Institute of Radio Engineering and Electronics (Saratov Division), Russian Academy of Sciences, 410019 Saratov, Russia Centro Mixto CSIC-UPV/EHU and Donostia International Physics Center, 20080 San Sebastian, Spain E-mail: firstname.lastname@example.org We have calculated light absorption spectra of planar metal surfaces with two-dimensional lattice of spherical nanovoids right beneath the surface. It is shown that nearly total absorption of light occurs at the plasma resonance in void lattice in the visible when the inter-void spacing and the void deepening into the metal are thinner than the skin depth, which ensures optimal coupling of void plasmons to external light. We conclude that the absorption and local-field properties of this type of nanoporous metal surfaces can be effectively tuned by nano-engineering the spherical pores and they constitute a very attractive system for various applications in future submicron light technology.
This work was supported by the Russian Foundation for Basic Research (grant N 02-02-81031) and the Russian Academy of Sciences Program „Low-Dimensional Quantum Nanostructures“. T. V. T. acknowledges the support from the President of Russia through the grant for young scientists MK-2314.2003.02 and from the National Foundation for Promotion of Science. F. J. G. A. acknowledges help and support from the University of the Basque Country UPV/EHU (contract N 00206.215-13639/2001) and the Spanish Ministerio de Ciencia y Tecnologa (contract N MAT2001-0946).
1. Introduction infinite metal having a negative permittivity. However, the huge resonant dips in the reflectivity spectra observed In general, planar metal surfaces absorb light very poorly.
in  suggest strong coupling of nanocavity plasmons to the The reason is their high free-electron density, which reacts incident light. Therefore, a better understanding of the effect to the incident light by sustaining strong oscillating currents of coupling between plasmons in metallic nanocavities and that, in turn, efficiently re-radiate light back into the external radiation becomes of great importance.
surrounding medium, whereas the light intensity inside the On the other hand, it has been shown in [4–6] that metal remains weak. Actually, the same phenomenon takes the spectra of plasma oscillations in spherical metallic place when light excites plasma oscillations in metallic nanoparticles with inner voids (nanoshells) are much richer particles, and light absorption is inhibited as a result at than those in metallic nanospheres. Both sphere-like the plasma resonances. In other words, the local-field plasmons (those mainly bound to the outer surface of enhancement inside or near the metallic particle appears the shell) and void-like plasmons (those mainly bound to to be quite moderate even at the plasma resonance. Localthe inner surface of the shell) can be excited in such a field enhancement factors up to 15 have been reported for particle. Optical properties of a single metallic nanoshell spherical metallic nanoparticles [1,2].
In apparent contradiction with the above arguments, sharp and nanoshell clusters can be effectivelly tuned by nanoand deep (down to –20 dB) resonant dips in the reflectivity engineering their geometry. As it has been theoretically spectra of light from nanoporous gold surface have been shown in , the local-field enhancement factor at the recently observed , which points to strong resonant light void-like plasmon resonance can reach ultra-high values for absorption on such a surface.
specific values of the metallic wall thickness in nanoshell:
It was presumed in  that this phenomenon is related to local-field enhancement factors exceeding 60 and 150 in the excitation of plasmon modes in spherical nanocavities gold and silver nanoshells, respectively, have been predicted, inside the metal, which couple much more effectively to the and this field enhancement is accompanied by sharply light than those in metallic spheres. As an intuitive explanaenhanced light absorption at the resonance.
tion of their observations, the authors of  employed a simIn this paper, we study the optical properties of ple model of plasmon modes supported by a spherical void nanoporous metal surface. We start with a simple model in an infinite metallic medium. Although that model gives of the resonant surface in order to examine the essential the eigen-frequency values that somehow can be fitted to the physics underling strong light absorption on such a surface.
frequencies of the resonances in the measured reflectivity Then we calculate the reflection/absorption spectra of spectra, it can not describe the coupling between plasmon nanoporous metal surfaces in the framework of a rigorous modes in nanocavity and the external radiation field. The electromagnetic scattering-matrix approach , taking into reason is that the plasmon modes in a void are non-radiative because their electromagnetic field cannot radiate into an account the actual porous structure of the surface.
Total resonant absorption of light by plasmons on the nanoporous surface of a metal rate, describes the Drude response of a homogeneous metal surface within inter-void regions to incident light by the equivalent electronic resistance Re = me/(e2Ne) and kinetic inductance Le = m/(e2Ne) (Fig. 1). In the vicinity of the lth plasma resonance, l, the lth term of the summation dominates the right hand side of Eq. (1) and we have m|l|2 l Zeff -i. (3) Figure 1. Nanoporous surface of metal and its equivalent circuit.
2e2 Ne l - - il l The surface impedance given by Eq. (3) leads to the following expression for the absorbance of light in the 2. Model of a resonant surface neighborhood of the lth plasma resonance 4ll Let us consider an electromagnetic plane wave incident A = 1 - rr, (4) from vacuum normally onto a planar surface of metal (l - )2 +(l + l)with a two-dimensional lattice of voids just underneath where the surface (Fig. 1). In order to examine the essential ml physics of energy transformation in the system we elaborate l = |l|2 (5) 2Z0e2 Ne l a simple equivalent model describing the resonant surface is the radiative damping of the lth plasmon mode. It should by its effective surface impedance Zeff defined by the be noted that the line of the absorption resonance given by relation E = Zeff(n B ), where E and B are the Eq. (4) has a Lorentzian shape with the full width at half tangential components of the total electric and magnetic maximum (FWHM) of 2(l + l). Free parameters | f |2/, fields, respectively, and n is the external normal to the l l and |l|2/ can be obtained by fitting the resonance planar metal surface. Making use of the impedance l frequency and FWHM yielded by this simple model to those boundary condition  and solving Maxwell’s equations in yielded by a rigorous electromagnetic modeling, which is the surrounding medium, it is easy to obtain the complex done in the next section of this paper.
amplitude reflection coefficient r =(Zeff - Z0)/(Zeff + Z0), Finally, at resonance ( = l), one finds where Z0 is the free-space impedance.
We describe plasma oscillations in the lattice of voids 4ll by an equivalent RLC circuit (Fig. 1) composed of the Ares (l + l)equivalent areal capacitance Cl = | f |20, where is the l thickness of the nanoporous surface layer, 0 is the electrical and it is readily seen that nearly total light absorption by constant, | f |2 is the dimensionless phenomenological forml the lth plasmon mode (i. e., Ares 1) occurs when l = l.
factor characteristic of a given lth multipole plasmon mode, The radiative damping l may be conceived as the coupling connected in parallel to Rl - Ll series (Fig. 1). The coefficient that controls the strength of interaction between equivalent areal electronic resistance and kinetic inductance the plasmon mode and light. For small l (i. e., l l) are defined as Rl = ml/(e2 Ne) and Ll = m/(e2 Ne), l l the coupling is weak and the plasmon mode absorbs light respectively, where l is the damping of the lth plasmon only weakly. In the opposite limit, l l, the strong mode due to all dissipative processes except radiative plasma oscillation currents flowing on the metal surface redamping, Ne is the total areal free-electron density in the radiate incident light back into the surrounding medium, surface skin layer, is the fraction of free electrons l which again reduces absorption drastically. Therefore, it participating in the plasma oscillations at the lth mode, is possible to realize the condition of total light absorption and e and m are the electron charge and mass, respectively.
by plasmons on nanoporous metal surface by varying the With this considerations, we can easily obtain the equivacoupling coefficient |l|2. The optimal value of |l|2 can be lent surface impedance in the form easily realized for void-like plasmon modes in the spherical 2 voids buried in a metal substrate. For example, the condition m m |l|2 l Zeff = (e - i) - i, e2Ne 2e2 l=1 Ne l - - il l = l can be easily satisfied for void-like plasmons in a l nanoshell by choosing a specific value of the shell-layer (1) thickness, as shown in .
where e2 Ne l l = (2) 3. Self-consistent electrodynamic | f |20m l modeling is the frequency of the lth plasmon mode, and |l|2 < is the phenomenological coefficient of coupling between Let us consider a periodic two-dimensional hexagonal the external light and the lth plasmon mode. The first lattice of spherical voids with the lattice vectors a and b, term in Eq. (1), where e is the free-electron scattering where |a| = |b| and a · b = |a|2 cos with = 60. We Физика твердого тела, 2005, том 47, вып. 174 T.V. Teperik, V.V. Popov, Garca de Abajo F.J.
the voids at distance h from the void bottoms form the interfaces between the periodic surface layer and either the surrounding medium or metal substrate, respectively. The total fields in the surrounding medium and in the substrate result from the superposition of propagating and evanescent plane waves with in-plane wavevectors Gpq = pA + qB, where A = 2(b n)/(a b) and B = 2(n a)/(a b) are the principal vectors of the reciprocal lattice, and p and q are integers. It should be noted that at frequencies below the bulk plasma frequency every plane wave in the metal substrate is evanescent. The total field inside the periodic surface layer is represented as a superposition of the incoming plane waves (both propagating and evanescent) and the field scattered from every void. In this way, the multiple light scattering between all voids in the surface layer is self-consistently accounted for. The in-plane summations of fields scattered from different voids performed in our case directly in real space provide a quite fast convergence.
The interaction between the combined electromagnetic field incident upon a given single void and the electromagnetic field scattered from this void is determined by its scattering matrix [9,10]. Because the scattering matrix of a single void is constructed in a spherical-harmonic representation, we decompose the combined field incident upon a given single void into spherical harmonics. Then, we transform the combined self-consistent field scattered from all voids into a plane-wave representation that is expressed as a sum over in-plane wavevectors Gpq, and apply the boundary conditions at the interfaces of the Figure 2. Absorption spectra of light incident normally onto a planar surface layer containing lattice of voids with the planar silver surface with a lattice of spherical voids right beneath surrounding medium and substrate. As a result we construct it (see inset). a — variation of the spectra with the inter-void the scattering matrix of the entire structure, which allows us spacing h, which is chosen to be equal to the void deepening, for to calculate the reflectance, R, and absorbance, A = 1 - R, the void diameter d = 300 nm. b — variation of the spectra with of the porous metal surface. Note that this approach can be the voids diameter for the inter-void spacing h = 5nm also taken straightforwardly extended to model an arbitrary number to be equal to the void deepenigh. The absorption of light on the of layers with periodically arranged spherical voids with the surface of bulk silver is shown by dash-dotted curve. Vertical same period but having different void radii in different layers arrows mark the energies of the fundamental plasmon modes (l = 1) of a single void in bulk silver. if one wishes. A detailed description of this method can be found in .
It is interesting to point out that the propagation of the electromagnetic field between voids is performed through assume that the lattice of voids is buried inside a metal the metal, so that each void interacts directly only with substrate at distance h from the planar metal surface to the its nearest neighbors, unlike what happens in a dielectric top of the voids, therefore we call h the void deepening.
environment. Accordingly, the Bragg resonances controlled We also assume that the inter-void spacing along the lattice by periodicity of the system are not exhibited in the vectors a and b is equal to the void deepening h (inset of calculated spectra. Therefore, only resonances originated Fig. 2, a). We consider that external light shines normally from the excitation of Mie plasmon modes in every single onto the metal surface.
void influenced by nearest void neighbors show up in the To calculate the light absorption on such a nanoporous spectra.
surface of metal we use a self-consistent rigorous electroFig. 2 shows the calculated absorption spectra of light dynamic method based on the scattering-matrix approach incident normally onto a nanoporous silver surface for the with making use of re-expansion of the plane-wave repre- case of a single periodic layer of close-packed voids buried sentation of electromagnetic fields in terms of the spherical inside the silver substrate (the inset of Fig. 2, a). We use harmonics . This approach involves the following steps.
experimental optical data  to describe the dielectric First of all we define a planar surface layer containing the function of silver to electric field in our calculations.
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