DIFFUSION IN SOLIDS
Steady-state diffusion experiments
(13.1)
(13.2)
r1, r2: inside and outside radii of the tube
C1, C2: corresponding concentrations of carbon at these surfaces
For steady state, the quantity of carbon passing through the tube
per unit time is constant and independent of r.
(13.3)
J = quantity of carbon passing through the tube wall per unit time
jr = local flux
I = length of the cylinder
(13.4)
or
(13.5)
According to Fick’ s firs law, the flux of hydrogen through the metal is:
(13.6)
According to Sievert’s law, for equilibrium between gas and the
metal:
(13.7a)
(13.7b)
K is the equlibrium constant for the reaction
(13.8)
p1 and p2 = partial pressures of hydrogen on both sides of the foil with a
thickness δ.
(13.9)
(13.10)
In relation to the diffusion of gases through solids, the term permeability, P,
is often defined by:
(13.11)
(13.12)
(13.13)
There are also other ways to define permeability:
(13.14)
P* = P = DS only at 1 atm pressure, or 0.1013 MPa
(13.15)
(13.16)
Transient Diffusion Experiments
In many cases it is not possible to carry out steady-state experiments in
order to determine diffusion coefficients in solids.
So;
(13.17)
Transient Diffusion Experiments
1. Thin film source: infinite and semi-infinite sink
(13.18)
(13.19)
(13.20)
(13.21)
(13.22)
Transient Diffusion Experiments
∼
2. Diffusion couple with constant D
∼
If D is independent of composition, and the bars extend far enough in the
positive and negative domains to be considered infinite, then Eq. (13.18)
applies with the boundary conditions:
(13.23a)
(13.23b)
(13.23c)
(13.23d)
Due to symmetry, the concentration at x = 0 immediately takes on the
average value of C1 and C2 (This average is Cs).
(13.24)
The concentration profile given by this equation is symmetrical about
∼
x = 0. This is known as Grube solution, and applies when D is not a
function of concentration.
Transient Diffusion Experiments
∼
3. Diffusion couple with variable D
(13.25)
∼
The solution to this equation is useful for obtaining D over a range of
compositions, but it doesn’t give a solution C(x,t), which is usually sought,
but rather lows ∼
D(C) to be calculated from an experimental plot of C(x). This
method of analyzing experimental data is called the Boltzmann – Matano
technique.
(13.26a)
(13.26b)
(13.27)
(13.28a)
(13.28b)
(13.29)
Since the concentration gradient goes to zero as C approaches C1, the
∼
right-hand side of Eq.(13.29) is simply D(dC/dλ). Then,
(13.30)
(13.31)
(13.32)
(13.33)
∼
A solution to Fick’s second law with constant D (or simply D):
(13.34)
θ is (C – Cs)
(13.36)
(13.37)
Boundary condition (13.35b) requires that c2 = 0, and when it is applied
(13.35c) c1cosλL = 0 results. This is satisfied by λ = (2n + 1)π/2L, where n is
an integer from 0 to ∞ .
(13.38)
The initial condition, θ(x,0) = θi = Ci –Cs, remains to be satisfied and when
substituted into Eq. (13.38):
(13.39)
If Fourier’ s analysis is applied:
(13.40)
(13.41)
This equation is useful for describing concentration profiles as a
function of time.
(13.42)
Average concentration
The relative change in average composition for diffusion into a slab:
(13.43)
This expression is good for diffusion into or out of the slab.
(13.44)
τ = 4L2/π2D
Time constant for diffusion
process
Microelectronic Diffusion Processing
Two steps of diffusion (for silicon processed into devices):
ƒ predeposition
ƒ drive-in diffusion
Predeposition step:
The silicon matrix is semi-infinite so the concentration profile of the
dopant is:
(13.47)
Cs = constant surface concentration
C0 = initial concentration
X = distance measured from the surface at x = 0
D = diffusion coefficient of the dopent in the silicon
t = time
The amount of dopant predeposited, usually expressed in terms of the
number of dopant atoms, is:
(13.48)
A = surface area of the silicon wafer
jx=0 = flux of dopant atoms at the surface (x=0)
The flux at the surface:
(13.49)
(13.50)
The amount predeposited
During predeposition, the diffusion depth of the dopant profile is also
calculated:
(13.51)
Drive-in diffusion:
It is done to allow diffusion from the surface further into the silicon.
During drive-in, the boundary conditions and the initial condition for
C(x,t) are:
(13.52a)
(13.52b)
C(x,0) = f(x’)
(13.52c)
(13.53)
f(x’) is the distribution of the solute from the predeposition step and
given by Eq. (13.47).
(13.54)
(13.55)
Cs and l are from the deposition step, but D and t are for the drive-in step.
‘Junction depth’ (Xj) is a measure of the depth of diffusion after drive-in
diffusion.
Since Cs>>C0 ,
C
2  l 
≅ 
C s π  Dt 
1/ 2
 x2 
exp  −

Dt
4


(13.56)
And xj is that value of x where C = C0
(13.57)
Homogenization of Alloys
Depending on the cooling rate, the dendritic arm spacing can be
between 5 µm and 400 µm, with typical values ranging from µm to
200 µm.
Within a local region undergoing solidification:
To get a quantitative picture of the microsegregation:
(13.59)
(13.60)
To describe the homogenization kinetics, a solution to Fick’s second law is
needed that satisfies:
(13.61a)
(13.61b)
(13.61c)
The solution (as given by the Crank):
(13.62)
(13.63)
(13.64)
Residual segregation
index
(13.65)
Example 13.7
Formation of Surface Layers
C = cation concentration, D = diffusivity of the cation
If the oxide layer is thin, the concentraiton profile of the cation is
approximated as linear, and can integrate Fick’s law as:
(13.66)
The flux is proportional to the rate of growth of the thickness of the oxide
layer:
(13.67)
(13.68)
Tammann scaling constant or Pilling and Bedworth constant (m2s-1)
(13.69)
∆m/A = kg (mass gained) m-2 (surface area)
ρ0 = concentration of oxygen in the oxide, kg of oxygen m-3 of oxide.
(13.70)
Practical parabolic scaling constant, (kg O2)2m-4s-1
(13.71)
(13.72)
In terms of free energy for an ideal solution:
(13.73)
(13.74)
(13.75)
(13.76a)
(13.76b)
(13.76c)
(13.76d)
For the compound in general:
(13.77)
(13.78)
(13.79)
Since the growth rate of the compound is the sum of the particle fluxes:
(13.80)
(13.81)
The Tammann scaling constant is related to kr by
(13.84)
If it is assumed that the anion exists as B2 in the gaseous state, and there is
ideal gas behaviour:
(13.85)
(13.86)
Solution: Since D*0 is much less than D*Ni,
Molar volume of Zr
(13.87)
Molar volume of ZrO2
Beyond the interface and into the metallic phase, for x > M,
(13.88)
If equilibrium is established at all times at the interface, and C is the oxygen
content of the metal in equilibrium with ZrO2,
(13.89a)
(13.89b)
(13.89c)
(13.90)
x = distance from the original interface
x’ = x – M
x’ = distance from the oxide – metal interface
(13.91)