Far‐Field Excitation of Acoustic Graphene Plasmons with a Metamaterial Absorber

Abstract

When a graphene sheet is placed near a metal surface, it supports a special type of highly confined and low‐loss electromagnetic mode called acoustic graphene plasmons (AGPs). AGPs squeeze infrared photons into extremely confined areas down to a subnanometric scale and provides a unique platform for strong light–matter interactions. However, the efficient excitation of AGPs is a challenge due to the large momentum mismatch between free‐space light and AGPs. With theoretical analysis and numerical simulations, it is shown that the far‐field excitation of AGPs is realized by integrating graphene in a metal–insulator–metal (MIM) metamaterial with magnetic resonance (MR). More than ten graphene plasmonic modes are excited in the midinfrared range, resulting in a multiresonant spectra with Fano‐like characteristics at each resonant wavelength. The proposal opens a new door to explore the strong plasmonic coupling between graphene and metallic metamaterials down to atomic scale for extreme nanophotonics. The potential applications range from ultracompact tunable metamaterials and ultrasensitive infrared spectroscopy to single‐molecule optics, quantum plasmonics, and others.

Introduction

Surface plasmons (SPs) are collective oscillations of free charge carriers along the surfaces of conductors. Their electromagnetic fields show characteristics of evanescent waves that decay exponentially with distance from the surface.^{[ ]} Surface plasmons provide a platform to manipulate light beyond the diffraction limit and achieve strong field enhancement, which enables the development of subwavelength optics and has been explored for various miniaturized photonic devices.^{[ ]} Graphene, a 2D material with a single layer of carbon atoms arranged in a honeycomb lattice has recently arisen as an attractive plasmonic material in the midinfrared-to-THz range.^{[ ]} The intrinsic plasmons in graphene exhibit strong spatial confinement, unprecedented tunability, and relatively low losses.^{[ ]} They have been exploited for various applications in the infrared and THz spectral ranges, such as optical modulators,^{[ ]} photodetectors,^{[ ]} nanotweezers,^{[ ]} bisosensors,^{[ ]} as well as tunable metamaterials^{[ ]} and waveguides.^{[ ]}

When a graphene sheet is placed near a metal surface, it can support a special type of graphene plasmonic mode called acoustic graphene plasmons (AGPs).^{[ ]} Characterized by a linear dispersion with large wavevectors, AGPs further enhance the confinement of light compared with ordinary graphene plasmons (GPs) with little contribution of damping from the metal. They can squeeze infrared photons into extremely confined areas down to a subnanometric scale, overcoming weak interaction of incident light and atomically thin graphene and providing a platform for demonstrating the ultimate limits of field confinement.^{[ ]} This extreme confinement of light by AGPs, along with field enhancement, effectively enhances light–matter interactions and is promising for ultrasensitive infrared spectroscopy,^{[ ]} sensing, nanoscale lasers,^{[ ]} and even quantum physics. However, the efficient excitation of AGPs is a challenge due to the large momentum mismatch of free-space light and AGP wave. In the THz range, the coupling of free-space light with AGPs has been demonstrated by near-field scattering of nanotips.^{[ ]} In the midinfrared range, metallic gratings have been used to excite AGPs.^{[ ]} A typical AGP system consists of a continuous graphene layer and periodic metallic gratings separated by a insulator nanogap and the launching of the AGP mode is realized by scattering at the metal edges to satisfy phase matching conditions, which generally has a poor coupling efficiency. The excitation efficiency of AGPs can be improved by placing a reflection mirror below the metal gratings to form a F–P resonant cavity in the vertical direction.^{[ ]} In such a design, the distance between the metal gratings and reflector is a quarter of effective optical wavelength at least. Very recently, the direct far-field excitation of AGP cavities by localized GP magnetic resonance (MR) has been demonstrated, but the efficiency is also yet to be improved.^{[ ]}

In this article, we propose a hybrid metamaterial structure to excite multiresonant AGPs where graphene is sandwiched in a metal grating–insulator–metal structure. Figure  shows the schematic of the proposed structure. The periodicity, width, and thickness of metallic grating are p, w, and h ₀, respectively. The distance between monolayer graphene and above (below) gold grating (slab) is s ₀ and the thickness of the insulator spacer is 2s ₀. The spacer dielectric material can be regarded as CaF₂, Al₂O₃, or 2D materials h-BN, etc. In metamaterials and plasmonics, the metal–insulator–metal (MIM) structure is well known to support strong MR which has explored to design high-impedance surfaces and metamaterial perfect absorbers.^{[ ]} It is also known that MIM nanocavities can compress light into the nanometric gap (at the expense of propagating distance due to high losses).^{[ ]} By far-field excitation of the strong MR, midinfrared free-space light is first squeezed into the nanoscale insulator gap. The metal plasmon mode then couples to the AGP mode with high efficiency due to the overlap of mode distributions in the near field. The resonant excitation of multiple AGP modes more than ten can be observed in the studied spectral range, resulting in multiresonant spectra with Fano-like characteristics at each resonant wavelength.

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Experimental Section

Numerical Simulations

The optical spectra and field distribution of the proposed structure are calculated with numerical simulations (commerically available finite element software COMSOL Multiphysics). In our calculation, we regard graphene as a continuous 2D surface current sheet characterized by its surface electric conductivity σ expressed by σ = σ _inter + σ _intra, where σ _inter and σ _intra exhibit contributions from the intraband and interband transition of electrons in graphene. They can be calculated by 1 ${\sigma }_{\text{intra}}=\frac{2i{e}^2{K}_{B}T}{\pi {\hslash }^2(\omega +i/\tau )}\ln \left(2\cos h\frac{E_F}{2K_{B}T}\right)$ 2 ${\sigma }_{\mathrm{inter}}=\frac{e^2}{4\hslash }(\frac{1}{2}+\frac{1}{\pi }\arctan \frac{(\hslash \omega -2E_F)}{2K_{B}T}-\frac{i}{2\pi }\ln \frac{{(\hslash \omega +2E_F)}^2}{{(\hslash \omega -2E_F)}^2+{(2K_{B}T)}^2})$ where T is temperature, K _B is Boltzmann constant, e is the quantity of elementary electric charge, ℏ means reduced Planck constant, and E _F is the Fermi energy of doped graphene. In this research we assume T = 300 K. We have $\tau =\mu E_f/e{V}_F^2$. $V_F={10}^6\mathrm{m}{\mathrm{s}}^{-1}$ is Fermi velocity. The mobility is $\mu =10000{\text{cm}}^2{(\mathrm{V}\cdot \mathrm{s})}^{-1}$. The metal is selected as gold whose optical permittivity is described by the Drude model, 3 ${\epsilon }_m={\epsilon }_{\infty }-{\omega }_p^2/({\omega }^2+i\tau \omega )$ where the parameters ω _p represent the bulk plasma frequency and τ represents the damping rate. For gold, ε _∞ = 1, ω _p = 1.37×10¹⁶ rad s⁻¹, and τ = 8.37×10¹³ rad s⁻¹, respectively.

Analytic Study of Waveguide Eigenmodes

Theoretical analysis is conducted to study the engenmode in a multilayer graphene plasmonic waveguide, which gives us more physics to understand the excitation of AGPs in the system. A five-layer waveguide system including metal slab, insulator layer, and graphene sheet is shown in Figure . The insulator layer is sandwiched between two semi-infinite thick metal slabs and the graphene layer is located in the middle of the insulator layer. The thickness of insulator layer is 2s ₀. The relative permittivity and permeability for each layer is ε _i and μ _i . We have ${\epsilon }_1={\epsilon }_4={\epsilon }_m$, ${\epsilon }_2={\epsilon }_3={\epsilon }_d$, and μ _i  = 1 for nonmagnetic materials in our research. A complex surface current density J = σ E is used to describe the electric properties of graphene sheet. The component of wavevector in the direction of z axis is defined by k _i z and ${\gamma }_i=i{k}_{i}z,i=\mathrm{1,2,3,4}$. The tangential component of the plasmonic wave vector is expressed by k _x or k _spp. We derive the GP dispersion relationship by solving Maxwells’ equations using boundary conditions to find out the transverse magnetic (TM) mode solution. In Figure , the magnetic field in media I, II, III, and IV can be expressed as 4 $\overrightarrow{H}=\{\begin{array}{rr}\hfill H_{1}exp(i{k}_{x}x+{\gamma }_{1}z)\widehat{y}& \hfill z>s_0\\ \hfill (H_{2}exp(i{k}_{x}x+{\gamma }_{2}z)+H_2^{\prime }exp(i{k}_{x}x-{\gamma }_{2}z))\widehat{y}& \hfill s_0<z<s_0\\ \hfill (H_{3}exp(i{k}_{x}x+{\gamma }_{3}z)+H_3^{\prime }exp(i{k}_{x}x-{\gamma }_{3}z))\widehat{y}& \hfill -s_0<z<0\\ \hfill H_{4}exp(i{k}_{x}x-{\gamma }_{4}z)\widehat{y}z<-s_0\end{array}$

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Among these expressions, the components of the wave vector satisfy this equation $k_x^2-{\gamma }_i^2={\epsilon }_i{\omega }^2/c^2$, where the parameter c represents vacuum light velocity and ω means wave angular frequency. After getting the expression of magnetic field in the multilayer waveguide, we can derive electric field expressions by this equation $\nabla \times \overrightarrow{H}=j\omega {\epsilon }_r{\epsilon }_0\overrightarrow{E}\mathrm{.}$ Therefore, the tangential components of electric and magnetic field can be represented by 5 $E_x=\{\begin{array}{rr}\hfill -{\gamma }_1{H}_{1}exp({\gamma }_1{s}_0)/i\omega {\epsilon }_1{\epsilon }_0,& \hfill z=s_0\\ \hfill (-{\gamma }_2{H}_{2}exp({\gamma }_2{s}_0)+{\gamma }_2{H}_2^{\prime }exp(-{\gamma }_2{s}_0))/i\omega {\epsilon }_2{\epsilon }_0,& \hfill z=s_0\\ \hfill (-{\gamma }_2{H}_2+{\gamma }_2{H}_2^{\prime })/i\omega {\epsilon }_2{\epsilon }_0,& \hfill z=0\\ \hfill (-{\gamma }_3{H}_3+{\gamma }_3{H}_3^{\prime })/i\omega {\epsilon }_2{\epsilon }_0,& \hfill z=0\\ \hfill (-{\gamma }_3{H}_{3}exp(-{\gamma }_3{s}_0)+{\gamma }_3{H}_3^{\prime }exp({\gamma }_3{s}_0))/i\omega {\epsilon }_3{\epsilon }_0,& \hfill z=-s_0\\ \hfill {\gamma }_4{H}_{4}exp({\gamma }_4{s}_0)/i\omega {\epsilon }_4{\epsilon }_0,& \hfill z=-s_0\end{array}$ 6 $H_y=\{\begin{array}{rr}\hfill H_{1}exp({\gamma }_1{s}_0),& \hfill z=s_0\\ \hfill H_{2}exp({\gamma }_2{s}_0)+H_2^{\prime }exp(-{\gamma }_2{s}_0),& \hfill z=s_0\\ \hfill H_2+H_2^{\prime },& \hfill z=0\\ \hfill H_3+H_3^{\prime },& \hfill z=0\\ \hfill H_{3}exp(-{\gamma }_3{s}_0)+H_3^{\prime }exp({\gamma }_3{s}_0),& \hfill z=-s_0\\ \hfill H_{4}exp({\gamma }_4{s}_0),& \hfill z=-s_0\\ \end{array}$

Using the boundary conditions, we can obtain the eigenmode equation as 7 $exp({\gamma }_2{s}_0)=\frac{2{\epsilon }_2^2{\gamma }_1+\alpha {\epsilon }_1{\gamma }_2^2+2{\epsilon }_1{\epsilon }_2{\gamma }_2+\alpha {\epsilon }_2{\gamma }_1{\gamma }_2}{2{\epsilon }_2^2{\gamma }_1+\alpha {\epsilon }_1{\gamma }_2^2-2{\epsilon }_1{\epsilon }_2{\gamma }_2-\alpha {\epsilon }_2{\gamma }_1{\gamma }_2}$

Here, the parameter $\alpha =i\sigma /\omega {\epsilon }_0$, with ε ₀ denoting vacuum permittivity. When α = 0, Equation () can be simplified to Equation (), which is the eigenmode equation for the three-layer slab waveguide (the MIM waveguide). 8 $\mathit{exp}({\gamma }_2{s}_0)=\frac{{\epsilon }_1{\gamma }_2+{\epsilon }_2{\gamma }_1}{{\epsilon }_1{\gamma }_2-{\epsilon }_2{\gamma }_1}$

Results and Discussion

Eigenmode of the Multilayer Graphene Plasmonic Waveguide

We first study the eigen AGP mode in the metal–insulator–graphene–insulator–metal (MIGIM) multilayer waveguide system. The spacer dielectric material is assumed to be lossless with the refractive index n = 1.4 (i.e., CaF₂). By solving Equation , we can get the dispersion relationship of the eigenmode. The derived propagation constant k _spp or k _x is a complex number for the reason of Ohmic loss when the GP mode propagates along the graphene sheet surface. The real part of k _spp decides the plasmon wavelength ${\lambda }_{\text{spp}}=2\pi /Re(k_{\text{spp}})$ and the imaginary part of k _spp determined the attenuation of the propagating wave. The calculated results are shown in Figure .

The black dashed curve is the dispersion of the plasmonic mode in supended graphene whose wave vector is typically tens of times that for free-space light.^{[ ]} If the middle insulator layer is thick, the screening effect of the metal is relatively weak and dispersion curve tends to that of suspended graphene film. As the thickness of the middle insulator layer s ₀ decreases, the wave vector increases, which means a decrease in plasmon wavelength and an increase in the spatial confinement for GPs. Meanwhile, dispersion becomes linear, which is typical for AGPs. The effective mode refractive index (EMRI) can be calculated by $n_{\text{eff}}=k_{\text{spp}}/k_0$, where $k_0(k_0=2\pi /{\lambda }_0)$ is vacuum wave vector. For example, when the frequncy is 59.96 THz (corresponding to the optical wavelength of 5 μm in free space), the real part of wavevector is $Re(k_{\text{spp}})=3.49\times {10}^7$ rad m⁻¹, momentum is $3.68\times {10}^{-27}$ J·s m⁻¹, and the EMRI is $n_{\text{eff}}=27.8$ for GPs in suspended graphene. For the AGPs in the multilayer graphene plasmonic waveguide with s ₀ = 100 nm, $Re(k_{\text{spp}})=6.83\times {10}^7$ rad m⁻¹, momentum is $7.21\times {10}^{-27}$ J·s m⁻¹, and $n_{\text{eff}}=54.4$. As s ₀ decreases to 2 nm, it increases to $Re(k_{\text{spp}})=1.89\times {10}^8$ rad m⁻¹, momentum is $1.99\times {10}^{-26}$ J·s m⁻¹, and $n_{\text{eff}}=150.2$, respectively.

Far-Field Excitation of AGPs

Now we study the far-field excitation of AGPs with a metamaterial absorber. We assume that the TM polarized plane wave is illuminated on the proposed metamaterial–graphene hybrid structure in Figure  at normal incidence. (If not specially mentioned in following article, the polarization of plane wave is the TM mode and incident direction is normal to this structure from above air medium.) Here p = 1620 nm, w = 30 nm, h ₀ = 30 nm, and dielectric thickness s ₀ = 15 nm and graphene's Fermi energy is E _F = 0.64 eV. The simulation results are shown in Figure . As the transmission is totally blocked by the optically thick gold film in the bottom, we only show the optical absorption spectra here (Figure ).

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If there is no graphene in the middle, the metal grating–insulator–metal structure forms a typical MIM metamaterial. It displays a relatively broad resonance at around 7.95 μm with a full width at half maximum (FWHM) at about 1.31 μm (see the blue solid curve in Figure ). The resonance arises from the MR of metal plasmons, which has been widely exploited for perfect absorbers or high-impedance surfaces.^{[ ]} Figure  shows the magnetic field and electric field distributions at the resonance wavelength, respectively. The magnetic field is strong enhanced and localized mainly in the middle insulator. If the graphene film is placed in the middle of the insulator layer, the absorption shows a multiresonant spectra with peaks and dips in the studied spectral range(the red solid line in Figure ). The resonances are related to the coupling between the MR of the metal plasmons and the Fabry–Pérot (F–P) modes of AGPs in the nanometer gap between the top and bottom metals (see Figure ) and they display a Fano-like lineshape.^{[ ]} It should be noted that coupling and hybridization of multiple spatially overlapped modes is a common route to achieve multiresonant plasmonic systems.^{[ ]} The resonant excitation of multiple AGP cavity modes has been observed in previous experiments^{[ ]} at the same time, but the excitation strength is quite weak. The effective cavity length satisfies the F–P equation which means that it should equal an integral multiple of half the GP wavelength. However, at normal incidence, only antisymmetric graphene plasmonic modes can be excited where the cavity length equals an integral multiple of the GP wavelength. The condition of excitation satisfies following equation 9 $\delta \varphi +Re(k_{\text{spp}})L=2m\pi ,m=\mathrm{1,2,3}\text{…}$ where δϕ is the phase shift and the integer m represent the order number. The above condition Equation () can be simplified to 10 $L_{\text{eff}}=mRe({\lambda }_{\text{spp}}),m=\mathrm{1,2,3}\text{…}$ where L _eff is the effective length of the cavity which is mainly decided by the width of gold rods in the grating. The frequency interval and bandwidth of the AGP resonances (typically from tens to a few hundred nanometers) are much narrower compared with the bandwidth of the metallic metamaterial's MR. As a result, more than ten AGP modes can be observed in the studied spectral range by resonant excitation(see Figure ). As an example, the field distributions of the ninth-order mode are shown in Figure  which are plotted at the wavelength of 9.995 μm. As shown clearly, the electromagnetic field of AGPs is tightly confined in the nanometer gap between the top and bottom metals. For simplification, we assume that the distance between graphene and the bottom metal mirror equals the distance between graphene and the top metal grating in the paper. However, AGPs can stilled be excited if the two distances are different in a asymmetric system (see Figure S1, Supporting Information).

Influence of Geometric Parameters

Now we investigate the influence of geometric parameters on the optical spectra of the multiresonant hybrid structure and the excitation of AGPs. As we have discussed earlier, the resonant wavelength of AGPs depends on the width of gold bars. The absorption in graphene versus the width of gold rods and wavelength calculated by numerical simulations is shown in Figure , where s ₀ = 15 nm and the slit width w = 30 nm. Instead of the total absorption of the hybrid structure, here, we study the absorption in graphene as the absorption peaks correspond to the resonant excitation wavelengths of AGPs in the F–P cavity. As shown in Figure , the calculated positions of AGP excitation by numerical simulations agree well with the dispersion relationship expression Equation () and excitation condition Equation (). For a fixed order of resonance, the resonance wavelength increases with the increase in gold bar width (i.e. the increase in cavity length). As an example, the tenth-order mode is excited at the wavelength of 8.92 μm when L = 1400 nm and the excitation wavelength redshifts to 9.48 μm as L increases to 1500 nm.

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In Figure , the bar width of metallic grating and the thickness of the insulator layer are fixed and it reveals the dependence of total absorption on slit widths. Particularly, Figure  shows the calculated absorption in the whole structure versus the slit width of top grating and wavelength by numerical simulations. Figure  shows the absorption spectra for five different slit widths from w = 30 nm to w = 70 nm. It is shown that the wide absorption peak produced by MR redshifts and becomes broader with the decrease in w which is due to the coupling effect of MR in nearby unit cells, and similar phenomena have been widely observed in plasmonic metamaterials.^{[ ]} But the positions of the AGP mode or multiresonance peaks don't move. This once again confirms that these resonance peaks are excited by resonant coupling of the MR of the metal plasmons to the F–P cavity modes of AGPs in the nanometer gap. These resonance wavelengths of the latter are decided by the cavity length and insulator thickness. But, on the other side, the width of silts does produce an impact on coupling efficiency or the strength of the AGP mode. As for the insulator thickness, it has a strong influence on the AGP mode in the MIGIM cavity. As shown in Figure , decreasing the thickness of the insulator layer leads to an increase in EMRI n _eff; thus, the resonances of AGP redshift, or different orders of graphene plasmonic modes may be excited at the same wavelength. For example, a tenth-order mode for s ₀ = 15 nm and a ninth-order mode for s ₀ = 19 nm are excited at the same wavelength λ₀ = 9.055 μm, whose field distributions are shown in Figure . AGPs can still be excited even the insulator is down to monolayer thickness (see Figure S2, Supporting Information, where s ₀ decreases to 1 nm).

Tunable Plasmons by Varying Graphene's Fermi Energy

One of the most important properties of GPs is the tunability, which can be realized by changing graphene's Fermi energy with electrical or chemical doping.^{[ ]} We fix the geometric parameters of the structure and study the absorption spectra (total absorption in Figure  and graphene absorption in Figure ) at different Fermi energies. It is shown that absorption spectra and resonant excitation of AGPs blueshift as the Fermi energy increases. Similar phenomena have been widely observed in graphene plasmonic structures. As the Fermi energy of graphene increases, the AGP wavelength increases and the confinement decreases. The simulation results for different Fermi energies are in agreement with theory prediction using Equation. and Equation (), as shown in Figure . The tunability of the proposed structure shows its potential to develop tunable optical devices for light modulation, infared spectroscopy, and other applicaitons. As an example, we designed a tunable perfect metamaterial absorber based on the proposed structure, and it shows that a contrast ratio of more than 600% can be realized (see Figure S3, Supporting Information). The thickness of the tunable absorber is less than λ₀/50. In contrast, conventional graphene-integrated tunable metamaterial absorbers generally consist of a quarter-wavelength cavity, leading to a much larger thickness.^{[ ]}

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Angular Dependence of Resonant Excitation and Optical Spectra

In Figure , the angular dependence of optical absorption spectra is studied. Here Fermi energy of graphene is 0.64 eV and the geometric parameters of the structure arethe same as those in Figure . The optical absorption spectra shows quite weak dependence on the incident angle of light and the resonant peaks don't shift as much as the incidence angles’ increase. In fact, the angular dependence of the absorption spectra in Figure  is quite similar to those of the reported metamaterial/plasmonic absorbers based on localized resonances. As we mentioned in Section , the multiresonant optical spectra of the proposed structure results from the coupling between the MR of the metal plasmons and the F–P modes of AGPs in the nanometer gap between the top and bottom metals. MR is a localized plasmonic resonance of the MIM metamaterial, which doesn't depend much on the incident angle. At the same time, the F–P modes of AGPs depend mainly on the cavity length, insulator thickness. and Fermi energy of graphene. Thus both the total absorption and the resonant excitation of AGPs are not sensitive to the incidence angle. To see the angular dependence more clearly, the absorption spectra of the hybrid multiresonant metamaterial at several different incident angles are shown in Figure .

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Conclusion

In conclusion, we have proposed a hybrid metamaterial structure to excite multiresonant AGPs where graphene is sandwiched in a MIM structure. Eigen AGP modes in the MIGIM multilayer waveguide system are studied analytically and the dispersion relationships agree well with FEM simulations. Numerical simulations were conducted to investigate the optical properties of proposed structures. It is shown that by the far-field excitation of MR, midinfrared free-space light can be squeezed into a nanoscale insulator gap and couples with the F–P cavity modes of AGPs. Resonant excitations of more than ten AGP modes are observed in the studied spectral range, resulting in multiresonant spectra with several absorption peaks and dips. The influence of geometric parameters is studied. The multiresonant spectra can be tuned by changing the Fermi energy of graphene and the optical sepctra show a relatively weak dependence on the incident angle. The thickness of the whole structure is less than λ₀/50.

Our proposed structure can be fabricated with the latest nanofabrication technology. The ultrasmooth metal surfaces can be fabricated by well-developed template-stripping procedures,^{[ ]} which have previously been explored for plasmonics and metamaterials.^{[ ]} In such procedures, gold layer is first deposited (e.g., by thermal evaporation) on a flat substrate (such as polished silicon wafers) and then template stripping is done to expose the ultraflat metal interface. Ultrathin and uniform insulator layers below and atop graphene in the nanogap can be fabricated by atomic layer deposition (ALD). We can use atomically thick 2D materials as the insulator layer and the insulator layers and graphene can be transferred to the top of the smooth metal surface layer by layer. The top array of gold ribbons can be fabricated via electron-beam lithography along with the lift-off process. The gold ribbons can be fabricated directly on the graphene layer or on a smooth template surface and then transferred to the top of graphene.^{[ ]} Our proposal opens a new door to explore the strong plasmonic coupling between graphene and metallic metamaterials down to atomic scale for extreme nanophotonics. The potential applications range from ultracompact tunable metamaterials and ultrasensitive infrared spectroscopies to single-molecule optics, quantum plasmonics, and others.^{[ ]}

Acknowledgements

This work was supported by the Science and Technology Planning Project of Hunan Province (2018JJ1033 and 2017RS3039), the National Natural Science Foundation of China (11304389 and 11674396), and National University of Defense Technology (ZK18-03-05).

Conflict of Interest

The authors declare no conflict of interest.

Authors Contribution

J.Z. conceived the idea and proposed the project. C.W., W.X., and J.Z. conducted the theoretical analysis. C.W., X.C, and J.Z. conducted numerical simulations. C.W. and J.Z. wrote the first draft of the manuscript. W.X., Y.Z., Z.Z., S.Q., and X.Y. contributed to the data analysis and revision of the manuscript. J.Z., Z.Z., S.Q., and X.Y. supervised the project. All authors discussed the results and commented on the manuscript.

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