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Crystal Structural Properties and Energy Bands, Schemes and Mind Maps of Law

An overview of crystal structural properties, including miller indexes for crystalline directions and planes, spacing of equivalent planes, and bragg's law. It also covers energy gaps in solids, derived from the scattering of periodic potentials, and discusses the atomic origins of energy bands and their relation to metals, insulators, and semiconductors.

Typology: Schemes and Mind Maps

2022/2023

Uploaded on 02/12/2024

shaunytheboi
shaunytheboi 🇺🇸

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Disclaimer: This file is meant to help students to learn the materials. It does not imply
that the final exam will be limited to the contents covered here. It is important to
review all HWs, sim labs, quizzes, previous exams, and other course materials.
Final review
EEE 352
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Download Crystal Structural Properties and Energy Bands and more Schemes and Mind Maps Law in PDF only on Docsity!

Disclaimer: This file is meant to help students to learn the materials. It does not imply

that the final exam will be limited to the contents covered here. It is important to

review all HWs, sim labs, quizzes, previous exams, and other course materials.

Final review

EEE 352

Crystal Structural Properties

Miller indexes for crystalline directions and planes

Spacing of equivalent planes

Bragg’s Law

, 1  4 , 3 , 2 

^ 

2 2 2

0

h k l

a

d

 

What is the spacing

of (004) plane?

2 d sin   n , n  1 , 2 , 3 , 

1/

1

1

1/

[100]
[220]

Crystal Planes and Directions

Crystal Planes and Directions

(113) planes

x

y

z

Basic Quantum Mechanics

The Schrödinger Equation

  • **Time-dependent
  • Time-independent**

t

E

i

x t x e

 ( , )  ( )

( ) ( ) ( )

( )

V x x E x

dx

d x

m

 

  

2

2 2

2

t

x t

x t V x t i

dx

d

m

  

( , )

( , ) ( , )

  

2

2 2

2

( x , t )  ( x ) w ( t )

Wavefunctions

Probability explanation and Wavefunction Normalization

  • Heisenberg Uncertainty Principle
  • Wavepacket and group velocity

Δx Δp  / 2

 E  t 

group particle

particle

phase

v

m

p

m

k

k

v

v

m

p

m

k

k

v

( ) d 1

x x

( x )

Electron in a Quantum Well

2

2

2

2

2

mE

k

k x

dx

d x

x

x = - a
V

0

x = a

I

II

III

Schrodinger
Equations

2

2

2

2

2

mV E

x

dx

d x

2

2

2

2

2

mV E

x

dx

d x

Wavefunctions

ikx ikx

Ce De

x x

Ae Be

  

x x

Fe Ge

  

( x  )finite 1

Boundary conditions

B  0 F  0

 

  

  0 0

1 2

1 0 2 0

x x x x

x x

x x x x

 

( )( )

( a ) ik ( a ) ik ( a )

Ae Ce De

   

( a ) ik ( a ) ik ( a )

Ae ikCe ikDe

   

ika ika a

Ce De Ge

  

ika ika a

ikCe ikDe Ae

 

For E < V

0

we have confined
states

Electron in a Quantum Well

2

2

2

2

2 2

2

mE

k

k x

dx

d x

Schrodinger
Equations

2

2

1

2

1 2

2

mE V

k

k x

dx

d x

Wavefunctions

ikx ikx

Ce De

ik x ik x

Ae Be

1 1

ik x ikx

Fe Ge

1 1

( x  )finite 1

Boundary conditions

 

  

  0 0

1 2

1 0 2 0

x x x x

x x

x x x x

 

( )( )

( ) ( ) ( ) ( ) 1 1 2 2 ik a ik a ik a ik a

Ae Be Ce De

    

ika ika ika ika

Ce De Fe Ge

1 1

 

For E > V

0

, we do not have confined states

2

2

1

2

1 2

2

m E V

k

k x

dx

d x

Yes. But not useful.

( )

2

( )

2

( )

1

( )

1

1 1 2 2

ik a ik a ik a ik a

ik Ae ikBe ik Ce ik De

    

ika ika ika ika

ikCe ikDe ik Fe ikGe

1 1

1 1

 

By adding symmetry requirement and the normalization condition, all parameters can be determined.

Model for Electronic Properties of Crystals

Free

electron

model

  • Potential: V=
  • Wavefunction: Plane wave
  • Periodic Boundary condition

Electron in

a Periodic

Potential

Potential: Periodic

Wavefunction: Bloch Theorem

Key conclusions: i) There are

bandgaps!!!! ii) 2N electronic

states in a band

Band

structure

Dispersion relation

Brillouin Zones (optional)

Electron effective mass

Explanation of insulators and

metals

Holes

Quasi Free Electron Model

(m* to replace m

0

DOS Distribution Functions Occupied DOS

3 1/

2 3

g E ( ) (2 m E )

1

1

 

exp[( )/ ]

( , )

E E k T

f E T

F B

( )

( ) ( )

exp[( ) / ] 1 F B

g E

g E f E

E E k T

 

1

1

1

 

 

exp[( )/ ]

( , )

E E k T

f E T

F B

*3 1/

2 3

( ) [2 ( )]

c e g

g E m E E

*3 1/

2 3

( ) [2 ( )]

v h v

g E m E E

ρ(E)

E

E c = E g

E v

= 0

g c (E)

g v (E)

Eg

1

1

 

exp[( )/ ]

( , )

E E k T

f E T

F B

E c

E v

HOLES

ELECTRONS

0.

E F

E

f ( E )

0 K

600 K

6000 K

(^01)

( )

( ) ( ) ( )

exp[( ) / ] 1

occ (^) e

e e

F B

g E

g E g E f E

E E k T

 

 

( ) ( )[1 ( )]

exp[( ) / ] 1

occ

v v

v

F B

g E g E f E

g E

E E k T

Intrinsic Semiconductor

Intrinsic & Extrinsic Semiconductors



( ) ln

e

h

B

g

F

m

m

k T

E
E T

Extrinsic Semiconductor

)* 13. 6 ( )

  • 1

( 2

0

eV

m

m

E E

r

c D

 

  • Impurity binding energy

0 0

0 D A

p n N N

 

  • (^) Charge neutrality condition    

D D D

N  N  f E

A A A

N  N f E

  • Number of ionized donors

Number of ionized acceptors

2

0 0

exp

g

i c v

B

E

n p n N N

k T

 

    

 

0 0

exp

g

i c v

B

E

n p n N N

k T

Extrinsic Semiconductors

  • It is reasonable to assume that ALL of the dopants are ionized at ROOM TEMPERATURE

Methods to determine the carrier concentration

**For HEAVY doping and FULL ionization of dopants, the position of the Fermi energy in** 

the gap varies as

A D i

i

A

F i B

N N N
N
N

E E k T   

  ln ,

D A i

i

D

F i B

N N N
N
N

E E k T   

  ln ,

Discuss the limitations

of these approximations

 

D D A A

N N , N N

1/ 2 2

2

0

0

2 2

i A D A D

i

n N N N N

p p n

n

  (^)        

        

     

1/ 2

2

0

2 2

D A D A

i

N N N N

n n N

       ^  

   (^)     

     

3/

( )/

(^0 )

1 2

,

4

F c B E E k T (^) e B

c c

m k T

n n N e N

 

    

 

3/ 2

( )/

(^0 )

1 2

,

4

v F B E E k T (^) h B

v v

m k T

p p N e N

 

    

 

Carrier Diffusion in Semiconductors

e edrift ediffusion e e e e

h hdrift hdiffusion h h h h

e N eD N

e N eD N

    

    

J J J E

J J J E

h e

J  J  J

h diff h h ediff e^ e

J  eDN JeDN

h drift hdiffusion edrift e diffusion

J  J J  J

At equilibrium

e

k T

D

B

n p

n p

Einstein Relation

TheThe pnpn JunctionJunction

Potential

energy

-x x p x n

p-TYPE n-TYPE

DEPLETION

REGION

V(x)

2

2

ln

B A D D A

bi

i D A

k T N N eW N N
V

e nN N

N d

-N a

-x p

x n

- - -

+ + +

E v

E c FERMI LEVEL E F

p-TYPE

REGION

n-TYPE

REGION

eV bi

a p d n

N xN x

ChargeNeutralityRequires :

 

 

A

x p

D

x n

dV eN
E x C x x
dx
dV eN
E x D x x
dx

1/

bi

A D

V
W

e N N

 ^ 