Optical absorption of a coated sphere above a substrate

Physica A 178 (1991) 195-205 North-Holland

Optical absorption above a substrate

of a coated sphere

R. Ruppin’ Department of Chemical Engineering London SW7 ZAZ, UK

and Chemical Technology,

Imperial College,

Received 23 April 1991

A general method for the calculation of the polarizability and optical absorption of coated spherical particle above a substrate is developed. The method is applied calculation of the infrared absorption spectrum of an oxide coated metal sphere dielectric or metallic substrate, and the optical absorption spectrum of a silver polystyrene sphere near a silver substrate.

a small to the near a coated

1. Introduction The problem of the interaction of a stratified sphere with an incident electromagnetic wave is encountered in various fields such as solid state physics [l-3], microbiology [4-61 and air pollution [7,8]. The exact theory for an isolated stratified sphere, which is a natural extension of the Mie theory [9], was first derived by Aden and Kerker [lo], and many examples of its applications have been reviewed by Kerker [ll]. In optical experiments the situation is often complicated by the presence of a substrate on which the sphere, or a collection of spheres, is deposited. In the present work we obtain the optical absorption properties of a small coated sphere above a substrate. The simpler case of a small homogeneous sphere above a substrate has previously been investigated by two different methods. One method utilizes the fact that the sphere surface and the air-substrate interface are constant coordinate surfaces in a bispherical coordinate system. Since the solutions of the Laplace equation are partially separable in this coordinate system, the problem can be reduced to the solution of a system of linear equations [12]. Another method involves the expansion of the potentials in the various regions in terms of Legendre functions and the introduction of image multipoles at the ’ Present address: Soreq Nuclear Research Centre, Yavne 70600, Israel 0378-4371/91/$03.50

@ 1991-

Elsevier Science Publishers B.V. (North-Holland)

R. Ruppin

196

I Optical

absorption

of sphere above substrate

mirror point of the sphere center relative to the air-substrate interface [13]. The latter technique has also been applied to the cases of a truncated sphere [13], a hemisphere [13-S] and a spheroid [16]. It can also be extended to the case of a coated sphere above a substrate, as will be shown in the present work. The theoretical formulation is developed in section 2 and examples of its application are presented in section 3.

2. Theory We consider a coated sphere of outer radius R and core radius R’ (fig. 1). The dielectric constants of the shell and core materials are c3 and Ed, respectively. The centre of the sphere is at a distance D from a substrate having a dielectric constant cZ. The medium surrounding the sphere has a dielectric constant &r. The centre of the sphere is chosen as the origin of the coordinate system, with the positive z axis normal to the surface in the direction of the substrate. In the electrostatic approximation, which is valid for spheres much smaller than the wavelength of the incident radiation, the potentials V, in the four regions have to satisfy the Laplace equation. As in ref. [13], we will use reduced potentials $, = -ViIEoR where E(, is the magnitude of the external field, and a reduced distance to the origin r = p/R, where (p, 13, 4) are spherical coordinates. The effect of the substrate is represented by image multipoles at (O,O, 20). The total potential in region 1 can be written in the form I+!+= Ycos 19cos 13,+ r sin 13sin 0, cos C#J +

2 r-'-'[AljP;(cos

0) +

B,,P;.(cos0) cos $1

j=l

+ c [AijV;(r, j=i

cos 0) + B,:V;(r,

R ---

cos 0) cos ~$1.

R’

--

-54 z

ES

@

Fig.

1. Geometry

El

of a coated

sphere

above

a substrate.

(1)

R. Ruppin I Optical absorption of sphere above substrate

197

The first sum is due to the multipoles induced in the sphere and the second sum is due to the image multipoles. The functions Vy are defined by

vp,

cos f3) = ( r2 - 4rr, cos 8 + 4ri)-(j+l)‘*

x PY((r cos 8 - 2r,)( r* - 4rr, cos 8 + 4?$“*)

)

(4

where rO = D/R, and Py is the associated Legendre function. The potentials in the other three regions can be expanded in the form

c

+

r-‘-‘[A,,P~(cos

e) + B,,P;(cos e)

c

e) +

j=l

i,b3= F3 +

r’[A ,,P,(cos ‘!

c.

I+$= F4 +

e) + B;jP;(cose)

r-i-l[A;jP;(cos

j=l

,,1r'[A,jZ';(cos

0) +

+]

(3)

)

e) cos 41

B,$(cos

j=l

+

c0s

B,P;(cost’)

c0s

$1

(4)

)

cos ~$1,

(5)

where all the coefficients appearing in these expansions will be determined from the continuity conditions at the boundaries between the various regions. The boundary conditions require that J, and E at,blan be continuous, where a+lan is the derivative of 9 in the direction normal to the surface. Applying the boundary conditions at z = D we obtain the following relations for the expansion coefficients [13]:

P=l,

&.Y=--

El E2

(6) Aij =

z 1

The boundary

(-l)‘Alj

)

(-l)j+‘Blj

Bij = F

2

conditions at the core-shell

.

2

interface yield the relations

198

R. Ruppin

Adi = A,, + 6 +A;,

t

Fj = Fd

I Optical absorption of sphere above substrate

,

Bdi = B,i + S -zi-‘B;j

e,jAdj = e3[jAxj - (j + 1)6 -“-‘A;,]

,

c4jBdj = E~[~B,, - (j + 1)s -2’-‘B;,]

,

, (7)

where 6 = R’IR. Multiplying the boundary conditions at the outer surface of the sphere by P,“(cos 19), with m = 0, 1, integrating over the surface, and using the identity [13] (I + j)!AJ~(cos 0) c ,=m (I + m)!( j - m)!(2rJ+j+’

V,m(r, cos 0) = (-l)‘+”

’

we obtain

ajk - f cos e,, s,,

C, (A,j + A;j - A,j) & j=l

-,L+

(-l)‘A;j

k!jjfiZTI(li,+l

&

=

O 7

(

C /=1

MJ’Asi-(~+lM~,l+

-&l

,I$

Cal)lAIj

&~(jfl)Alj)

(k

_

I~~~~~ti)k+j+l

/=I -,c

‘jk-

=

!T&~COS~I~,I

O )

tijk- $ sin f3,,6,, (k + j)!

(-l)‘+lBij

(k - l)!(j

2 [E~~B~,- .c3(j + l)Bjj /=I - 5.5, sin 0,, Sk, - E,

=o.

&

. .

. . (B3j -t Bii - BIj) I+&$+

i

&

- l)!(zrJ+‘+

+ E,( j + l)B,,]

,z(-l>“‘B;j

zk . $S$

(11)

= O’ ‘jk

(k + j)!

k

(k - l)!( j - 1)!(2r,,)k+ii’

2k + 1 (12)

Using (6) and (7) to express the coefficients AI,, Aij, B;, and B;, in terms of

R. Ruppin I Optical absorption of sphere above substrate Alj,

A,,

B,i

c

@,A

and Bjj, we find that eqs. (9)-(12)

3j

-

A lj)

j=l

_

_

z

reduce to the form

&I - f cos e,,Sk1

,$Alj k,jf~2~o~~Yj+l & =O .

c.

- 3 c1 cos

[e3QjASj + E,( j + l)AIj] &

80 6 kl

j=l

z 2 Alj

-&I

(13)

)

.

1

199

(k

_

l~fj~~~~)k+j+l .

1

2 (,SjB3j - Blj) ‘$f$

Sjk -

.

&

=O

(14)

3

2 sin 0, a,,

j=l

(k+ -z

2

Blj

1

(k

-

I)!(

j -

j)! 1)!(2rO)k+j+1

-= 1

2k +

1

O’

(15)

j(j + 1) ~jk - 3 E1sin 0, a& ,$ [EsQjBjj + E,( j + l)B,j] 2j+l El

-

- El El+ k =

(k+

3 E

j)!

j=l (k - I)!( j - 1)!(2yo)k+j+1

k 2k +

1 B1i = O ’

(16)

1,2, . . . . , where (17)

(18) The A coefficients, which correspond to the case of a perpendicular electric field, do not couple to the B coefficients, which correspond to the case of a parallel electric field, as could be expected from the symmetry of the system. The two systems of linear equations, (13) and (14) for the perpendicular polarization, and (15) and (16) for the parallel one, are solved by truncating them at a large enough value N of k and j. We then obtain a system of 2N linear equations for 2N coefficients (A lj, A2j for the perpendicular polarization, and Blj, B, for the parallel one). The polarizability of the sphere is given in terms of the dipole coefficients A,, and B,, by alI = -4+rrE,R3Blllsin $ ,

(19)

R. Ruppin

200

cYI

=

I Optical absorption

-4~r~,R~A,~/cos8,,

of sphere above substrate

(20)

The absorption cross section of the sphere is related to the imaginary part of the polarizability (Y”by CT= (4w/3cR2)af’, where o is the angular frequency, c is the velocity of light and CTis expressed in units of the geometric cross section

TRZ. For the case of a perfect metal substrate (F*-+ m), eqs. (13)-(16) can still be used, with the (E, - E~)/(E, + c2) factor replaced by -1. Similarly, the case of a sphere with a metallic core can be handled by replacing S, in eqs. (13) and (15) by (1- 8*j+’ ) and Qj in eqs. (14) and (16) by j -t (j + 1)8”+‘, which are their limiting values for .sq+ cc.

3. Numerical

results and discussion

As a first application of the method developed in section 2 we calculate the infrared absorption in a MgO coating on a metal sphere in the presence of a substrate. The frequency dependent dielectric constant of the MgO shell is given by %-

” = Ex+ 1 -

(o/+)~

%z - iy(w/f+)

’

(21)

where E, = 3.01, E” = 9.64, wT = 401 cm-‘, y = 0.019 [ 171. The calculations were performed for an outer radius R = 0.5km and inner to outer radius ratio 6 = 0.6. The absorption spectrum without a substrate is shown in fig. 2. The interpretation of the absorption peak and its dependence on 6 has been discussed in ref. [2]. There it was shown that in the limit of a very thin coating the peak is located at wL, the longitudinal optical frequency. As 6 increases the absorption peak moves to lower frequencies, and in the limit of 6 = 1 (i.e., a homogeneous MgO sphere) it reaches the Frohlich frequency [18,19], defined by E;(w~) = -2, where E; is the real part of the dielectric constant (21). The development of the spectrum as the sphere is brought close to a dielectric substrate (E, = 2.31) is shown in figs. 3 and 4. The main peak shifts slightly to the low frequency side and a subsidiary structure develops below it. These effects are stronger for the perpendicular polarization than for the parallel one. For the same sphere-substrate separations we have also calculated the absorption near a metallic substrate. The effects are qualitatively similar, but much stronger than in the dielectric substrate case (figs. 5 and 6). To exemplify the application of the theory in a different region of the electromagnetic spectrum, we discuss the absorption in small polystyrene

R. Ruppin I Optical absorption of sphere above substrate

201

16 -

12 -

b 6-

4-

0 550

Fig. 2. Absorption

600 650 FREQUENCY (CM-‘)

700

cross section of a MgO coated metal sphere with R = 0.5 km, R’ = 0.3 pm.

16 -

II

12 -

b 6-

4-

0 550

600

650

700

FREQUENCY (CM-‘)

Fig. 3. Absorption cross section of a MgO coated metal sphere near a dielectric substrate. R = 0.5 km, R’ = 0.3 pm, D/R = 1.1. Full curve: perpendicular polarization; dashed curve: parallel polarization.

202

R. Ruppin

I Optical absorption of sphere above substrate

16 -

:I\ I

12 -

I

’

I

b 0-

0 550

600

650

700

FREOUENCY (CM-‘)

Fig. 4. Same

as fig. 3, but for D/R = 1.02

lel L

12

;I

-

II 1I ‘\I I

b

I

8-

550

600

650

700

FREOUENCY (CM-‘)

Fig. 5. Absorption cross section of a MgO 0.5 pm, R’ = 0.3 pm, D/R = 1.1. Full curve: polarization.

coated metal perpendicular

spheres with a metal coating, which are configuration of this type has recently been who studied it using the scanning near-field calculations are performed for silver, which imaginary part of the dielectric constant, exhibits well-defined peaks. The calculations

sphere near polarization;

a metal dashed

substrate. R= curve: parallel

placed near a metal substrate. A employed by Fischer and Pohl [20], optical microscopy method. Our has a relatively small value of the so that the absorption spectrum are performed in the region of the

R. Ruppin I Optical absorption of sphere above substrate

203

Fig. 6. Same as fig. 5, but for D/R = 1.02.

surface plasmons of silver, and the optical constants are taken from the experimental data of Johnson and Christy [21]. These, however, refer to the bulk, and for small spheres have to be modified, so as to account for the limitation of the electron mean free path [22]. We introduce this size correction in the form suggested by Kreibig [23], according to which the dielectric constant of a small Ag sphere of radius R can be written in the form

&(w,

R) = Ebb

+i

>

.

(22)

Here .slb and car, are the real and imaginary parts of the bulk dielectric constant. or, = 1.38 x 1016s-r is the plasma frequency and ur = 1.4 X lo6 m/s is the Fermi velocity. Although eq. (22) applies to a full silver sphere, we will use it here for the spherical shell geometry, in the absence of a formula for the latter case. The absorption cross section of a silver coated polystyrene sphere with outer radius of 50 A and core radius of 30 8, (i.e., 6 = 0.6) is shown in fig. 7. In the presence of a silver substrate, with D/R = 1.1 (fig. 8), the main absorption peak shifts to the low energy side, the shift being larger for the perpendicular polarization than for the parallel one. The effect of decreasing spheresubstrate separation is demonstrated in fig. 9, which was calculated for DIR = 1.02. The shift of the main absorption peak increases, and additional structure appears in the spectrum. In conclusion, we have extended the homogeneous sphere theory of ref. [13] to the case of a stratified sphere. Numerical calculations for surface phonon

204

R. Ruppin

I Optical

absorption

of sphere above substrate

1 2-

b 1 -

1.5

2.0

2.5

3.0

3.5

4

4.0

ENERGY (eV)

Fig. 7. Absorption

cross

section

of a silver coated

polystyrene

sphere

with R = 50 A, R’ = 30 A.

2-

I ’ I ’

b 1 -

0 1.5

2.0

2.5

I 3.0

I 3.5

4.0

ENERGY (d)

Fig. 8. Absorption cross section R = 50 A, R’ = 30 A, D/R = 1 .l, polarization.

of a silver coated polystyrene sphere Full curve: perpendicular polarization;

near a silver substrate. dashed curve: parallel

absorption in the infrared and for surface plasmon absorption in the optical region have been performed. As the sphere approaches a substrate the main absorption peak shifts to the low energy side and a subsidiary structure appears in the spectrum. These effects are stronger for the perpendicular polarization than for the parallel one. Also, they are stronger for a metallic substrate than for a dielectric one.

R. Ruppin

I Optical absorption of sphere above substrate

205

2

b 1

o~--c-1.5

1

2.0

#

2.5 (“.;I”, ENERGY

I I

3.5

4.0

Fig. 9. Same as fig. 8, but for DIR = 1.02.

References 111 H. Weaver, R.W. Alexander,

L. Teng, R.A. Mann and R.J. Bell, Phys. Stat. Sol. (a) 20 (1973) 321. PI R. Ruppin, Surf. Sci. 51 (1975) 140. I31 T.P. Martin, Solid State Commun. 17 (1975) 139. A. Brunstig and P.F. Mullaney, Appl. Opt. 11 (1972) 675. t;‘; A. Brunstig and P.F. Mullaney, Biophys. J. 14 (1974) 439. 161 R.A. Meyer, Appl. Opt. 18 (1979) 585. [71 T.P. Ackerman and O.B. Toon, Appl. Opt. 20 (1981) 3661. PI J. Heintzenberg and R.M. Welch, Appl. Opt. 21 (1982) 822. [91 G. Mie, Ann. Phys. (Leipzig) 25 (1908) 377. WI A.L. Aden and M. Kerker, J. Appl. Phys. 22 (1951) 1242. IllI M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic Press, New York, 1969). I121 R. Ruppin, Surf. Sci. 127 (1983) 108. 1131M.M. Wind, J. Vlieger and D. Bedeaux, Physica A 141 (1987) 33. [I41 D.W. Berreman, Phys. Rev. 163 (1967) 855. [W R. Ruppin, Solid State Commun. 39 (1981) 903. [I61 P.A. Bobbert and J. Vlieger, Physica A 147 (1987) 115. [I71 J.R. Jasperse, A. Kahan, J.N. Plendl and S.S. Mitra, Phys. Rev. 146 (1966) 526. [I81 H. Frohlich, Theory of Dielectrics (Clarendon, Oxford, 1949). [I91 R. Ruppin and R. Englman, Rep. Prog. Phys. 33 (1970) 149. WI U.Ch. Fischer and D.W. Pohl, Phys. Rev. Lett. 62 (1989) 458. WI P.B. Johnson and R.W. Christy, Phys. Rev. B 6 (1972) 4370. P21 U. Kreibig and C. v. Fragstein, Z. Phys. 224 (1969) 307. t231 U. Kreibig, J. Phys. F 4 (1974) 999.