Band Theory1234.pptx

Band Theory

1. BAND THEORY 29-10-2022 1

2. Content Introduction Free Electron T heory Behaviour of an Electron in Periodic Potential Kronig -Penney Potential Band Theory Band Formation in Silicon Classification of Materials based on Energy Band Conclusion References 29-10-2022 2

3. Introduction Why Band Theory ? Conductors/ Semiconductors/ Insulators Physical properties of materials (Electrical resistivity, Optical absorption) Foundation in understanding all solid-state devices (Transistors, Solar cells) 29-10-2022 3

4. Free Electron Theory Classical free electron theory (Macroscopic) Proposed by Paul Drude in 1900 and elaborated (after the discovery of electron by J J Thomson) by Lorentz in 1909. Drude and Lorentz theory. Metals contains free electrons which are responsible for the electrical conductivity in metals and obeys the laws of classical mechanics. Paul Drude Hendrik Lorentz 29-10-2022 4

5. boundaries of the neighboring atoms slightly overlap with each other. Due to this overlapping, the valance electrons of all the atoms are free to move ( Free electron ) within the metal lattice, randomly in all direction with an average speed of the order 10 6 m/s. This is similar to the motion of gas molecules confined in a vessel. T he free electrons are responsible for electrical and thermal conduction in metals, they are also called as conduction electrons. 29-10-2022 5

6. Assumptions All metals contain large number of free electrons. The free electrons are treated as equivalent to gas molecules; the laws of classical kinetic theory of gases can be applied to them. Therefore these electrons have mean free path (λ), mean collision time (T), average speed (v). Since the motion of the electrons is random, the net current is zero in the absence of electric field. But when an electric field is applied, current is produced due to the drift velocity of the electrons. The electric field (or Potential) due to positive ionic cores is considered to be uniform throughout the metal and hence neglected. The force of attraction between the electrons & lattice ions and the force of repulsion between the electrons themselves are considered to be negligible. 29-10-2022 6

7. Success of classical free electron theory It verifies Ohm’s law It derives Wiedemann-Frenz theory It explains Electrical and Thermal conductivity of metals It explains optical properties of metals Drawbacks According to classical free electron theory Cv is independent of temperature, but the experimental value of Cv is directly proportional to temperature E xperimental value of mean free path (λ) is found to be 0.285 nm, which is 10 times less than the value obtained from classical free electron theory The actual relationship between Temperature and thermal conductivity deviates from classical free electron theory A ccording to classical free electron theory; bivalent & trivalent metals should posses much higher electrical conductivity than monovalent metals. This is contrary to the experimental observations that the monovalent element metals such as silver is more conducting than Zinc (bivalent) & aluminum (trivalent) 29-10-2022 7

8. Free Electron Theory Quantum free electron theory (microscopic) Developed by Sommerfeld in 1928. Free electrons move with a constant potential obeys quantum laws E lectron as a quantum particle Retains the vital features of classical free electron theory and included the Pauli Exclusion Principle & Fermi-Dirac statistics Arnold Sommerfeld 29-10-2022 8

9. The free electrons in a metal can have only discrete energy values. Thus the energies are quantized The electrons obey Pauli’s Exclusion Principle, which states that there cannot be more than two electrons in any energy level The distribution of electrons in various energy levels obey the Fermi-Dirac quantum statistics Free electrons have the same potential energy everywhere within the metal, because the potential due to ionic cores is uniform throughout the metal The force of attraction between electrons & lattice ions and the force of repulsion between electrons can be neglected Electrons are treated as wave-like particles 29-10-2022 9

10. Success of Quantum free electron theory The predicted value of specific heat capacity agrees well with the experimental results Thus quantum free electron theory properly explains the dependence of Thermal Conductivity on Temperature Electrical conductivity depends on electron concentration, mean free path and Fermi velocity, Explains why the conductivity of Copper is higher than Aluminum Drawback In real case, electron moves as a periodic potential function. 29-10-2022 10

11. Behaviour of an Electron in Periodic Potential Nearly Free electron theory 29-10-2022 11 1D periodic crystal potential

12. To find out the motion of electron in crystalline solid, Bloch has solved the Schrodinger equation for periodic boundary condition 29-10-2022 12 Felix Bloch

13. 29-10-2022 13 Travelling Part Periodic Function

14. Kronig -Penney Potential Rectangular array of potentials 29-10-2022 14 Ralph Kronig William Penney Braggs Law n λ = 2dSinθ 1D crystal θ=90 , λ =2 π /a n π /a = d

15. 29-10-2022 15 Allowed and forbidden energy band in E-k Diagram No solution for Schrodinger equation

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17. Free electron theory vs Nearly free electron theory 29-10-2022 17 Constant Potential Continuous Parabolic Curve Periodic Potential Discontinuous Curve

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19. 29-10-2022 19 Variation of potential energy of a conduction electron in the field of a ion cores of a linear lattice Distribution of probability density of standing wave + _ Travelling wave

20. Band Theory In 1928-1931, BLOCH/ WILSON/ PEIERLS move on to solids and invent band theory. they gave convincing explanation of metallic insulating and semiconducting behaviour of solids. A solid is made up of enormous number of closely packed atoms. When these atoms are isolated they have discrete set of energy levels as 1s,2s,2p etc. To form a solid many isolated atoms are brought together, then a continuously increasing interactions occurs between them so that the split energy levels form essentially continuous bands of energies. 29-10-2022 20 Rudolf Peierls Alan Wilson

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22. Band Formation in Silicon 29-10-2022 22

23. Classification of Materials based on Energy Band 29-10-2022 23

24. Properties Conductor Semi-conductor Insulator Definition A conductor is a material that allows the flow of charge when applied with a voltage A semiconductor is a material whose conductivity lies between conductor & insulator An insulator is a material that does not allow the flow of current Temperature Dependence The resistance of a conductor increases with an increase in temperature The resistance of a semiconductor decrease with increases in temperature. Thus it acts as an insulator at absolute zero Insulator has very high resistance but it still decreases with temperature Conductivity Conductors have very  high conductivity ( 10 -7  Ʊ/m ), thus they can conduct electrical current easily They have  intermediate conductivity   (10 -7  Ʊ /m  to  10 -13  Ʊ /m ), thus they can acts as insulator & conductor at different conditions They have very  low conductivity   ( 10 -13  Ʊ /m ) ,  thus they do not allow current flow Resistivity Low ( 10 -5   Ω/ m ) Normal ( 10 -5 Ω/ m  to  10 5   Ω/ m ) Very High ( 10 5   Ω/ m ) 29-10-2022 24

25. Conclusion Band theory specifies the types of solid’s, i.e ., we have the insulator, the conductors and the semi conductors . If a material takes more energy to conduct electricity through it then it is a Insulator, it also says that a material which takes very less or no energy (theoretically) is a conductor and material which takes more energy than a conductor and less energy than the Insulator is categorized as semi-conductor . 29-10-2022 25

26. THANK YOU 29-10-2022 26

27. Fermi-Dirac Statistics It deals with the system constituted by identical , non identifiable particles having odd half integer spins. These particles are called Fermions. The angular momentum of fermions are ħ/2, 3ħ/2, 5ħ/2,……. The spins of Fermions are 1/2, 3/2, 5/2,…. These particles obeys the Pauli exclusion principle. The wave function associated with Fermions are anti-symmetric. Pauli's Exclusion Principle No two electrons in the same atom can have identical values for all four of their quantum numbers. 29-10-2022 27

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