Theoretical Investigations of the Surface Polaritons Density of States on Silver Surfaces

THEORETICAL INVESTIGATIONS OF THE SURFACE POLARITONS DENSITY OF STATES...

THEORETICAL INVESTIGATIONS OF THE SURFACE POLARITONS DENSITY OF STATES ON SILVER SURFACES

C. Morari1, S. Astilean2

1 National Institute for Research and Development of Isotopic and Molecular Technologies, R-3400 Cluj-Napoca, P.O. 5, Box 700, Romania

2 Physics Faculty, 'Babes-Bolyai' University, Cluj-Napoca, R0-3400, Romania

Abstract

We present the implementation of a recently proposed method for solving the reduced Rayleigh equations at the metal-vacuum interface. Our implementation allows us to perform an extensive study of the surface plasmon polaritons properties (band structure and density of states are described). Several results are listed in order to illustrate the capabilities of our software.

Introduction

Conventional photonic crystals are artificial three-dimensional periodic dielectric structures that inhibit the propagation of electromagnetic waves of certain wavelengths [1]. In recent years, there has been a growing interest in employing surface-plasmon polaritons (SPP) propagating across rough surfaces for the same purpose.

In the present paper we report the results obtained for computing the density of states of the SPP's for a nanostructured silver surface. The theoretical model implemented by us was recently introduced by Kretschmann and Maradudin. It describes the propagation of the SPP's across periodic metal-vacuum interface and dielectric vacuum interface. Both band structure and density of states are reported, for several lattice parameters of the nanostructured surface. The results show the existence and the properties of the band gap in all the structures under investigation. Among the possible application of these results one of the most important is to give a deeper understanding of the phenomena involved in the SERS.

Method

Several methods were proposed in literature to compute the band structure of the SPP's [2,3]. Recently, Kretschmann and Maradudin proposed a nonperturbative approach based on the reduced Rayleigh equation, to study absolute photonic band gaps in the frequency spectrum of surface-plasmon polariton for periodic metal-vacuum interfaces [4]. In their model, a double periodic rough interface separates the metal (or dielectric) region from the vacuum region . In the relations above we have .

The profile function is describing an array of hemielipsoids with radius R and height, HR and it has the form (see Figure 1)

(1)

where S(XII) is the analytical equation that describe the profile given in Figure 1.

Figure 1: Schematic view of the x3=0 plane for a square array of hemispheres. kII is the wavevector of the surface plasmon polations. A detail on the profile function is given in the right side.

By asking to the electric field at the interface, governed by the reduced Rayleigh equation [5] to satisfy the Bloch-Floquet theorem, they arrived at the equations that governs the propagation of surface polariton across the interface. The solvability condition for the set of equations describing the eigenstates of SPP's is that the determinant of the coefficients vanishes

(3)

Eq. 3 provides the dispersion relations for the surface polariton, (kII). The corresponding density of states can be calculated using the relation

(4)

where the first summation index is running over the branches (k) computed by using Eq. 4.

Results

We performed a study of the dependence of the DOS with the filling factor, for a hexagonal lattice. Two ways for changing the filling factor were taken into account: (a) the lattice parameter was changed, while the radius of the profile function (R) was maintained constant; (b) the radius of the profile function (R) was changed for a given value of the lattice parameter. For all calculation the height of the profile function, H, was maintained constant: H=0.4. The results for the first set of calculations are summarised in figure 2, and those for the second set are given in figure 3. The parameters used for the calculations are also indicated. The value of the band gap was E = 0.260 eV (figure 3 left) and E = 0.275 eV (figure 3 right). The results are in good agreement with the results of Maradudin and Kretschman, showing that our implementation is correct.

Figure 2: The DOS for several lattice parameters, a. The numerical values of the lattice parameters are a=0.19 nm (left) and a=0.2nm (right). R=0.07.

The existence of a band gap due to the presence of nanostructures on the silver surface is proven theoretically. The position and the parameters of the band gaps depend both on lattice parameters and the shape of the profile function as shown in Figures 2 and 3.

Figure 3: The band structure of the SPP's for the DOS given in Figure 2.

A future application of these results will be to check, by experimental methods, the connection between the theoretical DOS and the SERS spectra on the nanostructured surfaces, as suggested by Gaponenko [6]. According to its studies, a connection between the peaks of the SPP's DOS and the SERS scattering amplitude must occur on nanostructured surfaces.

Conclusion

We implemented a new and efficient algorithm that allows us to compute the density of states for the surface plasmon polaritons at the metal-vacuum interface. The method is nonperturbative and has a good numerical accuracy. Our simulations show clearly the existence and the origin of the band gap that occurs into the SPP’s spectra on nanostrucutred surfaces.

References

1. J. D. Joannopoulos, P. R. Villeneuve, S. Fan, Nature 386, 143 (1997).

2. P. C. Chaumet, A. Rahmani, G. W. Bryant, Phys. Rev. B, 67 165404 (2003).

3. T. Sondergaard, S. I. Bolhevolnyi, Phys. Rev. B, 67 165405 (2003).

4. M. Kretschmann, A.A. Maradudin, Phys. Rev. B, 66, 245408 (2002).

5. G. C. Brown, V. Celli, M. Haller, A. Marvin, Surf. Sci. 136, 381 (1984).

6. S. V. Gaponenko, Phys. Rev. B, 65 140303 (2002).