Ac electrical conductivity

1. Consider a time dependent electric field E(t) acting on a metal.
Take the case when the wavelength of the field is large
compared to the electron mean free path between collisions:
l >> l
In this limit, the conduction electrons will “see” a 
homogeneous field when moving between collisions. Write:
E(t) = E(ω)e-iωt
That is, assume a harmonic dependence on frequency.
Next is standard Junior-Senior physics major E&M!
AC Electrical Conductivity in Metals
(Brief discussion only, in the free & independent electron approximation)
Application to the Propagation of
Electromagnetic Radiation in a Metal

2. electric field leads to
electric current j conductivity s = j/E metals
polarization P polarizability c = P/E dielectrics
dielectric function e =1+4pc
Response to the electric field mainly historically
both in metals and dielectrics described by used mainly for
i
i
i
i
i
i
q
dt
d
q
r
P
r
j


=
=
E
P
E
D ε
=
+
= p
4
)
(
4
4
div
4
div
ind
ext
ext


p
p
p
+
=
=
=
E
D e(w,0) describes the collective excitations of the
electron gas – the plasmons
e(0,k) describes the electrostatic screening
1
4
1
dP
j iω
dt
P j σ
χ i
E E iω ω
π
ε(ω) i σ(ω)
ω
= = 
= = =

= +
)
(
)
(
4
1
)
(
w
w
p
w
e
E
P
+
=
P
( , ) ( )
( , ) ~ ( )
( , ) ~ ( )
i t
i t
i t
t e
t e
t e
w
w
w
w w
w w
w w



=
E E
j j
P P

3. AC Electrical Conductivity of a Metal
w

w
w
w
w

w
w
w
w
w
w
w

w
w
w
w
i
m
ne
m
ne
i
e
e
t
e
t
t
e
t
dt
t
d
t
i
t
i

=

=

=
=
=


=


/
1
)
(
)
/
(
)
(
)
(
/
1
)
(
)
(
)
(
)
,
(
)
(
)
,
(
)
,
(
)
,
(
)
,
(
2
E
p
j
E
p
p
p
E
E
E
p
p
Newton’s 2nd Law Equation of
Motion for the momentum of one
electron in a time dependent
electric field. Look for a steady
state solution of the form:
m
ne
i

s
w
s
w
s
w
w
s
w
2
0
0
1
)
(
)
(
)
(
)
(
=

=
= E
j
AC conductivity
DC conductivity
)
(
4
1
)
( w
s
w
p
w
e i
+
=
2
2
2
2
1
)
(
4
w
w
w
e
p
w
p
p
m
ne

=
=
w
s
w
s
i

=
1
)
( 0
2
2
4
1
)
(
w
p
w
e
m
ne

=
Plasma Frequency
A plasma is a medium
with positive & negative
charges & at least one
charge type is mobile.
1

w
0
2 2
Re ( )
1
s
s w
w 
=
+

4. Even more simplified:
2
2
2
2
2
2
2
2
2
2
2
( ) 4
( ) 1 4 1
( )
4
( ) 1
p
p
d x
m eE
dt
eE
x
m
ne
P exn E
m
P ne
E m
ne
m
w
w
w p
e w p
w w
p
w
w
e w
w
= 
=
=  = 
= + = 
=
= 
Equation of motion of a Free Electron:
If x & E have harmonic time
dependences e-iωt
The polarization P is the dipole
moment per unit volume:
No electron collisions (no frictional damping term)
1

w

5. Transverse
Electromagnetic
Wave
Application to the Propagation of Electromagnetic Radiation in a Metal
T

6. The electromagnetic wave equation
in a nonmagnetic isotropic medium.
Look for a solution with the dispersion
relation for electromagnetic waves
2
2
2
2
2
2
2
)
,
(
)
exp(
/
)
,
(
K
c
i
t
i
c
t
=

+



=


w
w
e
w
w
e
K
r
K
E
E
E
K
(1) e real & > 0 → for w real, K is real & the transverse electromagnetic wave
propagates with the phase velocity vph= c/e1/2
(2) e real & < 0 → for w real, K is imaginary & the wave is damped with a
characteristic length 1/|K|:
(3) e complex → for w real, K is complex & the wave is damped in space
(4) e = →  → The system has a final response in the absence of an applied force
(at E = 0); the poles of e(w,K) define the frequencies of the free oscillations
of the medium
(5) e = 0 longitudinally polarized waves are possible
r
K
e
E


Application to the Propagation of Electromagnetic Radiation in a Metal

7. (1) For w > wp → K2 > 0, K is real, waves
with w > wp propagate in the media with
the dispersion relation:
The electron gas is transparent.
(2) For w < wp → K2 < 0, K is imaginary,
waves with w < wp incident on the medium
do not propagate, but are totally reflected
Transverse optical modes in a plasma
2
2
2
)
,
( K
c
=
w
w
e K
Dispersion relation for
electromagnetic waves
2
2
2
2
1
)
(
4
w
w
w
e
p
w
p
p
m
ne

=
=
2
2
2
2
K
c
p =
w
w
r
K
e
E


Metals are shiny due
to the reflection of light
w/wp w = cK
cK/wp
c
K
v
c
dK
d
v
ph
group

=

=
w
w
forbidden
frequency
gap vph > c → vph
This does not correspond to the velocity of
the propagation of any quantity!!
(2) (1)
w/wp
E&M waves are totally reflected
from the medium when e is negative
E&M waves propagate
with no damping when
e is positive & real
( )
e w
2
2
2
2
K
c
p +
= w
w

8. Ultraviolet Transparency of Metals
2
2
2
2
1
)
(
4
w
w
w
e
p
w
p
p
m
ne

=
=
Plasma Frequency wp & Free Space Wavelength lp = 2pc/wp
Range Metals Semiconductors Ionosphere
n, cm-3 1022 1018 1010
wp, Hz 5.7×1015 5.7×1013 5.7×109
lp, cm 3.3×10-5 3.3×10-3 33
spectral range UV IF radio
The Electron Gas is Transparent when w > wp i.e. l < lp
Plasma Frequency
Ionosphere
Semiconductors
Metals
The reflection of
light from a metal
is similar to the
reflection of radio
waves from the
Ionosphere!
reflects transparent for
metal visible UV
ionosphere radio visible

9. Skin Effect
When w < wp the electromagnetic wave is reflected.
It is damped with a characteristic length d = 1/|K|:
The wave penetration – the skin effect
The penetration depth d – the skin depth
r
K
r
e
e
E


=
 d
( )
( )










+


+
=







+
=
=
r
c
iKr
t
i
E
i
c
K
c
i
i
c
c
K
2
1
2
1
2
2
2
2
2
2
2
2
exp
)
exp(
)
1
(
2
4
4
1
psw
w
psw
s
w
p
w
s
w
p
w
e
w
The classical skin depth
( ) 2
1
2psw
d
c
cl =
d >> l
The classical skin effect
d << l: The anomalous skin effect (pure metals at low temperatures) the usual theory of
electrical conductivity is no longer valid; the electric field varies rapidly over l .
Further, not all electrons are participating in the wave absorption & reflection.
d’
l
( ) 2
1
2
1
'
2
'
2
'







=
sw
d
p
w
ps
d
l
c
c
3
1
2
2
' 







=
psw
d
lc
Only electrons moving inside the skin depth for most of the mean free
path l are capable of picking up much energy from the electric field.
Only a fraction of the electrons d’/l contribute to the conductivity

10. Longitudinal Plasma Oscillations
0
)
4
(
2
2
2
2
2
=
+



=

=


d
t
d
nde
Ne
NeE
t
d
Nm
p
w
p
A charge density oscillation, or a longitudinal plasma
oscillation, or a plasmon
m
ne
p
2
2 4p
w =
Oscillations at the
Plasma Frequency
Equation of
Motion
( ) 0
1 2
2
=

=
p
L
L
w
w
w
e
w/wp
cK/wp
Longitudinal Plasma Oscillations
wL= wp
Transverse Electromagnetic Waves
forbidden
frequency
gap
The Nature of Plasma Oscillations: Correspond to a
displacement of the entire electron gas a distance d with
respect to the positive ion background. This creates surface
charges s = nde & thus an electric field E = 4pnde.
Longitudinal Plasma Oscillations