organization and address National Bureau of Standards

Washington, DC 20234

12.

Technical contact(s) and phone

C. L. Wilson

J. L. Blue

921-3541

921-3395

13.

Narrative

A two-dimensional charge-sheet model for short-channel MOS transistors has been developed,

The unique feature of the model is that the effect of channel inversion layer charge is

included as a nonlinear integral boundary condition on the two-dimensional electrostatic

field in the transistor. The program is portable and has been run on a DEC-20, a VAX

11/780, a Cyber 275, and a Univac 1108.

14.

Keywords

Finite elements, MOS transistor, elliptic solvers, interactive graphics.

15.

Computer manuf’r and model DEC- 20

16.

Computer operating system

TOPS- 20

17. Programing language(s)

FORTRAN

18. Number of source program state- ments

19. Computer memory requirements

80K

20. Tape drives 0

21. Disk/Drum units 0

22. Terminals

1

23. Other operational requirements

Requires local vector-oriented graphics

24. Software availability

Available Limited 0

C. L. Wilson 921-3541

9- track magnetic tape.

In-house only

25. Documentation availability

Available Inadequate In-house only

Tital as in section 05.

26. FOR SUBMITTING ORGANIZATION USE

185-101 Standard Form 185

1974 July

U.S. Dept, of Commerce—NBS|

(FIPS. Pub. 30)

\h1I