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Let us continue our discussion on semiconductors
and semiconductor based optical sources in
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this lecture. So, what we were discussing
was the probability of occupancy of an energy
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state, which is given by a function which
is known as Fermi function is defined as fE
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is equal to 1 over 1 plus exponential E minus
Ef over kBT.
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Let us look at intrinsic semiconductor and
the probability of occupation of energy states
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in an intrinsic semiconductor.
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This kind of distribution is defined by this
Fermi function is known as Fermi Dirac a statistical
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distribution. So, in an intrinsic semiconductor
I have valence band and I have conduction
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band, then at absolute 0. I know that valence
band is completely filled and the conduction
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band is completely empty. That is the probability
of occupation of energy states here in the
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valence band is 1 while the probability of
occupation of energy states in conduction
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band is 0. So, if I look at this function
and I put T is equal to 0 here then for E
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smaller than Ef, what I have this is as 1
plus E to the power minus infinity. So, f(E)
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is equal to 1. So, I have the probability
of occupation 1. While for energies larger
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than Ef I have this because of T is equal
to 0 this becomes infinity. So, f(E) is equal
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to 0. So, what I see that at absolute 0 up
to this level the probability of occupation
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is 1 while above this level the probability
of occupation is nil.
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At elevated temperature what happens some
of the electrons go here. So, there are some
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states which are occupied by the electrons,
then and there are some vacancies created
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here. So, the probability of occupation in
the valence band goes down a little, and the
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probability of occupation in the conduction
band increases from 0. At E is equal to Ef;
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however, what I find that it is E to the power
0. So, 1 over 2 it is half. So, I define Fermi
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level as e is equal to Ef where f(E) f is
half, f at E is equal to Ef is half that is
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at some finite temperature T the level which
has 50 percent probability of occupation it
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is a Fermi level, and at absolute 0 the level
up to which the probability of occupation
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is one.
And above which the probability of occupation
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is nil. So, this is known as Fermi level.
In an intrinsic case semiconductor the Fermi
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level lies in the middle of the gap. If I
have n type semiconductor then there are large
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number of electrons here. So, large number
of states are occupied here. So, the Fermi
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level shifts up; correspondingly the Fermi
Dirac distribution looks like this, you have
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more electron states occupied here. So, you
have a larger value of f(E) here. While in
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p type semiconductor you have large number
of holes here. So, the Fermi level shifts
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down. Now if you make a junction using this
p type material and n type material then what
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happens is this.
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When you join them together then you know
in n type you have electrons excess carrier
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as electrons, which are around the positive
ion core and the holes in excess which are
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around the negative ion core. When you join
them together then electrons from here will
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move towards the p side and will recombine
with holes fill the vacancy and will leave
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behind the positive ion core while here they
will combine the hole. So, they will leave
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behind these negative ions, and this will
happen on until an equilibrium is reached.
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So, in equilibrium position near the junction
I have positive ions on this side and negative
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ions on this side this sets up an electric
field from n to p, this electric field creates
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a potential energy which looks like this.
So, this is the electric potential; if I plot
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here the energy of the electron as a function
of distance, now what happens is that in equilibrium
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you see in n type your Ef is near the conduction
band, and in p type your Ef is near the valence
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band. When equilibrium is reached then this
Ef is somewhere here and there is banding
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of this. So, this shifts downwards this shift
upwards and there is a potential created here
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because of this electric field.
So, what happens is that this potential or
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this electric field prevents electrons going
from here to there, and holds from here to
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there. Remember that this is electron energy
that is energy of electron is increasing in
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this direction, correspondingly the energy
of hole is increasing in this direction. So,
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this is a barrier for electrons and this is
a barrier for holes. So, you have a potential
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barrier created here and the height of the
barrier is eV naught. If you forward bias
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it that is if you apply the positive field
here.
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So, a positive potential here and 0 or negative
potential here then you are basically decreasing
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this potential barrier. Then electrons some
of the electrons can now flow from here to
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the from n type to the p type, and holes can
flow from p type to n type and there would
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be conduction. So, this is forward bias p-n
junction. If you reverse bias it that is if
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this is negative and this is positive then
you will further enhance this, and there would
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be no current flowing in the circuit or very
small current will be flowing in the circuit.
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So, if you look at the i-V characteristics
of a p-n junction. So, in forward bias if
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you increase the voltage the forward current
will increase; however, if you apply the negative
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voltage and increase this then you will have
a reverse saturation current flowing very
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small in amount and if you apply very large
voltage then there would be breakdown.
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The important thing now, is to understand
how photons interact with semiconductors because
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this will form the basis of our optical sources
and optical detectors right. So, if I have
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a semiconductor which has got valence band
and conduction band, there are large number
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of electrons available here and there are
states available here in conduction band for
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electrons. Now if a photon interacts with
this semiconductor and the energy of the photon
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is such that it is the difference of an energy
state here and an energy state available here,
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then this electron in this energy state will
absorb this energy of the photon and will
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go to this state in this conduction band.
In this way it will leave it will leave behind
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a vacancy here and an energy state here is
occupied. So, in this way it will create an
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electron hole payer; a photon would be absorbed
if the energy is larger than the bandgap then
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only it will be absorbed. So, absorption of
a photon by a semiconductor will create an
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electron hole pair, and this electron here
and hole here can now contribute towards conduction
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if you apply an appropriate electric field.
So, such kind of interaction is responsible
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for photo detection process to make photo
detectors. There is another way in which there
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can be interaction between a semiconductor
and a photon, and that is if some electrons
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are already there in conduction band that
is the states here are occupied, and holes
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are available in the valence band the vacancies
are there in the valence band, then there
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is a natural tendency of electrons to go back
to the lowest energy state to the lower state.
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So, an electron can go from here to the valence
band and recombine with hole, and it can do
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it at its own spontaneously, if the lifetime
of these states is very small. Then there
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would be recombination between electron and
hole spontaneously, and as a result of photon
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of energy difference between these two states
will be emitted. This kind of emission of
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photon is known as spontaneous emission, this
process is responsible for light generation
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in light emitting diodes.
There is another process a process similar
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to this you already have electrons available
in the conduction band and holes in the valence
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band. So, the electron will recombine with
the hole in the valence band, but if the energy
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states of the electrons here are such that
they are long lived they are known as metastable
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states. Then this recombination from here
to here transition of electron from here to
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the valence band is triggered by another photon
if this transition is triggered by another
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photon, then the emitted photon is in the
same phase and in the same direction as the
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triggering one, then this kind of emission
is known as stimulated emission, this emission
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is responsible for light generation in lasers.
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So, we have semiconductor p-n junction diode
based light sources is light emitting diode
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where the light generation is via the process
of spontaneous emission. So, what you do you
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forward bias a p-n junction diode by forward
biasing it, you are basically injecting the
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carriers that is electrons in the conduction
band and holes in the valence band. So, since
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you have now electrons available in conduction
band and holes in the valence band.
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So, this recombination of them is in is spontaneous
manner will give you light output, and this
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light is incoherent and this is the process
responsible for generation of light in light
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emitting diodes. However, if the light emission
is via the process of stimulated emission,
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then you amplify the radiation and if you
place two mirrors here and provide feedback
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in the K B T kBT then you can make it a laser
from amplifier you can make the oscillator
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and that is basically a laser. So, this is
the process responsible for light generation
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in laser diodes. So, let us now look at light
emitting diodes. So, these are various forms
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of light emitting diodes available commercially
available, the process responsible here for
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light emission is injection electroluminescence.
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So, I explain this with the help of a strongly
forward biased heavily doped p-n junction
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diode. So, if this p-n junction is heavily
doped, then there are large number of electrons
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in the conduction band and large number of
holes in the valence band such that this Fermi
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level goes even inside the conduction band
and this Fermi level shifts to inside the
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valence band and these are known as quasi
Fermi levels.
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The consequence of this heavy doping and strong
forward bias is that we are injecting lot
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of carriers, and lot of carriers are available
here now. So, if I look at in the junction
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region then in the junction region there is
simultaneous abundance of electrons and holes.
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Then these electrons will make transitions
will recombine with holes and give you light
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output. Photons coming out light radiation
and in this way an LED will work LED will
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emit light.
So, if you have a p-n junction diode you forward
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bias it and you get light out of it. A very
important parameter of a light emitting diode
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is its efficiency quantum efficiency, how
efficient it is in generation of light. To
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understand this let us go back to the very
basic phenomenon of light emission in such
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devices, I have just seen that light is emitted
by radiative recombination of electrons and
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holes. So, electron is in the conduction band
hole is in the valence band, electron recombines
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with hole there is a radiative recombination.
So, photon comes out, but this electron can
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recombine with hole.
By another process also, that is by releasing
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heat by releasing thermal energy instead of
releasing the photon such kind of combination
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is known as non-radiative recombination. So,
I now have two kinds of re-combinations one
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is radiative recombination where photon comes
out, another is non radiative recombination
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where the energy is released in the form of
thermal energy. But I know that light is generated
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by the process of radiative recombination.
So, when I inject current in a p-n junction,
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I create electron hole payers then not all
the electron hole payers recombine radiatively,
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some of them are recombined non-radiatively,
and this gives me some finite quantum efficiency,
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because some part is lost in heating up the
device.
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So, it will give me finite quantum efficiency
of the LED, I cannot have 100 percent quantum
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efficiency in an LED in such a way. So, to
work out the expression for quantum efficiency,
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let us look at the carrier concentration how
the carrier concentration decreases with time
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because electrons recombine with holes. So,
the number of electrons decrease in the conduction
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band and number of holes decrease in the valence
band.
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So, the number of carriers decrease according
to n is equal to n0 e to the power minus t
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over tau, where n0 is the number of carriers
at t is equal to 0, this tau is known as lifetime.
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I can find out from here what is the rate
of decrease of carrier concentration by simply
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doing dn over dt, if I do dn over dt from
here then it is minus n over tau. Minus simply
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indicates that the carriers are decreasing.
So, where tau is recombination lifetime and
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this is total recombination rate it involves
radiative as well as non-radiative. I do not
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know which process is responsible here in
what proportion, and tau is basically total
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recombination lifetime. Now if I say that
Rr is the rate which is due to radiative recombination,
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then Rr would be given by n over tau r I drop
the minus sign because it only indicates the
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decrease in the number of carriers then Rr
is equal to n over tau r where tau r is the
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radiative recombination lifetime and you can
also have decrease in the concentration of
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carriers by non-radiative recombination. So,
I define radiative recombination rate as Rnr
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is equal to n over tau nr, where tau nr is
the non-radiative recombination lifetime.
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So, the internal quantum efficiency I can
now calculate because my the light in LED
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is coming out only because of radiative recombination.
So, the internal quantum efficiency is given
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by eta internal is equal to Rr divided by
Rr plus Rnr. That this what fraction of re-combinations
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is via the process of radiative recombination.
Since R is equal to n over tau. So, if I put
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it there. So, I can write it down as tau over
tau r, where tau is the total recombination
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lifetime and is given by 1 over tau is equal
to 1 over tau r plus 1 over tau nr because
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r is equal to Rr plus Rnr. In general this
tau r and tau nr are comparable for a direct
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bandgap semiconductor such as gallium aluminum
arsenide or indium gallium arsenide phosphide.
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So, if they are comparable then eta internal
is typically 50 percent this is for homo junction
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LEDs homo junction means the junction between
the same material you know p type and n type
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materials are the same. However, for double
hetrojunction LED you can increase it up to
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80 percent 60 to 80 percent, how much power
is generated in the device.
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I know that total number of re-combinations
per second are Rr plus Rnr which is the total
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recombination rate and this is nothing, but
the total number of excess electrons injected
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per second.
So whatever excess electrons are injected
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per second they are getting recombined. So,
the number of electrons injected per second
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should be equal to the number of re-combinations
per second. How many excess electrons are
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injected per second? Well if i is the current
then i over e is the number of electrons injected
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per second. Now how many of these re-combinations
result in generation of photons well the fraction,
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which is corresponding to radiative recombination,
and that fraction is eta internal.
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So, the number of re-combinations which are
radiative are now eta internal and then the
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number of re-combinations that result in generation
of photons would be eta internal times i over
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e and each recombination these are the re-combinations
per second and each of these re-combinations
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will give you one photon, and I know the energy
of one photon is h nu. So, if I multiply this
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by h nu I will get the energy per unit time
that is power. So, the internally generated
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power would simply be eta internal times i
over e times h nu, and I can convert this
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into this particular fashion for convenience,
because h nu is equal to hc over lambda.
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So, your P internal is equal to eta internal
times i over e times h nu.
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So, this is eta internal i over e times hc
over lambda; if you look at hc over e then
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h. So, this is simply eta internal times h
6.6 into 10 to the power minus 34 joules second,
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c is equal to 3 into 10 to the power 8 meter
per second, and e is equal to 1.6 into 10
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to the power minus 19 coulomb times i over
lambda. So, this we had worked out previously
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also. So, this comes out to be 1.24 into 10
to the power minus 6, and 10 to the power
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minus 6. I observe as micrometer in lambda
downstairs. So, you can write it down like
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this. So, this is the internally generated
power.
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In the next lecture I would look into some
more characteristics of a light emitting diode
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and I will also look into a laser diode.
Thank you.