HW_2solution_2008

HW_2solution_2008 - ELEN 360/ENGR 262 Homework #2 Total 10...

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ELEN 360/ENGR 262 Homework #2 Total 10 points M.L.Shek, 2008/01/28 Solutions handed out in class2008/02/07 Since I am giving out this late this time, if you cannot hand in your homework on Jan 31, then you can email it to me before 5 p.m. of Feb 3 (Sunday). 1 . (3 points) Refer to Lecture 2, p.6 and 7. Derive (fill in the steps in my derivation) the electronic density of states of a sub-band for a 2D free electron gas and a sub-band of a 1D free electron gas. Please include the numerical constants which I left out. ψ k ( r ) = exp ( i k.r ) where the components of k satisfy k x = 2n π /L, n being a negative or positive integer. Each allowed k x value is separated from the next by 2 π /L. 2D : So each point ( k x , k y ) takes up a volume element (2 π /L) 2 in k -space. The number of states N' inside a circle radius k is N' = 2 ( π k 2 )/(2 π /L) 2 Use E k =( ħ 2 k 2 )/(2m*) to express N'/L 2 as a function of energy The number of states per unit area n'(E)= m*E / πħ 2 Density of states d n'(E)/dE = m* / πħ 2 1D: The number of states N' within +- k is N' = 2 ( 2 k )/(2 π /L) n'( k ) = 2 k /
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HW_2solution_2008 - ELEN 360/ENGR 262 Homework #2 Total 10...

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