Free Electron Fermi Gas Lecture3.pdf

Introduction to Condensed Matter
Physics, Phys4501
Lecture III: The Free Electron Fermi
Gas
by
Ali Mohammed
Department of Physics
Samara University
Samara, Ethiopia
December 13, 2022

Introduction Energy levels in one dimension Effect of temperature on the Fermi-Dirac distribution Free electron gas in three dimensions Heat capacity of the
OUTLINE
1 Introduction
2 Energy levels in one dimension
3 Effect of temperature on the Fermi-Dirac distribution
4 Free electron gas in three dimensions
5 Heat capacity of the electron gas

Introduction
Introduction
A free electron model is the simplest way to represent the
electronic structure and properties of metals.
According to this model, the valence electrons of the constituent
atoms of the crystal become conduction electrons and travel
freely throughout the crystal.
Therefore, within this model we neglect the interaction of
conduction electrons with ions of the lattice and the interaction
between the conduction electrons.
The classical theory fails to explain the heat capacity and the
magnetic susceptibility of the conduction electrons. (These are
not failures of the free electron model, but failures of the classical
Maxwell distribution function.)

Introduction
Introduction
An other difficulty with the classical model, that a conduction
electron in a metal can move freely in a straight path over many
atomic distances, undeflected by collisions with other conduction
electrons or by collisions with the atom cores.
Condensed matter is so transparent to conduction electrons:
a. A conduction electron is not deflected by ion cores arranged on a
periodic lattice because matter waves can propagate freely in a
periodic structure.
b. A conduction electron is scattered only in frequently by other
conduction electrons. This property is a consequence of the Pauli
exclusion principle.
By a free electron Fermi gas, we shall mean a gas of free
electron is subject to thc Pauli principle.

Energy levels in one dimension
Consider a free electron gas in one dimension, taking account of
quantum theory and of the Pauli principle.
An electron of mass m is confined to a length L by infinite
potential barriers as shown figure below.
The wave function ψn(x) of the electron is a solution of the
Schrödinger equation, Hψn(x) = Enψn(x) where En is the energy
of electron in the orbital.
Since we can assume that the potential lies at zero, the
Hamiltonian H = p2
/2m includes only the kinetic energy. In
quantum theory p may be represented by the operator −i~d/dx,
so that
Hψn(x) = −
~2
2m
d2
dx2
ψn(x) = Enψn(x)
We use the term orbital to denote a solution of the wave
equation for a system of only one electron.

The term allows us to distinguish between an exact quantum
state of the wave equation of a system of N interacting electrons
and an approximate quantum state which we construct by
assigning the N electrons to N different orbitals, where each
orbital is a solution of a wave equation for one electron.
The orbital model is exact only if there are no interactions
between electrons.
The boundary conditions for the wave function are
ψn(0) = ψn(L) = 0, as imposed by the infiniite potential energy
barriers.
They are satisfied if the wavefunction is sinelike with an integral
number n of half-wavelengths between 0 and L:
ψn(x) = A sin

πn
λn
x

;
1
2
nλn = L, where A is constant.

We see that the above wave function is a solution of the
Schrödinger equation, because
dψn(x)
dx
= A
nπ
L
x

cos
nπ
L
x

;
d2
ψn(x)
dx2
= −A
nπ
L
x
2
sin
nπ
L
x

Hence the energy En, is given by
En =
~2
2m
πn
L
2
.

According to the Pauli exclusion principle, no two electrons can
have all their quantum numbers identical. That is, each orbital
can be occupied by at most one electron.
In a linear solid the quantum numbers of a conduction electron
orbital are n and ms, where n is any positive integer and
magnetic quantum number ms = ±1
2 , according to spin
orientation.
A pair of orbitals labeled by quantum number n can
accommodate two electrons, one with spin up and one with spin
down.

More than one orbital may have the same energy. The number of
orbitals with the same energy is called the degeneracy.
Let nF denote the topmost filled energy level, where we start
filling the levels from the bottom (n=1) and continue filling higher
levels with electrons until all N electrons are accommodated.
It is convenient to suppose that N is an even number. The
condition 2nF = N determines nF , the value of n for the
uppermost filled level.
The Fermi energy EF is defined as the energy of the topmost
filled level in the ground state of the N electron system. From
previous equartion with n = nF , we have in one dimension:
EF =
~2
2m
nF π
L
2
=
~2
2m

Nπ
2L
2
.

Effect of temperature on the Fermi-Dirac distribution
The kinetic energy of the electron gas increases as the
temperature is increased: some energy levels are occupied
which were vacant at absolute zero, and some levels are vacant
which were occupied at absolute zero.
The distribution of electrons among the levels is usually
described by the distribution function, f(E), which is defined as
the probability that the level E is occupied by an electron.
Thus if the level is certainly empty, then, f(E) = 0, while if it is
certainly full, then f(E) = 1. In general, f(E) has a value between
zero and unity.
It follows from the preceding discussion that the distribution
functions for electrons at T = 0o
K has the form
f(E) =

1, E EF ,
0, E EF

That is, all levels below EF are completely filled, and all those
above EF are completely empty.
When the system is heated (T 0o
K), thermal energy excites
the electrons.
However, all electrons do not share this energy equally, as would
be the case in classical treatment, because the electrons lying
well below Fermi level EF cannot absorb energy.
If they did so, they would move to a higher level, which would be
already occupied, and hence exclusion principle would be
violated.
Recall in this context that the energy which an electron may
absorb thermally is of the order kBT, which is much smaller than
EF , this being of the order of 5 eV.

Therefore only those electrons close to the Fermi level can be
excited, because the levels above EF are empty, and hence
when those electrons move to a higher level there is no violation
of the exclusion principle.
Thus only these electrons which are small fraction of the total
number are capable of being thermally excited.
The distribution function at non-zero temperature is given by the
Fermi distribution function.
The Fermi distribution function determines the probability that an
orbital of energy E is occupied at thermal equilibrium.
f() =
1
e(−µ)/kBT + 1
The chemical potential (µ) can be determined in a way that the
total number of electrons in the system is equal to N.

At absolute zero µ = F , because in the limit T → 0 the function
f() changes discontinuously from the value 1 (filled) to the value
0 (empty) at = F = µ.
At all temperatures f() is equal to 1
2 when = µ, for then the
denominator of above equation has the value 2.
The high energy tail of the distribution is that part for which
− µ kBT; here the exponential term is dominant in the
denominator of Fermi-Dirac distribution, so that
f() ' e[(µ−)/kBT]
.
This limit is called the Boltzmann or Maxwell distribution.

Free electron gas in three dimensions
The Schrödinger equation in the three dimensions takes the form
−
~2
2m

∂2
∂x2
+
∂2
∂y2
+
∂2
∂z2

ψ(r) = ψ(r).
If the electrons are confined to a cube of edge L, the solution is
the standing wave
ψ(r) = A sin
πnx
L
x

A sin
πny
L
y

A sin
πnz
L
z

,
where nx , ny and nz are positive integers.
In many cases, however, it is convenient to introduce periodic
boundary conditions, as we did for phonons.
The advantage of this description is that we assume that our
crystal is infinite and disregard the influence of the outer
boundaries of the crystal on the solution.

We require then that our wave function is periodic in x, y, and z
directions with period L, so that
ψ(x+L, y, z) = ψ(x, y, z); and similarly for the y and z coordinates.
The solution of the Schrödinger equation which satisfies these
boundary conditions has the form of the traveling plane wave:
ψk (r) = A exp(ik · r),
provided that the component of the wave vector k are determined
from
kx =
2πnx
L
; ky =
2πny
L
; kz =
2πnz
L
where nx , ny and nz are positive or negative integers.

If we now substitute this solution to Schrödinger equation we
obtain for the energies of the orbital with the wave vector k
=
~2
2m
k2
=
~2
2m

k2
x + k2
y + k2
z

.
The wave functions equations are the eigenfunctions of the
momentum p = −i~∇, this can be readily seen by differentiating:
pψk (r) = −i~∇ψk (r) = ~kψk (r)
The eigenvalues of the momentum is ~k. The velocity of the
electron is defined by v= p/m = ~k/m.
In the ground state a system of N electrons occupies states with
lowest possible energies.
Therefore all the occupied states lie inside ak space, kF . The
energy at the surface of this sphere is the Fermi energy EF .

The magnitude of the wave vector kF and the Fermi energy are
related by the following equation:
F =
~2
2m
k2
F .
The Fermi energy and the Fermi wave vector (momentum) are
determined by the number of valence electrons in the system.
In order to find the relationship between N and kF we need to
count the total number of orbitals in a sphere of radius kF which
should be equal to N.
There are two available spin states for a given set of kx , ky and
ky .
The volume in the k space which occupies this state is equal to:
(2π/L)3
.

Thus in the sphere of (4πk3
F /3), the total number of states is
2
4πk3
F /3
(2π/L)3
=
V
3π2
k3
F = N,
where the factor 2 comes from the two allowed values of the spin
quantum number for each allowed value of k. Then;
kF =

3π2
N
V
1/3
,
this depends only of the particle concentration. We obtain then
for the Fermi energy:
F =
~2
2m

3π2
N
V
2/3
.
And the Fermi velocity; vF = ~
m

3π2
N
V
1/3
.

An important quantity which characterizes electronic properties
of a solid is the density of states, which is the number of
electronic states per unit energy range.
To find it we use Fermi energy and write the total number of
orbitals of energy ≤ :
N() =
V
3π2

2m
~2
3/2
.
The density of states is then
D() =
dN
d
=
V
2π2

2m
~2
3/2
1/2
,
Or equivalently
D() =
3N
2
.

So within a factor of the order of unity, the number of states per
unit energy interval at the Fermi energy D(F ) is the total number
of conduction electrons divided by the Fermi energy.
The density of states normalized in such a way that the integral
N =
Z F
0
D()d.
gives the total number of electrons in the system.
At non-zero temperature we should take into account the Fermi
distribution function so that
N =
Z ∞
0
D()f()d.
This expression also determines the chemical potential.

Heat capacity of the electron gas
The question that caused the greatest difficulty in the early
development of the electron theory of metals concerns the heat
capacity of the conduction electrons.
Classical statistical mechanics predicts that a free particle should
have a heat capacity of 3
2 kB, where kB is the Boltzmann constant.
If N atoms each give one valence electron to the electron gas
and the electrons are freely mobile, then the electronic
contribution to the heat capacity should be 3
2 NkB, just as for the
atoms of a monatomic gas.
But the observed electronic contribution at room temperature is
usually less than 0.01 of this value.
This discrepancy was resolved only upon the discovery of the
Pauli Exclusion Principle and the Fermi distribution function.

When we heat the specimen from absolute zero not every
electron gains an energy ∼ kBT as expected classically, but only
those electrons, which have the energy within an energy range
kBT of the Fermi level, can be excited thermally.
These electrons gain an energy, which is itself of the order of
kBT, as shown in Fig. below.
This gives a qualitative solution to the problem of the heat
capacity of the conduction electron gas.
If N is the total number of electrons, only a fraction of the order of
T
TF
can be excited thermally at temperature T, because only
these lie within an energy range of the order of kBT of the top of
the energy distribution.
Each of these N T
TF
electrons has a thermal energy of the order of
kBT.

The total electronic thermal kinetic energy U is of the order of
U ∼
= (N
T
TF
)kBT
The electronic heat capacity is
Cel =
dU
dT
= 2NkB(
T
TF
)
At room temperature Cel is smaller than the classical value 3
2 NkB
by a factor 0.01 or less, for TF = 5x104
.
We now derive a quantitative expression for the electronic heat
capacity valid at low temperatures kBT F .
The increase ∆U = U(T) − U(0) in the total energy of a system
of N electrons when heated from 0 to T is
∆U =
Z ∞
0
D()f()d −
Z F
0
D()d

Here f() is the Fermi-Dirac function
f(, T, µ) =
1
e(−µ)/kBT + 1
and D() is the number of orbitals per unit energy range. We
multiply the identity by F to obtain
N =
Z ∞
0
D()f()d =
Z F
0
D()d
Z F
0
+
Z ∞
F

D()f()F d =
Z F
0
D()F d
∆U =
Z ∞
F
( − F )D()f()d +
Z F
0
( − F )[1 − f()]D()d

The first integral on the right-hand side of gives the energy
needed to take electrons from F to the orbitals of energy F ,
and the second integral gives the energy needed to bring the
electrons to F , from orbitals below F .
The product D()f()d in the first integral of the above equation
is the number of electrons elevated to orbitals in the energy
range d at an energy .
The factor [1 − f()] in the second integral is the probability that
an electron has been removed from an orbital .
The heat capacity of the electron gas is found on differentiating
∆U with respect to T.
The only temperature-dependent term in above equation is f(),
hence we can group terms to obtain
Cel =
dU
dT
=
Z ∞
0
( − F )D()
df
dT
d.

Since we are interested only temperatures for which kBT F
the derivative df
dT is large only at the energies which lie very close
to the Fermi energy.
Therefore, we can ignore the variation of D() under the integral
and take it outside the integrand at the Fermi energy, so that
Cel = D(F )
Z ∞
0
( − F )
df
dT
d.
We also ignore the variation of the chemical potential with
temperature and assume that µ = F , which is good
approximation at room temperature and below. Then
df
dT
=
− F
kBT2
e(−F )/kBT
[e(−F )/kBT + 1]2
.

The above equation then be rewritten as
Cel = D(F )
Z ∞
0
( − F )2
kBT2
e(−F )/kBT
[e(−F )/kBT + 1]2
d
= D(F )
Z ∞
−F /kBT
x2
(kBT)3
kBT2
ex
(ex + 1)2
dx.
Since by letting x = ( − F )/kBT
Taking into account that F kBT, we can put the low integration
limit to minus infinity and obtain
Cel = D(F )k2
BT
Z ∞
−∞
x2
ex
(ex + 1)2
dx =
π2
3
D(F )k2
BT.

For a free electron gas we should use D(F ) = 3N
2F
; then for the
density of states to finally obtain
Cel =
π2
3
D(F )k2
BT
=
π2
3
(
3N
2F
)k2
BT =
π2
3
(
3N
2kBTF
)k2
BT
=
π2
2
NkB
T
TF
.
where we defined the Fermi temperature TF = F
kB
.
Recall that although TF , is called the Fermi temperature, it is not
the electron temperature, but only a convenient reference
notation.

THANK YOU!

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