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Equation of motion of an electron with an applied electric and magnetic field. dv me eE ev B dt This is just Newton’s law for particles of mass me and charge (-e). The use of the classical equation of motion of a particle to describe the behaviour of electrons in plane wave states, which extend throughout the crystal. A particle-like entity can be obtained by superposing the plane wave states to form a wavepacket. The velocity of the wavepacket is the group velocity of the waves. Thus 2 k2 E 2me d 1 dE k p v dk d k me me p k So one can use equation of mdv/dt dv v me eE ev B dt (*) = mean free time between collisions. An electron loses all its energy in time In the absence of a magnetic field, the applied E results a constant acceleration but this will not cause a continuous increase in current. Since electrons suffer collisions with phonons electrons v The additional term me cause the velocity v to decay exponentially with a time constant the applied E is removed. when The Electrical Conductivty In the presence of DC field only, eq.(*) has the steady state solution e v E me e e me Mobility for electron a constant of proportionality (mobility) Mobility determines how fast the charge carriers move with an E. Electrical current density, J J n ( e )v e v E me N n V Where n is the electron density and v is drift velocity. Hence ne J E me 2 Ohm’s law J E ne me 2 Electrical conductivity Electrical Resistivity and Resistance 1 R L A Collisions In a perfect crystal; the collisions of electrons are with thermally excited lattice vibrations (scattering of an electron by a phonon). This electron-phonon scattering gives a temperature dependent ph (T ) collision time which tends to infinity as T 0. In real metal, the electrons also collide with impurity atoms, vacancies and other imperfections, this result in a finite scattering time 0 even at T=0. The total scattering rate for a slightly imperfect crystal at finite temperature; 1 1 1 ph (T ) 0 Due to phonon Due to imperfections So the total resistivity ρ, me me me I (T ) 0 2 2 2 ne ne ph (T ) ne 0 Ideal resistivity Residual resistivity This is known as Mattheisen’s rule and illustrated in following figure for sodium specimen of different purity. Residual resistance ratio Residual resistance ratio = room temp. resistivity/ residual resistivity and it can be as high as 106 for highly purified single crystals. impure pure Temperature Collision time ( RT )sodium 2.0 x107 ( m)1 n 2.7 x10 m 28 pureNa 5.3x1010 ( m) 1 can be found by taking me m Taking residual 3 vF 1.1x106 m / s ; and m ne2 2.6 x1014 s at RT 7.0 x1011 s at T=0 l vF l ( RT ) 29nm l (T 0) 77 m These mean free paths are much longer than the interatomic distances, confirming that the free electrons do not collide with the atoms themselves. Thermal conductivity, K Due to the heat tranport by the conduction electrons Kmetals >> Knon metals Electrons coming from a hotter region of the metal carry more thermal energy than those from a cooler region, resulting in a net flow of heat. The thermal conductivity 1 K CV vF l 3 where CV is the specific heat per unit volume vF is the mean speed of electrons responsible for thermal conductivity since only electron states within about of F change their kBT occupation as the temperature varies. l is the mean free path; l vF and Fermi energy F 12 mevF2 2 2 2 1 1 N T 2 nk BT K CV vF2 kB ( ) F 3 3 2 V TF me 3me where Cv 2 T Nk B 2 TF Wiedemann-Franz law ne me 2 K 2 nk B2T 3me The ratio of the electrical and thermal conductivities is independent of the electron gas parameters; K 2 kB 8 2 2.45 x 10 W K T 3 e 2 Lorentz number K L 2.23x108W K 2 T For copper at 0 C