1. Solids and semiconductors
Physics 123
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Lecture XX

2. Bonding in solids
Atoms in solids organize themselves in crystal structures
Positions of atoms are determined by a balance of electrostatic attraction and repulsion
Minimum of potential energy U0 is called ionic cohesive energy and is equivalent to binding energy in nucleus
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Lecture XX

3. 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
Energy levels become energy bands
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Lecture XX

4. 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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Lecture XX

5. 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 V4pp2dp
Number of electrons that can live in this volume (number of available apartments)
2(spin)x(total volume)/(volume occupied by one electron)
Number of states per unit volume
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Lecture XX

6. 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?
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Lecture XX

7. Fermi-Dirac probability function
At T=0 all states below EF are occupied, above EF are free
When T increases some electrons get enough energy to get above EF
Fermi function – smoothened step
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Lecture XX

8. 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
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Lecture XX

9. Energy bands insulator semiconductor conductor Conduction band Eg
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
insulator
semiconductor
conductor
Conduction band Eg
Conduction band Eg
Valence band
Valence band
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Lecture XX

10. 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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Lecture XX

11. 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
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Lecture XX

12. 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 As Si Ga Si
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13. 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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Lecture XX

14. 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 p-type, thus creating the space charge. Mind that before materials were brought together they were electrically neutral
Q=-1e
Q=+1e As Si Ga Si
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Lecture XX

15. 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
+++
P-type
+++
current No
current + - -
---
Electron
flow +
---
n-type
electrons
n-type
LED: e+hole=light
Reverse bias
Forward bias
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Lecture XX

16. 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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