Optical and Transport Properties of Metals in Solid State Physics

phy 752 solid state physics 11 11 50 am mwf olin n.w
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Explore the macroscopic theory and Drude model in understanding the optical and transport properties of metals. Delve into Maxwells Equations, complex refractive index, and the reflection of electromagnetic waves at planar interfaces. Analyze the relationships and analytic properties of dielectric functions in this lecture series.

  • Physics
  • Solid State
  • Metals
  • Drude Model
  • Maxwells Equations
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  1. PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan for Lecture 23: Optical and transport properties of metals (Chap. 11 in GGGPP) Macroscopic theory Drude model Note: Debye-Waller discussion postponed to consideration of Chapter 9. 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 1

  2. 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 2

  3. Maxwells Equations (cgs Gaussian units) Assume pure time-harmonic freq and fixed geo u ency m etry : = = J 0 0 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 3

  4. Assume linear response of the electric field to produce the conductivity in terms of conductivity: For our geometry: with: 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 4

  5. Taking Fourier transform in space and time: For spatially uniform conductivity response, the q-dependence is trivial: N complex refractive index For = with q Nc 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 5

  6. Writing complex refractive index in terms of real functions N( )=n( )+ik( ) ( ) ( ) ( ) ( ) / / / / i c Nz i c nz c kz i c n z = = / z ( ) E z E e E e e E e e 0 0 0 c = skin depth: k In terms of dielectric func ( ) ( ) z E z D = tion: ( ) z E ( ) = + i 1 2 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 6

  7. In terms of complex conductivity: Reflection of electromagnetic wave at a planar interface at normal interface N( ) 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 7

  8. Reflectivity: Summary of relationships 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 8

  9. Analytic properties of dielectric function Here represents a small infinitesimal imaginary contribution 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 9

  10. Evaluation using Cauchy integral theorem 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 10

  11. In terms of principle parts integral over negative and positive frequencies 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 11

  12. In terms of positive frequencies only 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 12

  13. Analysis of reflectivity data 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 13

  14. Some more detailed notes: Analytic properties of the dielectric function (in the Drude model or from first principles -- Kramers-Kronig transform formula integral s Cauchy' Consider analytic an for function ( : ) f z 1 f(z) ( ) includes = = ( ) 0 dz f z f dz 2 z- i Im(z) Re(z) 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 14

  15. Kramers-Kronig transform -- continued Im(z) Re(z) =0 1 1 f(z) f(z ) f(z) ( ) = = + R f dz dz dz R 2 2 z- z - z- i i R includes rest 1 1 1 f(z ) f(z ) ( ) = = + ( ) R R f dz P dz f R R 2 2 2 z - z - i i R R 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 15

  16. Kramers-Kronig transform -- continued 1 1 f(z ) ( ) = + ( ) R f P dz f R 2 2 z - i R ( ) z R ( ) z R ( ) z R = + Suppose : f f if R I ( ) z R ( ) z R + - R 1 1 f if ( ( ) ( ) ) + = R I f if P dz R I R 2 2 z i 1 f (z ) ( ) = I R f P dz R R z - R 1 f (z ) ( ) = R R f P dz I R z - R 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 16

  17. u a us b Some practical considerat ions Principal integratio parts n : u b b lim s a a + s = + ( ) ( ) ( ) P du g u du g u du g u 0 u Example : u b b lim 1 1 1 s a a + s = + P du du du 0 u-u u-u u-u s s s u lim b u b u = + = ln ln ln s s 0 u a u a s s 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 17

  18. Kramers-Kronig transform -- continued 1 f (z ) ( ) = I R f P dz R R z - R 1 f (z ) ( ) = R R f P dz I R z - R Kramers This - Kronig transform is useful for the dielectric function ( ) ( ) z R when 1 f 0 ( ) ( ) z Must show that : 1. analytic is for 0 f z z I To be justified from Drude model -- 2. vanishes for f z 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 18

  19. Kramers-Kronig transform for dielectric function: ( ) 0 ( ) 0 1 ' 1 = d 1 ' R I P - ' ( ) with ( ) ; 1 ' 1 = d ' 1 I R P - = ' 0 0 ( ) ( ) ( ) ( ) = R R I I ( ) ( ) ' 2 ' P d = R I 1 ' 2 2 ' - 0 0 0 ( ) ( ) ' 2 1 - = I R ' 1 P d 2 2 ' 0 0 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 19

  20. Paul Karl Ludwig Drude 1863-1906 http://photos.aip.org/history/Thumbnails/drude_paul_a1.jpg 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 20

  21. Drude model: Vibrations of charged particles near equilibrium: = 2 0 i t r E r r m q e m m 0 E 1 q = i t r r r 0 For , e 0 0 2 0 2 m i Induced dipole : 2 E 1 q = = i t p r 0 q e 2 0 2 m i Displaceme nt field : r = = + r D E E P 0 ( ) i = 3 P p r i i http://img.tfd.com/ggse/d6/gsed_0001_0012_0_img2972.png 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 21

  22. Drude model: Vibration of particle of charge q and mass m near equilibrium: r http://img.tfd.com/ggse/d6/gsed_0001_0012_0_img2972.png i = m 2 0 t r E r r m q e m 0 Note that: > 0 represents dissipation of energy. 0 represents the natural frequency of the vibration; 0=0 would represent a free (unbound) particle 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 22

  23. Drude model: Vibration of particle of charge q and mass m near equilibrium: r http://img.tfd.com/ggse/d6/gsed_0001_0012_0_img2972.png i = m 2 0 t r E r r m q e m 0 E 1 q = i t r r r 0 For , e 0 0 2 0 2 m i Induced dipole : 2 E 1 q = = i t p r 0 q e 2 0 2 m i 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 23

  24. Drude model: Vibration of particle of charge q and mass m near equilibrium: r http://img.tfd.com/ggse/d6/gsed_0001_0012_0_img2972.png i = m 2 0 t r E r r m q e m 0 Displaceme D nt = field = : E = + r E P 0 ( ) 3 P p r p N f i i i i i i number dipole/vol ume N fraction of type dipoles i f i 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 24

  25. Drude model: Vibration of particle of charge q and mass m near equilibrium: r http://img.tfd.com/ggse/d6/gsed_0001_0012_0_img2972.png Drude model D expression permittivi for E P ty : p i 1 = = + = + E E N f 0 0 i i 2 E q = = i t p r 0 i q e i i 2 2 m i i i i 2 1 q i = + i t E E 1 i e N f 0 0 i 25 i 2 2 m 0 i i i 10/23/2015 PHY 752 Fall 2015 -- Lecture 23

  26. Drude model dielectric function: ( ) 0 2 1 q i = + 1 i N f i 2 2 m i 0 i i i ( ) 0 ( ) 0 = + R I i ( ) 0 2 2 2 q i i = + 1 i R N f ( ) i 2 m 2 + 2 2 2 0 i i i ( ) 0 2 q i = i i I N f ( ) i 2 m 2 + 2 2 2 0 i i i 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 26

  27. Drude model dielectric function: ( ) R 0 ( ) I 0 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 27

  28. Drude model dielectric function some analytic properties: ( ) 2 1 q 1 = + i N f i i 2 i 2 m i 0 0 i i ( ) 2 1 q 1 For i N f i i 2 m i 0 0 i 2 P 1 2 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 28

  29. Drude model dielectric function some analytic properties: ( ) 0 0 particle free a ing (represent 0 For = 2 1 q 1 = + i N f i i 2 i 2 m i i i of charge mass , q m 0 0 0 ( ) 0 2 2 1 1 q q 1 = + + 0 m i N f iNf ( ) 0 i i i 2 i 2 m 0 i 0 0 0 0 i i ( ) 0 ( ) 0 + b i Some D details : = = E J E D b i E ( ) = + = i = = i + H J E E b b t t 2 1 q ( ) = 0 Nf ( ) 0 i m 0 0 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 29

  30. Analysis for Drude model dielectric function: ( ) 0 2 1 2 q 1 = + i N f i 2 m i i 0 i i i ( ) 0 2 1 2 q z ( ) z = 1 = Let i f N f i 2 m z iz i 0 i i i For z i 2 1 q ( ) z vanishes large at i f N f z i 2 z m i 0 i 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 30

  31. Analysis for Drude model dielectric function continued -- Analytic properties: ( ) 2 1 2 q z ( ) z i = 1 = i f N f i 2 i m z iz 0 0 z i i ( ) z 2 = 2 i has poles at 0 f z iz P P P i 2 = 2 i i i z i P 2 2 ( ) z ( ) z ( ) z Note that 0 analytic is for 0 f P P 10/23/2015 PHY 752 Fall 2015 -- Lecture 23 31

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