L7
Free Electrons in Metal


Since the potential
energy U = 0, the
Schrödinger
equation for a free
electron has the
following form:
m is the mass of
an electron.
2

2

 ψ(r )  εψ(r )
2m
1
Solution of Schrödinger
equation


Wave functions
satisfying Schrödinger
equation are plane
waves
The
condition
of
normalization of the

function
is
performed
by
integration over the
volume L3 of specimen
ψ(r)  Ce
ψ
1
32
e
ik r
ik r
L
2
The boundary condition


the
wave
vector
components
satisfy
boundary conditions
which are
k x  2πn x L 

k y  2πn y L 

k z  2πn z L  


 = 0 at x = 0 and
x=L
1
 2


ψ
exp i
n x x  n y y  n z z 
the components of
 L

L3 2
the wave vector are
quantum numbers of
this problem
3
The energy level En

the energy levels are
quantized, and each
2
 2 2  2  2  2
2
2
is characterized by a
ε
k 
  nx  n y  nz
2m
2m  L 
set of three quantum
numbers (one for
each degree of
freedom) and the spin
quantum number ms.

4

quantum number space


The energy level En
called “energy state”
and represented by a
point in quantum
number space
corresponds to each
set of quantum
numbers
surface of equal
energy has the shape
of a sphere with
radius n
n 2  n x2  n 2y  n z2
5
quantum state

The number of
quantum states 
with energy equal
to or smaller than
En is determined
by the double
volume of the
sphere


4 3 8 2 2 232
η  2  n   nx  n y  nz 
3
3
8 2m 3 2
 V
E
3
3 2 
32
6
The density of states

differentiation of 
with respect to the
32


d
η
2
m
energy E provides
 g ( E )  4V

3
the number of
dE
2π 
energy states per
32
unit energy in the  V  2m  E1 2
4π 2   2 
energy interval dE,
i.e. the density of
state, g (E)
7
Density of state g(E) versus energy E




The density of states
plotted versus the
energy is a parabola.
The hatched area
within the curve is the
number of states filled
with electrons at
absolute zero
EF (0) is the Fermi level
TF is the Fermi
temperature
E F (0)
TF 
k
8
the Fermi surface


An isoenergetic surface in
k – space (k = p/ħ)
corresponding to the
energy Fermi EF is called
the Fermi surface. For free
electrons this surface has
the form of sphere.
The Fermi surface
separates the states filled
with electrons from the
unfilled states.
2
2 2
p
 k
EF 

2m
2m
9
What is Metal?



A metal is a system with a very large
number of energy levels.
Electrons fill these levels in accordance
with the Pauli Exclusion Principle,
beginning with E = 0 and ending with
EF.
At T = 0 К, the levels below the Fermi
energy are filled up and those above
the Fermi energy are empty.
10
Discrete structure?
The levels are discrete but so close
together that the electrons have
an almost continuous distribution
of energy.
 At 300 K, a very small fraction of
valence electrons are excited
above the Fermi energy.

11