Transcript Metals I: Free Electron Model

Metals I: 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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Free Electrons?
How do we know there are free electrons?
 You apply an electric field across a metal
piece and you can measure a current – a
number of electrons passing through a unit
area in unit time.
 But not all metals have the same current for a
given electric potential. Why not?
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:
E
V I
V  RI
L
 I
A
I
 L  Lj
A
V
 j
L
E  j
or j  E
Ohm’s Law: Free Electron Model
number

e  ne
volume
j  nevd
Conventional current
The electric field accelerates each electron for an average
time  before it collides with an ion core.
Ohm’s Law: Free Electron Model
F  eE  m a
vd
m

eE
 vd 
m
ne2
j  nevd 
E
m

Ohm’s Law: Free Electron Model
ne 

m
2
If electrons behave like a
gas…
8k BT
v 
m
The mean free time is related
to this average speed…

a


v
v
Then,
typical value
About 1014 s
ne 2 a m
 

m
8k B T
1
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)
Electrical
Thermal
2
Conductivities
where
ne 

mv
  12 n v kB
8k BT
v 
m
 4k B T

2

e
2
Lorentz number (Incorrect!!)
Wiedemann-Franz Law (1853)
(Ludwig) Lorenz Number
(derived via quantum mechanical treatment)
 1  kB
8 W  
L

 2.45 10
2
T
3e
K2
2
2
Free Electron Model: QM Treatment
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• Assume N electrons (1 for each ion) in a
cubic solid with sides of length L –
particle in a box problem.
• These electrons are free to move about
without any influence of the ion cores,
except when a collision occurs.
• These electrons do not interact with one
another.
• What would the possible energies of
these electrons be?
• We’ll do the one-dimensional case first.
0
L
Free Electron Model: QM Treatment
At x = 0 and at L, the
wavefunction must be zero,
since the electron is confined to
the box.
One solution is:
Free Electron Model: QM Treatment
Free Electron Model: QM Treatment
Free Electron Model: QM Treatment
If an electron is added, it goes
into the next available energy
level, which is at the Fermi
energy. It has little temperature
dependence.
Fermi-Dirac Distribution
1
f ( )  (   ) / k T
B 1
e
1
 (  ) / k T
F
B 1
e
For lower energies,
f goes to 1.
For higher energies,
f goes to 0.
Free Electron Model: QM Treatment
From thermodynamics,
the chemical potential,
and thus the Fermi
Energy, is related to the
Helmholz Free Energy:
  F ( N  1)  F ( N ) T ,V
where
F  U  TS
Free Electron Model: QM Treatment
where nx, ny, and nz are integers
Free Electron Model: QM Treatment
and similarly for y and z, as well
i kr 
k  e
2
4
,
, ...
L
L
2
4
k y  0, 
,
, ...
L
L
2
4
k z  0, 
,
, ...
L
L
k x  0, 
Free Electron Model: QM Treatment
2kF2
F 
2m
p k
v 
m m
Free Electron Model: QM Treatment
• Each value of k exists within a volume
 2 
V 

 L 
3
• The number of states inside the sphere
of radius kF is
4  k3
N Vs 3
F


 kF
3
2 V
2
L
 
1/ 3
 3 N 


 V 


2
 2 k F2  2  2 N 
F 

 3

2m
2m 
V 
2/3
• This successfully relates the Fermi energy to the electron density.
Free Electron Model: QM Treatment
1/ 3
 3 N 
vF  kF  

m
m V 
2
TF 
F
kB
Free Electron Model: QM Treatment
 2 k F2  2  2 N 
F 

 3

2m
2m 
V 
2/3
V  2m 
 N  2  2 F 
3  

dN
V  2m 
g ( ) 

 2
2
d 2   
3/ 2
3/ 2
 1/ 2
Free Electron Model: QM Treatment
3/ 2
V  2m 
N  2  2 F 
3  

 ln N  23 ln   constant
then
dN 3 N
g   

d 2 
The number of orbitals per unit
energy range at the Fermi energy
is approximately the total number
of conduction electrons divided
by the Fermi energy.
Free Electron Model: QM Treatment
This represents how many
energies are occupied as a
function of energy in the 3D
k-sphere.
As the temperature
increases above T = 0 K,
electrons from region 1
are excited into region 2.