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ECE230A_2009Fall_Quiz_2_Sol

# ECE230A_2009Fall_Quiz_2_Sol - ECE230A Fall 2009 Solution...

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ECE230A Fall 2009 Solution for Quiz #2 1. For the 1-D InGaAs/InAlAs superlattice, the effective mass for InGaAs and InAlAs are denoted as m 1 and m 2 , respectively. (a) Find the general solution of the Schordinger equation in regions I and II. (b) Write the boundary conditions that allow us to solve the coefficients and the eigenvalue (energy) E. Note that in this case, the effective mass of electron in InGaAs and InAlAs is different. ANS: (a) The Schordinger equation is: ϕϕ E x V m = + )] ( 2 [ 2 2 h In region I, V(x)=0, the Schordinger equation is: ϕ E dx d m = 2 2 1 2 2 h The general solution is: , where x i x i Be Ae x β + = ) ( 1 h E m 1 2 = I n the region II, V(x)=V 0 , so the Schordinger equation is: E V dx d m = + ] 2 [ 0 2 2 2 2 h The general solution is: , where x x De Ce x α + = ) ( 2 h ) ( 2 0 2 E V m = , E V 0 (b) Boundary conditions: Æ ) 0 ( ) 0 ( 2 1 = Velocity conserves in region I and II: 0 2 2 0 1 1 ) ( 1 ) ( 1 = = = x x dx x d m dx x d m So, 2 1 ) ( ) ( m D C m B A i = 1

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ECE230A Fall 2009 Solution for Quiz #2 2 Bloch Theroem: , where ikc e
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ECE230A_2009Fall_Quiz_2_Sol - ECE230A Fall 2009 Solution...

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