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Unidirectional transmission of visible region topological edge states in hexagonal boron nitride valley photonic crystals

Min Wu, Yibiao Yang, Hongming Fei, Han Lin, Yuhui Han, Xiaodan Zhao, and Zhihui Chen

Open Access Open Access

Abstract

Here we theoretically design valley photonic crystals (VPCs) based on two-dimensional (2D) hexagonal boron nitride (hBN) materials, which are able to support topological edge states in the visible region. The edge states can achieve spin-dependent unidirectional transmission with a high forward transmittance up to 0.96 and a transmission contrast of 0.99. We further study the effect of refractive index on transmittance and bandwidth, and it is found that with the increase of refractive index, both transmittance and bandwidth increased accordingly. This study opens new possibilities in designing unidirectional transmission devices in the visible region and will find broad applications.

© 2022 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Topological photonic waveguides can achieve robust unidirectional transmission of topologically edge states at the interface between two topological photonic crystals (TPCs), without being affected by the backscattering from defects or sharp bends [1,2]. It can be used in the development of quantum optics and optical communication devices [3–5]. There are several two-dimensional (2D) TPC systems that have been proposed, including the quantum Hall effect (QHE) system [6], the quantum spin Hall effect (QSHE) system [1,7] and the quantum energy valley Hall effect (QEVHE) system [8–10]. The design of TPCs realized by the QHE requires the magneto-optical materials and external magnetic fields to break the time-reversal symmetry. In comparison, the TPCs based on the QSHE and QEVHE systems can be achieved using all-dielectric materials to break the spatial reversal symmetry without external magnetic fields, which immediately expands the potential candidates and significantly simplifies the system. Based on QSHE, it is necessary to design and combine the two types of TPCs, namely, the trivial and non-trivial structures [1,7]. However, this strategy shows a narrow working bandwidth limited by the working principle. Because the working bandwidth needs to be in the shared bandgap of the two TPCs, which depends on the bandgap width of the two TPCs. In contrast, valley photonic crystals (VPCs) based on the QEVHE can achieve unidirectional coupling of circularly polarized light based on the spin-valley locking effect. The spin-valley locking effect is achieved by reducing the C6v symmetry of the honeycomb photonic crystal (PC) structure to open the Dirac cone and produce a spin-dependent bandgap [11], which also introduces a topological phase transition [3,12]. Then a topological waveguide can be built by combining two VPCs with mirror symmetry to achieve robust unidirectional transmission. Thus, the working bandwidth of a VPC system depends only on one VPC structure, which can be larger compared to other designs. In addition, the design process of a VPC system is much simpler by removing the requirements to achieve a topological non-trivial structure.

A unidirectional transmission device is essential in optical communication and information processing [13–21], it is important to explore the potential VPC designs working in the optical wavelength region, which should be manufacturable by the current complementary metal-oxide-semiconductor (CMOS) fabrication technique. In 2019, Shalaev et al. has demonstrated the VPC waveguides based on silicon material, which achieved a high forward transmittance up to 0.95 in the near-infrared (NIR) spectral range [8]. In addition, the device has a small footprint of tens of microns, which is suitable for integration with photonic chips. More recently, Peng et al. designed and manufactured a 2D hexagonal lattice silicon TPC to realize the splitting of Dirac cones to achieve topological edge stages in the visible spectral range [22]. However, the high loss of silicon material in the visible region prevents the structure from achieving high forward transmittance. Thus, it is necessary to actively explore novel dielectric materials with low absorption in the visible region to further push the working bandwidth into the visible range.

2D hexagonal boron nitride (hBN) exhibits optically stable, ultra-bright quantum emitters [23–25] in the visible region that makes hBN a promising nanophotonic platform for quantum computing and information processing. In addition, hBN has low absorption in the visible region and can be fabricated with current nanofabrication techniques. It has been reported experimentally that suspended 2D hBN PC cavities can achieve quality (Q) factors over 2000 [23], which confirms the feasibility of experimentally manufacturing high quality 2D hBN PC structures working in the visible to NIR region [25]. Thus, 2D hBN can be considered as an ideal candidate for building VPCs in the visible region. However, the relatively low refractive index of hBN material compared to commonly used semiconductor materials (e.g. silicon, GaAs) makes it challenging to design a VPC structure with topological bandgap for unidirectional transmission, and there is no VPC in visible region being demonstrated.

In this paper, we theoretically propose the first VPC structure in the visible region based on 2D hBN material by using the triangular shaped holes arranged in the triangular lattice structure. In this way, we are able to maximize the bandgap by reducing the area of a set of the triangular holes to zero to achieve maximized effective index contrast. In the meantime, the C3v rotation symmetry can be maintained, which is essential in achieving the spin-valley locking effect. Based on the spin-valley locking effect, the left-handed circularly polarized (LCP) and right-handed circularly polarized (RCP) light achieve valley-dependent unidirectional coupling in the VPC. By combining two VPCs with mirror symmetry, we construct waveguides in the shapes of “Z” and “Ω”, which realize robust unidirectional transmission of circularly polarized light. It is found the large bandgap holds the key in achieving high forward transmittance of the waveguides. The straight waveguide can achieve a high forward transmittance of 0.96 in the visible region due to the maximized bandgap, which is the highest in any unidirectional transmission devices in the visible region to the best of our knowledge. In addition, we further study the effect of different dispersive refractive indices of hBN being reported from references. It is found our design can generally work with hBN materials with different dispersive refractive indices. Thus, this study opens new possibilities in designing photonic devices in the visible wavelength region and will find broad applications in optical communication and information processing.

2. Conceptual design of hBN VPC

We propose a 2D hBN structure composed of two VPCs, namely VPC1 and VPC2 (mirror symmetrical about the x-axis) with a thickness of d = 220 nm. It is worth noting that the 2D hBN has a layered structure, and the overall thickness depends on the number of layers. In addition, the layered structure results in an anisotropic refractive index (nx = ny≠nz, nx, ny are in-plane refractive indices, nz is out-of-plane refractive index) [26,27], which is considered in the design. As shown in Fig. 1(a), the VPC is a 2D triangular lattice (the lattice constant a = 0.27 µm) with triangular shaped air holes (r = 0.13 µm, defined as the distance between the center and a vertex) embedded in 2D hBN material, which is suspended on a quartz substrate (the refractive index is 1.46). The dipole source is placed at the boundary between the VPC1 and VPC2 structures, and LCP and RCP light waves transmit along the opposite direction due to the spin-valley locking effect. Due to the low absorption of hBN material in the visible region, the structure allows highly efficient unidirectional transmission of visible light. The boundary between the VPC1 and VPC2 is enlarged, as shown in Fig. 1(b), in which the edge states propagate along the boundary.

 figure: Fig. 1.

Fig. 1. (a) Three-dimensional (3D) schematic diagram of the unidirectional transmission of LCP and RCP light sources in hBN VPC. (b) Enlarged view of the structural interface.

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3. Transmission mechanism of hBN VPC

In order to achieve the spin-valley locking effect, we first design a honeycomb PC with Dirac point, then by breaking the spatial inversion symmetry of the PC, the K’ and K valleys corresponding to spin up (LCP) and spin down (RCP) degenerate due to the valley-spin locking. The PC with a Dirac point is shown in Fig. 2(a), which is a honeycomb lattice with triangular shape air holes (shown in Supplement 1, the band structure is calculated by using commercial Lumerical FDTD Solutions software). The structure shows C6v symmetry. The band curves intersect at the K(K’) valley in the Brillouin zone (red dotted lines in Fig. 2(a)), resulting in a Dirac point. In order to break the Dirac cone dispersion and degenerate the Dirac point at K and K’ valleys, it is necessary to break the C6v symmetry of the lattice. We reduce the r of air holes in the unit cell in three directions to zero, which reduces the symmetry to C3v to form VPC1 and VPC2 (Fig. S1 in Supplement 1). The degeneracy of the K and K’ valleys is shown by the blue dotted curves in Fig. 2(a), which have a complete bandgap at the wavelength range of 0.381-0.417 a/λ (647 nm-709 nm) indicated by the blue shaded part. We found the central position and width of the bandgap slightly change with the thickness of the VPC, which is shown in Fig. 2(b). And we further simulated the influence of the thickness on the transmittance and bandwidth of the designed hBN VPC structure (shown in Supplement 1). Along with the increase of thickness of hBN VPC, the central wavelength redshifts with a slight increase in gap width. The phase distributions of Hz component of VPC1 excited by LCP and RCP at the K’ and K valleys are shown in Fig. 2(c), which show different handiness of the phase vortices. According to the right-hand system, this photon state of LCP light is defined as spin up and the photon state of RCP light is defined as spin down. In addition, due to the spin-valley locking effect, the unidirectional coupling of LCP and RCP (LCP to Г-K’ direction and RCP to Г-K direction) can be seen in the electric intensity distributions shown in Fig. 2(d) when the light sources are placed at the center of the design.

 figure: Fig. 2.

Fig. 2. (a) The band structures of a honeycomb lattice PC (solid red dotted line) and VPC1 (solid blue dotted line), the gray area is the light cone and the blue shaded area is bandgap. (b) The change of gap width with the different thicknesses in the hBN VPC. (c) The phase distributions of the valley intrinsic field Hz corresponding to K’ and K valleys excited by the LCP and RCP light sources, respectively. (d) The electric field intensity distributions of the coupling of LCP and RCP in VPC1 at K’ and K valleys, respectively.

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4. Unidirectional coupling transmission of hBN VPC

The minimum band model of bulk dispersions resorting to the photonic effective Hamiltonian through the $k \cdot p$ approximation can be expressed as

$$\hat{H} = {v_D}({{{\hat{\sigma }}_x}{{\hat{\tau }}_z}\delta {k_x} + {{\hat{\sigma }}_y}\delta {k_y}} )+ \lambda _{{\varepsilon _r}}^P{\hat{\sigma }_z}$$
where vDis the group velocity. ${\hat{\sigma }_i}$ and ${\hat{\tau }_j}$ are the Pauli matrices acting on sublattice and valley spaces, respectively. $\delta \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k}$ measures from the valley center K or K’ point. In the perturbation term $\lambda _{{\varepsilon _r}}^P{\hat{\sigma }_z},$ εris the permittivity tensor (hBN is the anisotropic medium) and $\lambda _{{\varepsilon _r}}^P$ is zero considering space and time inversion symmetry, which also corresponds to zero perturbation in the initial structure. Due to the change of the size of air holes, the spatial inversion symmetry is broken and a bandgap with a width of $2|{\lambda_{{\varepsilon_r}}^P} |$ is opened, in which $\lambda _{{\varepsilon _r}}^P$ is proportional to the difference between the integration of εr on the two triangles A and B (shown in Supplement 1), as $|{\lambda_{{\varepsilon_r}}^P} |\propto \left|{\int_B {{\varepsilon_r}dS - \int_A {{\varepsilon_r}dS} } } \right|.$ One can see that there are two potential ways to maximize the bandgap: 1) maximizing the area difference of A and B; 2) maximizing the refractive index contrast εr. In order to achieve maximum bandgap, we set rA = 0.13 µm, rB = 0 µm (Supplement 1). In addition, the high refractive index contrast (corresponds to the difference in εr) (here is hBN and air) is beneficial to achieve a large bandgap. Thus, generally high refractive index materials, such as silicon or GaAs, are more suitable for achieving a larger bandgap. However, they are highly lossy in the visible region.

Furthermore, the VPCs have a valley-dependent topological index [28,29]:

$${C_{\tau z}} = \frac{{{\tau _z}{\mathop{\rm sgn}} ({\lambda_{{\varepsilon_r}}^P} )}}{2}$$
which are ${C_K} = \frac{1}{2}$ and ${C_{K^{\prime}}} = \textrm{ - }\frac{1}{2}$, for K and K’ valleys, respectively. Therefore, even the total Chern number is equal to zero, we can define this new non-zero topological invariant, namely the valley Chern number: ${C_V} = {C_K} - {C_{K^{\prime}}}$, which is 1 for VPC1 and -1 for VPC2, respectively. According to the bulk-boundary correspondence [30], when the valley Chern number at the interface is not 0, the topological phase transition occurs.

Here, VPC1 and VPC2 can be combined to form a topological waveguide, so that the edge states (shown in the band diagram in Fig. 3(a) by the red and blue lines, corresponding to the LCP and RCP light) can propagate along the direction to the “locking” valley (Figs. 3(b) and 3(e)). From the band diagram of the edge states in Fig. 3(a), it can be seen that the bandgap from 0.381 a/λ to 0.417 a/λ of the C3v structure (Fig. 2(a)) shows a pass band. Therefore, the corresponding light waves in the wavelength range of 647 nm-709 nm can be transmitted at the edge. To further demonstrate the unidirectional coupling of the topologically protected edge states, we show that the LCP and RCP light are transmitted in opposite directions when the LCP or RCP light source is placed at the boundary between VPC1 and VPC2, as shown in Figs. 3(c) and 3(f). It shows that the valley edge states are selectively excited and coupled. At the same time, the phase distribution of the LCP or RCP light transmitted in the waveguides is π phase difference (Figs. 3(d) and 3(g)).

 figure: Fig. 3.

Fig. 3. (a) The band structure of the edge states of VPC1 and VPC2. (b) and (e) are schematics of the waveguide composed of the VPCs with mirror symmetry and the arrows indicate the LCP/RCP unidirectional coupling transmission to the left/right. (c) and (f) are electric field intensity distributions of LCP and RCP light. (d) and (g) are Ey phase distributions of LCP and RCP light.

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We combine VPC1 and VPC2 to form straight (Fig. 4(a)), Z-type (Fig. 4(b)) and Ω-type (Fig. 4(c)) waveguides, respectively. The corresponding electric field intensity distributions at the wavelength of 680 nm are shown in the bottom layer in each figure. The corresponding forward transmittance spectra are shown in Figs. 4(d)-(f). The straight waveguide shows the highest transmittance of 0.96, confirming the topologically protected unidirectional transmission due to the spin-valley locking effect. In comparison, we can identify some loss occurring at the sharp corner of the Z-type and Ω-type waveguides due to the relatively narrow bandgap resulting from the relatively low refractive index of hBN material. As a result, the peak transmittance is relatively lower with a narrower work bandwidth. In order to find out the effect of corners on the working bandwidth, we put power monitors at different positions of the Z-type waveguide in the Supplement 1. We found the corners are the main restricting factor of the bandwidth, which is similar to the previous work on silicon valley photonic crystals [8,16]. There are some scattering losses at the sharp corners. This can be seen from the relatively high intensity at the corners compared to the straight parts in Figs. 4(b) and (c), due to the interference of the scattered light. In this case, compared to straight waveguide of the hBN VPC, the transmittance of the Z-type and Ω-type waveguides is relatively low. However, the maximum transmittance of the waveguides is still higher than 0.75, which is much higher than normal photonic crystal waveguides with similar designs [19]. In addition, the entire device area is only 5.5 µm by 6.3 µm, showing the possibility of the application in highly integrated photonic chips or quantum optics. There is another arrangement to form the waveguide, which supports the edge states as shown in the Supplement 1. It shows that the arrangement of boundary does affect the edge states properties, as well as the transmittance spectrum.

 figure: Fig. 4.

Fig. 4. The transmission of light waves at the same wavelength (680 nm) in different types of waveguides: the structural diagram of (a) straight, (b) Z-type and (c) Ω-type waveguides, and the corresponding electric field intensity distributions. (d)-(f) are the corresponding forward transmittance. The grey region represents the unidirectional transmission working bandwidth, where the forward transmittance is higher than 0.5.

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Considering the experimental feasibility of the designed hBN VPC devices, we further studied the unidirectional transmittance of topologically protected edge states in hBN VPC waveguides with different experimentally measured in-plane dispersive refractive indices reported from different references (Fig. 5(a)-(c)) [26,31,32], and the out-of-plane refractive index is 1.84 [27]. We calculated the transmittance spectra of RCP light propagating in the straight waveguide (Fig. 5(d)) along the forward (right) and backward (left) directions. The forward (right) and backward (left) transmittances are TF and TB, respectively. The transmission contrast is defined as $C = |{({{T_F} - {T_B}} )/({{T_F} + {T_B}} )} |$, as shown in Figs. 5(d)-(f). As one can see, our design can generally work with different dispersive refractive indices. In addition, the hBN VPC with the high refractive index is beneficial to achieve the high peak transmittance and the broad working bandwidth. The working band shows blueshift when the refractive index is decreased. The highest peak transmittance is achieved when the in-plane refractive index is n1, which is 0.96 at 680 nm. The contrast is 0.99. The working bandwidth is defined as where the forward transmittance is higher than 0.5, which is 73 nm in the range of 641 nm-714 nm. The results confirm the spin-valley locking principle can be generally applied to dielectric materials with different refractive indices to achieve high forward transmittance.

 figure: Fig. 5.

Fig. 5. Different dispersive refractive indices of hBN from references (a) n1 [26]; (b) n2 [31]; (c) n3 [32], the transmission spectra (d), (e), and (f) of visible light (RCP) propagating in the VPC.

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We noticed that the maximum forward transmittance of the straight waveguide is less than 1 (shown in Supplement 1), which is different from the reported VPC waveguides based on high index materials, such as silicon [8,16]. The maximal working bandwidth of the waveguides depends on the width of the bandgap. Due to the relatively low refractive index contrast between hBN and air, there is a narrower bandgap compared to silicon structures [8,16], which results in a narrower working bandwidth. Here we found maximizing the bandgap is essential in achieving high forward transmittance, in particular for the waveguides with sharp bends (e.g. Z and Ω shape). Although the forward transmittance is lower than those achieved by using silicon [8,16], it is the highest in any unidirectional transmission devices in the visible region to the best of knowledge. Furthermore, the proposed fabrication method of hBN valley photonic crystals is mentioned in Supplement 1. Therefore, this design can generally contribute to the design of unidirectional transmission devices in the visible region and set up a new benchmark.

5. Conclusion

In this paper, we theoretically demonstrated that the VPC structure based on 2D hBN material can achieve unidirectional coupling of topologically protected edge states in the visible region due to the spin-valley locking effect. By combining two VPCs with mirror symmetry, we observed that the spin-valley locking effect leads to selective valley-dependent coupling of circularly polarized light in the VPC, thereby achieving unidirectional transmission. The waveguide based on the VPC structures can achieve the high forward transmittance of 0.96. In addition, we further studied the effect of different thicknesses and dispersive refractive indices of hBN being reported from references. It is found that our design can generally work with hBN materials with different dispersive refractive indices, both the transmittance and the bandwidth increased accordingly with the increase of refractive index of hBN within the visible region. Finally, the designed structure can be manufactured by the current CMOS nanofabrication technique experimentally. It is the first demonstration of hBN VPC working in the visible region. Thus, this study opens new possibilities in designing photonic devices in the visible wavelength region and will find broad applications in optical communication and information processing, as well as quantum optics devices.

Funding

Young Scientists Fund of the National Natural Science Foundation of China (11904255); National Natural Science Foundation of China (62175178); Key R&D Program of Shanxi Province (International Cooperation) (201903D421052); Applied Based Research Program of Shanxi Province (Youth Fund) (201901D211070); Central Guidance on Local Science and Technology Development Fund of Shanxi Province (YDZJSX2021A013).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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Supplementary Material (1)

NameDescription
Supplement 1       Unidirectional transmission of visible region topological edge states in hexagonal boron nitride valley photonic crystals: supplementary document

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (5)
Fig. 1.
Fig. 1. (a) Three-dimensional (3D) schematic diagram of the unidirectional transmission of LCP and RCP light sources in hBN VPC. (b) Enlarged view of the structural interface.

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Fig. 2.
Fig. 2. (a) The band structures of a honeycomb lattice PC (solid red dotted line) and VPC1 (solid blue dotted line), the gray area is the light cone and the blue shaded area is bandgap. (b) The change of gap width with the different thicknesses in the hBN VPC. (c) The phase distributions of the valley intrinsic field Hz corresponding to K’ and K valleys excited by the LCP and RCP light sources, respectively. (d) The electric field intensity distributions of the coupling of LCP and RCP in VPC1 at K’ and K valleys, respectively.

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Fig. 3.
Fig. 3. (a) The band structure of the edge states of VPC1 and VPC2. (b) and (e) are schematics of the waveguide composed of the VPCs with mirror symmetry and the arrows indicate the LCP/RCP unidirectional coupling transmission to the left/right. (c) and (f) are electric field intensity distributions of LCP and RCP light. (d) and (g) are Ey phase distributions of LCP and RCP light.

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Fig. 4.
Fig. 4. The transmission of light waves at the same wavelength (680 nm) in different types of waveguides: the structural diagram of (a) straight, (b) Z-type and (c) Ω-type waveguides, and the corresponding electric field intensity distributions. (d)-(f) are the corresponding forward transmittance. The grey region represents the unidirectional transmission working bandwidth, where the forward transmittance is higher than 0.5.

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Fig. 5.
Fig. 5. Different dispersive refractive indices of hBN from references (a) n1 [26]; (b) n2 [31]; (c) n3 [32], the transmission spectra (d), (e), and (f) of visible light (RCP) propagating in the VPC.

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Equations (2)

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H ^ = v D ( σ ^ x τ ^ z δ k x + σ ^ y δ k y ) + λ ε r P σ ^ z
C τ z = τ z sgn ( λ ε r P ) 2
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