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homework5sol - ECE407 Homework 4 Solutions(By Farhan Rana...

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1 ECE407 Homework 4 Solutions (By Farhan Rana) Problem 5.1 (1D lattice energy bands outside the FBZ) a) Lesson: The lesson is that if one chooses a value of the wavevector outside the FBZ for numerical solution then one does not obtain any new energy eigenvalues or wavefunctions that are not already in the FBZ. The reason for that, as discussed in the class, is that a solution found for a wavevector in the FBZ already contains superposition of plane waves, different from the starting wavevector value, by reciprocal lattice vectors. If one obtains a solution for a wavevector value, say k , outside the FBZ then it is identical to the solution obtained for a wavevector value, say k’ , that lies in the FBZ and is related to k via a relation of the form: k’=k+G , where G is that (unique) reciprocal lattice vector for which k’ is in the FBZ. This was also the motivation for zone-folding in which free-electron bands outside the FBZ were translated by appropriate reciprocal lattice vectors and placed in the FBZ. Also notice that if one were to solve for and plot energy bands outside the FBZ, as we did in this problem, then the bands will be periodic in k-space with the periodicity of the reciprocal lattice, in the sense that: ( ) ( ) k E G k E r r r = + where G r is any reciprocal lattice vector. This property, of course, holds in all dimensions, and even for the tight binding solutions (check!). Problem 5.2 (Ethene (or Ethylene) molecule: LCAO) a) As discussed in the lectures, the pz-orbitals (the ones that stick out of the plane of the molecule) are anti-symmetric with respect to reflections about the x-y plane. On the other hand, px-orbitals, py-orbitals, and s-orbitals are all symmetric with respect to reflections about the x-y plane. Consequently, the energy matrix elements of pz-orbitals with all other orbitals will be zero.
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