Observation of perovskite topological valley exciton-polaritons at room temperature

Abstract

Topological exciton-polaritons are a burgeoning class of topological photonic systems distinguished by their hybrid nature as part-light, part-matter quasiparticles. Their further control over novel valley degree of freedom (DOF) has offered considerable potential for developing active topological optical devices towards information processing. Here, employing a two-dimensional (2D) valley-Hall perovskite lattice, we report the experimental observation of valley-polarized topological exciton-polaritons and their valley-dependent propagations at room temperature. The 2D valley-Hall perovskite lattice consists of two mutually inverted honeycomb lattices with broken inversion symmetry. By measuring their band structure with angle-resolved photoluminescence spectra, we experimentally verify the existence of valley-polarized polaritonic topological kink states with a large gap opening of ~ 9 meV in the bearded interface at room temperature. Moreover, these valley-polarized states exhibit counter-propagating behaviors under a resonant excitation at room temperature. Our results not only expand the landscape of realizing topological exciton-polaritons, but also pave the way for the development of topological valleytronic devices employing exciton-polaritons with valley DOF at room temperature.

Achieving propagating topological exciton polaritons at room temperature is challenging. Here, the authors demonstrate room-temperature valley-polarized topological polaritons with valley-dependent propagation in a perovskite lattice formed by two mutually inverted honeycomb lattices with a bearded interface.

Introduction

Microcavity exciton-polaritons are bosonic quasiparticles resulting from the superposition of semiconductor excitons and microcavity photons in the strong coupling regime^1,2. Being part-light, part-matter, they fuse the advantages from their constituent components, such as a low effective mass, strong optical nonlinearity, etc^3,4. Leveraging these unique properties, exciton-polaritons have emerged as fascinating platforms, not only for investigating collective quantum phenomena at elevated temperatures, for instance, polariton Bose−Einstein condensation⁵, superfluidity⁶, quantum vortices⁷, but also for realizing high-performance optoelectronic devices, including low-threshold polariton lasers^5,8–11, all-optical polaritonic switches^12–14 and transistors^15,16. Recent advances in trapping potentials have further unlocked avenues to precisely manipulate exciton-polaritons with various intriguing Hamiltonians^17–22. In particular, inspired by the integer quantum Hall effect in condensed matter, the extension of topological concepts has largely sparked the emerging field of topological exciton-polaritons^23–31, endowing novel functionality of topological protection to exciton-polariton systems. Their hallmark feature is the appearance of non-trivial topological polariton edge states with immunity against perturbations, which not only provides exciting possibilities for investigating the interplay between bosonic phenomena and topology but also promises polaritonic devices a more robust future.

Early investigations into topological exciton-polaritons date back to the theoretical predictions based on various potential landscapes with or without a strong magnetic field, giving rise to localized or propagating edge states^32–35. Based on Su-Schrieffer-Heeger polariton lattices, localized topological exciton-polaritons were experimentally achieved with GaAs at cryogenic temperatures²³ and further pushed into novel semiconductors hosting more robust polaritons at room temperature³⁶, such as lead halide perovskites and organics^26,28. In addition to localized states, the pursuit of propagating topological exciton-polaritons with novel DOF is highly desirable, as they hold significant promise as information carriers capable of transmitting signals robustly against backscattering towards scalability and reliability. By breaking the time-reversal (TR) symmetry with a strong magnetic field, 1D chiral edge state in the quantum Hall phase was demonstrated in 2D GaAs honeycomb lattices but only at 4 K, constrained by the small exciton binding energy of GaAs. It exhibits a limited topological gap of 0.1 meV due to limited Zeeman splitting of excitons²⁴. Furthermore, strongly coupling a monolayer WS₂ to an analogous quantum spin Hall photonic crystal enabled the achievement of 1D helical topological exciton-polaritons with preserved TR symmetry^25,27. However, in the presence of limited cavity quality, they only work at cryogenic temperatures below 200 K for forming stable polaritons. Achieving 1D propagating topological exciton-polaritons at room temperature becomes a nontrivial but highly crucial task toward potential applications³⁷, which demands stable polaritons with a large topological gap opening and maintains a narrow polariton linewidth.

In addition to quantum Hall and quantum spin Hall systems, the quantum valley Hall effect appears as another promising mechanism to support propagating topological states, named valley kink states, which have witnessed substantial successes in various research fields, including electrons in condensed matter systems^38,39, electromagnetic waves in photonic systems^40–45 and acoustic/elastic waves in phononic systems^46–51. By breaking the inversion symmetry, the valley degeneracy is lifted, and a pair of counter-propagating valley kink states with opposite valley-polarization could arise at the non-trivial interface between two domains with opposite valley-Chern numbers^52,53, in the absence of inter-valley scattering. Importantly, they emerge without the need to break the TR symmetry and provide an extra degree of freedom (DOF) based on valleys. This additional DOF endows topological states with distinct valley polarizations and transport behaviors, holding great promise to store and carry information toward valleytronic applications^54–59.

In this study, by taking advantage of CsPbBr₃ perovskite microcavities hosting robust polaritons at room temperature, we report the experimental realization of valley-polarized topological polariton kink states and valley-dependent propagation in a 2D valley-Hall perovskite lattice working at room temperature. By designing honeycomb lattices with broken inversion symmetry, we experimentally construct a 2D valley-Hall perovskite lattice with a bearded interface consisting of two small pillars between two mutually inverted lattices. Through mapping the momentum-space and real-space photoluminescence spectra, we experimentally demonstrate compelling evidence to validate the emergence of the valley-polarized polaritonic topological kink states in both momentum space and real space. Furthermore, under a resonant pulsed excitation, we take advantage of the valley DOF and observe valley-dependent propagation of the topological kink states, where some polaritons can propagate a macroscopic distance of 8.2 µm at room temperature.

Results

Mechanism of the topological 2D valley-Hall perovskite lattice

The inherent symmetries of honeycomb lattices result in valley degeneracy, introducing an additional DOF akin to spins and paving the way for the burgeoning field of valleytronics. Motivated by this, we begin by theoretically examining a system of exciton-polariton micropillars arranged in a honeycomb lattice (lattice periodicity, a = 0.85 μm; micropillar diameter, d = 0.6 μm). The band structure of such a system along the high symmetry line (Γ→M→K→Γ) of the Brillouin Zone (BZ) is illustrated by the red dashed lines in Fig. 1a. Analogous to electronic graphene, this system also exhibits gapless Dirac points at the two valleys, which are protected by both TR and inversion symmetries and remain gapless unless either of these symmetries is disrupted. While TR symmetry breaking has been explored in polariton systems²⁴, our focus here is on breaking inversion symmetry by selecting different pillar diameters corresponding to the two sublattices (small micropillar diameter, d₁ = 0.5 μm; big micropillar diameter, d₂ = 0.7 μm). The brown solid lines in Fig. 1a showcase the band structure of the inversion symmetry broken system along the high symmetry line of the BZ, revealing the gapping of the Dirac point and the emergence of a bulk bandgap of approximately 5 meV.

Fig. 1. Theoretical mechanism of the topological valley-Hall polariton lattice.

a Band structure of exciton-polariton micropillars arranged in a honeycomb lattice. The dashed and solid lines correspond to the inversion symmetry-preserved and inversion symmetry-broken systems, respectively. The inset at the bottom shows the high-symmetry line of the BZ along which the band structures are calculated. b Two domains of the honeycomb lattice with broken inversion symmetry. One domain is a reflected copy of the other to create an interface at x = 0. The unit cells are shown above the lattice. c Numerically calculated Berry curvature of the two domains, indicating opposite Berry curvature in the two domains, resulting in a difference of $\mathrm{\Delta }{C}_{K\left(K^{\prime }\right)}=\pm 1$ in valley-Chern numbers. d Projected band structure of the system in (b) with the y-direction taken as periodic. Inside the bulk bandgap, counter-propagating valley-polarized modes located at the interface appear, which are protected by the non-trivial valley-Chern number. The spatial profile of one of the interface modes is shown on the right. a = 0.85 μm is the periodicity along the y-direction.

To achieve topologically protected valley kink states, we examine a structure comprising two domains of the inversion symmetry broken system, with one domain obtained by reflecting the other across the x = 0 axis, as depicted in Fig. 1b. While the entire structure maintains periodicity along the y-direction, periodicity is disrupted along the x-direction at x = 0, forming a bearded interface characterized by two small pillars connecting the two mutually inverted bulk domains, as highlighted by the red arrow in Fig. 1b. Despite having identical band structures, as shown in Fig. 1a, the domains exhibit distinct topological properties. To recognize these properties, we compute the Berry curvature $\mathcal{F}\left(k_x,k_y\right)$ for the two domains⁶⁰

$$\mathcal{F}\left(k_x,k_y\right)={\nabla }_{\mathrm{k}}\times \mathcal{A}\left(k_x,k_y\right)$$ 1

with $\mathcal{A}\left(k_x,k_y\right)=<u\left(k_x,k_y\right)\mid i{\nabla }_{\mathrm{k}}\mid u\left(k_x,k_y\right)>$ denoting the Berry connection comprising the Bloch modes $\left.∣u\right)\left(k_x,k_y\right)>$. Figure 1c illustrates the Berry curvature for the two domains, indicating non-zero values near the valleys with opposite characteristics for each domain. Additionally, a valley-Chern number can be defined as

$$C=\frac{1}{2\pi }\iint \mathcal{F}\left(k_x,k_y\right)d{k}_{x}d{k}_y$$ 2

with the integral taken over half of the BZ. For domain 1, $C_K^1=-C_{K^{\prime }}^1=1/2$, while for domain 2, $C_K^2=-C_{K^{\prime }}^2=-1/2$, resulting in a difference $\mathrm{\Delta }{C}_K=+1\mathrm{and}\mathrm{\Delta }{C}_{K\prime }=-1$ between the two domains. According to the bulk-boundary correspondence, the interface between the domains will host counter-propagating topologically protected states at the two valleys. To verify this, we compute the projected band structure of the system in a strip geometry, where the system is periodic along the y-direction. In Fig. 1d (left), the strip band structure is depicted, revealing topologically protected counter-propagating valley-polarized states inside the bulk bandgap. In addition, the spatial profile of one of the topological modes, shown on the right panel of Fig. 1d, further confirms their appearance at the interface.

Experimental characterization of bulk and topological valley kink states in the valley-Hall lattice

Experimentally, we fabricate the valley-Hall lattice by etching the spacer layer of CsPbBr₃ perovskite microcavity, in which the interface is well-aligned along the crystal axes of the perovskite, as illustrated in Fig. 2a (Refer to methods for more details). In order to achieve clear observation of 1D polaritonic topological states at room temperature, one needs stable polaritons with a large topological gap opening while maintaining a narrow polariton linewidth at room temperature. In the valley-Hall scheme, the topological gap opening is achieved by breaking the inversion symmetry and typically depends on the size of the polariton pillars, where a significant topological gap demands polariton pillars down to the nanoscale regime. However, when polariton pillars shrink in size, it becomes challenging to maintain a proper linewidth. Here, we overcome the challenges of fabricating nanoscale polariton pillars in perovskite microcavities and maintain a proper linewidth for achieving topological valley Hall states at room temperature. Figure 2b shows an atomic force microscopy (AFM) image of our valley-Hall lattice on the perovskite layer before the top DBR deposition, where the polariton pillars and bearded interface are clearly distinguished. The AFM image further reveals excellent sample homogeneity towards a uniform potential landscape, which plays a crucial role in achieving a narrow linewidth of the topological states. The inversion symmetry is broken by etching different pillar sizes down to the nanoscale regime for opening a large topological gap (small micropillar diameter, d₁ = 0.5 μm; big micropillar diameter, d₂ = 0.7 μm). To demonstrate the emergence of the topological valley kink states in the polariton valley-Hall lattice, we perform linearly-polarized momentum-space and real-space photoluminescence characterizations with a non-resonant continuous-wave laser excitation of 2.713 eV at room temperature (“Methods”). In valley Hall lattices, valleys are usually locked with particles’ momenta, whereas valley-polarized states possess different momenta in momentum space. In certain systems with strong spin-orbit coupling, such as monolayer transition metal dichalcogenides, these valleys could be locked with certain spins, leading to spin-valley-momentum locking, where valley-polarized states exhibit different momenta and spins simultaneously. However, in our perovskite polariton system, the spin-orbit coupling effect originates from the TE-TM splitting in microcavities, which is typically weak. The polariton states in our system are mainly linearly polarized, which is also validated by our theoretical calculation (Supplementary Note 1). We first probe the topological kink states in a valley-Hall lattice sample with a detuning of ~ − 115 meV, and similar results can also be observed in other samples with different detunings (Supplementary Note 3). Near the Dirac points (E = ~ 2.296 eV), we examine the behavior in the momentum space (Fig. 2c), and it exhibits a clear hexagonal shape, which corresponds to the BZ with typical K/ K’ valleys. Furthermore, we characterize the band structure of the valley-Hall lattice from the bulk area and the bearded interface area, framed by the white and red dashed lines in Fig. 2b, respectively. Figure 2d illustrates the energy-wavevector dispersion from the bulk area along the k_y direction at k_x = 0 μm⁻¹ (along K→Γ→K′). With the strong confinement in the polariton pillars, the s-mode polaritons couple together to form the s energy band, ranging from 2.262 eV to 2.320 eV, which constitutes the main region of the band structure. Within the s band, we experimentally observe a gap opening of ~ 5 meV at two nonequivalent K and K′ points (framed by black dashed lines), as a result of the broken inversion symmetry in the valley-Hall lattice. Subsequently, we move to the bearded interface and characterize its dispersion along K→Γ→K′ (white dashed line in Fig. 2c). As illustrated in Fig. 2e, we observe a larger gap opening of ~ 9 meV due to the limited bulk area collected. In addition to the gap opening of ~ 9 meV, a pair of topological kink states emerge inside the gap (indicated by the red arrow), which agrees well with our theoretical prediction (Fig. 1d and Supplementary Note 2).

Fig. 2. Experimental observation of topological valley exciton-polaritons at room temperature.

a Schematic representation of the perovskite topological valley-Hall lattice microcavity structure, where the lattice is created by patterning the ZEP layer and the lattice interface is well-aligned along the crystal axes of the perovskite. b Atomic force microscopy image of the 2D valley Hall lattice with excellent homogeneity before depositing the top DBR. The white and red dashed lines frame the emission collection areas in (d) and (e), respectively. The green and orange dashed lines represent the emission collection areas for the energy-resolved spatial images in Fig. 3c. c 2D momentum-space photoluminescence emission of the valley-Hall lattice near the Dirac points (E = ~ 2.296 eV), pumped by a non-resonant excitation of ~ 2.71 eV. The white and yellow dashed lines are at k_x = 0 μm⁻¹ and k_x = 3.2 μm⁻¹, respectively. d Momentum-space polariton energy dispersion of the bulk area at k_x = 0 μm⁻¹ along K→Γ→K′. The blue dashed lines represent the first BZ, and the black dashed lines highlight the bandgap opening inside the s band. e Momentum-space polariton energy dispersion of the topological bearded interface area at k_x = 0 μm⁻¹ along K→Γ→K′. The blue dashed lines represent the first BZ. The blue and red arrows indicate the bulk state (E = 2.285 eV) and the topological valley kink state (E = 2.296 eV), respectively. f Real-space photoluminescence images of the perovskite valley-Hall lattice at the energies of 2.285 eV (left, bulk state) and 2.296 eV (right, topological valley state), selected by a narrow band-pass (linewidth 1 nm) filter and pumped by a non-resonant excitation of ~ 2.71 eV. Source data are provided as a Source Data file.

Inside the first BZ (highlighted by the blue dashed lines), the dispersion slopes of the two topological kink states are opposite at different K and K′ valleys, leading to valley-dependent group velocities with opposite propagating directions. In other words, the topological kink states are valley-polarized and locked to one propagating direction in the absence of inter-valley scattering. The emergence of the topological kink states can be further clarified by detecting the real space profiles at the bulk state (E = 2.285 eV) and the topological state (E = 2.296 eV), indicated by blue and red arrows in Fig. 2e, respectively. Notably, the emission from the bulk state is mainly from the bulk valley domains (left panel in Fig. 2f), whereas the emission from the topological kink state appears mainly at the domain wall of the valley-Hall lattice (right panel in Fig. 2f).

To enhance the understanding of the valley-polarized topological kink states, we also collect the angle-resolved photoluminescence spectra along K′→M→K (yellow dashed line in Fig. 2c) by translating the Fourier lens. Figure 3a illustrates the dispersion from the bulk area along k_y at k_x = 3.2 μm⁻¹, revealing a similar bandgap opening of ~ 5 meV at ~ 2.296 eV without any states inside (highlighted by the black dashed lines). As a clear comparison, the band structure from the bearded interface exhibits a pair of additional dispersions with opposite slopes inside the gap, which correspond to the valley-polarized topological kink states (indicated by red arrow). The existence of the topological kink states can also be experimentally confirmed by comparing the energy-resolved spatial images collected along the bulk domain (top, Fig. 3c) and the domain wall (bottom, Fig. 3c), as indicated by the green and orange dashed lines in Fig. 2b, respectively. Specifically, only at the domain wall of the valley-Hall lattice, the topological kink states can be found inside the topological gap at 2.296 eV, highlighted by red dashed lines in Fig. 3c. Our results collectively validate the emergence of valley-polarized topological kink states localized at the domain wall in the polariton valley-Hall lattice.

Fig. 3. Experimental characterizations of topological valley exciton-polaritons at room temperature.

a Energy-resolved momentum-space polariton dispersion of the perovskite valley-Hall lattice collected from the bulk area at k_x = 3.2 μm⁻¹ along K→M→K′. The black dashed lines highlight the bandgap opening. b Energy-resolved momentum-space polariton dispersion of the perovskite valley-Hall lattice collected from the topological bearded interface area at k_x = 3.2 μm⁻¹ along K→M→K′. The red arrow indicates the topological valley kink state. c Energy-resolved spatial images were collected along the green dashed line (top) and orange dashed line (bottom) in Fig. 2b, respectively. The red dashed lines highlight the topological gap opening inside the s band. The topological valley kink states only emerge at the domain wall of the valley-Hall lattice. Source data are provided as a Source Data file.

Observation of valley-dependent propagation with topological exciton-polaritons

One of the unique advantages of valley-Hall polariton lattices is the valley DOF, with which the valley-polarized kink states are locked to one specific propagating direction in the absence of inter-valley scattering. We further experimentally demonstrate such valley polarization and valley-dependent topological polariton propagation in the linear regime by selectively and resonantly exciting the valleys in a transmission configuration at room temperature. As illustrated in Fig. 4a, a lattice sample with a detuning of ~ − 190 meV is resonantly excited with a linearly-polarized pulsed laser from the back of the microcavity, and we collect both the momentum space and real space spectra from its front side. In order to find out the exact valley positions, we also collect the photoluminescence dispersion under a non-resonant pumping from the bearded interface of the valley-Hall lattice along K→Γ→K′, revealing that the topological kink states located at E = ~ 2.231 eV with k_y = ± 3.8 μm⁻¹ and k_x = 0 μm⁻¹, as illustrated in Fig. 4b (left) and 4e (right). Subsequently, we tune the pumping laser beam to resonantly excite the K (K’) valley-polarized topological polaritons in the domain wall, respectively. As shown in Fig. 4b (right), when one K valley-polarized topological kink state at k_y = + 3.8 μm⁻¹ and k_x = 0 μm⁻¹ is resonantly excited, we observe scattering among the same valleys and negligible intervalley scattering, where the 2D momentum-space spectra exhibit bright spots only at equivalent K valleys but no signal at the K’ valleys (Fig. 4c). As only one of the K valleys is resonantly excited, the intensity at the excited valley will be much stronger than the scattered valleys, leading to the intensity asymmetry in the momentum space, as shown in Fig. 4c. In the meantime, as shown in Fig. 4d (left), these K valley-polarized topological polaritons exhibit long-range propagation along the domain wall in the + y-direction, where some polaritons can propagate as far as 8.2 µm at room temperature. Constrained by the limited energy and spatial resolution of our resonant pumping conditions (Methods), the bulk states in the bulk area are also weakly excited by the pumping laser, leading to some signals in the bulk (Fig. 4d).

Fig. 4. Experimental valley-dependent propagation of the topological exciton-polaritons.

a Schematic of propagating topological valley polaritons in the perovskite valley-Hall lattice. The resonant pumping laser beam is injected with an angle from the back of the microcavity, and the propagation of the topological valley polaritons along the domain wall of the valley-Hall lattice can be observed from the front of the microcavity. b (left) and (e) (right), Momentum-space polariton photoluminescence dispersions collected from the topological bearded interface area at k_x = 0 μm⁻¹ with a non-resonant pumping of ~ 2.71 eV. b (right) and (e) (left), Momentum-space spectra of the topological valley polaritons with a resonant excitation exactly at the topological energy state (E = 2.231 eV). c and (f) 2D momentum-space spectra of the K (K’) valley-polarized topological polaritons under resonant pumping conditions as in (b) (right) and (e) (left), respectively. d The corresponding experimental real-space images of the K (left) and K’ (right) valley-polarized topological polaritons resonantly excited at the topological energy state (E = 2.231 eV) as in (b) (right) and (e) (left), respectively. They show polariton propagations along the domain wall of the valley-Hall lattice in $\pm$y-direction, and some polaritons can propagate as far as 8.2 µm. Source data are provided as a Source Data file.

In sharp contrast, we observe distinct scenarios in the K’ valleys. As shown in Fig. 4e (left), we resonantly excite the K’ valley at k_y = − 3.8 μm⁻¹ and k_x = 0 μm⁻¹, which corresponds to the K’ valley-polarized topological polariton states. In the 2D momentum space, the other equivalent K’ valleys in the first BZ also light up, and no signal can be observed from the K valleys, suggesting the absence of intervalley scattering. Furthermore, the trajectory of the K’ valley-polarized topological polaritons exhibit similar propagation behaviors along the domain wall, but with an opposite direction of − y. Similar evidence can also be observed from the energy-resolved spatial images when K (K’) valley-polarized topological polaritons are excited respectively (Supplementary Note 4). For comparison, we also resonantly excite the bulk state (E = ~ 2.242 eV with k_y = 4.1 μm⁻¹ and k_x = 0 μm⁻¹) spectrally near the valley Hall state in the band structure, but spatially at the interface (Supplementary Note 5), we observe no polariton propagation along the domain wall, which further confirms that the observation of polariton propagation at the interface is from the topological valley Hall states. Our observations collectively suggest the existence of valley polarization and valley-dependent propagation in our polariton valley-Hall lattices at room temperature.

Discussion

In summary, we have demonstrated the realization of valley-polarized topological exciton-polaritons and their valley-dependent propagation in 2D valley-Hall perovskite lattices at room temperature. By designing a bearded interface between two mutually inverted polariton honeycomb lattices with broken inversion symmetry, we theoretically predict and experimentally verify the existence of topological exciton-polariton kink states. Our experimental results collectively demonstrate that a pair of valley-polarized topological kink states are spectrally locked to K (K′) valleys inside the topological gap and spatially confined at the bearded interface of the valley-Hall lattice. In addition, under resonant excitation, we further showcase their valley polarization and valley-dependent propagation of the topological kink state polaritons along the domain wall of the valley-Hall lattice, where some polaritons can propagate as far as 8.2 µm at room temperature. Although the current propagation length is limited due to the short lifetime and small group velocity of the topological valley Hall states, there are promising strategies for future improvement. For example, further optimizing the lattice quality could help to improve the polariton lifetime. In addition, one can further shrink the polariton pillar size to open a larger topological gap, which could lead to a higher group velocity of the topological valley Hall states and less bulk scattering. These two strategies together could help to extend the propagation distance of topological states. With the unique properties of exciton-polaritons, we anticipate exciting opportunities for designing advanced topological devices. For instance, our work introduces the valley DOF into the topological polariton system, one could possibly leverage non-Hermitian mechanisms by tailoring the gain and loss spectra for different valleys towards low-threshold valley-addressable topological lasers. Furthermore, with the strong confinement from the nanoscale pillars, we anticipate the enhancement of polariton nonlinearity in the valley Hall lattices, offering exciting opportunities for the realization of nonlinear switchable topological devices in the near future.

Methods

Perovskite valley-Hall lattice fabrication

The perovskite microcavity consists of a bottom DBR, an all-inorganic perovskite (CsPbBr₃) nanoplatelet, a spacer layer with a lattice pattern, and a top DBR. In detail, 15.5 pairs of TiO₂/SiO₂ layers are deposited on the silicon or silica wafer as bottom DBR substrates for reflection and transmission configurations, respectively, which are fabricated by an electron beam evaporator (Cello 50D). The 65 nm-thick single-crystalline CsPbBr₃ is synthesized via a chemical vapor deposition (CVD) method on mica substrates (the pressure in the CVD chamber is maintained at 42.5 Torr with the nitrogen flow rate of 30 sccm, and the CVD tube is heated to 590 °C within 5 min and maintained for 10 min, then cooled down to room temperature naturally)⁶¹, and then transferred onto the bottom DBR substrates through the dry-transfer process with Scotch tape⁶². Next, 85 nm-thick ZEP520A layers are spin-coated on the samples before depositing the top DBRs and then patterned into a 2D valley-Hall lattice by standard electron beam lithography. Lastly, 8.5 pairs of TiO₂/SiO₂ layers are deposited as top DBRs by the electron beam evaporator.

Optical spectroscopy characterizations

The energy-resolved momentum-space and real-space spectra are detected by a home-built angle-resolved spectroscopy setup with Fourier optics. The optical signal from the microcavity is collected by a 50 × objective lens (NA = 0.75) and sent to a 550 mm focal length spectrometer (Horiba iHR550) with a grating (600 lines/mm) and a liquid nitrogen cooled CCD (256 × 1024 pixels). For the photoluminescence measurements in Figs. 2 and 3, the perovskite lattice is non-resonantly pumped by a continuous wave laser (457 nm) with a pump spot of ~ 25 μm. The real-space photoluminescence images in Fig. 2 at different energies are measured with a band-pass filter (linewidth 1 nm, Semrock) on the detection path. For characterizing the valley-dependent topological polariton propagation in a transmission configuration as shown in Fig. 4, the topological valley Hall state is resonantly excited at the interface by a linearly-polarized pulsed laser (centered at E = 2.231 eV, 1 kHz repetition rate and 100 fs pulse duration) with a pump spot of ~ 4 μm. The pumping beam is spectrally filtered by a band-pass filter (linewidth 1 nm, Semrock), and the pumping density is set to be low to avoid nonlinear effect and sample degradation. In addition, a slit and an angle-variable lens are employed to adjust the incidence angle of the pulsed pumping source.

Theoretical calculations

To model the system, we utilize coupled Schrödinger equations within the mean-field approximation. These are formulated as follows:

$$i\hslash \frac{\partial {\varphi }_{ph}\left(\mathit{r},t\right)}{\partial t}=\left[-\frac{{\hslash }^2}{2m_{ph}}\left(\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}\right)+U\left(\mathit{r}\right)\right]{\varphi }_{ph}\left(\mathit{r},t\right)+\frac{g_0}{2}{\varphi }_{ex}\left(\mathit{r},t\right)$$ 3

$$i\hslash \frac{\partial {\varphi }_{ex}\left(\mathit{r},t\right)}{\partial t}=E_{ex}{\varphi }_{ex}\left(\mathit{r},t\right)+\frac{g_0}{2}{\varphi }_{ph}\left(\mathit{r},t\right)$$ 4

Here, ${\varphi }_{ph}$ and ${\varphi }_{ex}$ represent the mean-field wave functions of photons and excitons, respectively. $m_{ph}={1.7\times 10}^{-5}{m}_e$ denotes the isotropic cavity photon mass, with $m_e$ representing the free electron mass. $U\left(\mathit{r}\right)$ represents the in-plane photon confinement potential, as illustrated in Fig. 1b, while $g_0$ indicating the Rabi splitting. Due to the heavy exciton mass, excitons are considered dispersion-less, with energy $E_{ex}$. To derive the band structures in Fig. 1d, we apply the Bloch theorem and the plane wave expansion method. Additional parameters include $g_0$ = 120 meV and $E_{ex}$ = 2.407 eV.

Author contributions

R.S., Q.X., and T.C.H.L. supervised the project. F.J. and J.H.R. synthesized the perovskite materials with the help of W.W. F.J. fabricated the devices with the help of J.Q.W. and Z.H.Z. F.J. performed all the optical spectroscopy measurements. T.C.H.L. and S.M. conceived the lattice model with inputs from R.S. and Q.X. S.M. performed the theoretical calculations with inputs from T.C.H.L. and B.L.Z. R.S., F.J., and S.M. wrote the manuscript with the inputs from all the authors.

Peer review

Peer review information

Nature Communications thanks Nathaniel Stern, and the other anonymous reviewers for their contribution to the peer review of this work. A peer review file is available.

Data availability

Source data are provided in this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon request. Source data are provided in this paper.

Code availability

The codes are available from the corresponding author upon request.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-024-54658-4.