Lecture_30

Lecture_30 - PHYS 342 Fall 2011 Lecture 30: Free electron...

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PHYS 342 Fall 2011 Lecture 30: Free electron Density of States and the nearly free electron model Ron Reifenberger Birck Nanotechnology Center urdue University Purdue University Lecture 30 1
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Recap: E(k) for free electrons /3 2/3 2 22 2 3 F ee F E m k Vm      free- electron (k) E E(k) empty E F k x (or k y or k z ) 2 π 2 π - filled k - 2 a a F k F
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The filled states in 3-D k z Spherical Fermi Surface E  28 3 2.5 10 Sodium m V     k F illed unfilled E F k y k y filled filled k x k x 1/3 2/3 22 3  2 1/3 8 3 3 32 . 5 1 0 F V k m 2 34 8 2 1.05 10 2 . 5 1 0 F e mV E   3   91 9.04 10 m 31 19 29 .11 10 4.95 10 3.09 Je V  
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How many electrons with E < E F ? 2 number of electrons number occupied states  Counting states explicitly (previous lecture): 3 3/2 3 max max 2 48 2 3 2 2 3 e fp E m L M     /2 1 2 2 2 e m L where from previous lecture E M  3 3 4 3 2 8 ) 2 e F m L no units k A simpler way: each state requires (2 /L) 3 volume in reciprocal space 3 2 3 () 32 2 F E L 4 2 2 1 3 e F m nE u n i t o L sf m 
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Example : How many free electrons in a cm 3 iece of Na? 3 4 1 cm piece of Na?     3 33 3 3 32 3 81 38 3 2 2 F F F k L k L k L    9 3 3 1 1 29.6 9.04 0.01 10 mm 22 3 2. 9 10 41 free e in m c  5
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For a quantum electrons, it is useful to efine their position the location of define their position, the location of their charge, their location in energy • position: Ψ (r) Ψ * (r)dV KEY IDEA •charge:±±± ±-e Ψ (r) Ψ *(r)dV • energy: density of states
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3 3/2 3 4 The Free Electron Density of States    3 2 3 2 8 32 2 2 e k m L E L     3 1/2 2 /2 2 4 2 e m dL E dE 3 2 2 () 4 2 e E m L dg E d E g E  g(E) E 3 3 1/2 23 2 3 1/2 22 1/2 4 4 28 2 ee F Sodium e F F F mm LL g L EE m E E E   33 1 1/2 19 34 2 85 7 1 0 2 (0.01 ) 2 9.109 10 3.09 1.602 10 19.72 (1.054 10 ) 5.07 10 2.10 10 7.04 10 m   7 E Note: In some books, g(E) is called D(E) or (E) 7.5 40 3 22 3 10 ( 1 / ) 1.2 10 ( 1 / ) F F states at E in cm of Na J states at E in cm of Na eV
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Probability that a state is occupied () P(E) BB iB EkT nE gEe E e 
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