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Langmuir 2002, 18, 2436-2437
Optical Properties of Small Clusters of Silver and Gold Atoms Robert H. Doremus Materials Science and Engineering Department, Rensselaer Polytechnic Institute, Troy, New York 12180 Received August 24, 2001. In Final Form: December 18, 2001
Introduction The transition from atoms to metal leads to splitting of atomic electron levels to form a conduction band of closely spaced electronic levels of free electrons. The optical properties of metals involve both interactions of light with the free electrons and electron transitions from one band to another. As the particle size of metallic clusters decreases, the mean free path of the free electrons in the clusters is reduced, and the optical properties of the metal in the clusters change. Small noble metal (Au, Cu, Ag) particles show a plasma absorption band in visible light resulting from collective oscillations of the free electrons excited by light of a particular wavelength. As the particles become smaller than the mean free path of the free electrons, this band broadens. For very small particles (less than about 1 nm in diameter) this plasma absorption band completely disappears. For particles of gold of this size embedded in glass, the optical absorption in visible light depends on the negative fourth power of the wavelength.1 This dependence can be understood from the Mie equations for optical absorption of spheres2 and the optical properties of free electrons, modified for the small particle size.1 See ref 1 for more detailed discussion. Van Hyning and Zukoski have reported optical spectra of growing silver clusters in aqueous solution.3 Of particular interest for comparison with the gold results is their spectrum of very small clusters of silver (Figure 2, ref 3, Figure 1, ref 4). This spectrum also shows a dependence of optical absorption on the negative fourth power of the wavelength, as shown in Figure 1. In this Note a possible explanation for this dependence is discussed. Optical Properties of Spherical Particles The optical absorption R of a collection of uniform spheres that are much smaller than the wavelength of light and embedded in a medium of refractive index n is2,5
R)
18πQn32/λ (1 + 2n2)2 + 22
(1)
where Q is the volume fraction of particles and 1 and 2 are the real and imaginary parts of the complex dielectric constant * ) 1 - i2 of the particles. (1) Doremus, R. H.; Rao, P. J. Mater. Res. 1996, 11, 2834-2840. (2) Mie, G. Ann. Phys. 1908, 25, 377-445. (3) Van Hyning, D. L.; Zukowski, C. F. Langmuir 1998, 14, 70347046. (4) Van Hyning, D. L.; Klemperer, W. G.; Zukowski, C. F. Langmuir 2001, 17, 3128-3135. (5) van de Hulst, H. C. Light Scattering by Small Particles; John Wiley and Co.: New York, 1957; p 270.
Figure 1. Logarithm of optical absorption of small silver clusters as a function of logarithm of wavelength of light. Slope of the line is -4.27 by linear regression with an intercept of 4.15 and a correlation coefficient of 0.998.
When the optical properties of a metal depend on the free electrons in it, eq 1 gives an absorption band (the plasma absorption band) of Lorentzian shape.6 The absorption of this band is a maximum when the first term in the denominator of eq 1 is zero, or
1 ) -2n2
(2)
As the radii of noble metal particles decrease below the mean free path of electrons in them, the plasma absorption band broadens. At 25 °C mean free paths are as follows: silver, 57 nm; gold, 41 nm; copper, 42 nm.7 An equation for the dependence of the effective mean free path 1e of an electron in a spherical particle of radius R is6,8
1 1 1 + ) le R l
(
)
(3)
in which 1 is the bulk mean free path. (This eq 4 in ref 1 needs correction.) This reduction in mean free path causes a reduction in effective optical conductivity in the particle, which translates into an increase in the effective 2 of the particles, because 2 is inversely proportional to the conductivity in the classical Drude model for free electrons.9 In the Drude model 2 is approximately proportional to the cube of the wavelength (λ) of light for free electrons
2 ) Aλ3 + B
(4)
For silver approximate values of A and B are 1.5 and 0.08 (6) Doyle, W. J. Phys. Rev. 1958, 111, 1067-1077. (7) Kittel, C. Introduction to Solid State Physics, 2nd ed.; John Wiley and Co.: New York, 1956; p 240. (8) Doremus, R. H. J. Chem. Phys. 1965, 42, 414-417. (9) Mott, N. F.; Jones, H. The Theory and Properties of Metals and Alloys; Oxford Press: Oxford, 1936; p 105.
10.1021/la011350h CCC: $22.00 © 2002 American Chemical Society Published on Web 02/08/2002
Notes
Langmuir, Vol. 18, No. 6, 2002 2437
from the optical properties of Otter.10 The value of 1.5 for A corresponds to a mean free path of electrons of about 30 nm, which is smaller than the value calculated from electrical conductivity given above. This difference is probably caused by the anomalous skin effect and defects in the silver surface. The experiments of Otter give the lowest value of A and thus the highest mean free path, so they are judged to be the best values for silver; the optical properties measured by Otter are also found to be the best for gold and copper. The above discussion shows that the value of 2 increases as the particle size decreases, because A in eq 4 is inversely proportional to the mean free path. If the particle is so small that 22 . (1 + 2n2)2 in the denominator of eq 1, then this equation becomes
D ) 18πQn3/2λ
(5)
Since 2 is proportional to λ3 (eq 4), eq 5 shows an inverse (10) Otter, M. Z. Physik 1961, 161, 163-178.
fourth power dependence of absorption R on λ, as found in ref 1 for gold and in the spectra of refs 3 and 4 for silver (Figure 1). Discussion From the discussions in refs 3 and 4, it is estimated that the silver particles in the solution that gave the spectrum used to make the plot in Figure 1 were roughly 1 nm in diameter, which contain about 30 silver atoms. This size is similar to those in refs 1 and 11 that also gave a reciprocal λ4 dependence of absorption. This dependence should arise for free electron interaction with light when the mean free path of the electrons is of atomic dimensions (0.5 nm or less). Thus the small clusters containing about 30 gold and silver atoms have electronic properties characteristic of free electrons and not of atomic-like bandto-band or level-to-level transitions. LA011350H (11) Duff, D. G.; Baiker, A.; Edwards, P. P. Langmuir 1993, 9, 23012310.
