The modified first Fick's law for the diffusion flux density J can be written in terms of chemical potential as follows:

(5–3)

where:

For diffusion alone, the chemical potential can be written as , and its substitution in Equation 5–3 produces the first Fick's law in the form of Equation 9–1 (see Diffusion in the Theory Reference). When, in addition to diffusion, stress, temperature, and electric potential contribute to the process of migration, the expression for the generalized chemical potential of a particle can be written as:

(5–4)

where:

= particle volume

= hydrostatic stress =

Q = heat of particle transport

V = electric potential

Z = effective particle charge number

e = elementary charge

Substitution of Equation 5–4 in Equation 5–3 produces a flux J that includes both diffusion and migration of particles resulting from the hydrostatic stress gradient, temperature gradient, and electric field.

Using TB,MIGR, you can specify the option to model the flux of atoms (ions) (TBOPT = 0) or the flux of vacancies (TBOPT = 1).

The diffusion flux density J of atoms (ions) is coupled to the hydrostatic stress, temperature, and electric potential as follows:

(5–5)

where:

[D] = atomic diffusivity matrix

C = atomic concentration; normalized atomic concentration if and

= saturated concentration (input as MP,CSAT)

= atomic volume

An atom migrates in the direction of the hydrostatic stress gradient

Q = atomic heat of transport

Z = effective (or apparent) atom charge number

For electromigration applications, Z combines the electrostatic charge number (Ze) and the electron wind charge number (Zw). Ze represents the electrostatic force acting on the atom (ion) in the direction of the electric field. Zw represents the force resulting from the momentum exchange of the atom (ion) with the electrons drifting in the opposite direction of the electric field. The electron wind force is generally dominant, and Z is negative for atomic electromigration.

Alternatively:

(5–6)

where:

C = molar concentration

= molar volume

Q = molar heat of transport

Z = effective atomic charge number

F = Faraday constant

R = universal gas constant

In addition to migration effects, TB,MIGR with the atom flux option (TBOPT = 0) can be used for modeling the effects of temperature and hydrostatic stress on the coefficients of diffusivity:

(5–7)

where:

For the atomic flux option (TBOPT = 0), input the material data (TBDATA) as shown in the table below. The k and R notations in the table are defined as:

Constant	Meaning	Description	Equation
C1	Ea/k or Ea/R	Ea = activation energy	or
C2	/k or /R	= volume	or
C3	Q/k or Q/R	Q = heat of transport	or
C4	Ze/k or ZF/R	Z = effective charge number e = elementary charge F = Faraday constant	or
C5	Not used	--	--
C6	Not used	--	--
C7	h	Volume multiplier for diffusivity dependence on hydrostatic stress	or

k=1.38e-23               ! Boltzmann's constant, J/K
k_eV=8.62e-5             ! Boltzmann's constant, eV/K

Va=2.71e-29              ! atomic volume, m^3
Ea=0.8                   ! activation energy, eV
Qt=0.0094                ! heat of transport, eV
Ze=-23                   ! effective charge number
h=1                      ! hydrostatic stress multiplier for D

tb,migr,1,,,0            ! migration model with atomic flux option              
tbdata,1,Ea/k_eV      
tbdata,2,Va/k            
tbdata,3,Qt/k_eV        
tbdata,4,Ze/k_eV       
tbdata,7,h            

The diffusion flux density J of vacancies has the following form:

(5–8)

where:

[D] = vacancy diffusivity matrix

C = vacancy concentration; normalized vacancy concentration if and

= atomic volume

f = vacancy volume relaxation factor.

A vacancy has a volume f and migrates in the direction opposite to the hydrostatic stress gradient. [Kirchheim]

Q = vacancy heat of transport

Z = effective (or apparent) charge number of the vacancy.

A vacancy migrates in the direction opposite to the electron wind; therefore, Z is positive for vacancy electromigration.

TB,MIGR with the vacancy flux option (TBOPT = 1) can also be used to specify the vacancy generation-annihilation rate G:

(5–9)

where:

C = vacancy concentration; normalized vacancy concentration if and

= characteristic time of vacancy generation-annihilation

= equilibrium concentration of vacancies; normalized equilibrium concentration if and

(5–10)

where:

= equilibrium concentration in the absence of stress; normalized vacancy concentration if and

h = vacancy volume multiplier for the calculation of hydrostatic stress effects on the equilibrium concentration. Typically set to 1-f for vacancy electromigration.

e = natural logarithm base

For the vacancy flux option (TBOPT = 1), input the material data (TBDATA) as shown in the table below. The k and R notations in the table are defined as:

k=1.38e-23               ! Boltzmann's constant, J/K
k_eV=8.62e-5             ! Boltzmann's constant, eV/K

Va=1.68e-29              ! atomic volume, m^3
Ze=0.84                  ! effective vacancy charge number
f=0.6                    ! vacancy volume relaxation factor
h=1-f                    ! hydrostatic stress multiplier for Ce
Ce=1                     ! normalized vacancy equilibrium concentration 
ts=1.8e-3                ! characteristic time, s

tb,migr,1,,,1            ! vacancy flux option              
tbdata,2,Va/k            
tbdata,4,Ze/k_eV       
tbdata,7,h            
tbdata,8,f            
tbdata,9,Ce           
tbdata,10,ts    

Resources

The following resource offers more information about the vacancy flux model: