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02/21/2020 PHY 712 Spring 2020 -- Lecture 17 1

PHY 712 Electrodynamics12-12:50 AM Olin 103

Plan for Lecture 17:

Continue reading Chapter 7

1. Real and imaginary contributions to electromagnetic response

2. Frequency dependence of dielectric materials; Drude model

3. Kramers-Kronig relationships

02/21/2020 PHY 712 Spring 2020 -- Lecture 17 2

02/21/2020 PHY 712 Spring 2020 -- Lecture 17 3

For linear isotropic media and no sources: ;

Coulomb's law: 0

Ampere-Maxwell's law: 0

Faraday's law: 0

No magnetic monopoles:

t

t

D E B H

E

EB

BE

0 B

1

2

3

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02/21/2020 PHY 712 Spring 2020 -- Lecture 17 4

Plane wave solutions to sourceless Maxwell’s equations; extension of analysis to complex dielectric functions

0

2

0

4

2

2 2

2

1/ 22

2

2

2

For s

implicity assume that

Suppose the dielectric function is complex:

2

4 4 0

2

R I R i

I R

R

R

I

R I

R

R Ii n i

n n

n n n

in

n n

n

n n

1/ 22

ˆ ˆ ˆ

0 0

2

, R Ic c ci n ct i n ct nt e e e

k r k r k rE r E E

02/21/2020 PHY 712 Spring 2020 -- Lecture 17 5

Paul Karl Ludwig Drude 1863-1906

http://photos.aip.org/history/Thumbnails/drude_paul_a1.jpg

iii

ti

ti

ti

eim

qq

im

qe

mmeqm

rrpP

PEED

Erp

Errr

rrEr

3

0

220

02

220

000

200

:fieldnt Displaceme

1

:dipole Induced

1 ,For

02/21/2020 PHY 712 Spring 2020 -- Lecture 17 6

http://img.tfd.com/ggse/d6/gsed_0001_0012_0_img2972.png

Drude model: Vibrations of charged particles near equilibrium:

r

4

5

6

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rrEr 200 mmeqm ti

02/21/2020 PHY 712 Spring 2020 -- Lecture 17 7

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

Note that: > 0 represents dissipation of energy. 0 represents the natural frequency of

the vibration; 0=0 would represent a free (unbound) particle

rrEr 200 mmeqm ti

02/21/2020 PHY 712 Spring 2020 -- Lecture 17 8

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

eim

qq

im

qe

ti

ti

220

02

220

000

1

:dipole Induced

1 ,For

Erp

Errr

rrEr 200 mmeqm ti

02/21/2020 PHY 712 Spring 2020 -- Lecture 17 9

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

0

3

Displacement field:

number of dipoles/volume

fraction of type dipoles

i i i ii i

i

N f

N

f i

D E E P

P p r r p

7

8

9

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02/21/2020 PHY 712 Spring 2020 -- Lecture 17 10

Drude model: Vibration of particle of charge q and mass m near equilibrium:

rhttp://img.tfd.com/ggse/d6/gsed_0001_0012_0_img2972.png

0 0

20

2 2

2

0 0 2 20

Drude model expression for permittivity:

1

1 1

i ii

i tii i

i i i

i t ii

i i i i

N f

q q e

m i

q e N f

m i

D E E P E p

Ep r

E E

02/21/2020 PHY 712 Spring 2020 -- Lecture 17 11

Drude model dielectric function:

i ii

i

i

ii

I

i ii

i

i

ii

R

IR

i iii

ii

m

qfN

m

qfN

i

im

qfN

222220

2

0

22222

22

0

2

0

00

220

2

0

1

11

02/21/2020 PHY 712 Spring 2020 -- Lecture 17 12

R

0

I

0

Drude model dielectric function:

10

11

12

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02/21/2020 PHY 712 Spring 2020 -- Lecture 17 13

Drude model dielectric function – some analytic properties:

2

2

0

2

20

220

2

0

1

11 For

11

P

i i

iii

i iii

ii

m

qfN

ω

im

qfN

02/21/2020 PHY 712 Spring 2020 -- Lecture 17 14

Drude model dielectric function – some analytic properties:

00

000

20

00

220

2

0

000

220

2

0

1

11

mass , charge of particle free a ing(represent 0For

11

i

im

qiNf

im

qfN

mq

im

qfN

b

i iii

ii

i iii

ii

2

00

0 0

Some details:

1

b

b b

ii i

t t

q Nf

m i

D E J E

D EH J E E

02/21/2020 PHY 712 Spring 2020 -- Lecture 17 15

Analytic properties of the dielectric function (in the Drudemodel or from “first principles” -- Kramers-Kronig transform

Consider Cauchy's integral formula for an analytic function ( ):

1 ( ) 0

2 includes

f z

f(z)dz f z f dz

πi z - α

Re(z)

Im(z)

13

14

15

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02/21/2020 PHY 712 Spring 2020 -- Lecture 17 16

Kramers-Kronig transform -- continued

1 1

2 2R

R

Rincludes rest

f(z) f(z ) f(z)f dz dz dz

πi z - α πi z - α z - α

Re(z)

Im(z)

=0

f-αz

)f(zdzP

πi

-αz

)f(zdz

πi f

R

RR

R

RR )(

2

1

2

1

2

1

02/21/2020 PHY 712 Spring 2020 -- Lecture 17 17

Kramers-Kronig transform -- continued

-αz

)(zfdzPf

-αz

)(zfdzPf

-αz

zifzfdzP

πiiff

zifz fz f

f-αz

)f(zdzP

πi f

R

RRRI

R

RIRR

R

RIRRRIR

RIRRR

R

RR

1

1

2

1

2

1

: Suppose

)(2

1

2

1

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Kramers-Kronig transform -- continued

-αz

)(zfdzPf

-αz

)(zfdzPf

R

RRRI

R

RIRR

1

1

0

This Kramers-Kronig transform is useful for the dielectric function

when 1

Must show that: 1. is analytic for 0

2. vanishes for

R

I

f z

f z z

f z z

16

17

18

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02/21/2020 PHY 712 Spring 2020 -- Lecture 17 19

au

ubub

au

u-udu

u-udu

u-udu P

ugduugduugduP

s

ss

s

b

u s

u

a s

b

a s

b

u

u

a

b

a

s

s

s

s

lnlnln0

lim

1

1

0

lim1

:Example

)( )( 0

lim)(

:nintegratio parts Principal

ionsconsiderat practical Some

a us bu

02/21/2020 PHY 712 Spring 2020 -- Lecture 17 20

II

RR

: function dielectricFor

ionsconsiderat practical More

*

0

Analytic properties the dielectric function which justify

the treatment of 1

Must show that: 1. is analytic for 0

2. vanishes for (for 0)

I

I

zf z

f z z

f z z z

02/21/2020 PHY 712 Spring 2020 -- Lecture 17 21

Analysis for Drude model dielectric function:

zm

qfN

zzf

z

izzm

qfN

zzf

im

qfN

i i

ii

i

i iii

ii

i iii

ii

largeat vanishes 1

For

11 Let

11

0

2

2

220

2

0

220

2

0

19

20

21

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02/21/2020 PHY 712 Spring 2020 -- Lecture 17 22

Analysis for Drude model dielectric function – continued --Analytic properties:

2

2 20 0

2 2

22

1 1

has poles at 0

2 2

Note that 0 is analytic for 0

ii

i i i i

P i P P i

i iP i

P P

z q f z N f

m z iz

f z z z iz

z i

z f z z

02/21/2020 PHY 712 Spring 2020 -- Lecture 17 23

0for analytic is 0 that Note

22

0at poles has

11

22

22

220

2

0

PP

ii

iP

iPPiP

i iii

ii

zzfz

iz

izzzzf

izzm

qfN

zzf

)( pz

)( pz

( ) analyticf z

02/21/2020 PHY 712 Spring 2020 -- Lecture 17 24

Kramers-Kronig transform – for dielectric function:

IIRR

RI

IR

-dP

-dP

;with

'

1 1

''

1

'

1

''

11

00

00

Because of these analytic properties, Cauchy’s integral theorem results in:

22

23

24

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02/21/2020 PHY 712 Spring 2020 -- Lecture 17 25

Further comments on analytic behavior of dielectric function

0

0

0 0

"Causal" relationship between and fields:

, , ,

1 i

t t d G t

d G e

E D

D r E r E r

'

Some details: Consider a convolution integral such as

( ) ( ') ( ') ' where the functions ( ), ( ), and ( )

are all well-defined functions with Fourier transforms such as

( ( ')) ti

f t g t h t t dt f t g t h t

f f t e dt

)

1' ( ) (

2

It follows that: ) (

)

( ) (

tif t f e

gf h

d

02/21/2020 PHY 712 Spring 2020 -- Lecture 17 26

Further comments on analytic behavior of dielectric function

0

0

0 0 0

"Causal" relationship between and fields:

, , ,

( 1 1

) ( )2

= 1i i

t

e

t d G t

G G d G ed

E D

D r E r E r

2

2 20 0

2

0

2

2

/

2

1For 1

sin( (

where / 4

) )

i

i

i

i

i

ii i i i

ii

i i

i i

q N f

m i

q NG f e

m

02/21/2020 PHY 712 Spring 2020 -- Lecture 17 27

Some details

0 2

( 1 =1 1

) ( )2

i ize d f z e dzG

2

2 20 0

2 2

2 22 2

1 Let 1

has poles at 0

or 2 2 2 2

ii

i i i i

P i P P i

i i i iP i P i

z q f z N f

m z iz

f z z z iz

z i z i

25

26

27

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02/21/2020 PHY 712 Spring 2020 -- Lecture 17 28

zP

1) ( ) Res( )

2( iz

PP

G f z e dz i z

Note that: IRiz zize e e

Valid contour for 0

( ) 0 for 0G

2/2

0

Valid contour for 0

( )

sini ii

ii i i

G

q Nf e

m

02/21/2020 PHY 712 Spring 2020 -- Lecture 17 29

/2

0

2 2 2

2

2

sin( (

where / 4 assuming / 4 0

)

)iii

i i

i i ii i

i

i

q NG f e

m

1) ( ) Res( )

2( iz

PP

G f z e dz i z

2

2 20 0

2 2

2 22 2

1 Let 1

has poles at 0

or 2 2 2 2

ii

i i i i

P i P P i

i i i iP i P i

z q f z N f

m z iz

f z z z iz

z i z i

28

29