1. Introduction

The possibility to manufacture and nanoengineer the dimensions of periodic arrays of subwavelength semiconductor-based lines has led to a plethora of different optoelectronic applications, such as broadband reflectors [1–3], vertical-cavity surface-emitting lasers (VCSELs) [4,5], high-Q resonators [6], biosensors [7,8], ultrathin half-wave plates [9], metasurfaces [10,11], or grating couplers [12] and demultiplexers [13] for integrated photonic circuits. As all of the aforementioned devices require extremely low losses, their geometry-dependent optical properties have been studied in the near-infrared (near-IR), where absorption of semiconductors like Si, Ge or III-V compounds is negligible [14–16]. However, the understanding of size-dependent refractive and absorption properties of such structures in the absorptive near-ultraviolet/visible (near-UV/VIS) range is also crucial for light harvesting applications like photovoltaics, where the device performance strongly relies on the efficiency of transmission and absorption. Similarly, the manufacturing of fin field-effect transistors (finFETs) [17–19], which are periodically arranged nanoline arrays, relies on lasers of various near-UV/VIS wavelengths. For example, during laser annealing for dopant activation [20,21], for stress assessment using Raman spectroscopy [22–25] or scatterometry-based dimensional and shape metrology [26,27]. However, the introduction of these optical techniques in a routine in-line finFET integration scheme is still very challenging, as they rely on the efficiency of the light coupling and absorption inside nm-scale arrays, which are both strongly geometry-dependent as was shown previously [28,29]. Unfortunately, these previous studies not only focused on a very limited range of geometries, but most importantly, the absorption considered there had a negligible impact on the optical properties of semiconducting nanoline arrays.

To address these deficiencies, this paper provides a comprehensive physical understanding of the impact of the geometry on the polarization-dependent optical properties of periodic arrays of semiconducting nanolines in the near-UV/VIS/near-IR for a wide range of widths W and pitches Λ (i.e. the spatial periodicities). We focus on the study of W $\le $ 35 nm and Λ $\le $ 100 nm Si/SiO₂ nanoarrays, which are the relevant geometries for nanoelectronic applications [30], and can readily be extended to any semiconducting material embedded in a dielectric. As we show, the impact of geometry on the refractive properties of such structures varies significantly according to the spectral region, allowing to tailor them to values not only in-between, but also beyond these of Si and SiO₂. Similarly, we also demonstrate that the absorption of the array can be greater than that of the bulk semiconductor, due to the optical power flow being slowed down in the structure, which is commonly known as the slow light phenomenon [31–34].

This paper starts with the derivation of complex effective optical constants of an array with a fixed geometry from its complex band structure [31]. Next, the discussion focuses on the impact of dimensions on the optical properties of nanoelectronics-relevant Si/SiO₂ arrays in 3 different spectral regimes, where the refractive index (${n_{Si}}$) and extinction coefficient (${\kappa _{Si}}$) of Si satisfy respectively the following conditions: (i) ${n_{Si}} \gg {\kappa _{Si}}$, (ii) ${n_{Si}}\;{\mathbin{\lower.3ex\hbox{$\buildrel> \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} }\;{\kappa _{Si}}$ and (iii) ${n_{Si}} < {\kappa _{Si}}$. For semiconducting materials, like Ge or III-V compounds, these conditions are fulfilled as the considered wavelength ${\lambda _0}$ decreases from the near-IR to the near-UV range. Lastly, we propose a simple semi-analytical model describing the interactions of light with the aforementioned periodic arrays of semiconducting nanolines. The model not only validates our theoretical findings as it is in excellent agreement with reflectance spectra measured on actual finFET arrays with dimensions for the next-generation nanoelectronics, but also shows that the considered nanoarrays can be approximated by a homogeneous uniaxial birefringent medium. Moreover, contrary to conventional, computationally intensive methods like Rigorous Coupled-Wave Analysis (RCWA) [35] or Finite Element Method (FEM) [36], the model provides a simple analytical expression for reflectance allowing to study and hence understand the impact of geometry on the light interactions with nanoline arrays in a very intuitive fashion.

2. Numerical modelling

As we have demonstrated earlier [28,29], the optical properties of a periodic array of nanolines (Fig. 1(a)) can be understood by studying the band structure [31], i.e. the dispersion relation for the electromagnetic modes supported by the array. We propose to use this exact method rather than the approximate effective medium theory (EMT) [37,38], which is based on the weighted average of the optical constants. The latter can only be used for very small pitches (${\lambda _0} \gg $ Λ) and does not account for the slow-light-induced enhancement in absorption. A detailed comparison between the band structure framework and EMT can be found in Appendix A. Figures 1(b) and (c) show respectively the real and imaginary parts of the dispersion of the fundamental modes [39] propagating only along the z-direction, i.e. with a zero x-component of the wave vector, inside the Si/SiO₂ array of width W = 10 nm and pitch Λ = 40 nm. Both Transverse-Electric (TE) and Transverse-Magnetic (TM) polarizations are shown, corresponding respectively to the electric (black lines) and the magnetic (red lines) fields pointing in the y-direction. The calculation is done using a FEM approach [40,41] implemented in COMSOL [42], accounting for the strong dispersion of Si [14] in the considered spectral range. More details about the band structure simulations can be found in Appendix B. Note that only the fundamental modes are displayed since, in the given spectral range, the higher order modes are extremely evanescent, hence essentially not excitable, as confirmed with the integral overlap method [43].

 

Fig. 1. (a) Periodic array made of Si nanolines of width W embedded in SiO₂ with pitch Λ. (b) Real and (c) imaginary parts of the dispersion relation for the fundamental TE (black line) and TM (red line) electromagnetic modes, i.e. respectively with the electric and magnetic fields polarized along the y-direction, propagating along the z-direction inside the array from Fig. 1 (a), calculated for W = 10 nm and Λ = 40 nm. Green and gray dotted lines represent the Si and SiO₂ light lines, respectively. The horizontal solid black lines separate the spectrum into 3 spectral regions characterized by ${n_{Si}} \gg {\kappa _{Si}}$, ${n_{Si}}\;{ \mathbin{\lower.3ex\hbox{$\buildrel> \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} }\;{\kappa _{Si}}$ and ${n_{Si}} < {\kappa _{Si}}$ respectively. Inset: Normalized Poynting’s vectors of the fundamental TE and TM modes, calculated for ${\lambda _0}$ = 400 nm.

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The band structures of Fig. 1 help understand the light propagation inside the arrays both qualitatively and quantitatively. For example, qualitative information about the mode propagation can be found in the shape of the dispersion relation. The fundamental TE mode indeed follows both the real (Fig. 1(b)) and imaginary (Fig. 1(c)) parts of the Si dispersion (green dotted lines) because a significant fraction of its power is localized inside the Si nanoline (inset Fig. 1(c)). On the contrary, the real part of the fundamental TM mode resembles the SiO₂ dispersion (gray dotted line, Fig. 1(b)), and the mode absorption is very small (imaginary part of the fundamental TM mode, see Fig. 1(c)). This is because the TM mode propagates essentially in the lower-index SiO₂ (inset of Fig. 1(c)) [44,45]. Also, quantitative information about the effective optical properties of the array can be extracted from the band structures. For the propagation along the z-direction, the array’s effective refractive index ${n_{eff}}$ and effective extinction coefficient ${\kappa _{eff}}$ can indeed be readily obtained from the band structure as ${n_{eff}} = \frac{{Re\{ {k_z}\} }}{{{k_0}}}$ and ${\kappa _{eff}} = \frac{{Im\{ {k_z}\} }}{{{k_0}}}$, with $Re\{ {k_z}\} $ and $Im\{ {k_z}\} $ respectively the real and imaginary parts of the wave vector of the fundamental mode and ${k_0} = \frac{{2\pi }}{{{\lambda _0}}}$ the wave vector of the vacuum wavelength ${\lambda _0}$. Advantageously, as the array is single-moded, its optical properties can be fully described by ${n_{eff}}$ and ${\kappa _{eff}}$ of the fundamental mode.

The above extraction of the effective optical constants from the band structure now allows to study and understand the size-dependent optical properties of nanoarrays in a quantitative manner. Figures 2 and 3 show the impact of width (W = 5-35 nm) and pitch (Λ = 10-100 nm) on the effective optical constants of the TE and TM modes, respectively, in three distinctive spectral regions (solid horizontal black lines in Figs. 1(b) and (c)) where Si refractive index (${n_{Si}}$) and extinction coefficient (${\kappa _{Si}}$) satisfy the following conditions: $\; {n_{Si}} \gg {\kappa _{Si}}$, ${n_{Si}}\;{ \mathbin{\lower.3ex\hbox{$\buildrel> \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} }\;{\kappa _{Si}}$ and ${n_{Si}} < {\kappa _{Si}}$. Starting with TE when ${n_{Si}} \gg {\kappa _{Si}}$ and ${n_{Si}}\;{ \mathbin{\lower.3ex\hbox{$\buildrel> \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} }\;{\kappa _{Si}}$ (Figs. 2(a) and (b)), it can be observed that ${n_{eff}}\; $expectedly varies between the values of bulk Si and SiO₂ and decrease with the increase in the volume fraction of the low index material, e.g. as a function of Λ [29]. As we showed in Refs. [28,29], this behavior explains the enhanced transmission of light into the semiconducting nanolines observed in this range of wavelengths. Likewise, the effective extinction coefficient ${\kappa _{eff\; }}$behaves similarly when ${n_{Si}} \gg {\kappa _{Si}}$ and ${n_{Si}} < {\kappa _{Si}}$ (Figs. 2(d) and (f)). Unlike these intuitive results, two ranges of wavelengths also present quite unexpected optical properties. First, when ${n_{Si}} < {\kappa _{Si}}$, ${n_{eff}}$ is not only non-monotonic as a function of W and Λ, having a minimum moving diagonally, but it is also significantly smaller than both ${n_{Si}}$ and ${n_{Si{O_2}}}$ (Fig. 2(c)). This peculiarity is due to the fact that once ${\kappa _{Si}}$ exceeds ${n_{Si}}$, Si becomes optically “metallic”, i.e. it pushes the electric field outside of the nanoline. For that reason, when the distance in-between two adjacent nanolines is equal to roughly the cut-off distance [46] for the metallic waveguide $({{\Lambda } - \textrm{W}} )< \frac{{{\lambda _0}}}{{2{n_{Si{O_2}}}}}$, ${n_{eff}}$ increases with Λ following the dispersion of the metallic parallel-plate TE₁ mode [46], i.e. ${n_{T{E_1}}} = \sqrt {n_{Si{O_2}}^2 - {{\left( {\frac{{{\lambda_0}}}{{2({{\Lambda } - \textrm{W}} )}}} \right)}^2}} $, and hence is smaller than ${n_{Si{O_2}}}$. Below the cut-off the mode is forced into Si and the effective refractive index increases for decreasing Λ. Second, ${\kappa _{eff}}$ shows an unexpected behavior in the ${n_{Si}}\;{ \mathbin{\lower.3ex\hbox{$\buildrel> \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} }\;{\kappa _{Si}}$ spectral range (Fig. 2(e)) where it varies non-monotonically and is increased as compared to the bulk Si absorption. Interestingly, this increase is up to 2.5-fold larger than the one predicted with EMT (see Appendix A). This is due to the slow light phenomenon, i.e. the increase in the group index [39] (${n_g}$) of the mode, leading to the enhancement of the light-matter interactions, such as modal absorption $({{\kappa_{eff}} \propto {n_g}} )$ or gain [47–49]. As the slow light phenomenon is here due to a resonance [50] in the dispersion of ${n_{Si}}$ (“flat” region of Fig. 1(b)), it is only pronounced in the ${n_{Si}}\;{ \mathbin{\lower.3ex\hbox{$\buildrel> \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} }\;{\kappa _{Si}}$ region.

 

Fig. 2. (a-c) Normalized effective refractive index ${n_{eff}}$/${n_{Si}}$ and (d-f) normalized effective extinction coefficient ${\kappa _{eff}}/{\kappa _{Si}}$ of the TE mode as a function of width (5-35 nm) and pitch (10-100 nm), for a propagation along the z-direction inside periodic Si/SiO₂ arrays made of W-wide Si nanolines embedded in SiO₂ with a characteristic pitch Λ. The contour plots start at Λ = W + 5 nm. The results were calculated in three spectral regions of Si: (a, d) ${n_{Si}} \gg {\kappa _{Si}}$, (b, e) ${n_{Si}}\;{ \mathbin{\lower.3ex\hbox{$\buildrel> \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} }\;{\kappa _{Si}}$ and (c, f) ${n_{Si}} < {\kappa _{Si}}$, for the characteristic ${\lambda _0}$ = 970, 385 and 235 nm, respectively.

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Fig. 3. (a-c) Normalized effective refractive index ${n_{eff}}$/${n_{Si}}$ and (d-f) normalized effective extinction coefficient ${\kappa _{eff}}/{\kappa _{Si}}$ of the TM mode as a function of width (5-35 nm) and pitch (10-100 nm), for a propagation along the z-direction inside periodic Si/SiO₂ arrays made of W-wide Si nanolines embedded in SiO₂ with a characteristic pitch Λ. The contour plots start at Λ = W + 5 nm. The results were calculated in three spectral regions of Si: (a, d) ${n_{Si}} \gg {\kappa _{Si}}$, (b, e) ${n_{Si}}\;{ \mathbin{\lower.3ex\hbox{$\buildrel> \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} }\;{\kappa _{Si}}$ and (c, f) ${n_{Si}} < {\kappa _{Si}}$, for the characteristic ${\lambda _0}$ = 970, 385 and 235 nm, respectively.

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As observed in Fig. 3, the effective optical constants of the TM mode follow basically the same dependencies as their TE equivalents in most spectral ranges. However, three differences are to be noted. First, ${n_{eff}}$ and ${\kappa _{eff}}$ are generally lower as compared to the TE counterparts, leading to an even greater enhancement in transmission of TM polarized light. However, as already mentioned, unlike TE light which couples into the semiconducting nanoline, the TM mode propagates inside the surrounding dielectric. Second, the effective refractive index for ${n_{Si}} < {\kappa _{Si}}$ (Fig. 3(c)) is not only higher than in TE polarization but it exceeds the values of bulk Si and SiO₂. This behavior is due to the discontinuity of the electric field at the Si/SiO₂ interface and a very high extinction coefficient of Si [51]. Third, the slow light effects on ${\kappa _{eff}}$ are also observed when ${n_{Si}}\;{ \mathbin{\lower.3ex\hbox{$\buildrel> \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} }\;{\kappa _{Si}}$ (Fig. 3(e)), astonishingly, leading to up to more than two orders of magnitude absorption enhancement as compared to the EMT calculations (see Appendix A). However, unlike TE, it is here only observed for wide nanolines, where a more significant portion of the electric field is confined inside the absorbing Si.

3. Experimental verification

In order to confirm that the optical properties discussed in Section 2 correctly describe the interaction of light with the nanoline arrays, we propose a semi-analytical model. As can be seen in Fig. 4(a), we consider a periodic Si/SiO₂ array on a Si substrate made of tapered Si nanolines with height h and with top and bottom widths W_top and W_bot, respectively, embedded in SiO₂ with a pitch Λ. The tapering has to be considered in our model as it is present for virtually every structure manufactured with lithography and etch as used in nanoelectronics [52]. The array is illuminated by s- $({\bar{E}_s})$ or p-polarized $({{{\bar{E}}_p}} )$ light of wavelength$\; {\lambda _0}$, under an angle of incidence $\alpha $. Our model proposes to replace the array of Fig. 4(a) by the effective medium of Fig. 4(b) and uses the following three assumptions. First, the light propagation within the array is described with a single complex effective refractive index ${\tilde{n}_{eff}} = {n_{eff}} - i{\kappa _{eff}}$, calculated from the band structure. For Si/SiO₂ nanoarrays in the near-UV to near-IR, this assumption is met for W $\le $ 35 nm and Λ $\le $ 100 nm, as confirmed with the integral overlap method [43]. Second, we assume that the optical response, i.e. the transmission and reflection coefficients, of such an effective medium is governed by Fresnel’s coefficients [44] for flat and infinite interfaces, as discussed previously [29]. Third, the variation in ${\tilde{n}_{eff}}$ with depth due to the tapering is considered to be so small that reflections take place only at the top and bottom of the array. Under these circumstances, following the standard formula for multiple reflections in a thin film [53], the reflection coefficient of normally incident light reads

 

Fig. 4. (a) Scheme of a periodic Si/SiO₂ array made of tapered $h$-tall Si nanolines with the top and bottom widths of W_top and W_bot, respectively, embedded into SiO₂ spacer with a characteristic pitch Λ on a Si substrate. The array is illuminated with s- $({\bar{E}_s})$ or p-polarized $({{{\bar{E}}_p}} )$ light of ${\lambda _0}$ wavelength, under $\alpha $ angle of incident. (b) Model used to describe the optical response, such as reflectance, of the array from (a). The model utilizes the complex effective refractive index ${\tilde{n}_{eff}} = {n_{eff}} - i{\kappa _{eff}}$, calculated with the band structure. For simplicity only normal incidence case is shown.

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Figure 5(a) displays a comparison between the reflectance spectra calculated with Eq. (1) (lines) and measured experimentally (circles) using NOVA T600 scatterometry setup [54] on a periodic Si/SiO₂ nanoarray with dimensions tailored for the next-generation of transistors [30]. The impressive agreement not only confirms the validity of our approach, but most importantly, proves that the effective optical constants extracted from the band structure are correct. Advantageously, the simple analytical form of Eq. (1) allows to understand how and why a variation in the array geometry impacts its optical response. The dimensions of the array (W_top = 5 nm, W_bot = 17 nm, Λ = 90 nm and $h$ = 273 nm) were established using a fitting algorithm based on RCWA and confirmed with Transmission Electron Microscopy (TEM) (Fig. 5(b)). The appropriate ${\tilde{n}_{eff}}$ were calculated with the band structure framework described in Section 2, using the experimental W_top, W_bot and Λ values. Small discrepancies between the real (Fig. 5(b)) and modelled (Fig. 4(a)) geometries, i.e. necking present at the top and some rounding at the bottom of the nanolines, have a minor impact on the agreement between the experimental and theoretical spectra, as verified with RCWA. Notice that we purposely introduced a different polarization nomenclature for the light above the array, s/p in Fig. 4(a), as compared to TE/TM used for the light inside the structure (Fig. 1(a)). For Fig. 4(a), it may appear redundant as, due to the polarization alignment, s- (p-) light couples into the TE (TM) mode and hence the incident polarization could consistently be named using the TE/TM terminology. However, as the experimental arrays can be rotated azimuthally, i.e. around the z-axis (see Appendix C), after a rotation of 90 degree the impinging TE light would couple into the TM mode leading to confusion, which we avoid by labelling the incident polarizations s and p.

 

Fig. 5. (a) Theoretical (solid lines) and experimental (circles) reflectance spectra of W_top = 5 nm, W_bot = 17 nm, Λ = 90 nm and $h$ = 273 nm Si/SiO₂ arrays on a Si substrate. The spectra were obtained for s- and p-polarization, under normal ($\alpha $ = 0°) and oblique ($\alpha $ = 55°) incidence. (b) TEM image of the measured array.

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Moreover, as demonstrated in Fig. 5(a), the model also shows a very good agreement for oblique incidence ($\alpha $ = 55°), once the equations for ${r_1}$, ${r_2}$ and $\beta $ are modified according to the impact of a non-normal incidence on the reflectance of thin-films [53]. Interestingly, as the TE optical properties of the array are isotropic, only one ${\tilde{n}_{eff}}$ was used to calculate the reflectance under both normal and oblique incidence, whereas due to the anisotropy of the TM optical constants, the angular dependence of TM ${\tilde{n}_{eff}}$ had to be accounted for. For that reason, the nanoline array can be optically approximated with a homogenous uniaxial birefringent medium [55,56]. This approximation is valid not only for ${\lambda _0} \gg {\Lambda }$, but also when the impinging light can probe only the 1^st Brillouin zone [57], i.e. the x-component of the wave vector of the incident light ${k_x} = {k_0}{n_i}sin(\alpha )< \frac{\pi }{{\Lambda }}$, where ${n_i}\; $is the refractive index of the incident medium. For illumination from air this is true for all wavelengths and geometries studied in this paper. To calculate the effective optical constants for a given angle of incidence, one has to evaluate a band structure with a non-zero x-component of the wave vector ${k_x}$, and modify the definition of the effective refractive index to ${n_{eff}} = \frac{{\sqrt {Re{{\{ {k_z}\} }^2} + k_x^2} }}{{{k_0}}}$, while leaving the ${\kappa _{eff}}$ equation unchanged.

4. Conclusions

To summarize, we studied the size-dependent optical properties of periodic arrays of semiconducting nanolines in the near-UV to near-IR spectral range. Using the band structure framework, we demonstrated that the refractive properties of such structures can be tailored to values larger or smaller than these of the materials constituting the array once the extinction coefficient of the semiconductor exceeds its refractive index. Furthermore, we showed that thanks to the slow light, the absorption of the nanoarray can be altered to surpass that of the bulk semiconductor. Additionally, our calculations show that the absorption can be up to more than two orders of magnitude higher than the one predicted with the effective medium theory. Finally, we proposed and experimentally validated a semi-analytical model for the light interactions with periodic arrays of semiconducting nanolines. The fundamental insight established here will facilitate the development and manufacturing of nanoelectronic devices, such as finFETs, as well as the design of light harvesting applications like photovoltaics.

Appendix A: Comparison between the band structure calculations and the effective-medium theory (EMT)

The zeroth order effective-medium theory (EMT) [37,38], based on the weighted average of the optical constants, is a simple and useful tool to describe the optical properties of periodic arrays of deep-subwavelength gratings. However, once the pitch Λ of an array is no longer much smaller as compared to the incident wavelength (${\lambda _0}$), i.e. it does not meet the ${\lambda _0} \gg $ Λ criterion, EMT becomes invalid. As the widest periods considered in here are Λ ∼ ${\lambda _0}$/10 – ${\lambda _0}$/2, an exact band structure method [31] was used throughout the paper to calculate the optical constants of the nanoline arrays in the near-UV to near-IR.

Having experimentally proved the validity of the band structure approach (Fig. 5), we now use it to demonstrate and explain the limitations of EMT for our particular study of periodic arrays of W-wide Si nanolines embedded in SiO₂ with a pitch Λ. Note that for the band structure calculations we consider the light propagation along the z-direction (see Fig. 1(a)), whereas EMT is defined only w.r.t. the polarization and not the propagation direction. Figures 6 and 7 display the ratios between (a-c) the effective refractive indices ${n_{eff}}/{n_{EMT}}$ and (d-e) the extinction coefficients ${\kappa _{eff}}/{\kappa _{EMT}}$, calculated as a function of width (W = 5-35 nm) and pitch (Λ = 10-100 nm) with the band structure (${n_{eff}}$, ${\kappa _{eff}}$) and the effective medium theory (${n_{EMT}}$, ${\kappa _{EMT}}$), respectively for the TE and TM polarizations. Like before, we consider three spectral regimes of decreasing ${\lambda _0}$, where Si refractive index (${n_{Si}}$) and extinction coefficient (${\kappa _{Si}}$) satisfy the following relations: $\; {n_{Si}} \gg {\kappa _{Si}}$, ${n_{Si}}\;{ \mathbin{\lower.3ex\hbox{$\buildrel> \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} }\;{\kappa _{Si}}$ and ${n_{Si}} < {\kappa _{Si}}$. As expected, the agreement between the two methods is the best when the pitch-to-wavelength ratio is the closest to the ${\lambda _0} \gg $ Λ criterion, i.e. in the ${n_{Si}} \gg {\kappa _{Si}}$ regime (Figs. 6 (a, d) and 7 (a, d)). In this region only the effective extinction coefficients for wide nanolines and relaxed pitches are underestimated by EMT, whereas for tight arrays the approximate method is very accurate for both optical constants. Decreasing the wavelength in the ${n_{Si}}\;{ \mathbin{\lower.3ex\hbox{$\buildrel> \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} }\;{\kappa _{Si}}$ regime (Figs. 6(b) and (e) and Figs. 7(b) and (e)) introduces an error on the effective refractive index, but the main impact is observed for the absorption. Indeed, EMT underestimates the TE effective extinction coefficient up to a factor of 2.5 and the TM counterpart, astonishingly, up to more than two orders of magnitude. This is because EMT does not account for the slow-light-induced increase in absorption [48], which is crucial in the vicinity of a resonance [50] in ${n_{Si}}$. Finally, the differences in the optical constants observed in the ${n_{Si}} < {\kappa _{Si}}$ regime (Figs. 6(c) and (f) and Figs. 7(c) and (f)), are due to the fact that for high absorption and short wavelengths the electric field localization has a predominant effect on the effective refractive index and extinction coefficient. This localization is accounted for in the band structure calculations, whereas EMT only assumes a uniform distribution of the electric field across the array.

 

Fig. 6. (a-c) Ratio of the effective refractive indices (${{n}_{{eff}}}/{{n}_{{EMT}}}$) and (d-f) the effective extinction coefficients (${{\kappa }_{{eff}}}/{{\kappa }_{{EMT}}}$), evaluated with the band structure (${{n}_{{eff}}}$, ${{\kappa }_{{eff}}}$) and the effective-medium theory (${{n}_{{EMT}}}$, ${{\kappa }_{{EMT}}}$) for the TE polarization as a function of width (5-35 nm) and pitch (10-100 nm), for a propagation along the z-direction inside periodic Si/SiO₂ arrays made of W-wide Si nanolines embedded in SiO₂ with a characteristic pitch Λ. The contour plots start at Λ = W + 5 nm. The results were calculated in three spectral regions of Si: (a, d) ${{n}_{{Si}}} \gg {{\kappa }_{{Si}}}$, (b, e) ${{n}_{{Si}}}{ \mathbin{\lower.3ex\hbox{$\buildrel> \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} }{{\kappa }_{{Si}}}$ and (c, f) ${{n}_{{Si}}} < {{\kappa }_{{Si}}}$, for the characteristic ${{\lambda }_0}$ = 970, 385 and 235 nm, respectively.

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Fig. 7. (a-c) Ratio of the effective refractive indices (${{n}_{{eff}}}/{{n}_{{EMT}}}$) and (d-f) the effective extinction coefficients (${{\kappa }_{{eff}}}/{{\kappa }_{{EMT}}}$), evaluated with the band structure (${{n}_{{eff}}}$, ${{\kappa }_{{eff}}}$) and the effective-medium theory (${{n}_{{EMT}}}$, ${{\kappa }_{{EMT}}}$) for the TM polarization as a function of width (5-35 nm) and pitch (10-100 nm), for the propagation along the z-direction inside periodic Si/SiO₂ arrays made of W-wide Si nanolines embedded in SiO₂ with a characteristic pitch Λ. The contour plots start at Λ = W + 5 nm. The results were calculated in three spectral regions of Si: (a, d) ${{n}_{{Si}}} \gg {{\kappa }_{{Si}}}$, (b, e) ${{n}_{{Si}}}{ \mathbin{\lower.3ex\hbox{$\buildrel> \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} }{{\kappa }_{{Si}}}$ and (c, f) ${{n}_{{Si}}} < {{\kappa }_{{Si}}}$, for the characteristic ${{\lambda }_0}$ = 970, 385 and 235 nm, respectively.

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To summarize, the effective medium theory shows an acceptable agreement with the band structure method only for long wavelengths, i.e. when ${\lambda _0} \gg $ Λ, and far from resonances in ${n_{Si}}$, where the slow-light-induced absorption is negligible. Therefore, contrary to the rigorous band structure framework, it cannot be used to study the optical properties of semiconducting nanolines in the VIS and near-UV.

Appendix B: Details of the band structure simulations

The band structures of Fig. 1 were evaluated by solving an eigenvalue wave equation in a weak formulation [40,41] using Weak Form PDE module of COMSOL 5.4. The eigenvalue solved for was in this case the complex wave vector ${k_z}$ of the fundamental waveguide mode, which real and imaginary parts were later used to extract respectively the effective refractive index ${n_{eff}} = \frac{{Re\{ {k_z}\} }}{{{k_0}}}$ and effective extinction coefficient ${\kappa _{eff}} = \frac{{Im\{ {k_z}\} }}{{{k_0}}}$, where ${k_0} = \frac{{2\pi }}{{{\lambda _0}}}$ is the wave vector of the vacuum wavelength ${\lambda _0}$. All the calculations were performed in a uniformly meshed (mesh size = 1 nm) 1D simulation domain consisting of a unit cell of Si/SiO₂ nanoline array from Fig. 1(a) terminated by periodic boundary conditions.

The model can be acquired by contacting the corresponding author: andrzej.stefan.gawlik@gmail.com.

Appendix C: Impact of the azimuthal rotation on the reflectance spectra

The experimental setup used in this study allows not only to vary the angle ($\alpha $ = 0 and 55°) and polarization (s and p) of the incident light, but also the azimuthal rotation the sample (θ = 0 and 90°), which is equivalent to rotating the plane of incidence (POI) around the z-axis, as shown in Fig. 8(a). This rotation is the reason why we decided to use separate polarization nomenclatures for the light above and inside the array, s/p and TE/TM respectively. Indeed, although keeping the TE/TM terminology for the incidence light would make perfect sense for θ = 0° as e.g. the TE illumination would couple into the TE mode, it would lead to a confusion at θ = 90° where the TE incident light would excite the TM mode.

 

Fig. 8. (a) 3D scheme of a periodic Si/SiO₂ array made of tapered $h$-tall Si nanolines with the top and bottom widths of W_top and W_bot, respectively, embedded into SiO₂ spacer with a characteristic pitch Λ on a Si substrate. The array is illuminated with s- $({{\bar{E}}_{s}})$ or p-polarized $({{{{\bar{E}}}_{p}}} )$ light of ${{\lambda }_0}$ wavelength, under ${\alpha }$ angle of incident. The angle θ denotes the azimuthal rotation of the array or, equivalently, of the plane of incidence (POI). (b) Theoretical (solid lines) and experimental (circles) reflectance spectra of W_top = 5 nm, W_bot = 17 nm, Λ = 90 nm and $h$ = 273 nm Si/SiO₂ arrays on a Si substrate. The spectra were obtained for s- and p-polarization under oblique incidence (${\alpha }$ = 55°), with (θ = 90°) and without (θ = 0°) the azimuthal rotation of the sample/POI.

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Let us now highlight the main consequences of the sample rotation upon the reflectance spectra. Under normal incidence ($\alpha $ = 0°), the rotation of the POI by 90° causes s- (p-) polarized light to couple into the TM (TE) mode as opposed to the case discussed in the paper. However, as Fresnel’s equations [44] at normal incidence are identical for both s and p, no difference would be observed between e.g. the s-reflectance spectrum with θ = 0° and the p-spectrum with θ = 90°, as both would probe the same TE optical constants. Interestingly, under oblique incidence, it is no longer true as Fresnel’s equations become polarization-dependent and hence lead to different reflectances upon the azimuthal rotation. For example, with θ = 90° p-polarized light would probe the TE effective optical constants, but the reflectance would follow the equations for p-polarization, instead of the ones for s-polarization as in the θ = 0° case. Indeed, as can be seen in Fig. 8(b), the spectra collected experimentally (circles) at oblique ($\alpha $ = 55°) incidence differ significantly between the θ = 0° and 90° azimuthal rotations. The measurements were carried out on the same Si/SiO₂ array (W_top = 5 nm, W_bot = 17 nm, Λ = 90 nm and $h$ = 273 nm) as in Fig. 5. Like before, our simple semi-analytical model (solid lines) displays an excellent agreement with the experimental reflectances.

Subsequently, it is important to explain the difference in how the optical constants vary with the angle of incidence ($\alpha $) for θ = 0 and 90°. In Fig. 5, we demonstrated that for the θ = 0° case the optical properties of the array seen by the incident s- and p-polarized light are isotropic and anisotropic, respectively. The discrepancy between the two polarizations is a direct consequence of the different interface continuity conditions [55]. However, after setting θ = 90°, both polarizations would see angle-independent, i.e. isotropic, ${n_{eff}}$ and ${\kappa _{eff}}$. The difference is due to the fact that once the POI is aligned along the nanolines (θ = 90°), the electric field of the incidence light probes the same volume fraction of Si and SiO₂ irrespective of $\alpha $ and hence the effective optical constants for both polarizations are isotropic.

Lastly, in order to calculate the reflectance for an azimuthal rotation between 0 and 90° using our model, a decomposition of the problem into two orthogonal polarizations is required.

Funding

Fonds Wetenschappelijk Onderzoek (1S67318N).

Acknowledgments

The authors would like to thank the Logic Program of imec. The authors are also deeply indebted to Dr Chris Fietz for the very helpful technical discussions, Dr Mohamed Saib, Dr Joey Hung and Dr Roy Koret for their help with the scatterometry measurements, and Dr Alfonso Sepulveda Marquez for providing samples and TEM images.

Disclosures

The authors declare no conflicts of interest.

References

1. C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultrabroadband mirror using low-index cladded subwavelength grating,” IEEE Photonics Technol. Lett. 16(2), 518–520 (2004). [CrossRef]  

2. Y. Zhou, M. C. Y. Huang, C. Chase, V. Karagodsky, M. Moewe, B. Pesala, F. G. Sedgwick, and C. J. Chang-Hasnain, “High-Index-Contrast Grating (HCG) and Its Applications in Optoelectronic Devices,” IEEE J. Sel. Top. Quantum Electron. 15(5), 1485–1499 (2009). [CrossRef]  

3. C. J. Chang-Hasnain and W. Yang, “High-contrast gratings for integrated optoelectronics,” Adv. Opt. Photonics 4(3), 379–440 (2012). [CrossRef]  

4. M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high-index-contrast subwavelength grating,” Nat. Photonics 1(2), 119–122 (2007). [CrossRef]  

5. S. Kim, Z. Wang, S. Brodbeck, C. Schneider, S. Höfling, and H. Deng, “Monolithic High-Contrast Grating Based Polariton Laser,” ACS Photonics 6(1), 18–22 (2019). [CrossRef]  

6. Y. Zhou, M. Moewe, J. Kern, M. C. Y. Huang, and C. J. Chang-Hasnain, “Surface-normal emission of a high-Q resonator using a subwavelength high-contrast grating,” Opt. Express 16(22), 17282–17287 (2008). [CrossRef]  

7. T. Sun, S. Kan, G. Marriott, and C. Chang-Hasnain, “High-contrast grating resonators for label-free detection of disease biomarkers,” Sci. Rep. 6(1), 27482 (2016). [CrossRef]  

8. H. Yan, L. Huang, X. Xu, S. Chakravarty, N. Tang, H. Tian, and R. T. Chen, “Unique surface sensing property and enhanced sensitivity in microring resonator biosensors based on subwavelength grating waveguides,” Opt. Express 24(26), 29724–29733 (2016). [CrossRef]  

9. U. Levy, C.-H. Tsai, L. Pang, and Y. Fainman, “Engineering space-variant inhomogeneous media for polarization control,” Opt. Lett. 29(15), 1718–1720 (2004). [CrossRef]  

10. M. Khorasaninejad and F. Capasso, “Metalenses: Versatile multifunctional photonic components,” Science 358(6367), eaam8100 (2017). [CrossRef]  

11. B. Gerislioglu and A. Ahmadivand, “Photoluminescence spectroscopy of hybridized exciton-polariton coupling in silicon nanodisks with nonradiative anapole radiations,” arXiv preprint arXiv: 1911.07740 (2019) (n.d.).

12. P. Cheben, P. J. Bock, J. H. Schmid, J. Lapointe, S. Janz, D.-X. Xu, A. Densmore, A. Delâge, B. Lamontagne, and T. J. Hall, “Refractive index engineering with subwavelength gratings for efficient microphotonic couplers and planar waveguide multiplexers,” Opt. Lett. 35(15), 2526–2528 (2010). [CrossRef]  

13. G. Gao, M. Luo, X. Li, Y. Zhang, Q. Huang, Y. Wang, X. Xiao, Q. Yang, and J. Xia, “Transmission of 2.86 Tb/s data stream in silicon subwavelength grating waveguides,” Opt. Express 25(3), 2918–2927 (2017). [CrossRef]  

14. D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27(2), 985–1009 (1983). [CrossRef]  

15. T. N. Nunley, N. S. Fernando, N. Samarasingha, J. M. Moya, C. M. Nelson, A. A. Medina, and S. Zollner, “Optical constants of germanium and thermally grown germanium dioxide from 0.5 to 6.6 eV via a multisample ellipsometry investigation,” J. Vac. Sci. Technol., B 34(6), 061205 (2016). [CrossRef]  

16. S. Adachi, “Optical dispersion relations for GaP, GaAs, GaSb, InP, InAs, InSb, AlxGa1−xAs, and In1−xGaxAsyP1−y,” J. Appl. Phys. 66(12), 6030–6040 (1989). [CrossRef]  

17. X. Huang, W.-C. Lee, C. Kuo, D. Hisamoto, L. Chang, J. Kedzierski, E. Anderson, H. Takeuchi, Y.-K. Choi, K. Asano, V. Subramanian, T.-J. King, J. Bokor, and C. Hu, “Sub 50-nm FinFET: PMOS,” in International Electron Devices Meeting 1999. Technical Digest (Cat. No.99CH36318) (1999), pp. 67–70.

18. H. Zhou, Y. Song, Q. Xu, Y. Li, and H. Yin, “Fabrication of Bulk-Si FinFET using CMOS compatible process,” Microelectron. Eng. 94, 26–28 (2012). [CrossRef]  

19. S. Rigante, P. Scarbolo, M. Wipf, R. L. Stoop, K. Bedner, E. Buitrago, A. Bazigos, D. Bouvet, M. Calame, C. Schönenberger, and A. M. Ionescu, “Sensing with Advanced Computing Technology: Fin Field-Effect Transistors with High-k Gate Stack on Bulk Silicon,” ACS Nano 9(5), 4872–4881 (2015). [CrossRef]  

20. F. Cacho, H. Bono, R. Beneyton, B. Dumont, A. Colin, and P. Morin, “Simulation of pattern effect induced by millisecond annealing used in advanced metal-oxide-semiconductor technologies,” J. Appl. Phys. 108(1), 014902 (2010). [CrossRef]  

21. A. Colin, P. Morin, F. Cacho, H. Bono, R. Beneyton, D. Mathiot, and E. Fogarassy, “Optical–thermal simulation applied to the study of the pattern effects induced by the sub-melt laser anneal process in advanced CMOS technologies,” Appl. Phys. A 104(2), 517–527 (2011). [CrossRef]  

22. D. Kosemura, M. Tomita, K. Usuda, and A. Ogura, “Evaluation of Anisotropic Strain Relaxation in Strained Silicon-on-Insulator Nanostructure by Oil-Immersion Raman Spectroscopy,” Jpn. J. Appl. Phys. 51(2S), 02BA03 (2012). [CrossRef]  

23. T. Nuytten, T. Hantschel, D. Kosemura, A. Schulze, I. De Wolf, and W. Vandervorst, “Edge-enhanced Raman scattering in narrow sGe fin field-effect transistor channels,” Appl. Phys. Lett. 106(3), 033107 (2015). [CrossRef]  

24. T. Nuytten, J. Bogdanowicz, T. Hantschel, A. Schulze, P. Favia, H. Bender, I. De Wolf, and W. Vandervorst, “Advanced Raman Spectroscopy Using Nanofocusing of Light,” Adv. Eng. Mater. 19(8), 1600612 (2017). [CrossRef]  

25. T. Nuytten, J. Bogdanowicz, L. Witters, G. Eneman, T. Hantschel, A. Schulze, P. Favia, H. Bender, I. De Wolf, and W. Vandervorst, “Anisotropic stress in narrow sGe fin field-effect transistor channels measured using nano-focused Raman spectroscopy,” APL Mater. 6(5), 058501 (2018). [CrossRef]  

26. M. H. Madsen and P.-E. Hansen, “Scatterometry—fast and robust measurements of nano-textured surfaces,” Surf. Topogr.: Metrol. Prop. 4(2), 023003 (2016). [CrossRef]  

27. A. C. Diebold, A. Antonelli, and N. Keller, “Perspective: Optical measurement of feature dimensions and shapes by scatterometry,” APL Mater. 6(5), 058201 (2018). [CrossRef]  

28. J. Bogdanowicz, T. Nuytten, A. Gawlik, A. Schulze, I. De Wolf, and W. Vandervorst, “Nanofocusing of light into semiconducting fin photonic crystals,” Appl. Phys. Lett. 108(8), 083106 (2016). [CrossRef]  

29. A. Gawlik, J. Bogdanowicz, A. Schulze, T. Nuytten, K. Tarnowski, J. Misiewicz, and W. Vandervorst, “Enhanced light coupling into periodic arrays of nanoscale semiconducting fins,” Appl. Phys. Lett. 113(6), 063103 (2018). [CrossRef]  

30. A. P. Jacob, R. Xie, M. G. Sung, L. Liebmann, R. T. P. Lee, and B. Taylor, “Scaling Challenges for Advanced CMOS Devices,” in Scaling and Integration of High Speed Electronics and Optomechanical Systems, Selected Topics in Electronics and Systems No. Volume 59 (WORLD SCIENTIFIC, 2017), Vol. Volume 59, pp. 1–76.

31. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

32. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely Large Group-Velocity Dispersion of Line-Defect Waveguides in Photonic Crystal Slabs,” Phys. Rev. Lett. 87(25), 253902 (2001). [CrossRef]  

33. V. S. Volkov, S. I. Bozhevolnyi, L. H. Frandsen, and M. Kristensen, “Direct Observation of Surface Mode Excitation and Slow Light Coupling in Photonic Crystal Waveguides,” Nano Lett. 7(8), 2341–2345 (2007). [CrossRef]  

34. T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008). [CrossRef]  

35. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71(7), 811–818 (1981). [CrossRef]  

36. J. Jin, The Finite Element Method in Electromagnetics, 3rd ed. (Wiley, 1993).

37. S. M. Rytov, “Electromagnetic Properties of a Finely Stratified Medium,” Sov. Phys. JETP 2, 466 (1956).

38. T. K. Gaylord, W. E. Baird, and M. G. Moharam, “Zero-reflectivity high spatial-frequency rectangular-groove dielectric surface-relief gratings,” Appl. Opt. 25(24), 4562–4567 (1986). [CrossRef]  

39. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Springer US, 1983).

40. M. Davanço, Y. Urzhumov, and G. Shvets, “The complex Bloch bands of a 2D plasmonic crystal displaying isotropic negative refraction,” Opt. Express 15(15), 9681–9691 (2007). [CrossRef]  

41. C. Fietz, Y. Urzhumov, and G. Shvets, “Complex k band diagrams of 3D metamaterial/photonic crystals,” Opt. Express 19(20), 19027–19041 (2011). [CrossRef]  

42. Wave Optics Module User’s Guide, Version 5.4, COMSOL (2019).

43. C. Fietz, “Absorbing boundary condition for Bloch–Floquet eigenmodes,” J. Opt. Soc. Am. B 30(10), 2615–2620 (2013). [CrossRef]  

44. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).

45. A. Gawlik, J. Bogdanowicz, A. Schulze, J. Misiewicz, and W. Vandervorst, “Photonic properties of periodic arrays of nanoscale Si fins,” in Photonic and Phononic Properties of Engineered Nanostructures IX (International Society for Optics and Photonics, 2019), Vol. 10927, p. 1092727.

46. D. K. Cheng, Field and Wave Electromagnetics, 1st ed. (Addison-Wesley, 1983).

47. J. T. Robinson, K. Preston, O. Painter, and M. Lipson, “First-principle derivation of gain in high-index-contrast waveguides,” Opt. Express 16(21), 16659–16669 (2008). [CrossRef]  

48. J. Grgić, S. Xiao, J. Mørk, A.-P. Jauho, and N. A. Mortensen, “Slow-light enhanced absorption in a hollow-core fiber,” Opt. Express 18(13), 14270–14279 (2010). [CrossRef]  

49. S. Ek, P. Lunnemann, Y. Chen, E. Semenova, K. Yvind, and J. Mork, “Slow-light-enhanced gain in active photonic crystal waveguides,” Nat. Commun. 5(1), 5039 (2014). [CrossRef]  

50. R. W. Boyd and D. J. Gauthier, Slow and Fast Light, edited by E. Wolf, Progress in Optics (Elsevier, 2002), Vol. 43.

51. W. Cai and V. Shalaev, Optical Metamaterials Fundamentals and Applications (Springer, 2010).

52. D. Shamiryan, A. Redolfi, and W. Boullart, “Dry etching process for bulk finFET manufacturing,” Microelectron. Eng. 86(1), 96–98 (2009). [CrossRef]  

53. H. Fujiwara, “Spectroscopic Ellipsometry: Principles and Applications,” in (John Wiley & Sons, Ltd, 2007).

54. M. Saib, A. Moussa, A.-L. Charley, P. Leray, J. Hung, R. Koret, I. Turovets, A. Ger, S. Deng, A. Illiberi, J. W. Maes, G. Woodworth, and M. Strauss, “Scatterometry and AFM measurement combination for area selective deposition process characterization,” in Metrology, Inspection, and Process Control for Microlithography XXXIII (International Society for Optics and Photonics, 2019), Vol. 10959, p. 109591N.

55. C. Gu and P. Yeh, “Form birefringence dispersion in periodic layered media,” Opt. Lett. 21(7), 504–506 (1996). [CrossRef]  

56. I. Richter, P.-C. Sun, F. Xu, and Y. Fainman, “Design considerations of form birefringent microstructures,” Appl. Opt. 34(14), 2421–2429 (1995). [CrossRef]  

57. P. Yeh, Optical Waves in Layered Media, 1st ed. (John Wiley & Sons, Inc., 1988).