homework9

# homework9 - 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 9 Due on April 7, 2009 at 5:00 PM Suggested Readings: a) Lecture notes Problem 9.1: (Phonons bands in graphene) In this problem we are going to make the simplest model for the phonon bands in graphene. The graphene crystal is shown below. It has two atoms, A and B, in a primitive cell. So you expect to get 4 different phonon bands. We will only stick to the nearest-neighbor interactions but will include bond-stretching as well as bond-bending force constants. The lattice along with the nearest neighbor vectors are shown above. For each nearest-neighbor vector there is a vector that is perpendicular to it and given by: j j n z m r r × = ˆ The three vectors 3 2 1 , , m m m r r r can be used to model bond-bending movements. a y x a + = ˆ 2 1 ˆ 2 3 1 v a y x a = ˆ 2 1 ˆ 2 3 2 v a = 2.46 A A B 1 n r 2 n r 3 n r x a n

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