Post on 13-Jan-2016
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Graduate Lecture Series29 June – 3 July, 2009
Prof Ngee-Pong Chang
Lecture 2
Fermi Gas
Enrico Fermi
1901 - 1954
Paul Dirac
1902 - 1984
Ionisation Energy of Sodium 5.14 eV
Band Theory of Metals
Start with
isolated Sodium Atom
Ionisation Energy of Sodium 5.14 eV
Band Theory of Metals
Band Theory of Metals
Band Theory of Metals
Splitting of 3s level with 2 Sodium atoms
Bring two
Sodium Atoms
together
Band Theory of Metals
Splitting of 3s level with 6 Sodium atoms
Bring six
Sodium Atoms
together
Band Theory of Metals
Splitting of 3s levels with Sodium atoms in crystalline solid
Band Theory of Metals
Fermi-Dirac Distribution
electrons
holes
Probability of Occupancy
1.0
0.0
T=0
T > 0
Probability of Occupancy
Energy in
units
Filling up the Fermi Sea
In One Dimensional Box
10
20
30
²F
1-Dimensional Box
1-Dimensional Fermi Gas
Sum Over Spins
1-Dimensional Fermi Gas
Single Spin Orientation
Fermi Surface dependence on
number of electrons
2-Dimensional Fermi Gas
Single Spin Orientation
Fermi Surface dependence on
number of electrons
3-Dimensional Fermi Gas
Single Spin Orientation
Fermi Surface dependence on
number of electrons
1-Dimensional Fermi Gas
Single Spin Orientation
Total Energy
2-Dimensional Fermi Gas
Single Spin Orientation
Total Energy
3-Dimensional Fermi Gas
Single Spin Orientation
Total Energy
3-Dimensional Density of States
2-Dimensional Density of States
http://upload.wikimedia.org/wikipedia/en/c/c5/DOS_multdim.jpg
3-Dimensional Fermi Gas
Density of States
2-Dimensional Fermi GasDensity of
States
1-Dimensional Fermi GasDensity of
States
Conduction Band
Valence Band
Woodward Yang harvard lecture notes
CBO =conduction band offset
2-Dimensional Density of States
Quantum Wire
http://boulder.research.yale.edu/Boulder-2005/Lectures/Matveev/Boulder%20lecture.pdf
A graphene nanoribbon field-effect transistor (GNRFET). Here contacts A and B are at two different Fermi levels EF1 and . EF2
Ballistic Conductor
i
i’t’
t
Landauer formula
1927 - 1999
Conductance
µ
Gas Pressure on the Wall
A cos θ
v δ t
A
θ
Pressure due to
Non-Relativistic Degenerate Fermi Gas
Equation of state for Fermi Gas
since at T = 0
We have
for a metal
White dwarf as seen by Hubble Space Telescope
What's Inside a White Dwarf?
To say that white dwarfs are strange is an understatement. An earth-sized white dwarf has a density of
1 x 109 kg/m3.
In comparison, the earth itself has an average density of only 5.4 x 103 kg/m3.
That means a white dwarf is 200,000 times as dense!