Abstract
Investigations are made into the characterization of 1D metamaterials consisting of stacks of metal and dielectric. These stacks are modeled and designed to have a permittivity approaching zero. Simulation, fabrication and testing are conducted to verify the design of the layered material. These stacks are fabricated using magnetron sputtering and tested using Fourier Transform Infrared Spectroscopy (FTIR). Comparison between modeled and measured reflection and transmission are used to determine if the fabricated structure is behaving like a homogeneous material. Collected results indicate that a homogeneous structure was structures were formed, one with a possible low permittivity.
Introduction
Metamaterials are designed materials that can provide tailored material properties that are not found in nature. These materials hold promise for many different applications, from advanced imaging systems that can resolve beyond the diffraction limit to electromagnetic invisibility [3], [5], [12]. These unique properties are derived from the substances used and structure of the fabricated material. The dimensions of these structures are on the order of tens to hundreds of nanometers, so that they are orders of magnitude smaller than the micrometer-long infrared wavelengths they are interacting with, and the overall interaction of the material and the light can be seen as bulk properties, as if it were a homogeneous material [11]. This can be used to create an artificial permittivity, using a layered structure with metals, which can naturally have a negative permittivity when the wavelength of the incident light is shorter than the material’s plasma wavelength, and dielectrics, which have a positive permittivity [3],[4]. By combining these metals and dielectrics in a certain proportion, the positive and negative permittivity can effectively cancel each other out and result in an effective permittivity approaching zero [3]. This type of metamaterial is also commonly known as an epsilon near zero (ENZ) material. another type of metamaterial, a 2D "fishnet" metamaterials, which consist of periodic strips of metal and dielectric, can be used to create materials with a negative index of refraction. Unlike the ENZ materials which used thin layers of materials to create their material properties, these materials derive their properties from a negative permeability, created by resonant magnetic strips, and non-resonant electric strips, which have a negative permittivity from the metal uses in them [3],[1]. When these two pieces, negative permeability and permittivity are combined, a negative index is formed.
1D Metamaterial Design and Modeling
For a metal dielectric stack to have a low permittivity, the amount of metal and dielectric for this to occur need to be determined. Since this structure can be thought of as periodic, this can be done by taking one period of the structure, with a thickness, d, and finding the filling fraction, f fraction of the period thickness taken up by each material, such that
where ft and f2 and are the filling fractions for the dielectric and metal. The thickness of each material layer is then the filling times the period thickness. An illustration of this can be seen in Fig. 1.
Fig. 1 Example of a 1D metamaterial consisting of a metal dielectric stack d is the period of the material, and is made up of a layer of metal and a layer of dielectric The thickness of each layer is found from the filling fraction, f, multiplied by the period, d
Using the filling fractions and the permittivity of the structure can be determined using the Maxwell-Garnett approximation for the effective permittivity[6]
where eef is the effective permittivity of one period, with electric field of the incident light parallel to the layer and the light propagating into the structure, e’j, e’2 are the real part of the complex permittivity, e’+ie", of each layer, and fu f2 are the filling fractions. If a low, approaching zero, effective permittivity is desired, Equation (2) can be rearranged to form
providing everything needed to determine the filling fractions and thickness of each metal or dielectric layer, given the period of the stack and the permittivity of each material [3], [6].
Once the effective permittivity and filling fractions are set, the structure can then be modeled. The approach taken was to model the fabricated structure in two different ways, as either a layered material or as a homogeneous material. Using these treatments, the reflection and transmission for each case would be calculated, and the calculated results compared to the measured reflection and transmission. Comparisons can then be made between the two models and the measured reflection and transmission, and a determination could be made on whether the structure is behaving as a homogeneous or layered material. These two different methods of modeling can be seen in Fig. 2.
Fig. 2 Comparison of structures analyzed as layers (a) and a homogeneous media (b) Note how both structures have the same overall thickness
To calculate the reflection and transmission for these models, methods developed for layered media using the Fresnel equations, based on the methods described in [6], and [2]. This method takes into account non-normal incident angles and the TE and TM polarizations as well as the substrate.
The model for a homogeneous material uses an effective index, derived from the complex effective permittivity, which is the similar to the effective permittivity described in Equation (2), but with the complex parts of the permittivity included.
Results
Structures were then fabricated using a Denton Discovery 18 magnetron sputtering system to deposit thin film of metal and dielectric to create the structures. The metals were deposited using a DC and RF bias for the metals and dielectrics, respectively, without substrate heat. Three unique combinations of materials were used, titanium (Ti)-alumina (Al2O3), titanium-magnesium fluoride (MgF2), and nickel (Ni)-amorphous/poly silicon(Si). These material combinations were chosen based on their index and permittivity of each material, which would allow layers of at least 10nm, and period thicknesses of no larger than 500nm to be used in producing a ENZ in the range of 2.5-8^m. To the best of the authors’ knowledge, ENZ materials have not been attempted with these material combinations. The period, filling fractions, layer thicknesses and deposition parameters can be seen in tables for each fabrication. After fabrication these structures were then tested using a Bomem 157S FTIR, with the reflection and transmission measurements being made at 61.3° and 0° angle of incidence, respectively. These results are then plotted with the results from the layered and homogeneous model, where the models used literature values for the indices of the materials. Each combination will be described individually.
Titanium-A lumina
Table 1shows the settings used for the deposition of materials and filling fractions. A four layer, two period structure is fabricated. A scanning electron micrograph of a cleaved piece this structure can be seen in Fig. 3.
Table 1 Sputtering setting and layer parameters for Ti- Al2O3 structure
|
Period: 150nm |
Pre-sputter Voltage/Power |
Deposition Voltage/Power |
Deposition Thickness |
Filling Fraction |
|
Ti |
500V |
315V |
10nm |
.067 |
|
AI2O3 |
200W |
200W |
140nm |
.933 |
Fig. 3 Micrograph of fabricated Ti- Al2O3 structure This micrographs was taken of a cleaved piece, and shows a cross section of the structure, with two layers of Ti and Al2O3 visible on top of the silicon wafer used as substrate
This structure as then modeled as a layered and homogeneous material, and the reflection and transmission calculated. For these calculations, published data on the index of refraction of these materials was used, with the numbers coming from Kirillova and Charikov found in [9] and in [10], for Ti and Al2O3, respectively. The index for the crystalline silicon substrate was found using a dispersion relation found in [8]. Additionally, a thin 2nm thick silicon dioxide layer was put into the model above the silicon substrate. This was added to include the thin oxide that forms on a wafer due to exposure to air. The index for this layer came from data published in [10]. Using these numbers, this structure was calculated to give a zero permittivity at 4.4 5 |im. The modeled and measure d results are plotted and can be seen in Fig. 4.
Fig. 4 Plots of modeled and measured reflection and transmission from the Ti-Al2O3 structure Theses plots show a comparison between the modeled and measured data for both reflection and transmission
Looking at the collected data, it is seen that the modeled and measured values in both reflection and transmission curves have a similar shape and values. It can also be noted that the models are give results that are close to each other, indicating that the size of the structure in relation to the wavelength is such that homogeneous model is accurate in describing the structure. Another point of interest is the peak that can be seen in the transmission near 3 pm. This peak is due to the index of the structure changing and going to a value that is the square root of the substrate, allowing the structure to act as an antireflection coating. This is not directly indicative of an ENZ, however, it does suggest that the structure is behaving as a homogeneous layer, and has a changing index which could have a zero permittivity [6]. Though the modeled and measured results are similar, there still are differences, especially at the longer wavelengths. These differences in the model are the result of differences between the material parameters used in the modeling and the parameters of the deposited materials. These differences in material parameters are not unexpected, as the properties thin films often vary from literature values, and even among different facilities and deposition systems.
Titanium-Magnesium Fluoride
Table 2 shows the settings used for the deposition of materials and filling fractions. An eight layer, four period structure is fabricated. A scanning electron micrograph of a cleaved piece this structure can be seen in Fig. 5.
Table 2 Sputtering setting and layer parameters for Ti-MgF2 structure
|
Period: 167nm |
Pre-sputter Voltage/Power |
Deposition Voltage/Power |
Deposition Thickness |
Filling Fraction |
|
Ti |
400V |
340V |
12nm |
.072 |
|
MgF2 |
200W |
200W |
155nm |
.928 |
Fig. 5 Micrograph of fabricated Ti-MgF2 structure This micrographs was taken of a cleaved piece, and shows a cross section of the structure, with four layers of Ti and MgF2 visible on top of the silicon wafer used as substrate
The modeled reflection and transmission of the structure as layered and homogeneous were then calculated. The values used for the Ti, SiO2 and Si substrate in the Ti- Al2O3 structure were also used here. A dispersion equation for MgF2 found in [8] was used, where an average of the indices for the ordinary and extraordinary axes is used. Using this data, this structure was calculated to give a zero permittivity at 3.66pm. The results from the modeling and measurements are then plotted, and can be seen in Fig. 4.
Fig. 6 Plots of the measured and modeled reflection and transmission from the Ti-MgF2 structure Theses plots show a comparison between the modeled and measured data for both reflection and transmission
Looking at the above plots, there are a couple things that noted. First, the two modeled results line up very well, with a difference of no more than about 5% between them, indicating that the idea of this material being homogeneous is valid. Comparing the modeled results to the measured data, however, does not show much agreement, as the reflection curves have the same general shape, but values that differ by as much as 40%, and the transmission curve not showing the peak present in the models at 3.75^m, the peak from the structure acting as an antireflection coating. The lack of a peak does not mean that the fabricated structure is not behaving as a homogeneous material, but it does mean that the models, and the materials parameters they are using, are not accurate to the deposited materials. Further work into determining the material properties of the materials would allow for a better modeling and the determination of what is happening in this structure.
Nickel-Silicon
Table 3 shows the settings used for the deposition of materials and filling fractions. An eight layer, four period structure is fabricated. A scanning electron micrograph of a cleaved piece this structure can be seen in Fig. 7.
Table 3 Sputtering setting and layer parameters for Ni-Si structure
|
Period: 100nm |
Pre-sputter Voltage/Power |
Deposition Voltage/Power |
Deposition Thickness |
Filling Fraction |
|
Ni |
400V |
360V |
11nm |
.11 |
|
Si |
200W |
200W |
89nm |
.89 |
Fig. 7 Micrograph of Ni-Si structure This micrographs was taken of a cleaved piece, and shows a cross section of the structure, with four layers of Ni and Si visible on top of the silicon wafer used as substrate Note that a thin silicon dioxide layer on top of the wafer is visible in this micrograph
The modeling used index data for Ni from Lynch et al and found in [9], while the properties for amorphous Si were found in [10]. The values used for SiO2 and Si substrate in the Ti-Al2O3 structure were also used here. Using these numbers, the structure was calculated to give a zero permittivity at 3.16^m. The results from the modeling and FTIR are plotted, and can be seen in Fig. 8.
Fig. 8 Plots of the measured and modeled reflection and transmission from the Ni-Si structure Theses plots show a comparison between the modeled and measured data for both reflection and transmission
The comparison between the two models, and the collected and modeled data for the reflection and transmission shows that the models are not agreeing. The values for the two models are not as close as they were in previous structures, indicating that the structure may not have homogeneous behavior, while the lack of agreement between either model and the measured data is due to the differences between indices of the deposited materials and the literature values. Though there is little correlation between the models and data, an interesting feature can be seen near 5 pm in the reflection, where the minimum value is reached. This dip may be caused by the fabricated structure acting homogeneous and becoming an antireflection coating, however, it cannot be determined if this is the case, as there was essentially no transmission through the stack to indicate if this was happening. This lack of transmission is likely due to losses in the materials.
2D Metamaterial Design
In addition to investigating 1D metamaterials, patterned fishnet metamaterials capable of producing a negative index were also investigated. This procedure could be used to find the dimensions needed to create a negative index structure at infrared wavelengths. This structure was designed with the thought that it is able to be fabricated with standard photolithographic processes, using masks and techniques such as etch back of deposited layers. A diagram of this structure and, the dimensions needed, can be seen in Fig. 9.
Fig. 9 Illustration of the fishnet metamaterial (a) shows a top view and the electric and magnetic strips, along with the widths of each and the period of the structure, 2w (b) shows a side view with the layers of the structure and their thicknesses, t and d
The first step in the design is to pick the materials to be used as the metal and dielectric. A metal that has a negative permittivity at the design wavelength is needed, and one with low losses is desired [3]. A dielectric with a high index and permittivity are desired, as this would helps to keep the electric field between the magnetic strips and helps them to resonate [3]. To meet these requirements, the materials of gold and hafnium oxide are proposed, since Au has a high conductivity, negative permittivity in the infrared, and can easily be etched, and HfO2 due to its high permittivity and index.
After materials are the next step is the design of the negative permeability elements was done first, using a relation found in [3] describing the dimensions and material properties needed to have a magnetic resonance occur at a given wavelength. The equation, given below, is
where Xm, is the wavelength at which the strips are resonant, e’(Xm) is the real part of the metal permittivity at the resonant frequency, nd is the index of the dielectric spacer, d is the thickness of the dielectric spacer, t is the thickness of the metal, and k is defined as
Using these equations a MATLAB script is written to help determine the width of the resonator. Since it is more difficult to solve the above equations exactly, and the range of values for the sizes and thicknesses are restricted to certain ranges and are not continuous, due to fabrication and mask making restrictions, the MATLAB code was written to compute values for the right hand side of Equation (5), with t, w, and d being incremented. The values for Xm, nd, and e’(Xm) are all fixed, based on the wavelength the resonance is desired to occur at. The value of each combination is calculated following the right side of Equation (5), and then subtracted from the value of e’(Am) and the difference stored. The script then selects and displays the combinations that have the smallest difference, which are the structures that most closely fit the model in Equation (5).
If the proposed materials are run with this script, using the permittivity of Au found in [9] and index of HfO2 found in [7], a few different combinations of layer thickness and widths were found for a wavelength of 4.45^m, of which one of the best was w=1.12^m, t=78nm and d=93nm. The width of the electric strips is then chosen to be a width that will not resonate like the magnetic strips, and will not add a lot of loss, so a width thinner than that of the magnetic strips should then be chosen [3]. A width of 1^m is suggested, as this is the smallest dimension able to be produced at AFIT. The period of 2w=2.24^m is also suggested, for both x and y directions to keep the structure periodic.
A finite difference, time domain (FDTD) simulation was then run using the Lumerical® FDTD software, using periodic boundary conditions in the x and y directions to simulate this structure as one in an array of many. Two simulations were done, one for each polarization of light with respect to the structure, and averaged to give the response to unpolarized light. A screen shot of the setup used in the simulation can be seen in Fig. 10.
Fig. 10 Screen shots showing the layout of the Lumerical simulation, with (a) showing a top view of the structure, and (b) a side view
The results of the simulation can be seen in Fig. 11. These results show a peak in adsorption at 3.25^m, which is indicative of a resonance occurring and a possible negative index occurring [3].
Fig. 11 Plot of results from Lumerical simulation of the fishnet structure, showing the results of both polarizations and average values for the reflection, transmission and adsorption An adsorption peak is visible at 3.25^m
Conclusions and Future Work
Investigations into metamaterials at infrared wavelengths using novel material combinations have been made. 1D, ENZ metamaterials were fabricated and tested, with materials shown to exhibit homogeneous, and possibly ENZ behavior. The lack of fit between either the layered or homogeneous model and the collected data from the fabricated structures is due to differences between the optical properties of deposited materials and their descriptions found in literature. This difference is not surprising, as material values for thin films often have a wide variation and differ from lab to lab. A fishnet metamaterial is proposed, and a simulation of the structure, with literature values for the materials, indicates that a resonance in the structure is occurring, and a negative index may be formed at 3.25pm. A few areas exist for future work, one area of which is characterizing the thin films of materials deposited by sputtering, and providing indices of refraction that could be used to allow for more accurate modeling and design of ENZ and fishnet structures. These properties can be found through a few different means, though one of the easiest is through spectroscopic ellipsometry of a single layer of deposited material. Work could also be done in fabricating the proposed fishnet structure using standard lithography and etching holes in the deposited layers to form a fishnet structure that could be tested.
