Crystalline solids
A solid crystal consists of different atoms arranged in a periodic
structure.
Crystals can be formed via various bonding mechanisms:
•
•
•
•
Ionic bonding
Covalent bonding
Metallic bonding
Van der Waals forces /Hydrogen bonds
Crystalline Solids
Bonding mechanisms:
Ionic crystals: Example: Sodium Chloride : NaCl
Each sodium ion is attracted to 6 adjacent chlorine ions, repelled by
neighbouring sodium ions, attracted by other non-adjacent chlorine ions
and so on.
The net attractive potential per ion pair due to all the other ions in this
particular geometry is
e2
U = !" k
r
α: Madelung constant = 1.7476 for
the NaCl crystal.
All solids with the same crystal
geometry have the same Madelung
constant.
Crystalline Solids
Bonding mechanisms:
Ionic crystals: Example: Sodium Chloride : NaCl
Each sodium ion is attracted to 6 adjacent chlorine ions, repelled
by neighbouring sodium ions, attracted by other non-adjacent
chlorine ions and so on.
The net attractive + repulsive potential on a single sodium ion due to
all the other ions in this particular geometry is
e2 B
U = !" k + m
r r
The minimum energy is the ionic cohesive
energy (energy required to pull solid
apart):
e2 # 1 &
U 0 = !" k % 1 ! (
r0 $ m '
Crystalline Solids
Bonding mechanisms:
Properties of Ionic crystals:
•
Ionic cohesive energy is large, so they have high melting and
boiling points.
•
There are no free electron, so electrical conductivity is low.
•
Can absorb infrared but not visible radiation.
•
Easily soluble: polarized water molecule exerts a force on the ions
that can break the ionic bonds.
e2 # 1 &
U 0 = !" k % 1 ! (
r0 $ m '
Crystalline Solids
Bonding mechanisms:
Covalent crystals: Example: diamond
Each carbon atom in a diamond crystal is covalently bonded to four
adjacent atoms in a tetrahedral lattice.
Crystalline Solids
Bonding mechanisms:
Properties of Covalent crystals:
•
Atomic cohesive energy is larger than for ionic crystals, so
covalent solids are very hard and have high melting and boiling
points. Example: diamond
•
Electrons are tightly bound, so electrical conductivity is low (good
insulators).
•
Do not absorb visible radiation.
Crystalline Solids
Bonding mechanisms:
Metallic crystals: Example: iron, copper etc
Each atom loses its valence electrons to a common sea of electrons.
The crystal is made of positive ion centres.
Valence electrons are free to move about the positive ions in the crystal
Crystalline Solids
Bonding mechanisms:
Properties of metallic crystals:
•
Atomic cohesive energy is smaller than for crystals, but still strong.
•
Electrons are free to move, so electrical conductivity is high (good
conductors).
•
Can absorb and emit visible radiation close to the metal surface.
Band Theory of Solids
Symmetric and antisymmetric wavefunctions:
•
Two identical atoms that are far apart do
not interact.
•
Their electronic energy levels are those
of individual atoms.
•
As the atoms come closer together, the
wave functions start overlapping.
•
The joint wave functions is a symmetric
or an anti-symmetric combination of the
individual wave functions.
Symmetric combination: ! + = ! 1 + ! 2
Anti-symmetric combination: ! " = ! 1 " ! 2
Band Theory of Solids
Energy bands
•
The energies of the symmetric and antisymmetric
wavefunctions are different. So the individual
atom electronic energy levels split into two.
•
When many atoms are brought together, the
energies split into more levels that are closely
spaced.
•
For N = 1023 atoms in a crystal, the energies get
split into a very large number of levels, that are so
closely spaced that they form an energy band.
•
The separation between energy bands may be
large or small, depending on the atom. Energy
bands may overlap.
Band Theory of Solids
Energy bands
•
For N atoms in a crystal each energy band has N
energy levels.
•
Including the orbital angular momentum and the
spin quantum numbers, each band can hold
2(2l+1)N electrons.
Example: Sodium: electronic structure: 1s22s22p63s1
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•
•
•
The 1s, 2s and 2p bands are full.
The 3s band has one electron from each atom.
Total number of electrons in the 3s band: N.
The 3s band can hold 2N electrons, so it is half
full.
Band Theory of Solids
Energy bands
•
Electrons occupying lower lying energy bands are
tightly bound to the atom.
•
The electrons in the highest energy band
participate in conduction.
•
The highest occupied energy band is called the
valence band.
•
The lowest energy band with unoccupied states is
called the conduction band.
Band Theory of Solids
Metals
•
In metals the highest energy band is partially full.
•
This band is both the valence band and the conduction band.
•
There are many empty energy levels available nearby.
•
A small electric field can excite electrons into these empty levels.
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Electrons are thus free to move, hence metals are good conductors.
Band Theory of Solids
Insulators
• In insulators, the valence band is full. The conduction band is empty.
•
The two bands are separated by a large energy gap. The Fermi
energy lies between the gap.
•
A small electric field cannot excite electrons from the valence to the
conduction band.
•
Hence electrons are tightly bound: the material is a good insulator.
Band Theory of Solids
Semiconductors
• In semiconductors, the valence band is full at 0 temperature.
•
The valence band and conduction band are separated by a small
energy gap. The Fermi energy lies in between the gap.
•
At non-zero temperatures, electron can cross the energy gap into the
conduction band leaving holes that behave like positive charges.
Conductivity of semiconductors increases with temperature.
•
When an electric field is applied, conduction electrons move in one
direction and holes move in the other direction.
Semiconductor Devices
The band structure and conductivity of a semiconductor can be
modified by adding impurities to intrinsic semiconductors.
The process of adding impurities is called doping.
An intrinsic semiconductor can be doped with donor atoms or
acceptor atoms
Semiconductor Devices
n-type semiconductors:
Consider a semiconductor such as silicon doped with arsenic.
Silicon has 4 valence electrons. Arsenic has 5 valence electrons.
4 of the arsenic electrons participate in covalent bonds with silicon
atoms.
The remaining arsenic electron is almost free and has an energy just
below the conduction band
Semiconductor Devices
n-type semiconductors:
The arsenic donates a free electron and is called a donor atom.
Semiconductors doped with donor atoms are called n-type
semiconductors.
A small amount of energy can raise the free electron into the conduction
band.
The conductivity depends on the amount of impurity.
Semiconductor Devices
p-type semiconductors:
Consider a semiconductor such as silicon doped with indium.
Silicon has 4 valence electrons. Indium has 3 valence electrons.
The 3 indium electrons participate in covalent bonds with silicon
atoms leaving a single electron deficiency (hole).
The energy of the hole lies just above the valence band. Electrons from
the valence band can be excited leaving holes in the valence band.
Semiconductor Devices
p-type semiconductors:
The indium accepts an electron from the valence band and is called an
acceptor atom.
Semiconductors doped with acceptor atoms are called p-type.
The conductivity depends on the amount of impurity.
Semiconductor Devices
p-n junction:
When an p-type semiconductor is
joined to an n-type semiconductor,
a p-n junction is formed .
In the region around the junction,
electrons move from the n-side to
the p-side.
In this depletion region, a
net potential difference is created.
Semiconductor Devices
Diode:
When the p-n junction has a positive voltage applied to the p side
(forward bias), the net voltage barrier is decreased, and hence current
flow is increased.
When the p-n junction has a positive voltage applied to the n side
(reverse bias), the net voltage barrier is increased, and hence current flow
is decreased.
Thus diodes conduct in only one direction.
Semiconductor Devices
Diode:
Diodes can be used in a variety of ways such as current rectifiers, voltage
regulators, switches in circuits, photomultipliers, LEDs,solar cells.
Semiconductor Devices
Transistor:
A transistor consists of
an n type semiconductor between two p-type
semiconductors (pnp)
OR
a p type semiconductor between two n type
Semiconductors (npn).
A small base current can control a large
collector current.
Semiconductor Devices
Transistor:
Transistors, diodes, capacitors and resistors can be combined in a single
integrated circuit on a silicon wafer ‘chip’.
Many intergrated circuits are combined to build a computer.
The low power requirements of transistors and diodes make extreme
miniaturization possible without overheating.
The small size allows for faster response times, increasing
computational speed.