homework7sol

# homework7sol - ECE 407 Homework 7 Solutions(By Farhan Rana...

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1 ECE 407: Homework 7 Solutions (By Farhan Rana) Problem 7.1 a) The answer follows from elementary vector calculus result that that the gradient of any function is perpendicular to the surface of constant value of the function. In the present case, the velocity is related to the gradient of the energy function, () () k E k v k c r h r r r = 1 And therefore the velocity must be perpendicular to the surfaces of constant energy. Problem 7.2 a) First note that: B k B k r r r r . || = Now take the dot product on both sides of the crystal momentum equation with the magnetic field to get: () [] 0 0 . 0 . . . || = = = × = dt k d dt k B d dt k d B B k v B e dt k d B c r r r r h r r r r r r h r b) ( ) ( ) () () ( ) () () () () [ ] [ ] 0 . . 1 = × = = B t k v e t k v dt t k d t k E dt t k dE c c c k c r r r r r r h r h r r c) The complete argument follows in two steps: i) Since the energy of the electron remains unchanged and equal to its initial value o E , the motion in k-space of the electron must be confined to a constant energy surface corresponding to the initial energy o E of the electron. ii) Since the component of its crystal momentum parallel to the magnetic field also remains unchanged and equal to its initial value || k r , the motion in k-space of the electron must also be confined to the plane in k-space on which the components of all crystal momenta in the direction of the magnetic field is || k r (convince yourself that this later condition defines a plane in k-space that is perpendicular to the magnetic field).

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## This note was uploaded on 04/04/2009 for the course ECE 4070 taught by Professor Rana during the Spring '08 term at Cornell.

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homework7sol - ECE 407 Homework 7 Solutions(By Farhan Rana...

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