This is the 15th lecture of this
course in which we begin a new topic Excess Carriers.
This topic will be covered in two lectures.
First we will discuss the methods of generating Excess Carriers and then we will discuss the
concept of injection level, and the concept of Quasi Fermi-level. We will then discuss
an important parameter namely: the lifetime. Like the energy gap and mobility this is an
important parameter associated with any semiconductor.
We will then discuss about the direct and indirect recombination phenomena which govern
the lifetime and finally we will discuss surface recombination. Let us begin by discussing
the methods of generating Excess Carriers. The first method is the method of impact ionization.
Let us say we have a n-type semiconductor and we apply an electric field to this semiconductor,
there is a current "I" flowing through this and we will assume that the semiconductor is uniform.
We have already shown that if you plot the current as a function of voltage the picture
would look as something like this. That is the current will increase linearly for small
voltages but for large voltages it will saturate and this is because the drift velocity saturates.
If you increase the voltage further in the saturation region, it is found that beyond
a critical voltage a very high the current starts rapidly increasing again.
Since the velocity of carriers has saturated in this range of voltages the increase in
current here can only be attributed to change in the carrier concentration or increase in
the carrier concentration because the current depends on the carrier concentration and the
velocity.
So here the carriers are increasing in number beyond this limit because between any two
collisions the carriers are gaining energy from the electric field that we have applied
and the energy gained is so large that when the carrier collides with the silicon atom
it can break a bond and it can generate excess electron hole pairs. So the energy gained
by the carrier between two collisions is more than the energy gap. This is the situation
of impact ionization. The field involved here are of the order of 100KV/cm so E is > 100KV/cm
whereas the fields involved here are of the order of about 20KV/cm which means if this
silicon sample is 100µ in length where 100µ is 1/100th of a cm then this voltage would
correspond to 20KV/cm that is 20 × 103V/cm × 100µ that is 10-2cm which amounts to 200V
whereas this point is 100KV/cm which could be about 1000V. It is beyond such high voltages
you are going to see this kind of Excess Carrier generation.
In practice, please note that the power dissipated in the sample for such high voltages and these
currents is so high that the sample will melt unless you do the experiment in which this
high voltage is applied for a short time so that the total energy generated or dissipated
is small and the particular phenomenon of Excess Carrier generation due to impact ionization
can be observed. This is one way you create Excess Carriers.
Please note that Excess Carriers means carriers over and above the equilibrium value. So if
this an n-type sample and doping is 1015/cm3 of phosphorous atoms then under equilibrium
conditions it will have about 1015/cm3 of electrons and about 105/cm3 of holes of that
order. The exact value of hole concentration of holes is (ni)2/1015. So over and above
this we are creating carriers and those are Excess Carriers. Generally Excess Carriers
are created in pairs that are for every excess electron that is created you have an excess
hole. Let us discuss another method of generating excess carriers that is photo ionization or
photo generation.
In this case you have a sample which is very thin. Let us assume that you have applied
a small voltage to this so that a current flows. We will assume that this sample is
n-type and the thickness of the sample is about 3µ. Obviously this is an imaginary
sample because you cannot handle a 3µ thin sample of semiconductor. We will assume that
this is the third experiment.
We will not discuss about the practical aspects of how you can have a 3µ thin silicon sample.
Now let us also assume that its doping is as low as possible so that its resistivity
is high. If light is shown on this sample and the frequency of light is such that hµ
is greater than the energy gap of the sample and if the condition is satisfied where the
frequency is > Eg/h then what we will see is the following: Here I am sketching the
current I as a function of the frequency of the light and we will assume that the intensity
is kept constant so we are varying the frequency and the voltage also is kept constant. Now
if you see the current picture would look something like this. This is the point where
the frequency is exactly equal to Eg/h this is a critical frequency. We can call this
current as a dark current i0 because in this type of current even when the light is falling
its frequency is very small so no electron hole pairs are being generated.
Strictly speaking dark means that the intensity of light is 0 but here we are keeping the
intensity of light constant but the frequency is small so that no electron hole pairs can
be really generated. When the frequency exceeds this Eg/h the energy of the photons which
is incident on the sample is more than the energy gap then each photon can participate
in breaking of the silicon bond and creating an excess electron hole pair.
If the energy is more than Eg/h that is if your frequency is more then the number of
electron hole pairs will not increase but what will happen is that the extra energy
over and above the energy gap that the photon has will be dissipated as heat. So you look
at the energy band diagram, this is Ec and this is Ev the energy required for an electron
to jump from valance band to conduction band and created an electron pair is Eg so if Eg
is energy of the photon the entire energy is used for creating the electron hole pair.
If however the energy of the photon is more than Eg then the electron is going to jump
up to some other energy here. But the difference between this energy and this energy is the
energy given by the photon. Then this free electron to this particular energy state close
to the Ec and this energy will be dissipated as heat.
Please note that this is Eg and if the energy is more than Eg then the extra energy hµ
- Eg this energy is dissipated as heat. You do not get extra electron hole pairs because
of the extra energy and it is energy dissipated as heat. Therefore you note that if you have
some way of removing the heat from the sample so that the sample is kept at uniform temperature
then this is the picture that you will get. There are a certain number of excess electron
hole pairs created which are going to increase the conductivity of the sample and therefore
the current is increasing to this value. If the intensity of the sample is less then this
final current that you are getting will be less so this is increasing intensity. Please
understand the difference between intensity and frequency. Intensity of light means how
many of photons are falling on this sample per unit time per unit area. In other words
it can also be regarded as incident power.
Another point to note is that all the photons which are falling on the sample will not create
electron hole pairs because some of the light is going to be reflected and not the entire
light is going to absorbed. Another point to note is that the light which is absorbed
within a micron or so of this surface which is being illuminated. So within a couple of
microns the intensity of light is going to fall to 0. That is why only a thin layer near
the surface is going to absorb the light and create the excess electron hole pairs. This
is the method of creating excess electron hole pairs by photo generation or photo ionization.
Let us look at the third method of generating electron hole pairs called injection.
Consider a PN junction and supposing we apply a voltage a forward bias to this. In this
case what happens is the p-region will inject extra holes into n-region and the n-region
will inject extra electrons into the p-region. This is because this terminal is positive
and this is negative. This positive terminal attracts electrons from here and these electrons
injected into this region are excess electrons, over and above the equilibrium value of electrons
in the p-region. Similarly this negative terminal attracts these holes and p-region therefore
creates holes in this n-region which are over and above the equilibrium value. Of course
the sample as a hole will remain charge neutral because the holes which are being injected
from p to n will be neutralized by electrons which are injected from these terminals.
Now it is important to note that by neutralization we do not mean that the holes and electrons
are annihilated or it means the charge is 0. So this is a unique situation that can
happen in a semiconductor where you have excess electrons and equal number of excess holes
which together gives rise to 0 charges but both excess electrons and holes are present.
This is the method called as injection across a forward bias junction. We will show the
injection as something like this. If we slightly make the diagram bigger to show this the p-region is injecting holes and the n-region
is injecting electrons in this region.
The holes which are injected are recombining with electrons which are injected from this
terminal and these electrons are recombining with holes which are injected here so the
sample is charge neutral. So this is the picture. You see how excess electrons and holes have
been created by the application of a voltage or they are being injected from an external
contact. So this is the process of injection and this is the reason why you get a large
current in the forward bias junction. If you sketch the current voltage characteristics
of this diode then the forward bias you get a large current; this is the origin.
If you reverse the voltage polarity then there is a small current flowing and beyond a certain
critical voltage what happens is that the current increases rapidly. This increase in
current for reverse voltages also signifies generation of Excess Carriers. This is in
the breakdown region. Now these Excess Carriers either is created by impact ionization, a
process where high field is created in very narrow region called the depletion region
at the junction. The details of the operation of the PN junction will be discussed later
elaborately.
Now we are just mentioning the method of Excess Carrier creation. So, for this reverse voltage
a high electric field is created near the junction which may give rise to excess electron
hole pairs because of impact ionization or it could also be what is called tunneling.
Tunneling is another method by which excess carriers can be created. So an example of
Excess Carrier creation by tunneling is in a tunnel diode. We can probably show it here
as band to band tunneling.
So here you have a PN junction where this side is heavily doped and this side also is
very heavily doped. When you forward bias is junction it turns out that the characteristic
looks something like this. This is the curve if there is no tunneling and this portion
of the curve is because of tunneling.
What is happening is that because of tunneling a current starts increasing very rapidly but
the tunneling phenomenon drops off beyond a certain voltage. This entire current is
because of tunneling and then the normal process of injection that we have discussed takes
over. In fact this portion is nothing but this portion here which is expanded. So in
the reverse direction in such diodes again the current starts increasing rapidly because
of tunneling. There is Excess Carrier generation because of tunneling.
We will not discuss the details of how this tunneling occurs but we just want to mention
that this is one method of generation of Excess Carriers.
To summarize, Excess Carriers can be generated by impact ionization as discussed here that
is under the influence of high electric fields the Excess Carriers can be generated by photo
ionization that is illumination or with the help of photons. It can be generated by band
to band tunneling or it can be generated by injection. Excess Carriers can be generated
by injection mechanism.
Having discussed the various methods of generating Excess Carriers how do you analyze the devices
in the presence of Excess Carriers? That is the topic that we must consider. In this context
the first important idea we discuss is the concept of injection level.
The word injection is used in semiconductor devices to mean any method of generating Excess
Carriers. In our earlier discussion so far we used the word injection to denote creation
of Excess Carriers within a device because of contacts which are injecting the Excess
Carriers. So injection so far meant injection from outside into a semiconductor device.
But here after the word injection will be used to denote any method of generating Excess
Carriers. So the injection level refers to the extent to which the semiconductor is disturbed
from equilibrium because of Excess Carriers. Now here the first point to note is the Excess
Carriers are created in pairs. So this can be understood by the equation like this, you
write the charge balance equation we have p + Nd+ - n -Na- = 0 this is the picture
under equilibrium. Let us assume that you have some donor type impurities and some acceptor
type impurities. So under equilibrium this is the charge balance equation. Now if you
create Excess Carriers then the changed d = 0. Therefore what it means is dp - dn = Na-
- Nd+.
If we assume that the change in the ionization of the impurities is 0 the impurity ionization
state does not change which is a reasonable assumption. Then the left hand side can be
equated to 0 so this means dp can be assumed to be equal to dn so this is the situation
in most semiconductors. Here these impurities may mean deep levels or shallow levels and
although we have shown only two types of impurities one type of donor and other type of acceptor
in principle there can be several different impurities. All this will mean is the change
in the ionization of impurities is negligible. So in such a situation we can very easily
see that dp = dn. So this means Excess Carriers are generated in pairs.
Therefore what we will do is we will assume that the symbol d represents the Excess Carriers
over and above the equilibrium value either electrons or holes since they are equal in
number we use only a single symbol d. In other words concentration p in the presence of excess
carriers is equal to p0, which is the equilibrium concentration + d (p = p0+d) and concentration
of electrons is equal to equilibrium concentration of electrons + d (n = n0 + d).
Now how do you classify the disturbance from equilibrium because of Excess Carriers? Supposing
we show the concentration d on a log scale then this particular point shows that if d
is < 1/10 of the majority carrier concentration where this is majority carrier concentration/10.
So if d is < 1/10 of the majority carrier concentration it is clear that the concentration
of carriers which refer to the majority carriers the n-type semiconductor which is the electron
concentration then for d < majority carrier concentration/10 we have n = n0 + d which
is quantity < n0/10 because n0 is a majority carrier concentration which is equal to approximately
n0 itself. For d < majority carrier concentration/10 the majority carrier concentration is not
disturbed at all. However, the minority carrier concentration
is disturbed because the minority concentration is very small. For this case minority carrier
concentration will definitely be disturbed. Whenever you have Excess Carrier creation
minority carrier concentration will definitely be disturbed.
Therefore depending on whether the majority carrier concentration is disturbed or not
we classify the injection level into several parts. This particular region is called low
level or low injection level when the majority carrier concentration is not disturbed. On the other hand if d > 10
× the majority carrier concentration at this point if d is here in this region then both majority and minority
carrier concentrations are disturbed and what you will find is that the concentration of
electrons and holes will tend to become almost equal. So for d > majority carrier concentration
× 10 you have n = n0 + a quantity > 10 × n0 = d so d is a quantity approximately equal
to d itself. Here n ˜ d because delta is more than 10 × n0 so this plus this is this
quantity itself. Now (p = p0+d) ˜ d because p0 is much less than n0 in an n-type semiconductor.
This plus this would be again this itself. What is happening is n ˜ p. It is because
of Excess Carrier creation the Excess Carriers created as so large that the number of electrons
and number of holes in any unit volume are almost same. This particular condition is
classified as high injection level or high level.
This region will be the intermediate injection level region which is neither high nor low.
It turns out that the device analysis is simple either under low level conditions or under
high level conditions. In both these extreme cases simplifications are possible as when
the majority carrier concentration is not disturbed at all or when the electron and
hole concentrations in the device almost become equal and these are the two extremes.
So under these conditions the device can be analyzed in a simple way and that is why these
two regions are defined. Now let us consider this particular concept of this injection
level with a numerical example.
Let us assume an n-type semiconductor with a doping level of 1015 atoms of phosphorous/cm3.
Let us show the conditions of electrons and holes on a log scale. This is a concentration
on a log scale. Under equilibrium the majority carrier concentration is 1015/cm3 and the
minority carrier would
be ni2/1015/cm3. I am not considering the multiplication constant because when you take
ni2/1015 then your ni2 = 1020 × 2.25 so the 2.25 factor I am just ignoring here for simplicity
just to show this particular scale in a simple manner. So ni would be somewhere here exactly
in the middle and on a log scale ni is exactly in the middle so this is of the order of 1010/cm3
and for simplicity I am ignoring the constant. This is the picture under equilibrium 1015
and this is n0,105 this is p0 and this is ni. Now let us take low level injection. Supposing
you assume d = 1012/cm3 because of photo ionization or any other means.
Supposing you have excess electron and hole concentration or 1012/cm3 then the hole concentration would be 105 + 1012 ˜ 1012
itself so Excess Carrier concentration is equal to the minority carrier concentration.
Now let us locate at 1012 on this so it is 1/5 of this and each of this is a factor of
10 so this is 1010, 1011, 1012, this is the concentration p ˜ d for low level conditions. Now what is the majority carrier concentration?
The majority carrier concentration is 1012 + 1015 which is 1015 itself so the majority
carrier concentration is the same here. Now this is your n so let us call this d1 because
this is one particular condition we are considering. Here this is d1, p1 and this is n1. This is
the so-called low level condition so therefore p1 ˜ d1 and n1 ˜ n0 so here it is n1 ˜ n0
which is the equilibrium concentration.
Let us consider another case that is the high level. This means you have an Excess Carrier
concentration d2 which is 1016/cm3 or 1017/cm3. Now what are the electron and hole concentrations?
So the hole concentration is 1017 + 105 which is 1017 and the electron concentration is
1017 + 1015 which is again 1017 itself because 1015 is much less than 1017 so what you find
is electron and hole concentrations both are almost the same and this is 1017 so the value
is in /cm3 and here you have this is n2 ˜ p2 ˜ d2 which is the high level injection.
A log scale very clearly shows both the hole and electron concentrations for low and high
level. For the low level the minority carrier concentration is disturbed but majority carrier
concentration is almost same as under equilibrium. But for the high level both majority and both
electron and hole concentrations are almost the same and both are disturbed. So, that
is the concept of low and injection level. In this course we will be considering the
devices under low injection level because the analysis in that condition is very simple.
After this concept of injection level we need to consider the concept of the Quasi Fermi-level.
How do you show the effect on the energy band diagram of the Excess Carriers?
Quasi Fermi-level: Now recall that we said the electron concentration
in a semiconductor can be written in terms of the following equation n0 = Nc exponential
-(Ec - Ef/kT). And the hole concentration under equilibrium can be written as p0 = Nv
exponential - (Ef - Ev/kT). Now both n0 and p0 are characterized by the same Fermi-level.
The same Fermi-level Ef comes into both these formulae.
You can also write the electron and hole concentrations in terms of the deviation from the intrinsic
semiconductor. For example I could write n0 = ni exponential (Ef - Ei)/kT. And p0 = ni
exponential (Ei - Ef)/kT is another way of writing the same concentration. Now here the
Fermi-level is again the same that is used in both formulae and the diagram is as follows.
Ec Ev that is we are drawing the energy band diagram and this is Ei. And let us assume
an n-type semiconductor so this is Ef. So here Ef will be more than Ei, Ef > Ei.
Can we use similar equations also under non equilibrium conditions when Excess Carriers
are present. There is nothing that prevents us from using similar equations that is to
relate the electron concentration to the difference between the conduction band edge and a Fermi-level.
We can always use similar equations under non equilibrium conditions. What will happen
now is that instead of n0 you will have n which will be > n0.
Therefore the difference Ec - Ef should reduce. If your left hand side increases the right
hand side should also increase which means the difference should reduce. The Ef will
have to therefore change to a new value that is Efn this is called the Quasi Fermi-level
for electrons. It is that Fermi-level which when you substitute in this formula which
is similar to that under equilibrium you will get the concentration of the electrons and
similarly we can get the concentration of holes also as will see. So n = Nc exp- (Ec
- Efn) × kT or if you were to substitute here you will get n0 = ni exp (Quasi Fermi-level
of electrons - ei)/kT. Fermi-level under non equilibrium conditions is called Quasi
Fermi-level.
One important point is to be noted. When you want to express the hole concentration you
will note that since left hand side has increased the hole concentration is more because of
the presence of Excess Carriers because it is p0 + d then the right hand side should
also increase which means that this different should decrease. You have a new Fermi-level
for holes which is different from the Fermi-level under equilibrium.
This means that the Fermi-level should move closer to the valance band edge to have increased
hole concentration. But in the same semiconductor for increase electron concentration the Fermi-level
should move closer to the conduction band to have the higher concentration because this
difference should decrease. Obviously the same Fermi-level cannot move close to the
conduction band as well as to the valance band so this is what is important to note
under non equilibrium. The Quasi Fermi-levels for electrons and for holes are not the same
they are different.
The Quasi Fermi-levels for the electrons moves closer to the conduction band edge as compared
to the equilibrium whereas Quasi Fermi-level for holes moves closer to the valance band
edge as compared to equilibrium, this is an important difference.
Similarly we substitute Efp here instead of Ef and we will get the hole concentration,
this is the concept of Quasi Fermi-level, determining the concentrations in terms of
a new Fermi-level. Now what is the advantage of this? The advantage will be seen later
in analysis that many forms of equations under non equilibrium conditions for devices gets
simplified if you use this kind of representation or formula to relate the energy band parameters
and the concentrations. This point will be clear in the subsequent lectures when we take
up analysis of devices.
The Quasi Fermi-level also has lot of physical significance. Now before leaving this concept
of Quasi Fermi-level let us see what is the Quasi Fermi-level picture under low injection
level and under high injection level?
We will be assuming n-type semiconductor in this example. Under low injection level the
concentration of majority carriers the electrons will not change significantly but it remains
equal to the equilibrium value. Therefore the Quasi Fermi-level for electrons remains
equal to the equilibrium Fermi-level.
So if you take low injection level the Efn = Ef. However the Quasi Fermi-level for holes cannot be the same as under equilibrium
because definitely even if it is low injection level the minority carrier concentration is
always disturbed. So Efp will be closer to the valance band edge as it depends on how
much is your Excess Carrier concentration.
If Excess Carrier concentration is 1012 as we considered in this example for low level
then the Fermi-level would be definitely below Ei because when you have Fermi-level at Ei
the concentration is ni which in silicon is 1010. So if you want 1012 then it will be
below this which is the Quasi Fermi-level for holes. You see Efp and Efn are different
so Fermi-level has split. And further Efp - Ei will be less than Efn - Ei because
this deviation represents 1012 while this deviation from Ei represents 1015.
We are assuming an n-type semiconductor doping 1015. So low level d = 1012 therefore the
Fermi-level splits into two Quasi Fermi-levels. Now, what about high injection level? Under
high injection level the excess hole concentration will be same as excess electron concentration.
So if you take 1017 as we have discussed in this example here as d2 then the picture can
be shown separately to avoid confusion this is Ec, this is Ec, this is Ei and this is
Ef corresponding to equilibrium. The majority carrier concentration has increased to 1017
so you will have a new Quasi Fermi-level for electrons which is Efn. And you will have
a new Quasi Fermi-level for holes exactly at the same distance below so this is Efp
which is for d 1017/cm3 high level. So Efn - Ei ˜ Efp - Ei or Ec - Efn = Ev - Efp
the condition at high level. Therefore the Quasi Fermi-levels are situated
symmetrically about Ei under high injection level conditions. With this we have completed
the discussion on Quasi Fermi-level. Next we must consider the important parameter of
lifetime which describes the transient phenomena related to Excess Carriers. Let us discuss
this concept with the help of this figure of photo ionization.
Generation of Excess Carriers is because of light and we keep the voltage constant and
you have a current flowing. Supposing you switch on the light at some instant of time
how long does it take for the Excess Carriers to increase to their steady state value? And
when you switch off the impulse how long does the sample take to come back to equilibrium
conditions? We can show it on a graph.
We will plot the current I as a function of time. We will assume that it is the intensity
that we are changing suddenly, the intensity of the light is being changed suddenly. I
am going to plot the intensity here. Let us assume the intensity is 0 to start with and
at some instant it has suddenly increased so the light is switched on. It is maintained
on for sometime and then it is suddenly switched off. Here you have on and here you have off
and these are the instants. How do we sketch I as a function of T the observed current?
We will find that you will have a small current called the dark current i0. This is the dark
current and at this instant the current will start rising and will slowly reach the steady
state value something like charging of a capacitor, something similar to that. Then after it reaches
the steady state let us assume that the light is kept on until a steady state is reached.
It is for a sufficiently long duration it is on.
Now at this instant when it is switched off you find that the current will come back to
its dark current value slowly after decay. In other words your change in current will
not be instantaneous. The current will not reach a new steady state value instantaneously
when the intensity has increased so there is a delay and there is a time which elapses
before the sample can respond so Excess Carriers are not generated instantaneously.
The Excess Carriers are generated and their concentration starts increasing and then reaches
a steady state after sometime. It is this time that is of interest in the switching
performance of devices. We will show that these delays associated with the on-transient
and off-transient are related to an important parameter called the lifetime. Let us discuss
this situation further.
What is the reason for these transient durations? What is happening here is that the moment
we switch on the light the generation rate has changed suddenly but the recombination
rate has not changed at this instant. Since the generation rate is more than recombination
rate the carriers are being generated.
Excess Carriers are being generated and the carrier concentration is rising. As the Excess
Carrier concentration rises the recombination rate also goes on rising because recombination
rate depends on the hole and electron concentrations. Here we can say in this region g = g0 + g'
which is the excess generation rate but r which = r0 + r' this r is < g because r' < g'
in this region. I want to emphasize this fact that we are assuming that this particular
sample is absorbing the light in a uniform manner so that excess carrier concentration throughout
the sample is uniform so the generation rate is also said to be uniform. So this is the
third experiment it may not be actually true but in this experiment we are assuming uniform
conditions. So that any of these transients you see here are simply because of the differences
between the generation and recombination rates and not because of any carrier movement.
Since the carrier concentration is uniform carriers do not move from a one region to
another. Now in this portion what has happened is g = g0 because g' has become 0 because
the light has been switched off and therefore generation rate has dropped to 0. Recombination
rate however cannot drop to 0 immediately. So this is r0 + r' but as the Excess Carrier
concentration starts falling r' also starts falling. So in the steady state beyond this
you have r' = 0, g' is anyway 0 here so that is equilibrium condition.
In this region you have r' > g'. Since recombination rate is more than the generation rate you have a decay in Excess Carrier population.
Whereas here the generation rate is more than the recombination rate so you have a rise.
So in the steady state portion you have g' = r' so no change in carrier concentration
is seen because the recombination rate has risen and has become equal to the excess generation
rate.
However, this is not equilibrium because g' and r' both are non zero. Both g' and r' are
non zero whereas here r' is 0 and g' is 0 and that is why this is in equilibrium state
which is the condition. The rate of increase of carrier concentration in this region here
we can write dd /dt the rate of increase of Excess Carrier concentration is related to
g' - r'. The difference g' - r' is causing this.
On the other hand in this region you have dd x dt = - r because g' has become 0 and
there is an excess recombination and that is why carrier concentration is falling. So
here carrier concentration is rising. This is the situation from which we need to develop
the concept of lifetime. We will do this in the next class.
