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ECE230A_Homework-2_Solution

# ECE230A_Homework-2_Solution - ECE230A Fall 2009 Solution...

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Unformatted text preview: ECE230A Fall 2009 Solution for Homework #2 1 v 0 1. For the Kronig Penny model, show that (a) at the band edge. (b) h h m mE a a a a a p ka a dk d 2 2 where ) sin( ) cos( ) cos( ) cos( 2 2 2 2 = = = (c) Find the effective mass assuming P=3 /2, ka= , and a=3 /2. Indicate the band this effective mass refers to. (a) From the lecture note, ) cos( ) cos( ) sin( ka a a a P = + ......... , where . , , , 2 2 h h ba mV P b V b V mE = = = and group velocity k E dk dE V g = = h h 1 1 . From , [ ] ) sin( ) sin( ) cos( ) sin( 2 ka a k a a a a a P a a P = + ) sin( ) cos( ) sin( ) sin( 2 a a a a a P a a P ka a k + = .. and, m E mE 2 2 h h = = ... Substituting and into , ) sin( ) ) sin( ) cos( ( ) sin( 1 2 a am a a a a Pm ka a k E V g = = h h If a n k = , ) sin( ) sin( = = n ka , so at the band edge. = g V (b) ECE230A Fall 2009 Solution for Homework #2 From eq + = ) sin( ) cos( ) sin( ) sin( 2 2 2 a a a a a P a a P ka a k dk d = ) sin( ) sin( ) cos( ) cos( ) sin( ) cos( ) sin( ) cos( 2 2 2 2 a a a a a P ka a a a a a a P a a P ka a = + (c) 2 2 2 * dk E d m h = From (a), ) sin( ) ) sin( ) cos( ( ) sin( 2 2 a am a a a a Pm ka a k E dk dE = = h ) sin( ) ) sin( ) cos( ( ) cos( ) sin( ) ) sin( ) cos( ( ) sin( 2 2 2 2 2 2 2 a am a a a a Pm ka a a am a a a a Pm ka a k dk E d = = h h ) cos( ) sin( ) ) sin( ) cos( ( ) sin( ) ) sin( ) cos( ( ) cos( 2 2 2 2 2 2 * ka a a am a a a a Pm a am a a a a...
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ECE230A_Homework-2_Solution - ECE230A Fall 2009 Solution...

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