eee 3394 electronic materials chris ferekides spring 2014 week 2
TRANSCRIPT
EEE 3394Electronic Materials
Chris FerekidesSPRING 2014
WEEK 2
DEFECTSWhat are defects?
1. POINT DEFECTS:• Vacancy• Interstitial• Substitutional
DEFECTSWhat are defects?
2. Schottky Defect3. Frenkel Defect
4. LINE DEFECTS• Edge dislocation• Screw dislocation
F r enkel defect
S chottky defect
AC
D
Dislocation line
DEFECTSWhat are defects?
5. Planar Defects• Grain boundaries
Strained bond
Broken bond (danglingbond)
Grain boundary
Void, vacancySelf-interstitial type atomForeign impurity
Crystal Structure ???
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 EvkT
… 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