The migration model (TB,MIGR) is used to model electromigration effects. The atomic flux option is used in this example.
Most of the material properties for copper and SAC were taken from Wang and Liang.[2] Some copper material properties, such as the pre-exponential diffusivity coefficient, the activation energy of diffusion, and the change number, were selected from Chao et al.[3]
The material properties used for the solder are shown in the table below.
| SnAgCu (SAC) | ||
| Property | Value | MP Command Label |
| Young’s modulus (MPa) | 26.2E3 | EX |
| Poisson’s ratio | 0.35 | PRXY |
| Electrical resistivity (TOhm*μm @ 200°C) | 20.75E-14 | RSVX |
| Pre-exponential diffusivity ((μm)2/s) | 4.1E7 | DXX |
| Thermal conductivity (pW/(μm*K)) | 57E6 | KXX |
| Specific heat (pJ/(kg*K)) | 219E12 | C |
| Density (kg/(μm)3) | 7390E-18 | DENS |
| Coefficient of thermal expansion (1/°C) | 23E-6 | ALPX |
| Coefficient of diffusion expansion | 1E-5 | BETX |
| Property | Value | TBDATA Command Constant |
| Activation energy of diffusion Ea (eV) | 0.8 | C1 = Ea/kB_eV |
| Atomic volume V2 (μm)3 | 2.71E-11 | C2 = V2/kB |
| Heat of transport Q (eV) | 0.0094 | C3 = Q/kB_eV |
| Charge number Z2 | -23 | C4 = Z2/kB_eV |
The material properties used for the conductors are shown in the table below.
| Copper | ||
| Property | Value | MP Command Label |
| Young’s modulus (MPa) | 127.7E3 | EX |
| Poisson’s ratio | 0.31 | PRXY |
| Electrical resistivity (TOhm*μm @ 200°C) | 2.38E-14 | RSVX |
| Pre-exponential diffusivity ((μm)2/s) | 7.8E7 | DXX |
| Thermal conductivity (pW/(μm*K)) | 393E6 | KXX |
| Specific heat (pJ/(kg*K) | 385.2E12 | C |
| Density (kg/(μm)3) | 8900E-18 | DENS |
| Coefficient of thermal expansion (1/°C) | 17.1E-6 | ALPX |
| Coefficient of diffusion expansion | 1E-5 | BETX |
| Property | Value | TBDATA Command Constant |
| Activation energy of diffusion Qa (J/(K*mol)) | 210E3 | C1 = Qa/R |
| Atomic volume V1 (μm)3 | 1.182E-11 | C2 = V1/kB |
| Charge number Z1 | -4 | C4 = Z1/kB_eV |
The following constants are also defined.
| Constants | ||
|---|---|---|
| Boltzmann constant kB (pJ/K) | 1.3806488E-11 | |
| Boltzmann constant kB_eV (eV/K) | 8.6173324E-5 | |
| Universal gas constant R (J/(K*mol)) | 8.31445 | |
| Reference temperature for thermal strain calculation (°C) | 25 | |
| Reference normalized concentration for diffusion strain calculation | 1.0 | |
The properties which cannot be defined in the Mechanical user interface are defined by command snippets as shown below. The commands define electric and diffusion material properties along with the constants.
Conductors:
The following command snippet is used for the conductors (Solid 1 and Solid 2).
! Constants kB=1.3806488e-23*1.e12 ! Boltzmann constant, pJ/K kB_eV=8.6173324e-5 ! Boltzmann constant, eV/K R=8.31445 ! Universal gas constant, J/(K*mol) ! Copper (mat 1) et,matid,226,100111 ! structural-thermal-electric-diffusion mp,rsvx,matid,2.38e-8*1e-6 ! electrical resistivity, TOhm*um @ 200 C mp,dxx,matid,7.8e-5*1e12 ! pre-exponential diffusivity, (um)^2/s Qa=210e3 ! activation energy, J/(K*mol) V1=1.182e-29*1e18 ! atomic volume, um^3 Z1=-4 ! charge number tb,migr,matid !! migration model for Cu plate tbdata,1,Qa/R ! activation energy of diffusion/R tbdata,2,V1/kB ! atomic volume/kB tbdata,4,Z1/kB_eV ! charge number/kB_eV mp,cref,matid,1 ! reference concentration for diffusion strain calculation mp,betx,matid,1e-5 ! coefficient of diffusion expansion
Solder:
The following command snippet is used for the solder (Solid 3).
! Constants
kB=1.3806488e-23*1.e12 ! Boltzmann constant, pJ/K
kB_eV=8.6173324e-5 ! Boltzmann constant, eV/K
R=8.31445 ! Universal gas constant, J/(K*mol)
! Solder - SAC (mat 2)
et,matid,226,100111
mp,rsvx,matid,20.75e-8*1e-6 ! electric resistivity @ 200 degC, TOhm*um
mp,dxx,matid,4.1e-5*1e12 ! pre-exponential diffusivity, (um)^2/s
mp,cref,matid,1 ! reference normalized concentration for diffusion strain calculation
mp,betx,matid,1e-5 ! coefficient of diffusion expansion
! for back stress calculation
Ea=0.8 ! activation energy, eV
V2=2.71e-29*1e18 ! atomic volume, um^3
Q=0.0094 ! heat of transport, eV
Z2=-23 ! charge number
tb,migr,matid !! migration model for SAC solder joint
tbdata,1,Ea/kB_eV ! activation energy of diffusion/kB_eV
tbdata,2,V2/kB ! atomic volume/kB
tbdata,3,Q/kB_eV ! coefficient of thermomigration/kB_eV
tbdata,4,Z2/kB_eV ! charge number/kB_eV