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EMS172-2009-Homework2_Solutions

# EMS172-2009-Homework2_Solutions - UNIVERSITY OF CALIFORNIA...

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EMS 172 Profs. Y. Takamura and A. Moulé, 2009 UNIVERSITY OF CALIFORNIA, DAVIS Department of Chemical Engineering and Materials Science EMS 172: Electronic, Magnetic, and Optical Properties of Materials Homework 2 - Solutions Due Date: Tuesday, Oct. 13 th , 2009, in-class Note: You must show all the steps you used to obtain your answer in order to receive full credit. 1. Fermi-Dirac Distribution [14 pts] (a) At what temperature can we expect a 10% probability that electrons in a given material have an energy which is 2% above the Fermi energy? (Use E F =5.0 eV) When dealing with probability, we are referring to the Fermi-Dirac distribution. E=E F + 0.02*E F , E-E F =0.02*E F () () 1 / 10 62 . 8 0 . 5 02 . 0 exp 1 10 . 0 1 exp 1 5 + × × × = = + = T K eV eV T k E E E F B F () () () ) 9 ln( / 10 62 . 8 0 . 5 02 . 0 9 / 10 62 . 8 0 . 5 02 . 0 exp 10 . 0 1 1 / 10 62 . 8 0 . 5 02 . 0 exp 5 5 5 = × × × = × × × = + × × × T K eV eV T K eV eV T K eV eV Solving for T = 528 K. [1 pt for eqn, 1 pt for value, 1 pt for trying] (b) We stated that the Fermi-Dirac distribution function can be approximated by classical Boltzmann statistics if the exponential factor in the Fermi-Dirac distribution function is significantly larger than one. Using the relationship E-E F =nK b T for various values of n where n is an integer, calculate the value of the Fermi-Dirac distribution and the Boltzmann distribution. State at which value of n , T k E E B F exp can be considered to be significantly larger than one. Hint : Create a table with the calculated error between the full F(E) expression and the Boltzmann distribution for different values of

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EMS172-2009-Homework2_Solutions - UNIVERSITY OF CALIFORNIA...

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