1. Introduction

Due to the unique electrical and optical properties of the transition metal dichalcogenides (TMDCs), these atomically thin materials are widely used in designing miniaturized photonic and optoelectronic devices [1,2,3]. TMDCs with a general chemical formula of MX$_2$ consist of a honeycomb arrangement of $M$ atom layers sandwiched between two $X$ atom layers where all layers are stacked together by a weak van der Waals bond [4,5]. Compared with the other widely used two-dimensional material, graphene, with zero band gap; TMDCs have a nonzero band gap which makes them suitable in designing transistors, optical sensors, solar cells, and memories [6–9].

From TMDC’s family, molybdenum disulfide (MoS$_2$) with acceptable carrier mobility and low carrier dissipation is a suitable candidate for utilization in transistors, optoelectronic devices, and photovoltaics [10–12]. MoS$_2$ in its bulk state has an indirect bandgap while reducing its dimension to atomically sized thickness turns it into a material with a direct bandgap in the visible wavelength range. All these advantages of MoS$_2$ would be remained unused until less than $7\%$ of the incident light is absorbed in a freestanding MoS$_2$ monolayer in the wavelength range of $500$ to $800$ nm [13,14].

To increase MoS$_2$ absorption in one specific wavelength or a range of wavelengths utilizing different metallic or dielectric structures in one- (1D), two- (2D), or three-dimensions (3D) are proposed [15–19]. Absorbers with their application in optoelectronic and photothermal contexts play an important role in solar energy harvesting, sensing, and photodetectors. [20–22] Application of the absorbers depends strongly on their bandwidths. Narrowband absorbers are mostly used in sensors, the wideband ones are used in photovoltaics, solar cells, and thermophotovoltaic applications. [23,24].

As it is reported in [25], narrowband absorption efficiencies of $98.3\%$ are reached by stacking the defect layers of MoS$_2$ in a 1D defective photonic crystal (DPC) or defective quasi-photonic crystal (DQPC). Although this absorption value is significant, fabrication of such DPCs or DQPCs with more than hundreds of layers is not feasible. In another work, by using a low contrast 2D dielectric grating beneath the MoS$_2$, the narrowband absorption of the structure reaches a maximum of $91\%$ when the MoS$_2$ itself is cut out as ribbons. This way, the MoS$_2$ has lost its continuity that prevents the flow of carriers in photovoltaic applications [26].

To achieve wideband absorption peaks, in [14, 27], by the deployment of different TMDCs in a 1D structure or another general method, employing scatterers with different sizes and shapes in one unit cell of the structure [24,28–30], wideband absorption peaks are reported.

Apart from the TMDC-based absorbers, other absorbers which are utilizing metallic gratings or nanoparticles in their structures for narrowband or broadband absorptions are also being investigated. In [31] by utilizing gold gratings in the absorber structure, narrowband absorption is reached which was applied in polarization detection. The gold grating was fabricated by e-beam lithography followed by a lift-off process. In the broadband absorption regime, by utilizing gold ribbons on silica/chromium/silica multilayer, a polarization-sensitive tunable absorber is reported in [32]. The $200$ nm gold ribbons were reported to be patterned on the SiO2 layer by nanoimprint lithography.

In this work, in contrast to the previously suggested absorbers, we introduce an absorber with the capability of tuning the its bandwidth, which makes the designed absorber practically applicable in a wide range of applications. Both narrowband and wideband absorption peaks in both total absorption of the structure and the MoS$_2$ monolayer absorption are reached by utilizing 2D inclined metallic gratings on a MoS$_2$ monolayer that lands on a silica (SiO$_2$) substrate and resonant excitation of surface plasmon polaritons (SPPs) and localized surface plasmons (LSPs) in it. In our recently published paper, the dimension of the used gold grating period was set to one-tenth of the incident wavelength, and we reached $88\%$ total absorption in the whole visible spectrum. In contrast to that work, in the current paper, our gold grating’s period size is set to be of the order of the impinging light wavelength which results in the possibility of both narrowband and wideband absorption peaks in the structure. In our previous work [33], excitation of the "reflected mode" together with the LSP mode made the absorption of the structure broadband, while in our current study, selective excitation of SPPs, LSPs, or combination of SPPs and LSPs in the structure determine the absorption peak wavelength and full width at half maximum (FWHM). It is shown in [34] that applying curvature to the metallic ribbons, broadens the absorption spectrum of graphene. Here, we will show that the gap size between two successive ribbons of the structure is the determining parameter that either SPPs or LSPs excite in the structure and to excite simultaneously both SPPs and LSPs, ribbons are taken to be inclined.

The SPPs excite on a metal-dielectric interface only when a transverse magnetic (TM) polarized light illuminates the interface and the interface itself is covered by a grating, or prism to increase the wavevector of the incoming light. SPPs propagate at the metal-dielectric interface by the collective oscillation of the free electrons of the metal [35]. While the LSPs are produced by rearrangement of the free carriers of the metallic nanoparticles or nanovoids which is accompanied by the localized trap of the incoming light [35].

We will show that by changing the geometrical parameters, period of the structure, width, height, and inclination of the ribbons, the absorption spectrum can be controlled completely. In the case of SPP excitation, the absorption peak wavelength can be tuned by geometrical parameters. While in the case of LSP or SPP and LSP excitation, both absorption peak and FWHM of it can be set. Having control of the absorption peak wavelength and the peak’s FWHM are promising achievements in sensor applications and photovoltaics that we reached in this work.

2. Absorber geometry and parameters

The absorber structure with its inclined gold gratings on the MoS$_2$ monolayer is shown in Fig. 1. Ribbons of the gold grating are repeated in the x-direction with the period of p and are extended to infinity along the $z$-direction. The $h$, $w$, and $g$ are the height, width, and inclination of the metallic ribbons, respectively. The MoS$_2$ monolayer is placed on a SiO$_2$ substrate, and the incident light illuminates the structure normally with its wave vector along the $y$ direction. With its magnetic field perpendicular to the incident plane, $H_{z}$, the incoming light has the polarization of TM. Dielectric constants, the real and imaginary part of the refractive index of MoS$_2$, gold, and SiO$_2$ are taken from [14], [36], [37], respectively. In our simulations, along the $x$-direction, one unit cell is surrounded by periodic boundary conditions which is equivalent to the extension of the employed ribbons and the structure to infinity. In the $y$-direction, at one end of the simulation area, the periodic port illuminates the structure while, at the other end, the structure terminates by perfectly matched layers. Calculation of the structure’s absorption is done by reducing the reflected and transmitted light values through the structure from unity, while, MoS$_2$ absorption is reached by integrating over the absorbed heat in the MoS$_2$ domain.

 

Fig. 1. schematic of the absorber structure. Inclined gold gratings are demonstrated to stand on MoS$_2$ monolayer and they are placed on the silica substrate.

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To fabricate this structure, MoS$_2$ ultra-thin films in form of mono- to few-layers can be manufactured by different techniques. Such layers can be made directly on substrate mainly by chemical vapor deposition [38] or other techniques such as sputtering [39], sulfurization and electrochemical methods [40]. They can also be fabricated on another substrate or formed into exfoliated layers and then transferred on any demanding substrate. The edged gold long ribbons of the structure can be made in successive steps of gold layer deposition, electron beam lithography, and etching. Deposition can be suggested through e-beam evaporation or sputtering technique. It is possible to reach such octahedron-or-edged shape ribbon by plasma nano-machinery process [41] where different forms can be achieved with controlled lateral size.

We start our survey by designing absorbers with periods of the order of the incoming light wavelength. In Fig. 2(a) and Fig. 2(b), respectively, the total absorption of the structure and that of the MoS$_2$ under light illumination within the wavelength range of $600$ nm to $800$ nm are studied. Structure parameters are set as $h=100$ nm, $g=0$, and $p-w=100$ nm, where $p$ is swept over $400$ nm to $800$ nm. From the studied cases, the two structures with $p=600$ nm and $p=700$ nm, the blue and green solid lines in Fig. 2(a) and Fig. 2(b), encounter total absorption of $70\%$ at $\lambda =667$ nm and $78\%$ at $\lambda =757$ nm, respectively. Considering the MoS$_2$ absorption that reaches $40\%$ at $\lambda =667$ nm for the structure with $p=600$ nm, this structure is selected for further studies in the following figures.

 

Fig. 2. a) total and (b) MoS$_2$ absorption spectrum of the structure with $h=100$ nm, $g=0$, and $p-w=100$ nm, with $p$ values which are swept over $400$ nm to $800$ nm. In case of $p=700$ nm, total absorption of $70\%$ and MoS$_2$ absorption of $40\%$ is reached.

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3. Tunability of the absorption peak wavelength

Following the previously studied structure of Fig. 2, we started investigating the ribbon width’s effect on the absorption spectrum of the structure with $p=600$ nm, $h=100$ nm, and $g=0$. As can be seen in Fig. 3(a), the redshift of the absorption peak’s wavelength from $\lambda =617$ nm to $\lambda =668$ nm occurs by sweeping over specific values of $w$ from $w=200$ nm to $w=550$ nm. A maximum absorption value of $89\%$ is reached with $w=250$ nm at $\lambda =630$ nm, while in the covered wavelength range, none of the structures has absorption values less than $76\%$ which is a good achievement in this range of wavelength. Interestingly, by increasing the value of $w$ to sizes greater than $450$ nm, the absorption peaks start blue shifting which tells us about a change in the absorption peak’s nature. As can be seen in Fig. 3(b), the same procedure can be distinguished in MoS$_2$ absorption, in which from $w=200$ nm to $w=450$ nm absorption peak redshifts but by increasing the $w$ value further, the absorption peak blue shifts.

 

Fig. 3. a) total and (b) MoS$_2$ absorption spectrum of the structure with $p=600$ nm, $h=100$ nm, and $g=0$ with different values of $w$, from $w=200$ nm to $w=550$ nm. By increasing the $w$ up to $450$ nm, the absorption peak wavelength redshifts, while further increasing $w$ shifts the peak wavelength to blue.

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To investigate the reason for these red and slight blue shifts, we select the two cases of $w=250$ nm and $w=550$ nm and investigate their transverse magnetic, $H_z$, and inplane electric field components, $E_y$, of the exciting peaks at their relevant peak wavelengths, $\lambda =630$ nm and $\lambda =665$ nm in Fig. 4(a) and Fig. 4(c), and Fig. 4(b) and Fig. 4(d), respectively.

 

Fig. 4. a) and (b)transverse component of the magnetic field, $H_z$, distribution, (c) and (d) in-plane component of the electric field, $E_y$, distribution for the two cases of (a) and (b) $w=250$ nm at $\lambda =630$ nm and (c) and (d) $w=550$ nm at $\lambda =665$ nm. Other parameters of the structure are $p=600$ nm, $h=100$ nm, and $g=0$. Excitation of SPPs in the structure with $w=250$ nm can be deduced from (a) and (c), while LSP excitation is inferred from (b) and (d) within the structure with $w=550$ nm.

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It has worth mentioning that for more clarification, in Fig. 4, two unit cells of the structure are included to study the field distributions and in all of the figures, the maximum blue color which is denoted by "min" in the color bar refers to the maximum negative value of the field. By looking at the $H_z$ distribution of Fig. 4(a), within the structure with $w=250$ nm at $\lambda =630$ nm, excitation of surface waves in the form of SPPs both at the top and at the bottom of the gold gratings are obvious. Considering its $E_y$ distribution in Fig. 4(c) shows that these excited SPPs, in the $x$ direction, rearrange the charge distribution in the structure in a way that charges with different signs accumulate at sharp edges of a single ribbon which repeats throughout the whole structure. Increasing the width of the ribbons causes the accumulated opposite sign charges on the ribbon’s edges to experience more distance which causes the excitation of weaker SPPs that their excitation wavelength shifts to red. This redshift is what exactly happens when $w$ changes from $200$ nm to $450$ nm in Fig. 3 and we select $w=250$ nm as their representative.

By studying the field distribution of the structure with $w=550$ nm which is accompanied by less ribbon’s gap compared to the case of $w=250$ nm, and looking at its $H_z$ distribution in Fig. 4(b), localization of the incident wave at the gaps between the gold ribbons can be recognized. Excitation of LSPs can be seen in its $E_y$ field distribution as charges with opposite signs accumulate at the adjacent gold ribbons. Larger ribbon widths cause tighter gaps, which in turn causes the opposite sign charges to come closer which leads to the excitation of more strengthened LSPs and a blue shift in the spectrum of the absorption peaks.

Transformation of the excited modes in the structure from SPP to LSP by increasing the gold ribbon’s width causes the excited modes at the absorption peak wavelength to convert their shift from red to blue.

4. Tunability of the absorption peak FWHM

To investigate the effect of the gold ribbon’s inclination on the structure and MoS$_2$ absorption, in Fig. 5 we tested the ribbon’s inclination effect on three different cases: first, structures with excitation of SPPs with $w=250$ nm (Fig. 5(a) and Fig. 5(b)), second, structures with larger $w$ and excitation of LSPs with $w=550$ nm (Fig. 5(c) and Fig. 5(d)), and the last one, ribbons with $w=600$ nm (Fig. 5(e) and Fig. 5(f)). In all three cases, $p$ sets to be $600$ nm and $h$ varies from one structure to the other to achieve acceptable values of light absorption in the structure and MoS$_2$.

 

Fig. 5. (a)/ (c)/ (e) total and (b)/ (d)/ (f) MoS$_2$ absorption when $p=600$ nm with $w=250$ nm, $h=100$ nm/ $w=550$ nm, $h=150$ nm/ $w=600$ nm, $h=120$ nm for different values of $g$. Color legend of (e) is the same as (f). Increasing $g$ in (a) and (b)/ (c) and (d) causes blueshift/ broadening of the absorption peak with SPP/ LSP excitation. In (e) and (f), with excitation of both SPPs and LSPs, increasing $g$ widens the absorption peak. (g) and (h) $H_z$ field distribution at $\lambda =646$ nm (SPP excitation) and at $\lambda =710$ nm (LSP excitation) for the structure with $w=600$ nm and $g=200$ nm.

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The effect of the ribbon’s inclination increasing, on the shift of the absorption peak wavelength, is opposite to that of the ribbon’s width increase. In the case of SPP excitation, the ribbons’ width increase causes the excited charges with opposite signs in a single ribbon to separate more, which leads to a redshift of the absorption peak, while, by increasing the inclination, these charges excite closer and form stronger SPPs with a blue shift in their wavelength peak. In the case of LSP excitation, increasing the ribbons’ width causes smaller gaps between the gold ribbons which results in a closer accumulation of charges at the sharp edges of two successive ribbons, which means excitation of stronger LSPs with their wavelength peak that are shifting to blue. While increasing the inclination keeps these accumulated charges farther which is accompanied by the red shift of the absorption peak wavelength.

In the case of $w=250$ nm, and excitation of SPPs in the structure, by increasing the inclination from $g=5$ nm to $g=125$ nm, the blue shift of the absorption peak in the structure and MoS$_2$ is obvious in Fig. 5(a) and Fig. 5(b), respectively. In Fig. 5(a), by the inclusion of ribbons inclination, a maximum absorption value of $92\%$ in the structure with $g=10$ nm is achieved while, without any inclination, in Fig. 3(a) maximum value of absorption was $89\%$. Within the condition of excitation of SPPs in the structure, as the excitation of the SPPs is dictated by the strict condition of charge accumulation at sharp edges of a single gold ribbon, absorption peak only happens at specific wavelengths which limits their absorption bandwidth. The blue shift of the absorption peak with increasing inclination is what we discussed and occurs due to the excitation of stronger SPPs with closer opposite signed accumulated charges at a single ribbon.

In the case of $w=550$ nm and LSP excitation in the structure, inclination causes the peaks to get widened (Fig. 5(c) for total and Fig. 5(d) MoS$_2$ absorption). The more the inclination, the wider the absorption peak is. By changing $g$, FWHM tunability from $74$ nm for $g=80$ nm to $132$ nm for the structure with $g=140$ nm is reached (Fig. 5(c)), with maximum values of $69\%$ and $84\%$, respectively. Where maximum absorption of $91\%$ occurs within the structure with $g=100$ nm at $\lambda =720$ nm with FWHM of $80$ nm. MoS$_2$ absorption of Fig. 5(d) also reveals wider absorption peaks with the $g$ value increasing. A maximum MoS$_2$ absorption value of $54\%$ occurs at the structure with $g=100$ nm with the FWHM of $75$ nm. In our survey in finding the reason for such wideband absorption, we need to remind LSP excitation of Fig. 4(d), in which charges with negative signs accumulate on the walls of two successive gold ribbons (the gap walls). Of course, when these gap walls are straight only one wavelength will match the resonant excitation of LSPs, but when these walls are inclined other wavelengths find the chance of participation in resonant absorption which shows itself as a wideband absorption.

Another interesting design that we studied in Fig. 5(e) and Fig. 5(f), is the case of equality of the gold ribbon’s width to the period of the structure, i.e., $w=p=600$ nm, where the presence of the inclination prevents the ribbons to stick together. By increasing the inclination from $g=50$ nm to $g=300$ nm both total (Fig. 5(e)) and MoS$_2$ absorption (Fig. 5(f)) peaks become wider and wider. In the case of $g=200$ nm excitation of two distinct peaks can be recognized in the absorption spectrum which we marked with a circle, at $\lambda =646$ nm, and star, $\lambda =710$ nm, in Fig. 5(e). Significant total (MoS$_2$) absorption of $96\%$ ($60\%$) and $91\%$ ($56\%$) of the incoming light is reached at $\lambda =646$ nm and $\lambda =710$ nm.

To investigate the origin of these two peaks, the Hz distribution of the circle-signed peak at $\lambda =646$ nm, and the star-signed peak at $\lambda =710$ nm are included in Fig. 5(g) and Fig. 5(h), respectively. The $H_z$ distribution of Fig. 5(g) is the characteristic distribution of SPP excitation at the gold grating and silica substrate interface. While, from the $H_z$ distribution of Fig. 5(h), the origin of the star-signed peak can be recognized as LSP excitation.

By increasing the gold ribbon’s inclination, the blue shift of the circle-signed peak (SPP-based) together with the red shift of the star-signed peak (LSP-based) makes the bandwidth within which the structure absorbs the incoming light efficiently wider. The blue shift of the SPP peak with increasing the inclination of the ribbons occurs as the available area above the gold and silica interface gets tighter, and opposite sign charges are driven more together, which causes a blue shift in the excited SPP wavelength and absorption peak. While, by increasing the inclination, accumulated charges at the adjacent ribbon walls get farther which results in a redshift of the excitation wavelength of the LSPs, as we discussed also in Fig. 5(c).

In the case of $g=150$ nm, with total absorption of more than $90\%$, an FWHM of $95$ nm is reached which can be tuned by increasing the $g$ value to $200$ nm to an FWHM of $124$ nm.

For further investigation of the structure with $w=p=600$ nm and $g=200$ nm and checking the absorption peak FWHM tunability by the remaining structure parameters, the effect of the gold ribbon’s height, $h$, is studied in Fig. 6. As it is clear in Fig. 6(a) for total and in Fig. 6(b) for MoS$_2$ absorption, FWHM of the absorption peak reduces by increasing the ribbon’s height. As we discussed in Fig. 6(e) for the case of $w=p=600$ nm with ribbon inclination, two combined peaks, one with the origin of SPP and the other with LSP nature at longer wavelengths come together and form a wideband absorption. To check the validity of this model, we checked the $H_z$ distribution of the two peaks of the structure with $h=130$ nm, at $\lambda =653$ nm with $99\%$ of light absorption, and at $\lambda =700$ nm with $95\%$ absorption in Fig. 6(c) and Fig. 6(d), respectively. From Fig. 6(c) (Fig. 6(d)) nature of the enhanced absorption peak of the structure at $\lambda =653$ nm ($\lambda =700$ nm) can be recognized as SPP (LSP) excitation.

 

Fig. 6. (a) total and (b) MoS$_2$ absorption when $w=p=600$ nm and $g=200$ nm with different values of $h$. Increasing the $h$ value, decreases the bandwidth of the absorption peak. (c), (d) the $H_z$ field distribution at $\lambda =653$ nm (SPP excitation) and at $\lambda =700$ nm (LSP excitation) for the case of $h=130$ nm. (e) and (f) the $H_z$ field distribution at $\lambda =670$ nm and at $\lambda =673$ nm for the structures with $h=150$ nm and $h=160$ nm, respectively. These field distributions are illustrative of the combined excitation of SPPs and LSPs.

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By increasing the ribbons’ height, In Fig. 6(a) and Fig. 6(b), the SPP-based absorption peak shifts to red which occurs due to the added space to the ribbons with increasing the height of the ribbon and the possibility of charge accumulation to spread toward the edges of the ribbons. While the LSP peaks shift to the blue wavelengths by increasing the ribbons’ height which happens by the steeper gradient of the inclined ribbons that causes opposite charges to gather closer around the walls of successive ribbons’ gap, which is accompanied by the excitation of stronger LSPs and blue shift in the absorption peak.

By increasing the ribbons’ height to $h=150-160$ nm, the excited SPP and LSP peaks come together as they overlap at the same wavelength. Combined SPP and LSP excitation for the structure with $h=150$ nm at $\lambda =670$ nm and in the structure with $h=160$ nm at $\lambda =673$ nm can be recognized from the Hz field distribution of Fig. 6(e) and Fig. 6(f).

With sweeping over the ribbons’ height values from $h=100$ nm to $h=150$ nm, the FWHM of the absorption peak is tuned from $160$ nm to $110$ nm. The absorption peak FWHM of the structure with $h=130$ nm with a nearly perfect absorption value, above $95\%$, is $120$ nm which makes it a great choice in photovoltaic applications.

Within the studied absorber structure of this paper, with gold’s grating period of $600$ nm, we showed that excitation of SPPs occurs with the gold ribbon widths of $200$ nm to $450$ nm. For the excitation of LSPs, wider gold ribbons, $h=450$ nm to $550$ nm, with fewer gap sizes between them, are necessary. Increasing the inclined gold ribbon’s width to be equal to the grating’s period, $600$ nm, facilitates the excitation of both LSPs and SPPs. Tuning the absorption peak wavelength is possible by adjusting the ribbon’s widths at structures with narrowband SPP excitations. While FWHM control of the absorption peaks is possible with LSP excitation. Tunability of both absorption peak wavelength and FWHM of it is possible in the structures with simultaneous excitation of SPPs and LSPs. Within these structures, increasing the width and height of the gold ribbon causes the SPP absorption peak wavelength to redshift, while increasing the ribbon’s inclination leads to its blueshift of it. On the contrary, the excited LSPs blueshift with increasing the height of the ribbon and redshift with more ribbon inclination. This way, with the excitation of SPPs and LSPs together, with increasing the ribbon’s inclination, the absorption peak becomes wider, which makes the absorber applicable in solar cell and photothermal applications, while increasing the height of the ribbon causes the reduction of the absorption peak FWHM which makes the absorber suitable for sensing applications.

5. Conclusion

With the designed MoS$_2$-based absorber of this paper, which consists of 2D inclined gold gratings on a MoS$_2$ monolayer, we achieved high-efficiency absorptions with adjustable absorption peak wavelengths and FWHMs. We showed that within the selected grating period of the order of the incident wavelength, both SPPs and LSPs could be excited in the structure and enhance the absorbed light. Within the structure with $p=600$ nm, $h=100$ nm, and $g=0$, in which SPPs are dominant excited modes, by changing the ribbon width, $w$, from $200$ nm to $450$ nm, we reached the narrowband absorption peak’s wavelength shift from $\lambda =617$ nm to $\lambda =668$ nm with absorption efficiencies above $76\%$. By increasing the gold ribbon’s width further, the dominant excited mode changes to be in the form of LSPs. In the case of the structure with $p=600$ nm, $h=150$ nm, and $w=550$ nm, changing $g$ from $g=80$ nm to $g=140$ nm causes the FWHM of the absorption peak to get widened from $74$ nm to $132$ nm. We showed that simultaneous excitation of LSPs and SPPs in the structures with p = w is possible, in which by changing the other parameters of the structure, $h$, and $g$, we controlled the FWHM of the absorption peak. We showed that by sweeping over the ribbons’ height values from $h=100$ nm to $h=150$ nm within the structure with $p=w=600$ nm and $g=200$ nm the FWHM of the absorption peak is tuned from $160$ nm to $110$ nm. In one geometry with $p=w=600$ nm, $g=200$ nm, and $h=130$ nm, nearly perfect absorption occurs with the wideband FWHM of $120$ nm. Wavelength and FWHM control of the absorption peak, which is most important in sensor applications, photovoltaics, and modulators, can be tuned by adjusting the resonant wavelength of the excited SPPs and LSPs in the structure, which is a function of geometrical parameters of the structure.