Solids and semiconductors
Physics 123
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Lecture XX
1
Bonding in solids
• Atoms in solids organize themselves in crystal structures
• Positions of atoms are determined by a balance of electrostatic
2
attraction and repulsion
 e
B
U 
40 r

r
m
• Minimum of potential energy U0 is called ionic cohesive energy
and is equivalent to binding energy in nucleus
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2
Metals
• Metal have 1-2 e on the outer shell, they are loosely bound to the rest of the
atom and can be considered “free” to move within the boundaries of metal 
electron gas
• Electrons in potential well – boundaries on metal surface  L is very large
• Distance between energy levels inversely proportional to L2

h2
2
2
E  En1  En 2 
n

n
1
2
2
8mL

• Energy levels become energy bands
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Lecture XX
3
Metals
• Electrons are fermions, according to Pauli principle not more
than one electron can exit for each quantum state
• How much space does a free electron need to itself?
• dxdp>h
• In 3-D Phase space (3 spatial coordinates +3 momentum
coordinates)
• dxdydzdpxdpydpz =dVdP>h3
• electrons = balls in phase space each occupying h3 of space
• Actually two electrons can coexist in h3 – spin up and spin down
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Density of states
• Let’s calculate the number of states in unit volume between energy E to E+dE:
g(E)dE
• In momentum space think of a spherical layer of radius p and thickness dp
• Total phase space volume of this layer V4p2dp
• Number of electrons that can live in this volume (number of available
apartments)
2
4

p
dpV
2(spin)x(total volume)/(volume occupied by one electron) N  2
3
h
• Number of states per unit volume
N
4p 2 dp
g ( E )dE   2
V
h3
p2
E
 p  2mE
2m
m
dp 
dE
2E
8 2m 3 / 2 1/ 2
g (E) 
E
3
h
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5
Fermi energy
• Consider T=0K
• All electrons must fall into the lowest possible quantum state, but respect each
other’s privacy – Pauli principle
• Suppose you have n electron per unit volume, what is the highest energy that
they can have at T=0K - Fermi energy?
8 2m3 / 2 1/ 2
8 2m3 / 2 2 3 / 2
n   g ( E )dE  
E dE 
EF
3
3
h
h
3
0
0
EF
EF
h 3 
EF 
 n
8m   
2
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Lecture XX
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Fermi-Dirac probability function
• At T=0 all states below EF are
occupied, above EF are free
1, E  EF
f (E)  
0, E  EF
• When T increases some electrons
get enough energy to get above EF
1
f (E) 
exp(( E  EF ) / kT )  1
• Fermi function – smoothened step
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Density of occupied states
• g(E) – density of available states
• f(E)- probability to find electron with a certain value of E
• Number of occupied states per unit volume
8 2m 3 / 2
E 1/ 2
no ( E )  g ( E ) f ( E ) 
h3
exp(( E  EF ) / kT )  1
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Energy bands
• In conductors the highest energy band is
partially filled allowing electrons to move
freely – conduction band
• In insulators the highest energy band is
completely filled – valence band, there is an
energy gap between valence and conduction
band – Eg
• Semiconductors are similar to insulators,
but the energy gap is smaller
conductor
insulator
Conduction band
Eg
Valence band
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Lecture XX
semiconductor
Conduction band
Eg
Valence band
9
Intrinsic semiconductors
• Since in semiconductors the energy gap is small, thermal energy
can be enough for some electrons to jump to conduction band
• Resistivity of semiconductor decreases (unlike metals) with
temperature – more electrons in conduction band
• Electrons leave vacancies behind – holes, which act as effective
positive charge and also carry electric current
Conduction band
Eg
EF
Valence band
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Semiconductors
•
•
•
•
•
Most commonly used semiconductors Si (Z=14), Ge (Z=32)
Si electron structure: 1s22s22p63s23p2
Ge electron structure: 1s22s22p63s23p63d104s24p2
Semiconductors have 4 electrons on outer shell
In crystal structure each atom bonds with 4 neighbors to share
electrons
Si
Si
Si
Si
Si
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Semiconductors and doping
• Doping – introduction of impurities with valence 3 (Ga) or 5(As)
• As incorporates itself into the existing crystal structure sharing 4 of its e with
Si- neighbors, one e is free to move around – n-type doping
• Ga does the same, but instead of extra e it creates a vacancy – hole – p-type
doping
• Resistivity of doped semiconductor is much higher than that of intrinsic
material
Si
Si
Si
As
Si
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Si
Ga
Si
Si
Lecture XX
Si
12
P and n-type semiconductors
• Impurities create extra levels in the band structure
Conduction band
Conduction band
Acceptor level
Allows e to jump there
Donor level
Gives e to conduction band
Valence band
Valence band
p-type
n-type
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p-n junction
• Suppose that you bring p-type and n-type semiconductor in
contact
• Electrons from n-type will readily fill the vacancies provided in ptype, thus creating the space charge. Mind that before materials
were brought together they were electrically neutral
Q=-1e
Q=+1e
Si
Si
Si
As
Si
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Si
Ga
Si
Si
Lecture XX
Si
14
p-n diode
• The current flows through p-n junction if electrons have vacancies
to jump to, it does not flow in the opposite direction
– Not entirely true, there still is so called “dark” current, because of thermal
excitation to conduction band, this current grows with T
P-type +++
vacancies +++
+
+++
----n-type
electrons ---
P-type +++
+++
+++
--+
n-type -----
current
Electron
flow
LED: e+hole=light
Forward bias
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Lecture XX
No
current
Reverse bias
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Transistors
npn or pnp junction – no current is flowing – logical zero
•
• Small current (supply of electrons) on base (p in npn or n in pnp)
opens the transistor – larger current is flowing – logical one
current
P-type +++
+++
---
n-type
+++
P-type +++
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