 Namaste. Myself Dr. Mrs. Preeti Sunil Joshi working as assistant professor in VALCHAN Institute of Technology, Solapur. This session is related with the position of Fermi level in semiconductors. Learning outcomes are, by the end of this session, students will be able to state the definition of Fermi level and Fermi energy and describe the position of Fermi level in semiconductors. The contents include Fermi level and Fermi energy and position of Fermi level in semiconductors. We are next interested in knowing how the electrons are distributed among the various energy levels in the conduction band at a given temperature. We cannot apply Maxwell-Boltzmann distribution to electrons because they obey exclusion principle and also they are indistinguishable particles. The Pauli's exclusion principle postulates that only one fermion can occupy a single quantum state. Therefore, as fermions are added to an energy band, they will feel the available states in the energy band just like water feels a bucket. Here by fermions, we mean the electrons of an atom which are the particles with half spin and bound to Pauli exclusion principle. So, the states with the lowest energy are filled first followed by the next higher ones. Hence, the statistical distribution function that is applicable to the quantum particles is the Fermi Dirac distribution function. This function is named after Enrico Fermi and Paul Dirac, each of whom discovered the method independently. Before knowing what this distribution function is, let us first know the necessity of this 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 the energy band, 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. It is given by F of E is equal to 1 upon 1 plus E raise to E minus EF upon kT. The function F of E indicates the probability that a particular quantum state at the energy level E is occupied by an electron and EF is known as the Fermi level. In general, this EF may or may not correspond to an energy level but it provides a reference with which other energies can be compared. The function F of E is known as Fermi factor. These are the Fermi Dirac function details. The probability of electrons to occupy the energy level E increases with temperature. So, we first discuss about the distribution function and the related topics with reference to conductors. We shall find later these concepts are equally applicable to other cases. Here now we distinguish two situations, one at absolute zero and other at higher temperatures. This figure shows the conduction band of a conductor at zero Kelvin. At absolute zero, electrons occupy energy levels in pairs starting from the bottom of the band up to an upper level designated as EF, leaving the upper levels vacant. So, Fermi level can be therefore defined as the uppermost field energy level in a conductor at zero Kelvin. Correspondingly, Fermi energy can be defined as the maximum energy that a free electron can have in a conductor at zero Kelvin. To use an analogy, the electron distribution in the conduction band can be likened to water at rest in a container. The Fermi level corresponds to the top surface of the water. Let us now apply Fermi Dirac distribution function to the solid taking the value of temperature as zero Kelvin. For energy levels E lying below EF that is E is less than EF, E minus EF is a negative quantity therefore the probability will be equal to 1 which indicates that all the energy levels lying below the level EF are occupied. For energy levels E lying above EF, E minus EF is a positive quantity therefore the probability function becomes equal to zero which means that all the energy levels lying above this level EF are vacant at T is equal to zero Kelvin. Now for E is equal to EF the quantity E minus EF is equal to zero therefore F of E will be indeterminate. Indeterminate this implies that the occupancy of Fermi level at zero Kelvin ranges from zero to one. Corresponding energy function is shown in the figure. Now consider the case when temperature is above zero Kelvin. On heating the conductor electrons are excited to higher energy levels. In general EF is greater than KT therefore for most of the electrons lying deep in the conduction band the thermal energy is not sufficient to cause a transition to an upper unoccupied level. At normal temperatures only those electrons occupying the energy levels near the Fermi level can be thermally excited. These levels make a narrow band of width KT directly adjacent to the Fermi level. Therefore upon heating the solid electrons having a little below EF jump into the levels with the energy somewhat above EF and a new energy distribution of electrons is obtained thus as a result of thermal excitation the probability of finding the electrons in the levels immediately below EF will decrease. On the same hand the probability of finding the electrons in the levels immediately above EF increases. This fact is reflected in the graph that is in figure B as a blurring of the step plot. Now at temperature greater than zero Kelvin if we consider an electron at Fermi level then if we consider an electron at Fermi level then E is equal to EF that is probability becomes equal to half. This implies that the probability of occupancy of Fermi level at any temperature above zero Kelvin is 0.5 or 50%. So now we can define Fermi level as the Fermi level which has the probability of occupancy of 0.5. Fermi energy is the average energy possessed by electrons participating in the conduction in metals at temperatures above zero Kelvin. The Fermi Dirac distribution curves for different temperatures are shown in the figure. At T is equal to zero Kelvin there is an abrupt jump in the value of F of E from 1 to 0 at EF and when temperature is greater than zero Kelvin the change is gradual. The higher the temperature more gradual is the change. It is seen from the curves for different temperatures that they all pass through a crossover point C at which the probability of occupancy is 0.5. This is due to the fact that F of E has a value of 0.5 for any temperature greater than zero Kelvin. Students now please pause the video and try to solve this numerical. Check for the answer. So the solution is as we have from Fermi Dirac distribution function F of E is equal to 1 upon 1 plus e raised to E minus EF upon kT. E minus EF is equal to 0.5 electron volt and F of E is equal to 1%. So this is the given data and if we solve the numerical by substituting the values we get the temperature as 1262 Kelvin. Now let's know the position of Fermi level in semiconductors. So we have seen that the probability of occupation of energy levels in valence band and conduction band is called the Fermi level. At absolute zero temperature intrinsic semiconductor acts as a perfect insulator. However as the temperature increases free electrons and holes gets generated. Thus in intrinsic or pure semiconductor the number of holes in valence band is equal to the number of electrons in the conduction band. Hence the probability of occupation of energy levels in conduction band and valence band are equal. Therefore the Fermi level for the intrinsic semiconductor lies in the middle of forbidden band. In extrinsic semiconductors the number of electrons in the conduction band and number of holes in the valence band are not equal. Hence the probability of occupation of energy levels in conduction band and valence band are not equal. Therefore the Fermi level for extrinsic semiconductor lies close to the conduction or valence band. In entire semiconductor pentavalent impurity is added. Each pentavalent impurity donates a free electron. So addition of pentavalent impurity creates large number of free electrons in the conduction band. At room temperature the number of electrons in the conduction band is greater than the number of holes in the valence band. Hence the probability of occupation of energy levels by the electrons in the conduction band is greater than the probability of occupation of energy levels by the holes in the valence band. This probability of occupation of energy levels is represented in terms of Fermi level. Therefore the Fermi level in n-type semiconductor lies close to the conduction band and the donor levels are expected to be located very near to the bottom edge of the conduction band. Now for p-type semiconductor trivalent impurity is added. Each trivalent impurity creates a hole in the valence band and is ready to accept an electron. The addition of trivalent impurity creates large number of holes in the valence band. Again at room temperature the number of holes are greater than the number of electrons in the conduction band. Hence the probability of occupation of energy levels by the holes in the valence band is greater than the probability of occupation of energy levels by the electrons in the conduction band. This probability of occupation of energy levels is represented in terms of Fermi level. Therefore the Fermi level in p-type semiconductor lies close to the valence band and the acceptor level represents the ground level of the hole. Thank you.