Graduate Lecture Series
29 June – 3 July, 2009
Prof Ngee-Pong Chang
Lecture 2
Fermi Gas
Enrico Fermi
Paul Dirac
1901 - 1954
1902 - 1984
Band Theory of Metals
Start
with
isolated
Sodium
Atom
Ionisation Energy of Sodium 5.14 eV
Band Theory of Metals
Ionisation Energy of Sodium 5.14 eV
Band Theory of Metals
Band Theory of Metals
Bring
two
Sodium
Atoms
together
Splitting of 3s level with 2 Sodium
atoms
Band Theory of Metals
Bring
six
Sodium
Atoms
together
Splitting of 3s level with 6 Sodium
atoms
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
T=0
1.0
0.0
T>0
Probability of Occupancy
Filling up the Fermi Sea
30
In One Dimensional Box
Energy in
units
20
10
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 Gas
Density of
States
1-Dimensional Fermi Gas
Density 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/Boulder2005/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
Landauer formula
i
t
t’
i’
1927 - 1999
Conductance
Gas Pressure on the Wall
µ
θ
A
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!