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NIST SPECIAL PUBLICATION 400-95

U.S. DEPARTMENT OF COMMERCE/Technology Administration

National Institute of Standards and Technology

Semiconductor Measurement

User's Manual for the Program MONSEL-1: P

Simulation of SEM Signals for Linewidtl

zA
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Semiconductor Measurement Technology:

User's Manual for the Program MONSEL-1: Monte Carlo

Simulation of SEM Signals for Linewidth Metrology

Jeremiah R. Lowney

Semiconductor Electronics Division

Electronics and Electrical Engineering Laboratory

National Institute of Standards and Technology

Gaithersburg, MD 20899-0001

and

Egon Marx

Precision Engineering Division

Manufacturing Engineering Laboratory

National Institute of Standards and Technology

Gaithersburg, MD 20899-0001

August 1994

U.S. DEPARTMENT OF COMMERCE, Ronald H. Brown, Secretary

TECHNOLOGY ADMINISTRATION, Mary L. Good, Under Secretary for Technology

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, Arati Prabhakar, Director
National Institute of Standards and Technology Special Publication 400-95

Natl. Inst. Stand. Technol. Spec. Publ. 400-95, 43 pages (Aug. 1994)

CODEN: NSPUE2

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1994

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402-9325
Semiconductor Measurement Technology:

User's Manual for the Program MONSEL-I:

Monte Carlo Simulation of SEM Signals

for Linewidth Metrology

TABLE OF CONTENTS

Page

Abstract 1

Introduction 2

Installation 2

Random-Number Generator 2

General Specimen Shape 3

Structure of the Program 3

Computation of Cross Sections 5

Computation of Energy Loss 8

Computation of New Direction Cosines 8

Computation of Intersection of Trajectories with Boundaries 10

Material Constants 14

Summary 15

References 16

Appendix I 17

Appendix II 27

iii
LIST OF FIGURES

Page

1. Diagram of general specimen shape 4

2. Diagram of backscattered-electron detector 4

3. Diagram of an example of an interssection of an electron trajectory with

the edge of a trapezoidal line 12

4. Calculated transmission (solid) and backscatfering coefficient (dashed) for

half of a 0.5-u.m line with a 2-deg wall slope and a 50-nm jog 25 nm above

the bottom 13

iv
Semiconductor Measurement Technology:

User's Manual for the Program MONSEL-I

Monte Carlo Simulation

of SEM Signals for Linewidth Metrology

Jeremiah R. Lowney

Semiconductor Electronics Division

Electronics and Electrical Engineering Laboratory

National Institute of Standards and Technology

Gaithersburg, MD 20899

and

Egon Marx

Precision Engineering Division

Manufacturing Engineering Laboratory

National Institute of Standards and Technology

Gaithersburg, MD 20899

ABSTRACT

This user's manual is a guide to the FORTRAN code MONSEL-I which is a Monte Carlo

simulation of the transmitted and backscattered electron signals in a scanning electron

microscope (SEM) associated with a line specimen with a trapezoidal cross section. The line

is deposited on a multi-layer substrate. The primary purpose of the code is to determine the

actual linewidth from measured SEM signals. However, it can be used for many other

purposes such as transmission electron microscopy. Future extensions to model secondary

electron signals and multiple lines are planned.

Key words: backscattered electrons; electron transmission; linewidth; Monte Carlo simulation;

scanning electron microscope; X-ray lithography.

1
INTRODUCTION

The FORTRAN code MONSEL-I was written to simulate the linewidth measurements carried out on

a metal line on top of a multilayer substrate placed in a scanning electron microscope (SEM) used

in the transmission or backscattering mode. The simulation uses the Monte Carlo method, and the

simulated specimens were x-ray lithographic masks.

The trajectories of the electrons in the beam are represented by straight lines between localized

interaction sites, where the direction and the energy of the electron change. These changes are given

by formulas that involve a random number which represents the probability of a particular change.

Electrons can also dislodge other electrons (generated secondaries) at these interaction sites, and these

electrons have separate trajectories. Electrons are counted when they leave the line or the substrate, or when their energies become too low while still inside the line or substrate. Electrons that have

an energy of less than 50 eV when they leave the sample are called secondary electrons.

Future extensions of the code will model secondary signals more completely, as well as multiple lines on the same substrate. This present version, although aimed at a specific problem, should be useful

for a number of simulation cases. Suggestions for further extensions are encouraged.

INSTALLATION

The code is essentially a stand-alone FORTRAN-77 code except for the incorporation of a random

number generator (see below). The source code (approximately 73 kB) and sample input and output

data files are available in ASCII format on disk or by using electronic mail over the Internet. The

code is straightforward to run once the source has been compiled and linked by the user. The sample

input and output data files are included so that the user can check the programs for proper operation

as well as to become acquainted with the setup and use of the codes.

The code can run on nearly any machine, but hardware and software to speed up the execution should

be considered by the user. A typical calculation of 80 points across a line with 20,000 electrons per

point requires about 8000 s on a Cray Y-MP. 1 Such a calculation can take 20 h on a workstation.

The memory requirements are relatively small. The executable is only 1.12 MB in length, which

should fit on most machines. Although the code should perform correctly on all input data sets, there

is no assurance that the code will perform equally well on all such data sets. Therefore, the authors

and NIST assume no liability for possible losses resulting from the use of this code.

RANDOM-NUMBER GENERATOR

The heart of any Monte Carlo code is the random-number generator. The present code runs on a Cray Y-MP and several workstations at NIST. The code must be modified to call the random-number

Certain commercial equipment is identified in this report in order to describe the experimental procedure

adequately. Such identification does not imply recommendation or endorsement by NIST, nor does it imply that the equipment identified is necessarily the best available for the purpose.

2
generator (numbers are between 0 and 1) present on the user's computer. The Cray version, which

is being made available, calls RANSET(O) in subroutine RNDSET to initialize the random-number

generator and user function RANF() in function RNDFUN() to obtain the random numbers. The Cray names will have to be changed to agree with the user's supplied functions. Subroutine RNDSET and

function RNDFUN() contain the Cray routines and must be edited before compilation. These

subprograms should be "inlined" for efficient execution.

GENERAL SPECIMEN SHAPE

The shape of the specimen is shown in figure 1, which displays a cross section perpendicular to the

translationally invariant line. The substrate can have up to three different layers, and all layers and

the line can have four constituents. The layers in the present code are chromium, polyimide, and

silicon; they can be changed by editing MONSEL.DAT, which uses free-format data lines under

headers. The line is trapezoidal in cross section with a jog symmetrically located along both edges.

The jog can be at any height along the edge and can have any width. The line is presently gold

although it can be any material by editing MONSEL.DAT. The trapezoidal cross section can also

model an undercut if the angle is negative, but then the jog height and width must be set to zero. STRUCTURE OF THE PROGRAM

A call to INPT from MAIN inputs the data files MONSEL.IN and MONSEL.DAT along with the

physical constants. In the MAIN program, the electrons are subdivided into groups of 200 electrons

each to take advantage of vector processing. The variable Nl in the PARAMETER statements

throughout the code would have to be changed to change the value from 200, which is Nl/5 where

Nl is presently 1001. The main loop starts with a computation for the incident electrons and then

processes the electrons until all electrons are either backscattered, transmitted, or have been dropped

if their energy is below the cutoff energy. The incident electrons are in a beam of given energy,

diameter, and position, perpendicular to the plane of the sample.

The incident direction is normal to the substrate. A call to SBEAM distributes the electrons

according to a Gaussian distribution determined by the beam diameter. There is a test to see whether

the electrons hit the line or the substrate. The initial coordinates of each electron are computed in SBEAM. The initial elastic mean free path is computed in MFP.

In the repeating part of the main loop:

o a call to XSEC determines the scattering angle and the step to the new position, o a call to GEOM determines the new direction cosines from the old ones and the scattering angle, o a call to BOUND determines whether any boundaries have been crossed, dropping those electrons

that have been transmitted or backscattered, and those for which the energy at a boundary goes

below the cutoff energy, o a call to ELOSS calculates the energy loss of the electron when it reaches its new position, o a call to SCND determines whether any secondary electrons have been produced by ionization

of valence electrons, and

3
Lithographic

Line

Substrate Layer 1 Substrate Layer 2

Substrate Layer 3

Figure 1. Diagram of general specimen shape.

Beam

Figure 2. Diagram of backscattered-electron detector.

4
o the arrays are rearranged to eliminate the dropped electrons.

In the subroutine BOUND, the sequence of computations is:

for those electrons that started out in the substrate:

o transmitted electrons, i.e., those that end up under the substrate, are dropped after information on

their energy and location is recorded, if desired, as well as whether they hit a detector with a defined aperture, separation distance, and low energy cutoff, o the remainder may leave one of the layers of the substrate, o those that leave the substrate and miss the line are backscattered and dropped, o for those that hit the line, the energy at the interface is calculated by a call to SELOSS; if the

energy is below the cutoff they are dropped, and the remaining electrons are left at the

intersection with the line;

for the remaining electrons that started out in the line:

o those that end up above the line are backscattered and are dropped,

o those that remain within the height range of the line are checked to see whether they cross the

sides, o those that remain in the line are left alone, o those that cross the side and are moving upwards are either backscattered or reenter the line, o those that cross the side and are moving down end up in the substrate or back in the line, o for those that end up in the substrate or line the intersection coordinates are determined, the

energy at the interface is calculated by a call to SELOSS; if the energy is below the cutoff they

are dropped, and the remainder are set at the interface, o Note that backscattered electrons with energy below 50 eV are called secondaries, and that generated secondary electrons, which may have any energy, are deleted if they are generated by

an electron that crosses a boundary.

The output files are: MONOUT, which gives a detailed account of each computed position across

the line; MONSEL.OUT, which gives the backscattering coefficient, secondary yield (not fully

functioning yet), total transmitted, and detected transmitted signals as a function of position across

the line; MONBD.OUT, which gives the backscattered signal detected by a split-donut detector (see

fig. 2); and MONSEL.TR, which gives the x,y positions of all the electrons for trajectory plotting.

A practical number of electrons for trajectory plotting is 20. Examples of all the input and output

files are given in the Appendix I. COMPUTATION OF CROSS SECTIONS

Both the elastic cross section and the cross section for the generation of secondary electrons are considered.

5
The elastic cross section ae [cm ] is given by Browning's fit to the Mott cross section [1]:

Ge = 3xlO'^Z L7/(E + 0.005Z L7 ^+ 0.0007 Z

2/\/£) , 0)

where Z is atomic number and E is energy in kiloelectron volts.

The corresponding mean free path is K

= NAP°e

where NAis Avogadro's number, A is atomic weight, and p is density.

(2)

Browning obtained the differential elastic cross sections by assuming that they result from a sum of

screened Rutherford and isotropic scattering. One reason for choosing these two forms is that they

both have simple, analytic expressions for the differential cross sections, which can be readily

evaluated by the code at each scattering event. Monte Carlo codes generally have long running times

to achieve accuracy, and it is imperative that the numerical evaluations be done as quickly as possible.

The use of look-up tables (even after spline fitting) for the true differential Mott cross sections would

be very time-consuming. Browning adjusted the ratio r of the screened Rutherford to isotropic

scattering so that he obtained the same ratio of total forward-to-backward scattering cross section as from the true Mott cross sections:

r = 300 _ + z 3xl05 £

(3)

5 The screening parameter a in the Rutherford scattering was taken to 7 x 10°/£ for all Z

Secondary electrons may be generated when the energy is above a cutoff energy, Wc

, which may be

higher than the minimum energy for the scattering of electrons. Only ionization of valence electrons

has been considered so far. Other mechanisms will be included in later versions. For a fast secondary electron of energy W obtained by ionizing a valence electron by a primary electron of

energy E, Moller's differential cross section is given by [2]:

dvJdW = ne 1 1 1 W (E-W) {Anz 2 W{E-W)

0 fE

which can be integrated starting from a minimum value e m

= WJE to give

4

(4)

ne

(47ie0 ) 2 £

2

1 1

(1 -o

- In (5)

6
The probability for obtaining a value of the reduced energy of the secondary electron between em

and

e = W/E is the integral

P(z) = — f

8 d°se

dz (6)

which is a number that goes from 0 to 1 as 8 goes from e mto 0.5. Given a random number, R,

between 0 and 1, for this probability, the reduced energy 8 that corresponds to this number can be

determined.

The partial cross section up to an arbitrary energy 8 is obtained as well, and both it and eq (5) involve

a logarithmic term. This term presents difficulties when solving for the secondary-electron energies

in terms of random numbers generated by the Monte Carlo routine. Thus, a rational polynomial

function has been used, (A + Be + Ce + De )/(l + Ez + Fe ), to fit the following function j{z) to better than 1% throughout:

/(e) = 1 - 8/(1 -s) - s In [(1 -e)/e], 0 < e <0.5 , (7)

where A = 0.988798, B = 0.220312, C = -9.16690, D = 9.53217, E = 4.95919, and F = -10.5818.

The solutions for the cross sections, thus, only involve the solution of a cubic equation, which saves

considerable computer time instead of dealing with the logarithmic term.

In the subroutine SCND, the computations for the secondary electrons are carried out by assuming

that the new electron is the one with the lower energy, which has to be above the cutoff. The energy

of this electron is computed with the help of a random number generator, and so are the direction and

the starting location. The scattering angle and step of the primary electron are computed with the

formulas for elastic scattering for the next step.

The cross section for secondaries is, multiplying by the number of valence electrons Zv [3] in an

atom,

vse = 6.51xlO-20Zv /£ 2 1 1 (1-5J

In 1-s.

m

(8)

The mean free path for the production of secondary electrons is

A

NA^se

(9)

The probabilities of generating a secondary electron or undergoing elastic scattering are proportional

to the corresponding cross sections. A random number r between 0 and 1 is generated, and a secondary electron occurs if

r < o- se/(cj

e + ct S6) = Xe/(Xe se + XJSL" (10)

The total cross section is

7
Gtot = + CTse (10

and the total mean free path is

^tot = KKJ(K +^se) • (i 2)

The step is obtained from the corresponding mean free path X by multiplying X by another random

number,

d = - X\og(RND) , (13)

where RND is a uniformly distributed random number between 0.000001 and 0.999999. The

scattering angle is negligible in the event of the generation of a secondary electron, and for elastic

scattering the cosine of the scattering angle for screened Rutherford scattering is given by

/m , 2aRND nAx cos (9) = 1 - , (14)

1 +a-RND

where RND is another random number uniformly distributed between 0 and 1. For isotropic

scattering the scattering angle is uniformly distributed.

If the computed step takes the electron from one medium to another, as determined in the subroutine

BOUND, it is assumed that electron reaches the boundary without suffering a scattering event, but

the energy is decreased by using the modified Bethe formula [4]. Then the possibility of another

scattering event in the new medium is considered.

COMPUTATION OF ENERGY LOSS

The energy loss in eV/nm is obtained from Bethe's formula modified by Joy and Luo [4],

dEldS =

~ 785QpZ

ln[1.166(£ + 0.85J)/J] , (15)

AE

where J is the mean ionization potential in electron volts,

J = 9.76Z + 58.5/Z 019

, Z > 13; J = 11. 5Z, Z < 12 ( 16)

and S, p, and A are path length (nm), density (g cm"3

), and atomic weight (g), respectively. Equation

(15) is valid for energies E above 8 x 10 J in electron volts. COMPUTATION OF NEW DIRECTION COSINES

The new direction cosines A \ B \ and C, are defined in terms of the polar angle 0 and azimuthal

8
angle $ in a coordinate system where the old direction cosines are (1,0,0), that is, where the electron

is moving along the z-axis. There, 0 is the scattering angle and $ is selected at random. To change

from this coordinate system to the original one, a rotation about the x-axis by an angle 9 is followed

by a rotation about the new z-axis by an angle <|>. The matrix of this product of rotations is:

cos<j) sin<() 0

sin<j) cos(j) 0

0 0 1

1 0 0

0 cos<J) sinG

0 - sin6 cosB

cos<j) sin<j)cos0 sin(j)sin9

sin<j) cos())cos9 cos(j)sin9

0 - sinO cos9

(17)

Then, the unit vector that has the original direction cosines as components can be obtained from the vector along the z-axis by this transformation,

'A' cos(|) sin(|)cos9 sin(j)sin9

" 0' ' sin<|)sin0

B - sintj) cos(|)cos9 cos())sinG 0 cos<j)sin0

C 0 - sinO cos9 1 cos9

The new direction cosines in the original coordinate system can be obtained from those in the new

coordinate system by a similar transformation, that is

B

C

cos()) sin(|)cos0 sin<j)sin9

sin<j) cos(j>cos0 cos<|)sinG

0 - sinG cosO

a"

P

y

(19)

where the direction cosines in the new coordinate system are

Thus:

a' ' sinQsinO

'

P sin0cosO

Y

« cos©

(20)

B

C

acoscj) + sin<|)(Pcos9 + ysinG)

asin<() + cos(|>(Pcos9 + ysin9)

- PsinG + ycos9

(21)

Given the direction cosines A, B, and C and the components of the new direction cosines in the

rotated coordinate system, a(9,$), P(9,$), and y(9,$), cos9, sin9, sin<j), and cos<J> are first determined

from A, B, and C:

cos9 = C, sinG = (1 - C

2

)

Vl

, sinij) = ^/sin9, cos(j) = 5/sin9 . (22)

Then, A \ B\ and C ' are computed from a, p, and y. When 0 = 0, we are already in a coordinate

9
system with the electron going along the z-axis, the angle § is undetermined, and, for § = 0, the

above equation reduces to

A

1 a

B< P

C> y

(23)

Similarly, when 9 = n, the angle (|> is also undetermined, and

A> a

B> -P

c' -y

(24)

If 7i - <£ is needed instead of which is also a random angle, the sign of the ^-component does not

have to be changed, and:

A

1 a

B

1 P

c' -y

(25)

COMPUTATION OF INTERSECTION OF TRAJECTORIES WITH BOUNDARIES

This section corresponds to a trapezoidal line with no jog along the sides. The inclusion of the jog

follows the same procedures outlined here, but with the further complications due to the additional

ledge.

The base of the line is at z = 0, the top of the line is at z = h, centered about x = 0, and independent

of y. The width of the top is 2wj and the width of the base is 2w2. The thickness of the substrate

is b, so that the bottom of the substrate corresponds to z

b = -b.

The direction cosines of the step are a, (3, and y, which are components of a unit vector t. The final coordinates computed for that step are x^y^ and z/5 forming the position vector ~x

t

. The step is s

t and

the previous energy is E.

Assume first that the electron starts in the substrate. If the new position is below the substrate (z <

z

b

), the electron is transmitted and is no longer of interest. If there is a finite detector under the

sample, the position of the electron at the detector coordinate is determined and a check is made as

to whether it hits the detector itself.

10
If the electron crosses the top of the substrate, the coordinates of the point of intersection are found.

The backward step is s

b = z/y, and the coordinates of the intersection are x0 = x; - s

ba, y0 = yi - sbfi, and z

0 = 0. The step inside the substrate is s

0 = s

t - sb

, which is used to compute the energy of the

electron when it leaves the substrate.

The step to the top of the line, s

t = hly, and the x-coordinate at the top, x( = x0 + s

t a, are calculated.

Assume that the electron crosses into the vacuum at the right of the line, x0 > w2

. If xt > Wj, the

electron does not hit the line. If the energy is greater than the threshold, 50 eV, the electron is considered to be backscattered; otherwise it is a secondary electron. If the electron hits the line,

whether the remaining energy is greater than the cutoff energy is determined. If so, the intersection

with the right side of the line is found. The equation of the trajectory of the electron is ~x = ~x 0 + st. The equation of the side of the line is (x - w2)l(wj - w2

) = z/h. By letting w2

- w

} = Aw, the step is found to be:

Sj = - h(xQ

- w2)/(ah + yAw) = - (x0

- w2)/(a + yAw/h) . (26)

Similarly, the intersection with the left side of the line corresponds to the step

s

2 = - h(x0 + w2)/(ah - yAw) = - (x0 + w2)/(a - yAw/h) . (27)

If the electron leaves the substrate to the left of the line, x0 < - w2

, the same computations are made

with the appropriate changes. Otherwise, the electron crosses directly into the line, and the new step

in the line is determined.

If the electron starts out in the line, it can end up above the line, z

i > h. First the intersection with

the top, ~x

t = T/ - sb

t, is determined from the back step s

b = (z; - h)ly. The electron exits through the

right side if xt > Wj. The intersection with the right side, ~x*s = ~x*

t - ssJt is determined as above, and

s

sl = h(x

t - Wj)f(yAw + ah) . (28)

Similarly, the intersection with the left side of the line corresponds to the step,

s

s2 = - h(xt + Wj)/(yAw - ah) . (29)

The remaining energy where it leaves the line is calculated. Figure 3 shows a diagram of these

distances for an example trajectory.

If instead, the new electron position is in the substrate, the point where the electron enters the

substrate is determined, given, by ~x 0 = j?

f - s

b

t, where s

b = z/y (z

i < 0, y < 0). If it leaves through the

side, the exit point ~x

s = ~x 0

- s

st is determined. If it exits through the right side,

s

s] = h(x0

- w2)/(ah + yAw) . (30)

1

1
Figure 3. Diagram of an example of an intersection of an electron trajectory

with the edge of a trapezoidal line.

12
Figure 4. Calculated transmission (solid) and backscattering coefficient (dashed)

for half of a 0.5-|im line with a 2-deg wall slope and a 50-nm jog 25 nm above the

bottom.

13
If it exits through the left side,

s

s2 = h(x0 + w2)/(ah - yAw) . (31)

The remaining energy is determined as well as the new step in the substrate and new final position.

The remaining case corresponds to an electron with a z-component such that 0 < z - < h. To find out

whether it left the line, the distance from the point to the sides of the line, projected on the y = 0

plane is determined. If one of these distances is negative, the electron has left the line. A normal

to the right side of the line is 7V = h~T + Aw~K, and the scalar product with the vector from the new

position to the base of the line, (w2

- xjfT - zpt, gives

gj = h(w2

- xt)

- zfiw . (32)

The corresponding quantity for the other side is

g2 = h(w2 + x;)

- zfiw . (33)

If both are positive, the electron remains in the line. If one of these is negative, the intersection of

the trajectory of the electron with the corresponding side is computed and the step in the line and the

remaining energy are determined. If the electron is directed down, the location where the electron

hits the substrate is computed, as well as the new step. The intersection with the right side of the line

is now given by the back step

Sj = [z

; Aw + h(xi - w2)]/(yAw + ah) . (34)

Similarly, the intersection with the left side of the line corresponds to the back step

s

2 = [ZjAw - h(Xj + w2)]/(yAw - ah) . (35)

Figure 4 shows an example of the transmitted (solid) and backscattered (dashed) electron signals for

half of a 0.5-u.m gold line, 0.7-u.m thick, with a 2-deg wall slope and a 50-nm jog 25-nm above the

bottom of the line. The substrate is 2.5 u.m of silicon with a 5-nm layer of chromium on top. The

beam energy is 30 keV with a 10-nm diameter and 20,000 trajectories. The remaining parameters

are given by the last four lines of data of MONSEL.IN in Appendix I.

MATERIAL CONSTANTS

The constants for silicon are p = 2.328 g/cm , Z = 14, and A = 28.09.

For gold, they are p = 19.32 g/cm3

, Z = 79, and A = 196.97.

For chromium, they are p = 6.52 g/cm , Z = 24, and A = 52.01.

For polyimide, there are four components: carbon, hydrogen, nitrogen, and oxygen, p = 1.42 g/cm3

, Z(C) = 6, Z(H) = 1, Z(N) = 7, and Z(0) = 8. The corresponding values of A are: 12.011, 1.008,

14.008, and 16.

Since the material in one of the layers has several components, the way the cross sections and the

14
mean free paths are computed must be changed. Several cross sections (J

i exist for each component.

The mean free path that corresponds to a; is

Xt = !— (36) NA pcpt

where c, is the weight fraction for the ith component. For polyimide the weight fractions of carbon,

hydrogen, nitrogen, and oxygen are 0.691 1, 0.0264, 0.0733, and 0.2092, respectively. The total mean

free path is given by

MX = Zl/Xj . (37)

For the energy loss, the modified Bethe formula changes to —dF n c iZ; = - 78500 -E-ZJ-i log

ds E A ; 5

1.1 66 (£ + 0.85 J,-) (38)

where Ji is the mean ionization potential.

A similar approach has to be followed for secondary electrons. There are different cross sections

depending on which atom is involved. The mean free path for the production of secondary electrons

for each constituent is

A:

(39)

and the total mean free path for the production of secondary electrons is obtained from

VXse =U/Xise

. (40)

SUMMARY

This manual describes the operation of the FORTRAN code MONSEL-I, which computes the electron

transmission and backscattering signals from a trapezoidal line on a multi-layer substrate placed in a scanning electron microscope. The theoretical techniques involved are discussed in this manual as well as in Appendix II. Examples of input and output data appear in Appendix I. This code has been

designed primarily for the measurement of linewidths on x-ray lithographic masks, but can have wide

applications to simulate the interaction of electron beams and specimens.

15
REFERENCES

R. Browning, T. Eimori, E. P. Traut, B. Chui, and R. F. W. Pease, An elastic cross section

model for use with Monte Carlo simulations of low energy electron scattering from high atomic

number targets, J. Vac. Sci. Technol. B Vol. 9 (6), pp. 3578-3581 (Nov. 1991).

L. Reimer, Scanning Electron Microscopy (Springer-Verlag, New York, 1985), pp. 75-77.

M. Cardona and L. Ley, Photoemission in Solids I, Topics in Applied Physics 26 (SpringerVerlag, New York, 1978), pp. 265-274.

D. C. Joy and S. Luo, An Empirical Stopping Power Relationship for Low-Energy Electrons,

Scanning Vol. 11 (4), pp. 176-180 (1989).

16
APPENDIX I

INPUT AND OUTPUT DATA FILES

The following pages give sample input and output data. These files must be in the same directory

as the running code. The input file is MONSEL.IN. The entries are on data lines in free format

preceded by header lines. This is true for all input files. The units are shown in each header. The

first data line refers to the trajectory plot. If a trajectory plot is desired, the flag is true and the top,

bottom, and total width of the frame are given. The second data line specifies the height and the

width of the top of the line being measured, and the angle refers to the slope of the trapezoidal edge

(see fig. 1). The third data line specifies the height and width of the jog along the edges, and the

fourth data line specifies the thickness of the three layers that form the substrate. The fifth data line specifies the beam diameter, its initial and final positions, and the spacing between points. The sixth

data line specifies the number of trajectories per point and the incident beam energy. The seventh

data line specifies the distance between the sample and the transmission-signal detector, which is mounted concentric to the center of the line, the detector aperture (diameter), and cutoff energy. The

eighth data line specifies the distance between the sample and the backscattered - electron detector

(e.g., micro-channel plate or semiconductor diode detector), its inner and outer diameters (see fig. 2),

its collection efficiency, and cutoff energy. The ninth data line specifies the lowest energy an electron can have before being dropped, and the tenth data line specifies the lowest energy a generated

secondary electron may have. Setting both to 50 V eliminates collected secondaries.

MONSEL.DAT supplies the atomic numbers (Z), atomic weights (A), and densities (RHO) in grams

per cubic centimeter of the line and three substrate layers. The number of component elements of

each layer (NCE), weight fraction (WF), and the number of valence electrons (NVE), for each

component are also given. Any element may be entered, and the entries are in free format under the

headers.

MONOUT furnishes a detailed summary of the signals at each point. The first set of lines

recapitulates the input data, the next set summarizes the output. The letters L and R in parentheses

refer to the left half and right half of the backscattered-electron detector. The letters S and D in parentheses refer to the sum of both halves and the difference obtained by subtracting the right from

the left half, respectively. The energy distributions (between El and E2) of the electrons are given

in the following tables. The first is for total backscattered, total transmitted, and detected transmitted

electron signals. The second corresponds to the detected backscattered electron signal with the same

notation as above.

MONSEL.OUT gives the total backscattered, secondary, total transmitted, and detected transmitted

electron signals as a function of position. It is very useful if there is more than one point. The initial lines recapitulate the input data. MONBD.OUT is like MONSEL.OUT except that it refers to detected backscattered-electron signals.

The entries are: position, backscattered electron signal in the left half of the detector, same in the

right half, their sum, and the right half subtracted from the left.

MONSEL.TR provides the initial and final positions of each electron step for trajectory plotting. All

entries are in centimeters. The initial lines in the file are:

17
line 1) the half-width, top, and bottom of the plot frame;

line 2) the height, top half-width, and bottom half-width of the line; line 3) the bottom of the first, second, and third layers;

line 4) the height, half-width of inner separation, and width of the jogs.

The remaining lines are initial x,z positions of the electron followed by the corresponding final

position along each trajectory. The first few lines are shown for the same case as in the prior files but for only one electron.

18
MONSEL.IN

traj . plot: flag frame: top (urn) bottom (um) width (um)

F 1.0 -3.0 4.0

line: height (um) width (um) angle (deg.)

0.7 0.5 2.0

jog: height (um) width (nm)

0.35 20.0

substrate: layer 1 (um) layer 2 (um) layer 3 (um)

5 . Oe- 1 0.5 0.5

beam: diam. (nm) init . pos . (um) fin. pos . (um) spacing of pts. (nm)

10. 0 0.0 0.0 5.0

no. inc. el. beam energy (keV)

20000 30.0

trans, det . : dist. (mm) aperture (mm) cutoff en. (keV)

5.0 12 . 5 16.0

bcksc . det. : dist. (mm) id (mm) od(mm) ef f . cutoff en. (keV)

5.0 2.5 12. 5 1.0 16.0

minimum electron energy considered in calculation (eV)

50 . 0 minimum electron energy of generated secondary electron (eV)

50 . 0

19
MONSEL.DAT

Z A(g) RH0(g/cm3) NCE WF NVE

LINE: ELEMENT IS GOLD

19.32 1 79. 196.97 I- 11

FIRST LAYER: ELEMENT IS CHROMIUM

6 . 52 1 24.52.01 1 • 6

SECOND LAYER: COMPOUND IS POLYIMIDE (C, H, N, 0)

1.42 4

6 12.011 0.6911 4

1 1.008 0.0264 1

7 14 008 0.0733 3 8. 16. 0.2092 4

THIRD LAYER: ELEMENT IS SILICON

2.328 1 14. 28.09 1- 4

20
MONOUT

BEGIN OUTPUT

HEIGHT OF LINE IS 7.000E-01 urn HALF-WIDTH OF TOP OF LINE IS 2.500E-01 um HALF-WIDTH OF BOTTOM OF LINE IS 2.944E-01 um HEIGHT OF JOG IS 3.500E-01 um WIDTH OF JOG IS 2 . 000E+01 nm THICKNESS OF THE FIRST LAYER IS 5.000E-01 um THICKNESS OF THE SECOND LAYER IS 5.000E-01 um THICKNESS OF THE THIRD LAYER IS 5.000E-01 um BEAM DIAMETER IS 1.000E+01 nm BEAM STARTS AT X= 0.000E+00, ENDS AT X= O.OOOE+00 um NUMBER OF POINTS : 1 NUMBER OF INCIDENT ELECTRONS: 2 0000

INCIDENT BEAM ENERGY: 3.000E+01 keV

DIST. TO DET. FOR TRANS. ELEC . : 5.000E+00 mm APERTURE: 1.250E+01 mm CUTOFF ENERGY OF DETECTOR: 1 . 600E + 01 keV

DIST. TO DETECTOR FOR BACKSC . ELEC: 5.000E+00 mm

IN. DIAM. : 2.500E+00 mm, OUT. DIAM.: 1.250E+01 mm

EFFICIENCY: 1.000E+00, CUTOFF EN.: 1.600E+01 keV

MINIMUM ELECTRON ENERGY CONSIDERED: 0.050 keV

MINIMUM GEN. SECONDARY ELECTRON ENERGY: 0.050 keV

LOCATION 1 WITH BEAM AT X= 0 ..000000E + 00

NUMBER OF GEN. SECONDARY ELECTRONS IS 14 744 8 NUMBER OF BACKSC . ELECTRONS IS 14500

NUMBER OF DET. BACKSC . EL. (L) IS 2590

NUMBER OF DET. BACKSC . EL. (R) IS 2536

SUM OF DET. BACKSC. EL. (S) IS 5126

DIFF. OF DET. B. EL., L-R, (D) IS 54

NUMBER OF SEC. ELECTRONS IS 0 TOTAL NUMBER OF TRANS. ELECTRONS IS 1672

NUMBER OF DET. TRANS. ELECTRONS IS 627

BACKSCATTERING COEFFICIENT = 7.250E-01

DETECTED BACKSC. COEFF . (L) = 1.29 5E-01

DETECTED BACKSC. COEFF. (R) = 1.268E-01

DETECTED BACKSC. COEFF. (S) = 2.563E-01

DETECTED BACKSC. COEFF. (D) = 2.700E-03

SECONDARY ELECTRON COEFFICIENT = 0.000E+00

TOTAL TRANSMITTED ELECTRON COEFF. = 8.3 60E-02

DET. TRANSMITTED ELECTRON COEFF. = 3.135E-02

21
ENERGY DISTRIBUTIONS

El AND E2 (keV) NUMBER BACKS CAT NUMBER TRANS DET. Tl

0 . 00 1 00 63 4 0

1 . 00 2 00 30 7 0

2 . 00 3 . 00 28 5 0

3 . 00 4 . 00 43 16 0

4 . 00 5 . 00 45 10 0 5 . 00 6 . 00 65 20 0

6 . 00 7 . 00 70 24 0 7 . 00 8 00 74 29 0

8 . 00 9 . 00 96 48 0 9 . 00 10 . 00 126 56 0 10 . 00 11 . 00 130 60 0 11 . 00 12 . 00 151 73 0 12 . 00 13 . 00 158 101 0 13 . 00 14 . 00 213 109 0 14 . 00 15 . 00 252 139 0 15 . 00 16 . 00 257 157 0 16 . 00 17 . 00 316 168 121

17 . 00 18 . 00 409 170 125

18 . 00

19 . 00

20 . 00

21.00

22 . 00

23 . 00

24 . 00

25 . 00

26.00

27 . 00

28.00

29 . 00

19 . 00

20.00

21.00

22 . 00

23 . 00

24 . 00

25 . 00

26 . 00

27.00

28.00

29 . 00

30 . 00

438

501

636

726

920

1066

1254

1401

1434

1173

1122

1303

173

118

107

53

22

300

00

0

0

128

94

90

45

21

30

0

0

0

00

22
ENERGY DISTRIBUTIONS

EN. BET. El AND E2 (keV) NUM

o . 00 \ 1 . 00 2 . 0 0

2 . 00

3 . 0 0 4 0 0

4 . 00 5 0 0

5 . 0 0

c.D n n

g 00 7 n n 7 . 00 8 00

8 . 00 9 0 0

9 . 00 10 . 00

10 . 00 1

. 0 0

1. 00 12 . 00

12 . 00 13 . 00

13 . 00 14 . 00

14 . 00 15 . , 00

15 . 00 16 . 00

16 . 00 17 . . 00

17 . 00 18 . . 00

18 00 19 . 00

19 . 00 20 00

n n 21 . .00 22 . 00

22 . 00 23 . 00

23 . . 00 24 . 00

24 . . 00 25 . 00

25 . 00 26 . 00

26 . .00 27 . 00

27 . 00 28 . 00

28 . 00 29 . 00

29 . 00 30 . 00

BCKSC(L) NUM. BCKSC (R) SUM DIFF (L-R)

nu u n u nu nu u n u u u nu u nu u u u u U u

A

u u U

U u n n

U u n u U

u u nU u nu nu n n u n nu u

n n n n n n nw 0 n nu n o

D ^. 77 j- _> ^/

ft 7 1J. 7/ o

ft ^ ftO D J. D U Q ft 9 ft 1

-L QC_/ D u 1UU Z ^ X

- 9 1 119 129 248 -10

166 167 333 -1

182 197 379 -15

210 191 401 19

259 232 491 27

268 254 522 14

302 256 558 46

322 314 636 8 326 331 657 -5

23
MONSEL.OUT

HEIGHT OF LINE IS 7.000E-01 urn HALF-WIDTH OF TOP OF LINE IS 2.500E-01 um HALF-WIDTH OF BOTTOM OF LINE IS 2.944E-01 um HEIGHT OF JOG IS 3.500E-01 um WIDTH OF JOG IS 2.000E+01 ran THICKNESS OF THE FIRST LAYER IS 5.000E-01 um THICKNESS OF THE SECOND LAYER IS 5.000E-01 um THICKNESS OF THE THIRD LAYER IS 5.000E-01 um BEAM DIAMETER IS 1.000E+01 ran BEAM STARTS AT X= O.OOOE+00, ENDS AT X= 0.000E+00 um NUMBER OF POINTS: 1 NUMBER OF INCIDENT ELECTRONS: 20000

INCIDENT BEAM ENERGY: 3 . 000E+01 keV

DIST. TO DET. FOR TRANS. ELEC . : 5.000E+00 mm APERTURE: 1.250E+01 mm CUTOFF ENERGY OF DETECTOR: 1.600E+01 keV

DIST. TO DETECTOR FOR BACKS C . ELEC: 5.000E+00 mm

IN. DIAM. : 2.500E+00 mm, OUT. DIAM. : 1.250E+01 mm

EFFICIENCY: 1.000E+00, CUTOFF EN.: 1.600E+01 keV

MINIMUM ELECTRON ENERGY CONSIDERED: 0.050 keV

MINIMUM GEN. SECONDARY ELECTRON ENERGY: 0.05 0 keV

X (um) BS SEC TOT TR DET TR

0.0OOOE+O0 7.2500E-01 0.0000E+00 8.3600E-02 3.1350E-02

24
MONBD.OUT

HEIGHT OF LINE IS 7.000E-01 um HALF-WIDTH OF TOP OF LINE IS 2.500E-01 um HALF-WIDTH OF BOTTOM OF LINE IS 2.944E-01 um HEIGHT OF JOG IS 3.500E-01 um WIDTH OF JOG IS 2.000E+01 nm THICKNESS OF THE FIRST LAYER IS 5.000E-01 um THICKNESS OF THE SECOND LAYER IS 5.000E-01 um THICKNESS OF THE THIRD LAYER IS 5.000E-01 um BEAM DIAMETER IS 1.000E+01 nm BEAM STARTS AT X= 0.000E+00, ENDS AT X= 0.000E+00 um NUMBER OF POINTS: 1 NUMBER OF INCIDENT ELECTRONS: 20000

INCIDENT BEAM ENERGY: 3.000E+01 keV

DIST. TO DET. FOR TRANS. ELEC . : 5.000E+00 mm APERTURE: 1.250E+01 mm CUTOFF ENERGY OF DETECTOR: 1.600E+01 keV

DIST. TO DETECTOR FOR BACKSC . ELEC: 5 . 000E+00 mm

IN. DIAM. : 2.500E+00 mm, OUT. DIAM.: 1.250E+01 mm

EFFICIENCY: 1.000E + 00, CUTOFF EN.: 1 . 600E + 01 keV

MINIMUM ELECTRON ENERGY CONSIDERED: 0.050 keV

MINIMUM GEN . SECONDARY ELECTRON ENERGY: 0.050 keV

X(ura) BSDL BSDR BSDSUM BSDD(L-R)

0.0000E+00 1.2950E-01 1.2680E-01 2.5630E-01 2.7000E-03

25
MONSEL.TR

0 . 20000E-03

0 . 70000E-04

0 .50000E-04

0 .35000E-04

0 . 25576E-06

0 .25576E-06

0 . 25838E- 06

0 .26974E-06

0 .26981E-06

0 .26992E-06

0 . 26931E-06

0 . 26594E-06

0 . 25178E-06

0 . 22735E- 06

0 . 20027E- 05

0 . 14547E-06

0 . 11883E-06

0 . 11272E- 06

0 . 95091E-07

0 . 99394E-08

0 .23519E-07

0 .50896E-07

0.91678E-07

0 . 12471E-06

0 . 13976E-06

0 . 15320E-06

0 . 10000E-03

0 .25000E-04

- 0 . 10000E-03

0 . 26222E-04

0 . 10000E-03

0 . 69690E-04

0 . 69595E-04

0 . 69002E-04

0 . 68997E-04

0 . 68959E-04

0 . 68891E-04

0 . 68860E-04

0 . 68729E-04

0 . 68525E-04

0 . 68320E-04

0 . 67844E-04

0 . 67495E-04

0 . 67392E-04

0 . 66844E-04

0 . 65794E-04

0 . 65667E-04

0 . 65528E-04

0 . 65331E-04

0 .65166E-04

0 . 65094E-04

0 . 65040E-04

- 0 . 30000E- 03

0 . 29444E

04

- 0 . 15000E

03

0 . 20000E 05

o . 25576E- 06

0 . 25838E- 06

o . 2 69 74E06

o . 26981E- 06

0 . 26992E- 06

o 26931E- 06

o 26594E- 06

0 25178E- 06

0 22735E- 06

0 20027E- 06

0 14547E- 06

0 11883E- 06

0 11272E- 06

0 95091E- 07

-0 99394E- 08

-0 23519E- 07

-0 50896E- 07

-0 91678E- 07

-0 12471E- 06

-0 13976E- 06

-0 15320E- 06

-0 17525E- 06

0 . 69690E-04

0 . 69595E-04

0 . 69002E-04

0 . 68997E-04

0 . 68959E-04

0 . 68891E-04

0 . 68860E-04

0 . 68729E-04

0 . 68525E-04

0 . 68320E-04

0 . 67844E- 04

0 . 67495E-04

0 . 67392E-04

0 . 66844E-04

0 . 65794E-04

0 . 65667E-04

0 . 65528E-04

0 . 65331E-04

0 . 65166E-04

0 . 65094E-04

0 . 65040E-04

0 . 64948E-04

26
APPENDIX II

REPRINT OF A MONTE CARLO MODEL FOR SEM LINEWIDTH

METROLOGY, BY J.R. LOWNEY, M.T. POSTEK, and A.E. VLADAR,

PROCEEDINGS OF SPIE, Vol. 2196, pp. 85-96, 1994.

A Monte Carlo model for SEM linewidth metrology

Jeremiah R. Lowney, Michael T. Postek, and Andras E. Vladar

National Institute of Standards and Technology

Gaithersburg, MD 20899

ABSTRACT

A scanning electron microscope (SEM) can be used to measure the dimensions of the

microlithographic features of integrated circuits. However, without a good model of the

electron-beam/specimen interaction, accurate edge location cannot be obtained. A Monte Carlo code

has been developed to model the interaction of an electron beam with a line lithographically produced on a multi-layer substrate. The purpose of the code is to enable one to extract the edge position of

a line from SEM measurements. It is based on prior codes developed at NIST, but with a new

formulation for the atomic scattering cross sections and the inclusion of a method to simulate edge

roughness or rounding. The code is currently able to model the transmitted and backscattered

electrons, and the results from the code have been applied to the analysis of electron transmission

through a gold line on a thin silicon substrate, such as used in an x-ray lithographic mask. By

comparing the predictions of the code with measured data, it is possible to obtain edge positions to the order of ±10 nm, which is needed for the advanced lithography projected for the year 2000. The

uncertainty of this measurement is limited by the sample geometry and surface roughness and not by

the measurement process.

1. INTRODUCTION

As lithographically produced features on semiconductor wafers and masks become ever smaller (<0.1

urn), more demands are placed on the metrological tools used to measure their dimensions. The

scanning electron microscope (SEM) offers the advantage of a very small electron wavelength and

a beam diameter on the order of 1-10 nm depending upon instrument operating conditions. However,

it has the disadvantage that the electron beam expands greatly within a solid due to atomic scattering,

and thus, it is difficult to relate the collected signal electrons to their point of entry into the specimen.

Theoretical modeling of the interaction between the electron beam and specimen is required to obtain

meaningful dimensional information from the SEM signal images. To do so, accurate models for the

elastic and inelastic scattering of electrons by the host atoms are needed. This work has used a Monte

Carlo code that incorporates some of the most recent models for these effects. We have also carefully

studied both experimentally and theoretically the signal behavior near line edges and extended our

Contribution of the National Institute of Standards and Technology. Not subject to copyright.

27
previous work [1]. We have concentrated on transmitted and backscattered electron signals instead of the more common

secondary-electron signal (i.e., from electrons with less than 50 eV) because the former can be more

accurately modeled. Accurate first-principles theories for secondary electrons do not exist, and the

usual treatment is to assume that the generation of secondaries is proportional to the energy loss of

electrons as they traverse the sample. However, such a treatment is only semiquantitative because

the energies and trajectories of individual secondary electrons are not known precisely. Secondary

production affects the higher-energy transmitted and backscattered electrons only weakly, and

reasonably accurate first-principles theories exist for elastic scattering and overall energy loss. We

have applied these theories to calculate the transmitted and backscattered electron yields as a function

of position across a lithographically produced trapezoidal line on a multi-layer substrate. From the

results, we can obtain accurate and precise relationships between the measured SEM images and the

dimensions of the line. We have simulated edge roughness or irregularities by placing a symmetric

jog (i.e., a discontinuous change in the width of the trapezoid) along the trapezoidal line edges. The

results show the sensitivity of the SEM signals to this perturbation. We have also studied the

signal-to-noise ratio of the Monte Carlo calculations.

2. MONTE CARLO CODE

The code is designed to calculate the transmitted and backscattered electron distributions associated

with a multi-layer specimen. As presently written, the specimen is composed of a trapezoidal gold

line on top of a substrate consisting of silicon, polyimide, and chromium layers, representing an X-ray

lithographic mask [1]. The specimen dimensions and the incident electron-beam energy, diameter, and

positions are given as input. There is a provision for a jog of variable height and width to be placed

symmetrically along the trapezoidal line edges to simulate the effects of edge roughness. There is also provision for a detector of variable aperture, distance, and cut-off energy to be used for

measuring the electron transmission. A similar detector can be inserted for measuring the

backscattered electron signal. The minimum energy of an electron in the specimen before it is removed from consideration can be entered as input, and in this work it is set to 50 eV so that

outputted secondary electrons are not considered. There is provision for the generation of secondary

electrons with energies above an inputted cut-off, equal in this work to 50 eV. However, only

collisions with valence electrons are considered for secondary production because 1) the inclusion of

core electrons requires specific empirical relations and their effects are relatively small, and 2) the

treatment of plasmon-induced secondaries is very difficult and their effects, although large, are at low

energies (<25 eV). The total number of trajectories used per beam position can be varied and needs

to be around 20,000 for accurate results, which we use unless otherwise stated.

The code described here is an extension and modification of an earlier NIST Monte Carlo code [2]. The previous code used a modified screened-Rutherford model for elastic scattering, a simplification

of the Moller formula [3] for inelastic scattering, and Rao Sahib and Wittry's correction [4] to Bethe's formula for energy loss [5]. The primary subroutines in both codes are: input and output,

calculation of elastic mean free paths and scattering angles, calculation of energy loss due to inelastic

scattering, calculation of secondary production, and determination of when an electron crosses the

boundaries of either the trapezoidal line or multi-layer substrate. There is also a subroutine that distributes the input electron beam according to a Gaussian distribution.

28
The following discussion pertains to the new code. The elastic cross sections are now determined

by fitted formulas obtained by Browning [6] to approximate the highly accurate Mott cross sections

[7,8], which take into account the relativistic effects. First, Browning computed a relation for the

total elastic cross section oTas a function of atomic number Z and electron energy E [keV], which

includes the relativistic correction factor:

a T

= BxlO^Z^/^ + O.OOSZ^V^ + O.OOOTZ 2/^) cm

2

. 0)

Next he obtained the differential cross sections by assuming that they result from a sum of screened

Rutherford and isotropic scattering. One reason for choosing these two forms is that they both have

simple, analytic expressions for the differential cross sections, which can be readily evaluated by the

code at each scattering event. Monte Carlo codes generally have long running times to achieve

accuracy, and it is imperative that the numerical evaluations be done as quickly as possible. The use of look-up tables (even after spline fitting) for the true differential Mott cross sections would be very

time consuming. Browning adjusted the ratio r of the screened Rutherford to isotropic scattering so

that he obtained the same ratio of total forward-to-backward scattering cross section as from the true Mott cross sections:

r - 300| + . (2)

The main difference between his method and the actual use of Mott cross sections is that the

oscillatory structure present in the differential Mott cross sections as a function of angle is replaced

by a smooth average dependence. As long as the desired outputs result from partial integration over

solid angle or from many scattering events, as is true in this work, the results are not very sensitive

to this difference. However, if specific and accurate angular trajectories are desired, the actual Mott

cross sections would be needed.

The energy loss per electron step is caused by inelastic scattering such as by the excitation of

plasmons, inter- and intra-band transitions, and core excitations. We use Bethe's expression for the

loss rate [5], modified by Joy and Luo [9] to be more accurate at lower energies E [eV]:

dEldS =

" 785QpZ

ln[1.166(£+0.857)/J] eVInm , (3)

AE

where

J = 9.76Z + 58.5/Z 019

eV, Z>13; J = 11.5Z eV, Z<12 <4)

and S, p, and A are path length [nm], density [g cm"3

], and atomic weight [g], respectively.

This is also the formula that Browning used to calculate backscatter coefficients in his work. Other

treatments exist for modifying Bethe's formula so that it applies to low energy electrons, but our

choice of Joy and Luo's method is adequate for our purposes because we never consider electrons

29
with energies below 50 eV, when the value of the logarithm could become zero or negative and thus

be invalid.

The generation of secondary electrons has been dealt with only in an approximate way because the

focus of the code is to simulate transmission and backscattering of electrons at energies above 50 eV,

at which energies the contribution of secondary electrons is small. We have considered only the

contribution of valence electrons to secondary production because they constitute an important

contribution and can be modeled simply. Plasma contributions, which are at energies below 50 eV,

need not be included; and core-electron contributions, which are usually smaller, require

atomic-specific empirical relations as given by Gryzinski [3,10]. We have used the Moller formula

[3] for the valence electrons:

da/dW = ne (4m0 ) zE

1 1 1 W

2 {E-Wf W{E-W) (5)

where doldW is the differential inelastic cross section for an energy loss between W and W+dW, E

is the energy of the incident electron, and W is the energy of the secondary electron. We assume that E is much less than mQc =511 keV so that the relativistic terms can be ignored. We define a reduced secondary energy z-WIE and integrate Eq. (5) from a minimum secondary energy em

to the

maximum value of 8, 0.5, to obtain the total cross section:

c = ne (4mQ ) 2E

2

i 1 0-eJ

In (6)

The partial cross section up to an arbitrary energy 8 is obtained as well, and both expressions involve

a logarithmic term. This term presents difficulties when solving for the secondary-electron energies

in terms of random numbers generated by the Monte Carlo routine. Thus, we have used a rational

polynomial function, (A + Bz + Ce2 + De3

)/(\ + Ez + Fe2

), to fit the following function's) to better

than 1% throughout:

/(e) = 1 - e/(l -8) - 8 In [(1 -e)/e], 0 < s <0.5 , ( 7)

where A = 0.988798, B = 0.220312, C = -9.16690, D = 9.53217, E = 4.95919, and F = -10.5818.

The solutions for the cross sections, thus, only involve the solution of a cubic equation, which saves considerable computer time instead of dealing with the logarithmic term. Usually the logarithmic

term has been neglected in such calculations, but this shows how it can be included efficiently.

Finally, there is a subroutine that determines when any electron crosses a boundary, either between

layers or into empty space. The direction of the electron is needed upon crossing a boundary to know

where it is heading. For example, electrons may exit and reenter the sample or they may be detected.

The energy of a boundary-crossing electron must be calculated so that only the loss during the time

the electron is in the starting material is counted. If secondaries are produced during a

30
boundary-crossing step, they are eliminated since it is less than a full mean-free path. If an electron

has crossed a boundary into a new material or portion of the same material, the new step starts at the

point of entry. The treatment of boundaries is more difficult than one might expect because of the

need to model all possible scattering events. This prevents the modeling of generally rough surfaces.

Instead, we have tried to model the effect of surface roughness or rounding by placing a jog along

the trapezoidal line faces (symmetrically) that can vary in position and width. This one jog in principle accounts for a series of jagged edges.

3. RESULTS AND DISCUSSION

We show a general diagram of the specimens we can model in Fig. 1 and an SEM micrograph of an X-ray mask in figure 2. In a previous paper [1], we showed plots of the transmitted and

backscattered electron signals from 0.7 urn-thick gold lines on 2.5 um Si substrates separated by a

5 nm film of chromium. They were based on the version of the code we used at that time and were

representative of X-ray masks. We present in Figs. 3-5 a revision of Fig. 10 of Ref [1] for the same

specimens. These traces were obtained from 20,000 trajectories for an incident beam energy of 30

keV, beam diameter of 10 nm, and a transmission electron detector with a 16 keV cutoff energy and

a 1 .25 cm diameter, which was placed 5 mm below the sample. These parameter values are used

throughout this paper unless otherwise stated. The curve in figure 3 was obtained from a nonlinear

least-squares fit based on a rational polynomial function. The results of Figs. 3-5 are nearly the same

as those of Ref. [1], which shows that the improvements to the code have not altered the numerical

results significantly because the previous code had used internal adjustable factors to achieve good

agreement with measured data. This new code uses only first-principle models, which are not

adjusted. Again, we observe the short plateau-like feature near the middle of both the transmitted and

backscattered signals as the beam traverses the trapezoidal line edge. Note that the width of the jog

is zero. This feature is due to the "reflection-like" behavior of the electron beam when it impinges on a nearly vertical wall, and the plateau is due to the near independence of the signal on position

along the edge. This plateau provides a method to identify the edge if the noise in the experimental

signal is small enough.

Figures 6 and 7 show experimental data and theoretical simulation, respectively, of the detected

electron transmission across a 0.5 u,m gold line, on an X-ray mask with a field-emission SEM. The

plots show transmission plotted downward so that the images correspond in shape to the elevated line. The wall angle for the calculated signal is 2 deg. Agreement between theory and experiment is very

good, with the position of the plateau at nearly the same location in both figures.

The best fit between data and theory in our previous paper, as shown in Fig. 9 of Ref. [1], was

obtained with an edge slope of 4 deg. However, SEM photographs indicated that the actual slope was

only between 2 and 3 degs. We, thus, postulated that the extra "slope" was due to edge roughness.

We have investigated the effect of edge roughness by including a small jog along the edge of the

trapezoidal line. We show in Figs. 8 and 9 the changes in detected transmission and backscattering

associated with placing a jog 0.25, 0.5, and 0.75 of the distance above the substrate. The width of

the jog in each case was 20 nm, and the edge slope was 2 deg. The position of the jog in the signal

that correlates with the jog along the edge moves somewhat with the jog placement as expected. It also more than accounts for the difference between the needed effective 4-deg edge slope and the

measured 2-deg one. It is important to note that the plateau in the signals that occurs as the beam

31
traverses a clean sloped edge is similar to that created by a real jog along the edge. The first effect

is the result of electrons emerging from the surface due to the small glancing angle. The second is due to an additional overhang of edge material. The presence of the jog obscures the plateau-like

feature because the beam is not able to traverse the length of edge unobstructedly. The structure of

the signal modification caused by the jog is interesting. Its shape is serpentine because of the angular

beam trajectories that determine the signal. The transmission is relatively reduced and the

backscattering coefficient relatively enhanced when the beam is centered over the jog.

We show in Fig. 10 the transmission and backscattering from a line with a 2-deg wall slope and a

"foot," which is 25 nm high and 50 nm long. The feature is well represented in both signals. The

noise associated with the Monte Carlo calculations is shown for various numbers of trajectories in Fig.

1 1 . Transmission is plotted downward so that the figure conforms to the elevated line shape. A

histogram of the standard deviation of the signal about its mean as a percentage of the maximum

detected transmission is shown in Fig. 12. It is clear that about 10,000 trajectories are needed for a

noise level of less than 1%. We have used 20,000 in the calculations shown here where the noise

level is 0.5%. Above 50,000 there is hardly any visible noise left in Fig. 11, and a larger number

than this seems unnecessary.

4. CONCLUSIONS

The NIST Monte Carlo code for SEM linewidth metrology has been improved and extended by

incorporating the most recent models for electron scattering cross sections and adding a small jog

along the sides of a trapezoidal line to model roughness. Results are shown for the transmission and

backscattering of electrons from a gold line on a thin silicon substrate (with a chromium inter-layer).

Experimental data have confirmed the existence of a predicted plateau in both the transmission and

backscattering signals as the beam traverses the edge of the line. The addition of a jog along the edge was also investigated to study the effects of edge roughness and rounding. The X-ray lithography

mask is a very good model system for the development of accurate SEM standards because it provides

a measurement of both transmitted and backscattered electrons for comparison with predictions of

theoretical models.

5. REFERENCES

1) M. T. Postek, J. R. Lowney, A. E. Vladar, W. J. Keery, E. Marx, and R. D. Larrabee, "X-ray

Lithographic Mask Metrology: Use of Transmitted Electrons in an SEM for Linewidth

Measurement," J. Res. Natl. Inst. Stand. Technol., Vol. 98(4), pp 415-445, July 1993.

2) R. L. Myklebust, D. E. Newbury, and H. Yakowitz, "NBS Monte Carlo Electron Trajectory

Calculation Program," NBS Special Publication 460, Edited by K. Heinrich, D. Newbury, and

H. Yakowitz, pp. 105-125, U. S. Govt. Printing Office, Washington, DC, Dec. 1976.

3) L. Reimer, Scanning Electron Microscopy, Springer-Verlag, New York, pp. 75-77, 1985.

4) T. S. Rao-Sahib and D. B. Wittry, "X-ray continuum from thick elemental targets for 10-50 keV

electrons," J. Appl. Phys., Vol 45(11), pp 5060-5068, Nov. 1974.

32
5) H. A. Bethe, "Zur Theorie des Durchgangs schneller Korpuskularstrahlen durch Materie," Ann.

Phys. 5(5), pp 325-400, 1930.

6) R. Browning, T. Eimori, E. P. Traut, B. Chui, and R. F. W. Pease, "An elastic cross section

model for use with Monte Carlo simulations of low energy electron scattering from high atomic

number targets," J. Vac. Sci. Technol. B, Vol. 9(6), pp. 3578-3581, Nov. 1991.

7) L. Reimer and B. Lodding, "Calculation and Tabulation of Mott Cross-Sections for Large-Angle

Electron Scattering," Scanning, Vol. 6(3), pp. 128-151, 1984.

8) Z. Czyzewski, D. MacCallum, A. Romig, and D. Joy, "Calculations of Mott scattering cross

section, J. Appl. Phys., Vol. 68(7), pp. 3066-3072, Oct. 1990.

9) D. C. Joy and S. Luo, "An Empirical Stopping Power Relationship for Low-Energy Electrons,"

Scanning Vol. 11(4), pp. 176-180, 1989.

10) M. Gryzinski, "Classical theory of atomic collisions I. Theory of inelastic collisions, Phys. Rev.,

Vol. 138(2A), pp. 336-358, Apr. 1965.

33
J LITHOGRAPHIC 1

J LINE 1

SUBSTRATE LAYER 1

SUBSTRATE LAYER 2

SUBSTRATE LAYER 3

Fig. 1 Diagram of general specimen shape.

Fig. 2 SEM secondary image of lithographically produced gold lines on silicon substrate with chromium

interlayer.

34
0.2

0 1 2 3 4 5 WIDTH {jim)

Fig. 3 Minimum transmission (i.e., at line center) through lines and substrate calculated as a function

of linewidth. Squares are calculated values and lines are fits for total (solid) and detected (dash)

transmission. The wall slope of the lines is 4 deg.

0.7 -i 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

POSITION (fim)

Fig. 4 Calculated detected transmission as a function of position across half of line for labeled

linewidths in micrometers. The wall slopes are all 4 deg. Note "plateau-like" feature near the

middle of each line edge.

35
0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

POSITION (fim)

Fig. 5 Calculated backscattering coefficient for the same lines as in Fig. 4, again showing the

"plateau-like" feature.

Fig. 6 Measured transmission along axis of gold line showing plateaus in the data near the middle

the edges.

36
Fig. 7 Set of calculated lines for detected transmission along axis of gold line showing plateaus

observed in data.

0.7

0 0.1 0.2 0.3 0.4

POSITION (/Am)

Fig. 8 Calculated detected transmission as a function of position along half of 0.5 /im wide lines with

2-deg wall slopes and a 20 nm jog 0.175 (short dash), 0.35 (long dash) and 0.525 (solid) ^m

above the bottom. Note serpentine structure as the beam crosses the jog region.

37
0.2 0.3 0.4

POSITION (urn)

Fig. 9 Calculated backscattering coefficient as a function of position for the same lines as in Fig. 7 with

the same line-font designations. The features associated with the jogs are similar to those in

transmission.

0.8

0.7

0.6

0.5

i-9 <

2 0.4 H

o

CO 0.3 -

0.2 -

0.1 -

0

0 |— r— [— —r— —|—0.1 0.2 0.3

POSITION (/Am)

0.4

Fig. 10 Calculated transmission (solid) and backscattering coefficient (dash) for half of a 0.5 fim line

with a 2-deg wall slope and a 50 nm foot 25 nm above the bottom.

38
Fig. 1 1 Calculated detected transmitted signal across a 0.5 urn line with a 2-deg wall slope

for between 1000 and 200000 electron trajectories per point and 160 points per line.

Note how the noise level decreases with increasing number of trajectories.

2.50-

2.00-

1.50-

2 1.00H CO>

ID T3

n

c

<B

'Si

0.50-

o.oo-A

/ 7|

/ 71

5 10 20 50 100 200

Number of "primary" electrons in thousands

Fig. 12 Noise level (i.e., standard deviation as a percentage of maximum detected transmission) as a

function of the number of trajectories.

39

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