2006 2007 Teacher Guide ENG

8/16/2019 2006 2007 Teacher Guide ENG

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ES2B3 Electronic Materials

MODULE STRUCTURE

This part of the course consists of 5 lectures (Week 15-19) and alaboratory (Hall effect)

Lecture 1: A bit of Quantum theory

Lecture 2: Semiconductors

Lecture 3: Magnetic Electronic materials

Lecture 4: Optical Electronic materialsLecture 5: Electronic materials in an Electrical field


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ES2B3 Electronic MaterialsRecommended text for the course:

Copies in the library

Warning, the latest issues are priced at £150+

Older issues MUCH cheaper!


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ES2B3 Electronic MaterialsWell worth a read:


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ES2B3 Electronic MaterialsA good reference:


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ES2B3 Electronic MaterialsWell worth a read:


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Lecture 1: A bit of Quantum theory


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A Sense of Scale


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ES2B3 Electronic MaterialsQuantum Mechanics

To describe things that are very

small requires quantummechanics.

The Heisenberg uncertainty

principle:

– The more precisely we know the position of an object, the worse we

know its momentum.

To describe anything as small as an

atom requires the use of

quantum mechanics.

To understand electronic materials

we need to think at the atomic

level!

Heisenberg in 1925, at the age of 24


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Relativity

To describe things moving very fast requires the theory of relativity.

Special Relativity

– We cannot catch up with light.

– Mass is a form of energy.

E = m c2

General Relativity

– GR encompasses gravity anddescribes the expanding universeand black holes.

Not needed for the course, butneeds mentioning!

Einstein in 1905, at the age of 26


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• light and particles are both wavesand particles

• Want to think of particles or“systems” (e.g. a lump of

silicon!) in terms of waves• Talk about “probability

functions”

• Schr ödinger’s theory allows us to

start to do this!• Won’t go into Schrödinger in this

course in any detail!

Schrödinger


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Schrodinger’s atom

•Dispensed with the concept of the

particle

•Focussed on the wave-like

properties of matter

•Picture has electron standing waves

as orbits

•Actually in 3-D

•2-D pictures shown here


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ES2B3 Electronic MaterialsOur present theory of particle physics:

The Standard Model

This is a grand intellectualachievement of the second

half of the 20th Century

The theory is based onrelativistic quantum field

theory (QFT).

– The first QFT was the quantum

theory of electricity and

magnetism.

– Way, way beyond the scope of

this course!

Feynman ca. 1960


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The more precisely the position is

determined, the less precisely the

momentum is known in this instant,

and vice versa.

--Heisenberg, uncertainty paper,1927

2

1 x p x

Heisenberg’s uncertainty principle:

(“Uncertainty in position” times “uncertainty in momentum” is

greater than or equal to a constant)


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Consider an electron in the lowest-energy state of a hydrogen atom; its position is

known to an accuracy of about 0.05 nm (the radius of the atom). What is the

minimum range of its possible momenta? Velocity?

v= p/me

= 2.3106 m/s

me= 0.9110-30 kg.

Solution:

p /x

= 2.110-24 J·s/m

= 2.1

10

-24

kg-m/s

= 1.0510-34 J·s.

x p Heisenberg’s uncertainty principle (with = h/2 ).

Heisenberg Uncertainty Principle: Example


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Uncertainty Principle – Implications

• The uncertainty principle explains why (negative) electrons in

atoms don’t simply fall into the (positive) nucleus: If the

electron were “confined” too close to the nucleus (small x), it

would have a large p, and therefore a very large averagekinetic energy ( ( p)2/2m).

• The uncertainty principle does not say “everything is

uncertain”. Rather, it tells us very exactly where the limits ofcertainty lie when we make measurements of quantum

systems.


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ES2B3 Electronic Materials Bohr’s Atomic Model

• Niels Bohr proposed an

atomic model that suggested

that electrons orbit the

nucleus, much like the Earth

revolving around the Sun.

• These orbitals (the path by

which the electrons orbit thenucleus) had fixed distances

and fixed energies.

• The orbitals closest to the

nucleus have the lowest

energy, and the orbitalsfurthest away from the

nucleus have the greatest

energy (like a SPRING)


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ES2B3 Electronic MaterialsBohr’s Electron Shells and

Principal Quantum Numbers

• Bohr assigned his quantized electron orbits, which he called shells, anumber, n. This number is known as the principal quantum number. Asthe number grew larger, the shell got further and further away from thenucleus.

• Bohr also stated that:

– The energy of each shells grew as n grew larger. – Each shell can hold a maximum of 2n2 electrons

– As we try to determine where the electrons in an atom are, we shouldfill the shells from the innermost shell (n=1) to the outermost shell(n=4) (most of the time, although it can equal 5, 6, 7, 8, etc.)

• Any atom with its electrons in their lowest energy levels is said to be in itsground state


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Orbital occupancy for the first 10 elements, H through Ne.


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Using the Periodic Table to Fill Subshells


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Valance Shells, Energy, Ground states and Excited States

• A valance shell is the outermost occupied shell of an atom. Valenceelectrons are, therefore, the electrons occupying that shell.

• This valence electron determines the chemical and electricalcharacteristics of the element

• Core electrons are the electrons in the filled inner (core) shells.

• The energy of the atom increases as n increases, and as the shells arenot completely filled.

• The ground state refers to the configuration of electrons in an atom

that results in the lowest possible energy.• An excited state refers to when an electron “jumps” to a higher energy

shell even though the lower energy shells are not completely filled.


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Another Way of Looking

at the Hydrogen

Spectrum:

Electrons dropping from

one level to another giveout light with the energy

of each photon equal to

the difference in the

electronic energy levels


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What happens when we look at a solid? - Lithium (3

electrons)

• Single atom has 2 electrons in 1s, and 1 in 2s

• Know Lithium is a metallic solid

• Think of a solid; it will have N electrons (Lots:~1023) in N ψ2s

orbitals• As the atoms are brought together, the energy levels split into

N levels – finely separated

• This is a consequence of the Paul i exclusion principle

• Maximum energy spread between the energy levels when the

atoms are spaced at an inter-atomic distance, a• These N energy levels form an energy band


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From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)

http://Materials.Usask.Ca

a

InteratomicSeparation ( R)

SYSTEM

N Li Atoms

N Electrons

N Orbitals

2 N States

E 2p

1 s

2 s

2 p

E 2s

E 1s

System of N Li Atoms

solid

(N)

Solid

E T

E B solid

(1)

Isolated Atoms

E M P T Y

F U L L

E l e c t r o n E n e r g y i n t h e

S y s t e m o f N L i A

t o m s

Fig. 4.8: The formation of a 2 s-energy band from the 2 s-orbitalswhen N Li atoms come together to form the Li solid. The are N 2 s-electrons but 2 N states in the band. The 2 s-band therefore is

only half full. The atomic 1 s orbital is close to the Li nucleus andremains undisturbed in the solid. Thus each Li atom has a closed

K -shell (full 1 s orbital).


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• We call this the 2s energy band

• 1s band is, of course full!

• Because N is so big, then we think of the band as acontinuum

• About 10eV between the top and bottom of the band

• Other levels also split, and some overlap the 2s energy band (fig. 4.9)

• This forms a band that stretches from the bottom of the 2slevel up to the vacuum (free electron) level

• The higher levels (3d, 4s etc) have energies above the

vacuum level, so aren’t occupied


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Interatomic

Separation ( R)

E 2s

E 2p

E 3s

E 1s

R = a R = Isolated AtomsThe Solid

E = 0 (Vacuum Level)

Free electron

F U L L

E M P T Y



Fig. 4.9: As solid atoms are brought together from infinity, the

atomic orbitals overlap and give rise to bands. Outer orbitals overlapfirst. The 3 s orbitals give rise to the 3 s band, 2 p orbitals to the 2 p

band and so on. The various bands overlap to produce a single bandin which the energy is nearly continuous.

E l e c t r o n e n e r g y


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• At absolute zero, electrons fill all the lower levels, from EB up to EFO, the Fermi Level

• Need to know what reference you use for the energy levels• Fermi level measured relative to the bottom of the band,

and called the Fermi energy , 4.7eV for Li

• Energy needed to promote an electron from the Fermi level

to the vacuum level is the Work Function (Φ) of the metal• Above 0ºK, electrons are excited above the Fermi level by

heat

• Electrons aren’t bonded to a specific atom, and occupy a“gas” or “sea” around the atoms

• Means that the electrons are represented by a travelling ,not a localised wave-function

• Each electron has a wave-vector, k, and its momentum isħk


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Electron Energy

VacuumLevel

E F0

E B

E F0

Electron inside

the metal

Electron outside

the metal

0

2.5 eV

7.2 eV 0

4.7 eV

7.2 eV

Fig. 4.11: Typical electron energy band diagram for a metal All

the valence electrons are in an energy band which they only partially fill. The top of the band is the vacuum level where theelectron is free from the solid ( PE = 0).

Empty levels

Levels occupied

by electrons


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• If we think of a “sea” of electrons, then all of their energy

is kinetic• Energy of an electron increases with its momentum, p, as

p2/2me

• Electrons take on all momentum values until their energy

reaches EFO• Average momentum is zero, and there is no net current

• As Temperature increases, must consider what happens to

the distribution of electrons throughout the energy levels,

leads to the idea of the Fermi-Dirac function – Fig. 4.26


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ES2B3 Electronic Materials E

E F

0 1/2

1 f ( E )

T 1

T = 0

T 2 > T 1



Fig. 4.26: The Fermi-Dirac function, f ( E ), describes thestatistics of electrons in a solid. The electrons interact witheach other and the environment so that they obey the PauiliExclusion Principle.


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Semiconductors

• (What you’ve all been waiting for!)

• Si has 14 electrons

• The electrons in Si atoms strongly interact

when atoms brought close, so Si is a solid

• Ground levels full

• Only need to consider 3s and 3p levels


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• Each Si-Si bond has 2 paired electrons.

• Gives an energy band called the valence band (VB)

• This is full

• Also have an energy band with an energy gap, Eg abovethe valence band, and is called the conduction band

• So we have a CB and a VB across the whole solid


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E c

E v

CB

VB

E g Thermal

excitation

Fig. 4.18: Energy band diagram of a semiconductor. CB is theconduction band and VB is the valence band. At 0 K, the VB is full

with all the valence electrons.

E l e c t r o n e n e r g y


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Density of states in an energy band

• Following the same ideas we used for Li, we know there arelots of levels in a band (~1023)

• Single atom has a fixed number of nearest neighbours andmany distant neighbours

• Define the density of states g(E) such that g(E)dE is the

number of states (wavefunctions) in the energy interval E to E+dE per unit volume of the sample.

• The number of states per unit volume up to some energy E’is:

E v dE E g E S

0

l l


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Intrinsic semiconductors

• Intrinsic means perfect – i.e. with no impurities

• Si has a diamond structure, and will vibrate at temperaturesabove 0K

• Thermal vibrations can rupture bonds producing freeelectrons and holes

• Extrinsic Si has impurities added-

– Adding As effectively adds electrons to give n-type

– Adding B accepts electrons to give p-type

ES2B3 El M lorbitals


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Electron energy

Valence Band (VB)

Full of electrons at 0 K.

E c

E v

0

E c +

(c)(b)

B

Conduction Band (CB)

Empty of electrons at 0 K.

hyb orbitals

Si ion core (+4e )

Valence

electron(a)

Si crystal in 2-D

Fig. 5.1: (a) A simplified two dimensional illustration of a Si atom with

four hybrid orbitals, hyb. Each orbital has one electron. (b) A simplified

two dimensional view of a region of the Si crystal showing covalent bonds. (c) The energy band diagram at absolute zero of temperature.



Bandgap =E g

ES2B3 El i M i l


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• Electrons in the CB can be treated If free,though must use an effective mass, me

• Figure 5.3 shows the creation of an

electron-hole pair by a photon

• Figure 5.4 shows the creation of an

electron-hole pair by a thermal vibration

ES2B3 El i M i l


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e – hole

CB

VB

E c

E v

0

E c+

E g

FREE e – h > E g

HOLE

Electron energy

h

(a) (b)

Fig. 5.3: (a) A photon with an energy greater than E g can

excite an electron from the VB to the CB. (b) When a photon breaks a Si-Si bond, a free electron and a hole in the

Si-Si bond is created.



ES2B3 El t i M t i l


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End of lecture 1



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Heisenburg’s Uncertainty Principle

For an electron trapped in an infinite well , there is an uncertainty of a in itsposition, from x = 0 to x = a.

The momentum is

Taking the product of the uncertainties

Generally, -Heisenberg’s Uncertainty Principle

k p x

ha

a p x x p 2

2

1 x p x



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x = 0 x = a0

E 1

E 3

E 2

E 4

n = 1

n = 2

n = 3

n = 4

Energy levels in the well

1

2

3

4

(x) sin(np x/a) Probability density | ( x)|2

0 a a0

0 a x

V ( x)

0

V = 0

Electron

V = V =

x

Fig. 3.15: Electron in a one-dimensional infinite PE well. The energyof the electron is quantized. Possible wavefunctions and the probability distributions for the electron are shown.


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2006 2007 Teacher Guide ENG