## Abstract

In this work, we study topological edge and corner states in two-dimensional (2D) Su-Schrieffer-Heeger lattices from designer surface plasmon crystals (DSPCs), where the vertical confinement of the designer surface plasmons enables signal detection without the need of additional covers for the sample. In particular, the formation of higher-order topological insulator can be determined by the two-dimensional Zak phase, and the zero-dimensional subwavelength corner states are found in the designed DSPCs at the terahertz (THz) frequency band together with the edge states. Moreover, the corner state frequency can be tuned by modifying the defect strength, i.e., the location or diameter of the corner pillars. This work may provide a new approach for confining THz waves in DSPCs, which is promising for the development of THz topological photonic integrated devices with high compactness, robustness and tunability.

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

Photonic integrated circuits, which incorporate photonic devices such as lasers, detectors, waveguides and amplifiers on a single chip, are promising for extensive applications in the realms of optical computing, signal processing, communication and biosensing [1,2]. However, they also introduce new challenges, including the reduction of non-negligible losses and backscattering related to inevitable fabrication defects, the increase of the working bandwidth and the miniaturization of the devices, all of which have encumbered large-scale integration of stable photonic devices. Over the past decade, photonic topological insulators and the field of topological photonics in general [3–10] have advanced rapidly, which bring about new solutions for photonic integrated circuits to achieve superior performance of optical networks. A key characteristic of the photonic topological insulators is the presence of unidirectional transmission of edge states immune to backscattering, which can be harnessed to achieve versatile topological photonic devices, such as broadband one-way propagation waveguides [4,11], optical diodes [12,13], robust optical delay lines [6] and topological insulator lasers [14–20]. These topological devices can greatly enhance the functionality of photonic integrated circuits.

Generally, the *n*-dimensional topological insulators can host (*n*-1)D topological edge states, according to the so-called bulk-edge correspondence principle [21]. Recently, higher-order topological insulators (HOTIs) which support lower-dimensional topological states have been presented: in view of 2D case, a second-order topological insulator hosts 1D gapped edge states and 0D corner states (namely, second-order topological states) [22–35]. HOTIs have been demonstrated theoretically and experimentally in several kinds of systems, such as microwave circuits [24,25], waveguide arrays [26,27], photonic and sonic crystals [28–34]. However, recently reported photonic and sonic HOTIs typically need covers to confine the waves in the vertical direction to avoid leaking of energy into free space, whereas the design of the covers can hamper the direct observation of internal fields and can be adverse to sample preparation [36]. Therefore, a platform without covers would be desirable to facilitate the detection of higher-order topological states.

Considering the above needs, designer surface plasmon crystals (DSPCs) can confine the designer surface plasmons in the vertical direction, which provide an excellent platform to probe topological phenomena without covers [37–44]. Designer surface plasmons (also known as spoof surface plasmons) are leakage-free surface electromagnetic modes in the microwave and THz frequency ranges accomplished by manufacturing periodic subwavelength structures on metal surfaces, in analogy with surface plasmons of the metal structure at optical frequencies. Their properties, such as band dispersion, can be easily regulated by adjusting the underlying structural parameters, and this platform can thus be utilized to study intriguing topological phenomena. Up to now, various topological phenomena have been directly observed in the DSPCs already, such as the valley Hall effect by breaking the mirror or inversion symmetry [42,43] and the anomalous Floquet topological phase [44]. Therefore, it deserves further research to realize HOTIs with DSPCs in the THz band. Compared with the previously studied photonic HOTIs, the HOTIs implemented by DSPCs are expected to be more conducive to the miniaturization of devices and the detection of electromagnetic fields. In addition, the achievement of HOTIs in the DSPCs at THz regimes can inspire the development of THz devices [45] with superior performances, such as ultra-dense integration, high compactness and robustness to degradation of function caused by manufacturing defects and environmental perturbations.

In this work, we propose and design HOTIs based on the DSPCs to realize corner states with topological protection at the THz frequency band. The design of the lattice structure is based on the celebrated Su-Schrieffer-Heeger (SSH) model [46,47]. The topological phase transition is found by analyzing band diagrams with various structural parameters related to the coupling strength between adjacent pillars. The underlying mechanism for topological states can be characterized by the 2D Zak phase [22,48]. In fact, 1D topological edge states and 0D subwavelength corner states are found to coexist in specially designed 2D SSH lattices of DSPCs, which can be implemented on a single THz chip. We confirm the robustness of the corner states by introducing various defects, while frequency tuning of the corner state is also demonstrated by altering the structure of the corner metallic cylinder in the 2D lattice.

A perspective diagram of the DSPC is shown in Fig. 1(a), which is composed of subwavelength metallic cylinders arranged in a 2D lattice structure on a metallic surface. The design of the 2D SSH model is shown in the inset of Fig. 1(a), which meets *C*_{4} (4-fold rotation) symmetry. The black dashed box represents a selected unit cell. The structure possesses two distinct center-to-center distances between two adjacent cylinders indicating different coupling strengths: the intra-cell distance *d*_{1} (denoted by the red lines) and the inter-cell distance *d*_{2} (represented by the blue lines). The lattice constant is *a* = *d*_{1}+ *d*_{2} = 100 μm. The height and the diameter of all metal cylinders are *h* = 39 μm and *d* = 20 μm, respectively. The band diagrams of DSPCs with different configurations are illustrated in Figs. 1(b)-(d), which are calculated by the use of 3D finite element method. The gray area is the air light cone, representing all possible modes in the air. We only focus on the modes below the light cone which are truly surface electromagnetic modes localized on the metal surface. The boundary conditions of simulated area in *x* and *y* directions are set as Floquet periodic boundary conditions, and the one in the *z* direction is fixed as the perfect matched layer (PML). The metal is set to be perfect electrical conductor (PEC), due to its high conductivity on the order of 10^{7} S/m in the THz band.

We first numerically calculate the band structure of the DSPC with a simple classic square lattice, where *d*_{1} is equal to *d*_{2} as shown in Fig. 1(c). Along the *Χ*-*М* direction, a degeneracy occurs, i.e., there is no band gap in this case. Then, we calculate the bulk band diagrams of two deformed DSPC structures, where one is to decrease *d*_{1} and the other is to increase *d*_{1} with fixed lattice constant, as shown in Figs. 1(b) and 1(d), respectively. Compared with Fig. 1(c), the band gap appears in the structure with *d*_{1} = 22 μm and *d*_{2} = 78 μm, as shown in Fig. 1(b). The insets illustrate *E*_{z} field profiles of the eigenstates at high symmetric point (HSP) *X* of the two lowest bands. Field profiles explicitly show the even (denoted by the sign ‘+’) and odd (represented by ‘-’) parities of the bands at the HSPs. At point *X*, the parity of the first band is even, which is analogous to an *s* orbital; nevertheless, the parity of the second band is odd, which is analogous to a *p _{x}* orbital. Therefore, this kind of DSPC is analogous to an atomic insulator [30,49]. On the other hand, Fig. 1(d) exhibits the band diagram of the DSPC with

*d*

_{1}= 78 μm and

*d*

_{2 }= 22 μm, which looks like that in Fig. 1(b). However, the parities of the first and second bands at point

*X*are respectively odd and even, which are exactly opposite to those in Fig. 1(b). Accordingly, this type of structure differs from an atomic insulator. Strictly speaking, it can be called an obstructed atomic insulator. In the transition process from Figs. 1(b) to 1(d), the band gap changes from the initial opening to closing, and then to opening again, thereby achieving a band inversion. The difference in parities of the bands at the HSP

*X*between two distinct deformed structures clearly indicates that a band inversion has taken place.

The physical mechanism of the band inversion is then investigated. The presence of band inversion also indicates topological phase transition. The DSPC structure simultaneously fulfills time-reversal symmetry and *C*_{4} symmetry, which results in zero Berry curvature. Consequently, this DSPC is different from topological insulators with non-zero Berry curvature. In this DSPC structure, the topology can be characterized by integration of Berry connection over the first Brillouin zone, namely, the 2D Zak phase **Z** or 2D polarization **P** (**Z** = 2π**P**) calculated by [22,50]

*m*th band and

**is the wavevector. Owing to inversion symmetry of this DSPC, the calculation of the 2D polarization can be reduced to [51]**

*k**j*denotes

*x*or

*y*, and

*η*stands for the parity of the band at the HSP. The summation is carried out over all occupied bands. Therefore, topological phase can be determined by observing the even or odd parities of the

*E*

_{z}profile of the eigenmodes at the HSPs. Since our DSPC structure has

*C*

_{4}symmetry,

*P*is equivalent to

_{x}*P*. Considering the first bulk band, the parity of the eigenstate at the HSP

_{y}*Г*, like an

*s*orbital, is even. According to Eq. (2), when the parity of the eigenmode at point

*X*of the first band is even in the DSPC with

*d*

_{1}<

*d*

_{2},

**P**= (0, 0) which signifies that this structure is topologically trivial (similar to an atomic insulator); yet when the parity of the first band at

*X*is odd in our construction with

*d*

_{1}>

*d*

_{2},

**P**= (1/2, 1/2) manifesting that this DSPC is topologically nontrivial (considered as an obstructed atomic insulator). Those topological phases cannot be transformed adiabatically from one to the other, unless the bandgap closes or the symmetries are broken. Hence, our DSPC structure possesses a symmetry-protected topological property. In addition, according to bulk-edge correspondence, the presence of non-zero

*P*can protect the existence of edge states in the corresponding direction.

_{j}To verify the existence of the edge states, the dispersion curve of a strip-like supercell is calculated, as sketched in Fig. 2(a). The supercell is comprised of topologically nontrivial (*d*_{1} = 78 μm) and trivial (*d*_{1} = 22 μm) DSPCs, combined as shown in the inset. Note that the two DSPCs have distinct *P _{y}*, and thus edge states can occur along their interface predicted from the bulk-edge correspondence. The boundary condition in the

*y*direction is fixed as the continuity type of periodic boundary condition. If we adopt other boundary conditions, such as PML/PEC/scattering boundary condition (SBC), there will exist additional edge states localized at the interface between the DSPC and the PML/PEC/SBC to interfere with the analysis of topological edge states. All other settings are the same as those in Fig. 1. As expected, in Fig. 2(a), there exist edge states in the bandgap (denoted by the purple line), which are different from bulk states shown by the black dots. Figure 2(b) illustrates the

*E*

_{z}distribution of the eigenmode with a frequency of 1.27 THz. As can be seen, the field strength dramatically decreases along the direction perpendicular to the interface, which means that the eigenstate is localized at the boundary of two DSPC configurations with distinct

*P*, thereby signifying its topological origin.

_{y}To investigate topological corner states, we design a composite structure consisting of a topologically nontrivial DSPC (6×6 unit cells, *d*_{1} = 78 μm) encompassed by a trivial DSPC (*d*_{1} = 22 µm), which forms a rectangular frame with a width of 12 unit cells as shown in the right lower inset of Fig. 3(a). The eigenvalues of the structure are numerically calculated. The boundary conditions of the simulation area in the *x* and *y* directions are fixed as SBC, and the other parameters are identical with those in Fig. 1. Around 1.31 THz, there are four almost degenerated eigenmodes, whose electric field distributions are shown in the left inset of Fig. 3(a). *E*_{z} distributions localized at the corners demonstrate that they are corner states. The reason for the appearance of corner states is the filling anomaly in the *C*_{4}-symmetric topological insulator [49]. Accordingly, corner states can be characterized by a topological index

*of a 4-fold rotation operator $\widehat {{r_4}}$ at the HSP Π, where Π*

_{p}

_{p}_{ }= e

^{2πi(p-1)/4}, for

*p*= 1, 2, 3, 4. For a

*C*

_{4}-symmetric Bloch Hamiltonian, $[{\widehat {{r_4}},h(\mathrm{\Pi } )} ]= 0$, and thus the eigenstates of rotation operator $\widehat {{r_4}}$ at HSP Π are those of the Bloch Hamiltonian. In Eq. (3), [Π

*] = #Π*

_{p}*− #Г*

_{p}*, where #Π*

_{p}*stands for the number of bands below the band gap with the eigenvalue Π*

_{p}*. For a topologically nontrivial DSPC, [*

_{p}*Χ*

_{1}] = -1, [

*М*

_{1}] = -1, [

*М*

_{2}] = 0. As a result,

*Q*= 1/4, demonstrating 1/4 fractionalized corner states as shown in Fig. 3(a).

^{corner}As is obvious from Fig. 3(a), the eigenvalues with a frequency ranging from 1.20 THz to 1.58 THz exhibit a series of mode configurations: from bulk states (denoted by the black dots), edge states (presented by blue dots), corner states (indicated by red dots), and back to the bulk states. The electric field profile of the eigenstate at 1.22 THz spreads into the whole bulk which is a bulk state, as presented in Fig. 3(b). Two kinds of DSPCs, which make up the above combined structure, are on the basis of the identical square lattice with different selection of the unit cell. Therefore, the two types of DSPCs are topologically different, but they share a common bandgap where topological states occur. In this bandgap, there emerge gapped edge states, whose dispersion curves do not occupy the entire bandgap. From Fig. 3(c), the field strength at the boundary of the two DSPCs (indicated by the black dashed lines) is significantly strong, while the field strength in the bulk is extremely weak, indicating its nature of the edge state. In the bandgap, corner states also arise, which are highly localized at the four corners with distinct phase structures, denoted by the gray dots in the right lower inset of Fig. 3(a). As can be seen, the frequencies of the corner and edge states are asymmetric as to “zero energy” (that is, the middle of the band gap). This is because in simulation the next-nearest-neighbor coupling cannot be ignored, thereby inevitably breaking the chiral symmetry. Note that the electric field distributions of the four corner states exhibit different phases including a monopole, two dipoles, and a quadrupole. In addition, the ratio of the lattice constant over the corner state wavelength is *a*/*λ* = 0.44, which indicates that the combined DSPC structure presented above can support subwavelength corner states in the THz region. From the above discussion, it is evident that edge states and subwavelength corner states coexist in the combined DSPC structure.

A superior characteristic of the topological state is its robustness, meaning its immunity to structural defects. In order to verify the robustness of the topological corner state, we introduce two different types of defects, the change of the location and diameter of the cylinder, respectively, as shown by the blue/green rods in each inset of Fig. 4. For an ideal structure, the corner state frequency is around 1.31 THz. When shifting the pillar away from the original position as shown in Figs. 4(a) and (b), it is found that the corner state still exists. Similarly, when changing the diameter of the cylinder in Figs. 4(c) and (d), the corner state persists. Corner states are topologically protected by the nontrivial 2D Zak phase. Therefore, provided that the topological characteristic of the DSPC stays unchanged, the corner states remain intact despite the presence of some perturbations. Consequently, we confirm that the corner states, which are implemented in the combined DSPC structure, have topological protection against imperfections. We further investigate the influence of defects on the resonant frequency. The corner state frequency shifts slightly with defects introduced into the corner pillar where the corner state mainly resides, while it remains nearly unchanged with defects introduced into other inside pillars. The changes of the corner state frequency are depicted in Figs. 4(e) and (f). When the corner pillar deviates from its original position, the corner state frequency always has a “blueshift”, irrespective of the movement direction of the pillar. Yet, the corner state frequency decreases monotonously with the increase of the diameter *d* of the corner pillar. As can be seen, the tuning range by the corner pillar diameter is larger than that caused by the corner pillar location. Accordingly, we find that resonant frequency can be tuned by adjusting *Δg* or *d*. In this way, the tuning of the corner state frequency can be realized in this composite DSPC topological structure.

In conclusion, we have proposed and analyzed the formation of THz second-order topological insulators based on DSPC, where topological edge states and corner states are concurrent. Compared with previously reported photonic HOTIs, the proposed HOTI not only possesses tight and robust construction, but also does not require metallic covers to vertically confine electromagnetic waves. Due to the tight localization of the corner states, it can function as a high-Q cavity, which can contribute to the enhancement of the interaction between light and matter at THz regimes, with applications in optical sensing [52] and the detection of optical nonlinear effect [53]. Intriguingly, we find that the corner state frequency is tunable, which might be exploited to design tunable topological lasing if gain and loss are properly introduced. Moreover, in this type of THz topological insulators, subwavelength topological corner states and edge states coexist, promising to develop and integrate the corner cavities and 1D waveguides on a THz chip, which could stimulate the development of versatile topological photonic integrated circuits. In addition, topological robustness, in the form of immunity to fabrication defects and environmental perturbations, is an advantage with respect to manufacturing requirements of THz devices.

## Funding

National Key Research and Development Program of China (2017YFA0303800, 2017YFA030510); Program for Changjiang Scholars and Innovative Research Team in University (IRT_13R29); 111 Project (B07013); National Natural Science Foundation of China (11674182, 12074201, 91750204).

## Acknowledgments

We thank Zhi-Kang Lin from Soochow University for discussion and assistance.

## 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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