**Unformatted text preview: **F 1 k BT ∂η F ⎪ ∂η F 1/ 2 F
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⎭ Finally, using the differentiation property of FD
integrals, we find I2 = W m*k BT 2π F −1/ 2 (η F ) The trick of replacing − ∂ f0 ∂ E with + ∂ f0 ∂ E F and then moving the derivative outside of the integral is very useful in evaluating FD integrals. ECE- 656 3 Fall 2013 Mark Lundstrom 8/24/2013 ECE 656 Homework 1: Week 1 (continued) 4) It is important to understand when Fermi- Dirac statistics must be used and when non- degenerate (Maxwell- Boltzmann) statistics are good enough. The electron density in 1D is nL = N1DF −1/ 2 (η F ) cm -1 , where N1D is the 1D effective density of states and η F = ( E F − EC ) k BT . In 3D, n = N 3 DF 1/ 2 (η F ) cm -3 . For Maxwell Boltzmann statistics M
nL B = N1D exp (η F ) cm -1 n MB = N 3 D exp (η F ) cm -3 . M
Compute the ratios, nL nL B and n n MB for each of the following cases: a) η F = −10 b) η F = −3 c) η F = 0 d) η F = 3 e) η F = 10 Note that there is a Fermi- Dirac integral calculator available on nanoHUB.org. An iPhone app is also available. Solution: The iPhone app is called: “FD Integral” The nanoHUB.org app is at: nanohub.org/resources/11396 a) eta_F = - 10: M
nL nL B = F −1/ 2 (η F ) exp (η F ) = F −1/ 2 ( −10 ) exp ( −10 ) = 4.54 × 10−5 4.54...

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