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Abstract
A terahertz absorber that can be dynamically tuned via an external magnetic field is proposed. The absorber is composed of periodic gold-disks and an underlying graphene sheet on a dielectric/gold reflector substrate. Simulated results reveal that a new absorption peak appears under the perpendicularly applied external magnetic field. The new absorption peak under a 10 T magnetic field red-shifts from 14.22 THz to 4.47 THz as the Fermi level of graphene increases from 0.1 eV to 0.3 eV. At a fixed Fermi level, the new absorption peak blue shifts as the magnetic field increases. The new absorption peak can be enhanced by using multilayer graphene. The absorber is polarization independent. These results may promote the development and applications of flexibly tunable terahertz absorbers.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Terahertz (THz) technology has a wide range of potential applications in the fields of medical imaging [1], non-destructive testing [2] and communication [3]. Perfect absorbers have been intensely studied in recent years [4–12] because of the fundamental physics and application potentials in many areas such as cloaking [13,14], transducers [15] and electromagnetic compatibility [16]. Absorbers based on noble metals [17,18], silicon [19], germanium antimony telluride (GST) [20] and vanadium dioxide ($\mathrm {VO_{2}}$) [21–23] have been proposed and investigated.
Compared to THz absorbers made of noble metals or silicon, THz absorbers based on graphene [24–26] have the advantages of higher Q factors and dynamic control. Rasoul et al. [4] proposed a graphene THz absorber consists of graphene micro-ribbon array on top of gold ground plate separated by a dielectric spacer. Numerical results reveal that perfect absorption can be achieved at various values of spacer thickness around 3.6 THz for the first mode and around 7.3 THz for the second mode. The resonance frequency of the absorber can be tuned by varying the chemical potential ${\mu }_{c}$, which can be controlled by applying a gate voltage or chemical doping. In Ref. 5, Woo et al. demonstrated a graphene-based multi-band absorber operating in THz frequency. Multilayer graphene was transferred on 98-$\mathrm {\mu m}$-thick polymer substrate of which the back side was covered with 200-nm-thick gold. The graphene sheet resistance was modified using chemical doping method and excellent absorption of 0.95 and 0.97 were achieved at 0.5 and 1.5 THz respectively. Jiang et al. [16] designed and investigated a patterned graphene THz absorber, which consists of patterned graphene, polymide substrate and a reflective gold mirror. Both the simulation and experiment yielded a resonant frequency of $\sim$1.835 THz. A dual-band perfect absorber based on graphene disks in the THz range was proposed by Wang et al. [24]. By superposition of resonant responses of disks with different sizes, two nearly perfect absorption peaks at 7.1 THz and 10.4 THz were obtained. The frequency of the absorber can be effectively tuned by changing Fermi energy of graphene.
Although many achievements have been made in the area of graphene-based THz absorbers, the control method of many absorbers simply relies on changing the Fermi level of graphene by applying gate voltage or chemical doping. In fact, in addition to electric tuning method, external magnetic field can also be employed to modulate electrical and optical properties of graphene [27–31]. The tunability of graphene-based THz absorbers via external magnetic field is an important issue which is worthy of further investigations.
To address the above concerns, we designed a hybrid graphene-gold absorber consisting of gold disk array on top of a continuous single-layer graphene sheet, which is supported by a dielectric spacer with a gold layer at the bottom. Finite element method (FEM) is utilized to investigate the device performance. Numerical results reveal that without external magnetic field, the main absorption peak of the absorber locates at about 9 THz, while a new absorption peak appears under a 10 T external magnetic field perpendicular to graphene surface. The new absorption peak red-shifts from 14.22 THz to 4.47 THz as Fermi level of graphene rises from 0.1 eV to 0.3 eV. On the other hand, at fixed Fermi level, the new absorption peak blue-shifts as magnetic field increases. The tunable range via magnetic field shrinks towards lower frequency as the Fermi level rises. As a consequence, the absorption spectrum can be tuned not only by Fermi energy but also by external magnetic field, which facilitates the design, fabrication and applications of flexibly tunable THz absorbers.
2. Methods
As shown in Fig. 1, the proposed absorber is composed of a gold disk array on top of a continuous single-layer graphene sheet, which is supported by a dielectric substrate backed with a gold mirror. As a diamagnetic material, the magnetic susceptibility $\chi$ of gold is approximately $-1.4 \times 10^{-6}$ e.m.u. in c.g.s. units ($-1.76 \times 10^{-5}$ in SI units) [32]. Since the effect of gold on the applied magnetic field is negligibly small, we can assume the magnetic field inside the absorber to be the same as the externally applied one. The geometrical parameters are as follows: lattice constant $P=20\mathrm {\ \mu m}$, gold disk radius $R=6\mathrm {\ \mu m}$, gold disk thickness ${d}_{1}=50\mathrm {\ nm}$, dielectric thickness ${d}_{2}=1\mathrm {\ \mu m}$, back gold thickness ${d}_{3}=50\mathrm {\ nm}$. The relative permittivity of the dielectric $\varepsilon _{d}$ is set to be 2.1 [33–35]. The normal incident THz wave (wave-vector in the $-z$ direction) is polarized in the $x$ direction. As the thickness of back gold film is much larger than the skin depth of THz waves in gold, the absorption of the absorber $A=1-R$, where $R$ is the reflectivity of the absorber.
Fig. 1. 3D Schematic view of the proposed absorber under normal incident THz wave with electric field parallel with x direction.
To change Fermi level ${E}_{F}$ of graphene, a gate voltage ${V}_{g}$ can be applied between the graphene sheet and the back gold layer. The relation between ${E}_{F}$ and ${V}_{g}$ can be written as: [36–38]
In a magnetic field in the $z$ direction, the relative permittivity tensor of gold $\hat {\varepsilon }_{Au}(\omega )$ can be derived as [39]:
In a magnetic field perpendicular to the graphene surface in the $z$ direction, the relative permittivity tensor of graphene can be expressed as follows [27,45,46]:
In the calculations, the 3D finite element method (FEM) is used for the numerical simulations and the commercial software COMSOL Multiphysics [49] is employed. The 3D simulations were performed in a unit cell with periodic boundary conditions applied along the $x$ and $y$ directions. The THz wave with $x$ polarization incidents normally on the absorber in the $k(-z)$ direction. Perfectly matched layer (PML) boundary conditions are set at both ends of the computing space in $z$ direction. The meshes are refined several times to ensure the convergence of simulation results.
3. Results
For simplicity, we firstly investigate the device performance under no external magnetic field, as shown in Fig. 2(a). Without graphene, the maximum absorption of absorber is 94.3% at 8.89 THz, while the maximum absorption values for absorber with graphene of 0.1 eV, 0.2 eV and 0.3 eV Fermi energy are 96.9% at 9.15 THz , 87.5% at 9.41 THz and 77.2% at 9.41 THz respectively. To reveal the absorption mechanism, the real and imaginary parts of graphene permittivity at different Fermi levels are plotted in Fig. 2(b) and $z$ component of electric field distributions in $x-y$ plane above the device are shown in Figs. 2(c)–2(f), which indicate the charge distributions in gold plate and graphene. As shown in Fig. 2(c), without graphene, the charges oscillate back and forth in the $x$ direction, forming a dipolar mode, which couples strongly with incident THz radiation and results in the absorption peak. After a graphene layer being added, surface plasmon polaritons (SPPs) are formed on the graphene sheet shown in Fig. 2(d)–2(f). Notice that without gold nanodisks, the excitation of SPPs in graphene can not be realised because of the strong mismatch between wavenumbers in free space and graphene [50,51]. The gold disk can be recognized as a metal dipole antenna [52], and the Fourier transform of enhanced near field in gold disk provides high-momentum components, which match the wavevector of SPPs.
Fig. 2. (a) Absorption of the absorber without and with graphene at different Fermi energy values under 0 T external magnetic field. (b) Real and imaginary part of the relative permittivity of graphene at different Fermi energy values under 0 T external magnetic field. (c)-(f) Z component of electric field distributions at 10 nm above gold disk at 8.89 THz (no graphene), 9.15 THz (${E}_{F}=0.1 \mathrm {\ eV}$), 9.41 THz (${E}_{F}=0.2 \mathrm {\ eV}$) and 9.41 THz (${E}_{F}=0.3 \mathrm {\ eV}$) respectively.
When the Fermi level is 0.1 eV, the maximum absorption is 96.9%, 2.6% larger than that without graphene, which is owe to the extra energy dissipation of excited SPPs on graphene. However, as the Fermi level rises to 0.2 eV and 0.3 eV, the graphene sheet becomes more metallic since the real part of $\varepsilon _{g}$ becomes more negative, which is shown in Fig. 2(b). More radiation will be reflected back as the graphene sheet becomes more metallic, which explains the decrease of maximum absorption (87.5% and 77.2% when Fermi energy is 0.2 eV and 0.3 eV respectively) with Fermi energy increase.
In a 10 T external magnetic field in the $z$ direction, the device performances with no graphene and with graphene are investigated respectively. Without graphene, the absorption spectrum of device in 10 T magnetic field is almost identical as that of device in 0 T magnetic field, with maximum absorption of 94.3% at 8.89 THz, as shown in Fig. 3(a) (black curve). However, the absorption properties of device with graphene in 10 T magnetic field are quite different from that without magnetic field. As can be seen in Fig. 3(a), a new absorption peak appears in addition to the main absorption peak around 9 THz. So it is obvious that the new absorption features of the absorber in magnetic field are caused by the graphene layer instead of gold. This is not difficult to comprehend since the electronic mobility and cyclotron frequencies in graphene are much larger than that in gold due to the massless properties of Dirac fermions in single-layer graphene [53]. To explore the mechanism of the new absorption peak in external magnetic field, the imaginary part of the relative permittivity tensor element $\varepsilon _{gxx}$ of graphene at different Fermi energies are plotted in Fig. 3(b). Different from the imaginary part of $\varepsilon _{g}$ shown in Fig. 2(b), a peak appears in imaginary $\varepsilon _{gxx}$ in 10 T magnetic field (Fig. 3(b)). As the Fermi level rises from 0.1 eV to 0.2 eV to 0.3 eV, the peak position of imaginary $\varepsilon _{gxx}$ shifts from 14.36 THz to 7.18 THz to 4.78 THz. The peak positions of imaginary $\varepsilon _{gxx}$ roughly consist with that of the new absorption peaks. Since the imaginary part of $\varepsilon$ represents ohmic loss [54] , it is reasonable to believe that the new absorption peaks are caused by the peaks of imaginary $\varepsilon _{gxx}$. In other words, the graphene sheet turns out to be high loss material in external applied magnetic field at the peak frequency of imaginary $\varepsilon _{gxx}$, which result in the new absorption peak. To confirm the speculation, the electric field amplitude distributions at the new peak frequencies are plotted in Figs. 3(c)–3(e). The electric field concentrates mainly in graphene regions outside the gold disks, which indicates that the absorption is mainly caused by the ohmic dissipation in graphene. From Fig. 3(a), we can also see that as the Fermi level of graphene rises from 0.1 eV to 0.2 eV to 0.3 eV, the new absorption peak red shifts from 14.22 THz to 6.68 THz to 4.47 THz. The ultra-wide frequency tunable range is rarely seen in absorbers without external magnetic field. So it is a convenience that the new absorption peak can be tuned in a wide range by varying the Fermi level of graphene under a constant external magnetic field.
Fig. 3. (a) Absorption of the absorber under 10 T external magnetic field without graphene and with graphene at different Fermi energy values. (b) Imaginary part of the relative permittivity tensor element $\varepsilon _{gxx}$ of graphene at different Fermi energy values under 10 T external magnetic field. (c)-(e) Electric field amplitude distributions with graphene at 10 nm above gold plate at 14.22 THz (${E}_{F}=0.1 \mathrm {\ eV}$), 6.68 THz (${E}_{F}=0.2 \mathrm {\ eV}$) and 4.47 THz (${E}_{F}=0.3 \mathrm {\ eV}$) respectively.
On the contrary, the absorber performance can also be tuned by varying the external magnetic field while keeping the Fermi energy constant. As can be seen in Fig. 4(a), the new absorption peak blue-shifts from 2.91 THz to 14.22 THz as the magnetic field increases from 2 T to 10 T. Here in, the magnetic field proves to be an efficient way to dynamically tune the device in a wide frequency range. In Fig. 4(c), as the magnetic field increases from 4 T to 10 T, the new absorption peak blue-shifts from 2.78 THz to 6.68 THz. While in Fig. 4(e), as the magnetic field increases from 6 T to 10 T, the new absorption peak blue-shifts from 2.78 THz to 4.47 THz. As a consequence, the tunable range of device via magnetic field shrinks towards lower frequency range as the Fermi level rises, which can be interpreted by the peak positions of imaginary part of $\varepsilon _{gxx}$ shown in Figs. 4(b),4(d) and 4(f). Another interesting phenomenon is that the absorption of new absorption peak decreases as the peak position shifts away from the main absorption peak, which can be clearly seen in Figs. 4(c) and 4(e). In Fig. 4(c), the absorptions of new absorption peak at 6.68 THz, 5.38 THz, 4.08 THz and 2.78 THz are 62.7%, 33.9%, 16.3% and 6.18% respectively. The maximum value of electric field at 10 nm above the gold disks are $6.89 \times 10^{6} \mathrm {\ V} / \mathrm {m}$, $3.68 \times 10^{6} \mathrm {\ V} / \mathrm {m}$, $1.81 \times 10^{6} \mathrm {\ V} / \mathrm {m}$ and $8.61 \times 10^{5} \mathrm {\ V} / \mathrm {m}$ at above mentioned frequencies. Although the peak value of imaginary $\varepsilon _{gxx}$ increases as magnetic field decreases, the decrease of maximum electric field at the new absorption peak is more significant and dominates the decrease of absorption.
Fig. 4. (a)(c)(e) Absorption of the absorber under different external magnetic fields at 0.1 eV, 0.2 eV and 0.3 eV Fermi energy respectively. (b)(d)(f) Imaginary part of the relative permittivity tensor element $\varepsilon _{gxx}$ of graphene under different external magnetic fields at 0.1 eV, 0.2 eV and 0.3 eV Fermi energy respectively.
From practical point of view, the maximum absorption of absorber should be as close to 100% as possible. Multi-band perfect absorbers have been proposed and investigated [15,55]. Rectangular gratings or pillar arrays are covered by single-layer graphene and higher SPP modes can be excited on the structures, which result in multiple absorption peaks with more than 90% maximum absorption. However, the maximum absorption of our proposed absorber is unsatisfactory, especially the new mode, which can be seen in Figs. 3(a) and 4. Considering the implementability of real device fabrications, we intend to switch the single-layer graphene to multi-layer graphene (with 2,3 and 4 layers) and investigate the device performance. We focus the absorption peak of device under 10 T magnetic field and 0.3 eV Fermi energy and the simulation results are shown in Fig. 5. The maximum absorptions of the new absorption peak are 27.2%, 37.1%, 42.5% and 44.3% for graphene of 1 layer, 2 layers, 3 layers and 4 layers respectively. After inserting the 2nd layer graphene on the single-layer graphene, the absorption maximum increase is significant (9.9%). However, the 3rd and 4th added graphene layer only contributes 5.4% and 1.8% maximum absorption. The maximum absorption of the new mode tends to reach a saturation value with the graphene layer number increase. On the other hand, the main absorption peak shows a decrease trend as the layer number increases, which means the absorption is dominated by gold instead of graphene. In this sense, the main absorption peak can be termed as "gold peak" and the new absorption peak is "graphene peak". The red-shift of the "graphene peak" and the blue-shift of the "gold peak" might be interpreted as the stronger hybridization between the two modes as graphene layer number increases. Of course, the maximum absorption is far from ideal and the performance improvement is an interesting topic which is worthy of further investigations.
Fig. 5. Absorption of the absorber with 1-4 graphene layers at 0.3 eV Fermi energy under 10 T external magnetic field. The dashed line shows the new absorption peaks
So far, the polarization of incident THz wave has been restricted to x-direction. Optical response of the absorber for different polarization angles is also an important issue. Very recently, Xia et. al. proposed stacked anisotropic 2D material structures based on black phosphorus [56]. By crossly stacking even-layered nanostructures, polarization-independent absorption can be achieved. Unlike black phosphorus, which has intrinsic anisotropic in plane optical property, graphene is an intrinsic isotropic 2D material, so the device performance without external magnetic field is polarization insensitive. However, under an external magnetic field, the relative permittivity of graphene becomes a tensor with non-zero off-diagonal elements as shown in Eq. (6), which means graphene may not be treated as an isotropic material.
We simulate the device performance under normal incident THz wave with polarization angle $\theta$, as illustrated in Fig. 6(a). It is found that under 10 T magnetic field with graphene at different Fermi energies, the device absorption spectrums under y polarization incident wave ($\theta =90^\circ$) are identical with that under x polarization incidence. For clarity, device performances at different polarization angles are plotted in Figs. 6(b)–6(d). It turns out that the absorber performance is also polarization independent under an external applied magnetic field. To further reveal the mechanism, the $z$ component of electric field distributions of device (B = 10 T, ${E}_{F}=0.2 eV$) at 6.68 THz (Peak B in Fig. 3(a)) under different polarization angles are plotted, as shown in Figs. 6(e)–6(j). It can be seen that the electric dipole turns angle with respect to the incidence polarization angle. From the symmetry point of view, the polarization independent property of the system can be comprehended. Without external magnetic field, the system has ${C}_{4v}$ symmetry, which guarantees the polarization independent property. Although the symmetry reduces to ${C}_{4}$ in an perpendicular applied external magnetic field, the system still possesses 90$^\circ$ rotational symmetry along $z$ axis, which means the device still has polarization independent property in magnetic field.
Fig. 6. (a) 3D Schematic view of the absorber under normal incident THz wave with polarization angle $\theta$. (b)(c)(d) Absorption maps of the absorber (new absorption peak) under 10T external magnetic field with graphene at 0.1 eV, 0.2 eV and 0.3 eV Fermi energy respectively. (e)-(j) Z component of electric field distributions at 10 nm above gold disk at 6.68 THz (graphene at 0.2 eV Fermi energy under 10 T external magnetic field) of different polarization angles.
4. Conclusions
In summary, a tunable THz absorber which can be tuned by external magnetic field has been proposed and investigated by the FEM. Simulated results illustrate that in an external magnetic field perpendicular to graphene sheet, the new absorption peak can be tuned in an ultra-wide range as the Fermi level of graphene changes. On the other hand, the frequency of new absorption peak blue-shifts significantly as magnetic field increases while the Fermi level remains unchanged. The tunability of absorber is greatly enhanced when employing external magnetic field as a new control strategy. Besides, the maximum absorption of the new absorption peak can be enhanced by employing multilayer graphene. The absorber performance is polarization-independent with or without external magnetic field. This work may promote the development and applications of high performance flexibly tunable THz absorbers.
Funding
National Natural Science Foundation of China (11664020, 51962016, 61665004); Project for Distinguished Young Scholars of Jiangxi Province (20171BCB23098); Education Department of Jiangxi Province (GJJ151071, GJJ170944, GJJ170952); Natural Science Foundation of Jiangxi Province (20151BAB207056); Science Project of Jiujiang University (2015LGYB15).
Disclosures
The authors declare no conflicts of interest.
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