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Ellipsometry of randomly rough oxidized silicon surfaces
Solar Cells, 13 (1984) 179 - 183
179
E LLI P S OMETRY OF RANDOMLY ROUGH OXIDIZED SILICON SURFACES T. DAVID BURLEIGH Department o f Materials Science and Engineering, Massachusetts Institute o f Technology, Cambridge, MA 02139 (U.S.A.)
SIGURD WAGNER Department o f Electrical Engineering and Computer Science, Princeton University, Princeton, NJ 08544 (U.S.A.)
THEODORE F. CISZEK Solar Energy Research Institute, Golden, CO 80401 (U.S.A.)
(Received February 8, 1984 ; accepted July 20, 1984)
Summary This experimental study shows that ellipsometry can provide thickness and refractive index measurements of oxide films grown on r a n d o m l y rough surfaces. The sample roughness ranged f r om 0.01 to 10 pm, with measuring light o f wavelength 0.6328 pm. The calculated values of oxide thickness and refractive index deviate from those on flat reference samples. The deviations are reproducible, so t hat calibration curves may be established for rough v e r s u s s m o o t h samples.
Ellipsometry is e m p l o y e d routinely in the determination of the thickness and refractive index of layers o f SiO2 on silicon. Such layers are usually grown on silicon substrates with highly polished surfaces. In photovoltaic t e ch n o lo g y the characterization of dielectric layers on rough substrates also may be o f interest. Rough substrates are p r o d u c e d by the sawing, grinding and etching o f silicon wafers. Many polycrystalline s e m i c o n d u c t o r films are rough, although on a smaller scale t han sawed wafers. In this paper we present t he results o f an eUipsometric study o f SiO2/Si structures whose oxide had been grown on roughened silicon substrates. We find th at the values for thickness and refractive index deviate from those det e rmin ed on flat reference samples. However, the deviations are smooth functions o f the roughness, so t ha t the thickness and refractive index still m a y be determined with the help of calibration data such as those established in our work. The use of ellipsometry to measure thin films assumes perfectly s mo o th surfaces. Several workers have addressed the problems e n c o u n t e r e d 0379-6787/84/$3.00
© Elsevier Sequoia/Printed in The Netherlands
180
with e l l i p s o m e t r y on rough surfaces [ 1 - 7 ] . V o r b u r g e r and L u d e m a [7] provided an excellent review o f the topic. H o w e v e r , no group has y e t s h o w n the practicality o f converting the a p p a r e n t values o b t a i n e d by e l l i p s o m e t r y on rough surfaces to the c o r r e c t values o f refractive i n d e x and film thickness o n c e t h e surface roughness is k n o w n . We set o u t to establish an empirical c o r r e l a t i o n b e t w e e n the a p p a r e n t thickness of oxides on rough silicon substrates and the o x i d e thickness on s m o o t h substrates. T h e roughness o f o u r substrate surfaces ranged f r o m 0.01 to 10 pm, i.e. f r o m well below t o well above the wavelength o f the ellipsometer H e - N e laser light. A t o t a l o f 68 samples were measured. F o u r d i f f e r e n t t y p e s o f silicon wafer sample were tested, with five d i f f e r e n t grit sizes of surface polish and f o u r d i f f e r e n t c o n d i t i o n s o f t h e r m a l o x i d a t i o n . The silicon samples included cast multigrain, v a p o r - d e p o s i t e d multigrain and single-crystal material (Table 1). T h e surface finish varied f r o m d i a m o n d wheel cut (or as received) to alumina p o w d e r polishes with grit sizes ranging f r o m 12 to 0.05 p m (Table 2). In addition to a native ( r o o m t e m p e r a t u r e ) oxide, t h e r m a l o x i d e layers were grown in a t u b e f u r n a c e 10 cm in d i a m e t e r at 950 °C with a flow of 300 cm 3 of commercial-grade o x y g e n (Table 3). Annealing at this t e m p e r a t u r e usually p r o d u c e s slip and dislocation propagation which we assumed did n o t affect the rate o f o x i d a t i o n . Thus the real o x i d e thicknesses on the r o u g h wafers are assumed to be identical with t h o s e o n the s m o o t h r e f e r e n c e samples. The surface roughness was d e t e r m i n e d b e f o r e o x i d a t i o n f r o m the profile m e a s u r e d with a T e n c o r Alpha step profiler. We define as roughness the average peak-to-valley d i f f e r e n c e along a trace length o f 50 /2m. This length was twice the beam spot d i a m e t e r (25 gm) o f the laser e m p l o y e d in the ellipsometer. Obvious pits or scratches were avoided in the roughness determination. The thickness and refractive i n d e x o f the o x i d e were d e t e r m i n e d f r o m data t a k e n with a G a e r t n e r L 117 manual ellipsometer. An H e - N e laser served as the light source (h = 632.8 nm). T h e angle o f incidence was fixed at 70 °. Calculations were carried o u t o n a digital c o m p u t e r . In Fig. 1 t h e a p p a r e n t o x i d e thickness o f the rough samples is p l o t t e d against t h e o x i d e thickness o f the flat r e f e r e n c e samples. (The refractive i n d e x was set at 1.46, the value for s t o i c h i o m e t r i c native SiO2 films.) The rough oxide thickness begins to deviate f r o m t h a t o f the r e f e r e n c e samples when the surface peak-to-valley roughness exceeds a b o u t o n e - t e n t h o f the wavelength of the ellipsometric light source. Thin oxides are m o s t a f f e c t e d by roughness; the r o u g h e r the surface is, the greater is the a p p a r e n t thickness. Even for r o u g h samples, t h o u g h , the a p p a r e n t thickness a p p r o a c h e s t h a t of the r e f e r e n c e samples w h e n it is o f the o r d e r o f 100 nm. If the refractive i n d e x is n o t set at 1.46 b u t r a t h e r is c o m p u t e d f r o m the ellipsometric parameters, its value is larger t h a n the actual value while t h e c o m p u t e d oxide thickness is c o r r e s p o n d i n g l y smaller t h a n the actual value. The c o m p u t e d i n d e x o f r e f r a c t i o n reaches a m a x i m u m w h e n the roughness c o r r e s p o n d s to the wavelength o f the laser. Figure 2 shows t h e refractive i n d e x as a f u n c t i o n
181 TABLE 1 Silicon samples employed in the experiments Morphology
Resistivity (~ cm)
Manufacturer
Single crystal (111 ) Cast multigrain (Silso) Cast multigrain Vapor-deposited multigrain
25 1.5 0.08 278
T. F. Ciszek Wacker Solarex Dow-Corning
TABLE 2 Polishing method and surface roughness Surface polish
Peak-to-valley distance (nm)
As received or diamond cut 12 pm alumina 5 pm alumina 1/~m alumina 0.05 pm alumina
500 - 10000 1000 500 50 10
TABLE 3 Oxidation conditions and oxide thickness Temperature (°C)
Duration (min)
Thickness (nm)
Room temperature 950 950 950
10 55 120
4 24 55 120
o f roughness f o r o x i d e thicknesses o f 55 and 120 n m . T h e d a t a f o r an o x i d e thickness o f 24 n m are n o t s h o w n for clarity. The a m p l i t u d e o f t h e maxi m u m is inversely p r o p o r t i o n a l to t h e o x i d e t h i c k n e s s ; the t h i c k e r t h e o x i d e is, the l o w e r is the m a x i m u m . The oxide 120 n m t h i c k had a p e a k 50% smaller and t h e o x i d e 24 n m t h i c k had a peak 20% higher t h a n t h e o x i d e 55 n m t h i c k had. The refractive index versus roughness has been e x a m i n e d and m o d e l e d extensively b y several w o r k e r s [ 1 - 7 ]. T o o u r k n o w l e d g e , this w o r k is t h e first c o n c e r n e d with a large range o f roughnesses. Figures 1 and 2 d e m o n s t r a t e t h a t e l l i p s o m e t r y can be used for a r e p r o d u c i b l e c h a r a c t e r i z a t i o n o f layers o n surfaces with a wide range o f roughnesses. H o w e v e r , some c a u t i o n m u s t be used in applying the m e t h o d . T h e sample m u s t be r a n d o m l y rough. F o r e x a m p l e , parallel scratches f r o m a
182 IOOO
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._J
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w I ~ w
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0 I..z w
/
a. 13_ '~
/
/
o/
1
i
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OXIDE THICKNESS ON REFERENCE SAMPLE (niT,)
Fig. 1. M e a s u r e d a p p a r e n t o x i d e t h i c k n e s s o n r o u g h s a m p l e s v s . t h i c k n e s s o n flat r e f e r e n c e s a m p l e s as a f u n c t i o n o f r o u g h n e s s ( o x i d e r e f r a c t i v e i n d e x , 1 . 4 6 ) : / ~ , 1 0 0 0 n m ; ,~, 50 n m ; c:, 10 n m .
I
/\
5 ~: LI_ 2.C W tr b._ 0 X i,i a Z
'! tz
1,5
1.0
t LASER WAVELENGTF I
6 2 3 . 8 nm
I I i
i
I
I
10 IO0 |000 AVERAGE PEAK-TO-VALLEY ROUGHNESS (nm)
I I0000
Fig. 2. E l l i p s o m e t r i c a l l y d e t e r m i n e d r e f r a c t i v e i n d e x as a f u n c t i o n o f r o u g h n e s s f o r s a m p l e s w i t h v a r i o u s o x i d e t h i c k n e s s e s ( t h e m a x i m a c o r r e s p o n d t o t h e laser w a v e l e n g t h o f 6 3 2 . 8 n m ) : z::,, 55 n m ; o, 120 n m .
d i a m o n d s a w m a y act as a diffraction grating. This grating will p r o d u c e irrep r o d u c i b l e readings. S a m p l e s e t c h e d in N a O H s o l u t i o n b e f o r e o x i d a t i o n either had n o null p o i n t for the polarized light or else gave values far f r o m
183 t h o s e o f t h e r a n d o m l y p o l i s h e d samples. T h e principal e x p e r i m e n t a l diff i c u l t y o f t h e t e c h n i q u e results f r o m t h e r e d u c e d r e f l e c t a n c e o f r o u g h s a m p l e s ; t h e angle o f t h e null p o i n t is less precise t h a n f o r s m o o t h samples. T h e t h e o r e t i c a l m o d e l s f o r e l l i p s o m e t r y on s a m p l e s w i t h a small degree o f surface r o u g h n e s s have t r e a t e d t h e s a m p l e s as layers w i t h a g r a d e d i n d e x o f r e f r a c t i o n . Clearly, w h e n t h e r o u g h n e s s a p p r o a c h e s t h e m a g n i t u d e o f t h e m e a s u r i n g w a v e l e n g t h , a r e s o n a n c e will have to be i n t r o d u c e d into a n y m o d e l designed t o fit o u r e x p e r i m e n t a l results. In s u m m a r y , films o n r o u g h surfaces m a y be c h a r a c t e r i z e d b y ellipsome t r y if p r o p e r care is t a k e n in i n t e r p r e t i n g t h e results. T h e m e a s u r e d film t h i c k n e s s deviates f r o m t h e value on a flat c o n t r o l s a m p l e w i t h increasing r o u g h n e s s , and m o r e s t r o n g l y so f o r thin films. T h e d e v i a t i o n s o f b o t h thickness and r e f r a c t i v e i n d e x are r e p r o d u c i b l e and m a y be used f o r setting u p c a l i b r a t i o n curves.
Acknowledgments T h e a u t h o r s t h a n k Messrs. Mac J u n e a u and David Mills at H e w l e t t P a c k a r d in L o v e l a n d , CO, f o r t h e use o f t h e i r e l l i p s o m e t e r and c o m p u t e r program.
References 1 C. A. Fenster and F. L. McCrackin, Errors arising from surface roughness in ellipsometric measurement of the refractive index of a surface, Surf. Sci., 16 (1969) 85.
2 R. S. Sirohi, Ellipsometry with rough surfaces, Opt. Commun., 1 (1970) 304. 3 R. S. Sirohi, Optical constants of a rough surface by ellipsometry, J. Phys. D, 3 (1970) 1407. 4 I. Ohlidal and F. Lukes, Ellipsometric parameters of rough surfaces and of a system substantiate-thin film with boundaries, Opt. Acta, 19 (1972) 817. 5 E. L. Church and J. M. Zavada, Effects of surface microroughness in ellipsometry, J. Opt. Soc. Am., 66 (1976) 1136A. 6 K. Brudzewski, Effect of surface roughness on change of the polarization state of light reflected from silicon and germanium, Appl. Opt., 15 (1976) 115. 7 T. V. Vorburger and K. C. Ludema, Ellipsometry of rough surfaces, Appl. Opt., 19 (1980) 561.
179
E LLI P S OMETRY OF RANDOMLY ROUGH OXIDIZED SILICON SURFACES T. DAVID BURLEIGH Department o f Materials Science and Engineering, Massachusetts Institute o f Technology, Cambridge, MA 02139 (U.S.A.)
SIGURD WAGNER Department o f Electrical Engineering and Computer Science, Princeton University, Princeton, NJ 08544 (U.S.A.)
THEODORE F. CISZEK Solar Energy Research Institute, Golden, CO 80401 (U.S.A.)
(Received February 8, 1984 ; accepted July 20, 1984)
Summary This experimental study shows that ellipsometry can provide thickness and refractive index measurements of oxide films grown on r a n d o m l y rough surfaces. The sample roughness ranged f r om 0.01 to 10 pm, with measuring light o f wavelength 0.6328 pm. The calculated values of oxide thickness and refractive index deviate from those on flat reference samples. The deviations are reproducible, so t hat calibration curves may be established for rough v e r s u s s m o o t h samples.
Ellipsometry is e m p l o y e d routinely in the determination of the thickness and refractive index of layers o f SiO2 on silicon. Such layers are usually grown on silicon substrates with highly polished surfaces. In photovoltaic t e ch n o lo g y the characterization of dielectric layers on rough substrates also may be o f interest. Rough substrates are p r o d u c e d by the sawing, grinding and etching o f silicon wafers. Many polycrystalline s e m i c o n d u c t o r films are rough, although on a smaller scale t han sawed wafers. In this paper we present t he results o f an eUipsometric study o f SiO2/Si structures whose oxide had been grown on roughened silicon substrates. We find th at the values for thickness and refractive index deviate from those det e rmin ed on flat reference samples. However, the deviations are smooth functions o f the roughness, so t ha t the thickness and refractive index still m a y be determined with the help of calibration data such as those established in our work. The use of ellipsometry to measure thin films assumes perfectly s mo o th surfaces. Several workers have addressed the problems e n c o u n t e r e d 0379-6787/84/$3.00
© Elsevier Sequoia/Printed in The Netherlands
180
with e l l i p s o m e t r y on rough surfaces [ 1 - 7 ] . V o r b u r g e r and L u d e m a [7] provided an excellent review o f the topic. H o w e v e r , no group has y e t s h o w n the practicality o f converting the a p p a r e n t values o b t a i n e d by e l l i p s o m e t r y on rough surfaces to the c o r r e c t values o f refractive i n d e x and film thickness o n c e t h e surface roughness is k n o w n . We set o u t to establish an empirical c o r r e l a t i o n b e t w e e n the a p p a r e n t thickness of oxides on rough silicon substrates and the o x i d e thickness on s m o o t h substrates. T h e roughness o f o u r substrate surfaces ranged f r o m 0.01 to 10 pm, i.e. f r o m well below t o well above the wavelength o f the ellipsometer H e - N e laser light. A t o t a l o f 68 samples were measured. F o u r d i f f e r e n t t y p e s o f silicon wafer sample were tested, with five d i f f e r e n t grit sizes of surface polish and f o u r d i f f e r e n t c o n d i t i o n s o f t h e r m a l o x i d a t i o n . The silicon samples included cast multigrain, v a p o r - d e p o s i t e d multigrain and single-crystal material (Table 1). T h e surface finish varied f r o m d i a m o n d wheel cut (or as received) to alumina p o w d e r polishes with grit sizes ranging f r o m 12 to 0.05 p m (Table 2). In addition to a native ( r o o m t e m p e r a t u r e ) oxide, t h e r m a l o x i d e layers were grown in a t u b e f u r n a c e 10 cm in d i a m e t e r at 950 °C with a flow of 300 cm 3 of commercial-grade o x y g e n (Table 3). Annealing at this t e m p e r a t u r e usually p r o d u c e s slip and dislocation propagation which we assumed did n o t affect the rate o f o x i d a t i o n . Thus the real o x i d e thicknesses on the r o u g h wafers are assumed to be identical with t h o s e o n the s m o o t h r e f e r e n c e samples. The surface roughness was d e t e r m i n e d b e f o r e o x i d a t i o n f r o m the profile m e a s u r e d with a T e n c o r Alpha step profiler. We define as roughness the average peak-to-valley d i f f e r e n c e along a trace length o f 50 /2m. This length was twice the beam spot d i a m e t e r (25 gm) o f the laser e m p l o y e d in the ellipsometer. Obvious pits or scratches were avoided in the roughness determination. The thickness and refractive i n d e x o f the o x i d e were d e t e r m i n e d f r o m data t a k e n with a G a e r t n e r L 117 manual ellipsometer. An H e - N e laser served as the light source (h = 632.8 nm). T h e angle o f incidence was fixed at 70 °. Calculations were carried o u t o n a digital c o m p u t e r . In Fig. 1 t h e a p p a r e n t o x i d e thickness o f the rough samples is p l o t t e d against t h e o x i d e thickness o f the flat r e f e r e n c e samples. (The refractive i n d e x was set at 1.46, the value for s t o i c h i o m e t r i c native SiO2 films.) The rough oxide thickness begins to deviate f r o m t h a t o f the r e f e r e n c e samples when the surface peak-to-valley roughness exceeds a b o u t o n e - t e n t h o f the wavelength of the ellipsometric light source. Thin oxides are m o s t a f f e c t e d by roughness; the r o u g h e r the surface is, the greater is the a p p a r e n t thickness. Even for r o u g h samples, t h o u g h , the a p p a r e n t thickness a p p r o a c h e s t h a t of the r e f e r e n c e samples w h e n it is o f the o r d e r o f 100 nm. If the refractive i n d e x is n o t set at 1.46 b u t r a t h e r is c o m p u t e d f r o m the ellipsometric parameters, its value is larger t h a n the actual value while t h e c o m p u t e d oxide thickness is c o r r e s p o n d i n g l y smaller t h a n the actual value. The c o m p u t e d i n d e x o f r e f r a c t i o n reaches a m a x i m u m w h e n the roughness c o r r e s p o n d s to the wavelength o f the laser. Figure 2 shows t h e refractive i n d e x as a f u n c t i o n
181 TABLE 1 Silicon samples employed in the experiments Morphology
Resistivity (~ cm)
Manufacturer
Single crystal (111 ) Cast multigrain (Silso) Cast multigrain Vapor-deposited multigrain
25 1.5 0.08 278
T. F. Ciszek Wacker Solarex Dow-Corning
TABLE 2 Polishing method and surface roughness Surface polish
Peak-to-valley distance (nm)
As received or diamond cut 12 pm alumina 5 pm alumina 1/~m alumina 0.05 pm alumina
500 - 10000 1000 500 50 10
TABLE 3 Oxidation conditions and oxide thickness Temperature (°C)
Duration (min)
Thickness (nm)
Room temperature 950 950 950
10 55 120
4 24 55 120
o f roughness f o r o x i d e thicknesses o f 55 and 120 n m . T h e d a t a f o r an o x i d e thickness o f 24 n m are n o t s h o w n for clarity. The a m p l i t u d e o f t h e maxi m u m is inversely p r o p o r t i o n a l to t h e o x i d e t h i c k n e s s ; the t h i c k e r t h e o x i d e is, the l o w e r is the m a x i m u m . The oxide 120 n m t h i c k had a p e a k 50% smaller and t h e o x i d e 24 n m t h i c k had a peak 20% higher t h a n t h e o x i d e 55 n m t h i c k had. The refractive index versus roughness has been e x a m i n e d and m o d e l e d extensively b y several w o r k e r s [ 1 - 7 ]. T o o u r k n o w l e d g e , this w o r k is t h e first c o n c e r n e d with a large range o f roughnesses. Figures 1 and 2 d e m o n s t r a t e t h a t e l l i p s o m e t r y can be used for a r e p r o d u c i b l e c h a r a c t e r i z a t i o n o f layers o n surfaces with a wide range o f roughnesses. H o w e v e r , some c a u t i o n m u s t be used in applying the m e t h o d . T h e sample m u s t be r a n d o m l y rough. F o r e x a m p l e , parallel scratches f r o m a
182 IOOO
/ /
v
/
._J
(3_ ~E
/
c,') I (.9 100 z 0
w I ~ w
o///
10
0 I..z w
/
a. 13_ '~
/
/
o/
1
i
I
10
)00
tOO0
OXIDE THICKNESS ON REFERENCE SAMPLE (niT,)
Fig. 1. M e a s u r e d a p p a r e n t o x i d e t h i c k n e s s o n r o u g h s a m p l e s v s . t h i c k n e s s o n flat r e f e r e n c e s a m p l e s as a f u n c t i o n o f r o u g h n e s s ( o x i d e r e f r a c t i v e i n d e x , 1 . 4 6 ) : / ~ , 1 0 0 0 n m ; ,~, 50 n m ; c:, 10 n m .
I
/\
5 ~: LI_ 2.C W tr b._ 0 X i,i a Z
'! tz
1,5
1.0
t LASER WAVELENGTF I
6 2 3 . 8 nm
I I i
i
I
I
10 IO0 |000 AVERAGE PEAK-TO-VALLEY ROUGHNESS (nm)
I I0000
Fig. 2. E l l i p s o m e t r i c a l l y d e t e r m i n e d r e f r a c t i v e i n d e x as a f u n c t i o n o f r o u g h n e s s f o r s a m p l e s w i t h v a r i o u s o x i d e t h i c k n e s s e s ( t h e m a x i m a c o r r e s p o n d t o t h e laser w a v e l e n g t h o f 6 3 2 . 8 n m ) : z::,, 55 n m ; o, 120 n m .
d i a m o n d s a w m a y act as a diffraction grating. This grating will p r o d u c e irrep r o d u c i b l e readings. S a m p l e s e t c h e d in N a O H s o l u t i o n b e f o r e o x i d a t i o n either had n o null p o i n t for the polarized light or else gave values far f r o m
183 t h o s e o f t h e r a n d o m l y p o l i s h e d samples. T h e principal e x p e r i m e n t a l diff i c u l t y o f t h e t e c h n i q u e results f r o m t h e r e d u c e d r e f l e c t a n c e o f r o u g h s a m p l e s ; t h e angle o f t h e null p o i n t is less precise t h a n f o r s m o o t h samples. T h e t h e o r e t i c a l m o d e l s f o r e l l i p s o m e t r y on s a m p l e s w i t h a small degree o f surface r o u g h n e s s have t r e a t e d t h e s a m p l e s as layers w i t h a g r a d e d i n d e x o f r e f r a c t i o n . Clearly, w h e n t h e r o u g h n e s s a p p r o a c h e s t h e m a g n i t u d e o f t h e m e a s u r i n g w a v e l e n g t h , a r e s o n a n c e will have to be i n t r o d u c e d into a n y m o d e l designed t o fit o u r e x p e r i m e n t a l results. In s u m m a r y , films o n r o u g h surfaces m a y be c h a r a c t e r i z e d b y ellipsome t r y if p r o p e r care is t a k e n in i n t e r p r e t i n g t h e results. T h e m e a s u r e d film t h i c k n e s s deviates f r o m t h e value on a flat c o n t r o l s a m p l e w i t h increasing r o u g h n e s s , and m o r e s t r o n g l y so f o r thin films. T h e d e v i a t i o n s o f b o t h thickness and r e f r a c t i v e i n d e x are r e p r o d u c i b l e and m a y be used f o r setting u p c a l i b r a t i o n curves.
Acknowledgments T h e a u t h o r s t h a n k Messrs. Mac J u n e a u and David Mills at H e w l e t t P a c k a r d in L o v e l a n d , CO, f o r t h e use o f t h e i r e l l i p s o m e t e r and c o m p u t e r program.
References 1 C. A. Fenster and F. L. McCrackin, Errors arising from surface roughness in ellipsometric measurement of the refractive index of a surface, Surf. Sci., 16 (1969) 85.
2 R. S. Sirohi, Ellipsometry with rough surfaces, Opt. Commun., 1 (1970) 304. 3 R. S. Sirohi, Optical constants of a rough surface by ellipsometry, J. Phys. D, 3 (1970) 1407. 4 I. Ohlidal and F. Lukes, Ellipsometric parameters of rough surfaces and of a system substantiate-thin film with boundaries, Opt. Acta, 19 (1972) 817. 5 E. L. Church and J. M. Zavada, Effects of surface microroughness in ellipsometry, J. Opt. Soc. Am., 66 (1976) 1136A. 6 K. Brudzewski, Effect of surface roughness on change of the polarization state of light reflected from silicon and germanium, Appl. Opt., 15 (1976) 115. 7 T. V. Vorburger and K. C. Ludema, Ellipsometry of rough surfaces, Appl. Opt., 19 (1980) 561.