Download UNIT-4-free electron theory

UNIT-IV
Classical Free electron theory of metals:
The free electron theory of metals forms the basis of simple explanation of several properties of
metals, including the characteristic electrical and thermal conductivities. Proposed by Drude and
Lorentz, soon after discovery of the electrons, it does this by considering metallic crystal in terms of
a 'gas of free electrons' and then treating this gas for the properties of the metals. The laws of
classical kinetic theory of gases can be applied to a free electron gas also. Thus the free electrons
can be assigned an average speed
these are related as
, mean free path 'λ' and mean collision time '
and
.
Drude-Lorentz treated the classical gas of free electrons to discuss its steady state conditions when
it is under the influence of an electric field. Let us now assume that there are 'n' free electrons for
unit volume and that in the absence of any applied field, these are roaming randomly, so net
velocity is zero, 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 ith particle is given by
Since the right hand side of this equation is same for all electrons, we may write equation in another
form
The current density
From (2) & (3)
So that for constant field, this free electron model leads to a current which increase linearly with
time. Ohm's law(J=σE) failed. Nothing could be further from Ohm's law.
It is clear from equation (3) that the law can only be satisfied if
So, we introduced a
decelerating term due to electron-lattice collision which, when added to accelerating term
gives net
To obtain expression for
Suppose at an instant t=0, the average velocity of electron is
Just at that instant we switch the field off, so that
subsequently tends to zero.
Now let us assume that this process follows the simplest law of decay,
. Where 'τ' is the relaxation time.
(2) & (6) in (5)
This steady average velocity imposed by the field and proportional to it is called the drift velocity of
the electrons. The constant of proportionality,
is known as their mobility 'μ' and is defined as
drift velocity per unit electric field is given by
.
The current density
Ohm's law proved and compare with J =σE
.
Drawbacks of classical theory:
(i)Temperature dependence of resistivity:
From classical theory t
[Assuming the mean free path is of order of inter atomic separation and
that is therefore comparatively independent of temperature]
This is contrary to experimental observation
.
(ii) Heat capacity of electron gas: Classical free electron theory explains that all the free electrons
in metal can absorb thermal energy. According to the law of equipartition of energy, every free
electron has an average kinetic
so that in 1 kmol of a metal, in which there are NA free
electrons the total energy given by
From the above result the contribution to total specific heat by the free electrons is significant
because according to Dulong- Petit law total specific of solids is given by
But experimentally it
was proved that specific heat of free electrons is less almost negligible.
(iii) Calculation of mean free path:
The general equation for resistivity of a metal is given by the equation
The resistivity of copper at 200 C is 1.69x10-8 ohm-m and density of free electrons, n=8.5 x 1028m3.
Thus
The experimentally observed value for mean free path is ten times the above value. Classical theory
failed to explain the large values for mean free path.
(iv) Relation between electrical & Thermal conductivity (Wiedmann-Franz law):
The ratio of thermal conductivity(
to electrical conductivity ( is directly proportional to
absolute temperature. This is known as Wiedmann- Franz law.
From the classical free electron theory, by considering all free electrons participate in thermal
conduction, we obtained the expression for thermal conductivity as
.
For copper at 200C,
The value of Lorentz number does not agree with the value calculated from the classical formula
and thus the classical assumption that all free electrons participate in thermal conduction is not
correct.
Note: The main reason for all these failures is the assumption made i.e. all the free electrons
are equally participate in electric and thermal conduction. But it is impractical.
Quantum free electron theory of metals (Sommerfield
theory):
Salient features:
 Free electrons didn't possess all the energy values, they can take some discrete energy
values only.
 The distribution of electrons among the different energy levels is following Pauli's exclusion
principle. According to this principle each energy state can be occupied two electrons only
corresponding to spin +1/2 and -1/2.
 At T = 00 K, filling of energy states start from lowest energy level and the last energy level
filled by electrons is called Fermi energy(EF) level.
 With the increase of temperature the energy levels below EF are vacated and above EF are
occupied.
 The distribution electrons among the different energy levels at any temperature is given by
Fermi-Dirac distribution function f(E). It is defined as
.
Fermi- Dirac Distribution Function f(E):


This function gives probability of occupancy of a particular energy state by electron at a
given temperature.
It is defined as


It is applicable for all fermions.
For electrons

It is applicable only when the system is equilibrium.
Temperature dependence:
At T = 00 K, f(E) = 1 if E < Ef and f(E) = 0 if E > Ef.
With the increase of temp. f(E) below Ef decreases and
above Ef increases.
At E = Ef , f(E) = 1/2 for all temperatures.
The plots f(E) Vs E at different temperatures intersect at
the point (Ef , 1/2).




and for holes
Heat capacity of electron gas by taking quantum considerations:
The failure of classical theory in the explanation of heat capacity of electron gas is overcome in the
quantum theory as follows:
when we heat the specimen from absolute zero not every electron gains energy of the order of kBT
as expected classically, but only those electrons in orbital with energy range kBT of the fermi level
alone are excited thermally; these electrons gains an energy which itself is of the order of kBT and
go to higher unoccupied energy states . Thus the minimum free energy required for exciting all the
free electrons will be EF. thus the free electrons that will be excited at T=300K is given by :
ie at ordinary temperatures, less than 1% of the free electrons contribute to the heat capacity
Each of the electron may absorb a kinetic energy of the order
Hence the energy associated
with A kmol of metal will be
This value agrees with the experimental value.
Electrical conductivity for quantum mechanical considerations:
In equilibrium the valence electrons obeying classical free
electron theory do random motion with no preferential
velocity in any directions. It is now conveniently plotted the
velocities of these free electrons in velocity space such that the
points inside a sphere correspond to the end points of velocity
vectors. Fermi velocity (vF) is the maximum velocity that an
electron can assume i.e. vF is the actual velocity of electron at
Fermi level. The shape having vF as a radius represents ,
therefore , the Fermi surface. the velocity vectors cancel each
other pair wise at equilibrium and no net velocity of electron
exists .
It is very interesting to note that although all the electrons participate in the conduction
mechanism , the relaxation time of only those electrons which are at Fermi level occurs in the
conductivity . the conductivity is proportional to fermi surface area. Thus the metals with large
Fermi surface areas will have high electrical conductivity ; whereas the insulators with no Fermi
surface will have zero electrical conductivity.
Since , for a free electron
The equation of motion of each electron in the Fermi surface under the influence of static electric
field of intensity, E is
This means that in absence of collisions, the Fermi sphere will be in constant rate in k- space.
The following approach yields the quantum mechanical expression for electrical conductivity,
integrating the equation (2) of motion one gets
Thus if the electrical field is applied at time t=0 to a filled Fermi sphere centered at origin of kspace, then in a characteristic time
the sphere will have moved to a new centre at
The current density
Here 'n' is the number of electrons per unit volume. It is interesting to note that although all the
electrons participate in the conduction mechanism, the relaxation time of only those electrons
which are at Fermi level, occurs in the conductivity. So quantum mechanical treatment tells us that
the current is in fact, carried out by very few electrons only, all are moving at high velocity.
Thus equation (3) can be written as
This expression gives more accurate values for electrical conductivity.
The only quantity on the right side of Eq.(5) which depends on temperature is the mean free path
Since this mean free path inversely proportional to temperature at high temperatures, it follows
that