The effects of quasiperiodic grating on surface-plasmon polariton

Solid State Communications, Vol. 96, No. 2, pp. 73-78, 1995 Copyright Q 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-1098/k $9.50+.00 0038-lO!I8(95)00395-9

THE EFFECTS OF QUASIPERIODIC GRATING ON SURFACE-PLASMON POLARITON Tetsuji Tokihiro and Hiromi Ezaki

Department

of Applied Physics, University of Tokyo,

7-3-l Hongo, Bunkyoku,

Tokyo 113, Japan

(Received 27 March 1995; accepted 5 May 1995 by T. Tsuzuki)

Abstract A quasiperiodic the plasmon

induces new surface plasmon modes and peculiar

mode is equivalent

angle-resolved

Keywords:

grating

emission spectra from tunnel junction

A. quasicrystals

A. surfaces and interfaces

The discovery of quasicrystals cept of order, [l] [2). netic

models

have demonstrated

(continuous

with devil’s staircase

spectra

structure,

the direct

structure

consequences

its noncrystallographic other quasiperiodic

artificial

proposed

are necessary

to use classical

were proposed

Photons

photoetched

by Kohmoto,

another

good candidate:

is an electromagnetic

normal

po-

face is perfectly .smooth, free photons,

conservation

except

can acquire radiative

The

ness or a grating

Sev-

a quasiperiodic

in optical and

components

however, it

and is observed by op-

ways [6]. Thus,

modulation,

relation of the

if the surface

the quasiperiodicity

has

signifi-

relation and we can observe

the consequent

from optical measurements.

phenomena

is photon emission from metal-

tunnel junctions

cited by tunneling 73

by gratings,

cantly affects the dispersion

insulator-metal

films. In this paper, we

When the surface

[6] [7]. At the same time, the rough-

One of such phenomena

on

[5]. If the sur-

due to the energy

changes the dispersion

SPP in significant

waves on some

rules.

or is corrugated

tical measurements

to observe the

can

it does not couple to external

i.e., it is nonradiative

and wave-vector

for

mode that

both into the vac-

uum and into the inside of the metal

is roughened,

experiments

surface-plasmon

along the surface of a metal, but the ampli-

to observe

Sutherland,

The

to the quasiperiodicity.

tude of which decays exponentially

en-

helium was used by

[4] in the scattering

strips of aluminum

propagate

electronic

patterns.

propagating

Iguchi [3]. Third sound in superfluid Kono and Nakada

of

Unfortu-

on physical phenomena.

superlattices.

multilayers

it is difficult

X-ray diffraction

effects of quasiperiodicity eral authors

and so forth.

of quasiperiodicity

systems

SPP

and magfeatures

system with long range electron transfer.

of

lariton (SPP).

magnetization

has so complicated that

propose

with zero Lebesgue

eigenfunctions,

nately, actual quasicrystal and magnetic

peculiar

The energy spectrum

D. optical properties

physics

[2]. They have singular continuous

self-similar

measure),

electronic

electronic

show sharp peaks according

a new con-

in solid state

one-dimensional

quasiperiodicity ergy spectra

introduced

quasiperiodicity,

Many

to that of a quasiperiodic

light emission.

current

[8], where SPPs are ex-

and radiate free photons.

In

EFFECTS OF QUASIPERIODIC GRATING

74 the latter

part

phenomenon.

of the present

letter,

we examine

this

In the former part, we clarify the effects

of a quasiperiodic

grating

on the dispersion

relation

of

SPP.

Vol. 96, No. 2

function

C(r) expresses

assume

it a Lorenzian:

the shape of the striation. ((5)

E nst/rr(x2

choice makes the following calculations

We

+ C’).

This

fairly easy, and

we do not think that the other choices change the phys-

We consider electric

a semi-infinite

constant

c(w):

cc0 is the background plasma

frequency

e(w) = c, dielectric

(1 - w:/w’),

constant

of the metal.

C(r, Y), is vacuum.

and wp is the

The surface is defined

SPP can be obtained

to Maxwell’s

equations

nents of the associated H,(z, z;w),

i.e.,

and propa-

Then the only nonzero compomagnetic

and electronic

by I~~(I,z;w)

We as-

one-dimensional,

[(x, y) = c(z), and the SPP is p-polarized gating in the z-direction.

z >

as the so-

of this system.

sume that the grating is essentially

field is its y-component,

field can also be expressed

[S]. The wave equation

ical consequences

where

.z = ((z, y), where the region,

by the equation:

lution

metallic system with di-

for 11,(x, z;w) is

The problem

the approach systematic

* >

C(x),

[&+g +r(w)$f,(x, qw) = 0

2<

C(x), (1)

with the boundary

I,

2;

The surface profile function as c(x) = C,,, i(z - I,),

period

B + nscrsin8,

= (fi

integers

becomes

and these two distances as...,1

These periodic approximants B by 0, = tan-l(F,,/F,,+l),

are (n =

number which

(= Go). function

to 0 in the limit n -+ 00. The

approximant

We denote

is L = ,/-a

the resulting

surface profile

by Cl”)(z). we invoke

which

was often

present

context,

the

Rayleigh

used

for a periodic

this

hypothesis

hypothesis grating.

In the

is to assume

HJzr, z;w) has a Fourier series expansion

[6]

that

as

z; w) =

2

A,(k, w) exp[ilc,x

- a,(k, w)z]

p=-00

where I,

denotes

the posi-

in a quasiperiodic

we choose that x,

= x,,,~~

where u is a unit of length,

- 1)/2, and ml and ms range over all the

which satisfy

points

of the

for 2 > C(r), =

F B,(k,w)exp(+x p=-CU for 2 < ((x),

+ ~r(lE,w)z] (3)

C(r) may be expressed

the inequality:

0 < -ml

ms cos 0 < sin 0 + cos 8. Then the distance jacent

of the n-th

at the surface.

which is arranged

As an example,

= mracos

of periodic approximants

The angle 6’, approaches

f$(z, z;w)lz=c(~)+7(2)

where g is the normal derivative

manner.

Firstly, we use

Here F, is the n-th Fibonacci

= ~Y(~,=;w)lz=((z)+,

W)Iz=((z)- =

tion of a striation,

Here we adopt

condition:

H,(x, ZiW)],=((,)-

=hfv( c(w) an

to eqs.(l)

is given by F, = 1, Fr = 2 and Fn+s = F,, + F,,+,.

&(I,

tan0

grating.

by replacing

1,2,3;-.).

than it looks.

construction

quasiperiodic obtained

the solutions

which consists of two steps.

Secondly,

z;w) = 0

of obtaining

and (2) is more difficult

given as

rg+g +$H&,

so much.

where

kp E

dm,

k < f.

ZpafL,

a,,(k,w)

G

,/m

for kp’ > w’/c” (k; < w2/c2), &(k,w)

(-i,/m) f

k +

Re[&] > 0, Im[&] < 0 and 3L < Then,

keeping

lead to infinite number

the first order of 70, eqs.(l)-(3) of equations

for A,(k,w)

as

sin 0 +

between ad-

either 1, - a cos 0 or 1s z a sin 0 appear in a Fibonacci

1, 12, 11, 11, 121 1I, 129 111 11, 12, 1I, 11, 12,

x b,P, - kpkql(^(*)tp - e) A,tk w) = 0,

sequence ..‘*

The

where $“)(p - q) is a Fourier transform

of c(z):

(4)

Vol. 96, No. 2

$“‘(p - q)

E

EFFECTS OF QUASIPERIODIC

IL dxpqx) exp[-i2*(p; q,

;

r

TP

:sinI*71n+Jc+2(P

equal value voM/L.

’

integers

l)(p - q).

Here we have used some properties

above equation. notes

= 7zn+r(Fn+2 -

the second

(k,l),

F,,+*k + F,,l = m for a fixed integer fraction s e,)

line of the de-

which satisfy

m, the decimal

(iii) if m # m’

of (-l)“+*mF2,+~/F&,+2.

Now we take the plasmon establish system system.

an important

connection

between

tronic

2

C’“‘(P-

to:

P

- 1).

the atomic

(Note that

the

in a period L is exactly F,,+l.) so that the

the local modes are also quasiperiwhich deter-

of a one-dimensional [9]- [ll],

it is readily found that the

is equivalent

to a one-dimensional

with long-range

electron

sites are arranged transfer

quasiperiodic

This equivalence

transfer,

in a Fibonacci

is inversely

elecwhere

sequence

proportional

is important,

to the

because

it is

much easier to realize the present system than to realize

The solution

to

E(w) = 0 is the usual surface plasmon frequency, wsp = In case of E(w) # 0, eq.(6) is transformed

system

distance.

d(lrc,l - &+W;w),

(c(w) + l)/(c(w)

system

and electronic

(6) with E(w) f

system

of local

of a striation

eq.(7) with the equations

mine the eigenstates

the present

In this limit, eq.(4) can be transformed

q=-00

odic. Comparing

present

electronic

to the coefficients

are placed quasiperiodically, between

Since al

of the coefficients of the

In fact, the position

of the striations

interactions

00, so as to

quasiperiodic

with 2F,+2 variables.

is given by x = la/M with 1 E 0~.

electronic

limit c 4

and a one-dimensional

~WbMkw) =

up/d.

modes at 5 = la/M.

The striations

which

equation

plane waves, they correspond

(ii) for any integer p, E 0,

al and bl do not vanish

and bl are the Fourier transform

for any pair (rE,I) E R,,

one and only one pair (k,l)

the set of these F,,+s

Thus eq.(7) reduces to a simultaneous

linear algebraic

number

(F,,k - F,,+11)/F2,+2 = p te,,

We denote

Accordingly,

only if I E 0~.

has the same value (

[mod FZn+s], then E, # s,,,~, and (iv) E, is the decimal fraction

by 0~.

of (F,,k - F,+lI)/F2,+2

there .exists satisfies

of the

They are listed as; (i) when R,

the set of pairs of integers,

F,,+2 - l), and all of them have an

(5)

= F~.,+rlF2~+2 and %p-9

to obtain

mF2,+1 [mod

1, & do not vanish only if 1 s

M], (m = O,l,.*.,

with 72~

numbers

75

P

- 411

sin[rT2,+l(P- q)l

Fibonacci

yM <

WP - 4t _ ifl _ ]

= Fexp[-

GRATING

a quasiperiodic

electronic

From elementary sion relation

system.

operations

for the plasmon

on matrices, is obtained

the disper-

as

into

E’(w) - ($)l

= -($)2{(;

- i) + A,}2,

(8)

1

E(w)a, = 2, L(l ::-+,)

[(A - i •t ;(M + I))4

where X,, (m = 1,2,...,

@r(l'-I)/M +c.

,,+, 2~ sin K( I’ - I)/ M

of a Fn+2 x F,+z matrix S with matrix elements

b] ,

E(w)b, = i;

where i G g

(-$

5 i _< f), ar E C,“=;;r A, eiZn’piM,

bl= C,“=, A-, e -:Z~~PIM, F2n+2, 7 = 2a[/L

2,

I

cE;l

l(p)ei2”lp/M,

M

f

is small enough,

i{2sin[xrz,,+r(l-

I’)])-’

an (/,r’) element

of the matrix

i.e.

S depends

1-I’, one may feel that it represents

system.

But,

are concerning

of course

this is not true.

to a quasiperiodic

element of [S]r,r, is not a decreasing

important.

From eq.(8),

[S]r,,, =

for 1 # Y and [S],,) = 0. Since

difference

of 1 - I’, and contribution

and A E M/(1 - eeTM).

When the width of the grating

F,,+r), is the m-th eigenvalue

system.

only on the a periodic In fact, we The matrix

function as a function

of the boundary

terms

are

Am(k) s w(k) - wsp is given

76

EFFECTS OF QUASIPERIODIC

by

GRATING

finite width

Au(k)

= *(z)

[I - ;($‘{(;

- I) + A-,‘].

(9)

L. Equation 2&s

of the plotting

to decide numerically continuous

Here we have used the relations

Vol. 96, No. 2

or not.

line.

whether

It seems impossible

the spectrum

is singular

In any case, in the limit n +

co,

Aw < w.r and 6 < it has certainly

finite Lebesgue

measure

peculiar

of the present system.

1121. This is a

(9) shows that the plasmon has additional

bands and these bands are grouped

feature

into two sets When the polariton

with F,,+z bands, which are separated

dispersion and symmetric

with respect

effect is taken into account,

the

well by 7sw,,/27r~* relation

significantly

changes for small wave

to w.r. numbers.

For large wave numbers,

however, the disper-

Figure 1 shows the change of the upper set of bands sion relation is almost equal to the plasmon limit. Figure with respect

to the degree of approximation

n. A fre3 shows an example

of the dispersion

relation obtained

quency in the black regions belongs to the bands. Figure 2 shows the integrated behavior

from eq.(4).

It is displayed

is quite different

The grating

is the n = 5 approximant

[9] [lo] or the hierarchical

model [ll], in which gaps appear

the energy spectrum

scheme.

described

above

from that of the square-well and the energy

model

in the extended-zone

density of states for n = 14. The

has Lebesgue

region is 0 5 w 5 wSp/2. Many small

at k = mn/L

(m E Z). Roughly speaking,

measure 0. We only the modes near the gaps couple to free photon field, so

observe several band gaps and the integrated

density of that SPP can be observed

states

is quite smooth.

absolutely

continuous.

Therefore

the spectrum

seems

However, when we examine

Finally we discuss the photo-emission

from the tun-

the nel junction.

band gaps numerically,

in the optical measurements.

The geometry

considered

here is the same

we can find a lot of band gaps as that

discussed

by Laks and Mills [13]. On a metallic

at any place which are not seen in the figure due to the semi-infinite

= 1

5

10

14

FIG. 1. The energy band (AU(~))

of the SPP in terms

of the degree of periodic approximation

n. An energy in the

black regions belongs to the eigenenergies.

Though the spec-

substrate

with dielectric

FIG. 2. The integrated band for n = 14. minimum

density

constant

of states

es(w), an

of the energy

Emi, and E,,,,, denote respectively

and the maximum

of the energy band.

the

The ob-

vious band gaps are shown as A, B, and C. The gap C is

tra seem to ,have only a few gaps, there are lots of invisible

almost

invisible

small gaps on the figure.

band gap at I?.

in Fig.1.

The area of the inset shows the

EFFECTS OF QUASIPERIODIC

Vol. 96, No. 2

77

GRATING

(b)

1

k I

FIG. 3. The dispersion

0.2

curve of SPP at low energies.

The inset is the magnification

of the dispersion

constant

film of mean thickness

emission spec-

tunnel junction.

The peak

c*(w) (a) corresponds

and an metallic

6

0.6

of the angle-resolved

trum from metal-insulator-metal

d with dielectric

/

0.4

Emission Angle

curve. FIG. 4. An example

oxide layer of thickness

I

to the largest value of Ic(k)l, and both (b)

Ld with dielecto the fourth largest values. The peaks

and (c) correspond

tric constant

cl(w) are piled up. The metallic film has a

quasiperiodic

grating,

corresponding which is characterized

to the second and third largest values also

by the surappear in the figure, but they are weaker than (a)-(c).

face profile function

above. A dc voltage

(‘(z) discussed

is put across the junction,

which induces current

through

The fluctuations

the oxide barrier.

excite SPP and radiation spectra,

approach

[13]. The only difference surface

method

based

as that of Laks and Mills

in our calculation

has the quasiperiodic

surface was considered

in current

field. To calculate the emission

we used the same perturbation

on Green’s function

flow

grating,

is that

the

while a random

spectrum,

intensity

of emitted

The wave-vector plane.

that is, the angle dependence photons

photon ito) is in the IZ-

of it’).

is the n = 6 approximant.

the parameters

w.

axis gives the value of k.l(lol/]~(ol],

where Ic{lclis the z-component file function

with some frequency

of the detected

The horizontal

of the

for this calculation

cr = 3.0, a = 8pn,

dielectric the present

constant

frequency

w.

with frequency

the tunnel

current

isfies the dispersion

tered by the grating

w created

by the fluctuations

has the wave number relation

without because

rules.

a grating.

angle-resolved

Since

of the energy and

But it is possibly

scat-

to the wave number Q - Q and, if

rection of the angle 4. The provability is proportional

of

Q, which sat-

Q - q = Ii(‘)1 sin 4, it can radiate

it satisfies

to It(q) emission

A characteristic

to that of Ag around

point of view is as follows. The

conservation

The values of

are: L,(W) = es(w)

about the photo-emission

Q > ko, it can not radiate

to ]&l(P) - &)I*.

d = 0.1~1, and Ld = lOa. The

c1 corresponds

SPP

The surface pro-

= 3.5 - wp2/w(w + i7), ttw, = 9.2eV, tL7 = O.O2eV, fiw = 2.3eV,

explanation

from the perturbational

momentum

in Ref. [13].

Figure 4 shows a typical example of an angle-resolved emission

The intuitive

to the di-

of this scattering

Thus, roughly speaking, intensity

at $‘) is proportional

of a quasiperiodic

(formal) Fourier transform

the

function is that its

consists of dense C-functions.

In contrast,

the Fourier transform

of a periodic function

is a regular

array of 6-functions

and that of a random

function

is a smooth

function

in average.

In fact, Fig.4

shows dense spiky peaks. They really reflect the nature

78

EFFECTS OF QUASIPERIODIC

of the Fourier

transform

<(z). Therefore

of the quasiperiodic

function

we can conclude that the emission spec-

tra directly exhibit the quasiperiodicity

of the grating on

the surface.

GRATING

Vol. 96, No. 2

We are grateful to Professor ing our attention ported

Kazuo Ohtaka for draw-

to this problem.

This work is sup-

in part by a Grant-in-Aid

of Education,

from Japan

Ministry

Science and Culture.

REFERENCES [l] D.Schechtman,

D.Blech, D.Gratias,

and J.W.Cahn,

The Physics

edited by P.J.Steinhardt entific, Singapore, [3] M.Kohmoto,

of Quasicrystals,

and S.Ostlund,

(World Sci-

1987).

B.Sutherland,

and S.Nakada,

[S] J.Lambe

and

and R.M.Pierce,

and S.L.McCarthy,

Phys.

K.Iguchi,

Phys.

L.P.Kadanoff,

and C.Tang, Phys. Rev.

Lett. 50, 1870 (1983). [IO] S.Ostrund,

Phys. Rev. Lett. 69, 1185

Phys. Rev. Lett. 37,

923 (1976). [9] M.Kohmoto,

Rev. Lett. 58, 2436 (1987). [4] K.Kono

J.E.Rutiedge,

Rev. B34, 6804 (1986).

Phys. Rev. Lett. 53, 1951 (1984). [2] See, for example,

[7] S.Ushioda,

R.Pandit,

and E.D.Siggia,

DRand,

H.J.Schellnhuber,

Phys. Rev. Lett. 50, 1873 (1983).

(1992). [ll] T.Tokihiro, [5] D.L.Mills

and A.A.Maradudin,

Phys.

Phys. Rev. B40, 2889 (1989).

Rev. B12, [12] See, for example,

M.Reed and B.Simon,

Functional

2943 (1975). analysis I, (Academic, [6] A.A.Maradudin,

in

Surface

Polaritons

ed.

New York, 1980).

by [13] B.Laks and D.L.Mills, Phys. Rev. B21,5175

V.M.Agranovich Publishing therein.

and

Company,

D.L.Mills

(North-Holland

1982) and references

cited

(1980).