MSE 310
Elec. Props of Matls
Knowlton
Thermally Activated Processes
‰ General Description of Activated Process:
Unstable (Activated State)
U(X) = PE = mgh
Metastable
Stable
E
A
ΔU
A*
U
A*
UA
A
U
B
B
X
XA
XA*
XB
System Coordinate, X = Position of Center of Mass
Fig. 1.27: Tilting a filing cabinet from state A to its edge in state A*
requires an energy EA. After reaching A*, the cabinet
spontaneously drops to the stable position B. PE of state B is lower
than A and therefore state B is more stable than A.
Knowlton
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
MSE 310
Elec. Props of Matls
Knowlton
1
Thermally Activated Processes
- Arrhenius Behavior -
‰ Diffusion:
A
A*
B
U = PE(x)
U A*
A*
EA
UA= UB
A
B
Displacement
X
Fig. 1.28: Diffusion of an interstitial impurity atom in a crystal from
one void to a neighboring void. The impurity atom at position A
must posses an energy EA to push the host atoms away and move
into the neighboring void at B.
Knowlton
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
2
1
Thermally Activated Processes
- Arrhenius Behavior -
MSE 310
Elec. Props of Matls
Knowlton
‰ Diffusion: Thermally Activated Processes
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 phase1 ≠ μ phase 2 i.e. Δμ ≠ 0 or ∇μ ≠ 0
9 HOWEVER: Temperature increases the activity of a diffusing species.
9 Question: What is the probability that an atom will diffuse?
o Atoms are Bosons
o Use Maxwell-Boltzmann Statistics!
f (ε ) =
N (ε )d ε = f (ε ) g (ε )d ε
=
=
Ne − βε g (ε )d ε
∫e
− βε
g (ε )d ε
Ne− βε
∫e
− βε
g (ε )d ε
No. Impurities with E > E A
Total No. of Impurities
= Ae
− EA
kT
Knowlton
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MSE 310
Elec. Props of Matls
Knowlton
Thermally Activated Processes
- Arrhenius Behavior -
‰ Diffusion:
O'
After N jumps
L
Y
a
θ = 90°
θ = 0°
θ = 180°
O
y
X
x
θ = 270°
Fig. 1.29: An impurity atom has four site choices for diffusion to
a neighboring interstitial vacancy. After N jumps, the impurity
atom would have been displaced from the original position at O.
Knowlton
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
4
2
MSE 310
Elec. Props of Matls
Knowlton
Thermally Activated Processes
- Arrhenius Behavior -
‰ Diffusion:
9 Frequency of Jump is proportional to the probability
9 Coefficient of Diffusion is proportional to Jump Frequency
ν = ν 0 f (ε )
= Aν 0 e
−
D ∝ν
E
kT
∝ Aν 0 e
−
E
kT
9 Diffusivity or Diffusion Coefficient (Arrhenius Rate Equation):
Equation)
D = Do e
− EA
kT
• Eact is the activation energy for diffusion
• kb T is the thermal energy
• Do, 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
Knowlton
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MSE 310
Elec. Props of Matls
Knowlton
Thermally Activated Processes
- Arrhenius Behavior -
‰ Diffusion: Thermally Activated Processes
9 Diffusivity or Diffusion Coefficient:
D = Do e
Knowlton
− EA
kT
6
3
MSE 310
Elec. Props of Matls
Knowlton
Thermally Activated Processes
- Arrhenius Behavior -
‰ Diffusion: Diffusivity or Diffusion Coefficient:
9 Question:
Question How does one obtain:
ln D
o Activation energy
o Pre-exponential
D = Do e
− Eact
m=−
EA
kB
k BT
Linearize Diffusivity:
− Eact
⎛
⎞
ln D = ln ⎜ Do e kBT ⎟
⎝
⎠
Best way to calculate E A :
E
= − A + ln Do
k BT
m=
Has form of Equation of a Line!
y = mx + b
Thus:
y = ln D; x =
Δy ln D2 − ln D1
=
1 1
Δx
−
T2 T1
1
T
So :
E A = −k B ⋅ slope
1
E
; m = − A ; b = ln Do
T
kB
= −kB ⋅
ln D2 − ln D1
1 1
−
T2 T1
Knowlton
7
MSE 310
Elec. Props of Matls
Knowlton
Thermally Activated Processes
- Arrhenius Behavior -
‰ KINETICS: Diffusion and "Chemical Reactions: Thermally
Activated Processes
9 We will use Si as an example of a system with various diffusion
mechanisms.
9 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
As + V ⇔ AV Vacancy mechanism
As + I ⇔ AI Interstitialcy mechanism
As + I ⇔ Ai Kick-out mechanism
As ⇔ Ai + V Dissociative (Frank-Turnbull) mechanism
Knowlton
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Thermally Activated Processes
- Arrhenius Behavior -
MSE 310
Elec. Props of Matls
Knowlton
‰ KINETICS: Diffusion and "Chemical Reactions: Thermally
Activated Processes
9 Chemical reaction causes a change in concentration, CI, of interstitials
kf
As + I ↔ AI
kr
∂C
∂t
reac
I
= kr C AI − k f CI C As
k r & k f:
forward & reverse
Coefficients
of reaction
k = ko e
EA
kT
9 For the equation that describes the total kinetics, that is, the total change
in the concentration of C, or Ctotal:
∂Ctotal ∂Cdiff ∂Crctn
=
+
∂t
∂t
∂t
Knowlton
9
MSE 310
Elec. Props of Matls
Knowlton
Thermally Activated Processes
- Arrhenius Behavior -
Extra notes for those that are interested
‰ Arrhenius behavior is observed in many areas of science
9 Conduction in solids
Knowlton
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5
MSE 310
Elec. Props of Matls
Knowlton
Thermally Activated Processes
- Arrhenius Behavior -
Extra notes for those that are interested
‰ Other Examples of Arrhenius Rate Behavior:
9 Much of Kinetics shows this behavior
o
o
o
o
o
o
Carrier concentration and conduction in semiconductors and insulators
Mass Transport
Defect Formation
Rates of Chemical Reactions (Coefficient of Reaction Rate)
Creep Rate
Dislocation motion
Knowlton
11
MSE 310
Elec. Props of Matls
Knowlton
Thermally Activated Processes
- Arrhenius Behavior -
‰ Carrier Concentration in Semiconductors:
Intrinsic
EeEeeEc
Ec
e-
e-
Ee-
e- e- e-
ee-
e-
e-
e-
Ev
Ev
Ec
e-
Intrinsic
Ee-
e- Ed
Ec
n-type
e-
Ed
Ev
n-type
Knowlton
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6
MSE 310
Elec. Props of Matls
Knowlton
Thermally Activated Processes
- Arrhenius Behavior -
‰ Carrier Concentration:
n = no e
− EG
2 k BT
Intrinsic carrier concentration as a function
of 1/T (Arrhenius plot).
Knowlton
13
MSE 310
Elec. Props of Matls
Knowlton
Thermally Activated Processes
- Arrhenius Behavior -
‰ Carrier Conductivity in insulators:
1×10-1
σ ∝n
− EA
As3.0Te3.0Si1.2Ge1.0 glass
1×10-3
k BT
Conductivity 1/(Ωm)
σ =σe
24%Na2O-76%SiO2
Pyrex
1×10-5
1×10-7
12%Na2O-88%SiO2
1×10-9
1×10-11
PVAc
SiO2
PVC
1×10-13
1×10-15
1.2
1.6
2
2.4
2.8
103/T (1/K)
3.2
3.6
4
Fig. 2.28: Conductivity vs reciprocal temperature for various low
conductivity solids. (PVC = Polyvinyl chloride; PVAc = Polyvinyl
acetate.) Data selectively combined from numerous sources.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Knowlton
14
7
Thermally Activated Processes
- Arrhenius Behavior -
MSE 310
Elec. Props of Matls
Knowlton
‰ SiO2 Growth Kinetics Models: Deal-Grove Model
A & B/A
B = C1e
− E1
B
= C2 e
A
kbT
− E2
k bT
(oxidant diffusion)
(interface reaction rate)
Plots of B & B/A
using values in
table.
Knowlton
15
MSE 310
Elec. Props of Matls
Knowlton
Thermally Activated Processes
- Arrhenius Behavior -
‰ Chemical Vapor Deposition (CVD)
J1
J2
J SBL = J1
J rxn = J 2
= k s Cs
= − Dgas
=
Dgas
δS
∂C
∂x
(C
gas
− Cwaf. surf. )
= hg ( C gas − Cwaf. surf. )
Knowlton
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