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![Page 1: î l î ì l î ì î ì - users.wfu.eduusers.wfu.edu/natalie/s20phy712/lecturenote/Lecture17fornotes.pdf · kwws lpj wig frp jjvh g jvhgb b b blpj sqj 'uxghprgho 9leudwlrqv ri fkdujhg](https://reader033.vdocuments.mx/reader033/viewer/2022060801/60851bdea9234b07034dd29e/html5/thumbnails/1.jpg)
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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
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
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
02/21/2020 PHY 712 Spring 2020 -- Lecture 17 18
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