Surface Plasmon-polaritons on thin metal films

Surface Plasmon-polaritons on thin metal films

Dielectric – ε

Dielectric – ε

Metal – ε

References

Surface plasmons in thin films,

E.N. Economou, Phy. Rev. Vol.182, 539-554 (1969)

Surface-polariton-like waves guided by thin, lossy metal films,

J.J. Burke, G. I. Stegeman, T. Tamir, Phy. Rev. B, Vol.33, 5186-5201 (1986)

Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization, J. A. Dionne,* L. A. Sweatlock, and H. A. Atwater, Phy. Rev. B, Vol.73, 035407 (2006)

Geometries and materials for subwavelength surface plasmon modes, Rashid Zia, Mark D. Selker, Peter B. Catrysse, and Mark L. Brongersma, J. Opt. Soc. Am. A, Vol. 21, 2442-2446 (2004)

Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,

P. Berini, Phy. Rev. B, Vol.61, 10484 (2000)

Introduction:

When the film thickness becomes finite.

mode

overlap

Introduction:

Possibility of Propagation Range Extension

frequency

in-plane wavevector

Long-Range SP:

weak surface confinement, low loss

Short-Range SP:

strong surface confinement, high loss

Introduction: Extremely long-range SP ?

in-plane wavevector

frequency

Symmetrically coupled LRSP

Anti-symmetrically coupled LRSP

Introduction:

Dependence of dispersion on film thickness

practically forbidden

200 400 600 800

-1 -0.75 -0.5 -0.25 0.25 0.5 0.75 1

200 400 600 800

-1 -0.75 -0.5 -0.25 0.25 0.5 0.75 1

6 0 h = n m

250 500 750 1000 1250 1500

-1 -0.75 -0.5 -0.25 0.25 0.5 0.75 1

250 500 750 1000 1250 1500

-1 -0.75 -0.5 -0.25 0.25 0.5 0.75 1

1 0 h = n m

LOCAL THEORY FOR MULTIPLE-FILM SYSTEM

Maxwell equations with a local current-field relation as follows:

For electric (or TM) waves that H

= H

= E

= 0 in all of the media,

Local approximation to the current-field relation with

The local approximation satisfies the dielectric function in the metal,

Surface plasmons in thin films

E.N. Economou, Phy. Rev. Vol.182, 539-554 (1969)

the solution for any component of the fields can thus be represented in the form

A. Single Metal-Dielectric Interface

The solution for E

that remains finite at infinity is

The continuity of E and H fields across

the boundary gives the dispersion relation as

If the dielectric constant for the insulator ε

=1,

( )

( )

/ 1

/

metal insulator

R K

K ε ε

⎡ ⎤

≡ − ⎢ ⎥ =

⎢ ⎥

⎣ ⎦

m d x

m d

k c

ε ε ω

ε ε

= +

2 2

) 1 (

) (

p d

d p sp

x

k c

k ε ω ω

ε ω ω ω

− +

= −

=

It is obvious that retardation effects are important for q = (k/k

) < 1 and that they do not play any role for q>>1.

2 2

2 2 2 2 2

,

if

1

m m

c c

k k c k ω

ω ω ω ω ε

ε ε ω

= ⇒ = ± = ± + = −

Dispersion of bulk plasmon:

Retardation (radiative loss)

Therefore, we are interested in region IV.

B. Insulating Film between Two Semi-Infinite Metals

Branches I and II are adequately described by the longitudinal electrostatic theory when k >> k

,

If ε

=1, the low-frequency k < k

, part of I,

If k

d

<< l, as is usually the case, branch III is

If k

d

<< l, the k << k

portion of II is

and, at k = 0

C. Metal Film in Vacuum

When k < k

, for branch II (antisymmetric oscillation)

When k < k

, for branch I (symmetric oscillation)

When k > k

, both I and II approach

2 2 ,

, 2

i m

Ki m k c

= −ω ε

( )

( )

/ /

metal insulator

R K

K ε ε

⎡ ⎤

≡ − ⎢ ⎥

⎢ ⎥

⎣ ⎦

* Burke, PRB 1986 :

2 2 2

, 1, 2, 3.

n n o

S = k − ε k n =

D. Swihart's Geometry

When k << k

, for branch I

Branch II just R=1, corresponds to oscillations on the external interface (insulator/metal).

Branch III,

The intersection of branch III with the line ω=ck occurs, when

Branch IV behaves for small k as

E. Two Metal Films of Different Thicknesses

Where,

When k << k

,

branch II, III

The branch II and III

Where,

Branch II Branch III

Branch II

Branch III

F. Two Dielectric Films

For k<<k

branches I and II are given by

In the special case when d

=d

=d

they correspond to symmetric and antisymmetric solutions,

And, in the limit when

Branch III starts, when k= 0,

Branch III starts, when k= 0,

For k<k

Branch IV is, for small k,

It corresponds to an oscillation which couples the two

junctions, and, when one of the thicknesses of the

dielectric films becomes large, the coupling is broken

and the oscillation is confined to one of them.

G. Three Metal Films

When k<<k

branches I and II are given by

When k<<k

branches III and IV:

When k<<k

branches V and VI

Periodicity of alternating thicknesses d

and d

, implies that the eigensolutions obey the Floquet-Bloch theorem; namely,

The secular equation is

Solutions are found in the shaded regions.

When k<<k

curves III and IV can be taken as straight lines with phase velocities,

Any intermediate solution has phase velocity

H. Periodic structure of alternating metal and insulating films

The upper region within which solutions lie is bounded by I, a portion of IIa, and IIb.

If (k

d

d

)<< 1, Curve I is

while curve IIa is

Solutions near I are given by

When k>>k

the secular equation is

Conclusions of Economou

¾ Modes of SPO in multiple-film systems can be classified into two main groups.

9 One groupcontains those modes whose dispersion curves start from zero frequency at k=O, increase as k increases, but remain below the line ω= ck.

9 The other groupstarts at k= 0 from ω= ω_p, or a value slightly less than ω_p, and remains close to the line ω= ω_p. For very large k, all the dispersion curves of both groups converge asymptotically to the classical surface

plasmon frequency ω_p/root(2).

¾ In addition to these two groups, some uninteresting modes may appear with dispersion curves that lie just below the curve ω²= ω_p²+ c²k²which corresponds to the trivial solution of zero fields.

¾ For normal metals this description is valid only for high enough frequenciesso that oscillation damping is negligible.

On the other hand, for superconducting metals, the picture is valid not only for the high-frequency region but also for low frequencies.

¾ In Multiple-film structures radiative SPO exists, which should have observable effects in the radiation properties of these structures. In particular, there seems to be a possibility of obtaining intense radiation as the number of the films

increases.

Surface-polariton-like waves guided by thin, lossy metal films

J.J. Burke, G. I. Stegeman, T. Tamir, Phy. Rev. B, Vol.33, 5186-5201 (1986).

Burke, PRB 1986

Dispersion relations for waves guided by a thin, lossy metal film surrounded by dielectric media Characteristic of "spatial transients" :

Usual symmetric and antisymmetric branches each split into a pair of waves

Æ one radiative (leaky waves) and the other nonradiative (bound waves).

Symmetric modes : the transverse electric field does not exhibit a zero inside the metal film Antisymmetric modes : the transverse electric field has a zero inside the film.

ε

= - ε

– i ε

h

z

x ε

ε

ε

> ε

Dispersion relation for thin metal films (3 layers) obtained from the Maxwell equations

( )

( , , ) y 0 ( ) exp

i i

H x z t =e H f z ⎡⎣i βx−ωt ⎤⎦

[ ] ( )

[ ] [ ] ( )

[ ] ( )

3

2 0 2

1

exp ( ) in medium 3

( ) exp ( ) exp in medium 2 0

exp in medium 1 0

i h

B s z h z h

f z A s z h A s z z h

s z z

⎧ − − ≥

=⎪⎨ − + − ≤ ≤

⎪ ≤

⎩

2 2 2

0

j j j

s

= β − ε

[ ] [ ]

0

0

exp 0

exp ( )

( )

y x

y

j

j

z y

i H i

E H i x

i z

E

E H H i x

df z dz

f z ωε ωε β

ωε β β β

ωε ωε

⎧ = − ∂ = −

⎪ ∂

= ∇× → ⎪⎪⎨ =

⎪ − −

⎪ = =

⎪⎩

E H

[ ]

[ ] ( )

[ ] [ ]

{ } ( )

[ ] ( )

3

3 3

2

0 2 0 2

2

1

1 1

exp ( )

exp exp ( ) exp 0

exp 0

x h

s B s z h z h

i s

E H i x A s z h A s z z h

s s z z

ε

ωε β ε

ε

⎧− − − ≥

⎪⎪

− ⎪⎪

= ×⎨ − − − ≤ ≤

⎪⎪

⎪ ≤

⎪⎩

( )

( )

1 2 2 0

2 3 2 0

0 : exp[ ] 1

: exp[ ]

x x h

x x h

z H H s h A A

z h H H A s h A B

= = ⇒ − + =

⎧⎪⎨

= = ⇒ + − =

⎪⎩

From the boundary conditions,

2 1

1 2 2 0

1 2

2 3

2 3 2 0

3 2

0 : exp[ ]

: exp[ ]

x x h

x x h

z E E s h A A s

s

z h E E A s h A s B

s ε ε

ε ε

⎧ = = ⇒ − − =

⎪⎪⎨

⎪ = = ⇒ − − = −

⎪⎩

[ ] [ ] [ ] ( )

[ ] [ ] ( )

[ ] ( )

2 1

2 2 3

1 2

2 1

2 2

1 2 1

cosh sinh exp ( ) 3 :

( ) cosh sinh 2 : 0

exp 1: 0

j

s h s s h s z h j z h

s

f z s z s s z j z h

s

s z j z

ε ε ε ε

⎧⎛ ⎞

+ − − = ≥

⎪⎜ ⎟

⎝ ⎠

⎪⎪⎪

=⎨ + = ≤ ≤

⎪⎪ = ≤

⎪⎪⎩

From the equations at z = 0, A

, A

, and B can be determined by,

Therefore, anther equations at z = h gives the dispersion relation,

( ^{ε ε}

^s

^{+ ε}

^{s s}

) ^{tanh [ ] s h}

^{+ ε}

^{s ( ε}

^{s + ε}

^{s ) = 0}

2 2 2

0

j j j

s = β − ε k

Dispersion relation when

Burke, PRB 1986

When h >> c/ω_p(classical skin depth), tanh(S₂h) Æ 1,

SP₁ :

SP₃:

2

1

ε ω

= − ω

h → ∞

The solutions consist of decoupled surface-plasmon polaritons (SPP) :

propagating along the ε₁-ε_minterface

propagating along the ε₃-ε_minterface

1,3

( )

i

i m

h c

ε ε β ω

=

→ ∞ = ε ε

+

If we assume that β

>> β

and ε

= ε

+ i ε

Burke, PRB 1986

If β > ε ω

/ (light line), c

Hence S

is real.

But, there are two types of solutions for the semi-infinite media SPPs :

2 2 2

i i o

0

S = β − ε k >

(1) S_i

> 0, SPPs are nonradiative, or bound

(2) S_i

< 0, SPPs are grow exponentially with distance from the interface,

which are physically rejected because of their non-guiding property

If β < ε ω

/ (light line), c

Æ Therefore, SPPs, are bound at the semi-infinite media only when S

are real and positive.

For a finite film thickness,

the two allowed semi-infinite SPPs are coupled.

One of the SPPs could become leaky (radiative) in the ε

medium when ε

> ε

2

0

Si

<

Hence S

is imaginary.

Æ The field is a plane wave radiating away form the metal boundary.

[ ] ( )

[ ] [ ] ( )

[ ] ( )

3

2 0 2

1

exp ( ) in medium 3

( ) exp ( ) exp in medium 2 0

exp in medium 1 0

i h

B s z h z h

f z A s z h A s z z h

s z z

⎧ − − ≥

=⎪⎨ − + − ≤ ≤

⎪ ≤

⎩

Burke, PRB 1986

There are two types of solutions for S_i> 0 : symmetric (s), antisymmetric (a)

1 3

(1) :S >0 & S >0

h → ∞

The fields grow exponentially with wave front tilted to carry energy away from the metal

The fields decay exponentially into both ε₁and ε₃,

and the wave fronts are tilted in towards the metal film. -> nonradiative waves

The field in ε₁and the metal is guided by the interface, the filed in ε₃grows exponentially (leaky).

Waves guided by symmetric structures ( ε

= ε

)

There are four types of solutions satisfying :

We can estimate the properties of the solutions as follows:

0 h →

1 3

S S

± = ± (2) :S₁<0 & S₃ <0

1 3

(3) :S >0 & S <0

1 3

S S

± = ∓ (4) :S₁<0 & S₃ >0

1 3

(1) :S >0 & S >0

1 3

(2) :S <0 & S <0

1 3

(3) :S >0 & S <0

1 3

(4) :S <0 & S >0

The field in ε₃and the metal is guided by the interface, the filed in ε₁grows exponentially (leaky).

2 2 2

i i o 0

S =β −εk >

ε₁ εm

ε3

Leaky (ε₁) H_y

Leaky (ε3)

ε₁ εm

ε₃

H_y

Bound (a) ε1

εm

ε3

H_y

Bound (s)

ε1

ε_m ε3

Leaky (ε1) H_y

Bound (ε3) ε₁

εm

ε₃

H_y Leaky (ε3) Bound (ε₁)

Two nonradiative, Fano modes

Two radiative, leaky modes

Four SPP modes 2 SP solutions

1 SP solution

1 SP solution No solution

ε₁ ε_m ε₃

H_y

Bound (a) ε1

ε_m ε₃

H_y

Bound (s)

Two nonradiative, Fano modes

1 3

(1) : S > 0 & S > 0

Symmetric bound (s_b) Asymmetric bound (a_b)

Surface plasmon dispersion for thin films

Drude model ε

(ω)=1-(ω

/ω)

Two modes appear

L _-

L

(asymmetric)

Thinner film:

Shorter SP wavelength

Example:

λ

= 633 nm λ

= 60 nm

L

(symmetric)

Propagation lengths: cm !!!

(infrared)

0 50 100 150 200 1.5

1.6 1.7 1.8 1.9 2.0

a

s

β

/k

thickness (h : nm)

0 50 100 150 200

1E-7 1E-6 1E-5 1E-4 1E-3 0.01 0.1

1

a

b

s

β

/k

thickness (h : nm)

( ^{ε ε}

^s

^{+ ε}

^{s s}

) ^{tanh [ ] s h}

^{+ ε}

^{s ( ε}

^{s + ε}

^{s ) = 0}

2 2 2

0

j j j

s = β − ε k

2 1 3

1.55 m , 118 i 11.58 , 2.25

λ = μ ε = − + ε = ε =

Ex)

ε1

ε_m ε₃

H_y

Bound (a) ε1

ε_m ε₃

H_y

Bound (s)

s

a

Waves guided by asymmetric structures

Burke, PRB 1986

One antisymmetric mode is always obtained.

The "symmetric" solutions are of two types, nonradiative (bound) and nonradiative (leaky):

( ε

≠ ε

)

1 3 & 1 3

S S S S

± = ± ± = ∓

There are also four types of solutions satisfying

ε

ε

ε

H

Bound (a)

ε

ε

ε

H

Bound (s)

ε

ε

ε

Leaky (ε₁)

H

Bound (ε₃)

ε

ε

ε

H

ε

ε

ε

H

Growing (x)

z x

1 3

(1) :S >0 & S >0

1 3

(2) :S <0 & S <0 (3) :S1 >0 & S3 <0 (4) :S₁<0 & S₃ >0

2 2 2

i i o 0

S =β −εk >

ε

ε

ε

Leaky (ε₁)

H

Leaky (ε₃)

1 3

For example, 0 & 0

when - and -

R I

i iR iI i iR iI

S

S S iS i

β

β β β

< >

= =

2 2 2

Bound and Leaky modes when S

= β − ε

k

> 0 Growing modes

Waves guided by asymmetric structures

Burke, PRB 1986

Nonradiative mode

the fields in the dielectric decay exponentially away from the film and the wave fronts are tilted into the metal film, in order to remove energy from the dielectric media (for dissipation in the metal) as the wave attenuates with propagation distance.

Leaky (radiative) mode

The wave energy is localized in one of the dielectrics, say ε₁, at that dielectric-metal interface.

The wave amplitude decays exponentially across the film, and then grows exponentially into the other dielectric medium, ε₃in this case. In the ε₁medium, the wave fronts are tilted towards the film to supply energy from ε₁for both dissipation in the metal, and radiation into ε₃.

Leaky (radiative) mode

The analogous case of localization in ε₃, and radiation into ε₁.

Growing mode

The field amplitude grows both with propagation distance as exp(β_Ix), and into one of the dielectrics as exp(S_Rz). Since the wave-front tilt is into the film (as opposed to away from it for leaky waves), these waves are dependent on externally incident fields supplying energy to make the total wave amplitude grow.

( ε

≠ ε

)

Burke, PRB 1986

Leaky waves

They only have meaning in a limited region of space above the film and require some transverse plane (say x=O) containing an effective source that launches a localized wave in one dielectric near its metal-dielectric boundary. The field decays across the metal and couples to radiation fields in the opposite dielectric.

The ε₁field amplitude grows exponentially for only a finite distance z,In this sense, the solutions do not violate boundary conditions as z Æ- infinite.

θ

For fields radiated at an angle θ relative to the surface, the angular spectrum of the radiated plane waves is

Thus, the radiated power is

The wave attenuation due to radiation loss can be estimated from the solutions by calculating the Poynting vector for energy leaving, for example, the ε₁-ε_mboundary, per meter of wave front,

Dispersion relation for thin metal strips with finite widths

metal strip

dielectric

Finite film thickness and width

Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,

P. Berini, Phy. Rev. B, Vol.61, 10484 (2000)

( )

( )

( , , , ) ( , ) ( , , , ) ( , )

i z t

i z t

x y z t x y e x y z t x y e

β ω β ω

−

−

=

=

0 0

E E

H H

z-axis : propagation direction

1 2

r 0

( ε

) k

∇ × ∇ × H = H

From the Maxwell equations

1 1 2 2 1

r r 0 r

( ) ( ) ( ) 0

t

ε

ε

k β ε

∇ × ∇ × H − ∇ ∇ ⋅ H − − H =

t

i j

x y

∧

∂

∂

∇ = +

∂ ∂

( )

( )

t

H i

H j e

∧ ∧

= +

where H

Assume that all media be isotropic.

The magnetic field on x-y (transverse plane) satisfies

E

, H

: polarization direction

This eigenvalue problem can be solved by a numerical method

with proper boundary conditions, such as one of FDM, FEM, MoL, …

Here, we use the FDM (finite difference method).

y

FDM

0 50 100 150 200

1.50 1.52 1.54 1.56 1.58 1.60 1.62 1.64

β r/k

sa

ss

thickness of metal (nm)

0 50 100 150 200

1E-7 1E-6 1E-5 1E-4 1E-3 0.01

β

/k

sa

ss

o

thickness of metal (nm)

2 1 3

1.55 m , 118 i 11.58, 2.25, w 5 m

λ = μ ε = − + ε = ε = = μ

Ex)

y

x

x

y z

Plasmon slot waveguides : Metal-Insulator-Metal (MIM)

Plasmon slot waveguides:

Towards chip-scale propagation with subwavelength-scale localization

J. A. Dionne,* L. A. Sweatlock, and H. A. Atwater Phy. Rev. B, Vol.73, 035407 (2006)

Insulator ε

x

z

y

Metal ε

Metal ε

d

Insulator ε

x

z

y

Metal ε

Metal ε

d

Mode L+

Mode L-

Tangential (E

)

Electric Field Profiles

The curves for decoupled SP (infinite d) exhibit exact agreement

Asymmetric bound mode (L+) : a _b Symmetric bound mode (L-) : s _b

(infinite d) (infinite d)

(infinite d) (infinite d)

(infinite d)

conventional

waveguide modes In S

O

core

SP modes SP modes

D = 250 nm

(infinite d)

conventional

waveguide modes within ΔE ~ 1 eV

SP modes

D = 100 nm

S

The dispersion of the 50-nm-thick sample lies completely to the left of the decoupled SP mode.

Low-energy asymptotic behavior follows a light line of n = 1.5.

It suggests that polariton modes of MIM more highly sample the imaginary dielectric component.

In the low energy limit, the S

s _b SP : ^{D = 50 nm Æ 30 nm Æ 25 nm Æ 12 nm}

Insulator ε

x

z

y

Metal ε₂

Metal ε₂

d

a _b SP :

Purely plasmonic nature of the mode

The cutoff frequencies remains essentially unchanged, possibly by the Goos-Hanchen effect.

As waveguide dimensions are decreased, energy densities are more highly concentrated at the metal surface. This enhanced field magnifies Goss-Hanchen contributions significantly.

In the limit of d << s (skin depth), complete SP dephasing could result.

D = 50 nm Æ 30 nm Æ 25 nm Æ 12 nm

( D = 250 nm ) MIM (Ag/SiO

/Ag) TM-polarized propagation and skin depth

Forbidden band

a

a

s

s

Note that only a slight relation correlation between propagation distance and skin depth (σ).

The metal absorption is not the limiting loss mechanism in MIM structures.

( D = 12 nm, 20 nm, 35 nm, 50 nm, and 100 nm ) MIM (Ag/SiO

/Ag) TM-polarized propagation and skin depth

a _b s _b

σ ~ 20 nm

Evanescent within 10 nm for all wavelength Approximately constant in the Ag cladding.

Thus, MIM can achieve micron-scale propagation with nanometer-scale confinement.

Local minima corresponding to the transition between quasibound modes and radiation modes

Unlike IMI, extinction (prop. distance) is determined not by ohmic loss (metal absorption)

but by field interference upon phase shifts induced by the metal.

TE modes in MIM structures

(~ 4 eV: ~300 nm)

d = 250 nm d = 100 nm

EM energy density profiles of MIM structures (Ag/SiO2/Ag)

Geometries and materials

for subwavelength surface plasmon modes

Rashid Zia, Mark D. Selker, Peter B. Catrysse, and Mark L. Brongersma J. Opt. Soc. Am. A/Vol. 21, No. 12/December 2004, 2442-2446.

We demonstrate that, to achieve subwavelength pitches, a metal–insulator–metal geometry is required

with higher confinement factors and smaller spatial extent than conventional insulator–metal–insulator structures.

The resulting trade-off between propagation and confinement for surface plasmons is discussed,

and optimization by materials selection is described.

Consider the isotropic wave equation for

a generic three-layer plasmonic slab waveguide with metallic and dielectric regions,

For a guided surface-plasmon mode to exist,

where z is the propagation direction and thus k

is the conserved quantity.

If the radiation is unconfined in the y dimension (i.e., k

= 0),

Confinement of the MIM structure is limited by the decay length into the metallic regions, which can be approximated as follows for metals below the surface-plasmon resonance:

Note that this condition is met only near the surface plasmon resonance frequency.

Ultimate confinement of the IMI structure is limited by the decay length into the dielectric cladding. For confinement below the limit of a conventional dielectric waveguide (λ/2n),

2 π x (1/k

) < ( λ /2n)

( )

2 1/ 2

1/ 2 ,

metal

x metal d

metal d

k c c

ω ε ω ε

ε ε

⎛ ⎞

= ⎜ ⎝ + ⎟ ⎠ >

Power confinement factor (Γ) of field-symmetric TM modes

- MIM and IMI plasmonic waveguides - (Au–air, λ = 1.55 μm)

99.4%

2%

If plasmonic waveguides are intended to propagate light in subwavelength modes,

MIM geometries with higher confinement factors and shorter spatial extents are

much better suited for this purpose.

Appendix

2 2 2 2 ' "

2 2 3 3

1

p p

m m

i

ω τ i ω τ

ε ε ε ε

ω τ ωτ ω τ

⎛ ⎞ ⎛ ⎞

= + = ⎜ ⎜ ⎝ − + ⎟ ⎟ ⎠ + ⎜ ⎜ ⎝ + ⎟ ⎟ ⎠

Dispersion relation and Attenuation damping of surface

plasmons at Ag/air Ag/glass interface