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