TFY 4170 - Fysikk 2
Forelesning 21: Material physics
The band theory of solids. Conductors, insulators and
semiconductors. p-n junctions.
Mansfield & O’Sullivan: 20.7,20.8,20.9.
Material physics
!
Introduction.
!
!
Bonding.
!
!
Molecular and solid state
physics.
!
!
The free electron model
in solid state.
!
Classical free electrons
Quantum mechanical
free electrons.
!
!
!
!
The Fermi energy.
!
Density of states.
!
The Fermi distribution.
!
!
!
!
Heat capacity.
Band theory.
Conductors, insulators
and semiconductors.
p-n junctions.
Transistors.
The Hall effect.
Quantum statistics
Super conductors.
Ferromagnets.
Fermi-Dirac distribution at T>0K
The distribution function which describes the occupied density of
states at T>0K is called the Fermi-Dirac distribution:
F(E)
T>0
At T=0 we find:
T=0
At T>0K, the Fermi energy
is defined as:
EF
E
Conductivity
The thermal energy in insufficient to allow most electrons to change
energy level (i.e. they are too far from the Fermi level)
However, all electrons notice an external field.
The density of states is:
The energy is given by:
The distribution function for velocities is therefore:
Conductivity
E
-vF
vF
In addition, in an electric field all electrons get a drift velocity:
Ledningsevne
Conductivity
esultat:
un okkuperingen av tilstander nærme Fermi-nivået vF forandres
Ledningsevne
Result:
Only the
of veilengde
states close
the Fermi
can change
denoccupancy
midlere frie
ertobestemt
av level
hastigheten
til
Resultat:
elektronene
nærheten
Fermi-nivået
Kun
av
tilstander
nærmeby
Fermi-nivået
vF forandres
Theokkuperingen
‘meani free
path’
isavdecided
the electrons
near to the
Fermi level
den midlere frie veilengde er bestemt av hastigheten
ksempel: Kopper (Cu)
til elektronene i nærheten av Fermi-nivået
2 EF
F
Eksempel: Kopper (Cu)m
vF
vF 1.56 106 m/s
Example: Copper (Cu)v
vantemekanisk fri midlere veilengde
vF
2 EF
m
1.56 106 m/s
Quantum mechanical
mean
free path:
Kvantemekanisk
fri midlere
veilengde
l
vF
6
14
1.56l 10
2.5
10
6 m
14
v
1.56 10 2.5 1039nm
m
F
Elektronene
bremses
ektronene
bremses
av down
Electrons
are
slowed
byav
• Urenheter, defekter
• Urenheter,
defekter
defects
• Impurities,
• Vibrerende gitter-ion (ved T>0)
• Vibrerende
gitter-ion
T>0)
in the(ved
lattice
(at T>0)
• Vibrations
ne e 2 2
ne e
m m
39nm
T
1
T
1
Heat capacity
Quantum mechanics: Only a small proportion (kBT/EF) of the electrons can
be change state using the thermal energy
Classical: All electrons can be affected by the thermal energy
Simple estimate for the quantum mechanical heat capacity:
R: gas constant
(8.31 J/K mol)
A proper quantum mechanical calculation would show that:
The band model of solid materials
What happens when we put several similar atoms together?
The quantum states are not allowed to be the same (Pauli principle), so
one possibility is that the separate in energy.
E
relative separation
The more atoms we bring together, the more energy splittings we see.
N atoms give N energy splittings
The band model of solid materials
Sodium (Na): From atoms to metal
3s
Forbidden energy range
2p
2s
Continuum of levels
1s
Na atom
Na metal
The discrete atomic energy levels become broad continua with gaps.
The band model of solid materials
In the ground state, all energy levels up to the Fermi level are occupied.
EF
Forbidden energy range
Continuum of levels
Na metal
The highest occupied band is called the ‘valence band’.
If this band is only partially filled, then the material will be a metal.
The unoccupied part is called the ‘conduction band’ since it is responsible
for the electrical conductivity.
Conductors and insulators
Conductor
• The highest band is partially filled
• Electrons can be excited into unoccupied
states with a small energy budget
EF
Example: Sodium (Na)
Insulator
• The valence band is full
• There is a significant energy gap to reach the
next unoccupied band
• electrons are ‘stuck’
• the electrical conductivity is poor.
Example: Diamond (carbon)
EF
Bandgap for group IV elements
Element:
Bandgap (eV)
Material type
Diamond, C
6.0
Insulator
Silicon, Si
1.1
Semiconductor
Germanium, Ge
0.7
Semiconductor
Lead, Pb
0
Metal
Semiconductors
• the valence band is full
• The energy gap is not too big (~1eV)
EF
• at T>0, electrons are stuck in the valence band
and the sample is insulating
• at higher temperature, there is an increased
probability of exciting an electron to the next band
=> conductivity increases with T.
• electrons are negatively charged. We call them
n-type carriers.
T=0
T>0
Semiconductors
T=0
T>0
Notice that semiconductors can also carry a current in the valence band
when it is not completely filled, i.e. at T>0.
Promotion of electrons from the valence band leaves behind ‘holes’.
Holes represent a missing electron. We can consider them as a being
positively charged.
Holes are therefore called ‘p-type’ carriers.
Electrons and holes in
semiconductors
An n-type
particle in a box
A bubble in a
water bath is a bit
like a hole
Intrinsic semiconductors
Conduction band
Valence band
Intrinsic semiconductors: are very pure and have no (charged)
impurities. Each electron promoted to the conduction band leaves behind a
hole in the valence band
number of electrons = number of holes
The probability of finding an electron in the conduction band (energy = Ec)
is the same as the probability for finding a hole in the valence band at
energy = Ev.
Intrinsic semiconductors
The probability is given as:
Solution:
The Fermi level is in the middle of the gap.
Extrinsic (doped) semiconductors
!
!
!
!
There are a small number of impurities,
which dramatically change the
conductivity.
P, As impurities in silicon make it n-type
(i.e. they donate electrons)
B, Al, Ga impurities make silicon p-type.
The conductivity of doped
semiconductors is less temperature
sensitive than for intrinsic
semiconductors.
Extrinsic n-doped semiconductors
We will look at one particular case: that we add As impurities to a Ge
host crystal.
Germanium atoms have 4 valence electrons (4s24p2).
Arsenic atoms have 5 valence electrons (4s24p3)
This means that each As atom (replacing a Ge atom) adds an extra
electron to the crystal
This is called n-type doping; the system gets extra electrons.
The extra electron is not localised at the arsenic atom (it has some
freedom to move in the crystal).
Ed
intrinsic
extrinsic
Donor-level
Extrinsic p-doped semiconductors
We now look at putting boron atoms in a germanium crystal.
Germanium atoms have 4 valence electrons (4s24p2).
Boron atoms have 3 valence electrons (4s24p1)
It means that each boron atom has one electron too little, and thus it
creates ‘holes’ in the valence band
This is called p-doping; the system is hole-doped.
These holes are also somewhat mobile in the lattice.
Ea
intrinsic
extrinsic
acceptor
level
Extrinsic (doped) semiconductors
!
!
!
!
The position of the Fermi level in a doped
semiconductor depends on the relative
density of the n- and p- dopants.
The density of n- and p- dopants are no
longer equal. There is an excess of one type.
In an n-doped semiconductor, the Fermi level
is higher than for an undoped semiconductor.
In a p-doped semiconductor, the Fermi level
is lower than for an undoped semiconductor
Contact between conductors
Electrons have a ‘binding energy’. For a metal, the binding energy for an
electron at the Fermi level is called the ‘workfunction’, φ.
E=0
φ
EF
Conduction band
Thermionic emission: If the thermal energy is higher than the
workfunction (kbT>φ) electrons can escape from the metal. This is the
basis of an “electron gun” .
Contact between conductors
φa
EFa
φb
EFb
What happens if we bring two different metals into contact?
The lowest energy states must fill up first; this means that charge flows
form one metal to the other until the Fermi level is the same in both
materials
EFa
p-n junction
A similar thing happens if we make a contact
between p- and n-type semiconductors.
p-type
Ec
n-type
Ec
EFp
Ev
EFn
Ev
At the contact, there is a redistribution of
charge. It can create a charge accumulation
or a charge depletion. This has a dramatic
effect on the conductivity
p-n junction
What happens if we put a voltage across the p-n junction?
Ve
p-doped
n-doped
The conductivity depends on the direction of the current.
I
This gives rise to a diode effect!
V
Diode-action
Repetition – forelesning 21
In conductors, the valence band is partly
filled.
! In insulators, the valence band is full, and
there is an energy gap to the next
available band (typically, 5eV)
! In intrinsic semiconductors, the valence
band is also full, but the energy gap to the
next level is not too big (around 1eV)
! In extrinsic (doped) semiconductors, there
is an excess of p- or n- carriers.
!