Electrical Conductivity AND OHM PDF

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ELECTRICAL CONDUCTIVITY AND OHM’S LAW The Lorentz force is the force that act on an electron in electric field E and magnetic field B and it is given by: F=−e ( E+V × B ) where V is the velocity of electron. The momentum of a free electron is related to the wavevector by mv =hk 2π .So, from newton second law, Lorentz force (Kittel,2005). Where k = λ 1 dk dv become: F=m =h =−e E+ V × B . dt dt c

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On the top of Fig. 1 we have a picture of a Fermi sphere of radius k F. The typical electron has a very large velocity on the order of the Fermi velocity v F, but the average of all of the (vector) velocities is zero. When an electric field is applied in the bottom of Fig. 1 every electron in the system accelerates together in the ˆx direction, and the centre of the Fermi sea shifts. (The electric field in the figure is in the −ˆx direction, so that the force is in the +ˆx direction since the charge on the −eEt electron is −e) (H.Simon, 2013). The shift of Fermi sphere is given by δ k= h (1)

Figure 1: Fermi sphere ((H. Simon, 2013, p.34).

Because of collisions of elecrons with impurities, lattice imperfections and phonons, the shifted sphere may be maintained in steady state in E. In Fig.2 we can see that The electrons periodically collide with the defects mentioned above and lose their kinetic energy.As a result the electron velocity varies with time , the average time between collisions is 2 τ , where τ is the realtion time.The electron thus acquire an average drift velocity vd which is diractly propotional to E.( Amalraj,2012). From ohm,s law V=IR (2). We know that R is propotional to the Figure2: electron drift velocity vs. Time. (Amalraj,2012) length L and cross setional area A of conductor. So, R= ρ(L/ A) , where ρ is a constant called the electrial resistivity. So, equation (2) become: V=I ρ(L/ A) . now current density J is the current per unit area I/A. So, V/L=J ρ and V/L is the E (Units of V/m). So, E= J ρ and J=E/ ρ . The electrial conductivity σ

is the reciprocal of ρ . So, J= σ E. By using from (1) The −eE τ . If in a constant electric field E there are n incremental velocity is v= δ k/m= m electrons of charge q=-e per unit volume, the electric current density is j =nqv =ne2 τ E/m . This is Ohm’s law .(Kittel,2005). So, σ = ne2 τ /m, We expect the charge transported to be propotional to the charge density ne; the factor e/m is because of the acceleration in a given electric field is proportional to e and inversely proportional to the mass m .by substituting, ρ=m/n e 2 τ .The net resistivity1 is given by 1 Often ρ L is independent of the number of defects when their concentration i ρt is independent of temperature. (Kittel,2005). Figure3: graph of Electrical resistivity vs. Temperature (K) (Amalraj,2012)

ρ= ρL + ρt where ρ L is the resistivity caused by the thermal phonons, and ρt is the resistivity caused by scattering of the electron waves by static defects that disturb the periodicity of the lattice.( Kittel,2005). Umklapp Scattering When two phonons collide, the resulting phonon has the vector sum of their momenta. However, If the momentum is too great (outside the first Brillouin zone) then the resulting phonon moves in almost the opposite direction! This is the Umklapp scattering, and is dominant at higher temperatures- acting to reduce thermal conductivity as the temperature increases.[ CITATION Non18 \l 1033 ]. Fig.4 illustrates the difference between The normal scattering and Umklapp scattering.

Fig.5 the difference between the normal scattering and Umklapp scattering.

[ CITATION Non18 \l

1033 ]

References: - Amalraj. (2012). Electrical Conductivity and Ohm's Law-Mechanics of Materials-Handout, Exercises for Mechanics of Materials. Aligarh Muslim University. Retrieved from docsity: https://www.docsity.com/en/electrical-conductivity-and-ohm-s-lawmechanics-of-materials-handout/79815/ - H.Simon. (2013). The Oxford Solid State Basics. New york: Oxford University Press. - kittell. (2005). Introduction to solid state physics. John Wiley & Sons, Inc. - Non-metals: thermal phonons. (2018). Retrieved from University of Cambridge: https://www.doitpoms.ac.uk/tlplib/thermal_electrical/nonmetal_thermal.php

Written by: Nessreen al-qarni. ID:436201330...