In the previous post, we have introduced band diagrams and
you learned about the conduction and valence energy band. Now we will start
taking a deeper look at the charge carriers that can occupy energy states in
those bands. The occupation of energy states in the valence and conduction
bands of semiconductors determines the concentrations of charge carriers.
Concentration of charge carriers is a very important property
for understanding performance of solar cells. Let’s start with the equilibrium
condition. We define equilibrium of a system as a state in which the system is
unperturbed. Therefore, no external forces are applied on this system. These
forces could be: external voltage, magnetic field, illumination or mechanical
stress. We can define the thermal equilibrium of as ystem as a condition in
which its parameters do not change with time.
We use the word thermal because the equilibrium conditions
will change depending on the temperature. Several steps are necessary to take
into account to determine the concentration of carriers in a semiconductor. The
first one is to determine the available energy states for electrons in the
conduction and valence band. In the previous posts, we have described the
concept of energy levels and energy bands. We saw that only few of the levels
are effectively occupied by charge carriers. The density of energy states,
which we denote as ‘g’, is an important parameter that tell us the number of
allowed energy states per unit volume as a function of energy.
The further we move from the edges of the bands, the more energy
states are available. ‘g_C’ represents the density of states in the conduction
band. If you remember from the previous post, the conduction band represents
energy states of mobile electrons. ‘g_V’ on the other hand, represents the
density of states of holes in the valence band. Holes essentially occupy the
states in the valence band of missing electrons.
The electrons that have been excited to the conduction band.
You will learn much more about holes in future posts. Let’s look into the
equations that represent these densities. We can express the equation for the
density of states in the conduction band as follows. The equation is based on a
few constants like the Plank’s constant and the effective mass of an electron.
However, please note, that the dependence on energy is a square root
relationship. As the energy gets further away from the conduction band edge denoted
by E_C, the amount of allowed energy states increases. Now let’s continue with
the valence band. The equations are very similar to that of ‘g_C’ but now we
have to use the effective mass of holes instead of electrons. So now we know
the density of allowed energy states of mobile electrons and holes.
In order to calculate the total charge-carrier concentrations
we also need to know how many of energy states are really occupied. For this
purpose, we introduce what is called the occupation function. This function is
known as the Fermi-Dirac distribution function. The Fermi-Dirac distribution
function expresses the probability that an available energy state will be
occupied at a certain temperature. This function depends on the difference
between the energy level of interest and the so called Fermi level E_F. I will
explain the definition of the Fermi level in a couple of slides, but let’s
first take a look at the temperature dependence of this distribution function. At
zero Kelvin, the Fermi-Dirac distribution function is a step function. It
means, only the energy levels below the Fermi level are occupied.
Energy states of the conduction band that are above the Fermi
level are empty, so no electrons occupy these states. However, if the
temperature goes up, the probability of occupation of higher energy levels increases.
The more the temperature rises, the probability of occupation gets higher and more
electrons can fill these states. This is the result of thermal excitation. The
excited electrons are getting energy from the ambient heat. If the temperature
is higher than zero Kelvin, electrons can occupy energy levels above the
conduction band edge. Before we go any further, let me explain a very important
concept: the Fermi Level.
So, what is exactly the Fermi level? In general it represents
the total averaged energy of valence electrons of a material. This energy takes
into account the electro-chemical energy of all the electrons in the conduction
and valence band. From the Fermi-Dirac distribution function we can easily
calculate that the probability that the energy level corresponding to the Fermi
level is occupied is zero point five. The position of the Fermi level in an
intrinsic semiconductor is close to the middle of the band gap. Depending on
the position of the Fermi level in the band gap we can simplify the Fermi-Dirac
distribution function.
If the Fermi level is within 3 times k_b times T from both
the conduction band edge and the valence band edge, so in the pink area of the
band diagram we can use this simplified equation. This equation is known as the
Boltzmann Approximation. We have now discussed the density of states function
and the Fermi Dirac distribution function. In the next post we will use these
parameters to determine the charge carrier concentration in thermal
equilibrium. Now let’s continue with the valence band. The equations are very
similar to that of ‘g_C’ but now we have to use the effective mass of holes
instead of electrons. So now we know the density of allowed energy states of
mobile electrons and holes. In order to calculate the total charge-carrier
concentrations we also need to know how many of energy states are really
occupied.
For this purpose, we introduce what is called the occupation
function. This function is known as the Fermi-Dirac distribution function. The
Fermi-Dirac distribution function expresses the probability that an available
energy state will be occupied at a certain temperature. This function depends
on the difference between the energy level of interest and the so called Fermi
level E_F.
I will explain the definition of the Fermi level in a couple
of slides, but let’s first take a look at the temperature dependence of this
distribution function. At zero Kelvin, the Fermi-Dirac distribution function is
a step function. It means, only the energy levels below the Fermi level are
occupied. Energy states of the conduction band that are above the Fermi level
are empty, so no electrons occupy these states. However, if the temperature
goes up, the probability of occupation of higher energy levels increases.
The more the temperature rises, the probability of occupation
gets higher and more electrons can fill these states. This is the result of
thermal excitation. The excited electrons are getting energy from the ambient
heat. If the temperature is higher than zero Kelvin, electrons can occupy
energy levels above the conduction band edge. Before we go any further, let me
explain a very important concept: the Fermi Level. So, what is exactly the
Fermi level? In general it represents the total averaged energy of valence
electrons of a material. This energy takes into account the electro-chemical
energy of all the electrons in the conduction and valence band. From the
Fermi-Dirac distribution function we can easily calculate tha tthe probability
that the energy level corresponding to the Fermi level is occupied is zero point
five.
The position of the Fermi level in an intrinsic semiconductor
is close to the middle of the band gap. Depending on the position of the Fermi
level in the band gap we can simplify the Fermi-Dirac distribution function. If
the Fermi level is within 3 times k_b times T from both the conduction band
edge and the valence band edge, so in the pink area of the band diagram we can
use this simplified equation. This equation is known as the Boltzmann Approximation.
We have now discussed the density of states function and the Fermi Dirac
distribution function. In the next post we will use these parameters to
determine the charge carrier concentration in thermal equilibrium.
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