Abstract:
A mid-infrared tunable metamaterial comprises an array of resonators on a semiconductor substrate having a large dependence of dielectric function on carrier concentration and a semiconductor plasma resonance that lies below the operating range, such as indium antimonide. Voltage biasing of the substrate generates a resonance shift in the metamaterial response that is tunable over a broad operating range. The mid-infrared tunable metamaterials have the potential to become the building blocks of chip based active optical devices in mid-infrared ranges, which can be used for many applications, such as thermal imaging, remote sensing, and environmental monitoring.

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application claims the benefit of U.S. Provisional Application No. 61/286,560, filed Dec. 15, 2009, which is incorporated herein by reference. 
     This application is a continuation-in-part of U.S. patent application Ser. No. 12/965,659, filed Dec. 10, 2010, the benefit of which is hereby claimed and which is incorporated herein by reference. 
    
    
     STATEMENT OF GOVERNMENT INTEREST 
     This invention was made with Government support under contract no. DE-AC04-94AL85000 awarded by the U.S. Department of Energy to Sandia Corporation. The Government has certain rights in the invention. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates to electromagnetic metamaterials and, in particular, to metamaterials that are tunable in the mid-infrared portion of the electromagnetic spectrum. 
     BACKGROUND OF THE INVENTION 
     The ability of metamaterials to create artificial electromagnetic properties absent in nature has initiated intense research efforts for applications in frequency selective surfaces, sub-diffraction imaging, cloaking, and etc. See S. Linden et al., “Magnetic response of metamaterials at 100 terahertz,”  Science  306, 1351 (2004); X. Zhang, and Z. Liu, “Superlenses to overcome the diffraction limit,”  Nature Materials  7, 435 (2008); and J. Valentine et al., “An optical cloak made of dielectrics,”  Nature Materials  8, 568 (2009). The development of tunable metamaterials, which allow for real-time tuning of the electromagnetic response, is emerging as an important sub-topic in this field. Tunable metamaterials have the potential to become the building blocks of chip-based active optical devices, and as optical switches, modulators, and phase shifters. A typical way to make such tunable metamaterials is to integrate a natural reconfigurable material in the metamaterial structure and apply an external stimulus to achieve tuning. For example, tunable metamaterials have been demonstrated using electrical reorientation in liquid crystals and thermally/electrically induced insulator-to-metal phase transition in vanadium dioxide (VO 2 ). See D. H. Werner et al., “Liquid crystal clad near-infrared metamaterials with tunable negative-zero-positive refractive indices,”  Optics Express  15, 3342 (2007); M. J. Dicken et al., “Frequency tunable near-infrared metamaterials based on VO 2  phase transition,”  Optics Express  17, 18330 (2009); T. Driscoll et al., “Dynamic tuning of an infrared hybrid-metamaterial resonance using vanadium dioxide,”  Applied Physics Letter  93, 024101 (2008); and T. Driscoll et al., “Memory Metamaterials,”  Science  325, 1518 (2009). 
     Recently, active terahertz metamaterials based on variants of split-ring resonators (SRRs) on a doped gallium arsenide (GaAs) substrate have been realized by dynamically changing the carrier concentration of the underlying semiconductor using an electric bias voltage, which effectively tunes the strength of the resonance, producing an amplitude modulation effect or a phase modulation. This amplitude modulation is a result of “shunting” due to the presence of carriers in the doped substrate. See H.-T. Chen et al., “Active terahertz metamaterial devices,”  Nature  444, 597 (2006); H.-T. Chen et al., “A metamaterial solid-state terahertz phase modulator,”  Nature Photonics  3, 148 (2009); and U.S. Pat. No. 7,826,504 to Chen et al. However, all of these references disclose electrically tunable metamaterials at only terahertz frequencies (i.e., 0.1-3 THz). 
     Therefore, a need remains for a metamaterial that is tunable in a higher (e.g., infrared) spectral range. 
     SUMMARY OF THE INVENTION 
     The present invention is directed to a tunable metamaterial, comprising a doped semiconductor substrate having a large dependence of dielectric function on the carrier concentration and a semiconductor plasma resonance lying below an operating frequency range; an array of resonators on the doped semiconductor substrate; and an electrical circuit for applying a bias voltage between the doped semiconductor substrate and the resonator array for modulating the carrier concentration of the semiconductor substrate and tuning the resonance of the resonator array over the operating frequency range. In general, the resonator array can be tunable over a range of operating frequency range of 100 THz to 15 THz (wavelength range of 3-20 μm). For example, the semiconductor substrate comprises InSb, InAs, GaAs, GaSb, GaN, or Si that is doped n-type. In general, the resonator array can comprise a ring-like structure with one or multiple splits or a wire-like structure in a connected arrangement, such as a split-ring resonator, a cut-wire pair, or a fishnet-like structure, that is scalable through most of the short- to long-infrared. The metamaterial can further comprise a gate dielectric layer, such as HfO 2  or SiO 2 , between the resonator array and the doped semiconductor substrate. 
     The mid-infrared tunable metamaterials have the potential to become the building blocks of chip based active optical devices in mid-infrared ranges, which can be used for many applications, such as thermal imaging, remote sensing, and environmental monitoring. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The accompanying drawings, which are incorporated in and form part of the specification, illustrate the present invention and, together with the description, describe the invention. In the drawings, like elements are referred to by like numbers. 
         FIG. 1(   a ) is a top-view schematic illustration of a split-ring resonator (SRR) on a semiconductor substrate.  FIG. 1(   b ) is a side-view schematic illustration of the metamaterial. A voltage bias is applied between the metal resonator and doped semiconductor substrate to control the substrate carrier concentration.  FIG. 1(   c ) is an equivalent circuit of the electrically tunable metamaterial. 
         FIG. 2(   a ) is a graph of the real component of the complex dielectric function of indium antimonide (InSb) as a function of wavelength with different doping.  FIG. 2(   b ) is transmission spectra of SRRs on InSb substrates with different carrier concentration. The curves are the simulated transmission spectra of SRRs on an InSb substrate at carrier concentrations of 1×10 16  cm −3 , 1×10 17  cm −3 , 5×10 17  cm −3 , 1×10 18  cm −3 , and 2×10 18  cm −—3 . The inset shows the resonance wavelength of metamaterial versus carrier concentration in the InSb substrate. 
         FIG. 3(   a ) is a scanning electron microscopic image of an array of SRRs fabricated on an InSb substrate.  FIG. 3(   b ) is a graph of the resonance amplitude of the SRR array fabricated on the InSb substrate at the doping level at 5×10 17  cm −3  versus the intersection angle between incident light polarization direction and the gap of the split-ring resonator. The dots are the experimental measured values. The curve is a fit proportional to cos 2  θ.  FIG. 3(   c ) is the transmission spectra for the metamaterial with SRRs fabricated on the reference InSb wafer, and the other three doped substrates. The polarization direction of the incident light is parallel to the SRR gap.  FIG. 3(   d ) is a graph of the resonance shift when the carrier concentration is increased from the intrinsic level. The curve shows the results from a finite element simulation. The dots are the experimentally measured values. 
         FIG. 4  is a side-view schematic illustration of an electrically tunable metamaterial that uses a gate dielectric. 
         FIG. 5  provides a schematic cross-sectional view of an exemplary split-ring resonator array as described herein. 
         FIG. 6  provides results of theoretical calculations of the real and imaginary parts of the dielectric constant in n-doped GaAs for selected wavelengths in the range 6-14 μm. 
         FIG. 7  provides the results of our theoretical calculation of the depletion width as a function of the bias voltage in an exemplary device as described herein. 
         FIG. 8  provides a perspective view of one example of a fabricated device including a split ring resonator array, looking from the top. The figure also indicates, in schematic fashion, the application of a bias voltage between inner and outer contact regions. 
         FIG. 9  provides a plan view of one example of a unit cell of a split ring resonator array. The double-headed arrow in the figure indicates the orientation of the electric field of the incident polarized light. 
         FIGS. 10A and 10B  provide experimental results describing the performance of the device of  FIG. 9 . 
         FIGS. 11A ,  11 B, and  11 C provide further examples of a unit cell of a split ring resonator array. 
         FIG. 12  provides a plan view of a repeating array of unit cells of the type shown in  FIG. 11B . 
         FIGS. 13A ,  13 B, and  13 C provide theoretical transmission spectra, obtained from numerical simulations, for an asymmetric SRR device as described above, made with respectively the unit cells of  FIGS. 11A ,  11 B, and  11 C. In each figure, the transmittance is plotted versus the wavenumber for each of three bias voltages, namely 0V, −2V, and −5V. A negative sign indicates reverse bias. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1(   a ) shows a top-view schematic illustration of an exemplary SRR  11  on a semiconductor substrate  12 .  FIG. 1(   b ) is a side-view schematic illustration of an electrically tunable metamaterial  10 . A voltage bias is applied by an electrical circuit  15  between the metal resonator  11  and the doped semiconductor substrate  12  to control the substrate carrier concentration. The doped semiconductor substrate  12  can comprise a doped layer  13  grown on a semi-insulating substrate  14 . The metamaterial can further comprise an additional symmetric resonator on the backside of the doped layer or semiconductor substrate. In general, the tunable metamaterial can comprise an array of such resonators on the frontside and/or backside of the doped layer or semiconductor substrate. 
     As shown in  FIG. 1(   c ), the resonance of a SRR on a semiconductor substrate can be explained by an equivalent circuit model, where the ring inductance is described by L, the split gap and fringing capacitance are denoted by C, the resistor R models the dissipation in the split rings, and the resistor R d  models the dissipation due to the substrate free carrier absorption within the split gap (and in the regions in close proximity to the metal lines). See S. Linden et al., “Magnetic response of metamaterials at 100 terahertz,”  Science  306, 1351 (2004); and H.-T. Chen et al., “Active terahertz metamaterial devices,”  Nature  444, 597 (2006). When the metal lines act as gates, the underlying carrier concentration, in principle, can be changed by the application of an electrical bias affecting thus both R d  and C. The resistance R d  is inversely proportional to the AC conductivity of the doped semiconductor material due to free carrier losses. See P. Ikonen and S. Tretyakov, “Determination of generalized permeability function and field energy density in artificial magnetic using the equivalent-circuit method,”  IEEE Transactions on Microwave Theory and Techniques  55, 92 (2007). The capacitance C is related to the dielectric function of the semiconductor substrate due to the field lines fringing into the material. See L. I. Basilio et al., “Equivalent circuit models and optimization of a split-ring resonator,” IEEE International Symposium on Antennas and Propagation and USNC-URSI National Radio Science Meeting (2009); and J. F. O&#39;Hara et al., “Thin-film sensing with planar terahertz metamaterials: sensitivity and limitations,”  Optics Express  16, 1786 (2008). The resistance change in R d  influences the resonance strength while the capacitance change in C shifts the resonance frequency. 
     For terahertz tunable metamaterials, it is critical to choose a semiconductor substrate with the same plasma frequency as the resonance frequency of metamaterial resonator. Therefore, in previous tunable metamaterial work with doped GaAs, the SRRs were designed to work at terahertz frequencies and the plasma frequency of the doped GaAs substrate matched this designed frequency, the resistance change of R d  was the dominant mechanism that modified the resonance. See H.-T. Chen et al., “Active terahertz metamaterial devices,”  Nature  444, 597 (2006). Therefore, mainly an amplitude modulation of the resonance was observed, but a phase change was also possible. 
     For the mid-infrared tunable metamaterials of the present invention, the underlying semiconductor substrate preferably has a large dependence of dielectric function on the carrier concentration and the built-in semiconductor plasma resonance (set by material properties and carrier density) lies below the mid-infrared metamaterial operating range. The capacitance change (through a change in the real part of the dielectric function) becomes the dominant mechanism, resulting in a resonance shift instead of an amplitude modulation when the carrier concentration of the semiconductor substrate varies. 
     The resonance shift is determined by both the metamaterial geometry and the local dielectric environment. In the underlying semiconductor, the presence of free carriers can be described by the Drude model: 
               ɛ   =       ɛ   ∞     ⁡     [     1   -       ω   p   2         ω   2     +     i   ⁢           ⁢   ω   ⁢           ⁢     τ     -   1               ]         ,       ω   p     =       4   ⁢           ⁢   π   ⁢           ⁢     Ne   2           ɛ   x     ⁢     m   *                 
where ω p  is the plasma frequency, N is the carrier concentration, ∈ ∞  is the high frequency dielectric constant of the semiconductor, m* is the effective mass in the semiconductor, and τ is the scattering time. The model is based on treating electrons as damped harmonically bound particles subject to external electric fields. In general, any semiconductor that can achieve a plasma frequency near the desired metamaterial operating range with a damping constant on the order of the reciprocal operating frequency can be considered. The combination of low effective mass and high carrier concentration provides a high plasma frequency. The damping time in the semiconductor also impacts the amount of the effect that free carriers can have on the dielectric constant, where longer damping times lead to both greater tuning and lower losses. For operation near 10 μm wavelength, InSb-based semiconductors have a small electron effective mass and can produce useful effects with carrier concentrations on the order of 10 18  cm −3 . However, the Drude formalism is universal such that the semiconductor can generally also comprise indium arsenide (InAs), gallium arsenide (GaAs), gallium antimonide (GaSb), and gallium nitride (GaN)-based compound semiconductors, as well as silicon-based semiconductors.
 
     In order to provide a further tuning upon the application of a voltage, it is necessary to deplete this charged layer in an appropriate manner. Schottky contacts are difficult to implement at these doping levels and thus a metal-dielectric-semiconductor architecture can be used, very much like in an MOS transistor. 
     Various types of resonator elements can be used, including ring-like structures with one or multiple splits or wire-like structures in some connected arrangement, such as split-ring resonators (SRRs), cut-wire pairs (CWPs), or fishnet-like structures. 
     As an example of the present invention, InSb was used a substrate because it has a large dependence of dielectric function on doping levels, thereby enhancing this tuning effect through a change in the dielectric function of the substrate. This effect has been used previously for tunable subwavelength hole arrays, photonic crystals, etc. See B. S. Passmore et al., “Mid-infrared doping tunable transmission through subwavelength metal hole array on InSb,”  Optics Express  17, 10223 (2009); and W. Zawadski, “Electron transport phenomenon in small-gap semiconductors,”  Advances in Physics  23, 435 (1974). The exemplary mid-infrared tunable metamaterials were based on metallic split-ring resonators fabricated on doped InSb. Finite element simulations and measured transmission data showed that the resonance blue shifts when the semiconductor electron carrier concentration was increased while keeping the split ring geometry constant. A resonant wavelength shift of 1.15 μm was achieved by varying the carrier concentration of the underlying InSb epilayer from 1×10 16  cm −3  to 2×10 18  cm −3 . Therefore, active tuning of metamaterials in the mid-infrared can be achieved using metallic metamaterial resonators fabricated on semiconductor substrates having a large dependence of dielectric function on carrier concentration (e.g., doped InSb). 
     The real component of InSb&#39;s dielectric function at mid-infrared frequencies as a function of carrier concentration is displayed in  FIG. 2(   a ). This was calculated from published data using the Drude model, which includes scattering and measured values for carrier concentration dependent mobility and effective mass. See J. Manzanares-Martinez et al., “Temperature tuning of two-dimensional photonic crystals in the presence of phonons and a plasma of electrons and holes,”  Physical Review B  72, 035336-1 (2005); and E. Litwin-Staszewska et al., “The electron mobility and thermoelectric power in InSb at atmosphere and hydrostatic pressures,”  Physica Status Solidi  ( b ) 106, 551 (1981). As seen in this figure, the real part of InSb&#39;s dielectric function decreases as the carrier concentration increases and this trend becomes more significant at longer wavelengths. According to the LC circuit model, a decrease of the real component of the substrate&#39;s dielectric function will decrease the capacitance, and therefore shift the resonance to a higher frequency (i.e., a shorter wavelength). 
     A finite-element frequency domain solver was used to simulate the behavior of split-ring resonators on InSb substrates with varying carrier concentrations. The substrate comprised a 100-nm n-type doped layer grown on a semi-insulating InSb wafer. Gold SRRs were scaled from known designs such that the main resonance occurred at λ=10 μm (arm length of 660 nm, arm width of 130 nm, gap of 100 nm, and thickness of 80 nm). The computed wavelength dependent dielectric functions of InSb at different doping levels were used for the simulation. For the dielectric function of Au, a fitted Drude model based on ellipsometric data measured in the mid-infrared regime was used, with a plasma frequency of 1.27×10 16  rad/s and a collision frequency of 66 THz. A unit cell boundary condition was used to include the coupling effect between split-ring resonators, and the lattice constant between adjacent resonators was set as 1.34 μm. 
       FIG. 2(   b ) shows the simulated transmission spectra of metamaterials at normal incidence when the polarization direction of the excitation light was parallel to the gap as shown in the inset and the carrier concentration of the doped InSb layer was varied. The simulated metamaterial resonances (displayed as transmission minima) occur in the mid-infrared range. More importantly, the resonances shift monotonically from 11.9 μm to 10.4 μm when the carrier concentration increases from 1×10 16  cm −3  to 2×10 18  cm −3  (inset of  FIG. 2(   b )) due to a change in the bias voltage. Such resonance shifts are due to the differences in the dielectric functions of substrates when the carrier concentration in the doped layer varies as discussed above. There is very little change in the Q-factor of these resonances as the doping is varied, indicating that there is little effect of resistance shunting as observed at THz frequencies. See H.-T. Chen et al., “Active terahertz metamaterial devices,”  Nature  444, 597 (2006). Another way to look at the resonance shift is to study the transmission change at a specific wavelength. When the doping is varied from 2×10 18  cm −3  to 1×10 16  cm 3 , the metamaterial transmission at 10.4 μm changes from 12% to 52%, which corresponds to a modulation depth of 62.5%. No resonance was observed in the simulated transmission spectra of metamaterials when the polarization direction of the excitation light was orthogonal to the gap, as expected. 
     For the experimental study, metamaterial samples were fabricated on four different InSb substrates. One of the substrates, a (111) lightly doped InSb wafer without the doped epilayer, was chosen as a reference. The other three substrates consisted of a thin n-type doped layer grown on the reference wafer by molecular beam epitaxy. The thicknesses of the doped layers in the three substrates were 150 nm, 150 nm, and 750 nm, respectively. The corresponding carrier concentrations of the doped layers were 2×10 17  cm 3 , 5×10 17  cm −3 , and 2×10 18  cm −3 . The carrier concentrations were determined from doped InSb layers grown on SI—GaAs substrates by Hall measurements using the van der Pauw method at room temperature. The split-ring resonators were patterned on InSb substrates using standard nanofabrication techniques including electron-beam lithography, metal deposition, and lift-off. The metamaterial elements were patterned with a period of 1.34 μm to form a planar array of 2×2 mm 2 . The sample was spin coated with polymethylmethacrylate and baked at 170 degree for 30 minutes. The split-ring structures were exposed using an electron beam lithography system operating at 100 kV and 1 nA beam current. The dose used for the small structures was around 1000 μC/cm 2 . Electron beam evaporation was used to deposit 100 Å and 700 Å of Ti and Au, respectively. Lift-off was conducted in an acetone bath. A representative scanning electron microscope image of a split-ring resonator is shown in  FIG. 3(   a ). To minimize the scattering from surface roughness during the transmission measurement, the backside of the samples was polished using a grinder-polisher. 
     Transmission spectra of the fabricated metamaterials were measured using a Fourier-transform infrared spectrometer. Samples were analyzed at room temperature using a liquid-nitrogen cooled mercury cadmium telluride detector. A spectral resolution of 1 cm −1  was used and the data were averaged over 100 scans. An polarizer was placed in front of the sample so that polarization-dependent transmission could be recorded from θ=0 to 90 degrees in an increment of 15 degrees, where e is the intersection angle between incident light polarization direction and the gap of the split-ring resonator (i.e., θ=0 degree represents a polarization direction parallel to the gap; θ=90 degree represents a polarization direction orthogonal to the gap).  FIG. 3(   b ) shows the amplitude of the resonance at different intersection angles (measured on the sample with a carrier concentration of 5×10 17  cm −3 ) and reveals that the amplitude of the resonance varies as cos 2  θ, which represents the projection of the excited light intensity in the direction parallel to the gap. Such polarization dependence is consistent with the simulation results as discussed above and is indicative of excitation of the LC resonance of SRRs. 
     The position of the LC resonance is found to be strongly dependent on the carrier concentration of the semiconductor substrate.  FIG. 3(   c ) shows the normalized transmission spectra of the four metamaterial samples used in this study. Each spectrum was taken with incident light polarization parallel to the gap. A clear blue shift of the transmission peak is seen as the doping level of the InSb substrate is increased. The experimentally observed trend of the metamaterial resonance as a function of substrate doping is consistent with both the LC circuit model analysis as well as the finite element simulation. This trend is plotted in  FIG. 3(   d ) together with the simulation results. As shown, the experimentally observed shift is in very good agreement with the results from finite element simulations. The slight difference may be attributed to a number of factors, including the varying thickness of the doped layers and an uncertainty in the measured carrier concentrations. 
     As shown in  FIG. 1 , the carrier concentration can be modified by applying a voltage bias between a metallic Schottky gate connected to the array of resonator elements and an ohmic contact to the semiconductor substrate. A potential concern when using InSb as a substrate is the electron tunneling between gold SRRs and InSb substrate when a high doping level is used. Therefore, a metal-oxide-semiconductor (MOS) capacitor can be used as the gated device instead of the simple metal-semiconductor (Schottky) contact.  FIG. 4  is a side-view schematic illustration of an electrically tunable metamaterial  20  that uses a gate dielectric  16  between the resonator  11  and the semiconductor substrate  12 . In general, the thickness of the dielectric needs to be thin enough such that fields from the metamaterial resonator can still penetrate into the underlying semiconductor material. However, even when the gate dielectric is very thin, a very small absorption at the frequency of operation (i.e., where the metamaterials have their resonances) is effectively amplified by the Q of the metamaterial resonator. See D. J. Shelton et al., “Effect of thin silicon dioxide layers on the resonant frequency in infrared metamaterials,”  Optics Express  18 (2), 1085 (2010). This can have a significant degradation of the Q and thus the performance of such resonances. For example, in the thermal infrared, HfO 2  can be used as the gate dielectric, since its mid-infrared absorption is very low. For shorter infrared wavelengths, (e.g., less than 7 μm), SiO 2  can be used as the gate dielectric. Alternatively, heterostructures (AlInSb/InSb) can be used instead of InSb to achieve the gate isolation. 
     Metamaterial Layers on Gallium Arsenide Epilayers 
     We have also performed theoretical and experimental studies of various implementations of a split ring resonator (SRR) array formed on an n+ doped GaAs layer and forming a metal-semiconductor junction therewith.  FIG. 5  provides a schematic cross-sectional view of such an array, including semi-insulating GaAs substrate  30 , n+ GaAs epilayer  40 , and Al 0.3 Ga 0.7 As insulating barrier layer  50 , and gold metamaterial layer  60 , which also serves as an electrical gate. Depletion region  45  is also indicated in the figure. It is advantageous to include the barrier layer for reducing leakage current. 
     The results of our theoretical calculations of the real and imaginary parts of the dielectric constant in n-doped GaAs are shown in  FIG. 6  for selected wavelengths in the range 6-14 μm. It will be seen that as the dopant concentration increases beyond a threshold of about 10 18  cm −3 , the dielectric constant decreases rapidly. We utilize this effect by providing a dopant concentration near the threshold, e.g., a concentration of 5×10 18  cm −3 . Then we apply a negative bias to increase the width of the depletion region, thereby producing a relatively large change in the dielectric constant. 
     For example, at the stated dopant concentration we have produced a change Δ∈ in the dielectric constant of about 5.5 at a wavelength of 10 μm. As seen in the figure, the change becomes even greater for longer wavelengths. It should be noted that as also seen in the figure, the imaginary part of the dielectric constant also grows more rapidly with carrier depletion at the longer wavelengths, leading to greater optical loss. 
       FIG. 7  provides the results of our theoretical calculation of the depletion width as a function of the bias voltage in the arrangement as described above, at a doping level of 5×10 18  cm −3  and using Al 0.3 Ga 0.7 As barrier layers of respective thicknesses 30 and 60 nm. It will be seen that applying a modest negative bias can increase the depletion width by 10 nm, 20 nm, or even more. 
     An analytical estimate of the depletion width, based on a well-known theory of metal-insulator-semiconductor capacitors, is 
     
       
         
           
             
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     In the preceding expressions, ∈ 0  is the vacuum dielectric constant, ∈ GaAs =12.9 is the static dielectric constant of GaAs, ∈ AlGaAs =12.05 is the static dielectric constant of Al 0.3 Ga 0.7 As, V G  is the gate voltage, φ MS  is the flatband voltage, i.e., the difference between the work functions of the metal and the GaAs, W barrier  is the barrier-layer thickness, q is the electronic charge, and N D  is the donor dopant concentration. 
       FIG. 8  provides a perspective view of one example of a fabricated device, looking from the top. It will be seen that SRR array  70  is formed on substrate  80 . Conductive traces  90  are horizontal bus lines, which connect to inner contact region  100 , which functions as the gate contact. Outer contact region  110  is an ohmic contact. The figure also indicates, in schematic fashion, the application of a bias voltage between the inner and outer contact regions. 
     To make an exemplary device, we grew an undoped GaAs buffer layer and a 700 nm thick n+ GaAs epilayer (donor dopant concentration of 5×10 18  cm −3 ) on a semi-insulating GaAs substrate by molecular beam epitaxy. We then grew a 30 nm undoped Al0.3Ga0.7As barrier layer and a 5 nm GaAs cap layer. 
     Metal contacts were formed by optical lithography, metal deposition, and lift-off. The outer (ohmic) contact region was formed by electron-beam deposition in the sequence Ge (26 nm)/Au (54 nm)/Ni (14 nm)/Au (150 nm) followed by rapid thermal anneal at 380° C. for 30 seconds in argon. The inner (gate) contact region was then formed by PECVD deposition of a 70-nm silicon dioxide layer to prevent current from flowing through the metal gate and to provide mechanical protection to the barrier layer, followed by electron-beam deposition in the sequence Ti (10 nm)/Au (150 nm). The bus lines were used to connect the SRR array (see below) to the gate contact region. 
     An SRR array having an active area measuring 1 mm×1 mm was formed by a patterning step using electron-beam lithography and by metal deposition in the sequence Ti (5 nm)/Au (60 nm). Ohmic and metal gate contacts were then wire-bonded to a chip carrier. 
     One example of a unit cell of the SRR is provided in plan view in  FIG. 9 . The dimensions, as indicated in the figure, are G=110 nm, W=130 nm, L=720 nm. The double-headed arrow in the figure indicates the orientation of the electric field of the incident polarized light. 
       FIGS. 10A and 10B  provide experimental results describing the performance of the device of  FIG. 9 .  FIG. 10A  is a graph showing how the center frequency of the resonant peak in optical transmission varies with the bias voltage. The values of the center frequencies were obtained by Gaussian curve fitting. Error bars included in the plot represent experimental statistics.  FIG. 10B  provides the current-voltage curve of the device. 
     We have provided an example of Al 0.3 Ga 0.7 As as an insulating barrier material. This material is effective because it has a higher bandgap than the underlying layer of GaAs. However, such a material choice is merely exemplary and not limiting. For example, oxides such as Al 2 O 3  may also be effective as barrier materials. However, our numerical simulations indicated that the tuning effect, i.e., the shift in resonance with applied bias voltage, is greater with Al 0.3 Ga 0.7 As than with Al 2 O 3 . 
     We have provided an example of GaAs as an example of the semiconductor material in which the carrier density is to be varied. This choice should be understood as merely illustrative and not limiting. In particular, it may be desirable to select semiconductor materials that exhibit greater sensitivity of dielectric constant to changes in the carrier density. Generally, this is seen in semiconductors having smaller electron effective mass. InSb, for example, has an electron effective mass only 21% that of GaAs. For operation at still higher optical frequencies, it might be advantageous to employ semiconductor materials with higher carrier densities, such as indium tin oxide or aluminum-doped zinc oxide. 
     SRR Array Using an Asymmetric Unit Cell 
     Another example of a unit cell of the SRR, with variations, is provided in perspective view in  FIGS. 11A ,  11 B, and  11 C. A repeating array of unit cells of the type shown in  FIG. 11B  is shown in plan view in  FIG. 12 . 
     It will be seen that each unit cell includes separated arcs  120 ,  125  of a split ring and a portion  130  of a bus bar. As will be understood from inspection of the figure, this design is asymmetric because the respective arcs  120 ,  125  are unequal in length. The pattern dimensions are as follows: 
     Ring diameter, 770 nm; trace width, 150 nm; gap size, about 470 nm; smaller arc, 130°; larger arc, 160°; unit cell repeat distance, 4 μm. 
     The electric field of incident polarized light is oriented parallel to the gap between the two arcs, i.e., parallel to a diameter interposed between the two arcs. 
     Asymmetric structures, such as that described above, are of interest because the asymmetry can enhance the device&#39;s sensitivity to changes in the electrical permittivity of the substrate. For example, V. A. Fedotov et al., “Sharp Trapped-Mode Resonances in Planar Metamaterials with a Broken Structureal Symmetry,”  Phys. Rev. Lett . 99 (2007) 147401, reports an asymmetric SRR structure for use in the microwave range up to 14 GHz using a pair of arcuate metal arms having different lengths so as to resonate at slightly different frequencies. The two metal arms are out-of-phase within a narrow frequency range, and the transmission spectrum has a transparency window due to the destructive interference from the two resonator arms. Because this transparency window is cause by interference from two coupled resonators, it can be more sensitive to a substrate refractive index change. 
     Our asymmetric structure, described above, is similar to the Fedotov et al. structure, but has been adapted for use in electrically tunable mid-infrared devices. As such, it has been adapted for operation in a much higher frequency range, and in our implementations the metal array has the further function of serving as a gate electrode to modify the depletion width. More specifically, the depletion width in the underlying n-doped semiconductor epilayer changes with the electric gate bias as we have explained above. This induces a change of the permittivity of the substrate and leads to frequency tuning of the metamaterial resonance. 
       FIGS. 13A ,  13 B, and  13 C provide theoretical transmission spectra, obtained from numerical simulations, for an asymmetric SRR device as described above, made with respectively the unit cells of  FIGS. 11A ,  11 B, and  11 C. In each figure, the transmittance is plotted versus the wavenumber for each of three bias voltages, namely 0V, −2V, and −5V. A negative sign indicates reverse bias. 
     It will be seen that in the structure of  FIGS. 11A and 13A , both resonator arms are connected to an electrical bus line; in the structure of  FIGS. 11B and 13B , only the smaller arc is connected to the electrical bus line; and in the structure of  FIGS. 11C and 13C , only the larger arc is connected to the electrical bus line. 
     In each of the three figures, it will be seen that there are two resonances, situated at wavenumber values of approximately 1000 and 1200 cm −1 . A transparency window appears between the two resonances, which we attribute to destructive interference between the two resonances. We refer to this relatively transparent region as a “trapped mode”. In  FIG. 13A , representing the case in which both resonator arms are connected to the bus line, it will be seen that the tuning effect is strongest in the trapped mode region. 
     We also found that in the implementations in which only one resonator arm is connected to the bus line, effects attributable to localized control of the refractive index were observed. That is, the tuning behavior depended on which arm was biased. 
     When reverse bias is applied to a resonator arm, the refractive index in the substrate portion underlying that arm increases due to the expanded depletion region, leading to a shift in the resonance toward longer wavelengths (equivalently, lower wavenumbers). When both arms are biased, this shift is observed across both resonance regions. However, when only one arm is biased, our calculations show that the interference condition between the coupled resonators is changed, and that such a change is reflected in the spectral behavior.  FIG. 13B  shows that when the smaller arm is biased, the transmittance near the lower-energy resonance increases as the magnitude of the reverse bias increases. On the other hand,  FIG. 13C  shows that when the larger arm is biased, the transmittance near the lower-energy resonance decreases as the magnitude of the reverse bias increases. 
     Accordingly, it will be understood that when both arms are connected to the biasing circuit as in, e.g.,  FIG. 11A , the frequency-tuning effects resemble those seen in the case of a symmetric metamaterial. But when only one arm is connected (as in, e.g.,  FIG. 11B  or  FIG. 11C ), we observe an amplitude-modulation effect that is more pronounced than the frequency-tuning effect. We therefore believe that the asymmetric metamaterial design is flexible enough to provide, as desired, frequency tuning, amplitude modulation, or a combination of both, depending on the biasing scheme. 
     It should also be noted that two separate biasing lines are readily provided, so that one electrically connects only the long arcs, whereas the other connects only the short arcs. A switch is also readily implemented, so that an operator can select among biasing schemes to produce more diverse tuning and/or modulation behavior. 
     It should be noted further that although our examples relate to cases in which there are only two coupled resonators, this should not be understood as limiting. Rather, more complex structures are readily contemplated in which more than two coupled resonators are employed in order to produce desired changes in the spectral behavior in response to applied bias voltages.