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A two-layer thin film under electric current
Scripta METALLURGICA et M A T E R I A L I A
Vol. 29, pp. 579-582, i[993 Printed in the U.S.A.
A
TWO-LAYER
THIN
FILM
UNDER
ELECTRIC
Pergamon Press Ltd. All rights reserved
CURRENT
L.Klinger, A.Katsman, L . L e v i n D e p a r t m e n t of M a t e r i a l s Engineering, Technion, Israel I n s t i t u t e of Technology, H a i f a (Received M a r c h
23, 1993)
Layered thin film structures are an e s s e n t i a l element of modern m i c r o e l e c t r o n i c devices. An i n d i s p e n s a b l e r e q u i r e m e n t of these elements is stability in e l e c t r i c and temperature fields. At the o p e r a t i n g temperatures of layered t h i n films, grain b o u n d a r y d i f f u s i o n occurs both within each layer and b e t w e e n layers. Under the action of an electric current through the multilayer film, internal stresses arise in conducting layers as a r e s u l t of e l e c t r o m i g r a t i o n . R e l a x a t i o n of these stresses may lead to the g r o w t h of h i l l o c k s on the outer s u r f a c e (the case for w h i c h e l e c t r o m i g r a t i o n has been c o n s i d e r e d in [1-4]). Where a conductor does not posses an outer surface, r e l a x a t i o n can occur by the mutual p e n e t r a t i o n of atoms from one layer to another. A stress g r a d i e n t across the layered s t r u c t u r e i n t e n s i f i e s the interdiffusion between layers which, in turn, a c c e l e r a t e s h o m o g e n i z a t i o n of the layered structure. In the p r e s e n t work a two-layer, two-component thin film on an undeformed substrate under the action of electric current will be studied (Fig.l). We shall c o n s i d e r a model s i t u a t i o n in w h i c h the mutual p e n e t r a t i o n of m a t e r i a l is assumed to be taking p l a c e through two s y m m e t r i c a l l y s i t u a t e d a p e r t u r e s near the ends of the film. The surface b e t w e e n the apertures constitutes a barrier layer. In the limiting case of the b a r r i e r layer being absent, the a p e r t u r e size is equal to half the film length. We shall assume that the i n t e r p e n e t r a t i o n is c o n t r o l l e d by d i f f u s i o n creep (without t h r e s h o l d stress), and that no conventional diffusion homogenization occurs at the same time (e.g.because of the limited s o l u b i l i t y of the components).
apertures
Electrical current
Fig.l.
M a s s t r a n s p o r t in two-layer, t w o - c o m p o n e n t thin metal film under e l e c t r i c a l c u r r e n t
579 0 9 5 6 - 7 1 6 X / 9 3 $6.00 + .00 Copyright (c) 1993 Pergamon Press Ltd.
580
TWO-LAYER
THIN FILM
Vol.
29, No.
Initially , one layer is enriched with component A, the other - with component B. We shall assume, for the sake of simplification, that the diffusion coefficients and the atomic volumes of the two components are equal: DA=D B and nA=~B, but that the effective charges, z A and ZB, differ substantially: substances A and B are susceptible to electromigration to d i f f e r e n t degrees. The total atomic flow along the i-th layer (i=i,2) is d e t e r m i n e d by the gradient of the electric field, E, and by the internal stress gradient: D
~do
Ji =
(Ez i kT
)
(I)
dx
where n - the d i f f e r e n c e between atomic and vacancy (~V) volumes: n=~A-n v (we assume that the exchange of an atom for a vacancy is the elementary act of diffusion); z i - the average effective charge of atoms in the i-th layer: (2)
z i = ZAC i + ZB(l - Ci)
where C i is the average c o n c e n t r a t i o n of component A in the i-th layer. Initially CI=I , C2=0 and zI=ZA, z2=z B. The material balance in the region between the apertures is given by the condition, Ji=Const, i.e. the stress distribution is linear. In the aperture regions there are "sources" or "sinks" of material from the layer (depending on where the compressive stress is higher). Source or sink power is proportional to the difference in stresses in the layers (01 - 02) (as was said before, we assume that the mutual p e n e t r a t i o n is governed by diffusion creep) and to the thickness of the layer from which material issues. Let z A > z B. Then in the aperture near the anode end the substance passes from layer 1 to layer 2, and stationary stress d i s t r i b u t i o n provides a material balance: hl(dJl/dX ) =
hi(oI-O2)/~ (3a)
h 2 ( d J 2 / d x ) = -hi(oi-o2)/~ where ~ is the effective v i s c o s i t y and h i the thickness of the i-th layer. In the cathode aperture atoms pass from layer 2 to layer i: hl(dJi/dx ) =
h2(oi-o2)/~ (3b)
h 2 ( d J 2 / d x ) = -hI(oi-~2)/~ By this means, the rate of h o m o g e n i z a t i o n is d e t e r m i n e d by the stress difference (Ol-O2) in the aperture regions. This d i f f e r e n c e can be calculated from equations (1),(3). Taking into account the boundary conditions
(4)
Jilx=0 = Jilx=L = 0, i=i,2 we obtain: E(Zl-Z2) u = al-a 2 =
[ a [ K l C O S h ( x / a )-sinh(x/a) i' "Q, Q= ~ K 2 x + K3 | L-x L-x [bK4cosh ( )+sinh ( )] b b
x
(5)
5
Vol.
29, No.
S
T W O - L A Y E R THIN FILM
581
where a2=R2h2/(hl+h2) , b2~R2hl/(hl+h2), and R=(D~/kT)½ is the r e l a x a t i o n length. The c o e f f i c i e n t s , Ki, are d e t e r m i n e d from the fact of the continuity of the f u n c t i o n u and its d e r i v a t i v e s at points x=l and x=L-l. The total q u a n t i t y of m a t e r i a l e x c h a n g e d b e t w e e n the layers per unit time is equal to the total flow: 1 hl f Zex ](Ol-O2)dx
L h2 f = ](o2-ol)dx -
0
(6)
-
;~L-I
Using eqn. (5), f u r t h e r c a l c u l a t i o n yeilds:
Iex
=
hlh2 (z I - z 2 ) - G (hl+h 2 )
(7)
where (L/l)-2+(cosh~-l)/~sinh~+(coshB-l)/BsinhB G= (L/l)-2+coshe/esinha+coshB/BsinhB
and ~=I/a, B=i/b. The exchange of materials described alters the c o m p o s i t i o n of the layers, i.e.it results in h o m o g e n i z a t i o n . The average c o n c e n t r a t i o n s of c o m p o n e n t s in layers 1 and 2 can be found w i t h the aid of the kinetic equations: dCl/dt =
(Iex/Lhl) (C 2 - Cl)
dC2/dt =
(Iex/Lh2) (CI - C2)
(8) w h i c h take into a c c o u n t t h a t through one a p e r t u r e c o m p o n e n t A enters the layer and t h r o u g h the other it exits. It should be noted that the e x c h a n g e flow Iex d e p e n d s on the d i f f e r e n c e in c o n c e n t r a t i o n s (CI-C2) , since the d i f f e r e n c e of e f f e c t i v e charges, in a c c o r d a n c e w i t h (2), is equal: z I -z 2 = The kinetics reaction:
(9)
(z A - ZB) (C I -C2)
of h o m o g e n i z a t i o n
is thus
d e s c r i b e d as
a second
(i0)
d ( C i - C 2 ) / d t = -(CI-C2)2/T
where [=T0/G is T0=kTL/((ZA-ZB) "D-E ) .
the
characteristic
order
homogenization
time,
The h o m o g e n i z a t i o n time, ~, depends on the e f f e c t i v e v i s c o s i t y which in turn determines the v a l u e of c o e f f i c i e n t G. For ~=0 the relaxation length R=0 and G=I. In that case h o m o g e n i z a t i o n o c c u r s under conditions of free e l e c t r o m i g r a t i o n w i t h the e f f e c t i v e charge (ZA-ZB). If v i s c o s i t y is increased, and the r e l a x a t i o n length R comes to be c o m p a r a b l e to the film length, the h o m o g e n i z a t i o n time increases s u b s t a n t i a l l y (Fig.2). The i n f l u e n c e of the r e l a t i o n s h i p b e t w e e n the layer thicknesses, hl/h2, on • is weak. It governs the values to which the component
582
TWO-LAYER THIN FILM
Vol. 29, No. 5
concentrations tend during homogenization (Fig.3). When the aperture size, i, is increased, the homogenization time decreases (Fig.4). "Open" homogenization (when mutual penetration occurs over the entire interface) corresponds to an aperture size of I=L/2. As can be seen from Fig°4, the influence of the aperture size on ~ increases with the viscosity. The mechanism of homogenization under the action of electric current, described in this paper, and the kinetic regularities found can be used for analyzing the stability of layered thin films under given operating conditions.
Reference l.I.A.Blech, J.Appl. Phys., 47, 1203 (1976) 2.E.Glicman, A.Vilenkin, Proceed. 5th Inter. Conf. Quality Electr. Comp., Bordeau-France, 7-10 October, p.234 (1991) 3.M.A.Corhonin, P.Borgesen, C.-Y.Li, MRS Bulletin, 17, N7, 61, (1992) 4.A.Katsman, L.Klinger, L.Levin, E.Glickman, Proceed. 6th Israel Materials Engineering Conference, IMEC-VI, February, 1993 11-
.
0.8
9 0.6 ~
01
0'4t~0.2
C2
hl
=h2
0
53-
hi =0.1h2
1
3
4
0~"--,, (;2 , , 0 60 120
5
180
L/'r.,,
[JR
Fig.2. Homogenization time as a function of relaxation length R for constant film length L and different aperture size i. Fig.3. Concentration of component A in the layers 1 and 2 as a function of time for different ratios of layer thicknesses. Here 21=R=L.
o~I I
0.0
Fig.4. Homogenization time as a function of the aperture size 1 for different R; hl=h 2
0.1
0.2
0.3
Itk
0.4
0.5
Vol. 29, pp. 579-582, i[993 Printed in the U.S.A.
A
TWO-LAYER
THIN
FILM
UNDER
ELECTRIC
Pergamon Press Ltd. All rights reserved
CURRENT
L.Klinger, A.Katsman, L . L e v i n D e p a r t m e n t of M a t e r i a l s Engineering, Technion, Israel I n s t i t u t e of Technology, H a i f a (Received M a r c h
23, 1993)
Layered thin film structures are an e s s e n t i a l element of modern m i c r o e l e c t r o n i c devices. An i n d i s p e n s a b l e r e q u i r e m e n t of these elements is stability in e l e c t r i c and temperature fields. At the o p e r a t i n g temperatures of layered t h i n films, grain b o u n d a r y d i f f u s i o n occurs both within each layer and b e t w e e n layers. Under the action of an electric current through the multilayer film, internal stresses arise in conducting layers as a r e s u l t of e l e c t r o m i g r a t i o n . R e l a x a t i o n of these stresses may lead to the g r o w t h of h i l l o c k s on the outer s u r f a c e (the case for w h i c h e l e c t r o m i g r a t i o n has been c o n s i d e r e d in [1-4]). Where a conductor does not posses an outer surface, r e l a x a t i o n can occur by the mutual p e n e t r a t i o n of atoms from one layer to another. A stress g r a d i e n t across the layered s t r u c t u r e i n t e n s i f i e s the interdiffusion between layers which, in turn, a c c e l e r a t e s h o m o g e n i z a t i o n of the layered structure. In the p r e s e n t work a two-layer, two-component thin film on an undeformed substrate under the action of electric current will be studied (Fig.l). We shall c o n s i d e r a model s i t u a t i o n in w h i c h the mutual p e n e t r a t i o n of m a t e r i a l is assumed to be taking p l a c e through two s y m m e t r i c a l l y s i t u a t e d a p e r t u r e s near the ends of the film. The surface b e t w e e n the apertures constitutes a barrier layer. In the limiting case of the b a r r i e r layer being absent, the a p e r t u r e size is equal to half the film length. We shall assume that the i n t e r p e n e t r a t i o n is c o n t r o l l e d by d i f f u s i o n creep (without t h r e s h o l d stress), and that no conventional diffusion homogenization occurs at the same time (e.g.because of the limited s o l u b i l i t y of the components).
apertures
Electrical current
Fig.l.
M a s s t r a n s p o r t in two-layer, t w o - c o m p o n e n t thin metal film under e l e c t r i c a l c u r r e n t
579 0 9 5 6 - 7 1 6 X / 9 3 $6.00 + .00 Copyright (c) 1993 Pergamon Press Ltd.
580
TWO-LAYER
THIN FILM
Vol.
29, No.
Initially , one layer is enriched with component A, the other - with component B. We shall assume, for the sake of simplification, that the diffusion coefficients and the atomic volumes of the two components are equal: DA=D B and nA=~B, but that the effective charges, z A and ZB, differ substantially: substances A and B are susceptible to electromigration to d i f f e r e n t degrees. The total atomic flow along the i-th layer (i=i,2) is d e t e r m i n e d by the gradient of the electric field, E, and by the internal stress gradient: D
~do
Ji =
(Ez i kT
)
(I)
dx
where n - the d i f f e r e n c e between atomic and vacancy (~V) volumes: n=~A-n v (we assume that the exchange of an atom for a vacancy is the elementary act of diffusion); z i - the average effective charge of atoms in the i-th layer: (2)
z i = ZAC i + ZB(l - Ci)
where C i is the average c o n c e n t r a t i o n of component A in the i-th layer. Initially CI=I , C2=0 and zI=ZA, z2=z B. The material balance in the region between the apertures is given by the condition, Ji=Const, i.e. the stress distribution is linear. In the aperture regions there are "sources" or "sinks" of material from the layer (depending on where the compressive stress is higher). Source or sink power is proportional to the difference in stresses in the layers (01 - 02) (as was said before, we assume that the mutual p e n e t r a t i o n is governed by diffusion creep) and to the thickness of the layer from which material issues. Let z A > z B. Then in the aperture near the anode end the substance passes from layer 1 to layer 2, and stationary stress d i s t r i b u t i o n provides a material balance: hl(dJl/dX ) =
hi(oI-O2)/~ (3a)
h 2 ( d J 2 / d x ) = -hi(oi-o2)/~ where ~ is the effective v i s c o s i t y and h i the thickness of the i-th layer. In the cathode aperture atoms pass from layer 2 to layer i: hl(dJi/dx ) =
h2(oi-o2)/~ (3b)
h 2 ( d J 2 / d x ) = -hI(oi-~2)/~ By this means, the rate of h o m o g e n i z a t i o n is d e t e r m i n e d by the stress difference (Ol-O2) in the aperture regions. This d i f f e r e n c e can be calculated from equations (1),(3). Taking into account the boundary conditions
(4)
Jilx=0 = Jilx=L = 0, i=i,2 we obtain: E(Zl-Z2) u = al-a 2 =
[ a [ K l C O S h ( x / a )-sinh(x/a) i' "Q, Q= ~ K 2 x + K3 | L-x L-x [bK4cosh ( )+sinh ( )] b b
x
(5)
5
Vol.
29, No.
S
T W O - L A Y E R THIN FILM
581
where a2=R2h2/(hl+h2) , b2~R2hl/(hl+h2), and R=(D~/kT)½ is the r e l a x a t i o n length. The c o e f f i c i e n t s , Ki, are d e t e r m i n e d from the fact of the continuity of the f u n c t i o n u and its d e r i v a t i v e s at points x=l and x=L-l. The total q u a n t i t y of m a t e r i a l e x c h a n g e d b e t w e e n the layers per unit time is equal to the total flow: 1 hl f Zex ](Ol-O2)dx
L h2 f = ](o2-ol)dx -
0
(6)
-
;~L-I
Using eqn. (5), f u r t h e r c a l c u l a t i o n yeilds:
Iex
=
hlh2 (z I - z 2 ) - G (hl+h 2 )
(7)
where (L/l)-2+(cosh~-l)/~sinh~+(coshB-l)/BsinhB G= (L/l)-2+coshe/esinha+coshB/BsinhB
and ~=I/a, B=i/b. The exchange of materials described alters the c o m p o s i t i o n of the layers, i.e.it results in h o m o g e n i z a t i o n . The average c o n c e n t r a t i o n s of c o m p o n e n t s in layers 1 and 2 can be found w i t h the aid of the kinetic equations: dCl/dt =
(Iex/Lhl) (C 2 - Cl)
dC2/dt =
(Iex/Lh2) (CI - C2)
(8) w h i c h take into a c c o u n t t h a t through one a p e r t u r e c o m p o n e n t A enters the layer and t h r o u g h the other it exits. It should be noted that the e x c h a n g e flow Iex d e p e n d s on the d i f f e r e n c e in c o n c e n t r a t i o n s (CI-C2) , since the d i f f e r e n c e of e f f e c t i v e charges, in a c c o r d a n c e w i t h (2), is equal: z I -z 2 = The kinetics reaction:
(9)
(z A - ZB) (C I -C2)
of h o m o g e n i z a t i o n
is thus
d e s c r i b e d as
a second
(i0)
d ( C i - C 2 ) / d t = -(CI-C2)2/T
where [=T0/G is T0=kTL/((ZA-ZB) "D-E ) .
the
characteristic
order
homogenization
time,
The h o m o g e n i z a t i o n time, ~, depends on the e f f e c t i v e v i s c o s i t y which in turn determines the v a l u e of c o e f f i c i e n t G. For ~=0 the relaxation length R=0 and G=I. In that case h o m o g e n i z a t i o n o c c u r s under conditions of free e l e c t r o m i g r a t i o n w i t h the e f f e c t i v e charge (ZA-ZB). If v i s c o s i t y is increased, and the r e l a x a t i o n length R comes to be c o m p a r a b l e to the film length, the h o m o g e n i z a t i o n time increases s u b s t a n t i a l l y (Fig.2). The i n f l u e n c e of the r e l a t i o n s h i p b e t w e e n the layer thicknesses, hl/h2, on • is weak. It governs the values to which the component
582
TWO-LAYER THIN FILM
Vol. 29, No. 5
concentrations tend during homogenization (Fig.3). When the aperture size, i, is increased, the homogenization time decreases (Fig.4). "Open" homogenization (when mutual penetration occurs over the entire interface) corresponds to an aperture size of I=L/2. As can be seen from Fig°4, the influence of the aperture size on ~ increases with the viscosity. The mechanism of homogenization under the action of electric current, described in this paper, and the kinetic regularities found can be used for analyzing the stability of layered thin films under given operating conditions.
Reference l.I.A.Blech, J.Appl. Phys., 47, 1203 (1976) 2.E.Glicman, A.Vilenkin, Proceed. 5th Inter. Conf. Quality Electr. Comp., Bordeau-France, 7-10 October, p.234 (1991) 3.M.A.Corhonin, P.Borgesen, C.-Y.Li, MRS Bulletin, 17, N7, 61, (1992) 4.A.Katsman, L.Klinger, L.Levin, E.Glickman, Proceed. 6th Israel Materials Engineering Conference, IMEC-VI, February, 1993 11-
.
0.8
9 0.6 ~
01
0'4t~0.2
C2
hl
=h2
0
53-
hi =0.1h2
1
3
4
0~"--,, (;2 , , 0 60 120
5
180
L/'r.,,
[JR
Fig.2. Homogenization time as a function of relaxation length R for constant film length L and different aperture size i. Fig.3. Concentration of component A in the layers 1 and 2 as a function of time for different ratios of layer thicknesses. Here 21=R=L.
o~I I
0.0
Fig.4. Homogenization time as a function of the aperture size 1 for different R; hl=h 2
0.1
0.2
0.3
Itk
0.4
0.5