THE COMPLEX [MATH]-PLANE, A NEW THEORETICAL CONCEPT IN REFLECTION SPECTROSCOPY1. APPLICATION TO A FILM (NON-ABSORBING, ISOTROPIC)/SUBSTRATE SYSTEM

(1)

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THE COMPLEX [MATH]-PLANE, A NEW

THEORETICAL CONCEPT IN REFLECTION

SPECTROSCOPY1. APPLICATION TO A FILM

(NON-ABSORBING, ISOTROPIC)/SUBSTRATE

SYSTEM

K. Naegele

To cite this version:

K. Naegele. THE COMPLEX [MATH]-PLANE, A NEW THEORETICAL CONCEPT

IN REFLECTION SPECTROSCOPY1. APPLICATION TO A FILM (NON-ABSORBING,

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JOURNAL DE PHYSIQUE Colloque C5, supplkment au no 1 1, Tome 38, Novembre 1977, page C5-225

THE COMPLEX ;-PLANE, A

NEW THEORETICAL CONCEPT

IN REFLECTION SPECTROSCOPY

1. APPLICATION TO A FILM

(NON-ABSORBING, ISOTROPIC)/SUBSTRATE SYSTEM

K. D. NAEGELE

Institut fiir Physikalische Chemie und Quantenchemie der Freien Universitiit Berlin, Fabeckstrasse 32, 1000 Berlin 33, Federal Republic of Germany

Rbum6. - Les proprjCtCs de rtflexion d'une surface, recouverte d'un film, peuvent &tr%reprC- sentks par un numero r dans un plan mathCmatiquecomplexe. Dans cette representation, r dkrit l'etat de rkflec2vit6 du systeme de surface. Une fonction qui caractgise le changeme%t de r%ectivit6 (R. T. F.) lie r A I'aide d'une matrice au facteur de dCphasage X = exp(- 4 nidnz cos azll). La discussion des propriktts de cette matrice aboutit B des conclusions importantes concernant une nouvelle methode pour determiner I'epaisseur et les constantes optiques du film. Si I'on ne varie q u ~ 1'ep:isseur du film d, la fonction R. T. F. devient une transformation bilineaire et conforme entre r et X. Cela a pour rQultat l'utilisation d'une nouvelle technique mathematique permettant de sur- monter les problemes du tail-fit lorsqu'on applique des relations de dispersion aux rCflectivitCs d'un systkme film (isotrope, non absorbant)/substrat.

Abstryt.

-

The reflection properties of a iilm covered surface ca%be represented by a complex number r in a mathematically complex plane. In this representation, r descr%es the State of reflecti- vity of the surface %ystemAA reflectivity-transfer-function (RTF) connects r with the phase-factor

A

X = exp(- 4 nidllnz cos az) by means of a matrix of reflectivity-transformation. The discussion of this matrix leads to important conclusions concerning a new reflection-spectroscopic method to determine the thickness and optical constant of a film. The RTF becomes a conformal bilinear transformation between ;and

2

if only the thickness d of the film is varied. This leads to a new method to overcome the tail-fit problem of the application of dispersion relations to reflection- spectroscopic data of film (isotropic, non-absorbing)/substrate systems.

1. Introduction.

-

The principal aim of this paper is to ktroduce a new theoretical concept - the complex r-plane - into the experimental technique of reflection spectroscopy. This theoretical concept allows unification of different experimental techniques and is based upon one basic idea :

Every result of a reflection-spectroscopic experiment which consists in the determination of both the reflectivity R and the phase cp can be characterized by a single complex number

;:

which is defined as

state of reflectivity.

We speak of a state of reflectivity t o underline the fact that the result is independent of the experimental technique which is employed.

Although describing the reflection properties of a system by a point in a (mathematically) complex plane is a new idea for reflection spectroscopy, this concept is well known in other optical methods such as ellipsometry. Jzzam and Bashara [l, 2, 31 introduced the complex X-plane in order to describe the state of pola-

rization of completely polarized lighias a simple point in this plane. The basic idea of the X-plane concept is

the possibility of conlpressing all four polarization- parametzrs Ex, E,, 6,, 6 , into one single complex number

X.

The utility of their concept (and its success !) is also due to the fact that this complex number is the quotient of the two components of the pertinent Jones [4]

-

vector as demonstrated by equation (1) and (2).

A 0.

E, and Ex are the two components of the Jones-vector. As is generally known, the Jones-calculus [4] allows the straightforward calculation of the change of the

state of polarization of completely polarized light when it passes through orland is reflected by optical devices. Quick information about the optical properties of surfaces can be gathered using this method.

In reflection spectroscopy, on the other hand, the aim of the experimental method is t o determine

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C4-226 K. D. NAEGELE reflectivity R,,, and phase cp which are combined in

+e form of a complex Fresnel reflection coefficient

rabs

A

=

1

;abs

I

. e x ~ ( i ~ ) = &.ex~(iq) =

E,,

Ei

are the amplitudes of the reflected and incident electric field vectors, S,, Si are the phases, respectively. Due to the inherent experimental difficulties in measuring absolute reflectivities R,,, and absolute phases q , relative reflectivities R,,, and phase differences Aq are generally determined (I).

This means that a quotient of two Fresnel reflection coefficients is experimentally obtained

Here again the four parameters, characterizing a reflection experiment, are compressed into a single complex number.

The index abs refers to the determination of the absolute reflectivity Rab, and absolute phase q.

The index ref refers to reference and explains the experimental method : reflectivity and phase are determined with regard to an experimental reference state, which must be well defined, easily reproducible and unambiguous (2). ,

It should be pointed out that ellipsometry is not a relative method but yields finally

-

from the point of view of thz optical properties of a surface

-

a relative

A *

quantity p = rll/r,, to,o.

A It is the quantity r,,, which forms the complex

r-plane. And it is the analogy with the complex

A

X-plane that every state of reflectivity is characterized by the quotient of two complex numbers : the Fresnel reflection coefficients of the surface system under investigation, first when our system is in an unknown state to be investigated, second when our surface is in a state which is already known. The second state is our reference state and can be represented in electroche- mistry by a film-free electrode or in ultra-vacuum expe- riments by an uncovered targe: It is the behavior of this single point in the complex r-plane during an experiment which definitely describes the surface reflection properties.

A A

The concept of the r-plane (strictly speaking r,,,-plane allows a number of important conclusions which shall be discussed in this paper :

i) Every surface system from which light is reflected can be described by a state of rejectivity

(1) Relative reflection spectroscopy seems, therefore, to be a

term which describes the real experimental situation more correctly.

( 2 ) The difficulties in finding such a state shall not be discus-

qed here.

ii) A Rejection-Transfer-Function (RTF) can be given in analogy to the polarization transfer-function of Bashara and Azzam [5]. As will be discussed in chapter 3, the RTF is a well k y w n relation, but its interpretation in the light of the r-plane concept will yield interesting new results and insights.

iii) In the case of a film (non-absorbing)/substrate system the RTF behaves like a biliniar conformal transformation.

In addition to this

iv) a new method of determining the thickness of the layer and

v) a new criterium to overcome the tail-fit problem when Kramers-Kronig analysis of the reflectivity data is undertaken in order to evaluate the relative phases will be proposed and discussed briefly.

In this work, we will restrict ourselves to the case of reflection spectroscopy on non-absorbing film/substrate systems. The case of absorbing films and strati- fied media will be treated in a future publication.

2. The basic reflectivity equation.

-

For a three- phase-(123) system with dejined phase-boundaries the Fresnel reflection coefficient is given by equation (5)

where

Equation (5) 2olds for parallel and perpendicular polarized light. D, becomes real in the case of a non- absorbing film. In equation (5) the following notation is used :

r\

r,, : Fresnel reflection coefficient for the phase boundary 12,

A

r,, : Fresnel reflection coefficient for the phase boundary 23,

d : thickness of film 2, i : imaginary unit,

A

: wavelength of incident light,

a : angle of incidence,

A1

e, : complex dielectric constant of film 2,

el : real dielectric constant of ambient medium 1. In accordance with equation (4), equation (5) can be written in the form of equation (7a), (7b), (7c)

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'THE COMPLEX ;-PLANE, A NEW THEORETICAL CONCEPT IN REFLECTION SPECTROSCOPY C5-227

where

Equation (7c) makes use of equation (9), which holds for d = 0 of equation (5) :

A A

Equations (7a), (7b), (7c) can be witten in the gene- ral form of equation (10).

A

The meaning of the particular A,;s is resumed in table I.

A

Meaning of the A,{s of equation (10) for equations (7a), (7b), (7c)

Equation (10) is the central equation of this paper and will be referred to as the Reflection-Transfer- Function (R. T. F.). We would like to point out that equation (10) does not contain any new information since equations (5), (7), (9) have been in use for a long time. The form of equation (10) and its pertinent interpretation, however, is new, as it connects - and here again we make use of the analogies between polarization and reflection a certain input quantity

2

with an output quantity r by means of a matrix

Equation :

--

A A , 1 x 2 22 1 222

-

I\ A

A = (Aij) as can be seen by comparison of equation (10) with equations (6), (7), (8) of reference [I].

The resemblance between equation (8) of reference

[I] and equation (10) ends here due to the completely different physical backgrounds of both equations. The changes in the state of polarization of completely polarized light can be followed mathematically by simple matrix/vector algebra in the frame of the Jones calculus,

I. e.

-

A

o = ? * I (1 1)

h A

w

I, 0, f a r e input-vector, output-vector and transfor- mation matrix, respectively. The transformation ma-

*

trix

^T

of the Jones calculus represents the optical

7a

1

7b

1

7c

device T. The physical meaning of the state ofpolariza- tion before and after the mutual action of completely polarized light with the optical device T is obvious from equations (11) and (12) and needs no further explanation. A * r 2 3 / r 1 3 h A r 1 2 / r 1 3 A " r 1 2 ' r 2 3 1

In the case of reflection, on the other hand, an ana-

-

A A A

logous interpretation of r, X, A is impossible. It makes no physical sense to interpret the phasefactor

X^ by

means of an input-vector ! This becomes even more evident after taking the fact into account that there is no change in the state of reflectivity when light falls on a phase boundary. A r2 3 h '-1 2 * A A r 1 2 . r 2 3 ' r 1 3 A r13

A change of the state of reflectivity is obtained, however, when the thickness of Jilm d is varied, in other words : A r 2 3 A r12 A A A r 1 2

+

r23

-

r13 A r1 3

Equation (10) reflects the fact, that the state of reflectivity

;'

is changed when the thickness of film d (all optical constants, angle of incidence and wavelength are assumed to be constant) and therefore

?

is varied.

This is the way equation (10) has to be interpreted. In the light of the work of Azzam and Bashara and bearing in mind the foregoing interpretation of and

A

X, one can define a matrix of reflectivity-transformation as is done in equation

_-

(13). This notation seems to be reasonable as

A^

contains only Fresnel reflection coefficients (see table I).

-

A^

: matrix of reflectivity transformation _*

The properties of this matrix

A^

and the two quanti- ties

<

?will be discussed in the following chapters.

,-.

3. The matrix of reflectivity-transformation

-

A^.

-

The matrix Zconnects a film of optical constant

n^,

and thickness d, characterized by

-

2,

with the relative reflec-

A A

tivity r. The inverse matrix A - is even more important,

_-

A

as

A^-'

connects a given experimental quantity r to a quantity ?which contains all the information about the

-

film

(&,

_-

d). The inverse matrix

2-1

exists only when det

(6

# 0.

From equations (7a), (7b), (7c) one obtains for det

,-,

A

( A ) = 0 after some minor transformations :

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K. D. NAEGELE

TABLE _,-, I1

6 e A

Conditions for det (A,) = 0 and det ( A l l ) = 0

A r Z 3 = 0 A ( 1 5 ~ ) r , , =

+

1 A (15b) r 1 2 =

-

1 A (154 r13 = 0 A A (154 r 2 3 = r 1 2 A (154 r 1 3 = ' . I ,

-

(15!f)

The discussion of these conditions is summarized

in table I1 ( 3 ) . _w

To ilIustrate equations (15a-15f), det

(A^,)

and w

det ( i l l ) are presented as functions of the angle of incidence for different cases of optical constants in figure 1 and figure 2.

The behaviour of a pure dielectric three-phase sys- tem ( k , = k , = k 3 = 0) is demonstrated in figure 1.

Angle of lncldence /Degrees

Remarks trivial Brewster condition grazing incidence total reflection thin film, trivial

Brewster condition

,.,

FIG. 1. - Det (2 as function of the angle of incidence a1 for

A A

the system nl = 2 ; n2 = (1, 0) ; n3 = (3, 0).

A

: Real part of

,-, .-,

A

det (AL). A : Real part of det

(All).

: Imaginary part of

s M Equation A r 2 3 1 1 = 0 A A r1211 = r 1 3 1 1 A r 1 2 =

+

1 A r 1 2 =

-

1 A r1311 = 0 A A r1211 =

-

r 2 3 j J no 15a and 15f 156 ' 15c 15d and 15e

det

(A?).

0 :Imaginary part of det

(41).

aB23 = 28.32O ;

tot = 30.000 ; ~ ~= 156.31' 3 Condition A A E , = E~

G2

= 90' a , = 90' A 8, = e3 Condition A A E , = &a A n A

2

= tan a , n2 a , = 90'

a^,

= 90' A & I = z3 A

-

n3

-

tan a , nl Equation A r23, =

o

A A r l ~ l = r 1 3 1 A r,,, =

+

1 A r 1 2 , =

-

1 A r 1 3 , = 0 A A r12, =

-

r23L

Angle of lncldence /Degrees

Remarks trivial

total reflection grazing incidence

thin film, trivial

N

FIG. 2. - Det

(3

as function of the angle of incidence a1

A A

for the system nl = 1 ; nz = (2, 0) ; ns = (3, 3). 0 : Real part

M w

of det

-

(&).

: Real part of det

(All).

A

: Imaginary part of

,-,

A

det (A,). A : Imaginary part of det (All). as12 = 63.84O ;

ap = 82.62O.

It becomes clear that for perpendicular polarization at a , = a,,, (condition eq. (15b) and for parallel polarization at a , = aB23 (conditions eq. (15a), (15f), a , = a,,, (condition eq. (15)) and a , = aB13 (condi-

w

A

tions eq. (15d), (15e)) no inverse matrix A-I exists, i. e. no determination of thickness is possible under these conditions.

In figure 2, however, it is shown that for the case

A A

.-.

w

nl = 1, n2 = (2, O), n, = (3, 3) both

A^;

'

and

2i

'

do

exist for all angles of incidence. No total-reflection takes place, as n,

<

n,. Brewster conditions are not fulfilled as phase 3 is assumed to be absorbing. It should be noted, however, that the real part of det

_,.,

A * A r\

(3) Note that r12 = 0 doesnot fulfill the condition d ~ t ( A ) = 0. ( A I I ) becomes zero at a,,, This is due to the fact that in this case the light must travel A

(6)

THE COMPLEX ;-PLANE, A NEW THEORETICAL CONCEPT IN REFLECTION SPECTROSCOPY CS-229

-

whereas the imaginary part of det (A?]) becomes zero at necessary to look for a second possible eigenvalue.

an angle of incidence _{By setting}

_C'=

₃

_{equation (10) becomes after some}

where the condition for Aq = q l Z 3

-

vl3

= 900 is _: fulfilled (4).

The fact that no thickness can be determined for

z2

+

(222 - &1)/&1.8 -I- (- 212/&1) = 0 . (20) grazing incidence does not need any special discussion.

-

The roots of equation (20) are given by equation (21) It is interesting to note that for det (A) = 0 equation A

(10) becomes independent of

X^

: X1,1 = ( 4 1 _A

-

A^222)/(2221)

w

A A A

r = A 1 1 / A 2 , = for det

(A)

= 0 . (16)

+

((A,,

-

A^22)2/(4

2)

+

&2/A^21)0'5

.

(21)

By inserting the following equivalent expressions In accordance with equation (161, ;becomes for the _{of equation (22) into equation (21)}_{one can easily} different conditions of equations (15a)-(15d) _{show that no definite solutions X^,., exist as indefinite}

A A A A r = l for rZ3 = 0 or r12 = r13 (17a) A A A r = 1 for r,, = 1 A A A (17b) r =

-

l/r, for r12 =

-

1 h IZ A A (1 7 4 r = w for r,, = 0 or r,, = - r,,

.

(17d) Resuming the results we can state that under the

+,

condition of det (2) = 0, specified by equations (1 50)-

(15f), the relative reflectivities from equations (17a-d) are independent of the thickness offilm d.

Another interesting case of equation (10) is the condition

as equation (10) degenerates into an ordinary linear form :

P

=

(2,,/Z2,).2

+

(2,2/A2,) a (19)

Table 1 reveals the different condition where equation (18) becomes effective. These conditions and the pertinent relations for pare presented in table I11 ( 5 )

A

The different conditions for A,, = 0 Realization

Table 111 shows that if the Brewster-condition for the phase boundary 12 is fulfilled, ;will be equal to the phase-factor 2 (Eigenvalue). If one excludes the trivial

h

case E, = e2 we can state that this will be the case only for parallel polarization. As the condition a, = tan-' (g2/n,) reflects the fact that an eigenvalue exists, it is

I _A rI2, = 0 A r1211 = 0 A r231 = 0 A r2,,, = 0

quotients of the form 010 are found

A A A A tan a, = n,/nl

*

rlzIl = 0

-

7 1 3 ~ ~ = r2311

.

(22) A el = g2 A A = E~ and n2/n, = tan a l A A = ~E. A A A A c2 = E~ and n3/n2 = tan a2

This, at first glance, quite surprising result is easily understood if one takes into account that X^is a phase- factor containing

A

and d and cannot be expressed in any way only by pure Fresnel reflection coefficients

A A A

of two-phase boundaries r,,, r13, r2,.

We, therefore, cannot determine the eigenvalues by means of equation (21) ! The determination of the eigenvalues can be achieved, however, by using numerical methods. The eigenvalue condition

A 6 r, = X

,-.

A rll = X A rl = 1 A rll = 1

is equivalent to the following equation (24)

A A

r - X = O . (24)

Equation (10) and equation (24) yield an expression which can be easily calculated on a digital computer if d,

L

and the optical constants n,,

n^,

and

n^,

are held constant.

, "

a, : angle of eigenvalue.

By varying the angle of incidence a, in F(xl) the eigenvalues are characterized by the angles of incidence where both the real and imaginary part of 2(a,)

become zero.

The result of a numerical calculation is presented in figure 3.

Figure 3 affirms that for a given couple of optical

r\ A

constants n,, n,, n, the eigenvalue is only characterized by one angle of incidence where the Brewster condition a, = tan-I (n^,/n,) is fulfilled. This eigenangle is independent of thickness d, wavelength 1 and optical

A

constant n, = (n,, k,) as it is shown for the case

A ,".

nl = 1 ; n2 = 2 ; n, = (3,3) in figure 4.

No eigenangle can be detected for perpendicular

A

polarization (the case E, = E, is excluded). (4) This condition is the analogue to the well-known criterium

for the existence of the principal angle ! The fact that the eigenangle and consequently the

A A

( 5 ) Note that Y~~ = 0 implies also =

o

(see equation (16)) eigenvalue are independent of

&

could be experimen-

A A A

(7)

AEGELE

Angle 01 lncldencs I Cmgreer

h A

FIG. 3.

-

(r

-

X) as function of the angle of incidence a1 for

A A

the system n l = 1 ; n2 = (2, 0 ) ; n3 = (3, 3) ; d / l = 0 . 1 . e :

A * A A

Real part of (r

-

X ) I . W : Real part of (r

-

X ) I I . 0 : Imagi-

A A h A

nary part of (r - X),. A : Imaginary part of (r - X ) I I .

Optical constant ( s e e below )

FIG. 4.

-

Eigenangle aelsen as function of the different optical

A * constants w, x, y, z . 4 : nl = 1 ; n2 = (w, 0 ) ; n 3 = (3, 3). A A A : nl = 1 ; n2 = (2,O) ; n3 = (x, 3).

A

: nl = 1 ; n2 = (2,O) ; A A A n3 = (3, y). 0 : nl = z ; n2 = (2, 0 ) ; n3 = ( 3 , 3).

of two identical films of equal thickness on two different substrates should be equal at a, = tan-' (n^,/n,). There is an infinite number of eigenvalues as the eigenangle condition does not depend on the thickness d, i. e. every thickness d results in an own eigenphase-

C\

factor Xeisen (6).

4. The bilinear properties of the Reflectivity-Transfer- Function (R. T. F.).

-

One interesting property not yet discussed is the fact that equation (10) represents,

h *

in the case i t , , n,, n2, a,,

&=

const., a bilinear trans-

formation between r and X. Since this paper discusses the case of a non-absorbing film (k, = 0) a quick examination of equations (6) a n d (8) makes clear tha by increasing the thickness d, X^moves along the unit- circle in the complex T-plane. In accordance with the

( 6 ) Note that the reflectivity R123/R13 is under the condition

a1 = ~ E I G E N always equal to one.

properties of a bilinear transformation

f

must move on a circle in the >plane, too. The significance of

5,

now becomes clear :

5,

represents the thickness when one circle has been accomplished. Radii and centers

A

of all the possible circles depend on n,, n,, n, and a,. These dependences are shown schematically in figure 5a and 5b.

Optlcal constant [ s e e below )

FIG. 5a.

-

Radii of equi-angle-of-incidence circles as function of different optical constants w , x, y, z.

*

Dashed symbols represent the standard nl = 1 ; n2 = ( 2 , 0 ) ;

A A A n3 = (3, 3). 0 : rrl = 1 ; ~2 = ( w , 0 ) ; 113 =: (3, 3). : nl = 1 ; A A A A n2 = (2, 0 ) ; n3 = ( x , 3).

+

: nl = 1 ; n2 = (2, 0 ) ; n3 = (3, )3. A A

A

: nl = z ; n2 = (2, 0 ) ; n3 = (3, 3).

I

1

Real part of ( ?123/ iI3

FIG. 56.

-

Loci of centers of equi-angle-of-incidence circles as function of different optical constants w, x , y, z. Symbols as in

figure Sa.

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THE COMPLEX ;PLANE, A NEW THEORETICAL CONCEPT IN REFLECTION SPECTROSCOPY C5-231 5. The equi-thickness contours. - We use equa- which yields finally equation (28) :

lion (10) : the angle of incidence is varied for fixed n,,

A i. D,

n,, n,, d and

A.

The results of some calculations are d = - x

2. 7r

presented in figure 6.

x ln

((Y12

-

+

F12.;23-?13.?)).

t

I- (28)

FIG. 6. -Some equi-thickness curves for the system nl = 1 ;

h A

n2 = (2, 0) n3 = (3, 3) ; l, = 6 000 A. Note that for reasons of perspicuity the d = 600 A and d

-

800 A contours are not shown for parallel polarization. Dashed points represent

perpendicular polarization.

The result is a closed curve consisting of two different branches

a) the part, where the reflectivity is parallel ; b) the part, where the reflectivity is perpendicular. All branches have one point ( 1 , 0) in common, the locus of all the other points where the pertinent parallel and the perpendicular branches join together is the equi-angle-of-incidence contour for a, = 00, a circle.

6. The determination of thickness d and optical

A

constant n2. - The inversion of equation (1 1) yields a relation where 2 is a function of

The &,'s of equation (26) have the meanings as described by table I. Equation (26) offers the possibility of determining

g2

and d simultaneously (7). By

inserting equation (8) for

X^

into equation (26) one obtains

(7) For restrictions see chapter 3.

The right hand side of equation (28) contains only

z2

as an unknown parameter, whereas the left hand side consists of a single real quantity, the thickness d. Equation (28) offers the possibility of determining d and

n^,

simultaneously as the correct

n^,

is found when the right hand side of equation (28) becomes real. Equation (28) holds for parallel and perpendicular polarization. An analogous method has been proposed by Azzam and Bashara for the case of ellipsometry [6]. Equation (28) is valid for an absorbing ( k , # 0) and a non-absorbing ( k , = 0) film. Note that in this case of well-defined phase-boundaries,

n^,

and d are completely separable. This may not be the case for very thin films (d

<

10

A)

as has been shown by Plieth and Naegele [7]). The practical importance of equation (28) and an error analysis shall be discussed else- where [ 8 ] here we give only the results of a calculated example of a non-absorbing film which are presented

in figure 7.

FIG. 7. - Computer simulation of an evaluation of the thick-

,,

ness dand the optical constant n2: = n2 by means of equation (28)

A A

nl = 1 ; n2 =(2, 0 ) ; ns = (3, 3 ) ; d = 1200 A ; 1. = 6000 A ;

a1 = 4 5 O .

(9)

C5-232 K. D. NAEGELE mental barrier. Very often this assumed reflectivitytail

is fitted on a known experimental constant

(G

determined by ellipsometry at a single wavelength. We propose that the circle mapping properties of a non- absorbing film/substrate system should be used as a reasonable fitting criterium. We only want to give here a brief outline of the proposed method, as the algorithm of the method and experimental results shall be discussed elsewhere [8] :

a) R, ,,/R,, + h. v curves must be measured experimentally by at least four different thicknesses.

b) A Kramers-Kronig analysis of the experimental reflectivity data is performed where the criterium for the tail-fit is the best-circle-form of our four, five,

A

...

n, r (I = const.) points at every wavelength. The proposed method has one advantage over the other methods proposed in the literature, as no experimental standard value is necessary. It is restricted, for the time being, to non-absorbing, dielectric film/ substrate systems.

8. Conclusions. - Stimulated by the complex concept of Azzam and Bashara, which has proved its extreme usefulness in ellipsometry, an analogous theoretical concept for reflection spectroscopy, the c ~ m ~ l e x ? ~ l a n e concept has been developed.

(8) A circle is detined by three points, the fourth point can

serve as a check a s to whether the circle exists.

The principal equation of these authors (see eq. (8) of ref. [3]) finds its analogue in equation (10) of this paper. Nevertheless, there are strong differences between both concepts : the X-plane concept describes changes in the state of polarization whereas the ;-plane concept describes changes in the state of reflectivity. A state of reflectivity? is defined in this work by the quotient of two Fresnel reflection coefficients of equal polarization. This notation is best suited for the commonly used relative reflection spectroscopy.

The discussion of the Reflectivity-Transfer-Function (R. T. F.) yields some interesting results which can be explained by well-known optical effects such as total reflection, Brewster angle and principal angle. By varying the thickness d of the film and under the condition A, a, = const. the obtained relative reflectivities

A A

r,,,/r,, form a circle. This interesting mapping property is used to propose a new tail-fit method for Kramers- Kronig-analysis of experimental R,,,/R,, + h. v data. As the validity of equation (10) assumes well-defined phase boundaries, it is interesting to note that

$,

and d can be completely separated. This property is used to propose a new reflection-spectroscopical method to simultaneously determine thickness and

A

the optical constant n2.

Acknowledgments. - The author is indebted to the S t i f t u n Stipendien Fonds of the Verband der Chemis- chen Industrie for a Liebig fellow-ship. Assistance of Miss Steinki and Mrs. Naegele during the preparation of the drawings is gratefully acknowledged.

References [I] AZZAM, R. M. A. and BASHARA, N. M., J. Opt. SOC. Am.

62 (1 972) 222.

[2] AZZAM, R. M. A. and BASHAKA, N. M., J. Opt. SOC. Am. 62 (1972) 336.

131 AZZAM, R. M. A. and BASHARA, N. M., Appl. Opt. 12 (1973) 62.

[4] SHURCLIFF, W. A., Polarized Light, p. 27ff, p. 118ff, p. 165ff (Harvard University Press, Cambridge, Mass.) 1962.

[5] AZZAM, R. M. A. and BASHARA, N. M., Opt. Commun. 7 (1973) 317.

[6] AZZAM, R. M. A., ZAGHLOUL, A.-R. M. and BAS- HARA, N. M., J. Opt. S O ~ . Am. 65 (1975) 252. [7] PLIETH, W. J. and NAEGELE, K. D., SurJ Sci. (1977) in

press.

[8] NAEGELE, K. D., to be published. [9] STERN, F., Solid State Phys. 15 (1963) 331.

[lo] CARDONA, M. and GREENAWAY, D. L., Phys. Rev. 133A (1964) 1685.

DISCUSSION

A. HUGOT-LE GOFF. - Quel systeme est propose pour mettre en ceuvre le dCpouillement preconist par l'auteur ?

K. NAEGELE. - L'auteur travaille actuellement sur la prkcipitation de I'oxyde de plomb (PbO,) sur du platine. Les mCthodes utilisees sont la spectroscopie de reflectance et I'ellipsomCtrie. Les rksultats obtenus par la mkthode de spectroscopie de reflectance et l'application, par la suite, des relations de dispersion sont cornparks aux rksultats ellipsomttriques. L'auteur

espere pouvoir dCmontrer l'utilitt de la mtthode proposte pour dtterminer I'epaisseur de la couche d

,v

et les constantes optiques n , = n,

-

ik,.