The enhanced light transmission of periodic arrays of subwavelength holes, attributed to the excitation of surface plasmons in the metal film, has been widely studied in the past few years [1–9]. It has been shown both experimentally and theoretically that the geometric characteristics of the nanostructures drastically influence the spectral behavior of the transmission as the periodicity, the hole diameters or the film thickness [10–14]. In addition to the geometrical parameters, the transmission strongly depends both on the dielectric functions of the metal and of the surrounding medium. Indeed it is a general property of nanostructured metals that their optical properties are intimately related to their dielectric function. For example, for noble metal aggregates, the relative importance of the interband and the intraband transitions is determinant for the static and dynamical spectral characteristics of the resonant Mie scattering [15–21]. To our knowledge, an extensive analysis of the influence of the metal dielectric function on the enhanced light transmission (ELT) has not been done. However, on the experimental side, a recent study has shown the existence of an atypical behavior of the first order peak on the substrate side (1, 0)_SS, in the case of noble metals [22]. It manifests as a nonlinear spectral shift of this peak, as a function of the period of the array, in the vicinity of the interband optical transition threshold.

In the present work, we report a detailed study of the influence of the dielectric function of noble metals on the enhanced transmission observed on periodic subwavelength nano-apertures. Different dielectric functions of the metal are considered in our modeling of the static optical response of the nanostructures. On one hand, we consider a theoretical dielectric function of the metal based on the Lindhard description of core and valence electrons [23]. On the second hand, we determine the experimental dielectric function of gold deduced from reflection and transmission measurements on a gold film. This dielectric function is fitted by the Lindhard dielectric function and it is used to investigate several features of the transmission near the interband threshold from the d-band to the Fermi level (d→E_F). In particular, we explain the nonlinear spectral shift, as a function of the period of the array, near (d→E_F). In addition, we show the existence of an apparent resonance and which does not shift when modifying the geometrical parameters of the nanostructure. It is shown to be related to the transparency spectral region, proper to noble metals, due to the interplay between the real part of the conduction electron dielectric function (Drude) and the imaginary part of the d-electron dielectric function. Let us notice that in the case of silver we have considered the optical transitions from the d band to the Fermi level (d→E_F) and from the conduction band p to the empty band s (p→s) in the calculation of the dielectric function.

We have studied the experimental transmission of several gold nanostructures with various periods. The samples are made on a 250 nm gold film, deposited on an Al₂O₃ substrate, milled with a Focused Ion Beam technique. The resulting nanostructures consist of squared grids of holes with a lattice constant a varying in the range 260-300 nm. For each array, the diameter d of the holes has been chosen such that the filling factor $\left(f=\frac{\pi d^2}{4a^2}\right)$ is ~27%. Typical Scanning Electronic Microscopy images have been reported previously [24]. The linear transmission spectra are measured by exciting the sample with a white light source in a quasi collimated configuration (Numerical Aperture = 0.1) and by imaging the nanostructures on the entrance slit of a spectrometer. The output numerical aperture is 0.3. These spectra are recorded by a liquid nitrogen cooled Coupled Charged Device camera. Figure 1 represents the linear zero order transmission at normal incidence, normalized to unity, of arrays with different periods. All spectra exhibit two main resonances which behave differently as a function of the period. The longer wavelength resonance (located at 647 nm for a=260 nm) shifts to longer wavelengths when the period of the array increases. Such spectral behavior has been shown to be characteristic of the surface plasmon mode associated to the metal/substrate interface [1–2, 4, 7–8, 24]. Let us recall that the spectral position of the resonance is approximately given by [2]: ${\lambda }_{max}=\frac{a}{\sqrt{i^2+j^2}}\sqrt{\frac{{\epsilon }_{I,\mathit{III}}{\epsilon }_M}{{\epsilon }_{I,\mathit{III}}+{\epsilon }_M}}$, where (i, j) are integer mode indexes associated to the array. ε_{I, III} are respectively the dielectric functions of either the incident medium (air) or the substrate and ε_M is the one of the metal. In the present case, this long wavelength resonance corresponds to the (1, 0)_SS peak on the substrate side. In contrast, the transmission has a more complex resonant behavior in the spectral range 500-600 nm. For the periods a=260 nm and a=280 nm, a fixed resonance at 515 nm is present. It apparently shifts to 530 nm when a increases. In fact, as will be shown in the following, this resonance is composed of the (1, 0)_AS mode on the air side and a spectrally fixed resonance at 515 nm, associated to the transparency region of gold. When the period still increases [not shown on Fig. 1(a)], a shoulder corresponding to the (1, 1)_SS mode on the substrate side occurs [2]. For example it occurs at 580 nm for a period a=320 nm.

 

Fig. 1. Linear transmission spectra normalized to unity, of nanostructures with different periods a designed in a 250 nm thick gold film evaporated on an Al₂O₃ substrate.

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To explain this complex behavior of the transmission in the 500-600 nm spectral region, we have modeled our nanostructures using an analytical model [25]. It consists in an approximate solution of Maxwell’s equations for a p-polarized electromagnetic wave illuminating a one-dimensional nanostructure [26–28]. The nanostructure consists in a periodically modulated metallic film of thickness h (region II). The period of the modulation is a. The nanostructure is surrounded by two media: air (region I with a dielectric function ε_I) and substrate (region III with a dielectric function ε_III). In this method, based on the Rigorous Wave Coupled Analysis (RCWA), the field in the nanostructured region is developed in Bloch waves modes. The dielectric function is expressed in Fourier series as:

where n is an integer, g = 2π/a is the period of the reciprocal lattice. The Fourier harmonics are given by:

where ε_h is the dielectric function of the holes, ε_m the dielectric function of the metal and f the filling factor defined earlier.

To extract analytical expressions of the diffraction efficiencies, the developments have been truncated at first order. This allows us calculating the zero order (T ₀, R ₀) and first order (T ₁, R ₁) transmission and reflection coefficients, as well as the absorption defined as A = 1 - R ₀ - 2R ₁ - T ₀ - 2T ₁. The coefficients T_n and R_n are given by [25]:

where t_n and r_n are the transmitted and reflected amplitudes of the diffracted fields and θ the angle of incidence of the light on the nanostructure. Let us stress that the one dimensional truncated RCW model does not allow retrieving the exact spectral position as well as the amplitudes of the nanostructure resonances. In addition, it is best suited for low filling factors [26]. Nevertheless, it has been shown to catch the main physical features involved in the transmission process of two-dimensional periodic nanostructures [28]. A more accurate description consists in solving numerically the 2D Maxwell equations with the proper boundary conditions [29].

Let us focus now on the effect of the metal dielectric function. Figure 2 shows different complex dielectric functions of gold that we used to model the static response of our nanostructures. The black lines correspond to the dielectric function of gold ε_exp (real part (solid line); imaginary part (dashed line)) extracted from measurements of the linear reflectivity and transmission of a 29 nm gold film. In this procedure, we use a Fabry-Pérot-like model. This experimental dielectric function is close, although not perfectly superimposed, to the one tabulated (green circles) in ref [30]. The red triangles correspond to the real and imaginary part of the theoretical Lindhard dielectric function of gold ε_theo, obtained with the self-consistent field theory [23, 31]. It contains the two contributions associated to the conduction electrons (Drude-like dielectric function ε _Drude) and to the d-band electronic states ε_inter from which optical interband transitions to the Fermi level may occur (d → E_F threshold = 2.35 eV = 527 nm).

where N, e, m and f_o are the density, the charge, the mass and the distribution function of the electrons in the metal. ħω_p is the energy of the volume plasmon. γ_ee and γ_f stand for the interband and intraband damping processes. P_ll’ and ħω_l’l are the matrix element and the energy of the transition between the bands l and l’. In the spectral region considered here, the interband dielectric function of gold is well described by a two-band model [32] corresponding to the d electrons [l=d in Eq. (5)] and the p conduction electrons [l’=p in Eq. (5)]. We have used the band parameters of B. R. Cooper et al. [33]. In practice, the imaginary part of the interband dielectric function is first computed and the real part is obtained by the Kramers-Kronig relations. Since our integration is performed in a limited frequency range, a constant has been added to Re[ε _theo] so that it fits to Re[ε _exp]. The inset of Fig. 2 shows the detailed dielectric functions near the interband threshold. In the following, we used ε _theo to model the linear zero order transmission of the gold nanostructures.

 

Fig. 2. Spectral dielectric function of gold: tabulated values [30] (green circles) ε _exp, determined experimentally on a 29 nm gold film (black lines) and ε _theo calculated with the Lindhard formalism (red triangles). The closed symbols or solid line display the real part of these dielectric functions while the open ones or dashed line represent their corresponding imaginary parts.

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Figure 3(a) represents the experimental linear transmission spectrum of a nanostructure, at normal incidence, a = 300 nm, d = 177 nm, h = 250 nm deposited on an Al₂O₃ substrate. Figure 3(b) displays the corresponding calculated transmission using either ε_theo (red line) or ε_exp (blue line). Let us notice that for the transmission spectrum obtained with ε_theo, the spectral bandwidths of the resonances are underestimated as compared to the one calculated with ε_exp. This is probably due to the fact that damping mechanisms such as the ones related to crystalline defects of the metallic film are not considered in the Lindhard dielectric function. In our calculation, we take into account only the damping due to electron-electron scattering. In the interband part of ε _theo, γ_ee is described by a Fermi liquid like model [16, 19], where the electron-electron scattering time depends quadratically both on the temperature and on the excess energy with respect to the Fermi level. In the intraband part of the dielectric function, a phenomenological damping constant γ_f=25 meV is considered.

 

Fig. 3. (a) Experimental linear transmission of a gold nanostructure with the following parameters: a = 300 nm, d = 177 nm, h = 250 nm, Al₂O₃ substrate. (b) Calculated zero order transmission spectra using the Lindhard dielectric function ε_theo and the experimental one ε_exp.

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In Fig. 4(a), we have represented the spectral positions of the maxima of transmission (closed symbols) and absorption (open symbols), calculated with our model, as a function of the period of the array. For each period, the diameter of the “holes” is modified so that the filling factor is kept constant (f ≈ 0.26) as required in the definition of the Fourier coefficients ε_n in Eq. (2). The other parameters of the gold nanostructures are: film thickness h = 110 nm, Al₂O₃ substrate with ε_III= 3.1. The long wavelength resonance (1, 0)_SS first shift linearly to the shorter wavelengths when the period decreases. Below a period of ~ 300 nm the maxima of both the transmission and absorption associated to this (1, 0)_SS peak shifts sub-linearly to the blue and tends to saturate close to the interband threshold d → E_F. In the spectral region 500-600 nm, the spectral behavior of the transmission is more complex. Two resonances are present. One corresponds to the (1, 0)_AS order peak represented by the green circles. It is situated near 600 nm for a period of 550 nm. Like the (1, 0)_SS peak it is blue shifted when the period decreases and ultimately also tends to saturate near the interband threshold as clearly seen in the inset of Fig. 4(a). In addition, for all periods, there is a fixed resonance, represented by the blue circles, situated slightly lower than the interband threshold i.e. at 510 nm for the transmission peak and at 490 nm for the associated absorption peak. This transmission peak results from the interplay of the large absorption above the interband threshold and the large reflectivity associated to the intraband transitions. The combination of these two effects manifests as an apparent transparency window. In Fig. 4(b) we have represented the transmission spectra of the nanostructures for the periods a = 450, 300 and 150 nm. For clarity purpose, a vertical offset has been added for each spectrum. In these spectra one can trace back the blue shift of both (1, 0)_SS and (1, 0)_AS. In addition, they spectrally converge towards the interband threshold. The large fixed resonance at ~510 nm also appears clearly. The spectral evolution of the two resonances (1, 0)_SS and (1, 0)_AS, when varying the period a, is characteristic of the surface plasmon dispersion observed in noble metal nanostructures. The fact that their maximum shifts sub-linearly and ultimately saturates to the interband threshold when the period decreases is due to the contribution of the d electrons to the real part of the dielectric function ε _theo displayed in Fig. 2.

 

Fig. 4. (a) Spectral positions of the maxima in transmission (closed symbols) and absorption (open symbols) as a function of the period of the array of gold nanostructures, with a constant filling factor f = 0.26, a thickness h = 110 nm and ε _III = 3.1. The red symbols correspond to the (1, 0)_SS order peak. The blue symbols represent the transparency window and the green ones display the positions of the (1, 0)_AS order peak. Inset: detailed view of the splitting of the resonance at ~510 nm. (b) Transmission spectra of the preceding nanostructures for different periods a. A vertical offset has been added for clarity. The metal dielectric function is ε _theo.

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To better see the relative importance between the core d electrons and the conduction p electrons, we have represented in Fig. 5(a) and 5(b) the same quantities as in Fig. 4(a) and 4(b) but considering only the real part of the Drude dielectric function. Both (1, 0)_SS and (1, 0)_AS resonances shift quasi-linearly to the blue, without saturation effect near 530 nm when the period decreases. In other words, it is the slight increase of Re(ε _theo), due to the interband transitions, which induces the important nonlinear shift of these resonances when varying the period of the array. To check the consistency of this interpretation and the validity of the RCW model using the Lindhard dielectric function, we have analyzed the position of the (1, 0)_SS order peak, obtained self-consistently from the dispersion relation $\lambda =\frac{a}{\sqrt{i^2+j^2}}\sqrt{\frac{{\epsilon }_{I,\mathit{III}}{\epsilon }_M\left(\lambda \right)}{{\epsilon }_{I,\mathit{III}}+{\epsilon }_M\left(\lambda \right)}}$, as a function of the period of the nanostructure. The results are shown in Fig. 5(c) (red circles). In this figure we have also reported the position of the (1, 0)_SS order peak obtained with our theoretical model, using the Lindhard dielectric function (green line). The two curves do not overlap because we have considered only the real part of the Lindhard dielectric function in the above dispersion relation. The blue line in Fig. 5(c) corresponds to the (1, 0)_SS order peak positions using the Drude dielectric function in the theoretical model. It is completely unable to reproduce the correct behavior below 600 nm (a ≈ 300 nm), showing the predominance of the core d electrons in this spectral region.

 

Fig. 5. (a) Spectral positions of the maxima of transmission as a function of the period. The structures parameters are the same as in Fig. 4 but the dielectric function of gold does not include the interband transitions. (b). Corresponding transmission spectra for different periods. (c). Spectral positions of the (1, 0)_SS peak determined by the surface plasmon dispersion relation given in the text (red circles), calculated with our model, using ε _theo (green line) and the real part of the Drude-like dielectric function (blue line).

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Figure 6 represents the transmission spectra of a gold array of holes with the following parameters: a = 300 nm, d = 180 nm, h = 110 nm and ε _III = 3.1, for different dielectric functions of the medium I (ε_I) and of the medium filling the holes (ε_h). In this simulation, we have ε_I = ε_h. The (1, 0)_SS peak shifts to the red when ε_h increases. The amplitude of this peak also increases when ε_h approaches the value of the substrate ε _III. These results are consistent with previous observations [24]. On the other hand, the resonance near the transparency region splits in two peaks. The one centered at 510 nm is robust when modifying ε_h. It corresponds to the transparency window described in Fig. 4, which is the manifestation of the intrinsic electronic properties of gold, independently of the geometry of the nanostructure. The other peak, associated to the (1, 0)_AS resonance, shifts to the long wavelengths when increasing ε_h as expected from the above dispersion relation.

 

Fig. 6. Linear zero order transmission spectra of gold nanostructures with a = 300 nm, d = 180 nm, h = 110 nm, ε_III = 3.1 for different dielectric functions ε_I.

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To explore the pertinence of the above study for other noble metals, we have applied our model to the case of copper and silver. Figure 7(a) displays the calculated dielectric function of copper (red triangles) and the one given in ref [30] (green circles). Figure 7(b) shows the calculated transmission spectra using either ε_theo or ε_tab in the case of a copper nanostructure with the parameters given in the figure caption. These spectra display essentially the same characteristics as in the case of the gold nanostructures. The longer wavelength resonance corresponds to the first order (1, 0)_SS peak on the substrate side. The shorter wavelength resonance corresponds to the transparency window, situated near the interband threshold of copper (E_d→E_F =2.1eV), resulting from the superposition of the strong absorption and the high reflectivity of copper. In Fig. 7(c), the (1, 0)_SS order peak also displays a non linear shift with the period of the nanostructure with a saturation occurring near the interband threshold. Moreover, the second peak splits in two part, one corresponding to the (1, 0)_AS peak on the air side, the other nearly independent of the period associated to the transparency window of copper.

 

Fig. 7. (a) Real (closed symbols) and imaginary parts (open symbols) of the dielectric function of copper calculated in the self-consistent field method (red triangles), or tabulated [30] (green circles). (b) Calculated transmission spectra with ε _theo and ε _tab of a copper nanostructure with a = 350 nm, d = 200 nm, h = 110 nm ε _III = 2.25. (c) Spectral positions as a function of the period of the (1, 0)_SS peak (red), (1, 0)_AS peak (green), and the transparency window (blue).

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Finally, we have also investigated the case of silver. Figure 8(a) displays the dielectric function tabulated in Ref. [30] (green circles), and the calculated one (red triangles). In the spectral region considered here, the dielectric function of silver is well described by a three band-model [34]. Following Rosei [35], we have taken into account the interband transition from the filled d band to the Fermi level (E_d→EF =3.99eV) and the transition from the conduction band p to the empty band s (E_p→s = 3.85 eV). Figure 8(b) shows the transmission spectra of a silver nanostructure with a=400 nm, f=0.26, h = 110 nm and ε _III = 2.25 using either ε _theo or ε _tab. The spectra exhibit the (1, 0)_SS order peak on the substrate side at longer wavelengths and the (1, 0)_AS order resonance on the air side located at 450 nm when using ε_theo. As in the case of the other noble metals, an apparent resonance shows up in the interband region which corresponds to the transparency window of silver. A detailed view of the calculated zero order transmission in the interband region is presented in the inset of Fig. 8(b). Two peaks are present in this resonance. They correspond to the superposition of the high reflectivity due to intraband processes and the large absorption associated to each interband transition thresholds of silver. The peak located at 316 nm corresponds to the d→E_F interband threshold and the one at 325 nm to the p→s interband one. In Fig. 8(c), the (1, 0) order peaks both exhibit a non linear shift and converge toward the p→s interband threshold when decreasing the period of the array. The spectral position of the transparency windows are, as previously mentioned for gold and copper, independent of the period.

 

Fig. 8. (a) Real (closed symbols) and imaginary parts (open symbols) of the dielectric function of silver calculated in the self-consistent field method (red triangles), or tabulated [30] (green circles). (b) Calculated transmission spectra with etheo and etab of a silverr nanostructure with a = 400 nm, f = 0.26, h = 110 nm ε _III = 2.25. Inset: detailed view in the spectral region of the transparency windows. (c) Spectral positions as a function of the period of the (1, 0)_SS peak (red), (1, 0)_AS peak (green), and the transparency window associated to the absorption due to p→s transition (black) and due to d→E_F transition (blue).

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In conclusion, our analysis shows the crucial importance of the metallic dielectric function to predict the spectral behavior of the light transmission through noble metal nanostructures. It is particularly the case near the interband transitions threshold where the core electrons play a major role. Two main spectral features have been unraveled. The first one corresponds to the strong nonlinear shift, as a function of the lattice period, of the (1,0)_SS and (1,0)_AS peak positions when approaching the interband threshold. Secondly, a large resonance appears near the interband threshold of the noble metals. It is associated to a transparency window resulting from the superposition of the large absorption associated to the core d-electrons and the high reflectivity due to the conduction electrons.