Objectives
1. Fermi Dirac Distributed Function to be reviewed.
2. Necessity of Fermi Dirac Distribution Function to be reviewed
Figure 1.1 Internal Structure of a Typical Atom
Atoms constitute the
building blocks of all materials in existence. In these atoms, there is a
central portion called nucleus shown in above figure. Which consists of protons
and neutrons, around which revolves the particles called electrons. Next, it is
to be noted that all the electrons constituting the considered material do not
revolve along the same path. However this even does not mean that their
revolutionary paths can be random. That is, each electron as show in Figure 1.0
of a particular atom has its own dedicated path, called orbit, along which it
circles around the central nucleus as shown in Figure 1.1. It is these orbits
which are referred to as energy levels of an atom.
Fermi Dirac Distribution Function:
Distribution functions are nothing but the probability
density functions used to describe the probability with which a particular
particle can occupy a particular energy level. When we speak of Fermi-Dirac distribution function,
we are particularly interested in knowing the chance by which we can find a
fermion in a particular energy state of an atom. Here, by fermions, we mean the
electrons of an atom which are the particles with ½ spin, bound to Pauli
Exclusion Principle.
Necessity of Fermi Dirac
Distribution Function
In fields like electronics, one particular factor which is of
prime importance is the conductivity of materials. This characteristic of the
material is brought about the number of electrons which are free within the
material to conduct electricity.
As per energy band theory, these are the number of electrons which
constitute the conduction band of the material considered. Thus in order to
have an idea over the conduction mechanism, it is necessary to know the
concentration of the carriers in the conduction band.
Fermi Dirac Distribution
Expression
Mathematically the probability of finding an electron in the energy state E at the temperature T is expressed as
K
is the Boltzmann constant
T is the absolute temperature
Ef is the Fermi level or the Fermi energy
Now, let us try to understand the meaning of Fermi level. In order to accomplish this, put
in equation (1).
By doing so, we get,
This means the Fermi level is the level at which one can expect the electron to be present exactly 50% of the time.
Fermi Level in Semiconductors
Intrinsic semiconductors are the pure semiconductors which have no impurities in them. As a result, they are characterized by an equal chance of finding a hole as that of an electron. This inturn implies that they have the Fermi-level exactly in between the conduction and the valence bands as shown by Figure 1.2a.
Figure 1.2: Fermi levels of (a) Intrinsic Semiconductor (b) N-Type Semiconductor (c) P-Type Semiconductor
Next,
consider the case of an n-type semiconductor. Here, one can
expect more number of electrons to be present in comparison to the holes. This
means that there is a greater chance of finding an electron near to the
conduction band than that of finding a hole in the valence band. Thus, these
materials have their Fermi-level located nearer to conduction band as shown by Figure1.2b
Following on the same grounds, one can expect the Fermi-level in the case of p-type semiconductors to be present near the valence band
(Figure 1.2c). This is because, these materials lack electrons i.e. they have
more number of holes which makes the probability of finding a hole in the
valence band more in comparison to that of finding an electron in the
conduction band.
Effect of temperature on Fermi-Dirac Distribution Function
Figure 1.3: Fermi-Dirac Distribution Function at Different Temperatures
At T = 0 K, the electrons will have low energy and thus occupy lower energy states. The highest energy state among these occupied states is referred to as Fermi-level. This in turn means that no energy states which lie above the Fermi-level are occupied by electrons. Thus we have a step function defining the Fermi-Dirac distribution function as shown by the black curve in Figure 1.3. However as the temperature increases, the electrons gain more and more energy due to which they can even rise to the conduction band. Thus at higher temperatures, one cannot clearly distinguish between the occupied and the unoccupied states as indicated by the blue and the red curves shown in Figure 1.3.
Continuity Equation:
https://youtu.be/1N8qy1iiUrM
The
fundamental law governing the flow of charge is called the Continuity Equation.
The continuity equation as applied to semiconductor described how the carrier
concentration equation in a given elemental volume of the crystal varies with
time and distance. The variation in density is attributable two basic causes.
i)
The rate of generation and loss by
recombination of carriers within the element
ii)
Drift of carriers into or out of the element.
The
continuity equations enable us to calculate the excess density of electrons or
holes in time and space.
As shown
following figure 1.4 consider an infinitesimal N-Type semiconductor bar of
volume of area A and length dx and the average minority carrier (hole)
concentration p, which is very small compared with the density of majority
carriers. At time t, if minority carriers (holes) are injected, the minority
current entering the volume at x is Ip and leaving at x+dx is Ip+
dIp which is predominantly due to diffusion. The minority carrier
concentration injected into one end of the semiconductor bar decreases
exponentially, with distance into the specimen, as a result of diffusion and
recombination, Here, dIp is the decrease in number of coulombs per
second within the volume.
Figure 1.4:
Relating to continuity equation
equals the decrease in the number of holes per
second within the elemental volume A ∝ x. As the current density
Decrease
in hole concentration per second, due to current Ip.
We know
that there is an increase of holes per unit volume per second given by G = p0/τp
due to recombination but charge can neither be created nor destroyed. Hence,
increase in holes per unit volume per second, dp/dt, must equal the algebraic
sum of all the increase in hole concentration. Thus,
Similarly, the continuity equation
for electrons states the condition of dynamic equilibrium for the density of
mobile carrier electrons and is given by
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