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homework4sol - ECE407 Homework 4 Solutions(By Farhan Rana...

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1 ECE407 Homework 4 Solutions (By Farhan Rana) Problem 4.1 (1D lattice) a) See plot below. b) The size of the bandgap that opens at ka= ±π is approximately 0.4 eV which equals 2V 1 as predicted by the nearly free electron model. c) See the matrix given below for this part. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = + + + + M M M M O O 2 1 1 2 2 1 1 2 2 1 2 1 1 1 2 2 1 1 2 2 1 1 1 2 1 2 0 0 G c G c c G c G c k E G c G c c G c G c G k e V V V G k e V V V V k e V V V V G k e V V V G k e d) See the plot attached below. a) V 1 =0.2 eV and V 2 = 0.0 eV
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2 d) The bandgaps now open at ka=0 between the second and the third energy band – of magnitude ~0.8 eV – and at ka= ±π between the first and the second energy band – of magnitude ~0.4 eV. The values of the bandgaps are in decent agreement with the nearly free electron model. Problem 4.2 (Energy bands for a hexagonal lattice in 2D) a) The K-point ( 0 , a 3 4 π ) is strongly coupled to two other equivalent K-points in the FBZ, as shown in the picture above (all three K-points have been labeled with subscripts 1,2, and 3 for convenience). The
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