HW5soln

HW5soln - ECE215B/Materials206B Fundamentals of Solids for...

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ECE215B/Materials206B Fundamentals of Solids for Electronics E.R. Brown/Spring 2008 1 Homework #5(a) Solutions Problem 1. Cyclotron resonance in spheroidal band We start with the semiclassical dynamic equation, dk e U B dt k = × G G G = = . From the given spheroidal form for U(k), 2 22 ,, y xz xT yT zL k Uk U kmkmkm ∂∂∂ === = == . So if we confine B to the x, y plane ˆˆ x y BB B xy =+ G the semi-classical equation becomes zy x L ek B dk dt m = y zx L dk ek B dt m = z x yy x kB kB dk e dt mm TT  =−   (1) Note: 2 ˆ ˆ ˆ 1 ˆ 0 yx TTL k k k y x z mmm LL T T y x z kB U Bx y z k m m ×= = + + G G = We seek oscillatory solutions in k space (would be oscillatory in real space too, but do not need that for the present problem). Hence, ;;; 00 0 0 jt y x yz dk dk dk kk e j k j k j k dt dt dt ω ωωω = = GG and (1) becomes 0 y x z L eB jk k m 0 x L eB j m + 000 0 y x zxy eB eB k k + We can write this in elegant matrix form as 0 0 0 y x y x L x oy L z eB j m k eB m k eB eB j ± ±− From linear algebra we know this matrix equation has non-trivial solutions for the column vector k G if and only if the matrix is singular, i.e., the determinant of the matrix vanishes. So [] 2 2 0 y y
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This note was uploaded on 01/17/2012 for the course ECE 215B taught by Professor Brown during the Spring '08 term at UCSB.

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HW5soln - ECE215B/Materials206B Fundamentals of Solids for...

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