Hello. Today we will start with this very
well known approximation that went about explaining
many of the phenomena in metals for nearly
about a century and this is called the free
electron approximation. Although most of the
materials are insulating in nature, it is
a metals that are of central interest because
many interesting phenomena happen in metals.
The earliest ideas about metals before the
birth of quantum mechanics was introduced
in very interesting paper by Drude, D R U
D E which constitutes one of the basic postulates
of the free electron approximation. Now, what
does this approximation tell us. I will tell
you the approximations the set of approximations
and then I will work out one or two problems
based on these approximations. Then you will
realize how this approximation is used to
calculate certain transport phenomena.
So, the basic tenet of this approximation
is that the electrons that form the conduction
electrons in a metal come from the atoms.
The valance electrons of the atoms in the
solid wonder about living their parent atoms
and they wonder about throughout the solid
and then they basically form like a behave
like a free electron gas.
And, what one does in free electron theory
is to assume that this is like a free electron
gas of a density typically of Avogadro’s
number. Number of electrons is typically Avogadro
number per mole. And then these electrons
are basically classical you can use kinetic
theory, Boltzmann statistics and all that
thermodynamic principles to explain their
behavior. So, that is the fundamental tenet
of free electron gas model.
Later on of course, when quantum mechanics
came in people modified these classical assumptions
and introduced quantum mechanics and we will
come to that as we go along. Now, the one
thing that one in addition to this free electron
assumption of a gas of electron classical
gas one has to also assume and find out what
how the electron moves about in the solid
and since there are these ions heavy ions
sitting static at their in positions in this
approximation, the electrons will collide
with them. The ions carry a positive charge
then electron carries a negative charge, so
the collision is fairly strong and although,
the Coulomb interaction between electron and
the ions are not included in this approximation.
so, are the electron interactions.
Remember, the electrons are negatively charge;
they also interact with each other via Coulomb
interaction. So, both these interactions electron
ion interaction and electron interaction are
neglected. Only thing that happens is collision
of electrons with the ions heavy ions.
Now, the other assumption is that between
two collisions the electron basically moves
freely. Now, if there is a free electric field
applied to the system then of course, the
electron will move under the influence of
this field. So, between 2 collisions it is
under the influence of the field if there
is a field applied, both electric or magnetic
and sometimes both.
Now, the collisions are assumed to be instantaneous,
this is the only way the electron will achieve
equilibrium in the solid because there is
no other interaction allowed. So, the only
interaction that it has is this collision
and through these collisions the electrons
will reach their equilibrium.
So, another very important approximation is
that immediately after each collision the
electron emerges with the velocity that has
no relation to it is velocity before the collision.
So, an electron comes with the certain velocity
collides with an ion and then it loses memory
of its previous velocity completely.
So, it emerges as if it has it has just it
is velocity is completely new in the sense
that it has no memory of it is previous velocity
and it is completely the direction of it is
emergency is completely random. So, the velocity
gets randomized after each collision. So,
this is a very strong assumption that it has
no memory of its state, its configuration
before the collision as it emerges from one
collision and every collision.
So, this with this assumption one proceeds
further and then assumes of course, that between
2 collisions, there is a certain time that
is that electron remains free and on an average
this time is called the relaxation time tau.
So, between; so, there electron collides with
a probability of tau in collides the probability
1 by tau in unit time. So, the probability
that it collides in a time delta t is delta
t by tau. So, under these assumptions one
can now calculate the transport property of
an electron as Drude did . So, let me just
show you how this calculation goes.
So, what we want to calculate is the DC conductivity
under this free electron approximation along
with the relaxation time approximation. So,
how does one do it?
Well, it is fairly simple. Now we know that
if there is a cons, if there is an electrical
field E, then it is connected to the current
density by the resistivity right, j is the
current density, E is the electric field which
also implies that j equal to sigma times E
where sigma is the conductivity choose inverse
of rho. E is electrical field and j is current
density, which is basically the amount of
charge 
that flows across a unit area normal to the
flow direction in unit time. So, let me just
show you what it is.
So, suppose I choose a cylinder whose area
here is one A, this area is A currently is
flowing in this direction. In that case, the
amount of current that flows in unit time
across this area will be given by j. So, that
is the definition of j.
So, let us just calculate j in this geometry.
So, how many electrons will pass through this
area in unit time. Well, it is simply those
electrons that have within a distance of the
velocity v because, that is the distance one
covers in one second. So, that, those many
electrons will cross this area A in unit time.
So that means, the current density j is v
times the area A times, the density of electrons.
So, these many electrons are going to cross,
but I need the charge density. So, I have
to multiply by this per unit area ok. So,
this is my current density for the system
where you have a current that is flowing along
this due for example, along this cylinder
and this is then the current density. So,
A and A will cancel will give me n e v. So,
let me repeat it is the number of electrons
that cross this area per unit time divided
by the area. So, number of electrons which
will cross in unit time are those which are
within a distance of v from this end.
So, this distance is v, all of these electrons
will cross it in unit time. So, that is exactly
what I have done here. I multiplied this volume
v times A the magnitude of v times the area
is the volume times is the density of electrons
that gives the gives me the number of electrons
in this area, in this volume between this
surface and a surface here at a distance v.
So, this gives me this number divided by the
area times the charge of an electron which
is minus. So, that is what it is.
Now, what is v? Now v of course, in relaxation
time average is something that you acquire
between 2 collisions. So, once a collision
takes place your basically losing your memory
of your velocity. Then what happens is the
there is an electric field. So, after emerge
from the electron emerges from one collision
this electric field drives it along the direction
of the field.
So, that is what I have to calculate and in
relaxation time approximation I can easily
calculate it, because that is the charge times
the field by mass times tau. So, that is the
velocity. So, once that is the velocity that
you acquire on an average. So, I put in average
here and that is what I will put here. So,
j equal to then minus n e square tau by m
into E.
Now, from this relation I can see read of
easily that sigma is equal to n e square.
This plus this becomes plus because the 2
negative signs n e square tau by m. So, that
is the formula that one uses quite a lot even
today more than 100 years after it was written
down and it what is for many metals quite
well. Of course, it has it is bit falls it
will not working in certain a large number
of materials also metals particularly, but
it is certainly a good formula to work to
start working on working with.
Now, tau is typically of the order of 10 minus
14 to 10 minus 15 second in a typical metal
typically. Now one then go further and one
can actually find out the equation of motion
of an electron in this relaxation time approximation.
So, let us just try to do that.
Now, let us what we do. We have this solid
and zillions of electrons was scattering of
this solid. So, their coming is scattering
of and as I say they lose their memory the
moment they scatter. Memory of their previous
velocity as this scatter after the emerge;
in emerge from one scattering ok. So, under
these assumption what I need to know is suppose
an electron scatters at time t then in the
next interval of delta t which is very small
comparable to tau what is the momentum of
this electron.
So, suppose electron has a collision at time
t. Its momentum at time t was just emerging
from the collision at time t its momentum
was P of t. It is has to be a function of
time. Then there is a force to be electrical
field which is f of t times delta t will be
the normal momentum of the will be the momentum
of the electron if you follow just Newton’s
law classical physics.
Now of course, this is fine, but the fact
is that not all electrons will survive, because
the next collision this momentum history will
be obliterated. So, you have to find out how
many electron the survival probability in
time delta t that it has not suffered another
collision in time delta t. Now, what is the
probability of suffering a collision of suffering
a collision in delta t. This is simply delta
t by tau. Then it is survival probability
that it has not suffered collision is 1 minus
delta t by tau and these of the electrons
that electron which has not suffered collision
will pick up this momentum in the time interval
of dt.
So, that will be my momentum at t plus dt,
remember dt dt is very small. There are corrections
to it which will let us not bother about it.
The corrections are can be shown to be of
the order of dt square, but from this equation
let us just expand and keep terms only up
to dt linear in dt because dt is small. So,
these P of t plus ft dt minus delta t by tau
P of t and this term is delta t square order
so, these are of the order of delta t square.
So, those I neglect.
So, then what I do is that I just arrange
rearrange terms is equal to ft minus Pt by
tau. Now, I take a limit delta t dt small.
So, going to 0 means in the with respect to
delta tau it is small time. So, in that limit
I can write this as dP dt minus P plus Pt
by tau equal to f of t. Now, f of t is minus
E times this so, that gives me the equation
of motion in the, for the electron under relaxation
time approximation.
Now, you can easily check that if there is
a steady state then I can neglect this, because
there is no more time dependence of the momentum
and then of course, we will in the steady
state we can figure of that we will again
go back get back these equation that the sigma
equal to n e square tau by m, because you
can just check for yourselves. So, this equation
is actually a central equation in the Drude
model that you can write down a an equation
for the momentum of the electron under this
approximation. So, this is called the relaxation
time approximation and this is how it works.
Next class we will work out certain properties
of this equation and find out the Hall Effect
for example, and the conductivity at finite
frequency and so on.
Thank you.
