Metals: Free Electron Model
Physics 355
Free Electron Model
Schematic model of metallic
crystal, such as Na, Li, K, etc.
The equilibrium positions of
the atomic cores are
positioned on the crystal lattice
and surrounded by a sea of
conduction electrons.
For Na, the conduction
electrons are from the 3s
valence electrons of the free
atoms. The atomic cores
contain 10 electrons in the
configuration: 1s22s2p6.
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Paul Drude
(1863-1906)
• resistivity ranges from 108 m (Ag) to
1020 m (polystyrene)
• Drude (circa 1900) was asking why? He
was working prior to the development of
quantum mechanics, so he began with a
classical model:
• positive ion cores within an electron
gas that follows Maxwell-Boltzmann
statistics
• following the kinetic theory of gasesthe electrons in the gas move in
straight lines and make collisions
only with the ion cores – no electronelectron interactions.
Paul Drude
• He envisioned instantaneous collisions in
which electrons lose any energy gained
from the electric field.
• The mean free path was approximately
the inter-ionic core spacing.
(1863-1906)
• Model successfully determined the form
of Ohm’s law in terms of free electrons
and a relation between electrical and
thermal conduction, but failed to explain
electron heat capacity and the magnetic
susceptibility of conduction electrons.
Ohm’s Law
Experimental observation:
V I
E
Ohm’s Law: Free Electron Model

j  nevd
number
e  ne
volume
Conventional current
Ohm’s Law: Free Electron Model
12
Predicted
behavior
10
Resistivity
8
B
High T: Resistivity
limited by lattice
thermal motion.
6
4
2
0
0
20
40
60
80
100
120
Temperature
The mean free path is actually many
times the lattice spacing – due to the
wave properties of electrons.
Low T: Resistivity
limited by lattice
defects.
Wiedemann-Franz Law (1853)
(Ludwig) Lorenz Number
(derived via quantum mechanical treatment)
 1  kB
8 W  
L

 2.45  10
2
2
T
3e
K
2
2
Free Electron Parameters
F
Li
N/V
×1022 /cm3
4.70
kF
×108 /cm
1.11
vF
×108 cm/s
1.29
eV
4.72
TF
×104 K
5.48
Na
2.65
0.92
1.07
3.23
3.75
Cu
8.45
1.36
1.57
7.00
8.12
Au
5.90
1.20
1.39
5.51
6.39
Be
24.20
1.93
2.23
14.14
16.41
Al
18.06
1.75
2.02
11.63
13.49
Pb
13.20
1.57
1.82
9.37
10.87
Electron Heat Capacity
Naïve Thermodynamic Approach:
• You could start out by considering the average thermal energy of a free
electron at some temperature T.
E  32 k BT
• Then,
U  32 Nk BT
• And the electronic heat capacity would then be:
dU 3
Cel 
 2 Nk B
dT
• However, when we go out and measure, we find the electronic
contribution is only around one percent of this.
Electron Heat Capacity
D(F)
• The electron energy levels are
mostly filled up to the Fermi
energy.
• So, only a small fraction of
electrons, approximately T/TF,
can be excited to higher levels
– because there is only about
kBT of thermal energy
available.
T 
3
• Therefore, U  Nk BT  
2
 TF 
which goes as T2.
• …and the heat capacity,
Cel = dU/dT goes as T, which
is the correct result.
Sommerfeld Constant
• The Sommerfeld constant is proportional to the density of states at the
Fermi energy, since
1
3
   2 k B2 g ( F )
• Now, we look at this and say, “Obviously!” – because only the electrons
very close to the Fermi energy can absorb energy.
The Sommerfeld constant is
also related to another, rather
important, concept in Solid
State Physics – effective mass.
 2 k F2  2  2 N 
F 

 3

2m
2m 
V 
2/3
Effective Mass
~m
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Electrons interact with
• periodic lattice potential
• phonons
• other electrons
 ~ meffective
Sommerfeld Constant
 observed  calculated m / m
e
mJ/ mol  K 
mJ/ mol  K 
2
2
Observed values come
from the linear heat
capacity measurements.
Calculated values are
determined using the
conduction electron
density and from
assuming me = m,
Heavy Fermions
Heavy fermions are intermetallic compounds containing noncomplete 4f- or 5f-electronic shells. The orbital overlap with
ligand atoms in the lattice leads to strong correlation effects in
the system of delocalized electrons. As a result, the effective
mass of the electrons can increase by orders of magnitude as
 observed
compared to the free electron mass.

mJ/ mol  K 2
Heavy fermion materials
exhibit very interesting
ground states - such as
unconventional
superconductivity, smallmoment band magnetism and
non Fermi liquid behavior.
CeAl 3
CePb 3
UBe 13
CeCu 2Si 2
CeCu 6
U 6 Fe
1700
1200
1100
1000
1000
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
Heavy Fermions
A large value of the
Sommerfeld parameter
indicates that heavy
fermion materials have
a high density of states
at the Fermi Energy.
g  
Electrical Conduction
Electrical Conduction
Hall Effect
In 1879, while working on his doctoral thesis,
Hall was pursuing the question first posed by
Maxwell as to whether the resistance of a coil
excited by a current was affected by the
presence of a magnet.
Does the force act on the conductor or the
current?
Hall argued that if the current was affected by
the magnetic field then there should be "a state
of stress... the electricity passing toward one
side of the wire."
Hall Effect
Initially, v  vx x̂  v y ŷ  vz ẑ
E  Ex x̂
B  Bz ẑ

  
 d 1 
F  m  v  e(E  v  B)
 dt  
net force in
x direction
net force in
y direction
 d 1
Fx  m  vx  e( Ex  vy B )
 dt  
 d 1
F y  m    v y  e( v x B )
 dt  
Hall Effect
As a result, electrons
move in the y direction
and an electric field
component appears in the
y direction, Ey. This will
continue until the Lorentz
force is equal and
opposite to the electric
force due to the buildup of
electrons – that is, a
steady condition arises.
B
Hall Effect
mv x
  e( E x  v y B )
mv y
  e( E y  v x B )


eB
C 
m
eE x
vx  
 C v y
m
eE y
vy  
 C vx
m
Hall Effect
vy  
eE y
m
 C vx  0
 Ey  m
C vx
e
e
vx  
Ex
m
vx
 Ex  m
e
eB
E y  C Ex  
Ex
m
Hall Effect
The Hall coefficient is defined as:
eB
Ex
Ey
1
m
RH 
 2

jx B
ne
ne 
Ex B
m
For copper:
n = 8.47 × 1028 electrons/m3.
Hall Effect
Hall Effect: Electrons & Holes
• The Hall Effect experiment suggests that a carrier can have a
positive charge.
• These carriers are “holes” in the electron sea - the absence of an
electron acts as a net positive charge. These were first explained
by Heisenberg.
• We can’t explain why this would happen with our free electron
theory.
• Note: the conditions we derived for the steady state can be
invalid for several conditions (for example, when there is a
distribution of collision times). But in general, it is a very
powerful tool for looking at properties of materials.
Hall Effect: Applications
For a 100-m thick Cu
film, in a 1.0 T
magnetic field and
through which I = 0.5 A
is passing, the Hall
voltage is 0.737 V.
Hall Effect: Applications
Hall-Effect Position Sensors
Hall-Effect position sensors have replaced
ignition points in many distributors and
are used to directly detect crank and/or
cam position on distributorless ignition
systems (DIS), telling the computer when
to fire the coils. Hall-Effect sensors
produce a voltage proportional to the
strength of a magnetic field passing
through them, which can come from a
permanent magnet or an electric current.
Since magnetic field strength is
proportional to an electric current, HallEffect sensors can measure current. They
convert the magnetic field into millivolts
that can be read by a DMM.
Recap: Free Electron Model
Some successes:
1. electrical conductivity
2. heat capacity
3. thermal conductivity
Some failures:
1. physical differences between conductors, insulators,
semiconductors, semi-metals
2. positive Hall coefficients – positive charge carriers ??