ZHOU Linyong，HUANG Peizhen

State Key Laboratory of Mechanics and Control of Mechanical Structures，Nanjing University of Aeronautics and Astronautics，Nanjing 210016，P.R.China

Abstract:Based on the weak formulation for combined surface diffusion and evaporation-condensation，a governing equation of the finite element is derived for simulating the evolution of intergranular microcracks in copper interconnects induced simultaneously by stressmigration，electromigration and thermomigration.Unlike previously published works，the effect of thermomigration is considered.The results show that thermomigration can contribute to the microcrack splitting and accelerate the drifting process along the direction of the electric field.The evolution of the intergranular microcracks depends on the mechanical stress field，the temperature gradient field，the electric field，the initial aspect ratio and the linewidth.And there exists a critical electric fieldχc，a critical stress fielda critical aspect ratioβc and a critical linewidthWhen，the intergranular microcrack will split into two or three small intergranular microcracks.Otherwise，the microcrack will evolve into a stable shape as it migrates along the interconnect line.The critical stress field，the critical electric field and the critical aspect ratio decrease with a decrease in the linewidth，and the critical linewidth increases with an increase in the electric field and the aspect ratio.The increase of the stress field，the electric field or the aspect ratio and the decrease of the linewidth are not only beneficial for the intergranular microcrack to split but also accelerate the microcrack splitting process.

Key words:stressmigration;electromigration;thermomigration;surface diffusion;finite element method;intergranular microcrack

0 Introduction

There is a long-standing interest in the reliability of interconnect lines.The interest originates to a large extent from the fact that interconnect lines are abundantly used in the ongoing miniaturization of microelectronic circuits and microelectromechanical systems.Interconnect lines have bamboo grain structure with dimensions in the submicron range and the material within the interconnect line is subjected to severe mechanical and electric loading.In addition，interconnect lines inevitably exist some drawbacks，such as voids and microcracks.Joule heating due to current crowding around voids or microcracks will cause high local temperature gradient，leading to local non-uniformity in material parameters in the vicinity of voids or microcracks.Local variation in material behaviors could change interconnect life by several orders of magnitude in some conditions[1-6].Therefore，the effect of the thermomigration driven by temperature gradient should be taken into account in the study of the reliability of interconnect lines.However，little attention has been devoted on the microcrack evolution induced simultaneously by thermomigration，stressmigration and electromigration.

There have been a number of analytical studies on the grain boundary void growth and the motion of an elliptical inclusion in interconnect lines with bamboo microstructures under stress field，gradient stress field or/and electric field[7-11].The void morphological evolution induced by electromigration under high current density has also been studied[12-13].Many theoretical studies indicated that a flat surface under the stress could evolve into a wavy shape via surface diffusion[14-16].Ru[2-3]analyzed the instability of electromigration induced mass transport in interconnect lines and examined the role of thermomigration caused by electromigration and found that thermomigration is the leading driving force for linear instability.

Experimental observations of damage defects in interconnect lines induced by stressmigration，electromigration and thermomigration have been reported by many workers[1，17-24].The earlier experiments mainly focused on the void formation，migration，growth in grains or grain boundaries caused by electromigration and the stressmigration.As the temperature effect attracts people’s attention widely，Raghavan et al.[21]observed the diffusion of copper through dielectric interconnect line under electric field and elevated temperature and the results showed that both the electric field and the temperature field can strongly affect the lifetime of the dielectric barrier.As for the thermomigration，Du et al.[22]researched the stabilization of the surface morphology of stressed solids using simultaneously applied electric fields and temperature gradients.Tan et al.[23]investigated the effect of temperature and stress gradients on accelerated electromigration test for Cu narrow interconnect lines.Bastawros et al.[24]experimentally studied the electric current induced damage evolution at the crack tip in interconnect lines.And the effect of thermal stress on the electromigration failure time was analyzed by measuring the change of electric resistance during the electromigration test[1].

There are also many numerical analyses on stressmigration and electromigration due to diffusion processes[25-34].Bower et al.[25-29]showed two dimensional finite element simulations of void evolution due to strain and electromigration induced surface diffusion.Fridline and Kraft simulated the shape changes of voids in bamboo lines induced by stressmigration[30]and electromigration driven[31]surface diffusion.He et al.[32-34]analyzed the intragranular microcracks evolution under surface diffusion induced by stressmigration and electromigration.Du and He et al.[35-36]respectively researched the effect of linewidth on evolution of intergranular and intragranular microcracks in interconnects induced by stressmigration and electromigration.The above works mainly focused on the evolution of voids or microcracks in a grain.

Experimental observations have shown that voids or microcracks often occur on grain boundaries and intergranular microcracks are common sources of failure in bamboo films[37-41].In addition，intergranular microcracks are likely to grow much more rapidly than those within grains.During service，these microcracks continue to grow and change their shape by stress and electric current driven mass transport，and eventually reach a size that causes an unacceptable increase in the resistance of the line.It is proposed that the intergranular microcrack changes shape due to surface diffusion under the combined action of surface energy，elastic energy，electric current as well as temperature gradient.Thus，there is a great interest in identifying the mechanisms responsible for nucleating and propagating of the intergranular microcracks in interconnect lines，and in finding ways to reduce the likelihood of failure.

In this paper，a model of intergranular microcracks in interconnect line under the stress field，the electric field and the temperature gradient field caused by Joule heating is described.The weak formulation incorporating surface diffusion and evaporation-condensation induced simultaneously by electromigration，stressmigration and thermomigration has not been reported in the literature up to now.The present work extends previous models by accounting in detail for the effects of thermomigration and the grain boundary.The aim of the work is primarily to develop a finite element program to simulate the morphological evolution of the intergranular microcrack under surface diffusion induced simultaneously by thermomigration，stressmigration and electromigration.

In the next section，an analysis model of twodimensional intergranular microcrack and the governing equations of surface diffusion are introduced.And then the weak statement is briefly described，which governs the concurrent processes of surface diffusion and evaporation-condensation and a finite element method induced by stressmigration，electromigration and thermomigration is developed.The major focus of the study include the simulation and analysis of the evolution procedure under surface diffusion induced by electromigration，stressmigration and thermomigration.The corresponding effect is discussed in detail.At last the main conclusions are presented.

1 Model and Governing Equations

1.1 Geometric model and boundary conditions

Fig.1 schematically illustrates the cross section of an intergranular microcrack in an interconnect line，which is found in the grain boundary and is symmetrical in shape.The interconnect line is subjected to biaxial tensile stressesσx=σy=σand remains elastic.Meanwhile，the interconnect line is subjected to an electric field with a voltage V0.It is assumed that the distribution of the voltage and the stress in the boundary is uniform and the interconnect line deforms in a state of plain strain and that both the electric and the stress fields in the interconnect line have no component normal to the plane of the figure.Diffusion through the bulk is assumed to be negligible.Therefore，in the absence of grain boundary diffusion，the only mode of mass transport is diffusion along the microcrack surface.We have assumed that the surface energy does not interfere with the electric field，the stress field and the temperature field.

Fig.1 Simplified model of intergranular microcrack in interconnect line under stress and electric fields

For simplicity，the intergranular microcrack is characterized by the aspect ratioβ=a/h0，where a is the initial major semiaxis of the microcrack，h0the initial minor semiaxis，L the length of the interconnect line，and h the linewidth.The surface tensionγsand the grain boundary tensionγbmust be in equilibrium at the triple point（the microcrack tip）as shown in Fig.1.This equilibrium requirement leads to a discontinuity in the tangent of free surface at the junction which is often referred to as the dihedral angle 2θand

1.2 Governing equations

The intergranular microcrack changes its shape due to the atom diffusion over its surface.Surface diffusion is driven by the variation in chemical potential，which causes atoms migrate from regions of high chemical potential to those of low chemical potential.Under the effect of the curvature，the stress field and the electric field，the chemical potentialμ can be represented as[4]

whereμ0is the reference value ofμ，Ωthe atomic volume，ωthe elastic strain energy density，Z*the effective charge coefficient，V the electric potential，e the charge of an electron and k the curvature of the surface.

Based on Herring’s classical theory[42]，the atomic flux of the surface diffusion J is proportional to the driving force F，which is

where M is the mobility of atoms on the surface and related to the self-diffusivity by the Einstein relation.M=ΩDδ/kT，in which D is the self-diffusivity on the surface andδthe effective thickness of atoms participating in matter transport.The driving force F is defined by the decrease in free energy per unit volume of matter moving per unit distance on the surface.The relationship of the driving force and the chemical potential is

As the matter is deposited onto or removed from a free surface，the normal velocity vnsof the free surface is generated.According the mass conservation，it is required that

where?·J stands for the surface divergence of the flux.Letδrnsbe the virtual normal displacement on the surface andδI be the virtual mass displacement，we have

For a given system，the chemical potential gradient induces the driving force F.Then through Eqs.（3，5），the corresponding J and vnscan be obtained，which update the displacement of the point on the intergranular microcrack surface for a small time incrementΔt.Repeating the procedure for many time increments，the evolution of the unstable shape for intergranular microcracks can be traced.

2 Formulations of Finite Element Method

2.1 Weak statement for combined surface diffusion and evaporation-condensation

The surface diffusion problem，which conserves solid mass as formulated above，is difficult to implement in a finite element setting.However，once the evaporation-condensation process is introduced into microstructure evolution，the overall mass conservation is easier to treat by the finite element method[43].

Two concurrent processes are imagined to be on a microcrack surface，which include surface diffusion and evaporation-condensation.The kinetic law for evaporation-condensation is similar to that for the surface diffusion.The free energy reduction per unit volume of atoms crossing an interface（denoted by p）is called the driving force on the interface in the evaporation-condensation process and vnvis called the volume of matter added to per unit area of the microcrack surface per unit time.When a structure is not far from the equilibrium，the normal velocity of the interface vnvis proportional to the driving force[44].The following kinetic law is used.

where m is the interface mobility for the evaporationcondensation process.

The resulting equations of the surface normal velocity vnand the surface virtual displacementδrndue to the combined action of evaporation-condensation and surface diffusion are

Associated with the virtual motion，the free energy changes byδG.According to the two kinds of driving forces，F(xiàn) and p，matter relocation and exchange on the microcrack surface area element，δA，reduce the free energy by an amount of(F·δI+pδrnv)d A.Thus，the weak formulation for combined surface diffusion and evaporation–condensation can be obtained as

Substituting Eqs.（3，5-9）into Eq.（10）yields

The finite element method introduced by Sun et al.[41]is used for solving the above integral equations.The difference between Refs.[32，33，43]and this paper is that the coupling with mechanical stress field，temperature gradient field and electric field is considered.That is，the kinetic and mass conservation laws expressed in the left hand side of Eq.（11）are the same，but the energy term in the right hand side of Eq.（11）has three extra terms，including the strain energy from mechanical stress，temperature gradient field，and the electric potential.

2.2 Finite element method

The system consists of two coupled subsystems：the microcrack surface and the solid body.The motion of the microcrack surface is affected by the strain energy induced by the mechanical stress field，the temperature gradient field and the electric potential within the solid body，which in turn is affected by the shape changes due to the surface motion.Each subsystem is discretized into finite elements as follows.

Here，the shape evolution of an elliptical microcrack is studied.Its shape change can be treated as a two-dimensional problem and the microcrack is represented by an area and surface by a curve.Thus，an assembly of linear elements can be used to approach a microcrack surface in two-dimensional problems.Each element has four and three degrees of freedom to describe the motion of element and the diffusion on the element，respectively.These coordinates，together with mass displacement I at all nodal points form generalized coordinates q1，q2，q3，…，qn-2，qn-1，qn，where n is the number of the total degrees of freedom.The generalized velocities areThe velocity and virtual motion of any point on the surface can be interpolated by the corresponding values at the nodes.Integrating the weak statement Eq.（11）element by element，a bilinear form in q˙andδq is got.The right-hand side of Eq.（11）is the total free energy change associated with the virtual motion，which is

Eq.（12）allows us to compute the generalized forces f1，f2，…，fn.Collecting the coefficient ofδqiyields

The free energy variation associated with a given element is computed by

whereω*is the strain energy density caused by temperature gradient stress.The force components acting on the two nodes of the element due to the element surface energy，strain energy density and electric potential are

whereω1andω2represent the strain energy density of nodes 1 and 2，the strain energy density of nodes 1 and 2 caused by temperature gradient field，and V1and V2the nodal values of the electric potential function，respectively.

We choose to discretize the solid body and use the standard finite element procedure to solve the electric field，the stress field and the temperature gradient field problem on the current configuration，including the computation of the strain energy density caused by mechanical stress field and temperature gradient field，the electric potential and project the results onto the microcrack surface nodes.Then，the velocity of all the discrete points on the microcrack surface is computed by the preceding finite element method under the electric field，stress field and temperature gradient field and let them advance by an amount vnΔt in the direction normal to the surface，whereΔt is an appropriately chosen time increment.The simulation of the intergranular microcrack is carried out by repeating the preceding procedures.

3 Numerical Simulation and Discussion

In this section，the behavior of intergranular microcracks caused by surface diffusion induced by stressmigration，electromigration and thermomigration is studied.For convenience，the dimensionless time t^=tMγs/h40is introduced in the calculation.

It is known from diffusional theory that the difference in chemical potential constitutes a thermodynamic force，causing atoms to migrate from regions of high potential to those of low potential.The actual evolution of intergranular microcracks in this paper depends on the dimensionless temperature stress and mechanical stressσ^=σh0/γs，the aspect ratioβ and the dimensionless linewidth h^=h∶h0.The relative magnitude of the two forces，the electromigration driving force and the surface tension，is given byχ=V0|e|Z*h0/ΩγsL.When the effects of the grain boundary diffusion and evaporation-condensation are neglected，mass conservation requires that the total area of the microcrack during evolution remains constant.In the present study，the actual change of this area during simulation can therefore be taken as a measure of the computing accuracy.As the interface of the microstructure moves，some elements shorten and others elongate.A short element substantially reduces the time step and a very long element poorly models a curved geometry.One minimum and maximum elements are prescribed，a very short element is eliminated，and a very long one is divided.Large numbers of numerical calculations indicate that the finite element method used is robust，accurate and efficient.

3.1 Effect of temperature gr adient field

Joule heating due to current crowding around the microcrack will cause high local temperature gradient，leading to local non-uniformity in material parameters in the vicinity of microcracks.In this section，the effect of Joule heating caused by current crowding induced temperature gradient in the vicinity of the evolving microcrack is focused，neglecting the effect of mechanical stress field.The resistivity of the materialρhas a linear temperature dependence[37]，which can be expressed as

whereρ0is 1.75μΩ·cm for copper，αthe material constant of about 0.004/K at the reference temperature and TR=300 K for copper.

Figs.2（a—d）show the evolution of the intergranular microcrack with h^=10，β=4 for different values ofχ=0.2，0.4，0.6 and 0.8.In the morphological evolution graphics，the solid lines stand for the results under the electric field and the temperature gradient field，and the dotted lines stand for those only under the electric filed.As shown in Figs.2（e，f），when the current flows through the interconnect line，a temperature gradient field is created inside the conductor，leading to a temperature stress field.It can be seen from Fig.2（e）that the gradient of the temperature field at the microcrack tip is higher than that at the middle of the microcrack，so the corresponding strain energy densityω*generated by the temperature stress at the microcrack tip is bigger than that at the middle of microcrack.That is，，as shown in Fig.2（f）.The electromigration driving force is proportional to the gradient of the electric potential.It is known that(?V/?s)C＞(?V/?s)B＞(?V/?s)Ain the initial time.Thus，the direction of the electromigration driving force at the right side of the microcrack is the same as that of the thermomigration driving force，which accelerates the microcrack migrating along the electric field direction（see Fig.2（a））.But，the two forces at the left side of the microcrack are in the opposite directions，which are beneficial for the intergranular microcrack to split as shown in Figs.2（b—d）.In addition，the intergranular microcrack simultaneously grows in linewidth，which tends to accelerate the failure process.The result is consistent with the viewpoint of Ref.[4]and the effect of the thermomigration driven by the temperature gradient should be taken into account in the study.Therefore，in the following numerical simulations，the effect of the temperature gradient field on the evolution of the intergranular microcracks is considered.

Fig.2 Evolution of intergranular microcrack with h^=10,β=4

3.2 Effect of electric field

Fig.3 shows the evolution of the intergranular microcrack withσ^=15，β=4，h^=10 for different values ofχ=0.2，0.4，0.6 and 0.8.Here，t^frepresents the splitting time.As shown in Fig.3（a），the curvature of each point on the microcrack surface is different.The pronounced difference in curvature along the microcrack perimeter induces mass redistribution，with mass being removed from relatively flat microcrack surfaces and depositing in the microcrack tips.The atoms in point C would move to point B，and then move from point B to point A under the electromigration driving force.Moreover，the atoms in point A would move to B and C under the driving force induced by strain energy density.When there is a rest of atoms on point B，a bulge forms.With an increase in the electric field，atoms unceasingly gather on the bulge，while the intergranular microcrack migrates along the interconnect line under the electromigration driving force.When the bulges on the upper and lower surfaces are connected at B′（see Fig.3（b）），the microcrack cavity might split into two small intergranular microcracks.When the bulge reaches a certain size，the atoms will emit from the bulge because the chemical potential here is promoted and is higher than that of the vicinity.If the process takes dominant position，the bulge will revers and eventually the energy in the microcrack surface tends to be the same under surface diffusion as shown in Fig.3（a）.

Fig.3 Evolution of intergranular microcrack withσ^=15,β=4,h^=10

Following the above analysis，there must exist a critical electric fieldχc.Whenχ＜χc，the intergranular microcrack will evolve into a stable shape as it migrates along the interconnect line.In contrast，whenχ≥χc，the intergranular microcrack splits into two small parts.Fig.4 shows the critical electric fieldχcas a function of the linewidth under different aspect ratios forσ^=15.The slope of these curves at any point represents the magnitude of the dependence.It can be seen from the variation tendency of the curve，the critical electric field decreases gradually with the decreases of linewidth，demonstrating that the increasing current density accelerates the microcrack splitting.It also can be found that the larger of the aspect ratio or the electric field，the easier for the intergranular microcrack to split.That is，the increase of the aspect ratio and the electric field are beneficial for the intergranular microcrack to split.

Fig.5 shows the relationship of the splitting time t^fwithχ.It is noticeable that the splitting time decreases asχincreases.This behavior indicates that the increase inχaccelerates the microcrack splitting process.Comparing the curves with different aspect ratios in Fig.5，it can be seen that the larger of the aspect ratio，the shorter time of the intergranular microcracks needed to split.That is，an increase of the aspect ratio also accelerates the microcrack splitting process.

Fig.4 χc as a function of the linewidth

Fig.5 a function of the electric fieldχ

3.3 Effect of mechanical stress field

Fig.6 shows the evolution of the intergranular microcrack withχ=0.2，β=4，=10 for different values of=15，20，25 and 30.Through the analysis of Fig.6，it can be concluded that there must exists a critical stress.When，the intergranular microcrack will evolve into a stable shape and migrate along the interconnect line.In contrast，when，the intergranular microcrack will split into two small parts.With the increase ofσ^，the strain energy density in surface diffusion driving force plays an important role and the left crack tip does not recede and the area of the small part increases（see Figs.6（c）and（d））.

Fig.6 Evolution of intergranular microcrack forχ=0.2,β=4

Fig.7 shows the critical stressof the microcrack splitting as a function ofThis figure indicates thatdecreases with a decrease ofand an increase of the aspect ratioβ.It is noticeable that the increase of the aspect ratioβor the decrease of h^is beneficial to microcrack splitting.In addition，the splitting time decreases as the increases of stress field and aspect ratio for a given electric field，indicating that large values of stress field and aspect ratio accelerate the microcrack split as shown in Fig.8.

Fig.7 as a function of the linewidth h^

Fig.8 as a function of the stress fieldσ^

3.4 Effect of linewidth

Fig.9 Evolution of the intergranular microcrack forσ^=15,χ=0.2,β=4

Fig.9 shows the evolution of the intergranular microcrack withσ^=15，χ=0.2，β=4 for the decrease of linewidth（h^=10，7，6 and 5）.It can be seen that there exists a critical linewidth.When，the microcrack will split into two small parts and with an decrease in linewidth，the morphological evolution is more distinct.Fig.10 reveals the critical linewidth as a function of the electric field.It is noticeable that the critical linewidth increases with an increase in the electric field.When the values of the electric field and the aspect ratio exceed a certain number for a given aspect ratio，the critical linewidth tends to be infinite，which means that the microcrack will split directly.Meanwhile，F(xiàn)ig.11 shows the splitting time as a function of the linewidth.It can be seen that the splitting time decreases with a decrease in the linewidth for a given electric field，stress field and aspect ratio.That is，reducing the microcrack linewidth can accelerate the microcrack splitting.

3.5 Effect of aspect ratio

Fig.12 shows the evolution of the intergranular microcrack withχ=0.2，=15=10 for different values ofβ=4，8，12 and 16.It also can be seen that there exists a critical aspect ratioβc.Whenβ＜βc，the intergranular microcrack will evolve into a stable shape as it migrates along the interconnect line.Whenβ≥βc，one or two pair of bulges form on the upper and lower microcrack surface and the intergranular microcrack splits into two or three small parts.

Fig.10 as a function of the electric fieldχ

Fig.11 as a function of the linewidth h^

Fig.12 The evolution of the intergranular microcrack for^=15,χ=0.2,=10

Fig.13 reveals the critical aspect ratio as a function of the linewidth.It is noticeable that the critical aspect ratio decreases with the decrease of linewidth.For a given electric field，the microcrack splitting will be much easy to occur with linewidth decreasing.Moreover，the critical aspect ratioβcdecreases with the increase of the electric field，and the increase of the applied electric field is therefore beneficial to microcrack splitting.Fig.14 shows that the splitting time decreases with the increase of the aspect ratio for a given electric field and stress field.The slope of these curves at any point represents the magnitude of the dependence.It can be seen from the variation tendency of the curve，the splitting time dependence of the aspect ratio is relatively weakened whenβ≥8.Comparing the three curves in Fig.14 respectively，it can be found that the larger of the aspect ratio or the electric field，the faster for the microcrack to split.That is，the increase of the aspect ratio and electric field are beneficial for the microcrack to split.

Fig.13 βc as a function of the linewidth h^

Fig.14 as a function of the aspect ratioβ

4 Conclusions

In this paper，the governing equation of the finite element incorporating surface diffusion and evaporation-condensation induced by stressmigration，electromigration and thermomigration is derived and the corresponding program is developed to simulate the morphological evolution of the intergranular microcrack in a copper interconnect line under surface diffusion caused by the mechanical stress field，the electric field and the temperature gradient field.The main results obtained are summarized as follows.

（1）Electromigration induces the temperature gradient field in the vicinity of intergranular microcracks in the interconnect lines，which is beneficial for the intergranular microcrack to split and accelerate the drifting process along the direction of the electric field.The effect of the thermomigration driven by temperature gradient should be taken into account in the evolution of the intergranular microcracks.

（2）There exist a critical electric fieldχc，a critical stress field，a critical aspect ratioβcand a critical linewidthWhenχ＜χc，，the intergranular microcrack will migrates along the interconnect lines stably，and more interesting results show that the microcrack will split into two small parts whenχ≥χc，，β≥βc，Moreover，with the increase of the aspect ratio，the microcrack even might split into three small parts.

（3）All ofχc，，βcdecrease gradually with the decrease of the linewidth.increases with the increase of the electric field and the aspect ratio，and tends to be infinite when the value of the electric field and the aspect ratio exceeds a certain value for a given stress field.

（4）The splitting time decreases with the increase of the mechanical stress，the electric field and the aspect ratio，but increases with the increases of the linewidth.