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Physics of Thin Films

PES 449 / PHYS 549

Kinetics and Diffusion

Ohring: Chapter 1, Sections 1.6, 8.1 - 8.2

Kinetics

= how fast it will happen

we will concentrate on mass transport

= atoms diffusing through a solid

Diffusion in one dimension - Fick's 1st and 2nd Laws

Fick's 1st Law = "stuff moves from where you have lots to where you have little"

equation of Fick's First Law

graphs showing Fick's First Law

 

Now let us consider the flux of atoms into and out of a particular volume in the solid.

The fluxes (J) may be different at different positions (x) in the material.

Mathematically:

equation relating fluxes

model of mass flow

rate of increase of matter in the region = rate of flow in - rate of flow out

= (J1 - J2 ) Æx = - J / ¶x

or this is also = rate of change of concentration in the volume

so . . .

derivation of Fick's Second Law

Solve this equation as a boundary value problem (see example in Ohring p. 34-36.

Atomic view

for simplicity consider a cubic lattice

atomic diffusion is easier into a vacancy

There is always a potential energy barrier to diffusion (activation energy).

What do we expect mathematically for the flux to the right (from position1 to 2):

equation of flux to right

Similarly we can find the flux to the left:

derivation of diffusion coefficient

(note: if we had used the gas constant, R, instead of Boltzmann constant, k, then the energy would be the diffusion energy/mole)

COMMENTS:

  • diffusion increases with temperature

Diffusion Coefficients

  • self diffusion (element A in A): DA
  • vacancy diffusion: DV
  • chemical diffusion (element A in B): DAB
  • grain boundary diffusion: Dgb
  • surface diffusion: Ds

    actually D is typically NOT CONSTANT, D is a function of:

    • position
      • jump frequency depends on local atomic arrangement and defects
      • temperature gradients
    • direction in lattice
    • time
      • defects, concentrations, temperature may vary with time

    high diffusivity paths:

  • grain boundaries
  • three dimensional dislocation networks
  • surfaces

    These are all more open structures with higher jump frequencies and lower energy barriers.

    • Do and ED are different for these paths

    Usually the cross sectional areas of these are small compared to the rest of the film.

    In general (but not necessarily at high temperature): DS > Dgb > D

Arrhenius Plot

for determining activation energies

 start with Fick's First Law:

equations for Arrhenius plot

Arrhenius plot with slope -E/k

Other effects on diffusion

Diffusion can be changed by stress fields, electric fields, other energy gradients (interfaces)

[note: Ohring changes notation here: uses diffusion free energy per mole GD instead of diffusion energy per atom ED so all of the k's become R's.]

Examine what happens when we apply a field:

diffusion in a field

 

 equations for diffusion in a field

How fast do atoms diffuse?

equations of Nernst-Einstein

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