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School of Mathematical and Physical Sciences
PHYS1220
Solid State Physics
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Quantum mechanics has played a very important role in the
development of our understanding of the electronic and physical nature
of solid materials.
Technology has progressed to sub-micron dimensions in electronic
microchips and in this regime quantum effects are important.
Generally, solid material may be:


amorphous, with no periodic structure, e.g. glass.
crystalline, with a orderly arrangement of atoms or molecules in
the form of a periodic lattice.
Crystalline Structure
Lattice
Basis
+
Sept. 2002
Solids Slide1
School of Mathematical and Physical Sciences
PHYS1220
Crystals
Crystalline materials are theoretically tractable due to the presence of
long range order in the lattices.
There are 14 different standard crystal lattices, three of which are
shown below.
Face centred cubic lattice.
Simple cubic lattice.
Body centred cubic lattice.
Sept. 2002
Solids Slide2
School of Mathematical and Physical Sciences
PHYS1220
Example - NaCl Crystal
Face centred cubic structure that repeats.
Similar sharing of electrons as in molecular
bonding. Each Na+ ion is surrounded by 6
Cl- ions and vice versa.
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
Potential function has both attractive and
repulsive components (as for molecules):
1 e2 B
U  
 m
40 r r
 is known as the Madelung constant and has a value of 1.75 for NaCl; m
is a small integer.
NaCl has an equilibrium distance ro between ions at which the ionic
cohesive energy (energy/ion to disassemble solid into separate ions) is:
 e2  1 
U0  
1  
40 r0  m 
Sept. 2002
Solids Slide3
School of Mathematical and Physical Sciences
PHYS1220
Metalic Solids
The bulk of the elements are metallic in nature.
In general metallic elements form crystalline structures which are
relatively close-packed such as hexagonal close packing, body
centred cubic, or face centred cubic
The outermost electrons of metallic atoms are weakly bound. When
these atoms come together to form crystalline structures, the loosely
bound electrons are relatively free to move among the atoms as an
electron ‘gas’.
Since the atoms ‘lose’ their outermost electrons, they are essentially
positive ions.
Sept. 2002
Solids Slide4
School of Mathematical and Physical Sciences
PHYS1220
Metallic Solids (ctd)
According to current metallic bond theory the metalic structure is
held together by the electrostatic attraction between the positive
ions and the negatively charged electron ‘gas’.
The electron ‘gas’ is also thought to be responsible for the high
electrical and thermal conductivities, surface luster and other
metallic properties.
Since the outermost electrons are free to move among the ions,
they do not belong to any single atomic bond. It is therefore
possible to alloy different metals provided their atoms are similar
in size.
Sept. 2002
Solids Slide5
School of Mathematical and Physical Sciences
PHYS1220
Free electron theory of metals
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Electrons that are free to move in a piece of metal may have an
extremely large number of energy states available to them.
It is as if the electrons are trapped in a finite well whose width L is of
the size of the piece of metal and, therefore, very large compared to
atomic dimensions.
The electron energy is therefore quantised (as in the case of a one
dimensional finite well) and, since L is very large, the allowed energy
levels are very closely spaced.
n 2h 2
For infinite square well: En 
8mL2
The “density of states” g(E) provides a statistical means of dealing
with the large number of states which are available.
g(E) represents the number of states per unit volume per unit energy
interval.
Sept. 2002
Solids Slide6
School of Mathematical and Physical Sciences
PHYS1220
Free electron theory of metals (ctd)
g(E)dE represents the number of states per
unit volume that have energy between E and
E + dE
The density of states g(E) is given by
g (E ) 
8 2m
h

3
3
2
1
E
2
For a 1cm cube of Copper, the number of states in the range 5.05.5eV is:
N ~ g(E)VΔE 
8
3
2(9.1x1031 kg) 2
(6.63x1034 J.s) 3
1
(1x106 m3 )(5.25x1.6 x1019 J) 2
x (0.5x1.6 x1019 J) ~ 8x10 21
Sept. 2002
Solids Slide7
School of Mathematical and Physical Sciences
PHYS1220
Occupancy of States - Fermi-Dirac statistics
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In keeping with the exclusion principle, two electrons cannot be in the
same state if they have the same set of quantum numbers.
A given state can therefore accommode only two electrons, one with
spin +1/2 and the other with spin -1/2.
A consequence is that, at zero Kelvin,
the energy states are filled from the
zero energy level, with two electrons
per state, until all electrons are
accounted for. The last occupied state
is called the Fermi level and the
corresponding energy is called the
fermy energy EF.
Sept. 2002
Solids Slide8
School of Mathematical and Physical Sciences
PHYS1220
Occupancy of States - Fermi-Dirac statistics (ctd)
EF is determined by integrating the density of state expression for
energy over the interval from E =0 up to E= EF.
h 3 N
EF 


8m   V 
2
2
3
N/V is the number of conduction electrons per unit volume in the
metal
Copper has EF = 7.0eV and Eaverage = 4.2eV. Energy from thermal
motion is 3/2kT=0.04 eV.
As the temperature increases, it is expected that the electrons will
gain thermal energy and occupy energy states above the Fermi
level. The probability of an energy state E being occupied is
given by the Fermi factor f(E)
f (E) 
Sept. 2002
1
e  E  E F  kT  1
Solids Slide9
School of Mathematical and Physical Sciences
PHYS1220
Fermi-Dirac Probability Function
f (E) 

At T = 0 K,
1
e  E  E F  kT  1
f(E) =

1
E < Ef
0
E > Ef
i.e. all states are occupied up to the
Fermi level (probability f(E) =1)
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Note that at higher temperatures (e.g. T = 1200K in the figure) the
change in the Fermi factor is not very large.
The equation for f(E) shows that for any temperature T, f(E) = 0.5
when E = Ef.
A state of energy E = Ef has therefore a 50% chance of being
occupied.
Sept. 2002
Solids Slide10
School of Mathematical and Physical Sciences
PHYS1220
Density of Occupied States
The distribution of electrons in the
allowed states is abtained by
multiplying the availability of states
g(E) by the probability of occupancy
f(E).
The product g(E)f(E) is the density of
occupied states.
8 2m
no ( E )  g ( E ) f ( E ) 
h3

3
2
E
1
2
e  E  E F  kT  1
no(E)dE is the number of electrons per unit volume with energy
between E and E+dE in equilibrium at temperature T.
Sept. 2002
Solids Slide11
School of Mathematical and Physical Sciences
PHYS1220
Density of Occupied States
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Note that the number of electrons promoted to
higher energy states is relatively small and these
are the ones closest to the Fermi level. Electrons
in lower energy states are not promoted
thermally.
Compare with x-ray production where the lower
energy electrons are ejected and a high energy
electron relaxes, by emitting an x-ray, to fill the
vacancy.
The energy at the Fermi level corresponds to very high electron
speeds. E.g. for Copper:
h 3 N
EF 


8m   V 
2
2
3
N
where 
 8.4  1028 m 3
 V  Copper
 7.0eV  KE  1 mvF2
2
v F  1.6 106 m / s
Sept. 2002
Solids Slide12
School of Mathematical and Physical Sciences
PHYS1220
Band Theory of Solids
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The electrical and thermal properties of metals can be explained
relatively well in terms of the free electron model.
However the model does not explain why some solids are metals
and others insulators or semiconductors.
It also does not explain differences in the conductivities of these
materials.
In the free electron model, the electrons are assumed to move in a
well of uniform potential (i.e. a field-free region). No account is
taken of the influence of periodic arrangement of ions in the crystal.
A travelling ‘electron wave’ in the crystal is subjected to a periodic
potential. It is the interaction of the ‘electron wave’ with the periodic
potential that results in differences in the properties of solid
materials.
Sept. 2002
Solids Slide13
School of Mathematical and Physical Sciences
PHYS1220
Band Theory of Solids (ctd)
In order to explain the
properties of metals, insulators
and semiconductors, the
periodic potential experienced
by electrons must be accounted
for.
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The figure shows the periodic
potential U that arises from
interaction of electrons with the
periodic array of ions in a
crystal.
For a free electron, U =0. Negative potentials imply bound electrons.
Solution of the Schrodinger equation for a periodic potential results in
energy bands and bandgaps, as shown in the figure.
Only electrons that are in the highest energy band close to the Fermi
level can move freely in the solid.
Sept. 2002
Solids Slide14
School of Mathematical and Physical Sciences
PHYS1220
Band structure of Solids (ctd)
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When two atoms approach
approach each other, the
wavefunctions of the
outermost electrons overlap.
Taking hydrogen as an
example the two 1s states,
which have the same energy
when the atoms are far apart, split into two states of different energy
in keeping with the Pauli exclusion principle. The same happens to the
2s state as shown in figure (a).
If six atoms are brought together, each state splits into six states
of different energy as shown in figure (b).
Sept. 2002
Solids Slide15
School of Mathematical and Physical Sciences
PHYS1220
Energy Bands in Solids (ctd)
If N atoms are brought together, each state splits into N states of
different energy (i.e. the number of states equals the number of
atoms) because of the overlapping wavefunctions.
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Since a sample of solid material contains a
large number of atoms (e.g. in 1cm3 of Cu,
there are ~ 1023 atoms), each state splits
into energy levels so close together that
energy bands are formed. Each band
consists of essentially a continuous range of
allowed energies. The energy bands are
separated by energy gaps known as
bandgaps. The width of a band depends on
the lattice spacing.
The properties of a solid are determined by the energy bands, the extent
to which they are occupied by electrons and the size of the band gap.
Sept. 2002
Solids Slide16
School of Mathematical and Physical Sciences
PHYS1220
Conductors, Insulators and Semiconductors
We are now in a position to address the question of why some solids
are conductors, while others are insulators or semiconductors.
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For solids that are good conductors, the highest
energy band occupied by electrons is only partially
filled as illustrated in the figure.
Taking sodium as an example, the 1s, 2s and 2p
bands are full. For a sample of N sodium atoms, the
3s band has 2N available states but there are only N
electrons (one 3s electron/atom) to fill the 3s band.
Consequently, the 3s band is only half-filled.
Conductor
Partially filled band
Partially filled
band
If a potential difference is applied across the
sample, the electrons in the partly filled band can
accelerate and gain energy because there are
unoccupied states of higher energy available.
It is therefore easy for a current to flow, making
sodium a good conductor.
Sept. 2002
Sodium energy bands
Solids Slide17
School of Mathematical and Physical Sciences
PHYS1220
Conductors, Insulators and Semiconductors (ctd)
In the case of insulators the highest band
occupied by electrons, called the valence band,
is completely filled.
Insulator
The next higher band, called the conduction
band, is completely empty and there is a
bandgap of typically 5 to 10eV between the
valence and conduction bands.
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At room temperature (~300K), electrons have an average kinetic
energy of ~ 0.04eV and can therefore not overcome the
bandgap.
Also, if a potential difference is applied across the sample, the
electrons in the valence band cannot accelerate and increase
their energy since there are no empty states readily available.
Such materials are therefore insulators.
Sept. 2002
Solids Slide18
School of Mathematical and Physical Sciences
PHYS1220
Conductors, Insulators and Semiconductors (ctd)
Conductor
Insulator
Semiconductor
The band structure of pure (intrinsic) semiconductors is similar to
that of an insulator except that the valence and conduction bands are
separated by a smaller bandgap Eg of typically 1eV (e.g. Eg for Si is
1.11eV at 300K).
At room temperature a few electrons have sufficient energy to
overcome the bandgap. At higher temperatures, more electrons are
able to do so resulting in lower resistivity. The resistivity of
semiconductors generally decrease with increasing temperature
(resistivity of Si is -.07/oC and that of Ge is -0.05/oC) in contrast with
that of metals which generally increases.
Sept. 2002
Solids Slide19
School of Mathematical and Physical Sciences
PHYS1220
Conductors, Insulators and Semiconductors (ctd)
When an electron in the valence band of a
semiconductor makes a transition to the
conduction band, it leaves behind a vacant
state known as a ‘hole’.
When a potential difference is applied
across the semiconductor sample, the
electrons in the conduction band result in
a current flow.
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Conduction band
Hole
Valence band
However the electrons in the valence band also contribute to the
current by filling the empty states (or holes) left behind by electrons
that have made transitions to the conduction band.
Both electrons and holes contribute to conduction, and the
resistivity decreases.
Sept. 2002
Solids Slide20
School of Mathematical and Physical Sciences
PHYS1220
Conductors, Insulators and Semiconductors (ctd)
At T = 0K, the Fermi energy of
insulators and semiconductors are
mid-way between the top of the
valence band and the bottom of the
conduction band.
The situation does not change
significantly at 300K

For a solid sample of 1021 atoms, the number of electrons promoted
across the band gap at 300K is:
12 across 1.1eV band gap.
 Semiconductor: Total  10
 Insulator: Total  none across 5 eV band gap.
21 electrons are available for conduction.
 For a conductor all 10
Sept. 2002
Solids Slide21
School of Mathematical and Physical Sciences
PHYS1220
Doped Semiconductors: n-type doping
The band structure and
resistivity of intrinsic
semiconductors can be
modified by the controlled
addition of ‘impurity’ atoms
(typically 1 part in 106 or 107).
This process is known as
doping.
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n-type doping
Silicon is an important semiconductor material with 4 valence electrons.
When silicon is doped with arsenic (or some other element which also
has 5 valence electrons) the arsenic atoms occupy silicon sites in the
lattice.
Of the five valence electrons from each arsenic atom, four form
covalent bonds with adjacent silicon atoms but the fifth electron can
move relatively freely as in a conductor. This increases the conductivity
of the doped silicon sample.
Sept. 2002
Solids Slide22
School of Mathematical and Physical Sciences
PHYS1220
Doped Semiconductors: n-type doping
Silicon that has been doped with a pentavalent atom such as arsenic is
known as an n-type semiconductor because conduction is due to
negative charges (electrons).
Since each pentavalent atom essentially ‘donates’ an electron to the
lattice, it is called a donor atom.
Sept. 2002
Solids Slide23
School of Mathematical and Physical Sciences
PHYS1220
Doped Semiconductors: p-type doping
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If silicon is doped with a trivalent
element such as gallium, the
three valence electrons form
covalent bonds with adjacent
silicon atoms, but a vacency
(hole) exists at the gallium site.
p-type doping
An electron from a silicon atom can move into the hole at the gallium site
leaving a hole at the silicon site. This hole can then be filled by an
electron from another silicon atom , etc. The hole is equivalent to a
positive charge. This increases the conductivity of the doped silicon
sample.
Silicon samples doped with trivalent atoms such as gallium are known as
p-type semiconductors because conduction is due to positive holes.
Note that both n-type and p-type semiconductors are electrically neutral.
Sept. 2002
Solids Slide24
School of Mathematical and Physical Sciences
PHYS1220
Semiconductor Doping - Energy Band Picture
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In doped semiconductors, ‘impurity’ states are formed between the valence
and conduction bands.
In n-type semiconductors the ‘impurity’ energy level lies very close (~
0.05eV for silicon compared to thermal energy of 0.04eV at 300K) to the
conduction band. Electrons are readily promoted to the conduction band
from the ‘impurity‘ level which is, therefore, known as the donor level.
In p-type semiconductors the ‘impurity’ level lies just above the valence
band. Electrons are readily accepted from the valence band leaving holes
behind. The ‘impurity’ levels are therefore known as acceptor levels.
Sept. 2002
Solids Slide25
School of Mathematical and Physical Sciences
PHYS1220
Semiconductor Devices
Semiconductor doping is an important process that, under carefully
controlled conditions, can be used to produce well-defined regions of
different conductivities in a doped semiconductor material.
This has resulted in the development of semiconductor devices such as
the diode and transistor, and miniaturisation of electronic circuitry
which has revolutionised the electronics industry.
In 1965, Dr Gordon Moore predicted that the number of transistors on
a manufactured chip would double every year. In 1986, the 386
processor contained 275,000 transistors. The current Pentium 4
processor contains 42,000,000 transistors. Ultimately the maximum
density of components will be determined by quantum effects in the
semiconductor materials.
Sept. 2002
Solids Slide26
School of Mathematical and Physical Sciences
PHYS1220
Semiconductor Devices - Diodes
A diode consists of a semiconductor
substrate which has been doped so that
one end is p-type and the other n-type.
In the absence of a voltage across the
diode, some electrons from the n-type
region drift into the p-type region where
they combine with some holes. The n-type
region is left with a positive charge.
Similarly, holes drift from the p-type region
into the n-type region combining with
electrons. The p-type region is left with a
negative charge.
A potential difference is established which
prevents further diffusion of holes and
electrons.
Sept. 2002
Solids Slide27
School of Mathematical and Physical Sciences
PHYS1220
Semiconductor Devices - Diodes (ctd)
If a potential difference is applied
across the diode such that the p-side is
positive and the n-side negative, a
current flows in the diode if the
voltage exceeds 0.6V for Si and 0.3V
for Ge. The diode is then said to be
forward biased.
If the p-side is made negative and the
n-side positive, the diode is said to be
reverse biased and virtually no current
flows.

Diode symbol:
P
n
However if the diode is reverse biased and the voltage is high enough, a
large current flows due to ionisation of atoms in the material. The
voltage at which this occurs is relatively constant for a large range of
current. Diodes which are designed to have this property are known as
Zener diodes. They are used in voltage regulation circuits.
Sept. 2002
Solids Slide28
School of Mathematical and Physical Sciences
PHYS1220
Semiconductor Diodes - A Few Applications.
Rectifiers
Since a diode p-n junction conducts in one direction but not in
the reverse direction (provided the voltage is not too high),
diodes can be used to change ac voltages into dc voltages.

Figure (a) shows a simple rectifier circuit. The diode conducts during
the positive half of the ac cycle but not the negative half. The result
is as shown in figure(b).
Sept. 2002
Solids Slide29
School of Mathematical and Physical Sciences
PHYS1220
Semiconductor Diodes - A Few Applications (ctd)
Light Emitting Diodes (LED)
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Conduction
band
When a diode is forward biased, electrons
Emitted photon
from the n region cross to the p region
Valence
where they combine with holes. In the band band
model, this is equivalent to an electron
making a transition from the conduction to
the valence band.
In the process, a photon may be emitted with energy approximately
equal to that of the bandgap Eg. In some materials the value of Eg
results in photons whose wavelengths are in the visible part of the
spectrum.
LEDs are commonly used in various types of visible displays such as
those of calculators, clocks, car dashboards, VCRs, CD, DVD players,
etc.
Infrared LEDs are used in remote controls.
Sept. 2002
Solids Slide30
School of Mathematical and Physical Sciences
PHYS1220
Semiconductor Diodes - A Few Applications (ctd)
Photodiodes and Solar Cells
One way of looking at photodiodes and
solar cells is to think of them as LEDs but
in reverse.

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Conduction
band
Valence
band
Incident photon
When light of the right wavelength (depending of the value of the
bandgap Eg) falls on a p-n junction, photons are absorbed creating
electrons and holes (in the band model this is equivalent to an
electron making a transition from the valence to the conduction
band).
If the p-n junction is connected to an external electrical circuit, the
electrons and holes move creating a current in the circuit. The
junction acts as a source of emf.
Photodiodes are used as optical radiation detectors.
Sept. 2002
Solids Slide31
School of Mathematical and Physical Sciences
PHYS1220
Semiconductor Devices - Transistors
Basically, a transistor consists of one type of doped semiconductor (ntype or p-type) sandwiched between two doped semiconductors of the
opposite type.
Small base current is
used to control large
collector current
Transistor Symbols
Sept. 2002
Arrow shows
direction of
conventional current
flow during normal
operation
Solids Slide32
School of Mathematical and Physical Sciences
PHYS1220
Transistor Operation
IC
DC current gain hFE of transistor
is given by
hFE
L
Collector
Collector current I C


Base current
IB
R
vi
The output voltage Vout is given by
Vout  VCC  ICR L
(supply voltage)
R
B
I
B
Base
Emitter
n

vcc
vou (vCE
)
t
Simple circuit illustrating transistor action
(U sin g Ohm' s law)
 VCC  hFEIBR L

Output voltage is controlled
by small base current.
(e.g.) For VCC = 10V, hFE = 200 (typical), RL =4.7kW
Vout =10V if IB = 0A
Vout = 0V if IB ~11mA
Sept. 2002
Solids Slide33