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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 23 — Nov. 18, 2013
  • pp: 27587–27601
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Mie resonance-enhanced light absorption in periodic silicon nanopillar arrays

Francisco J. Bezares, James P. Long, Orest J. Glembocki, Junpeng Guo, Ronald W. Rendell, Richard Kasica, Loretta Shirey, Jeffrey C. Owrutsky, and Joshua D. Caldwell  »View Author Affiliations


Optics Express, Vol. 21, Issue 23, pp. 27587-27601 (2013)
http://dx.doi.org/10.1364/OE.21.027587


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Abstract

Mie-resonances in vertical, small aspect-ratio and subwavelength silicon nanopillars are investigated using visible bright-field µ-reflection measurements and Raman scattering. Pillar-to-pillar interactions were examined by comparing randomly to periodically arranged arrays with systematic variations in nanopillar diameter and array pitch. First- and second-order Mie resonances are observed in reflectance spectra as pronounced dips with minimum reflectances of several percent, suggesting an alternative approach to fabricating a perfect absorber. The resonant wavelengths shift approximately linearly with nanopillar diameter, which enables a simple empirical description of the resonance condition. In addition, resonances are also significantly affected by array density, with an overall oscillating blue shift as the pitch is reduced. Finite-element method and finite-difference time-domain simulations agree closely with experimental results and provide valuable insight into the nature of the dielectric resonance modes, including a surprisingly small influence of the substrate on resonance wavelength. To probe local fields within the Si nanopillars, µ-Raman scattering measurements were also conducted that confirm enhanced optical fields in the pillars when excited on-resonance.

© 2013 Optical Society of America

1. Introduction

While many reports are concerned with single nanoparticles or random arrangements of nanoparticles, periodic arrays provide a means for studying the optical properties in well controlled systems [23

23. D. A. Genov, A. K. Sarychev, V. M. Shalaev, and A. Wei, “Resonant field enhancements from metal nanoparticle arrays,” Nano Lett. 4(1), 153–158 (2004). [CrossRef]

27

27. B. S. Simpkins, J. P. Long, O. J. Glembocki, J. Guo, J. D. Caldwell, and J. C. Owrutsky, “Pitch-dependent resonances and near-field coupling in infrared nanoantenna arrays,” Opt. Express 20(25), 27725–27739 (2012). [CrossRef] [PubMed]

], and enable the investigation of near- and far-field interactions between adjacent nanostructures, which may lead to the controlled manipulation of optical signals and light-matter interactions [28

28. J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010). [CrossRef] [PubMed]

]. Array periodicity provides an additional dimension for improving the performance of optical and optoelectronic devices through phase-coherent coupling of the localized electromagnetic fields between adjacent nanoparticles or through multiple scattering or grating-diffraction effects. For instance, within specific wavelength regions, periodic arrays of nanoparticles embedded in photovoltaic devices of simple architectures have been reported to produce higher energy harvesting efficiencies than their nanoparticle-free counterparts [14

14. S. A. Mann, R. R. Grote, R. M. Osgood, and J. A. Schuller, “Dielectric particle and void resonators for thin film solar cell textures,” Opt. Express 19(25), 25729–25740 (2011). [CrossRef] [PubMed]

,22

22. Y. Yu, V. E. Ferry, A. P. Alivisatos, and L. Cao, “Dielectric core-shell optical antennas for strong solar absorption enhancement,” Nano Lett. 12(7), 3674–3681 (2012). [CrossRef] [PubMed]

,29

29. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010). [CrossRef] [PubMed]

31

31. L. Cao, P. Fan, A. P. Vasudev, J. S. White, Z. Yu, W. Cai, J. A. Schuller, S. Fan, and M. L. Brongersma, “Semiconductor nanowire optical antenna solar absorbers,” Nano Lett. 10(2), 439–445 (2010). [CrossRef] [PubMed]

].

We find that short, vertical nanopillars exhibit intense Mie resonances in the visible spectral region governed by a transverse-mode eigencondition that reliably predicts the resonant wavelength. This resonant wavelength essentially scales linearly with pillar diameter and is nearly independent of pillar height within the range studied here. Consistent with finite-difference time-domain (FDTD) computations of the resonantly strengthened internal optical fields in the pillars, we measure an enhanced Si Raman line from the arrays, and find that the enhancement exceeds that reported from taller Si nanowires [38

38. M. Khorasaninejad, N. Dhindsa, J. Walia, S. Patchett, and S. S. Saini, “Highly enhanced Raman scattering from coupled vertical silicon nanowire arrays,” Appl. Phys. Lett. 101(17), 173114 (2012). [CrossRef]

]. We also find that the resonant wavelength is modulated by the density of the nanoarray through both air- and substrate-mediated interpillar interactions. Although similar pitch-induced effects have been thoroughly investigated for arrays of plasmonic nanoantennas [26

26. R. Adato, A. A. Yanik, C.-H. Wu, G. Shvets, and H. Altug, “Radiative engineering of plasmon lifetimes in embedded nanoantenna arrays,” Opt. Express 18(5), 4526–4537 (2010). [CrossRef] [PubMed]

,27

27. B. S. Simpkins, J. P. Long, O. J. Glembocki, J. Guo, J. D. Caldwell, and J. C. Owrutsky, “Pitch-dependent resonances and near-field coupling in infrared nanoantenna arrays,” Opt. Express 20(25), 27725–27739 (2012). [CrossRef] [PubMed]

,39

39. R. Kullock, S. Grafström, P. R. Evans, R. J. Pollard, and L. M. Eng, “Metallic nanorod arrays : negative refraction and dipolar interactions,” J. Opt. Soc. Am. B 27(9), 1819–1827 (2010). [CrossRef]

41

41. W. L. Auguie, X. M. Bendaña, W. L. Barnes, and F. J. García de Abajo, “Diffractive arrays of gold nanoparticles near an interface: critical role of the substrate,” Phys. Rev. B 82(15), 155447 (2010). [CrossRef]

], to our knowledge, this is the first experimental report of pitch-induced shifts in the optical resonances of arrays of dielectric nanocavities.

The arrays were characterized experimentally by integrating bright-field reflection (BFR) spectroscopy and µ-Raman spectroscopy. FDTD computations reproduce our measured spectra with superb fidelity, lending confidence in the computed field distributions that provide detailed and distinguishing mode characteristics critical for their identification. The BFR spectra exhibited sharp dips due to an intense Mie resonance (mode 1), which FDTD computations showed to be confined within a pillar and to bear similarities to the lowest-order hybrid mode in microwave pillboxes [32

32. D. Kajfez and P. Guillon, Dielectric Resonators (Artech House, 1986).

]. For larger diameters, a second order resonance (mode 2) appears at bluer wavelengths. Simulations show that the same modes are excited for pillars in free space or affixed to a substrate as in the experiments, with the substrate providing a surprisingly small influence on the resonance wavelength. Although excited at normal incidence with polarization parallel to the substrate (in the x-direction), it is Ez, the field parallel to the cylinder axis, that achieves greatest electromagnetic field magnitude for our geometry. The field intensity within a pillar is computed to exhibit substantial enhancements (|E/E0| ~3), which leads to enhancements in the Si 520 cm−1 Raman line when the cavity resonance spectrally coincides with the Raman pump laser wavelength. This is consistent with work by Cao et al. [42

42. L. Cao, B. Nabet, and J. E. Spanier, “Enhanced Raman scattering from individual semiconductor nanocones and nanowires,” Phys. Rev. Lett. 96(15), 157402 (2006). [CrossRef] [PubMed]

], where Raman enhancements within single, isolated Si nanowires were observed. In addition, the Raman enhancements measured in this work are two times larger than those obtained for periodic arrays of longer (~0.8 – 1.1 µm) Si nanopillars under longer wavelength excitation (632.8 nm), as reported by Khorasaninejad et al. [38

38. M. Khorasaninejad, N. Dhindsa, J. Walia, S. Patchett, and S. S. Saini, “Highly enhanced Raman scattering from coupled vertical silicon nanowire arrays,” Appl. Phys. Lett. 101(17), 173114 (2012). [CrossRef]

], and ~three times the maximal Raman enhancement reported for longer Si nanopillars (~0.4 – 2.3 µm in length), also with 632.8 nm excitation, as reported by Wells et al. [6

6. S. M. Wells, I. A. Merkulov, I. I. Kravchenko, N. V. Lavrik, and M. J. Sepaniak, “Silicon nanopillars for field-enhanced surface spectroscopy,” ACS Nano 6(4), 2948–2959 (2012). [CrossRef] [PubMed]

].

Remarkably, the simulations show that the resonant wavelength of the fundamental mode (mode 1) is essentially independent of pillar height for short pillars (over a range of 100~215 nm), despite a strong z-dependence to the field amplitudes within a pillar. Indeed, we measure that the resonant condition is determined essentially by the pillar diameter, given by a relation nka = κ, where k is the free space wavevector, a is the pillar radius, and the eigenvalue κ is ~2.6 for mode 1, and 5.6 for mode 2, similar to specific leaky transverse modes in a single semiconductor nanowire [33

33. L. Cao, J. S. White, J.-S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. 8(8), 643–647 (2009). [CrossRef] [PubMed]

,43

43. G. Chen, J. Wu, Q. Lu, H. R. Gutierrez, Q. Xiong, M. E. Pellen, J. S. Petko, D. H. Werner, and P. C. Eklund, “Optical antenna effect in semiconducting nanowires,” Nano Lett. 8(5), 1341–1346 (2008). [CrossRef] [PubMed]

,44

44. L. Cao, J.-S. Park, P. Fan, B. Clemens, and M. L. Brongersma, “Resonant germanium nanoantenna photodetectors,” Nano Lett. 10(4), 1229–1233 (2010). [CrossRef] [PubMed]

]. However, the resonant wavelength is modulated by the density of nanopillars within the array, blue-shifting the resonant wavelength by up to ~15% as the pitch is reduced from 3 to 1.5 times the diameter, which we attribute to interparticle coupling through air [35

35. A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B 82(4), 045404 (2010). [CrossRef]

]. Furthermore, the shift exhibits weak oscillations as the pitch is varied, which are attributable to a substrate mediated interaction.

The demonstration of the resonant confinement of the incident laser light within shallow nanostructures patterned directly on a substrate of equal refractive index suggests that this approach provides potential for enhancing optical processes in patterned nanostructures atop both active and passive devices in a controlled manner, in support of the work of Spinelli et al. [13

13. P. Spinelli, M. A. Verschuuren, and A. Polman, “Broadband omnidirectional antireflection coating based on subwavelength surface Mie resonators,” Nat Commun 3, 692 (2012). [CrossRef] [PubMed]

].

2. Experimental details

The nanopillar arrays (100 x 100 pillars) were fabricated in silicon via electron-beam lithography and a standard liftoff process, whereby square periodic arrays of Cr nanodots were created and used as a hard mask for reactive ion etching (RIE). During the RIE process, a 3:4 ratio of CHF3 and SF6, respectively, for 14.5 min resulted in Si nanopillars of approximately 217 nm in height. Additional information concerning the fabrication of similar nanopillars can be found in the literature [25

25. J. D. Caldwell, O. Glembocki, F. J. Bezares, N. D. Bassim, R. W. Rendell, M. Feygelson, M. Ukaegbu, R. Kasica, L. Shirey, and C. Hosten, “Plasmonic nanopillar arrays for large-enhanced Raman scattering sensors,” ACS Nano 5(5), 4046–4055 (2011). [CrossRef] [PubMed]

,45

45. F. J. Bezares, J. D. Caldwell, O. Glembocki, R. W. Rendell, M. Feygelson, M. Ukaegbu, R. Kasica, L. Shirey, N. D. Bassim, and C. Hosten, “The role of propagating and localized surface plasmons for SERS enhancement in periodic nanostructures,” Plasmonics 7(1), 143–150 (2012). [CrossRef]

,46

46. J. D. Caldwell, O. J. Glembocki, F. J. Bezares, M. I. Kariniemi, J. T. Niinistö, T. T. Hatanpää, R. W. Rendell, M. Ukaegbu, M. K. Ritala, S. M. Prokes, C. M. Hosten, M. A. Leskelä, and R. Kasica, “Large-area plasmonic hot-spot arrays: sub-2 nm interparticle separations with plasma-enhanced atomic layer deposition of Ag on periodic arrays of Si nanopillars,” Opt. Express 19(27), 26056–26064 (2011). [CrossRef] [PubMed]

]. A SEM micrograph of an array of ~209 nm diameter nanopillars on a 300 nm pitch, imaged at 45°, is presented as an inset in Fig. 1(b)
Fig. 1 (a) and (b) show the experimental BFR intensity of Si pillar arrays, with pillar height of 217 nm, as a function of wavelength (bottom axis) for a constant pitch of 300 nm and a constant diameter of 133 nm, respectively. (a) displays spectra for nanopillar arrays with diameters of 115, 133, 152 and 209 nm, while (b) refers to arrays with pitches of 225, 275, 325, 375 and 425 nm. (c) and (d) present the results obtained from FDTD simulations of arrays matching the dimensions of (a) and (b), respectively. The inset in (a) is a SEM image of the 425 nm pitch array. A direct comparison reveals excellent agreement between experiment and simulations.
. Each array was designed to have a different nanopillar diameter and/or pitch, with diameters ranging from ~115 to 260 nm and gaps between pillar walls ranging from ~60 to 165 nm, resulting in periodicities ranging from 175 to 425 nm. The overall dimensions of the arrays ranged from ~17 to ~42 μm square.

For the BFR measurements, two microspectroscopy systems were used to record the reflectance spectra strictly from individual arrays without signal contamination from the surrounding substrate. In the first, a quartz-tungsten-halogen lamp fully illuminated an individual array through a microscope objective (20X, NA = 0.45). We employed standard Koehler illumination, but apertured the system to restrict incoming radiation to ± 10° about the sample normal. The same objective collected the reflected and scattered light, which was then focused onto the entrance slit of an imaging spectrometer (300 mm focal length, 50 groove/mm grating) mounted directly onto a side port of an inverted microscope. Use of an imaging spectrometer equipped with a charge-sensitive device camera permitted recording spectra only from within the array under study. The raw spectra were normalized by the spectrum reflected from nearby portions of the unpatterned silicon substrate, which is featureless in the spectral band of interest. Hence our reported reflectance percentages are relative to a silicon wafer. A second, commercial system (Craic Technologies), was used to acquire spectra at wavelengths ranging from ~400 to 1700 nm. As above, a single objective (40X, NA = 0.60) illuminated and collected the light, but here the incident angles were unrestricted, and a square aperture in the image plane selected the region of interest within an array. The two systems produced very similar spectra where their wavelength ranges overlapped.

µ-Raman spectra of each nanopillar array were collected using a custom-built experimental setup. The excitation of 8 mW CW at 514 nm from an argon ion laser was focused to a diameter of ~1 μm at the sample surface through a microscope objective (100X, NA = 0.70). The back-scattered light was collected through the same objective and coupled with optical fiber to an Ocean Optics QE6500 thermoelectrically cooled CCD array detector.

Finite-difference time domain (FDTD) as well as Finite Element Method (FEM) simulations were performed using the LumericalTM and COMSOL MultiPhysicsTM software packages, respectively. The two packages solve Maxwell’s equations in either the time or the frequency domains, respectively. Both methods produced nominally identical results and thus only the results from the FDTD simulations are presented unless otherwise noted. In all cases, we used optical constants from Palik [47

47. E. D. Palik, Handbook of Optical Constants of Solids, 2nd ed. (Academic Press, 1998).

] and the geometries and boundary conditions were constructed in such manner as to eliminate spurious reflections.

3. Results and discussion

3.1 The Mie resonance in nanopillar arrays: overview

Figure 1 displays representative BFR spectra collected from arrays of nanopillars of 217 nm height with (a) varying nanopillar diameter D, at a constant pitch P = 300 nm, and (b) with varying array pitch at a constant 133 nm pillar diameter. The remarkably close correspondence between these measured spectra and our FDTD simulations are evident by comparing the computed spectra in Figs. 1(c) and 1(d). It should be noted that quantitatively comparable spectra (not shown) were also computed using the full-wave FEM capability of COMSOL. In all cases, a pronounced reflectance resonance can be identified (mode 1) that disperses strongly to longer wavelengths as pillar diameter increases. For larger diameters, a higher order mode (mode 2) appears at bluer wavelengths, and also strongly disperses with diameter. In addition, mode 1 is seen in Figs. 1(b) and 1(d) to comprise more than one nearly degenerate submode component, which appear as a resolved doublet at the smallest pitch of 225 nm. Hereafter, we define the resonant wavelength λr as the minimum in the reflectance spectrum of this complex of modes, which FDTD computations show is associated with a mode internal to the nanopillars, as discussed below.

The quality of a resonance mode can be defined as Q = λ/Δλ, where λ and Δλ are the wavelength and FWHM, respectively, of the resonance band obtained from a Lorentzian fit. On one hand, mode 1 exhibits a quality factor Q that remains roughly constant with increasing diameter and resonant wavelength, Fig. 1(a), with a value of ~5, which must be considered a lower bound due to possible broadening by unresolved components, and due to the inevitable inhomogeneities in nanopillar diameter within a given nanopillar array that would artificially broaden the resonance as well. On the other hand, mode 2 is narrower, with a Q of ~10. This suggests that high-Q, sub-wavelength resonators may be achievable for larger diameterstructures through the introduction of higher order modes, while gaining spectral bandwidth through a lower-Q fundamental mode shifted to longer wavelengths. This is one of the reasons behind the broadband absorption response recently demonstrated by Spinelli et al. [13

13. P. Spinelli, M. A. Verschuuren, and A. Polman, “Broadband omnidirectional antireflection coating based on subwavelength surface Mie resonators,” Nat Commun 3, 692 (2012). [CrossRef] [PubMed]

], and clearly has practical implications for broadband UV-Vis-NIR absorbers, enhanced photovoltaics, and optical emitters and detectors, as the authors discussed. It is also notable that the computed on-resonance reflectance for mode 1 can acquire sub-percent values, which are nearly reached experimentally. Hence nanopillar arrays of this nature may prove fruitful as an alternative approach to achieving “near-perfect absorbers” [48

48. J. Hendrickson, J. Guo, B. Zhang, W. Buchwald, and R. Soref, “Wideband perfect light absorber at midwave infrared using multiplexed metal structures,” Opt. Lett. 37(3), 371–373 (2012). [CrossRef] [PubMed]

51

51. H. Shi, J. G. Ok, H. Won Baac, and L. Jay Guo, “Low density carbon nanotube forest as an index-matched and near perfect absorption coating,” Appl. Phys. Lett. 99(21), 211103 (2011). [CrossRef]

].

The pitch dependences of λr for a number of representative pillar diameters with a height of 217 nm are summarized in Fig. 2(a)
Fig. 2 Characteristics of the Mie resonance wavelengths λr as determined from minima in reflectance spectra. (a) Pitch dependence for arrays of various constant diameters where pillar height was 217 nm. The straight solid lines are linear least squares fits. (b) The dependence on nanopillar diameter of the resonant wavelength, expressed as the wavelength in Si (i.e., λr/nSi) for mode 1 (circles) and mode 2 (squares). The pillars had a height of 272 nm and were arrayed with a constant gap of 188 nm separating the pillar walls. Solid lines through the points are least-squares linear fits forced to include the origin. The dashed curve gives the prediction of an empirical formula from Ref [56]. for the lowest-order hybrid-mode in an isolated cavity.
, together with linear least-squares fits. The non-negligible effect due to the interactions among pillars (discussed below) is evident for each diameter by the observed slope in the pitch dependence. Figure 2(b) displays the diameter dependence of the mode 1 and 2 resonant wavelengths λr by plotting λr/nSi versus diameter D for a series of arrays with pillars of 272 nm height and with a constant gap of 188 nm between the circumferential walls of the pillars. Note that P = D + gap. The size of this gap was kept constant to avoid complications from possible interpillar interactions at small pitch, as mentioned above. Normalizing λr by the wavelength dependent refractive index nSi [52

52. C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83(6), 3323–3336 (1998). [CrossRef]

] (which gives the wavelength in Si) removes the effects of refractive index dispersion. The resonant frequency for both modes is seen to conform to a linear dependence on diameter, as demonstrated by the linear least-squares fits (solid lines) in Fig. 2(b). The linearity of λr/nSi with diameter suggests casting data in a “universal” plot as displayed in Fig. 3
Fig. 3 “Universal” plot of all mode 1 data of Fig. 2. The pillars in arrays with a 188 nm constant gap (open circles) had a height of 272 nm while the data for pillars with varying diameters (filled symbols) belongs to pillars with a height of 217 nm. The dashed line derives from the diameter dependence of 178 nm high pillars arrayed at random on the substrate in order to escape effects of array periodicity (inset).
, which graphs(λr/nSi)/D versus P/(λr/nSi) (i.e., the ordinate is the resonant wavelength in Si divided by pillar diameter, and the abscissa is the array pitch in units of the wavelength in Si). The normalizations cause all mode 1 data of Fig. 2 to cluster in revealing ways (the symbols are retained between the two figures). Also shown in Fig. 3 is an inset plotting (λr/nSi)/D versus D for pillars arranged at random on the Si substrate, which have been used to reveal the resonant properties of arrayed plasmonic nanostructures independent of effects induced by periodicity [26

26. R. Adato, A. A. Yanik, C.-H. Wu, G. Shvets, and H. Altug, “Radiative engineering of plasmon lifetimes in embedded nanoantenna arrays,” Opt. Express 18(5), 4526–4537 (2010). [CrossRef] [PubMed]

,27

27. B. S. Simpkins, J. P. Long, O. J. Glembocki, J. Guo, J. D. Caldwell, and J. C. Owrutsky, “Pitch-dependent resonances and near-field coupling in infrared nanoantenna arrays,” Opt. Express 20(25), 27725–27739 (2012). [CrossRef] [PubMed]

]. The dashed lines in both the inset and main figure mark the average value of 1.2 for (λr/nSi)/D as determined from the random arrays. This value indicates that the resonance condition within the silicon nanopillars for non-periodic arrays occurs at wavelengths that are 1.2 times the nanopillar diameter. This value is also approached in the periodic arrays (filled symbols) in Fig. 3 as pitch increases. In addition, the open circles in Fig. 3 show that the mode 1 resonance for pillars separated by a constant gap between pillar walls appears to oscillate about this same limiting value. These facts taken together suggest that the interpillar interactions play an important role at small pitches, leading to a substantial blue shift in the resonant wavelength.

While dielectric cavities are not in general amenable to simple theoretical treatments [32

32. D. Kajfez and P. Guillon, Dielectric Resonators (Artech House, 1986).

], the close linear dependence of λr/nSi on diameter in Fig. 2 suggests a practical guideline for predicting the Mie resonances of pillars such as these in the form of an eigenvalue expression analogous to that used to describe leaky-mode scattering resonances in infinite cylinders [33

33. L. Cao, J. S. White, J.-S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. 8(8), 643–647 (2009). [CrossRef] [PubMed]

,53

53. H. C. van der Hulst, Light Scattering by Small Particles (Dover Publications, 1981).

,54

54. Y. Yu and L. Cao, “Coupled leaky mode theory for light absorption in 2D, 1D, and 0D semiconductor nanostructures,” Opt. Express 20(13), 13847–13856 (2012). [CrossRef] [PubMed]

] or confined modes in a cylinder with highly reflecting endfaces [32

32. D. Kajfez and P. Guillon, Dielectric Resonators (Artech House, 1986).

] kra=a[(nSik)2kz2]1/2=κ, where kr is the radial wavevector, k is the free space wavevector 2π/λr, a is the cylinder radius, and kz is the wavevector describing the variation in field amplitudes along the cylinder axis (as ordinarily imposed by an external boundary condition). The eigenvalue κ depends on the nature of the mode. For our pillars, kz can evidently be neglected, so that nSika = κ pertains. Evaluating κ from the fits of modes 1 and 2 in Fig. 2(a) gives κ1 = 2.6 and κ2 = 5.6, respectively. Interestingly, these eigenvalues compare to those for two low-order leaky-mode resonances that are excitable in infinite cylinders illuminated at normal incidence, for which kz = 0, and for which κ1 = 2.31 and κ2 = 5.45 for Si in this wavelength range [54

54. Y. Yu and L. Cao, “Coupled leaky mode theory for light absorption in 2D, 1D, and 0D semiconductor nanostructures,” Opt. Express 20(13), 13847–13856 (2012). [CrossRef] [PubMed]

]. Each of these particular infinite-cylinder eigenvalues apply to both an accidentally degenerate transverse magnetic (TM) and a transverse electric (TE) mode, respectively. As indicated by FDTD simulations, discussed next, a near-degeneracy for analogous TM and TE modes in our finite-height pillars may play a role in the doublet observed for mode 1.

3.2 Modal character: FDTD simulations

To more fully characterize the nature of the resonant modes observed in the optical spectra, we examined field patterns computed with FDTD methods. Figures 4(a)
Fig. 4 Optical field patterns on-resonance for mode 1 for various nanopillars. (a) |E|2 for the shorter-wavelength doublet of mode 1 (λr = 597 nm) for a 133-nm diameter pillar (215-nm height) in a tight 225-nm array, for which the spectrum is given in Fig. 1(d). Dashed arrow schematically indicates the direction of (E) in computed field-amplitude plots (not shown), reminiscent of an HEM11 mode in an isolated cavity. (b) |E|2 for the longer-wavelength doublet (λr = 640 nm) showing fields primarily outside the pillar. (c) and (d) show |E|2 for the shorter- (569 nm) and longer- (623 nm) wavelengths of the resonant doublet for an array of pillars in air (125-nm diameter, 215-nm height, 425-nm pitch), respectively. (e) The field componentHy (not intensity squared) corresponding to the mode of 4(a), and showing the strong magnetic-dipolar aspect of this mode. Hy is presented on a horizontal plane through a pillar in a substrate array (100-nm diameter, 215-nm height, 300-nm pitch).
and 4(b) show |E|2 for the short- and long-wavelength modes, respectively, of the doublet observed at the tightest pitch of 225 nm in the spectra of Figs. 1(b) and 1(d) for 133 nm diameter pillars. Additional simulations for larger pitch arrays (not shown) show that the deepest reflection feature is always associated with the pattern of Fig. 4(a), where fields are concentrated within the pillar. Below we discuss how this mode bears strong similarities to the HEM11 magnetic-dipole mode known for dielectric cavities at microwave frequencies. The fields associated with the longer-wavelength component of the doublet predominantly reside exterior to the pillar as plotted in Fig. 4(b), and approximately conform to the field pattern of an electric dipolar mode. Additional computations (not shown) of field patterns for mode 2 reveal that it derives from the same type of internal mode as the short-wavelength component of mode 1, Fig. 4(a), but with an extra node in the radial direction, consistent with the linear dependence on diameter of the resonant frequency as plotted in Fig. 2(b).

To clarify the role of the substrate, we also computed the resonant properties of nanopillar arrays suspended in air. Figure 5(a) shows far-field FEM simulation results for arrays with diameters ranging from 100 to 200 nm (at a constant pitch of 300nm) and Fig. 5(b) presents results for arrays with pitches ranging from 225 to 425 nm (at a constant diameter of 125 nm). The resonant modes of these substrate-free arrays are similar to those of arrays anchored to a substrate in terms of: (i) the resonant wavelength and its strong dependence on diameter, Fig. 5(a)
Fig. 5 Far-field FEM simulation of 215 nm tall nanopillar arrays in air with (a) pitches of 225, 275, 325, 375 and 425 nm at a constant diameter of 125 nm and (b) diameters of 100, 125, 150 and 200 nm at a constant pitch of 300 nm. The prominent resonant peak red-shifts significantly with diameter variation while the shift is only marginal for pitch variation. However, the nearly degenerate resonant components split with increasing pitch from 225 to 425 nm.
; (ii) the occurrence of a submode sensitive to pitch, Fig. 5(b); and (iii) the field patterns, Figs. 4(c) and 4(d). However, the two cases differ in that for the substrate-free array, it is the longer wavelength component of the doublet that has electric-field strength predominantly within the pillar, and it is at larger pitches that the splitting is best resolved. Despite these interesting differences, Fig. 5(b) shows that a substrate-free array exhibits a doublet with acentral wavelength very near the resonant wavelength of pillars of the same diameter and approximate height arrayed on a substrate. For example, the resonance wavelength computed for a 125 nm pillar in air, Fig. 5(b), is centered near 600 nm, while that of a pillar with nearly equal diameter (133 nm) on a substrate is computed to be quite similar, as seen in Fig. 4(a) and 4(b), where the doublets appear at 597 nm and 640 nm. Similarly, in Fig. 2(b), a 125 nm diameter pillar is seen to resonate at λ/nSi~150 nm, which corresponds to λ~617 nm, again very close to the air simulation for a pillar of the same diameter. Evidently, the substrate has only a minor effect on the resonance wavelength.

To further test the similarity of our mode to an HEM11 cavity mode, we plot in Fig. 2(b) the resonant wavelength predicted by an empirical formula from the literature that was derived by fitting the resonance wavelength obtained with numerically computed surface-integral methods [56

56. D. Karfez and A. A. Kishk, “Dielectric resonator antenna: possible candidate for antenna arrays,” Proc. VITEL 2002, International Symposium on Telecommunications: Next Generation Networks and Beyond (2002).

]. This empirical prediction for an isolated cavity falls surprisingly close to our data for cavities on a substrate, and is also essentially linear with the same slope. Furthermore, a weak height dependence is predicted, consistent with our measurements. For example, we find no identifiable influence of height on resonance wavelength in Fig. 3, which compiles data for pillars that varied in height from 178 to 272 nm. Additional FDTD computations (not shown) reveal that the resonant wavelength of these arrays remains remarkably constant, shifting less than ~10 nm even down to pillar heights of ~100 nm (the shortest examined).

As noted above, the simulations surprisingly find that the substrate does not significantly affect the resonance wavelength observed in our nanopillars. However, the substrate clearly plays a significant role in the overall reflective response, as the simulations also show that an array in air produces reflective peaks through resonant scattering, while pillars on the Si substrate produce reflective dips at resonance. For resonant particles on a high index substrate, a contributing cause of this reduction in back-reflected power is the strong forward coupling into the high index substrate, as recently shown by Spinelli et al. [13

13. P. Spinelli, M. A. Verschuuren, and A. Polman, “Broadband omnidirectional antireflection coating based on subwavelength surface Mie resonators,” Nat Commun 3, 692 (2012). [CrossRef] [PubMed]

]. The substrate also affects the lifting of the degeneracy of the submodes comprising mode 1, as shown by comparing |E|2 for substrate-anchored and substrate-free pillars in Fig. 4. While the strongly confined internal mode appears to be pulled toward the substrate, Fig. 4(a), and no doubt radiates in the forward direction into the higher index substrate [13

13. P. Spinelli, M. A. Verschuuren, and A. Polman, “Broadband omnidirectional antireflection coating based on subwavelength surface Mie resonators,” Nat Commun 3, 692 (2012). [CrossRef] [PubMed]

,14

14. S. A. Mann, R. R. Grote, R. M. Osgood, and J. A. Schuller, “Dielectric particle and void resonators for thin film solar cell textures,” Opt. Express 19(25), 25729–25740 (2011). [CrossRef] [PubMed]

,31

31. L. Cao, P. Fan, A. P. Vasudev, J. S. White, Z. Yu, W. Cai, J. A. Schuller, S. Fan, and M. L. Brongersma, “Semiconductor nanowire optical antenna solar absorbers,” Nano Lett. 10(2), 439–445 (2010). [CrossRef] [PubMed]

,35

35. A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B 82(4), 045404 (2010). [CrossRef]

], evidently the internal junction between the pillar and the substrate presents a boundary sufficiently reflective to support a dielectric cavity mode quite similar to the lowest hybrid mode in an isolated cylinder. In fact, it appears that interpillar interactions are at least as strong as any influence from the substrate, especially for arrays with smaller pitch. In addition, the field profiles in Figs. 4(a)-4(d) suggest that the substrate plays a role in enhancing of the confinement of the mode, either internally or externally to the nanopillar. The substrate-free modes appear somewhat less localized than the modes on the substrate.

While we have drawn connections between the character of our observed resonances and the character of various known resonant modes in dielectric structures such as magnetic Mie resonances, transverse modes in infinite-cylinders, and microwave cavities, we note that further investigation is needed to clarify more completely how best to describe the set of nearly degenerate components comprising mode 1. Such analysis is beyond the scope of the present manuscript.

3.3 Interparticle interactions: effects of array density

Interparticle interactions in arrays of nanoplasmonic resonators have drawn widespread interest [27

27. B. S. Simpkins, J. P. Long, O. J. Glembocki, J. Guo, J. D. Caldwell, and J. C. Owrutsky, “Pitch-dependent resonances and near-field coupling in infrared nanoantenna arrays,” Opt. Express 20(25), 27725–27739 (2012). [CrossRef] [PubMed]

,57

57. E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003). [CrossRef] [PubMed]

62

62. M. Meier, A. Wokaun, and P. F. Liao, “Enhanced fields on rough surfaces: dipolar interactions among particles of sizes exceeding the Rayleigh limit,” J. Opt. Soc. Am. B 2(6), 931 (1985). [CrossRef]

], but have received relatively little attention in arrays of dielectric nanoresonators. In plasmonic arrays, the interaction is usually treated by considering, at the location of any given particle, the total self-consistent field produced by both the incident radiation and the sum of all fields emanating from the surrounding array of resonators. The net effect, which can be coherently enhanced in periodic arrays, is to produce shifts in the resonant frequency and alterations of the quality factor Q. In analytical treatments, the resonators themselves are approximated as radiating dipoles, which can be especially suitable for plasmonic particles because, in principle, the dipole resonance is supported even as the particle size approaches zero. In contrast, dielectric resonators require a minimum size ~λ/2n to support an internal standing wave. Nonetheless, for high-index materials like silicon, where λ/2n can be considerably less than the free space wavelength, pitch-dependent effects akin to plasmonic arrays have recently been predicted for periodic arrays of dielectric spheres in air and periodic arrays of tellurium nanocubes on BaF2 substrates [63

63. J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108(9), 097402 (2012). [CrossRef] [PubMed]

] resonating in the lowest magnetic- and electric-dipole Mie modes [35

35. A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B 82(4), 045404 (2010). [CrossRef]

]. For our arrays, the criterion allowing a strict dipole approximation is likely not met [35

35. A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B 82(4), 045404 (2010). [CrossRef]

], and in any event a detailed quantitative analysis is beyond the scope of this work. Here we qualitatively describe the measured effects on resonant wavelength induced by array density.

As noted above, significant effects on the spectra of our nanopillar dielectric resonances [Fig. 1(b)], occur as pitch is varied. In addition to changes in resonant lineshape, the resonant wavelength λr for a given pillar diameter shifts blue by up to 15-20% as pitch is reduced, as is evident in Fig. 2(a). When plotted as in the universal graph of Fig. 3, the shift appears to accelerate as the array density increases. Such a blue shift is characteristic of periodic arrays when in the “evanescent” regime, which occurs when radiation at the resonant wavelength diffracts into a propagating wave in-plane, with evanescent tails out of plane [60

60. B. Lamprecht, G. Schider, R. T. Lechner, H. Ditlbacher, J. R. Krenn, A. Leitner, and F. R. Aussenegg, “Metal nanoparticle gratings: influence of dipolar particle interaction on the plasmon resonance,” Phys. Rev. Lett. 84(20), 4721–4724 (2000). [CrossRef] [PubMed]

62

62. M. Meier, A. Wokaun, and P. F. Liao, “Enhanced fields on rough surfaces: dipolar interactions among particles of sizes exceeding the Rayleigh limit,” J. Opt. Soc. Am. B 2(6), 931 (1985). [CrossRef]

]. Since, for normal incidence, this diffraction condition is given by P≤λr/n, it can in principle pertain either to interactions mediated through air (n = 1), or to interactions mediated through both air and Si. For all arrays investigated here, for which λr/nSi<Pr/nair, the evanescent regime condition is met for any interactions mediated through air. Note that the constant gap data presented in Fig. 3 (open circles) appears immune to the overall blue-shift observed at small pitch for constant diameter data (filled symbols), which suggests that the overall blue shift might be dominated by a gap-mediated effect (as opposed to a specific periodicity effect), as would be the case if coupling originated with lateral evanescent fields that arise from satisfying boundary conditions in Maxwell’s equations, or from laterally confined cavity modes. However, our study does not permit unambiguously distinguishing periodic- from gap-mediated interactions.

Finally, as noted previously, we have discovered that array density affects differently the resonant properties of each submode comprising mode 1. For example, as seen in Figs. 1(b) and 1(d), the submode resonant wavelengths and amplitudes disperse differently with pitch, leading to a resolved doublet at the smallest pitch investigated. The substrate evidently plays a role as well, since the simulations for substrate-free pillars show an overlap of submodes at the smallest pitch, Fig. 5(b). A role of the substrate in moderating air-mediated interactions is not surprising, given the report of a gradual suppression of diffractive coupling between periodically-arrayed plasmonic Au nanoparticles [41

41. W. L. Auguie, X. M. Bendaña, W. L. Barnes, and F. J. García de Abajo, “Diffractive arrays of gold nanoparticles near an interface: critical role of the substrate,” Phys. Rev. B 82(15), 155447 (2010). [CrossRef]

] as the distance was reduced between the particles and a higher index substrate underneath. Therefore, it is anticipated that interparticle coupling effects are also directly tied to the asymmetry of the refractive index of the surrounding environment, and thus must be taken into account for any device design.

3.4 µ-Raman signal enhancements

The reflectance spectra discussed above reveal the far-field optical response to cavity resonances in the nanopillar arrays. In order to probe local field intensities, we measured the ~521 cm−1 Si µ-Raman scattering intensity as a function of nanopillar diameter at 514 nm excitation, as presented in Fig. 7(a)
Fig. 7 Si Raman enhancement (solid blue) measured on arrays of 272 nm tall pillars under (a) 514 nm (green dashed line) and (b) 785 nm (red dashed line) incident, normalized to the intensity of the Si substrate, as a function of nanopillar diameter. The top axis corresponds to the array pitch values. The Raman intensities are maximal in both cases when the resonance spectral position of mode 1 (solid red line) coincides with the corresponding excitation wavelength. (c) |E|2 simulation results for an array of nanopillars with a constant gap of 190 nm and diameters of 109 (left) and 200 nm (right) at 514 nm incident.
, and at 785 nm excitation as displayed in Fig. 7(b). These measurements employed the same array as used for Fig. 2(b), with the interpillar gap fixed at 188 nm and a nanopillar height of 272 nm. In Fig. 7(a), the Raman scattering enhancement (blue circles) exhibits a distinct maximum for a nanopillar diameter of ~109 nm, where the Mie resonance wavelength of mode 1 (red circles) coincides with the pump wavelength (horizontal dashed line). We attribute this enhancement to the resonant increase in the internal optical fields in the nanopillars. This is clearly indicated by field intensity distributions from the simulations under 514 nm incident, shown in Fig. 7(c), corresponding to arrays of nanopillars with diameters of 109 nm (left, for mode 1) and 200 nm (right, for mode 2), a height of 272 nm and a gap of 190 nm. On resonance, a ~30-fold overall enhancement of the Si µ-Raman scattering signal intensity is observed at 514 nm incidence, relative to the intensity measured from the unpatterned substrate. This enhancement is approximately 3 times the enhancement obtained off-resonance, i.e. from nanopillar arrays for which the resonant mode is not supported near the 514 nm incident laser wavelength. The enhancement from non-resonant arrays may arise from an improved escape probability for photons originating in the substrate between pillars and scattered by the nanopillar texture [64

64. A. I. Zhmakin, “Enhancement of light extraction from light emmiting diodes,” Phys. Rep. 498(4–5), 189–241 (2011). [CrossRef]

], or originating within the pillars themselves and guided preferentially away from the substrate [65

65. T. M. Babinec, B. J. M. Hausmann, M. Khan, Y. Zhang, J. R. Maze, P. R. Hemmer, and M. Loncar, “A diamond nanowire single-photon source,” Nat. Nanotechnol. 5(3), 195–199 (2010). [CrossRef] [PubMed]

]. For larger diameters, 190 – 200 nm, the Raman intensity increases 15-fold relative to the surrounding unpatterned silicon, due to the coincidence of the laser wavelength with the resonance condition of mode 2. This corresponds to a 2x increase in Raman intensity over that which was observed from arrays with resonances spectrally detuned from the incident pump laser wavelength. The resonant enhancements in the Raman signals for both, modes 1 and 2, are attributable to the resonantly enhanced field strengths within the nanopillars, as indicated by the corresponding plots of |E|2 obtained from FDTD field simulation, Fig. 7(c). It is important to make a distinction between this mechanism and that reported in Ref [38

38. M. Khorasaninejad, N. Dhindsa, J. Walia, S. Patchett, and S. S. Saini, “Highly enhanced Raman scattering from coupled vertical silicon nanowire arrays,” Appl. Phys. Lett. 101(17), 173114 (2012). [CrossRef]

]. for periodic arrays of Si nanowires (0.8-1.1 µm in height), where the increase in enhancement from larger diameter nanopillars was attributed to interference effects within the arrays and not due to a higher order cavity resonance as reported here.

4. Conclusion

In conclusion, we have shown that periodic arrays of Si nanopillars interact with incident electromagnetic radiation to produce Mie resonances that are consistent with dielectric nanocavities. This leads to the confinement of a large portion of the incident EM field within the subwavelength Si nanopillars as evidenced by FDTD and FEM simulations as well as µ-Raman measurements. The resonances are observed to blue-shift linearly with decreasing nanopillar diameter, and less so with decreasing array pitch. The strong dependence of the resonance wavelength on the diameter of the Si nanopillars indicates that the local Mie resonance and the resonance-enhanced extinction in the high index Si nanopillars dominate the overall optical properties of the arrays. The pitch-induced shifts can be considered as the relatively weak, coherent coupling between these Si nanopillar Mie resonators, similar to effects studied widely for arrays of plasmonic particles. In addition, the dependence of these resonances on the nanoparticle and array parameters demonstrates that these resonances can be tuned to preferred wavelengths with specific linewidths and bandwidth for a given application. This can be useful in the development of photonic and plasmonic/photonic hybrid devices such as higher-efficiency photovoltaic solar cells and enhanced optical emitters and detectors.

Acknowledgments

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T. R. Jensen, M. D. Malinsky, C. L. Haynes, and R. P. Van Duyne, “Nanosphere lithography : tunable localized surface plasmon resonance spectra of silver nanoparticles,” J. Phys. Chem. B 104(45), 10549–10556 (2000). [CrossRef]

60.

B. Lamprecht, G. Schider, R. T. Lechner, H. Ditlbacher, J. R. Krenn, A. Leitner, and F. R. Aussenegg, “Metal nanoparticle gratings: influence of dipolar particle interaction on the plasmon resonance,” Phys. Rev. Lett. 84(20), 4721–4724 (2000). [CrossRef] [PubMed]

61.

D. Weber, P. Albella, P. Alonso-González, F. Neubrech, H. Gui, T. Nagao, R. Hillenbrand, J. Aizpurua, and A. Pucci, “Longitudinal and transverse coupling in infrared gold nanoantenna arrays: long range versus short range interaction regimes,” Opt. Express 19(16), 15047–15061 (2011). [CrossRef] [PubMed]

62.

M. Meier, A. Wokaun, and P. F. Liao, “Enhanced fields on rough surfaces: dipolar interactions among particles of sizes exceeding the Rayleigh limit,” J. Opt. Soc. Am. B 2(6), 931 (1985). [CrossRef]

63.

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108(9), 097402 (2012). [CrossRef] [PubMed]

64.

A. I. Zhmakin, “Enhancement of light extraction from light emmiting diodes,” Phys. Rep. 498(4–5), 189–241 (2011). [CrossRef]

65.

T. M. Babinec, B. J. M. Hausmann, M. Khan, Y. Zhang, J. R. Maze, P. R. Hemmer, and M. Loncar, “A diamond nanowire single-photon source,” Nat. Nanotechnol. 5(3), 195–199 (2010). [CrossRef] [PubMed]

66.

F. J. Lopez, J. K. Hyun, U. Givan, I. S. Kim, A. L. Holsteen, and L. J. Lauhon, “Diameter and polarization-dependent Raman scattering intensities of semiconductor nanowires,” Nano Lett. 12(5), 2266–2271 (2012). [CrossRef] [PubMed]

OCIS Codes
(140.4780) Lasers and laser optics : Optical resonators
(290.4020) Scattering : Mie theory
(350.4238) Other areas of optics : Nanophotonics and photonic crystals
(050.6624) Diffraction and gratings : Subwavelength structures

ToC Category:
Photonic Crystals

History
Original Manuscript: July 26, 2013
Revised Manuscript: October 1, 2013
Manuscript Accepted: October 17, 2013
Published: November 4, 2013

Citation
Francisco J. Bezares, James P. Long, Orest J. Glembocki, Junpeng Guo, Ronald W. Rendell, Richard Kasica, Loretta Shirey, Jeffrey C. Owrutsky, and Joshua D. Caldwell, "Mie resonance-enhanced light absorption in periodic silicon nanopillar arrays," Opt. Express 21, 27587-27601 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-23-27587


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