Surface plasmons in small layer metal particles

Solid State Communications, Vol. 20, pp. 545—547, 1976.

Pergamon Press.

Printed in Great Britain

SURFACE PLASMONS IN SMALL LAYER METAL PARTICLES A.A. Lushnikov, V.V. Maksimenko and A.J. Simonov Karpov Institute of Physical Chemistry, Moscow, USSR (Received 11 March 1976 byL. Hedin) Positions of two surface plasma resonances in a layer metal particle containing a dielectric foreign nucleus are calculated within the random phase approximation. The results obtained give a possible explanation of a recent experimental observation by Qenzel et aL’ of two peak structure in the photoabsorption cross-section of small silver particles.

1. INTRODUCTION RECENT experimental studies1’2 of surface pla~monsu~ small metal particles contain some contradictions concerning the size dependence of the dipole plasma peak position. Smithard2 observed that the plasmon fre. quency decreases with increase in particle size while Genzel et aL’ in their experiments obtained quite the opposite results. Moreover, for particles whose diameter is less than 20A the last group discovered a two peak structure the plasma resonance. Almost all existing theoretical models1’35 predict increasing the plasmon resonance frequency (wy) with decrease in particle size. No one of these models, however, is able to explain the latter fact. In this note we wish to propose a possible explanation of the two peak structure of the plasma resonance discovered in experiments by Genzel et al.’ and point out possible reasons for decreasing c~,with decrease in particle size observed by Smithard.2 Our basic idea is that the particles used in both these experimental studies were not continuous. l’his assumption is not too unexpected, the for particle formation process described in references wassmall heterogeneous, which the particles1 and grow2 on foreign nuclei. Thismeans makesthat us assume the particle to have the nut structure shown in Fig. 1, i.e. the particle is a small dielectric nucleus covered with metal shell. The presence of the foreign

also to the lowering of the position of the external plasmon peak. The size stiffness, however, acts against this effect.7 So, the position of the external plasma peak is determined by an interplay between the two counteracting effects. Below, we outline the method of calculation of the plasma peak positions in a layer metal particle and compare our results with available experimental data. 2. POSITIONS OF THE PLASMA PEAKS In order to make our presentation more concise we restrict ourselves by considering the model of free elec. tron gas, i.e. put Ed = Em = 1 (Cd and 6m are the dielectric constants of the surrounding dielectric and the ion lattice inside the metal shell respectively). So the partide is replaced by a model in which the conduction electrons move inside a rectangular potential well forming by positive ion background. Consider first the case a > b ~ r 0 where a and b are external and internal particle radii respectively and r0 is Thomas—Fermi screening length. The positions of two plasma peaks 1.~1,2 are defined by the condition5of existence of nontrivial e~ solution of the equation V(r) = V~’Q(r r’)V~’n(r’) V(r’) d3r’ (1) —

where Q(r

—

r’)

—~

=

J

—

—

r’11, n(r) = n

0 [O(a r) O(b r)] is the conduction electron density, 0 is the Heavyside step function and the system of units is used h = m = I (m is the electron mass). Expanding equation (1) into spherical harmonics we obtain the following equation for VL(r) (in what follows we consider the dipole oscillation only L = 1) —

nucleus leads to the appearance of the second branch of surface plasmons related to surface charge oscifiations Ofl the internal boundary of the metal shell. It is quite evident that the probability of excitation of the internal plasmon should fall on increasing the particle size, for the size of the foreign nucleus is fixed whereas the thickness of the metal shell grows. In decreasing the particle size the metal shell becomes thinner making thereby observation available of the second plasma peak in photoabsorption experiments. the case thin 6 decrease of thickness ofAs theinmetal shellofleads fflms, 545

—

—

~,

(r)

=

2 3

2~

~ ~

—

~

a

V1(a) + 1

WO —

a V1(b) ~2 ~.2

(2) where wo = (4irnoe2 )I /2 is the bulk plasmon frequency.

SURFACE PLASMONS IN SMALL LAYER METAL PARTICLES

546

/

Vol. 20, No.6

______

Fig. 1. Internal structure of the particle C$~e~and ~m are the dielectric constants of the nucleus, surrounding dielectric and ion lattice inside the metal respectively.

___________________________________________________

10

20

30

I~ettingin this equation r = a and r = b we come to the Fig. 2. Size dependence of surface plasmon frequencies. system of two algebraic equation the condition of exisCurves 3 and 4 and curves 2 and 1 correspond to the tence of its solution determines two plasmon frequencies: internal and external plasmons (accounting for the size stiffness effect and without it respectively). ~m = 4~9, ed=eS=2.25,b/rO=4,ro=l.3A. 3J (3) Wi,2 = [1 ; ~s~/l+ 8x 2 where where x = b/a. We also write down two eigenfunctions, correspondW 2, W_ 2, 1 = ~8 w~v~ 1 = ~8 w’1v081 ing to these eigenfrequencies: —

Vk(r)

r =

+

1 a2 / ~ ~1 3~) —

0

(4)

~ are the excitation energies of the electron gas and V is an external field exciting effectively the plasmons. We choose it in the form (4): V 1 or V2 for excitation of

where k = 1, 2. external and internal plasmons respectively. The sum At x 0,the frequency Wi = w~/~ while W~ = rule W_1 is expressed in terms of the static electric v’~7T)wo.These values are characteristic for surface polarizabiity and dipole plasma oscillations of continuous metal sphere ~ V HV 0> 6 (~o/~/~) and for spheric hole in a bulk metal (v’~7~)wo) 1 (v’~7~o.~o). The dependencies of ~ i and w2 on a at where H is the Hamiltonian of conduction electrons and fixed b are shown in Fig. 2 by dashed lines (curves 1, 2). [ ] stands for the commutator. Once the solution of equation for the effective field ~k (r) in the static limit 3. SUM RULE CALCULATIONS (w =0) 2p small atoand values, someofeffects essentiallyAtrelated the bsize stiffness Fermibecome gas of conducO,~(’)= Vk(r) e Q(r ri)n(ri)Ok(rl) d3r 1 (7) 7 This fact leads to increasing the surface tion electrons. plasmon frequency in small continuous metal particles is known the static polarizabiity ~(0) can be found with decrease in the particle radius. In small layer parfrom the relation:5 tides the position of the external plasma peak is deter~2 mined by an interplay between two counteracting a(0) = j Vk(r)n(r)~k(r)d3r effects: decrease of the thickness of the metal shell tends to reduce w 1 while the size stiffness acts in opposite where PF is the Fermi momentum. Equation (7) may be direction. In order to calculate the plasma peak positions solved exactly, but the solution is too cumbersome and in small particles we use the sume rule technique deis not presented here. scribed in detail in reference 5. The averaged squared The frequencies c~512calculated in this way are plasma frequencies ~ are defmed as the ratios of drawn in Fig. 2. energy weighted and reciprocal energy weighted sum 4. DISCUSSION rules: = W1/W_1 (5) In Fig. 2 the dependences of plasma peak positions on the ratio a/r0 at fixed b are shown (curves 3 and 4) -~

—

—

—

__~~~EJ

—

—p-

Vol. 20, No.6 for Em

=

4.9~~s =

SURFACE PLASMONS IN SMALL LAYER METAL PARTICLES =

2.25, b

= 4r

0 and r0 = 1.3 A is the dielectric constant of the foreign nucleus). Curves I and 2 are the result of calculations in the limit of r0 = 0. The most remarkable feature of the depen(e~

—

.

plasma peak in photoabsorption experiments.’ The photoabsorption cross-section, calculated in the limit of r0 -÷0 has the form: 3

a

2 2

aW07r

.

dence of w1,2 on a/r0 is the presence of the maximum on the curve 4. This is the result of an interplay between two counteracting effects: increase of the size stiffness and the effective repulsion of two plasma levels appearing decreasing the metal shell thickness. Now we thatinthe dipole p1~smafrequency might increase or see decrease with decrease in the particle size depending on the size of the foreign nucleus. This fact sheds some light on possible reasons for difference in experimental observations of Genzel et al.1 and Smithard:2 the foreign nuclei might be smaller in the latter case. Our assumption about the nut structure of the small metal particle explains also the appearance of the second

547

=

[f~(x)6(w

—

w1)

+f2(x)~(w

c

—

(02)]

(8

where 3)[1 ±(1 + 8x3)U2]. (1 —x As is seen from equation (8) there are two peaks in the cross-section and the intensity of the higher peak is lower than of the lower one. At small x = b/a the photoabsorption cross-section of excitation of the internal plasmon is smaller in x3 times as compared to the cross-section of the external plasmon. This agrees well with the fact that the second peak disappears in increasing the particle size. fi,2(x)

=

REFERENCES 1. 2.

GENZEL L., MARTIN T.P. & KREIBIG U., Z. Phys. B21, 339 (1975). SMITHARD M.A., Solid State Commun. 13, 153 (1973).

3.

GANIERE J.-D., RECHSTEINER R. & SMITHARD M.A., Solid State Commun. 16, 113 (1975).

4. 5.

CINI M. & ASCARELLI P., J. Phys. F: Metal Phys. 4, 1998 (1974). LUSHNIKOV A.A. & SIMONOV A.J., Z. Phys. 270, 17(1974).

6.

JAKLEVIC R.C., LAMBE J., MIKKOR M. & VASSELL W.C., Phys. Rev. Lett. 26,88(1971).

7.

LUSHNIKOV A.A. & SIMONOV A.J., Phys. Lett. 44A, 45(1973).