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Tunable and multifunctional terahertz devices based on one-dimensional anisotropic photonic crystals containing graphene and phase-change material

Xiangfei Gao, Zebin Zhu, Jing Yuan, and Liyong Jiang

Open Access Open Access

Abstract

In the past few years, designing tunable and multifunctional terahertz devices has become a hot research area in terahertz science and technology. In this work, we report a study on one-dimensional anisotropic photonic crystals (1D APCs) containing graphene and phase-change material VO2. We numerically demonstrate the band-pass filtering, perfect absorption, comb-shaped extraordinary optical transmission and Fano-like resonance phenomenon in pure 1D APCs and 1D APCs with a VO2 defect layer under different conditions of a tangential wave vector. The performance of these phenomena in the terahertz region can be modulated by changing the chemical potential of graphene. The band-pass filter and perfect absorber functions of 1D APCs with a VO2 defect layer can be freely switched by changing the phase of VO2. We employ the equivalent-permittivity model and dispersion-relation equation to give reasonable explanations on these behaviors.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

In the past few decades, one-dimensional photonic crystals (1D PCs) have been widely studied due to the advantages of easy fabrication, low cost, and high compatibility. 1D PCs are usually made of alternating dielectric layers with different refractive indexes. The prominent band-gap feature of a 1D PC makes it as an ideal reflector for potential applications related to mode coupling and enhancement [1]. By inserting a defect layer into the 1D PC, strong photonic localization feature can be obtained and makes the 1D PC as an ideal band-pass filter for potential applications related to mode coupling and selection [2]. For either reflector or band-pass filter, wide band gap, all-angle response, and polarization insensitivity are main targets in the design. The size of the Bragg band gap in dielectric 1D PCs can be scaled up by increasing the refractive-index ratio of alternating layers while the boundaries of the Bragg band gap are usually blue-shifted towards higher frequencies as the incident angle increases. Meanwhile, for a given incident angle, the TE band gap is usually wider than the TM band gap. This is because the TM Bragg band gap in dielectric 1D PCs will be gradually closed up due to the Brewster effect [3].

In order to obtain broad-band and all-angle responses in conventional 1D PCs, researchers have proposed several effective methods, such as constructing the 1D PC heterostructure in the frequency domain or in the incident-angle domain [4,5], as well as using quasi-period 1D PCs [6,7]. Alternatively, researchers have developed some 1D metamaterial-based PCs which show dispersionless and polarization insensitive band gaps, such as the zero-average-refractive-index band gap in the 1D PC composed of alternating layers of positive- and negative-index media [8,9], the zero-effective-phase band gap in the 1D PC heterostructure composed of alternating layers of epsilon-negative (ENG) and mu-negative (MNG) materials [10–12], and the phase-variation-compensation band gap in the 1D PC composed of alternating layers of hyperbolic metamaterials and dielectrics [13,14].

Except for above efforts, tunable 1D PCs have also drawn a lot of attentions in the past few years. Typical materials used in tunable 1D PCs include graphene and phase-change materials (such as VO2 and Ge2Sb2Te5) [15–18]. Graphene is a uniaxial anisotropic material which shows different conductivities in the surface and normal planes. The conductivity of graphene in the surface plane can be freely modulated via electrostatic biasing. In the terahertz band (0.1–10 THz), its permittivity satisfies the Drude-model and thus can support the surface plasmons [19]. In previous works, researchers have investigated the tunable transmission and absorption properties in 1D graphene-dielectric multilayers and 1D dielectric multilayers containing graphene defect layers [20–34]. In many studies, graphene is simplified as a conductor sheet without thickness or an isotropic material during the transmission calculation based on the transfer matrix method (TMM) [20–24]. When the studies involve the full-wave or hyperbolic dispersion relations of 1D graphene-dielectric multilayers, the graphene should be rigorously treated as an anisotropic material with a finite thickness [28–34] and the 1D graphene-dielectric multilayers can also work as “metallic-dielectric” metamaterials [28–32]. As a typical phase-change material, VO2 will change from the monoclinic-lattice structure (insulator phase) to the tetragonal-lattice structure (metal phase) when the temperature reaches around 68 °C, accompanying with a fast and reversible change of permittivity before and after the phase transition [35]. In some previous works, researchers have reported band-pass filters or absorbers when VO2 is usually applied as a defect layer under insulator phase or surface coating under metal phase in conventional 1D dielectric PCs [15,16].

In principle, the coaction of graphene and phase-change material can not only modulate the performance but also switch the functions of a particular device. Recently, Hajian et al. proposed a VO2-hBN-graphene-based metamaterial which shows bi-tunable transmission and absorption characteristics in the middle infrared region [36]. Wang et al. proposed a dual-controlled switchable broadband terahertz absorber based on graphene-VO2 metamaterial [37]. Unfortunately, to the best of our knowledge, a similar idea has not been applied in 1D PCs yet. In previous studies related to tunable 1D PCs, the graphene or phase-change material is used alone [15–18,28–34], which provides only one available pathway for modulating the performance of proposed device. In this paper, we would like to report novel terahertz devices based on the coaction of graphene and phase-change material in the 1D anisotropic PC (1D APC) which is defined as a 1D PC incorporated with anisotropic material such as graphene. We will numerically demonstrate tunable and multifunctional terahertz devices in pure 1D APCs and 1D APCs with a VO2 defect layer under different conditions of tangential wave vector.

2. Models and methods

2.1 1D APC models without or with a VO2 defect layer

Figure 1 depicts the 1D APC models investigated in this paper. As shown in Fig. 1(a), a pure 1D APC is composed of alternating N layers monolayer graphene and Si arranged in the z direction. The permittivity and thicknesses of graphene and Si are represented by (ɛg, dg) and (ɛs, ds) respectively. As a comparison, Fig. 1(b) shows the model of a 1D APC with a single layer defect constructed by VO2 in the center. ɛv and dv represent the permittivity and thickness of VO2. In our study, the silicon layer is treated as a dispersionless and lossless material with a fixed permittivity of 11.7 according to the experimental database [38].

 figure: Fig. 1.

Fig. 1. (a) Model of a pure 1D APC made of monolayer graphene and Si alternatively arranged in the z direction. (b) Model of a 1D APC with a VO2 defect layer in the center.

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The surface conductivity of graphene includes the intraband electron-photon scattering term and interband electron transition term which can be derived from Kubo formula [39]:

$${\sigma _\textrm{g}}({\omega ,{\mu_c},\Gamma ,T} )= {\sigma _{\textrm{intra}}}({\omega ,{\mu_c},\Gamma ,T} )+ {\sigma _{\textrm{inter}}}({\omega ,{\mu_c},\Gamma ,T} )$$
$${\sigma _{\textrm{intra}}}({\omega ,{\mu_c},\Gamma ,T} )= i\frac{{{e^2}{k_B}T}}{{\pi {\hbar ^2}(\omega + i2\Gamma )}}[\frac{{{\mu _c}}}{{{k_B}T}} + 2\ln \left( {{e^{ - \frac{{{\mu_c}}}{{{k_B}T}}}} + 1} \right)]$$
$${\sigma _{\textrm{inter}}}({\omega ,{\mu_c},\Gamma ,T} )= i\frac{{{e^2}}}{{4\pi {\hbar ^2}}}\textrm{ln}[\frac{{2|{{\mu_c}} |- (\omega + i2\Gamma )\hbar }}{{2|{{\mu_c}} |+ (\omega + i2\Gamma )\hbar }}]$$
Where ω is the angular frequency, μc is the chemical potential, Γ is the scattering rate, T is the Kelvin temperature, e is the charge of an electron, ℏ is the reduced Planck constant, and KB is the Boltzmann constant. In the terahertz region, graphene behaves like a Drude-type material and the intraband conductivity plays a dominated role. For practical application, large-scale monolayer graphene can be fabricated by the chemical vapor deposition (CVD) method. In our study, a typical value of Γ = 1 THz is used for the scattering rate at room temperature T = 300 K [28,33,34].

As an anisotropic material, the out-of-plane permittivity of graphene sheet is 1 and the in-plane relative permittivity of graphene sheet can be written as [34,40]:

$${\varepsilon _g} = 1 + \frac{{i{\sigma _g}}}{{\omega {\varepsilon _0}{d_g}}}$$
Where ɛ0 is the vacuum permittivity. The thickness of graphene sheet is assumed to be dg = 0.34 nm. The permittivity of graphene sheet can be modulated by changing its chemical potential. Figure 2(a) shows the real part and imaginary part of ɛg as a function of frequency when the chemical potential of graphene varies from 0.4 eV to 1.0 eV with a step of 0.2 eV. In the lower frequency region of terahertz band, the real part of graphene’s permittivity is negative and the imaginary part is very high, which indicates the metallic property of graphene in this region. In Section 3B, we will show the metallic band gap of 1D APC based on the metallic property of graphene. Meanwhile, both real and imaginary parts of ɛg are sensitive to the chemical potential when the frequency is below 2 THz, which can be used to design terahertz sensor based on 1D APC in Section 3D. As shown in Fig. 2(b), both real and imaginary parts of graphene’s refractive index increase with graphene’s chemical potential, which can be used to explain the tunable band gaps, transmission peaks and absorptions peaks of 1D APC without or with a defect layer in Section 3B-D.

 figure: Fig. 2.

Fig. 2. The variation of complex permittivity (a) and refractive index (b) of monolayer graphene as a function of frequency for different chemical potentials. The insets show the zoomed-in view for selected plots.

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The phase transition temperature (∼68 °C) of VO2 is close to room temperature, which makes it an appropriate candidate for a defect layer of photonic crystals. There is a drastic change of electrical conductivity when insulator-metal phase transition (or metal-insulator phase transition) of VO2 occurs during the heating (or cooling) process. The properties of VO2 film in incomplete phase transition in the terahertz band are usually described by the Effective Medium Theory (EMT) [41], in which the composite permittivity is determined by:

$${\varepsilon _v} = \frac{1}{4}\left\{ {{\varepsilon_D}({2 - 3f} )+ {\varepsilon_M}({3f - 1} )+ \sqrt {{{[{{\varepsilon_D}({2 - 3f} )+ {\varepsilon_M}({3f - 1} )} ]}^2} + 8{\varepsilon_D}{\varepsilon_M}} } \right\}$$
Where f is the volume fraction of the metal component. ɛD and ɛM are the permittivity of the insulator phase and the metal phase of VO2 film, respectively. In this work, the VO2 film is considered in the complete insulator phase when f = 0 and the metal phase when f = 0.95 [42]. Figure 3(a) indicates the permittivity of VO2 film in the terahertz region under 30 °C (insulator phase). It can be seen that the real part of the permittivity remains at 9, and the imaginary part is negligible. As a comparison, under 90 °C (metal phase), the real part of the permittivity slightly changes with a negative value while the imaginary part varies fast when the frequency is below 2 THz [Fig. 3(b)].

 figure: Fig. 3.

Fig. 3. The variation of complex permittivity of VO2 as a function of frequency when it is in the insulator phase (a) and metal phase (b), respectively.

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2.2 Equivalent permittivity and transfer matrix method for the graphene-Si 1D APC

The equivalent permittivity tensor of the graphene-Si 1D APC can be written as [31]:

$$\varepsilon = \left[ {\begin{array}{{ccc}} {{\varepsilon_t}}&0&0\\ 0&{{\varepsilon_t}}&0\\ 0&0&{{\varepsilon_n}} \end{array}} \right]$$
Where ɛt and ɛn are the components of the equivalent permittivity tensor parallel and perpendicular to the x-y plane, respectively. Since the electric field in the z direction excites negligible current on the graphene sheet, ɛt and ɛn can be further written as [31]:
$${\varepsilon _n} = {\varepsilon _s}\textrm{, }{\varepsilon _t} = {\varepsilon _s} + \frac{{i{\sigma _g}}}{{\omega {\varepsilon _0}{d_s}}}$$

The complex equivalent permittivity tensor of the graphene-Si 1D APC is shown in Fig. 4. In the lower frequency region of terahertz band, we can see negative real part of tangential equivalent permittivity, which is more sensitive to the chemical potential than the imaginary part.

 figure: Fig. 4.

Fig. 4. The variation of complex tangential equivalent permittivity (a) and normal equivalent permittivity (b) of the graphene-Si 1D APC as a function of frequency for different chemical potentials.

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We employ the Transfer Matrix Method (TMM) [34] to calculate the transmission and absorption characteristics of 1D APC. Generally, the characteristic matrix of the i-th layer in a 1D PC can be expressed by:

$${m_i} = \left[ {\begin{array}{{cc}} {\textrm{exp} \left( { - j\sqrt {{{({{n_i}{k_0}} )}^2} - k_x^2} {d_i}} \right)}&0\\ 0&{\textrm{exp} \left( {j\sqrt {{{({{n_i}{k_0}} )}^2} - k_x^2} {d_i}} \right)} \end{array}} \right]$$
Where kx is the tangential wave number of the electromagnetic wave. di and ni are the thickness and refractive index of the i-th layer, respectively.

The transfer matrix at the interface between the i-th and j-th layers can be expressed by:

$$\begin{array}{{c}} {{m_{ij}} = \frac{1}{2}\left[ {\begin{array}{{cc}} {1 + \frac{{\sqrt {n_j^2 - n_x^2} }}{{\sqrt {n_i^2 - n_x^2} }}}&{1 - \frac{{\sqrt {n_j^2 - n_x^2} }}{{\sqrt {n_i^2 - n_x^2} }}}\\ {1 - \frac{{\sqrt {n_j^2 - n_x^2} }}{{\sqrt {n_i^2 - n_x^2} }}}&{1 + \frac{{\sqrt {n_j^2 - n_x^2} }}{{\sqrt {n_i^2 - n_x^2} }}} \end{array}} \right]\textrm{ }({\textrm{TE - polarization}} )}\\ {} \end{array}$$
$$\begin{array}{{c}} {{m_{ij}} = \frac{1}{2}\left[ {\begin{array}{{cc}} {\frac{{{n_j}}}{{{n_i}}} + \frac{{{n_i}\sqrt {n_j^2 - n_x^2} }}{{{n_j}\sqrt {n_i^2 - n_x^2} }}}&{\frac{{{n_j}}}{{{n_i}}} - \frac{{{n_i}\sqrt {n_j^2 - n_x^2} }}{{{n_j}\sqrt {n_i^2 - n_x^2} }}}\\ {\frac{{{n_j}}}{{{n_i}}} - \frac{{{n_i}\sqrt {n_j^2 - n_x^2} }}{{{n_j}\sqrt {n_i^2 - n_x^2} }}}&{\frac{{{n_j}}}{{{n_i}}} + \frac{{{n_i}\sqrt {n_j^2 - n_x^2} }}{{{n_j}\sqrt {n_i^2 - n_x^2} }}} \end{array}} \right]\textrm{ }({\textrm{TM - polarization}} )}\\ {} \end{array}$$
Where nx = kx/k0 = ki(j)sinθi(j)/k0 is the normalized tangential wave number. θi(j) is the incident angle at the interface of the i(j)-th layer. Based on Eqs. (8)-(10), the total transfer matrix of the graphene-Si 1D APC can be written as:
$$M = {m_{\textrm{0g}}}{({{m_g}{m_{gs}}{m_s}{m_{sg}}} )^{N - 1}}{m_g}{m_{gs}}{m_s}{m_{\textrm{s0}}}$$
Where mg (ms) represents the characteristic matrix of graphene (Si) layer. m0g, mgs, msg and ms0 are the transfer matrixes at the interfaces of air/graphene, graphene/Si, Si/graphene and Si/air, respectively. N is the number of periods.

The electrical field of incident beam, transmission beam, and reflection beam can be represented by EI, ET and ER, which satisfy the following relationship:

$$\left[ {\begin{array}{{c}} {{E_I}}\\ {{E_R}} \end{array}} \right] = M\left[ {\begin{array}{{c}} {{E_T}}\\ 0 \end{array}} \right] = \left[ {\begin{array}{{cc}} {{M_{11}}}&{{M_{12}}}\\ {{M_{21}}}&{{M_{22}}} \end{array}} \right]\left[ {\begin{array}{{c}} {{E_T}}\\ 0 \end{array}} \right]$$

Thus, the transmission and reflection coefficients can be written as:

$$t = \frac{{{E_T}}}{{{E_I}}} = \frac{1}{{{M_{11}}}},r = \frac{{{E_R}}}{{{E_I}}} = \frac{{{M_{21}}}}{{{M_{11}}}}$$

Finally, the transmittance, reflectance, and absorption of the 1D APC are calculated by:

$$T = {|t |^2},R = {|r |^2},A = 1 - T - R$$

3. Results and discussion

3.1 Dispersion relations of the graphene-Si 1D APC

The dispersion relations of the graphene-Si 1D APC can be described by the following equation under the long-wavelength approximation condition [28,43]:

$$\left\{ {\begin{array}{{c}} {k_x^2 + k_z^2 = {\varepsilon_t}k_0^2\textrm{ }({\textrm{TE - polarization}} )}\\ {\frac{{k_x^2}}{{{\varepsilon_n}}} + \frac{{k_z^2}}{{{\varepsilon_t}}} = k_0^2\textrm{ }({\textrm{TM - polarization}} )} \end{array}} \right.$$
Where kx, ky and kz are the x, y and z components of the wave vector. It is not difficult to find that, if Re(ɛt) > 0 at a particular frequency, kx must satisfy |kx|2 < ɛnk02 to support a real value of kz for the propagation of electromagnetic wave inside the 1D APC. Under this condition, the dispersion curve is a circle and an ellipse for the TE and TM polarizations, respectively. Otherwise, if Re(ɛt) < 0 at a particular frequency, kx must satisfy |kx|2 > ɛnk02 to support a real value of kz for the propagation of electromagnetic wave inside the 1D APC. Then, the dispersion curve should be a hyperbola for the TM polarization, while the propagation of TE-polarized THz wave is forbidden. As a result, there is a cut-off frequency (denoted as fc when Re(ɛt) = 0) for the TM-polarized beam switched from elliptical propagation to hyperbolic propagation states, as well as for the TE-polarized beam switched from propagation to forbidden states inside the 1D APC. As shown in Fig. 5, the cut-off frequencies of the 1D APC (ɛs = 11.7 and ds = 10 μm) are about 1.025 THz, 1.275 THz, 1.485 THz and 1.665 THz when the chemical potentials are 0.4 eV, 0.6 eV, 0.8 eV and 1.0 eV, respectively. In the following sections, we will show the designs of tunable and functionality-switchable terahertz devices when |kx|2 < k02 and |kx|2 > k02 respectively.

 figure: Fig. 5.

Fig. 5. The isofrequency curves of the graphene-Si 1D APC in the kx-kz coordinate with normalized wave vectors. The graphene’s chemical potentials are (a) μc = 0.4 eV, (b) μc = 0.6 eV, (c) μc = 0.8 eV and (d) μc = 1.0 eV, respectively.

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3.2 Tunable band gaps in the graphene-Si 1D APC when |kx|2 < k02

When |kx|2 < k02, both TE and TM waves can directly propagate inside the graphene-Si 1D APC. For the case of normal incidence (kx = 0), we calculated the complex photonic bands and transmission performance of the 1D APC as shown in Fig. 6. As seen from the real part of photonic bands in Fig. 6(a), except for the metallic band below the cut-off frequency (about 1.11 THz), there are another two Bragg band gaps with a center frequency located at about 4.6 THz for Gap-I and 8.8 THz for Gap-II, respectively. In particular, the frequencies at the lower edges of the two Bragg band gaps are equal to the Fabry-Perot limit fp = mc/(2dsns) for the single Si layer (without graphene sheet), where m = 1 and 2 [34]. As the graphene’s chemical potential grows from 0.4 eV to 1.0 eV, the lower edges of the metallic and Bragg band gaps approximately overlap, while the higher edges move toward the higher-frequency region and result in an enlargement of the band gaps.

 figure: Fig. 6.

Fig. 6. Real part (a) and imaginary part (b) of the photonic bands of the graphene-Si 1D APC when kx = 0 and different chemical potentials are considered. (c) Corresponding transmission (T), reflection (R) and absorption (A) spectra of the graphene-Si 1D APC. The total layer number of 1D APC is 20. The green area represents the metallic band. The insets show the zoomed-in view for selected band gaps.

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As shown in Fig. 6(b), the non-zero regions of the imaginary part of photonic bands coincide with the band-gap edges in Fig. 6(a). Meanwhile, it means that the position and width of band gaps can be determined by the non-zero regions in the imaginary part of photonic bands. Meanwhile, the area of non-zero region actually represents the energy dispassion in the band gap. The metallic band gap shows a much larger non-zero region than Bragg band gaps due to higher absorption. In order to verify the complex photonic bands, we calculated the corresponding transmission, reflection and absorption spectra of the graphene-Si 1D APC. As shown in Fig. 6(c), the band edges of both metallic (green area) and Bragg band gaps can be clearly identified from the transmission and reflection spectra, while the absorption spectra demonstrate larger absorption in the metallic band gap than the Bragg band gaps. The tunable blue-shifted band edges with respect to the chemical potential of graphene are also in good agreement with the complex photonic bands in Figs. 6(a) and 6(b). The tunable band gaps of the 1D APC can be well interpreted by Fig. 2(b). On one hand, as the chemical potential increases, the real part of the graphene’s refractive index will gradually increase and thus the enlarged difference of refractive indexes between graphene and Si will broaden the band gaps of 1D APC. On the other hand, as the chemical potential increases, the imaginary part of the graphene’s refractive index will also gradually increase and thus the non-zero region of either metallic band gap or Bragg band gaps will be enlarged due to increased absorption.

For a fixed chemical potential μc = 0.8 eV, Fig. 7 further shows the photonic bands and transmission performance of the graphene-Si 1D APC at different values of incident angle. It is found that the lower Bragg band gap (Gap-I) is less sensitive to the incident angle and polarization as compared to the upper one (Gap-II). It is well-known that, for a conventional dielectric 1D PC, the existence of Brewster angle for TM polarization will result in a close of band gap and thus bring obvious difference of photonic bands between TE and TM polarizations. However, since the maximum internal incident angle at the Si/Graphene interface is smaller than the Brewster angle, the influence of Brewster angle can be avoided. The dispersionless and polarization-insensitive Bragg band gap in the graphene-Si 1D APC is useful to design high-quality omnidirectional reflectors and filters.

 figure: Fig. 7.

Fig. 7. For the 20-layer graphene-Si 1D APC at different incident angles: (a) Real part of photonic bands; (b) Color map of transmission; and (c) Transmission spectra. The chemical potential of graphene is 0.8 eV. The green area represents the metallic band.

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3.3 Tunable band-pass filter and perfect absorber based on the graphene-Si 1D APC containing a VO2 defect layer when |kx|2 < k02

We consider a layer of VO2 with an optimized thickness of 28 μm embedded in the middle of the graphene-Si 1D APC as a defect layer. When VO2 is in the insulator phase (30 °C), we can clearly find two defect states in two Bragg band gaps by comparing the transmission spectra of 1D APC without or with a VO2 defect layer when μc = 0.8 eV, as shown in Fig. 8(a). As graphene's chemical potential grows from 0.4 eV to 1.0 eV, the defect state in Gap-I is located at 4.49 THz, 4.55 THz, 4.59 THz and 4.63 THz, respectively, as shown in Fig. 8(b). The corresponding transmittance-absorbance of the defect state is 39.64%–50.89%, 20.21%−55.99%, 8.13%−48.03% and 2.64%−34.17%, respectively. These transmission and absorption peaks are not high enough for applications in neither THz filters nor absorbers. As a comparison, the defect state in Gap-II shows obviously higher transmission and lower absorption than that in Gap-I, as shown in Fig. 8(c). When the chemical potential of graphene increases from 0.4 eV to 1.0 eV with a step of 0.2 eV, the defect state gradually moves from 8.66 THz to 8.72 THz with a step of 0.02 THz. The corresponding transmittance-absorbance of the defect state is 85.70%−13.79%, 79.57%−19.40%, 73.77%−24.41% and 68.24%−28.80%, respectively. As a result, the defect state in Gap-II shows potential application in tunable THz band-pass filter with a tunability of ∼0.10 THz/eV.

 figure: Fig. 8.

Fig. 8. (a) Comparison of transmission spectra in the whole frequency region for the 1D APC without or with an insulator-phase VO2 defect layer. The yellow areas represent two Bragg band gaps. (b,c) T, R and A spectra in two sub-frequency regions for the 1D APC with an insulator-phase VO2 defect layer. The thickness of VO2 is 28 μm.

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When VO2 is in the metal phase (90 °C), the performance of 1D APC is quite different from that in the insulator phase. As shown in Fig. 9, the transmission is always equal to zero while nearly perfect absorption can be found within a broad bandwidth. As the chemical potential of graphene increases from 0.4 eV to 1.0 eV with a step of 0.2 eV, the absorption peaks located in the range of 4.25–4.32 THz reach 98.49%, 99.28%, 94.34% and 87.00%, respectively, while the absorption peaks located in the range of 8.59–8.65 THz reach 90.28%, 96.28%, 99.35% and 99.90%, respectively. Such perfect absorption is not related to the defect state and it is actually caused by the raised lower-edge absorption of the band gap. The lower-edge absorption can reach as high as 100% when VO2 layer acts as a perfect reflector in the metal phase.

 figure: Fig. 9.

Fig. 9. T, R and A spectra in two sub-frequency regions for the 20-layer graphene-Si 1D APC containing a metal-phase VO2 defect layer. The thickness of VO2 is 28 μm.

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During the design, it is found that the performance of band-pass filter or perfect absorber is closely dependent on the thickness of VO2 layer, as shown in Fig. 10. For the case of VO2 in the insulator phase [Fig. 10(a)], when dv is less than 0.1 μm, there are no defect states in the band gaps (e.g., dv = 0.05 μm). When dv is in the range of 0.1–5 μm, a clear defect state only appears in Gap-II (e.g., dv = 4.5 μm). When dv is in the range of 5–102 μm, a clear defect state appears in each band gap (e.g., dv = 28 μm). As dv further increases from 102 to 112 μm, double defect states will appear in Gap-II (e.g., dv = 110 μm). As dv further increases, double defect states will appear in each band gap (e.g., dv = 150 μm). As a comparison, for the case of VO2 in the metal phase [Figs. 10(b) and 10(c)], the absorption peak will be affected only when dv is less than 1 μm. It is not difficult to understand that a relatively thick defect layer can act as a perfect reflector and result in nearly 100% lower-edge absorption.

 figure: Fig. 10.

Fig. 10. Performance of the band-pass filter (a) and perfect absorber (b,c) at different thicknesses of VO2. The chemical potential of graphene is 0.6 eV. The yellow areas represent two Bragg band gaps.

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As shown in Fig. 11, the performance of filter and absorber in the lower Bragg band gap also shows less sensitivity to the incident angle and polarization as compared to that in the upper one. In the practical application, the dispersionless and polarization-insensitive performance of a band-pass filter is quite useful in nonlinear wave mixing process, where the critical all-angle phase matching condition should be satisfied. The dispersionless and polarization-insensitive performance of a perfect absorber is very important for electromagnetic screen, which is not easy to be obtained in conventional perfect absorbers.

 figure: Fig. 11.

Fig. 11. Performance of the band-pass filter (a) and perfect absorber (b) at different incident angles. The chemical potential of graphene is 0.6 eV and the thickness of VO2 is 28 μm.

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3.4 Tunable frequency comb and Fano-like resonance in the graphene-Si 1D APC when k02 < |kx|2

When k02 < |kx|2, the incident THz beam from the free space is impossible to directly propagate inside the graphene-Si 1D APC due to a mismatch of wave vector. Instead, it can be coupled into the 1D APC by adding a silicon grating on the surface of 1D APC [ Fig. 12(a)]. The wave-vector compensation equation is written as kx= k0sinθ0+2mπ/ΛSi, where ΛSi is the grating constant and m is a positive nonzero integer. Under this condition, the surface plasmon polaritons (SPPs) can be excited on the air/graphene, graphene/Si and Si/graphene interfaces.

 figure: Fig. 12.

Fig. 12. For the 20-layer graphene-Si 1D APC when k02 < |kx|2: (a) Sketch of the grating-based wave-vector compensation method; (b) Real part of photonic bands; (c,e) Transmission spectra in the whole and sub frequency regions; (d) Electrical-field distribution inside the graphene-Si 1D APC at 9.77 THz. (f) Color map of transmission; (g) Reflection spectra; (h) Color map of reflection. kx = 2k0 for (b)-(e) and kx = 4k0 for (g). The inset in (c) shows the intensity-frequency relationships of the EOT peaks.

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We take the TM wave as an example to show the tunable transmission and band gaps in the graphene-Si 1D APC. For the case of k02 < |kx|2 < ɛnk02, the propagation of THz beam satisfies the elliptical dispersion relationship. Figure 12(b) shows the photonic bands of the 1D APC when kx = 2k0. It can be found that the cut-off frequency of metallic band gap is at about 1 THz and there is only one Bragg band gap with a center frequency located at about 5.5 THz. As the chemical potential of graphene increases, the higher edge of Bragg band gap moves toward the higher-frequency region and results in an enlargement of the band gap, which is consistent with the case of kx = 0.

Figure 12(c) shows the corresponding transmission coefficient of the graphene-Si 1D APC. Except for the tunable band gaps in the range of 5–6 THz, we can also observe comb-shaped extraordinary optical transmission (EOT) in the rest region. The EOT phenomenon in the 1D APC is due to the SPPs coupling between graphene and Si layers [44], which can be verified from the electrical-field distribution inside the graphene-Si 1D APC [Fig. 12(d)]. The intensity of EOT peaks linearly increases with frequency and decreases with chemical potential. The linearity is varying from 1.950 to 0.807 when the chemical potential of graphene increases from 0.4 eV to 1.0 eV (see inset). The comb-shaped EOT peaks with ultra-narrow linewidth have potential applications in super-continuum laser generation [45]. From Fig. 12(e) we estimate the tunable sensitivity of EOT peaks as ∼0.17 THz/eV. From Fig. 12(f) we can find that the band gap and EOT are sensitive to |kx| for a fixed chemical potential of 0.4 eV.

For the case of ɛnk02 < |kx|2, the propagation of THz beam satisfies the hyperbolic dispersion relationship. Figure 12(g) shows the reflection coefficient of the 1D APC when kx = 4k0. It can be clearly found that the reflection curve shows a Fano-like resonance below the cut-off frequency. The reflection coefficient which is larger than 1 is due to the abnormal reflection caused by SPPs coupling between graphene and Si layers. As compared to the EOT peak for the case of k02 < |kx|2 < ɛnk02, the Fano-like resonance for the case of ɛnk02 < |kx|2 shows a much higher tunable sensitivity (∼1.63 THz/eV), which is useful in terahertz sensing. From Fig. 12(h) we can find that the Fano-like resonance is also sensitive to |kx| for a fixed chemical potential of 1.0 eV.

3.5 Performance comparison and practical application

Table 1 summarizes the above designs of tunable and multifunctional terahertz devices under different conditions of tangential wave vector. Table 2 further compares our proposed terahertz filter and absorber with previous works based on 1D PCs or metamaterials. As compared to the tunable terahertz filter based on 1D PC [33], the filter we have designed shows comparable performance in peak value, bandwidth and tunability. As compared to the tunable terahertz absorbers in previous works based on 1D PC [23] or metamaterials [37,46], our design shows comparable absorption peak value but poor performance in bandwidth due to the narrow frequency response of the single defect layer. A possible method to design broadband absorber is to construct multiple defect layers inside the 1D PC [33].

Tables Icon

Table 1. Summary of tunable and multifunctional terahertz devices in the present study

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Table 2. Comparison of tunable terahertz devices based on 1D PCs and metamaterials

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Nevertheless, as compared to previous works, the main novelty of our design is realizing switchable functions between terahertz filter and absorber, which have many potential applications in the terahertz science and technology. For example, the bi-functional filter and absorber can be applied to the stabilization of terahertz wireless communication systems [47]. Meanwhile, some issues and challenges should be pointed out in the practical application of our designs. Firstly, the 1D APC could be realized by alternating graphene and silicon. When the amorphous silicon is used in the practical application, the loss may be much higher than crystal silicon. We have recalculated Figs. 8 and 9 and found that high loss of silicon will only obviously influence the performance of filter. The transmission peak will be reduced to half of the maximum value when the extinction coefficient of silicon reaches about 0.0035. Secondly, it is difficult for the large-scale graphene sheet to be directly fabricated on the silicon substrate. To solve this problem, we can first deposit a several nanometer h-BN layer on the silicon layer and then fabricate the graphene sheet on the h-BN layer by the CVD method [32]. Due to the existence of h-BN insulating layer, an external gate voltage can be applied to modulate the graphene’s chemical potential [48]. Thirdly, a high-quality VO2 film with thickness of more than several micrometers is difficult to be fabricated. In the application, we can choose a thin VO2 film with thickness of about several hundreds of nanometers. This will not obviously influence the performance of the device (see Fig. 10). Finally, how to apply an external gate voltage onto each layer of graphene and how to integrate the gate voltage/temperature control together will be a challenge in fabrication and application.

4. Conclusions

In summary, we have numerically demonstrated the band-pass filtering, perfect absorption, comb-shaped extraordinary optical transmission and Fano-like resonance phenomenon in the pure graphene-Si 1D APC and the graphene-Si 1D APC with a VO2 defect layer under different conditions of tangential wave vector. When |kx|2 < k02, the pure graphene-Si 1D APC shows a metallic band gap and two Bragg band gaps in the THz frequency region. As the VO2 defect layer is introduced and switched between the insulator phase and metal phase, the graphene-Si 1D APC can work as a band-pass filter and a perfect absorber respectively. When k02 < |kx|2 < ɛnk02, the pure graphene-Si 1D APC shows comb-shaped EOT phenomenon and it can work as a frequency comb. When ɛnk02 < |kx|2, the pure graphene-Si 1D APC shows Fano-like resonance in the reflection spectra which can be used for near-field terahertz sensing. The working frequency and performance of these different functions at different values of kx can be freely modulated by changing the chemical potential of graphene. These novel terahertz devices which show good performance, high tunability and switchable functions will have potential applications in terahertz imaging, sensing, and communication [49–51].

Funding

National Natural Science Foundation of China (61205042, 61675096); Natural Science Foundation of Jiangsu Province (BK20141393); Six Talent Climax Foundation of Jiangsu (XYDXX-027); Fundamental Research Funds for the Central Universities (30919011106); Open Research Fund of State Key Laboratory of Bioelectronics (Sk1b2021p06).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are available in Refs. [23,33,37,46].

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Data availability

Data underlying the results presented in this paper are available in Refs. [23,33,37,46].

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Figures (12)
Fig. 1.
Fig. 1. (a) Model of a pure 1D APC made of monolayer graphene and Si alternatively arranged in the z direction. (b) Model of a 1D APC with a VO2 defect layer in the center.

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Fig. 2.
Fig. 2. The variation of complex permittivity (a) and refractive index (b) of monolayer graphene as a function of frequency for different chemical potentials. The insets show the zoomed-in view for selected plots.

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Fig. 3.
Fig. 3. The variation of complex permittivity of VO2 as a function of frequency when it is in the insulator phase (a) and metal phase (b), respectively.

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Fig. 4.
Fig. 4. The variation of complex tangential equivalent permittivity (a) and normal equivalent permittivity (b) of the graphene-Si 1D APC as a function of frequency for different chemical potentials.

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Fig. 5.
Fig. 5. The isofrequency curves of the graphene-Si 1D APC in the kx-kz coordinate with normalized wave vectors. The graphene’s chemical potentials are (a) μc = 0.4 eV, (b) μc = 0.6 eV, (c) μc = 0.8 eV and (d) μc = 1.0 eV, respectively.

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Fig. 6.
Fig. 6. Real part (a) and imaginary part (b) of the photonic bands of the graphene-Si 1D APC when kx = 0 and different chemical potentials are considered. (c) Corresponding transmission (T), reflection (R) and absorption (A) spectra of the graphene-Si 1D APC. The total layer number of 1D APC is 20. The green area represents the metallic band. The insets show the zoomed-in view for selected band gaps.

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Fig. 7.
Fig. 7. For the 20-layer graphene-Si 1D APC at different incident angles: (a) Real part of photonic bands; (b) Color map of transmission; and (c) Transmission spectra. The chemical potential of graphene is 0.8 eV. The green area represents the metallic band.

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Fig. 8.
Fig. 8. (a) Comparison of transmission spectra in the whole frequency region for the 1D APC without or with an insulator-phase VO2 defect layer. The yellow areas represent two Bragg band gaps. (b,c) T, R and A spectra in two sub-frequency regions for the 1D APC with an insulator-phase VO2 defect layer. The thickness of VO2 is 28 μm.

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Fig. 9.
Fig. 9. T, R and A spectra in two sub-frequency regions for the 20-layer graphene-Si 1D APC containing a metal-phase VO2 defect layer. The thickness of VO2 is 28 μm.

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Fig. 10.
Fig. 10. Performance of the band-pass filter (a) and perfect absorber (b,c) at different thicknesses of VO2. The chemical potential of graphene is 0.6 eV. The yellow areas represent two Bragg band gaps.

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Fig. 11.
Fig. 11. Performance of the band-pass filter (a) and perfect absorber (b) at different incident angles. The chemical potential of graphene is 0.6 eV and the thickness of VO2 is 28 μm.

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Fig. 12.
Fig. 12. For the 20-layer graphene-Si 1D APC when k02 < |kx|2: (a) Sketch of the grating-based wave-vector compensation method; (b) Real part of photonic bands; (c,e) Transmission spectra in the whole and sub frequency regions; (d) Electrical-field distribution inside the graphene-Si 1D APC at 9.77 THz. (f) Color map of transmission; (g) Reflection spectra; (h) Color map of reflection. kx = 2k0 for (b)-(e) and kx = 4k0 for (g). The inset in (c) shows the intensity-frequency relationships of the EOT peaks.

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Tables (2)

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Table 1. Summary of tunable and multifunctional terahertz devices in the present study

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Table 2. Comparison of tunable terahertz devices based on 1D PCs and metamaterials

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Equations (15)

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σ g ( ω , μ c , Γ , T ) = σ intra ( ω , μ c , Γ , T ) + σ inter ( ω , μ c , Γ , T )
σ intra ( ω , μ c , Γ , T ) = i e 2 k B T π ℏ 2 ( ω + i 2 Γ ) [ μ c k B T + 2 ln ⁡ ( e − μ c k B T + 1 ) ]
σ inter ( ω , μ c , Γ , T ) = i e 2 4 π ℏ 2 ln [ 2 | μ c | − ( ω + i 2 Γ ) ℏ 2 | μ c | + ( ω + i 2 Γ ) ℏ ]
ε g = 1 + i σ g ω ε 0 d g
ε v = 1 4 { ε D ( 2 − 3 f ) + ε M ( 3 f − 1 ) + [ ε D ( 2 − 3 f ) + ε M ( 3 f − 1 ) ] 2 + 8 ε D ε M }
ε = [ ε t 0 0 0 ε t 0 0 0 ε n ]
ε n = ε s ,  ε t = ε s + i σ g ω ε 0 d s
m i = [ exp ( − j ( n i k 0 ) 2 − k x 2 d i ) 0 0 exp ( j ( n i k 0 ) 2 − k x 2 d i ) ]
m i j = 1 2 [ 1 + n j 2 − n x 2 n i 2 − n x 2 1 − n j 2 − n x 2 n i 2 − n x 2 1 − n j 2 − n x 2 n i 2 − n x 2 1 + n j 2 − n x 2 n i 2 − n x 2 ]   ( TE - polarization )
m i j = 1 2 [ n j n i + n i n j 2 − n x 2 n j n i 2 − n x 2 n j n i − n i n j 2 − n x 2 n j n i 2 − n x 2 n j n i − n i n j 2 − n x 2 n j n i 2 − n x 2 n j n i + n i n j 2 − n x 2 n j n i 2 − n x 2 ]   ( TM - polarization )
M = m 0g ( m g m g s m s m s g ) N − 1 m g m g s m s m s0
[ E I E R ] = M [ E T 0 ] = [ M 11 M 12 M 21 M 22 ] [ E T 0 ]
t = E T E I = 1 M 11 , r = E R E I = M 21 M 11
T = | t | 2 , R = | r | 2 , A = 1 − T − R
{ k x 2 + k z 2 = ε t k 0 2   ( TE - polarization ) k x 2 ε n + k z 2 ε t = k 0 2   ( TM - polarization )
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