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Driving force for grain boundary migration during electromigration: Effect of elastic anisotropy
Scripta Materialia, Vol. 36, No. 4, pp. 489-493, 1997 Elsevier Science Ltd Copyright 0 1997 Acta Metalhqica Inc. Printed in the USA. All rights reserved 1359-6462/97 $17.00 + .OO
Pergalmon
PII Sl359-6462(96)00397-l
DRIVING FORCE FOR GRAIN BOUNDARY MIGRATION DURING ELECTROM~GRATION: EFFECT OF ELASTIC ANISOTROPY A. Katsman, L. Klinger and L. Levin Department of Materials Engineering, Technion, Haifa 32000, Israel (Received July 12, 1996) (Accepted October 4, 1996) Introduction Mass transport along grain boundaries during electromigration in thin metal films may lead to the generation of internal stresses. Due to crystal anisotropy these stresses may differ in adjacent grains of different orientations relative to the grain boundary and to the direction of electromigration. In such a case the stress gradient across the grain boundary may cause coherency strain energies in adjacent grains to differ resulting in a driving force for grain boundary migration. This driving force is analyzed in the present work. Strains Caused by Electromigration A redistribution of atoms and vacancies along interconnect lines during electromigration gives rise to inelastic strains in the film [ 11: E; = E; = ;[(Cv
i = 1,2,3,
-C..)dt,
(1)
where C, and C,, are respectively the current and the equilibrium vacancy concentrations z is the average lifetime of the vacancies. The evolution of these strains is determined by the time dependence of the vacancy concentration, which can be described by the diffusion equation comprising a sink/source term :
ac, aJ” _=--at
at
where J, , the vacancy flux, is described by
489
C”
-cw 7
’
(2)
BOUNDARY
490
J,=
MIGRATION
DURING ELECTROMIGRATION
Vol. 36, No. 4
-D,
in which D, is the diffusion coefficient of the vacancies, E is the electric field, z* is the effective charge, Q and &2.,are, respectively, the atomic and the vacancy volumes, OH= (crir + oz2 + ts33)/3is the internal hydrostatic stress caused by the redistribution of atoms during electromigration. In a thin film attached to a massive undeformed substrate (in the X-Y plane) a plane stress state is initiated, in which only two components of the stress tensor, oll and oz2 are not equal to zero. In an isotropic crystal: 011 =022=-&/(1-V)
(4)
where Y is Young’s modulus v is the Poisson ratio. Using eqs.( l)-(4) the equation for the evolution of inelastic strains can be derived. After a certain transient time Tt - (kT/QY)(l’/D), where 1 is the average distance between vacancy sinks and D=D,C,, is the self-diffusion coefftcient, the inelastic strains are governed by a diffusion-like equation:
where D, =D&/kT,
Y=2Y/3(1-v).
A similar equation for the evolution of internal stresses
during electromigration, but taking account of possible stress relaxation, was obtained earlier [2]. By this means, the inelastic strains develop according to the diffusion law with an effective diffusion coefficient D,. The steady state distribution of the strains is linear. Assuming zero strain at the cathode end (which is displaced due to the formation and coalescence of voids), yields: EP =E 0
,,(l-x/L),
(6)
where L is the film length. These strains arise in adjacent grains having, in the general case, different orientations relative to the direction of diffusion, to the grain boundary plane, and to the substrate plane, which may lead to different internal stresses in these grains. Let us consider two adjacent grains in a film attached to a massive substrate, with the grain boundary perpendicular to the direction of electromigration, x. For the sake of simplicity we will consider the case of a tilt grain boundary when the grain boundary plane contains the tilt axis. The two neighboring grains are rotated round the common crystallographic axis through different angles, c1t and a2 (Fig. 1.). As was mentioned above, electromigration causes inelastic strains in the material, which are described by eq.(5) and, in the steady-state case, by eq.(6). The inelastic strains, in turn, lead to elastic strains and corresponding internal stresses. The components of the total strains in the XY plane are equal to zero due to the rigidity of the substrate: Ey
where
=Ei
+E&
=0,
i,j=1,2,
(7)
~61 me the elastic strains. Since the external surface is free of external forces, the stress tensor
components,
033,
013,
and
023,
equal to zero:
Vol. 36,~No.4
BOUNDARYMIGRATIONDURINGELECTROMIGRATION
491
Electrical current Figure 1. The scheme of two adjacent grains in a film attached to a massive substrate, with the grain boundary perpendicular to the direction of electromigration, x. The grains are rotated round a common axis, y, through different angles, a, and az.
033
=013
=<223
=o
(8)
Using Hooke’s law for the stress tensor components in each of grains, and eqs.(7),(8), the components of the stress tensor can be found:
+c(“) 2233
1 E&(X)
)
(9)
where m is the ordinal number of the grain (m = 1 or 2). The strain along the z-axis: (In) _ E33
cl;; +c(“) 2233 &P(X),
c(m) 3333
(10)
The components of the elastic moduli tensor of grain m in the chosen coordinate system differ from those in the crystallographic coordinate system:
cy=Cf + (-l)i-j+’ 2Asin2a,cos2a,(l-6i,)(1-6j,),
(11)
where q is the cardinal number of the axis of rotation, A = Cp, - Cpz - 2C& is the anisotropy parameter. As can be seen, when the tilt axis is perpendicular to the substrate plane, the components of the stress tensor are independent of the grain’s angle of rotation; i.e. the stresses arising during electromigration are the same in both adjacent grains. When the axis of rotation is a tilt axis [OlO] (or a twist axis [lot>]), the components of the stress tensor are functions of the angle of rotation. The strain along the z-axis is also a function of that angle. When the strains along the z-axis in grain 1 differ from those in grain 2, additional stresses may arise: If the grains can slip along the grain boundary, no additional stress appears. If they cannot, additional stresses arise inside the grains. Since the edges of the film are assumed to be free, the additional stresses in grain 1 are equal in value to those in grain 2 but of opposite sign [3]. These additional stresses are not considered in the present paper.
492
BOUNDARY MIGRATION
DURING ELECTROMIGRATION
Vol. 36, No. 4
The coherency strain energy density of grain m is given by WE =YO(cL,)&zp
(12)
(C;, +2C;,)(C;, -CpZ -ISA,) where Y’(n,)
=
is the effective modulus. This function coincides C;, -Aaiam with that governing the coherency strain energy density for a one-dimensional composition modulation in a crystal lattice (neglecting shear strains) [4]. The Driving Force for Grain Boundary Migration.
The driving force for grain boundary migration is defined as the Gibbs energy change of the system caused by grain boundary displacement. The elastic energies of adjacent grains arising during electromigration, may be different due to crystal anisotropy and different orientations of grains relative to the substrate plane. In that case the driving force for grain boundary migration is given by: AG”“‘“/0=AWE =[Y’(ai)-Y’(u,)]a;
(13)
The said difference is a function of the angles through which the grains rotate around the common axis. Using the expression for Y”(a,), that difference can be approximated by the expression: AWE =K*A*(sin*2crl
where K =
-sin22aZ)E:
(14)
(CP, +2cPz)(G +G) 6(CP,1= Discussion
As can be seen from eq.( 14), the driving force is zero for isotropic crystals (A = 0) and for symmetrical grain boundaries (a, = a~) for any type of crystal. However, crystals are generally anisotropic, A # 0, and most grain boundaries or, at least, some parts of a boundary, are not symmetrical. In the latter cases, a special driving force for grain boundary migration arises in accordance with eqs (13),( 14). Let us compare this driving force with other possible driving forces for grain boundary migration. Using elastic moduli for copper (C ,, = 170GPa, Cl2 = 123GPa, A = -103GPa) yields K=0.998. The = oJY, where ot is the threshold stress after which maximum value of up,Emax,canbe calculated as E,,,~~ the stresses relax (for example, by hillock growth in the relaxation zone d > @. Since, in thin metal films, crt - 1O”Y, the inelastic strains can reach values of about 0.00 1. The driving force, eq.(14), as a function of the angle of rotation, aI,for a2 = 0 and E, = 10-3 , is presented in Fig.2. The maximum value of the driving force corresponding al = 45“, AW = 0.1 MPa, is comparable to the driving force for grain boundary migration due to grain boundary curvature, AG”““/R = 2r/R, where r is the grain boundary free energy and R is the average radius of curvature, AG”“/Q = 0.1 MPa, for typical values of r = 0.5.1/mzand R=lO~‘m.
Vol. 36, No. 4
BOUNDARY MIGRATION DURING ELECTROMIGRATION
493
Figure 2. The difference between the coherency strain energy densities in grains 1 and 2 as a function of the angle of rotation, al, of grain 1 round the common axis, y, when uz = 0, q, = 10”.
Thus, the driving force due to elastic anisotropy may be compaable to, or even exceed, that caused by grain boundary curvature. The driving force described may thus be significant in the migration of grain boundaries during electromigration. It may cause low-temperature recrystallization in the regions close to the stress relaxation zone and inside this zone and may also lead to an increase in the number of symmetrical, grain boundaries during electromigration in the near-anode zone of the interconnect lines. The direction of grain boundary migration under the action of the driving force described is determined by the values of the angles of rotation of neighboring grains and by the sign of the anisotropy parameter, A. When the latter is negative, (as for most of metals), the compressive stress generated during the electromigration is larger in the grain rotated through the greater angle relative to the grain boundary. Accordingly the grain boundary will move in the direction of this grain. Conclusion Electromigratison in thin metal films leads to internal stresses. Elastic crystal anisotropy may cause a distinct gradient of these stresses across the grain boundary if the adjacent grains have different orientations relative to the grain boundary and to the substrate. The stress gradient gives rise to a driving force for grain boundary migration. This driving force is proportional to the crystal anisotropy parameter and to the dilatation caused by electromigration. At typical values of the parameters the driving force is comparable to the driving force caused by the curvature when the radius of curvature is about lOp,m. The anisotropy driving force may cause low temperature recrystallization, especially in the regions where internal stresses are about the threshold value, or exceed it, and it may lead to an increase in the number of symmetrical gram boundaries during electromigration in the near-anode zone of interconnect lines. References R. Kirchheim, Acta Metall. Mater., 42 (1992) 309. L. Klinger, E. Glickman, A. Katsman and L. Levin, Mater. Sci. Eng, B23 (1994) 15. A. Katsman, L. Klinger, L. Levin, E. Rabkii and W. Gust, in Materials Engineering Conference, edited by A. Rosen and R. Chaim (The 7th Israel Mat. Eng. Conf. Proc., Haifa, Israel, 1994), pp.272-277. J.W. Cabn, Actametall., 10, 179 (1962).
Pergalmon
PII Sl359-6462(96)00397-l
DRIVING FORCE FOR GRAIN BOUNDARY MIGRATION DURING ELECTROM~GRATION: EFFECT OF ELASTIC ANISOTROPY A. Katsman, L. Klinger and L. Levin Department of Materials Engineering, Technion, Haifa 32000, Israel (Received July 12, 1996) (Accepted October 4, 1996) Introduction Mass transport along grain boundaries during electromigration in thin metal films may lead to the generation of internal stresses. Due to crystal anisotropy these stresses may differ in adjacent grains of different orientations relative to the grain boundary and to the direction of electromigration. In such a case the stress gradient across the grain boundary may cause coherency strain energies in adjacent grains to differ resulting in a driving force for grain boundary migration. This driving force is analyzed in the present work. Strains Caused by Electromigration A redistribution of atoms and vacancies along interconnect lines during electromigration gives rise to inelastic strains in the film [ 11: E; = E; = ;[(Cv
i = 1,2,3,
-C..)dt,
(1)
where C, and C,, are respectively the current and the equilibrium vacancy concentrations z is the average lifetime of the vacancies. The evolution of these strains is determined by the time dependence of the vacancy concentration, which can be described by the diffusion equation comprising a sink/source term :
ac, aJ” _=--at
at
where J, , the vacancy flux, is described by
489
C”
-cw 7
’
(2)
BOUNDARY
490
J,=
MIGRATION
DURING ELECTROMIGRATION
Vol. 36, No. 4
-D,
in which D, is the diffusion coefficient of the vacancies, E is the electric field, z* is the effective charge, Q and &2.,are, respectively, the atomic and the vacancy volumes, OH= (crir + oz2 + ts33)/3is the internal hydrostatic stress caused by the redistribution of atoms during electromigration. In a thin film attached to a massive undeformed substrate (in the X-Y plane) a plane stress state is initiated, in which only two components of the stress tensor, oll and oz2 are not equal to zero. In an isotropic crystal: 011 =022=-&/(1-V)
(4)
where Y is Young’s modulus v is the Poisson ratio. Using eqs.( l)-(4) the equation for the evolution of inelastic strains can be derived. After a certain transient time Tt - (kT/QY)(l’/D), where 1 is the average distance between vacancy sinks and D=D,C,, is the self-diffusion coefftcient, the inelastic strains are governed by a diffusion-like equation:
where D, =D&/kT,
Y=2Y/3(1-v).
A similar equation for the evolution of internal stresses
during electromigration, but taking account of possible stress relaxation, was obtained earlier [2]. By this means, the inelastic strains develop according to the diffusion law with an effective diffusion coefficient D,. The steady state distribution of the strains is linear. Assuming zero strain at the cathode end (which is displaced due to the formation and coalescence of voids), yields: EP =E 0
,,(l-x/L),
(6)
where L is the film length. These strains arise in adjacent grains having, in the general case, different orientations relative to the direction of diffusion, to the grain boundary plane, and to the substrate plane, which may lead to different internal stresses in these grains. Let us consider two adjacent grains in a film attached to a massive substrate, with the grain boundary perpendicular to the direction of electromigration, x. For the sake of simplicity we will consider the case of a tilt grain boundary when the grain boundary plane contains the tilt axis. The two neighboring grains are rotated round the common crystallographic axis through different angles, c1t and a2 (Fig. 1.). As was mentioned above, electromigration causes inelastic strains in the material, which are described by eq.(5) and, in the steady-state case, by eq.(6). The inelastic strains, in turn, lead to elastic strains and corresponding internal stresses. The components of the total strains in the XY plane are equal to zero due to the rigidity of the substrate: Ey
where
=Ei
+E&
=0,
i,j=1,2,
(7)
~61 me the elastic strains. Since the external surface is free of external forces, the stress tensor
components,
033,
013,
and
023,
equal to zero:
Vol. 36,~No.4
BOUNDARYMIGRATIONDURINGELECTROMIGRATION
491
Electrical current Figure 1. The scheme of two adjacent grains in a film attached to a massive substrate, with the grain boundary perpendicular to the direction of electromigration, x. The grains are rotated round a common axis, y, through different angles, a, and az.
033
=013
=<223
=o
(8)
Using Hooke’s law for the stress tensor components in each of grains, and eqs.(7),(8), the components of the stress tensor can be found:
+c(“) 2233
1 E&(X)
)
(9)
where m is the ordinal number of the grain (m = 1 or 2). The strain along the z-axis: (In) _ E33
cl;; +c(“) 2233 &P(X),
c(m) 3333
(10)
The components of the elastic moduli tensor of grain m in the chosen coordinate system differ from those in the crystallographic coordinate system:
cy=Cf + (-l)i-j+’ 2Asin2a,cos2a,(l-6i,)(1-6j,),
(11)
where q is the cardinal number of the axis of rotation, A = Cp, - Cpz - 2C& is the anisotropy parameter. As can be seen, when the tilt axis is perpendicular to the substrate plane, the components of the stress tensor are independent of the grain’s angle of rotation; i.e. the stresses arising during electromigration are the same in both adjacent grains. When the axis of rotation is a tilt axis [OlO] (or a twist axis [lot>]), the components of the stress tensor are functions of the angle of rotation. The strain along the z-axis is also a function of that angle. When the strains along the z-axis in grain 1 differ from those in grain 2, additional stresses may arise: If the grains can slip along the grain boundary, no additional stress appears. If they cannot, additional stresses arise inside the grains. Since the edges of the film are assumed to be free, the additional stresses in grain 1 are equal in value to those in grain 2 but of opposite sign [3]. These additional stresses are not considered in the present paper.
492
BOUNDARY MIGRATION
DURING ELECTROMIGRATION
Vol. 36, No. 4
The coherency strain energy density of grain m is given by WE =YO(cL,)&zp
(12)
(C;, +2C;,)(C;, -CpZ -ISA,) where Y’(n,)
=
is the effective modulus. This function coincides C;, -Aaiam with that governing the coherency strain energy density for a one-dimensional composition modulation in a crystal lattice (neglecting shear strains) [4]. The Driving Force for Grain Boundary Migration.
The driving force for grain boundary migration is defined as the Gibbs energy change of the system caused by grain boundary displacement. The elastic energies of adjacent grains arising during electromigration, may be different due to crystal anisotropy and different orientations of grains relative to the substrate plane. In that case the driving force for grain boundary migration is given by: AG”“‘“/0=AWE =[Y’(ai)-Y’(u,)]a;
(13)
The said difference is a function of the angles through which the grains rotate around the common axis. Using the expression for Y”(a,), that difference can be approximated by the expression: AWE =K*A*(sin*2crl
where K =
-sin22aZ)E:
(14)
(CP, +2cPz)(G +G) 6(CP,1= Discussion
As can be seen from eq.( 14), the driving force is zero for isotropic crystals (A = 0) and for symmetrical grain boundaries (a, = a~) for any type of crystal. However, crystals are generally anisotropic, A # 0, and most grain boundaries or, at least, some parts of a boundary, are not symmetrical. In the latter cases, a special driving force for grain boundary migration arises in accordance with eqs (13),( 14). Let us compare this driving force with other possible driving forces for grain boundary migration. Using elastic moduli for copper (C ,, = 170GPa, Cl2 = 123GPa, A = -103GPa) yields K=0.998. The = oJY, where ot is the threshold stress after which maximum value of up,Emax,canbe calculated as E,,,~~ the stresses relax (for example, by hillock growth in the relaxation zone d > @. Since, in thin metal films, crt - 1O”Y, the inelastic strains can reach values of about 0.00 1. The driving force, eq.(14), as a function of the angle of rotation, aI,for a2 = 0 and E, = 10-3 , is presented in Fig.2. The maximum value of the driving force corresponding al = 45“, AW = 0.1 MPa, is comparable to the driving force for grain boundary migration due to grain boundary curvature, AG”““/R = 2r/R, where r is the grain boundary free energy and R is the average radius of curvature, AG”“/Q = 0.1 MPa, for typical values of r = 0.5.1/mzand R=lO~‘m.
Vol. 36, No. 4
BOUNDARY MIGRATION DURING ELECTROMIGRATION
493
Figure 2. The difference between the coherency strain energy densities in grains 1 and 2 as a function of the angle of rotation, al, of grain 1 round the common axis, y, when uz = 0, q, = 10”.
Thus, the driving force due to elastic anisotropy may be compaable to, or even exceed, that caused by grain boundary curvature. The driving force described may thus be significant in the migration of grain boundaries during electromigration. It may cause low-temperature recrystallization in the regions close to the stress relaxation zone and inside this zone and may also lead to an increase in the number of symmetrical, grain boundaries during electromigration in the near-anode zone of the interconnect lines. The direction of grain boundary migration under the action of the driving force described is determined by the values of the angles of rotation of neighboring grains and by the sign of the anisotropy parameter, A. When the latter is negative, (as for most of metals), the compressive stress generated during the electromigration is larger in the grain rotated through the greater angle relative to the grain boundary. Accordingly the grain boundary will move in the direction of this grain. Conclusion Electromigratison in thin metal films leads to internal stresses. Elastic crystal anisotropy may cause a distinct gradient of these stresses across the grain boundary if the adjacent grains have different orientations relative to the grain boundary and to the substrate. The stress gradient gives rise to a driving force for grain boundary migration. This driving force is proportional to the crystal anisotropy parameter and to the dilatation caused by electromigration. At typical values of the parameters the driving force is comparable to the driving force caused by the curvature when the radius of curvature is about lOp,m. The anisotropy driving force may cause low temperature recrystallization, especially in the regions where internal stresses are about the threshold value, or exceed it, and it may lead to an increase in the number of symmetrical gram boundaries during electromigration in the near-anode zone of interconnect lines. References R. Kirchheim, Acta Metall. Mater., 42 (1992) 309. L. Klinger, E. Glickman, A. Katsman and L. Levin, Mater. Sci. Eng, B23 (1994) 15. A. Katsman, L. Klinger, L. Levin, E. Rabkii and W. Gust, in Materials Engineering Conference, edited by A. Rosen and R. Chaim (The 7th Israel Mat. Eng. Conf. Proc., Haifa, Israel, 1994), pp.272-277. J.W. Cabn, Actametall., 10, 179 (1962).