JP2016156971A - Topological photonic crystal - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a topological photonic crystal that uses a common material such as silicon, and can be produced without requiring a special process difficult to control.SOLUTION: In a state (a) in which dielectric cylinders disposed in a regular hexagonal shape are arranged in a honeycomb shape, a topological photonic insulation state is obtained by establishing a/R<3. Where, 2R is the length of the diagonal line of the regular hexagon, and a0 is a distance between the center of a regular hexagon and the center of its adjacent regular hexagon.SELECTED DRAWING: Figure 1

Description

  The present invention relates to a topological photonic crystal, and more particularly to a topological photonic crystal that can be produced using a normal material such as silicon.

  With the discovery of the quantum Hall effect (QHE), new progress in physical property research centered on topology has been made (Non-Patent Documents 1 to 11). The topological state is not only interesting from an academic point of view, but is also expected to have a significant impact on applications. This is because the strong surface (or edge) state protected by the bulk topology provides new possibilities for spintronics and quantum computation (Non-Patent Documents 12 to 17).

  However, only a few substances have been confirmed to have a topological state so far, and most of them exhibit topological properties only at very low temperatures. This problem hindered detailed research and manipulation of materials essential for practical applications.

  An object of the present invention is to design a topological photonic crystal that can be formed without using a special material or process.

According to one aspect of the present invention, the dielectric cylinder includes a plurality of dielectric cylinders standing parallel to each other on a plane, and the periphery of the dielectric cylinder has a dielectric constant different from that of the dielectric constituting the dielectric cylinder. In the region filled with the dielectric, the plurality of dielectric cylinders are divided into a plurality of sets of dielectric cylinders arranged in a regular hexagon of the same size, and a plurality of pairs corresponding to the plurality of sets of dielectric cylinders The center of the regular hexagon is disposed at a lattice point of a triangular lattice, the vertex of the regular hexagon is disposed on a lattice line connecting two adjacent lattice points, and the diagonal length of the regular hexagon is 2R. , when the distance between the centers of the regular hexagon and the regular hexagon adjacent thereto and a _0, a _{0 / R} <topological photonic crystal 3 is satisfied is provided.
Here, the region may be sandwiched between two parallel metal plates.
The metal plate may be made of gold.
The dielectric cylinder may be made of silicon.
The dielectric filling the periphery of the dielectric cylinder may be air or vacuum.

  According to the present invention, a topological photonic crystal can be provided that uses a common material such as silicon and can be formed without requiring a special and difficult-to-control process.

The figure explaining the design of a photonic crystal. (A) is a figure which shows the outline of a structure of the triangular photonic crystal of the "artificial atom" comprised by the dielectric cylinder which each extends in six z directions. Diamonds and regular hexagons drawn with thin broken lines indicate unit cells of a honeycomb lattice and a triangular lattice, respectively. A regular hexagon drawn with a slightly thick black solid line indicates an artificial atom, and a hexagon drawn with a slightly thick black broken line indicates an interstitial region of the artificial atom. (B) TM (transverse magnetic) mode of the magnetic field H _x ± iH diagram conceptually illustrating a pseudo spin state of the left hand and the present photonic system associated with the right-hand circularly polarized light of _y. (A) _p in an artificial atom x / _{p y} and
It shows the distribution of the electric field E _z photonic trajectory. (B) Distribution of the electric field E _z having positive and negative angular momentum

as well as

FIG.
FIG. 4 shows a photonic band, TM mode dispersion for a 2D photonic crystal with ε _d = 11.7 (silicon), ε _A = 1 (air, of course vacuum), and R = 1.5d. Show the relationship. (A) a ₀ /R=3.16 (inset represents Brillouin zone of triangular lattice), (b) a ₀ / R = 3 and (c) a ₀ /R=2.78 Each case is shown. In the figure, as shown in the vertical bar at the right end of the figure, the halftone density curve is shown by distinguishing the d _± band and the p _± band by density, and in particular, in the figure for (c) a ₀ /R=2.78. The halftone density curve continuously transitions between the densities representing the d _± band and the p _± band, respectively, so as to represent the hybrid of both. The figure which shows the real space distribution of the Poynting vector in (GAMMA) point below a photonic gap. (A) When a ₀ /R=3.16, which is a trivial state. (B) In the case of a ₀ /R=2.78 which is a topological state. The figure which shows a helical topological edge state. (A) The dispersion relation of a ribbon-shaped photonic crystal having an infinite length in one direction and 45 and 6 artificial atoms in the topological region and the obvious region in the other direction, respectively. The right side is an enlarged view near the photonic crystal gap including the topological edge state, and the thick curve is the topological edge state. A and B are two points where the electric field E _z is shown. Parameters used herein, the topological region is _a 0 /R=2.9, for obvious areas except a _a 0 /R=3.125 is the same as FIG. (B) The distribution of the electric field Ez at points A and B is shown. On the right side, the time-average Poynting vector in the unit cell on the topological region side adjacent to the interface
(Black arrow).
The figure which shows notionally the structure of the topological photonic crystal of the Example of this invention. (A) A photonic crystal having a finite height, and two horizontal gold plates are arranged symmetrically at both ends of a large number of dielectric (silicon) cylinders. (B) Energy density distribution due to electric field Ez in a square sample of topological photonic crystal sandwiched between two parallel gold plates
In an embodiment of the present invention, a 0 _{/ R} = 3 conceptually shows an arrangement of silicon cylinder in the case of. In an embodiment of the present invention, a 0 _{/ R>} 3 conceptually shows an arrangement of silicon cylinder in the case of. In an embodiment of the present invention, a 0 _{/ R} <diagram conceptually showing the arrangement of a silicon cylinder in the case of 3.

  A photonic crystal is an optical material in which atoms arranged in a solid are replaced by a medium having a periodic dielectric constant and / or magnetic permeability (Non-patent Document 18). It is called a meta-material, such as a negative refractive index and an electromagnetic property that cannot be obtained in nature such as a magnetic lens (Non-patent Document 19). Recently, the topological photonic state characterized by the unique edge propagation mode of electromagnetic waves is a gyromagnetic material under an external magnetic field, and a bi-anisotropic metamaterial that combines an electric field and a magnetic field. It has also been found that this can be realized by a coupled resonator optical waveguide (CROW) (Non-Patent Documents 20 to 32).

  According to one aspect of the present invention, a two-dimensional (2D) photonic crystal made only of a normal dielectric such as silicon is provided. In the following, a configuration using silicon exclusively as such a dielectric will be described as a representative example, but it should be noted that the generality is not lost.

A honeycomb lattice is equivalent to a triangular lattice of regular hexagonal clusters composed of six neighboring sites, and the large regular hexagonal unit cells should be considered instead of the diamond-shaped unit cells composed of two sites. As a result, the Dirac cone at the K and K ′ points in the first Brillouin zone of the honeycomb lattice is folded and double degenerates to the Dirac cone at the Γ point. Here, it is interesting to consider that the two 2D irreducible representation corresponding to the odd and even parity in spatial inversion operation in Γ point of the C ₆ symmetry group is present. Based on these properties, the inventor modified the honeycomb lattice so as to maintain the shape and size of regular hexagonal clusters and the C ₆ symmetry that they satisfy, as shown in FIG. Realized a topological nontrivial state. Strictly speaking, by solving the Maxwell equation, the inventor found that the electromagnetic modes in hexagonal clusters operating as “artificial atoms” in the configuration of the present invention are similar to the electron orbits in solids. It has been found that a photonic band is formed with an s-waveform, a d-waveform and a p-waveform. Also revealed pseudo TR symmetry born by the Maxwell equations combine C ₆ crystal symmetry by generally having time-reversal symmetry (time reversal (TR) symmetry) and design. This pseudo-TR symmetry exhibits the same behavior as TR symmetry in an electronic system, and causes Kramers doubling in the photonic system of the present invention. This is similar to the case of the topological crystal insulator proposed for the three-dimensional system (Non-patent Document 33). This is closely related to the one-to-one correspondence between right-handed / left-handed rotational polarization and electron up / down spin in TM mode. Through evaluating the Berry curvature of the photonic band and the edge state of the finite system, the lattice constant of the “artificial atom” triangular lattice is reduced topologically from the value corresponding to the original honeycomb lattice of the silicon cylinder. Succeeded to show that the condition appears. The topological state of the present invention, which can be made of silicon alone, which does not require any external strong magnetic field nor rotating magnetic field or bi-anisotropic material, is very promising for future applications.

<Artificial atoms and pseudospin>
Consider the TM mode of electromagnetic waves in a dielectric material (ie, finite in-plane H _x , H _y and out-of-plane E _x components; others are zero). For simplicity, the dielectric constant in the frequency range considered is considered a real number independent of frequency. In this case, the master equation for the electromagnetic wave mode at the frequency ω is given as follows from the Maxwell equation (Non-patent Document 34):

Here, ε (r) is a position-dependent dielectric constant, and c is the speed of light. The transverse component of the magnetic field is Faraday related

Given by. Here, the permeability μ ₀ assumes a vacuum permeability. The master equation (1) can be considered as a generalized Schroedinger equation, and therefore Bloch theory on the electronic system is periodic in space as ε (r) is schematically shown in FIG. Is applied to the system of the present invention. However, it should be noted that the master equation (1) describes an electromagnetic wave, not an electron having a spin degree of freedom. The most significant difference is in the response to the time reversal operation.

In the 2D photonic crystal according to the present invention, as shown in FIG. 1A, a dielectric (for example, silicon) having a relative dielectric constant ε _d in a direction parallel to the z-axis has a different dielectric constant ε _A. Embedded in a dielectric material (eg air). The following discussion is for a state that is uniform along the z-axis, and this assumption reduces the problem to 2D.

Starting from a plurality of dielectric cylinders arranged in a honeycomb grid, this arrangement is modified so that regular hexagonal clusters of ₆ adjacent cylinders and C ₆ symmetry are maintained. When such a deformation is performed, it is more convenient to consider the arrangement of the dielectric cylinders as a regular hexagonal “artificial atom” triangular lattice. There are two 2D irreducible representations E ′ and E ″ for the C ₆ symmetry group associated with the triangular lattice. E ′ and E ″ are the simple functions x / y and xy / (x ² −y ² ). It corresponds to such odd and even space parity (Non-patent Document 35). As can be seen from the electric field E _z in Fig obtained by solving the master equation (1) 2 (a), p x / p artificial atoms with the same symmetry as the symmetry of the electron orbit normal atoms in solids _y and

Has an orbit.

  Here, two sets of eigenwave functions are defined as follows.

The “angular momentum” j _z of these wave functions is ± 1 and ± 2. The pseudo-TR operation defined by the following anti-unitary operator is constructed.

Here, K is a complex conjugate operator and corresponds to the TR operation on the TM mode governed by the master equation (1). Σ _z is a 2 × 2 Pauli matrix.

as well as

So that under the two sets of eigenwave functions shown in equation (2)

Is brought about. It should be emphasized that the TR symmetry of the Maxwell equation is typically boson-like and is characterized by K ² = 1. There is a hidden creation TR symmetry that is typically fermion-like and characterized by the TR operator T defined by equation (3) and satisfying equation (4). The pseudo-TR operator T ensures that Kramers duplication in this Maxwell system is very similar to Kramers duplication in electronic systems (Non-patent Document 35). From the eigenwave function in equation (2), the two quasi-spin states are shown in FIG. 1 (b) and FIG. 2 (b) by positive and negative angular momentum of the electric field _Ez field, or equivalently. It is evident that the in-plane magnetic field defined by H _x ± iH _y is given by left-handed and right-handed rotational polarization.

  In the photonic system, the pseudo-spins that have been discussed so far are the electric field and magnetic field bonding / antibonding states (Non-Patent Documents 24 and 25), and the left / right-handed polarization of electromagnetic waves (Non-Patent Document 28). And clockwise / counterclockwise circulation of light in CROW (Non-Patent Documents 29 and 30). Effective spin-orbit coupling (SOC) is achieved by split-ring resonators and metamaterials based on Ω particles, by piezoelectric / piezomagnetic superlattices, and asymmetry of coupling loops in CROW. Each position contributes. The first two of these implementations involve a complicated metamaterial structure, so a delicate manufacturing process is required. The last one uses only dielectric fiber such as silicon, so it is easy to make, but if one of the coupling loops breaks, it works as a magnetic impurity and the quantum spin Hall effect ( QSHE) Helical edge state is destroyed.

<Topological photonic crystal>
The inventor has determined that the unit vector given by FIG.

as well as

The photonic band dispersion described by the master equation (1) under the periodic boundary condition along is calculated. As shown in FIG. 3, double degeneracy in band dispersion appears at the Γ point, which is confirmed to be p _± and d _± states, which is consistent with the consideration from symmetry. When the lattice constant is large, the photonic band below (or above) the gap is occupied by the p _± (or d _± ) state (eg, FIG. 3 (a) showing the case of a ₀ /R=3.16. )). Quantitatively speaking, when a ₀ = 1 μm, the gap Δω = 7.7 THz at Ω = 138.6 THz is obtained. Wherein these frequencies is inversely proportional to the lattice constant a _0.

When the lattice constant a ₀ / R is decreased to 3, the p and d states degenerate at the Γ point, and two Dirac cones appear as shown in FIG. This is because when the lattice constant becomes like this, the system is equivalent to the individual cylinders arranged in a honeycomb lattice, the doubly degenerated Dirac cone is a honeycomb based on rhomboid unit cells at two sites. It is none other than the Dirac cone at the K and K 'points in the Brillouin zone of the grid. Dirac dispersion in photonic systems has been discussed for both square and triangular lattices (Non-Patent Documents 36 and 37). Unlike these past systems, there are two Dirac cones associated with the pseudo-TR symmetry guaranteed here by the structure incorporated in the structure of the artificial atom, which leads to a non-trivial topology in this system. Is possible.

If the lattice constant a ₀ / R is further reduced, the band gap (global band gap) is reopened over the entire Brillouin zone as shown in FIG. 3C for the case of a ₀ /R=2.78. Around the Γ point, the electric field E _z on the low (high) frequency side of the band gap exhibits the d _± (p _± ) characteristics, and the order is reversed as the distance from the Γ point is increased. That is, in the present application of the photonic lattice, band reverse happens when reducing the lattice constant a _0.

  To characterize the system before and after band reversal, the Poynting vector that averages the energy flow in the electromagnetic system of this application over the period τ = 2π / ω

The real space distribution of was confirmed. As shown in FIG. 4 (a) for a ₀ /R=3.16 (the Poynting vector with pseudo spin-up is not specified), the Poynting vector circulates around individual atoms. In this case, electromagnetic energy flows around individual atoms, which is characteristic of the normal “insulated” state. In the case of a ₀ /R=2.78, that is, after band inversion, the Poynting vector is greatly enhanced in the region between artificial atoms, as shown in FIG. 4 (b). This is a conspicuous difference from the case of FIG. 4A, which means that the insulation state is not a conventional one.

In order to further clarify this situation, let us consider a ribbon-like topological photonic crystal. Here, a photonic crystal having an obvious band gap in the same frequency region is placed outside the ribbon. This prevents the topological edge state from leaking into the free space. It should be noted here that a cluster of 6 cylinders is the basic block of the present design and should not be broken in a meaningful discussion. As shown in FIG. 5A, an edge state having a double degeneracy appears as a thick curve in the bulk dispersion gap. Examining the real space distribution of the electric field E _z at a typical wave number near the Γ point (k _x = ± 0.04 × (2π) / a ₀ in the enlarged display (right side) in FIG. 5A. As shown in FIG. 5 (b), the in-gap state is located at the edge and decays exponentially towards the bulk (the other two states are on the opposite side of the ribbon). Located on the edge, but not explicitly shown here). As can be seen from the inset on the right side of FIG. 5 (b), the Poynting vector shows a finite downward / upward electromagnetic energy flow for the pseudo spin down / spin up state even when averaged for artificial atoms. This unambiguously shows the electromagnetic energy flow in the opposite direction associated with the two pseudo-spin states at the edge, which is characteristic of the QSHE state. The distribution of Poynting vectors in the bulk band shown in FIG. 5A for the ribbon system is not so different from the distribution for the infinite system shown in FIG. In the quasi-classical picture, QHE has been described by a circular motion in the bulk of electrons and a skip motion at the edge under a strong external magnetic field (Non-Patent Document 38). Here, it is emphasized that the Poynting vector is a physical quantity in the electromagnetic system, and therefore the distribution shown in FIGS. 4, 5 and 6 can be observed experimentally for the photonic QSHE state. QSHE in the present system can be confirmed by also calculating the Z ₂ topological index based on the k · p model established in the vicinity of point gamma.

  For the experimental implementation of the topology state of the present application, the finite height of the silicon cylinder along the z-axis direction was taken into account. A square sample sandwiched between two parallel gold plates separated by a distance H = 10 mm was examined (FIG. 6A). Topological sample size

And all the edges of the four sides are surrounded by obvious photonic crystals. R was set to 3.65 mm and 3 mm for the topological region and the obvious region, respectively. Note that the R values in the two regions are different from each other because the values of the lattice constant a ₀ are matched in both regions for the convenience of calculation at the time of design. This is because. In addition, d = 2.4 mm and a ₀ = 10 mm for both regions. With this structure, the topological frequency was 13.47 ± 1.2 GHz. A linear electromagnetic wave source having linearly polarized light with a frequency ω = 13.47 GHz was placed in parallel with the silicon cylinder. Since an electromagnetic wave source having linearly polarized light can be decomposed into two circularly polarized lights (two pseudo spins), the system shows a helical topological edge state as shown in FIG.

  Below, with reference to FIG. 7A-FIG. 7C, the arrangement | positioning of the silicon | silicone cylinder which implement | achieves the trivial area | region of a topological photonic crystal and a topological area | region in a present Example is demonstrated more concretely. Each of these figures is a view of a large number of silicon cylinders erected on a plane from above (from the z direction according to the notation of FIG. 1), and a black circle represents each silicon cylinder. . The dielectric around the silicon cylinder is air.

In FIG. 7A, silicon cylinders are arranged in a honeycomb lattice shape, and these cylinders are equally spaced from each other. That is, the diagonal line of a light-colored regular hexagon with each silicon cylinder in the set of 6 silicon cylinders as a vertex is half of the diagonal length (note that this length is 2R), that is, the center of the regular hexagon. When the distance R is extended by a distance R from the top, the center position of the nearest silicon cylinder belonging to the set of six adjacent silicon cylinders is reached. When this diagonal line is further extended by R, it reaches the center of the set of six adjacent silicon cylinders. Since the distance between the centers of such a pair of adjacent silicon cylinders is a ₀ , this arrangement corresponds to a ₀ / R = 3 described above.

FIG. 7B is a modification of the arrangement of FIG. 7A, and maintains the regular hexagonal shape and size described above in each of the groups of six silicon cylinders (ie, R is constant). it is an enlargement by increasing the distance a ₀ in. In this case, obviously, a ₀ / R> 3. Therefore, this arrangement is self-evident.

The expansion of this distance can be explained as follows using a triangular lattice (regular triangular lattice) configured by using the center of the regular hexagon as a lattice point. In the arrangement of FIG. 7A, the centers of these regular hexagons are located on the lattice points of the triangular lattice, and the vertices of the regular hexagon are located on lattice lines connecting adjacent lattice points. When this triangular lattice is proportionally expanded (isometrically expanded) in the plane of this triangular lattice, the lattice points move so that the mutual distances are uniformly expanded (in this movement, the shape, size, and orientation of the regular hexagon) Therefore, the interval a ₀ between adjacent lattice points becomes larger from the initial 3R. In addition, the regular hexagon moves together with the movement of the lattice point originally placed. As a result, a ₀ / R> 3. In this movement, the direction of the regular hexagon does not change, that is, a parallel movement is performed. Thereby, the vertex of a regular hexagon maintains the state located on the lattice line which connects the adjacent lattice point.

FIG. 7C is obtained by modifying the arrangement of FIG. 7A in the direction opposite to that of FIG. 7B. That is, it is reduced by reducing the distance a ₀ between each set while maintaining the shape and size of the regular hexagon formed by the six silicon cylinders. In this case, it is clearly a ₀ / R <3. Therefore, a topological state appears in this arrangement.

As in the case of FIG. 7B, if the reduction is expressed based on the triangular lattice having the hexagonal centers as lattice points, the triangular lattice is proportionally reduced in the plane of the triangular lattice, contrary to FIG. 7B. When (isometrically reduced), the lattice points move so that the mutual distance is uniformly reduced (the shape, size, and orientation of the regular hexagon do not change during this movement), so the distance a between adjacent lattice points a ₀ is smaller than the original 3R. In addition, the regular hexagon moves in parallel with the movement of the lattice point originally placed. As a result, a ₀ / R <3.

Here, when the distance between regular hexagons composed of six cylinders is reduced from the honeycomb lattice arrangement of silicon cylinders (generally, dielectric cylinders), when a ₀ / R = 2 is reached, the adjacent regular hexagons are Since they are in contact with each other, it is actually necessary to satisfy 2 <a ₀ / R <3. Since R is the distance from the center position of the dielectric cylinder, it is natural that the range of this ratio can be further increased if the thickness of the cylinder is not negligible. Note that it becomes narrower.

  As described above in detail, according to the present invention, a topological photonic crystal is used without using a common material such as a material used for a normal semiconductor device, and without requiring a special and difficult control process. Can be realized. Further, since the topological photonic crystal realized in this way can operate at room temperature, there are few restrictions on practical use.

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Claims (5)

A plurality of dielectric cylinders standing parallel to each other on a plane are provided, and a region where the periphery of the dielectric cylinder is filled with a dielectric having a dielectric constant different from that of the dielectric constituting the dielectric cylinder is provided. ,
In the region, the plurality of dielectric cylinders are divided into a plurality of sets of dielectric cylinders arranged in regular hexagons of the same size,
The centers of the plurality of regular hexagons corresponding to the plurality of sets of the dielectric cylinders are disposed at lattice points of a triangular lattice, and the vertices of the regular hexagons are disposed on lattice lines connecting two adjacent lattice points. And
A topological photonic crystal in which a ₀ / R <3 is established, where the diagonal length of the regular hexagon is 2R and the distance between the centers of the regular hexagon and the regular hexagon adjacent thereto is a ₀ .   The topological photonic crystal according to claim 1, wherein the region is sandwiched between two parallel metal plates.   The topological photonic crystal according to claim 2, wherein the metal plate is made of gold.   The topological photonic crystal according to claim 1, wherein the dielectric cylinder is made of silicon.   The topological photonic crystal according to any one of claims 1 to 4, wherein the dielectric filling the periphery of the dielectric cylinder is air or vacuum.