Drude Model
for dielectric constant of metals.
•
•
•
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Conduction Current in Metals
EM Wave Propagation in Metals
Skin Depth
Plasma Frequency
Ref : Prof. Robert P. Lucht, Purdue University
Drude model
z Drude model : Lorenz model (Harmonic oscillator model) without restoration force
(that is, free electrons which are not bound to a particular nucleus)
Linear Dielectric Response of Matter
Conduction
Conduction Current
Current in
in Metals
Metals
The equation of motion of a free electron (not bound to a particular nucleus; C = 0),
r
r
r
2
r
uu
r
r
uur
me d r
d r
dv
− e E ⇒ me
+ me γ v = − e E
me 2 = − C r −
τ dt
dt
dt
1
−14
(
:
relaxation
time
10
τ
=
≈
s)
Lorentz model
(Harmonic oscillator model)
γ
If C = 0, it is called Drude model
The current density is defined :
r
r
⎡ C ⎤
J = − N ev
with units of ⎢
2
⎣ s − m ⎥⎦
Substituting in the equation of motion we obtain :
r
r ⎛ N e2 ⎞ r
dJ
+γJ =⎜
⎟E
dt
m
⎝ e ⎠
Conduction
Conduction Current
Current in
in Metals
Metals
Assume that the applied electric field and the conduction current density are given by :
r r
E = E0 exp ( −i ω t )
r r
J = J 0 exp ( −i ω t )
Local approximation
to the current-field relation
Substituting into the equation of motion we obtain :
r
d ⎡⎣ J 0 exp ( −i ω t ) ⎤⎦
r
r
r
+ γ J 0 exp ( −i ω t ) = − i ω J 0 exp ( −i ω t ) + γ J 0 exp ( −i ω t )
dt
⎛ N e2 ⎞ r
=⎜
⎟ E0 exp ( −i ω t )
⎝ me ⎠
Multiplying through by exp ( +i ω t ) :
r
⎛ N e2 ⎞ r
( −i ω + γ ) J 0 = ⎜
⎟ E0
⎝ me ⎠
or equivalently
r ⎛ N e2 ⎞ r
( −i ω + γ ) J = ⎜
⎟E
m
⎝ e ⎠
Conduction
Conduction Current
Current in
in Metals
Metals
For static fields (ω = 0 ) we obtain :
r ⎛ N e2 ⎞ r
r
=
J =⎜
E
σ
E
⎟
⎝ meγ ⎠
⇒
N e2
= static conductivity
σ =
meγ
For the general case of an oscillating applied field :
r ⎡
r
⎤r
σ
J =⎢
⎥ E = σω E
−
1
/
i
ω
γ
(
)
⎣
⎦
σ ω = dynamic conductivity
For very low frequencies, (ω γ ) << 1, the dynamic conductivity is purely real and
the electrons follow the electric field .
As the frequency of the applied field increases, the inertia of electrons introduces
a phase lag in the electron response to the field , andthe dynamic conductivity is complex.
For very high frequencies, (ω γ ) >> 1, the dynamic conductivity is purely imaginary
and the electron oscillations are 90° out of phasewith the applied field .
Propagation
Propagation of
of EM
EM Waves
Waves in
in
Metals
Metals
Maxwell ' s relations give us the following wave equation for metals :
r
r
2
r
1 ∂ E
1 ∂J
∇2 E = 2 2 +
P = 0, J ≠ 0
c ∂t
ε 0 c 2 ∂t
r ⎡
⎤r
σ
J =⎢
⎥E
i
−
1
/
ω
γ
(
)
⎣
⎦
But
Substituting in the wave equation we obtain :
r
r
r
⎤ ∂E
1 ∂2 E
1 ⎡
σ
2
∇ E= 2 2 +
⎢
⎥
c ∂t
ε 0 c 2 ⎣1 − ( iω / γ ) ⎦ ∂t
The wave equation is satisfied by electric fields of the form :
r r
r r
E = E0 exp ⎡i k ⋅ r − ωt ⎤
⎣
⎦
(
)
where
k =
2
⎡ σ ω μ0 ⎤
+
i
⎢
⎥
ω
γ
1
/
−
c2
i
(
)
⎣
⎦
ω2
c2 =
1
ε 0 μ0
Skin
Skin Depth
Depth
Consider the case where ω is small enough that k 2 is given by :
k =
2
⎡ σ ω μ0 ⎤
⎛ π⎞
+
i
⎢
⎥ ≅ i σ ω μ0 = exp ⎜ i ⎟ σ ω μ0
2
c
⎝ 2⎠
⎣1 − ( iω / γ ) ⎦
ω2
Then , k% ≅
kR = kI =
σ ω μ0
⎡ ⎛π ⎞
⎛ π⎞
⎛ π⎞
⎛ π ⎞⎤
exp ⎜ i ⎟ σ ω μ0 = exp ⎜ i ⎟ σ ω μ0 = ⎢cos ⎜ ⎟ + i sin ⎜ ⎟ ⎥ σ ω μ0 = (1 + i )
2
⎝ 2⎠
⎝ 4⎠
⎝ 4 ⎠⎦
⎣ ⎝4⎠
σ ω μ0
2
⎛c⎞
nR = ⎜ ⎟ k R =
⎝ω ⎠
σ c 2 μ0
=
2ω
σ
= nI
2ωε 0
In the metal , for a wave propagating in the z − direction :
r r
r
⎛ z⎞
E = E0 exp ( −k I z ) exp ⎡⎣i ( k R z − ωt ) ⎤⎦ = E0 exp ⎜ − ⎟ exp ⎡⎣i ( k R z − ωt ) ⎤⎦
⎝ δ⎠
The skin depth δ is given by :
1
=
δ =
kI
2
σ ω μ0
=
2ε 0 c 2
σω
For copper the static conductivity σ = 5.76 × 107 Ω −1m −1 = 5.76 ×107
C2 − s
→ δ = 0.66 μ m
J −m
Plasma
Plasma Frequency
Frequency
Now consider again the general case :
k2 =
⎡ σ ω μ0 ⎤
+
i
⎢
⎥
c2
⎣1 − ( iω / γ ) ⎦
ω2
⎧⎪
⎫⎪
⎫⎪
σ c 2 μ0
σ c 2 μ0
iγ ⎧⎪
1
n = 2 k =1+ i ⎨
i
=
+
⎬
⎨
⎬
ω
iγ ⎩⎪ ω ⎡⎣1 − ( iω / γ ) ⎤⎦ ⎭⎪
⎪⎩ ω ⎡⎣1 − ( iω / γ ) ⎤⎦ ⎭⎪
γ σ c 2 μ0
2
n = 1− 2
ω +iω γ
2
c2
2
The plasma frequency is defined :
⎛ N e2 ⎞ 2
N e2
μ
=
c
⎟
0
meε 0
⎝ meγ ⎠
ω p2 = γ σ c 2 μ0 = γ ⎜
The refractive index of the medium is given by
n = 1−
2
ω p2
ω2 + iω γ
Plasma
PlasmaFrequency
Frequency
If the electrons in a plasma are displaced from a uniform background of
ions, electric fields will be built up in such a direction as to restore the
neutrality of the plasma by pulling the electrons back to their original
positions.
Because of their inertia, the electrons will overshoot and oscillate
around their equilibrium positions with a characteristic frequency
known as the plasma frequency.
E s = σ o / ε o = Ne (δ x ) / ε o : electrostatic field by small charge separation δ x
δ x = δ xo exp(− iω p t ) : small-amplitude oscillation
d 2 (δ x )
m
= (− e ) E s
2
dt
⇒
−
mω 2p
=−
Ne 2
εo
⇒
∴ ω 2p
Ne 2
=
mε o
Plasma
Plasma Frequency
Frequency
⎛c
n =⎜
⎝ω
2
iσ c 2 μo
σ c 2 μ oγ
⎞
k ⎟ = 1+
= 1− 2
ω (1 − iω / γ )
ω + iωγ
⎠
2
n2 = (nR + inI )2 = 1 −
n2 = 1 −
ω 2p
ω
2
ω 2p
ω 2 + iωγ
by neglecting γ , valid for high frequency (ω >> γ ).
For ω < ω p , n is complex and radiation is attenuated.
For ω > ω p , n is real and radiation is not attenuated(transparent).
Plasma
Plasma Frequency
Frequency
λc = λ p =
2πc
ωp
Born and Wolf, Optics, page 627.
Plasma
Plasma Frequency
Frequency
Dielectric constant of metal : Drude model
ε (ω ) = ε R + iε I = n 2
= (nR + inI ) 2 = 1 −
ω p2
ω 2 + iωγ
= (nR2 − nI2 ) + i 2nR nI
⎛
ω p2 ⎞ ⎛ ω p2γ ⎞
= ⎜1 − 2
+i
⎜ ω + γ 2 ⎟⎟ ⎜⎜ ω 3 + ωγ 2 ⎟⎟
⎝
⎠ ⎝
⎠
ω >> γ =
1
τ
⎛ ω p2 ⎞ ⎛ ω p2
ε (ω ) = ⎜⎜1 − 2 ⎟⎟ + i ⎜⎜ 3
⎝ ω ⎠ ⎝ω /γ
⎞
⎟⎟
⎠
Ideal case : metal as a free-electron gas
• no decay (infinite relaxation time)
• no interband transitions
2
⎛
ωp ⎞
τ →∞
→ ε (ω ) = ⎜1 − 2 ⎟
ε (ω ) ⎯⎯⎯
γ →0
⎜ ω ⎟
⎝
⎠
ω
εr = 1−
ω
2
p
2
0
Plasma waves (plasmons)
Note: SP is a TM wave!
Plasmo
ns
Plasma oscillation = density fluctuation of free electrons
+
-
+ +
-
-
k
+
-
+
Plasmons in the bulk oscillate at ωp determined by
the free electron density and effective mass
Ne 2
drude
=
ωp
mε 0
Plasmons confined to surfaces that can interact
with light to form propagating “surface plasmon
polaritons (SPP)”
1 Ne 2
ω particle =
3 mε 0 SPP modes
Confinement effects result in resonant
in nanoparticles
drude
Dispersion relation for EM waves in electron gas (bulk plasmons)
• Dispersion relation:
ω = ω (k )
Dispersion relation of surface-plasmon
for dielectric-metal boundaries
Dispersion relation for surface plasmon polaritons
εd
εm
TM wave
Z>0
Z<0
• At the boundary (continuity of the tangential Ex, Hy, and the normal Dz):
Exm = Exd
H ym = H yd
ε m Ezm = ε d Ezd
Dispersion relation for surface plasmon polaritons
(−ik zi H yi ,0, ik xi H yi )
k zi H yi = ωε i E xi
(−iωε i E xi ,0,−iωε i E zi )
k zm H ym = ωε m E xm
k zd H yd = ωε d E xd
Exm = Exd
H ym = H yd
k zm
εm
H ym =
k zd
εd
H yd
k zm
εm
=
k zd
εd
Dispersion relation for surface plasmon polaritons
• For any EM wave:
⎛ω ⎞
2
k 2 = ε i ⎜ ⎟ = k x2 + k zi2 , where k x ≡ k xm = k xd
⎝c⎠
SP Dispersion Relation
ε mε d
kx =
c εm + εd
ω
Dispersion
relation for surface plasmon polaritons
Dispersion relation:
1/ 2
x-direction:
ω ⎛ ε mε d ⎞
k x = k 'x + ik "x = ⎜
⎟
c ⎝ εm + εd ⎠
⎛ω ⎞
z-direction: k = ε i ⎜ ⎟ − k x2
⎝c⎠
For a bound SP mode:
2
2
zi
ε m = ε m' + iε m"
ω⎛
kzi must be imaginary: εm + εd < 0
⎛ω ⎞
⎛ω ⎞
k zi = ± ε i ⎜ ⎟ − k x2 = ±i k x2 − ε i ⎜ ⎟
⎝c⎠
⎝c⎠
2
k’x must be real: εm < 0
So,
ε < −ε d
'
m
1/ 2
⎞
k zi = k 'zi + ik zi = ± ⎜
⎟
c ⎝ εm + εd ⎠
ε
2
i
2
⎛ω ⎞
⇒ kx > ε i ⎜ ⎟
⎝c⎠
+ for z < 0
- for z > 0
Plot of the dispersion relation
• Plot of the dielectric constants:
ωp2
ε m (ω ) = 1 − 2
ω
• Plot of the dispersion relation:
ε mε d
kx =
c εm + εd
ω
k x = k sp =
• When ε m → −ε d ,
⇒ k x → ∞, ω ≡ ωsp =
ωp
1+ ε d
ω
(ω 2 − ω p )ε d
c
(1 + ε d )ω 2 − ω p
2
2
Surfaceplasmon
plasmon dispersion
dispersion relation:
Surface
relation
ω⎛ ε ε
k x = ⎜⎜ m d
c ⎝ εm + εd
ω
ω 2 = ω p2 + c 2 k x2
ck x
1/ 2
⎞
⎟⎟
⎠
εd
1/ 2
⎞
k zi = ⎜
⎟
c ⎝ εm + εd ⎠
Radiative modes
(ε'm > 0)
ω⎛
ε i2
real kx
real kz
ωp
Quasi-bound modes
(−εd < ε'm < 0)
ωp
imaginary kx
real kz
1+ εd
z
x
Dielectric:
εd
Bound modes
(ε'm < −εd)
Metal: εm = εm' +
εm"
Re kx
real kx
imaginary kz