Applying Thermo: an Example
of Kinetics - Diffusion
MSE 310
Elec. Props
of Mat’ls
MSE 310
Elec. Props
of Mat’ls
‰ Fundamental Physics of Force and Energy/Work:
‰ Thermo & Physics as applied to Diffusion:
9 Energy and Work:
9 Consider a force/forces acting on an atom producing
atomic motion.
9 The applied force is given by the previous equation:
o In general:
o The work is given by:
F = −∇Y
dw = − F ⋅ dr (5)
Work or Energy are scalar potentials (e.g., voltage).
Force is a Vector field.
If a potential is constant, there is no FIELD!
Work is done by a Force!
o Combining equations 4 and 5, we have:
dw = − F ⋅ dr = −
∂Y
dr
∂r
v = μ mob F
(6)
(8)
9 The constant of proportionality is called the mobility.
9 Consider:
o Flux of atoms, A, diffusing at an average velocity, v,
through a homogeneous distribution of B atoms.
o The flux of A atoms through B is equal to the product of:
o Key Point: If the field potential is not changing, then no
work would be done.
• Number of A atoms per unit volume (i.e., concentration, CA)
• Average velocity of the A atoms, vA.
o This is given by:
Enough fundamentals! Let’
Let’s apply this physics and thermo!
Knowlton
(7)
9 The motion of the atom will be often interrupted by other
atoms and collisions occur.
9 Thus: the velocity of the diffusing atom over a time
period larger than the time between collisions is an
average velocity.
9 The velocity is proportional to the applied force and can
be written as:
(One can argue that Eqns. 4 and 5 are really one in the same.)
o
o
o
o
Applying Thermo: an Example
of Kinetics - Diffusion
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Knowlton
J A = C AvA
(9)
2
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MSE 310
Elec. Props
of Mat’ls
Applying Thermo: an Example
of Kinetics - Diffusion
‰ Thermo & Physics as applied to Diffusion (cont.):
‰ Thermo & Physics as applied to Diffusion (cont.) –
Derivation of Ficks 1st Law:
9 By combining the last two equations, we obtain:
9 In the following slides, Fick’s 1st law, in which the
concentration gradient is obtained from the chemical
potential, is explicitly derived.
9 Key point: Diffusion in solids is based on this delineation.
(10 )
A
J A = C A μ mob
FA
9 Substituting the Force Field Eqn. 7 (field gradient):
A
J A = −C A μ mob
∇YA
(11)
o This last equation is a general form of Fick’s 1st Law of
Diffusion.
o That is, the flux of A atoms through a homogeneous
distribution of B atoms is due to the gradient of some
potential field.
o NOTE: This potential field can be any of the ones shown in
the previous table.
9 Because the gradient of a potential field follows the
superposition principle, the more general form of Fick’s
1st Law is:
A
J A = −C A μ mob
∑ ∇Yi, A
Applying Thermo: an Example
of Kinetics - Diffusion
MSE 310
Elec. Props
of Mat’ls
9 The chemical potential of atoms A is given by the
thermodynamic relation:
⎛ ∂Ε A ⎞
⎟
⎝ ∂N A ⎠T , P , N
(13)
μA = ⎜
≠A
where ΕA is a free energy of the A atoms in B.
9 Examples of Free Energy, E:
o
o
o
o
(12 )
i
Gibbs, G = G(T, P, N)
Helmhotz, F = F(T, V, N)
Enthalpy, H = H(S, P, N)
Omega potential or Grand Potential, Ω = Ω(T, V, μ)
o where the sum indicates the superposition of potential field
gradients.
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MSE 310
Elec. Props
of Mat’ls
Applying Thermo: an Example
of Kinetics - Diffusion
MSE 310
Elec. Props
of Mat’ls
‰ Thermo & Physics as applied to Diffusion (cont.):
‰ Thermo & Physics as applied to Diffusion (cont.):
9 Substituting eqn. 13 into eqn. 11, the flux with respect to
the chemical potential gradient is obtained:
A
J A = −C A μ mob
∇μ A
9 The mathematical description of the activity is given by:
a A = γ AC A
(14 )
9 Case I: (Henry’s Law) a range of CA much smaller than
the concentration of B atoms (CB), γA becomes constant.
o The chance of interaction between A and B atoms is small
since A is so dilute in B.
o The primary interaction of A is with B.
o This phenomenon is known as Henry’s law.
o Mathematically, as CA approaches zero, the activity
coefficient of A, γA, is given by:
9 The chemical potential of the A atoms may be written as a
function of the chemical activity (i.e., aA) of A in a
distribution of B:
(16 )
γA =
o where aA is the activity of A among B, k is Boltzmann’s
constant and μΑ is the chemical potential of A in the pure
state.
9 The aA may be described as the amount that the chemical
potential of A deviates from the ideal or pure state (i.e.,
ideality).
9 The ideality can be interpreted as the absence of A-A
interaction upon adding an extra A atom to the system.
9 Thus, the enthalpy change, ΔH, of the system is zero.
Knowlton
(17 )
o where γA is the activity coefficient and CA is the
concentration of the A atoms.
9 Assuming one-dimensional diffusion, equation 14
simplifies to:
A ∂μ A
J A = −C A μ mob
(15)
∂x
μ A = μ φA + kT ln a A
Applying Thermo: an Example
of Kinetics - Diffusion
aA
≅ constant
CA
(18)
9 Case II: (Raoult’s law) CA >> CB (i.e., B rather than A
atoms follow Henry’s law), activity coefficient of A is 1.
o The A atoms have a small probability of interacting with B
atoms.
o The primary interaction of A atoms is with other A atoms.
o Hence, the solution of A is effectively pure.
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MSE 310
Elec. Props
of Mat’ls
Applying Thermo: an Example
of Kinetics - Diffusion
MSE 310
Elec. Props
of Mat’ls
‰ Thermo & Physics as applied to Diffusion (cont.):
‰ Thermo & Physics as applied to Diffusion (cont.):
9 Since the ratios of both γA and CA factor to 1, then:
9 In this case, and the activity coefficient is given by:
γA =
aA
≅1
CA
(19 )
A
J A = − μ mob
kT
o This condition is known as Raoult’s law.
o Raoult’s law is predominant for most diffusion processes in
Si since CSi >> Cdopant
9 For either Henry’s or Raoult’s law, Fick’s 1st and 2nd law
may still be derived from the chemical potential.
9 This eventually can be seen by combining equations 15,
16, and 17 into the following form:
A
J A = −C A μ mob
(
φ
∂ μ A + kT ln γ AC A
∂x
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kT ∂ (γ AC A )
∂x
γ AC A
( 22 )
9 Einstein’s relation states that the diffusivity, D, of an atom
is proportional to its mobility where the constant of
proportionality is kT.
9 Mathematically & in terms of A atoms, this is written as:
( 23)
ο μmob units = square of the distance per unit time per unit
energy.
o kT units = energy.
o Thus, DA units = square of the distance per unit time.
) ( 20 )
( 21)
o For Henry’s and Raoult’s law, γA is a constant or one,
respectively.
o In either case, γA may be taken out of the differential.
∂C A
∂x
A
DA = kT μ mob
o Since the chemical potential of a pure substance is constant,
its derivative is zero.
o Furthermore, k and T are constant.
o Under these circumstances and taking the derivative of the
natural logarithm, equation 20 becomes:
A
J A = −C A μ mob
Applying Thermo: an Example
of Kinetics - Diffusion
9 Invoking Einstein’s relation with respect to equation 23,
Fick’s 1st law is obtained:
J A = − DA
∂C A
∂x
( 24 ) Fick’s 1st law
Note: It is the concentration gradient that drives the flux of
atoms from one area to another.
Fick’
Fick’s 1st law = Steady State Diffusion ⎡⎣C ≠ f ( t )⎤⎦
• C, at every point, does not change wrt time.
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MSE 310
Elec. Props
of Mat’ls
Applying Thermo: an Example
of Kinetics - Diffusion
MSE 310
Elec. Props
of Mat’ls
‰ Thermo & Physics as applied to Diffusion (cont.):
‰ Diffusion: Solving Fick’s 2nd Law
9 Solve for C(x,t)
9 Infinite Solutions
9 Need Boundary Conditions
NONNON-STEADY STATE DIFFUSION
9 It has been found that as the concentration of A atoms in
B changes with time, the concentration changes with
position.
9 Known as Fick’s 2nd law, it has the following form:
∂C A
∂J
=− A
∂t
∂x
o Fixed Surface Concentration (Infinite Source)
( 25 )
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( 26 )
∂C A
∂ 2C A
= DA
∂t
∂x 2
9 Two Primary Boundary Conditions:
• Solution: Complimentary Error Function
o Redistribution of a constant total number of diffusing atoms
(Finite Source)
9 Assume that DA is concentration independent.
9 For diffusion in S/Cs, the concentration of dopant atoms
is very small, thus assumption may be comfortably made.
9 Substituting equation 24 into equation 25, Fick’s 2nd law,
is given by:
∂C A
∂ 2C A
= DA
∂t
∂x 2
Applying Thermo: an Example
of Kinetics - Diffusion
• Solution: Gaussian Function
Fick’s 2nd law
9
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Applying Thermo: an Example
of Kinetics - Diffusion
MSE 310
Elec. Props
of Mat’ls
MSE 310
Elec. Props
of Mat’ls
‰ Diffusion: Solving Fick’s 2nd Law
‰ Diffusion: Solving Fick’s 2nd Law
o Fixed Surface Concentration (Infinite Source)
∂C A
∂ 2C A
= DA
∂t
∂x 2
9
1st
• Solution: Complimentary Error Function
⎛ x ⎞
C ( x, t ) = Cs − ( Cs − Co ) erf ⎜
⎟
⎝ 2 Dt ⎠
⎛ x ⎞
= Co + ( Cs − Co ) erfc ⎜
⎟
⎝ 2 Dt ⎠
of 2 Primary Boundary Conditions:
o Fixed Surface Concentration (Infinite Source)
• Solution: Complimentary Error Function
• Boundary Conditions:
For t = 0, C = Co at 0 ≤ x ≤ ∞
For t > 0, C = Cs at x = 0
C = Co at x = ∞
⎛ x ⎞
C ( x, t ) = Cs − ( Cs − Co ) erf ⎜
⎟
⎝ 2 Dt ⎠
⎛ x ⎞
= Co + ( Cs − Co ) erfc ⎜
⎟
⎝ 2 Dt ⎠
Knowlton
Applying Thermo: an Example
of Kinetics - Diffusion
erfc ( z ) = 1 − erf ( z )
= 1−
2
π
∫
z
0
e − y dy
2
11
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MSE 310
Elec. Props
of Mat’ls
Applying Thermo: an Example
of Kinetics - Diffusion
MSE 310
Elec. Props
of Mat’ls
Applying Thermo: an Example
of Kinetics - Diffusion
‰ Diffusion: Thermally Activated Processes
‰ Diffusion: Solving Fick’s 2nd Law
9 Temperature Plays a significant role in diffusion
9 Temperature is not the driving force.
9 Remember:
DRIVING FORCE = GRADIENT of a FIELD
VARIABLE
9 Remember: Driving force for diffusion is a
difference in the chemical potential, μ, eqn. 13.
∂C A
∂ 2C A
= DA
∂t
∂x 2
9 2nd of 2 Primary Boundary Conditions:
o Redistribution of a constant total number of diffusing atoms
(Finite Source)
• Solution: Gaussian Function
• Boundary Conditions:
μ phase1 ≠ μ phase 2 i.e. Δμ ≠ 0 or ∇μ ≠ 0
9 HOWEVER: Temperature increases the activity of
a diffusing species.
9 Diffusivity or Diffusion Coefficient:
Coefficient
⎡ ⎛ x ⎞2 ⎤
C ( x, t ) = C ( 0, t ) exp ⎢ − ⎜
⎟ ⎥
⎢⎣ ⎝ 2 Dt ⎠ ⎥⎦
⎛ x2 ⎞
= C ( 0, t ) exp ⎜ −
⎟
⎝ 4 Dt ⎠
D = Do e
− Eact
k BT
9 Eact is the activation energy for diffusion
9 kb T is the thermal energy
9 Do, the pre-exponential factor, contains a number
of physical constants and properties including:
o entropy of formation of the defect
o attempt frequency for jumps into available
neighboring sites
o lattice constant
o crystal structure dependence
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MSE 310
Elec. Props
of Mat’ls
Applying Thermo: an Example
of Kinetics - Diffusion
MSE 310
Elec. Props
of Mat’ls
‰ Diffusion: Thermally Activated Processes
‰ Diffusion Mechanisms = Processes = Reactions
9 Diffusivity or Diffusion Coefficient:
D = Do e
− Eact
Applying Thermo: an Example
of Kinetics - Diffusion
9 We will use Si as an example of a system with various
diffusion mechanisms.
9 Two types of diffusion mechanisms:
k BT
o Direct diffusion mechanisms: diffusion without the aid of
point defects.
• Interstitial diffusion
o Indirect diffusion mechanisms: diffusion with the aid of
point defects
As + V ⇔ AV Vacancy mechanism
As + I ⇔ AI Interstitialcy mechanism
As + I ⇔ Ai Kick-out mechanism
As ⇔ Ai + V Dissociative (Frank-Turnbull) mechanism
kf
As + I AI
kr
∂ CIreac
= kr C AI − k f CI C A
∂t
k r & k f:
forward & reverse
Coefficients
of reaction
s
∂Ctotal ∂Cdiff ∂Crctn
=
+
∂t
∂t
∂t
Knowlton
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Knowlton
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Applying Thermo: an Example
of Kinetics - Diffusion
MSE 310
Elec. Props
of Mat’ls
‰ Diffusion Mechanisms = Processes = Reactions
9 Indirect diffusion mechanisms: diffusion with the aid of
point defects
1.0
P
B
Ga, Al
0.8
sI
0.6
0.4
As
Acceptors (p-type dopants)
Donors (n-type dopants)
Fit to Donors
0.2
0.0
0.6
0.7
0.8
0.9
Sb
1.0
1.1
1.2
r/rI (atomic radius)
f2.1 interstitial
vs radius.o
5/98
DXeffs ,eq =
DXeffs =
DXeffs
DXeffs ,eq
Knowlton
eq
eq
C XI
DXI C XV
DXV
eff
eff
+
= DXI
+ DXV
eq
CX s
C Xeqs
Equilibrium
eq
eq
C XI
DXI CI C XV
DXV CV
eff C I
eff CV
Nonequilibrium
+
= DXI
+ DXV
C Xeqs CIeq
C Xeqs CVeq
CIeq
CVeq
=
eff
eff
CI
DXV
CV
C
C
DXI
+
= sI eqI + sV Veq
eff
eff
eff
+ DXV
CIeq DXeffI + DXV
CVeq
CI
CV
DXI
17
9