Spectral absorptance of a metal-semiconductor-metal thin-multilayer structured thermophotovoltaic cell

Kazuma Isobe; Ryota Okino; Katsunori Hanamura

doi:10.1364/OE.410828

1. Introduction

Thermo-photovoltaic (TPV) power generation has been developed to harvest radiative energy from unused thermal resources such as the thermal radiation from high-temperature steel manufacturing processes. Molten steel at a temperature of 1500–1800 K is a convenient radiation source for photocurrent generation since most of the TPV cells have high quantum efficiencies at wavelengths shorter than approximately 2.5 µm. Fabrication processes of low-bandgap semiconductors, such as gallium antimonide (GaSb), have been developed since the end of the 20th century to promote industries relating to TPV power generation [1 –3]. Wang et al. showed a TPV cell made of GaInAsSb/GaSb with a bandgap energy level of 0.5 eV that realized a high internal quantum efficiency (IQE) of 90% at a wavelength of 1.6 µm [4]. Several researchers proposed a thin-film TPV cell with several hundred nanometers to reduce the bulk non-radiative carrier recombination loss after photocurrent generation, resulting in improved IQE and a lower saturation current [5 –8]. Simultaneously, the reduction of surface recombination loss is predicted with a well-formed passivation layer on the thin-film TPV cell. In addition to developing the TPV cell, enhancement of the irradiation intensity is an essential issue to improve the power generation density of TPV [9]. Simultaneously, spectral control of incident radiation using optical metamaterials has been progressing to increase conversion efficiency and reduce cooling costs for the TPV cell. This is because each TPV cell has suitable external quantum efficiencies for a specific wavelength range [4,10 –12]. A perfect blackbody [13] and a selective thermal emitter [14 –16] were studied, among others, as optical metamaterials. These optical metamaterials consist of micrometer- or nanometer-sized structures, which are smaller than the wavelength of incident light. Thus, they have spectral absorptance and reflectance values that do not exist in nature [16 –18]. A metal–insulator–metal (MIM) multilayer, in which the insulator layer is between a substrate and top metal layers, is an optical metamaterial that can absorb or emit radiation only at a specific wavelength [16,19]. A metal–dielectric–metal (MDM) multilayer, in which the insulator layer of a MIM is replaced with a dielectric layer, has similar spectral absorptance to MIMs [20]. Here, the intermediate layer must be transparent to excite resonant modes originating from magnetic polariton (MP) and Fabry–Pérot interference. These are the dominant mechanisms of absorption. With this requirement, a metal–semiconductor–metal (MSM) multilayer has also been proposed as a metamaterial-TPV cell through numerical simulations [8,21]. A photovoltaic semiconductor such as GaSb is sandwiched between the top and substrate metal electrodes. The GaSb thickness is reduced to several hundred nanometers to conserve fabrication materials and diminish thermal resistance. Moreover, the top electrode has a shape with periodic islands, a one-dimensional grating, or a periodic fishnet, which excite resonant modes. Then, incident light with a specified wavelength is trapped inside the MSM due to MP or cavity resonance. Consequently, the incident light is converted into electricity at the depletion layer of the GaSb with a high spectral efficiency of nearly unity. A resonant phenomenon related to a thin-film TPV cell has a potential to significantly enhance the power generation density. Its theoretical performance using far- or near-field radiation as incident radiation was calculated in past studies [5,6,8]. In this work, a MSM thin-multilayer consisting of a gold fishnet-type top electrode, a thin GaSb layer, and a gold backside electrode (fishnet MSM) was manufactured using nanofabrication technology to initiate the formation of a metamaterial-TPV cell. Its normal spectral absorptance at near-infrared wavelengths was measured to demonstrate high absorptance only around a specific wavelength. Moreover, the experimental results were compared with those of simulations to validate the analytical model that indicates the peak wavelengths of the fishnet-MSM. Then, the contributions of several absorption modes, i.e., the cavity resonance, MP, and Fabry–Pérot interference, were evaluated to reveal the requirement for the electrode width of the fishnet.

2. Methodology of sample fabrication

In the current study, the MSM thin-multilayer structure depicted in Fig. 1 was manufactured for measurement of spectral absorption to confirm the analytical model of previous work [21]. The thickness of the GaSb layer should be greater than 100 nm because it is reported that the thickness of the depletion layer of the p–n junction is approximately 100 nm [22]. Usually, an undoped or a zinc-doped GaSb is used as the p-type GaSb semiconductor [23,24], while a tellurium- or a sulfur-doped GaSb serves as the n-type GaSb semiconductor [25,26]. Additionally, a gold–germanium (Au–Ge) alloy is always used for the flat backside electrode to make ohmic contact with the n-type GaSb semiconductor [27 –29]. For evaluating the simulation result experimentally with a simple manufacturing process, only an undoped p-type GaSb semiconductor with a thickness of about 100 nm was grown because the optical absorptance of p-type GaSb is much the same as that of the n-type material. The real parts of both of their refractive indices are within 5% of each other [30]. Additionally, pure gold, which has a lower reflectance compared with that of Au–Ge [29], is used for the backside electrode. For the top fishnet-type gold electrode, a square aperture size in the x- and y- directions is defined as w_x × w_y, the pitch lengths of the fishnet in x- and y-directions are Λ_x and Λ_y, and the height of the electrode is h.

Fig. 1. The fabrication process of the MSM thin-multilayer with a gold electrode, p-GaSb semiconductor, and backside gold electrode. At first, (a) a 100 nm thickness of the p-type GaSb crystal layer was epitaxially grown on an InAs substrate. (b) An acid-resist wax with a 3 mm thickness was attached to the p-GaSb surface. (c) The InAs substrate was removed using HCl. (d) The surface of a thin GaSb layer was bonded to a gold surface using a VDW force. (e) The acid-resist wax was dissolved using a TCE. (f) Using an EB writer, a fishnet pattern was formed as an EB mask on the p-GaSb surface. (g) Gold was deposited on the mask using an EB vapor deposition process, and the mask was lifted off with remaining the fishnet gold electrode on the p-GaSb.

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Figure 1 also illustrates a fabrication process of the MSM thin-multilayer structure with a gold electrode, p-GaSb semiconductor, and backside gold electrode. In the first step, as shown in Fig. 1(a), an undoped GaSb crystal layer was grown to a thickness of 100 nm on an indium arsenide (InAs) substrate with a thickness of 500 µm using molecular beam epitaxial (MBE) equipment modified from a vacuum evaporator (VX-30; Eiko Corporation, Tokyo, Japan). It is impossible to have epitaxial growth of the GaSb layer directly on the gold electrode surface, while it is possible to do this on the InAs crystals due to a negligible mismatch (less than 0.6%) between the GaSb and InAs crystal lattice constants [31,32]. For the future development of the manufacturing process to form a p–n junction inside the ultra-thin GaSb layer, n-GaSb will need to be grown to a thickness of 50 nm on InAs before growing a p-GaSb layer to 50 nm. Additionally, a reactive ion etching using BCl₃, Cl₂ and Ar gases will be required to precisely adjust the total thickness of the GaSb layer [33]. In the second step, depicted in Fig. 1(b), an acid-resist wax (Apiezon Wax W; M&I Materials, Manchester, United Kingdom) layer with a 3 mm thickness was made on the p-GaSb surface to support a thin GaSb film after removal of the InAs substrate. A droplet of a liquid wax solution made from 5 g of wax dissolved in 20 ml of trichloroethylene (TCE), was deposited and spread uniformly on the surface of the p-GaSb. After that, the liquid solution was dried at room temperature for 30 minutes. Then, another droplet was deposited and spread uniformly on the surface of the previous wax layer. This procedure was repeated until the wax thickness exceeded 3 mm. This wax layer was baked at 100 °C to completely volatilize TCE. In the third step, shown in Fig. 1(c), the InAs substrate was entirely removed with a wet etching process using a 37% hydrochloric acid (HCl), since there is a significant difference in the etching rates for InAs, 120 µm/h, and GaSb, 0.005 µm/h [32,34]. In the next step, depicted in Fig. 1(d), the surface of the thin GaSb layer supported on a thick resist wax substrate was bonded to the surface of a 200 nm thick layer of sputtered gold on a silicon substrate using a van der Waals (VDW) bonding method [35]. In the fifth step, shown in Fig. 1(e), the thick resist wax substrate was entirely removed using TCE. Figure 2(a) shows an image of a thin GaSb–gold bilayer manufactured by the abovementioned procedure. The thin GaSb layer was bonded on a part of gold–silicon substrate. In the subsequent step, illustrated in Fig. 1(f), the surface of the GaSb was coated using an electron beam (EB) resist (ZEP520A-7; Zeon Corporation, Tokyo, Japan) using a spin-coater, and then, the resist was baked for 5 minutes. Using an EB writer (F7000-VD02; Advantest Corporation, Tokyo, Japan), a fishnet pattern was made as an EB mask through an EB lithography process, where the pitch lengths, Λ_x and Λ_y, and the square aperture size, w_x × w_y, were designed to be 400 nm and 300 nm, respectively. This aperture size was sufficient to set the first resonant peak at 1.6 µm, where GaSb has high internal quantum efficiency. After the lithography, an inverse pattern of the fishnet structure was made using a developer (ZED-N50; Zeon Corporation, Tokyo, Japan) and a rinse (ZMD-B; Zeon Corporation, Tokyo, Japan). Next, as shown in Fig. 1(g), gold was deposited on the mask with an inverse pattern of the fishnet structure using an EB vapor deposition process. Finally, the mask was lifted off using N-methyl-2-pyrrolidone and an ultrasonic cleaner. Through these processes, a fishnet-structured gold electrode was manufactured on a thin GaSb semiconductor, as shown in Fig. 2(b) and 2(c).

Fig. 2. (a) Image of a thin p-GaSb–gold bilayer. (b) Image of a fishnet gold–GaSb–gold thin-multilayer. (c) FE-SEM image of a fishnet gold electrode.

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3. Numerical simulation

In order to estimate the absorptance of the MSM thin-multilayer, numerical simulations using a finite difference time domain (FDTD) method were conducted. In the FDTD method, Maxwell’s electromagnetic field equations [36] were numerically solved. Only one period of the cyclic fishnet MSM thin-multilayer, i.e., one cavity, was modelled in the computational area. A planar Gaussian wave was irradiated perpendicular to the surface of the multilayer because of small angular dependency [8,18]. Then, the absorptance was estimated by dividing the intensity of the absorbed wave by that of the incident wave [21]. The spatial and time resolution of the current simulations were 5 nm and 5.0 × 10⁻¹⁸ s, respectively. A second-order perfectly matched layer (PML) condition [37,38] was applied to both the top and bottom ends of the computational area, while other side boundaries were set to a periodic boundary condition to express planes with semi-infinite areas. Optical properties, i.e., the complex permittivities of gold [39] and undoped p-GaSb [40], were introduced into the simulation using a piecewise linear recursive convolution (PLRC) method [41]. The complex permittivity of gold was fitted to the Drude model with the same parameters described in the previous study [21]. Figure 3 shows a comparison of the complex refractive indices of GaSb with a reference value [40] and a fitting model used in the simulation. In the current simulation using the PLRC method, the complex permittivity, i.e., the square of the complex refractive index, of GaSb, ɛ _GaSb, needed to be fitted to the Lorentz oscillator model as shown below:

(1)$${\varepsilon _{\textrm{GaSb}}}(\omega ) = {\varepsilon ^{\prime}_{\textrm{GaSb}}} + i{\varepsilon ^{\prime\prime}_{\textrm{GaSb}}} = {\varepsilon _\infty } + \frac{{{f_\textrm{l}}\omega _\textrm{l}^2}}{{\omega _\textrm{l}^2 - {\omega ^2} - i{\mathrm{\Gamma} _\textrm{l}}\omega }}, $$

where, ɛ _∞ = 8.272 is the relative background permittivity, f _l = 5.892 is the strength of the Lorentz oscillator with a center frequency ω _l = 3.703 × 10¹⁵ rad/s and Γ _l = 1.501 × 10¹⁵ rad/s as the damping factor. Fitting of permittivity in this study was mainly conducted above 0.8 µm for the real part, which is essential in evaluating power generation performance. There is a slight discrepancy in the imaginary part at a longer wavelength than 1.73 µm. The imaginary part functions to enlarge the full width at half maximum (FWHM) of the resonant oscillation phenomena, which form peaks of absorptance. However, the imaginary part does not affect the peak wavelength. Therefore, the discrepancy in the imaginary part is acceptable enough to analyze the absorptance peak and resonant modes.

Fig. 3. Complex refractive indices of GaSb in [40] and fitted by the Lorentz oscillator model. Solid lines correspond to the real part, while dashed lines correspond to the imaginary part.

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4. Surface measurement of the sample

Figure 2(c) shows a field emission-scanning electron microscopic (FE-SEM) image of the fishnet structure. The bright portion of the image corresponds to the fishnet structure made of gold. Due to the backscattering and proximity effects of EB in the lithography process [42], the length of one side of a square cavity appears slightly smaller than 300 nm, and the corners of the fishnet apertures are rounded. In this work, compensation for these effects during the EB lithography could not be perfectly achieved.

The surface distribution of the fishnet was observed using atomic force microscopy (AFM) to determine the geometric parameters of the fishnet MSM thin-multilayer for the numerical simulation. Figure 4(b) shows the detailed displacements of cantilevers on two line segments A–B and C–D in Fig. 4(a). From the relative vertical displacement, the thickness of the fishnet electrode was found to be about 30 nm, although both the ceilings of the fishnet and the surface of GaSb were rough. In the horizontal direction, it was found that the cyclic fishnet structure period in the A–B direction was about 3% longer than that of C–D direction. This warping was not intended in the design process and originated from thermal expansion of the substrate or lattice orientation of GaSb. Moreover, the sidewalls of the fishnet became sloped with widths of approximately 30 nm. Here, the radius of curvature of the cantilever of AFM (OMCL-TR800PSA-1; Olympus Corporation, Tokyo, Japan) was about 15 nm at its tip. Since the tip radius was smaller than the fishnet height, physical interference between the ceiling of the fishnet and the tip caused slightly larger displacement in vertical direction at distant points from the sidewall. Thus, the sidewalls of the fishnet structure were estimated as vertical walls. Finally, the geometric parameters of the manufactured fishnet structure on the GaSb layer were found to be w_x = 300 nm, w_y = 310 nm, Λ_x = 400 nm, Λ_y = 410 nm, and h = 30 nm. It was better to deposit a thicker electrode than 30 nm to enhance the cavity resonance mode; however, it is still acceptable to utilize its function to control the absorptance. Furthermore, the thickness of the GaSb layer, d, was measured using a step gauge and found to be approximately 115 nm.

Fig. 4. (a) The surface distribution of the fishnet-MSM thin-multilayer scanned using AFM. (b) Sectional views of surface distributions at two line segments A–B and C–D.

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5. Results and discussion

Figure 5(a) compares four absorptances obtained from sample measurements and numerical simulations. The absorptance measurements were conducted using a spectrophotometer (MSV-370; JASCO Corporation, Tokyo, Japan). In the simulation, the geometric parameters of a MSM thin-multilayer were equal to the results of AFM measurements. The corners and edges of the cavities were assumed to have an ideal rectangular shape. The black and red lines express the spectral absorptance obtained from the experiment without and with a fishnet electrode. Without a fishnet electrode, only a peak originating from the interference of thin-film at 0.75 µm is apparent. With a fishnet electrode, there are three peaks, at 0.72, 0.91, and 1.66 µm. In this work, we focus on two notable peaks at 1.66 and 0.72 µm (the first and second peaks). The most significant peak at 1.66 µm, which the fishnet electrode emphasized, reaches an absorptance of 0.95. It has a high absorptance at the peak while it diminishes to less than 0.2 at wavelengths greater than 2.0 µm.

Fig. 5. (a) Spectral absorptances of the fishnet-MSM thin-multilayers obtained from the experimental and FDTD numerical simulations. The green and light blue lines show simulation results with original or adjusted Lorentzian coefficients of GaSb, respectively. Two resonant wavelengths of an equivalent resonant cavity shown in (b) are depicted in (a) by dashed lines.

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There is a discrepancy in the peak wavelength of the simulation results depicted by the green line and the experimental results in Fig. 5(a). This inconsistency primarily comes from the optical properties of the p-GaSb grown in the experiment. The precise refractive index of the p-GaSb cannot be measured due to the complicated fabrication process of the fishnet-MSM. Alternatively, the carrier density of p-GaSb was measured, which was 2.646 × 10¹⁹ cm⁻³ and the mobility was 2.34 × 10² cm²/Vs in these experiments. The real part of the refractive index of p-GaSb tends to become smaller with increasing carrier density [30]. Here, this carrier density was ∼10 to ∼ 100 times higher than in other studies [30,43]. This is because the rate of epitaxial growth was too fast to make high quality GaSb. Considering the relationship between the carrier density and refractive index [30], the refractive index in the experiment was possibly about 10% lower than the simulation. The magnitude of the estimation of the refractive index is comparable to the errors of the first and second peak wavelengths, approximately 9%. The light blue line shows another simulation result roughly considering a reduced refractive index. Here, the parameter ɛ _∞ in Eq. (1) was taken as 6.100. Despite the approximated adjustment to the refractive index, the first peak wavelength drastically shifted closer to the experimental result. This indicates the cause of the difference between experimental and original simulation results.

We discuss an equivalent physical model to describe the absorbing mechanism for the first and second peaks. Here, the fishnet electrode has a topologically identical structure to a bundle of square waveguides. Moreover, when one side of the square waveguide is closed, it can be regarded as a resonance cavity with an open end. From these associations, the equivalent cavity model depicted in Fig. 5(b) was proposed to describe the peak wavelength [18,21,44]. In the model, the p-GaSb fills the open cavity made of gold. The resonant wavelength of the cavity mode, λ _Resonant, is derived as follows:

(2)$${\lambda _{\textrm{Resonant}}} = \frac{{2{n_{\textrm{GaSb}}}(\lambda )}}{{\sqrt {{{\left( {\frac{{{N_x}}}{{{w_x} + 2\Delta {w_x}}}} \right)}^2} + {{\left( {\frac{{{N_y}}}{{{w_y} + 2\Delta {w_y}}}} \right)}^2} + {{\left( {\frac{{{N_z}}}{{2d + \Delta d}}} \right)}^2}} }},$$

where, n _GaSb is the refractive index of GaSb noted in [40], Δw_x, Δw_y, and Δd are penetration depth of electric field inside the gold surface in the x-, y-, and z-directions respectively. N_i (i = x, y, and z) is the order of resonant modes for the x-, y- and z-directions, respectively. Here, N_x and N_y are natural numbers, and N_z needs to be an odd number [44]. In this study, as fitting parameters, these Δw_x, Δw_y, and Δd values were set to 45 nm, 45 nm, and 35 nm, respectively, to account for the attenuation length of the electric field at the gold surface. These parameters were necessary for the ultra-thin metal wall because the electromagnetic field penetrates slightly deeper than the depth derived from the extinction coefficient [45]. Here, the wavelength of the first resonant mode (Cavity Mode (CM) 1), calculated using Eq. (2), is 1.67 µm when the combined orders of the resonant modes (N_x, N_y, N_z) are (1, 0, 1) or (0, 1, 1) and n _GaSb is 3.82. For the fishnet-MSM, there was a 30 nm thick vacuum space inside the fishnet aperture, and the domain below the fishnet electrode was not a conductive wall. Here, the thickness of the vacuum space inside the aperture, h, was negligible since it was optically thin compared with the incident wavelength. Moreover, a part of incident radiation was partially diffracted at the atmosphere, fishnet gold, and GaSb layer triple-phase boundary. Then, it could propagate along the interface of the gold and GaSb layers as an evanescent wave. When the gold electrode width, Λ − w, was small enough, the evanescent wave interfered with another wave from the next cavity with a phase difference of π, and they canceled each other out. Thus, the GaSb domain just below the fishnet gold functioned as an imaginary conductive wall. However, Λ − w of 100 nm is not small enough to function as a perfect imaginary conductive wall. It caused a slight shift of the first peak to a 9% longer wavelength in the simulation

The resonant cavity model also suggests a peak at 0.72 µm in the experiments. The wavelength of the second resonant mode of the resonance cavity model (CM2) calculated using Eq. (2) is 0.74 µm when the combination of the orders of the resonant modes (N_x, N_y, N_z) are (1, 0, 3) or (0, 1, 3) and n _GaSb is 4.25. These combinations of the order numbers were chosen because the resonant modes (1, 0, N_z) or (0, 1, N_z) tend to be apparent in the resonance of a cavity with an open end [44]. While the absorptances at each peak are smaller than the first peak, both experimental and simulation results correspond to the indicated wavelength within a 3% error. Although the geometric conditions were of an ideal resonant cavity and fishnet-MSM dimensions, the correspondence of the wavelength for these two modes confirms the effectiveness of the equivalent resonance cavity model. The model has been shown useful in approximate estimation of distinct absorptance peaks of the fishnet-MSM.

This imperfection in the cavity resonance also affected the experimental results. The red line in Fig. 5(a) depicts a small shoulder beside the first peak at 1.3 µm. This shoulder implies another absorption mode. Here, the available range of the equivalent resonant cavity mode was examined through additional simulation. Figure 6 shows the relationship among absorptance, wavelength of the incident wave and electrode width, Λ − w, with a constant cavity width w_x = w_y = 300 nm. In this examination, the GaSb thickness, d, and the fishnet electrode thickness, h, were both set to 100 nm. When Λ − w is larger than the thickness of the GaSb layer, the first peak at 1.5 µm split into three peaks.

Fig. 6. Spectral absorptance of the fishnet-MSM multilayer, obtained through numerical simulation, depicted by a contour. The thickness of the semiconductor layer d, the height of the fishnet structure h and the cavity width w_x = w_y are set to 100 nm, 100 nm, and 300 nm, respectively. The dashed lines indicate the wavelength of the MP, interference, and CM1.

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The peak at the longest wavelength is excited by the MP originating from an equivalent LC circuit resonance [8,16,46]. Figure 7(a) shows a schematic diagram of the LC circuit model for the fishnet-MSM, consisting of two capacitors and inductors formed at the GaSb layer and the space inside the gold aperture. For the transverse magnetic (TM) mode with an angular frequency of ω, the mutual and kinetic inductances between the bottom gold and y directional top electrodes, L _m1 and L _k1, are defined as follows:

(3)$${L_{\textrm{m1}}} = 0.5{\mu _0}\frac{{({\Lambda _x} - {w_x})d}}{{{w_y}}},$$

Fig. 7. Magnetic field contours (H_y/H _incident), electric field vectors (arrows) and a schematic diagram of an equivalent LC circuit model. (a) A schematic diagram of the equivalent LC circuit model around the fishnet-MSM. Wavelengths of incident radiation were (b) 2.25 and (c) 1.38 µm.

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(4)$${L_{\textrm{k1}}}(\omega ) ={-} \frac{{{\Lambda _x} - {w_x}}}{{{\varepsilon _0}{\omega ^2}{w_y}\delta }}\frac{{\varepsilon ^{\prime}_{\textrm{Au}}}}{{\varepsilon ^{\prime^2}_{\textrm{Au}}} + \varepsilon ^{\prime\prime^2}_{\textrm{Au}}},$$

where, µ ₀ and ɛ ₀ are the magnetic permeability and electric permittivity of vacuum, while ${\varepsilon ^{\prime}_{\textrm{Au}}}$ and ${\varepsilon ^{\prime\prime}_{\textrm{Au}}}$ are the real and imaginary parts of the permittivity of gold. The parameter δ is the penetration depth of the electric field for the gold surface, expressed as follows:

(5)$$\delta = \frac{{2{c_0}}}{{3\omega }}\sqrt {\frac{2}{{ - {\varepsilon ^{\prime}_{\textrm{Au}}} + \sqrt {\varepsilon ^{\prime^2}_{\textrm{Au}} + \varepsilon ^{\prime\prime^2}_{\textrm{Au}}}}} }, $$

where, c ₀ is the speed of light. Similarly, the mutual and kinetic inductances between the bottom gold and x-directional top electrodes, L _m2 and L _k2, are defined as follows:

(6)$${L_{\textrm{m2}}} = 0.5{\mu _0}\frac{{{w_x}d}}{{{\Lambda _y} - {w_y}}},$$

(7)$${L_{\textrm{k2}}}(\omega ) ={-} \frac{{{w_x}}}{{{\varepsilon _0}{\omega ^2}({\Lambda _y} - {w_y})\delta }}\frac{{\varepsilon ^{\prime}_{\textrm{Au}}}}{{\varepsilon ^{\prime^2}_{\textrm{Au}}} + \varepsilon ^{\prime\prime^2}_{\textrm{Au}}}.$$

The capacitance values between top and bottom gold layer, C _m, and between two y directional electrodes, C _g, are defined as follows:

(8)$${C_\textrm{m}}(\omega ) = c^{\prime}{\varepsilon ^{\prime}_{\textrm{GaSb}}}{\varepsilon _0}\frac{{({\Lambda _x} - {w_x}){w_y}}}{d},$$

(9)$${C_\textrm{g}} = {\varepsilon _0}\frac{{h{w_y}}}{{{w_x}}},$$

where, c’ = 0.38 is a numerical factor to account for the non-uniform electric charge distribution at the gold surface. Since the relation between the structure and c’ has not been formulated rigorously, it was decided as a fitting parameter. Although the top fishnet electrode is physically connected, two y directional electrodes can be regarded as electrically isolated. The Drude relaxation time of gold is 3.0 × 10⁻¹⁴ s, and its reciprocal is 3.3 × 10¹³ Hz [47]. The incident wave oscillates the free electrons at a higher frequency, e.g., 1.5 × 10¹⁴ Hz for 2.0 µm, than a group oscillation of free electrons in gold. Thus, the electric charge in the fishnet gold does not become neutral, and a capacitor, C _g, is formed. Finally, the total impedance of the LC circuit model is described as follows:

(10)$${Z_{\textrm{total}}}(\omega ) = i\omega \left( {\frac{{({L_{\textrm{m1}}} + {L_{\textrm{k1}}})({L_{\textrm{m2}}} + {L_{\textrm{k2}}})}}{{({L_{\textrm{m1}}} + {L_{\textrm{k1}}} + {L_{\textrm{m2}}} + {L_{\textrm{k2}}}) - {\omega^2}{C_\textrm{g}}({L_{\textrm{m1}}} + {L_{\textrm{k1}}})({L_{\textrm{m2}}} + {L_{\textrm{k2}}})}} - \frac{2}{{{\omega^2}{C_\textrm{m}}}} + \frac{{({L_{\textrm{m1}}} + {L_{\textrm{k1}}})({L_{\textrm{m2}}} + {L_{\textrm{k2}}})}}{{{L_{\textrm{m1}}} + {L_{\textrm{k1}}} + {L_{\textrm{m2}}} + {L_{\textrm{k2}}}}}} \right).$$

When Z _total(ω _r) becomes zero, an electromagnetic wave at the resonant frequency, ω _r, will be significantly absorbed.

Figure 7(b) shows the distribution of an electromagnetic field surrounding the fishnet MSM at the resonant wavelength, 2.25 µm, for the electrode width, Λ − w, of 190 nm. The arrows correspond to the electric field vectors, while the contour depicts the logarithmic intensity of y-directional magnetic fields. The magnetic field is localized and enhanced so that is ten times stronger than the incident wave in the GaSb region between the top and bottom gold layers. Simultaneously, the left and the right sides of the electrode at the center of the image are positively and negatively charged, respectively. The tips of the vectors near the right bottom corner of the center electrode point to the corner. It is also seen that the left and the right sides of the gold substrate just below the top electrode are negatively and positively charged, respectively. Moreover, the distribution of electric charges alternatively switches at an interval of a half the period of the electric wave. These electric charges come from movement of free electrons inside the gold layer, and promotes two capacitors, C _m, at the right and left halves of the electrode. Thus, the equivalent circuit was established and radiation at 2.25 µm was absorbed due to magnetic resonance.

The second peak for the electrode width of 190 nm in Fig. 6 corresponds to optical interference inside the GaSb layer. Here, a part of the incident wave diffracts at the bottom corner of the electrode and propagates as a surface wave in a direction parallel to the gold surface. For the electrode width, Λ − w, of 190 nm, the central peak stands at 1.73 µm. Here, the refractive index of GaSb at 1.73 µm is approximately 3.82. Thus, it corresponds to the wavelength of 453 nm inside the GaSb layer. The shortened wavelength is close to the pitch lengths of the fishnet, Λ = 490 nm. Similarity is also observed in the results with a longer pitch length. Therefore, the wavelength of the interference peak increases linearly. This trend implies that the surface wave interferes with itself, depending on the pitch length of the periodic structure.

The third peak for the electrode width of 190 nm in Fig. 6 corresponds to CM1, whose resonant wavelength was affected by the reflection peak of the interference mode, and shifted so that it became slightly shorter than 1.6 µm. Figure 7(c) shows the distribution of an electromagnetic field around the fishnet-MSM at the resonant wavelength, 1.38 µm, for an electrode width, Λ − w, of 190 nm. Contrary to the case of MP, a strong magnetic field is observed at the center of the equivalent cavity. Its intensity is more than ten times greater than that of the incident wave. Additionally, an electric field is also promoted at the center, while it is comparably weak below the fishnet electrode. Thus, the abovementioned imaginary conductive wall is formed below the fishnet electrode to confine the electromagnetic energy inside the fishnet-MSM.

However, these separated peaks were attenuated with increasing electrode width. It is considered that these peaks dissipated due to their overlap with each other’s resonant mode, especially the horizontal interference mode excited around the strip-wired gold structure [21]. Furthermore, the larger the area occupied by the gold wire, the less radiation contributed to cavity resonance. Therefore, an electrode having a width of 100 nm with d = 115 nm was suitable for making the separated peaks inconspicuous while keeping the principal peak apparent.

6. Conclusions

As a metamaterial-TPV cell, a fishnet-MSM thin-multilayer was manufactured using wet etching, VDW bonding, EB lithography, and liftoff techniques. Then, its normal absorptance was measured to validate the equivalent cavity resonance absorption model for the fishnet-MSM. Although the square cavity of the top fishnet electrode shrunk slightly and became rounded because of backscattering and the proximity effect during EB lithography, the top fishnet electrode was shaped as designed, and the methodology to fabricate the fishnet-MSM was established. Here, the experimental absorptance spectra slightly departed from the FDTD simulation results, especially for the two distinct peak wavelengths. Another simulation result with an adjusted refractive index revealed that the cause was a decreased refractive index of the experimentally grown GaSb. An equivalent resonance cavity model was proposed to clarify the principle of absorbing radiation at two distinct peak wavelengths. The first and second eigenmode wavelengths indicated by the model roughly corresponded to the two peaks in the simulation results. Through further simulation, the equivalent resonance cavity model hardly functioned as indicator of the resonant wavelength with the fishnet electrode width broader than the GaSb thickness. This was because the magnetic resonance and interference of the surface wave surrounding electrode partially canceled the cavity resonance. Consequently, the model was verified as a useful indicator of the peak wavelength for a fishnet-MSM, when its fishnet electrode width is narrow.

Funding

Japan Society for the Promotion of Science (17H03184, 20H02084).

Acknowledgements

The authors would like to thank the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant in Aide (Number: 17H03184, 20H02084) for their financial support. A part of this work was conducted at the Takeda Sentanchi Supercleanroom, The University of Tokyo, supported by “Nanotechnology Platform Program” of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan, Grant Number JPMXP09F19UT0144. The absorptance measurements were conducted at the Center for Analytical Instrumentation Chiba University. Additionally, surface analysis using AFM was conducted at the Open Facility Center, Material Analysis Division, Tokyo Institute of Technology. All simulations in this study were conducted using the TSUBAME 3.0 supercomputer provided by Global Scientific Information and the Computing Center at Tokyo Institute of Technology.

Disclosures

The authors declare no conflicts of interest.

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Spectral absorptance of a metal-semiconductor-metal thin-multilayer structured thermophotovoltaic cell

Abstract

1. Introduction

2. Methodology of sample fabrication

3. Numerical simulation

4. Surface measurement of the sample

5. Results and discussion

6. Conclusions

Funding

Acknowledgements

Disclosures

References

Cited By

Figures (7)

Equations (10)

Optics Express