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Week1HWSolutions - Mark Lundstrom SOLUTIONS ECE 656...

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Mark Lundstrom 8/24/2013 ECE-­‐656 Fall 2013 1 SOLUTIONS: ECE 656 Homework 1: Week 1 Mark Lundstrom Purdue University 1) Working out Fermi-­‐Dirac integrals just takes some practice. For practice, work out the integral I 1 = M E ( ) f 0 E ( ) dE −∞ where f 0 E ( ) = 1 1 + e E E F ( ) k B T and M E ( ) = W 2 m * E E C ( ) π H E E C ( ) where H E E C ( ) is the unit step function. Solution: I 1 = W 2 m * E E C ( ) π 1 1 + e E E F ( ) k B T dE E C Note that the unit step function in M E ( ) makes the lower limit of the integral E C . I 1 = W 2 m * π E E C ( ) 1/2 1 + e E E F ( ) k B T dE E C Now make the change in variables, η = E E C ( ) k B T and η F = E F E C ( ) k B T to find: I 1 = W 2 m * π k B T η ( ) 1/2 1 + e η η F k B T ( ) d η 0
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Mark Lundstrom 8/24/2013 ECE-­‐656 Fall 2013 2 ECE 656 Homework 1: Week 1 (continued) I 1 = W 2 m * π k B T ( ) 3/2 η 1/2 1 + e η η F d η 0 Now we can recognize the FD integral of order ½: η 1/2 1 + e η η F d η 0 = π 2 F 1/2 η F ( ) so the result becomes I 1 = W m * 2 π k B T ( ) 3/2 F 1/2 η F ( ) , which is the final answer. 2) For more practice, work out the integral in 1) assuming non-­‐degenerate carrier statistics. Solution: We could approximate the Fermi function as f 0 E ( ) = 1 1 + e E E F ( ) k B T e E F E ( ) k B T and then work out the integral I 1 = W 2 m * E E C ( ) π 1 1 + e E E F ( ) k B T dE E C W 2 m * E E C ( ) π e E F E ( ) k B T dE E C
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