In the last lecture we discussed the free
electron theory of metals 
in this connection we noted that metals constitute
a particularly simple kind of solids in which
most of the conduction properties and other
related thermal behaviour all these are determined
by the. So, called conduction electrons 
which behave very much like an ideal gas atoms
or molecules except that the electron gas
obeys in subjected to Pauli exclusion principle
and therefore, satisfy fermi dirac distribution.
So, even though the metal is a solid crystalline
solid it is mainly the electron gas which
decides these physical properties like electrical
transport heat transport of heat specific
heat all these properties are determined by
of course, there is a role from the ions the
conduction electrons are formed by ionisation
of the atoms of the metal. So, that you have
positive ions. And then into which there is
a free electron gas which is free to wander
around as long as it is within the metal it
is confined to the metal as a whole the metallic
bond is something that binds the electron
gas to the metal it is not able to escape
it and become completely free.
So, except that they are free to wander around
inside the metal under the influence of applied
electric or magnetic fields applied heat thermal
gradients and so on. So, it is this behaviour
of this electron gas which is profoundly different
from.
That of an ideal gas atom or molecule because
the molecules are atoms of an ideal gas a
classical ideal gas or satisfied the or governed
by the Maxwell Boltzmann 
distribution this is familiar already to all
of us, but the electron gas is subjected to
the fermi dirac distribution. This is because
the electrons are quantum mechanical particles
and fermi dirac distribution is different
from the Maxwell Boltzmann distribution because
of the quantum behaviour of the electrons
which are determined by the pauli exclusion
principle the essence of the pauli exclusion
principle is that. If there is an energies
level and an electron occupies this energy
level then no other electrons can come and
occupy the same energy level. So, that is
why it is called the exclusion principle and
this profoundly affects the way the electrons
are distributed in energy and we saw the precise
form of the fermi dirac distribution which
at absolute zero the distribution function
goes like this as a value one here to zero
and it is like this and this is known as the
fermi energy ef this is f zero kelvin.
So, all these states within for energies less
than the fermi energy the states are completely
occupied each state being occupied by a given
single electron and all these states above
the fermi level are completely empty. So,
the fermi energy at absolute zero is the highest
energy level which is occupied in the case
of a metal. And therefore, this will modify
the way there are electrons are distributed
in energy and this is again given by the dispersion
curve of the electron, which is the e versus
k curve and this is governed by the kinetic
energy of the electrons which is h cross square
k square by 2 m. And therefore, this will
be a parabolic curve which will look like
this. So, that is a and states up to the fermi
energy are filled these are all these states
are completely filled.
So, what happens is that we discussed last
time the behaviour of the contribution of
these electrons to the specific heat, because
when there is a thermal excitation the electrons
are going to observe this heat. And therefore,
there's going to be specific heat contribution
due to this electron gas. Now this contribution
we saw is like the hound in the hound of baskervilles
is the dog that did not bark at night. So,
the electronic in specific heat does not appear,
it is not a dominant contribution that is
the overall result of this, that is because
this fermi energy is at the order of 10 to
the power four Kelvin. Whereas normal thermal
elicitations are of the order of thermal excitation
is at the order of ten to the power two kelvin.
So, it is a very small quantity in comparison
to this. So, if you have a small temperature
window here which say this is the energy initial
energy, and the thermal excitation suppose
it takes the electron to this now this is
the initial. And final states are all already
occupied and therefore, the electron cannot
go into this state. So, even though you excite
it these electrons which are deep within the
energy scheme the occupied energy level they
are unable to participate in the thermal excitation
it is only the electrons which are the fringe
which are here in a small skin layer around
the fermi energy these are the fraction of
electrons which will be able to contribute
to the specific heat by being thermally excited.
So, it is this fraction and this fraction
as you can see is about a hundredth this ratio
of this temperatures is one in hundredth.
So, it is only a fraction of point zero one
or one in hundred of total of number of electron,
which can get excited and therefore, contribute
to this specific heat it is for this reason.
So, the fraction of electrons excited is of
the order of t by t f, where t f is given
by k b t f equals e f zero or e f. So, this
fraction is only at the order of 0.0.
One and this fraction of electron each electron
will be excited by an amount k b t by Boltzmann's
equipartition theorem therefore, the total
contribution is T f, that is the mean energy
of these electrons, which are excited since
this goes as t square. So, the specific heat
c electronic the specific heat which is the
e by d t is proportional to the absolute temperature.
So, that is what we write as c e equals it
goes as gamma t where gamma is the electronic
heat capacity coefficient. So, it is the Pauli
exclusion principle and the fermi dirac distribution
which profoundly modify the behaviour of the
electronic system, and prevent it from absorbing
thermal excitation energy to a large extent
and confine only a small fraction t by T f
of the total number of electron to be thermally
excited. And therefore, contribute only a
term or the order of gamma t as we will see
later the lattice the crystal lattice of a
ions in a metal will have a contribution which
goes as t cube the cube of the absolute temperature
and therefore, the total specific heat will
be of the form gamma t plus beta t cube.
So, at high temperatures it is this term which
will dominate therefore, this will be negligible,
and you cannot even detect it it is only when
you go to temperatures as the order of one
Kelvin, which is an extremely low temperature
it is only at such low temperature. These
two terms will become comparable and then
you can detect the electronic contribution.
So, this is the important concept that we
developed last time.
Now, we move on to discuss how this picture
of conduction electron gas in a metal is going
to lead to the very well-known behaviour of
metals namely that they are very good electrical
conductors. So, we would like to know how
and why a metal like silver or gold or copper
they are very good conductors of electricity
this is a very important characteristic of
a metal which we would like to understand
in the frame work of the free electron gas
picture. So, this is our next aim. So, what
do we do we just take this conduction electron
gas consider it and then apply an electric
field a dc electric field.
So, let us first start looking by looking
at the behaviour of a single electron of electronic
charge electron of mass m and charge minus
e. So, let us look at what happens to these
electron when we apply a dc electrical field
of strength e. So, we know that we can this
is a very simple situation. And we will to
start with use a classical picture which was
due to which was first proposed by a person
named Drude. So, this is known as the Drude
theory of electrical conductivity this is
an extremely simple picture where I have a
particle of mass m, but a charged particle
carrying the charge minus e and therefore,
in an electric field the force on it will
be minus e e and that will be equal to this
is the force.
So, Newton's law of motion tells us that this
should be 
because this v d, because v d is known as
the drift velocity of the electron why do
we call it drift velocity. This is because
normally if you do not have an applied electric
field what happens to these electrons, they
are still moving around they they are very
much like as we said they are very much like
the atoms on a ideal gas. So, they are not
keeping quite. So, they are free to move around.
And therefore, they do move does it mean that
they there will be a conductivity there will
be electrical conduction whenever an electron
moves somewhere there should be a current
and therefore, there should be a conduction,
but this question is answered because in the
classical picture these electrons are free
to move around.
But they more around in perfectly a random
fashion very much like what is said in the
kinetic theory of gases. So, they are moving
around a given electron is moving around in
all possible directions randomly with equal
probability. Therefore, this electron is very
much like a drunkard what does a drunkard
do a drunkard stands here he is under the
influence of liquor. So, you watch him he
is moving a few steps in this way and then
talks to himself and comes, and moves a few
steps this way and then this way. So, what
happens even after a few hours, if you watch
him he if he is standing in a place is moving
this moving this way moving this way moving
everywhere all the time, but the where is
the net displacement he is where is was a
few hours ago. So, it is a drunkard who walks
all the time, but with no net displacement
there is no net displacement. So, in the same
way the electrons when they are simply diffusing
like the atoms of a gas then the net velocity
in any given direction when there is no field
vanishes identically it is zero and therefore,
when there is these current density is just
given by minus ev. So, this velocity is zero.
So, it vanishes. So, there is no conduction
even though the electrons are moving around
they are bumping around in all possible directions,
but nothing happens, if you are cannot focusing
on a particular direction and trying to measure
the conduction conductivity in that direction.
So, it vanishes in the absence of an applied
electric field, but when you put an applied
electric field in then this electric field
forces the electron to move in a direction
opposite to the applied electric field. Therefore,
there is a net drift in a given direction
that is why this is called a drift velocity,
and this gives you the rate at which this
distribute this this drift velocity changes
with time and gets accelerated by the applied
electric field. So, that is the equation of
motion well if this is all there is to head.
Let us see what happens therefore, integrating
this we will see that v d is integral minus
e e by m d t. Therefore, this is by t plus
a constant v zero the initial speed which
is zero to start with there was no velocity
when there was no electric field. So, if we
start from rest this is the net, and the j
the current density will go as e square e
by m into t from this equation so; that means,
there will be a current build up. And as time
passes on the current will go on increasing
monotonically, and it will eventually if you
wait long enough it can even blow up and become
infinitely large, but we all know that this
does not happen in any conductor there is
a finite current. If you apply a certain voltage
producing a certain electric field it produces
a certain amount of current which is given
by ohm's law, this is the observation that
we are all familiar with, but this model does
not explain that instead it predicts a current
density which goes on increasing monotonically
with time.
If you wait long enough you can get an infinite
current from a finite electric field which
is up surd, this is because there is something
that we ignored you are not taken into account
these electrons this is the behaviour of one
electron. And even if you have ten thousand
or ten to the power 24 electrons the behaviour
can be described by a simple addition or super
position of these current contributions.
But this gas is being when it is moving when
it is drifting under the influence of an electric
field, there are other things that are happening
on the path these electrons gets scattered
by various obstacles on their way. For example,
in a in a metallic lattice there are many
impurities impurity atoms, there are also
the positive ions and then there are defects
of various kinds like dislocations stacking
walls grain boundaries and. So, on all these
act as scattering centres. So, this scattering
can arise from impurities also these atoms
are ions in a crystalline solid are not at
rest there are vibrating all the time there
are thermal vibrations at any finite temperatures
and these thermal vibrations increase as temperature
increases.
So, it is a even if you think that these vibrations
are simple harmonic there will be an effect
due to these vibrations vibrating atoms and
therefore, they can act as scattering centres
the vibrating ions in the crystal lattice
in the metallic crystal lattice. So, these
thermal vibrations when they are quantise,
they are called phonons we will discuss them
a little later for our present discussion,
it is enough to know that these are quantised
thermal vibrations of the solid.
So, there can be scattering due to phonons,
which will increase with temperature unlike
the impurities the phonon scattering will
depend on the temperature. So, these scattering
events have to be considered in order to decide
what will be the drift velocity of a given
electron the way this scattering is taken
into account is by its thinking that suppose
there is no scattering of a given electron
is scattered at a particular instant of time.
Then the entire distribution is affected the
distribution of the electrons momentarily,
but then this distribution if you leave this
like this. And look at only the scattering
even immediately after the scattering the
entire distribution will relax back to its
original value there is an equilibrium distribution.
And then that is momentarily disturbed by
the scattering of the electrons and then after
a little time this disturbed distribution
will relax back to the original equilibrium
distribution function. So, this is model which
is called the relaxation time model.
So, if this takes as an amount of frame tau
tau is known as the relaxation time the characteristic
time in which the drifting electron relax
back to an equilibrium configuration, when
there will be a limiting velocity not a unlimited
velocity like that. Then this is described
mathematically by any equation of this form
these are simple first order differential
equation which as you all know will produce
a solution which gives you a velocity which
decays exponentially with a characteristic
time. So, this will a drift velocity which
goes as. So, that is why this tau is known
as the characteristic time of relaxation through
which describes this exponential relaxation
process. So, this can now be combined. So,
there are two processes one the applied electric
field accelerates the electron, and then the
electrons which gets scattered by the various
scattering centres in the solid they produce
a relaxation at the distribution function
towards an equilibrium or limiting value.
And therefore, we have to consider both of
these equations together to describe the rate
of change in time of the drift velocity.
So, when you do this you get an equation a
combined equation which is of this form eE
by m like this plus an additional term. So,
that equation is that is the equation will
describe the time rate of change, and when
you solve this first order differential equation
this will give you a steady state solution
which will give you something like. And therefore,
if there are if there is a number n electrons
if if n is the electron concentration then
j is n e v d, and this will be n e square
tau by m times e and since by ohm's law this
is equal to sigma e where sigma is the conductivity.
So, we get the electrical conductivity as
n e square tau by m; that is the drude expression
for the electrical conductivity of a metal
having a concentration n of conduction electrons
each carrying a charge e. And a mass having
a mass m, which are drifting under the influence
of an electric field getting scattered by
the various scattering centres inside the
metal. And relax with a characteristic in
time tau towards an equilibrium value. So,
for such a situation the drude theory, which
is a purely classical theory which does not
take into account the quantum nature as electrons
this is a very old theory, but which gives
a remarkably accurate expression for the electrical
conductivity. If you we already saw how we
can calculate the electron concentrations
using fermi dirac distribution, and if you
plug in the value one finds a very nice way
to describe the electrical resistivity or
conductivity behaviour of simple metals well.
This is all very well, but the question is
can we use classical behaviour a classical
description the answer is no as we already
saw in the connection with the electronic
heat capacity. So, we have to require that
the electrons obey fermi dirac statistics.
So, we have to write the equilibrium distribution
function in the presence of scattering, and
in the presence of an applied electric field
in order to do this we make use of a formalism
which was again developed by Boltzmann.
This is known as the Boltzmann transport equation
the Boltzmann transport equation says tells
us what happens to the distribution function
in the presence of an applied electric field,
and also in the presence of scattering mechanisms.
So, we talk about again the distribution function
f of e, which is the fermi dirac standard
fermi dirac distribution function 
but we will call it f zero when it is when
there are no applied electric fields, and
there are now scattering mechanism. We will
call it f zero, that is the equilibrium distribution
function, which has we know has the form one
by we saw this last time.
So, this is a standard equilibrium distribution
function in the absence of applied electric
fields and scattering mechanisms, but now
the Boltzmann's transport equation tells us
how to write the distribution function in
the presence of fields and collisions due
to scattering. So, the distribution function
changes the f of e changes with time. And
now we have to it is convenient to distinguish
between the influence of fields fields can
be electric fields it can be magnetic fields
it can be even temperature gradient. So, depending,
if it is an electric field the transfer to
the electrons is determined by the electrical
conduction mechanism, if it is a thermal gradient
then this is determined by the thermal conduction.
So, you can have via this formalism we can
at the same time describe electrical as well
as thermal conduction and many other processes
as you see which come under the general category
of transport processes, that is why this equation
is known as the transport equation. So, the
change in the distribution function with time
has two contributions one due to fields and
another due to collisions. So, we will evaluate
them separately.
So, how do we do this. So, this will be implying
this f nought minus I can write this as in
terms of the energy using the energy momentum
relationship. Therefore, I can write de by
d k x here, which will give me h cross v x,
you can check this up times e ex by h cross
d t where v x is the corresponding speed.
So, this is k x square by two m. So, this
simplifying this we will find now differentiating
this d f by d t field.
And now f nought is the equilibrium distribution
function in the absence of the fields, and
therefore that will not change the fields
do not affect the equilibrium configuration
the value the way they are distributed under
equilibrium in steady state. So, the change
is coming only from this and that is given
as please note that I am writing the x component
of the applied electric field in terms of
e in this form, and the energy is written
by represented by e in this form. So, please
distinguish these two let us keep these two
separately not mix them up. So, this gives
you this term and the df by dt due to collisions
you have already seen how it goes by the velocity
and therefore, this is a similar form very
much similar to what happens in the case of
the drift velocity.
So, the distribution from this describe this
equation describes the exponential relaxation
at the distribution function to the equilibrium
value f nought with the characteristic time
tau. So, these two have to be combined in
order to get the total rate of change. So,
that will give me f as taking f in this, and
combining these two equations the results
here, we arrive at the net distribution function
in the presence of the applied field into...
So, we have to now use this distribution function
the new distribution function to describe
the average behaviour of various quantities
such as the current density.
So, the evaluation of the current density
proceeds in the same way as before j x equals
e by four pi cube f v x d k x d k y d k z
integral a triple integral in k's place, where
f is what we have on the other side. Now this
has two contribution from f naught and d f
naught by de now this contribution due to
the part involving f nought vanishes, because
it is the equilibrium configuration. And it
is a as we have already seen under steady
state equilibrium in the absence of applied
fields this contribution to the current density
vanishes because the electron has a random
motion. So, it is only the other term which
contributes to this in order to evaluate this
integral the usual procedure is to consider
this volume element in k's place which can
be written rewritten. We rewrite this part
as d s times d k n, where d s is an element
of area of constant energy surface and k n
d k n is a length element in the direction
normal to this constant energy. So, we evaluate
this integral using this relationship.
So, that I can write d k x d k y d k z as
one by h cross v x d s de, so replacing this
and calculating this we arrive at the final
result j x equals evaluating all this e square
e x by four pi cube h cross tau integral v
x square by v d s de into d f by d t. Now
we left ex we would like to not only calculate
j x, but we will also like to calculate it
along with three principle directions xyz.
So, we would like to evaluate j y and j z.
Under the influence of electric fields directed
along the y and z directions setting ex to
be equal to e y to be equal to e z, that is
we apply the same electric field and we assume
that this metal is a cubic metal having cubic
symmetry. So, that j x equal to j y equal
to j z equal to j in other words we for the
moment we ignore the anisotropic of a solid
and consider the metal as an isotropic conductor,
which has the same behaviour in all the three
directions. If we do this and simplify this
integral we get the relation connecting j
to e, and and using ohm's law j equal to sigma
e we can write the conductivity as e square
by 12 pi cube h cross to tau integral v square
by v.
So, which is v d s and evaluating this and
using the relation n equal to four pi by three
k f cube divided by four pi cube, that is
the electron concentration. We get back we
find that simplifying we find again the same
relation the old drude formula for the electrical
conductivity this means that the application
of the fermi dirac distribution does not change
the form in the drude's formula. And we get
this this expression gives you a very nice
way to determine the a calculate the electrical
conductivity of a metal. We will continue
in the next lecture to see how we can describe
other transport process like thermal conduction
using the same formulation.
