Comment on "Silver nanoparticle array structures that produce remarkably narrow plasmon line shapes" [J. Chem. Phys. 120, 10871 (2004)].

Recently, Zou, Janel, and Schatz sreferred to as ZJS belowd have described remarkably narrow plasmon resonances in linear arrays of silver nanospheres. Without questioning the novelty and significance of these results, I would like to point out that the above-referenced paper contains two incorrect statements. The first statement is about my previous work. Namely, ZJS write that in a previous study I have considered “...infinite one-dimensional arrays in the quasistatic approximation.” In fact, there was no quasistatic approximation made in Ref. 2. The approximation that was made was the dipole approximation. These two approximations are distinctly different. For example, even in the electrostatic limit, the dipole approximation is grossly inaccurate for two touching conducting spheres excited by a constant external electric field parallel to the axis connecting the spheres’ centers. On the other hand, electromagnetic interaction of small impurities in a crystal or of dye molecules in large molecular aggregates cannot be understood within the quasistatics, although the dipole approximation may be very accurate in this case. Perhaps, the source of confusion is that in Sec. II A of Ref. 2 I wrote “The object under investigation is a linear infinite chain with step a consisting of pointlike dipole units smonomersd....” Also, in the Introduction of Ref. 2, I have suggested that the physical system to which the considered model is applicable is a molecular aggregate. Later, in Sec. V, I have considered a particular example in which the polarizability of a dipole, a, was given by the quasistatic polarizability of a small sphere with the appropriate radiative correction. However, the theoretical formalism of Ref. 2 did not put any restrictions on a. And, regardless of the form of a, the interaction of dipoles was described with full account of retardation effects. In fact, ZJS also work in the dipole approximation, although they validate their results by comparison with a more general T-matrix solutions. The situation is somewhat more complicated, however, because ZJS use, in addition, an approximation proposed by Doyle in 1989 sRef. 5d in the context of effective-medium theory of the so-called extended Maxwell–Garnett composites, i.e., composites in which inclusions are not small compared to the wavelength. More specifically, Doyle has studied electromagnetic properties of a homogeneous host with randomly distributed spherical inclusions. The essence of the approximation is to consider only dipole-dipole interactions of the inclusions but to assign them dynamic dipole polarizability a. The latter is given by formula s2d below; it is defined as the linear coefficient between the amplitude of incident plane wave and the total dipole moment of polarizable sphere of arbitrary size sassuming, the sphere is isolatedd and, in that sense, is exact. It can be seen that the Doyle’s approach only concerns the choice of a within the dipole approximation. Thus, it is fully consistent with the general formalism developed in Ref. 2. It should be noted that the accuracy and limits of applicability of the Doyle’s approximation have not been systematically investigated. In one critical study of extended Maxwell–Garnett composites Ruppin has shown that the Doyle’s approximation is consistent with the asymptotes obtained in the limit of small volume fraction of inclusions, and, in that limit, allows one to consider inclusions with size parameters of at least x,0.5. Thus, the Doyle’s approximation can be useful for moderate size parameters. However, if the spherical inclusions are in close proximity of each other, the secondary scattered waves incident upon each of them are no longer plane waves. But the dynamic polarizability used by Doyle is exact only with respect to incident plane waves. Besides, coupling of higher multipole modes excited in spherical inclusions can become significant. Therefore, it is quite obvious that the use of Doyle’s approximation does not fix, in principle, the deficiencies of the dipole approximation. The second statement concerns the possibility of cancellation of the imaginary part of denominator in the expression P=aE0 / s1−aSd fEq. s5d, or, in a more specific form, Eq. s7d of Ref. 1g. This is discussed on p. 10874 of Ref. 1. ZJS consider the case when the incident wave vector is perpendicular to a linear chain of polarizable dipoles with the period D. The polarization of the incident wave is also perpendicular to the chain. It is stated that the resonance width, which is related in Ref. 1 to the imaginary part of the denominator of the above equation, vanishes when g ø8p3A /D3, where g and A are parameters which specify the polarizability of an isolated sphere. Namely, ZJS use the formula a=−A / sv−vp+ igd, where v is frequency of incident radiation, vp is the surface plasmon frequency and g is the relaxation parameter. Assuming that the result ImS=−8p3 /D3, which is given in Ref. 1 for l slightly larger than the interparticle distance D, is correct, one immediately can see that the cancellation takes place exactly at g =8p3A /D3. For smaller values of g, the imaginary part of THE JOURNAL OF CHEMICAL PHYSICS 122, 097101 s2005d