MSE 510
Thermally Activated Processes

General Description of Activated Process:
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MSE 510

Diffusion:
Thermally Activated Processes
- Arrhenius Behavior -
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MSE 510
Thermally Activated Processes
- Arrhenius Behavior -
 Diffusion: Thermally Activated Processes
 Temperature Plays a significant role in diffusion

Temperature is not the driving force.

Remember:
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DRIVING FORCE = GRADIENT of a FIELD VARIABLE
 Remember: Driving force for diffusion is a difference in the chemical potential.
µ phase 1
≠ µ phase 2
i.e.
µ
0 or
µ
0
 HOWEVER: Temperature increases the activity of a diffusing species.

Question: What is the probability that an atom will diffuse? o Atoms are Boltzons o Use Maxwell-Boltzmann Statistics !
N
ε ε = f
ε ε ε
=
Ne
− βε g
ε ε e
− βε g
ε ε f
=
=
Ne
− βε
e
− βε g
ε ε
No. Impurities with E
>
E
A
Total No. of Impurities
=
Ae
− E
A k T
3

MSE 510

Diffusion:
Thermally Activated Processes
- Arrhenius Behavior -
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MSE 510
Thermally Activated Processes
 Diffusion:
- Arrhenius Behavior -

Frequency of Jump is proportional to the probability

Coefficient of Diffusion is proportional to Jump Frequency
ν ν
=
0
ν f
0
ε
−
E kT
D
∝
∝
ν
ν
−
E kT
 Diffusivity or Diffusion Coefficient (Arrhenius Rate Equation) :
D
=
D e o
−
E
A kT
• E act
is the activation energy for diffusion
• k b
T is the thermal energy
• D o
, the pre-exponential factor , contains a number of physical constants and properties including:
» entropy of formation of the defect
» attempt frequency for jumps into available neighboring sites
» lattice constant
» crystal structure dependence
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MSE 510
Thermally Activated Processes

Diffusion:
- Arrhenius Behavior -
Thermally Activated Processes

Diffusivity or Diffusion Coefficient :
D
=
D e o
−
E
A kT
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MSE 510 Thermally Activated Processes
- Arrhenius Behavior -

Diffusion: Diffusivity or Diffusion Coefficient :

Question: How does one obtain: o Activation energy ln D o Pre-exponential ln D ln

D e o
−
E act
D
=
D e o
Linearize Diffusivity:
−
E act


= −
E
A
+ k T
B ln D o
Has form of Equation of a Line!
y
= mx b
Thus: y
= ln D x
=
1
T m
= −
E
A k
B
=
D o
Best way to calculate E
A
: m
=
∆
∆ y x
= ln D
1
2
−
− ln
1
T
2
T
1
D
1
So :
E
A
= − ⋅ slope ln D
2
− ln
1
−
1
T
2
T
1
D
1 m
= −
E
A k
B
1
T
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MSE 510 Thermally Activated Processes
- Arrhenius Behavior -

KINETICS: Diffusion and "Chemical Reactions: Thermally
Activated Processes

We will use Si as an example of a system with various diffusion mechanisms.

Two types of diffusion mechanisms: o Direct diffusion mechanisms: diffusion without the aid of point defects.
• Interstitial diffusion o Indirect diffusion mechanisms: diffusion with the aid of point defects
A s
+ ⇔
AV
A s
+ ⇔
AI
Vacancy mechanism
Interstitialcy mechanism
A s
+ ⇔
A i
A s
⇔
A i
+
V
Kick-out mechanism
Dissociative (Frank-Turnbull) mechanism
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MSE 510 Thermally Activated Processes
- Arrhenius Behavior -

KINETICS: Diffusion and "Chemical Reactions: Thermally
Activated Processes

Chemical reaction causes a change in concentration, C
I
, of interstitials
A s
I k k f
+ ↔ r
AI
∂
C
I reac
∂ t
= k C r AI
− k C C f I A s k r
& k f
: forward & reverse
Coefficients of reaction k
= k e o
E
A kT

For the equation that describes the total kinetics, that is, the total change in the concentration of C, or C total
:
∂
C
∂ total t
=
∂
C diff
∂ t
+
∂
C rctn
∂ t
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MSE 510 Thermally Activated Processes
- Arrhenius Behavior -
 Arrhenius behavior is observed in many areas of science
 Conduction in solids
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MSE 510 Thermally Activated Processes
- Arrhenius Behavior -
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
Other Examples of Arrhenius Rate Behavior:

Much of Kinetics shows this behavior o Carrier concentration and conduction in semiconductors and insulators o Mass Transport o Defect Formation o Rates of Chemical Reactions (Coefficient of Reaction Rate) o Creep Rate o Dislocation motion
11

MSE 510 Thermally Activated Processes
- Arrhenius Behavior -

Carrier Concentration in Semiconductors:
Intrinsic E e-
Intrinsic
E e- e -
E c
E c e - e - e - e - e - e -
E v
E e- e - e - e -
E c e - e - e -
E d
E v
E e-
E c
E d n-type e -
E v n-type
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MSE 510 Thermally Activated Processes
- Arrhenius Behavior -

Carrier Concentration: n
= n e
−
E
G
2
Intrinsic carrier concentration as a function of 1/T (Arrhenius plot).
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MSE 510 Thermally Activated Processes
- Arrhenius Behavior -

Carrier Conductivity in insulators:
σ ∝ n e
−
E
A
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MSE 510 Thermally Activated Processes
- Arrhenius Behavior -

SiO
2
Growth Kinetics Models: Deal-Grove Model
A & B/A
B
=
C
1 e
−
E
1
B
=
C
2 e
−
E
2
A
(oxidant diffusion)
(interface reaction rate)
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Plots of B & B/A using values in table.
15

MSE 510 Thermally Activated Processes
- Arrhenius Behavior -

Chemical Vapor Deposition (CVD)
J
1
J
2
J rxn
=
J
2
= k C s
J
SBL
=
J
1
= −
D gas
∂
C
∂ x
=
=
D gas
δ
(
C gas
−
C waf. surf.
S
( gas
−
C waf. surf.
)
)
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