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NBSIR 81-1652

A MEASUREMENT METHOD FOR

DETERMINING THE OPTICAL AND

ELECTRO-OPTICAL PROPERTIES

OF A THIN FILM

Donald R. Larson

National Bureau of Standards

U.S. Department of Commerce

Boulder, Colorado 80303

K)0-

100

,1156

81-1652

1981

c. 2

December 1981

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Library, Ml Atfmffi. SM; . NBSIR 81-1652 FEB 2 4 1982

f)ob dec. - (x-u.

&C/DP

A MEASUREMENT METHOD FOR

DETERMINING THE OPTICAL AND

ELECTRO-OPTICAL PROPERTIES

OF A THIN FILM

Donald R. Larson

Electromagnetic Technology Division

National Engineering Laboratory

National Bureau of Standards

U.S. Department of Commerce

Boulder, Colorado 80303

December 1981

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

NATIONAL BUREAU OF STANDARDS. Ernest Ambler, Director

CONTENTS

Page

1. Introduction 1

2. A Mathematical Model for the Optical Transmittance

of a Planar Multilayered Structure 2

2.1 Wave Equation Development 2

2.2 Total Field Impedance and ReflectionCoefficient Formulation 5

2.3 Scattering Matrix Formulation 12

3. Determination of the Optical Parameters of a Thin Film Exhibiting Interference Extrema 21

3.1 Determining Index of Refraction and Thickness

in a Spectral Region of Zero Absorption 22

3.2 Determining the Optical Parameters of an Absorbing Layer 25

3.3 Determining the Magnitude of an Induced Change

in the Complex Refractive Index....... 27

4. A Comparison of Calculated and Experimental

Transmittance Spectra, Values of Refractive

Index, and Thickness 27

4.1 Comparison of an Experimentally and a Theoretically Determined Transmittance

Spectrum of a Sapphire Wafer 29

4.2 Gold on Sapphire Transmittance Comparison 31

4.3 Comparison of Refractive Index Values

of Epitaxial Silicon from 0.6 to 1.6 ym 33

4.4 Comparison of Two Thickness Determinations 37

5. Results for a-Si:H 38

5.1 Refractive Index of a-Si:H 39

5.2 A Model for the Modulator Structure:

Evidence of Gold-Silicon Interaction 42

5.3 Determination of the Change in Refractive

Index of a-Si : H with Applied Electric Field 46

6. Conclusions 48

7. References 49

Appendix A. FORTRAN Implementation of the

Transmittance Model 51

Appendix B. Derivation of Equation (3.9) 54

Appendix C. Index Determination Error Resulting

From Dispersion 58

A Measurement Method for Determining the

Optical and Electro-Optical Properties of a Thin Film

Donald R. Larson

National Bureau of Standards*

Boulder, Colorado 80303

A method of determining the complex refractive index of a thin film on a nonabsorbing substrate is developed. The optical

transmittance spectrum of the structure is measured and the index

is determined by matching this spectrum numerically. An iterative

procedure for finding the magnitude of an induced change in re- fractive index is also presented. In nonabsorbing spectral

regions, the index and film thickness are determined directly.

The optical transmittance of sapphire and thin films of gold

and epitaxial silicon, both on sapphire, is examined. The re- fractive index of epitaxial silicon on sapphire, SOS, is determined and compares favorably with the results of other investigators.

The measurement method is applied to a thin film of hydrogenated amorphous silicon, a-Si:H, deposited by a capacitively

coupled rf glow discharge. The index is tabulated for various

wavelengths and a field induced change in index comparable to GaAs

is measured.

Key Words: Electro-optic modulation; hydrogenated amorphous silicon; optical transmittance; refractive index; scattering matrix;

thin film; transmittance extrema.

1. Introduction

This work shows the development of a method for determining the complex

refractive index, the thickness and the change in the index with an applied

voltage of a thin layer on a transparent substrate. A check of the applicability of the published refractive index values for sapphire and gold is

made. The method is applied to epitaxial silicon, and the results are com- pared with published results of other investigators. The method is then

applied to hydrogenated amorphous silicon, a-Si:H, and the index results are

*

El ectromagnetic Technology Division, National Engineering Laboratory.
compared with the results of another recent investigation of the same type of

material. A field induced change in the index is measured and compared with

other materials. A reflectance analysis is ignored due to associated experimental difficulties.

2. A Mathematical Model for the Optical Transmittance

of a Planar Multilayered Structure

This section is devoted to developing a mathematical model of the optical

transmittance of a multilayered thin film structure. Two mathematical

approaches to the modeling are examined: the reflection coefficient and total

field impedance formulation as done by Johnk [1], and the scattering matrix

formulation as done by Azzam and Bashara [2]. The experimental applications

to be discussed in section 3 will dictate the assumptions which can be made to

simplify the model.

2.1 Wave Equation Development

The well known Maxwell equations are

V • e E = P v

,

V • B = 0,

$ *

V x E = - 3B/3t,

(2.1a)

(2.2a)

(2.3a)

and

* * , V X B/u = J + 3(eE)/3t. (2.4a)

Assuming a charge-free, isotropic region with linear material parameters

of e (permittivity), u (permeability) and a (conductivity). Maxwell's equations may be written as:

$

V . E

= 0 , (2.1b)

V • H

= 0 , (2.2b)

$ $

V x E = - u 3H/3t, (2.3b)

2
and

VxH=oE+e 3E/3t. (2.4b)

$ * Assuming further that E and B are in sinusoidal steady state or are complex

time harmonic fields with angular frequency u, Maxwell's equations may be

rewritten as follows:

V • E = 0, (2.1c)

v • H = 0, (2.2c)

V x E

= -jtoyH, (2.3c)

and

V x H = ju)(e - j -) E. (2.4c)

tO

To obtain the wave equation, the curl of both sides of eq (2.3c) is taken.

V x (V X E) = V X (-jtoyH) (2.5)

Since the permeability y is assumed isotropic, it is a constant with respect

to the curl operation and may be moved outside along with -j and to, resulting

in eq (2.6).

V x (V x E) = - jtoy V x H (2.6)

Curl of H is also related to E via eq (2.4c). Substituting the right side of

eq (2.4c) into the right side of eq (2.6) yields:

V x (V x E) = -jtoy [ jto( e - j-)] E. (2.7)

to

The left side of eq (2.7) may be rewritten using a vector identity for the

curl operation and the right side may be simplified algebraically. This gives

3
V( V • E) - V (2.8) 2 E = yw 2 (e - j-) E.

0)

Equation (2.1c) shows that the divergence of E is zero, therefore eq (2.8)

becomes, after some simple alegbraic manipulation, the wave equation.

V

2 E + uu> 2 U - j-) E = 0 (2.9)

0)

To put eq (2.9) in a standard wave equation form, let

Y = ju>[y(e - J-)]^

2

, (2.10)

U>

and then eq (2.9) becomes

V

2 E - y

2 E = 0 . (2.11)

In a rectangular coordinate system, E can be represented by component unit

vectors.

+ *

E = a E + a E + aE

xx y y z z (2.12)

If it is assumed that E is a uniform plane wave propagating in the z direction, (normal incidence) then E doesn't change with a change in x or y position.

3E 3E

3x 3y

(2.13)

If it is further assumed that only E x

(z) exists, eq (2.11) becomes — E (z) - y

2 E v (z) » 0 . (2.14)

dz2 x x

The solution to this second order, ordinary differential equation is well

known and is given by

E (z) = E, + -yz m

+ E

" „ YZ

(2.15)

4
It can be shown that Em

+ e" yZ is a positive z traveling wave and thus the Em+

notation. Similarly, Em

" eYZ represents a negative z traveling wave. For H^,

from eq (2.3c) and the previous assumptions

-3E (z)

8z

= juHiHy(z) . (2.16)

In view of eq (2.15), eq (2.16) may be written as

Hy(z) = yE

+ e“

YZ

yE " e YZ m m

(2.17)

The application of the above results to the total field impedance formulation

and the scattering matrix formulation will now be considered.

2.2 Total Field Impedance and Reflection Coefficient Formulation

Equation (2.15) may be written as

E <z) = E +e'

YZ

[1 + r(z)] s (2.18a)

A

where

r(z) = e 2yz

. (2.18b)

Since this is the ratio of the negative-going and positive-going waves, r(z)

is termed the reflection coefficient. Let

n = jwy/y • (2.19)

Now eq (2.17) may be rewritten as

H (z)

y

[1 - r(z)] . n( 2 . 20 )

5
"A total -field impedance Z(z) is defined at any z location by the ratio of the

total electric field to the total magnetic field" [3]. That is

E (z) l+r(z)

Z, . = — = n (2.21)

,z)

H !z) l-r(z)

Several relations will now be determined which will facilitate the calculation

of the total EM-field at any location as it travels through a planar multilayered structure. The translation of the reflection coefficient will be con- sidered first. The reflection coefficient at a location z is denoted by r(z).

If another z location, z' , in the same material is considered, r(z') is given

by

r(z') = Em

“ e 2YZ '/Em

+

, (2.22)

and in terms of r(z)

r(z') = r(z) e 2y ^ z " z ^

. (2.23)

Solving for r(z) in eq (2.21) yields another useful relation.

r(z) =

Z

- ( -z)-~..J2 (2.24)

Z ( z ) + n

To translate across an interface, examination of the total field impedance is

made. If 0

1 and 02 are two points on each side of an interface, since the

tangential E and H field components are continuous across the interface or boundary, then

E.(°i) — = ZlOj)

y°i>

E (0 2 ) Z(0 2 ) = —

H <0 2 ) (2.25)

Equations (2.25) and (2.24) give a means of determining the reflection coefficient on each side of a boundary.

The above formulation is now applied to a specific structure. The formulation will be used to describe the transmittance of a thin layer of a-Si : H on

a sapphire ( A1 2 O 3 ) substrate (see fig. 2-1). It will be assumed that the

6
Region

Figure 2-2. Transmittance spectrum from 0.6 to 1.2 urn. Full scale is 100

percent. Interference effects of substrate unresol vable and cause "aliasing".

7
interference effects in the substrate will be undetectable experimentally.

The substrate's large optical thickness creates interference extrema which are less than an angstrom apart in wavelength and are thus unresol vable by the

monochrometer used in this work. Also, substrate nonuniformities may "washout" these extrema. When a computer-generated display of the thin film

extrema is plotted over several thousand angstroms, the fine structure of the

substrate may cause aliasing, i.e., noisy results (see fig. 2-2) and must be

removed. This removal will be accomplished by integrating the predicted

transmittance over one period of the interference effects. Anothe- nethc*"i

will be discussed in section 2.3. The material parameters, y, e, o fo»

optical frequencies. These are related to the complex refractive

At optical frequencies a material is usually described by n and k.

Since no material other than air is assumed present to the right of the

structure depicted in figure 2-1, there will be no reflected wave and from

eq (2.18b)

and

Y = j 2 tt

(

n- jk ) / A = j 2 ttN/x . (2.27)

r 3 (0+

) = 0 . (2.28)

This implies, using eq (2.24), that

Z 3 (0+

) = 13 • (2.29)

Further, from eqs (2.29) and (2.25),

Z 2 (0“) = Z 3 (0 +

) = n 3 (2.30)

and this implies that

03-02

(2.31)

O3+O2

8
Then, using eq (2.23) to translate to z

= -d

2

,

2Y2("d2“0) _ “^2^2

r 2 (-d 2 ) = r 2

( 0 ) e = r 2

( 0 ) e (2.32)

and hence the impedance at this location is given by

l+r 2 (-d 2 ) Z 2 (-d 2 ) = n 2

• (2.33)

l-r 2 (-d 2 )

As noted before, at an interface

Z 2 (-d 2+

) = Zi (-d 2

"

) (2.34)

and so

Z 2 (-d 2

) — n

r i

( -d 2

”

) = . (2.35)

Z 2 (-d 2 )+ni

The above procedure is continued until ro(-di-d 2 ) is found.

If the incident EM-field is given by Em

+

o, the total field expression at

the incident point is found using eqs (2.18a) and (2.18b).

+ Yo(^1 +c*2) E (-d } -d 2 ) = Em

e [1 + r 0 (-d 1 -d 2 )] (2.36)

xq m o

Now, since the tangential field components are continuous across an

i nterface,

E (-d 1 -d 2 ) = E ( -d ! -d 2

) . (2.37)

x 0 M

Em

+

1 is then found in terms of Em

+

Q from eqs (2.18) and similarly Em

+

2 and

Em

+

3

. The intermediate result is then

9
2

T(x) (2.38)

The above expression contains the description of the interference effects of

the substrate. These effects are removed from the model by finding an average

TU) for one period of these substrate effects. This may be accomplished by

using Gaussian quadrature or another numerical integrating technique.

Gaussian quadrature is used in this analysis and will now be outlined. The

averaged transmittance is denoted by Tave

(x) and is given by

Tave

(x)

x 2

/ T( X)dX, (2.39)

X' = +

X 2 (2.29b)

where X 2 and Xi are the wavelength locations of adjacent substrate interference maxima. Using the following change of variable,

(

X

2 -X })x + ( X 2+X i) X = (2.40)

Equation (2.39) is changed to a standard form for Gaussian quadrature.

1 T ave U) = 1;2 |

T(x)dx (2.41)

The integral is now approximated by a summation.

Tave U) =V2 A

i T(xi

} (2.42)

10
"n" is specified and the and x.,- are given for several "n"s in table 2.1.

Figure 2-3 is the resulting T ave U) after applying the above averaging technique to figure 2-2. The above analysis was programmed using BASIC on a mini

computer and generated figures 2-2 and 2-3. The preceding analysis is more

easily done in the FORTRAN computer language due to that language's complex

arithmetic capabilities. A transmittance spectrum for a wavelength interval can be generated for any planar structure by the easy extension of the above

analysis and using a computer to evaluate the equations.

Table 2.1

Table of Legendre Polynomials

n = 1

X

1 = 0 A

! = 2

n = 2

X

1 = -x2 = 0.5773502692 A

! = a

2 = 1

n = 3

X

1 = -x = 0.7745966692 A

1 = A

3 = 0.55555. . .

X

2 = 0 A

2 = 0.8888. . .

n = 4

X

1 = -x = 0.8611363116

4 A

1 = A = 0.3478548451

4 X

2 = -x3 = 0.3399810436 A

2 = A

3 = 0.6521451549

n = 5

X

1 = -x

5 = 0.9061798459 A

1 = A

5 = 0.2369268851

X

2 = -x, = 0.5384693101 A

2 = A = 0.4786286705

4 X

3 = 0 A

3 = 0.5688888. . .

n = 6

X

1 = -x

6 = 0.9324695142 A

! = Ac = 0.1713244924

b X

2 = -x

5 = 0.6612093865 A

2 = A

5 = 0.3607615730

X

3 = -x, = 0.23861918614 A

3 = A = 0.4679139346

4

11
Figure 2-3. Result when the transmittance spectrum of figure 2-2 is averaged

over the substrate interference period, using six point Gaussian

quadrature. Full scale is 100 percent.

2.3 Scattering Matrix Formulation

The transmittance model will now be developed using the scattering matrix

formul ation.

Starting with eq (2.15),

E = E + e“

YZ

+ E

‘ x m m

(2.15)

this wave equation may be rewritten as a 2x1 column matrix.

E +(z) is defined by

E (z) = x

E +

( z)

E'(z)

E+(z)

and E~(z) is defined by

E"(z) ,YZ

(2.43)

(2.44)

(2.45)

12
Since the equations used have been linear, the E field at some location

zi can be related to the E field at another location Z 2 via a 2x2 matrix

transformation. Following the notation used by Azzam and Bashara [2],

E +

( 2 1

) s n s 12 E+

(z 2 )

E'lzj) S21 s 22 E'(z 2 )

(2.46)

with matrix S called the scattering matrix.

To specialize the S matrix to an interfacial region, let z\ and Z2 be on either side of an interface. For this special case, S will be renamed I. The

elements of matrix I may be found by considering the system depicted in figure

2-4. The matrix representation of this system is as follows, dropping the z variable notation in favor of subscripts

+

—

-f- El 111 1 12 E 2

_Ef_ } 21 1 22 0

When eq (2.47) is expanded, it is seen that

(2.47)

(2.48a)

(2.48b)

but using the Fresnel reflection and transmission coefficients defined by

Ei

+

* l/riz Ej (2.49)

E x +

= l/t 12 E 2 + (2.50)

yields the following results.

Iii = l/ti 2 2.51)

13
Region

Figure 2

-

Figure 2

-

t +Z

System used in the determination of the interface matrix elements

In and 1 2 1

.

Region

#1 #2

+ Z

. System used in the determination of the interface matrix elements

I i 2 and 122-

14
(2.52)

r 12

1 2 1 = tl2

To find I 22 and 1 1

, the direction of propagation assumed in figure 2-4 is

simply reversed (see fig. 2-5). The matrix representation of this new system

is

0 1

1 1 1 12

Ei I 2 i I22

The applicable Fresnel coefficients are now given by eqs (2.54) and (2.55).

E 2 +

= r 21 E 2

“ (2.54)

Ei" = t 21 E 2

" (2.55)

When the matrix product in eq (2.53) is performed, relations (2.56) and (2.57)

resul t.

E 2

e 2

“ (2.53)

1

1 iE 2 + 1 12 e 2 “ 0 (2.56)

I 21 E 2 + ^22^2 = E 1 (2.57)

Some algebra and the use of the Fresnel coefficient identities found in [4],

1-r 12

2 1 2

(2.58)

^12

and

r 2

“ _r 12 (2.59)

gives the following results

15
Region

#

0

n #2

.z. +z

Figure 2-6. System used in the determination of the interface matrix

elements.

12

22

r 12

•12

•12

(2.60)

(2.61)

The interface matrix, I, is now completely determined and is given by

I =

•12

1 r 12

r 12 1

(2.62)

Matrix S is also called the layer matrix L if the system under consideration

consists of locations zi and Z2 that are at either end of an isotropic layer.

Figure 2-6 illustrates this system and eq (2.63) describes it.

16
(2.63)

Performing the matrix multiplication indicated on the right side of eq (2.63)

give eqs (2.64) and (2.65).

E

+

= L n E 2+ + L 12 E 2

- (2.64)

Ei" = L 21 E 2+

+ L 22 E 2

" (2.65)

Equations (2.44) and (2.45) for z = zi and z = Z 2 yield

+ “YZi -yz 2 _ yz 2 E e = L nE m

e + L 12 E me (2.66) m

1

m

1 ^ m

and

_ tzi + -yz 2 _ yz 2 E e = L 21 E m

e + E me . (2.67) m

z

m m

The layer matrix under consideration must be valid for any isotropic

layer. Ln and L21 may be found if the assumption is made that the layer is

infinite and only a forward traveling (positive z traveling) wave is present.

That is.

Ef = E 2 = 0 . (2.68)

Using these two conditions with eqs (2.66) and (2.67) yields

+ -YZl + -YZ 2 E m

e = L ,

E m

d (2.69)

and

+ -YZ2

0 = L 21 E m

e . (2.70) z 1 m

These two equations reduce to

17
Ln e (2.71)

-yUi-z 2 ) = e

and

L21 = 0 . (2.72)

Similarly, L12 and L22 may be found if the assumption is made that the layer

is again infinite but only a negative z traveling wave exists. The results

are

L 1 2 ~ 0 (2.73)

L22 - e y(zi-z 2 ) = e yd

(2.74)

All elements of the layer matrix have been determined and L is given by

L

= e Yd 0

0 e -yd (2.75)

and y can again be defined as

Y = j 2n N/X (2.27)

Any multilayered structure can now be described by using the product of the

interface and layer matrices to give the scattering matrix. This formulation

is now applied to thin layer of a-Si : H on a lossless sapphire substrate. Originally, the substrate will be considered semi-i nf ini te and later the effect

of the substrate-ai r interface will be included.

This will be done in such a manner as to neglect the substrate interference effects. This structure (see fig. 2-7) may be described by eq (2.46)

with matrix S given by

S = 1 12 L 2 1 2 3 , (2.76)

wi th

18
Region

#1

air

#2

a-Si :H

#3

sapphi re

I T

Figure 2-7. Thin film on an infinite substrate.

r 32 f l

54

r 34

r 32 r

t

,14

<

f

34

r 32 t

t 34

,

r 34

l

34

T

i

•

•

•

•

•

Figure 2-8. Thin film on a finite substrate. Rays used in geometrical

optics analysis portrayed.

19
1 12

1 ti2

1 r 12

r ' 2

1 J

L 2

Y2 d 2 e 0

-y 2 d 2 0 e

and

1 23 [

r 23 !

The transmittance of this structure is given by relation (2.80).

(2.77)

(2.78)

(2.79)

Ti = (2.80)

The asterisk denotes the complex conjugate.

From previous assumptions, the substrate may be analyzed using geometrical optics, therefore to "fold-in" the effects of the substrate-ai

r

interface, ray tracing and the Fresnel coefficients are used for the EM

radiation in the substrate. I denotes intensity in figure 2-8. The total

transmitted intensity is the infinite sum of I i , I 2 , •••

I = I

I . (2.81)

j = l J

Ti is the transmittance for an infinite substrate and 1 1 , 1 2 , etc. can be

found using the Fresnel coefficients.

h (2.82)

I 2 ” *0 T 1 I r 34 r 32 1 34 I (2.83)

20
It can be written as

I T

Io T 1

t 34 1 + 3 (2.84)

where the expression inside the brackets is a geometric series which may be

summed to give eq (2.85) for lj.

I 0 T 1 ! 1 34 I ^

I = 7(2.85)

1 - I

34 r 32 l

The total average transmittance is then

>T

2 ave

= T.

•34

1 - r 34 r 32

( 2 . 86 )

The matrix S can be extended to any arbitrary multilayered structure by annexing I and L matrices. The above analysis is readily programmed using BASIC or FORTRAN to provide a transmittance spectrum for any wavelength interval for

which the material parameters, n, k and thickness, are available. Such a FORTRAN program appears in appendix A.

It should be noted that either integrating out the substrate interference

effects or neglecting them via the ray tracing technique may be applied to

either formulation and are mathematically equivalent under the assumptions of

this section. Either formulation may be used for determining the optical

constants of a layer, which will be considered in the next section.

3. Determination of the Optical Parameters of a Thin Film Exhibiting Interference Extrema

This section outlines a method for determining the complex refractive

index and thickness of a single thin film on a lossless substrate, from its

transmittance spectrum. The method is based on the transmittance model developed in section 2 and requires that the material thickness be large enough so that transmittance extrema are present in the spectral region of interest. If

the material under examination has zero or immeasurably low absorption in a particular wavelength region, several major simplifications may be made in the

model

21
3.1 Determining Index of Refraction and Thickness

in a Spectral Region of Zero Absorption

The structure under consideration appears in figure 3-1. It is assumed

that the layer and its substrate are lossless and that the layer is sufficiently thick to exhibit interference caused transmittance extrema. Using the

scattering matrix formulation, the averaged transmi ttance, Tave , is given by

eq (3.1a), where S is given by eq (3.1b) (also refer to section 3.2).

TiT 2 Tave I-R2R3

(3.1a)

where

S = I 01 L 1I 12 , (3.1b)

T 1 = , (3.2a)

SnSu*

and

T 2 - t 2 3 t 23*, ( 3.3a)

R 2 = r 23 r 23 *, (3.4a)

S12S12*

R 3 = . (3.5a)

s n s n*

For the optical system depicted in figure 3-1, eqs (3.2a) through (3.5a)

become

16n 0 2 n 1 2

2 2 2 2

4irn 3 d 3 (n 0 +n x

) (n i+n 2 ) + (n 0 -n x

) ( n i-n 2 ) + 2(

n

0 2 -n } 2

) (

n

3 2 -n 2

2 )cos( ) A

(3.2b)

22
air a-S1:H sapphire

n o

n

l n2

air

n

3 =n

o

H d h

Figure 3-1. Structure depicting a thin film on a thick but finite substrate.

Note labeling of material regions.

4n 2 2 T 2 -

2

5 (n 2+n 0 )

R 2

n 2 -n 0

n 2 +n 0

(3.3b)

(3.4b)

and

O O O O

4TTn

1 d]^ (n 0 +ni) (

x -n 2 ) + (n^n^ (n!+n 2 ) + 2(n 0 2 -n 1 2 )(n 1 2 -n 2

2 )cos( r—

)

R

= —7i

. o o o o 4nn 1 d 1 (n o+n x

) (n!+n 2 ) + ( n 0 -n x

) (n L -n 2 ) + 2(n 0 2 -n 1 2) (n! 2 -n 2

2 )cos( ^—

)

(3.5b)

23
Since Tave

is a function of wavelength, its spectral extrema may be found by

differentiating eq (3.1a) with respect to wavelength and letting the result go

to zero. For the case when the refractive index of the layer (m) is greater

than that of the substrate (

n

2 ) , relations (3.6) and (3.7) result when dn/dx

is zero.

For a maximum 2n x d = mx m = 0, 1 2 • ••

.

(3.6)

For a minimum 2n x d = (m+l/2)X m = 0, 1, 2* . (3.7)

If m is less than n 2 , relation (3.6) applies for transmittance minima and eq

(3.7) applies for transmittance maxima. For the materials examined in this

work, m is greater than nz. If eq (3.7) is substituted into the cosine terms

in the expressions for Ti and R 3 , it is seen that:

cos -n

-d-- = cos ( ( 2m+l

)

tt) = -1 . (3.8)

X

This allows T ave(min) to be reduced algebraically to

4n 0 n! 2 n 2 T (min) = . (3.9)

n 0 2 ni 2 + n 0 2 n 2

2 + n^*4 + n^ 2 n 22

The intermediate steps leading to eq (3.9) may be found in appendix B. This

implies that the unknown index, ni, may be determined directly from

T ave(xmin), if the substrate refractive index is known. Solving for ni yields

n 1 = -(n 0 2 - 4n 0 n 2 /T + n 2

2

) + [(n 0 2 -4n 0 n 2 /T + n 2

2

) 2

- 4n o 2 n 2 2 f/2] j

l/2

(3.10)

It should be noted that the above expression for m is not dependent on

m, the order of the extremum or the layer thickness. If m is known, the

thickness of the layer may be found using eq (3.7) in a rearranged form and

using the ni just determined.

24
d = (3.11

n 1

Appendix C examines the error introduced in the index determination of a dispersive material when eq (3.7) is used, an often negligible error.

The analysis in section 2, coupled with the simplifying assumptions of

this section, allowed an expression for m to be found in terms of no, n 2 and

Tave(min). It is useful to note that no information on m is found if the

previous analysis is carried out using Tave

(max) instead of Tave

(min). The

resulting expression is:

Tave(max)

2n 0 n 2

n 0 2+n 2

2 (3.12)

Equation (3.12) is valuable in checking the experimental procedure and data.

The peak experimental transmittance should agree with the value given by eq

(3.12). The amount of agreement is a quality measure for the experimental

data.

3.2 Determining the Optical Parameters

of an Absorbing Layer

A determination of the optical constants of an absorbing layer requires

the layer thickness to be known. Many optical and semiconductor materials are optically transparent in some spectral region and the procedure outlined in

section 3.1 may be used to determine the layer thickness. A nonzero k causes

the interference minimums to shift to shorter wavelengths and the maximums to

shift to longer wavelengths. If n is approximately 3.7, a value of k greater

than 0.05 causes a shift of 10 A in extrema location, one which may be experimentally resolvable. If k is less than 0.02, the location of the extrema can

still be determined by eqs (3.6) and (3.7), as the shift is less than 10 A and

may be unresol vable.

For this work, it will be assumed that the thickness d can be determined

via the procedure outlined in the previous section. That is, the material

under examination is not absorbing in a particular wavelength region and eqs

(3.10) and (3.11) can be used. With the thickness and order determined, the

25
complex refractive index can be determined in the wavelength region of

interest via an iterative procedure. This procedure is now outlined.

The transmittance spectrum of the structure must be observed and recorded

for the particular wavelength region desired and must exhibit extrema. Equations (3.6) and (3.7) are used to find values of "n" at the extrema locations

and these values may be extended to wavelengths between extrema via linear

interpolation. A theoretical transmittance spectrum is then calculated. The

peak amplitude of the synthesized and experimental transmi ttances are compared

and the value of k is adjusted to increase the agreement. If the value of k

is increased, the peak transmittance will decrease, "n" may also be adjusted

to account for a possible shift in extrema position due the adjustment of k

and to increase the agreement in extrema location. An increase in n causes

the extrema to shift to longer wavelengths. A new transmittance spectrum is

calculated with the revised n and k and again compared to the experimental

data. The process is repeated until the theoretical and experimental agreement is within the error of the experimental data measurement.

Goodman [5] has observed that the peak-to-val 1 ey ratio of the transmittance spectrum may be used to determine n if both the layer and substrate

are lossless and nondi spersive. Although these two restrictions are not made

here, the peak-to-val 1 ey ratio as well as the extrema position may be used as an agreement criteria for the experimental and theoretical transmittance

compari son. The above iterative procedure may be programmed or done by hand although

the transmittance spectrum calculation is best done using a computer.

Determination of n and k can also be done for a layer buried in a multilayered structure via the preceding iterative method. The other layers need

to be well characterized and the layer under examination must be of known

thickness. The initial guess for n from eqs (3.6) and (3.7) may be further

from its actual value but more iterations resolve this complication.

In concluding this section, it should be pointed out that the determination of the layer thickness is very critical. Denton, Campbell and Tomlin [6]

point out that in Tomlin's method of determining n and k from reflectance and

transmittance measurements, a 5 percent error in the layer thickness can make

the results far from reasonable. The same thing is true for the procedure

outlined in this section, particularly when applied to multilayered structures.

26
3.3 Determining the Magnitude of an Induced Change

in the Complex Refractive Index

The model developed in section 2 may be used to calculate a theoretical

transmittance spectrum of a stratified structure consisting of an arbitrary

number of layers. If the complex refractive index of one of those layers is

changed slightly, the transmittance spectrum will also change. This section

will outline a procedure for determining this change in index from a change in

the transmittance spectrum. The change in index will be assumed to have been

accomplished by the application of energy to the structure.

The "modulation spectrum" will be defined as follows.

T

T is the transmittance without applied energy and Twis the transmittance when

energy is being applied to the structure.

Finding the change in n and k is done via an iterative process. The

iterative process involves generating a modulation spectrum from an initial

guess of the change in index and comparing the resulting theoretical modulation spectrum with the experimental modulation spectrum. The changes in n and

k are then adjusted to increase the agreement between experiment and theory

and a new modulational spectrum is calculated. This new spectrum is compared

to experiment. This process is repeated until the desired agreement is

achieved and the values for the change in n and k are determined.

Figure 3-2 illustrates a modulation spectrum from a layer showing interference effects (see fig. 3-3). A change in k results in amplitude modulation

as the absorptance changes. This effect may be seen in figure 3-2 at the

shorter wavelength end of the spectrum. A change in n causes a shift in the

interference peaks and phase modulation results. This is the dominant effect

seen at longer wavelengths in figure 3-2. These observations aid the programmer in using the iterative process.

4. A Comparison of Calculated and Experimental Transmittance

Spectra, Values of Refractive Index, and Thickness

This section outline the procedures followed to compare experimentally

determined transmittance spectra of sapphire and of gold on sapphire with the

transmittance spectrum of each, calculated from the scattering matrix model

27
Figure 3-2. Calculated modulation spectrum from 0.6 to 1.2 ym.

Figure 3-3. Calculated transmittance spectrum from 0.6 to 1.2 ym.

28
developed in section 2. These calculations are made using values of refractive index from the literature. The comparison of calculated with experimental transmittance provides a check of the applicability of the published

refractive index data of these two materials with the actual refractive index

of the materials used in this work.

The experimental procedure and the results obtained when the measurement

method is applied to an epitaxial silicon layer is also described. The index

of refraction results are compared with published values and the thickness

result is compared with a measurement made with a surface profilometer. This

provides a verification of the measurement method.

4.1 Comparison of an Experimentally and Theoretically

Determined Transmittance Spectrum of a Sapphire Wafer

A sapphire wafer is used as the substrate of every structure considered

in this work, and its measured transmittance spectrum should agree with a theoretical spectrum calculated from published index data. By comparing the

measured spectra with the calculated spectra based on published data, some

idea of the experimental limitations is obtained.

The physical dimensions of the substrate were 0.635 cm x 0.635 cm

x 0.0254 cm and the sample was provided polished on both sides. A transmittance spectrum from 0.6 to 1.2 pm is considered here. Figure 4-1 is a schematic of the experimental setup. A minicomputer is used to control the

grating monochrometer wavelength setting, take and store the data, and provide

a plot and printout of the results. Reflective optics are used to focus the

light from the tungsten source into and as it emerges from the monochrometer. By using reflective optics, the focal point is constant with wave- length which minimizes the effort needed to keep the light focused on the

sample and the detector throughout the spectral range examined. A silicon p-n

junction detector is used in conjunction with a lock-in amplifier to obtain

the data. The analog signal is processed by an A-D converter and sent to the

computer. Figure 4-2 is a plot of the incident spectrum (I), the transmitted

spectrum (T), and the ratio of these two or the transmittance spectrum (N).

The minicomputer was then programmed in BASIC to calculate a transmittance

spectrum using the scattering matrix formulation of the model. The refractive

index used for sapphire was obtained from the supplier and was originally

determined by I. H. Malitson [7]. Figure 4-3 is a plot of the calculated

29
Figure 4-1. Equipment array used to obtain experimental transmittance

spectra

.

Figure 4-2. (I) incident, (T) transmitted, (N) transmittance of sapphire

from 0.6 to 1.2 urn. Transmittance (N) full scale is 100 percent.

30
spectrum made to the same scale as the plot of the experimental transmittance

spectrum, figure 4-2 ( N )

The experimental and theoretical spectra agree extremely well in the

region where a good signal -to-noi se ratio exists. The published values of

index for sapphire are concluded to be in good agreement with the actual index

of the sapphire wafer used.

4.2 Gold on Sapphire Transmittance Comparison

In section 5 the modulator structure will be modeled. This structure

consists of an a-Si : H layer sandwiched by two gold films with a sapphire

substrate. Since the gold films are thin enough to be semitransparent, 200 A

each, the refractive index is dependent on the rate of deposition [8]. To

model the modulator transmittance, the transmittance of the gold layers must

be known.

A 200 A film of gold was deposited on a sapphire wafer and the transmittance of this structure was obtained following the procedure outlined in

section 4.1. Figure 4-4 is a plot of the incident (I), transmitted (T), and

transmittance ( N ) spectra from 0.6 to 1.2 urn. The transmittance was then

compared to a transmittance spectrum plotted using the model and refractive

index data from the literature [9]. The transmittance amplitude and structure were initially in reasonable agreement but to increase the agreement, the

thickness was decreased to 185 A and the imaginary part of refractive index was changed. Several iterations of changing the index parameter, "k" , and

comparing theoretical and the experimental transmittance spectra produced a theoretically generated plot which is in good agreement with experiment (see

fig. 4-5). The refractive index data used to generate figure 4-5 is listed in

table 4.1 (DL) along with data published by Heavens [10] and Weiss [9]. The

variation in refractive index, "k", is an indication that the gold layers used

in the modulator should be examined on each modulator device, the "k" appears

to be very deposition process dependent. The examination of the gold film on sapphire will be understood to be part of the experimental procedure in the

examination of the modulator structure in section 5.

31
Figure 4-3. Calculated transmittance spectrum from 0.6 to 1.2 urn for

sapphire. Full scale is 100 percent.

Figure 4-4. Gold on sapphire experimental spectra from 0.6 to 1.2 pm. (I)

incident, (T) transmitted, (N) transmittance. Transmittance (N)

full scale is 100 percent. The (T) scale is 33% of the (I) scale.

32
4.3 Comparison of Refractive Index Values of

Epitaxial Silicon from 0.6 to 1.6 ym

A comparison of epitaxial silicon on sapphire (SOS) with hydrogenated

amorphous silicon on sapphire ( a - S i : H ) is of general interest due to their

physical similarities and applications. The rough, uncoated side of an SOS

wafer was polished using a polishing wheel and a diamond slurry. Four

samples, 0.635 cm square by approximately 0.0254 cm thick were thus processed. The silicon was etched from one of these samples and a transmittance

spectrum was observed and compared with the calculated sapphire spectrum.

Good agreement is seen when figures 4-6 and 4-2 are compared, indicating the

polishing described above is sufficient. The transmittance spectrum of an SOS

sample was then obtained using the same experimental procedure as outlined in

section 4.1 and in figure 4-1. The results are plotted in figure 4-7 for the

0.6 to 1.2 ym range and in figure 4-8 for the 1.0 to 1.6 ym range. Both

figures depict the incident (I), transmitted (T) and transmittance (N)

spectra. A germanium p-n junction detector was used for both spectra and was required for the 1.0 to 1.6 micrometer range as silicon detectors cut off

around 1.2 ym.

Table 4.1

Refractive Index of Gold

Wei ss

X ym n k^ n .603 .2 2.9 .2

.65 .165 3.37 .165

.7 .13 3.84 .13

.75 .14 4.245 .14

.8 .15 4.65 .15

.85 .16 4.995 .16

.9 .17 5.34 .17

.95 .175 5.69 .175

1.0 .18 6.04 .18

1.25 .27 7.33 .27

DL Heavens

k n k

2.78 .23 2.97

3.37 .19 3.5

3.84 .17 3.97

4.2 .16 4.42

4.55 .16 4.84

4.8 .17 5.30

5.0 .18 5.72

5.2 .19 6.10

5.4 *

6.25 * *

33
Table 4.2

Refractive Indices of Epitaxial Silicon

DL Hul then Kuhl et al

.

X Min n _k n n k

1.543 3.3900 0

* * 3.458 *

1.405 3.3956 0

* * 3.466 *

1.298 3.4221 .0005 * 3.474 *

1.2 3.4274 .0010 * * 3.483 *

1.123 3.4542 .0014 * 3.495 *

1.052 3.4669 .0020 3.54 .0012 3.508 *

.992 3.4871 .0026 3.55 .0029 3.526 *

.938 3.5034 .0034 3.56 .0046 3.540 *

.893 3.5315 .0043 3.58 .0073 3.554

.848 3.5399 .0061 3.60 .0112 3.569

.813 3.5724 .0079 3.63 .0119 3.582 *

.777 3.5849 .0092 3.66 .0141 3.596 .0102

.749 3.6203 .01 3.68 .0171 3.626 .011

.722 3.6484 .0118 3.71 .0199 3.65 .0125

34
Figure 4-5. Calculated gold on sapphire transmittance spectrum from 0.6

to 1.2 um. Full scale is 100 percent.

Figure 4-6. Experimental transmittance spectrum of sapphire from 0.6

to 1.2 urn. Full scale is 100 percent.

35
Figure 4-7. Experimental SOS spectra from 0.6 to 1.2 pm. Transmittance (N)

full scale is 100 percent.

Figure 4-8. Experimental SOS spectra from 1.0 to 1.6 pm. Transmittance (N)

full scale is 100 percent.

36
In examining these transmittance spectra, eq (3.12) is useful. In a non- absorbing region of the spectrum, eq (3.12) is applicable and may be used to

indicate which spectral regions are actually absorbing and which are not. The

spectral regions where eq (3.10) may be used are thus determined. Equation

(3.10) is then used to calculate the refractive index of the silicon at the

wavelength of a transmittance minimum. The results obtained by following the

above procedure and the measurement method outlined in section 3 are listed in

table 4.2 along with the published results of other investigators, [11,12].

The transmittance spectrum illustrated in figure 4-7 shows definite absorption

effects in the region from 0.6 to 0.95 urn. Since the imaginary part of the

refractive index, k, is nonzero, the iterative method outlined in section 3

must be employed for the 0.6 to 0.95 ym region. This was done on a computer

using the FORTRAN language, a graphics terminal, hard copy unit, and PLOT-10

plotting software. The results are also listed in table 4.2, along with the

results of the other investigators. For the real part of the refractive

index, n, Hulthen [11] and Kuhl et al . [12] agree to 1.5 percent, and both

agree with the results presented here to 2 percent. For the imaginary part of

the index of refraction, Hulthen and Kuhl only agree to 37 percent, and they

agree with the present results to 69 percent. The absorption, and thus the k,

is very material dependent so the above agreement is still considered

reasonable.

4.4 Comparison of Two Thickness Determinations

The documentation provided with the SOS sample states that the silicon

layer is approximately 1 ym thick. The thickness of the sample examined above

was calculated from the index n determined at a wavelength of 1.405 ym using

eq (3.7). The order, m, was required for this calculation and was determined

by assuming the index was nondispersive in the spectral region from the transmittance minimum at 1.405 ym to the transmittance maximum at 1.543 ym.

Equations (3.6) and (3.7) can be combined and yield eq (4.2) for an adjacent

transmittance minimum and maximum.

(m + V2) \

= m X p (4.2)

From eq (4.2), m, an integer, is found to be 5 for the transmittance peak

located at 1.543 ym. The v and p subscripts in eq (4.2) indicate the

37
Figure 4-9. Chart recording of

silicon layer profile from a surface

prof il ometer. Scale is 2 x 10 4 per

major division.

wavelength of the transmittance minimum (V) and maximum (P). The calculated

thickness is 1.1379 ym. To check this thickness determi nation , a surface

profil ometer was used. A portion of the silicon layer was etched from the

substrate to form a step, required for a thickness determination. The surface

profil ometer employs a chart recorder to display the profile of the surface as

a diamond stylus is mechanically drawn across the sample. Figure 4-9 is a portion of this record. The step height determined from this record is 1.05

urn. The agreement is then 8.4 percent, a reasonable value for a prof i 1 ometer.

5. Results for a-Si :H The measurement method described in section 3 and applied to epitaxial

silicon on sapphire in section 4 is now applied to hydrogenated amorphous

silicon deposited on sapphire by the glow-discharge method [13]. The refractive index results are tabulated and compared with the refractive index of

epitaxial silicon and of a-Si:H [14]. The model for a modulator structure is

38
presented and the inconsistency which appears is examined. The required

change in refractive index is then determined from some preliminary optical

modulation data. A more detailed and complete description of the optical

modulation capabilities of a-Si :H will be presented in a paper to be published

by others.

5.1 Refractive Index of a-Si:H

A sample [15] of a-Si : H was prepared by the glow discharge technique.

This technique involves exciting a plasma via an RF antenna in a chamber con- taining silane (SiHi+) gas. The silane decomposes to silicon and hydrogen and

the silicon deposits with some hydrogen on a heated substrate in the chamber.

The substrate is a sapphire wafer and is kept at approximately 250°C during

the deposition.

The sapphire wafer used was polished on both sides by the manufacturer;

this facilitated the desired optical transmission measurements. The same experimental procedure is followed for a-Si : H sample as for the SOS sample.

Figures 5-1 and 5-2 contain the incident (I), transmitted (T) and transmittance (T) spectra of a-Si : H on sapphire from 0.6 to 1.2 pm and from 1.0 to

1.6 urn respectively. The source is filtered by a red plastic filter for the

0.6 to 1.2 ym spectrum and a silicon wafer which has been polished on both

sides for the 1.0 to 1.6 ym spectrum. The filters block the harmonics coming

through the grating monochrorneter . A layer thickness of 1.150 ± 0.002 ym was determined using eq (3.11). In figure 5-1, absorption effects are seen by a decrease in the transmittance peak at 0.821 ym from the value predicted by eq

(3.12) and these effects increase dramatically as the 0.6 ym wavelength is

approached. The computer aided iterative method is used to find n and k for

the 0.6 to 0.821 ym region. The results are presented in table 5.1. The

values presented in table 5.1 are used to calculate, via linear interpolation,

the real part of the refractive index at other wavelengths for comparison with

the real part of the refractive index of a-Si : H determined by Zanzucchi et al

[14]. The results of Zanzucchi's work indicate that the substrate temperature

maintained during the silicon deposit effects the refractive index of the

silicon. Table 5.2 presents Zanzucchi's results for two samples prepared with

substrate temperatures which bound the substrate temperature used in the preparation of the sample examined in here. The real part of the refractive

index for this material is also bounded by that of Zanzucchi.

39
Figure 5-1. (I) incident, (T) transmitted, and (N) transmittance spectra

for a-Si :H and sapphire for 0.6 to 1.2 pm. Transmittance (N)

full scale is 100 percent.

Figure 5-2. (I) incident, (T) transmitted, and (N) transmittance spectra

for a-Si:H and sapphire for 1.0 to 1.6 pm. Transmittance (N)

full scale is 100 percent.

40
Table 5.1

Experimental Results of the Refractive Index of a-Si :

X n k

0.673 3.8090 0.017

0.693 3.7714 0.0108

0.712 3.7198 0.005

0.737 3.6899 0.003

0.761 3.6444 0.001

0.790 3.6114 0.0008

0.821 3.5744 0.0005

0.857 3.5445 0.0003

0.898 3.5186 0.0001

0.944 3.5077 0

0.996 3.4690 0

1.055 3.4448 0

1.124 3.4255 0

1.204 3.3967 0

1.298 3.3802 0

1.405 3.3539 0

1.541 3.3442 0

41
Table 5.2

Comparison of the Real Part of the Refractive Index

195°C

X a-Si : H (Z)

0.65 3.77

0.7 3.68

0.75 3.60

0.8 3.50

0.85 3.40

0.9 3.40

0.95 3.40

1.00 3.36

250°C 325°C

a-Si : H (L) a-Si : H (Z)

3.85* 3.99

3.7524 3.88

3.6653 —

3.5995 3.66

3.5503 3.69

3.5181 3.70

3.5032 -- 3.4674 3.76

*extrapol ated

(L) This work

(Z) Zanzucchi et al . , [14]

5.2 A Model for the Modulator Structure: Evidence of Gold-Silicon Interaction

To determine the change in refractive index with applied electric field,

the transmittance of the modulator structure must be modeled.

The modulator consists of a layer of a-Si:H sandwiched by two gold electrode layers with a sapphire substrate. The modulator is seen to be a stratified thin film device and the model examined in section 2 can be applied. The

refractive indices and thickness of the a-Si:H were determined in section 5.1,

the refractive indices of the gold layers were determined as outlined in sec- tion 4 and sapphire substrate indices were taken from the literature. The

thicknesses of the gold layers were taken from the crystal thickness monitor

and then adjusted to produce the proper transmittance amplitudes as discussed

in section 4. Figure 5-3 is a plot of the expected transmittance spectrum for

the wavelength region from 0.6 to 1.2 micrometers. Figure 5-4 is the actual

transmittance spectrum of the modulator. The amplitudes of the maxima are down by a factor of two from the expected value.

42
Figure 5-3. Calculated transmittance spectrum of the modulator structure from

0.6 to 1.2 ym. Full scale is 100 percent.

Figure 5-4. Experimental transmittance spectrum of the modulator from 0.6 to

1.2 ym. Full scale is 30 percent.

43
Two possible causes of this reduction in transmittance became apparent in

a search of the literature. Chopra [16] and Knights [17] describe surface

roughness in gold and a-Si : H films, respectively. The roughness is due to the

deposition techniques used for these two materials. Knights also states that

the a-Si:H layer may be anistropic due to variations in the amount of hydrogen

incorporated during deposition. A rough surface would cause scattering of the

incident light and reduce the measured transmittance. A study done by Hiraki

[18] on crystalline silicon reveals that the silicon migrates through the gold

layer and forms silicon crystallites in the gold layer. The net effect would

again be an increased scattering and reduced transmittance. Hiraki's study of

silicon and gold also indicated that the gold will migrate into the silicon at room temperature. The gold is then a dopant and would increase the absorption

in the silicon.

Two possible corrections to the device model are suggested by the above

discussion of Hiraki's work. The first is to average the transmittance of the

device over a variation in the gold and a-Si:H thicknesses, but keeping the

sum of the three thicknesses constant. The other possible correction is an increase in the imaginary part of the refractive index of the a-Si:H to introduce more absorption.

The effectiveness of these two corrections can be measured by the resulting reduction in peak transmittance to the experimental values and a preservation of the shape of the resulting transmittance spectrum. Since the a-Si :

H

layer sandwiched by two gold layers forms a Fabry-Perot i nterferometer, the

shape of the transmittance spectrum can be termed "the device finesse."

Figure 5-5 is a theoretical plot of the transmittance spectrum of another

modulator device. The plot was generated by averaging 15 transmittance

spectra. The thickness of the a-Si : H and gold layers were varied but their

combined sum was kept constant for each of the 15 spectra. A weighting system

was used to approximate a Gaussian thickness distribution. When figure 5-5 is

compared with figure 5-4, it is seen that both the peak amplitude and the

finesse are in good agreement.

Figure 5-6 is a theoretical plot of the transmittance spectrum of the

modulator device in which the a-Si : H absorption is increased by increasing k,

the imaginary part of the refractive index. Figure 5-6 also compares well

with figure 5-4 in both peak transmittance and finesse.

44
Figure 5-5. Calculated transmittance spectrum of the modulator structure

from 0.6 to 1.2 ym. Fifteen thicknesses averaged. Full

scale is 30 percent.

Figure 5-6. Calculated transmittance spectrum of the modulator structure

from 0.6 to 1.2 ym, k for a-Si :H increased over previously

measured value. Full scale is 30 percent.

45
Since increasing the k for a-Si : H produces acceptable results and uses much less computer time, it will be used in the next section as a model for

determining a change in refractive index of a-Si:H.

5.3 Determination of the Change in Refractive

Index of a-Si:H with Applied Electric Field

The model developed in section 3 and corrected in the previous section is

used to determine the change in refractive index of a-Si : H with applied electric field. This is done by assuming a change in index and calculating a modulation spectrum using eq (3.13). The calculated and experimental modulation spectra are compared and the magnitude of the index change is adjusted to

increase the agreement between the calculated and actual spectra. This iteration continues until the desired agreement is achieved.

Figure 5-7 is a plot of an experimental modulation spectrum. This is

obtained by using the same procedure as outlined in section 4, but with the

"lock-in" amplifier tied to a voltage pulse applied to the modulator as opposed to being tied to a mechanical chopper. The transmission spectrum is

recorded and the pulse train from the lock-in is switched from the modulator

to the chopper and a transmission spectrum is again recorded. The ratio of

modulated to chopped transmission spectra forms the modulation spectrum.

Figure 5-8 is a plot of the calculated modulation spectrum after adjustment

for acceptable agreement with experiment. Table 5.3 contains the required

change in refractive index and is compared to several other electro-optic

materials as tabulated by Hammer [19]. As expected, a change in k produces an amplitude modulation and a change in n produces a phase modulation. In the

spectral region from 0.6 to 0.8 ym; the change in k dominates while from 0.8

to 1.2 ym, k is approximately zero and the change in n dominates.

46
Figure 5-7. Calculated modulation spectrum from 0.6 to 1.2 ym.

Full scale is ± 0.2 percent.

Figure 5-8. Experimental modulation spectrum from 0.6 to 1.2 ym.

Full scale is ± 0.2 percent.

47
Table 5.3

Change in Refractive Index of Four Materials

Material An Ak

a-Si :

H

1 x 10' 6 1.75 x 10

z n° 1.1 x 10~ 5 not given

GaAs 2.8 x 10" 5 not given

Li Nb0

3

1.6 x 10' 4 not given

Applied field E = 10

6 V/m

6. Conclusions

As was pointed out in sections 3 and 5, eq (3.12) provides a means of

determining the quality of a measurement. The transmittance maxima obtained

experimentally, in regions where eq (3.12) holds, can be compared to the

result of eq (3.12) and the error between these two values, will be the

approximate error in the Tdve (min) amplitude measurement. An error of ± 2

percent in Tave(min) will yield an error of ± 1.3 percent in the value of m

(assuming the index of sapphire, n 2 , is 1.7500). In calculating the thickness

of the layer from eq (3.11), another source of error is in the determination

of the wavelength of the extrema. An error of ± 0.5 percent in the wavelength

of the transmittance extremum determination will produce an error of ± 0.5

percent in the thickness. These two errors produce a 1.8 percent error in the

calculation of the index at other wavelengths when eqs (3.6) and (3.7) are used to do so. Appendix C also gives an indication of the measurement error introduced by assuming the material is nondi spersi ve. It has also been

pointed out that the value for refractive index, m, calculated from eq (3.10)

is not dependent on the wavelength of the minimum, the order of the minimum or the layer thickness. These two points are the main attributes of the measurement method. Since the values of refractive index obtained in section 4 for

SOS are in good agreement with published values, it can be concluded that the

measurement method produces reasonable results. The transmittance model

revealed a significant interaction between gold and a - S i : H , an interaction

which was not previously suspected.

48
7. References

[1] Johnk, Carl T. A. neering electromagnetic fields and waves. New

York: Wiley; 1975 . “383.

[2] Azzam, R. M. A.; Be ra, N. M. Ellipsometry and polarized light.

Amsterdam: North- H:: id; 1977. 332-340.

[3] Johnk, Carl T. A. cit., 378.

[4] Azzam, R. M. A.; Ba ra, N. M. Op. cit., 283.

[5] Goodman, Alvin M, :al interference method for the approximate

determination of re reive index and thickness of a transparent layer.

Applied Optics 17 —2787 ; 1978 October 1.

[6] Denton, R. E.; Campbe : , R. D.; Tomlin, S. G. The determination of the

optical constants in films from measurements of reflectance and

transmittance at n: incidence. J. Physics, D: Applied Physics

5:852-863; 1972.

[7] Malitson, Irving H -fraction and dispersion of synthetic sapphire.

J. Opt. Soc. Amer 1377-1379; 1962 December.

[8] Wise, Edmund M. Go recovery, properties and applications.

Princeton: Van Nos: 1; 218-219; 1964.

[9] Weiss, K. Optische nstanten dicker metal Ischichten. Z. Naturforsch

3a : 143; 1948.

[10] Heavens, 0. S. Op 1 properties of thin solid films. New York:

Academic Press; 200 ,

: 955 . [11] Hulthen, Rolf. Opt constants of epitaxial silicon in the region

1-3.3 eV. Physica ipta 12:342-344; 1975.

[12] Kuhl Ch.; Schlottere 1 . H. ; Schwidefsky, F. Optical investigation of

different silicon films. J. Electrochem. Soc.: Solid-State Science and

Technology 121:1496 1500; 1974 November.

[13] Chittick, R. C.; Alexander, J. H.; Sterling, H. F. The preparation and

properties of amorphous silicon. J. Electrochem. Soc. 116:77-81; 1969.

[14] Zanzucchi, P. J.; Wronski, C. R.; Carlson, D. E. Optical and

photoconductive properties of discharge-produced amorphous silicon.

J. Appl . Phys. 48 : 5225 - 5236 ; 1977.

[15] Prepared by Paul Werner of the Electromagnetic Technology Division of

the National Bureau of Standards.

49
[16] Chopra, Kasturi L. Thin film phenomena. New York: McGraw-Hill; 1969.

164.

[17] Knights, J. C. Growth morphology and defects in plasma-deposited

a - S i : H . J. Non-Crystal ine Solids 35&36 : 159-170; 1980.

[18] Hiraki, A. Low temperature migration of silicon through gold film in

Si-Au system. Progress in the Study of Point Defects. Doyama,

Yoshida, eds. Tokyo: Univ. of Tokyo Press; 1977. 393-440.

[19] Hammer, J. M. Topics in applied physics. Integrated Optics. Tamir,

ed. New York: Springer-Verlag; 1975. 150.

50
ooo

ooo

ooo

ooo

ooooooo

oooooooooooooo

Appendix A. FORTRAN Implementation of the Transmittance Model

PROGRAM SILICON (INPUT, OUTPUT ,TAPE2=0UTPUT)

THIS MODELS THE TRANSMISSION AND REFLECTION OF LIGHT INCIDENT

ON A MULTILAYERED, THIN FILM STRUCTURE WITH A TRANSPARENT SUBSTRATE.

SCATTERING MATRIX IS EMPLOYED AND THE RESULT HAS BEEN AVERAGED

OVER THE SUBSTRATE THICKNESS. FOR A NON-NORMAL ANGLE

OF INCIDENCE, PERPENDICULAR OR PARALLEL POLARIZATION MAY BE SELECTED.

RESULTS DISPLAYED BY PL0T10 (TEKTRONIX)

LAYER PARAMETERS INPUT INTO ARRAY P (LAYERS , L , 5

)

IN FORMAT N=N1 *L+N2 ,K = K1 *L+K2 : ,D=THICKNESS Nl ,N2 ,K1 ,K2 ,D

COMPLEX S(2 2,2) , R(2 , 6 ), E ( 6 ) ,T (2 . 6 )

, ANGLE (4 ) .

COMPLEX T1,T2,T3 R1 ,R2 , R3 , A (2 , 2 ,2 ) ,DP DS,N(4)

COMPLEX TO ,IMAG,EC0N(4)

DIMENSION NKDATA (4),TEF(48)

DIMENSION P (4 9 ,6) ,REF(4 ,30)

AIR , AMORPHOUS SILICON , SAPPHIRE , AIR

ENTER THICKNESS'

DATA P(1 ,1 ,6) ,P(2,1 ,6) ,P(3,1 6)/0 0 , 1 .OE-6 ,6 . 35E-4/

DATA P(4 ,1 ,6)/0 /

ENTER INDEX DATA: WAVELENGTH N ,K

DATA NKDATA/2 10,2,2/

DATA TEF/6E-7 1

. ,0. ,1 .2E-6,1 . ,0. , #6E-7,3.95, .06.6.5E-7,3.9 , .032.7E-7.3.812, .01

,

#7.5E-7,3.716,2.75E-3,8e-7 3 .64 , 1 . 1 2E -3 ,8 5E-7,3.597 .00018

#9E-7 ,3 .563 0..

#9 5E 7,3 537,0. .IE-6 3.516 ,0.0,1 .25E-6 3.45,0.0,

#6E-7 ,1 .74,0. ,1 . 2E-6 , 1 .76,0. ,6E-7 1 . ,0. , #1 .2E-6 ,1 . OJ

FUNCTION DEFINITIONS

U(C B)=(SQRT(B*B+(1+C)**2)-SQRT(B*B+(1-C)**2) )/2

V ( C , B) = ( SQRT(B*B+ ( 1 +C) **2 )+SQRT (B*B+ (1-C)**2) )/2

RSIN (C , B ) = ATAN (U ( C . B) /SQRT (

1 -U (C B)*U(C.B)))

AIMSIN ( C ,B)=ALOG( V(C ,B )+SQRT( V(C B)* 2-l))

LAYERS=2

LAY2=LAYERS+2

LAY 1 =LAYERS+1

DEFINE CONSTANTS (SQRT(-I) AND PI)

IMAG=CMPLX (0 ,1.)

PI=3. 1415926

FILL THE P MATRIX

K= 1 DO 30 1= 1 .LAY2

NKD=3*NKDATA(I)

DO 20 J = 1 , NKD

REF(I , J )=TEF(K)

K = K+1

20 CONTINUE

30 CONTINUE

9

51
40

50

C

C

C

80

90

C

CC

C

C

C

CCC

C

NKDMAX=1

DO 50 J=1.LAY2

NKD=NKDATA( J )-1

IF (NKD .GE. NKDMAX)NKDMAX=NKD

K = 1 DO 40 1=1 ,NKD

;j,I,1)=REF(J,K+3)

J ,1 ,2 ) = ( REF ( J ,K+1 ) -REF ( J K+4))/(f : J ,K)-REF(J ,K+3)

)

J , 1 , 3 ) =REF ( J K+1 ) -P ( J ,1,2) *REF(

J

J » I 3)=lREF(J l K+2) r REF(J i ,K+5))/(H 'J,K)-REF(J K+3)

)

J > J >5

PP

P

P

PK=K+3

:REF(J,£+2)-P(J,I,4)*REF(j7k

CONTINUE

CONTINUE

FILL REST OF P DO 90 1 = 2 .NKDMAX

DO 80 J=1 ,6

PCI , I , J ) =P (

1 ,1 ,J P (2 , 1 ,6 ) =P (2 1 ,6 P 3,i,J

'

CONTINUE

CONTINUE

'=p 3,1

!

'=P(4,1,

DEFINE THE COMPLEX ARCSIN=ARCSIN ( REAL P I SET UP FOR PLOT- 10 AND FRAME THE PLOT

CALL INITT (960)

CALL TERM (3,4096)

CALL DWINDOlO . ,1023 ,0 . ,1023.)

CALL CHRSIZ (4)

CALL MOVEA(0 . ,0.

)

CALL DRAWA (0 .590 ) CALL DRAWA 816 . ,590.

)

CALL DRAWA (81 6 . ,0 ) CALL DRAWA(0 . ,0.

)

********* PLOT THE SPECTRUM *********

K IS A DATA POINTER

K=1

DO 1000 M=1 ,601

WAVE=.599E-6+1E-9*M

IF (WAVE .GT.P(2,K,1)) K=K+1

DO 100 1= 1 ,LAY2

ABS0RP=-1 *(P(I,K,4) *WAVE+P (I ,K ,5 )

)

C THE COMPLEX REFRACTIVE INDEX FOR LAYER I

N ( I ) =CMPLX(P ( I ,K ,2 ) *WAVE+P (I ,K , 3 )

, ABSORP

)

100 CONTINUE

C ANGLE (1 )=INCIDENT ANGLE

ANGLE (1 )=CMPLX(0. ,0.)

DO 110 1 = 2 ,LAY2

W=N ( 1-1 ) *CSIN (ANGLE ( 1-1 ) )/N(I)

C=REAL(W)

B=AIMAG(W)

Cl =RSIN ( C B)

B1=AIMSIN(C,B)

ANGLE ( I )= CMP LX (Cl ,B1

)

110 CONTINUE

9

52
C CALCULATE THE INTERFACE AND LAYER MATRICES' ELEMENTS

C * R.T.EXP(JB) .EXP(-JB) * DO 000 1=2 ,LAY2

DP=N ( I

) *CCOS( ANGLE (1-1 ) )+N ( I- 1 ) *CCOS( ANGLE ( I

)

)

DS=N ( 1-1 ) *C CO S( ANGLE (1-1 ) )+N ( I

) *CCOS( ANGLE ( I )

)

R (

1 ,I)=(N(I) *CCOS( ANGLE ( I- 1 ) )-N ( 1-1 ) *CC0S( ANGLE ( I ) ) ) /DP

R 2,I)=(N(I-1 )*CCOS(ANGLE(I • 1 ) )-N (I ) *CCOS( ANGLE ( I ) ) ) /DS

T (

1 ,I)=2*N(I-1 ) *CCOS( ANGLE ( I- 1 ) )/DP

T(2,I)=2#N(I-1 ) *CCOS( ANGLE ( I 1 ) ) /DS

E ( I ) =CEXP ( IMAG*2 a PI*N (I)*CCOS(ANGLE(I) )*P(I 1 ,6)/WAVE)

ECON ( I ) = CEXP ( CONJG ( IMAG ) *2 *P I*N ( I

) *CCOS ( ANGLE (I)

)

#*P(I 1 ,6 ) /WAVE)

200 CONTINUE

C

C *** CALCULATE SCATTERING MATRIX ***

C

210

225

250

275

300

C

C

C

c

c

DO 210 1=1 2

SCI , 1 , 1 ) = 1 /T( I 2)

S 1,1 ,2)=R(I,2)/T(I,2)

S 1,2,1 =S(I,1,2

SCI 2,2)=S(I 1 1)

CONTINUE

DO 300 1 = 3 ,LAY

DO 275 J=1 2

AC J , 1 ,1)=(S(J,1 , 1

) *E(I-1 )+S

A(J ,1 ,2) = (S(J ,1 ,1 )*R(J ,I)*E

A J,2,1 = S J 2,1 )*E(I-1 )+S

A(J.2.2)=(S(J 2,1 )*R(J,I)*E

DO 250 LI = 1 ,2

DO 225 L2=1 2

S( J ,L1 . L2 ) = A( J ,L1 ,L2 ) CONTINUE

CONTINUE

CONTINUE

CONTINUE

(J,1,2)*R(J ,I)*ECON

(1-1 )+S(J.1 ,2 ) *ECON

(J,2,2)*R(J,I)*EC0N

(1-1 )+S(J ,2 2

) *ECON

1-1

1-1

1-1

1-1

*** CALCULATE TAVE AND RAVE *** SELECT DESIRED POLARIZATION

J=1 FOR PAR , J=2 FOR PER

J = 2

)/T(J I)

)/T J,I)

)/T(J I)

)/T(j,I)

C

T0=N(LAY1 ) *CCOS(ANGLE(LAY1 ) ) / (N ( 1 ) *CCOS( ANGLE( 1 ) )

)

T 1 = 1 .0/(S( J , 1 .1 )<CONJG(S(J 1 1 ) ))

T2=T(J.LAY2) *CONJG (T ( J ,LAY2 )

)

T3=T1*TO*CONJG(TO)

R1=T1*S(J ,2,1 ) “CONJG (S(J ,2.1 )

)

R2=R(J LAY2)*C0NJG(R(J,LAY2))

R3=SC J , 1 ,2)*C0NJG(S(J 1 ,2) )*T1

T AVE=REAL (T1 *T2 /(1-R2*R3))

RAVE=REAL( (R1 -R1*R2«R3+T1*T3*R2)/(1-R2*R3)

)

C

C PLOT TAVE

C

Y = 590. OCTAVE

X=1 .36E9*(WAVE-.6E-6)

CALL DRAWA(X,Y)

1000 CONTINUE

CALL FINITT (0. ,0.)

STOP

END

-END OF FILE-

?

53
Appendix B. Derivation of Equation (3.9)

The following equations model the structure depicted in figure B-l.

Tave

= WU-R2 M,

where

T 1 = 1/tSnSu*),

T 2 = t 2 3 1 2 3* j

R 2 = r 23 r 23 *,

R 3 = S 12 S 12 */(S 11 S 11 *)

;

and

s " lo l Li 1 12

air

n

a-Si :

H

n

l

sapphire

rial rn =n

3 o

H d h"

Figure B-l. System modeled in appendix B.

54
where

1 r r 1

’

and

L

= with 3 = 2tt Nd/X .

This implies that:

n 1 1 r 0

" J 3

e 0

1 r 12

S

= - ^0 it 12

r 01 1 -J 3i

Lo e

r 12 ^

resulting in:

jBi ~ J 3

S

i = (e + roir 12 e )/(toit 12 )

and

J 3 1 -j 3

S 1

= (r 12 e + r 0 i e )/(toiti 2 )

•

Since we assume that ki = 0,

3 1 ~ 2 u n^d^/X,

t 01 = 2n 0 /(n 0 +n 1 ),

t 12 = 2n 1 /(n 1 +n 2 ),

55
r o i = ( n o -n i )/ (

n

0 +n x

)

,

and

r 12 = (n 1 -n 2 )/(n 1 +n 2 )

•

Therefore

,

T i = 16n 0 2 n 1 2 /[ (n 0 +n 1

) 2 (n !+n 2 ) 2 + (

n

0 -n x

) 2

( n i-n 2 ) 2

+ 2(n 0 2 -n 1 2

) (n 1 2 -n 2

2

) cos (4nn 1 d 1 /X)] = 16n 0 2 n 1 2 /B,

R 3 = [( n o +n i

)

2 (n 1 -n 2 ) 2 + ( n 0 -n x

) 2 (n 1 +n 2 ) 2

+ 2(n 0 2 -n 1 2

) (n 1 2 -n 2

2

) cos (47rn 1 d 1 /x)]/B,

and for a lossless substrate, k 2 = 0, so

and

T 2 = 4n 2

2 /(n 2+n 4n 22 / (n 2+n 0

) 2

2 R 2 = [

( n 2-n 0 ) / ( n 2+n 0 ) ]

.

The cosine term in the expressions for Ti and R 3 may be simplified since we

are assuming that Tave

is at a minimum or valley point. At a valley, assuming

m is greater than n 2 .

2nj_ dj_ = (m + 1/2)

Xy m = 1, 2,***.

Therefore

,

cos(4Trn 1 d 1 /x) = cos ( ( 2m+l

)

tt ) = -1 ,

56
since 2m+l is an odd integer. This allows the expression for Tave

to be

simplified greatly.

Tave UV

} = T

V

= 4n 0 n 1 2 n 2 /[n 0 2 n 1 2 +n 0 2

n 2 2 +n 1

1+ +n 1 2 n 2

2

]

.

The known or measured values are no, n 2 and Tave

(Ay), so the equation can be

solved for m.

n

1 = {-(n 0 2 -4n 0 n 2 /T

v + n 2

2

) /2 + [n 0 2 -4n 0 n 2/T

y + n 2 2

) 2

- ^n Q 2 n z

2 ^2/2}^

Since

di = (m+1/2) Xv /(2 ni ) ,

we can determine the thickness d

1 if m is known.

57
Appendix C. Index Determination Error Resulting

From Dispersion

Equation (3.7) was found by making the assumption that the thin layer was both nonabsorbing and nondispersive. The method outlined in section 3 for

determining the refractive index incorporates eq (3.10), a result stemming

from the use of eq (3.7). The error introduced in the index determination by

the probable dispersive nature of the thin film is now examined.

The structure being considered here consists of an unsupported thin film

in air (see fig. C-l). The refractive index will be modeled as a linear

function of wavelength. The transmittance of this structure is given by eq

(C.l).

16 n

2

(n-1) 4 + (n+1) 4 - 2 ( n- 1

)

2

( n+1

)

2 cos (4-rcnd/A)

(C.l)

Assuming the index n is changing with wavelength A, the transmittance extrema

is found by letting the derivative of the transmittance with respect to

wavelength go to zero. This is given by eq (C.2).

1 n 1

b

- d H

Figure C-l. Structure modeled by eq (C.l).

58
3T _ 16 nA 3n/ax - 8n^(B-C)

(C.2)

where

A = n

4 + 6n 2 + 1 - (n 4 - 2n 2 + 1) cos (4-rTnd/x),

B = 4n

an

[n 2 + 3 - (n 2 + 1) cos (4-iTnd/A),

and

C = 4iTd(n 4 - 2n 2 + 1)(— ) sin ( 4tt nd/X )

xax x

2

If an/ax is zero, eq (C.3) results.

T = 4n 2 (C.3)

[ ( n-1

)

2 + (n+1) 2

] 2

From eq (C.3), the index n may be found from the value of the transmittance,

as given in eq (C.4)

For many semiconductors and other optical materials the assumption that an/ax

is zero is not valid. Equation (C.2) is transcendental and the roots (with

respect to wavelength) are found numerically for a fixed an/ax using linear

bisection. The index n

y and the wavelength at a transmittance minimum (root

just determined) are used to find the transmittance using eq (C.l). The value

for the transmittance is then used in eq (C.4) to determine an approximate

value for the index, n a

. The percent error is then given by

(C.4)

T

% error = (n - n )/n x 100%.

V d V

59
Table C.l contains the percent error for four values of an/ax. The thickness

is 1 pm, the order (heading), index, and an/ax (right column, index change per

meter) are also given. All calculations were performed to 12 decimal places.

As can be seen in the table, the worst error is for a an/ax of -1.0 E+06 at

the 1.5 order minimum. This error of 1 percent occurs at a unrealistic value

of dispersion and wavelength for nearly all optical materials but is still

less than the probable experimental error in any transmittance measurements.

It can be concluded from an examination of table C.l, that the approximation (an/ax = 0) introduces negligible error into the index determination

made in this work.

Table C-l.

Percent error introduced into refractive index

calculation for listed values of index, order (m)

and dispersion (3n/3X, X in meters)

1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5

Index 3.912 3.903 3.739 3.734 3.719 3.709 3.700 3.694

X error 1 . 3E-02 1.6E-03 4.1E-04 1.5E-04 6 . 8E-03 3 . 4E-03 1 . 9E-03 1.2E-03 +3 .0E+04

Index 2.054 2.617 2.909 3.249 3.440 3.587 3.703 3.797

X error 1.0E+00 2.1E-01 6.9E-02 2.8E-02 1.4E-02 7.7E-03 4.6E-03 2.9E-03

Index 2.340 3.001 3.264 3.433 3.330 3.636 3.702 3.734

X error 4.5E-01 8.1E-02 2.4E-02 9.9E-03 4.6E-03 2.5E-03 1.4E-03 9.0E-04

Index 3.516 3.605 3.645 3.660 3.693 3.693 3.700 3.706

X error 1 . 0E-02 1 . 4E-03 3.7E-04 1.4E-04 6.2E-03 3.2E-03 1.0E-03 1.1E-03 -3.0E+04

60
U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

2. Performing Organ. Report No. 3. Publ ication Date NBSIR 81-1652 December 1981

4. TITLE AND SUBTITLE

A Measurement Method for Determining the Optical and Electro-Optical Properties

of a Thin Film.

5. AUTHOR(S)

Donald R. Larson

6. PERFORMING ORGANIZATION (If joint or other than NBS. see in struction s)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8 . Type of Report & Period Covered

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

10. SUPPLEMENTARY NOTES

^ 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 method of determining the complex refractive index of a thin film on a nonabsorbing

substrate is developed. The optical transmittance spectrum of the structure is measured

and the index is determined by matching this spectrum numerically. An iterative

procedure for finding the magnitude of an induced change in refractive index is also

presented. In nonabsorbing spectral regions, the index and film thickness are determined

directly.

The optical transmittance of sapphire and thin films of gold and epitaxial silicon, both

on sapphire, is examined. The refractive index of epitaxial silicon on sapphire, SOS, i determined and compares favorably with the results of other investigators.

The measurement method is applied to a thin film of hydrogenated amorphous silicon,

a-Si:H, deposited by a capacitively coupled rf glow discharge. The index is tabulated

for various wavelengths and a field induced change in index comparable to GaAs is

measured.

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

Electro-optic modulation; hydrogenated amorphous silicon; optical transmittance;

refractive index; scattering matrix; thin film; transmittance extrema.

13. AVAILABILITY

Unlimited

| | For Official Distribution. Do Not Release to NTIS

Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402.

|_X

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

14. NO. OF

PRINTED PAGES

65

15. Price

$ 8.00

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