Electrical  —  The end of one lead is grounded while a current of (2.85x10¹²)/2 pA (for the half model) is step-applied to the end of the other lead. The VOLT degree of freedom is coupled to distribute the current evenly. This produces a current density of 8.9x10⁷ pA/(μm)² (or 8.9x10⁷ A/m²) in the copper leads. This is a typical current density for components vulnerable to electromigration.

nsel,s,loc,x,600        ! Ground the nodes at the end of one conductor
d,all,volt,0
nsel,s,loc,x,-600       ! Apply a uniform current at the end of the other conductor
cp,1,volt,all           ! Couple the VOLT degree of freedom
nd=ndnext(0)
alls

f,nd,amps,2.85e12/2     ! Applied current (pA) (half model)

Thermal  —  A convection boundary condition with a film coefficient of 20 pW/(μm)²(°C) to a bulk temperature of 50°C is specified for all surfaces except the symmetry plane. This film coefficient is the same in units of Watts and meters; it is a low value and is two or three times what is produced by natural convection. A more realistic way to model heat dissipation out of the solder ball would be to include thermal conduction through the materials that surround the solder ball.

An initial temperature of 50°C is specified. The temperature offset from absolute zero to zero is set to 273 degrees.

nsel,s,ext            ! Select nodes on the surface of the model
nsel,u,loc,z,0        ! Unselect nodes on the symmetry plane
sf,all,conv,20,50     ! hc in pW/(µm)(°C), tbulk in deg C
nsel,all
toffs,273             ! Temperature offset from absolute zero to zero
...
ic,all,temp,50        ! Model temperature at start of transient

Structural  —  The UZ displacements are constrained on the symmetry surface.

nsel,s,loc,z,0   
d,all,uz,0           ! Symmetry plane

The UY displacements at the top and bottom surfaces of the model are constrained to simulate resistance to thermal and diffusion expansion by the passivation and other interface layers.

nsel,s,loc,y,-265    ! Select nodes at top and bottom of conductors
nsel,a,loc,y,265     
d,all,uy,0           ! Fully constrain the normal DOF to simulate the
                     ! the structural restraint by the adjacent layers

The UX displacements are constrained at the lower left end of the copper plate.

nsel,s,loc,x,-600   
d,all,ux,0

Diffusion  —  An initial unit normalized concentration is specified. No concentration sink is specified at the ends of the copper conductors. Diffusion through the copper is very low, and electromigration essentially occurs at the blocked interface between the solder and the copper.

ic,all,conc,1        ! Specify the initial normalized concentration to be 1