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Motion of an elliptical void in interconnects embedded in matrix under gradient stress field
Thin Solid Films 519 (2011) 4256–4261
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Thin Solid Films j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / t s f
Motion of an elliptical void in interconnects embedded in matrix under gradient stress field H.J. Xie, X. Wang ⁎, S. Li, Z. Li School of Naval Architecture, Ocean and Civil Engineering, (State Key Laboratory of Ocean Engineering), Shanghai Jiaotong University, Shanghai 200240, PR China
a r t i c l e
i n f o
Article history: Received 29 June 2010 Received in revised form 14 February 2011 Accepted 15 February 2011 Available online 24 February 2011 Keywords: Interconnects embedded in a matrix Motion of an elliptical void Orthotropic elastic model Thermodynamics potential
a b s t r a c t We present an analytical solution for the motion of an elliptical void in representative interconnects embedded in a matrix with different line aspect (volume) ratio, under gradient stress field. An orthotropic elastic model is used to represent representative interconnects embedded in a matrix. The effects of stress gradient, stress states, an equivalent void size, the orthotropic material characteristic, and the shape parameter of the void on the motion velocity of an elliptical void are described and discussed. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Interconnection film lines are used in a chip, which are thin wires of copper or aluminum alloy. The mechanical reliability of interconnection film lines remains a serious concern for the microelectronics industry. Over the past decades, the width of metal interconnection film lines used in a chip has gradually decreased because of the miniaturization of microelectronics devices [1,2]. In processing of small-scale-metallic lines, various voids and other structural defects in the metallic line are often generated, which have detrimental effects on the function of microelectronic devices [3–8]. The lines are often subjected to severe gradient stresses induced by thermal deformation mismatching between the line and the surrounding matrix, which induces voids in the interconnection lines embedded in matrix to motion. The void coalescence in stress concentration region as a result of void motion is one of the major damage mechanisms. The dynamics of stress-driven void in metallic solids has been generating great research interest because of the failure of metallic interconnects in integrated circuits [9–15]. Therefore, the void motion-induced failure poses a great concern both in microelectronic devices and in common engineering structures. In actual devices, metallic interconnects in a chip are embedded in an insulation matrix. Elastic body composed of metallic interconnection film lines and insulation matrix is an effective orthotropic elastic body. Thus, one of the key challenges
⁎ Corresponding author. E-mail address: [email protected] (X. Wang). 0040-6090/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2011.02.057
is the formulation of a robust continuum theory for the motion of void in an effective orthotropic elastic body under applied stress field. Analytical solutions for the motion of circular and elliptical voids by electro-migration (EM)-driven have been given by authors [16–18]. By utilizing different numerical simulations [9,10,19], the stressdriven void motion has also been described. However, an analytical solution for the motion of an elliptical void in representative interconnection film lines embedded in a matrix for different line aspect (volume) ratios, under a gradient stress field, has not been presented so far. It is seen from previous works [2–9] that when the driving force for void migration is low, a steady state can be reached and the void shape remains unchanged after that, and when the driving force is large, the void shape may appear in an instable state and evolves into crack-like configurations or even breaks up. Although the initial void shape is a non-steady state at the early stage and is generally different from the steady-state shape, the initial void shape evolves towards the steady state shape at low driving force. Based on surface diffusion, the investigation on void shape evolution under uniform biaxial and triaxial stress states has been presented in Ref. [4], where the void does not move, and changes its shape only. The shape in steady state is a family of ellipses, and will collapse to a slit when the applied stress reaches a critical value. Therefore, in this paper, we can assume that an initial void under low driving force evolves into a stable elliptical shape, and local surface geometry changes arising from the surface diffusion of void cannot change the elliptic shape of the void, so that the applied stress gradient does not change the symmetry of void. According to the foregoing assumption, we give an analytical solution for the motion of the stable elliptical void in representative
H.J. Xie et al. / Thin Solid Films 519 (2011) 4256–4261
interconnection film lines embedded in a matrix for a different line aspect (volume) ratio, under gradient stress field. An orthotropic elastic model is found to represent representative interconnection film lines embedded in a matrix, and the void size is considered as very small compared to the dimension of orthotropic elastic model composed of finite-size interconnection film lines and their surrounding matrix. It is seen from the result that the motion characteristics of an elliptical void in representative interconnection film lines embedded in a matrix are mainly affected by stress gradient, stress states, an equivalent void size, the orthotropic material characteristic, and the shape parameter of the void, which is signification for concerning the mechanical reliability of interconnection film lines in a chip and some microelectronics. 2. Govern equations and solving method Fig. 1 shows that an elliptic void appears in the integrated circuits composed of a metal interconnection line and a matrix surrounding the line under gradient stress field induced by thermal mismatch. In Fig. 1, the global coordinate system OXY is fixed at the symmetric centre of a metal interconnection line embedded in matrix, the local (moving) coordinate system oxy is fixed at the centre of an elliptic void, and xL represents a moving distance from the centre of an elliptic void to the symmetric centre of an interconnection line embedded in matrix. The relation between two coordinate systems is given by xL = X − x and y = Y. The shape of an elliptic void having the same area as a circle of radius ρ can be described by x=ρ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1+m cos θ; 1−m
y=ρ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−m sin θ 1+m
ð1Þ
where θ is the angle between the vector radius ρ of ellipse and x-axes, and m is the shape parameter of the void. The shape parameter m = 0 corresponds to a circle, m → + 1 to x-direction slit and m → − 1 to ydirection slit, respectively. The void in the materials may move by the atomic migration of a diffuse surface layer from regions of high chemical potential to those of low chemical potential along the void surface [9]. Because the surface diffusion is generally much faster than the bulk one, only the surface diffusion is considered as the major mechanism for the void motion. In the absence of electrical current, the chemical potential inducing atomic diffusion along the void surface is written as [9] μ = μ 0 −γ s κΩ + W ρ Ω
ð2Þ
where μ0 is the reference value of the potential, γs is the free surface energy of void, κ is the surface curvature of the void, positive for
qY = pX, ( p >0)
EX = Ei Vi + Em ð1−Vi Þ; EY =
Ei Em Ei ð1−Vi Þ + Em Vi
vXY = vi Vi + vm ð1−Vi Þ; GXY =
Gi Gm Gi ð1−Vi Þ + Gm Vi
ð3Þ
where for plane stress, Ei,vi, Gi and Em,vm,Gm represent the Young's modulus, Poisson ratio and shear modulus of a metal interconnection line and a matrix, respectively, for plain strain Ei = Ei/(1 − 2vi) and Em = Em/(1 − 2vm), and Vi = bi/(bi + bm) represents an interconnection line volume ratio to the total volume of metal interconnection line and matrix. In Fig. 2, J represents the atomic flux on the void surface, the number of atoms per time crossing unit length on the void surface, which satisfies Nernst–Einstein equation as follows D δ ∂μ Dδ J=− s s = s s ΩKT ∂s KT
∂κ ∂Wρ γs − ∂s ∂s
! ð4Þ
where K represents the Boltzmann's constant, T the absolute temperature, Ds the lattice diffusion coefficient on the free surface of void, δs the diffusion layer thickness of the void surface and s the arc length of the free surface, in which all locate on the void surface. It is seen from Eq. (4) that the atomic flux on the void surface is dependent on the variation of the free energy and the strain energy on the void surface. The mass conservation equation of void surface diffusivity is given by [9] dJ V =− n ds Ω
ð5Þ
From Fig. 2, it is seen that Vn represents the normal velocity of the void surface, and is written as Vn = V sin φ
xL Interconnection lines
SiO2 matrix ( Em , vm , Gm)
convex, Wρ is the strain energy stored in the bulk elastic material associated with an atom, and Ω is the atomic volume. Thus, the key point to obtain an analytical expression for the chemical potential inducing atomic diffusion along an elliptic void surface is to solve the strain energy along the surface of the elliptic void in an interconnection line embedded in matrix (Fig. 1). Here, the width of the matrix surrounding a metal interconnection line is not equal to the width of the metal interconnection line. Because the size of an elliptic void in the interconnection line embedded in a matrix is very small as compared with the width of the interconnect, its effect on the EM-induced stress along the interconnect/passivation interface is negligible, so that the typical element of an interconnection line embedded in matrix (Fig. 1) is approximately considered as an infinite orthotropic elastic body with an elliptic void, shown in Fig. 2. The corresponding effective elastic modulus of the orthotropic material along the coordinate axis X and along the coordinate axis Y are, respectively, given by [20]
Y
SiO2 matrix ( Em , vm , Gm )
( Ei , vi , Gi )
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y
where V represents the horizontal velocity of the void surface. Substituting Eq. (6) into Eq. (5) and utilizing dy = − (sin φ)ds, Eq. (5) can be rewritten as
0.5bm x
O
bi
ð6Þ
X
dJ V = dy Ω
ð7Þ
0.5bm
Fig. 1. An elliptical void in an interconnection line embedded in a matrix, under gradient stress field along the global coordinate axis X.
Generally, the morphologically stable shape for a void under a linearly distributed stress gradient field is not an ellipse. But, when the average value of gradient stress is lower, the elliptical shape for a void may be an energetically favorable steady state. To consider the effect of the local surface changes of a moving elliptical void on the atomic flux along the ellipse, we integrate Eq. (7)
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qYm
pxL
qY 1
Y
px
Infinite effective orthotropic electric body
y
xL
A
O
Vn
J
X o
J
x
V
Fig. 2. An effective computing model for the motion of an elliptical void in an interconnection line embedded in a matrix, under gradient stress field, qY = qYm + qY1.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi along the elliptic surface from y to y = ρ ð1−mÞ = ð1 + mÞ, as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1−mÞ = ð1 + mÞ V dy ∫dJ = ∫y Ω ρ
ð7aÞ
Because the horizontal velocity V of the void surface is a constant along the void free surface when the elliptic shape of void does not vary during motion, from Eq. (7a), we have rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! V 1−m J= ρ −y + C Ω 1+m
ð8Þ
where C is the undetermined constant which can be solved by the continuous condition of the atomic flux at the apex A of the elliptic void. Utilizing Eqs. (4) and (8), the continuous condition of the atomic flux at the apex A of elliptic void is expressed as " # " # rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! Ds δs ∂κ ∂Wρ V 1−m ρ −y + C γs − = Ω 1+m KT ∂s ∂s A
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
y=ρ
ð1−mÞ = ð1 + mÞ
ð9Þ Because ∂ κ/∂ s in Eq. (4) at the apex A of elliptic void equals zero, solving Eq. (9) gives A D δ ∂Wρ C=− s s K T ∂s
ð10Þ
where WρA is the strain energy density at the apex A of the elliptic void, and is expressed as a function of the circumferential stress σ θA at the apex A and the local modulus at the point, as follows
A
Wρ =
2 σ θA
ð11Þ
2Ei
where Ei is the local Young's modulus on the void surface for plane stress, or for plain strain Ei = Ei/(1 − 2vi). Due to the symmetrical requirement of void surface diffusion shown in Fig. 2, the atom flux at y = 0 should be zero at all times, i.e., ½ J y = 0
Vρ = Ω
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−m +C=0 1+m
ð12Þ
in an effective orthotropic elastic body under remote gradient stress field, and the effective elastic modulus in Eq. (3) induced by an interconnection line volume ratio to the total volume of metal interconnection line and matrix. So far the analytic expression for the stress field at the surface of the elliptic void in an orthotropic elastic body under gradient stress field shown in Fig. 1 is not presented in literatures. Here, utilizing a loading superposition method, a gradient stress field, qY = pX, applied on an effective orthotropic body with an elliptic void can be described from a superposition of an antisymmetric stress field of stress gradient with respect to the y-axis of the moving elliptic void and a uniform stress field, as shown in Fig. 2, i.e., ð14Þ
qY = qYm + qY1
Based on the linear theory of elasticity, the circumferential stress σAθ at the apex A of void in an effective orthotropic elastic body under coupled loadings qY = qYm + qY1 is given by A
A
A
σ θ ðqY = pX Þ = σ θ ðqYm = pxL Þ + σ θ ðqY1 = pxÞ
Because an antisymmetric gradient stress field, qY1 = px, with respect to the y-axis of the moving elliptic void makes the circumferential stress σAθ at the apex A of the void in an effective orthotropic elastic body to equal zero, the circumferential stress σθA at the apex A of the void in an effective orthotropic elastic body under remote gradient stress field can be replaced by the circumferential stress σθA at the apex A of the void under a uniform applied stress field, i.e., A
A
σ θ ðqY = pX Þ = σ θ ðqYm = pxL Þ
A
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 1 + m ∂σ θ 1−m ∂s
ð16Þ
Thus, the circumferential stresses at the short axis apex A of an elliptic void in an infinite orthotropic body under gradient stress field are written as [21] 1 1 E 2 E 2 A σ θ = − X pxL = − X pðX−xÞ EY EY
ð17Þ
On the other hand, from Eq. (17), the circumferential stress gradient at the apex A of the elliptic void in an effective orthotropic elastic body under remote gradient stressed field is written as 1 A A ∂σ θ ∂σ E 2 =− θ =− X p EY ∂s ∂x
ð18Þ
Substituting Eqs. (17) and (18) into Eq. (13) gives an analytical solution for the moving velocity of elliptic void in an interconnection line embedded in a matrix under gradient stress field, as follows V=
ΩDs δs EX p2 KTEi EY ρ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1+m x 1−m L
ð19Þ
In addition, applying a uniform stress field qXm on the two ends of the effective orthotropic elastic body along X-direction, we have [21] 8 " 1 #1 9 1 < 1−m 1 EX 2 EX −2vXY GXY 2 = E 2 = qXm 1 + 2 + − X pxL : ; 1 + m 2 EY 4GXY EY
A σθ
Substituting Eqs. (10) and (11) into (12), gives ΩDs δs σ θ V= KTEi ρ
ð15Þ
ð20Þ ð13Þ
It is seen from Eq. (13) that the horizontal velocity of a void in an interconnection line embedded in a matrix is mainly dependent on the circumferential stress σAθ and its gradient at the apex A of the void
The corresponding motion velocity is written as V=
ΩDs δs p KTEi ρ
8 1 " 1 #1 9 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 < 1+m @ 1−m 1 EX 2 E −2vXY GXY 2 = EX 2 E −qXm 1 + 2 + X + X pxL A : ; EY 4GXY EY 1−m 1 + m 2 EY
ð21Þ
H.J. Xie et al. / Thin Solid Films 519 (2011) 4256–4261
It is seen from Eq. (21) that the motion velocity of the elliptic void in an interconnection line embedded in a matrix under gradient stressed field is directly proportional to the distance between the centre of the void and the zero point of the gradient stress, inversely proportional to the equivalent void size ρ, and are affected by the shape parameter of the void, the elastic modulus of the effective orthotropic elastic body and the applied stress field.
4259
0.02 1a. m=-0.5, p 0.01 1b. m=-0.5, p 0.02 2a. m=0.5, p 0.01 2b. m=0.5, p 0.02
V 0.01
1b 1a
0.00 2a
3. Numerical examples and discussions
-0.01 2b
In example calculations, dimensionless parameters related to every calculating terms are, respectively, taken as
-0.02 -30
-20
-10
0
10
20
V=
KTρ p E E V; p = ρ; qXm = qXm = EX ; aE = X ; aG = X ; xL = xL = ρ Ei ΩDs δs Ei EY GXY
ð22Þ Substituting Eq. (22) into Eq. (21) gives the normalized velocity of the elliptic void as V=
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i0:5 1+m 1−m h 0:5 0:5 − 1+2 0:5aE + 0:25ðaG −2vXY Þ aE qXm + aE p xL p 1−m 1+m
ð23Þ Here, the SiO2 and Cu are characterized as an isotropic linear elastic matrix and an isotropic elastic interconnection line, respectively. The material properties are, respectively, taken as [2]: Em = 72 GPa; Gm = 31 GPa; vm = 0:16; Ei = 128 GPa; Gi = 47 GPa; vi = 0:36
ð24Þ
Substituting Eq. (24) into Eq. (3), and taking the interconnection line volume ratio as 0.3, the material properties of effective orthotropic body are given by EX = 60 GPa;
EY = 83 GPa;
GXY = 34:5 GPa;
ð25Þ
vXY = 0:22
It is seen from Fig. 3 that when the position of void is at a compression stress region, the void moves with increasing velocity to the region of high compressive stress, and when the position of the void is at a tension stress region, the void moves with increasing velocity to the region of high tension stress. As shown in Fig. 3, the motion velocity of the void is linear proportional to the distance between the centre of void and the zero point of gradient stress, and a slit like void along the direction of the stress gradient (the shape parameter m → 1) moves quickly than the perpendicular one (the shape parameter m → − 1).
30
xL
2
Fig. 4. The influence of stress gradient on the normalized velocity V of void at a different distance xL , where qXm = 0, aE = 0.73,aG = 1.74.
From Fig. 4 it is seen that the moving velocity of void increases as the amplitude of stress gradient increases, the influence of stress gradient on the moving velocity of the void generally increase as the distance between the centre of the void and the zero point of the gradient stress increases. Fig. 5 describes the influence of a uniform stress field qXm along the X-direction on the moving velocity of void. As shown in Fig. 5, a uniform tension stress field qXm can enhance the moving velocity of the void, a uniform compressive stress field will decrease the moving velocity of the void, and the influence of a uniform stress field qXm on the moving velocity of a slit like void along the direction of the stress gradient (the shape parameter m → 1) is less than that of the perpendicular one (the shape parameter m → − 1). As shown in Fig. 6, the moving velocity of void in an orthotropic elastic body is dependent on the orthotropic characteristics of the effective elastic body (the volume ratio of an interconnection line embedded in matrix). The moving velocity of void in an orthotropic elastic body decreases when the Young's modulus in X direction of the orthotropic body, EX is larger than the Young's modulus in Y direction EY. In additional, from the previous theory work [22] and the experiment investigation using x-ray diffraction [23], it is seen that a linearly distributed stress gradient along an interconnect embedded in a silicon dioxide matrix cannot completely be relaxed in the case for the interconnect of a single grain or when the interconnection deformation is in the elastic state. It is noteworthy that the present solution is derived based on the model of an infinite orthotropic material, as some pure theoretical works, and its realistic application in interconnect systems will exist in scale effects. Utilizing the method of numerical analyses, authors
0.008
V
0.0045
1. m=-0.8 2. m=-0.4 3. m=0. 4. m=0.4 5. m=0.8
0.006 0.004 0.002
5
V 0.0030
4 3
0.0015
2 1
0.000
0.0000
2b
2c
1a. m=-0.5, qXm 0.01 0 1b. m=-0.5, q X m 0.01 1c. m=-0.5, qXm 0.01 2a. m=0.5, q Xm 0 2b. m=0.5, q X m 0.01 2c. m=0.5, qXm
1c
1b
1a
-0.002 -0.0015 -0.004 -0.0030
-0.006
2a -0.0045
-0.008 -30
-20
-10
0
10
20
30
xL Fig. 3. The variation of normalized velocity V as a function of normalized motion distance xL of void along the X direction, where qXm = 0, p = 0:01, aE = 0.73,aG = 1.74.
-30
-20
-10
0
10
20
30
xL Fig. 5. The influence of a uniform stress field qXm on the normalized velocity V of void at a different distance xL , where aE = 0.73,aG = 1.74.
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0.010
V 0.005
1a. m=-0.5, EX / EY 1b. m=-0.5, EX / EY 2a. m=0.5, EX / EY 2b. m=0.5, EX / EY
0.5
2b
2.0 0.5 2.0
1b 2a
0.000
1a
-0.005 -0.010 -30
-20
-10
0
10
20
30
xL Fig. 6. The influence of orthotropic material characteristic on the normalized velocity V of the void at a different distance xL , where qXm = 0, aG = 1.74.
[24,25] have proved that the scale effect is negligible when the ratio of inclusion (void) size to the line width of interconnect is less than 0.15. Thus the present solution can be used to calculate representative values for the void motion in a realistic interconnect when the void size is much less than the line width of interconnect. In order to show that stress gradients do indeed play an important role in affecting the void motion in a realistic interconnect, the representative parameters for a realistic interconnect embedded in a silicon dioxide matrix are given in Table 1. Substituting the representative parameters in Table 1 into Eq. (19), the moving velocity of a circular void in a realistic interconnection embedded in a matrix under a given stress gradient is calculated as V = 1.2 μm/hour. On the other hand, the motion velocity of a circular hole induced by electro-migration is written as [26] VE = 2
Ds δs Z eE0 KTρ
ð26Þ
where E0 is the electric field strength, e is the charge of an electron, and Z* is the phenomenological “effective valence” of an atom. From Eq. (26) and Table 1, it is seen that when the electric field strength applied on an interconnection E0 = 23.6 (V/m), the motion velocity of a void induced by electro-migration is equal to the motion velocity of the void induced by a given stress gradient. Because the motion velocity of a hole induced by electro-migration is along the electric field direction, the hole will stop motion when the electric field direction is opposite the increase direction of stress gradient. It is seen from the above calculating value that stress gradients do indeed affect void motion in a realistic interconnection. But, when electric field strengths used in practice are around hundreds V/m, the effect of stress gradients on void motion in interconnection is less than that of electric field strengths on the void motion. In practice, the strict stress field along the surface of a void in interconnections surrounded by matrix should be solved from the
Table 1 Representative parameters for a realistic interconnect embedded in a silicon. Line width, bi (m)
Line length, L (m)
Volume ratio of line
Radius of circular void ρ (m)
bi = 0.5 × 10− 6 Atomic volume, Ω(Cu) (m3)
L = 300 × 10− 6 Surface diffusion coefficient, Ds (m2 s− 1) Ds = 1.5 × 10− 5 Stress gradients p (MPa/m) p = 3.0 × 106
0.3 Diffusion layer thickness δs (m)
ρ = 0.1 × 10− 6 KT constant (m2 kg s− 2)
δs = 6.1 × 10− 9 Effective valence for Cu Z * Z*=2
KT = 4.14195 × 10− 21 Charge of an electron e (C) e = 1.6 × 10− 19
Ω = 1.18 × 10− 29 Initial location of void, xL (m) xL = 50 × 10− 6
matrix–interconnect–matrix model with different elastic properties in matrix and in the interconnection line. But, it is very difficult to obtain an analytical expression for stresses needed in the present derivation from the elasticity system where a small elliptical void is embedded in matrix–interconnect–matrix model with different elastic properties in matrix and in the interconnection line. Based on a simplified equivalent elastic model consisting of matrix and interconnection, authors [2,27,28], respectively, predicted stress driven void growth and evolution in representative interconnect microstructures by applying Eshelby's method in micromechanics. It is seen from their investigations that the thermal stress in interconnection is dependent on the aspect ratio of the line cross section. In order to enable us to obtain an analytical expression for stresses along an elliptical void surface in interconnects, the solution of a homogenized orthotropic elastic system consisting of the matrix and interconnection is applied in the present study. Although some simplifications and approximations are involved in the solving model, the closed-form solution with a very general manner can be obtained in this paper. The general closed-form solutions are presented in dimensionless form, in order to eliminate the scale effects of void in interconnects embedded in a matrix, so that the results can be applied directly to visualize the effect of gradient stress field on the void motion induced failure both in microelectronic devices and in common engineering microstructures. 4. Conclusion Elliptic voids in an interconnection line embedded in a matrix move with increasing velocity to the region of high tension stress or the region of high compressive stress according to the initial position of the void. The moving velocity of void increases as the amplitude of stress gradient increases. A uniform tension stress field qXm along Xdirection can enhance the moving velocity of void, and a uniform compressive stress field will decrease the moving velocity of the void. In general, because the morphologically stable shape for a void in a linearly distributed stress gradient does not remain an ellipse, a clear comparison of predictions in the present paper with results of previous works is difficult. However, this does not influence the purpose of this paper to clearly visualize the roles of some parameters of interest in the void motion in representative interconnects embedded in a matrix and to provide a benchmark and reference for future experimental and computational studies. Acknowledgement The authors wish to thank the National Science Foundation of China under numbers: 10872127 and 10932007, and two reviewers for valuable suggestions and comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
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Thin Solid Films j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / t s f
Motion of an elliptical void in interconnects embedded in matrix under gradient stress field H.J. Xie, X. Wang ⁎, S. Li, Z. Li School of Naval Architecture, Ocean and Civil Engineering, (State Key Laboratory of Ocean Engineering), Shanghai Jiaotong University, Shanghai 200240, PR China
a r t i c l e
i n f o
Article history: Received 29 June 2010 Received in revised form 14 February 2011 Accepted 15 February 2011 Available online 24 February 2011 Keywords: Interconnects embedded in a matrix Motion of an elliptical void Orthotropic elastic model Thermodynamics potential
a b s t r a c t We present an analytical solution for the motion of an elliptical void in representative interconnects embedded in a matrix with different line aspect (volume) ratio, under gradient stress field. An orthotropic elastic model is used to represent representative interconnects embedded in a matrix. The effects of stress gradient, stress states, an equivalent void size, the orthotropic material characteristic, and the shape parameter of the void on the motion velocity of an elliptical void are described and discussed. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Interconnection film lines are used in a chip, which are thin wires of copper or aluminum alloy. The mechanical reliability of interconnection film lines remains a serious concern for the microelectronics industry. Over the past decades, the width of metal interconnection film lines used in a chip has gradually decreased because of the miniaturization of microelectronics devices [1,2]. In processing of small-scale-metallic lines, various voids and other structural defects in the metallic line are often generated, which have detrimental effects on the function of microelectronic devices [3–8]. The lines are often subjected to severe gradient stresses induced by thermal deformation mismatching between the line and the surrounding matrix, which induces voids in the interconnection lines embedded in matrix to motion. The void coalescence in stress concentration region as a result of void motion is one of the major damage mechanisms. The dynamics of stress-driven void in metallic solids has been generating great research interest because of the failure of metallic interconnects in integrated circuits [9–15]. Therefore, the void motion-induced failure poses a great concern both in microelectronic devices and in common engineering structures. In actual devices, metallic interconnects in a chip are embedded in an insulation matrix. Elastic body composed of metallic interconnection film lines and insulation matrix is an effective orthotropic elastic body. Thus, one of the key challenges
⁎ Corresponding author. E-mail address: [email protected] (X. Wang). 0040-6090/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2011.02.057
is the formulation of a robust continuum theory for the motion of void in an effective orthotropic elastic body under applied stress field. Analytical solutions for the motion of circular and elliptical voids by electro-migration (EM)-driven have been given by authors [16–18]. By utilizing different numerical simulations [9,10,19], the stressdriven void motion has also been described. However, an analytical solution for the motion of an elliptical void in representative interconnection film lines embedded in a matrix for different line aspect (volume) ratios, under a gradient stress field, has not been presented so far. It is seen from previous works [2–9] that when the driving force for void migration is low, a steady state can be reached and the void shape remains unchanged after that, and when the driving force is large, the void shape may appear in an instable state and evolves into crack-like configurations or even breaks up. Although the initial void shape is a non-steady state at the early stage and is generally different from the steady-state shape, the initial void shape evolves towards the steady state shape at low driving force. Based on surface diffusion, the investigation on void shape evolution under uniform biaxial and triaxial stress states has been presented in Ref. [4], where the void does not move, and changes its shape only. The shape in steady state is a family of ellipses, and will collapse to a slit when the applied stress reaches a critical value. Therefore, in this paper, we can assume that an initial void under low driving force evolves into a stable elliptical shape, and local surface geometry changes arising from the surface diffusion of void cannot change the elliptic shape of the void, so that the applied stress gradient does not change the symmetry of void. According to the foregoing assumption, we give an analytical solution for the motion of the stable elliptical void in representative
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interconnection film lines embedded in a matrix for a different line aspect (volume) ratio, under gradient stress field. An orthotropic elastic model is found to represent representative interconnection film lines embedded in a matrix, and the void size is considered as very small compared to the dimension of orthotropic elastic model composed of finite-size interconnection film lines and their surrounding matrix. It is seen from the result that the motion characteristics of an elliptical void in representative interconnection film lines embedded in a matrix are mainly affected by stress gradient, stress states, an equivalent void size, the orthotropic material characteristic, and the shape parameter of the void, which is signification for concerning the mechanical reliability of interconnection film lines in a chip and some microelectronics. 2. Govern equations and solving method Fig. 1 shows that an elliptic void appears in the integrated circuits composed of a metal interconnection line and a matrix surrounding the line under gradient stress field induced by thermal mismatch. In Fig. 1, the global coordinate system OXY is fixed at the symmetric centre of a metal interconnection line embedded in matrix, the local (moving) coordinate system oxy is fixed at the centre of an elliptic void, and xL represents a moving distance from the centre of an elliptic void to the symmetric centre of an interconnection line embedded in matrix. The relation between two coordinate systems is given by xL = X − x and y = Y. The shape of an elliptic void having the same area as a circle of radius ρ can be described by x=ρ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1+m cos θ; 1−m
y=ρ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−m sin θ 1+m
ð1Þ
where θ is the angle between the vector radius ρ of ellipse and x-axes, and m is the shape parameter of the void. The shape parameter m = 0 corresponds to a circle, m → + 1 to x-direction slit and m → − 1 to ydirection slit, respectively. The void in the materials may move by the atomic migration of a diffuse surface layer from regions of high chemical potential to those of low chemical potential along the void surface [9]. Because the surface diffusion is generally much faster than the bulk one, only the surface diffusion is considered as the major mechanism for the void motion. In the absence of electrical current, the chemical potential inducing atomic diffusion along the void surface is written as [9] μ = μ 0 −γ s κΩ + W ρ Ω
ð2Þ
where μ0 is the reference value of the potential, γs is the free surface energy of void, κ is the surface curvature of the void, positive for
qY = pX, ( p >0)
EX = Ei Vi + Em ð1−Vi Þ; EY =
Ei Em Ei ð1−Vi Þ + Em Vi
vXY = vi Vi + vm ð1−Vi Þ; GXY =
Gi Gm Gi ð1−Vi Þ + Gm Vi
ð3Þ
where for plane stress, Ei,vi, Gi and Em,vm,Gm represent the Young's modulus, Poisson ratio and shear modulus of a metal interconnection line and a matrix, respectively, for plain strain Ei = Ei/(1 − 2vi) and Em = Em/(1 − 2vm), and Vi = bi/(bi + bm) represents an interconnection line volume ratio to the total volume of metal interconnection line and matrix. In Fig. 2, J represents the atomic flux on the void surface, the number of atoms per time crossing unit length on the void surface, which satisfies Nernst–Einstein equation as follows D δ ∂μ Dδ J=− s s = s s ΩKT ∂s KT
∂κ ∂Wρ γs − ∂s ∂s
! ð4Þ
where K represents the Boltzmann's constant, T the absolute temperature, Ds the lattice diffusion coefficient on the free surface of void, δs the diffusion layer thickness of the void surface and s the arc length of the free surface, in which all locate on the void surface. It is seen from Eq. (4) that the atomic flux on the void surface is dependent on the variation of the free energy and the strain energy on the void surface. The mass conservation equation of void surface diffusivity is given by [9] dJ V =− n ds Ω
ð5Þ
From Fig. 2, it is seen that Vn represents the normal velocity of the void surface, and is written as Vn = V sin φ
xL Interconnection lines
SiO2 matrix ( Em , vm , Gm)
convex, Wρ is the strain energy stored in the bulk elastic material associated with an atom, and Ω is the atomic volume. Thus, the key point to obtain an analytical expression for the chemical potential inducing atomic diffusion along an elliptic void surface is to solve the strain energy along the surface of the elliptic void in an interconnection line embedded in matrix (Fig. 1). Here, the width of the matrix surrounding a metal interconnection line is not equal to the width of the metal interconnection line. Because the size of an elliptic void in the interconnection line embedded in a matrix is very small as compared with the width of the interconnect, its effect on the EM-induced stress along the interconnect/passivation interface is negligible, so that the typical element of an interconnection line embedded in matrix (Fig. 1) is approximately considered as an infinite orthotropic elastic body with an elliptic void, shown in Fig. 2. The corresponding effective elastic modulus of the orthotropic material along the coordinate axis X and along the coordinate axis Y are, respectively, given by [20]
Y
SiO2 matrix ( Em , vm , Gm )
( Ei , vi , Gi )
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y
where V represents the horizontal velocity of the void surface. Substituting Eq. (6) into Eq. (5) and utilizing dy = − (sin φ)ds, Eq. (5) can be rewritten as
0.5bm x
O
bi
ð6Þ
X
dJ V = dy Ω
ð7Þ
0.5bm
Fig. 1. An elliptical void in an interconnection line embedded in a matrix, under gradient stress field along the global coordinate axis X.
Generally, the morphologically stable shape for a void under a linearly distributed stress gradient field is not an ellipse. But, when the average value of gradient stress is lower, the elliptical shape for a void may be an energetically favorable steady state. To consider the effect of the local surface changes of a moving elliptical void on the atomic flux along the ellipse, we integrate Eq. (7)
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qYm
pxL
qY 1
Y
px
Infinite effective orthotropic electric body
y
xL
A
O
Vn
J
X o
J
x
V
Fig. 2. An effective computing model for the motion of an elliptical void in an interconnection line embedded in a matrix, under gradient stress field, qY = qYm + qY1.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi along the elliptic surface from y to y = ρ ð1−mÞ = ð1 + mÞ, as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1−mÞ = ð1 + mÞ V dy ∫dJ = ∫y Ω ρ
ð7aÞ
Because the horizontal velocity V of the void surface is a constant along the void free surface when the elliptic shape of void does not vary during motion, from Eq. (7a), we have rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! V 1−m J= ρ −y + C Ω 1+m
ð8Þ
where C is the undetermined constant which can be solved by the continuous condition of the atomic flux at the apex A of the elliptic void. Utilizing Eqs. (4) and (8), the continuous condition of the atomic flux at the apex A of elliptic void is expressed as " # " # rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! Ds δs ∂κ ∂Wρ V 1−m ρ −y + C γs − = Ω 1+m KT ∂s ∂s A
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
y=ρ
ð1−mÞ = ð1 + mÞ
ð9Þ Because ∂ κ/∂ s in Eq. (4) at the apex A of elliptic void equals zero, solving Eq. (9) gives A D δ ∂Wρ C=− s s K T ∂s
ð10Þ
where WρA is the strain energy density at the apex A of the elliptic void, and is expressed as a function of the circumferential stress σ θA at the apex A and the local modulus at the point, as follows
A
Wρ =
2 σ θA
ð11Þ
2Ei
where Ei is the local Young's modulus on the void surface for plane stress, or for plain strain Ei = Ei/(1 − 2vi). Due to the symmetrical requirement of void surface diffusion shown in Fig. 2, the atom flux at y = 0 should be zero at all times, i.e., ½ J y = 0
Vρ = Ω
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−m +C=0 1+m
ð12Þ
in an effective orthotropic elastic body under remote gradient stress field, and the effective elastic modulus in Eq. (3) induced by an interconnection line volume ratio to the total volume of metal interconnection line and matrix. So far the analytic expression for the stress field at the surface of the elliptic void in an orthotropic elastic body under gradient stress field shown in Fig. 1 is not presented in literatures. Here, utilizing a loading superposition method, a gradient stress field, qY = pX, applied on an effective orthotropic body with an elliptic void can be described from a superposition of an antisymmetric stress field of stress gradient with respect to the y-axis of the moving elliptic void and a uniform stress field, as shown in Fig. 2, i.e., ð14Þ
qY = qYm + qY1
Based on the linear theory of elasticity, the circumferential stress σAθ at the apex A of void in an effective orthotropic elastic body under coupled loadings qY = qYm + qY1 is given by A
A
A
σ θ ðqY = pX Þ = σ θ ðqYm = pxL Þ + σ θ ðqY1 = pxÞ
Because an antisymmetric gradient stress field, qY1 = px, with respect to the y-axis of the moving elliptic void makes the circumferential stress σAθ at the apex A of the void in an effective orthotropic elastic body to equal zero, the circumferential stress σθA at the apex A of the void in an effective orthotropic elastic body under remote gradient stress field can be replaced by the circumferential stress σθA at the apex A of the void under a uniform applied stress field, i.e., A
A
σ θ ðqY = pX Þ = σ θ ðqYm = pxL Þ
A
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 1 + m ∂σ θ 1−m ∂s
ð16Þ
Thus, the circumferential stresses at the short axis apex A of an elliptic void in an infinite orthotropic body under gradient stress field are written as [21] 1 1 E 2 E 2 A σ θ = − X pxL = − X pðX−xÞ EY EY
ð17Þ
On the other hand, from Eq. (17), the circumferential stress gradient at the apex A of the elliptic void in an effective orthotropic elastic body under remote gradient stressed field is written as 1 A A ∂σ θ ∂σ E 2 =− θ =− X p EY ∂s ∂x
ð18Þ
Substituting Eqs. (17) and (18) into Eq. (13) gives an analytical solution for the moving velocity of elliptic void in an interconnection line embedded in a matrix under gradient stress field, as follows V=
ΩDs δs EX p2 KTEi EY ρ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1+m x 1−m L
ð19Þ
In addition, applying a uniform stress field qXm on the two ends of the effective orthotropic elastic body along X-direction, we have [21] 8 " 1 #1 9 1 < 1−m 1 EX 2 EX −2vXY GXY 2 = E 2 = qXm 1 + 2 + − X pxL : ; 1 + m 2 EY 4GXY EY
A σθ
Substituting Eqs. (10) and (11) into (12), gives ΩDs δs σ θ V= KTEi ρ
ð15Þ
ð20Þ ð13Þ
It is seen from Eq. (13) that the horizontal velocity of a void in an interconnection line embedded in a matrix is mainly dependent on the circumferential stress σAθ and its gradient at the apex A of the void
The corresponding motion velocity is written as V=
ΩDs δs p KTEi ρ
8 1 " 1 #1 9 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 < 1+m @ 1−m 1 EX 2 E −2vXY GXY 2 = EX 2 E −qXm 1 + 2 + X + X pxL A : ; EY 4GXY EY 1−m 1 + m 2 EY
ð21Þ
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It is seen from Eq. (21) that the motion velocity of the elliptic void in an interconnection line embedded in a matrix under gradient stressed field is directly proportional to the distance between the centre of the void and the zero point of the gradient stress, inversely proportional to the equivalent void size ρ, and are affected by the shape parameter of the void, the elastic modulus of the effective orthotropic elastic body and the applied stress field.
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0.02 1a. m=-0.5, p 0.01 1b. m=-0.5, p 0.02 2a. m=0.5, p 0.01 2b. m=0.5, p 0.02
V 0.01
1b 1a
0.00 2a
3. Numerical examples and discussions
-0.01 2b
In example calculations, dimensionless parameters related to every calculating terms are, respectively, taken as
-0.02 -30
-20
-10
0
10
20
V=
KTρ p E E V; p = ρ; qXm = qXm = EX ; aE = X ; aG = X ; xL = xL = ρ Ei ΩDs δs Ei EY GXY
ð22Þ Substituting Eq. (22) into Eq. (21) gives the normalized velocity of the elliptic void as V=
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i0:5 1+m 1−m h 0:5 0:5 − 1+2 0:5aE + 0:25ðaG −2vXY Þ aE qXm + aE p xL p 1−m 1+m
ð23Þ Here, the SiO2 and Cu are characterized as an isotropic linear elastic matrix and an isotropic elastic interconnection line, respectively. The material properties are, respectively, taken as [2]: Em = 72 GPa; Gm = 31 GPa; vm = 0:16; Ei = 128 GPa; Gi = 47 GPa; vi = 0:36
ð24Þ
Substituting Eq. (24) into Eq. (3), and taking the interconnection line volume ratio as 0.3, the material properties of effective orthotropic body are given by EX = 60 GPa;
EY = 83 GPa;
GXY = 34:5 GPa;
ð25Þ
vXY = 0:22
It is seen from Fig. 3 that when the position of void is at a compression stress region, the void moves with increasing velocity to the region of high compressive stress, and when the position of the void is at a tension stress region, the void moves with increasing velocity to the region of high tension stress. As shown in Fig. 3, the motion velocity of the void is linear proportional to the distance between the centre of void and the zero point of gradient stress, and a slit like void along the direction of the stress gradient (the shape parameter m → 1) moves quickly than the perpendicular one (the shape parameter m → − 1).
30
xL
2
Fig. 4. The influence of stress gradient on the normalized velocity V of void at a different distance xL , where qXm = 0, aE = 0.73,aG = 1.74.
From Fig. 4 it is seen that the moving velocity of void increases as the amplitude of stress gradient increases, the influence of stress gradient on the moving velocity of the void generally increase as the distance between the centre of the void and the zero point of the gradient stress increases. Fig. 5 describes the influence of a uniform stress field qXm along the X-direction on the moving velocity of void. As shown in Fig. 5, a uniform tension stress field qXm can enhance the moving velocity of the void, a uniform compressive stress field will decrease the moving velocity of the void, and the influence of a uniform stress field qXm on the moving velocity of a slit like void along the direction of the stress gradient (the shape parameter m → 1) is less than that of the perpendicular one (the shape parameter m → − 1). As shown in Fig. 6, the moving velocity of void in an orthotropic elastic body is dependent on the orthotropic characteristics of the effective elastic body (the volume ratio of an interconnection line embedded in matrix). The moving velocity of void in an orthotropic elastic body decreases when the Young's modulus in X direction of the orthotropic body, EX is larger than the Young's modulus in Y direction EY. In additional, from the previous theory work [22] and the experiment investigation using x-ray diffraction [23], it is seen that a linearly distributed stress gradient along an interconnect embedded in a silicon dioxide matrix cannot completely be relaxed in the case for the interconnect of a single grain or when the interconnection deformation is in the elastic state. It is noteworthy that the present solution is derived based on the model of an infinite orthotropic material, as some pure theoretical works, and its realistic application in interconnect systems will exist in scale effects. Utilizing the method of numerical analyses, authors
0.008
V
0.0045
1. m=-0.8 2. m=-0.4 3. m=0. 4. m=0.4 5. m=0.8
0.006 0.004 0.002
5
V 0.0030
4 3
0.0015
2 1
0.000
0.0000
2b
2c
1a. m=-0.5, qXm 0.01 0 1b. m=-0.5, q X m 0.01 1c. m=-0.5, qXm 0.01 2a. m=0.5, q Xm 0 2b. m=0.5, q X m 0.01 2c. m=0.5, qXm
1c
1b
1a
-0.002 -0.0015 -0.004 -0.0030
-0.006
2a -0.0045
-0.008 -30
-20
-10
0
10
20
30
xL Fig. 3. The variation of normalized velocity V as a function of normalized motion distance xL of void along the X direction, where qXm = 0, p = 0:01, aE = 0.73,aG = 1.74.
-30
-20
-10
0
10
20
30
xL Fig. 5. The influence of a uniform stress field qXm on the normalized velocity V of void at a different distance xL , where aE = 0.73,aG = 1.74.
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0.010
V 0.005
1a. m=-0.5, EX / EY 1b. m=-0.5, EX / EY 2a. m=0.5, EX / EY 2b. m=0.5, EX / EY
0.5
2b
2.0 0.5 2.0
1b 2a
0.000
1a
-0.005 -0.010 -30
-20
-10
0
10
20
30
xL Fig. 6. The influence of orthotropic material characteristic on the normalized velocity V of the void at a different distance xL , where qXm = 0, aG = 1.74.
[24,25] have proved that the scale effect is negligible when the ratio of inclusion (void) size to the line width of interconnect is less than 0.15. Thus the present solution can be used to calculate representative values for the void motion in a realistic interconnect when the void size is much less than the line width of interconnect. In order to show that stress gradients do indeed play an important role in affecting the void motion in a realistic interconnect, the representative parameters for a realistic interconnect embedded in a silicon dioxide matrix are given in Table 1. Substituting the representative parameters in Table 1 into Eq. (19), the moving velocity of a circular void in a realistic interconnection embedded in a matrix under a given stress gradient is calculated as V = 1.2 μm/hour. On the other hand, the motion velocity of a circular hole induced by electro-migration is written as [26] VE = 2
Ds δs Z eE0 KTρ
ð26Þ
where E0 is the electric field strength, e is the charge of an electron, and Z* is the phenomenological “effective valence” of an atom. From Eq. (26) and Table 1, it is seen that when the electric field strength applied on an interconnection E0 = 23.6 (V/m), the motion velocity of a void induced by electro-migration is equal to the motion velocity of the void induced by a given stress gradient. Because the motion velocity of a hole induced by electro-migration is along the electric field direction, the hole will stop motion when the electric field direction is opposite the increase direction of stress gradient. It is seen from the above calculating value that stress gradients do indeed affect void motion in a realistic interconnection. But, when electric field strengths used in practice are around hundreds V/m, the effect of stress gradients on void motion in interconnection is less than that of electric field strengths on the void motion. In practice, the strict stress field along the surface of a void in interconnections surrounded by matrix should be solved from the
Table 1 Representative parameters for a realistic interconnect embedded in a silicon. Line width, bi (m)
Line length, L (m)
Volume ratio of line
Radius of circular void ρ (m)
bi = 0.5 × 10− 6 Atomic volume, Ω(Cu) (m3)
L = 300 × 10− 6 Surface diffusion coefficient, Ds (m2 s− 1) Ds = 1.5 × 10− 5 Stress gradients p (MPa/m) p = 3.0 × 106
0.3 Diffusion layer thickness δs (m)
ρ = 0.1 × 10− 6 KT constant (m2 kg s− 2)
δs = 6.1 × 10− 9 Effective valence for Cu Z * Z*=2
KT = 4.14195 × 10− 21 Charge of an electron e (C) e = 1.6 × 10− 19
Ω = 1.18 × 10− 29 Initial location of void, xL (m) xL = 50 × 10− 6
matrix–interconnect–matrix model with different elastic properties in matrix and in the interconnection line. But, it is very difficult to obtain an analytical expression for stresses needed in the present derivation from the elasticity system where a small elliptical void is embedded in matrix–interconnect–matrix model with different elastic properties in matrix and in the interconnection line. Based on a simplified equivalent elastic model consisting of matrix and interconnection, authors [2,27,28], respectively, predicted stress driven void growth and evolution in representative interconnect microstructures by applying Eshelby's method in micromechanics. It is seen from their investigations that the thermal stress in interconnection is dependent on the aspect ratio of the line cross section. In order to enable us to obtain an analytical expression for stresses along an elliptical void surface in interconnects, the solution of a homogenized orthotropic elastic system consisting of the matrix and interconnection is applied in the present study. Although some simplifications and approximations are involved in the solving model, the closed-form solution with a very general manner can be obtained in this paper. The general closed-form solutions are presented in dimensionless form, in order to eliminate the scale effects of void in interconnects embedded in a matrix, so that the results can be applied directly to visualize the effect of gradient stress field on the void motion induced failure both in microelectronic devices and in common engineering microstructures. 4. Conclusion Elliptic voids in an interconnection line embedded in a matrix move with increasing velocity to the region of high tension stress or the region of high compressive stress according to the initial position of the void. The moving velocity of void increases as the amplitude of stress gradient increases. A uniform tension stress field qXm along Xdirection can enhance the moving velocity of void, and a uniform compressive stress field will decrease the moving velocity of the void. In general, because the morphologically stable shape for a void in a linearly distributed stress gradient does not remain an ellipse, a clear comparison of predictions in the present paper with results of previous works is difficult. However, this does not influence the purpose of this paper to clearly visualize the roles of some parameters of interest in the void motion in representative interconnects embedded in a matrix and to provide a benchmark and reference for future experimental and computational studies. Acknowledgement The authors wish to thank the National Science Foundation of China under numbers: 10872127 and 10932007, and two reviewers for valuable suggestions and comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
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