1. Introduction
Metal nanostructures support localized surface plasmons and propagating waves, known as surface plasmonpolaritons. The variety of applications of plasmonic nanostructures includes molecular sensing and imaging, alloptical devices, subwavelength waveguides and integrated circuits, and metamaterials with optical magnetism and negative refractive index. Advanced fabrication and computation methods allow the engineering of optimal nanostructure geometries of metal dielectric composites aimed at particular application. The fundamental properties of the metal and the artificially designed geometries of the structural units enable the needed functionality for the nanostructures. Thus, the dielectric function of metal is a key factor for the design and optimization of plasmonic nanostructures. However, the dielectric function of nanostructured metallic elements differs from ideal bulk metal [
1–31. A. Kawabata and R. Kubo, “Electronic properties of fine metallic particles. II. Plasma resonance absorption,” J. Phys. Soc. Jpn. 21, 1765 (1966). [CrossRef]
] especially in the imaginary part. The difference depends on many factors, including the surrounding media and the dimensions and shapes of the elements. This is why the dielectric function of nanostructured metal has been under comprehensive study for many years [
33. U. Kreibig and M. Vollmer, Optical properties of metal clusters, (SpringerVerlag: Berlin, Heidelberg, 1995).
]. Size effects are more significant as the dimensions become comparable to the electron mean free path, which is about 50 nm for silver [
33. U. Kreibig and M. Vollmer, Optical properties of metal clusters, (SpringerVerlag: Berlin, Heidelberg, 1995).
]. The sizedependent contribution to the electron relaxation rate involves several mechanisms. Among them are the quantum size effect and the chemical interface effect caused by static and dynamic charge transfer between a particle and the surrounding material [
4–94. H. Hövel, S. Fritz, A. Hilger, U. Kreibig, and M. Vollmer, “Width of cluster plasmon resonances: bulk dielectric functions and chemical interface damping,” Phys. Rev. B 48, 18178 (1993). [CrossRef]
].
Previous studies of Ag dielectric functions in nanostructures were focused mostly on spherical nanoparticles or spheroids. Structures such as nanowires and nanostrips are less studied, although they are of great interest for nanoscale waveguiding and metamaterial applications. Here we report on experiments involving paired nanostrips fabricated with ebeam lithography. Special, asymmetric plasmon modes in such coupled strips result in circular currents and consequently a magnetic response at the magnetic plasmon resonance. Paired strips arranged in a subwavelength grating can produce a material with an optical permeability other than unity. Negative permeability of the effective layer containing a grating of paired strips has been reported for the whole visible spectral range in our recent papers [
10–1110. H.K. Yuan, U. K. Chettiar, W. Cai, A. V. Kildishev, A. Boltasseva, V. P. Drachev, and V. M. Shalaev, “A negative permeability material at red light,” Opt. Express 15, 1076 (2007). [CrossRef] [PubMed]
].
The dielectric function of a bulk noble metal at the frequency ω is well described by the DrudeLorentzSommerfeld formula for the conduction electrons using an additional contribution from interband electron transitions between the valence and conduction bands:
where γ
_{∞} is the phenomenological relaxation constant of bulk metal, χ_{ib} is the interband term of the metal susceptibility, and
ωp=4πne2me
is the plasma frequency, with n, e, m_{e} being the density of free electrons, electron charge and effective electron mass, respectively.
In classical theory, damping in the dielectric function above is due to the collision of electrons with electrons, phonons, and lattice defects or grains boundaries; the total damping term is the sum of the individual rates,
γ
_{∞}=
ν_{ee}+
ν_{ep}+
ν_{ed}. The collision rate of electrons with defects and grains boundaries,
ν_{ed}, could be responsible for the diversity in the literature data for
ε″
_{m} (
ω) of Ag. Note that the smallest
ε″
_{m} (
ω) was obtained by Johnson and Christy for thin (about 34 nm) largearea films [
1212. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370 (1972). [CrossRef]
]. Those films were deposited at very fast deposition rates (50 Å/s), after which they were polished and annealed. In opposition to largearea films, we obtained smooth nanostructure surfaces at a very low deposition rates of about 0.5 Å/s, although different rates up to 80 Å/s were tested in our experiments.
Fig. 1. (a) and (b): Real part and imaginary part of the dielectric function of bulk silver from various sources in the literature, J&C [
12], L&H [
13], and Weber [
14]; (c) The same as (b) but only the visible range; (d) Imaginary part of the bulk Ag dielectric function: experiment Ref. [
12] (red crosses), DrudeLorentz approximation (dashed orange line), Drude term (blue solid line).
The classical freepatheffect model implies interactions of the electrons with the particle surface that result in an additional, sizedependent term in the damping constant:
where the A parameter, the coefficient of the Fermi velocity to particle radius ratio, includes details of the scattering process. As follows from the classical model, the particle surface roughness should increase the relaxation rate.
A significant source of the electron damping rate is the chemical interface effect. A direct comparison of the plasmon resonance width for free particles and that for particles embedded in a matrix allows one to extract the influence of the interface effect on the A parameter [
44. H. Hövel, S. Fritz, A. Hilger, U. Kreibig, and M. Vollmer, “Width of cluster plasmon resonances: bulk dielectric functions and chemical interface damping,” Phys. Rev. B 48, 18178 (1993). [CrossRef]
]. In the case of Ag particles in a matrix Al
_{2}O
_{3}, experiments give
A=
0.6 for the particles on a substrate and
A=
1.6 for fully imbedded particles in a matrix, in contrast to
A=
0.25 for spherical particles in vacuum [
55. A. Hilger, M. Tenfelde, and U. Kreibig, “Silver nanoparticles deposited on dielectric surfaces,” Appl. Phys. B 73, 361 (2001). [CrossRef]
]. This additional contribution of the chemical interface damping was first mentioned by Persson [
6–76. B. N. J. Persson, “Surface resistivity and vibrational damping in adsorbed layers,” Phys. Rev. B 44, 3277 (1991). [CrossRef]
], as
A=
A
_{size}+
A
_{interface}, to explain experimental findings by Charle with coauthors [
88. K. Charlé, F. Frank, and W. Schulze, “The optical properties of silver microcrystallites in dependence on size and the influence of the matrix environment,” Ber. Bunsenges. Phys. Chem. 88, 350 (1984).
].
The quantum mechanical solution for a rectangular prism (the approximate shape of the strips) predicts anisotropy of the sizedependent term. The sizedependent contribution to the relaxation rate for a rectangular prism with the field applied along the
x axis is [
1515. W. A. Kraus and G. C. Schatz, “Plasmon resonance broadening in small metal particles,” J. Chem. Phys. 79, 6130 (1983). [CrossRef]
]
This result suggests anisotropy of the dielectric function of the metal strips if the size effect is significant.
The measured transmission and reflection spectra of several samples are matched with those simulated by a frequency domain finite element method (FEM). The sizedependent term of ε″_{m} is clearly identified with relatively large Aparameter. It follows from the above considerations that surface roughness could potentially affect the metal dielectric function and distort the nanostructure geometry. Surface roughness was introduced in the modeling of the paired strips. Surprisingly, the geometrical effect of roughness is mostly responsible for increased losses at the plasmon resonances of the nanostructure, while the surface roughness does not affect the Ag permittivity. Anisotropy in ε″_{m} observed in the experiments indicates a significant contribution from the quantum size effect and the chemical interface effect.
Note that both sizedependent permittivity of Ag and surface roughness affect the effective parameters of metamaterials. Relative to prototypes simulated with the bulkmetal permittivity, larger surface roughness acts to significantly decrease the real part of effective permeability μ′ at the resonances, as we will see from the data presented.
2. Experiment
Five grating samples with different nanostrip widths and roughness have been fabricated and studied. Initial test samples with different silver surface roughness values were fabricated by varying the silver deposition rate from 80 Å/s to 0.5 Å/s. Results for those samples showed that a lower deposition rate provided lower roughness. To further investigate the situation of relatively low roughness, for the experiments presented herein we deposited four samples with a rate of about 0.5 Å/s and one with 2 Å/s. A complete set of the parameters for all five samples is presented in
Table 1. Among the samples there are three samples of similar widths but different roughness values (# 1–3), and three samples of different widths fabricated in the same session (# 3–5).
Fig. 2. Paired strips geometry with different roughness. (a) Ideal structure, δ=0, (b) δ=1.5nm, and (c) δ=2.0nm.
Electron beam lithography techniques have been used to fabricate the samples. First, the geometry of the periodic strips was defined in resist by use of an electron beam writer (JEOL JBX6000FS) on a glass substrate initially coated with a 15nm film of indiumtinoxide (ITO). Then, a stack of lamellar films was deposited with vacuum electron beam evaporation. Finally, a liftoff process was performed to obtain the desired silver strips.
Fig. 3. Example of field emission scanning electron microscope and atomic force microscope images (sample #2).
2D surface roughness at the boundaries of silver strips is simulated by random displacements taken at discrete sets of surface points. The points of a given set are equidistantly distributed along the closed boundary defined by the ideal crosssection of a corresponding metallic strip. The distance between the points,
ξ, gives to the correlation length of the roughness. Lengths from about 3nm to 9nm have been preliminary tested; of those, a correlation length of about 7 nm gives the best agreement with experimental results. The final correlation length for each set is slightly adjusted to provide an even number of points along a given crosssection boundary. The local surface displacements are simulated using finite sets of random numbers (
δ_{p},
p=0,1…
p
_{max}) with the uniform distribution between −0.5 and 0.5. Each set is scaled to provide roughness with a required root mean square (RMS) value,
δ=s∑δp2pmax
, where
s is an individual scaling factor and
p
_{max} is the total number of points in a given set. Each
p
^{th} point is then displaced at a distance of
sδ_{p} perpendicular to the initially ideal boundary. A 2D generating line of a given rough surface is then arranged as a smooth closed curve that passes through the displaced points. Each curve is created automatically through an intrinsic splineinterpolation procedure and is guaranteed to have continuous second derivatives [
1616. COMSOL Multiphysics. Command Reference, Comsol AB: Stockholm, Sweden, 2007.
]. The simulations are carried out using finite element method with 5
^{th}order elements and adaptive meshing. The smooth splineinterpolated boundaries are essential for reducing topological difficulties of adaptive meshing. Nonetheless, a few bad realizations are still very occasionally giving topologically degenerate mesh cases; such boundaries are replaced by topologically better realizations with the same RMS and correlation length. For each value of
δ, 20 valid random realizations are simulated to obtain statistically representative data.
Figure 2 depicts the selected examples of roughness realizations with
δ=1.5 nm (
Fig. 2(b)), and
δ=2nm (
Fig. 2(c)); a corresponding ideal structure is shown in
Fig. 2(a).
The transmission and reflection spectra of the samples were measured at normal incidence with a spectroscopy system appropriate for small area (160×160 µm) samples. The system contains an ultrastable tungsten lamp (B&W TEK BPS100), a Glan Taylor prism polarizer, a spectroscopic collection device (SpectraCode), a spectrograph (Acton SpectraPro 300i), and a liquidnitrogen cooled CCDdetector (Roper Scientific). The transmission and reflection spectra were normalized to a bare substrate and a calibrated silver mirror, respectively. To test that the collection area is less than the sample area the reflection spectra were first collected from a calibrated aperture in a highly reflective foil (Tedd Pella). The reflected signal was typically less than 1%. Reflection and transmission spectra were collected for both the TM and TE polarizations of incident light. In the resonant (TM) polarization, the magnetic field is aligned with the largest dimension of the structure – the infinite length of the strips. Only one component of the magnetic field should ideally be present in this case. In the nonresonant (TE) polarization, the single component of the electric field is aligned with the strip length, giving no resonant effects.
3. Results and discussion
Fig. 4. TM polarization spectra (a)–(c): (a) simulated (dashed lines) spectra of transmission (blue), reflection (red), and absorption (orange) for one of the samples (sample #4) calculated with different the RMS surface roughness vs. the experimental data (solid lines); (b) simulated effective optical density D obtained for the RMS surface roughness (orange, δ=1.7 nm, purple, δ=2.3 nm, and blue, δ=2.9 nm) vs. the experimental data (red). (c) simulated effective optical density D obtained for the RMS surface roughness (orange, δ=1.7, blue, δ=2.3, and purple, δ=2.9). TE polarization spectra (d): experimental (solid lines) and simulated (dashed lines) spectra of transmission (blue), reflection (red), and effective optical density (orange).
Table 1. Parameters of the samples. 

One can see that
ε″
_{m} for the three samples with similar widths show almost no differences, indicating that the roughness does not affect the dielectric function. However, the distorting effect of roughness on the geometry is exhibited by the differences in the absorption spectra and the strong difference in the retrieved permeabilities of the effective layer, as is illustrated in
Fig. 6(b).
The results of our experiments on the Ag dielectric function in coupled strips lead us to several important conclusions. First of all, the spectra of
ε″
_{m} for the TE polarization (electric field parallel to the strips) suggest good quality of the Ag crystal structure. Indeed
ε″
_{m} for samples with large widths are between the literature values from Johnson and Christy, and Lynch and Hunter [
12–1312. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370 (1972). [CrossRef]
] for bulk Ag carefully fabricated with subsequent polishing and annealing. As we mentioned above, a very probable reason for the diversity in the bulk Ag dielectric function is the term in the electron relaxation rate describing crystal defects and grain boundaries,
ν
_{ed}. The strips have a reasonable
ε″
_{m} for the TE polarization, comparable with bulk, largearea films.
Fig. 5. The experimental spectra of the imaginary part of the Ag dielectric function in the coupled strips for five samples in comparison with the bulk dielectric function from two sources J&C, Ref. [
12] and L&H, Ref. [
13] for TE (a) and TM (b) polarizations.
A sizedependent increase of ε″_{m} is detected for both polarizations. The Aparameter is larger for the TM polarization, which is indicative of a quantum size effect contribution that predicts anisotropy. The crystal defects could be also sizedependent. Chemical interface damping could be the reason for the size dependence at the TE polarization and anisotropy of the dielectric function.
According to Persson [
77. B. N. J. Persson, “Polarizability of small spherical metal particles: influence of the matrix environment,” Surface Science 281, 153 (1993). [CrossRef]
], an electron hits the particle surface, passes through the barrier, and occupies an adsorbate state. After some residence time, the electron may return to the particle. The tangential component of the internal current at the surface also contributes to A
_{interface} via friction processes. These processes can be either elastic or inelastic, depending on the energy positions of the involved adsorbate levels relative to the Fermi energy. The energy transfer can be between zero and the whole “plasmon quantum energy.” Moreover the tangential component provides a greater contribution to the A
_{interface} than the normal component by factor 4–10 [
99. A. V. Pinchuk, U. Kreibig, and A. Hilger, “Optical properties of metallic nanoparticles: influence of interface effects and interband transitions,” Surface Science 557, 269 (2004). [CrossRef]
].
The chemical interface damping is proportional to the surfacetovolume ratio. Thus, one expects for TM polarization the tangential component to be proportional to the inverse thickness and the normal component proportional to be proportional to the inverse width of the strips, (S/V)_{tangential} ∝1/t, (S/V)_{normal} ∝1/w. For the TE polarization, one has only the tangential component, (S/V)_{tangential} ∝c
_{1}/t+c
_{2}/w. Our results also show an abnormally large Aparameter for Ag strips relative to known Aparameters for spherical nanoparticles. The Aparameter is about 4.2 for the TM polarization calculated with an effective width, w
^{1}
_{eff}=0.5(w
^{1}
_{t}+w
^{1}
_{b}).
The interface effect depends on particle shape in a way similar to that of the quantum size effect dependence. The theories presented in [
55. A. Hilger, M. Tenfelde, and U. Kreibig, “Silver nanoparticles deposited on dielectric surfaces,” Appl. Phys. B 73, 361 (2001). [CrossRef]
,
77. B. N. J. Persson, “Polarizability of small spherical metal particles: influence of the matrix environment,” Surface Science 281, 153 (1993). [CrossRef]
,
99. A. V. Pinchuk, U. Kreibig, and A. Hilger, “Optical properties of metallic nanoparticles: influence of interface effects and interband transitions,” Surface Science 557, 269 (2004). [CrossRef]
] deal with spherical particles. However one can estimate an expected value using an observation by Kraus and Schatz [
1515. W. A. Kraus and G. C. Schatz, “Plasmon resonance broadening in small metal particles,” J. Chem. Phys. 79, 6130 (1983). [CrossRef]
] that most of the shape dependence is the direct result of the shape dependence of the average length,
L_{av}=
V/
S, defined as the ratio of the particle volume to the projected area of the particle perpendicular to the direction of the applied field. For the tangential component, it is expected that there should be a projected area of the particle parallel to the applied field. This shape factor, calculated for a spherical quantum well and rectangular quantum box, increases by about 1.5 for the rectangular prism [
1515. W. A. Kraus and G. C. Schatz, “Plasmon resonance broadening in small metal particles,” J. Chem. Phys. 79, 6130 (1983). [CrossRef]
]. Thus, taking the experimental value of
A=
A
_{size}+
A
_{interface}=1.6 for spheroidal particles [
55. A. Hilger, M. Tenfelde, and U. Kreibig, “Silver nanoparticles deposited on dielectric surfaces,” Appl. Phys. B 73, 361 (2001). [CrossRef]
] and multiplying by the factor 1.5, one might expect
A=2.4 for the rectangular prism. This rough estimate is still significantly less than our experimental values. On the other hand, the sizedependent contribution to the width for the rectangular prism with the field applied along the
x axis is
The quantum mechanical treatment [
1515. W. A. Kraus and G. C. Schatz, “Plasmon resonance broadening in small metal particles,” J. Chem. Phys. 79, 6130 (1983). [CrossRef]
] predicts
A
_{size}=1.5. In the case of spheroidal particles, the interface effect enhances the Aparameter by a factor of 6 [
55. A. Hilger, M. Tenfelde, and U. Kreibig, “Silver nanoparticles deposited on dielectric surfaces,” Appl. Phys. B 73, 361 (2001). [CrossRef]
]. This means that large values of the Aparameter are possible and likely due to the chemical interface effect.
Fig. 6. (a) Size dependent term of the relaxation rate (eV) versus the inverse effective width of the strips for TM (blue squares) and TE (red circles) polarizations. (b) the real part of effective permeability vs. the RMS of surface roughness.
It would be realistic to assume another damping channel caused by grain boundaries, especially in the case of TE polarization. Indeed, it is hard to expect that the strips along their long axis have properties of an ideal metal with a bulk relaxation constant.