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 inter...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 fieldχ_c,a critical stress field■,a critical aspect ratioβ_c and a critical linewidth■.When■or■,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.展开更多
Based on the bulk free energy density and the degenerate mobility constructed by the quartic double-well potential function,a phase field model is established to simulate the evolution of intragranular microvoids due ...Based on the bulk free energy density and the degenerate mobility constructed by the quartic double-well potential function,a phase field model is established to simulate the evolution of intragranular microvoids due to surface diffusion in a stress field.The corresponding phase field governing equations are derived.The evolution of elliptical microvoids with different stressesΛ,aspect ratiosβand linewidths hˉis calculated using the mesh adaptation finite element method and the reliability of the procedure is verified.The results show that there exist critical values of the stressΛc,the aspect ratioβc and the linewidth hˉc of intragranular microvoids under equivalent biaxial tensile stress.When Λ≥Λ_(c),β≥β_(c) or h≤h_(c),the elliptical microvoids are instable with an extending crack tip.WhenΛ<Λ_(c),β<β_(c) or hˉ>h_(c),the elliptical microvoids gradually cylindricalize and remain a stable shape.The instability time decreases with increasing the stress or the aspect ratio,while increases with increasing the linewidth.In addition,for the interconnects containing two elliptical voids not far apart,the stress will promote the merging of the voids.展开更多
The effect of interconnect linewidth on the evolution of intragranular microcracks due to surface diffusion induced by electromigration is analyzed by finite element method.The numerical results indicate that there ex...The effect of interconnect linewidth on the evolution of intragranular microcracks due to surface diffusion induced by electromigration is analyzed by finite element method.The numerical results indicate that there exists critical values of the linewidth hc,the electric fieldχc and the aspect ratioβc.When h>hc,χ<χc orβ<βc,the microcrack will evolve into a stable shape as it migrates along the interconnect line.When h≤hc,χ≥χc orβ≥βc,the microcrack will split into two smaller microcracks.The critical electric field,the critical aspect ratio and the splitting time have a stronger dependence on the linewidth when h≤6.In addition,the decrease of the linewidth,the increase of the electric field or the aspect ratio is beneficial to accelerate microcrack splitting,which may delay the open failure of the interconnect line.展开更多
Based on the thermal kinetic and mass conservation, a series of controlling equations for the finite element are derived and related programs are developed to simulate the damage microcrack healing process controlled ...Based on the thermal kinetic and mass conservation, a series of controlling equations for the finite element are derived and related programs are developed to simulate the damage microcrack healing process controlled by surface diffusion. Two kinds of typical models for microcrack splitting are proposed, i.e., the grain boundary energy existing on the crack surface and residual stresses applying on the crack surface. And the conditions of microcrack splitting in the two models are given as a function of the microcrack aspect ratio. The microcrack with traction-free surfaces will directly evolve into a spheroid.展开更多
基金supported by the Natural Science Foundation of Jiangsu Province of China (No.BK20141407)the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions
文摘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 fieldχ_c,a critical stress field■,a critical aspect ratioβ_c and a critical linewidth■.When■or■,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.
基金supported by the Natural Science Foundation of Jiangsu Province of China (No. BK20141407)the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions。
文摘Based on the bulk free energy density and the degenerate mobility constructed by the quartic double-well potential function,a phase field model is established to simulate the evolution of intragranular microvoids due to surface diffusion in a stress field.The corresponding phase field governing equations are derived.The evolution of elliptical microvoids with different stressesΛ,aspect ratiosβand linewidths hˉis calculated using the mesh adaptation finite element method and the reliability of the procedure is verified.The results show that there exist critical values of the stressΛc,the aspect ratioβc and the linewidth hˉc of intragranular microvoids under equivalent biaxial tensile stress.When Λ≥Λ_(c),β≥β_(c) or h≤h_(c),the elliptical microvoids are instable with an extending crack tip.WhenΛ<Λ_(c),β<β_(c) or hˉ>h_(c),the elliptical microvoids gradually cylindricalize and remain a stable shape.The instability time decreases with increasing the stress or the aspect ratio,while increases with increasing the linewidth.In addition,for the interconnects containing two elliptical voids not far apart,the stress will promote the merging of the voids.
基金supported by the Natural Science Foundation of Jiangsu Province of China (No. BK20141407)a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions
文摘The effect of interconnect linewidth on the evolution of intragranular microcracks due to surface diffusion induced by electromigration is analyzed by finite element method.The numerical results indicate that there exists critical values of the linewidth hc,the electric fieldχc and the aspect ratioβc.When h>hc,χ<χc orβ<βc,the microcrack will evolve into a stable shape as it migrates along the interconnect line.When h≤hc,χ≥χc orβ≥βc,the microcrack will split into two smaller microcracks.The critical electric field,the critical aspect ratio and the splitting time have a stronger dependence on the linewidth when h≤6.In addition,the decrease of the linewidth,the increase of the electric field or the aspect ratio is beneficial to accelerate microcrack splitting,which may delay the open failure of the interconnect line.
基金The project supported by the National Natural Science Foundation of China(19972053 and 59889101)the National Outstanding Young Scientist Fund of China(59925104)
文摘Based on the thermal kinetic and mass conservation, a series of controlling equations for the finite element are derived and related programs are developed to simulate the damage microcrack healing process controlled by surface diffusion. Two kinds of typical models for microcrack splitting are proposed, i.e., the grain boundary energy existing on the crack surface and residual stresses applying on the crack surface. And the conditions of microcrack splitting in the two models are given as a function of the microcrack aspect ratio. The microcrack with traction-free surfaces will directly evolve into a spheroid.
