EEE 3394 Electronic Materials Chris Ferekides Fall 2014 Week 3.
-
Upload
arline-edwards -
Category
Documents
-
view
216 -
download
2
Transcript of EEE 3394 Electronic Materials Chris Ferekides Fall 2014 Week 3.
EEE 3394Electronic Materials
Chris FerekidesFall 2014Week 3
HELP SESSIONS
• FRIDAY: @12:10 pm … (2 hrs)
• SATURDAY @ 10:05 am …(2 hrs)
• In ENG 003 … Basement of Kopp Building (ENG)
Kinetic Molecular Theory
What is it? What do we need it for?• Links the “macroscopic” properties of
gases and solids to the kinetic energy of atoms/molecules;
• Explains the pressure of gases … heat capacity of metals … average speed of electrons in semiconductors etc.
• Assumes that atoms/molecules of gases, liquids, solids are in constant motion when above absolute zero temperature
RTNN
PVA
KMT of gases … from Newton’s 2nd Law
…dp/dt=Force
Empirical Result
See assumptions in text …. ..molecules in constant motion .. collision
time negligible compared to free motion .. collisions are elastic .. no effect from external forces etc.
Consider N molecules inside a cubic volume of side a
The change in momentum of a molecule that collides with one of the walls is …
Force exerted by gas on a wall is equal to the rate of change in momentum …
The total pressure is equal to the total force per unit area …
Due to random motion and collisions, mean square velocity in x direction same as in y and z directions … average velocity is 1/3 of vx
3VvNm
P2
3
2x
3
2xN
2x3
2x2
2x1
2 avmN
amv....mvmvmv
aforce Total
P
amv
)v2a(
2mvΔtΔp
F2x
x
x
x2mvpvy
a
Gas atoms
Area A
a
Square Container
a
Face A
Face B
vx
Derivation
Compare …
…where k is Boltzman’s constant
Therefore …the mean square velocity is proportional to T! … adding heat to a gas … raises its temperature and total internal energy!
Rise in internal energy per unit temperature – HEAT CAPACITY
22
vm21N
32
3vNm
PV
kT23
TNR
23
vm21
KEA
2
RTNN
PVA
Derivation
Heat Capacity
... Energy (U) increase per unit temperature (T)
Molar Heat Capacity Cm:
heat capacity of one mole
… for a monatomic gas kTN23
vm21
NU A2
A
dTdU
C
… above based on constant volume … because all added energy is considered to contribute to the temperature rise and not volume expansion (i.e. doing work to increase volume)
R23
kN23
dTdU
C A
Maxwell’s Principle of Equipartition of Energy
... assigns 1/2kT to each “independent way” (degrees of freedom) a molecule can absorb energy
For example:3 degrees of freedom …
5 degrees of freedom …
kT21
3U
kT21
5U
Degrees of Freedom:Monatomic gas – 3 translational…
Diatomic gas – 5 … 3 + 2 rotationalSolid – 6 … 3 kinetic energy of vibration… + 3 potential energy of “spring” i.e. bond stretchingtherefore … Cm=3R
vxvz
vy
x
Iy
y axis out of paper
z
y
Ix= 0
Iz
x
y
z
(a)
Molecular Velocity and Energy Distribution
Term “average velocity” used to this point … therefore a range of velocity values exists…
i.e. VELOCITY DISTRIBUTION
Velocities from zero (at collision) to larger values …
The Velocity Distribution is described by the Maxwell-Boltzmann distribution function
2kT
mv
22
3
v
2
evkT2π
mN4πn
0
0.5
1
1.5
2
2.5
0 500 1000 1500 2000Speed (m/s)
1000 K (727 °C)
298 K (25 °C)
v*vav
vrms
v*vavvrms
Rel
ativ
e nu
mb e
r of
mol
ecu l
esp e
r un
it v e
loci
ty
( s/ k
m)
With nE being the number of molecules per unit volume per unit energy at an energy E!
… last term is know as the BOLTZMANN factor
Atoms have a range of energies BUT a mean energy of 3/2kT !
And another important GENERAL relationship – the PROBABILITY that a certain molecule in a given system will have an energy E
kT
E
212
3
21E eE
kT1
Nπ
2n
kT
E
E CeN
nEnergy, E
T1
T2 > T1
EA
Average KE at T1.
Average KE at T2
Num
ber o
f ato
ms p
er u
nit e
n erg
y, n E
Maxwell-Boltzmann Distribution for Translational Energies (monatomic gas)
Thermally Activated Processes
Arrhenius Behavior …where the rate of change is proportional to:
The Energy EA is “characteristic” of the particular process
What are the consequences of high EA or raising the temperature?
kTEA
e
Thermally Activated Processes
Fig 1.29
D is p la c e m e n t
U = P E (x )
U A *
U A = U B
E A
A B
A *
A A * B
X
Diffusion of an interstitial impurity atom in a crystal from one voidto a neighboring void. The impurity atom at position A must possesan energy EA to push the host atoms away and move into theneighboring void at B.
Fig 1.30
q = 0°
q = 90°
q = 180°
q = 270°
x
yO
A fter N ju m p s
X
L
Y
a
O '
An impurity atom has four site choices for diffusion to aneighboring interstitial vacancy. After N jumps, the impurity atomwould have been displaced from the original position at O.
Thermally Activated Processes
DIFFUSION … ??
EA for P diffusion in Si is 3.69 eV
D is the diffusion coefficient … andDO is a constant (10.5 cm2/s)Rms distance in t seconds is …
WATCH out for the units … Start using eV for energy …And K for TemperaturekT at room temp. is 0.0258 eVD(RT)=1.08x10-61cm2/s …in 5 minutes …L(RT)=8.04x10-26 μmL(200C)=1.74x10-14 μmL(800C)=0.00171 μmL(1100C)=0.134 μm
kTE
O
A
eDD
2DtL
Thermally Activated Processes
DIFFUSION … ??
EA for P diffusion in Si is 3.69 eV
D is the diffusion coefficient … andDO is a constant (10.5 cm2/s)Rms distance in t seconds is …
WATCH out for the units … Start using eV for energy …And K for TemperaturekT at room temp. is 0.0258 eVD(RT)=1.08x10-61cm2/s …in 5 minutes …L(RT)=8.04x10-26 μmL(200C)=1.74x10-14 μmL(800C)=0.00171 μmL(1100C)=0.134 μm
kTE
O
A
eDD
2DtL
nv = vacancy concentration
N = number of atoms per unit volume
Ev = vacancy formation energy
nv N exp Ev
kT
… also a thermally activated process
Equilibrium Concentration of Vacancies
Phase and Phase DiagramPhase: a HOMOGENEOUS portion of a chemical system that has same structure, composition and properties everywhere.
Phase Diagram: A Temp vs Phase diagram in which various phases of a system are identified by lines and regions.
100% Cu 100% Ni
Isomorphous ??… same morphology everywhere
For pure Cu (or Ni) T remains constant as liquid solidifies (or solid melts)
Not for alloy; i.e. temperature does not remain constant as liquid solidifies (or solid melts)
Initial crystal formation – nucleation
Liquidus and Solidus lines ??
Phase Diagrams – T vs. Composition
What Happens @
L0:all liquid
L1:nucleation begins …what is the composition of the solid?go to S1what is the composition of the liquid?go to L1
X:both solid and liquidwhat are the compositions of the solid and liquid?go to S2 and L2what fraction is solid and what fraction is liquid?Use Lever Rule
S3:“opposite” of L1; i.e. nearly all solid!What is the composition of the solid and liquid?go to S3 and L3
S4:ALL solid w composition of 20% Nickel
53.3%0.130.280.200.28
CCCC
WLS
OSL
46.7%0.130.280.130.20
CCCC
WLS
LOS
1000
1100
1200
1300
0 20 40 60
LIQUID
SOLID(a-PHASE)
S2
S1
S3
L2
L3
C0wt.% Ni
L0
L1
X
Pure Cu
S4
L(20%Ni)
L(20%Ni)S(36%Ni)
L(13%Ni)S(28%Ni)
S(20%Ni)
Liquid
LIQUID
US
SOLID
US
TEM
PERA
T URE
( °C)
Phase Diagram
• Solvus Curve:defines the solubility limit boundary …
• Eutectic Point/Temperature:Composition of alloy that results in the lowest melting point temperature
• TWO solid phases (different compositions and Structures):Pb-rich and Sn rich …
………… HOW DO you READ this diagram ?
P u re P b1 0 0
S O L I D U S
L IQ U ID
+ L
6 1 .9 %18 3 °C 9 7 .5 %
8 06 04 02 0C o m p o sitio n in w t.% S n
1 0 0
2 0 0
3 0 0
4 0 0
0
1 9 .2 %
0
LL
M
NO
P
Q
R
L
R '
Q '
R ''
P u re S n
S O L ID U S S O L I D U SL+E
A
B
C D
Tem
per
atu
re(o
C)
SOL
VU
S
The equilibrium phase diagram of the Pb-Sn alloy. The microstructureson the left show the observations at various points during the cooling ofa 90%Pb-10%Sn from the melt along the dashed line (the overall alloycomposition remains constant at 10 %Sn)
Phase Diagrams – Binary Eutectic
Point L:All liquid … composition: 10% Sn
46.7%0.130.280.130.20
CCCC
WLS
LOS
P u re P b
1 0 0
S O L I D U S
L IQ U ID
+ L
6 1 .9 %18 3 °C 9 7 .5 %
8 06 04 02 0C o m p osition in w t.% S n
1 0 0
2 0 0
3 0 0
4 0 0
0
1 9 .2 %
0
LL
M
NO
P
Q
R
L
R '
Q '
R ''
P u re S n
S O L ID U S S O L I D U SL+E
A
B
C DT
emp
erat
ure
(oC
)
SOL
VU
S
The equilibrium phase diagram of the Pb-Sn alloy. The microstructureson the left show the observations at various points during the cooling ofa 90%Pb-10%Sn from the melt along the dashed line (the overall alloycomposition remains constant at 10 %Sn)
Point M:First solid appears – nucleation begins; (L + a); small amount of а-phase
What is the composition ofthe а-phase?
Go across to thesolvus line and read it!
Point N:Both L and a;
What is the composition ofthe а-phase?
Go across to thesolidus line and read it! – 0.07
What is the composition ofthe L-phase?
Go across to theliquidus line and read it! 0.015
Point N:What is the phase content of the alloy? i.e. what fraction is a and what fraction is L?
USE LEVER RULECa=0.07
CL=0.15
CO=0.10
37.5%0.070.150.070.10
CCCC
WaL
aOL
Point O:Nearly all solid a;
What is the composition of the last “drops” of liquid?
Point P:All solid a;
Composition: 10% Sn
Point R:All solid a and β; Composition of a? Composition of β
3% Sn … 98% Sn
How much is β?And how much is a?
USE LEVER RULE …
92.6%0.030.980.100.98
CC
CCW
αβ
Oβα
Point QFirst nuclei of β begin to formWhat are the compositions?
Pb-Sn Binary Eutectic: 10% Sn