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Measurements and interpretation of the electrical resistivity and Hall coefficient in polycrystalline gold films: Part I
Thin Solid Films, 35 (1976) 137-141 © Elsevier Sequoia S.A., Lausanne--Printed in Switzerland
137
M E A S U R E M E N T S A N D I N T E R P R E T A T I O N OF T H E E L E C T R I C A L RESISTIVITY A N D H A L L C O E F F I C I E N T IN P O L Y C R Y S T A L L I N E G O L D FILMS: P A R T I*
D. GOLMAYO AND J. L. SACEDON C.I.F. "' L. Torres Quevedo", Serrano 144, Madrid 6 (Spain)
(Received August 25, 1975; accepted December 8, 1975)
By measuring the resistivity of gold films as a function of temperature, the phonon contribution can be separated from the intrinsic one. The temperature dependence of the phonon terms can be fitted to a Sondheimer plot with the bulk electron mean free path and the value of the scattering parameter p varying with temperature. The phonon term at room temperature is not very different from the bulk. The intrinsic term in the resistivity is important, and in very thin films even more important than the phonon contribution. Electron microscopy including automatic image processing has shown that the product of grain size and intrinsic resistivity is constant. This agrees with the mosaic theory fairly well.
INTRODUCTION
This study reports resistivity measurements of polycrystalline gold films in the temperature range from 4.2 to 300 K, and measurements of the grain sizes of the films, in order to show the effects of structure on the electrical conduction of these films; such effects are not small compared with the size effects calculated by Sondheimer 1. In a later paper 2, we shall report a related study of the Hall coefficient. EXPERIMENTAL
We used polycrystalline gold films (100-1500 A) obtained on glass by electron beam evaporation in an ultrahigh vacuum system at room temperature; they were not annealed. Thickness was measured with an optical microdensitometer using the Moretz 3 method. Although the basic method is well known, its applications and accuracy are not. In order to obtain a calibration curve, we made pairs of * Paper presented at the Third International Conference on Thin Films, "Basic Problems, Applications and Trends", Budapest, Hungary, August 25-29, 1975, but appearing in the Conference Proceedings only in abstract form.
138
D. GOLMAYO, J. L. SACEDON
films simultaneously, one deposited on glass to measure the optical density and the other on optically smooth and flat quartz to measure the thickness by interferometry. Figure 1 shows the optical density of the films as a function o f their thickness. The precision is about 8% in a 100 A measurement and 0.4% in a 1000 A measurement. The method is simple and non-destructive. The grain sizes of the films were measured by an automatic image analyser from microphotographs.
///
3D 201 _
.~ 15
2D
D
10 1D 5
I
500
1000
I
1500 ~,
Fig. 1. Optical density vs. thickness. The deflection of the microdensitometer is also shown as a function of thickness. The experimental points can be approximated by a straight line. RESULTS ON RESISTIVITY
The resistivity of the films as a function of temperature is shown in Fig. 2. It can be seen that the resistivity falls slightly near 40 K. Thus we can write p ( T ) = pf( T ) + p i
(1)
i.e. Matthiessen's rule is valid for these films, pf is due to electron-phonon interac-
tions and Pl is due to structural effects and does not change with temperature. INTERPRETATION OF THE RESISTIVITY
From the expression of the resistivity of a film derived by Sondheimer, P = Po F(p, d, 2o), and taking into account that p020 is a constant at a given temperature 4, we have calculated the resistivity for each value of d using the experimental values ofpo and values ofp varying from 0 to 1. We used the complete expression for p and not approximations. Figure 3 shows the calculated resistivity, for various values ofp and two values of d, versus temperature. On this curve we have also shown the experimental
RESISTIVITY AND
HALL
COEFFICIENT
IN
Au
98 7
139
FILMS. I
165~
--
300
A
6
o >. >
5
(z
3
2
1
II
I
I
~0 50 ~00 200 Fig. 2. Variation o f resistivity with temperature for several fi|ms.
I 300 (K)
values of pf ---- p - - Pi for films of the same thickness. Similar results were obtained for the other films. Therefore, by comparing the Sondheimer theory only with the phonon term, and by supposing that the scattering coefficient changes with temperature (a particularly abrupt change occurring near 40 K), we have found that the variation of resistivity with temperature can be interpreted quantitatively in terms of this size effect theory. The intrinsic term Pi is then due to structural effects, and we shall show that it is a function of the grain size. Following Jayadevaiah and Kriby s, we have used the mosaic theory of Volger 6 to explain the conduction phenomenon in the polycrystalline films. The film resistivity is given by
P = PI +flP2
(2)
where fl = 12/11, 12 << 11 and P2 > P l . Ii and Pl are the dimensions and resistivity of the crystalline grains, and 12 and P2 those of the intercrystalline regions. The grain size ll differs from film to film, and taking into account that these variations are much greater than those in 12, we suppose that 12 is constant. Furthermore, Pl is dominated by electron-phonon interactions, whereas P2 is dominated by electron--defect interactions and is much greater than Pl. Therefore, we assume that the contribution Pl due to the crystallites will suffer size
140
D. G O L M A Y O , J. L. S A C E D O N
p:
06
2.5 1 ~
2.0
P
= 0'9o d = 565A
1.5 i
1.0
r--0.
0.5: 100
200
300
( KJ
p=0 p:0.5 } p:0,9
2+5
2.0
7 T
d = 1350 .~ 1.0
0,5 100 Fig. 3. Calculated resistivity
200
300
(K]
temperature from Sondheimer's theory. The points represent the experimental resistivity of a film of the same thickness minus its intrinsic resistivity: p - p ~ . vs.
effects, but the intercrystalline contribution P2 will not, since the electron mean free path in this region is much smaller than that in the other region; thus for the thicknesses employed here there will be no size effect. We then suppose that P2 does not change with either temperature or thickness. By comparing eqns. (1) and (2) we obtain pf(T)
= Pl(T)
Pl
=
tiP2 ~- (12/ll) P2
(3)
R E L A T I O N B E T W E E N I N T R I N S I C RESISTIVITY A N D G R A I N SIZE
From eqn. (3) we conclude that Pi ll must be constant since l 2 and P2 are constant. We have measured 11 to check this conclusion and have found that the experimental results are in agreement with the theoretical prediction (Table I). (In fact, ptll-~constant.) This means that the intrinsic resistivity of a polycrystalline film is a linear function of 1/ll.
141
RESISTIVITY AND HALL COEFFICIENT IN A u FILMS. I TABLE I THICKNESS, GRAIN SIZE, INTRINSICAND PHONON RESIS]'IVITIESAND [lPi FOR SEVERAL FILMS
Thickness
Grain size
Intrinsic resistivity
Phonon resistivity at 300 K
li pi
(A)
(A)
(lO - s fl m)
(10 - s ~2 m)
(10 - s A fl m)
7.09 3.91 3.60 1.45 1.36 1.66 0.64 0.69 3.20
2.30 2.36 2.31 2.20 2.27 2.24 2.26 2.40 2.44 2.20
165 300 335 550 565 700 1005 1315 1350
bulk
124 130 317 312 280 734
484 470 470 425 464 470
CONCLUSIONS
The electrical resistivity of a polycrystalline film is described by the mosaic model as the sum of contributions of both the crystallites and the intercrystalline regions, multiplied by a shape factor. The first contribution changes with temperature and suffers a size effect. It can be accommodated in the Sondheimer theory. The second term is comparable in magnitude with the phonon one but, unlike the first, it cannot be accommodated in a size effect theory. It is an inverse function of the grain size of the film. Therefore any study on film resistivity has to consider the intrinsic resistivity, which is strongly dependent on the conditions of preparation of the films. REFERENCES 1 2 3 4 5 6
E . H . Sondheimer, Phys, Rev., 80 (1950) 401. D. Golmayo and I. L. Sacedon, Thin Solid Films, 35 (1976) 143. R . C . Morezt, H. M. Jhonson and D. F. Parsons, J. Appl. Phys., 39 (1968) 12. P. Broquet and V. Nguyen Van, S u r f Sci., 6 (1967) 98. T.S. Jayadevaiah and R. E. Kriby, Appl. Phys. Lett., 15 (1969) 150. J. Volger, Phys. Rev., 79 (1950) 1023.
137
M E A S U R E M E N T S A N D I N T E R P R E T A T I O N OF T H E E L E C T R I C A L RESISTIVITY A N D H A L L C O E F F I C I E N T IN P O L Y C R Y S T A L L I N E G O L D FILMS: P A R T I*
D. GOLMAYO AND J. L. SACEDON C.I.F. "' L. Torres Quevedo", Serrano 144, Madrid 6 (Spain)
(Received August 25, 1975; accepted December 8, 1975)
By measuring the resistivity of gold films as a function of temperature, the phonon contribution can be separated from the intrinsic one. The temperature dependence of the phonon terms can be fitted to a Sondheimer plot with the bulk electron mean free path and the value of the scattering parameter p varying with temperature. The phonon term at room temperature is not very different from the bulk. The intrinsic term in the resistivity is important, and in very thin films even more important than the phonon contribution. Electron microscopy including automatic image processing has shown that the product of grain size and intrinsic resistivity is constant. This agrees with the mosaic theory fairly well.
INTRODUCTION
This study reports resistivity measurements of polycrystalline gold films in the temperature range from 4.2 to 300 K, and measurements of the grain sizes of the films, in order to show the effects of structure on the electrical conduction of these films; such effects are not small compared with the size effects calculated by Sondheimer 1. In a later paper 2, we shall report a related study of the Hall coefficient. EXPERIMENTAL
We used polycrystalline gold films (100-1500 A) obtained on glass by electron beam evaporation in an ultrahigh vacuum system at room temperature; they were not annealed. Thickness was measured with an optical microdensitometer using the Moretz 3 method. Although the basic method is well known, its applications and accuracy are not. In order to obtain a calibration curve, we made pairs of * Paper presented at the Third International Conference on Thin Films, "Basic Problems, Applications and Trends", Budapest, Hungary, August 25-29, 1975, but appearing in the Conference Proceedings only in abstract form.
138
D. GOLMAYO, J. L. SACEDON
films simultaneously, one deposited on glass to measure the optical density and the other on optically smooth and flat quartz to measure the thickness by interferometry. Figure 1 shows the optical density of the films as a function o f their thickness. The precision is about 8% in a 100 A measurement and 0.4% in a 1000 A measurement. The method is simple and non-destructive. The grain sizes of the films were measured by an automatic image analyser from microphotographs.
///
3D 201 _
.~ 15
2D
D
10 1D 5
I
500
1000
I
1500 ~,
Fig. 1. Optical density vs. thickness. The deflection of the microdensitometer is also shown as a function of thickness. The experimental points can be approximated by a straight line. RESULTS ON RESISTIVITY
The resistivity of the films as a function of temperature is shown in Fig. 2. It can be seen that the resistivity falls slightly near 40 K. Thus we can write p ( T ) = pf( T ) + p i
(1)
i.e. Matthiessen's rule is valid for these films, pf is due to electron-phonon interac-
tions and Pl is due to structural effects and does not change with temperature. INTERPRETATION OF THE RESISTIVITY
From the expression of the resistivity of a film derived by Sondheimer, P = Po F(p, d, 2o), and taking into account that p020 is a constant at a given temperature 4, we have calculated the resistivity for each value of d using the experimental values ofpo and values ofp varying from 0 to 1. We used the complete expression for p and not approximations. Figure 3 shows the calculated resistivity, for various values ofp and two values of d, versus temperature. On this curve we have also shown the experimental
RESISTIVITY AND
HALL
COEFFICIENT
IN
Au
98 7
139
FILMS. I
165~
--
300
A
6
o >. >
5
(z
3
2
1
II
I
I
~0 50 ~00 200 Fig. 2. Variation o f resistivity with temperature for several fi|ms.
I 300 (K)
values of pf ---- p - - Pi for films of the same thickness. Similar results were obtained for the other films. Therefore, by comparing the Sondheimer theory only with the phonon term, and by supposing that the scattering coefficient changes with temperature (a particularly abrupt change occurring near 40 K), we have found that the variation of resistivity with temperature can be interpreted quantitatively in terms of this size effect theory. The intrinsic term Pi is then due to structural effects, and we shall show that it is a function of the grain size. Following Jayadevaiah and Kriby s, we have used the mosaic theory of Volger 6 to explain the conduction phenomenon in the polycrystalline films. The film resistivity is given by
P = PI +flP2
(2)
where fl = 12/11, 12 << 11 and P2 > P l . Ii and Pl are the dimensions and resistivity of the crystalline grains, and 12 and P2 those of the intercrystalline regions. The grain size ll differs from film to film, and taking into account that these variations are much greater than those in 12, we suppose that 12 is constant. Furthermore, Pl is dominated by electron-phonon interactions, whereas P2 is dominated by electron--defect interactions and is much greater than Pl. Therefore, we assume that the contribution Pl due to the crystallites will suffer size
140
D. G O L M A Y O , J. L. S A C E D O N
p:
06
2.5 1 ~
2.0
P
= 0'9o d = 565A
1.5 i
1.0
r--0.
0.5: 100
200
300
( KJ
p=0 p:0.5 } p:0,9
2+5
2.0
7 T
d = 1350 .~ 1.0
0,5 100 Fig. 3. Calculated resistivity
200
300
(K]
temperature from Sondheimer's theory. The points represent the experimental resistivity of a film of the same thickness minus its intrinsic resistivity: p - p ~ . vs.
effects, but the intercrystalline contribution P2 will not, since the electron mean free path in this region is much smaller than that in the other region; thus for the thicknesses employed here there will be no size effect. We then suppose that P2 does not change with either temperature or thickness. By comparing eqns. (1) and (2) we obtain pf(T)
= Pl(T)
Pl
=
tiP2 ~- (12/ll) P2
(3)
R E L A T I O N B E T W E E N I N T R I N S I C RESISTIVITY A N D G R A I N SIZE
From eqn. (3) we conclude that Pi ll must be constant since l 2 and P2 are constant. We have measured 11 to check this conclusion and have found that the experimental results are in agreement with the theoretical prediction (Table I). (In fact, ptll-~constant.) This means that the intrinsic resistivity of a polycrystalline film is a linear function of 1/ll.
141
RESISTIVITY AND HALL COEFFICIENT IN A u FILMS. I TABLE I THICKNESS, GRAIN SIZE, INTRINSICAND PHONON RESIS]'IVITIESAND [lPi FOR SEVERAL FILMS
Thickness
Grain size
Intrinsic resistivity
Phonon resistivity at 300 K
li pi
(A)
(A)
(lO - s fl m)
(10 - s ~2 m)
(10 - s A fl m)
7.09 3.91 3.60 1.45 1.36 1.66 0.64 0.69 3.20
2.30 2.36 2.31 2.20 2.27 2.24 2.26 2.40 2.44 2.20
165 300 335 550 565 700 1005 1315 1350
bulk
124 130 317 312 280 734
484 470 470 425 464 470
CONCLUSIONS
The electrical resistivity of a polycrystalline film is described by the mosaic model as the sum of contributions of both the crystallites and the intercrystalline regions, multiplied by a shape factor. The first contribution changes with temperature and suffers a size effect. It can be accommodated in the Sondheimer theory. The second term is comparable in magnitude with the phonon one but, unlike the first, it cannot be accommodated in a size effect theory. It is an inverse function of the grain size of the film. Therefore any study on film resistivity has to consider the intrinsic resistivity, which is strongly dependent on the conditions of preparation of the films. REFERENCES 1 2 3 4 5 6
E . H . Sondheimer, Phys, Rev., 80 (1950) 401. D. Golmayo and I. L. Sacedon, Thin Solid Films, 35 (1976) 143. R . C . Morezt, H. M. Jhonson and D. F. Parsons, J. Appl. Phys., 39 (1968) 12. P. Broquet and V. Nguyen Van, S u r f Sci., 6 (1967) 98. T.S. Jayadevaiah and R. E. Kriby, Appl. Phys. Lett., 15 (1969) 150. J. Volger, Phys. Rev., 79 (1950) 1023.