RUGOSITÉTHE PROBLEM OF SURFACE ROUGHNESS IN ELLIPSOMETRY AND REFLECTOMETRY

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RUGOSITÉTHE PROBLEM OF SURFACE

ROUGHNESS IN ELLIPSOMETRY AND

REFLECTOMETRY

I. Ohlídal, F. Luke, K. Navrátil

To cite this version:

THE PROBLEM OF SURFACE ROUGHNESS

IN ELLIPSOMETRY

AND

REFLECTOMETRY

I. OHL~DAL,

F.

L U K E ~ and K.

NAVRATIL

Dept. of Solid State Physics, Faculty of Science, J. E. Purkyne University, 61 137 Brno, Czechoslovakia

R6sum6.

-

Une classification des surfaces rugueuses a kt6 effectuke. Trois mod61es d'un systkme substrat-film avec des interfaces rugueuses ont BtB dkhis. On prksente aussi une approche thkorique pour trouver une solution aux equations ellipsometrique et de rkflectance (composants cohbrents et incohkrents gkneralement

A

incidence non normale pour ces derniers). Les principaux rksultats sont donnks sous forme mathkmatique. Ces resuItats sont utilises dans une discussion et une inter- pretation des r6sultats experimentam.

Abstract.

-

The classifications of rough surfaces has been performed. Three models of a system substrate-thin film with randomly rough boundaries have been defined. Also is presented the theoretical approach to the solution of the problem to find the ellipsometric parameters and reflec- tance (both coherent and incoherent components, generally at non-normal incidence) and the main results are given in mathematical fom. These results are used in a discussion concerning their application at interpretation of experimental results.

1. Introduction. - The contribution of ellipsometry and measurements of light intensity reflected or trans- mitted by a certain system to the study of different physical, chemical or electrochemical processes occur- ing at the surfaces of solids either clean or covered with thin films has increased considerably during last years [I-131. Both methods are often used even at the study of ion-implanted materials [14] and at the study of structural changes near the surface of crystals concluding from their surface treatment [15-171. Of course, we cannot miss their significance in solving classical problems of determination of optical cons- tants and thickness of bulk materials and thin films.

The application of optical methods at the study of solids is generally influenced by surface roughness. It is important to solve the problems connected with this influence from both theoretical and experimental points of view at least for the following reasons :

1) the precision of optical measurements has increa- sed considerably in the course of several years, so that it is necessary to consider the influence of surface roughness on measured optical values, especially in ultraviolet and visible ;

2) many real systems are more or less rough- etched and cleaved surfaces, the boundaries between the substrate and films prepared by anodic [18] or thermal oxidation [19], to give several examples.

We consider here some fundamental properties of ellipsometric parameters and of reflectance which characterize randomly rough surfaces and the systems substrate-thin film with randomly rough bounda- ries.

2. Theory.

-

2 . 1 CLASSIFICATION OF MODELS OF ROUGH SURFACES.

-

The theoretical procedure applied

to the study of the influence of surface roughness on optical properties of solids depends on the chosen model of rough boundary. We may choose the follow- ing classification system with respect to the form of rough boundaries :

1) periodically rough boundaries described by the function z(x, y ) = z(x

+

L,,

y -I-

L,)

; L, and L, are so-called periods of the boundary. The systems with cylindrical symmetry have mostly been considered [20]. Such boundaries are described by the function z(x) = z(x f L) ;

2) randomly rough surfaces generated by a random (stochastic) process. The function z(x, y) representing this boundary may be considered a random function of the coordinates

x,

y.

A rough surface is then, like a

random process, determined by a set of functions of coordinates x and y and by the theorems which charac- terize the statistical properties of this set. Each func- tion of this set is called the realization of the random function ;

3) randomly rough surfaces formed by rough sur- faces of special shape. A boundary represented by a

smooth surface covered with randomly spaced geome- trical forms (e. g. hemispheres) may serve as an example. The geometrical forms may either be iden- tical or they may have even different shape. The review concerning these surfaces was published by Beckmann and Spizichino [20]. These models are less general than those generated by a random process (see case 2). Nevertheless, in some cases it is more suitable to appro-

rS

AND K. NAVRATIL

ximate the real surface by some of these special models since the mathematical problems are often simpler in this case. These special models represent a simplifica- tion of the models from the case 2 which are the most general.

The second criterion used in the classification of rough boundaries is based on the ratio of the linear dimensions of surface irregularities with respect to the wavelength of incident light I. Let h be the mean value of the height coordinates of surface irregularities in the z direction (we consider three-dimensional Cartesian coordinate system with axes x, y, z

-

figu- re 1). The plane (x, y) coincides with the mean plane of

FIG. 1. - Monochromatic wave incident on a mean plane

M characterizing a rough surface : 81 the angle of incidence ; 0 2 and 83 are the angles characterizing the geometry of light scattering ; k l and kz are the wave vectors of incident and

scattered waves.

the rough surface and t is the mean value of the linear dimensions of surface irregularities within this plane. The suitable exact mathematical definition of both parameters will be given later for a case of a randomly rough surface. Then we may define another three groups of models of rough boundaries, as follows :

b) h 4 1 , t x l ;

m

In case of periodically rough surfaces it holds t = L.

The combinations of the models 1-3 and a-c give us all possible models of rough surfaces which may appear in practice.

Theoretically, we could consider the cases h x I, t 4 I and h % A, t x I as well. We have not found any sign of such surfaces in practice, neither have we found any mention of them in literature.

Real rough surfaces or boundaries belong mostly to models 2c or 2b [21-321. These models may appear especially in the study in ultraviolet or visible of sur-

faces which originate at mechanical grinding and polishing, at chemical etching or at anodic and thermal oxidation. Rather rarely do we meet surfaces belong- ing to the type 2a [33-361. The surfaces which may be approximated by models sub 1 and 3 appear quite exceptionally [37, 381.

We intend to consider here the systems (clean sur- faces and a substrate covered with a film) belonging to model 2c. The procedure which may be used at the general and exact solution of the interaction between incident light and a single boundary represented by model 2b is given in principle in the paper by Zipfel and DeSanto [39]. The mathematica1 procedure in this case is very complicated. Nevertheless, we have shown earlier [la] that theoretical results obtained for the

system 2c may often be used at the interpretation of experimental results obtained for the system belonging to model 2b as well.

2.2 MODELS OF THE SYSTEM SUBSTRATE-THIN FILM WITH RANDOMLY ROUGH BOUNDARIES. - TO Create areasonable model of a system substrate-thin film it is necessary to consider general relation between both boundaries (air-film and film-substrate) from both the geometrical and the statistical points of view. Generally, there exists certain statistical dependence between random functions describing both boun- daries. It is determined by the physical process responsible for the formation of the rough surface. The situation is relatively simple provided that we can express the relation between both random functions mathematically in an explicit form considering the mechanism of forming the rough surface and/or thin film. Then it is possible to use the theorems about the transformation of random variables and to express the statistical properties of one boundary by means of those characterizing the other one. We do not intend to consider here just this general form of a thin film which may be called coherent thinfilm. We concentrate our attention to the following special cases of the general model most often occuring in practice :

I) thin film with randomly rough boundaries which are identical geometrically and statistically as well (Fig. 2). This model is called identical thin Jilm (ITF). Its local thickness is d, = a c o s a, where

2

is the distance between the mean planes representing both boundaries and a is the angle between the mean plane and the tangent plane constructed in the considered point of the boundary. The magnitude of a changes randomly along the boundary. The symbols a and

Po

will be defined later. The ITF or a film which is very close to it may be generated during the vacuum eva- poration or cathode sputtering of a film on a rough surface ;

11) thin film with constant local thickness d, (Fig. 3) in any point of both the upper and the lower surfaces. Let us call this model uniform thinfilm (UTF). Such a

THE PROBLEM O F SURFACE ROUGHNESS IN ELLIPSOMETRY AND REFLECTOMETRY : C5-79

FIG. 2. - Geometrical picture of the system rough surface

-

identical thin film (ITF) : a) general view ; b) locally is an ITF represented by a planparallel thin iilm with thickness d~

=cd

cos a (a is a variable) ; 1. ambient, 2. thin film, 3. substrate.

The remaining symbols are explained in the text.

FIG. 3. - Geometrical picture of the system rough surface

-

uniform thin film (UTF) ; a) general view ; b) locally is an UTF represented by a planparallel thin film with d~ = d, The remain-

ing symbols have the same meaning as in figure 2.

of metals or semiconductors assuming that the oxida- tion velocity does not depend considerably on crystal orientation.

In both cases it is possible to find an unambiguous mathematical relation between the random functions describing both boundaries ;

111) thin film with both boundaries independent statistically and geometrically as well. The local thick- ness dL may differ considerably even in that case when local slopes on both boundaries are quite small (Fig. 4). This is different when compared with cases I

FIO. 4. - Geometrical picture of the system rough surface- general thin film (GTF) ; a) general view ; b) locally is a GTF represented by a wedge-shaped thin film with thickness d~ ( d ~ varies randomly). The remaining symbols have the same meaning

as in figure 2.

and 11. We call this model general thin Jilm (GTF). Such a film may originate, for instance, at polishing an ITF or an UTF system [211 or at thermal oxidation of some materials as was proved on GaAs [19].

Let us emphasize that we intend to consider all models 1-111, provided that both boundaries fulfil all assumptions defining the model of a rough surface 2c as well.

It is evident that all models 1-111 of thin films with rough boundaries exist from the geometrical point of view even in case when both boundaries are periodi- cally rough.

2.3 DEFINITIONS AND ASSUMPTIONS.

-

Let US

assume that a plane monochromatic wave falls on the mean plane of a rough surface and/or of the upper boundary of a system substrate-thin film under the angle 8,. Its mathematical form is

where

r =

xx,

-t yy,

+

zz,;

x,, yo, z, are the unit vectors along three axes of the Cartesian coordinate system, o is the anmJar fre-

-+

quency, A is the amplitude of incident wave, El is the

complex vector of electric field. The time factor exp(- iwt) will be omitted. After the interaction of the incident wave with arbitrary system with rough boun- daries there appears a non-zero light flux even in direc- tions different from the specular one (Fig. 1). The optical characteristics, i. e. (the state of polarization and the flux reflected into the specular and non-spe- cular directions as well are determined not only by the optical parameters of the system but also by geometri- cal and statistical properties of the boundaries.

Now we introduce the most important general assumptions which define unambiguously and correc- tly the physical model of rough surfaces and which specify the geometrical conditions for which our theore- tical results are valid. Let us assume :

1) rough surfaces are locally smooth (model 2c), i. e., the radius of curvature r, is much greater than 3, at any point of the boundary. Especially it holds [20]

4 nr, cos cp, B 3, ; (2)

cp, is the angle between k, and n, k, being the wave vector of the incident wave, n being the local normal to the surface. The condition (2) is equivalent to the assumption that the tangent plane to the surface in its arbitrary point deviates only slightly from this surface over the region the linear dimensions of which are much larger than 3, ;

c5-80 I. OHL~DAL, F. LUKES AND K . NAVRATIL

dence 8, of the incident wave with respect to the mean plane of the boundary lies within the interval

<

0, 850

>

so that it is possible to neglect the shado- wing and multiple reflections among the surface irre- gularities ;

3) the linear dimensions 2 X, 2 Y of the irradiated

surface in the mean plane (an rectangular) are much greater than A ;

4) the boundaries are generated by a stationary normal random process ;

5) at the detection of light are fulfilled the conditions of the Fraunhofer diffraction ;

6) all materials considered in our systems are opti- cally homogeneous and isotropic. The substrate and the film may be absorbing, the ambient is always nonabsorbing.

2 . 4 THE CALCULATION OF ELLIPSOMETRIC PARAME- TERS AND OF THE REFLECTANCE OF SYSTEMS WITH RAN- DOMLY ROUGH BOUNDARIES BY MEANS OF THE SCALAR THEORY. - The electric field in the point of observa- tion P is given by the Helmholtz-Kirchhoff integral ~401

where j = p, s, p(s) being the component in the plane of incidence (normal to it).

8

and are the local electric field on the rough surface and its derivative according to the direction of the local normal to the surface. S is the irradiated part of the boundary and

f

is defined by the equation

f = exp(ik, R,) exp(- ik, r)/R, , ₍₄₎

where Ro is the distance of the origin situated on the boundary from the point P, k, is the wave vector of diffracted wave

Considering eq. (1) we can express the local electric field in an arbitrary point of the surface by equation

.&

is the local complex Fresnel amplitude of the rough surface, A, is the amplitude of the incident wave. We assume that A , is the same for both p and s compo- nents. Let us note that eqs. (3)-(5) hold for all systems with rough boundaries. In case of the system formed by a substrate covered with one or more films the above equations relate to the upper boundary (ambient-film). The differences between particular systems are expres- sed by explicit form of the Fresnel amplitudes

gj.

It follows from eqs. (3)-(5) that &P) must be consi- dered a random quantity since

gj,

k, r and k, r are random functions. At the determination of reflectance

and of the ellipsometric parameters we detect the light Aux Fj',

where 9, is the solid acceptance angle of the detector and k o is the proportionality constant.

From the physical point of view only the mean value of flux

<

Fj'

>

=

>

is significant. The symbol

<

>

denotes the statistical mean value of the corresponding random function. &(P) must fulfil, like each random function, the following equation

+

D

{

E j p )

}

9 (7)

where

D denotes the so-called variance of the corresponding random value. It follows from eqs. (6) and (7) that total light flux for bothp and s components is generally given as a sum of two terms. Therefore it holds

where FTj, Fcj and Fij are the total, the coherent and the incoherent light fluxes. Total reflectance RTj cha- racterizing an arbitrary system with randomly rough surfaces is given by equation

R T j = R c j + R i j = ( F c j + F i j ) / F o ; (10) RCj and Rij are the coherent and incoherent compo- nents of total reflectance and F, is the incident light flux.

Let us consider now the ellipsometric parameters Y

(azimuth) and A (phase difference). Let us have a standard ellipsometer and let us choose the PCSA arrangement. Then we have the polarizer 5, the com- pensator

e,

the sample S and the analyzer A successi- vely between the light source and the detector. We also assume that A and Y are determined by means of the method suggested by McCrackin et al. [41]. The angle between the fast axis of the compensator and the plane of incidence is then either 71.14 or - n/4. It may be proved that the complex amplitude of linearly polarized wave incident on the detector is determined by equation

A A

THE PROBLEM OF SURFACE ROUGHNESS IN ELLIPSOMETRY AND REFLECTOMETRY C5-81

analyzer and the plane of incidence. The value A' is where Pi is the angle contained between the polariza- defined [42] by equation tion plane of the polarizer and the plane of incidence.

z

A ' = 2 P i

- -

Equation (12) holds only when the compensator is an

2 (I2) exact quarter-wave plate 141-431.

The light: flux registered by the detector is equal to

A

FA = F , cos2 A~

+

F,

sin2 A,

-

sin 2 Ai(cos A ' Re { F,,,

] -

sin A' Im (

?,,

))

,

(13)

We look after the minimum of light flux FA by changing the Pi and A,. The necessary conditions of the existence of the minimum of the function FA in the point [Po, A,] are the following

Equation (14) yields tan 2 P

- -

=

-

Im { F P , S

I

(

3

Re { ?p,s ]

'

2

1

if,,.

I

tan 2 A, = ---

.

Fs

-

F,

It is necessary to consider the angles Po and A, as the values equivalent to the ellipsometric parameters in general case of systems with rough boundaries. It is evident that these ellipsometric parameters Po and A, have not so simple geometrical meaning as A and Y characterizing the systems with smooth boundaries.

Let us now consider the case when we may neglect the terms connected with the incoherent light flux in eqs. (15) and (16). This means that we may write

This special case is very important since eqs. (15) and (16) are then considerably simplified. It is possible to show

1441 that it holds then

2 X, 2 Yare the sides of the irradiated rectangular,

is a random function describing the rough surface or the upper bou~dary of the system substrate-thin film.

* * A

The values aj, bj, cj depend explicitly on the local Fresnel amplitudes Rj 1201. These depend on the values 14, I;, on optical parameters of the system considered and on 8,. Putting eq. (17) into eqs. (15) and (16) and integrating over the variables x, y and

a,

we obtain

<

2,

>

tan $ exp(iA) = ---

< 2 s > '

A = - A h , 2 m z , $ = A , * m z ,

c5-82 I. OHL~DAL, F. LUKES AND K. N A V R ~ ~ T I L

here it holds [44, 451

h

<

e j

>

=

,

'

j

11,

j"

A

exp(iv, z) [z: sin 8,

-

R j cos 6, J w(z, z:, zi) dz dzi dzi

.

-

m

The function w(z, zk, 2:) is a simultaneous distribution function of the variables I, I:, I; ; v, =

-

(4 n/A) no cos 8,. no is the refractive index of the ambient. It follows from the latter equations that the meaning of the ellipsometric parameters of systems formed by a substrate covered with films with randomly rough surfaces considered in this special case is the same as for systems with smooth boundaries. We proved [45] that eqs (15) and (16) turn into eq. (19) when it holds

'a: is the solid acceptance angle of the detector, T is the correlation length of rough surface and a is the standard deviation of height irregularities. It holds

a is the mean value of the function I(x, y).

The inequality (21) was derived at the assumption that the correlation coefficient of the rough surface C(z)

could be expressed by means of equation

Here z is the distance between two points on the surface within the mean plane.

The magnitude of a positive number E is determined by the experimental accuracy of the values Po and A,.

Considering assumption 4) in paragraph 2.3. we get the equation

tan

Po

=

45

o/T assuming that eq. (23) holds. In case of randomly rough surfaces h and t are substituted by a and T.

Taking into account the assumption about small slopes on the surface (tan

Po

<

0.1) we can express the local Fresnel amplitudes of the rough surface or of a substrate covered with a film with the help of the MacLaurin series with respect to the variables I:, I;. If we limit ourselves to the terms of the zeroth, first and second orders the calculation of the value

<

sj

>

is relatively easy in case of a simple rough surcace and ITF and UTF sys- tems. The final form expressing the ellipsometric parameters of all these three systems is

1

tan $o exp(iAo) + - tan2

Po

$iX(0l)

+

2iy($l) 2

tan

+

exp(iA) = Z,(el> ,

1

1

+

- tan2

Po

&x(el>

+

%;y(6,>

2 ZS(01)

A , and Y, are the ellipsometric parameters of the same system but with smooth boundaries,

etc.

A h h

The relations for

&(el),

R;x(81), R;,, are given in our previous papers [21,44]).

The coherent component of reflectance Rj of the above mentioned systems is given at oblique incidence by the following equation

A

THE PROBLEM OF SURFACE ROUGHNESS IN ELLIPSOMETRY AND REFLECTOMETRY C5-83

herent component of reflectance at oblique incidence is too complicated. Here we give only the expression holding a t normal incidence ; then we get [20, 211

It follows from that shown above that the ellipsometric parameters characterizing a rough surface or the ITF and UTF systems are functions of the optical parameters of the corresponding system, of refractive index of the ambient, of angle of incidence

el,

of the standard deviation tan

Po

and of 2. when the above given assump- tions are fulfilled. It is essential that in this case A and Y do not depend on standard deviation o. Both coherent and incoherent components of reflectance of systems considered here depend also on the ratio a/A. The coherent component

R,

does not depend on tan

Po

at normal incidence as follows from equation [20,46]

This means that the spectral dependence of

R,

measured at normal incidence and the dependence of A and Y on 8 , may be used as complementary methods to the determination of both fundamental characteristics of a ran- domly rough boundary, a and tan

Po.

Let us consider a special case of the GTF system now. We assume that the slopes on both boundaries are very small (tan Po,, tan

Po2

<

0.001). Then the local thickness d, of a GTF may be expressed in the form

-

d is the mean thickness of the film, I , and I, are random functions describing the upper and the lower boun- daries. The procedure described in detail earlier [44] leads to the following equations :

tan

#

exp(iA) =

A A 4 11.

v, =

-

-

no

cos 0 ,

,

1 (30)

A

n is the complex refractive index of the film.

The ellipsometric parameters of a GTF always depend on cr, and o, and in case of greater slopes even on tan

pol

and tanPo2. The formulae for

R,

component characterizing a GTF are included in our earlier papers 121,451, The formula expressing the

Ri

component of GTF at normal incidence was derived in another paper [47].

2 . 5 CALCULATION OF ELLIPSOMETRIC PARAMETERS AND OF REFLECTANCE OF SYSTEMS WITH RANDOMLY ROUGH SURFACES WITHIN THE FRAMEWORK OF THE VECTOR THEORY. - Hitherto we have assumed that the interaction of plane polarized light wave with a rough surface does not change the direction of the polarization vector - for instance, p component before the interaction turns into the g component after the interaction. Generally, it is natural to assume that this does not hold. This possibility may be represented only within the framework of the vector theory of diffraction. The complex vector of the electric field in the far region may be represented with the help of the Stratton-Chu-Silver (SCS) integral [48]

A

+

c5-84 I. OHL~DAL, F. LUKES AND K. NAVRATIL

A and Y and also the coherent component of reflectance R, characterizing a simple surface and an ITF and a UTF systems. It holds

tan $ exp(iA) = tan

9,

exp(iAo)

+

f

tanZ

A[(%,

+

2iy)/g,

-

2[1

+

tan 1(1, exp(iAo)7 cotz 0,]

1

+

) tanz po[(3,

+

%Y)/z,

-

2[1

+

tan y?, exp(iA,)] cotZ

el]

-

(32) The

4

components of all three systems are given by equations

R,, = exp(- 16 nz ng cosz 0, oZ/1') x

R,, = exp(- 16 nz ng cosz 9, oz/Az) x

Having performed the numerical analysis we have found that there appear greater differences between the results derived by means of the scalar theory and those obtained by means of the vector theory only for the parameter A, namely when tan

Po

;y 0.1 (we still consider only the systems with tan

Po

5

0.1) and within the

regions of 9, where A changes rapidly [45]. In the case of GTF there exist differences between the results obtained from scalar and vector theories only when tan

Po,

and tan

Po,

cannot be neglected.

3. Numerical analysis of theoretical results. - The numerical analysis of the theoretical results is very important before we start interpeting experimentally found ellipsometric parameters and reflectance since it gives us good ideas about the dependence of these values on the parameters characterizing surface roughness. Since we performed this analysis fairly in detail in our earlier papers [la, 21, 44, 471 and there is not space enough to summarize it here we refer the reader to those papers.

4. The use of theoretical results at interpretation of experiment. - In our earlier papers [18, 44, 46, 471

we proved the correctness of theoretical results by means of the comparison with experiment. We found that the above theoretical results might be used even in such case that instead of T 9 A (this is equivalent to the inequality (1)) only T

2

A holds [18, 471.

Let us consider here how the determined depen- dences A = f1(9,) and Y = f2(9,) and of Rc = f (A) and/or RT = f (a,) may be used in optical analysis. We use the relations derived within the framework of the scalar theory [see eqs. (25), (26a)I.

4.1 SIMPLE BOUNDARY. - In this case correspond- ing to a clean surface we can determine the optical constants of the material and the value tan

p,

from experimentally found dependencies A = fl(O,) and

Y = f2(Ol) correctly for any A. The measured depen- dences PC =

f

(2) (PC = RCIRO) or PT =

f

(a,) ( p , = RT/Ro) may be used to determine o [18]. Unfortunately, we can scarcely use this procedure since clean surface - at least clean from the point of view of ellipsometric measurements performed in ultraviolet, visible or even in near infrared - may be realized at most in ultrahigh vacuum.

4.2 SUBSTRATE-THIN FILM.

-

We consider here

only the system absorbing substrate-nonabsorbing

film covered with a very thin film

(4

d

<

10 nm). Then it is meaningful to consider only the ITF or the UTF system for practical reasons. We concentrate our attention to the system which we have studied in more detail, i. e. when the inequalities n , % k, (k, 50.01) or n, > k,

(k,

% 0.1) are fulfilled. We require all

measurements to be performed on the same sample. Our system is then characterized by 6 parameters from the optical point of view : refractive index n, and absorption index k, of the substrate, the refractive index of the film n and its mean thickness B(for ITF) or thickness d (for UTF), tan

Po

and o. The problem to analyze this system optically may be divided into four parts :

1) The most general case occurs when we do not know any of these 6 parameters : the interpretation of the dependence p, = f (A) or p, = f (a,) gives us sufficiently correct magnitude of o. The interpretation of the dependences A = f,(0,) and Y = f2(el) yields relatively precise magnitude of n, and of tan

Po.

The values k,, n and z ( o r d) cannot be determined from the experimentally found A, Y , p, = f (A) and/or pT = f (a,) values sufficiently precisely due to their mutual correlation. This problem was discussed in detail by Ibrahim and Bashara [50] and by Loescher et al. [51] for a system with smooth boundaries. Our conclusions may be demonstrated on the system rough Si substrate-natural oxide film. Rough surface of single-crystalline Si sample was prepared by mecha- nical grinding 1471. Experimentally found dependences A = fl(gl) and Y = f2(9,) at A = 546.1 nm are given in figure 5. The magnitudes of the parameters n, and tan

p,

were determined with the help of the least-squares method. We found n, = 4.05 f 0.02, tan

p,

= 0.050

t.

0.005. If we assume that the refrac- tive index of natural oxide film on Si lies within the interval n E

<

1.44 ; 1.54

>

and its thickness

THE PROBLEM OF SURFACE ROUGHNESS I N ELLIPSOMETRY AND REFLECTOMETRY C5-85

o and tan

Po.

Unfortunately, neither in this case can we determine these values with sufficient accuracy from the dependences A = fl(O1), Y = f2(01), pc = f (A) and/or p, = f (a,). Again we need to know one of the values n or h(or d) with sufficient accuracy. We can demonstrate it again on the system rough Si surface-thin natural oxide film. The rough surface was prepared with the help of anodic oxidation from smooth single-crystalline sample at constant voltage [18]. The curves A = f,(B,) and Y = f2(01) are shown in figure 7. The application of the least-squares

0 4

70 72 74 76 78 80 817 82 l o method yielded tan

Po

= 0.100

+

0.005 and FIG. 5. - A = fi(8) and a , ~ = fz(8) (0 = 01) for a rough Si d = (5.5

+

0.5) nm [we assumed n E

<

1.44,1.54

>].

surface (the surface was mechanically treated) covered with a The possibility how the difference

natural oxide film (ITF or UTF [la]) : A-x x x x , a,~-0000 of the parameter Y (see Fig. 7) was discussed in our

(experimental results) ; A - - - -, a , ~

-

(theoretical curves) :

nl = 4.05, kl = 0.028, n = 1.46, d = 2.5 nm, tan PO = 0.05. - I N , / I I

magnitude of k, from above experimental values. We obtained k , = 0.05

+

0.05. In the case that we did not use the limitation of n and d the error of k , value was approximately three times as high. o was deter- mined from the dependence p, = g(a,) measured at normal incidence (A = 632.8 nm) ; it is shown in figure 6. The interpretation of these results by means of eq. (27) (we can limit the series to several initial terms) yields o = (47 f 1) nm and tan

Po

= 0.03

+

0.01.

FIG. 6. - p~ = f(uo) for a rough Si surface (the surface was mechanically treated) at normal incidence of light; IZ = 632.8 nm.

It is necessary to add other suitable experimental values of optical characteristics which could supply us with another independent information about the system considered here if we want to determine the magnitudes of all 6 parameters with sufficient accuracy. We have found that A and Y studied in different am- bients (air, liquids) fulfil these requirements in many cases.

2) The case when we know optical constants of the substrate with suficient accuracy in the whole spectral ,range considered in the experiment : then we want to find the magnitudes of the parameters n, bandlor d,

FIG. 7.

-

A = fl(0), a , ~ = f2(0), (0 = 01) for a rough Si surface (anodiacally oxidized) covered with a natural oxide film (ITF or UTF) : A- x x X x , y-0 00 0 (experimental results) ;

A

---

-, a , ~

-

(theoretical curves) : nl = 4.05, kl =

0.028, n = 1.46, d = 5.5 nm, tan Bo = 0.10.

papers [18,45]. a was determined from the dependence pc = f (A) (Fig. 8) and we found o = (26.5

+

0.4) nm. We can determine all 4 parameters with sufficient accuracy in the case that the above measured optical characteristics ( A = f,(O,) and Y = f2(01) in air, pc = f (A), p, = f (a,)) will be completed with the determination of A and Y in another ambient (one or more suitable liquids).

3) The case when n and

a

(or d) are known with suficient accuracy : then we want to determine n,, k,, tan

Po

and o. In this case we can determine all 4 para- meters with sufficient accuracy. From the dependences A = f,(O,) and Y = f2(0,) we determine n,, k, and tan

Po

; p, = f (A) or p, = f (a,) supplies us with o.

Of course, we hardly meet such a case in practice since n and a ( o r d) can be determined mostly only by means of ellipsometric measurements.

4) The case when optical constants of the system are known with suficient accuracy (this occurs often in practice) : we want to determine o, tan

Po

and z(or d). From experimental dependences A = f,(O,) and

c5-86 I. O H L ~ A L , F. LUKES AND K. NAVRATIL

FIG. 8. - R/R0 = f(u/A) (pc = R/Ro) for a rough Si surface. Open circles correspond to experimental values, full curve

represents the calculated dependence.

have used this procedure to study the influence of constant voltage on tan

Po

and o at anodic oxidation of Si single-crystal. We used the arrangement similar to that described by Manara et al. [7]. The oxide film was dissolved in H F acid. The results of our experi- ment are presented in table I.

o and tan

Po

of Si single-crystal anodically oxidized at constant voltage U

The situation is considerably complicated for the systems discussed above at the interpretation of the dependence A = f,(O,) and Y = f,(O,) when n, w k, or n,

<

k,. In such case even n, is more or less a correlated parameter. For that reason it can be deter- mined with much lower accuracy than n, in cases discussed above (n, % k, or n,

>

k,).

The optical analysis of the above systems but with greater d(or d) may supply us with more information when

R,

and

R,

measured at both normal and non- normal incidence are properly interpreted. On the other hand sometimes the interpretation of ellipsome- tric parameters may be more complicated. Generally, then it is necessary to use special procedures in indivi-

dual cases. Their choice depends especially on the magnitude of d(or d) and on the magnitude of optical constants of both substrate and film. The deterrnina- tion of the values n, z ( o r d) and o of systems nonab- sorbing substrate-nonabsorbing ITF or UTF from the dependence

R,

= f (A) when nB> 214 was des- cribed [46, 471 as well as of GTF [52].

5. Influences which most often combine with surface

roughness. - It often happens that the influence of roughness combines with the influence of defects in the structure. This influence has been observed at cleavage of single-crystal [22], at mechanical grinding and polishing [53], at ion-implantation [54] and pro- bably even at anodic oxidation 1451. The defects of structure often lead to an increase in optical constants [15-171 so that the cumulative influence of these defects and of surface roughness is very complicated. This means that without special supplementary experimental techniques it is almost impossible to interpret the optical measurements discussed here unambiguously.

Another complication which may appear especially at interpretation of ellipsometric parameters of semi- conductors is the influence of surface electric field on optical constants [55, 561.

6. Comparison with results concerning other models.

-

Unfortunately, we cannot discuss this important topic here for the scarcity of place. We apologize for that to the reader and we refer him to our previous papers 121, 44, 471.

7. Conclusion.

-

The purpose of this paper is to classify the rough surfaces and/or boundaries and to demonstrate their influence on ellipsometric para- meters and reflectance in case of a normal (Gaussian) randomly rough surface. The essential property of such surface is local smoothness.

We have considered one surface (clean sample) and three models of the system substrate-thin film :

identical (ITF), uniform (UTF) and general (GTF) films. Beside the formulation of fundaments of theore- tical approach to the solution of the problem and presentation some results in mathematical form we have attempted to outline even some practical aspects of theoretical results, especially we have demonstrated several procedures of how to solve the problems appea- ring in practice. We have concentrated our attention to the case when the ambient is air. But all our results are valid even in case that the ambient is a liquid. Of course, there appear some additional problems in connection with the fact that another boundary, namely air-liquid, appears. In some cases an inter- action between liquid and solid may exist-chemical reaction, adsorption of molecules or a change of the electric field close to the interface.

THE PROBLEM OF SURFACE ROUGHNESS IN ELLIPSOMETRY AND REFLECTOMETRY C5-87

a) extension of the theory o n t o surfaces with grea- metric parameters corresponding to incoherently ter slopes (tan

Po

>

0.1) ; this problem i s connected scattered light ;

with the i ~ ~ f l u e n c e of shadowing and multiple reflec- _{d) theoretical and experimental studies of surfaces} ti0ns On both ellipsometric parameters and reflectance ; _{with non-normal (non-Gaussian) randomly rough}

b) detailed study of systems with m o r e than one surfaces ;

film ; _{e) study of randomly r o u g h surfaces generated b y a}

c) theoretical and experimental studies of ellipso- non-stationary random process.

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DISCU 'SSION M. THEYE.

-

Up to now you have considered only

surface roughness with very small values for

P.

Would you expect the same kind of results for surface irregu- larities with larger slopes, in particular a much smaller sensitivity to roughness of the A parameter ?

F. LUKES. - The theoretical results given in our paper were obtained as the first approximation of general formulae derived within the framework of our approach. The consideration of higher approximations of course, there will be a smooth transition at going from higher to lower approximation in obtained dependencies of A and $ on tan

Po.

But we may expect and our results prove it (see i. e. Ohlidal I., Lukes F., Opt. Acta 19 (1972) 817 figures 3, 5, 6) that increases superlinearly with increasing tan

Po.

K. NAEGELE.

-

I would like to ask your opinion on a problem commonly known to all electrochemists working with reflection spectroscopic methods : when one is measuring at one wavelength one can easily observe that the intensity of light of the reference state decreases continuously. This is generally ascribed to scattering due roughening of the surface.

B. Cahan told us at the Zurich ISE-meeting that he observed drestic changes in $ and A too.

Now my questions : 1) Is it still possible in the light of your talk to speak of a constant dielectric constant of the substrate during an experiment? 2) Is it possible to describe the influence of progressive roughening of a surface by means of the off-diagonal elements of the jones-matrix of the surface or by another matrix

method (Muller calculus) ?

F. LUKES.

-

1) Within the framework of our theory of the influence of surface roughness on optical parameters (both ellipsometry data and reflectances either coherent or incoherent).

We assume that the optical constants of all media involved are really constant.

2) We have not yet tried any other procedure at mathematical development of our studies beside that one presented here, especially we have not used the matrix concept.

0. HUNDERI.

-

When the slopes are large, you also inevitably have local regions of high curvature, and thus one of your basic assumptions breaks down ; you can no longer define a local reflectivity when the local curvature is much smaller than the wavelength of your light.

F. LUKES.

-

I agree completely with the com-

ment ; it is necessary first substitude the condition 4 nr, cos 9, 2.

by another one and to include, eventually, the shadow- ing and multiple reflections.

E. YEAGER.

-

The roughness encountered in elec- trochemical studies is often very great. For example for metals such as platinum and gold subjected to electro- chemical conditions to provide clean surfaces, it is not unusual to have a ratio of true to apparent surface area of two. The surface evolution must be great. What hopes do you and Dr. Digman hold for quantitative treatments for optical studies of roughness for such surfaces ?

F. LUKES. - The theory of the influence of surface roughness on ellipsometric parameters and reflectance developed in our papers for randomly rough surfaces within the framework of either scolar or vector theory at certain assumptions (one of the most important is the assumption of small slopes, i. e. of small Po) makes possible to evaluate the parameters characterizing the surface roughness (o, tan Po) quantitatively at certain circumstances mentioned in our paper. This assertion is demonstrated in present paper and also in some pre- vious papers cited here for several special systems, especially for Si single crystal-SiO, film.