Dispersion relation of surface plasmons

(1)

Dispersion relation of surface plasmons

“Metal Optics”

Prof. Vlad Shalaev, Purdue Univ., ECE Department, http://shay.ecn.purdue.edu/~ece695s/

“Surface plasmons on smooth and rough surfaces and on gratings”

Heinz Raether(Univ Hamburg), 1988, Springer-Verlag

“Surface-plasmon-polariton waveguides”

Hyongsik Won, Ph.D Thesis, Hanyang Univ, 2005.

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Metal Optics: An introduction

Photonic functionality based on metals?!

(3)

Note: SP is a TM wave!

What is a plasmon?

(4)

Plasmon-Polaritons

Plasmons in the bulk oscillate at ω

_p

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 (localized plasmons)

+ + +

- - -

+ - +

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

=

2 2 2 2

) 1

(

) (

p d

d p sp

x

k c

k ε ω ω

ε ω ω ω

− +

= −

=

(5)

Local field intensity depends on wavelength

(small propagation constant, k) (large propagation constant, k)

(6)

Ekmel Ozbay, Science, vol.311, pp.189-193 (13 Jan. 2006).

Some of the challenges that face plasmonics research in the coming years are

(i) demonstrate optical frequency subwavelength metallic wired circuits

with a propagation loss that is comparable to conventional optical waveguides;

(ii) develop highly efficient plasmonic organic and inorganic LEDs with tunable radiation properties;

(iii) achieve active control of plasmonic signals by implementing electro-optic, all-optical, and piezoelectric modulation and gain mechanisms to plasmonic structures;

(iv) demonstrate 2D plasmonic optical components, including lenses and grating couplers, that can couple single mode fiber directly to plasmonic circuits;

(v) develop deep subwavelength plasmonic nanolithographyover large surfaces;

(vi) develop highly sensitive plasmonic sensors that can couple to conventional waveguides;

(vii) demonstrate quantum information processing by mesoscopic plasmonics.

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To derive the Dispersion Relation of Surface Plasmons,

let’s start from the Drude Model

of dielectric constant of metals.

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Dielectric constant of metal

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

(9)

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

( : relaxation time )

The conduction current density

2 2

14

0 1 10

G G G

G JJG G JJG

JG

C

d r m d r d v m

m C r e E m v e E

dt dt

dt

s J

τ τ

τ γ

⁻

=

= − − − ⇒ + = −

= ≈

=

{ }

.

For a harmonic wave ( ) Re ( ) , the current desity varies at the same rate ( ) Re ( )

( ) ( ) ( )

( / )

( ) (

( / )

2

1

1 G

JG JG JJG

JJG JJG

JG

JG JJG

o i t

Nev

d J Ne

J E

dt m

E t E e

J t J e

i J Ne E

m ne m

J E

i

ω

τ

ω

ω ω ω

τ

ω ω

τ ω

−

⎛ ⎞

+ = ⎜ ⎟

⎝ ⎠

=

⎛ ⎞

− + = ⎜ ⎟

⎝ ⎠

= − )

Lorentz model

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

Drude model

Drude model: Metals

Local approximation to the current-field relation

(10)

those localized in interband (modified Drude model)

Maxwell Equations

( )

_{o B}E t

( )

H D t J J

t t

ε ε

∂ ∂

∇× = + = +

∂ ∂

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

o B

o B o B

o

H E t J i E E i E

t i

ε ε ωε ε ω ω σ ω ω ωε ε ω σ ω ω

ωε

⎡ ⎤

∇× = ∂ ∂ + = − + = − ⎢ ⎣ − ⎥ ⎦

: Static (ω =0) conductivity

(11)

2 2

2 2 2 2

2 2 3 3

( ) ( )

(1 )

1 1 (1 )

1

EFF B

o

o o

B B

o o

p p

B

i

i i i

i i ε ω ε σ ω

ε ω

σ σ ωτ

ε ε

ε ω ωτ ε ω ω τ

ω τ ω τ

ε ω τ ωτ ω τ

= +

⎛ ⎞ +

= + ⎜ ⎝ − ⎟ ⎠ = + +

⎛ ⎞ ⎛ ⎞

= ⎜ ⎜ ⎝ − + ⎟ ⎟ ⎠ + ⎜ ⎜ ⎝ + ⎟ ⎟ ⎠

2

where,

2_p ^o

: bulk plasma frequency ( 10 for metal)

o o e

ne eV

m ω σ

ε τ ε

= = ∼

Dielectric constant of metal : Drude model

Plasma frequency

(12)

From Microscopic origin of ω-response of matter Al¹³: 1S²2S²2P⁶3S²3P¹

Aluminum

(13)

Ag⁴⁷: 1s²… 4s²4p⁶4d¹⁰5s¹ Au⁷⁹: 1s²… 5s²5p⁶5d¹⁰6s¹

(14)

Ag⁴⁷: 1s²… 4s²4p⁶4d¹⁰5s¹

The fully occupied 4d band of Agcan influence the dynamical surface response in two distinct ways:

• First, the s-d hybridization modifies the single-particle energies and wave functions.

As a result, the nonlocal density-density response function exhibits band structure effects.

• Second, the effective time-varying fields are modified due to the mutual polarization of the s and d electron densities.

Here we focus on the second mechanism for the following reason:

At the parallel wave vectors of interest, the frequency of the Ag surface plasmon lies below the region of interband transitions. Thus, the relevant single-particle transitions all occur within the s-p band close to the Fermi energy where it displays excellent nearly-free-electron character.

A. Liebsch, “Surface Plasmon Dispersion of Ag,” PRL 71, 145-148 (1993).

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Measured data and model for Ag:

ω γ ε ω

ω

ε ω

₃

2 2

2

"

, 1

' = −

^p

=

^p

ω γ ε ω

ω ε ω

ε ' =

_∞

−

₂²^p

, " =

₃²^p

Drude model:

Modified Drude model:

Contribution of bound electrons Ag: ε

_∞

= 3 . 4

200 400 600 800 1000 1200 1400 1600 1800

-150 -100 -50 0 50

Measured data:

ε ' ε "

Drude model:

ε ' ε "

Modified Drude model:

ε '

ε "

ε

Wavelength (nm)

ε '

-ε

_d

Bound SP mode : ε’

_m

< -ε

_d

( ) ^ω

ε

Wavelength (nm)

(16)

Ideal case : metal as a free-electron gas

• no decay (γ Æ 0 : infinite relaxation time)

• no interband transitions (ε

_B

= 1)

2 2 2 2 2

1

2 2 3 3 2

( ) ( )

1

^B

1

EFF B

o

p p p

B EFF

i

^τ_ε

ε ω ε σ ω

ε ω

ω τ ω τ ω

ε ε

ω τ ωτ ω τ

^→∞⁼

ω

= +

⎛ ⎞ ⎛ ⎞ ⎛ ⎞

= ⎜ ⎜ ⎝ − + ⎟ ⎟ ⎠ + ⎜ ⎜ ⎝ + ⎟ ⎟ ⎠ ⎯⎯⎯ → = − ⎜ ⎜ ⎝ ⎟ ⎟ ⎠

2 1 ^p 2 r

ε ω

= − ω ⁰

(17)

Dispersion relation of surface-plasmon

for dielectric-metal boundaries

(18)

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

( ) k

ω ω =

• Dispersion relation:

(19)

Dispersion relation for surface plasmons

TM wave

ε _m ε _d

xm xd

E = E H

_ym

= H

_yd

ε

_m

E

_zm

= ε

_d

E

_zd

• At the boundary (continuity of the tangential E

_x

, H

_y

, and the normal D

_z

):

Z > 0

Z < 0

(20)

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

ε

ε ⁼

) ,

0 ,

( − i ωε

_i

E

_xi

− i ωε

_i

E

_zi

)

, 0 ,

( − ik

_zi

H

_yi

ik

_xi

H

_yi

xi i yi

zi

H E

k = ωε

xd d yd

zd

H E

k = ωε

(21)

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

(22)

1/ 2

' "

^{m d}

x x x

m d

k k ik

c

ε ε ω

ε ε

⎛ ⎞

= + = ⎜ ⎝ + ⎟ ⎠

x-direction:

For a bound SP mode:

k

_zi

must be imaginary: ε

_m

+ ε

_d

< 0

k’

_x

must be real: ε

_m

< 0 So,

2 1/ 2

'

ⁱ

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

m

ε = ε + ε

'

m d

ε < − ε

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

(23)

( ) ( )

( )

2 1

" 2 4

2

" 2 2

1

" 2 2

'

"

2 1

" 2 4

2 2 1

" 2 2

' '

) 2 (

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

⎭ ⎬

⎫

⎩ ⎨

⎧ + +

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

+

= +

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ + +

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

+

= +

d m e

e

d m m

d m x d

d m e

e m

d m x d

k c k c

ε ε ε

ε

ε ε ε

ε ε

ε ω

ε ε ε

ε ε

ε ω

( ) ( )

^' ² ^" ² ^'

2

,

_e _m _m _d _m

where ε = ε + ε + ε ε

1/ 2

' " m d

x x x

m d

k k ik

c

ε ε ω

ε ε

⎛ ⎞

= + = ⎜ ⎝ + ⎟ ⎠

metals, of

most in

and ,

,

0

^' ^' ^"

'

m m

d m

m

ε ε ε ε

ε < > >>

,

( )

^' ²

"

2 / 3 '

'

"

2 / 1 '

' '

2

_m

m d

d d m x

d m

d x m

k c k c

ε ε ε

ε ε ε ω

ε ε

ε ε ω

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

≈ +

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

≈ +

' "

m m

i

m

ε = ε + ε

(24)

Plot of the dispersion relation

2 2

1 )

( ω

ω ω

ε

_m

= −

^p

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

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

− +

= −

=

(25)

Surface plasmon dispersion relation:

2 / 1

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

= +

d m

x

c

k ε ε

ε ε ω

ω

_p

d p

ε ω

+ 1

Re k _x

real k

_x

real k

_z

imaginary k

_x

real k

_z

real k

_x

imaginary k

_z

d

ck

x

ε

Bound modes Radiative modes

Quasi-bound modes

Dielectric:

ε_d

Metal: ε_m= ε_m^'+ ε_m^"

x z

(ε'

_m

> 0)

(−ε

_d

< ε'

_m

< 0)

(ε'

_m

< −ε

_d

)

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

⎛ ⎞

= ⎜ ⎝ + ⎟ ⎠

(26)

Very small SP wavelength

λ

_vac

=360 nm

X-ray wavelengths at optical frequencies

Ag

SiO

₂

(27)

2 / 1

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

= +

d m

d x m

k c

ε ε

ε ε ω

2 2 2 2

2 2 3 3

1 1

p p

m

ω τ i ω τ

ε ω τ ωτ ω τ

⎛ ⎞ ⎛ ⎞

= − ⎜ ⎜ ⎝ + ⎟ ⎟ ⎠ + ⎜ ⎜ ⎝ + ⎟ ⎟ ⎠

Dispersion relation for bulk and surface plasmons

d p sp

d p d

p d

p

m

ω ω ω

ε ω ε

ε ω ω

ω ε ε ω

= + + ⇒

=

⇒

−

=

−

=

−

= 1 , 1 1

When

2 2

D

Cut-off frequency of SP

(28)

Measured data and model for Ag:

ω γ ε ω ω

ε ω

₂

2 2

2

"

, 1

' = −

^p

=

^p

ω γ ε ω ω

ε ω

ε

2

2 2

2

"

,

' =

_∞

−

^p

=

^p

Drude model:

Modified Drude model:

Contribution of bound electrons Ag: ε

_∞

= 3 . 4

200 400 600 800 1000 1200 1400 1600 1800

-150 -100 -50 0 50

Measured data:

ε ' ε "

Drude model:

ε ' ε "

Modified Drude model:

ε '

ε "

ε

Wavelength (nm)

ε '

-ε

_d

Bound SP mode : ε’

_m

< -ε

_d

( ) ^ω

ε

Wavelength (nm)

(29)

( )

^" ¹ ^" ' ^{1 2}^' ^{3 / 2} ¹^"' 2

1 2 1

The length after which the intensity decreases to 1/e :

2 , where

i x x

2( )

L k k

c

ε ε ε

ω

ε ε ε

−

⎛ ⎞

= = ⎜ ⎝ + ⎟ ⎠

Propagation length

(30)

2 2 2 2 ' "

2 2 3 3

1

p p

m m

i

m B

ω τ i ω τ

ε ε ε ε

ω τ ωτ ω τ

⎛ ⎞ ⎛ ⎞

= + = ⎜ ⎜ ⎝ − + ⎟ ⎟ ⎠ + ⎜ ⎜ ⎝ + ⎟ ⎟ ⎠

Propagation length : Ag/air, Ag/glass

(31)

Penetration depth

'

1 2

At large (

x

), z

i

1 .

x

k k

ε → − ε ≈ Strong concentration near the surface in both media.

'

At low ( k

_x

ε

1

>> 1),

^'

_{Larger E}

_z

_component

1

in air :

z x

E i

E = − ε

( : , : - )

z x

E = ± iE air + i metal i

Smaller E

_z

component

' 1

1 in metal :

z x

E i E ⁼ ε

Gooood waveguide!

(32)

(33)

Another representation of SP dispersion relation

c k

_p _p

m m m

/

2

"

'

ω

ν ε

ε ε

=

−

= +

=

(34)

Excitation of surface plasmons

(35)

d d

m d x m

k c

k c ω ε

ε ε

ω _> ₌

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

= +

2 / 1 '

' '

x

sin

sp p

c k

ω θ

= ε

o d

c k ω ⁼ ε

'

k

x

ω

excitation

'

sin

x p sp sp

k k

c

ε ω θ

⎛ ⎞

= ⎜ ⎝ ⎟ ⎠ =

1

p sp

d

ω ω

= ε +

Excitation of surface plasmon by prism

o p

c k ω ⁼ ε

ε

_p

ε

_d

1

p sp

p

ω ω

= ε +

ω

p

2 2 2 2

p c k

ω = ω +

(36)

Generalization :

Surface Electric Polaritons and Surface Magnetic Polaritons

: Energy quanta of surface localized oscillation of electric or magnetic dipoles in coherent manner

Surface Electric Polariton (SEP) Surface Magnetic Polariton (SMP)

+q -q +q -q N S N S

E K K

H

Coupling to TM polarized EM wave Coupling to TE polarized EM wave

Common Features

- Non-radiative modes → scale down of control elements - Smaller group velocity than light coupling to SP

- Enhancement of field and surface photon DOS

(37)

Generalization :

Surface Electric Polaritons and Surface Magnetic Polaritons

1 2 2 1 1 2

2 2 0

2 1

' ' ( ' ' ' ' ) " "

' ' ' , '

SEP

ε ε ε μ ε μ

k O

ε μ

β ε ε ε μ

⎛ ⎞

= − − + ⎜ ⎝ ⎟ ⎠

,1 ,2

1 1 2 2

, 2 2 0

2 1 1 2

' ( ' ' ' ' ) " "

, ,

' ' ' ' ' '

SEP SEP

i

SEP i

ε ε μ ε μ

k O

ε μ γ γ

γ ε ε ε μ ε ε

⎛ ⎞

= − − + ⎜ ⎝ ⎟ ⎠ =

Dispersion Relation & Decay Constants

( ^{"/ ' ,} ^"/ ^' ) ^{(1,1) ,}

For ε ε μ μ << {

J. Yoon, et al., Opt Exp 2005.

(38)

In Summary

1/ 2 d m

SPP

d m

k c

ω ε ε

ε ε

⎛ ⎞

= ⎜ ⎝ + ⎟ ⎠

Dispersion relations

2 2

2 2 2 2

2 2

( ) 1

1 /

p p

m

p

ω i ω γ

ε ω ω γ ω γ ω

ω ω

= − + + + ⎛ ⎞ ⎜ ⎟ ⎝ ⎠

≈ −

Permittivity of a metal

(39)

Type-A

- Low frequency region (IR) - Weak field-confinement

- Most of energy is guided in clad - Low propagation loss

► clad sensitive applications

► SPP waveguides applications

Type-A : low k

H. Won, APL 88, 011110 (2006).

(40)

Type-B

- Visible-light frequency region - Coupling of localized field

and propagation field - Moderated field enhancement

►

Sensors, display applications

► Extraordinary transmission of light

Nano-hole

Type-B : middle k

(41)

Type-C

- UV frequency region - Strong field confinement - Very-low group velocity

► Nano-focusing, Nano-lithography

► SP-enhanced LEDs

n-GaN

p-GaN (20nm, 120nm) Ag (20nm)

Λ

Light emission

QW QW

2

0

1 1

( ) 2 ( )

R f i ρ ω

τ ω ε

= = p E ⋅

= SE Rate :

Electric field strength

of half photon (vacuum fluctuation)

Photon DOS

(Density of States)

Type-C : high k

(42)

Importance of understanding the dispersion relation :

Broadband slow and subwavelength light in air

(43)

Mode conversion: negative group velocity

Re k _x

SiO

₂

Si

₃

N

₄

1

_SiO2 p

ε ω +

4

1

_Si3_N p

ε ω +

SiO₂ Si₃N₄ Al

= 0 dk d ω

< 0

dk

d ω

(44)

Appendix

(45)

(46)

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.