Physics PY4118 Physics of Semiconductor · PDF filePhysics PY4118 Physics of Semiconductor...

Physics PY4118Physics of Semiconductor Devices

Introduction

1.1Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland

ROINN NA FISICE

Department of Physics

PY4118 Physics of Semiconductor Devices

Book of Modules:� Module Objective: To introduce students to the physics of

semiconductors and semiconductor devices.

� Module Content: Topics in quantum mechanics and solid state physics as applied to electronic components such as diodes, transistors and optoelectronic devices. Specific topics include crystal lattice structure, charge carrier generation and recombination, mobility. Semiconductor PN junction characteristics are covered and applied to diodes, photo and solar cells, LEDs and semiconductor lasers. Bipolar transistors are also studied, as well as different field-effect transistors.


ROINN NA FISICE



Topic 1 – Quantum of Atoms

Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland

ROINN NA FISICE

Department of Physics 1.3


Topic 2 – Hybrid Orbitals


ROINN NA FISICE



Topic 3 – Crystals & Symmetry


ROINN NA FISICE



Topic 4 – Coupled States


ROINN NA FISICE



Topic 5 – Bands


ROINN NA FISICE


Arising from many coupled states


Topic 6 – Band Filling


ROINN NA FISICE



Topic 7 – Electrons and Holes


ROINN NA FISICE



Topic 8 – Carrier Motion


ROINN NA FISICE



Topic 9 – Generation & Recombination


ROINN NA FISICE



Topic 10 – Continuity Equation


ROINN NA FISICE


Gdx

ndD

dx

dFn

dx

dnF

nn

dt

dn p

npn

p

n

n

ppp ++++−

−=2

2

0 µµτ

Gdx

pdD

dx

dFp

dx

dpF

pp

dt

dp npnp

np

p

nnn ++−−−

−=2

2

0 µµτ

What equations govern the way charges flow?


Topic 11 – The Diode


ROINN NA FISICE



Topic 12 – Metals with Semiconductors


ROINN NA FISICE


cE

vE

VACE

FEFEmetal

n-doped

VACEmqφ

sqφBqφ

smbi qqqV φφ −=

biqV

biqV


Topic 13 – Transistors


ROINN NA FISICE


S G D

Source Gate Drain

n-channeln+ n+

Semi-insulating


Topic 14 – Heterostructures


ROINN NA FISICE



Topic 15 – Processing?


ROINN NA FISICE



The Hydrogen Atom


ROINN NA FISICE



The Hydrogen Atom� We will start with the quantum mechanical

description of the hydrogen atom

� This will be used to explain a great deal of how semiconductors form

� We start with the 3D Schrödinger equation


ROINN NA FISICE



3D Schrödinger equation

( ) 02

22

2

2

2

2

2

=−+∂∂

+∂∂

+∂∂

ψψψψ

VEm

zyx h

04

2

sin

1

sinsin

11

0

2

22

2

22

2

2

2

=

++

∂∂

+

∂∂

∂∂

+

∂∂

∂∂

ψπεφ

ψθ

θψ

θθθ

ψ

r

eE

m

r

rrr

rr

h

r

eV

0

2

4πε−=


ROINN NA FISICE


Spherical coordinates


H Atom: separation of variables

04

sin2

sinsinsin

0

2

2

22

2

2

22

=

++

∂∂

+

∂∂

∂∂

+

∂∂

∂∂

ψπε

θφψ

θψ

θθ

θψ

θ

r

eE

mr

rr

r

h

Set: ( ) ( ) ( ) ( )φθφθψ ΦΘ= rRr ,,

And then divide by: ( ) ( ) ( )φθ ΦΘrR


ROINN NA FISICE




2

2

0

2

2

22

22

1

4

sin2

sinsinsin

φπεθ

θθ

θθθ

∂Φ∂

Φ−=

++

∂Θ∂

∂∂

Θ+

∂∂

∂∂

r

eE

mr

r

Rr

rR

h

The term on the right has no dependence( )θ,r

Thus, we can write the equation as:


ROINN NA FISICE




2

0

2

2

22

22

4

sin2

sinsinsin

lmr

eE

mr

r

Rr

rR

=

++

∂Θ∂

∂∂

Θ+

∂∂

∂∂

πεθ

θθ

θθθ

h

2

2

21lm=

∂Φ∂

Φ−

φ

A constant

The second equation has a easy general solution…


ROINN NA FISICE



H Atom: φ equation

02

2

2

=Φ+∂

Φ∂lm

φ

But, we can also say that:

φlimAe=Φ

( )φπφ +==Φ 2ll imimAeAe

Thus: int12 =→= l

mime lπ

Or: ℤ=lm

Discuss symmetry of solution:φlim

Ae=Φ


ROINN NA FISICE



H Atom: r,θ equations

==

++

∂Θ∂

∂∂

Θ+

∂∂

∂∂

2

0

2

2

22

22

4

sin2

sinsinsin

lmr

eE

mr

r

Rr

rR

πεθ

θθ

θθθ

h

For this equation, we divide by: θsin

∂Θ∂

∂∂

Θ−=

++

∂∂

∂∂

θθ

θθθ

πε

sinsin

1

sin

4

21

2

2

0

2

2

22

lm

r

eE

mr

r

Rr

rR h

Only:

Only:

r

θ


ROINN NA FISICE



H Atom: r,θ equations

( )1sinsin

1

sin 2

2

+=

∂Θ∂

∂∂

Θ− ll

ml

θθ

θθθ

Both must be constant!

A solution exists, when the constant is: ( ) =→+ lll 1

( )14

21

0

2

2

22 +=

++

∂∂

∂∂

llr

eE

mr

r

Rr

rR πεhOnly:

Only:

r

θ


ROINN NA FISICE



H Atom: r equation

First multiply by:2

r

R

( )0

1

4

212

0

2

2

2

2=

+−

++

∂∂

∂∂

Rr

ll

r

eE

m

r

Rr

rr πεh

This equation can be transformed into the Laguerre equation,

and this would take a few lectures

Instead, the solution will be discussed, and a link provided for

the detailed calculation.

https://faculty.washington.edu/seattle/physics227/reading/reading-26-27.pdf


ROINN NA FISICE




H Atom: r equation

Solutions available when: 22

0

2

4

32 n

meEn

hεπ−=

( )rRnl

ln >

1.28ROINN NA FISICE



H Atom: r equation

( )rRnl


ROINN NA FISICE



H Atom: θ equations

( )1sinsin

1

sin 2

2

+=

∂Θ∂

∂∂

Θ− ll

ml

θθ

θθθ

( ) 0sin

1sinsin

12

2

=Θ

−++

∂Θ∂

∂∂

θθθ

θθlm

ll

But this equation is a function of and lm

Thus linking the angular equations.

We use: ( ) ( ) ( )φθφθ ΦΘ=,Y

l


ROINN NA FISICE



Quantum Numbers:

22

0

2

4

32 n

meEn

hεπ−=Principal Quantum Number: n

Orbital angular momentum quantum number: l

Magnetic quantum number: lm

Rules:

lm

nl

n

l ≤

<

> 0

And all are integers

=

gfdpsl

43210

Names of orbits


ROINN NA FISICE



H Atom: angular equation

The angular

equation can be

reduced to a

Legendre equation

(you saw this in 2nd

year E&M), with

following solutions:



Radial wavefunction

Does this mean the electron exists in the nucleus?

( )rR10


ROINN NA FISICE



Probability distribution functions

The probability that the electron exits in is:τd

( ) τψψτ ddrP*=

( )( )( ) τdRR ΦΦΘΘ= ***

( )( )( )φθθ dddrRrR ΦΦΘΘ= **2* sin

The probability that the electron exits in is:dr

( ) ∫∫ ΦΦΘΘ=ππ

φθθ2

0

*

0

**2 sin ddRdrRrdrrP

RdrRr*2=

Why do the integrals cancel?


ROINN NA FISICE



Probability distribution functions

( )rP10

And the electron has a finite probability within the nucleus


ROINN NA FISICE



Probability



Visualise Orbitals

� The best way today is:� Write a program� Download an app.

http://www.orbitals.com/orb/ov.htm


ROINN NA FISICE



s – orbitals (l=0)


ROINN NA FISICE



p – orbitals (l=1)


ROINN NA FISICE



d – orbitals (l=2)


ROINN NA FISICE



Some f and g – orbitals (l=3&4)

g g gg

f ff


ROINN NA FISICE



Question??

� If the course is on semiconductors, why are we worrying about the Hydrogen atom??

� Good question.

� We will discuss how electron orbits lead to crystal structures.

� But first…larger atoms…1.42


ROINN NA FISICE



Larger Atoms


ROINN NA FISICE



H vs larger atoms

H atom All other atoms

multi-electron atoms

1.44


Quantum numbers

1.45


Energy Ordering


ROINN NA FISICE



Aufbau Principle

electrons occupy lowest energy levels available


ROINN NA FISICE


PY4118 Physics of Semiconductor Devices 1.48


ROINN NA FISICE



Rules for filling orbitals

Bottom-up

(Aufbau’s principle)

Fill orbitals singly before doubling up

(Hund’s Rule)

Paired electrons have opposite spin

(Pauli exclusion principle)


ROINN NA FISICE



PhosphorusSymbol: P

Atomic Number: 15

Full Configuration: 1s22s22p63s23p3

Valence Configuration: 3s23p3

Shorthand Configuration: [Ne]3s23p3

↑↓↑↓↑↓↑↓

↑↓↑↓↑↓↑↓

↑↓↑↓↑↓↑↓

↑↓↑↓↑↓↑↓ ↑↓↑↓↑↓↑↓ ↑↓↑↓↑↓↑↓

↑↑↑↑ ↑↑↑↑ ↑↑↑↑

↑↓↑↓↑↓↑↓ ↑↓↑↓↑↓↑↓ ↑↓↑↓↑↓↑↓ ↑↓↑↓↑↓↑↓ ↑↓↑↓↑↓↑↓ ↑↓↑↓↑↓↑↓ ↑↑↑↑ ↑↑↑↑ ↑↑↑↑

1s 2s 2p 3s 3p

Box Notation

1.50

PY4118 Physics of Semiconductor Devices 1.51


Molecular Bonds


ROINN NA FISICE



Bonding Theories

� Valence Bond (VB) theory

� Focuses on atomic orbits

� Molecular Orbital (MO) theory

� Focuses on the modules orbitals (thus more complicated)


ROINN NA FISICE



Valence Bond Theory� Valence Bond Theory is an attempt to explain the

Covalent bond from a Quantum Mechanical view

� All orbitals of the same type (s, p, d, f) have the same energy

� According to this theory, a bond forms when two atomic orbitals

(s/s s/p p/p) “overlap”

� The space formed by the overlapping orbitals has a capacity for

two electrons that have opposite spins, +1/2 & -1/2 (exclusion

principle)

� Note: Each orbital forming the bond has at least one unfilled slot

to accommodate the electron being shared from the other

bonding orbital


ROINN NA FISICE



Valence Bond Theory� Valence bond theory (con’t)

� The bond strength depends on the attraction of the nuclei for

the shared electrons

� The greater the orbital overlap, the stronger (more stable) the

bond

� The extent of the overlap depends on the shapes and directions

of the orbitals

� An s orbital is spherical, but p and d orbitals have more electron

density in one direction than in another

� Whenever possible, a bond involving p or d electrons will be

oriented in the direction that maximizes overlap


ROINN NA FISICE



Valence Bond Theory

Hydrogen, H2 1s1

Hydrogen Fluoride, HF [He]2s22p5

To maximize overlap, half-filled H 1s and F 2p orbitals overlap along the long axis of the 2p orbital

Fluorine, F2 [He] 2s22p5

In F2, the half-filled 2 px orbital on one F atom points end to end toward the half-filled 2px of the other F to maximize overlap


ROINN NA FISICE



Molecular Orbital Theory (1)

� Electrons are shared in a molecule when:� They have similar energy levels.� They overlap well.� They are close together.


ROINN NA FISICE





ROINN NA FISICE


There is an optimum

bond length




ROINN NA FISICE




� Two similar atomic orbitals combine to form two molecular orbitals, one bonding (σ) and one antibonding (σ*).


ROINN NA FISICE



Molecular Orbital Theory (5)� The electrons fill the molecular orbitals of molecules

like electrons fill atomic orbitals in atoms� Electrons go into the lowest energy orbital available

to form lowest potential energy for the molecule.� The maximum number of electrons in each

molecular orbital is two. (Pauli exclusion principle) � One electron goes into orbitals of equal energy, with

parallel spin, before they begin to pair up. (Hund'sRule.)

(With an atom, we used the Aufbau Principle)


ROINN NA FISICE





ROINN NA FISICE


MO Diagram for O2

Why is there no He2?




ROINN NA FISICE


MO Diagram for H2O

Physics PY4118 Physics of Semiconductor · PDF filePhysics PY4118 Physics of Semiconductor...

Documents

Transcript of Physics PY4118 Physics of Semiconductor · PDF filePhysics PY4118 Physics of Semiconductor...