Introduction

Electromagnetic absorbers are greatly utilized in different devices. There are different types of absorbers such as microwave [1], metamaterial (MTM) [2], and plasmonic absorbers [3]. The function of plasmonic absorbers is based on the surface plasmon polariton (SPP) or localized surface plasmon (LSP) [4, 5]. Plasmonic absorbers are widely proposed to be utilized in sensors [6], solar cells [7], modulators [8], switches [9], etc.

Absorption tunability has been a great consideration for many researchers [10], and in this case, liquid crystal (LC) has been widely chosen due to its flexible anisotropic property [11]. LC has been utilized in MTM [12] or plasmonic absorbers [13] to make a tunable absorber by applied electric field [14] or Argon laser if azo dye is added to LC to make an all optical tunable absorber [15]. Furthermore, liquid crystal can be used in different frequency regions. In [16], a microwave-tunable absorber has been proposed that can be tuned by external bias voltage applied to the LC. Shrekenhamer et al. have fabricated a single-band terahertz MTM absorber operating in 2–3.5-THz region with the maximum absorption of 0.8 at 2.62 THz that is tunable about 0.12 THz by both frequency and voltage rate of the applied voltage [17]. A near-infrared tunable plasmonic absorber has been fabricated with two perfect absorption peaks and tunability of 30 nm [15]. A tunable plasmonic broadband absorber has been proposed that is sensitive to the size and angle of nanorods and LC acts as the host medium for nanorods to create a multi-resonance effective medium [13].

Effective medium has been a matter of discussion for many years and there have been many investigations to model the effective medium [18]. Maxwell-Garnett [19], Garcia [20], and Bruggman [21] are well-known methods to identify the effective parameters of the effective medium. Gao et al. have investigated the nonlinear behavior of silver-dielectric nanocomposite with effective permittivity variations under different applied electric filed as well as metal shape, volume fraction, and incident angle [22]. In [23], an effective medium has been fabricated to create large third-order susceptibility. Furthermore, the implementation of effective medium in birefringence material has been done in [24].

Graphene is a two-dimensional material with interesting optical properties, which can be utilized in different opto-electronic devices [25]. Graphene does not support SPP at visible and near-infrared wavelengths and its absorption is 2.3% as Nair et al. have measured it [26]. In addition, graphene is capable of supporting SPP at mid-infrared wavelengths [27, 28]. Therefore, there are graphene-based plasmonic absorbers [29]. In [30], graphene is utilized to enhance the absorption due to the strong localized electric field caused by the magnetic resonances by the silver grating. Lu et al. have used graphene to enhance the absorption of ring groove structure in near-infrared region [31]. Zare et al. have proposed graphene-based plasmonic absorbers to improve absorption [32, 33].

In this work, a plasmonic absorber that consists of titanium nitride (TiN) nanospheres hosted by nematic LC has been proposed. It has been proved that TiN-nanospheres are capable of supporting localized surface plasmon resonance (LSPR) [34]. The proposed absorber has three near unity tunable absorption peaks in the wavelength range of 0.5 to 1.2 μm. Nematic type of LC is responsive to the applied voltage [14]. Hence, it can be a good candidate to make the proposed structure tunable. In addition, there are other geometric parameters to tune the structure with a strong absorption. In the proposed structure, the absorption tunability is enhanced to 76 nm. Also, there is an absorption peak in near-infrared region which is enhanced to more than 90% as a single sheet of graphene is on the silica layer sandwiched between the top gold layer and the nanocomposite.

The paper is organized as follows. The geometry and characteristics of the proposed absorber is presented in “Structure Geometry and Materials” section. In “Simulation Procedure and Results” section, the simulation procedure is explained and the results are depicted. The paper is concluded in “Conclusion” section.

Structure Geometry and Materials

The unit cell of the proposed absorber is depicted in Fig. 1. The absorber is composed of TiN-nanospheres in the liquid crystal host medium that is sandwiched between two gold layers. There is a silica layer with a thickness of t2 = 5 nm between the composite layer and the top gold layer. The gold substrate is thick enough with the thickness of t4 = 200 nm to avoid any transmittance. The length of the top square aperture is a = 150 nm. The radius of nanospheres is r = 10 nm and the thicknesses of the top gold layer and the composite layer are t1 = 10 nm and t3 = 100 nm, respectively. The period of the structure is P = 340 nm and the unit cell is repeated along x- and y-axes.

Fig. 1
figure 1

The schematic view of the proposed absorber

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The LC nanocomposite is a combination of nematic LC and TiN-nanospheres to create LSPR. Nematic LC is an anisotropic medium that its permittivity is described by a second rank tensor as the following formula [13]:

$$ \varepsilon =\left(\begin{array}{ccc}{\varepsilon}_o{\cos}^2\phi +{\varepsilon}_e{\sin}^2\phi & 0& \left({\varepsilon}_o-{\varepsilon}_e\right)\sin \phi \cos \phi \\ {}0& {\varepsilon}_o& 0\\ {}\left({\varepsilon}_o-{\varepsilon}_e\right)\sin \phi \cos \phi & 0& {\varepsilon}_e{\cos}^2\phi +{\varepsilon}_o{\sin}^2\phi \end{array}\right) $$
(1)

where εo = 3 and εe = 4 are ordinary and extraordinary permittivities, respectively [35], when z-axis is considered as the reference. The tilt angle of the LC molecules is denoted by ϕ, which is the angle between the optical and z-axes in case the optical axis is in x-z plane. LC molecules are oriented according to the applied voltage. When liquid crystal is illuminated by a linearly polarized wave, it is possible to define an effective permittivity as below [36]:

$$ {\varepsilon}_{\mathrm{eff}}={\left(\frac{\cos^2\phi }{\varepsilon_o}+\frac{\sin^2\phi }{\varepsilon_e}\right)}^{-1} $$
(2)

In the proposed absorber, the effective permittivity of the nanocomposite is analyzed and approximated by the Maxwell-Garnett formula. According to [37], the effective permittivity of the effective medium is given as follows:

$$ {\varepsilon}_{\mathrm{eff}}=\left(\frac{F}{3}\right)\left(\left(1+2p\right){\varepsilon}_{\mathrm{TiN}}+2\left(1+p\right){\varepsilon}_{\mathrm{LC}}\right) $$
(3)

The local field factor (F) is defined as follows:

$$ F=\frac{3{\varepsilon}_{\mathrm{LC}}}{\left({\varepsilon}_{\mathrm{TiN}}\left(1-p\right)+{\varepsilon}_{\mathrm{LC}}\left(2+p\right)\right)} $$
(4)

where p is the volume fraction, εLCand εTiN are the LC and TiN-nanosphere permittivities, respectively. Moreover, Drude model can describe the permittivity of TiN-nanospheres which is approximated as follows [38]:

$$ {\varepsilon}_{\mathrm{TiN}}={\varepsilon}^{\prime }+\left(\frac{\varepsilon^{{\prime\prime} }+A{\omega}_{\mathrm{p}}^2{V}_{\mathrm{F}}}{\omega^3r}\right) $$
(5)

where ε′ and ε″ are the real and imaginary parts of the bulk TiN permittivity, respectively. Johnson and Christy data are utilized to describe the bulk permittivities of gold [39] and TiN [40]. Also, Lemarchan data is used to describe the silica permittivity [41]. In Eq. (5), ωp = 2.15 × 1015(rad/s) is the plasma frequency [42], VF = 1.39 × 106 (m/s) is the TiN Fermi velocity [43], r is the radius of TiN-nanospheres, and A = 5 is to show the additional damping of electron and the quantity is considered as mean free path of electron [38].

The real and imaginary parts of the effective permittivity of the nanocomposite structure for ϕ = 0∘, ϕ = 45∘, and ϕ = 90∘ are shown in Fig. 2.

Fig. 2
figure 2

The real (a) and imaginary (b) parts of the nanocomposite structure for different values of ϕ when p = 0.001

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Graphene is simulated by a sheet and characterized by the surface conductivity. According to Kubo formulation, the graphene conductivity is given as follows [44]:

$$ {\sigma}_{\mathrm{g}}\left(\omega, {\mu}_{\mathrm{c}},\Gamma, \mathrm{T}\right)={\sigma}_{\mathrm{intra}}+{\sigma}_{\mathrm{inter}},\kern0.5em $$
(6)

where σintra is the intraband electro-photon scattering that is described as the following:

$$ {\sigma}_{\mathrm{intra}}=-j\frac{e^2{k}_{\mathrm{B}}T}{\pi {\mathrm{\hslash}}^2\left(\omega -j2\Gamma \right)}\left[\frac{\mu_{\mathrm{c}}}{k_{\mathrm{B}}T}+2\ln \left(\exp \left(-\frac{\mu_{\mathrm{c}}}{k_{\mathrm{B}}T}\right)+1\right)\right] $$
(7)

and σinter is the interband transition which is formulated as follows:

$$ {\sigma}_{\mathrm{inter}}=-j\frac{e^2}{4\pi \mathrm{\hslash}}\ln \left[\frac{2\left|{\mu}_{\mathrm{c}}\right|-\left(\omega -j2\Gamma \right)\mathrm{\hslash}}{2\left|{\mu}_{\mathrm{c}}\right|+\left(\omega -j2\Gamma \right)\mathrm{\hslash}}\right] $$
(8)

Equation (8) is approximated when kBΤ <  < |μc|, ℏω. ℏ, μc, and Γ = 1/2τ are reduced Planck constant, chemical potential, and phenomenological scattering rate, respectively. The momentum relaxation time is denoted by τ and T is temperature. In our simulation, T = 300 K, μc = 0.15 eV and Γ = 0.659 meV [45].

Simulation Procedure and Results

The finite element method (FEM) is utilized to calculate the absorption of the structure. The structure is illuminated normally by a plane wave along the z-axis and the periodic boundary condition (PBC) is applied along x- and y-axes. The absorption is calculated according to A = 1 − R − T, where T and R are the transmission and reflection powers, respectively. The absorption spectra of the proposed structure, when it is filled with air, pure LC, and TiN-nanosphere/LC hybrid, are shown in Fig. 3. It is obvious that without the nanocomposite structure, the absorption is low and there is no resonance in our desirable wavelength range. It is worth noting that by utilizing LC, three minor resonances appear in the analyzed wavelength region. The minor resonances are regarded as the intrinsic resonances of the structure. By utilizing TiN-nanosphere/LC hybrid, three more than 90% absorption peaks appear at 0.66-, 0.76-, and 0.95-μm wavelengths, when ϕ = 90∘, a = 150 nm, r = 10 nm, and p = 0.001. The resonance of the TiN-nanosphere/LC hybrid is due to the LSPR on the surface of TiN-nanospheres. The electric filed distribution of TiN-nanospheres at the wavelength of 0.6 μm in the xy plane is shown in Fig. 4 that five TiN-nanospheres are in each row and the distance between the adjacent nanospheres is denoted by d = 50 nm. It is obvious that the LSP resonance of nanospheres near the aperture is more than the peripheral ones.

Fig. 3
figure 3

Absorption spectra of the proposed structure filled with air, pure LC, and TiN-nanosphere/LC hybrid when ϕ = 90∘, p = 0.001, a = 150 nm, and r = 10 nm

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Fig. 4
figure 4

(a) An array of TiN-nanospheres in LC with r = 10 nm, a = 150 nm, and ϕ = 90°. (b) Electric field distribution of LSP resonance of TiN-nanospheres at λ = 0.6 μm

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In order to show the tunability of the structure, the absorption spectra for different values of ϕ that is caused by the applied voltage are depicted in Fig. 5. It is obvious that by increasing ϕ, there is a red shift in the absorption spectrum. This phenomenon is in accordance with the red shift of the real and imaginary parts of the effective permittivity resonance of the Fig. 2. Therefore, the resonance of the structure depends on the resonance wavelength of the LC nanocomposite structure.

Fig. 5
figure 5

Absorption spectra for different angles of LC optical axis (ϕ = 0∘, 45∘, and 90∘) as p = 0.001, a = 150 nm, and r = 10 nm

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In our simulations, the tunability of the absorption wavelength is 76 nm in average, which is fully controllable by the applied voltage and can be enhanced if the LC birefringence, Δn(θ) = ne − no, is chosen to be the maximum value [46]. When ϕ = 0o, the absorption peaks are at λ1 = 0.6 μm, λ2 = 0.69 μm, and λ3 = 0.83 μm, while the absorption peaks occur at λ1' = 0.66 μm, λ'2 = 0.76 μm, and λ'3 = 0.95 μm as ϕ = 90o.

In order to investigate the effect of nanosphere radius, volume fraction, and length of the aperture in top gold layer, our simulations are executed with different geometry values. The real and imaginary parts of the effective permittivity of the TiN-nanosphere/LC hybrid for ϕ = 90∘ and radii of 5, 10, and 15 nm are demonstrated in Fig. 6.

Fig. 6
figure 6

The real (a) and imaginary (b) parts of the effective permittivity of TiN-nanosphere/LC hybrid for different nanosphere radii when ϕ = 90∘ and p = 0.001

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It can be seen that the resonance of the TiN-nanosphere/LC hybrid remains the same by varying the nanosphere radius and only there is a small change in the amplitude of the resonance. Therefore, the radius of nanospheres is not considered as a degree of freedom to tune the absorber. The absorption spectra for different nanosphere radii are depicted in Fig. 7.

Fig. 7
figure 7

The absorption spectra for different nanosphere radii when ϕ = 90∘, a = 150 nm, and p = 0.001

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The real and imaginary parts of the effective permittivity for different values of volume fraction (p) are shown in Fig. 8. The amplitude of the LC nanocomposite resonance is enhanced as the concentration of the TiN-nanospheres is increased.

Fig. 8
figure 8

The real (a) and imaginary (b) parts of the effective permittivity for volume fractions of p = 0.001, p = 0.005, and p = 0.01 as ϕ = 90∘ and r = 10 nm

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It is worth noting that as the volume fraction increases, the real part of the nanocomposite permittivity is decreased at the wavelengths near 0.7 μm. The decrement of the TiN-nanosphere/LC hybrid permittivity causes a blue shift in the absorption peaks that are smaller than 0.7 ‐ μm wavelengths. However, there is a red shift for the absorption wavelengths greater than 0.7 μm. The absorption spectra for the mentioned volume fractions are shown in Fig. 9.

Fig. 9
figure 9

The absorption spectra for p = 0.001, p = 0.005, and p = 0.01 as ϕ = 90∘, a = 150 nm, and r = 10 nm

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The length of the aperture in the top gold layer is one of the most important factors in the design of the proposed structure. The absorption spectra for different square lengths are depicted in Fig. 10. The absorption behavior of the structure is changed by different values of the aperture length. When a = 150 nm, there are three more than 90% absorption peaks in the absorption spectrum. For the case of a = 200 nm, also, there are three absorption peaks but the absorption is decreased to 75%. The structure has two absorption peaks as a = 250 nm that are lower than 90%.

Fig. 10
figure 10

The absorption spectra for different aperture lengths when ϕ = 90∘, p = 0.001, and r = 10 nm

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The absorption of the structure can be enhanced by utilizing a single sheet of graphene on the silica layer, as shown in Fig. 11. The light and graphene interaction is influenced by the near field enhancement of LSP created on the TiN-nanospheres. In other words, LSPR of TiN-nanospheres boosts the light absorption in graphene, so the absorption value of the whole structure is enhanced to 90% due to the light and graphene interaction.

Fig. 11
figure 11

The absorption spectra of the structure of Fig. 1 as a single sheet of graphene with μc = 0.15 is used on the silica layer sandwiched between top gold layer and LC nanocomposite when ϕ = 90∘, a = 200 nm, r = 10 nm, and p = 0.001

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Conclusion

In this paper, a tunable plasmonic absorber based on TiN-nanosphere/liquid crystal (LC) nanocomposite is studied. It is shown that the flexible property of LC permittivity plays a great role to make the device tunable. In addition, volume fraction is another degree of freedom to make the absorber tunable. The nanosphere radius of the nanocomposite structure is not considered as a parameter to tune the absorber due to its insignificant effect. The length of the top aperture is another parameter of the structure which is decisive. Overall, the proposed absorber has three more that 90% absorption peaks at visible and near-infrared wavelengths. Moreover, it is shown that the decrement of the absorption caused by the structure dimensions can be compensated by a single sheet of graphene over the nanocomposite.