Surface plasmon dispersion relation for spherical metal particles

Surface plasmon dispersion relation for spherical metal particles

SATISH B OGALE, V N BHORASKAR and P V PANAT Department of Physics, University of Poona, Pune 411 007

MS received 28 October 1977; revised 30 March 1978

Abstract. Using the hydrodynamieal model, we have obtained the surface plasmon dispersion relation for spherical metallic particles in the following two cases: (1) a sharp surface cut off in electron density and (2) a diffused electron density at the surface. The diffused density is modelled with a step function. The diffuse nature of the electron density at surface of the metal particle is necessary to understand the experimental result for particles with small radii. Shift in the absorption frequency is estimated and found to be small.

Keywords. Hydrodynamical model; diffused surface; surface plasmons.

1. Introduction

Recently considerable interest has grown in the surface plasmon modes of small spherical metallic particles, with the experimental investigations of Smithard (1973) and Gariere et al (1975). According to these experiments, the Mie optical absorption peak shifts to longer wavelengths with the decrease of the radius of sphere. Gariere e t a l (1975), using a homogeneous sphere model and dielectric response theory have shown that the experimental results are exactly opposite to the theoretical expectations.

They have come to a similar conclusion using classical theory also. Ruppin(1976) has taken a step in the right direction by considering the diffused nature of the electron density at the metal surface of the particles. Assuming a linear density profile, he has shown that the absorption frequencies are lowered with the decrease in sphere radius, in qualitative agreement with the experimental results. His theory, however, does not include thepressure effects and hence is purely static. The quantum hydrodynamical model of Bl'bch (1933, 1934) as developed by Ritchie and Wilems (1969) can incorpo- rate the pressure effects in the calculations of collective modes in a natural way.

Ritchie and Wilems (1969) have obtained electron-plasmon, photon-plasmon vertices for the metallic slab system using this model. In view of the simplicity of the model, and the possibility of finding the above vertices, we have embarked on a programme to study the spherical metal surface in the quantum hydrodynamical model and have obtained the surface plasmon dispersion relations (non retarded) for the following two cases of interest (a) sharp electron density cut off at the surface (b) diffused surface for which we have used the double step model of Boardman et al (1976). The double step model haS an advantage that the solutions of the concerned differential equation can be obtained analytically and at the same time the results are qualitatively similar to those obtainable for more realistic density distributions such as Gaussian or

P.~2

135

exponential profiles (Boardman et al 1975). This conclusion also finds the support in recent work of Ctmningham et al (1974) on the effect of depletion layers on the surface plasmon polariton of semiconductors.

2. Calculations of dispersion relation

2.1. Sharp cut off

We have a uniform positive neutralising background for our electron gas in a spherical boundary. We use (1) Euler equation for momentum transfer in which imbalance of electrostatic force and gradient of pressure give the acceleration. We use Fermi pressure to take into account the Pauli principle; (2) Equation of continuity; and (3) Poisson equation. Thus the equations of motion for the hydrodynamic velocity v and the potential $ in electrostatic approximation are,

n (r, t)

Ov -- f d~ (n')

r n ~ = e V ~ - V J n--;

0

(1)

V~,p = 4~e

In

(r, t) - z l(r)].

We use the pressure ~(n), the Fermi pressure, as

(2)

~(n) = a~ (3':)sis n 5/8 5m

l(r) is the + v e ion density. D / D t represents a co-moving time derivative. The hydrodynamic equations can be simplified by introducing the velocity potential

(r, t) according to the definition,

v = - r e . (3)

This transforms eq. (1) to the following form

Ot m 0 n' m

The equation of continuity can be written down as

On = v ' I n v , ] . (s)

8t

We shall now follow the standard procedure of linearisa~ion of these equations and expand the hydrodynami~l variables as,

~(r,O

= no (r) + n~ (r, O + n~ (r, O + ^{. . .}

(6)

~ (r, t) = ~ o ( r ) + ~ (r, t) + ~a (r, t) + ... (7)

÷ (r, t) = ~x (r, t) + ~, (r, t) + ... (8) Here it has been assumed that n o ~, n 1 >. na. Using these expansions in eqs (2), (4) and (5) and collecting quantities of the same order vve get in zeroth order,

5 no, I,

V'~o = 4.n.e (n o -- Z/o). (10)

And in first order,

Here

a ~ t = _ f,h + 56

O---t m 3m no 1/-'---8 nl (11)

VaSt = 4 7ten 1 (12)

Ont _ V" [no (r)V(~bl)]. (13)

Ot

~*

(3,r*p1'

~

5 m In our pzxtic~ar problem,

0 r > R .

Hence eqs (11), (12) and (13) take still simpler forms viz.,

_e x

+

Ot m 3 n o

V * ~ ~ 47ten1

and

(14)

(16)

Here V F = h (3~n0)x/S/m, the speed o f the most energetic electrons in the Fermi sea at

T=O.

Bt

Eliminating ~b 1 between (15) and (17) and then using (16) we get, 4- ~ - - ~2V2 nl = 0.

Here

4~rn o e ~ VF z

~ z - - _ _ a n d f l S : ~ .

m 3

08)

n~(r) ---- ~ ! , m R, (r) Ytm (0,¢). (19)

Substituting (19) into (18) with K 2 = (oJz" -- % s ) / ~ we get the equation for Rl (r) as,

d2R~dr ~ 4- 2r__drr4_dRt

[IC a - 1(1+1)1~.~.j R~ = O. (20) The solution of eq. (20) with the boundary condition of finiteness of Rz at origin is

Rz = Auz (Kr).

Here

u~(Kr) = j ~-~ l~+~2 (Kr),

the modified Bessel's function. From (16) we note that Cx(r, t) can also be separated as,

¢1 (r, t) = ~1 (r) exp ( - i o,, 0 (21)

expanding ¢l(r) as,

~1 (r) = ~ l , m ~t (r) r,m (0, ¢).

Using (16), and (22) we get the equation for ¢z(r) to be,

(22)

where

d2~b, 2 d~b, l ( l + l ) . L

Cu,(Kr) d# +-r drU-- r ~ v t =

C = 4~eA.

(23)

Addition of KS¢t on both sides of (23) gives,

d~z d_2 d~ 4_ [/~ _/(/4_1)]~b~

= Cu, (Kr) 4_ R'~v

dr~ ) ~ r r s d

⁽²⁴⁾

It is obvious from (18) that space and time can be separated as nl(r ) e x p ( - - i ~ d ) and since the problem has spherical symmetry, expanding ni(r) in spherical har- monics we get for the interior,

The interior solution of eq. (24) clearly is,

~t(r) : - - K--- C ~ us ( K r ) -t- B r i •

(25)

For exterior there are no real charges and therefore nl=-0 and the equation for ~t(r) simply becomes V ~ = 0 which clearly has the exterior solution as,

~1 (r) -- r 1+1 F (26)

Putting continuity condition on 4~(r) and O~dOr, we get by solving,

B-- C /I+1 ~ [

^{K R} ( K R ) + us

(KR)] (27)

~R, t2-TTis (#--~T) ,6

Thus we have got the complete solution for the interior.

nil(r) = ~t,m AU, (Kr) Y~m (0,4) (2S)

And,

f-"'<K')+ "" ](')'1 -

~ ( r ) = ,4m /C a \ 2 1 + l / k (1+I) R

× XYu~ (0, 4). (29)

Putting the usual hydrodynamical condition that the electronic velocity normal to the surface must vanish we get the surfaee-plasmon dispersion relation.

Since,

;¢1 = e v41 - - Vnr

j~2

(3o)

m n o

In our case the condition simply becomes,

This gives,

e 0~_~ _ ~ _ 0n~ _ -- 0 at r ---- R. (31)

m Or n o Or

KR %2 l ( l + 1)) (32)

which is the dispersion relation.

The dispersion relation given in eq. (32) is valid when the medium in which the embedded spNere is vacuum.

If on the other hand, ^~= is a dielectric constant of a medium in which the metal sphere is embedded, eq. (32) can easily be modified to,

u , ( ~ ) l

K R 1 [ l+P/~(e=l+l+~'~p~ ~ /J

Equation (33) reduces to a well known cold plasma result,

¢Oill ~°Jlm s

= as K R - , oo (34)

e . l + l + l

Further, as 1-~ oo eq. (34) reduces to, . ' = . . ' / ( ~ = + 1 )

which is the well known result of Stem and Ferrell (1960) for plane interface.

Actually, the model of a metal sphere with sharp density cut off in the electron density is rather artificial. If one analyses the expression (32) for oJ4 versus R, one can see that o,, depends very weakly on R, increasing as R decreases. This clearly is a wrong behaviour. If one looks closely at the surface, the electron distribution oozes out, going to zero within the distance of the order of one or two Fermi wave- length. For exact profile calculation, one will have to solve the equation of Kohn and Sham (1965) and Hohenberg and Kohn (1964). This is a rather complicated density profile to handle. Benett (1970) and Feibelman (1971) investigated the plane surface with a linear density profile and have obtained a qualitatively correct beha- viour of 0)4 versus R. The same problem can be tackled with different density pro- files and the qualitative behaviour of the dispersion relation does not change with the nature of the profile. We assume that this is true for the sphere as well. For this case, the double step model of Boardman et al (1975) is rather easy to handle.

2.2. Double step model

We take the density of electron as

n(r) = n o r < R

= an o R < r < R + A

= 0 r > R + A

In this case one has to solve the basic eq. (18) in three different regions. The radial solution in the three regions for density fluctuation will be,

R,(r) ⁼ ~lu,(Kr); r < R

= Bxuj(K~r) + Cxv,(/txr), R < r < R + A --- O: r > R + A

F o r

K1 s = (oJ~ a - - co~pl)/flx 2 < 0 a n d / t ~ < 0, u~ and v~ are modified Bessel's functions

2Krl ll+tt/s~

(Kr) and \ 2 K r /

L~-tl/s~ (Kr)

respectively. With this choice of a density fluctuation, one can solve Poisson's equation (analogus to eq. 23) and can obtain

.d 1

if, (r) = - - ~ u,(Kr) +

Dr'; r < R

B1 ~vl(Klr)+ Erl +r-~i; R <r <R + A -- _ 1(,1--- 2 u,(Kz) --

~--- G/r~+l; r > R -]- A.

Here ~p~ and ~ are parameters corresponding to density an 0. The boundary con- ditions that fix the constants are as follows. The density fluctuation is continuous at

r : R

and the normal component of velocity must be continuous at R. Then we have the continuity of ~t and

d~ddr

at

r:-R

and at

r = R + A .

These six conditions are sufficient to express all the constants of the previous equations in terms of one. To get the dispersion relation, as usual, we impose the condition similar to eq. (31) at

r = R + A

and we obtain,

qt

t [ 2 °Jp,. 1 ( l - t - l ~ ]

^--

l(l +

1) 1 u, (/(1 [R + A]) ~ 2

(2t + l) (R + A) KI' ~ --

~R--+---A/{ R 1'+21Iu't(K1R)(2l

~ 1)

-~t -- Rl Ul(Kt R) C°~ ] a pt +

I [2

^{÷ % .}

_,a]

_{K1 ~21+1/}

]

^-

z(l+l) 1 v,fK~[R+A]).

~ , _ 2 (2t-t--l) ' ( R + A ) K~"

where,

(,:+1) k

__ ~I u,(KR)__u,'?)I( R__I'+' " I ~p.

q, = (KIR)' [pu,,(KR) u,(KIR ) - - u,

(KR)

^u,"

(KxR)'I

(35)

and

P ~ no / n01 / with no 1 = an o

91

u,(KR) (K1R)

u,

(iclR)

This expression, certainly not very transparent, reduces to eq. (32) when A -+ 0,

O. --~" 1 .

In this case K 1 -* K, /~i - * / / , no I -* no.

Hence q~-* 0 a n d qz ~ 1.

The extension o f the result in eq. 0 5 ) to the case when the sphere is embedded in a dielectric is straightforward.

3. Discussion

The plasmon dispersion relations for both the cases discussed earlier are quite compli- cated and require a large computer programming and due to its inadequacy, we could not carry out intensive calculations. We have obtained a set o f computer results which is sufficient to indicate the purpose behind this paper a n d the success achieved therein.

F o r o u r theoretical calculations we have taken the Smithard's (Smithard 1973) case o f silver particles embedded in glass. F o r this case

f l = 3 . 6 × 10-Sc and , = = 2 . 3 .

In figure 1 we have shown the variation o f cot/o~p with R, f o r / = 1 a n d / = 2 modes.

It can be clearly seen that o,t increases as R decreases in contradiction with the experi- ments.

In

figure

2 we have shown the variation o f odwj, with R for ~ ---- 0.12 and A ---- 2A, 3A and 5~k. Our calculations indicate that the trend o f decrease in o t with de-

o

crease in R sets in at A ~ 2.5A; however, a sufficiently diffused electron surface profile with A = 5A is necessary to obtain an agreement with experimental results.

T h o u g h the total inhomo~enity at the surface o f silver particles exists over a radial

Sharp surface Cutoff

O~B, Computer accurOcy in w t/(~p

u S e d ~ ± 0 . 0 1

0 , 4 I I I I I I

0 qro EO 3 0 4 0 5 0 6 0

R%

Figure 1. Dependence o f ~'ll~w o n the radius o f a silver sphere embedded in glass (era=2"3) for l = l a n d 1 = 2 modes. Here cot is the surface plasmon frequency for the sharp electron density cut off at the surface.

Diffused sur face Ct = O" 12 , L = 1 Mode

0 . 6 Computer OccuroCy ,n

WllUPp used= ± 0.01

• Experimental values 0 , 5

F5

~- ^{0 4} U

0-2 I I I I l I

10 2 0 3 0 4 0 SO 6 0

RoA

Figure 2. Dependence of cot/to v on the radius of a silver.sphere with diffused surface, embedded in glass (era=2"3) for i=1 mode with A = 2 A, 3A and 5A [==0"12].

dimension of 4-5,&; for a = 0"12 the model value of 5/k is not very realistic. One would rather expect a value A "~ 2.5 to 3A. We feel that more refined experi- mental investigations are necessary for arriving at definite conclusions. The present

o

model value of A --- 5A may therefore be considered as partially empirical.

The question arises here that the shift in the frequency may occur because o f photon plasmon coupling. This interaction part o f the Harniltonian will naturally be

n'

= - = n 0

e f

A.v@ld~

c

where A is a vector potential for photon field. By applying a second order perturba- tion theory and expressing ~b 1 and A in second quantised form one can obtain the

shift. Wilems (1968) has obtained the shift for the case o f slab. His results o f shift are

~ a S e

2K~,

where a is the width o f the slab and Ap is wavelength corresponding to plasmon fre- quency oJp. With the dimensional argument, we expect the shift in o u r case to be

cG2R 8 A oJl ~ - - a l .

where a t is a constant. With ct ~ 0.1, R , ~ 50 ~,, l = 1 a n d ~p , ~ 3000 A, we find AoJt ~ , a t × l0 s. The constant a 1 can be f o u n d by exact calculations. However, we do not expect it to be large enough so that AoJt , ~ oJ~ which is o f the order o f 10 ~5. F r o m the theoretical calculations and the results given in this paper, the following conclusions can be drawn.

(1) The double step model gives a simple representation o f a ddffused surface, which is easy to use and which has sufficient parametric freedom. (2) The surface profile must be taken into account to obtain the correct p l a s m o n dispersion for the spatially dispersive electrostatic modes o f small metal spheres.

Acknowledgements

The authors are thankful to Prof. M R Bhiday for valuable suggestions and en- couragement. One o f the authors (SBO) is grateful to D A E for financial support.

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