## Abstract

Topological photonics provides a novel route for designing and realizing optical devices with unprecedented functionalities. Topological edge states, which are supported at the boundary of two photonic systems with different band topologies, enable robust light transport immune to structural imperfections and/or sharp bends in waveguides. Furthermore, the topological edge states are expected to revolutionize cavity-based optical devices such as lasers. Optical devices with built-in topological protection with a small footprint are fascinating as on-chip optical devices for low-loss and functional photonic integrated circuits. Semiconductor photonic crystals are promising platforms enabling the miniaturization of topological optical devices. Herein, we review the recent realizations of semiconductor topological photonic crystals. In particular, we discuss topological waveguides in valley photonic crystals, which have received increasing attention because of their simple realization. In addition, we provide recent demonstrations of topological nanocavities, which are another key component of topological nanophotonics. Progress in semiconductor topological photonic crystals will propel the use of topological photonic devices in various applications as well as deepen the understanding of topological photonic phenomena at the wavelength scale.

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

## 1. Introduction

Topological photonics [1–7], which has been receiving significant attention, offers a new approach for controlling light waves based on the concept of band topology. Photonic states with topological origin can equip optical devices with novel functionalities that cannot be realized by conventional approaches. Haldane and Raghu theoretically demonstrated that a photonic crystal (PhC) with broken time reversal symmetry can exhibit photonic analogue of the quantum Hall effect [8,9]. Photonic chiral edge states appearing at the perimeter of such a PhC can transport light unidirectionally without backscattering even in the presence of structural imperfections and sharp bends. Following the first demonstration of robust waveguiding using a photonic chiral edge state in the microwave regime [10], topologically protected wave propagation utilizing topological edge states in photonic quantum Hall, photonic quantum spin Hall, and photonic quantum valley Hall systems has been observed not only in microwaves [11–20] but also in optical and near-infrared ranges. [21–24]. Robust single mode lasing from topological edge states in microcavity arrays has been demonstrated [25–28].

The capability of guiding light robustly and other intriguing functionalities enabled by topological protection are highly attractive for photonic integrated circuit technology. Currently, photonic integrated circuits are crucial in a wide range of photonic applications including data communication, signal processing, sensing, and quantum photonics [29]. For further progress, it is expected to miniaturize photonic devices while maintaining or even improving their performances. Conventional photonic devices are often sensitive to local structural perturbations, which will become more prominent when the devices are miniaturized further. Hence, topological photonics may address the demand for high-performance photonic circuits densely integrating various optical components.

In this context, semiconductor PhC slabs are promising platforms. The large
refractive index of a semiconductor enables a large in-plane photonic
bandgap at near-infrared wavelengths while confining light within the
semiconductor layer by total internal reflection. Low-loss PhC waveguides
[30,31], slow-light devices [32–34], and nanocavities with high quality factors
(*Q*) [35] have been
developed. PhC waveguides and nanocavities can confine photons strongly,
enhancing light-matter interactions dramatically. The enhanced
light-matter interactions enable various nanophotonic devices, including
low-power photonic processing components [36], nanolasers [37–39], and quantum photonic devices [40–43]. Harnessing
topological states of light in semiconductor PhCs could allow diverse
nanophotonic devices with novel functionalities to be realized.

Herein, we review the recent progress in semiconductor topological PhCs primary operating in the near-infrared regime, focusing on the introduction of experimental realizations. First, we discuss topological waveguides in semiconductor PhCs in Section 2. Topological waveguides can be realized using an edge state hosted in photonic analogues of quantum Hall, quantum spin Hall, and quantum valley Hall systems. Among them, we mainly discuss topological waveguides in the valley quantum Hall system, which have recently begun to attract attention because of their ease of realization. The waveguides in the first two systems will be discussed in detail in other review papers in this special issue. Section 3 is dedicated to reviewing topological nanocavities, which include topological nanobeam cavities and nanocavities based on corner states. Finally, we briefly discuss other related topics and provide the prospects in Section 4. This paper will not comprehensively explain the theory of topological photonics itself, the progress in topological photonics in other platforms, and emerging fields such as active and nonlinear topological photonics. Many excellent review papers have been published regarding topological photonics [1–7]. Review articles focusing on active [44] and nonlinear topological photonics [45] are available as well. Readers can refer to these review papers.

## 2. Topological waveguides in semiconductor PhCs and their applications

We discuss semiconductor topological PhC waveguides formed by connecting two two-dimensional (2D) PhCs possessing different band topologies side by side. The boundary between the two PhCs can support one-dimensional (1D) topological edge states, which are exploited as a waveguide mode. The properties of the edge states depend on the topological phase of the bulk PhCs: the photonic quantum Hall, quantum spin Hall, and quantum valley Hall phases. In this section, we introduce the features of the edge state in each phase and the corresponding experiments in sequence.

#### 2.1 Topological PhC waveguide in photonic quantum Hall system

Chiral edge states hosted in photonic quantum Hall systems differ
significantly from conventional waveguide modes. The chiral edge
states can travel only in one direction along the boundary of two PhCs
forming a waveguide with no backscattering, even in the presence of
structural imperfections [8,9]. To obtain chiral
edge states, the time-reversal symmetry of the system should be
broken. The strong gyromagnetic effect of vanadium-doped
calcium-iron-garnet enabled the realization of chiral edge state in a
microwave PhC [10]. However, in
the optical domain, only a weak magneto-optical effect is available,
making the realization of chiral edges states challenging. Bahari
*et al.* adopted a hybrid approach to break the
time-reversal symmetry of a semiconductor PhC [46]. They bonded a semiconductor PhC slab with
InGaAsP quantum wells on an yttrium iron garnet film, which is known
as one of the best magneto-optical materials at near-infrared
wavelengths. Figure 1(a)
shows a traveling-wave cavity constructed by surrounding a topological
PhC by a trivial PhC. An external magnetic field perpendicular to the
plane opens a topological bandgap with a 42-pm width at a wavelength
of 1.55 µm in the topological PhC, resulting in the appearance
of a chiral edge state at the boundary between two PhCs. Despite the
small bandgap, nonreciprocal lasing showing a unidirectional output
was demonstrated at room temperature [46]. However, a narrow topological bandgap due to the weak
magneto-optical effect hinders the further studies. In fact, one-way
propagations in passive devices have not yet been demonstrated.

#### 2.2 Topological PhC waveguides in the photonic quantum spin Hall system

Helical edge states in the photonic quantum spin Hall phase can be realized while preserving the time reversal symmetry. Two edge states propagating along opposite directions coexist at the same boundary. They have opposite binary degrees of freedom, that is, opposite photonic pseudospin. The propagation direction of edge states depends on the pseudospin, resulting in robust unidirectional waveguiding by the edge states unless the pseudospin flips.

Helical edge states have been demonstrated in a semiconductor ring-resonator array [22,23,28]. The photonic pseudospins in the system are represented by degenerate whispering gallery modes circulating in opposite directions in each ring resonator. Introducing couplings with a complex phase between two adjacent ring resonators is required to make the system topological. Photonic quantum spin Hall systems can be realized using bianisotropic metamaterials [12,14,47,48]. However, these metamaterials need sophisticated structures in subwavelength scale, which are difficult to be realized for the optical domain.

Wu and Hu [49] proposed another
approach for emulating the quantum spin-Hall phase by exploiting the
crystalline symmetry of PhCs. Although they discussed a PhC structure
consisting of dielectric rods in air, the scheme can be directly
applied to semiconductor PhCs because it requires neither a
magneto-optical material nor a metamaterial. One year later from the
proposal, Barik *et al.* reported a design of the
topological PhC in a semiconductor slab [50]. They arrayed triangular-shaped air holes in a
semiconductor slab as shown in Fig. 1(b). In Fig. 1(b), the blue region corresponds to the PhC
having a topological bandgap, whereas the PhC in the yellow region has
a trivial bandgap. Therefore, the boundary between two PhCs can host
helical edge states. A quantum optical interface utilizing the
spin-momentum locking of helical edge states was demonstrated [51]. Circularly polarized photons
emitted from a single InAs quantum dot (QD) coupled predominantly with
one of the counterpropagating edge states, depending on the handedness
of photon polarization. In addition, topological protection on the
edge states allowed the robust single-photon routing against sharp
bends, which indicates that topological waveguides are attractive
platforms for chiral quantum optics [52]. Chiral light-matter interactions can be observed in
topological ring resonators utilizing helical edge states as well
[53,54].

Note that the topological edge states created by this scheme are inherently leaky because the edge states are above the light line. Optical modes above the light line cannot be confined within a PhC slab and will eventually be radiated out through coupling with the free-space modes. This radiation loss limits the propagation length of the edge states [51,55]. Meanwhile, this feature allows the edge state to be observed directly by measuring the radiation from the waveguide [56,57]. The leaky nature elucidates the surface emitting laser using helical edge states [58]. Interestingly, owing to the spin-momentum locking feature of the edge state and the gain saturation, one of helical modes becomes dominant in the lasing process, resulting in vortex laser emission. A surface emitting laser without using an edge state, topological bulk laser, has been also reported [59]. The band edge states of the topological and trivial PhCs have different symmetries, inducing the light reflection at the boundary between two PhCs. This band-inversion-induced reflection is the optical feedback mechanism in the topological bulk laser.

#### 2.3 Topological PhC waveguides in the photonic quantum valley hall system

The other structure garnering significant attention recently is valley PhCs (VPhCs), which are photonic analogues of the quantum valley Hall system. VPhCs can be realized by breaking the spatial inversion symmetry of PhCs originally possessing photonic Dirac cones in their band structures. VPhCs can host topological edge states, referred to as valley kink states. The valley kink states are in below the light line, allowing guiding light without radiation loss in principle.

### 2.3.1 Valley PhCs

Triangular and honeycomb lattices are the famous basic structures of VPhCs. They have symmetry-protected photonic Dirac cones at the K and K’ points in the Brillouin zone. The photonic valley phase emerges when the degeneracies at the Dirac cones are lifted by breaking spatial inversion symmetry of the lattices. VPhC was originally discussed in a triangular-lattice PhC made of silicon posts [60]. Immediately after the proposal, robust wave propagation utilizing valley kink states has been demonstrated in microwave PhCs [17–20] and in an array of optical waveguides [24]. It is noteworthy that the effect of reduced symmetry in triangular and honeycomb lattices has been investigated previously, aiming to open a complete 2D photonic bandgap [61–63]. Triangular-shaped air holes arranged in a triangular lattice reduce the crystal symmetry and lift the Dirac degeneracy for the TM mode at the K points, resulting in a complete gap. The complete gap was experimentally confirmed in a silicon PhC slab [62].

Herein, we explain some of the fundamental features of VPhC using a honeycomb-PhC with air holes as an example. Figure 2(a) shows the unit cell of a honeycomb lattice with equilateral triangular air holes located at sublattices A and B as well as the corresponding first Brillouin zone. When the holes at sublattices A and B are of the same size, the PhC has photonic Dirac cones for the TE mode at the K and K’ points (see dashed black curves in Fig. 2(c)), which are protected by time-reversal and spatial inversion symmetries of the original lattice. For structures with different hole sizes at A and B (see Fig. 2(b)), owing to the broken spatial inversion symmetry, the degeneracies at the Dirac points are lifted and a photonic bandgap is opened (see the red curves in Fig. 2(c))), resulting in the emergence of photonic valley Hall phase. The Berry curvature becomes finite at around the K and K’ points and with opposite signs. Interchanging the hole sizes at the sites A and B flips the signs of the Berry curvature at the K and K’ points. This means that the two VPhCs shown in Fig. 1(b) possess different band topologies. The topological invariant characterizing the band topology of a VPhC is known as the valley Chern number [60], which is defined as the integral of the Berry curvature only within half of the Brillouin zone, not over the entire Brillouin zone as for the Chern number characterizing the photonic quantum Hall phase. The sign of valley Chern number reflects the sign of the Berry curvature at the valley of interested. The difference in valley Chern number gives the number of edge states at the K(K’) valley. The valley Chern number is quantized to 1/2 or −1/2 provided that the difference between the hole sizes at the sites A and B is small. Hence, according to the bulk-edge correspondence [64], an interface can host one topological edge state at the most at each valley.

Figure 3(a)
illustrates zigzag and bearded interfaces, which can support
valley kink states. The red and blue curves in Fig. 3(b) show typical dispersion
curves for the kink states in the zigzag and bearded interfaces,
respectively. Note that the dispersions change when the larger
holes face each other at the interface. Both interfaces can
support an edge state below the light line (dashed line in
Fig. 3(b)), which is
in a stark contrast to the helical edge states in topological PhCs
based on crystalline symmetry. Although the dispersions are
symmetric with respect to the zone center
(*k _{x}* = 0) because
of the time-reversal symmetry, backscattering at sharp corners and
certain types of defects is well suppressed provide that the
inter-valley scattering is negligible [60]. Thus, a VPhC waveguide is expected to
function as a compact and low-loss waveguide in integrated
photonic circuits.

The dispersion of the kink state of a bearded interface shows the band degeneracy at the zone boundary. This degeneracy is protected by glide plane symmetry with respect to the boundary of two VPhCs [65–67]. Around the degeneracy point, the dispersion tends to be flat, giving a large group index. This feature was employed to design a single-mode slow-light waveguide in a silicon VPhC [68]. Robust light propagation through sharp corners in a slow-light VPhC waveguide with large group indices of ∼60 was demonstrated using two-dimensional finite-difference time-domain calculations. Another theoretical study suggests that the valley-protected slow light is more robust than trivial waveguides against the disorder-induced backscattering provided that the disorder level is small [69].

### 2.3.2 Silicon-based VPhC waveguides

Since silicon photonics is a technology underpinning the recent development of photonic integrated circuits [29], the realization of topological nanophotonic devices in silicon platforms is of great importance. Figure 4(a) shows a silicon VPhC with a trapezoidal-shape zigzag interface [70]. The VPhC was designed such that the edge state can guide light at telecommunication wavelengths and was patterned in a 270-nm-thick silicon membrane suspended in air. The transmittance along the interface was almost the same as that through a straight interface, indicating that the back reflection at the sharp turns was suppressed owing to the topological protection [70]. Furthermore, the optical tuning of the transmission of a silicon VPhC waveguide was demonstrated [71]. Free carriers excited in the VPhC by photopumping induced a change in the complex refractive index of silicon, resulting in a shift in transmission spectrum and reduced transmission by ∼85%.

Silicon-on-insulator (SOI) is a widely used platform in silicon
photonics. Collins *et al*. fabricated a VPhC
structure on an SOI platform and confirmed the gap opening induced
by spatial inversion symmetry breaking [72]. The first demonstration of robust light
propagation in an SOI VPhC waveguide was reported in [73] (Fig. 4(b)). The structural asymmetry with respect
to the center of the Si slab layer, in principle, induces a finite
TE/TM coupling. However, the coupling is weak enough to preserve
the topologically protected light transport at the interface. The
spin-momentum locking effect of valley kink states enables the
selective excitation of valley kink states of two different
interfaces. Figure 4(c) shows a device realizing the selective excitation
[73]. A subwavelength
micro-disk coupled with two silicon waveguides was located close
to the entrance of a VPhC having two bearded interfaces along the
K and K’ directions. When light is introduced from silicon
waveguides, a phase vortex is excited in the microdisk. The
direction of the phase winding depends on the direction from which
the light is introduced [74]. The phase vortex can excite the edge state
selectively. In fact, in the experiment, either of the edge states
along the K or K’ direction was excited with high
selectivity depending on the input direction, whereas almost no
selectivity was observed in a control experiment using two silicon
waveguides instead of VPhC waveguides.

Moreover, resonators and beam splitters in SOI VPhCs have been
demonstrated [75]. The
topological protection at sharp corners allows the construction of
a tortuous cavity, as shown in Fig. 4(d), while maintaining relatively high
*Q* factors. A loaded *Q* at the
telecom band of 1.6 × 10^{4},
corresponding to the intrinsic *Q* of
2 × 10^{4}, was experimentally
obtained. Figure 4(e) shows a valley-dependent beam splitter. The incident
light propagating along the K’ direction excited only the
kink states propagating along the K’ direction after the
splitting. This result indicates that the beam splitter functions
based on the conservation of valley pseudospin. In the
demonstration, topological refraction [20] was utilized for efficient coupling between
the VPhC waveguide and silicon waveguides at the input and output
ports. The coupling efficiency between the VPhC waveguide and the
silicon waveguide was estimated to be at least 40%. The
development of efficient couplers between a topological waveguide
and a conventional waveguide will be more important for the use of
topological nanophotonic devices in integrated photonic circuits
[70,76].

Although robust light propagation has been demonstrated in recent experiments discussed above, the direct quantification of topological protection has not been reported. A recent experiment using a phase-sensitive near-field microscope technique quantitatively demonstrated the effectiveness of topological protection [77]. In the experiment, the light field propagating in a silicon VPhC waveguide with sharp turns was mapped in the momentum space, enabling the extent of backscattering in the waveguide to be estimated. The results show that VPhC waveguides utilizing valley kink states were about 100 times more robust against backscattering compared with trivial W1 PhC waveguides. This is encouraging when considering the application of topological waveguides in integrated photonics.

### 2.3.3 VPhC waveguides using III–V semiconductors

PhCs fabricated using III*–*V semiconductors
with high-quality light emitters and optical gain materials are
attractive for active/quantum topological nanophotonics. The first
demonstration of a VPhC with light emitters was a GaAs VPhC slab
embedding high-density InAs QDs [78]. Figure 5(a) (top) shows a scanning electron microscopy (SEM)
image of a GaAs VPhC with a Z-shaped zigzag interface. The QDs
around the input grating located on the upper left were optically
excited at low temperature. Subsequently, the emission from the
output grating at the bottom right was measured. Although the QD
ensemble showed a broad emission from 920 to 1,250 nm, in the
spectrum measured above the output grating, strong emission was
observed only in the range from 1,020 to 1,120 nm, which
corresponded to the wavelength range of the edge state. The PL
image captured for the transmission wavelength range is shown in
Fig. 5(a) (bottom).
In addition to clear emission from the output grating, no
significant scattering at the corners was observed. These results
indicate that the light emitted from the QDs was efficiently
guided along the interface, even in the presence of sharp turns in
the topological waveguide.

Topological lasers with a small footprint are key for future
applications. Noh *et al*. recently realized a
valley PhC laser at a telecommunication wavelength [79]. The authors fabricated a
triangular-shaped ring resonator comprised of a zigzag interface
between two topologically distinct VPhCs in a III-V membrane
containing InGaAsP quantum wells (Fig. 5(b) (top)). The valley edge state below the
light line forms high-*Q* resonant modes that
extends throughout the ring cavity. By optimizing the structure
such that the difference between the highest and the
second-highest *Q* of these modes is maximized, the
authors successfully demonstrated single-mode lasing at 1.55
µm (Fig. 5(b)
(middle)). Figure 5(b) (bottom) shows the emission patterns at different
pumping powers. Below the threshold, emission was observed over a
broad area corresponding to the pump beam size. Above the
threshold, strong emission was observed mainly from the vertices
of the triangular cavity. The robustness of the lasing against
defects is an issue to be investigated in future research.

Valley kink states are also attractive as photonic states available
for chiral quantum photonics [52]. Chiral interactions between a single quantum emitter
and valley kink states have been investigated using a
triangular-shaped ring resonator like Fig. 5(b) (top) [80]. Owing to the suppressed backscattering at the sharp
corners of the interface, the cavity hosted two counterpropagating
cavity modes preserving the spin-momentum locking property. Photon
routing with directionality as high as ∼90% was
demonstrated via the chiral interaction between single QDs and the
cavity modes [80]. A high
directionality of emission from a single QD embedded in a
topological ring cavity formed by a bearded interface has been
also reported [81]. As
shown in Fig. 3(b),
two branches of interface modes appeared in a bearded interface.
Cavity modes originating from the band at lower frequencies had
high *Q* factors up to 125,000 (in simulation),
whereas modes from the band at higher frequencies had smaller
*Q* factors and their fields tended to localize at
the corners. This result is consistent with the argument that only
the modes belonging to the band at low frequency possesses
topological protection when the small holes face each other at the
interface [68].

### 2.3.4 Semiconductor VPhCs for terahertz (THz) waves

The applications of semiconductor VPhCs are not limited to the
near-infrared range. The robust transport of THz waves through
sharp corners has been observed in a VPhC THz chip fabricated
using a high-resistivity 190-µm-thick Si membrane [82]. The error-free real-time
transmission of an uncompressed 4K high-definition video through
the VPhC chip was demonstrated, indicating the potential of valley
kink states as excellent information carriers for on-chip THz
communication. Zeng *et al*. realized a topological
THz quantum cascade laser, which is the first electrically driven
VPhC laser. The authors formed a triangular ring cavity utilizing
a valley kink state in a wafer containing multiple quantum wells,
providing optical gain in the THz range via quantum cascade
process and realizing multi-mode lasing [83]. In contrast to a laser using trivial guiding
modes, the laser emission wavelengths in the topological cavity
are insensitive to defects around the cavity. Furthermore, the
lasing wavelengths are fixed along the perimeter of the ring
cavity. These results show that the wave localization at the
corners and defects is sufficiently suppressed due to the
topological protection of the valley kink state forming the cavity
modes.

## 3. Topological nanocavity in semiconductor PhC systems

Nanocavities are another key component of integrated photonics. The topological concept can offer a different approach from conventional ones for designing PhC nanocavities. For example, a single-mode optical cavity can be designed deterministically because the presence and number of topological edge states are predictable by knowing the band topologies of two PhCs in contact. In contrast to topological ring cavities utilizing 1D topological edge states, a topological nanocavity has a mode volume close to the diffraction limit, resulting in a significant enhancement in its light-matter interaction. This enhanced interaction will enable the realization of efficient lasers, nonlinear optical devices, and sensors using topological nanocavities. In addition, coupling between topological nanocavities and topological waveguides may offer a new opportunity for exploring intriguing functionalities combining the topological features for both. In this section, we discuss two types of topological nanocavities: nanocavities based on a 0D topological state hosted in a 1D system and nanocavities based on a higher-order topological state appearing at a corner in a 2D PhC system.

#### 3.1 Topological nanocavity in 1D PhC systems

Topological edge states in a 1D system are spatially localized at the
edge of the system or at the boundary between two systems with
different band topologies, resulting in a state functioning as a
standing-wave optical cavity. A representative model of a topological
1D system is the Su-Schrieffer-Heeger (SSH) model initially introduced
to describe the electron transport in polyacetylene molecules [84]. 1D arrays of optical waveguides
or microcavities have been employed to implement a photonic SSH chain
[25–27,85,86]. Lasers using the edge states in microcavity-based
photonic SSH chains have been demonstrated [25–27]. Constructing such an SSH chain
using PhC nanocavities is expected to yield a localized state with
much a smaller mode volume. Han *et al*. realized a
nanophotonic SSH chain with an array of L3 PhC nanocavities in a
230-nm-thick InP slab containing InGaAs quantum wells [87]. The coupling between adjacent
nanocavities was controlled by changing the distance between two
cavities (Fig. 6(a)).
Topological-edge-state lasing was confirmed based on the emission
wavelength and of the near-field distribution of the lasing mode. The
robustness of the topological-edge-state lasing was also investigated
experimentally.

PhC nanobeams are another platform that can support a topological
localized state. A topological nanocavity in a GaAs PhC nanobeam and
its lasing oscillation have been reported [88]. Figure 6(b) schematically shows the topological nanocavity. The
nanocavity is formed at the interface between two topologically
distinct PhC nanobeams. The unit cells on both sides of the interface
contain two air holes of different sizes. In each cell, one hole is
located at the center and the other at the edge. The band topology in
1D systems can be characterized by the Zak phase. Since the unit cells
have inversion symmetry, the Zak phase of each photonic band in the
system is either as 0 or π [89]. The Zak phase of the first band for the blue region in
Fig. 6(a) having a
larger hole at the edge is π while the Zak phase for the red
unit cell is 0. The nontrivial nature of the band can be recognized by
a change in symmetry of the field distribution across the Brillouin
zone [44,88]. The presence of a single in-gap mode at the
interface between two nanobeam PhCs is guaranteed if the two PhCs have
different quantized Zak phases [90]. This unique feature differentiates the topological
nanocavity from conventional defect-based PhC nanocavities, in which
the single-mode character is not guaranteed in general.
Figure 6(c) shows an SEM
image of a GaAs topological nanobeam cavity containing InAs QDs as the
gain media along with a calculated mode distribution of the in-gap
topological mode. Experimentally, the edge mode was identified as a
sharp peak in µ-PL spectra measured at a cryogenic temperature.
The cavity mode was observed as a single in-gap mode even when the
photonic bandgap became narrow as the size difference between the two
holes decreased. Lasing oscillation under quasi-CW optical pumping was
realized when the pumping power increased (Fig. 6(d)). The experimental
*Q* factor of the mode was 9,600. A spontaneous
emission coupling factor of 0.03 for the laser was reported. This
large coupling factor is a consequence of the enhanced light-matter
interactions in the high *Q* nanocavity.

Topological physics is also applied to design a high-Q nanocavity in a
line-defect PhC waveguide. An effective quasi-periodic bichromatic
potential for light can be created by introducing additional air holes
in a W1 waveguide with a period of *a*’, which
differs from the period *a* for the surrounding bulk
PhC, along the waveguide axis (Fig. 7(a) and (b)). The system is described by the
Aubry-André-Harper model. An extremely high *Q*
factor exceeding 10^{9} with a mode volume of 1.6
(*λ*/*n*)^{3} was
designed in a PhC waveguide with such an effective bichromatic
potential [91]. An experimental
*Q* of ∼10^{6}, which was limited by the
scattering loss caused by fabrication imperfections, was obtained in a
silicon PhC slab at 1,564 nm (Fig. 7(c)) [92]. A recent theoretical study [93] showed the nontrivial topology of the optical
spectrum in a structure with bichiromatic potential and demonstrated
the formation of a localized mode at the edge of the potential, which
corresponds to a one-way edge state appear in the two dimensional
momentum space including one synthetic dimension, i.e.,
*β*
=*a*’/*a*.

#### 3.2 Topological nanocavity based on corner states

Another approach for constructing a topological nanocavity is to
utilize a topological corner state appearing as a higher-order
topological state in a 2D system. The conventional bulk-edge
correspondence [64] indicates
that an *n*-dimensional topological system can host
(*n*−1)-dimensional edge states at the boundary.
By contrast, the higher-order topological phase recently discovered in
condensed matter systems can support topological edge states with more
than one dimension lower than that of the system [94,95]. Hence, the higher-order topological state in a 2D system
becomes a corner state that is a localized state at the corner of a
topological structure. Such a localized state of photons can be
utilized as a nanocavity mode. The corner state formed in a 2D
platform may facilitate coupling with topological waveguides, enabling
the development of novel photonic circuits using topological photonic
components.

The corner states originally discussed in condensed matter physics are
induced by non zero bulk quadrupole polarization. The photonic
counterpart has been realized in microwaves using a
gigahertz-frequency reconfigurable microwave circuit [96] and in optical frequencies using
a microring cavity array [97].
The unit-cell plaquette consists of a square with four resonators. A
synthetic π flux threading each plaquette is introduced by
rendering one of the couplings between adjacent resonators negative
while keeping the others positive, resulting in the formation of
quadrupole corner states. However, no simple mechanism exists for
introducing negative coupling in the PhC platform. Recently, a 2D PhC
exhibiting the quadrupole topology owing to the glide symmetry has
been proposed, and quadrupole corner states were observed in the
microwave regime [98]. Since
such a quadrupole corner state can be supported only in a photonic
bandgap between higher-order photonic bands, a careful design to
increase the *Q* factor will be necessitated for
nanocavity applications.

A bulk insulator without bulk quadrupole polarization can host another
type of corner state. B. -Y Xie *et al.* considered a
square-lattice PhC with dielectric rods and theoretically demonstrated
the appearance of topological corner states with no negative coupling
[99]. The corner states appear
in the bandgap between the first and second photonic bands and are
originated from non-zero bulk dipole polarization instead of non-zero
bulk quadrupole polarization.

The first experimental observation of the corner state
in the optical regime using a PhC platform was reported in a GaAs PhC
slab as depicted schematically in Fig. 8(a) [100]. The unit cells for the topological and trivial lattices
are indicated by red and blue squares in Fig. 8(b), respectively. Each contains two
square-shaped air holes with sides of different lengths,
*d*_{1} and *d*_{2}. One
is located at the center, whereas the other at the corner of the unit
cell as shown in Fig. 8(b). When
*d*_{1} = *d*_{2},
the first and second bands degenerate along the X-M line in the first
Brillouin zone and no photonic bandgap appears. Breaking the size
symmetry between two holes creates a photonic bandgap between the
first and second bands for the TE mode. Figure 8(c) shows the band structure calculated using
2D plane wave expansion method for the trivial and topological PhCs
with
(*d*_{1},*d*_{2}) = (0.7*a*,
0.1*a*), where *a* is the period.
Because the unit cells can be exchanged with each other by shifting
one of them by half a period along the *x-* and
*y-* directions, the trivial and topological PhCs have
the same band structure. However, their band topologies characterized
by the 2D Zak phase [99,101] are different. The 2D Zak phase ($\theta_{x}^{zak},
\theta_{x}^{zak}$) associated with the first band is
(0, 0) for the trivial PhC and ($\pi, \pi$) for the topological PhCs. This is
reflected in the field distributions at the band edges.
Figure 8(d) shows the
magnetic field (*H _{z}*) distributions at the X
point for the first and second bands in the trivial and topological
PhCs. The distributions in the first and second bands for the trivial
PhC are symmetric and antisymmetric with respect to the centerline
along the

*y*direction, respectively. Meanwhile, each distribution for the topological PhC shows the opposite symmetry. This band inversion is also confirmed at the M point. The same topological transition can be induced by changing the hole position instead of the hole size [102].

When a 90° corner is formed by the topological and trivial PhCs,
an in-gap localized state at the corner emerges. 3D FDTD simulations
indicated that the mode possesses a *Q* factor over
10,000 and a small mode volume of
∼0.4(λ/*n*)^{3}. A typical field
distribution of the corner mode is shown in Fig. 8(a). The simple 2D-SSH model
including only the nearest-neighbor couplings predicts a corner state
embedded in a bulk band. Long-range interactions are essential to
isolate the corner state from the bulk modes in frequency and to
localize the corner state spatially [102]. Long-range interactions in photonic systems also induce
a new class of topological corner states, which have been observed in
a photonic kagome lattice [103].

The authors of [100] fabricated the structure in a 180-nm-thick GaAs slab,
embedding five layers of InAs QDs as internal light sources for
characterizing the nanocavity. An SEM image of the structure is shown
in Fig. 8(e). Because of
fabrication limitations, *d*_{1} was set to
zero. In µ-PL measurements, a sharp peak was observed at a
wavelength of 1,079 nm, which corresponds the corner state
(Fig. 8(f)). Specially
resolved PL measurements confirmed that the emission was highly
localized at the corner. The observed *Q* factor was
∼2,500 [100].
Independent observations of the corner states in PhCs with dielectric
rods have been reported in the microwave regime [104,105].
Photonic corner states realized with no negative coupling have been
also observed in optical-waveguide arrays [106,107].

Nanocavity lasers based on the corner state have been demonstrated at cryogenic temperatures [108] and room temperature [102]. The authors of [102] patterned a square-shaped topological PhC surrounded by a trivial PhC into a 230-nm-thick membrane with InGaAsP quantum wells. By optically pumping the area of a corner, lasing oscillation was observed at room temperature (Fig. 9(a)). The spontaneous emission coupling factor of the laser was ∼0.06. A topological nanocavity in a VPhC platform was also experimentally demonstrated (Fig. 9(b)) [109]. A large difference between the hole sizes at the sites A and B indicated in Fig. 3(a) induces a frequency gap between the bulk modes and the valley kink state. Consequently, a state spatially localized at three corners, which the authors termed as the “triad mode”, is formed within the frequency gap. The nanocavity formed in an InGaAsP slab with quantum wells emitting at 1.5 µm exhibited lasing oscillation at room temperature with pulsed optical pumping. The far-field pattern from an isolated corner shows a doughnut shape (Fig. 9(b), right bottom), suggesting the laser beam has an optical singularity [109].

## 4. Summary

In this paper, we reviewed the recent progress in semiconductor topological PhCs, focusing on waveguides and nanocavities harnessing topological edge states. We also discussed their applications to lasers and quantum photonics. Table 1 summarizes semiconductor-based topological PhC waveguides and nanocavities we discussed.

Topological waveguides and nanocavities in semiconductor PhC platform will be fundamental elements for various topological nanophotonic devices, potentially advancing integrated photonics technology. Besides, semiconductor topological PhCs can provide fascinating platforms for exploring the new realms of topological photonics such as nonlinear topological photonics [45,110–112] and non-Hermitian topological photonics [113–116] on-chip. The progress in these fields could realize novel nanophotonic devices including robust signal processing devices, compact optical switches, and sensors with high sensitivity. Utilizing additional dimensions would be one of the future directions of the field of semiconductor topological PhCs. The introduction of synthetic dimensions in the PhC platform is worth investigating as it may enable the realization of nonreciprocal light propagation without a magneto-optical material [117–119]. Compact and high-performance optical isolators are highly demanded in the current integrated photonic circuit technology. The realization of semiconductor topological three-dimensional (3D) PhCs at the optical domain is one of the future challenges. Novel nanophotonic applications leveraging topological physics of gapped and gapless photonic phases in 3D systems [120–123] are expected to be discovered. Furthermore, topological 3D PhCs with optical gain would provide a playground for 3D non-Hermitian topological physics.

Semiconductor topological PhCs will gather further attention as a platform of topological nanophotonics. We envision that further advance in semiconductor topological PhCs will open a new frontier of nanophotonics and will pave the way to integrated topological photonics.

## Funding

Ministry of Education, Culture, Sports, Science and Technology (17H06138, JP15H05700, JP15H05868); Core Research for Evolutional Science and Technology (JPMJCR19T1); Nippon Sheet Glass Foundation for Materials Science and Engineering; Asahi Glass Foundation.

## Acknowledgements

The authors thank T. Yamaguchi, H. Yoshimi, F. Liu, K. Wakabayashi, H. Hatsugai, T. Ozawa, S. Takahashi, T. Baba, K. Kobayashi, and K. Ikeda for fruitful discussions regarding some of the topics discussed herein.

## Disclosures

The authors declare no conflicts of interest.

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