Fermi dirac distribution

Fermion, Boson

1. Presented by- Md Ahsan Halimi Scholar No: 1 9 -3-04-1 05 Dept. of ECE, NIT Silchar “ Introduction of Fermi Dirac Distribution Function ” 1

2. Contents Some Basic Concept Fermi-statistics and Bose Statistics Postulates of Fermi particles Fermi Dirac Distribution Function Conclusion Ref e re n c e s 2

3. Some basic concepts Fermi level :- Fermi level  is the highest energy state occupied by electrons in a material at absolute zero temperature . Fermi energy:-This is the maximum energy that an electron can have at 0K. i.e. the energy of fastest moving electron at 0K. It is given by, Fermi velocity ( ):- It is the velocity of electron at Fermi level. The band theory of solids gives the picture that there is a sizable gap between the Fermi level and the conduction band of the semiconductor. At higher temperatures, a larger fraction of the electrons can bridge this gap and participate in electrical conduction.   3

4. Fermi-statistics and Bose Statistics The wave function of a system of identical particles must be either symmetrical (Bose) or antisymmetrical (Fermi) in permutation of a particle of the particle coordinates (including spin). It means that there can be only the following two cases: 1. Fermi-Dirac Distribution 2. Bose-Einstein Distribution The differences between the two cases are determined by the nature of particle. Particles which follow Fermi-statistics are called Fermi-particles (Fermions) and those which follow Bose-statistics are called Bose- particles ( Bosones ) . Electrons, positrons, protons and neutrons are Fermi-particles , whereas photons are Bosons . Fermion has a spin 1/2 and boson has integral spin . Let us consider this two types of statistics consequently. 4

5. Different types of systems considered Distinguishable particles >(Fermions when spin is not considered) Indistinguishable particles that obey Pauli exclusion principle > (Fermions ) Indistinguishable particles that doesn't obey Pauli exclusion principle >( Bosons ) 5

6. Postulates of Fermi Particle Particles are indistinguishable. Particles obey Pauli principle. Each quantum state can have only one particle. Each particle has one half spin. be the quantum states associated with energy level. is the no. of particles associated with energy level. For a particular value of N, there is only one distribution   6 N2 Nn N1 ……………….

7. Fermi -Dirac distribution function (Derivation) Consider now the i th energy level with degeneracy g i. For this level, the total no. of ways of arranging the particles is : Consider all energy level, the permutation among themselves Now the N i particles can have N i ! Permutations We now apply, the other two assumptions, namely conservation of particles and energy. 7

8. Contd… Stirling approximation (x>>1) Lagrangian multiplier method for lnQ Now we proceed in the standard fashion, by applying Stirling’s approximation to lnQ, and then using the method of Lagrange multipliers to maximize Q . 8  

9. Contd… For i=j, And because energy level is continuous, g(E) dE is the number of available states in the energy range E and E+dE Number of particles between E and E+dE is given by N(E) dE = f (E)*g(E) dE f(E) is the probability that a state at energy E is occupied by a particle   9

10. Contd… Density of states tells us how many states exist at a given energy E . The Fermi function f ( E ) specifies how many of the existing states at the energy E will be filled with electrons. Whereas (1- f ( E ) ) gives the probability that energy state E will be occupied by a hole. The function f ( E ) specifies, under equilibrium conditions , the probability that an available state at an energy E will be occupied by an electron. It is a probability distribution function . 10

11. Contd … 11

12. 12 Fermi-Dirac distribution: Consider T  0 K For E > E F : For E < E F : E E F 0 1 f ( E )

13. 13 Classical limit For sufficiently large  we will have (  -  )/kT>>1 , and in this limit (5.47) This is just the Boltzmann distribution . The high-energy tail of the Fermi-Dirac distribution is similar to the Boltzmann distribution. The condition for the approximate validity of the Boltzmann distribution for all energies  is that (5.48)

14. Fermi -Dirac distribution function:- 14

15. References Statistical Physics ( 2nd Edition), F. Mandl, Manchester Physics, John Wiley & Sons, 2008, ISBN 9780471915331. H.J.W . Muller-Kirsten, Basics of Statistical Physics, 2nd ed., World Scientific, ISBN: 978-981-4449-53-3. 15

16. Thank You 16