Download Free Electron Gas in Metals

Condensed Matter Physics:
Quantum Statistics &
Electronic Structure in Solids
Read: Chapter 10 (statistical physics) and Chapter
11 (solid-state physics)
Homework due Wednesday Nov. 12th
Chapter 10: 1, 3, 6, 9, 11, 20, 23, 25, 29, 35
1
Quantum Statistics



In quantum physics, identical particles must
be treated as indistinguishable – this affects
counting of states
In quantum physics there are two types of
particles – fermions (half-integer spin, e.g.
electrons) and bosons (integral spin, e.g.
photons)
Bosons and Fermions obey different statistics
2
Different statistics
Fermions


Bosons
Obey the Pauli Exclusion
Principle – no two
particles can have the
exact same quantum
numbers
Distribution function

1
f FD  E  
 E  EF 
e
kT
1
3
Differences
fermions



Only one particle per
quantum state allowed
Cannot all “condense” to
E=0
Must fill up to the
“Fermi Energy”
bosons



ABE does not depend
strongly on temperature
The only states that
have any probability at
low temperatures are
those at E=0 for which
the exponential
approaches 1
This is Bose-Einstein
Condensation
4
Free Electron Gas in Metals



Solid metals are bonded by the metallic bond
One or two of the valence electrons from
each atom are free to move throughout the
solid
All atoms share all the electrons. A metal is a
lattice of positive ions immersed in a gas of
electrons. The binding between the electrons
and the lattice is what holds the solid
together
5
Fermi-Dirac “Filling” Function
Probability of electrons to be found at various energy levels.
• For E – EF = 0.05 eV  f(E) = 0.12
For E – EF = 7.5 eV  f(E) = 10 –129
1
f FD  E  
 E  EF 
e
kT
1
• Exponential dependence has HUGE effect!
• Temperature dependence of Fermi-Dirac function shown as follows:
6
Free Electron Gas in Metals
The number of electrons in the interval
E to E+dE is therefore
n( E )dE 
1
e
( E  EF ) / kT
1
V
  8m 
 2 
2 h 
3/ 2
1/ 2
E dE
The first term is the Fermi-Dirac distribution
and the second is the density of states g(E)dE
7
Free Electron Gas in Metals
From n(E) dE we can calculate many global
characteristics of the electron gas. Here are
just a few
 The Fermi energy – the maximum energy
level occupied by the free electrons at
absolute zero


The average energy
The total number of electrons in the
electron gas
8
Free Electron Gas in Metals
The total number of electrons N is given by

N   n( E )dE  V
0
  8m 
 2 
2 h 
3/ 2

1/ 2

0
E dE
e
( E  EF ) / kT
1
The average energy of a free electron is given by
1
E 
N


0
V   8m 
En( E )dE 
 2 
N 2 h 
3/ 2


0
3/ 2
E dE
e( E  EF ) / kT  1
9
Free Electron Gas in Metals
At T = 0, the integrals are easy to do.
For example, the total number of electrons is
N V
V
  8m 
3/ 2
 2 
2 h 
  8m 
 2 
2 h 

EF
0
3/ 2
1/ 2
E dE
2 3/ 2
EF
3
10
Free Electron Gas in Metals
The average energy of an electron is
V   8m 
E 
 2 
N 2 h 
3/ 2
V   8m 

 2 
N 2 h 
This implies

EF
0
3/ 2
3/ 2
E dE
2 5/ 2
EF
5
EF = k TF defines
3
E  EF the Fermi
5
temperature
11
Summary of metallic state




The ions in solids form regular lattices
A metal is a lattice of positive ions immersed
in a gas of electrons. All ions share all
electrons
The attraction between the electrons and the
lattice is called a metallic bond
At T = 0, all energy levels up to the Fermi
energy are filled
12
Heat Capacity of Electron Gas
By definition, the heat capacity (at constant
volume) of the electron gas is given by
dU
CV 
dT
where U is the total energy of the gas. For a gas
of N electrons, each with average energy <E>,
the total energy is given by
UN E
13
Heat Capacity of Electron Gas
Total energy

U  N E   E n( E )dE
0
V
  8m 
 2 
2 h 
3/ 2


0
E 3/ 2 dE
( E  EF ) / kT
e
1
In general, this integral must be done
numerically. However, for T << TF, we can use
a reasonable approximation.
14
Heat Capacity of Electron Gas
At T= 0, the total energy of the electron gas is
3 
U  N E  N  EF 
5 
For 0 < T << TF, only a small fraction kT/EF
of the electrons can be excited to higher energy
states
Moreover, the energy of
each is increased by
roughly kT
15
Heat Capacity of Electron Gas
Therefore, the total energy can be written as
 kT 
3
U  NEF     NkT
5
 EF 
where  = 2/4, as first shown by Sommerfeld
The heat capacity of the
electron gas is predicted to
be
dU  2
T
CV 

Nk
dT
2
TF
16