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Grain-boundary scattering and surface plasmon attenuation in noble metal films
~
Solid State Communications, Printed in Great Britain.
Vol.49,No.4,
pp.343-345,
1984.
0038-]098/84 $3.00 + .00 Pergamon Press Ltd.
GRAIN-BOUNDARY SCATTERING AND SURFACE PLASMON ATTENUATION IN NOBLE METAL FILMS Sambles, J.R. Department of Physics, University of Exeter, Exeter, Devon
(Accepted 4 November 1983 by C.W. McCombie)
It is shown how grain-boundary and not surface scattering may explain the large range in values found for the attenuation coefficient of surface plasmons propogating on thin noble metal films.
Recently there has been substantial interest in the attenuation coefficients for surface plasmons propagating on noble metal films. These coefficients show a large degree of scatter. Although some scatter might be expected as a consequence of different surface roughnesses due to chemical attack of silver and particularly copper, no such scatter would be anticipated for gold. Schlesinger and Sievers 1,2 have drawn attention to this problem and, although they attribute the high room temperature resistivity of their silver and gold films to grain-boundary scattering, they surprisingly chose to use a surface scattering model to interpret their attenuation coefficient results. Furthermore although their data points lie within the limits of smooth and rough surface scattering this is not so for data from other studies.
6wZh
where A is the F e r m i surface area. Using known values we then estimate A~ and taking Wg as d we may then calculate the relative magnitudes of the two terms for a 40 run film at room temperature. This we have done as tabulated in table I. Clearly even with p R 0 (totally diffuse scattering) and Rg - 0.1 the grain-boundary contribution may be quite substantial. However it is known that p for gold is very close %o 16 and recent studies on silver produced in ultra-hlgh vacuum 7 suggest the same (as do results for silver in alr8). Furthermore graln-boundary reflection coefficients for these metals 9,10, lle in the range 0.25 to 0.3. Consequently, one is forced to conclude that the dominant extra scattering
Thin metal films produced by evaporation on to amorphous or even single crystal substrates held at room temperature are known to be highly polycrystalline unless, as in the case of silver on mica, epitaxial growth occurs. Such polycrystalline films have grain sizes of the order of the film thickness up to a maximum of perhaps a few hundreds of nanometers. In circumstances where the grains are smaller or of the order of the mean free path one may write the d.c. resistivity as: 3,4
Table 1
Rg
8 (A®ffiO.5Wg)
0.0
where p~ is the bulk resistivity, A~ is the bulk mean free path, Wg is the mean grain size, Rg is the grain-boundary reflection coefficient, p is the Fuchs 5 surface specularity pa/ameter, d is the film thickness and Ag is the effective mean f r e e p a t h given by
One may estimate the relative significance of the grain-boundary a n d surface t e r m s in the following manner. Being cubic metals the product ~ is expressed as
8 (A --Wg)
0
O. 250
0.250
0.250
0.05
0.053
O. 240
0.230
0.210
O.lO
0.Iii
0.229
0. 208
0.167
O.15
0,176
O.217
O.184
0 .ll8
O. 20
0. 250
O. 203
O.156
0.063
0.25
0.333
O.188
0.125
0.00
0.30
0.429
O.170
O.O89
Table
of ~ = ~
and B = g
for
343
8 (A®f2Wg)
various
R
g
and ~
values.
1 - ~
1-Rg
344
GRAIN-BOUNDARY
SCATTERING
mechanism in thin films used in either the Kretschm~ann or O t t o 12 geometry will be g r a i n - b o u n d a r y a n d not s u r f a c e s c a t t e r i n g . If we examine Schleslnger a n d S i e v e r s 1,2 w o r k w e find room temperature restivity enhancements of i.i a n d 1 . 3 6 for t h i n films o f s i l v e r a n d g o l d respectively. A s s u m i n g R g = 0.25 for s i l v e r and 0.3 for gold and nearly specular surface s c a t t e r i n g leads to W g v a l u e s o f 2 6 4 a n d 67 n m respectively these both appear acceptable values although the silver is s o m e w h a t large implying perhaps a smaller grain-boundary scattering coefficient. Even though these are rather crude estimates because of the simplification of the model it is c l e a r that s u r f a c e s c a t t e r i n g n e e d not b e at a l l s i g n i f i c a n t in t h e s e samples. In v i e w o f t h i s w e are forced to examine the influence of grain-boundary scattering on the surface plasmon attenuation c o e f f i c i e n t ~s. Following Schlesinger and S i e v e r s '1,2 we w r i t e for t h e s u r f a c e p l a s m o n o n a b a r e m e t a l surface
~aF ~S = WpaC where F is the total electron scattering f r e q u e n c y a n d ~ p is t h e b u l k p l a s m a frequency. These authors modify F from the bulk scattering r a t e b y a d d i n g in a s u r f a c e term, w e c h o o s e t h e other viewpoint and add in only the g r a i n - b o u n d a r y term. Now
r=rg~r®
1+
~
a n d t h u s in t h i s m o d e l a s is p r o p o r t i o n a l to p. More generally one should incorporate both grain-boundary and surface scattering so that F = F g = ( l - p ) F s and
~z
3 [_~_]~] + (l-p)rs}
P
w h e r e l 3 Fs =
3 V_~, 8
VF being
t h e fermi v e l o c i t y
c
Now while the surface term can never contribute more than Fs to F the grain boundary term may t a k e o n a n y v a l u e d e p e n d i n g o n Wg. This may t h e n a l l o w a n e x p l a n a t i o n o f the large s p r e a d in a s v a l u e s reported, a t t r i b u t a b l e n o w to a large spread in grain-slze related to sample production. H o w e v e r it is s t a r t l i n g t o find a s enhanced by a factor of i0 14,15 over the e x p e c t e d v a l u e as t h i s s u g g e s t s v e r y s m a l l g r a i n s and a similar enhancement of the room temperature resistivity. Schoenwald Bursteln and Elson 14 a n d M c M u l l e n 15 p e r f o r m e d their measurements on films evaporated on to glass or quartz eubstratee. In b o t h c a s e s t h e a m o r p h o u s n a t u r e of the substrate will encourage small grain-growth resulting in a grain-boundary enhanced resistivity. M c M u l l e n 15 coHm%ents t h a t the grains in b o t h c o p p e r and gold films o n q u a r t z a r e l e s s t h a n l ~ m in d i a m e t e r . A s shown b y S c h l e s l n g e r a n d S i e v e r s 1,2 e v e n t o t a l l y r o u g h surface scattering cannot explain the large enhancements recorded in the above studies.
AND
SURFACE Further greater
PLASMON ATTENUATION w i t h the than
Vol.
49, No.
4
observational wavelength much the surface roughness, surface-roughness-coupled radiation loss is a l m o s t c e r t a i n l y not the cause. O n e is forced to conclude that grain-boundary scattering is, in these cases, d o m i n a t i n g the a t t e n u a t i o n o f the s u r f a c e plasmon. Zhishln, Moekaleva, Shomina and Y a k o v l e v ' s 16 results are also of some i n t e r e s t s i n c e t h e y s t u d i e d a v a r i e t y of s a m p l e s of the same material. T h e i r s i l v e r and g o l d s a m p l e s w e r e p o l y c r y s t a l l i n e films s p u t t e r e d 16 o r e v a p o r a t e d 17 on t o g l a s s plates. Here, a l t h o u g h the attenuation is less severe than previous d a t a l@ it is s t i l l t o o g r e a t t o b e e x p l a i n e d b y s u r f a c e s c a t t e r i n g and o n c e a g a i n g r a i n - b o u n d a r y s c a t t e r i n g n e e d b e invoked. T h e i r c o p p e r data, for films on glass, also showed enhanced attenuation, but somewhat surprisingly a highly polished mirror sample gave even greater a t t e n u a t i o n - t h i s is not l i k e l y t o b e due to grain-boundary scattering. H o w e v e r since t h e s u r f a c e r o u g h n e s s o f the p o l i s h e d s a m p l e is not known, or t h e d e g r e e o f s u r f a c e d a m a g e no u s e f u l c o n c l u s i o n s m a y b e drawn. Clearly from the above examples it is of some interest to be able to distinguish between attenuation due to resistivity scattering and t h a t due t o r a d i a t i o n loss c o u p l i n g v i a s u r f a c e roughness. O n e w a y of p a r t i a l l y distinguishing these contributions is s i m p l y b y m e a s u r i n g the d.c. r e s i s t i v i t y of t h e films. S c h l e s i n g e r and S i e v e r s 1,2 h a v e d o n e t h i s a n d c l e a r l y for g o l d samples the a s enhancement b y ~i. 2 is of the s a m e o r d e r as t h e g r a i n - b o u n d a r y enhancement of the resistivity. For silver films the a s enhancement is ~] .5 s i g n i f i c a n t l y g r e a t e r than the i. 1 found for the resistivity, this is s u g g e s t i v e t h a t in t h e s e p a r t i c u l a r s i l v e r films a r a d i a t i o n loss m e c h a n i s m is a l s o significant. In v i e w o f t h e a b o v e e s t i m a t i o n o f g r a i n size of 2 6 4 nm it is q u i t e conceivable that surface r o u g h n e s s ( m a c r o s c o p i c ) i n d u c e d r a d i a t i o n loss ls s i g n i f i c a n t in t h e c a s e of s i l v e r films. This is a l m o s t c e r t a i n l y not the c a s e for the m u c h s ~ a l l e r g r a i n e d g o l d films. It is clear from the above that in measurements of surface plasmon attenuation coefficients on thin metal films detailed information should be obtained on the d.c. r e s i s t i v i t y and in p a r t i c u l a r on t h e g r a i n size o f t h e samples. T h e g r a i n s i z e is i m p o r t a n t in t w o respects. F i r s t l y it a l l o w s an e s t i m a t i o n of the resistivity scattering and consequential d a m p i n g of t h e p l a s m o n and s e c o n d l y it a l l o w s an e s t ~ ,mation of the roughness-coupled radiation loss. If the r o u g h n e s s - c o u p l i n g loss is small then the d.c. resistivity should correlate c l o s e l y w i t h t h e p l a s m o n damping. In v i e w of the above comments it would clearly be of interest to repeat McMullens w o r k 15 using epitaxially grown silver films on mica substrates. F o r s u c h films the g r a i n s i z e m a y be several microns which should therefore largely s u p r e s s the g r a l n - b o u n d a r y c o n t r i b u t i o n . C l e a r l y m u c h o f the e a r l y w o r k o n s u r f a c e p l a s m o n a t t e n u a t i o n is o f little v a l u e since the d a t a p e r t a i n s £o p a r t i c u l a r s a m p l e s p r o d u c e d in particular environments with unknown g r a l n - b o u n d a r y s c a t t e r i n g a n d s u r f a c e roughness. It is hoped that further studies will be accompanied by good characterization of the m o r p h o l o g i e s o f the m e t a l films.
Vol. 49, No. 4
GRAIN-BOUNDARY SCATTERING AND SURFACE PLASMON ATTENUATION
References
].
Schlesinger, Z. and Sievers, A.J. (1982). Sol.St.Comm. 4__33,671.
11.
Kretschmann, 24!, 313.
2.
Scbleslnger, Z. and Sievers, A.J. (1982). Phys.Rev. B, 26, 6444.
12.
Otto, A. (1968). Z.Phys. 21__~6, 398.
13.
3.
Mayadas, A.F. and Shatzkes, M. (1970). Phys.Rev.B, l, 1382.
McKay, J.A. and Rayne, J.A. (1976). Phys. Rev. B. 16, 5377.
14.
4.
Mayadas, A.F., Sbatzkes, M. and Janak, J.F. (1969). Appl.Phys. Lett. 14, 345. Fuchs, K., (1938) Proc.camb.Phll. Soc. 3_44, i00.
Schoenwald, J., Burstein, E. and E/son, J.M. (1973). S 0 1 . S t . C o ~ . 1/2, 185.
15.
McMullen, J.D. ( 1 9 7 5 ) S o l . S t . C o m m . I_~7, 331.
16.
Zhizhin, G.N., Moskaleva, M.A., Schomina, E.V. and Yakov/ev, V.A, (1980). Fiz.Metal.Metallove~ 50, 7 3 4 ( E n g . t r a n s : Phys. Met. MeEall.
5. 6.
Sambles, J.R., Elsom, K.C. and Jarvis, D.J. (1982). Phil.Trans. Roy. Soc. Lond.A 304, 365.
7.
Schumacher, D. and Stark D. (1982). Surf. Sci. 123, 384.
8.
Coulson, I., Jarvis, D.J., and Sambles, J.R., (1983) to be published.
9.
C o m e l y , R.H. and Ali, T.A. (1978). J.Appl,Phys. 49, 4094.
I0.
Tochitskii,E.I. and Belyavskii,N.M. (1980). Phys. St. Sol.(a) 61_m k21.
E. (1971). Z.Phys.
5-0, S l ( 1 9 e o ) ) . 17.
Zhlzhln, G.N. Moskaleva, M.A., Shomina, E.V. and Yakovlev, V.A. (1982) Ch.3 in 'Surface Polarltons' Ed. Agranovich, V.M. and Mills D.L. (North Holland ).
345
Solid State Communications, Printed in Great Britain.
Vol.49,No.4,
pp.343-345,
1984.
0038-]098/84 $3.00 + .00 Pergamon Press Ltd.
GRAIN-BOUNDARY SCATTERING AND SURFACE PLASMON ATTENUATION IN NOBLE METAL FILMS Sambles, J.R. Department of Physics, University of Exeter, Exeter, Devon
(Accepted 4 November 1983 by C.W. McCombie)
It is shown how grain-boundary and not surface scattering may explain the large range in values found for the attenuation coefficient of surface plasmons propogating on thin noble metal films.
Recently there has been substantial interest in the attenuation coefficients for surface plasmons propagating on noble metal films. These coefficients show a large degree of scatter. Although some scatter might be expected as a consequence of different surface roughnesses due to chemical attack of silver and particularly copper, no such scatter would be anticipated for gold. Schlesinger and Sievers 1,2 have drawn attention to this problem and, although they attribute the high room temperature resistivity of their silver and gold films to grain-boundary scattering, they surprisingly chose to use a surface scattering model to interpret their attenuation coefficient results. Furthermore although their data points lie within the limits of smooth and rough surface scattering this is not so for data from other studies.
6wZh
where A is the F e r m i surface area. Using known values we then estimate A~ and taking Wg as d we may then calculate the relative magnitudes of the two terms for a 40 run film at room temperature. This we have done as tabulated in table I. Clearly even with p R 0 (totally diffuse scattering) and Rg - 0.1 the grain-boundary contribution may be quite substantial. However it is known that p for gold is very close %o 16 and recent studies on silver produced in ultra-hlgh vacuum 7 suggest the same (as do results for silver in alr8). Furthermore graln-boundary reflection coefficients for these metals 9,10, lle in the range 0.25 to 0.3. Consequently, one is forced to conclude that the dominant extra scattering
Thin metal films produced by evaporation on to amorphous or even single crystal substrates held at room temperature are known to be highly polycrystalline unless, as in the case of silver on mica, epitaxial growth occurs. Such polycrystalline films have grain sizes of the order of the film thickness up to a maximum of perhaps a few hundreds of nanometers. In circumstances where the grains are smaller or of the order of the mean free path one may write the d.c. resistivity as: 3,4
Table 1
Rg
8 (A®ffiO.5Wg)
0.0
where p~ is the bulk resistivity, A~ is the bulk mean free path, Wg is the mean grain size, Rg is the grain-boundary reflection coefficient, p is the Fuchs 5 surface specularity pa/ameter, d is the film thickness and Ag is the effective mean f r e e p a t h given by
One may estimate the relative significance of the grain-boundary a n d surface t e r m s in the following manner. Being cubic metals the product ~ is expressed as
8 (A --Wg)
0
O. 250
0.250
0.250
0.05
0.053
O. 240
0.230
0.210
O.lO
0.Iii
0.229
0. 208
0.167
O.15
0,176
O.217
O.184
0 .ll8
O. 20
0. 250
O. 203
O.156
0.063
0.25
0.333
O.188
0.125
0.00
0.30
0.429
O.170
O.O89
Table
of ~ = ~
and B = g
for
343
8 (A®f2Wg)
various
R
g
and ~
values.
1 - ~
1-Rg
344
GRAIN-BOUNDARY
SCATTERING
mechanism in thin films used in either the Kretschm~ann or O t t o 12 geometry will be g r a i n - b o u n d a r y a n d not s u r f a c e s c a t t e r i n g . If we examine Schleslnger a n d S i e v e r s 1,2 w o r k w e find room temperature restivity enhancements of i.i a n d 1 . 3 6 for t h i n films o f s i l v e r a n d g o l d respectively. A s s u m i n g R g = 0.25 for s i l v e r and 0.3 for gold and nearly specular surface s c a t t e r i n g leads to W g v a l u e s o f 2 6 4 a n d 67 n m respectively these both appear acceptable values although the silver is s o m e w h a t large implying perhaps a smaller grain-boundary scattering coefficient. Even though these are rather crude estimates because of the simplification of the model it is c l e a r that s u r f a c e s c a t t e r i n g n e e d not b e at a l l s i g n i f i c a n t in t h e s e samples. In v i e w o f t h i s w e are forced to examine the influence of grain-boundary scattering on the surface plasmon attenuation c o e f f i c i e n t ~s. Following Schlesinger and S i e v e r s '1,2 we w r i t e for t h e s u r f a c e p l a s m o n o n a b a r e m e t a l surface
~aF ~S = WpaC where F is the total electron scattering f r e q u e n c y a n d ~ p is t h e b u l k p l a s m a frequency. These authors modify F from the bulk scattering r a t e b y a d d i n g in a s u r f a c e term, w e c h o o s e t h e other viewpoint and add in only the g r a i n - b o u n d a r y term. Now
r=rg~r®
1+
~
a n d t h u s in t h i s m o d e l a s is p r o p o r t i o n a l to p. More generally one should incorporate both grain-boundary and surface scattering so that F = F g = ( l - p ) F s and
~z
3 [_~_]~] + (l-p)rs}
P
w h e r e l 3 Fs =
3 V_~, 8
VF being
t h e fermi v e l o c i t y
c
Now while the surface term can never contribute more than Fs to F the grain boundary term may t a k e o n a n y v a l u e d e p e n d i n g o n Wg. This may t h e n a l l o w a n e x p l a n a t i o n o f the large s p r e a d in a s v a l u e s reported, a t t r i b u t a b l e n o w to a large spread in grain-slze related to sample production. H o w e v e r it is s t a r t l i n g t o find a s enhanced by a factor of i0 14,15 over the e x p e c t e d v a l u e as t h i s s u g g e s t s v e r y s m a l l g r a i n s and a similar enhancement of the room temperature resistivity. Schoenwald Bursteln and Elson 14 a n d M c M u l l e n 15 p e r f o r m e d their measurements on films evaporated on to glass or quartz eubstratee. In b o t h c a s e s t h e a m o r p h o u s n a t u r e of the substrate will encourage small grain-growth resulting in a grain-boundary enhanced resistivity. M c M u l l e n 15 coHm%ents t h a t the grains in b o t h c o p p e r and gold films o n q u a r t z a r e l e s s t h a n l ~ m in d i a m e t e r . A s shown b y S c h l e s l n g e r a n d S i e v e r s 1,2 e v e n t o t a l l y r o u g h surface scattering cannot explain the large enhancements recorded in the above studies.
AND
SURFACE Further greater
PLASMON ATTENUATION w i t h the than
Vol.
49, No.
4
observational wavelength much the surface roughness, surface-roughness-coupled radiation loss is a l m o s t c e r t a i n l y not the cause. O n e is forced to conclude that grain-boundary scattering is, in these cases, d o m i n a t i n g the a t t e n u a t i o n o f the s u r f a c e plasmon. Zhishln, Moekaleva, Shomina and Y a k o v l e v ' s 16 results are also of some i n t e r e s t s i n c e t h e y s t u d i e d a v a r i e t y of s a m p l e s of the same material. T h e i r s i l v e r and g o l d s a m p l e s w e r e p o l y c r y s t a l l i n e films s p u t t e r e d 16 o r e v a p o r a t e d 17 on t o g l a s s plates. Here, a l t h o u g h the attenuation is less severe than previous d a t a l@ it is s t i l l t o o g r e a t t o b e e x p l a i n e d b y s u r f a c e s c a t t e r i n g and o n c e a g a i n g r a i n - b o u n d a r y s c a t t e r i n g n e e d b e invoked. T h e i r c o p p e r data, for films on glass, also showed enhanced attenuation, but somewhat surprisingly a highly polished mirror sample gave even greater a t t e n u a t i o n - t h i s is not l i k e l y t o b e due to grain-boundary scattering. H o w e v e r since t h e s u r f a c e r o u g h n e s s o f the p o l i s h e d s a m p l e is not known, or t h e d e g r e e o f s u r f a c e d a m a g e no u s e f u l c o n c l u s i o n s m a y b e drawn. Clearly from the above examples it is of some interest to be able to distinguish between attenuation due to resistivity scattering and t h a t due t o r a d i a t i o n loss c o u p l i n g v i a s u r f a c e roughness. O n e w a y of p a r t i a l l y distinguishing these contributions is s i m p l y b y m e a s u r i n g the d.c. r e s i s t i v i t y of t h e films. S c h l e s i n g e r and S i e v e r s 1,2 h a v e d o n e t h i s a n d c l e a r l y for g o l d samples the a s enhancement b y ~i. 2 is of the s a m e o r d e r as t h e g r a i n - b o u n d a r y enhancement of the resistivity. For silver films the a s enhancement is ~] .5 s i g n i f i c a n t l y g r e a t e r than the i. 1 found for the resistivity, this is s u g g e s t i v e t h a t in t h e s e p a r t i c u l a r s i l v e r films a r a d i a t i o n loss m e c h a n i s m is a l s o significant. In v i e w o f t h e a b o v e e s t i m a t i o n o f g r a i n size of 2 6 4 nm it is q u i t e conceivable that surface r o u g h n e s s ( m a c r o s c o p i c ) i n d u c e d r a d i a t i o n loss ls s i g n i f i c a n t in t h e c a s e of s i l v e r films. This is a l m o s t c e r t a i n l y not the c a s e for the m u c h s ~ a l l e r g r a i n e d g o l d films. It is clear from the above that in measurements of surface plasmon attenuation coefficients on thin metal films detailed information should be obtained on the d.c. r e s i s t i v i t y and in p a r t i c u l a r on t h e g r a i n size o f t h e samples. T h e g r a i n s i z e is i m p o r t a n t in t w o respects. F i r s t l y it a l l o w s an e s t i m a t i o n of the resistivity scattering and consequential d a m p i n g of t h e p l a s m o n and s e c o n d l y it a l l o w s an e s t ~ ,mation of the roughness-coupled radiation loss. If the r o u g h n e s s - c o u p l i n g loss is small then the d.c. resistivity should correlate c l o s e l y w i t h t h e p l a s m o n damping. In v i e w of the above comments it would clearly be of interest to repeat McMullens w o r k 15 using epitaxially grown silver films on mica substrates. F o r s u c h films the g r a i n s i z e m a y be several microns which should therefore largely s u p r e s s the g r a l n - b o u n d a r y c o n t r i b u t i o n . C l e a r l y m u c h o f the e a r l y w o r k o n s u r f a c e p l a s m o n a t t e n u a t i o n is o f little v a l u e since the d a t a p e r t a i n s £o p a r t i c u l a r s a m p l e s p r o d u c e d in particular environments with unknown g r a l n - b o u n d a r y s c a t t e r i n g a n d s u r f a c e roughness. It is hoped that further studies will be accompanied by good characterization of the m o r p h o l o g i e s o f the m e t a l films.
Vol. 49, No. 4
GRAIN-BOUNDARY SCATTERING AND SURFACE PLASMON ATTENUATION
References
].
Schlesinger, Z. and Sievers, A.J. (1982). Sol.St.Comm. 4__33,671.
11.
Kretschmann, 24!, 313.
2.
Scbleslnger, Z. and Sievers, A.J. (1982). Phys.Rev. B, 26, 6444.
12.
Otto, A. (1968). Z.Phys. 21__~6, 398.
13.
3.
Mayadas, A.F. and Shatzkes, M. (1970). Phys.Rev.B, l, 1382.
McKay, J.A. and Rayne, J.A. (1976). Phys. Rev. B. 16, 5377.
14.
4.
Mayadas, A.F., Sbatzkes, M. and Janak, J.F. (1969). Appl.Phys. Lett. 14, 345. Fuchs, K., (1938) Proc.camb.Phll. Soc. 3_44, i00.
Schoenwald, J., Burstein, E. and E/son, J.M. (1973). S 0 1 . S t . C o ~ . 1/2, 185.
15.
McMullen, J.D. ( 1 9 7 5 ) S o l . S t . C o m m . I_~7, 331.
16.
Zhizhin, G.N., Moskaleva, M.A., Schomina, E.V. and Yakov/ev, V.A, (1980). Fiz.Metal.Metallove~ 50, 7 3 4 ( E n g . t r a n s : Phys. Met. MeEall.
5. 6.
Sambles, J.R., Elsom, K.C. and Jarvis, D.J. (1982). Phil.Trans. Roy. Soc. Lond.A 304, 365.
7.
Schumacher, D. and Stark D. (1982). Surf. Sci. 123, 384.
8.
Coulson, I., Jarvis, D.J., and Sambles, J.R., (1983) to be published.
9.
C o m e l y , R.H. and Ali, T.A. (1978). J.Appl,Phys. 49, 4094.
I0.
Tochitskii,E.I. and Belyavskii,N.M. (1980). Phys. St. Sol.(a) 61_m k21.
E. (1971). Z.Phys.
5-0, S l ( 1 9 e o ) ) . 17.
Zhlzhln, G.N. Moskaleva, M.A., Shomina, E.V. and Yakovlev, V.A. (1982) Ch.3 in 'Surface Polarltons' Ed. Agranovich, V.M. and Mills D.L. (North Holland ).
345