1. Introduction

Photonic crystals (PCs) are materials that consist of periodically varying permittivity and/or permeability [1, 2]. In conventional photonic crystals, the structure of the crystal is fixed at fabrication time, which results in designable though not tunable optical properties. There has been a great deal of effort to make photonic crystals tunable. In particular, tunability at visible frequencies is desirable for many applications. Many tuning mechanisms have been investigated: In [3] liquid crystals were used to modify the bandgap of a three-dimensional photonic crystal, in [4] mechanical deformation was used to achieve a tunable superlens, and in [5] a microelectromechanical system was used to deform a photonic crystal in a waveguide to create a tunable filter. In this article we consider soft photonic crystals whose structure may be deformed by different actuation schemes. The implementation we discuss comprises freestanding vertically aligned nanowires (NWs) as building blocks of a photonic crystal. We focus on a design where the NWs are connected to a voltage source, thereby making it possible to bend them electrostatically, which will change the crystal structure. This is an example of a nano-optoelectromechanical system (NOEMS) that is closely related to nanoelectromechanical systems (NEMS) that have been studied extensively previously, e.g. in [6, 7].

Many different types of NWs can be considered as the elementary building elements. Vertically aligned arrays of free-standing carbon nanofibres have been fabricated by many groups, see e.g. [8]. Carbon nanofibre or nanotube arrays have been discussed previously as building blocks for static photonic crystals [9]. Other possible material platforms include zinc oxide nanowires, investigated in [10] and dielectric or semiconducting NWs such as GaAs [11]. The lengths of different NWs can vary from hundreds of nanometres to several microns, and the diameters fall in the range 30–100 nm. The lattice constant is limited by the fabrication techniques of the arrays, but typically center-to-center distance of a few hundred nanometers is achievable using catalytic growth of nanowires from substrates prepared e.g. by electron beam lithography.

In this article we focus on the representative cases of dielectric nanowires (ε_r=10) and on perfect metallic (PEC) nanowires. We do not consider magnetic structures, hence µ_r=1. We investigate the optical behaviors of one-, two- and three-dimensional structures as a function of lattice deformation. One-dimensional structures are only used as a guide for interpreting the results in higher dimensions. The two- and three-dimensional geometries are modeled using a finite-difference time-domain (FDTD) method [12].

2. Modeling

We start by considering the one-dimensional Kronig-Penney model with a dielectric function as shown in Fig. 1. The dielectric constant ε(x) is periodic with a period 2a (the unperturbed case d=0 has a lattice constant of a). To get a qualitative idea of how a shift d in the lattice basis affects the dispersion relation, we determine the dispersion relation using the transfer matrix method [13]. To simplify the expression we let the width of the scatterers b→0 while keeping ε₂b=C=constant. This yields the implicit dispersion relation

where c denotes the speed of light, n ₁=√ε 1 is the refractive index and k the Bloch wave vector. For C=0.5a, n ₁=1 and d=0, 0.01a, 0.05a the resulting dispersion relations are depicted in Fig. 2.

 

Fig. 1. The 1D model considered in this paper. A one-dimensional Kronig-Penney model with a basis of two scattering centers where every second is displaced a distance d. Gray denotes every second row in the unperturbed system. The dashed rectangle indicates a unit cell.

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Fig. 2. Result from the transfer matrix analysis of the Kronig-Penney model. Solid line is Re(k) and dashed is Im(k). Blue line is for d=0, red d=0.01a, green d=0.05a.

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For the unperturbed case, d=0, there is no bandgap at k=π/2a, which is expected since the lattice period is a and the first gap is expected to appear at k=π/a, which in the reduced zone becomes k=0. For small values of d we can see a small gap opening at k=π/2a. This is depicted by k becoming imaginary for frequencies in the gap. The higher frequency gaps that are closed for d=0 at k=π/2a can be seen to open for d>0. The effect of the shift is much stronger at higher frequencies and k obtains a large imaginary part. The increasing sensitivity to a shift can be seen from equation (1) where the disturbance appears only as the combination 2dω/c.

In two and three dimensions, we consider two ways to displace the NWs as depicted in Fig. 3(a) and (b): the actuation is either in the propagation direction (longitudinal case), or perpendicular to the direction of propagation (transverse case). This gives rectangular lattices with unit cells a×2a (transverse, a being the unperturbed lattice constant) and 2a×a (longitudinal). Cylinder radius is set to r=0.05a. This correspond to a realistic situation for a photonic crystal made from NWs designed to work in the visible region. In the remaining we set a=500 nm and r=25 nm. We use an FDTD algorithm (in the MEEP implementation [14]) to investigate a setup consisting of m rows of dielectric or metallic NWs, see Fig. 3 and 4. The computational grid is terminated by perfectly matched layers (PMLs) in the direction of propagation to avoid artifacts caused by reflections from the edges of the computational cell. In the transverse direction we use periodic boundary conditions to simulate an infinite system in this direction. For the three-dimensional case the vertical direction is terminated by PMLs where it is appropriate. The systems are excited by current sources with Gaussian frequency spectra on the left and the flux is recorded to the right of the scatterers as indicated in Fig. 3 and 4. A normalization run, with recorded flux S₀, is done where no NWs are present. This is used to normalize the fields to yield a measure of the transmission coefficient T=S_PC/S₀, where SPC is the flux from a computation including the NWs. We investigate the E-polarized case, where the electric field is parallel to the NWs, since this is the most sensitive one for a PC consisting of rods, and focus on a system with 10 rows of NWs in the direction of light propagation for the two-dimensional and 4 rows for the three-dimensional case.

 

Fig. 3. The 2D models considered. Green lines denote sources, blue lines regions of observation, gray areas PMLs, red cylinders represent bent NWs and gray cylinders represent unperturbed NWs. Figure (a) show the case where the cylinders are shifted along the direction of propagation, Fig. (b) show the case where the cylinders are shifted perpendicular to the direction of propagation. Periodic boundary conditions are used in the direction perpendicular to light propagation.

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Fig. 4. The 3D model considered. The green plane denote source, blue plane regions of observation, gray areas PMLs, red cylinders represent bent NWs and gray cylinders unperturbed NWs. The transparent top rectangle correspond to the lid.

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Most of our simulations have been carried out in two dimensions. Since bending of the NWs results in a lattice deformation that varies along the axes of the wires, it is necessary to perform three-dimensional simulations to assure that the tunability survives the non-uniform deformation and to confirm that the effect is sufficiently strong in a typical experimental setup. In the three dimensional simulations we set the NWs height to H=1 µm. The substrate is simulated by a PEC. To confine the light signal to the vicinity of the photonic crystal in the vertical direction, a PEC lid is placed a distance g=300 nm above the tips of the NWs. The substrate covers the whole computational cell. The system is excited by a uniform field which is recorded at different heights, h_i=100–900 nm above the substrate. The lid extends distances a/2 outside the first and last NW in the PC slab. The current source extends from the substrate to the lid. Only metallic NWs are considered in three dimensions.

The bending of the NWs in the three-dimensional simulations is described by the Euler-Bernoulli beam equation

$\frac{d^2}{d{x}^2}\left(EI\frac{d^2}{d{x}^2}w\left(x\right)\right)=P\left(x\right)$,

where E denotes the Young’s modulus and I the second moment of inertia of the NWs. The load P(x) is evaluated by solving the Poisson equation to determine the charge distributions in the NWs in the presence of applied external voltages. The two equations are coupled: the beam equation determines the device geometry which influences the charge distributions and, consequently, the bending forces on the NWs. For metallic NWs most of the charge is located in the tip of the NW and the load is therefore concentrated to the tip, P(x)≈F₀δ(x-H). The NW profile is then approximately given by the analytic expression

This is the profile used for the NWs in the three-dimensional simulations. The main approximation in the electromechanical analysis is that the cross sections and Young’s moduli of the NWs are independent of bending, which is valid if the minimal radius of curvature is large, i.e. the maximum deflection is small compared to the NW length.

3. Results

In the two-dimensional model we consider a lattice distortion that changes the crystal symmetry from a simple square to rectangular with a basis. This can be done in two ways, either bending the NWs in the direction of propagation, or orthogonal to the direction of propagation.

The bandstructure for a rectangular PC (an infinite lattice) corresponding to the system depicted in Fig. 3 with dielectric constant ε=10 and r=0.05a can be obtained using the plane wave expansion method. Similarly to what is seen in one dimension, there are bandgaps opening at the new Brillouin zone edge at the X_x point, see Fig. 5 where a as before is set to 500 nm. However, the effect is much weaker than in one dimension. This is expected from the fact that the filling fraction of the scatterers is small. In Fig. 5 the first three gaps that appear are depicted. We denote them by I (λ≈2000 nm, corresponding toω2a/2π c≈0.5), II (λ≈710 nm, corresponding to ω2a/2π c≈1.4) and III (λ≈550 nm, corresponding to ω2a/2π c≈1.8). As in one dimension, gaps open faster for higher frequencies. Therefore we focus on frequencies above the first band gap in two and three dimensions.

 

Fig. 5. The opening of the first three bandgaps at the Brillouin zone edge in the X_x point in a rectangular lattice of size 2a×a. The bandstructure is calculated using the plane wave expansion method for a PC with ε=10, r=0.05a and a=500 nm. The lower inset show a closeup of the first gap and the right inset the Brillouin zone with high symmetry points marked.

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A lattice distortion in the direction of light propagation is seen to have similar effects in two-dimensional crystals as in the simple one-dimensional model for the bandstructure. The transmission for the longitudinal case, as computed with FDTD, through photonic crystals with ten rows of NWs are shown in Fig. 6 for metallic scatterers and in Fig. 7 for dielectric scatterers (ε=10). The decrease in transmission that appears for both metallic and dielectric systems around 600 and 700 nm respectively is related to the opening of bandgap II in Fig. 5.

Note that the two systems show quite similar sensitivity to the disturbance even though they have very different dielectric functions. The appearance of the dip at different wavelengths is due to the difference in dielectric constant in the two systems. The drop in transmission around 1050 nm in the dielectric and 900 nm in the metallic system is present in all systems. It is located at a wavelength that indicates that it is the first bandgap in the unperturbed system. In the one-dimensional case this gap was very large, but in two dimensions it only shows as a small decrease in transmission in the dielectric case. In the metallic systems, however, the gap causes a drop of four orders of magnitude in the transmission. In both the dielectric and metallic system this gap closes with the shift, similar to what can be seen in one dimension. The widths of the gaps O and II (see Fig. 5–7) as functions of the shift, d, are shown as insets in Fig. 6 and Fig. 7. The gap widths are normalized by the center frequencies of the gaps in the unperturbed systems and the width is defined as FWHM[log₁₀(T)], where FWHM denotes the full width at half minimum.

The results for a transverse shift in a metallic system are depicted in Fig. 8. The bandgap that opens in the longitudinal direction around λ=600 nm does not open at all in the transverse case: the Brillouin zone edge does not change in the x-direction, and hence a gap is not expected to appear here. In the first ordinary bandgap, where λ is approximately between 700 and 1000 nm there is a new feature appearing. The transmission minimum exhibit a surprising behavior as a function of the transverse lattice deformation: as the deformation increases, the relatively wide minimum of the unperturbed system splits up into two sharp minima with a peak separating them. Sharp edges like this that move with the shift are interesting features to use in applications with e.g. tunable filters.

 

Fig. 6. Transmission through a PC for different lattice deformations in the longitudinal direction from a two dimensional simulation. The lattice is composed of ten rows of NWs modeled as perfect metals with a diameter of 50 nm and a lattice spacing of 500 nm. The curves correspond to a shift d (see Fig. 3(a)) of: 0 nm (blue), 20 nm (green), 40 nm (red), 60 nm (light blue), 80 nm (magenta). The curve is plotted against vacuum wavelength instead of frequency to make the interpretation more clear. The inset shows the gap width divided by gap center frequency at zero shift as a function of the shift for the first ordinary gap, denoted O (blue), and the second new gap, denoted II (green).

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Fig. 7. Transmission through a PC for different lattice deformations in the longitudinal direction from a two dimensional simulation. The lattice is composed of ten rows of NWs modeled here as dielectrics with ε=10 with a diameter of 50 nm and a lattice spacing of 500 nm. The curves correspond to a shift d (see Fig. 3(a)) of: 0 nm (blue), 20 nm (green), 40 nm (red), 60 nm (light blue), 80 nm (magenta). The curve is plotted against vacuum wavelength instead of frequency to make the interpretation more clear. The inset shows the gap width divided by gap center frequency at zero shift as a function of the shift for the first ordinary gap, denoted O (blue), and the second new gap, denoted II (green).

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Fig. 8. Transmission through a PC for different lattice deformations in the transverse direction from a two dimensional simulation. The lattice is composed of ten rows of NWs modeled here as perfect metals with a diameter of 50 nm and a lattice spacing of 500 nm. The curves correspond to a shift d (see Fig. 3(b)) of: 0 nm (blue), 20 nm (green), 40 nm (red), 60 nm (light blue), 80 nm (magenta). The curve is plotted against vacuum wavelength instead of frequency to make the interpretation more clear.

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To understand this unexpected behavior we investigate how transmission depends on the thickness of the PC slab. In a bandgap the transmission drops exponentially with PC thickness, while a propagating mode in the PC is not affected by the thickness. In Fig. 9 transmission at wavelengths λ ₁≈950 nm (a wavelength longer than where the peak in the gap appears, see Fig. 8) and λ ₂≈830 nm (a wavelength within the new peak, that is, a wavelength affected by the shift in the lattice, see Fig. 8) in PC slab systems with d=80 nm is displayed against PC slab thickness. The plots are normalized to transmission 1 in systems consisting of 4 rows of NWs for clarity. It can be seen that for λ₁ the decrease in transmission is exponential, while λ₂ is more or less not affected by the PC width. λ2 is thus not located in a band gap. These wavelengths are propagating undamped in the PC. The small constant decrease in transmission for frequencies around λ₂ seen in Fig. 8 is instead due to coupling between the air mode and the mode in the PC.

Encouraged by the results found in the two-dimensional simulations we consider now the full three-dimensional nature of the problem, Fig. 4. We choose the constant F₀/2EI in equation (2) so that the tips of the NWs are displaced a distance d to compare to the two-dimensional system. When the flux recording regions are placed 500 nm away from the last NW and at different heights, h_i, the results are depicted in Fig. 10. As expected, the transmission closest to the substrate is not much affected by the bending. At h=500 nm it can be seen that the system is quite sensitive to bending and transmission increases several times. Higher up, the transmission again is not very affected. This is also expected as there is a 300 nm gap between the tip of the NWs and the metallic lid. It can further be noted that close to the substrate surface the transmission curves are quite smooth and show strong suppression at well defined frequencies. Even though the PC slab is only four NWs thick there is almost a 100-fold change in transmission between the bottom of the stop band and the pass bands. Close to the tips of the NWs, the curves are irregular and the two stop bands are hardy resolvable at all. In the middle regions at 300 nm height the dip around λ=440 nm is still resolvable but only slightly affected by the bending of the NWs. The wider gap between λ=350 nm and 390 nm is, however, more affected and the transmission increase. At 500 nm height the narrow stop band around λ=440 nm is no longer detectable but the wide gap is still there and also quite strongly affected by the bending.

 

Fig. 9. Transmission as a function of PC width in propagation direction in two dimensional simulation for wavelengths λ ₁≈950 nm (a wavelength longer than where the peak in the gap appears, see Fig. 8, blue) and λ ₂≈830 nm (a wavelength within the new peak, that is, a wavelength affected by the shift in the lattice, see Fig. 8, green) in PC slab systems with a perpendicular shift of d=80 nm. Transmission is normalized to 1 for a system of 4 NWs. For a system of around 20 NWs the transmission is down at the noise floor of the computation and does not decrease further with wider PCs.

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Fig. 10. Results from a three dimensional simulation of 4 PEC NWs for the flux recorded at different heights, h_i above the substrate and different bending of the NWs. Blue is no bending, green 20, red 40, light blue 60 nm in the tip. Unperturbed lattice constant is 500 nm.

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The performance of the device we consider can be characterized by the voltage required to electrostatically bend the NW and the speed limit with which this can be done. The required voltage for metallic nanowires can be estimated to be approximately 20 V [15]. The tuning speed is limited by the resonance frequency of the NW. It can be estimated e.g. for carbon nanofibres via the cylinder cantilever model where

$f=\frac{1}{\sqrt{2\pi }}\frac{r}{H^2}\sqrt{\frac{E}{\rho }}$,

where ρ is the density, which has been reported to be in the range 0.015–1.8 g cm^-3 [16] and E is the Young’s modulus. With the values used in the simulation, a Young’s modulus of E=300 GPa and a density of 1.8 g cm^-3 we obtain f≈73 MHz as an upper bound on the frequency. The switching speed is determined by the Q factor of the NW resonator. A too high Q-value makes the NWs vibrate which is not desirable. This can be controlled electrically e.g. by a dissipating element that lowers a too high Q-value. Based on recent measurements [17] we estimate that a switching time of [f/Q]^-1≈1 µs is readily achievable.

For dielectric NWs, the load P(x) arises fromNW polarization, and similar changes in geometry require larger applied voltages than in the case of metallic NWs.

4. Discussion

We have shown that the transmission of light at optical frequencies can be controlled by electrostatically deforming NWs in a PC made from vertically aligned NWs. The sensitivity of the system can be understood schematically from a simple one-dimensional model which can be solved analytically to yield the dispersion relation. Here it is seen that higher frequencies are stronger affected by a displacement. In a realistic system there will be a trade-of between sensitivity to tuning and sensitivity to defects, which limits the maximum frequency.

Many of the features in the transmission in two dimensions can be ascribed to changes in the band diagram. From an application point of view many features appear very promising: sharp edges in some parts of the transmission that shift with changing deformation and transmission that drop very much with a small shift are attractive features for filters and switches.

In three dimensions we see a strong dependence in the transmission on the height above the substrate where the detection is made. If detection is carried out in the middle region, around 500 nm height above the substrate in our system with 1 µm NWs, sensitivity is maximized in the system. This is a combined effect: At a larger distance from the substrate the system is insensitive due to the air gap between the tips of the NWs and a confining lid which allows the field to penetrate. At lower height the deflection of the NWs is small and transmission is left unchanged. In a measurement situation this sensitivity can be handled either by high precision detection or by a more elaborate system design, but we estimate the change to be well detectable in an experimental situation. The switching speed and voltage requirement are within a technologically interesting range.

We like to thank Yury Tarakanov, Caroline Huldt, Peter Apell and Andreas Isacsson for stimulating discussions. We acknowledge financial support by the Swedish foundation for Strategic Research (SSF).