Lecture_30

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

Info iconThis preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
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  
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 
Background image of page 4
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
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 6
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
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Probability that a state is occupied () P(E) BB iB EkT nE gEe E e 
Background image of page 8
Image of page 9