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OPTICAL PROPERTIES OF ULTRAFINE Au
PARTICLES PREPARED BY GAS EVAPORATION
C. Granqvist, O. Hunderi
To cite this version:
C. Granqvist, O. Hunderi. OPTICAL PROPERTIES OF ULTRAFINE Au PARTICLES
JOURNAL DE PHYSIQUE Colloque C2, suppl6ment au no 7 , Tome 38, Juillet 1977, page C2-143
OPTICAL PROPERTIES
OF ULTRAFINE Au
PARTICLES PREPARED BY GAS EVAPORATION
(*)C. G. GRANQVIST and 0. HUNDERI (**)
Physics Dept., Chalmers University of Technology Fack, S-402 20 Gothenburg, Sweden
R h m b .
-
La transmission optique par particules extrafines d'or (diamktres 3-4 nm), prBparBes par Bvaporation dans l'air, fut mesurBe dans l'intervalle de longueur d'ondes0,3-2,5 pm. Les rBsultats furent interpr6tBs avec l'aide des thCories pour un medium effectif d'aprks les travaux de Maxwell-Garnett, Bruggeman et Hunderi. Pour la fonction dielectrique nous avons utilisB des rBsultats experimentaux pour or massif, qui furent modifies en regard de l'influence de la limitation du libre parcours moyen des Blectrons de conduction. La couche &ant considBrBe comme un melange de sphkres indBpendantes, de chaines linBaires infinies et de masses compactes et en regardant l'interaction dip8le-dip8le pour ces configurations nous avons obtenu des r6sultats de transmission en accord avec les experiences pour un certain pourcentage, indCpendernrnent de la longueur d'onde.
Abstract. - Optical transmittance of ultrafine gas evaporated gold particles (diameters
3-4 nm) was measured in the wavelength range 0.3-2.5 pm. The data were interpreted within effective medium theories based on works by Maxwell-Garnett, Bruggeman and Hunderi. For the dielectric function we used measured bulk results, which were modified to account for boundary scattering of the electrons. Assuming the samples to consist of a mixture of independent spheres, infinite linear chains and closepacked clusters, and incorporating the dipole-dipole interaction for these configurations, we obtained computed transmittance data which agreed with the measurements to within a few percent at all wavelengths.
Optical properties of fine noble metal grains have been studied for a long time. In the previous works the particles were prepared as colloidal solutions [I-31, in a photosensitive glass [I], by island growth in discontinuous films 14-63 or in sputtered Au-SiO, cermet films
[?I.
Here we report on some results from our extensive investigation of the optical absorption in spherical and individually isolated gold particles produced by gas evaporation [8]. In contrast with most other techniques this method allows reliable determinations of particle sizes and size distributions and avoids effects from a medium surrounding the metallic grains. Some complication enters because the gas evaporated particles display aggregation effects. Our experimental results can be brought into excellent quantitative agreementwith effective medium theories which incorporate dipole-dipole interactions. For the metallic particles we used the ellipsometrically determined [9] dielec- tric permeability for bulk Au, which was modified to account for a size limited mean free path in the Drude part. Some preliminary results from our study have been reported in reference [lo].
The particles were prepared by a technique devised by Harris et al. [ll]. Gold was evaporated
from a tungsten boat at a pressure of 2 t o n of air,
(*) Supported by the Swedish National Science Research Council.
(**) Presently at Dept. of Physical Metallurgy, Norwegian Institute of Technology, Trondheim, Norway.
and the spontaneously nucleated particles were collected on a glass surface positioned above the vapour source. The electrical resistance of the layer was very high, showing that the particles were isolated at least on a macroscopic scale. The deposit thickness t was obtained by focusing an
optical microscope at the.top and bottom surfaces ;
a typical value was 5 pm. The mass per unit area W/A was obtained by weighing the sample before and after wiping off the deposit from part of the substrate. The volume fraction of metal (the filling factor
fi
was derived from f = (W/A) ( ~ t ) - ' where p is the density of bulk gold. Experimentalf
s for four samples are given in table I, where it is seen that 0.3 % is a characteristic value.Electron microscopy showed well rounded objects which normally were aggregated into chains or clusters. A typical bright field electron micro- graph is depicted in figure la. Electron diffraction displayed broadened Debye rings signifying f .c.c.
TABLE I
Data for the Au particle samples
Sample W/A f [glm21 (%)
C2-144 C.G. GRANQVIST AND 0 . HUNDERI
0
2
4
6
x
C n m l
FIG. 1. - Part (a) shows a high resolution electron micrograph obtained for sample 20 B in a Philips EM 300 electron rnicros- cope. The corresponding diameter distribution is shown in part (b). The circles denote staple midpoints in a size histogram and the curve represents the log-normal distribution function (eq. 1)
with I and o as shown in the figure.
Au and in addition one diffuse ring due to WOs. Dark field electron microscopy revealed that the tungsten oxide, which originated from the evapora- tion source, formed tiny particles interdispersed among the gold granules. No coating on the Au particles was evident, but it must be surmised that a thin layer of, presumably, a gold oxide provided the desired electrical insulation of neighbounng grains.
The size distributions were log-normal, i.e. the number of particles An per logarithmic diameter interval A(ln x) could be well approximated by
(1) where
x
is the statistical median for the diameters and a denotes the (geometric) standard deviation. This functional form is characteristic for coales- cence growth of particles [8]. The accuracy ofeq. (1) is illustrated in figure lb, where Z and a
were adjusted to provide a good fit to the experimental size histogram. Table I contains 3 s and d s for four samples.
Optical transmittance at normal incidence w a s measured in t h e wavelength range 0.33 < h
<
2.5 pm with a UNICAM SP 700 double beam spectrophotometer. A glass with particle coating was put in one of the light beams and a clean substrate in the reference beam. Figure 2 shows transmittance against wavelength for four samples of different thicknesses. The overall repro- ducibility is seen to be excellent. A distinct transmittance minimum, which is not present for bulk specimens, occurs consistently around 0.56 to 0.60 pm.0
0 0 5 1 .O 1.5 2.0
Wavelength [ p m ]
FIG. 2. - Measured wavelength dependent transmittances for four gold deposits.
For our samples x 4 h showing that one can rely on an effective medium treatment for the composite of particles (dielectric function E ) in a surrounding medium (dielectric constant E,). The simplest relation for an average dielectric permeability E was given. by Maxwell-Garnett (MG) [12] who found
OPTICAL PROPERTIES OF ULTRAFINE Au PARTICLES C2-145
The subscript j denotes particles in the jth column of a size histogram centered at a diameter xi, and the fi's are a set of fractional filling factors normalized by
2
fJ = f. aj is proportional to the polarizability ofj
the particles of size j, and for an ellipsoidal shape it is given by
where the L, 's are the depolarization factors for the particles. To first order eq. (3) reduces to eq. (2) for spheres, i.e. when L 1 = L2 = Lj = 1 /3. Though an improvement over the simple MG model, the MG-PvS theory should be restricted to small filling factors.
Another self-consistent effective medium theory, which in principle should be valid for all f's as it treats the components in the mixture on an equal basis, was originally formulated by Bruggeman (BR) [14]. The effective medium permeability is here given by
which has obvious formal similarities with eq. (3). Finally, the apparent dielectric function from the backscattered fields, as shown recently by Hunderi (HU) [IS], yields still another expression,
which should hold for small filling factors. For the samples of present interest the f's are so small that the MG-PvS, BR and HU theories give indistin- guishable "results.
For the,,"dielectric function of the gold particles we started from ellipsometrically determined bulk data [9], E ~ ~ ~ ( w ) , which were modified to account for size dependent scattering in the Drude (free electron) part according to
This construction clearly leaves the interband part in E ~ ~ ~ ( w ) unchanged. The two Drude terms in eq. (7) can be written
and
where W , is the plasma frequency, r b is the electron lifetime for bulk Au and 7j is given by
where v, is the Fermi velocity. Eq. (10) presumes diffuse boundary scattering of the conduction electrons, in which case the mean free path is equal to the particle radius. From Winsemius [9] the bulk properties for Au are fiw, = 8.55 eV, h/rb = 0.108 eV and vF/c = 4.7 X
The above equations allow a calculation of the effective dielectric permeability from which the transmittance is obtained by the standard equations for a thin film on a substrate. The dotted curve in figure 3 shows theoretical results with srn = 1 and
0
0 0.5 10 15 2 0
Wavelength [ p m ]
RG. 3. - Transmittance versus wavelength for ultrafine gold particles. The solid curve shows measured data for sample 20 B. The dotted curve represents computed results for spherical particles without dipole-dipole interaction, i.e. with LP = 113 in
the effective medium theories. The dashed curve was obtained for interacting spheres (LT # 113) described by 6 = 0.32 and
6 = 0.45.
parameters chosen to correspond to sample 20 B (cf. Table I). Comparing with the measurements (solid- curve) it is found that the qualitative behaviour of the two sets of 'data agree, but that there are important quantitative discrepancies : the location of the computed transmittance minimum occurs at a too small wavelength, and the overall magnitude of the theoretical transmittance is roughly a factor two too high. To understand the relevance of these features we have investigated [16] the role of dielectric coatings on the particles and of deviations from sperical shapes. Both these effects are capable of displacing the transmittance dip towards larger wavelengths, but detailed computations disclose unambiguously that neither can be reconciled with all the experimental evi- dence.
Instead we have focused on dipole-dipole interac- tions, which can be important in spite of the very small f's because the gas evaporated particles must be touching (cf. Fig. la). Our semi-quantitative description of the coupling is founded on a recent
C2-146 C.G. GRANQVIST AND 0. HUNDERI article by Clippe, Evrard and Lucas [17], who
calculated the resonance frequency for several geometrical configurations of identical touching spheres. From their work we have extracted a set of effective depolarisation factors, L
T
; which make the approach formally identical to that for ellipsoi- dal particles (described instead by a triplet of Li's). For the two most interesting cases, infinite linear chains of particles and closepacked clusters, the L*'s are (0.133, 0.435, 0.435) resp. (0.0865, 0.0865, 0.827). To get a final comparison of theory and experiments we considered the sample to consist of a mixture of aggregates with geometrically well determined configurations, i.e. we required that the filling factor would beEf
for single spheres,5f
for infinite chains, and (1-&C)f for f .c.c. clusters,where
6,
2 0 and5
+
5
5 1. By adjusting the twoparameters
6
and we arrived at a best fit shown as the dashed curve in figure 3. It is seen that agreement with the measured data to within a few percent is found over the entire wavelength inter-val. We would like to stress, that we feel that our approach to treat dipole-dipole interactions cer- tainly involvs more than mere curve fitting. The
only parameters entering the calculations are
5
and f ; whereas we used experimental values for t, f, size distribution and bulk dielectric function. With only two parameters we fit well a large number of characteristic features of the spectra ; for the absorption band centered at about 600 nm we get the right position, width and absolute transmittance value ; we also reproduce accurately features of both the infrared and the ultraviolet transmittance. In conclusion, we have measured the optical transmittance of ultrafine gold particles prepared by gas evaporation. The results could be describedquantitatively by effective medium theories which
incorporate dipole-dipole interactions. The dielec- tric function for the particles was given by bulk data which were modified to account for a size limited mean free path in the Drude part, whereas the interband part was kept unchanged. It is remarkable that quantum size effects, apparently, do not need to be invoked. A detailed report on this work will appear in reference [16].
References
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. - . ,
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