Abstract
Backscattering suppression in silicononinsulator (SOI) is one of the central issues to reduce energy loss and signal distortion, enabling for capability improvement of modern information processing systems. Valley physics provides an intriguing way for robust information transfer and unidirectional coupling in topological nanophotonics. Here we realize topological transport in a SOI valley photonic crystal slab. Localized Berry curvature near zone corners guarantees the existence of valleydependent edge states below light cone, maintaining inplane robustness and light confinement simultaneously. Topologically robust transport at telecommunication is observed along two sharpbend interfaces in subwavelength scale, showing flattop high transmission of ~10% bandwidth. Topological photonic routing is achieved in a beardedstack interface, due to unidirectional excitation of valleychiralitylocked edge state from the phase vortex of a nanoscale microdisk. These findings show the prototype of robustly integrated devices, and open a new door towards the observation of nontrivial states even in nonHermitian systems.
Introduction
Silicononinsulator (SOI) provides a CMOScompatible platform to fasten and enlarge data transfer both between and within silicon chips, by using optical interconnects to replace their electronic components^{1}. Miniaturization of SOI devices can achieve highly integrated photonic structures comprised of numerous optical components in a single chip, but increase inevitable backscattering that leads to energy loss and signal distortion. Consequently, optical backscattering suppression is of fundamental interest and great importance for compact SOI integration. The discovery of topological photonics offers an intriguing way for robust information transport of light^{2,3}, particularly for their capacities in backscattering–immune propagation and unidirectional coupling. Such robustness is derived from the nontrivial bulk topology, enabling reflectionfree transport between two topologically distinct domains^{2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17}, such as by using a magnetooptical effect, 3D chiral structures, and bianisotropic metamaterials. As a target to integrated topological nanophotonics, some alldielectric strategies have been proposed recently. An array of coupling resonator optical waveguides was designed to implement topological SOI structures at superwavelength period^{18}, and has been exploited to the topologicalprotected lasing effect^{19,20}. Later, a subwavelengthscale strategy attracted much attention to reduce the size of topological devices, e.g., applying C_{6v} crystalline symmetries to realize alldielectric topological structures above a light cone^{21,22,23}, which have been achieved under inplane unidirectional propagation over sufficient distances^{24} and observed topological states through outofplane scattering^{25}. These developments of topological nanophotonics open avenues to develop onchip optical devices with builtin protection, such as robust delay lines, onchip isolation, slowlight optical buffers, and topological lasers.
Valley pseudospin provides an additional degree of freedom (DOF) to encode and process binary information in graphene and twodimensional transition metal dichalcogenide (TMDC) monolayers^{26,27,28,29}. Analogous to valleytronics, exploration of valley physics in classical waves (e.g., photonics^{30,31,32,33,34,35,36,37} and phononics^{38,39,40,41}) renders powerful routes to address the topological nontrivial phase by emerging an alternative valley DOF. To retrieve the topological valley phase, a general method is to break spatialinversion symmetry for accessing the opposite Berry curvature profiles near Brillouin zone corners, i.e., the K and K′ valley. Advanced in nanofabrication techniques, precise manufacture of the inversionsymmetrybroken nanophotonic structures is easy to implement nowadays^{42}. Consequently, valley photonic crystal (VPC) is a reliable candidate for SOI topological photonic structures, in particular for a subwavelength strategy that still remains much of a challenge in topological nanophotonics. Furthermore, the topological valley phase below a light cone ensures high–efficient light confinement in the plane of a chip, such that the photonic valley DOF naturally makes a balance between inplane robustness and outofplane radiation. This is a crucial condition to design topological photonic structures for chips. Realization of topological valley transport in the SOI platform is desirable for integrated topological nanophotonics.
In this work, we experimentally demonstrate a valley topological nanophotonic structure at telecommunication wavelength. Our design is based on a standard SOI platform that allows integration with other optoelectronic devices on a single chip. Valleydependent topological edge states can operate below light cone, benefiting to the balance between inplane robust transport and outofplane radiation. Broadband robust transport is observed in sharpturning interfaces constructed by two topologically distinct valley photonic crystals, with a footprint of 9 × 9.2 μm^{2}. In addition, we achieve topological photonic routing with high directionality, by exploiting unidirectional excitation of valley–chiralitylocked edge state with a subwavelength microdisk.
Results
Silicononinsulator valley photonic crystals
In this work, our nanophotonic structures are prepared on SOI wafers with 220nmthickness silicon layers. As depicted in Fig. 1a, the valley photonic structure comprises two honeycomb photonic crystals (VPC1 and VPC2). The VPC layer is asymmetrically placed between the SiO_{2} substrate and top air region along the z axis (see the inset of Fig. 1a). Instead of a freestanding membrane, the use of a zasymmetric SOI slab can improve the compatibility with other types of building blocks (e.g., microdisk later in this work). Figure 1b gives the details of VPC. The unit cell of VPC1 (red) contains two nonequivalent air holes, i.e., the smaller one d_{1} = 81 nm and the bigger one d_{2} = 181 nm. On the other hand, the diameter of two air holes is altered to form another type of VPC (blue), i.e., VPC2 with d_{1} = 181 nm and d_{2} = 81 nm. Here, VPC2 is the inversionsymmetry partner of VPC1. Thus, VPC1 and VPC2 have the same band structure, as shown in Fig. 1c. The details about the VPC design can be seen in Supplementary Note 1. Because the two air holes have different diameters that break the inversion symmetry, a bandgap (1360 nm ~1492 nm) emerges for TElike polarizations.
Due to the bulkedge correspondence, we will first study the bulk states of the first TElike band (labeled as “TE1” in Fig. 1c), before discussing the topology of a TElike gap. The electromagnetic fields in the zcentral plane (labeled as “z = 0” in the inset of Fig. 1a) can mainly reflect the optical properties of the VPC slab, so that we will focus on the field patterns at z = 0 plane in the following discussion. Take the eigenstates at the K valley as examples. The simulated H_{z} phase profile at z = 0 plane is plotted in Fig. 1d. We can see that the phase profile of VPC1/VPC2 increases anticlockwise/clockwise by 2π phase around the center of a unit cell. Such optical vortex is related to valley pseudospin in an electronic system, and thus can be termed as a photonic valley DOF. In TMDCs, the valleypolarized excitons can be selectively generated through control of the chirality of light^{43,44}. Similarly, the photonic valley is also locked to the chirality of excited light, i.e., leftcircularly polarized (LCP) light couples to the K′valley mode, while rightcircularly polarized (RCP) light couples to the Kvalley mode^{32}. To demonstrate this valley–chirality locking property remaining in the SOI slab, we give the distribution of the polarization ellipse of the inplane electric field in Fig. 1e. Here, the polarization ellipse is generally defined as ellipticity angles^{45} \(\chi = \arcsin [ {2 {E_x}  {E_y} \sin \delta / ( { {E_x} ^2 +  {E_y} ^2} )} ]/2\), where \(\delta = \delta _y  \delta _x\) is the phase difference between E_{y} and E_{x}. For VPC1, the RCP response (χ = π/4) exists in the singularity point of the phase vortex at the K valley (red center in Fig. 1e), and vice versa for VPC2 (blue center in Fig. 1e). Figure 1f shows the temporal evolution of the RCP and LCP responses, respectively. Such valley–chirality locking gives the possibility to manipulate photonic valley modes. For example, when we place a circularpolarized dipole source in the singular point of the phase vortex, the photonic valley Hall effect can be observed (see Supplementary Note 2).
The above observation of the optical vortex and the photonic valley Hall effect can be related to a topologically nontrivial phase. Next, we numerically calculate the Berry curvature distribution and the corresponding topological invariant to insightfully confirm the topological valley phase in our proposed VPC slab^{46}. See the section Methods for more details on numerical simulation. Figure 2a shows the Berry curvature of the TE1 band for VPC1, calculated with the parallel gauge transformation^{47}. The Berry curvature of VPC is mainly distributed near two valleys, i.e., a singular sink at the K valley while peaking at K′ (red line in the inset of Fig. 2a). On the contrary, VPC2 reverses the Berry curvature distribution of the two valleys (blue line in the inset of Fig. 2a). In general, the global integration of Berry curvature over the whole Brillouin zone, the socalled Chern number, is zero under the protection of timereversal symmetry. Instead, the valleydependent integration of Berry curvature gives rise to a nonzero value, i.e., the valleydependent index C_{K} (C_{K′}) ≠ 0. Thus, we can use the valley Chern index C_{V} = C_{K}–C_{K′} to characterize the topology of the whole VPC system, providing a new route to retrieve a topologically nontrivial phase.
We should note that the VPCs open a large TElike gap (~10%) to guarantee broad bandwidth operation. Thus, the Berry curvature for both valleys will overlap with each other. As a consequence, the valley Chern index is not a welldefined integer, i.e., \(0 < \left {C_V} \right < 1\). The edge dispersion will not gaplessly cross from the lower band to the upper band. Regardless of this side effect, the difference in the sign of valley Chern index will ensure the protection of the topological valley phase, as long as the bulk state at the K valley is orthogonal to the K′ valley. Therefore, such novel design still enables broadband robust transport along the ΓK/ΓK′ direction against certain perturbations (such as sharpbend corners or 10% random bias of hole diameter), as the intervalley scattering is suppressed due to the vanishing field overlapping between two valley states.
As an intuitive example, we construct an interface by using two VPCs with the opposite valley Chern index. As schematically shown in Fig. 2b, the bearded interface is stacked with the bigger holes. The upper domain (VPC1 in Fig. 1) has valley Chern index C_{V} < 0, while the valley Chern index of the lower domain (VPC2 in Fig. 1) exhibits the opposite sign (C_{V} > 0). The valleydependent edge states (green lines in Fig. 2c) for the beardedstack interface, include one with negative velocity at the K valley and the other with positive velocity at the K′ valley. The simulated patterns shown in Fig. 2e confirm that the propagating light at λ = 1430 nm will smoothly detour by 120^{o} bending (60^{o} sharp corner). Such propagation is valid for other wavelengths inside the bandgap, leading to optical broadband operation. Note that a little bit modulation of the signal in Fig. 2e is mainly caused by the outofplane radiation loss in the openslab system. Note also that the TE/TM coupling in such zasymmetric slab is too weak (see Supplementary Note 3) to affect valleydependent interface transport, i.e., the robust transport in 120^{o} bending and unidirectional coupling. In the next section, we will experimentally characterize this broadband robust transport phenomenon.
Topological robust transport
To experimentally demonstrate topological robust transport of the valleydependent edge states, we employ an advanced nanofabrication technique to manufacture the flat, Z, and Ωshape VPC interfaces. The scanningelectronmicroscope (SEM) images of fabricated samples are shown in Fig. 2d. The devices were prepared on a SOI wafer, with a nominal 220nm silicon layer and 2.0μm buried oxide layer. After the definition of a 370nmthickness positive resist through electronbeam lithography, inductively coupled plasma etching step is applied to pattern the top silicon layer, such that the VPC structure and its coupling waveguide was formed. Then the resist was removed by using an ultrasonic treatment process. See Methods for more details of the nanofabrication process. These processes are able to precisely achieve our designed structures even in close proximity (separation of about 40 nm in the topological interface).
Next, we will characterize the broadband robust transport of the VPC edge states at the Z/Ωshape interface. The experimental setup is shown in Supplementary Figure 5. The TEpolarized continuous waves at the telecommunication wavelength were coupled to the 1.7μmwidth input waveguide by using a polarizationmaintaining lensed fiber, and then launched into the VPC sample from the left end of the topological interface. After passing through the VPC devices, the propagating wave was coupled to the output waveguide at the right end and then collected by another lensed fiber. The corresponding transmission spectra were detected by using an optical powermeter, with tuning the operation wavelength of excited waves. Note that all the transmission spectra are normalized to the 1.7μmwidth silicon strip waveguide located in the same writing field near the VPC samples. See Methods for more details of optical characterizations. Figure 2f shows the measured transmission spectra in the wavelength range of 1320–1570 nm, for flat, Z, and Ωshape topological interfaces. In the bandgap region (yellow), the spectra are kept on the flattop hightransmittance platform, even for a sharpbending geometry (green and red lines). This intriguing property indicates the broadband robust transport in the frequency interval from 1360 to 1492 nm, due to the suppression of intervalley scattering. Although there is some noise in Fig. 2f due to Fabry–Perot resonance between the entrance and exit facets of the strip waveguide and dark current noise of the detector, these experimental spectra are in good agreement with simulations (Fig. 2g). Note that the SOI platform with a compact footprint of 9 × 9.2 μm^{2} enables to integrate many photonic components on a single chip. In other words, the proposed SOI VPC with a subwavelength periodicity (about λ/4) can develop a highperformance topological photonic device with a compact feature size of less than 10 μm.
Unidirectional coupling
Unidirectional coupling is another important property of topological photonic structures to manipulate the flow of light. We should emphasize that the realization of robust transport does not definitely correspond to unidirectional coupling. For example, it is inaccessible to achieve highefficiency unidirectional coupling from a single circularly polarized source to zigzagstack valleydependent interfaces, due to protection of the inversion symmetry with regard to the yaxis center of the interface. In such yodd/evenlike edge states, the circularpolarized point of the polarization ellipse is predominant at the lowintensity positions. For highefficiency unidirectional coupling, breaking the inversion symmetry of the interface is required to engineer the generation of vortex fields in the edge states. Therefore, we chose beardedstack VPC interface (Fig. 2b) with the inversion symmetry broken. In this case, the vortex fields around bearded holes at the upper domain will interact with the lower domain, and thus generate chiralflow edge states (see Supplementary Note 3). As depicted in Fig. 3a, such chirality ensures the rightward (leftward) excitation by using the RCP (LCP) source. Simulated results in Fig. 3b confirm that the proposed valleydependent topological interface can realize unidirectionality through control of the source chirality. In fact, similar results have been studied in a photonic crystal W1like waveguide, by shifting one side of the waveguide by half a lattice constant^{48,49}.
We should emphasize that the introduction of a topological nontrivial phase can guarantee high directionality under chiral source excitation in relatively broadband operation, while the case of the topologically trivial system commonly operates in a narrowband as it is sensitive to the source position with frequency variation. To quantitatively determine unidirectional coupling, we define the directionality for a given frequency as \(\kappa _0 = \left( {T_L  T_R} \right)/\left( {T_L + T_R} \right)\), where T_{L} and T_{R} are the transmittances detected at the left and right end, respectively. Furthermore, we would like to analyze the global efficiency of unidirectional coupling inside the photonic bandgap, and thus define an averaged parameter related to the frequencydomain integration of \(\kappa _0\) in the whole bandgap, i.e., \(\left\langle {\kappa _0} \right\rangle _{{\mathrm{gap}}} =  {{\int}_{{\mathrm{gap}}} {\kappa _0d\omega } } /\Delta \omega _{{\mathrm{gap}}}\), where \(\Delta \omega _{{\mathrm{gap}}}\) is the bandwidth of a photonic bandgap. \(\left\langle {\kappa _0} \right\rangle _{{\mathrm{gap}}} = 1\) represents that the chiral source couples to pure left/rightforward edge states for all frequencies in the bandgap. Here, we analyze the global directionality in the center on two dominant factors, i.e., relative positions (D_{x} and D_{y}) and diameter (d_{e}) of bearded edge holes (Fig. 3c). The separation (δ_{Si}) of the silicon region between the two bearded holes is variable as \(\delta _{{\mathrm{Si}}} = \sqrt {D_x^2 + D_y^2}  d_e\). A simulated phase map of Fig. 3d shows that the maximum \(\left\langle {\kappa _0} \right\rangle _{{\mathrm{gap}}}\) emerges at the point of D_{x} = 192.5 nm and D_{y} = 111 nm, which is in correspondence with the valleydependent topological interface. It is interesting that the proposed design based on valley topology can certainly find the point of high directionality, while the general method requires massive simulations, just like what we do in Fig. 3d.
On the other hand, considering a fixed relative position that D_{x} = 192.5 nm and D_{y} = 111 nm, the global directionality \(\left\langle {\kappa _0} \right\rangle _{{\mathrm{gap}}}\) as a function of separation δ_{Si} is also retrieved in Fig. 3e, when tuning the diameters d_{e} of bearded edge holes. We can see that \(\left\langle {\kappa _0} \right\rangle _{{\mathrm{gap}}}\) will stand on a highdirectionality platform (above 0.9), when the separation is less than 50 nm. Qualitatively, this is because such extreme separation will enhance the interaction of vortex fields between upper and lowerdomain bearded holes, and thus strengthen the valley–chirality coupling of the topological interfaces.
Topological photonic routing
Experimental realization of unidirectional coupling of topological edge states shows many promising applications in light manipulation. Recently, it has been demonstrated in the microwave region as a valley filter^{36} and realized in a chipscale system as a topological quantum optics interface^{24}. For the onchip strategy, the latter one use chiral quantum dots under a strong magnetic field at ultralow temperature^{24,50}. In this work, we aim to develop an alloptical strategy, for unidirectional excitation of the valley–chirality locking edge states in the SOI platform. To do this, a subwavelength microdisk serving as a phase vortex generator^{51}, is introduced into the topological interface. Figure 4a shows the schematic of a designed device, combining SOI VPC and a microdisk. The fabricated sample around a microdisk can be seen in Fig. 4b. There are two 373nmwidth strip silicon waveguides (labeled as “WVG1” and “WVG2”) at the left of the sample. When incident light couples to the WVG1/WVG2 input waveguide, it will generate an anticlockwise/clockwise phase vortex at the designed microdisk with a closetodiffractionlimited scale (630nm diameter). Due to valley–chirality locking, the edge state near the K/K′ valley can be selectively routed to the upper/lower topological interface, through control of the chirality of the optical vortex inside a microdisk. This shows a prototype of the onchip photonic routing device, with the advantage of an ultracompact (in submicrometer scale) coupling distance.
Farfield microscopy is used to verify the photonic valley–chirality locking property and the topological routing effect. A 20× objective is used to predominantly collect the outofplane radiation from two nonuniform grating couplers (labeled as “G1” and “G2” in Fig. 4a) and then imaged by using an InGaAs CCD. For a given incidence waveguide, an asymmetric radiation is obvious between G1 and G2. For example, the microscope images are presented in Fig. 4d for the WVG1 incidence at λ = 1400 nm. In this case, the propagating light was routed to the upper interface and radiated from the G1 port. The asymmetry of photonic routing is reversed when the incidence port is flipped to the other interface (Fig. 4e). For comparison, we also fabricated a control sample that replaced the SOI VPC by a twostrip silicon waveguide (Fig. 4c). The nearequal routing profiles demonstrate low directionality in Figs. 4f, g. Valleydependent unidirectional routing is already visible to be distinguished from the control experiment.
The intensity of each grating coupler was collected from CCD, and the intensities I_{G1} and I_{G2} scattered from the upper (G1) and lower (G2) ports can be extracted with a high signaltonoise ratio, after subtracting the noise that mainly arises from background radiation. The extracted intensities I_{G1} and I_{G2} reflect the amount of light transmission that is coupled to the upper and lowerpropagating valleydependent edge states, respectively. To experimentally qualify directional coupling efficiency of routing devices, we define the experimental directionality as \(\kappa _{\exp } = \left( {I_{G1}  I_{G2}} \right)/\left( {I_{G1} + I_{G2}} \right)\). The fullband directionality can be measured by tuning the operation wavelength of the excited waves. Figure 4h shows the directionality spectra as a function of wavelength. In the bandgap, a strong and broadband directionality was observed. For anticlockwisephasevortex excitation, the incident light couples to valleydependent edge states propagating along the upper interface (red line in Fig. 4g). The directionality of the topological routing device is up to 0.5 within a broadband region. Note that the maximum \(\kappa _{\exp }\) is up to ~0.895, implying a 18:1 extinction ratio between G1 and G2. When the handedness of the excitation flips, so do the propagation directions of the valleydependent edge states (blue line in Fig. 4h). For comparison, the lowdirectionality spectra for the control experiment were depicted in Fig. 4i. There are a few discrete wavelengths to reach \( {\kappa _{\exp }} \) > 0.5. A more experimental description is presented in Supplementary Note 5.
Discussion
In summary, we have successfully applied the valley DOF to topologically manipulate the flow of light in a silicononinsulator platform. Benefiting from the belowlightcone operation, the valleydependent topological edge state can balance inplane robust transport and outofplane radiation, which is important to the open system, such as a photonic crystal slab. Topological robust transport and topological photonic routing are experimentally demonstrated and confirmed at telecommunication wavelength. Our study paves the way to explore the photonic topology and valley in the SOI platform, which is a promising system in taking advantage of the topological properties into nanophotonic devices, particularly important for backscattering suppression and unidirectional coupling. Furthermore, our subwavelength strategy enables to design compactsize topological SOI devices that allow integration with other optoelectronic devices on a single chip. It shows a prototype of the onchip photonic device, with promising applications for delay line, routing, and dense wavelength division multiplexing for information processing based on topological nanophotonics. Finally, the platform of the SOI topology opens a new door toward the observation of nontrivial states even in nonHermitian photonic systems.
We are aware of a related work on experimental demonstration of valleydependent edge states through airbridge slab structures with sharpturning profiles^{37}.
Methods
Numerical simulation
All of the simulation results in this work are retrieved from a 3D asymmetric slab instead of a 2D model^{52}. The band structures and the corresponding eigenfield patterns were calculated by MIT Photonic Bands^{53} (MPB) based on the planewave expansion (PWE) method, while all of the optical transport calculations were implemented by MIT Electromagnetic Equation Propagation^{54} (MEEP) based on the finitedifference timedomain (FDTD) method. In all 3D simulations, the maximum scale of the discrete grid is smaller than 24 nm, making the resolution large enough to ensure the convergence. For Berry curvature calculations, the original data of eigenfield \({\bf{\psi }}\left( {x,y,z} \right) = \left[ {\sqrt {\varepsilon _z\left( {x,y,z} \right)} u_k^{Ez}\left( {x,y,z} \right),\sqrt {\mu _z\left( {x,y,z} \right)} u_k^{Hz}\left( {x,y,z} \right)} \right]^T\) are obtained from MPB, by scanning the whole Brillouin zone with step δk = 0.005(2π/a). Here, \(u_k^{Ez}\) and \(u_k^{Hz}\) are the periodic parts of E_{z} and H_{z}, respectively. Then the Berry curvature can be calculated by \({\bf{\Omega }} = i{\bf{\nabla }}_{\bf{k}} \times \left\langle {\bf{\psi }} \right{\bf{\nabla }}_{\bf{k}}\left {\bf{\psi }} \right\rangle\).
Sample fabrication
The experimental samples were manufactured by employing a top–down nanofabrication process on a SOI wafer (with a nominal 220nm device layer and a 2.0μm buried oxide layer). First, a 370nmthickness positive resist (ZEP520A) was spun with a rotating speed of 3500 min^{−1} on the wafer, and dried for 10 min at 180 °C. The VPC patterns were defined by electronbeam lithography (EBPG5000 ES, Vistec) in the resist, and developed by dimethylbenzene for 70 s. Second, inductively coupled plasma (ICP) etching step was applied to etch the VPC structures and coupling waveguides on the top 220nmthickness silicon layer. Then the resist was removed by using an ultrasonic treatment process at room temperature. The final step was to cut up and polish the facets of samples, in order for high–efficient incident coupling.
Optical characterization
The experimental results of the transmission spectra and farfield microscopy images were realized with three tunable continuouswave lasers (Santec TSL550/710) at telecom wavelength (1260 ~1640 nm). The incident light was first launched into a fiber polarization controller to select the TE wave, and then coupled to the input waveguide with the aid of a polarizationmaintaining lensed fiber. After passing through VPC devices, the propagating waves coupled to the output waveguide at the right end of the topological interface. For robust transport measurement (Fig. 2), the output signals were collected by another lensed fiber and detected by an optical powermeter (Ophir NovaII). For photonic routing measurement (Fig. 4), the propagating waves coupled out in the zdirection thanks to the gratings at the end of the waveguide. The outofplane radiation was collected by a 20× microscope objective and then imaged by using an InGaAs CCD (Xenics Bobcat640GigE). More details on the experimental setup are provided in Supplementary Note 4.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. 61775243, No. 11761161002, No. 11704422, and No. 11522437), Natural Science Foundation of Guangdong Province (No. 2018A030310089, No. 2018B030308005), Science and Technology Program of Guangzhou (No. 201804020029), and Project funded by the China Postdoctoral Science Foundation (No. 2018M633206).
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All authors contributed extensively to this work. X.T.H. and J.W.D. conceived the idea. X.T.H. performed the numerical simulations and designed the experiment. E.T.L. and H.Y.Q. fabricated the samples. J.J.Y. performed the measurements. J.J.Y., X.T.H., F.L.Z. and J.W.D. did the experimental data analysis. X.T.H., X.D.C. and J.W.D. wrote the paper. All the authors contributed to discussion of the results and paper preparation. J.W.D. supervised the project.
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He, XT., Liang, ET., Yuan, JJ. et al. A silicononinsulator slab for topological valley transport. Nat Commun 10, 872 (2019). https://doi.org/10.1038/s4146701908881z
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DOI: https://doi.org/10.1038/s4146701908881z
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