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Free Electron theory :Quantum Mechanical Treatment Basic Assumption : A metal crystal consists of positive ions whose valence electrons are free to move between the ions as if they constituted an electron gas Density of Available states D(E) and Density of filled States N(E) D(E) : total number of available electron states per unit energy range at E N(E): Number of range at E states that are filled with electrons per unit energy Density of Available states D(E) and Density of filled States N(E) Parameter of free electron gas at absolute zero Parameter of free electron gas at absolute zero Parameter of free electron gas at absolute zero AVERAGE VELOCITY AND VELOCITY OF ELECTRON AT FERMI LEVEL Average Velocity Velocity at Fermi Level 3 1 E0 EF mv 02 5 2 2 3 2 N EF 2m V 2 E0 So v 0 m 1 2 1 2 mv F 2 1 2 v F (3 n) 3 m 2/3 N n V Fermi Energy at temperature T is given by 2 kT 2 EF EF 0 1 ( ) 12 EFo So fermi energy is not constant but decreases as temperature rises. Above expression is applicable only when kT<<EF . Electrical Conductivity On the basis of free electron model, electrons at 0 K fill a sphere of radius kFo in the wave number space kFo is known as Fermi wave vector. Fermi surface is the surface of maximum energy. On application of field E, each electron acquires a certain additional velocity dv. This is equivalent to the displacement of the Fermi Sphere by dk. Accordinglly, F eE dk m d dt dt Ee d dt m This additional velocity is acquired in the characteristic time e dv E m Current density J nedv ne 2 E m ne 2 But J E E m Hence electrical conductivi ty ne 2 m Mean free path Fo vFo , where vFo is the velocity at fermi surface, vFo (3 2 n)1/ 3 m Electrical Conductivity Hence electrical conductivi ty ne 2 m Mean free path Fo vFo , where vFo is the velocity at fermi surface, ne 2 Fo mvFo Since only quantity on the RHS which depends upon temperature is F F 1 T Thus 1 T Electrical Conductivity The kinetic energy of the electron is 1 3 mc 2 kT 2 2 3kT and c m At room temperature , the drift velocity imparted to the electrons by the applied electric field is very much smaller than the average thermal velocity. The average distance travelled by an electron between two successive collisions in the presence of applied field is known as free mean path . The time taken by an electron between two successive collisions is known as mean collision time of the electron c. Hence the time taken by the electrons in traversing the distance will be decided mainly by rms velocity. Now c c m 3kT Mobility • Mobility of the electron μ is defined as the steady state drift velocity<vd> per unit electric field. d e E m ne 2 e ne. m m ne 1 m m 2 ne ne Where (resistivit y ) ne ne 2 m m •The electrical conductivity σ depends on two factors ,the charge density n and their mobility. These two quantities depend on temperature. •In metals n is constant and μ decreases slightly with temperature and hence with increase of temperature ,the conductivity decreases. •In semiconductors the exponential increase of n with temperature is responsible for increase of conductivity with temperature. • In insulator n remains constant and above certain temperature μ increase exponentially resulting in dielectric breakdown. For the thermal conductivity of a metal in terms of temperature. Relaxation Time Relaxation Time can be defined as the time taken for the drift velocity to decay to 1/e of its initial value Let assume that the applied field is cut off after the drift velocity of the electron has reached its steady value. Drift velocity after this instant is governed by d d m dt d d dt m d d (t ) d (0) exp( t / ) at t (0) d (t ) d e Differenti ating equation d d (t ) d dt Vd(0)is the steady state drift velocity Collision time vd 0 the change in average velocity on collision is opposite. Hence the rate of change of average velocity is given by Q.1/Tut 9 The relaxation time and root mean square velocity of the electron at room temperature are 2.5x10-14 s and 1x105 m/s. Calculate the value of mean free path of the electron. Q.2/Tut 9 The resistivity of a metal at temperature 20°C is 1.69 x 10-8 ohm m and concentration of the free electrons in metal, ne = 8.5x1028/m3. Calculate root mean square velocity (c), relaxation time (τ), mean free path (λ), mobility of electrons (μ) and value of electrical conductivity (σ) on the basis of classical free electron theory. Semiconductor For intrinsic semiconductors like silicon and germanium, the Fermi level is essentially halfway between the valence and conduction bands. Although no conduction occurs at 0 K, at higher temperatures a finite number of electrons can reach the conduction band and provide some current.