Size effects in epitaxial aluminium films

Thin Solid Films, 43 (1977) 267-273 0 Elsevier Sequoia %A., Lausanne-Printed

SIZE EFFECTS

IN EPITAXIAL

E. DOBIERZEWSKA-MOZRZYMAS Institute of Physics, (Poland)

Wroclaw

(Received

16,1976;

November

267

in the Netherlands

ALUMINIUM

FILMS

AND F. WARKUSZ

Technical accepted

University,

December

Wybrzbze

Wyspiariskiego

27, 50-370

Wroofaw

8, 1976)

Size effects in epitaxial Al films were investigated. By employing the conductivity theories so far developed, resistivity pF uersus grain diameter D curves were calculated with the fraction p of electrons specularly scattered at the external surface and the grain boundary scattering coefficient R as parameters. Resistivity measurements were made on Al films with different grain diameters in the range 1300-18000 A. A qualitative agreement is found between the theoretical pF-D relationships and the experimental pF-D curves. An analysis of the calculated and measured results shows that p reaches low values (approaching zero), whereas the values of R (especially for thin films) can be much higher.

1. INTRODUCTION

The electrical conductivity of a thin film is lower than that of the bulk:This is due not only to electron scattering at the film surface and grain boundaries but also to impurities, vacancies and dislocations. These defects are most often formed in thin films. The theories so far of the electrical conductivity of films involve the size effect theory developed by Fuchs’ and extended by Sondheimer2 (the FS theory). Both Fuchs and Sondheimer have discussed the phenomena of isotropic electron scattering and surface scattering (surface scattering is determined by the fraction of electrons specularly scattered at the external surface and is called the external she effect). Mayadas et ~1.~ have developed a theory of film conductivity that takes into account the grain boundary scattering phenomenon (the internal size effect). During the last few years experiments reported by many workers4-9 have proved that both the conductivity and the resistivity of metallic films are dependent on their thickness or grain diameter. The conductivity in terms of the internal and external size effects has been considered in refs. 4 and 8-l 1. The temperature dependence for polycrystalline and monocrystalline Al films has been investigated in the range 4.2-300 K, and the experimental results have been analysed in terms of the FS theory12Tr3. The anisotropy of the size effect in monocrystalline Al foils has been shown by Sato and Yonemitsui4.

268

E. DOBIERZEWSKA-MOZRZYMAS, F. WARKUSZ

2. THEORETICAL ANALYSIS

In this paper we consider the resistivity as a function of electron scattering both at the film surfaces and at the grain boundaries by using methods other than those suggested in refs. 4 and 8-11. It was assumed that the variation of the film resistivity PF with the external size effect is determined by Cottey's ~5 function F(t/2,p), while the variation of the film resistivity with the internal size effect is described by the Mayadas_Shatzkes 3, 4 expressions G ( D / 2 , R ) . Hence P~

(1)

Pv - F(I~)G(~)

where PB is the resistivity of the bulk and F(/~) is a function that takes into account the effect of surface scattering. This function becomes 3f

1

1

where/~ = t/(1 - p ) 2 , t is the thickness of the film, 2 is the electron mean free path and p is the fraction of electrons specularly scattered at the external surface. The function G(e) takes into account the effect of grain boundary scattering and has the following form: G ( c 0 = 3 ~(1-

(1 1+:~+2 -c~ ! 31n ) } ~

(3)

ct = 2 R / D ( 1 - R ) , where D is the grain diameter and R is the grain boundary scattering coefficient. Both eqns. (2) and (3) hold when ~ > 1 and ~ < 1. This is particularly true for relatively thick films. Films with very small grain diameters and smaller thicknesses show an island structure that is characterized by an electrical conductivity mechanism which is different from that of continuous films. For/~ > 1 and c~< 1, the functions F(#) and G(~) become

3 1 F(/t) = 1 - 8/1 q 5/z2

(4)

G(e) = 1 - ~ e + 3 c ~ e . . . .

(5)

and

Neglecting the squares of kL and c~ in eqns. (4) and (5) and introducing these equations into eqn. (1), we obtain two alternative expressions for the resistivity: PB Pv = (1 - 3/8/~)(1 - 3/2~)

(6)

or

3 31 9c~ PF = P, +2~Pn + 8 ~ P B + i ~ P .

(7)

The last term ofeqn. (7) is very small because of the assumption that/~ > 1 and c~< 1. Therefore we can ignore it, and eqn. (7) will become

SIZE EFFECTS IN EPITAX1ALA1 FILMS

269 (8)

PF = PB + PD + Pt

w h e r e p o = 3;~Rpa/2D(1 - R ) is the resistivity t h a t d e p e n d s on the internal size effect a n d p, = 3,~(1 - p ) p n / 8 t is the resistivity that d e p e n d s on the external size effect. It can be seen f r o m eqn. (8) that these resistivities are additive, in a c c o r d a n c e with M a t t h i e s s e n ' s rule, The total resistivity in eqn. (8) is the same as that given in refs. 8 a n d 16. E q u a t i o n (1) was used to i n t e r p r e t the e x p e r i m e n t a l results; we p l o t t e d theoretical curves o f the film resistivity PF as a f u n c t i o n o f grain d i a m e t e r D for Al films o f two different ranges o f thickness: t = 600/~, a n d t = 2000 a n d 3500 ,~. T h e b u l k resistivity Pn is 2.65 × l0 - 6 f~ cm, a n d the electron m e a n free p a t h 2 is 600 A. F i g u r e 1 shows the resistivity v a r i a t i o n s f o r p = 0. l a n d p = 0.8, with R = 0.2. T h e resistivity as a function o f grain d i a m e t e r w i t h p = 0.1 a n d R = 0.2 a n d 0.4 is p l o t t e d in Fig. 2. A n a l y s i s o f these curves allows us to m a k e the following conclusions. (1) T h e film resistivity increases r a p i d l y when the grain d i a m e t e r is smaller t h a n the electron m e a n free path. (2) F o r a given grain d i a m e t e r the resistivities o f thin a n d thick films show significant differences when p is near zero (see Fig. l, curves 3 a n d 4). These differences b e c o m e less p r o n o u n c e d when p a p p r o a c h e s unity (cfi Fig. l, curves 1 a n d 2). (3) F o r larger values o f R, the resistivity increases as the grain d i a m e t e r decreases (Fig. 2).

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Fig. 1. The theoretical curves of resistivity as a function of grain diameter for two ranges of film thickness : t = 600 A (curves 2, 4) and t = 2000 A (curves 1, 3) (p = 0.1 and 0.8 ; R = 0.2; 2 = 600 A).

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Fig. 2. The theoretical curves of resistivity as a function o f grain d i a m e t e r for two ranges o f film t h i c k n e s s : t = 600 A (curves 4, 5), t = 2000 A (curves 2, 3) and t = 3500 A (curve 1) (R = 0.2 a n d 0.4; p = 0.1 ; 2 = 600 A).

3. EXPERIMENTAL

Monocrystalline and textured A1 films were obtained by the methods previously developed by Dobierzewska-Mozrzymas 17'18 and, additionally, with the use of both parallel and perpendicular electric fields during the evaporation process i9. Crystallization was carried out over a wide substrate temperature range at various evaporation rates in an external electric field; films were formed with average grain diameters varying from approximately 1300 to 18 000 A.

Fig. 3. A n electron m i c r o g r a p h o f an AI film d e p o s i t e d o n an N a C I s u b s t r a t e (t = 3500 A; Tsubstrat e = 400 °C; 13 = 6 0 A S- 1 ; E l = 95 V c m - 1). (This p h o t o g r a p h was t a k e n at the L a b o r a t o r y o f Electron Microscopy, Wrool'aw Technical University.) Fig. 4. A n electron m i c r o g r a p h o f a n AI film d e p o s i t e d on a q u a r t z s u b s t r a t e (t = 570 A ; Tsubstrate = 3 6 0 ° C ; v = 100 A s - l ) .

SIZE EFFECTS IN EPITAXIAL A1 FILMS

271

Resistivity measurements and calculations were carried out for films with thicknesses in the range 600 + 150/~ (comparable with the electron mean free path) and also for films with thicknesses ranging from approximately 2000 to 3500 /~ (significantly greater than the electron mean free path). The average grain diameter was determined, from electron micrographs. Films on an NaCl substrate were investigated by transmission microscopy, while those on a quartz substrate were studied using a carbon replica technique (these film-substrate combinations are represented in Figs. 3 and 4 respectively). To establish the average grain diameter, the size of about 300 grains from a given part of the specimen was determined (these diameters were measured at different points on the film to obtain an average). The largest grains were found to occur in films evaporated using a perpendicular electric field of intensity exceeding 100 V c m - 1 For films approximately 600/~ thick the average grain diameters fell within the range 1300-4000 ,~, whereas for films with thicknesses within the range 2000-3500 ,~ the average grain diameters ranged from 2000 to 18000 ,~. The room temperature resistivity of the films was measured outside the vacuum chamber by the four-point (in-line) probe method.

E~cm~

Fig. 5. The experimental curves of resistivity as a function of grain diameter for two ranges of film thickness: t = 6 0 0 + 150 A ( A , A1/(100)NaC1; C), Al/quartz); t = 2000-3500 ,~ ( A , Al/(100)NaCl; O, Al/quartz).

4.

DISCUSSION OF RESULTS

Figure 5 shows the experimental resistivity v e r s u s grain diameter curves for two

272

E. DOBIERZEWSKA-MOZRZYMAS, F. WARKUSZ

ranges of film thicknesses: 600 + 150 A and 2000-3500 ,~. The shape of the curves for corresponding grain diameter ranges is similar; the resistivity increases with a decrease in the grain diameter. It should be noted that this effect becomes more pronounced as the film thickness decreases. The resistivity values for the same grain diameter but for various thicknesses show significant differences, as can be seen from the shift in the plots. For thick films, the resistivity values range from 3.50 x 10- 61) cm (corresponding to a grain diameter of 1800 A) to 2.47 x 10- 6 f~ cm (a grain diameter of 4000 A). With a further increase in the grain diameter (up to 18 000 A) the resistivity remains constant after reaching a value of about 2.50 x 10- 6 f~ cm. As the thicknesses were only measured to an accuracy of about 10~/oo, the resistivity results obtained are in good agreement with those reported by Bassewith and Mitchell 12 and by Mayadas 13. Bassewith and Mitchell ~2 have reported a resistivity value for monocrystalline AI films with a grain diameter of 30 jam and for polycrystalline AI films with a grain size of 2500 ,~ of from 2.66x 10 _6 to 2.85 X 1 0 - 6 ~ cm. For monocrystals the values were lower. In our investigations on films with a thickness that was comparable with the electron mean free path, the resistivity decreased as the grain diameter increased. The resistivity values were 7.19 x 10- 6 f~ cm when the grain diameter was 1300 A, and 4.37 x 10- 6 f~ cm when the grain diameter approached 4000 A. For a thickness of 600 + 150 A we could not obtain films that had grains which were any larger. As can be seen from Figs. 1 and 5, there is a qualitative agreement between the theoretical and experimental resistivity grain size dependences. The considerable shift in the resistivity curves corresponding to the two ranges of thickness may be due to a lower value of the parameter p. Curves 1 and 2 of Fig. 1 were constructed with p = 0.8, while curves 3 and 4 (which have evidently shifted) are plotted with p = 0.1. Figure 2 represents the resistivities that were calculated with p = 0.1 and with two values of R (0.2 and 0.4). It can be seen that, as R increases, the film resistivity also increases for smaller grain diameters. A comparison of the theoretical and experimental curves (Fig. 5) shows that for thin films the resistivities that we measured are higher than the calculated values. This may be due to the fact that the boundary scattering coefficient for thin films is higher than that for thick films or higher than the value that we used in the calculations. The resistivity was measured outside the vacuum chamber after deposition of the films. Under these conditions, the grains that form the film are covered with an A120 3 film (on average 60 A thick); for thin films (600 A) such a film may contribute to a considerable increase in the resistivity. 5. CONCLUSIONS

Both the theoretical and the experimental results suggest that the resistivity of A1 films depends on the grain diameter. This dependence is more pronounced for thin films. The considerable variation in the resistivities of thick and thin films for a given grain diameter indicates that the surface scattering coefficient p is very low (close to zero) for thin films. The high resistivities of thin films may result from higher grain boundary scattering coefficients R due to the oxidation of the grain surface.

SIZE EFFECTS IN EPITAXIAL AI FILMS

273

ACKNOWLEDGMENTS

The authors are grateful to Professor C. Wesotowska for encouraging this work. They also wish to thank Dr. T. Ohly for help with the electrical measurements. REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

K. Fuchs, Proc. Cambridge Philos. Soc., 34 (1938) 100. E.H. Sondheimer, Adv. Phys., 1 (1952) 1. A.F. Mayadas, M. Shatzkes and J. F. Janak, Appl. Phys. Lett., 14 (1969) 343. A.F. Mayadas and M. Shatzkes, Phys. Rev., Sect. B, 1 (1970) 1382. V.P. Nagpal and V. P. Duggal, Thin Solid Films, 9 (1972) 213. S. Arajs, B. F. Oliver and J. T. Michalak, J. Appl. Phys., 38 (1967) 1676. P.V. Andrews, Phys. Lett., 19 (1965) 558. P. Wissmann, Thin Solid Films, 5 (1970) 329. A.F. Mayadas, R. Feder and R. Rosenberg, J. Vac. Sci. Technol., 6 (6) (1969) 690. E.E. Mola and J. M. Heras, Thin Solid Films, 18 (1973) 137. C.R. Tellier and A. J. Tosser, Thin Solid Films, 33 (1976) L19. A. Bassewith and E. N. Mitchell, Phys. Rev., 183 (3) (1969) 182. A.F. Mayadas, J. Appl. Phys., 39 (9) (1968) 4241. H. Sato and K. Yonemitsu, Phys. Status Solidi, B, 73 (1976) 723. A.A. Cottey, Thin Solid Films, 1 (1967) 297. F. Thiema and W. Kirstein, Thin Solid Films, 30 (1975) 371. E. Dobierzewska-Mozrzymas, Acta Phys. Pol. A, 47 (1975) 93. E. Dobierzewska-Mozrzymas, T. Ohly and F. Warkusz, J. Cryst. Growth, 32 (1976) 129. E. Dobierzewska-Mozrzymas, Komunikat Inst. Fiz. 301, 1975.