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**APPLIED PHYSICS CODE : 07A1BS05 I B.TECH CSE, IT, ECE & EEE UNIT-3**

NO. OF SLIDES : 24

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**UNIT INDEX UNIT-3 S.No. Module Lecture No. PPT Slide No. 1 L1-2 4-12 2**

Introduction, Classical free Electron theory of metals. L1-2 4-12 2 Mean Free path, Relaxation time and drift velocity. L3 13-15 3 Quantum free electron theory of metals. L4 16 4. Fermi Level, Fermi Dirac Distribution L5-6 17-18

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**Electron scattering and resistance. L7 19**

5 Electron scattering and resistance. L7 19 6 Classification of materials L8 20-23 7 Effective mass of electron L9 24

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**INTRODUCTION Lecture-1**

The electron theory of solids aims to explain the structures and properties of solids through their electronic structure. The electron theory of solids has been developed in three main stages.

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**(i). The classical free electron theory:**

Drude and Lorentz developed this theory in According to this theory, the metals containing free electrons obey the laws of classical mechanics. (ii). The Quantum free electron theory: Sommerfeld developed this theory during According to this theory, the free electrons obey quantum laws.

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(iii). The Zone theory: Bloch stated this theory in According to this theory, the free electrons move in a periodic field provided by the lattice. This theory is also called “Band theory of solids”.

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**The classical Free Electron Theory of Metals (Drude - Lorentz theory of metals**

Lecture-2 postulates : (a). In an atom electrons revolue around the nucleus and a metal is composed of such atoms. (b). The valence electrons of atoms are free to move about the whole volume of the metals like the molecules of a perfect gas in a container. The collection of valence electrons from all the atoms in a given piece of metal forms electrons gas. It is free to move throughout the volume of the metal

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(c) These free electrons move in random directions and collide with either positive ions fixed to the lattice or other free electrons. All the collisions are elastic i.e., there is no loss of energy. (d). The movements of free electrons obey the laws of the classical kinetic theory of gases. (e). The electron velocities in a metal obey the classical Maxwell – Boltzmann distribution of velocities.

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**(f). The electrons move in a completely uniform potential field due to ions fixed in the lattice.**

(g). When an electric field is applied to the metal, the free electrons are accelerated in the direction opposite to the direction of applied electric field.

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**Success of classical free electron theory:**

(1). It verifies Ohm’s law. (2). It explains the electrical and thermal conductivities of metals. (3). It derives Wiedemann – Franz law. (i.e., the relation between electrical conductivity and thermal conductivity) (4). It explains optical properties of metalsl.

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**Drawbacks of classical free electron theory:**

The phenomena such a photoelectric effect, Compton effect and the black body radiation couldn’t be explained by classical free electron theory. According to the classical free electron theory the value of specific heat of metals is given by 4.5Ru is the Universal gas constant whereas the experimental value is nearly equal to 3Ru. Also according to this theory the value of electronic specific heat is equal to 3/2Ru while the actual value is about 0.01Ru only.

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**3.Electrical conductivity of semiconductor or insulators couldn’t be explained using this model.**

4. Though K/σT is a constant (Wiedemann – Franz Law) according to the Classical free electron theory, it is not a constant at low temperature. 5. Ferromagnetism couldn’t be explained by this theory. The theoretical value of paramagnetic susceptibility is greater than the experimental value.

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**Mean free path Lecture-3**

The average distance traveled by an electron between two successive collisions inside a metal in the presence of applied field is known as mean free path.

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Relaxation Time The time taken by the electron to reach equilibrium position from its disturbed position in the presence of an electric field is called relaxation time.

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Drift velocity In the presence of electric field, in addition to random velocity there is an additional net velocity associated with electrons called drift velocity. Due to drift velocity, the electrons with negative charge move opposie to the field direction.

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**Quantum free electron Theory Lecture-4**

According to quantum theory of free electrons energy of a free electron is given by En = n2h2/8mL2 According to quantum theory of free electrons the electrical conductivity is given by σ = ne2T/m

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Fermi Level Lecture-5 “The highest energy level that can be occupied at 0K” is called Fermi level. At 0K, when the metal is not under the influence of an external field, all the levels above the Fermi level are empty, those lying below Fermi level are completely filled. Fermi energy is the energy state at which the probability of electron occupation is ½ at any temperature above 0k.

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**Fermi-Dirac statistics Lecture-6**

According to Fermi Dirac statistics, the probability of electron occupation an energy level E is given by F(E) = 1/ 1+exp (E-EF/kT)

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**Electrical Resistivity Lecture-7**

The main factors affecting the electrical conductivity of solids are i) temperature and ii) defects (i.e. impurities). According to Matthiesens’s rule, the resistivity of a solid is given by ρpure= ρpure+ ρimpurity where ρpure is temperature dependent resistivity due to thermal vibrations of the lattice and ρimpurity is resistivity due to scattering of electrons by impurity atoms.

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**CLASSIFICATION OF MATERIALS Lecture-8**

Based on ‘band theory’, solids can be classified into three categories, namely, insulators, semiconductors & conductors.

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**Bad conductors of electricity **

INSULATORS Bad conductors of electricity Conduction band is empty and valence band is full, and these band are separated by a large forbidden energy gap. The best example is Diamond with Eg=7ev.

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**SEMI CONDUCTORS Forbidden gap is less**

Conduction band an d valence band are partially filled at room temperature. Conductivity increases with temperature as more and more electrons cross over the small energy gap. Examples Si(1.2ev) & Ge(0.7ev)

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**CONDUCTORS Conduction and valence bands are overlapped**

Abundant free electrons already exist in the conduction band at room temperature hence conductivity is high. The resistively increases with temperature as the mobility of already existing electrons will be reduced due to collisions. Metals are best examples.

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**EFFECTIVE MASS Lecture-9**

Def : When an electron in a periodic potential of lattice is accelerated by an electric field or magnetic field, then the mass of the electron is called effective mass. It is denoted by m* m* = ћ2/(d2E/dk2)

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Lecture 24: Electrical Conductivity

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