Topological super-modes engineering with acoustic graphene plasmons

Jiang Fan; Zhiguo Sun; Yao Lu; Weiwei Luo; Weiwei Luo; Mengxin Ren; Wei Cai; Wei Cai; Jingjun Xu

doi:10.1364/OE.480044

1. Introduction

Topological states of matter have been affecting our fundamental understanding of the physical system, which have been demonstrated beyond condensed matter. Recently, topological photonics [1–3], utilizing the topological properties of light in the reciprocal space, has shown many novel phenomena. In particular, the robustness of interface states to immune the error from disorders and defects, while keeping electromagnetic waves propagating along the domain wall between the photonic crystals (PCs) with disparate topology, has attracted wide attention [4–7]. Moreover, topological laser, waveguides, and cavities have been demonstrated [8–12].

On the other hand, graphene, a typical two-dimensional material, supports plasmons with long lifetime, strong field localization, and easy tunability [13–16]. Moreover, by coupling the charge density oscillations of graphene with their mirror images, acoustic graphene plasmons (AGPs) can be formed within the gap between graphene and metal film [17,18]. The larger wave vector of AGPs enables higher spatial confinement and field enhancement, which can even be down to the atomic scale [19]. Therefore, AGPs provide a promising platform for enhanced light-matter interaction and miniaturized optoelectronic devices [20–22].

It is natural to incorporate topological concepts into AGP systems, which would endow topological photonic devices with extreme field confinement and tunability. In fact, robust topological interface states have been proposed in various graphene plasmon (GP) systems [23–26]. The valley hall effect and corner states were realized in metagate-tuned graphene [27–29]. In addition, one-dimensional (1D) topological structures based on AGPs have recently been reported [30–32], which were described by the Su-Schrieffer-Heeger (SSH) model [33,34]. However, most of these current works focus on isolated interface states formed between two kinds of crystals with different topological natures. By hybridizing isolated interface states, topological super-modes can be formed. The adjustable geometric parameters, including unit element size, number, and period provide rich degrees of freedom for modulating the dispersion of super-modes and, therefore, light-matter interactions [35,36]. Yet, as far as we know, there is a lack of enough consideration of hybrid topological states in AGPs.

In this work, two kinds of 1D AGP crystals (AGPCs) with different topological properties are designed. Their difference in topology is verified by the topological invariant quantity (Zak phase [37]) calculations and symmetry analysis of the electric fields of the band-edge state. Moreover, by hybridizing the isolated topological interface states within these two AGPCs, super-modes are realized, which are well described by the tight-binding model. Delicate adjustment of the geometric parameters generates flat-band-like dispersion of the super-modes, indicating a possible high density of optical states. Finally, the proposed super-modes are detected from absorption spectra simulations. The variation of absorption peaks in frequency stemming from topological phase transition is observed. Our studies open an avenue for engineering the topological properties of hybrid AGP interface states and are promising for applications in integrated plasmonic devices, including optical sensors and light modulators.

2. Results

First of all, the unit cells of two types of AGPCs (labeled as A and B) are presented in Fig. 1(a) and Fig. 1(b). Both of them have the same period length $L=200$ nm. In each unit cell, monolayer graphene is encapsulated in dielectric fims ($\epsilon _d$), which are placed on the substrate with permittivity $\epsilon _s$. The top dielectric spacer with a thickness of $d = 3$ nm is used to separate the monolayer graphene from silver rods. The height of these rods is $h = 75$ nm, and widths are $w_a =L/2$ and $w_b = L/4$ for the unit cell of A- and B-type AGPCs, respectively. Due to the coupling of the graphene and their image in the silver rods, AGPs can exist in this graphene-dielectric-metal systems [31]. Full-wave electromagnetic simulations are performed based on the finite element method to explore the optical properties of AGPCs. In the calculations, the optical conductivity of monolayer graphene is described by the Drude model [38] as $\sigma _g(\omega )=4\sigma _0E_F/\hbar \pi (\gamma -i\omega )$, where $\sigma _0=\pi e^2/2h$. To obtain a greater effective refractive index contrast, $E_F=0.2$ eV, which is the Fermi energy of graphene, is easy to achieve in experiments by doping. $\omega$ is the frequency of light, and the relaxation rate $\hbar \gamma = 3$ meV. For simplicity, $\epsilon _s = \epsilon _d = 1$ are utilized.

Fig. 1. Band structures of two complementary 1D plasmonic crystals. Schematic of the unit cell of A-type (a) and B-type (b) AGPC. Here, waveguiding properties along $x$-axis are considered. In each unit cell, graphene is encapsulated by dielectrics, and $d=3$ nm, $h=75$ nm. $L=w_a+2w_b=200$ nm denotes the period length of the unit cell. $w_a=2w_b$ represents the width of the metallic rods. (c)-(d) Plasmonic band structure of A-type (c) and B-type (d) AGPCs. Red (blue) dots imply the Zak phase of this band is $\pi$($0$).

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The calculated band structures of these two AGPCs are shown in Fig. 1(c) and Fig. 1(d), where TM modes are considered. The first five bands are given for each crystal, where four band gaps appear. The Zak phase is calculated to understand their intrinsic topological properties. For the $n$th isolated band (without cross point), Zak phase can be defined as [39]:

(1)$$\theta_n^{\text{zak}} = \int_{-\pi/L}^{\pi/L} [i \int_{u.c}dx\epsilon(x) u_{n,k}^{*}(x) \partial{k} u_{n,k}(x)] \,dk$$

where $L$ is the period length of a unit cell, $\epsilon (x)$ is the permittivity of repeating unit, $u_{n,k}(x)$ is the periodic-in-cell part of Bloch electrical field eigenfunction of a state at the $n$th band with Bloch vector $k$. For A-type AGPC, the Zak phases of the first five bands are 0, $\pi$, 0, 0, and $\pi$, respectively. In contrast, $\pi$, 0, $\pi$, $\pi$, and 0 are the Zak phases of the first five bands for B-type AGPC.

On the other hand, the Zak phase of each band can also be verified intuitively by investigating the symmetry of electric fields at two specific symmetric points of the Brillouin zone [40]. Generally speaking, when the electric fields at the two points $k = 0$ and $k = \pm \pi /L$ have the same symmetry, the Zak phase of this band is quantized as 0. On the contrary, the Zak phase is quantized as $\pi$. Taking the third and fourth bands for example, we can see that the electric fields of the third (Fig. 2(a)) and the fourth band (Fig. 2(b)) at $k = 0$ and $k =\pi /L$ possess the same symmetry for A-type AGPC, though even and odd distributions exists for the third and fourth bands, respectively. These results imply that the Zak phases of the third and fourth band are both 0 for A-type AGPC. In contrast, the electric field distributions at $k = 0$ and $k =\pi /L$ show obvious different symmetries for the third (Fig. 2(c)) and fourth bands (Fig. 2(d)) for B-type AGPC, implying the Zak phases of $\pi$. These results are consistent with the accurate calculations of the Zak phase by using Eq. (1).

Fig. 2. Electric field symmetric properties of the plasmonic bands for A-type and B-type AGPC. (a)-(b) Electric field $E_y$ of the third band (a) and the fourth band (b) for A-type AGPC. (c)-(d) Electric field $E_y$ of the third band (c) and the fourth band (d) for B-type AGPC. The electric fields are chosen along the middle line in the spacer, which is between the graphene and metallic rods. Red (blue) lines give the electric field $E_y$ of the band-edge state, $k=0$ ($k=\pi /L$) in Fig. 1(c)-(d).

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According to the bulk-edge correspondence, it is known that exotic topologically protected states exist at the interface between two structures with different topological phases [41–44]. For the cases considered between these two AGPCs here, we can predict the existence of interface states in the first and third bandgaps. Here, we choose the third bandgap for investigation. The analysis for the other one is similar. Nextly, super-modes are studied by coupling of topological interface states existing between A- and B-type AGPC, meaning that there should be at least two interfaces existing in a supercell. To describe two interfaces within a supercell, we can consider systems like $A_{m/2}B_nA_{m/2}$ or $B_{n/2}A_mB_{n/2}$, with repeating unit numbers of $m$ and $n$ for A- and B-type AGPCs, respectively. The superlattice system could be regarded as a 1D dimerized chain.The SSH model can give a befitting description of these systems, which was originally used to describe a coupled spin-free Fermi subsystem in a 1D dimer lattice. The Hamiltonian without polarization is given [45,46]:

(2)$$H=\sum_{n = 1}^{N} \hbar \omega_0 (p_{n}^{{\dagger}} p_{n}+q_{n}^{{\dagger}} q_{n})+(\sum_{n = 1}^{N} v p_{n}^{{\dagger}} q_{n} + \sum_{n = 1}^{N-1} wp_{n+1}^{{\dagger}} q_{n} + h.c.)$$

where $p_{n}^{\dagger }$($q_{n}^{\dagger }$) and $p_{n}$($q_n$) are the creation and annihilation operators of electrons on different sites ($p$ and $q$), respectively. $v$ denotes the hopping amplitude in the unit cell of the dimerized lattice, while $w$ denotes the hopping amplitude between the unit cells in the SSH model. After some algebra and Fourier transformation [46], the Hamiltonian can be written in a compact form,

H(k)=\begin{bmatrix} \hbar \omega_0 & v \\ w & \hbar \omega_0 \end{bmatrix}

with two eigenvalues

(3)$$E_\pm{=}\hbar \omega_0\pm \sqrt{v^2 + w^2 +2vw\cos(k\Lambda)}$$

here, $k$ is the Bloch wave vector, and $\Lambda$ is the length of the supercell. For the case of $|v|\neq |w|$, a bandgap $\Delta E = 2|v-w|$ appears. Otherwise, the bandgap is closed. Meanwhile, we can define the Zak phase of the eigenstates as same as Eq. (1), and by some algebra, we can obtain the relationship of the Zak phase and hopping amplitude [47]:

(4)$$\gamma=\frac{\pi}{2}[\mathrm{sgn}{\lvert v\rvert}-\mathrm{sgn}{\lvert v\rvert -\lvert w\rvert}]$$

$\gamma =0$ ($\lvert v\rvert -\lvert w\rvert >0$) and $\gamma =\pi$ ($\lvert v\rvert -\lvert w\rvert <0$) correspond to trivial and nontrivial bands, respectively. As we know, a topological phase transition should experience a bandgap closure from one phase to another. Specifically, the topological property changes with the band crossing, accompanied by the exchange of electric field symmetry. Based on these, the band shape can be modified and the dispersion of the super-modes can be engineered by controlling the intra-(inter-)hoping amplitude $v(w)$ via design of the geometric parameters of the supercell.

Next, we turn to the concrete implementation of topological AGPs super-mode, which can be formed by the coupling of topological edge modes. In Fig. 3(a), a system of $B_2A_4B_2$ is considered. The two individual interface states of the supercell have the same frequency $\omega _{0}$. Hybridization of these two interface states leads to two new modes, as shown in Fig. 3(b). These two hybrid modes are presented in an extended Brillouin zone, and are well-fitted by the tight-binding model. From the electric field distribution in Fig. 3(c), one mode shows symmetric electric field distributions around the center of the supercell, with frequency $\omega _{s}$. The other one with frequency $\omega _{as}$ has antisymmetric electric field distributions. Moreover, for both super-modes, field enhancement appears close to the interface between A and B, indicating the appearance of topological interface states. Because of the properties of AGPs, the electric fields in both modes are tightly confined within the gap between graphene and silver rods. Besides, lifetimes of these super-modes are calculated in Fig. 3(d). Benefiting from the unique properties of AGPs, these two modes have long lifetimes at the picosecond (ps) scale. Moreover, the antisymmetric mode has longer lifetime than that of the symmetric mode.

Fig. 3. Topological acoustic plasmon super-mode (TAPS) in a period structure formed by supercell of $B_2A_4B_2$. (a) The schematic illustration of the structure, and the TAPS can be excited by a $p$-polarized plane wave. The inset shows the effective dimerized model, which is described by Eq. (2). $\Lambda$ is the length of the supercell. (b) Band structure of the TAPS in an extended Brillouin zone. The grey region shows the edge of the energy band in Fig. 1(c) and 1(d). Blue circles (red circles) indicate antisymmetric mode (symmetric mode) in the bandgap. The light blue (light red) line shows the fitted dispersion of super-modes by Eq. (3). (c) Cross view of the electric field $E_y$ of the $B_2A_4B_2$ structure. The top (bottom) panel shows the electric field at frequency $\omega _s$ ($\omega _{as}$) at $k=0$, and the blankness denotes metallic rods. The green triangles are used to indicate the interfaces between A and B. (d) The lifetime of the super-modes. Blue and red colors correspond to antisymmetric and symmetric modes, respectively.

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Then, we consider how to excite and tune the super-modes. There are always two ways to excite optical modes: far-field light and a localized source such as dipole. Compared with a localized source, far-field light excitation is much simpler. However, due to the requirement of symmetry match, only antisymmetrical mode $\omega _{as}$ can be excited. Figure 4(a) and Fig. 4(b) show the absorption spectra of the super-modes for $A_2B_nA_2$ and $B_2A_mB_2$, respectively. Some interesting features are presented. Firstly, only one peak appears in the band gap region, which corresponds to the antisymmetric mode at $k=0$. Secondly, the frequency of absorption peaks of $A_2B_nA_2$ and $B_2A_mB_2$ oscillates with the increase of $m(n)$. However, these oscillations show different trends. A parameter $\alpha =(\omega _{as}-\omega _0)/\omega _0$ is defined, characterizing the oscillation amplitude, and is calculated at $k=0$ and $k=\pi /\Lambda$, respectively, as shown in Fig. 4(c) and Fig. 4(d). Firstly, for all the investigated values of $m(n)$, both configurations show opposite signs of $\alpha$ between the case of $k=0$ and $k=\pi /\Lambda$, indicating topological non-trivial states with $\lvert v\rvert <\lvert w\rvert$. Meanwhile, with the increase of $m(n)$, the oscillation amplitude of supercell $A_2B_nA_2$ lies always on one single side of $\omega _0$. In contrast, the amplitude for the configuration $B_2A_mB_2$ oscillates across $\omega _0$. This difference can be understood intuitively as follows. For both configurations, each additional unit cell of the A-type (B-type) AGPC leads to a phase accumulation of $\pi$ [35], thus the convexity of the super-mode band changes (see Appendix). For the case of $B_2A_mB_2$, the change of the hoping amplitude $w$ implies an exchange of the two super-mode bands, and $\alpha$ oscillates across $\omega _0$ at $k=0$. While in the case of $A_2B_nA_2$, change of the hoping amplitude $v$ modifies only the convexity of the super-mode band, and $\alpha$ oscillates always on one side of $\omega _0$. And the variation of $v$ and $w$ could be verified by fitting the dispersion of super-modes with the SSH model by using Eq. (3), and the resulting bands can be shown.

Fig. 4. Excitation and control of topological acoustic plasmon super-mode. (a) Absorption spectra of super-modes for $A_2B_nA_2$ configurations as the increase of $n$. Colored lines represent the parity of $n$. (b) The same spectra as (a) but with $B_2A_mB_2$ configurations. The green dashed line denotes $\omega _0$. (c) and (d) The relation between the coupling factor $\alpha$ and $m(n)$ at $k=0$ and $k=\pi /\Lambda$, respectively. The variation trend of anti-symmetric modes indicate the non-trivial topological properties of the optical excited mode. Red dots (blue dashed line) denote $B_2A_mB_2$ ($A_2B_nA_2$) configuration.

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3. Conclusion

In summary, we propose two kinds of 1D plasmonic crystals supporting AGPs. By reconfiguring these two crystals to form a superlattice, the hybridization of the topological interface states is realized, and super-modes appear. Flat-band like symmetric and antisymmetric super-modes with long lifetimes are observed. Furthermore, the anti-symmetric super-mode can be directly detected by far-field light through absorption spectra. And the variation in absorption through changing the distance between the two interface states inside the supercell can be attributed to the topological phase transition of super-modes. Our scheme provides a possible way for designing topological superlattice based on AGPs structures, which should be beneficial to implement integrated sensors and modulators.

4. Appendix

The parameter $\alpha =(\omega _{as}-\omega _0)/\omega _0$ is used to show the change of antisymmetric modes. Specifically, the sign of $\alpha$ indicates the antisymmetric mode is the upper band or under band at $k=0$ (Fig. 4(c)) or $k=\pi /\Lambda$ (Fig. 4(d)), and the value means the coupling strength. As shown in Fig. 5, the increase of $m$ ($n$) means the decrease of the hopping amplitude $w$ ($v$). And in Table 1, one can confirm that the configurations we given have different symmetry at $k=0$ and $k=\pi /\Lambda$, which means that it is non-trivial in topology. Moreover, only the configuration of $B_2A_mB_2$ exchange the symmetry as the $m$ increasing. According to the fitting parameter, the increase in $m$ causes the decrease of $w$, and exchange of super-mode bands occurs. Meanwhile, there is another variation that affects the result. In a superlattice, the emergence of each unit in Fig. 1 leads to a phase accumulation of $\pi$. And the phase accumulation reverses the sign of $v$ or $w$. In the configuration $A_2B_nA_2$, the change of smaller parameter is only in accord with the latter cases. Thus, the change of smaller parameter only affects the convexity at the $k=0$.

Fig. 5. The dispersion of super-modes. (a)-(c) The dispersion of the configuration $A_2B_nA_2$ as the increase of the number of $n$; (d)-(f) The dispersion of the configuration $B_2A_mB_2$ as the increase of the number of $m$. The blue (red) solid lines shows fitting curves of the upper (under) super-mode, and the blue (red) hollow circle shows the simulation results of super-modes. The value of $v$ and $w$ is the fitting hopping amplitudes by the SSH model.

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Table 1. The symmetry of electric fields of super-modes

View Table

Funding

National Key Research and Development Program of China (2022YFA1404800); Guangdong Major Project of Basic and Applied Basic Research (2020B0301030009); National Natural Science Foundation of China (12004196, 12074200, 12127803, 12174202, 12222408, 92050114); Changjiang Scholars and Innovative Research Team in University (IRT13_R29); the 111 Project (B07013); Fundamental Research Funds for the Central Universities (010-63221426).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Configuration	Frequency(THz) at $k = 0$	Frequency(THz) at $k = π / Λ$
$A_{2} B_{2} A_{2}$ ( $A_{4} B_{2}$ )	26.91(even)/27.00(odd)	26.58(odd)/27.26(even)
$A_{2} B_{3} A_{2}$ ( $A_{4} B_{3}$ )	26.68(even)/27.22(odd)	26.86(odd)/27.04(even)
$A_{2} B_{4} A_{2}$ ( $A_{4} B_{4}$ )	26.80(even)/27.10(odd)	26.66(odd)/27.16(even)
$A_{2} B_{5} A_{2}$ ( $A_{4} B_{5}$ )	26.71(even)/27.12(odd)	26.75(odd)/27.14(even)
$A_{2} B_{6} A_{2}$ ( $A_{4} B_{6}$ )	26.73(even)/27.16(odd)	26.74(odd)/27.11(even)

$B_{2} A_{2} B_{2}$ ( $A_{2} B_{4}$ )	26.20(even)/27.50(odd)	26.16(odd)/27.59(even)
$B_{2} A_{3} B_{2}$ ( $A_{3} B_{4}$ )	26.48(odd)/27.33(even)	26.61(even)/27.27(odd)
$B_{2} A_{4} B_{2}$ ( $A_{4} B_{4}$ )	26.80(even)/27.10(odd)	26.66(odd)/27.16(even)
$B_{2} A_{5} B_{2}$ ( $A_{5} B_{4}$ )	26.78(odd)/27.12(even)	26.90(even)/27.05(odd)
$B_{2} A_{6} B_{2}$ ( $A_{6} B_{4}$ )	26.90(even)/26.96(odd)	26.80(odd)/27.04(even)

Topological super-modes engineering with acoustic graphene plasmons

Abstract

1. Introduction

2. Results

3. Conclusion

4. Appendix

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Tables (1)

Equations (5)

Optics Express