Topological photonic crystal of large valley Chern numbers Download: 651次
1. INTRODUCTION
Photonic topological states, rooted in the studies of topological insulators in electronic systems, have opened up an intriguing way to control the motion of electromagnetic (EM) waves. There has been a lot of interest in the topological edge states resulting from the topological phases that can route the wave propagation, overcoming the backscattering and remaining robust against defects [1
Symmetry plays a key role in the design of the topological phases of photonic crystals (PCs). In the honeycomb structured magnetic PCs, for example, breaking the time-reversal (TR) symmetry will gap a pair of Dirac cones at high symmetry and points, so that the Berry curvature has the same sign at these points and yields a nonzero topological invariant number, the Chern number, . If the Dirac points are away from the high symmetry points, the Berry curvature will have more extremes around the and points, and the Chern number will be greater than one [6,7]. Similarly, the QVH phase is related to the breaking of inversion symmetry, which introduces a binary degree of freedom (DOF) in the PCs, an analog of the valley DOF spintronics in the electronic system. Valley labels the energetically degenerate yet inequivalent points in momentum space [20]. This new DOF also opens the Dirac cones at high symmetry and points, but the Berry curvature at these two points has an opposite sign; therefore, the Chern number of the bandgap will be zero. However, the valley Chern number , defined at the valley points, is nonzero [22,24]. The topological valley phase gets rid of the limitation of a bias magnetic field, opening a path toward the topological phase in all-dielectric PC [19
In this work, we report what we believe, to the best of our knowledge, is a new type of valley phase where the valley Chern number can be , 2, or 3, depending on the configuration of the unit cell. The variations of the valley Chern number are achieved by expanding or shrinking one set of rods of the hexamer. These new QVH phases are characterized by the Berry curvature in the first Brillouin zone, and further proven by the edge states at the domain wall according to the bulk-edge correspondence [34,35]. Note that the number of edge states is the same as the difference in the valley Chern numbers across the domain wall. The robust wave transmissions of these QVH phases are demonstrated by the -shaped domain wall. Having band gaps with larger valley Chern numbers greatly expands the phases available for topological photonics.
2. TOPOLOGICAL VALLEY STATES WITH LARGE VALLEY CHERN NUMBERS
2.1 A. Dirac Points Away from the High Symmetry Point
Consider that the PC structure is fabricated by the artificial molecules, as shown in Fig.
Fig. 1. (a) Schematic of 2D PC structure, composed of hexamers of six ferrite rods and embedded in the air background. The white lines denote the edge of a unit cell. (b) Band structure of the PC at . A Dirac point is away from the high symmetry points in the first Brillouin zone. (c) 3D band structure of the PC. Three pairs of Dirac points are between the two bands in the Brillouin zone. (d) 3D band structure of Ref. [13]. One pair of Dirac points presents at the point.
Figure
2.2 B. Valley Phase of Large Valley Chern Number
To gap the Dirac points, one can break the time-reversal symmetry or the space-inversion symmetry. After the symmetry-breaking operation, the Dirac points are opened, and each degeneracy lifting contributes a Berry flux of magnitude in each band [2–
Here, we gap the Dirac points by breaking the inversion symmetry of the system. Shrinking or expanding the distance of the set of rods (marked in red) and keeping the other set (marked in blue) unchanged, the rotation symmetry of the system changes from original to . Meanwhile, the inversion symmetry of the system is broken; that is, the unit cell does not keep its original form under the transform of () to () for two-dimensional (2D) systems. This operation opens a full bandgap at the Dirac point, and the valleys appear at the point. As displayed in Fig.
Fig. 2. (a) Band structures of the PC at and . (b) Berry curvature in the first Brillouin zone. Curvature is opposite around the and points, and the valley Chern number .
To show the difference, the Berry curvature and the valley Chern number were numerically calculated. The Berry curvature of the nth band is defined as [4,36], where is the Berry connection and is the Bloch state. The efficient algorithm [37] was used in the calculation of the Berry curvature over the Brillouin zone. In the zone, each peak of the curvature contributes to the Chern number [8], and the valley Chern number is obtained by counting the number of peaks. As an illustration, Fig.
A further demonstration of the system having a larger valley Chern number is to check the number of topological edge states at the boundary. According to the bulk-edge correspondence [34,35], the number of edge states between two topologically distinct domains should be the difference of the valley Chern number across the boundary. We construct a domain wall where two domains have opposite valley Chern numbers: ( and ) and ( and ). The two domains have identical parameters, as in Fig.
Fig. 3. Topological edge state of valley Chern number . (a) Projected band structures for the valley Chern difference at point across the domain wall. Insets are the distributions of at the given points and in the band structure. (b) Transmission spectra of -shape corners in the frequency range of 12.69–12.85 GHz. The red and yellow regions correspond to the single-mode and multimode regions, respectively. The insets are the field distributions at the frequencies in the single-mode and multimode regions.
3. VALLEY PHASES OF DIFFERENT VALLEY CHERN NUMBERS
The valley phases of the PC depend on the expansion or shrinking of one set of rods. One can identify the different phases by their eigenstates or their valley Chern numbers. As an example, we keep the blue set of rods at , while expanding the red set of rods to , as shown in the inset of Fig.
Fig. 4. Band structures, phase and power flow distribution, and corresponding Berry curvature of the PC, where is fixed at , (a) and valley Chern number , (b) and , and (c) and . Arrow inserted in the phase distribution indicates the Poynting vector.
Fig. 5. Electric field distribution in momentum space. (a) Chiral sources carry positive and negative OAM. Colors are the phase of electric field excited by the source in the center. Arrows show the direction of OAM: the counterclockwise arrow represents the positive OAM and the clockwise arrow represents the negative one. (b) Field excited by the chiral source with positive OAM, where the field is strongly localized at point . (c) Field excited by the source with negative OAM, where the field is strongly localized at point . Panel insets are the close look near the point or .
A direct reflection of the valley phase is the Berry curvature, as shown in Fig.
Changing the value of one subset may transform the valley state to other phase. Supposing we reduce the dimension of the red subset to . Once again, the full bandgap and valleys present in the band structure [Fig.
Further shrinking the dimension of the red set will strengthen the coupling between the six rods, and all the rods in the unit cell function as a whole, the hexamer. As displayed in Fig.
The different phases of the large valley Chern number are related to the band inversion happening in the band structure, which can be identified by the opening–closing–reopening of the full bandgap. Figure
Fig. 6. Variation of the valley frequency with varying when is fixed. The valley Chern number remains unchanged in the region of the same color.
The valley Chern numbers can further be proven by the number of the topological edge states at the domain wall. Again, we make the domain wall between the two PCs with opposite valley Chern numbers. Figure
Fig. 7. Topological edge state of valley Chern number of and 3. Panels (a) and (b) are the projected band structures for the valley Chern number difference across the domain wall, respectively, and 3 at point. The number of edge states present in the bandgap is the same as the valley Chern number difference. The panel insets are the distributions of at the given points in the band structure. The fields are all localized at the domain wall. (c) and (d) Transmission spectra of -shaped topological domain wall for valley Chern number difference and 3 at point, respectively. Simulated field distributions are inserted. The yellow curves represent the energy flows.
4. CONCLUSION
In conclusion, we have proposed and theoretically demonstrated what we believe, to the best of our knowledge, is a new type of valley Hall phase in 2D photonic crystal, which is made of the hexamers of dielectric rods and has a large valley Chern number. By simply shrinking or expanding one set of rods in the hexamer, we realize a valley phase transition from the valley Chern number of one to three. The multiple edge states further demonstrate our valley phases with large valley Chern numbers, which are perfectly compatible with the bulk-edge correspondence. Robustness of the edge modes is demonstrated by the wave transmission along the domain wall of the -shaped form. We believe our study provides new opportunities in topological photonics, according to practical requirements.
5 Acknowledgment
Acknowledgment. The authors acknowledge assistance from the project funded by Priority Academic Program Development of Jiangsu Higher Education Institutions and Jiangsu Provincial Key Laboratory of Advanced Manipulating Technique of Electromagnetic Wave.
Article Outline
Xiang Xi, Kang-Ping Ye, Rui-Xin Wu. Topological photonic crystal of large valley Chern numbers[J]. Photonics Research, 2020, 8(9): 090000B1.