Dispersion relation of SPPs on thin metal films - IMI (insulator-metal-insulator) structure -

Dispersion relation of SPPs on thin metal films - IMI (insulator-metal-insulator) structure -

Dielectric – 

₃

Dielectric – 

₁

Metal – 

₂

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

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

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

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



_m

= - 

_R

– i 

_I

h

z

x



₁



₃



₁



₃

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 xt 

   

     

   

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 i z h

f z A s z h A s z i z h

s z i z

    

      

  



2 2 2

0

j j j

s

   

k

 

 

0

0

exp 0

( )

( )exp

y x

j y

z y

j

j

j

i H i

E H i x

z

i E

E H

d

H i x

f z dz

f z

  

   

 

     

 



   

  

  



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

s

E iH i x A s z h A s z z h

s s z z



  



   



 

      



 



 ^s

^j

^{ k}

^zj

^

^j

^{ k}

^xj



 

 

1 2 2 0

2 3 2 0

0 : exp[ ] 1

: exp[ ]

y y h

y y h

z H H s h A A

z h H H A s h A B

      



     



From the boundary conditions at z = 0 and h,

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









      



       



       

     

   

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









 

    

 

 



    

  



A

_h

, A

_o

, and B can be determined by,

The E

_x

equation at z = h gives the dispersion relation,

 ^{ }

^{1 3 2}

^s

²

^{ }

²²

^{s s}

^{1 3}

 ^{tanh   s h}

²

^{ }

^{2 2}

^{s  }

^{3 1}

^{s  }

^{1 3}

^{s   0 s}

^j2

^{ }

^2j

^{ }

^j

^k

⁰²

 

( , , ) y 0 ( ) exp

i i

H x z t e H f z i xt 

   

     

   

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 i z h

f z A s z h A s z i z h

s z i z

    

      

  



 

   

   

  ^{ }

   

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

s

E iH i x A s z h A s z z h

s s z z



  



   



 

      



 



 

 

 

1 2 2 0

2 1

1 2 2 0

1 2

2 3 2 0

exp[ ] 1

0 :

exp[ ]

: exp[ ]

y y h

x x h

y y h

H H s h A A

z s

E E s h A A

s

z h H H A s h A B





     

 

    



     

2 3

2 3 2 0

3 2

: _{x x h} exp[ ] s

z h E E A s h A B

s



      

Dispersion relation when

Burke, PRB 1986

When h >> c/_p(classical skin depth), tanh(S₂h)  1,

SP₁ :

SP₃ :

2

1

^p2 m

 

  

h  

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

propagating along the ₁-_minterface

propagating along the ₃-_minterface

1,3

( )

^{i m}

i

i m

h c

 

 



    



If we assume that 

_R

 

_I

and 

_m

 

_R

 i 

_I

 ^{ }

^{1 3 2}

^s

²

^{ }

²²

^{s s}

^{1 3}

 ^{tanh   s h}

²

^{ }

^{2 2}

^{s  }

^{3 1}

^{s  }

^{1 3}

^{s   0}

 

2



m



 

zm zd

m d

k k

 

 

   

 

2 2 2

0

j j j

s ^  ^  k

Burke, PRB 1986

If    

_i

/ (light line), c

Hence S

_i

is real.

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

2 2 2

i i o

0

S     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

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

_i

are real and positive.

   

     

   

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     k

 ^{ }

^{1 3 2}

^s

²

^{ }

²²

^{s s}

^{1 3}

 ^{tanh   s h}

²

^{ }

^{2 2}

^{s  }

^{3 1}

^{s  }

^{1 3}

^{s   0}

Dispersion relation when h  

Burke, PRB 1986

1 13 3

If   / c      / , c For a finite film thickness,

the two allowed semi-infinite SPPs are coupled.

Either one of the propagation vectors, 

₁₃

, of the coupled SPPs may have



₁

> 

₁₃

> 

₃

. (if 

₁

> 

₃

)

This coupled SPP mode with 

₁₃

becomes radiative into the 

₁

medium when 

₁

> 

_3.

2 2

3

0 but

1

0

S



S



Hence S

₁

is imaginary.

 The field is a plane wave radiating away from the metal-

₁

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

   

     

 



2 2 2

0

j j j

s ^  ^  k

 ^{ }

^{1 3 2}

^s

²

^{ }

²²

^{s s}

^{1 3}

 ^{tanh   s h}

²

^{ }

^{2 2}

^{s  }

^{3 1}

^{s  }

^{1 3}

^{s   0}

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 ₁and ₃,

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

The field in ₁and the metal is guided by the interface, the filed in ₃grows exponentially (leaky mode)

Waves guided by symmetric structures  

1

 

3



There are four types of solutions satisfying :

We can estimate the properties of the solutions as follows:

0 h 

1 3

S S

  

1 3

(2) :S 0 & S 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 ₃and the metal is guided by the interface, the filed in ₁grows exponentially (leaky mode)

2 2 2

i i o 0

S   k 

1

m

3

Leaky (1) H_y

Leaky (₃)

1

m

₃

H_y

Bound (a)

1

m

3

H_y

Bound (s)



1

m

3

Leaky (1) H_y

Bound (3)

1

m

3

H_y Leaky (3) Bound (1)

Two nonradiative, Fano modes

Two radiative, leaky modes

Four SPP modes

2 SP solutions

1 SP solution

1 SP solution No solution

2 2 2

0

j j j

s ^  ^  k

 ^{ }

^{1 3 2}

^s

²

^{ }

²²

^{s s}

^{1 3}

 ^{tanh   s h}

²

^{ }

^{2 2}

^{s  }

^{3 1}

^{s  }

^{1 3}

^{s   0}

: tanh(s h2 ) ~ 1 : tanh(s h2 ) ~ 0

   

     

   

3

2 0 2

1

exp ( ) 3

( ) exp ( ) exp 2 0

exp 1 0

i h

B s z h i z h

f z A s z h A s z i z h

s z i z

    

      

  



₁

m

3

H_y

Bound (a)

1

m

3

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

₁

(ω)=1-(ω

_p

/ω)

²

Two modes appear

L

_-

(asymmetric)

Thinner film:

Shorter SP wavelength

Example:



_HeNe

= 633 nm



_SP

= 60 nm ( h = 5 nm)

L

₊

(symmetric)

Propagation lengths: cm !!!

(at infrared)

1

m

3

Hy

Bound (a)

1

m

3

Hy

Bound (s)

s

_b

a

_b

Thinner film:

longer SP

wavelength

0 50 100 150 200 1.5

1.6 1.7 1.8 1.9 2.0

a

_b

s

_b



r

/k

0

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

b



i

/k

0

thickness (h : nm)

 ^{ }

^{1 3 2}

^s

²

^{ }

²²

^{s s}

^{1 3}

 ^{tanh   s h}

²

^{ }

^{2 2}

^{s  }

^{3 1}

^{s  }

^{1 3}

^{s   0}

2 2 2

0

j j j

s     k

2 1 3

1.55 m , 118 i 11.58 , 2.25

          

Ex)

1

m

3

Hy

Bound (a)

1

m

3

H_y

Bound (s)

s

_b

a

_b



R



^I

Waves guided by asymmetric structures

Burke, PRB 1986

This antisymmetric mode always exists.

The symmetric solutions are of two types, leaky and growing modes:

 

1

 

3



1 3 & 1 3

S S S S

     

There are also four types of solutions satisfying

₁

_m

₃

H_y

Bound (a)

₁

_m

₃

H_y

Bound (s)

₁

_m

₃

Leaky (₁) H_y

Bound (₃)

₁

_m

₃

H_y Leaky (₃) Bound (₁)

₁

_m

₃

H_y Growing (z)

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  k 

₁

_m

₃

Leaky (₁) H_y

Leaky (₃)

1 3

For example, 0 & 0

when - and -

R I

i iR iI i iR iI

S

S S iS i



  

 

 

Growing modes

This symmetric mode

Exists only when  << 1.

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 ₁, at that dielectric-metal interface.

The wave amplitude decays exponentially across the film, and then grows exponentially into the other dielectric medium, ₃in this case. In the ₁medium, the wave fronts are tilted towards the film to supply energy from ₁for both dissipation in the metal, and radiation into ₃.

Leaky (radiative) mode

The analogous case of localization in ₃, and radiation into ₁.

Growing mode

The field amplitude grows both with propagation distance as exp(_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.

 

1

 

3



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 ₁field amplitude grows exponentially for only a finite distance z,In this sense, the solutions do not violate boundary conditions as z - infinite.



For fields radiated at an angle  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 ₁-_mboundary, per meter of wave front,

LOCAL THEORY FOR MULTIPLE-FILM SYSTEM

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

For electric (or TM) waves that H

_z

= H

_x

= E

_y

= 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, multilayer 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

_x

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 

_i

=1,

 

 

/ 1

/

metal insulator

R K

K





 

    

 

 

m d x

m d

k c

 



 

 

^{2 2}

2 2

) 1 (

) (

p d

d p sp

x

k c

k   





 





 



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

_p

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

2 2

2 2 2 2 2

,

_p

if

_m

1

^p2

m m

c c

k k c k 

    

  

        

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

_p

,

If 

_i

=1, the low-frequency k < k

_p

, part of I,

If k

_p

d

_i

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

If k

_p

d

_i

<< l, the k << k

_p

portion of II is

and, at k = 0

C. Metal Film in Vacuum

When k < k

_p

, for branch II (antisymmetric oscillation)

When k < k

_p

, for branch I (symmetric oscillation)

When k > k

_p

, both I and II approach

2 2 ,

, 2

i m

Ki m k c

  

 

 

/ /

metal insulator

R K

K





 

   

 

 

* Burke, PRB 1986 :

2 2 2

, 1, 2, 3.

n n o

S  k   k n 

D. Swihart's Geometry

When k << k

_p

, 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 =ck occurs, when

Branch IV behaves for small k as

E. Two Metal Films of Different Thicknesses

Where,

When k << k

_p

,

branch II, III

The branch II and III

Where,

Branch II Branch III

Branch II

Branch III

F. Two Dielectric Films

For k<<k

_p

branches I and II are given by

In the special case when d

₁

=d

₂

=d

_i

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

_p

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

_p

branches I and II are given by

When k<<k

_p

branches III and IV:

When k<<k

_p

branches V and VI

Periodicity of alternating thicknesses d

_m

and d

_i

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

The secular equation is

Solutions are found in the shaded regions.

When k<<k

_p

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

_p²

d

_i

d

_m

)<< 1, Curve I is

while curve IIa is

Solutions near I are given by

When k>>k

_p

the secular equation is

Conclusions of Economou

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

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

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

plasmon frequency _p/root(2).

 In addition to these two groups, some uninteresting modes may appear with dispersion curves that lie just below the curve ²= _p²+ c²k²which corresponds to the trivial solution of zero fields.

 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.

 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.

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

 

 









0 0

E E

H H

z-axis : propagation direction

1 2

r 0

( 

^

) k

    H  H

From the Maxwell equations

1 1 2 2 1

r r 0 r

( ) ( ) ( ) 0

t



^ t t



^ t t t

k  

^ t

    H     H   H 

t

i j

x y









  

 

( )

( )

^{i z t}

t

H i

x

H j e

y ^{ }

 

 



where H

Assume that all media be isotropic.

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

E

_o

, H

_o

: 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

 r/k

o

sa

_b^o

ss

_b^o

thickness of metal (nm)

0 50 100 150 200

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



i

/k

0

sa

_b^o

ss

b

o

thickness of metal (nm)

2 1 3

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

            

Ex)

y

x

x

y z