• Conduction Current in Metals

Drude Model

for dielectric constant of metals.

• 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 Current in Metals Conduction Current in Metals

2

2

: v

:

e

The current density is defined J N e with units of C

s m

Substituting in the equation of motion we obtain

dJ N e

J E

dt γ m

⎡ ⎤

= − ⎢ ⎣ − ⎥ ⎦

⎛ ⎞

+ = ⎜ ⎟

⎝ ⎠

r r

r r r

The equation of motion of a free electron (not bound to a particular nucleus; ),

( : relaxation time )

2 2

14

0

1 10

r r r

r

uur r uur

e e e

C m

d r d r d v

m C r e E m m v e E

dt dt

dt

s τ γ

τ γ

=

= − − − ⇒ + = −

= ≈

Lorentz model

(Harmonic oscillator model) If C = 0, it is called Drude model

Conduction Current in Metals Conduction Current in Metals

( ) ( )

( )

( ) ( ) ( )

0 0

0

0 0 0

:

exp exp

: exp

exp exp exp

Assume that the applied electric field and the conduction current density are given by

E E i t J J i t

Substituting into the equation of motion we obtain

d J i t

J i t i J i t J i t

dt

ω ω

ω γ ω ω ω γ ω

= − = −

⎡ − ⎤

⎣ ⎦ + − = − − + −

r r r r

r r r r

( )

( )

( )

( )

2 0

2

0 0

2

exp

exp :

e

e

e

N e E i t

m

Multiplying through by i t

i J N e E

m

or equivalently i J N e E

m ω

ω ω γ

ω γ

⎛ ⎞

= ⎜ ⎟ −

⎝ ⎠

+

⎛ ⎞

− + = ⎜ ⎟

⎝ ⎠

⎛ ⎞

− + = ⎜ ⎟

⎝ ⎠

r

r r

r r

Local approximation to the current-field relation

Conduction Current in Metals Conduction Current in Metals

( )

( )

( )

2 2

0 :

:

1 /

, 1,

e e

For static fields we obtain

N e N e

J E E static conductivity

m m

For the general case of an oscillating applied field

J E E dynamic conductivity

i

For very low frequencies the dyn

ω ω

ω

σ σ

γ γ

σ σ σ

ω γ

ω γ

=

⎛ ⎞

= ⎜ ⎟ = ⇒ = =

⎝ ⎠

⎡ ⎤

= ⎢⎣ − ⎥⎦ = =

<<

r r r

r r r

.

, ,

amic 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 conducti

( )

.

, 1,

90 .

vity is complex

For very high frequencies the dynamic conductivity is purely imaginary and the electron oscillations are out of phasewith the applied field

ω γ

°

Propagation of EM Waves in Metals

Propagation of EM Waves in Metals

( )

( )

2 2

2 2 2

0

2 2

2 2 2

0

' :

1 1

1 /

:

1 1

1 /

Maxwell s relations give us the following wave equation for metals

E J

E c t c t

But J E

i

Substituting in the wave equation we obtain

E E

E c t c i t

The wave equation is s ε

σ ω γ

σ

ε ω γ

∂ ∂

∇ = +

∂ ∂

⎡ ⎤

= ⎢⎣ − ⎥⎦

⎡ ⎤

∂ ∂

∇ = ∂ + ⎢⎣ − ⎥⎦ ∂

r r

r

r r

r r

r

( )

( )

0

2

2 0 2

2

0 0

: exp

1

1 /

atisfied by electric fields of the form

E E i k r t

where

k i c

c i

ω

σ ω μ ω

ω γ ε μ

⎡ ⎤

= ⎣ ⋅ − ⎦

⎡ ⎤

= + ⎢⎣ − ⎥⎦ =

r

r r r

0

,

0 ≠

= J

P

Skin Depth Skin Depth

( )

( )

2

2

2 0

0 0

2

0

0 0 0

0

:

1 / exp 2

, exp exp cos sin 1

2 4 4 4 2

R I 2 R

Consider the case where is small enough that k is given by

k i i i

c i

Then k i i i i

k k n

ω σ ω μ

ω σ ω μ π σ ω μ

ω γ

σ ω μ

π σ ω μ π σ ω μ π π σ ω μ

σ ω μ

⎡ ⎤ ⎛ ⎞

= + ⎢⎣ − ⎥⎦ ≅ = ⎜⎝ ⎟⎠

⎡ ⎤

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

≅ ⎜⎝ ⎟⎠ = ⎜⎝ ⎟⎠ = ⎢⎣ ⎜ ⎟⎝ ⎠+ ⎜ ⎟⎝ ⎠⎥⎦ = +

= = =

%

( ) ( ) ( )

2 0

0

0 0

2 2

, :

exp exp exp exp

R I

I R R

c c

k n

In the metal for a wave propagating in the z direction

E E k z i k z t E z i k z t

σ μ σ

ω ω ωε

ω ω

δ

⎛ ⎞ = = =

⎜ ⎟⎝ ⎠

−

⎛ ⎞

= − ⎡⎣ − ⎤⎦ = ⎜⎝− ⎟⎠ ⎡⎣ − ⎤⎦

r r r

2 0 0

2

7 1 1 7

2

1 2

:

5.76 10 5.76 10 0.66

I

The skin depth is given by c

k

C s

For copper the static conductivity m m

J m

δ δ ε

σ ω μ σ ω

σ ^{− −} δ μ

= = =

= × Ω = × − → =

−

Plasma Frequency Plasma Frequency

( )

( ) ( )

2

2 0

2

2 2

2

2 2 0 0

2

2

2 0

2

2

2 2 2

0

:

1 /

1 1

1 / 1 /

1

:

p

e

Now consider again the general case

k i

c i

c c

c i

n k i i

i

i i

n c

i

The plasma frequency is defined

c N e c

m σ ω μ ω

ω γ

σ μ γ σ μ

ω ω ω γ γ ω ω γ

γ σ μ ω ω γ

ω γ σ μ γ

γ

⎡ ⎤

= + ⎢⎣ − ⎥⎦

⎧ ⎫ ⎧ ⎫

⎪ ⎪ ⎪ ⎪

= = + ⎨ ⎬ = + ⎨ ⎬

− −

⎡ ⎤ ⎡ ⎤

⎪ ⎣ ⎦⎪ ⎪ ⎣ ⎦⎪

⎩ ⎭ ⎩ ⎭

= − +

⎛ ⎞

= = ⎜ ⎟

⎝ ⎠

2 0

0

2 2

1 2

e

p

N e m

The refractive index of the medium is given by

n i

μ ε

ω ω ω γ

=

= − +

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.

/ ( ) / : electrostatic field by small charge separation exp( ) : small-amplitude oscillation

( )

( )

2 2 2

2 2

2

s o o o

o p

s p p

o o

E Ne x x

x x i t

d x Ne Ne

m e E m

dt m

σ ε δ ε δ

δ δ ω

δ ω ω

ε ε

= =

= −

= − ⇒ − = − ⇒ ∴ =

Plasma Frequency

Plasma Frequency

( / )

( )

2 2 2

2

2 2

2 2

2

1 1

1

1

o o

p

R I

i c c

n c k

i i

n n in

i

σ μ σ μ γ

ω ω ω γ ω ωγ

ω ω ωγ

⎛ ⎞

= ⎜ ⎝ ⎟ ⎠ = + − = − +

= + = −

+

by neglecting , valid for high frequency ( ).

For , is complex and radiation is attenuated.

For , is real and radiation is not attenuated(transparent).

2 2

1

p p

n

n n

ω γ ω γ

ω ω ω ω ω

= − >>

<

>

Plasma Frequency

Plasma Frequency

Plasma Frequency Plasma Frequency

Born and Wolf, Optics, page 627.

p p

c

c ω λ π

λ = = ²

Plasma Frequency

Plasma Frequency

2

2 2

2

2 2

2 2

2 2 3 2

( )

( ) 1

( ) 2

1

R I

p

R I

R I R I

p p

i n

n in

i

n n i n n

i

ε ω ε ε

ω

ω ωγ

ω ω γ

ω γ ω ωγ

= + =

= + = −

+

= − +

⎛ ⎞ ⎛ ⎞

= − ⎜ ⎜ ⎝ + ⎟ ⎟ ⎠ + ⎜ ⎜ ⎝ + ⎟ ⎟ ⎠

Dielectric constant of metal : Drude model

γ τ

ω >> = ^{1 2 2}

2 3

( ) 1

/

p p

ω i ω

ε ω ω ω γ

⎛ ⎞ ⎛ ⎞

= − ⎜ ⎜ ⎟ ⎟ + ⎜ ⎜ ⎟ ⎟

⎝ ⎠ ⎝ ⎠

Ideal case : metal as a free-electron gas

• no decay (infinite relaxation time)

• no interband transitions

2

0 2

( ) ^τ _γ ( ) 1 ω ^p

ε ω ε ω

ω

→∞

→

⎛ ⎞

⎯⎯⎯ → = − ⎜ ⎜ ⎝ ⎟ ⎟ ⎠

2

1 ^p 2 r

ε ω

= − ω ⁰

Note: SP is a TM wave!

Plasma waves (plasmons)

Plasmo ns

Plasmons in the bulk oscillate at ω

determined by the free electron density and effective mass

Plasmons confined to surfaces that can interact with light to form propagating “surface plasmon polaritons (SPP)”

Confinement effects result in resonant SPP modes in nanoparticles

+ + +

- - -

+ - +

k

Plasma oscillation = density fluctuation of free electrons

0 2

ω ε

m Ne

drude

p

=

0 2

3 1 ω ε

m Ne

drude

particle

=

Dispersion relation for EM waves in electron gas (bulk plasmons)

( ) k

ω ω =

• Dispersion relation:

Dispersion relation of surface-plasmon

for dielectric-metal boundaries

Dispersion relation for surface plasmon polaritons

TM wave

ε _m ε _d

xm xd

E = E H

= H

ε

E

= ε

E

• At the boundary (continuity of the tangential E

, H

, and the normal D

):

Z > 0

Z < 0

Dispersion relation for surface plasmon polaritons

zm zd

m d

k k

ε ⁼ ε

xm xd

E = E

ym yd

H = H

xm m ym

zm

H E

k = ωε

yd d

zd ym

m

zm

k H

k H

ε

ε ⁼

) ,

0 ,

( − i ωε

E

− i ωε

E

)

, 0 ,

( − ik

H

ik

H

xi i yi

zi

H E

k = ωε

xd d yd

zd

H E

k = ωε

Dispersion relation for surface plasmon polaritons

• For any EM wave:

2

2 2 2

i x zi x xm xd

k k k , where k k k

c ε ^{⎛ ⎞} ω

= ⎜ ⎟ = + ≡ =

⎝ ⎠

m d x

m d

k c

ε ε ω

ε ε

= +

SP Dispersion Relation

1/ 2

' "

x x x

m d

k k ik

c

ε ε ω

ε ε

⎛ ⎞

= + = ⎜ ⎝ + ⎟ ⎠

x-direction:

For a bound SP mode:

k

must be imaginary: ε

+ ε

< 0

k’

must be real: ε

< 0 So,

2 1/ 2

'

zi zi zi

m d

k k ik

c

ε ω

ε ε

⎛ ⎞

= + = ± ⎜ ⎝ + ⎟ ⎠

z-direction:

2

2 2

zi i x

k k

c ε ^{⎛ ⎞} ω

= ⎜ ⎟ −

⎝ ⎠ Dispersion relation:

Dispersion relation for surface plasmon polaritons

' "

m m

i

ε = ε + ε

'

m d

ε < − ε

2 2

2 2

zi i x x i x i

k k i k k

c c c

ω ω ω

ε ^{⎛ ⎞} ε ^{⎛ ⎞} ε ^{⎛ ⎞}

= ± ⎜ ⎟ ⎝ ⎠ − = ± − ⎜ ⎟ ⎝ ⎠ ⇒ > ⎜ ⎟ ⎝ ⎠

+ for z < 0 - for z > 0

Plot of the dispersion relation

2 2

1 )

( ω

ω ω

ε

= −

m d x

m d

k c

ω ε ε

ε ε

= +

• Plot of the dielectric constants:

• Plot of the dispersion relation:

d p sp

d m

ω

ω ε

ω ε

ε

= +

≡

∞

→

⇒

−

→

•

1 , k

, When

x

2 2 2 2

) 1

(

) (

p d

d p sp

x

k c

k ε ω ω

ε ω ω ω

− +

= −

=

Surface plasmon dispersion relation:

2 / 1

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

= +

d m

d m

x

c

k ε ε

ε ε ω

ω

ω

d p

ε ω

+ 1

Re k _x

real k

real k

imaginary k

real k

real k

imaginary k

d

ck

ε

Bound modes Radiative modes

Quasi-bound modes

Dielectric:

ε_d

Metal: ε_m= ε_m^'+ ε_m^"

x z

(ε'

> 0)

(−ε

< ε'

< 0)

(ε'

< −ε

)

2 2 2 2

p c kx

ω

ω

Surface plasmon dispersion relation

2 1/ 2 i zi

m d

k c

ε ω

ε ε

⎛ ⎞

= ⎜ ⎝ + ⎟ ⎠