PHYS 260 Lecture 1: unit 6b- free electron theory of metals and band theory
1
Classical Free electron theory of Metals:
Metals achieve structural stability by letting their valance electrons roam freely through
the crystal lattice. These valence electrons are the equivalents of the molecules of an
ordinary gas. Since electrons are negatively charged particles their motion corresponds
to a flow of electricity or electric current. It is assumed that the electrons are moving
about at random and colliding frequently with the residual ions. Hence the laws of
classical kinetic theory of gases can be applied to a free electron gas also. Thus the
electrons can be assigned a mean free path
λ
a mean collision time
τ
and an average
speed c. In the absence of an externally applied potential difference, there are on an
average as many electrons wandering through a given cross-section of the conductor in
one direction as there are in the opposite direction. Hence the net current is zero. In
between two collisions, the electron may move with the uniform velocity. During every
collision both the direction and magnitude of the velocity get changed in general. The
Zigzag motion is the thermal motion of the electron.
In 1900, P.Drude made use of the electron gas model to explain theoretically electrical
conduction in metals. According to his theory the kinetic velocities of the electrons are
assumed to have a root mean square velocity c given by kinetic theory of gases in
conjunction with the law of equipartition of energy c is obtained as follows for unit
volume of the metal.
m
A
V
cmN
CP
2
2
3
1
3
1==
ρ
Where V
m
is molar volume and N
A
is Avogadro’s number.
Now PV
m
= TRcmN
UA
=
2
3
1
Now
Thus m c
2
= TK
N
TR
B
A
U
3
3=
2
ρ
Where R
U
and K
B
are universal gas constant and Boltzmann’s constant respectively. The
kinetic energy of the electron is
TKcm
B
2
3
2
1
2
=
And --------------------- 1
At 20
o
c c =
2
1
31
23
1011.9
293101383
×
×××
−
−
c = 1.154
×
10
5
m/s.
Equation (1) indicates that the root mean square velocity of the electron is directly
proportional to the square root of the absolute temperature of the metal.
At room temperature the drift velocity imparted to the electrons by an applied electric
filed is very much smaller than the average thermal velocity. The time
τ
taken by the
electrons in traversing the distance
λ
Will thus be decided not by the drift velocity due to the field but by the much greater
velocity c due to the kinetic thermal motion now
== TK
m
c
B
3
λ
λ
τ
the vital test of the validity of any theory of electrical conductivity is whether or not it
can account for ohm’s law. This law is usually expressed as V = RI. Where I is the
current flowing through a conductor as the result of applying a potential difference V
across its ends, at a given temperature R the resistance is a constant for a given
conductor. In order to simplify the analysis let us rewrite the law in terms of the current
density J =
A
I
and field strength along the conductor E = -dv/dx.
Thus we have
AJI
R
V==
,
E
AEl
Ael
V
AR
J
I
ρ
===
, E
E
J
σ
ρ
== ____(A)
the resistivity and conductivity
σ
of a conductor of length
ι
and area of cross-
m
TK
c
B
3
=
3
section A are defined by
ι
σ
ρ
RA
== 1
Let us now assume that there are n free electrons per cubic meter and that in the
absence of any applied field these are darting about in all direction with no net velocity
just like gas molecules in a container when a field Ex is applied in the
X-direction all electrons are accelerated in the x-direction with the acceleration of the i
th
electron a
ix
given by
−= m
eE
a
x
ix
Alternatively this equation may be written as
−=
m
eE
dt
dv
x
E
ix
x
Where V
ix
is the x-component of the velocity of the i
th
electron, and the subscript Ex
means the acceleration arising from the applied field. Since the right-hand side of this
equation is the same for all electrons we may write above equation in another from
2−−−−
−=
>< m
eE
v
dt
d
x
E
x
x
Where <V
x
> is the average velocity of all n electrons as given by
<V
x
> =
∑
=
n
iix
V
n
1
1
We have used <V
x
> to denote the average value of V.
The current density is given by j
x
=nq<v
x
>___________(B)
Differentiating equation (B) with respect to time dt
dj
nev
dt
d
x
E
x
x
−=
><
x
x
E
dt
dj ∝This is the only thing we get from application of ohm’s law.