Letter pubs.acs.org/journal/apchd5
Subdiffraction Confinement in All-Semiconductor Hyperbolic Metamaterial Resonators Kaijun Feng,† Galen Harden,† Deborah L. Sivco,‡ and Anthony J. Hoffman*,† †
Department of Electrical Engineering, University of Notre Dame, Notre Dame, Indiana 46556, United States Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, United States
‡
S Supporting Information *
ABSTRACT: The strong optical anisotropy of hyperbolic metamaterials has enabled remarkable optical behavior such as negative refraction, enhancement of the photonic density of states, anomalous scaling of resonators, and super-resolution imaging. Resonators fashioned from these optical metamaterials support the confinement of light to dimensions much smaller than the diffraction limit. These ultrasmall resonators can be used to increase light−matter interactions for new applications in photonics. Here, we present subdiffraction mid-infrared resonators based on all-semiconductor hyperbolic metamaterials. Importantly, these resonators are fully compatible with epitaxial growth techniques and can be engineered to incorporate quantum well intersubband transitions that are degenerate with the mode of the resonators, enabling an entirely new generation of quantum optoelectronic devices. The strongest optical confinement achieved is λ/33 for a free-space wavelength of 10 μm, and the measured Q-factors are in the range of 14−17. The dispersion of the resonance mode is presented through both experimental data and numerical solutions, and greater than 10% tuning of the resonance frequency (106 cm−1) is demonstrated. Radiation patterns and radiative Q-factors are also mapped out using experimental results. Finally, the resonator structures are investigated with finite element simulations and the field profile indicates the presence of a strong vertical polarization, which is essential for coupling to intersubband transitions in quantum well structures. These extreme subdiffraction resonators could be useful for engineering novel light-matter interactions and devices in the mid-infrared. KEYWORDS: Hyperbolic Metamaterial, Nanoresonators, Semiconductor Metamaterial, Mid-Infrared
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Recently, there has been significant interest in designing optoelectronic devices that strongly couple mid-infrared (midIR) photons in resonators to intersubband transitions (ISTs) in quantum well heterostructures.13−16 These light−matter interactions are enhanced when the mode volume is strongly confined and can be used to engineer the effective spontaneous emission rate of ISTs, which can be further enhanced by the large photonic density of states in HMMs.11,13,17,18 Subwavelength metallic structures patterned on the surface of a substrate have been used to confine optical modes to small volumes; however, coupling of the confined field to the resonant metallic structures increases optical loss and can limit the interaction of the optical mode with quantum structures in the underlying dielectric material.13,19 Strong subdiffraction confinement of the mode over the full volume of the HMMs is possible because the hyperbolic dispersion allows highmomenta extraordinary rays. Subdiffraction confinement has been demonstrated in Type II HMM resonators comprising alternating subwavelength layers of Ag and Ge9 and in Types I and II HM resonators fabricated in hBN.8 For the Ag/Ge
ptical metamaterials incorporate subwavelength inhomogeneities in order to engineer the interaction of light with the material. Often this engineered interaction enables control of the optical field that cannot be realized, or is difficult to realize, using conventional materials. For example, optical metamaterials with strong anisotropy have garnered attention because they support intriguing optical behavior such as negative refraction,1−3 subdiffraction imaging,4−7 subdiffraction confinement,8,9 and enhanced photonic density of states.10,11 This optical behavior is enabled by an optical permittivity tensor with principal components of opposite sign, which results in hyperbolic dispersion.3 Because of this unique dispersion, the materials are referred to as hyperbolic metamaterials (HMMs). HMMs have been engineered in a variety of material systems including semiconductor heterostructures and metal/dielectric stacks,1−3 and recently, hexagonal boron nitride (hBN) was demonstrated as a naturally occurring hyperbolic material (HM).8 While naturally occurring materials offer benefits such as lower-loss and stronger optical confinement,8,12 the ability to engineer HMMs using semiconductor-based technologies opens a path toward a new generation of optical devices. © XXXX American Chemical Society
Received: March 27, 2017 Published: June 19, 2017 A
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HMM resonators, the strongest reported confinement was λ/12 and the quality factors, Q, were approximately 4. While the confinement and Q of the hBN resonators exceeded those of the HMM resonators (λ/86 and Q = 283, respectively), engineering optoelectronic devices with hBN is much less developed than engineering with III−V semiconductors, and incorporating ISTs into these materials has not been demonstrated. In this manuscript, we demonstrate subdiffraction confinement of mid-IR light in all-semiconductor, Type I HMM resonators. We investigate Type I HMM resonators because the optical loss is lower than Type II HMMs1 and Type I resonators are able to support a strong vertically polarized electric field. The all-semiconductor HMM is grown via molecular beam epitaxy (MBE) and the III−V semiconductor materials employed in this work are routinely used for engineering IST-based active regions in mid-IR quantum cascade lasers.20 For resonators fabricated in the HMM, the strongest confinement, λ/33, is significantly larger than previous work on HMM resonators,9 and the maximum Q ∼ 17 is an improvement of a factor of 4. Additionally, the resonators support strong vertical polarization of the electric field, which is crucial for coupling to intersubband transitions, over almost the entire resonator volume. The bulk HMM is realized by growing interleaved, subwavelength layers of Si-doped n+-InGaAs and i-AlInAs via MBE on an intrinsic InP substrate. Each layer is 50 nm thick and the total thickness of the HMM is 1 μm. While InGaAs/ AlInAs heterostructures can form quantum wells, the thick InGaAs regions ensure that the ISTs energies (20.1 meV or 162 cm−1 for the first two levels) are far away from the frequency of the subdiffraction localized modes (approximately 1000 cm−1). As such, coupling between the localized mode and ISTs is not expected for these engineered resonators. To determine the effective permittivity of the HMM, the unpatterned HMM is characterized by measuring angledependent transmission spectra for transverse magnetic (TM) and transverse electric (TE) polarized light, as shown in Figure 1a. The dip in transmission around 1190 cm−1 for the TM polarization is due to an increase in the absorption coefficient, and the minimum of the dip occurs when the permittivity of the isotropic InGaAs layer, εInGaAs, is 0.1 εInGaAs = 0 at the plasma frequency of the InGaAs layers, ωp,InGaAs. Therefore, the minimum in the transmission is used to extract the free-carrier density of the sample, nd. For the HMMs in this work, nd depends on the spatial location of the measurement on the wafer, and we measure values that range from 7.8 to 8.3 × 1018 cm−3 with the larger nd toward to the center of the wafer. The principal components of the effective permittivity tensor, ε̂, are calculated using an effective medium theory with frequencydependent permittivities for the n+-InGaAs and i-AlInAs layers and the recovered nd.21,22 The permittivity for the n+-InGaAs layers includes effects due to free-carriers and optical phonons, while the permittivity for the i-AlInAs layers includes only optical phonons. Figure 1b shows the calculated principal components of ε̂ for TM polarized light and nd = 8.3 × 1018 cm−3, which was recovered from the transmission measurements in Figure 1a. The extraordinary component, εe, is normal to the surface of the sample and in the growth direction, and the ordinary component, ε0, is in the growth plane as shown in the inset of Figure 1a. Two regions of strong anisotropy exist where the signs of the principal components of ε̂ are opposite. From 886
Figure 1. (a) Measured transmission spectra of the HMM for transverse electric (dashed, blue) and transverse magnetic (solid) polarizations at several incident angles. Orientations of TE and TM are defined in the inset. (b) Derived real parts of the extraordinary (e) and ordinary (o) components of the hyperbolic metamaterial (HMM). The free carrier density of InGaAs nd = 8.3 × 1018 cm−3. Inset shows schematic of the isofrequency contour for Type I HMM in the pertinent region.
to 1190 cm−1, the HMM exhibits Type I hyperbolic dispersion (shaded in blue), where Re(εe) < 0 and Re(εo) > 0. The spectral location and bandwidth of this region can be controlled via the doping of the InGaAs layers and the ratio of the InGaAs to AlInAs layer thicknesses.1,21,23 Arrays of subdiffraction resonators of various size and pitch are fabricated in the HMM using standard fabrication techniques. Negative resist electron-beam lithography (EBL) is used to define the resonator arrays and the full thickness of the HMM is etched using inductively coupled plasma reactive ion etching (ICP-RIE).24,25 Figure 2a is a scanning electron microscope (SEM) image of an array of 0.47 μm wide resonators with a 2.5 μm pitch. The inset is a magnified image of a single resonator. The width of the resonator is defined as the average of the top, midpoint, and bottom widths of the structure. In the manuscript, we refer to the arrays as width × pitch. We fabricate and characterize 25 arrays with different doping densities and resonator/array geometries. Coupling to localized modes is observed in angle-, wavelength-, and polarization-dependent reflection and transmission measurements using a Fourier transform infrared spectrometer (FTIR). Figure 2b is a schematic of specular reflection on a resonator array and shows the definition of the TM and TE polarizations. The incident beam is focused to a spot with a diameter of approximately 500 μm, covering 10000 to 250000 resonators depending on the array geometry. Figure 3a shows the measured TM reflectivity for four different resonator arrays at an incident angle θi = 45°. The recovered free-carrier density for the unpatterned HMM as described previously is the same for the four arrays, nd = 8.3 × 1018 cm−3. For each of the arrays, a minimum in the reflectivity is observed around 1100 cm−1 due to coupling to a localized subdiffraction mode in the resonator. Additionally, coupling to a higher-order B
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resonators arrays with two different free-carrier densities. Contrary to conventional optical resonators, as the size increases for these HMM resonators, the resonance frequency also increases. This anomalous scaling rule is a result of the closing of the hyperbolic isofrequency contour (Figure 1b, inset) as frequency increases. Since all the resonators have the same height, kz of the confined mode remains constant. As the width of the resonator increases, kx decreases and results in a shift to higher frequency (see Supporting Information). Additionally, for a fixed array geometry, increasing the doping density increases the frequency of the modes, as is shown in Figure 3b. For the arrays reported in this work, a tuning range of 67 cm−1 is achieved by changing the width of the resonator only and 106 cm−1 when accounting for free-carrier density and geometry. The tuning of the resonance is limited by both fabrication and the hyperbolic dispersion. At small aspect ratios (thinner resonators), the undercut of the dry etch becomes a significant issue and limits the smallest dimension of the resonators. This limitation can be improved by optimizing the fabrication process. For wider resonators, the resonance frequency approaches the epsilon-near-zero (ENZ) point of εe, thus, pinning the resonance frequency (see Supporting Information). To understand the frequency tuning with aspect ratio and free carrier density, we numerically solved the resonance condition for spheroid subwavelength particles:8,26
Figure 2. (a) SEM images of fabricated hyperbolic metamaterial resonator array (0.47 × 2.5 μm). (b) Schematic of reflection measurement of the arrays.
⎛ 1⎞ εj = εm⎜⎜1 − ⎟⎟ Lj ⎠ ⎝
(1)
Here, εj and Lj are the permittivity and geometric factor in a certain axis direction, respectively; and εm is the permittivity of the surrounding medium. Lj is calculated as a function of the aspect ratio. Since the resonator is surrounded by both InP and air, not a single medium as in eq 1, εm is treated as an effective permittivity. We use εm as a fitting parameter in a least-squares fitting algorithm with the measured resonance frequency versus aspect ratio. The experimental data and recovered dispersion curves with εm = 1.61 are plotted in Figure 3b. Despite the obvious difference in geometric shape of the fabricated resonators and approximated spheroids, there is good agreement between the experimental results and calculations. The localized modes are studied using finite element simulations in COMSOL Multiphysics. Here, the resonators are modeled as rectangular structures (Figure 4a) with an anisotropic permittivity. A comparison of simulated (dashed blue) and measured reflection spectra (solid blue) for a 0.35 × 2 μm array is shown in Figure 3a. As with experiment, there are two dips in the transmission (920 and 1042 cm−1) that correspond to coupling to localized modes. Except for the difference in mode strength, there is excellent agreement between the measurement and the simulation, and other simulated arrays agree similarly well with the experimental results. We attribute the difference in mode strength primarily to imperfections in the fabrication process and differences between the simulated and actual resonator geometry. For potential applications in mid-infrared cavity quantum electrodynamics, an important consideration for these resonators is the vertical component of the electric field, Ez, which can be calculated using the numerical models.15,27,28 Due to selection rules, only E z couples to an intersubband transition.29,30 Figure 4a shows the magnitude of Ez in a cross-section through the center of a resonator for a 0.91 × 5
Figure 3. (a) Measured and simulated TM reflection spectra of four resonator arrays. Arrays are labeled with the width × pitch. Spectra of 0.71 × 4, 0.91 × 5, and 1.11 × 6 μm arrays are shifted for 0.05 in reflectivity for clarity. Dashed line is the simulated spectrum of 0.35 × 2 μm array. (b) Summary of resonance frequencies of 25 different resonator arrays with different sizes and doping levels. Solid lines represent analytical calculation results for dispersion. The aspect ratio is defined as width to height.
mode is observed in the spectra as the broad, low-energy shoulder centered around 950 cm−1. Figure 3b shows the measured resonance frequencies of the highest energy mode as a function of aspect ratio (width to height) for 25 different C
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Figure 4. (a) Cross-sectional plot of simulated Ez distribution for TM (0,0,1) and TM (2,2,1) mode, respectively, in a 0.91 μm wide resonator. (b) Simulated volume-averaged |Ez| as a function of resonator geometry. Legends shows different resonator height. The inset compares the strength of Ez and Ey components relative to normalized E field.
Figure 5. (a) Measured angle dependent transmission spectra of the 0.46 × 2 array. The incident angles range from 0 to 70° with 5° spacing. (b) Measured 1/Qrad,θ of three different resonator arrays, plotted in a polar coordinate system. The radiation patterns of the Hertzian dipole resonances are mapped out based on least-squares fitting. The data points are plotted in different quadrants for clarity. Simulated charge distribution of a 0.46 μm resonator is shown in the center.
μm array at the frequencies that correspond to the two dips in the measured reflection spectra, Figure 3a. Using the naming convention from ref 8, we label the mode near 1100 cm−1 as TM (0,0,1) and higher energy mode near 950 cm−1 as TM (2,2,1). The Ez field is the strongest for the TM (0,0,1) mode. Additionally, characteristic cross-hatch patterns that are due to the hyperbolic dispersion are observed in the Ez plots for both modes.8 Additional simulations of the orientation of the electric field for the TM (0,0,1) mode are carried out for various resonator geometries. Figure 4b shows the volume-averaged vertical component of the electric field ⟨|Ez|⟩vol inside of the resonators as a function of resonator width and height. For a resonator of fixed height, ⟨|Ez|⟩vol increases as the resonator width decreases. A factor of 8 improvement in ⟨|Ez|⟩vol is possible by reducing the resonator width from 1.1 to 0.2 μm for a 0.4 μm tall resonator. The magnitude of ⟨|Ez|⟩vol can also be increased by reducing the height of the resonator; this effect is strongest for narrow resonators. These trends of increasing ⟨|Ez|⟩vol can be understood by considering the charge distribution on the surface of the resonators. As seen in the charge density plots in Figure 5b and in the Supporting Information, the surface charge distribution is similar to a dipole and is largest around the edges of the top and bottom of the resonators. As the width of the resonator is decreased, the average charge density increases, resulting in a stronger vertical component of the field, Ez. Additionally, decreasing the height of the resonator reduces the distance between the top and bottom surface charges, which also results in a stronger Ez field. The inset of Figure 4b compares |Ez| and |Ey| to the total electric field |E0| for a 0.35 μm wide resonator, indicating a strong vertical field over the majority of the resonator volume. To evaluate the effective mode volume Veff of the resonators, we calculate the volume where the optical field is greater than half of the maximum value.13 We find that Veff is always approximately half of the physical volume of the resonators. For example, for a 0.35 μm resonator, Veff is calculated to be 0.057 μm3 (see Supporting Information).
Finally, we analyze coupling to the optical modes and demonstrate that the radiation pattern of the subdiffraction resonators is similar to that of Hertizian dipole antennas. Figure 5a shows the measured angle-dependent transmission spectra of a 0.46 × 2 μm array from 0 to 70°. The Q-factors of the resonators are obtained by measuring the full width at halfmaximum (fwhm) of the transmission spectra. For all resonators, the quality factor, Q, is evaluated to be within the range of 14−17, which is a factor of 4 larger than previous work on Type II HMM resonators.9 To further investigate the radiation pattern of the resonators, coupled mode theory is utilized to decompose the total Q as 1/Q = 1/Qrad + 1/Qabs, where Qrad is the radiative component of Q and Qabs is the absorptive component Q that arises mainly from optical loss associated with free-carrier absorption.9,31 For all of our HMM resonators, Q is dominated by Qabs, which is verified by the following Qrad calculations. We map out the radiation pattern of several resonator arrays using the angle-dependent transmission measurement. We first extract the angular components of the radiative quality factor, Qrad,θ. Figure 5b plots 1/Qrad,θ versus angle for three resonators (symbols) and a fit to the characteristic sin2(θ) radiation pattern of a Hertzian dipole antenna.32 The Hertzian radiation pattern describes the measured results well. To verify this result, we calculate the charge density distribution of a 0.46 μm wide resonator using COMSOL and plot the results in the center of the radiation patterns. A strong longitudinal subwavelength dipole is found at the resonance frequency. In general, the Hertzian dipole is a better approximation for high aspect ratio resonators in that the radiation pattern is more consistent with the sin2(θ) distribution and the radiated power is larger. Finally, to estimate Qrad, we calculate the radius of the D
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radiation pattern of an equivalent isotropic radiator with the same radiation power (see Supporting Information). The results are summarized in Table 1. Qrad is found to be close to Qrad,55°. A similar result can also be readily deduced from the gain of Hertzian dipoles G = 1.5.33
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AUTHOR INFORMATION
Corresponding Author
Table 1. Calculated Qrad for Various Resonator Arrays
*E-mail: ajhoff
[email protected].
array dimensions (μm)
Qrad
ORCID
0.46 × 2 0.61 × 3 0.81 × 4
155 192 217
Author Contributions
Kaijun Feng: 0000-0001-9079-2731 The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.
Our work demonstrates subwavelength mid-infrared resonators using all-semiconductor Type I HMMs. The characteristic traits of HMM resonators are observed, including the anomalous scaling rule and subdiffraction confinement. The resonators support fields that are strongly polarized in the vertical direction and are promising for increasing light-matter interactions with ISTs in engineered quantum wells. The Q of the resonators are an improvement over previous Type II HMM resonators due to reduced optical absorption. The radiation pattern of the resonators is studied using the angular components of Qrad, and the well-known Hertzian dipole radiation pattern is mapped out. Future work with these HMM resonators includes resonantly coupling the extreme subdiffraction optical mode to intersubband transitions that are integrated into the HMM design. These extreme subdiffraction HMM resonators provide a promising approach for engineering novel light-matter interaction and optical devices in the midinfrared regime.
Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported in part by NSF ECCS-1508961. REFERENCES
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METHODS Fabrication. The interleaved layers of n+-InGaAs and iAlInAs are grown on undoped InP substrate by MBE. Each layer is 50 nm and 10 pairs of InGaAs/AlInAs layers are grown in total. The negative e-beam resist HSQ is used to define the resonator arrays with EBL. Following the developing process at an elevated temperature, a Cl2 based ICP etch is utilized to remove the entire InGaAs/AlInAs layers where it is not covered by the HSQ patterns. FTIR Spectroscopy. A Bruker Vertex 80 V FTIR spectrometer is used to collect the reflection and transmission spectra in this work. A Globar is used as the mid-infrared light source and a Thallium Bromoiodide (KRS-5) holographic wire grid polarizer is used to create the TM and TE polarized light. The reflected or transmitted light is collected using a liquid nitrogen cooled mercury cadmium telluride (MCT) detector. For reflection measurements, the spectra are normalized to the reflection spectra of a gold mirror. Numerical Simulations. Finite element simulations of the resonators arrays are performed using the RF module of the commercial software COMSOL Multiphysics. Each component of the simulation consists of a single resonator along with the periodic boundary condition. The effective permittivity calculated with effective medium theory is used to describe the electromagnetic property of the resonators.
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The dispersion of HMM resonators, calculation of charge density and effective mode volume, derivation of coupled mode theory and radiative Q-factor (PDF).
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsphotonics.7b00309. E
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