1. Introduction

As a rising star semiconductor material, perovskites have shown great potential in optoelectronic applications including photovoltaics, lasers, photodetectors, and light emitting diodes, attributed to their advantages such as long carrier lifetimes, long diffusion length, high fluorescence quantum yield, wavelength tunability etc [1–4]. As a promising semiconductor material, quasi-two-dimensional (quasi-2D) perovskites have drawn extensive attention due to their excellent optical properties and simple preparation processes [5,6]. Benefit from their unique multi-quantum well structure, which enable existence of stable excitons at room temperature, quasi-2D perovskites have made rapid progress in the field of light emitting devices such as lasers and light-emitting diodes [7,8]. Furthermore, compared with their three-dimensional counterparts, the quais-2D perovskites also show better solvability, film formation, optical stability, etc [9,10].

The emission characteristics of a laser depend on the quality of the cavity [11]. High-quality (Q) factor photonic nanocavities can strongly confine photons in mode volume (V) of optical-wavelength dimension. Then, a strong light–matter interaction can be obtained [12]. On the other hand, a high-Q factor can also contribute to the stability and efficiency of a laser. Therefore, high-Q factor is a key goal in the design and optimization of lasers [13]. Meanwhile, topological edge states, although not unidirectional owing to reciprocity, can, in principle, exhibit robustness to a certain class of perturbations [14,15]. Utilizing topological edge states is a promising approach for fabricating high-Q factors, low-loss and low cost lasers [16]. Normally, topological edge states with diverse topological properties appear at the boundary between spatial domains in photonic crystals [17]. Various schemes have been developed to implement such states in two-dimensional photonic crystal lasers, such as quantum Hall (QH), quantum spin Hall (QSH), and quantum valley Hall (QVH) effect [18]. However, the photonic topological insulator based on QH effect should be achieved by using gyromagnetic materials in strong magnetic fields and low-temperature environment [19]. To realize QSH effects, metamaterials with complex structure are required to implement photonic spin-orbit interaction [20]. In contrast, QVH effects can be achieved by using two photonic crystal structures with identical bandgaps, which makes it relative simpler and less restrictive [21,22].

Up to now, the Q factor of these cavities is still limited by electric field leakages in the topological edge states [23]. Topological laser based on the quantum valley Hall effect using hexagonal valley photonic crystal backbone structure, with a simulated Q factor of approximately 2 × 10⁴, has been designed, and further exploration and optimization of these structures to improve Q factor are very essential [24,25]. Improving the Q factor of photonic crystal lasers typically involves tuning of various factors, such as air hole positions [26], the shape of air holes [27], which are very important for developing high performance micro or nano lasers [12]. By optimizing the ratio of thickness to lattice constant to improve sidewall verticality due to the lower aspect ratio of the etched holes, the Q factor has increased by an order of magnitude [28]. By changing the ratio between the periodicity constants of two one-dimensional lattices, the Q factor has been increased by up to one order of magnitude compared to previous studies [29]. Currently, some methods have been reported to suppress the electric field leakage in topological lasers, such as altering the size proportions of photonic crystals [30]. An important design rule of photonic crystal cavities that the form of the cavity electric-field distribution should slowly vary, in order to suppress leakages, can also been applied to improve Q factors in the valley Hall cavities [31].

Here, we propose a high-Q quasi-2D perovskite topological laser based on valley hall photonic crystals that consist of hexagonal lattice with primitive cells containing a hexagonal nanoholes in semiconductor membrane. By adjusting the position of the air holes relative to the pillar, radiation leakage in topological edge states is reduced to a large extent and the Q factor can be as high as 3.6 × 10⁴ which is more than four times higher than that of the cavity with air hole located at the center of the pillar. Despite the sharp corners of the cavity, we find that the electric field distribution still exhibits uniform in the presence of a certain class of perturbations. Finally, green laser emission with a wavelength of ∼ 550 nm using bromine-based quasi-2D perovskite (PEA)₂FA₇Pb₈Br₂₅ has been proposed. The findings in this work hold the promise of advancing the development of high-power, stable perovskite lasers.

2. Topological properties of the quasi-2D perovskite photonic crystals

Firstly, we started from a hexagonal photonic crystal composed of air holes inside the semiconductor as shown in Fig. 1(a). The substrate is a novel semiconductor quasi-2D perovskite with n = 8, (PEA)₂FA₇Pb₈Br₂₅, with metal coatings on both its top and bottom surfaces. The lattice constant a is set to 180 nm and the thickness H of the substrate is set to 100 nm. The air hole pattern in each unit cell is a hexagonal shape with internal angles of 120° (Fig. 1(b)). The six edges have two different lengths which are d₁ (blue lines) and d₂ (red lines). In order to open a bandgap, a perturbation should be introduced to break the ΓK-mirror symmetry of the unit cell, which can lift the degeneracy of the valleys. Here, the perturbation was achieved by setting d₁ = 0.26*a, d₂ = 0.58*a so that a wide open bandgap range can be obtained. As shown in Fig. 1(c), the degeneracy between the first band and the second band at the valleys are lifted and the bandgap between two lower blue lines confirms that the ΓK-mirror symmetry are broken. The grey shading indicates the light cone. Furthermore, we plot the absolute value of the out-of-plane electric field |E_z| (colour maps) within each unit cell at the K points in Fig. 1(d), where the white arrows represent the time-averaged Poynting vectors (energy flux). The fields at the K′ valley could be obtained by the time-reversal operation, which reverses the direction of the energy flux. It is noted that the direction of Poynting vectors at the lower and upper bands is exactly opposite.

 

Fig. 1. Construction characteristics of the hexagonal valley photonic crystal. (a) Each unit cell of the valley photonic crystal contains a hexagonal hole perforated through the top metal and the semiconductor layer in a metal–semiconductor–metal structure. The lattice period a is 180 nm, and the thickness H of the substrate is 100 nm. (b) Top-view unit cell of the lattice. Three of the edges of hexagonal air hole marked by blue outlines have the length d₁ and the other three marked by red outlines have the length d₂. (c) Calculated band structure of the hexagonal valley photonic crystal along the special directions of the First Brillouin zone with d₁ = 0.26*a, d₂ = 0.58*a. (d) Plots of the absolute value of the out-of-plane electric field |Ez| (colour maps) within each unit cell at the lower band and upper band, showing the two degenerate states at the K valley with energy flux represented by white arrows.

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According to bulk-boundary correspondence [32], for pattern I, C_K= -1/2, C_K’=1/2; for pattern II, C^’_K= 1/2, C^’_K’=-1/2. It is evident that the differences in valley Chern numbers between the two domains are ΔC_K= C_K-C^’_K= -1; ΔC_K’=C_K’-C^’_K’=1. As the Chern number for pattern I and pattern II are opposite, non-trivial edge states should emerge in the first bandgap at a domain wall between two different topological photonic crystals as shown in Fig. 2(a). For a boundary between domains of pattern I and pattern II which have opposite hole orientations, energy is well localized in topological edge states. In Fig. 2(b), the projected band diagram has a gap spanned by edge states with opposite group velocities in each valley. These results verify the unidirectional propagation feature of the edge states in the hexagonal valley photonic crystal waveguides. These states are topologically protected under the condition that inter-valley scattering is negligible. This condition is due to the overall symmetry of the valley photonic crystals, and similar conditions also apply to other photonic topological edge states that do not rely on magnetic materials.

 

Fig. 2. Band structures of the hexagonal valley photonic crystal. (a) Supercell comprising two inequivalent valley photonic crystal domains separated by a domain wall (highlighted by a red box). Out-of-plane electric field |Ez| (colour maps) for the edge modes. (b) Projected band diagram for the supercell.

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3. Resonant cavity based on quasi-2D perovskite photonic crystals

Based on the above results, we constructed quasi-2D perovskite topological laser cavities with pattern I and pattern II. A triangular loop of side length 21*a formed by a triangular domain wall was fabricated on a quais-2D perovskite square film with an edge length of approximately 8 µm. As shown in Fig. 3(a), the center of the internal air hole is defined as point A, and the center of the external quais-2D perovskite pillar is defined as point B. Their separation is denoted as d. Figure 3(b), (c) show the calculated electric-field distribution of the quasi-2D perovskite topological laser cavity and its profile along the triangular topological edge state, respectively. It can be seen that the electric field is inhomogeneous, with apparent radiation leakage along the bottom edge when center points of air hole and pillar are overlap with each other. This may be because there are areas with larger and smaller gaps in the distribution of the triangular topological edge states. This provides an important hint for suppressing radiation losses: the spatial variation of the triangular topological edge state should be more uniform. Then we tune the separation d between two center points at the cavity edges in order to make the electric-field profile vary more gently. Adjusting the position of the air holes in the Y-direction, when d is approximately 11 nm, results in a leak-free, uniformly shaped triangular electric field (Fig. 3(d)). The electric field intensities distributed uniformly along the domain wall, even at the sharp corners as shown in Fig. 3(e). Comparing the electric field diagrams, it can be observed that adjusting the air holes can reduce leakage and decrease radiation losses. By adjusting the air holes, the lattice spacing of the bottom edge of the triangle is modified, altering the conditions for Bragg reflection. Such reflection is determined by a summation of partial reflections at a series of air holes near the topological edges. In summary, changing the condition for Bragg reflection results in a more uniform distribution of the topological edge states in the triangular ring, leading to a more uniform electric field.

 

Fig. 3. 2D perovskite topological laser cavity with tuning air-holes. (a) Schematic diagram of the 2D perovskite topological laser cavity. (b) Typical eigenmode electric field (|Ez|) profiles of the 2D perovskite topological laser cavity without tuning air-holes. (c) The profile along the triangular topological edge state without tuning air-holes. (d) Typical eigenmode electric field (|Ez|) profiles of the 2D perovskite topological laser cavity with tuning air-holes at the optimal distance. (e) The profile along the triangular topological edge state with tuning air-holes at the optimal distance.

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Meanwhile, the Q factor of the topological edge state is promoted by suitably tuning the air holes. From the electric field plot, it can be observed that adjusting the positions of the air holes can make the distribution of the topological edge state region more uniform. At the same time, due to finite size effects [33], the characteristics of the topological edge state, such as energy distribution, may change with variations in size. The red lines in Fig. 4(a) represent the calculated results of the Q factor of different d. It can be seen that the Q factor first increases and then decreases with increasing d. Initially, the spatial distribution of the topological edge state becomes uniform with increasing d, because the radiation losses induced by the inhomogeneous spatial distribution of topological edge states are reduced. By tuning the air holes, the topological edge state leaking was suppressed, resulting in a significant improvement of Q factor. However, increasing d further, the transverse losses caused by the finite size effect becomes larger, which leads to a decrease in Q factor. At d = 11 nm, the two effects reach equilibrium and the topological laser reaches a peak Q factor of approximately 3.6 × 10⁴ which is more than four times higher than that of the cavity without tuning the air holes. Modifying the position of the air holes has greatly reduced the losses and improved Q factor. As shown in Fig. 4(b), calculated Q factors of the structure’s eigenmodes at the optimal distance for adjusting the air holes. The maximum Q factor is at 567 THz, within the optimal frequency range of quais-2D perovskite luminescence.

 

Fig. 4. 2D perovskite topological laser cavity with tuning air-holes. (a) Illustration of the change in Q factors after adjusting the length(d) of the air holes. (b) Calculated Q factors of the structure’s eigenmodes at the optimal distance for adjusting the air holes.

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The topological protected laser cavity is also robust against special defects which have regular shapes. Defects such as air hole connection or missing air holes, are very common in a laser cavity. As shown in Fig. 5(a), defects were intentionally introduced to disrupts the periodic arrangement structure of the laser cavity. The defects have a rectangle shape, with a length of 370 nm and a width of 320 nm, respectively. The internal material is quasi-2D perovskite, but the result also applies to other internal materials. In the introduced defects, two complete air columns were missing, and six surrounding air columns were partially missing. As shown in Fig. 5(b) and Fig. 5(c), the defects placed at the side or the corner of the topological edge states do not affect the uniformity of electric field. After adding edge defects and corner defects, the Q factors are 36663 and 35429, respectively, which are close to the Q factor of 36045 without defects. Also, placing special defects randomly in the cavity, as shown in Fig. 5(d) and Fig. 5(e) does not affect the uniformity of electric field, and the Q factors are almost unchanged. This is because the two valley boundary states at the interface are decoupled from each other, and the excited boundary states suppress backscattering during transmission. Therefore, even when encountering a defect, the boundary state can still transmit unidirectionally with negligible losses in the waveguide. However, when the disorders traverse the sample, or multiple defects are placed randomly, as shown in Fig. 5(f), there is serious leakage in the electric field and the Q factors decrease by at least 1.0 × 10⁴. In summary, by adjusting the positions of the air holes, a quasi-2D perovskite topological laser with high Q factor which exhibits robustness to a certain class of perturbations was obtained.

 

Fig. 5. 2D perovskite topological laser cavities with defects. (a) Schematic top-view of the cavity with a defect which has a rectangle shape, with a length of 370 nm and a width of 320 nm, respectively. (b) Typical eigenmode electric field (|E_z|) profiles at around 567 THz, with a side defect. (c) Typical eigenmode electric field (|E_z|) profiles at around 567 THz, with a corner defect. (d) Typical eigenmode electric field (|E_z|) profiles at around 567 THz, with a defect inside the topological edge state of the triangular structure. (e) Typical eigenmode electric field (|E_z|) profiles at around 567 THz, with a defect outside the topological edge state of the triangular structure. (f) Typical eigenmode electric field (|E_z|) profiles at around 567 THz, with multiple defects placed randomly.

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For comparison, as shown in Fig. 6(a), we constructed a cavity with the same valley photonic crystal design, but replaced the topological waveguide with a photonic crystal wave-guide of size-graded holes(with size scale factors b1 = 0.87, b2 = 0.77), with all holes having the same orientation. This reflects the tendency of traditional waveguide modes to undergo localization, unlike the valley edge modes. In Fig. 6(c), we show that in a comparable cavity based on a conventionally designed photonic crystal defect waveguide, the lasing modes behave very differently: they tend to be localized and exhibit inhomogeneous electric field. Moreover, the Q factor of the traditional waveguide laser without defects is 2.4 × 10⁴, but it decreases to 8.2 × 10³ when a defect is introduced.

 

Fig. 6. Trivial laser with triangular loop cavity formed by a conventional photonic crystal waveguide. (a) Schematic diagram of the 2D perovskite trivial laser cavity. (b) Typical eigenmode electric field (|E_z|) profiles of the 2D perovskite trivial laser cavity without defects. (c) Typical eigenmode electric field (|E_z|) profiles of the 2D perovskite trivial laser cavity with a defect.

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4. Conclusion

In this paper, we demonstrated a triangular ring quasi-2D perovskite topology cavity with high-Q factor by utilizing topological edge states. By adjusting the positions of the air holes, the radiation losses in topological edge state regions are suppressed to a large extent and the Q value (∼3.6 × 10⁴) was four times higher than that with air holes located at the center of the pillar. Simulations show that the laser cavity is robust against a certain class of perturbations of the photonic crystal. Compared with traditional photonic crystal lasers, quasi-2D perovskite topology photonic crystal lasers have shown many advantages, such as insensitive to the special defects and impurities, various cavity shapes, small radiation losses and lower lasing thresholds. Our cavity design based on a two-dimensional valley photonic crystal structure provides an effective way for developing high-Q topological lasers and facilitate the development of low-power nanoscale lasers at room temperatures.