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homework1 - Department of Electrical and Computer...

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1 Department of Electrical and Computer Engineering, Cornell University ECE 407: Physics of Semiconductor and Nanostructures Spring 2009 Homework 1 Due on Jan. 27, 2009 at 5:00 PM Suggested Readings: a) Revise Fourier transforms from your favorite book(s). b) Lecture notes c) Chapter 6 in Kittel (Introduction to Solid State Physics) Problem 1.1: (Drude Model: Motion in Magnetic Fields and the Hall Effect) In the lecture notes, we considered electron motion in electric fields. In this problem we will include the magnetic field as well. Consider the metallic sample shown below. The metal has an electron density equal to n , and electron scattering time τ . A uniform magnetic field in the z-direction, given by z B B o ˆ = v , is applied to the sample. In addition a uniform electric field in the x- direction, given by x E , is also applied by connecting the sample to an external voltage/current source via leads, as shown. In the presence of the fields, the force on the electrons is given by the Lorentz expression: ( ) B v E e F r r r r × + = And the electron average velocity satisfies the equation: ( ) τ v m B v E e dt v d m r r r r r × + = a) Suppose the scattering rate is zero (i.e. ignore scattering). And also assume that x E is zero. Now solve the equation given above and find ( ) t v x and ( ) t v y , the components of the electron average velocity, assuming the initial conditions that ( ) A t v x = = 0 and ( ) 0 0 = = t v y . Hint: It would be easiest to break the vector equation into its x and y components. And then you will get two coupled linear differential equations for the two components of the electron average velocity. d x y z current
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2 b) The components of the electron average displacement, ( ) t u x and ( ) t u y are related to the average velocities by the relations: ( ) ( ) t v dt t u d x x = ( ) ( ) t v dt t u d y y = Once you have obtained ( ) t v x and ( ) t v y , integrate them once more, assuming the initial conditions that ( ) ( ) 0 0 0 = = = = t u t u y x , to get the components of the electron average displacement. You will see that the electron motion is oscillatory with an angular frequency given by m eB o c = ω . This frequency c ω is called the electron cyclotron frequency. You will also find that the electron moves in a circular path. What is the radius of the electron orbit? Looking down on the sample from the top, is the
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