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ELECTRICAL PROPERTIES Department of Physics K L University 1 Session 1 15-Sep-15 2 Contents  Free Electron Model  Bloch theorem, Kronig- Penny model, Brillouin Zones  Energy band theory, Band structures in Conductors, Semi conductors and Insulators  Electrical properties of conductors- Ohms, Mathiessen rule, conductivity, Mobility  Electrical properties of Semi conductors, Factors effecting the carrier concentration, Conductivity and Mobility of charge carriers  Electric properties of Insulator-Dielectrics- Types of Dielectrics, Dielectric Constant, Polarization, Types of Polarizations, Frequency Dependence of Polarization, Ferro, Piezo Electrics. Free Electron Model  To explain the structure and properties of solid  To explain bondings in solids, behavior of conductors, semiconductors and insulators, electrical and thermal conductivities of solids, magnetism, elasticity, through their electronic structure  Development of Free Electron Theory: 1. The classical free electron theory (Drude and Lorentz Model) 2. The quantum free electron theory (Sommerfeld Model) 3. Band Theory (Brillouin Zone Theory) The Classical Free Electron Theory  Postulates: 1. The valance electrons (electron gas) are free to move about the whole volume of the metal like the molecules of a perfect gas in a container. 2. Electrons suffer collisions among themselves, with ion core and with boundaries of the specimen. 3. All these collisions are ELASTIC, i.e., there is no loss of energy. Electrons obey classical kinetic theory of gases. Postulates (Cont.) 4. Velocities of electrons in metal obey classical Maxwell-Boltzmann distribution of velocities. Root mean square velocity of electron is Vrms = (3KBT/m)1/2 Where, KB is Boltzmann constant, T is absolute temperature and m is mass of the electron. 5. As Vrms is RANDOM, it does not contribute to any current. Only directed motion of electrons, imparted by an external force causes current. The Classical Free Electron Theory (Cont.)  Neglecting electron–electron interaction between collisions is “independent electron approximation.”  In the absence of external fields, random motion of electrons is observed.  In the presence of external fields, electrons acquire some amount of energy from the field and are directed to move towards higher potential. As a result, the electrons acquire a constant velocity known as DRIFT VELOCITY (Vd). Trajectory of a conduction electron The Classical Free Electron Theory (Cont.)  Time taken for the drift velocity to decay (1/e) of its initial value is known as RELAXATION TIME (τ).  The mean time between successive collisions is called MEAN COLLISION TIME (τc).  The average distance travelled by an electron between any two successive collisions in the presence of external field is called MEAN FREE PATH (λ).  Mathematically, mean free path λ = Vrms . τc. Success of Classical Free Electron Theory 1. Explains the concept of resistance in metals 2. Verifies Ohm‟s law 3. Explains high electrical and thermal conductivity of metals 4. Establishes relation between electrical and thermal conductivities of metals (Wiedemann – Franz law) k/σ = L.T; wher, k is thermal conductivity, σ is electrical conductivity, L is Lorentz number, T is temperature (in K) 5. Explains optical properties of metals Drawbacks of Classical Free Electron Theory  Classical theory failed to explain: 1. Many phenomenon observed in materials such as photoelectric effect, Compton effect and black body radiation, etc. 2. Electrical conductivity of semiconductors and insulators. 3. Specific heat capacity of solids. 4. The concept of ferromagnetism. 5. The theoretical value of paramagnetic susceptibility is greater than the experimental value. Quantum Free Electron Theory  Postulates:  Sommerfeld retained the concept of free electrons moving in a uniform potential within the metal.  Treated electrons obeying laws of quantum mechanics instead of those of classical mechanics.  Electron within the boundaries of the metal is considered as electron trapped in a potential well.  Energy levels of electrons are explained by distribution functions besides the laws of quantum mechanics.  Fermi-Dirac statistics was used instead of Maxwell-Boltzmann statistics. Schrödinger Time Dependent Wave Equation The Schrödinger time dependent wave equation is 2    2  V  i 2m t  2      2  V   i   2m

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