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Abstract
Graphene plasmons, the electromagnetic waves coupled to charge excitations in a graphene sheet, have attracted great interest because of their intriguing properties, such as electrical tunability, long plasmon lifetime, and high degree of spatial confinement. They may enable the manufacture of novel optical devices with extremely high speed, low driving voltage, low power consumption and compact sizes. In this paper, we propose a graphene-based metasurface which can support a topologically protected graphene plasmon mode with the ability of ultrastrong field localization. We show that such a plasmonic metasurface, constructed by depositing a graphene sheet on a periodic silicon substrate, would exhibit different bandgap topological characteristics as the filling factor of the periodic substrate changes. By setting suitable Fermi levels of graphene at two different areas of the metasurface, topological interface plasmon modes can be excited, resulting in over 8 orders of magnitude enhancement of the plasmon intensity. The topologically protected plasmon mode is robust against the perturbation of the structural parameters, and its frequency can be tuned by adjusting the gate-voltage on the graphene sheet. This highly integrated platform could provide a pathway for low-power and actively controllable nonlinear optics.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Surface plasmon polartons are a type of surface wave existing at the interface between two materials where the real part of the permittivity changes signs across the interface. During the last decade, Surface plasmon polartons on the metal or doped-semiconductor surface have been investigated extensively due to their promising applications in many diverse fields, including optical sensing [1,2], light harvesting [3], optical nanoantennas [4], and photonic metamaterials [5]. Recently, as one of the novel plasmonic materials, graphene has attracted great interest because of its two-dimensional nature and extraordinary electronic and optical characteristics [6]. The behaviors of graphene plasmons (GPs) can be dynamically tuned via electrical gating, opening an exciting way for the control of light at nanoscale [7,8]. In addition, GPs possess strong field confinement and their wavelengths are much smaller than free space wavelength, which are useful for developing compact nonlinear optical devices [9]. However, disorder in graphene-based plasmonic structures will induce a scattering potential and result in larger energy dissipation, which may significantly degrade the performance of the GP-based devices. In addition, higher field enhancement of GP is desired for the design of low-power nonlinear nano-optical devices.
Topology, a property in a system that describe the quantized behavior of the wavefunctions on its bulk bands, has aroused much interest because it promises to offer unique device functionalities. The Jackiw-Rebbi solution describes that topological interface state exists in a quantum system where the topological properties on each side of an interface are different [10–13]. For one-dimensional (1D) periodic systems, Zak phase (geometric phase) is the most common topological invariant to characterize the topological properties of the dispersion bands [14]. It was demonstrated that when two 1D semi-infinite atomic chains with different Zak phases are connected, a topological interface state emerges and the density of states reach maximum at around the connection point [15]. Similar topological interface states have been observed in 1D optical systems [16].
In this paper, we propose a graphene-based metasurface where a graphene sheet is placed on a silicon substrate with periodic grooves. We demonstrate that the bandgap topological characteristics of this plasmonic metasurface depend on the filling factor of the periodic substrate. Topological GP mode forms at the interface separating two metasurface regions possessing different bandgap topological characteristics. Such a topological interface GP mode leads to significant enhancement of the plasmon intensity. We also show that the topological GP mode is robust against addition of disorder to the considered metasurface and its frequency can be dynamically tuned by changing the Fermi level of graphene. Our results may find potential applications in the design of deep-subwavelength and low-power nonlinear devices.
2. Model and theoretical methods
When the grooves in the silicon substrate are deep enough, the metasurface can be seen as a plasmonic waveguide made up of alternating connected graphene-Si and graphene-air waveguides, supporting the propagation of GPs. The unit cell of the periodic system has a symmetric center at the middle of the graphene-Si waveguide, as shown in Fig. 1(b). The green dashed line is the interface between the incident waveguide and the periodic waveguide. By applying the Maxwell’s equations and proper boundary condition, the dispersion relation of the GP in a graphene-Si or graphene-air waveguide can be written as [17]
Fig. 1. (a) Schematic of the graphene-based metasurface. (b) Side view of the graphene-based metasurface. The blue dashed frame is a unit cell of the periodic waveguide and the red dashed line marks its symmetry center. The green dashed line represents the interface between the incident graphene-Si waveguide and the waveguide with periodic substrate. (c) Real and (d) imaginary parts of the effective indices of the plasmon modes in the graphene-Si and graphene-air waveguides under different Fermi level.
By solving the Kubo equation of the graphene [12], the surface conductivity σg of the graphene sheet can be obtained [18]. At terahertz frequencies and under room temperature, σg is dominated by the intraband transition and it obeys a Drude dispersion model
where Ef is the Fermi level and $\mathrm{\tau }= \mathrm{\mu }{\textrm{E}_\textrm{f}}\textrm{/eV}_\textrm{f}^\textrm{2}$ is the relaxation time. In our simulations, we set the Fermi velocity ${\textrm{V}_\textrm{f}}{\; = \; 1}{\textrm{0}^\textrm{6}}\; \textrm{m/s}$ and the carrier mobility $\mathrm{\mu }\; = \; 10000\; \textrm{cm/Vs}$ [19]. Under suitable graphene doping and low temperature environment, the relaxation time $\mathrm{\tau }{\; }$can increase to reduce the loss of GP mode. substituting Eq. (3) into Eq. (2), the effective index of the GP can be written asThe electric field distribution along a 1D periodic structure satisfies the Bloch equation
Fig. 2. (a) Plasmonic band structures for PC1 (Q = 0.2 and ${E}_{f}^{{PC1}}$ = 0.5 eV) and PC2 (Q = 0.29 and ${E}_{f}^{{PC2}}$ = 0.59 eV). The period of the two PCs is Λ = 1 µm. The Zak phase of each band is labelled in blue, and the numbers of the gaps are listed in red. The yellow and green strips inside the gaps represent the different gap topological properties. (b) The center frequency and (c) the width of the 3rd plasmonic gap as functions of the filling factor Q and the Fermi level Ef. (d) The eigenfrequencies of the band-edge states bounding the 3rd gap (color regions) as functions of Q and Ef. The red and blue curves correspond to the band-edge states with different symmetry. The variations of Q and Ef in (d) are in accordance with the white line path in (b) and (c). The insets show the electric field distributions of the symmetric and antisymmetric modes corresponding to the blue and red points, respectively. The field distributions are obtained by Finite-Difference Time-Domain (FDTD) simulations.
Next, we investigate the topological properties of these two PC2. The Zak phase for each isolated band of the PC can be calculated from [25–28]
3. Relationship between the field symmetry of the band-edge states, the reflection phase and the Zak phase
Next, we demonstrate that the symmetry of the band-edge states can also be used to determine the Zak phase, similar to the results in electric systems [15,30]. Figures 3(a) and 3(b) show the third and fourth bands of PC1 and PC2 with the band-edge states marked by red points A1‒D1 and A2‒D2, respectively. The Zak phases corresponding to these bands are labelled in blue and they can be obtained by investigating the symmetries of the electric field distributions of the band-edge states. Figures 3(c) and 3(d) show the absolute value of the FDTD simulated electric field |E|/|E0| for these band-edge states in one unit cell of the infinite PC1 and PC2, respectively. In our simulations, the graphene plasmons are excited by a polarized electromagnetic source. The purple dashed line marks the symmetric center of a unit cell of the PC1 or PC2. If |E| = 0 at the symmetric center, the band-edge state is an antisymmetric mode. Otherwise, if |E| reaches maximum at the symmetric center, the band-edge state is a symmetric mode. It is seen from Fig. 3(c) and 3(d) that the band-edge state B1 is antisymmetric and B2 is symmetric, while the band-edge state C1 is symmetric and C2 is antisymmetric. This results from the switching of the two band-edge states at the band crossing point, as shown in Fig. 2(d). According to the researches of Kohn [31] and Zak [29] for 1D systems with inversion symmetry, the Zak phase of a band is 0 when the two band-edge states have the same symmetry. If the two band-edge states are different in symmetry, the Zak phase of the band is π. It can be seen from Figs. 3(c) and 3(d) that the Zak phase of the fourth band of PC1 is π because the symmetry of the band-edge modes A1 is different from that of B1, whereas the Zak phase of the fourth band of PC2 is 0 because the band-edge states A2 and B2 are both symmetric modes. Similarly, the Zak phases of the third bands of PC1 and PC2 can also be obtained from the symmetry of the band-edge states.
Fig. 3. The band structures of (a) PC1 and (b) PC2 around the third gap. The Zak phase of each band is labelled in blue. Points A1‒D1 and A2‒D2 correspond to the band-edge states in the third and fourth bands of PC1 and PC2, respectively. (c), (d) Electric field distributions of the band-edge states in (a) and (b) in one unit cell of the PCs. The purple dashed line is the symmetric center of a unit cell of the PC. (e), (f) The reflection phase spectra correspond to the third gaps of PC1 and PC2.
Another method to determine the Zak phase of a band is to calculate the reflection phase at the boundary of the PC. Figure 3(e) and 3(f) show the FDTD simulated reflection phase spectra within the third gap of PC1 and PC2, respectively. It is seen that the reflection phase φ is 0 or π, corresponding to a symmetric or antisymmetric band-edge state. Therefore, the Zak phase of a band, depending on the symmetry of the band-edge states, can also be determined from the reflection phase φ of the band-edge states. It is seen that the reflection phase increases monotonically from -π to 0 for PC1 with the increasing frequency, whereas for PC2, it increases from 0 to π. The different frequency dispersions of the reflection phase φ within the two band gaps indicate that the two band gaps are topological inequivalent.
4. Tunable topological plasmon mode with ultrastrong field localization in a graphene-based metasurface
Topological interface states can form when two systems with different topological characteristics are combined together [10–12]. As discussed before, the third bandgap of PC1 with Q = 0.2 and ${E}_{f}^{{PC1}}$ = 0.5 eV and that of PC2 with Q = 0.29 and ${E}_{f}^{{PC2}}$ = 0.59 eV are almost overlapped and they are topologically inequivalent. Thus, a topological interface GP mode is expected to emerge in the heterostructure composed of PC1 and PC2. Figures 4(a) and 4(b) show the schematic of the heterostructure PC1-PC2. Here, both PC1 and PC2 have ten complete periods and an extra half period. Figure 4(c) shows the reflection spectra of PC1, PC2 and PC1-PC2. It is seen that the reflectance of the GP remains high within the third bandgap from about 18.8 to 20.3 THz for both PC1 and PC2. While for the heterostructure PC1-PC2, a narrow reflection dip corresponding to the topological interface GP mode appears at 19.44 THz. Figure 4(d) shows the electric field distribution |E|/|E0| at the frequency of the interface GP mode. It is seen that the electric field at around the interface (x = 0 µm) between PC1 and PC2 is more than 60000 times larger than that of the incident plasmon.
Fig. 4. (a) Schematic of the heterostructure PC1-PC2. The Fermi level of graphene in the two PC can be separately tuned by gate voltage. (b) The side view of the heterostructure PC1-PC2. (c) The reflection spectra of PC1, PC2 and PC1-PC2 around the third bandgaps of PC1 and PC2. The number of periods for PC1 and PC2 are both 10.5. (d) The electric field distribution |E|/|E0| corresponding to the topological interface GP mode in the heterostructure PC1-PC2.
Next, we investigate the dependence of the field confinement ability on the number of periods of the PCs for the heterostructure PC1-PC2. Figures 5(a) and 5(b) show the side view of two different heterostructures where the interface between PC1 and PC2 locates at the graphene-air waveguide (heterostructure A) or at the graphene-Si waveguide (heterostructure B). Np is the number of periods of the PCs. Figures 5(c) and 5(d) show the electric field distributions at the graphene waveguide corresponding to the interface GP modes for the heterostructures with different Np. It is seen that strong localization of the electric field exists in all graphene-air waveguides. For heterostructure A, the electric field reaches maximum in the graphene-air waveguide at the interface between PC1 and PC2, but for heterostructure B, the electric field is strongest in the two graphene-air waveguides close to the interface. It is also seen from Fig. 5(c) that as the number of period Np increases from 10 to 14.5, the normalized electric field |E|/|E0| at the interface decreases from 68000 to 1950. Here, E0 is the electric field of the incident graphene plasmon at the interface between the incident graphene-Si waveguide and the periodic waveguide, as shown in Fig. 1(b). Similar phenomenon can also be seen in heterostructure B, as shown in Fig. 5(d). It means that over eight orders of magnitude enhancement of the plasmon intensity |E|2|/|E0|2 can be achieved inside the considered metasurface at the micron scale. The decrease of the field enhancement with the increase of Np is the result of the loss of the graphene-based waveguide. Considering the electric field intensity of the graphene plasmon is much stronger than that of the wave in free space [17,32]. The field confinement ability of our metasurface is much better than most of the previous plasmonic structures where the magnitude enhancement of the electric field is generally smaller than 100 [33–35]. Therefore, the proposed metasurface are useful for the design of low-power nonlinear devices.
Fig. 5. (a), (b) Side view of the heterostructure PC1-PC2 with two different configurations A and B. For heterostructure A, the interface between PC1 and PC2 is at the graphene-air waveguide; for heterostructure B, the interface is at the graphene-Si waveguide. (c), (d) Field distributions of the topological interface GP modes in the heterostructure A and B, respectively. Np is the number of periods. The green lines mark the interface between PC1 and PC2.
As shown in Figs. 2(b) and 2(c), the positions of the bandgaps for the PC can be tuned by varying the Fermi level of graphene. However, the variations of the Fermi level will not lead to the band crossing process if the structural parameters of the PC are fixed. Figure 6 shows the overlapped region of the third gap and the interface GP mode of heterostructure A as functions of ${E}_{f}^{{PC1}}$ and ${E}_{f}^{{PC2}}$. It is seen that the interface GP mode always exists because the third gap of PC1 and that of PC2 are topologically different. Moreover, both the interface GP mode and the overlapped bandgap shift to higher frequencies as both ${E}_{f}^{{PC1}}$ and ${E}_{f}^{{PC2}}$increases. During this tuning process, the interface GP mode remains near the center of the overlapped gap. The tuning rate is $\textrm{d}(\mathrm{\omega }\textrm{/2}\mathrm{\pi} \textrm{)/d}E_{f}^{{PC1}\; {and}\; {PC2}}\textrm{ = 1}\textrm{.7}\; \textrm{THz/eV}$. It is also obtained from our simulation results that the value of |E|/|E0| at the interface between PC1 and PC2 remains at about 60,000 during this tuning process. Such properties are useful for the design of tunable devices.
Fig. 6. The overlapped region of the third gap and the interface GP mode as functions of the Fermi levels of graphene in PC1 with Q = 0.20 and in PC2 with Q = 0.29. Here the number of periods Np = 10.5.
Since the interface GP mode arises from the combination of two PCs with different topological characteristics, it is protected by topology. It should be noted that the topological interface GP modes in our PC heterostructures are different from the topologically protected one-way edge state in systems where the time reversal symmetry is broken [36]. We then investigate the influence of structural disorder on the topological interface GP mode in the PC heterostructures. Here, we introduce structural disorder into the two PCs by randomly changing the filling factor Qn in the nth unit cell with a fixed period length Λ = 1 µm. The average filling factor $\bar{ Q} = \mathop \sum \nolimits_1^{n} \frac{{{{Q}_{n}}}}{{n}}$ are kept invariant for both PC1 ($\bar { Q}\; = \; {0}{.20}$) and PC2 ($\bar{ Q}\; = \; 0{.29}$). We use a coefficient of variation$\; \mathrm{\sigma }\textrm{= } \sqrt {\frac{{\sum\nolimits_1^n {{{({Q_n} - \overline Q )}^\textrm{2}}} }}{{n{{\overline Q }^2}}}} \times 100\% $ to characterize the level of the structural disorder. Figures 7(a) and 7(b) show the dependence of the topological interface GP mode on the coefficient of variation$\; \mathrm{\sigma }$ in the heterostructures A and B. It is seen that the frequencies of the interface GP modes in heterostructure A or B remain at 19.44 THz and 19.55 THz as σ varies, respectively. The increase of σ leads to the decrease of the transmission peak and the broadening of the interface GP mode.
Fig. 7. The topological interface GP mode in (a) heterostructure A and (b) heterostructure B with different levels of structural disorder. (c), (d) The electric field distributions correspond to the interface GP mode in (a) and (b). σ is the coefficient of variation. Here, $\bar{{ Q}}$ = 0.2 and ${E}_{f}^{{PC1}}$ = 0.5 eV for PC1 and $\bar{{ Q}}$ = 0.29 and ${E}_{f}^{{PC2}}$ = 0.59 eV for PC2.
Figures 7(c) and 7(d) show the simulated electric field distributions correspond to the topological interface GP modes in Figs. 7(a) and 7(b). It can be seen that the field distributions are similar for the PC heterostructures with different σ, and the strongest field localization exists in the graphene-air waveguides which are close to the interface. With the increasing of σ from 0% to 8%, the electric field intensity of the topological interface GP mode decreases gradually. These results confirm that the topologically protected interface GP mode is robust against the disorders.
5. Conclusion
In conclusion, we have proposed a graphene-based metasurface containing plasmonic crystals and demonstrated that it may support the topologically protected plasmon mode. We show that a topological transition of the plasmonic crystal occurs as its filling factor changes continuously. The bandgap topological characteristics can be determined by the symmetry of the band-edge states or the dispersion of the reflection phase. By combining two plasmonic crystals with different topological characteristics, an interface GP mode emerges inside the bandgap and it leads to ultrastrong field confinement. We also show that over four orders of magnitude enhancement of the electric field intensity, i.e., over eight orders of magnitude enhancement of the plasmon intensity, can be achieved in the considered graphene-based metasurface with subwavelength dimensions. In addition, the frequency of the topological plasmon mode can be tuned by changing the Fermi levels of the graphene. Such an interface GP mode is topologically protected and consequently, is robust against the structural disorder. The proposed metasurface may find applications in designing low-power, tunable, subwavelength-scale nonlinear optical devices.
Funding
Science and Technology Program of Guangzhou (2019050001); Natural Science Foundation of Guangdong Province (2015A030311018, 2017A030313035).
Disclosures
The authors declare no conflicts of interest.
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