1. Introduction

A polariton is an electromagnetic wave coupled to a polarization excitation in matter [1, 2]. When this coupled excitation is bound to the interface between two media it is called a surface polariton [3]. A surface polariton is an evanescent electromagnetic wave that propagates along the interface and with an amplitude decaying exponentially into the two media. Surface polaritons are named surface plasmon polaritons (SPPs) when one of the media is a metal and the other a dielectric. The continued scaling down of integrated circuits has led to an increasing interest in surface plasmon optics or plasmonics as a mean to guide electromagnetic waves confined into sub-wavelength distances to surfaces [4]–[10]. A limitation of plasmonic circuits is the relatively large damping of SPPs on metals at optical and UV frequencies, which leads to a short propagation length [4]. A way to overcome this limitation and to increase the propagation length is to couple surface polaritons on opposite sides of thin films [11, 12]. These coupled modes, named long-range surface plasmon polaritons in the case of metallic films, have a reduced damping due to the exclusion of the electric field from the absorbing layer and the larger penetration of this field into the surrounding non-absorbing dielectric.

Long-range surface polaritons (LRSPs) have been thoroughly investigated in thin metallic films [11]–[16]. There is, however, little work on surface polaritons supported by thin films of absorbing dielectrics. In the pioneeringwork of Yang et al. [17], it was demonstrated the excitation of LRSPs in thin films of vanadium at infrared wavelengths at which this material behaves as a strongly absorbing dielectric. In subsequent works, the same group demonstrated the excitation of these modes in other materials such as palladium, organic films and islandised metallic films [18]–[20]. Takabayashi et al. measured the propagation of guided modes in thin slabs of Si at 633 nm [21]. However, at this wavelength the absorption of Si is weak and the modes investigated by Takabayashi and co-workers were more likely TM₀ modes rather than LRSPs. The phenomenon of enhanced optical transmission through arrays of subwavelength holes in thin metallic films [22], which is interpreted in terms of the excitation of surface modes, has triggered several studies on similar arrays structured in thin absorbing layers [23]–[26]. Very recently, the excitation of surface polaritons in Au nanoparticles has been also demonstrated in the extreme UV by means of electron energy-loss spectroscopy [27]. At these frequencies Au behaves as a strongly absorbing dielectric.

Here, we demonstrate that it is possible to excite long-range surface polaritons (LRSPs) in ultra-thin layers of amorphous Si (a-Si) at frequencies at which a-Si is a strongly absorbing dielectric, i.e., frequencies at which Si has a positive real component of the permittivity and a large imaginary component. These modes on films of a-Si have similar or even superior characteristics to those on gold films, with longer propagation lengths and similar confinement to the thin film. These important characteristics of LRSPs in Si layers will allow the scaling down of surface polariton circuits using well-known standard Si manufacturing processes. Moreover, Si is the most thoroughly investigated semiconductor and the ability to deposit with high precision atomically flat layers of this semiconductor, combined with the possibility of exciting surface modes on ultra-thin layers, could have important implications in several fields, including photovoltaics, near-field microscopy, optical data storage, optical lithography and sensing [28].

The manuscript is organized as follows. In section 2 we review theoretically the excitation of surface polaritons in thin films and compare the propagation length and the mode confinement in gold and a-Si. In section 3 we demonstrate experimentally with attenuated total reflection measurements the excitation of LRSPs in thin films of a-Si deposited on SiO₂ substrates. These measurements are compared to calculations using Fresnel theory. The conclusions are presented in section 4.

2. Surface modes in thin films

The system that we are going to study is a thin film parallel to the x-y plane with thickness d and extending from 0<z<d. The film is made of a material with a complex permittivity ε=ε_r+iε_i and it is surrounded by a non-absorbing and homogeneous medium with permittivity ε_d (see Fig. 1). Surface polaritons at the two interfaces of the thin film can couple when its thickness is smaller than the penetration length of the field. This coupling hybridizes the two surface polariton modes into two new modes called long-range surface polaritons (LRSPs) and short-range surface polaritons (SRSPs), respectively. LRSPs have a tangential component of the electric field to the film (E_x component in the system of coordinates defined in Fig. 1), which is asymmetric with respect to the middle plane of the film, i.e., the z=d/2 plane in Fig. 1. This field component vanishes inside the thin film and LRSPs have longer propagation lengths than that of surface polaritons on single interfaces due to their weaker interaction with the absorbing material. On the other hand, SRSPs have a tangential electric field component symmetric to the plane z=d/2 and a maximum field amplitude in the absorbing film, which leads to a shorter propagation length.

The general solution for the H_y magnetic field component of a TMpolarized electromagnetic mode bounded at the interfaces, i.e., exponentially decaying into the dielectric, and propagating in the x direction, is given by [3]

where A_i (i=1…4) are the field amplitudes, which are determined by the electromagnetic boundary conditions, and α_d and α are given by

where k ₀ is the wave vector in vacuum and k=k_r+ik_i is the complex propagation constant of the mode in the x direction. According to Eqs.(1)–(3), for modes bounded to the thin film is required that Re(αd)>0. This condition is only fulfilled for non-absorbing films (ε=ε_r), when k>k ₀ ε ^1/2 _d (see Eq.(4)). If the thin film is absorbing, Re(α_d) is positive when Im(k)>0 [17, 18]. The confinement of the electromagnetic field to the surface of the film is defined by the intensity decay length into the surrounding dielectric

The field intensity decays in the x-direction with a characteristic propagation length L_x given by

Imposing the electromagnetic boundary condition to Eqs.(1)–(3) we find the dispersion relations for LRSPs

Similarly to LRSPs, the dispersion relation of SRSP modes is given by

The real (k_r) and imaginary (k_i) components of the complex propagation constant of LRSP and SRSP modes are displayed in Figs. 2(a) and 2(b) as a function of the thickness of the layer with blue solid (LRSPs) and red dashed (SRSPs) curves. The values of k _r and k_i in these figures are normalized by the wave number in the surrounding dielectric k ₀n_d, where we have assumed n_d=√ε_d=1.45. For the calculations of Figs. 2(a) and 2(b) we have considered the permittivity of the thin layer to be purely imaginary with a value of ε=18.5i, which corresponds to the permittivity of a-Si at λ ₀=318 nm. The values of the permittivity of a-Si in the UV, visible and near-infrared are displayed in the inset of 2(b) [30].

 

Fig. 1. Schematic representation of the electric field component along the x-direction of a long-range (red dashed curve) and of a short-range (blue solid curve) surface polariton in a thin layer of a material of permittivity ε surrounded by a dielectric of permittivity ε_d.

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Fig. 2. (a) real k_r and (b) complex k_i components of the wave number of long-range surface polaritons (blue solid curves) and short-range surface polaritons (red dashed curves) in a-Si films at λ₀=318 nm as a function of the thickness of the film. The film is surrounded by a dielectric with a refractive index n_d=√ε_d=1.45. k_r and k_i are normalized to the wave number in the surrounding dielectric. The black curve and the inset in (a) represent the normalized wave number of the TM₀ mode in a non-absorbing thin film surrounded by a medium with n_d=1.45. Inset of (b): wavelength dependence of the real (black curve) and imaginary (red curve) components of the permittivity of a-Si.

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For LRSPs and small values of film thickness, k_r is greater than k₀n_d, i.e., k_r/(k₀n_d)>1 in Fig. 2(a)), in contrast to the smaller values of k_r of SRSPs. In this situation it is necessary to enhance the momentum of incident light to couple to this long-range guided mode. Another important difference between LRSP and SRSP modes is the dependence of k_i for small values of d (Fig. 2(b)). For LRSPs k_i tends to zero as d vanishes, in contrast to the divergence of k_i of SRSPs for small d.

At this point it is useful to review the different types of long-range guided modes supported by thin films symmetrically surrounded by a non-absorbing dielectric. These modes can be classified according to the values of the complex permittivity of the thin layer. Long-range surface plasmon polaritons exist when the permittivity has a negative real value and a small imaginary value (|ε_r|≫ε_i⋍0) and |ε_r|>ε_d [3]. TM₀ modes are supported by films of positive permittivity in which ε_r≫ε_i⋍0 and ε_r>ε_d [29]. As mentioned above, for strongly absorbing thin films in which |ε_r|≾ε_i (independent of the sign of ε _r) and ε_i≫1, there are also guided modes that can propagate many wavelengths [19]. These modes arise from the coupling of surface polaritons at the opposite sides of the absorbing layer. Note that there is an important difference between a generic TM₀ guided mode and LRSPs. A TM₀ mode is a bulk mode resulting from the interaction of the fields at the two surfaces of the thin film by means of total internal reflection; it is not a mode due to the coupling between two surface modes at opposite interfaces. In fact, for the permittivity range in which TM₀ modes exist, i.e., ε_i0, there are no surface modes at the interface between two semi-infinite media [19]. To illustrate further the similitude and difference of TM₀ modes in non-absorbing layers and LRSPs in thin absorbing films, we have plotted in Fig. 2(a) with a black solid curve and in the inset of Fig. 2(a) the normalized wave number of the TM₀ mode in a layer of medium with n=3.0, i.e., with a refractive index equal to the real value of the refractive index of a-Si at λ ₀=318 nm, surrounded by a dielectric with n_d=√ε_d=1.45. The wave numbers of the TM₀ and the LRSP modes are similar for values of the film thickness smaller than 25 nm, indicating also the similar characteristics of these two modes for these thickness. However, the wave numbers differ significantly for larger values of d. For d→∞ the wave number of the LRSP converges to that of a surface polariton on a single interface, while the wave number of the TM₀ mode converges to k₀n.

The propagation length L_x at λ₀=318 nm of LRSPs calculated for a-Si films surrounded by a dielectric with n_d=1.45 is displayed in Fig. 3(a)(blue solid curve) as a function of the thickness of the film. For comparison we have also plotted in Fig. 3(a) the propagation length of LRSPs in similar thin films of gold (red solid curve) at λ₀=318 nm, and for a-Si and gold at λ₀=650 nm (blue and red dashed curves respectively). In general L _x decreases as the thickness of the film increases, approaching the values of surface polaritons at single interfaces. A remarkable behavior is found by comparing the propagation lengths in these two media (see also inset of Fig. 3(a)): L_x in the case of a-Si films is close to or in general longer than the propagation length in Au in spite to be commonly accepted that metals are superior for the propagation of long-range polaritons. The values of the permittivity of Au and a-Si are displayed in table 1 for λ₀=318, 450 and 650 nm, together with the values of L_x for a film of thickness d=5 nm. It should be pointed out that the guided mode in a-Si at λ₀=650 nm, with a remarkable long propagation length, corresponds to the TM₀ mode in the weakly absorbing layer at this wavelength.

The decay lengths of these modes in the surrounding dielectric L_z are displayed in Fig. 3(b) as a function of the thickness of the thin layer and in table 1 for d=5 nm. These decay lengths illustrate the mode confinement to the film. Again, very similar values of L_z are found for Au and a-Si in the UV and in the visible. As the thickness of the thin layer is reduced, the mode is excluded from this layer decaying a longer distance into the dielectric.

The figure of merit (FOM) describing the quality of waveguiding structures can be defined as the ratio between the propagation length and the decay length into the surrounding dielectric L_x/L_z [31, 32]. Large values of this FOM indicate the preferable situation of a long propagation length and strong field confinement. The FOM is displayed in Fig. 3(c) for silicon (thin blue curves) and gold (thick red curves) at λ₀=318 nm (full curves) and at λ₀=650 nm (dashed curves). In general, due to the strong increase of L_x for small values of d, the FOM increases in spite of the also larger values of L_z. At λ₀=318 nm the performance of a-Si thin films is better than that of gold. At λ₀=650 nm similar FOMs are obtained only for the thinnest layers. The FOM represented in Fig. 3(c) correspond to calculations considering thin films without roughness. Surface roughness, inherent to real systems, will reduce L_x. However, as we will see later, it is possible to sputter very thin layers of a-Si (~10 nm) with high precision and with a roughness in the sub-nanometer range.

In order to illustrate further the reason why long propagation lengths are possible for LRSPs modes on thin films of strongly absorbing media, we have calculated the Poynting vector of the

Table 1. Characteristic lenghts of long-range guided modes in a film with a thickness of 5 nm of amorphous Si and gold surrounded by a dielectric with refractive index n_d=√ε_d=1.45. Calculated values of propagation length L_x and decay length L_z into the dielectric at three different wavelengths λ₀=318, 450 and 650 nm. The permittivities of the materials at these wavelengths are also listed.

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electromagnetic field in the case of an a-Si thin film in a SiO₂ medium at λ₀=318 nm. This calculation is presented in Fig. 4. Note that, for clarity, different scales have been used for the different layers. The poynting vector is strongly reduced inside the absorbing layer and it extends into the non-absorbing SiO₂ surrounding medium. Therefore, the electromagnetic energy density is mainly confined in the loss-less material and the surface polariton can propagate with reduced losses.

 

Fig. 3. (a) LRSP propagation length L_x normalized by λ₀ in a film of thickness d of gold (thick red curves) and silicon (thin blue curves) surrounded by a dielectric with refractive index n_d=1.45. The solid curves correspond to λ₀=318 nm, while the dashed curves are the calculations for λ₀=650 nm. The inset represents a close view of L_x at λ₀=318 nm for small values of d. (b) decay length of the field intensity L_z (normalized by λ₀) in the dielectric for the same materials and wavelength as in (a). (c) Figure of merit of thin-film guiding structures defined as L_x/L_z for the same materials and wavelengths as in (a).

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3. Long-range surface polaritons on thin a-Si films

We have investigated experimentally the excitation of LRSPs at UV and visible frequencies in thin films of a-Si. Two samples were prepared, the first one is an a-Si film with a thickness of 13 nm sputtered onto a SiO₂ substrate and covered by 350 nm of SiO₂. The second sample consists of an a-Si film with thickness of 20 nm covered by 500 nm of SiO ₂. A scanning electron microscope side view image of the first sample is shown in Fig. 5.

The excitation of LRSPs in the films of a-Si was investigated with attenuated total reflection measurements. Due to momentum mismatch, it is not possible to couple free space light to LRSPs in a thin film below a critical thickness. A prism with a higher permittivity than the dielectric surrounding the thin film can be used to match the momentum of the incident light to the momentum of LRSPs. In our experiments we used a prism of F2 Schott optical glass placed on top of the samples. The specular reflection at the prism-SiO₂ interface was measured for different angles of incidence and at different wavelengths. The incident beam is totally internally reflected at angles larger than the critical angle of the prism-SiO₂ interface. However, the evanescent transmitted field can resonantly couple to LRSPs at certain angles and wavelengths. This coupling to LRSPs leads to the reduction of the reflected intensity.

 

Fig. 4. LRSP’s Poynting vector component tangential to an a-Si thin-film. The thickness of the a-Si is d=5 nm with permittivity ε=18.5i at λ₀=318 nm, and the background is SiO₂. Note that, for clarity, the scales for the SiO₂ layer and the a-Si film are different.

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The experiments were done in a computer controlled rotation stage set-up that allowed the simultaneous rotation of sample and detector. We used a pulsed diode laser (Picoquant PDL 800-B) emitting at a wavelength λ₀=375 nm and a Si photodiode as detector to investigate the UV excitation of LSRPs. We also used a collimated beam from a halogen lamp (Yokogawa AQ4303) and a fiber coupled spectrometer (Ocean Optics USB2000) to perform broadband measurements at optical frequencies. The angle of incidence was varied by rotating the sample in steps of 0.05 degrees.

 

Fig. 5. (a) Side view image, made with a scanning electron microscope, of an a-Si thin film sputtered onto a SiO₂ substrate and covered by a SiO₂ layer.

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Figure 6(a) displays a reflectivity measurement of a p-polarized beam of the halogen lamp on the sample formed by the a-Si film of 20 nm. Calculations of the reflectivity are presented in Fig. 6(b). The increase of reflectivity close to θ=62° at long wavelengths corresponds to the total internal reflection at the critical angle on the prism-SiO₂ interface. The excitation of guided modes in the a-Si is manifested by the narrow band of low reflectivity around 63.5° and λ0⋍550 nm. This band actually covers the transition between a LRSPs due to high absorption in a-Si at short wavelengths (ε≈17+15i at λ₀=450nm) to the TM₀ mode in a weakly absorbing a-Si layer at long wavelengths (ε≈16+2.4i at λ₀=700nm). As previously pointed by Yang et al. in Ref. [19] and as can be appreciated in Fig. 6, there is not a drastic transition between LRSPs and the TM₀ mode as the absorption in the thin film increases.

 

Fig. 6. (a) Measurements and (b) calculations of the reflected spectra of p-polarized light incident at an angle θ on the system formed by F2 glass, a SiO₂ layer with a thickness of 500 nm, a 20 nm layer of a-Si and a SiO₂ substrate.

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The resonant frequency of excitation to LRSPs can be tuned by varying the coupling between surface polaritons at the opposite interfaces of the absorbing film. This tuning can achieved by changing the thickness of the film and the SiO₂ layer on top. Figure 7 displays measurements at λ₀=375 nm (circles) and calculations (curves) of the reflection of the a-Si layer with a thickness of 13 nm covered by a SiO₂ layer of 350 nm. The theoretical curves were obtained by leaving the thickness of the SiO₂ and a-Si layers as free parameters to fit to the measurements. The permittivities of a-Si and SiO₂ were determined by ellipsometry measurements (not shown here) on a-Si layer with the same thickness and on a SiO₂ substrate, and were kept fixed in the fits. Within the experimental accuracy, the values of the permittivity of the thin film of a-Si were in agreement with those of bulk a-Si [30], and electronic quantum confinement perpendicular to the layer plane could be neglected. The values of the thickness of the SiO ₂ layer, 13 nm, and of the a-Si layer, 345 nm, obtained from the fits are in agreement with the SEM image of Fig. 5.

 

Fig. 7. Experimental (open circles and triangles) and theoretical (curves) reflectivity at λ₀=375 nm of the system formed by F2 glass, a SiO₂ layer with a thickness of 345 nm, an a-Si layer with a thickness of 13 nm and a SiO₂ substrate. The green circles and solid black curve correspond to p-polarized light, while the blue triangles and red dashed curve correspond to s-polarized light.

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The measurement and calculation of the p-polarized reflection (green circles and black curve in Fig. 7) have a sharp dip at an angle θ=63.6°, larger than the critical angle of total internal reflection (θ_c⋍62°). At θ=63.6° the momentum of the evanescent transmitted beam through prism-SiO₂ interface matches the momentum of the LRSP and coupling is possible, giving rise to the reduction of the reflection. For λ₀=375 nm the absorption in a-Si is very strong (ε=10.8+19.9i) and this minimum in the reflection can be attributed to coupling to LRSPs. No particular features are found in the s-polarization (blue triangles and red dashed curve in Fig. 7), as for this polarization and configuration it is not possible to excite LRSPs.

An important aspect which determines the propagation length of long range guided modes in thin films is the surface roughness. We have determined this roughness by performing atomic force microscopy measurements (not shown) on a similar layer of sputtered a-Si with a thickness of 12.5 nm on top of a SiO₂ substrate. These measurements lead to a root mean square value of the surface roughness of only 0.35 nm, which was mainly determined by the substrate. The effect of the roughness in thin films of a-Si on the propagation length of LRSPs is therefore negligible. Note that, due to the formation of islands, reported values of the surface roughness of similar thin films of gold are one order of magnitude larger [33]. Consequently, ultra-thin layers of a-Si have better characteristics than gold layers for the propagation of long-range surface polaritons.

As mentioned above, the position and depth of the resonance shown in Fig. 7 are very sensitive to variations in thickness of the a-Si and SiO₂ layers. If we consider a system assuming a 10 nm a-Si film and a 300 nm SiO₂ top layer, it is possible to obtain perfect coupling to LRSPs as is shown in the calculation displayed in Fig. 8. The blue solid curves in Fig. 8(a) and 8(b) represent calculations of the attenuated total reflection of p-polarized light (λ₀=375 nm) incident onto this system of optimized dimensions. For this structure the reflection vanishes at θ=62.9° indicating a 100% coupling efficiency to LRSPs. Figure 8(c) displays a calculation of the magnetic field amplitude, i.e., the only field component in the y-direction, for a p-polarized incident beam at 62.9°. Only the incident field is present in the prism (upper part of the plot), as can be appreciated by the absence of interference that would point to specular reflection in the prism-SiO² interface. This vanishing reflection arises from the perfect coupling of the evanescent field transmitted into the SiO₂ layer to LRSPs in the thin a-Si layer. The maximum field amplitude of the LRSP in Fig. 8(c) is at the surface of the a-Si film and decays evanescently from this layer into the surrounding dielectric. For clarity, we display on Fig. 8(d) with a red solid curve a vertical cut at x=0 of the contour plot 8(c). To make the a-Si layer visible, we have scaled its thickness in this figure. The oscillatory amplitude on the upper part of the figure corresponds to the incident field in the prism. We can see a small dip in the magnetic field amplitude in the center of the a-Si thin film and the exponential amplitude decay in the surrounding SiO₂ characteristic of a surface wave. The electric field amplitude component along the x direction at x=0 is also represented in Fig. 8(d) with a black dashed line. This field amplitude is asymmetric with respect to the middle plane of the thin film, as it is expected for long range surface polaritons [19].

 

Fig. 8. (a) and (b) display the reflectivity at λ₀=375 nm of a sample formed by a prism (n _prism=1.67) on top of a SiO₂ layer with a thickness of 300 nm, an a-Si layer with a thickness of 10 nm and a SiO₂ substrate (n=1.48). The reflectivity is calculated for different values of the imaginary part (a) and the real part (b) of the permittivity of the thin layer. The full blue curves in (a) and (b) are calculated for ε=10.8+19.9i, i.e., the dielectric constant of a-Si at λ₀=375 nm. In figure (a) ε=10.8+1.0i (red dashed curve) and ε=10.8+0.1i (green dot-dashed curve). In figure (b) ε=0+19.9i (red dashed curve) and ε=-10+19.9i (green dot-dashed curve). (c) Calculation of the magnetic field amplitude for a p-polarized beam (λ0=375 nm) incident at 62.9° onto the sample with the same dimensions as in (a) and (b). Magnetic field amplitude (red full curve) and x-component of the electric field (black dashed curve) at x=0 are shown in (d). Note that, for clarity, the SiO₂ layer and the silicon film are scaled.

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In order to highlight the relevance of the imaginary permittivity of the thin film for the excitation of LRSPs, we have calculated the attenuated total reflection for different values of ε_i keeping ε_r constant. These results correspond to ε=10.8+1.0i for the red dashed curve in Fig. 8(a) and ε=10.8+0.1i for the green dot-dashed curve. The LRSP resonance vanishes as the absorption in the thin film is reduced. If, on the other hand, the real component of the permittivity is varied the LRSPs remains unchanged. This is illustrated in Fig. 8(b) where the attenuated total reflection of the thin layer of absorbing material is calculated for values of the permittivity of ε=10.8+19.9i (blue curve), ε=0+19.9i (red dashed curve) and ε=-10+19.9i (green dot-dashed curve). The results of Fig. 8(a) and Fig. 8(b) can be summarized by saying that LRSPs are excited in thin films of strongly absorbing layers embedded in non-absorbing dielectrics independently of the value of the real part of the permittivity of the absorbing layer. This is in contrast to TM₀ modes and long-range surface plasmon polaritons, which mainly depend on this value of the real part of the permittivity.

4. Conclusions

We have demonstrated the excitation of long-range surface polaritons LRSPs in ultra-thin films of amorphous silicon at visible and UV frequencies. These modes have similar characteristics to long-range surface plasmon polaritons and TM₀ modes with the important difference that LR-SPs are sustained by thin layers of strongly absorbing media while surface plasmon polaritons and TM₀ are supported by thin layers of metals and non-absorbing dielectrics respectively. Excitation of long-range surface modes in absorbing Si opens the possibility to surface polariton optics compatible with standard Si processing technology.