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Insulator-conductor transition in very thin metal films
Thin Sohd Fdms, 126 (1985) 155--159
155
GENERAL FILM BEHAVIOUR
INSULATOR-CONDUCTOR FILMS*
T R A N S I T I O N IN V E R Y T H I N M E T A L
E. HARNIK, S. KOVNOVICH AND T. CHERNOBELSKAYA Racah Institute of Physics, Hebrew Universtty, Jerusalem 91904 (Israel) (Received August 8, 1984; accepted September 14, 1984)
The insulator-conductor transition in very thin metal films is investigated by acoustoelectric measurements of the relation between the charge carrier density and the sheet resistance at various substrate temperatures from room temperature down to about 90 K. Near room temperature a rather sharp mobility edge is observed but at lower temperatures, in quench-condensed films, percolation effects mask the manifestation of the Anderson transition. This difference in behaviour is attributed to the change in surface mobility of the metal islands and its effect on the microscopic structure of the film.
1. INTRODUCTION The insulator-conductor transition in very thin metal films has been treated as a percolation problem with considerable success 1. The question whether the Anderson model can be applied to this transition has been subject to some controversy because of the inhomogeneous semicontinuous structure of these films in the transition region. However, Licciardello 2 has argued that, even in such systems, theory predicts a sharp mobility edge and an abrupt change in the nature of electrical conductivity near a critical sheet resistance R D of about 30000f~/E]. Indeed, increasing experimental evidence indicating a change in the electrical properties of such films near this value of R D has been accumulating 3'4. These experiments, however, do not provide an accurate determination of the critical resistivity or detailed information on the sharpness of the Anderson transition. Recently, enquiring into the latter question in disordered two-dimensional systems, Kaveh s showed that the various stages of the Anderson transition at a finite temperature can be traced on a curve of conductivity (or resistivity) v e r s u s carrier density n. In this paper we report and discuss acoustoelectric measurements ofn as a function o f R D in very thin metal films at various substrate temperatures from room temperature down to about 90 K. Near room temperature the results indicate the * Paper presented at the Sixth International Conference on Thin Films, Stockholm, Sweden, August 13-17, 1984. 0040-6090/85/$3.30
© ElsevierSequoia/Printed in The Netherlands
[56
E. H A R N I K , S. K O V N O V I C H A N D T. C H E R N O B E L S K A Y A
presence of a rather s h a r p m o b i l i t y edge but at lower temperatures, in quenchc o n d e n s e d films, a p e r c o l a t i o n t r a n s i t i o n is observed, which m a s k s the manifestation of the A n d e r s o n transition. 2. EXPERIMENTAl+ METHOI)
The e x p e r i m e n t a l m e t h o d is based on the acoustoelectric d e t e r m i n a t i o n of carrier m o b i l i t y in thin c o n d u c t i n g layers +' ~. A r e c t a n g u l a r strip of the metal is v a c u u m d e p o s i t e d o n t o a piezoelectric ( L i N b O 3 ) s u b s t r a t e in the space between two mterdigital transducers, and the a t t e n u a t i o n :~{dB) of surface acoustic waves (SAWbl and the a c o u s t o e l e c t r i c voltage |/]c induced by S A W s a l o n g the tilm are m e a s u r e d at various values of R F . The value of n is derived from the a c o u s t o e l e c t r i c expression ~' for the carrier m o b i l i t y I~ and the general expression R~ 1 = enl~. W e o b t a i n for the charge carrier density per unit a r e a the expression P n
l-exp(-0+23~)
-
(1)
ecw
Li,~.
where P is the S A W p o w e r incident on the film, r is the S A W phase velocity a n d w is the lateral d i m e n s i o n of the film. In fact, the e x p e r i m e n t is s o m e w h a t m o r e complicated. Because of a s e c o n d a r y a c o u s t o e l e c t r i c current of t h e r m a l origin s. two s e p a r a t e m e a s u r e m e n t s of 1]~, one with the S A W p r o p a g a t i o n direction reversed, have to be carried out a n d half the a l g e b r a i c difference of the two results has to be determined, t a k i n g into a c c o u n t the ratio of the P values associated with the two t r a n s d u c e r s ; . To reduce the e r r o r involved, a yx (instead of the m o r e c o m m o n y z ) k i N b O 3 plate was used as the substrate, since the ratio of s e c o n d a r y to p r i m a r y a c o u s t o e l e c t n c current is c o n s i d e r a b l y lower. The e x p e r i m e n t was carried out in a c o m m e r c i a l Bendix c o a t e r unit. The film was v a p o u r d e p o s i t e d in a large n u m b e r of successive steps in a v a c u u m of a b o u t 5 x 10 ~' Torr. N e a r r o o m t e m p e r a t u r e , because of structural changes, with the i n t e r r u p n o n of each d e p o s i t i o n step the film p a r a m e t e r s m e a s u r e d (R b, :~ a n d I/]e) u n d e r g o large relaxations at a g r a d u a l l y decreasing rate. The m e a s u r e m e n t s were t a k e n in s i t u when a q u a s i - s t a t i o n a r y state was reached in which the changes were slow e n o u g h to permit the d e t e r m i n a t i o n of the p a r a m e t e r s , one after the other, w i t h o u t the i n t r o d u c t i o n of significant errors. This p r o c e d u r e does not in a n y way affect the results because the p o i n t s c o r r e s p o n d i n g to film p a r a m e t e r s m e a s u r e d at later stages of the r e l a x a t i o n process are found to lie on the same n r e r s u s R a curve. The m a i n u n c e r t a i n t y in n originates from a p r o b a b l e systematic e r r o r in the values of the incident S A W p o w e r P, which m a k e s our n values t o o high. T h e relative a c c u r a c y of n and, therefore, the conclusions reached below are not affected by this s h o r t c o m i n g of the experiment. 3.
RESULTS A N D DISCUSSION
A typical n l,ersus RFn curve o b t a i n e d at r o o m t e m p e r a t u r e with a gold film is shown m Fig. l(a). In its general form, it r e p r o d u c e s the m a i n features of K a v e h ' s theoretical curve 5. The p r o m i n e n t feature of this curve is the h o r i z o n t a l p l a t e a u of
INSULATOR--CONDUCTOR TRANSITION IN VERY THIN METAL FILMS
157
Cu
Au c I
I
I
Ro
(n/o)
Ro
(a)
10 (104 ~ / o )
I
15
Ag 2tO
(b)
Fig. 1. (a) Typical room temperature sheet resistance dependence of the carrier density in a very thin gold film: (b) room temperature carrier density versus sheet resistance curves showmg the Anderson transmon m a number of very thm metal films
nearly constant n extending over almost two orders of magnitude of R[]. When it is borne in mind that in a two-dimensional system the Fermi energy is proportional to n and that/~ ~ ( R [ ] n ) - 1, replotting these data as I~ v e r s u s n shows that this section represents a rather sharp increase in ~ at a nearly constant value of the Fermi energy corresponding to n = no, i.e. a mobility edge (Fig. 2), which begins to level off at an R value of 30 000 f~/I--q or more. A "close-up" of the transition region in a few metal films is shown in Fig. l(b). The value of R[] at which the slope of the curve changes as R[] decreases can be readily determined. It ranges from 30 000 to about 50 000 f~/[] in the various films investigated, with most values within (35 ___4) x 10 3 ~-2/[-]. This is in agreement with the lowest unactivated conductivity found in quench-condensed metal films 3, i.e. films with resistivities higher than about 30 000 ft/l--q show activated conductivity while those with resistivities below this value show metallic behaviour. All these results suggest that Fig. l(a) represents an Anderson transition. When the substrate temperature is lowered until the relaxation effects disappear (below about 250 K), the character of the n v e r s u s R D curves changes
f
10 o
::t lO-Y
J 10-2
I 1
nc
I 2
I 3
n x 7014 c m -2
Fig. 2. Carrier mobility versus carrier density curve calculated from the data of Fig. l(a)
158
E. H A R N I K . S. K O V N O V I C H A N D T. C H E R N O B E L S K A Y A
completely. Such curves plotted on a log log scale are shown m Fig. 3 for silver films at two substrate temperatures. For the convenience of presentation, the curves are arbitrarily shifted along the ordinate axis. It can be seen that the experimental points lie well on two straight lines of widely different slopes. The slope of the line at the high R D end varies from film to film, whereas that of the second line is fairly constant with d ( l o g n ) / d ( l o g R D) ~ - 1. The value of R[] at the intersection R v of the two lines varies, at a constant substrate temperature, from film to film between 10 5 and about 106 ~/[~. Although this covers the entire range of its variation, statistically Rp increases with decreasing temperature. The rather sharp change in slope at Rp indicates a change in the nature of the electracal conduction process. We suggest that this represents a percolation transition and that R v corresponds to the percolation threshold. Accordingly, the experimental results can be explained in the following way. It has been shown previously 7 that in very thin metal films the carriers are positively charged, i.e. charge transport is effected predominantly by electron tunnelling to positively charged islands. Now, when the percolation threshold is reached and passed, infinite clusters appear, connecting the two electrodes and opening up a parallel conduction path for a current of negatively charged carriers (percolating electrons). The acoustoelectric current, in which carriers of both signs move along with the SAW in the same direction, will be adversely affected by the appearance of this new current component, and its rate of increase with decreasing R~, and hence that of Vae, will be slowed down. According to eqn. ( 1), this will lead to an increase in the slope ofn v e r s u s R ~ starting at R v, as observed.
180 x:/
.
\
~
90 K
J 3
i
4
i
i
5 6 log R~ IJ~,~,J
i
__
Fig 3. Carrier density versus sheet resistance curves in very thin Sliver films at two reduced substrate temperatures. The curves are arbitrarily displaced along the ordinate axis The difference between the character of the n t'ersus R 1 curves near room temperature and that at a reduced substrate temperature is thought to be associated with the mobility of the metal islands and its effect on the microscopic structure of the film. As indicated by the relaxations mentioned above, near room temperature the metal islands are highly mobile and deformable and, responding to the forces acting on them, they tend to agglomerate into larger islands separated by wider gaps 9. This continues until quite abruptly coalescence sets in, gaps close and the islands join into large clusters. In contrast, on cooled substrates, in the absence of relaxations the film grows mainly around fixed nucleation centres in the form of a
INSULATOR--CONDUCTOR TRANSITION IN VERY THIN METAL FILMS
159
large number of rather small islands separated by narrow gaps, and the formation of large infinite clusters is determined by random processes rather than by coalescence forces. This model is supported by the following observation. If a quench-condensed metal film of sheet resistance about 100 f~/l--q is left in vacuum to heat up slowly to room temperature, its sheet resistance becomes very high, larger than 108 f~/D. Therefore, in quench-condensed films the conditions are more appropriate for the manifestation of the percolation transition. Thus, in such films the percolation threshold appears first, i.e. at higher R D values, and masks the manifestation of the Anderson transition in our experiment. However, in films grown near room temperature the percolation threshold is delayed close to the onset of coalescence, and the Anderson transition becomes observable. REFERENCES 1 A. Kapltulnik and G. Deutscher, Phys. Rev. Lett., 49 (1982) 1444. 2 D . C . Licciardello, Comments Solid State Phys., 8 (1977) 61. 3 R . C . Dynes, J. P. Carno and J. M. Rowell, Phys. Rev. Lett., 4C (1978) 479. 4 M . J . Burns, W. C. McGinms, P. W. Simon, G. Deutscher and P. M. Chaikin, Phys. Rev. Lett., 47 (1981) 1620. 5 M. Kaveh, J. Phys. C, 15 (1982) L181. 6 J . H . Cafarella, A. Bers and B. E. Burke, 1972 Ultrasome Symp. Proe., in IEEE Publ. 72 CHO 708-8, 1972, p. 181 (Institute of Electrical and Electronic Engineers, New York). 7 E. Harnik, S. Kovnovich and T. Chernobelskaya, Thin Solid Ftlms, 89 (1982) 39. 8 E. Harnik, S. Kovnovlch and T. Chernobelskaya, Phys. Status Solidi A, 69 (1982) K 179. 9 K . L . Chopra, Thm Ftlm Phenomena, McGraw-Hill, New York, 1969, Chapter 4.
155
GENERAL FILM BEHAVIOUR
INSULATOR-CONDUCTOR FILMS*
T R A N S I T I O N IN V E R Y T H I N M E T A L
E. HARNIK, S. KOVNOVICH AND T. CHERNOBELSKAYA Racah Institute of Physics, Hebrew Universtty, Jerusalem 91904 (Israel) (Received August 8, 1984; accepted September 14, 1984)
The insulator-conductor transition in very thin metal films is investigated by acoustoelectric measurements of the relation between the charge carrier density and the sheet resistance at various substrate temperatures from room temperature down to about 90 K. Near room temperature a rather sharp mobility edge is observed but at lower temperatures, in quench-condensed films, percolation effects mask the manifestation of the Anderson transition. This difference in behaviour is attributed to the change in surface mobility of the metal islands and its effect on the microscopic structure of the film.
1. INTRODUCTION The insulator-conductor transition in very thin metal films has been treated as a percolation problem with considerable success 1. The question whether the Anderson model can be applied to this transition has been subject to some controversy because of the inhomogeneous semicontinuous structure of these films in the transition region. However, Licciardello 2 has argued that, even in such systems, theory predicts a sharp mobility edge and an abrupt change in the nature of electrical conductivity near a critical sheet resistance R D of about 30000f~/E]. Indeed, increasing experimental evidence indicating a change in the electrical properties of such films near this value of R D has been accumulating 3'4. These experiments, however, do not provide an accurate determination of the critical resistivity or detailed information on the sharpness of the Anderson transition. Recently, enquiring into the latter question in disordered two-dimensional systems, Kaveh s showed that the various stages of the Anderson transition at a finite temperature can be traced on a curve of conductivity (or resistivity) v e r s u s carrier density n. In this paper we report and discuss acoustoelectric measurements ofn as a function o f R D in very thin metal films at various substrate temperatures from room temperature down to about 90 K. Near room temperature the results indicate the * Paper presented at the Sixth International Conference on Thin Films, Stockholm, Sweden, August 13-17, 1984. 0040-6090/85/$3.30
© ElsevierSequoia/Printed in The Netherlands
[56
E. H A R N I K , S. K O V N O V I C H A N D T. C H E R N O B E L S K A Y A
presence of a rather s h a r p m o b i l i t y edge but at lower temperatures, in quenchc o n d e n s e d films, a p e r c o l a t i o n t r a n s i t i o n is observed, which m a s k s the manifestation of the A n d e r s o n transition. 2. EXPERIMENTAl+ METHOI)
The e x p e r i m e n t a l m e t h o d is based on the acoustoelectric d e t e r m i n a t i o n of carrier m o b i l i t y in thin c o n d u c t i n g layers +' ~. A r e c t a n g u l a r strip of the metal is v a c u u m d e p o s i t e d o n t o a piezoelectric ( L i N b O 3 ) s u b s t r a t e in the space between two mterdigital transducers, and the a t t e n u a t i o n :~{dB) of surface acoustic waves (SAWbl and the a c o u s t o e l e c t r i c voltage |/]c induced by S A W s a l o n g the tilm are m e a s u r e d at various values of R F . The value of n is derived from the a c o u s t o e l e c t r i c expression ~' for the carrier m o b i l i t y I~ and the general expression R~ 1 = enl~. W e o b t a i n for the charge carrier density per unit a r e a the expression P n
l-exp(-0+23~)
-
(1)
ecw
Li,~.
where P is the S A W p o w e r incident on the film, r is the S A W phase velocity a n d w is the lateral d i m e n s i o n of the film. In fact, the e x p e r i m e n t is s o m e w h a t m o r e complicated. Because of a s e c o n d a r y a c o u s t o e l e c t r i c current of t h e r m a l origin s. two s e p a r a t e m e a s u r e m e n t s of 1]~, one with the S A W p r o p a g a t i o n direction reversed, have to be carried out a n d half the a l g e b r a i c difference of the two results has to be determined, t a k i n g into a c c o u n t the ratio of the P values associated with the two t r a n s d u c e r s ; . To reduce the e r r o r involved, a yx (instead of the m o r e c o m m o n y z ) k i N b O 3 plate was used as the substrate, since the ratio of s e c o n d a r y to p r i m a r y a c o u s t o e l e c t n c current is c o n s i d e r a b l y lower. The e x p e r i m e n t was carried out in a c o m m e r c i a l Bendix c o a t e r unit. The film was v a p o u r d e p o s i t e d in a large n u m b e r of successive steps in a v a c u u m of a b o u t 5 x 10 ~' Torr. N e a r r o o m t e m p e r a t u r e , because of structural changes, with the i n t e r r u p n o n of each d e p o s i t i o n step the film p a r a m e t e r s m e a s u r e d (R b, :~ a n d I/]e) u n d e r g o large relaxations at a g r a d u a l l y decreasing rate. The m e a s u r e m e n t s were t a k e n in s i t u when a q u a s i - s t a t i o n a r y state was reached in which the changes were slow e n o u g h to permit the d e t e r m i n a t i o n of the p a r a m e t e r s , one after the other, w i t h o u t the i n t r o d u c t i o n of significant errors. This p r o c e d u r e does not in a n y way affect the results because the p o i n t s c o r r e s p o n d i n g to film p a r a m e t e r s m e a s u r e d at later stages of the r e l a x a t i o n process are found to lie on the same n r e r s u s R a curve. The m a i n u n c e r t a i n t y in n originates from a p r o b a b l e systematic e r r o r in the values of the incident S A W p o w e r P, which m a k e s our n values t o o high. T h e relative a c c u r a c y of n and, therefore, the conclusions reached below are not affected by this s h o r t c o m i n g of the experiment. 3.
RESULTS A N D DISCUSSION
A typical n l,ersus RFn curve o b t a i n e d at r o o m t e m p e r a t u r e with a gold film is shown m Fig. l(a). In its general form, it r e p r o d u c e s the m a i n features of K a v e h ' s theoretical curve 5. The p r o m i n e n t feature of this curve is the h o r i z o n t a l p l a t e a u of
INSULATOR--CONDUCTOR TRANSITION IN VERY THIN METAL FILMS
157
Cu
Au c I
I
I
Ro
(n/o)
Ro
(a)
10 (104 ~ / o )
I
15
Ag 2tO
(b)
Fig. 1. (a) Typical room temperature sheet resistance dependence of the carrier density in a very thin gold film: (b) room temperature carrier density versus sheet resistance curves showmg the Anderson transmon m a number of very thm metal films
nearly constant n extending over almost two orders of magnitude of R[]. When it is borne in mind that in a two-dimensional system the Fermi energy is proportional to n and that/~ ~ ( R [ ] n ) - 1, replotting these data as I~ v e r s u s n shows that this section represents a rather sharp increase in ~ at a nearly constant value of the Fermi energy corresponding to n = no, i.e. a mobility edge (Fig. 2), which begins to level off at an R value of 30 000 f~/I--q or more. A "close-up" of the transition region in a few metal films is shown in Fig. l(b). The value of R[] at which the slope of the curve changes as R[] decreases can be readily determined. It ranges from 30 000 to about 50 000 f~/[] in the various films investigated, with most values within (35 ___4) x 10 3 ~-2/[-]. This is in agreement with the lowest unactivated conductivity found in quench-condensed metal films 3, i.e. films with resistivities higher than about 30 000 ft/l--q show activated conductivity while those with resistivities below this value show metallic behaviour. All these results suggest that Fig. l(a) represents an Anderson transition. When the substrate temperature is lowered until the relaxation effects disappear (below about 250 K), the character of the n v e r s u s R D curves changes
f
10 o
::t lO-Y
J 10-2
I 1
nc
I 2
I 3
n x 7014 c m -2
Fig. 2. Carrier mobility versus carrier density curve calculated from the data of Fig. l(a)
158
E. H A R N I K . S. K O V N O V I C H A N D T. C H E R N O B E L S K A Y A
completely. Such curves plotted on a log log scale are shown m Fig. 3 for silver films at two substrate temperatures. For the convenience of presentation, the curves are arbitrarily shifted along the ordinate axis. It can be seen that the experimental points lie well on two straight lines of widely different slopes. The slope of the line at the high R D end varies from film to film, whereas that of the second line is fairly constant with d ( l o g n ) / d ( l o g R D) ~ - 1. The value of R[] at the intersection R v of the two lines varies, at a constant substrate temperature, from film to film between 10 5 and about 106 ~/[~. Although this covers the entire range of its variation, statistically Rp increases with decreasing temperature. The rather sharp change in slope at Rp indicates a change in the nature of the electracal conduction process. We suggest that this represents a percolation transition and that R v corresponds to the percolation threshold. Accordingly, the experimental results can be explained in the following way. It has been shown previously 7 that in very thin metal films the carriers are positively charged, i.e. charge transport is effected predominantly by electron tunnelling to positively charged islands. Now, when the percolation threshold is reached and passed, infinite clusters appear, connecting the two electrodes and opening up a parallel conduction path for a current of negatively charged carriers (percolating electrons). The acoustoelectric current, in which carriers of both signs move along with the SAW in the same direction, will be adversely affected by the appearance of this new current component, and its rate of increase with decreasing R~, and hence that of Vae, will be slowed down. According to eqn. ( 1), this will lead to an increase in the slope ofn v e r s u s R ~ starting at R v, as observed.
180 x:/
.
\
~
90 K
J 3
i
4
i
i
5 6 log R~ IJ~,~,J
i
__
Fig 3. Carrier density versus sheet resistance curves in very thin Sliver films at two reduced substrate temperatures. The curves are arbitrarily displaced along the ordinate axis The difference between the character of the n t'ersus R 1 curves near room temperature and that at a reduced substrate temperature is thought to be associated with the mobility of the metal islands and its effect on the microscopic structure of the film. As indicated by the relaxations mentioned above, near room temperature the metal islands are highly mobile and deformable and, responding to the forces acting on them, they tend to agglomerate into larger islands separated by wider gaps 9. This continues until quite abruptly coalescence sets in, gaps close and the islands join into large clusters. In contrast, on cooled substrates, in the absence of relaxations the film grows mainly around fixed nucleation centres in the form of a
INSULATOR--CONDUCTOR TRANSITION IN VERY THIN METAL FILMS
159
large number of rather small islands separated by narrow gaps, and the formation of large infinite clusters is determined by random processes rather than by coalescence forces. This model is supported by the following observation. If a quench-condensed metal film of sheet resistance about 100 f~/l--q is left in vacuum to heat up slowly to room temperature, its sheet resistance becomes very high, larger than 108 f~/D. Therefore, in quench-condensed films the conditions are more appropriate for the manifestation of the percolation transition. Thus, in such films the percolation threshold appears first, i.e. at higher R D values, and masks the manifestation of the Anderson transition in our experiment. However, in films grown near room temperature the percolation threshold is delayed close to the onset of coalescence, and the Anderson transition becomes observable. REFERENCES 1 A. Kapltulnik and G. Deutscher, Phys. Rev. Lett., 49 (1982) 1444. 2 D . C . Licciardello, Comments Solid State Phys., 8 (1977) 61. 3 R . C . Dynes, J. P. Carno and J. M. Rowell, Phys. Rev. Lett., 4C (1978) 479. 4 M . J . Burns, W. C. McGinms, P. W. Simon, G. Deutscher and P. M. Chaikin, Phys. Rev. Lett., 47 (1981) 1620. 5 M. Kaveh, J. Phys. C, 15 (1982) L181. 6 J . H . Cafarella, A. Bers and B. E. Burke, 1972 Ultrasome Symp. Proe., in IEEE Publ. 72 CHO 708-8, 1972, p. 181 (Institute of Electrical and Electronic Engineers, New York). 7 E. Harnik, S. Kovnovich and T. Chernobelskaya, Thin Solid Ftlms, 89 (1982) 39. 8 E. Harnik, S. Kovnovlch and T. Chernobelskaya, Phys. Status Solidi A, 69 (1982) K 179. 9 K . L . Chopra, Thm Ftlm Phenomena, McGraw-Hill, New York, 1969, Chapter 4.