Proof of the universality of mode symmetries in creating photonic Dirac cones

Kazuaki Sakoda

doi:10.1364/OE.20.025181

1. Introduction

Photonic Dirac cones, or the linear dispersion relation around certain points in the Brillouin zone, have been attracting considerable interest during the last five years [1–12]. Haldane et al. [1, 2] pointed out the presence of photonic Dirac cones on the Brillouin-zone boundary of two-dimensional triangular-lattice photonic crystals due to their structural symmetry and discussed unidirectional propagation of surface modes caused by time-reversal symmetry breaking. Ochiai et al. extended the discussion to honeycomb-lattice photonic crystals [3]. Zhang proposed optical simulation of Zitterbewegung, or trembling motion, in particle physics by propagating an optical pulse of the Dirac point frequency [4]. A pseudo-diffusive transmission was found by Sepkhanov et al. [5] and numerically demonstrated by Diem et al. [6].

Recently, Huang et al. reported that Dirac cones can also be created in the Brillouin-zone center of two-dimensional dielectric photonic crystals by accidental degeneracy of two modes [7]. They showed for square- and triangular-lattice photonic crystals that combinations of a non-degenerate mode and a doubly degenerate mode yield a Dirac cone together with a quadratic dispersion surface. Because the Dirac point in the Brillouin-zone center is equivalent to a zero effective refractive index [7], it has much potential for various applications like scatter-free waveguides [13] and lenses of arbitrary shapes [14].

On the other hand, we showed by tight-binding approximation and group theory that Dirac cones can also be created in the Brillouin-zone center of metamaterials, which are characterized by well-defined electromagnetic resonant states localized in their unit structures, by accidental degeneracy of two modes [8–12]. First, we analyzed one-dimensional regular arrays of metallic unit cells to show that one-dimensional photonic Dirac cones can be created by accidental degeneracy of A₁ and B₁ modes and that of A₂ and B₂ modes [8]. We also showed that a one-dimensional Dirac cone can be created by the controlled symmetry reduction of doubly degenerate modes of two-dimensional square-lattice metamaterials [9]. In addition, we proved that the combination of A₁ and E modes of square-lattice metamaterials and the combination of A_1g and T_1u modes of simple-cubic-lattice metamaterials create an isotropic Dirac cone with quadratic dispersion surfaces [10]. We further proved that the combination of E₁ and E₂ modes of triangular-lattice metamaterials yields two-dimensional double Dirac cones, or a pair of identical Dirac cones [11].

Quite recently, we systematically examined the relation between the symmetry of modes and the shapes of dispersion curves generated by accidental degeneracy for both dielectric photonic crystals and metamaterials with localized electromagnetic resonant states by numerical photonic-band calculation and tight-binding approximation, respectively [12]. We found that the two calculations gave the same results, which strongly suggested the presence of universality of mode symmetries that enabled the creation of photonic Dirac cones irrespective of the details of the sample structure, irrespective of the presence or absence of localized resonant states, and irrespective of the approximation used in the calculation.

In this paper, anticipating that the presence or absence of photonic Dirac cones is solely determined by the combination of mode symmetries, we develop a new theory that does not depend on the details of the sample structure, but depends only on the spatial symmetry of the accidentally degenerate modes.

This paper is organized as follows. In Section 2, a degenerate perturbation theory is formulated for the vector electromagnetic field of periodic systems and it is applied to the problem of the creation of Dirac cones in the Brillouin-zone center by accidental degeneracy of two modes. A necessary condition is derived by which we can easily select candidates of mode combinations that enable the creation of the Dirac cone. In Section 3 to Section 6, this method is applied to four periodic systems, that is, the one-dimensional lattice of C_2v symmetry, the square lattice of C_4v symmetry, the triangular lattice of C_6v symmetry, and the simple-cubic lattice of O_h symmetry. The condition for obtaining Dirac cones and/or double Dirac cones is clarified for each case. A summary of the present study is given in Section 7.

2. Theory

The eigen equation for the magnetic field of periodic systems is given by

ℒ H_{k n} \equiv \nabla \times [\frac{1}{ε (r)} \nabla \times H_{k n}] = λ_{k n} H_{k n},

where n and k denote the band index and a wave vector in the first Brillouin zone, respectively, ε is the periodic dielectric constant, and operator ℒ is defined by the first equality. We assume that the magnetic permeability of the system is equal to that of free space, since we do not deal with magnetic materials. Then the k dependent eigenvalue is given by

λ_{k n} = \frac{ω_{k n}^{2}}{c^{2}},

where c is the speed of light in free space and ω_kn denotes the eigen angular frequency. ε should be real for ω_kn to be real. According to Bloch’s theorem, the eigen function, H_kn, is a product of an exponential factor and a vector field with the lattice-translation symmetry:

H_{k n} (r) = e^{i k \cdot r} u_{k n} (r),

u_{k n} (r + a) = u_{k n} (r),

where a is the elementary translation vector.

When we assume that ε(r) is real and independent of frequency and impose a periodic boundary condition to make our problem well-defined, ℒ is a Hermitian operator in the Hilbert space of complex vector fields, for which the inner product of two vector fields is defined as

〈 H_{1} | H_{2} 〉 \equiv \frac{1}{V} \int_{V} d r H_{1}^{*} (r) \cdot H_{2} (r),

where V is the volume on which the periodic boundary condition is imposed [15]. We normalize the eigen functions as

〈 H_{k n} | H_{k^{'} n^{'}} 〉 = δ_{k k^{'}} δ_{n n^{'}} .

Then, u_kn is an eigen function of operator ℒ_k defined by

ℒ_{k} \equiv e^{- i k \cdot r} ℒ e^{i k \cdot r} = (\nabla + i k) \times [\frac{1}{ε (r)} (\nabla + i k) \times],

which is a Hermitian operator in the Hilbert space of complex vector fields with the lattice-translation symmetry. So, {u_kn| n = 1, 2,…} is a complete set. We normalize it as

{〈 u_{k n} | u_{k n^{'}} 〉}_{0} \equiv \frac{1}{V_{0}} \int_{V_{0}} d r u_{k n}^{*} (r) \cdot u_{k n^{'}} (r) = δ_{n n^{'}},

where V₀ denotes the volume of the unit cell. Thus, in particular, for k = 0,

{u_{0 n} | n = 1, 2, \dots}

is an orthonormal complete set. Therefore, we can express any eigen function u_kl of operator ℒ_k by a linear combination of eigen functions {u_0n} of operator ℒ₀. Thus, for small k in the vicinity of the Γ point, we can calculate λ_kl perturbatively using {u_0n} as a basis set. Because we are interested only in the presence or absence of terms linear in k, we neglect the quadratic term of the perturbation operator:

Δ ℒ_{k} \equiv ℒ_{k} - ℒ_{0} \approx Δ ℒ_{k}^{(1)} + Δ ℒ_{k}^{(2)},

where

Δ ℒ_{k}^{(1)} = i k \times [\frac{1}{ε (r)} \nabla \times],

Δ ℒ_{k}^{(2)} = \nabla \times [\frac{1}{ε (r)} i k \times] .

We assume according to the situation of our problem that {u_0l| l = 1, 2, … ,M} are degenerate and denote their eigenvalue by $λ_{0} (= ω_{0}^{2} / c^{2})$ . By the degenerate perturbation theory, the first-order solution for u_kl (l = 1, 2, … ,M) is obtained by diagonalizing the matrix whose ij (1 ≤ i, j ≤ M) element is given by

C_{i j}^{(k)} = {〈 u_{0 i} | Δ ℒ_{k} | u_{0 j} 〉}_{0},

which can bring about eigenvalue corrections linear in k. Thus, our problem on the creation of photonic Dirac cones is reduced to examining whether the eigenvalues of matrix

C_{k} = (C_{i j}^{(k)})

are non-zero.

For this purpose, we examine C_k using the spatial symmetry of {u₀_l}. We assume that the periodic structure that we deal with is invariant by symmetry operations of point group 𝒢. We denote the symmetry operations and their matrix representations by ℛ and R, respectively. First, we should note that

R^{t} = R^{- 1} and detR = \pm 1,

since R is an orthogonal matrix. Second, from the nature of inner and outer products, we have

{〈 u_{0 i} | Δ ℒ_{k}^{(1)} | u_{0 j} 〉}_{0} = - i k \cdot P_{i j},

where P_ij is defined as

\begin{array}{l} P_{i j} & = & \frac{1}{V_{0}} \int_{V_{0}} d r u_{0 i}^{*} (r) \times [\frac{1}{ε (r)} \nabla \times u_{0 j} (r)] \\ \equiv & {〈 u_{0 i} | Δ ℒ | u_{0 j} 〉}_{0} . \end{array}

We also have

\begin{array}{l} V_{0} {〈 u_{0 i} | Δ ℒ_{k}^{(2)} | u_{0 j} 〉}_{0} & \equiv & \int_{V} d r {\nabla \times [\frac{1}{ε (r)} i k \times u_{0 j} (r)]} \cdot u_{0 i}^{*} (r) \\ = & \int_{S_{0}} d S {[\frac{1}{ε (r)} i k \times u_{0 j} (r)] \times u_{0 i}^{*} (r)}_{n} \\ + \int_{V_{0}} d r [\frac{1}{ε (r)} i k \times u_{0 j} (r)] \cdot [\nabla \times u_{0 i}^{*} (r)], \end{array}

where S₀ denotes the surface of V₀ and the first integral on the right-hand side is the surface integral of the normal component of the integrand. This surface integral is equal to zero because of the periodicity of ε and u. By the same transformation as in Eq. (15), we finally obtain

{〈 u_{0 i} | Δ ℒ_{k}^{(2)} | u_{0 j} 〉}_{0} = i k \cdot P_{j i}^{*} .

So, we have

C_{i j}^{(k)} = i k \cdot (- P_{i j} + P_{j i}^{*}) .

Thirdly, we can prove [15]

ℛ (\nabla \times) ℛ^{- 1} = detR (\nabla \times) .

By a similar calculation, we can also prove

ℛ (\times) ℛ^{- 1} = detR (\times) .

Next, by introducing five pairs of ℛ⁻¹ℛ (= identity operator) to the definition of P_ij, we obtain

\begin{array}{l} k \cdot P_{i j} & = & \frac{k}{V_{0}} \cdot \int_{V_{0}} d r ℛ^{- 1} ℛ u_{0 i}^{*} (r) ℛ^{- 1} ℛ \times ℛ^{- 1} ℛ [\frac{1}{ε (r)} ℛ^{- 1} ℛ \nabla \times ℛ^{- 1} ℛ u_{0 j} (r)] \\ = & \frac{{(detR)}^{2}}{V_{0}} k \cdot \int_{V_{0}} d r ℛ^{- 1} [ℛ u_{0 i}^{*}] (r) \times {\frac{1}{ε (r)} \nabla \times [ℛ u_{0 j}] (r)} \\ = & \frac{(R k)}{V_{0}} \cdot \int_{V_{0}} d r^{'} [ℛ u_{0 i}^{*}] (r^{'}) \times {\frac{1}{ε {(r)}^{'}} \nabla^{'} \times [ℛ u_{0 j}] (r^{'})} \\ = & (R k) \cdot {〈 ℛ u_{0 i} | Δ ℒ | ℛ u_{0 j} 〉}_{0}, \end{array}

where the prime denotes r′ = Rr and by definition,

[ℛ u_{0 i}] (r) \equiv R u_{0 i} (R^{- 1} r) .

In Eq. (22), we used the facts that ε is invariant by ℛ and that R does not change the size of the volume element, since it is an orthogonal matrix.

So, unless the product of k, u_0i, and u_0j contains a term invariant for all ℛ ∈ 𝒢, k · P_ij, and consequently, $C_{i j}^{(k)}$ vanish. In this case, the linear term is absent in the dispersion curve, so Dirac cones do not exist. Therefore, the presence of an invariant term in the above-mentioned product is a distinct necessary condition for the Dirac cone.

Because both u_0i and u_0j are eigen functions of ℒ₀, which commutes with ^∀ℛ ∈ 𝒢, they are irreducible representations of group 𝒢, so their transformation by ℛ is well known. The transformation of vector k by ℛ can also be found easily. Thus, we can select the candidates of combinations of mode symmetries that enable the creation of Dirac cones by using this necessary condition. The exact shapes of dispersion curves can also be clarified by examining the structure of C_k. Some examples will be given in next four sections.

3. One-dimensional lattice of C_2v symmetry

We start with the one-dimensional lattice of C_2v symmetry. An example of such structures was previously analyzed by tight-binding approximation in Ref. [8]. There are four one-dimensional representations (A₁, A₂, B₁, B₂) for the C_2v point group [16]. As can be verified easily, vector k, which should be regarded as a one-dimensional vector (or scalar), has the B₁ symmetry. We can examine by the well-known reduction procedure [16] whether the product of k, u₀_i, and u_0j contains an invariant term, or the totally symmetric A₁ representation. As is summarized in Table 1, { $C_{i j}^{(k)}$ } is non-zero only for the combinations of (A₁, B₁) modes and (A₂, B₂) modes.

Table 1. Types of dispersion curves generated by accidental degeneracy of two modes (Mode 1 and Mode 2) for one-dimensional lattices of the C_2v symmetry. Symmetries of the magnetic field of the two modes on the Γ point are given in the left column. The middle column shows whether { $C_{i j}^{(k)}$ } vanishes by symmetry (×) or it can be non-zero (○). Shapes of the dispersion curves are given in the right column, where D and Q denote Dirac cone and quadratic dispersion surface, respectively.

View Table | View all tables in this article

By examining all symmetry transformations ℛ ∈ C_2v, we can prove that matrix C_k has the following form for the combination of the A₁ (u₀₁) and B₁ (u₀₂) modes.

C_{k} = (\begin{matrix} 0, & b k \\ b^{*} k, & 0 \end{matrix}) .

In Eq. (24),

b = i e \cdot [- {〈 u_{01} | Δ ℒ | u_{02} 〉}_{0} + {〈 u_{02} | Δ ℒ | u_{01} 〉}_{0}^{*}],

where e is a unit vector in the direction of the one-dimensional alignment of unit cells. Then, the secular equation to determine the first-order correction, Δλ, to the eigenvalue of Eq. (1) is given by

| \begin{matrix} - Δ λ, & b k \\ b^{*} k, & - Δ λ \end{matrix} | = 0 .

Thus, its solutions are given by

Δ λ = \pm | b | k,

which leads to the following dispersion relation in the vicinity of the Γ point:

ω_{k} \approx ω_{0} \pm \frac{| b | c^{2} k}{2 ω_{0}} .

So, there is a one-dimensional Dirac cone. This result is consistent with Ref. [8], in which the creation of the linear dispersion was verified for the accidental degeneracy of an A₁ mode and a B₁ mode by tight-binding approximation based on the resonant states of unit cells of periodic metamaterials. For the combination of the A₂ and B₂ modes, we have the same relations as Eq. (24), so there is also a Dirac cone.

Let us make a remark here. If two modes have the same symmetry, they are mixed and their dispersion curves repel each other when their eigen frequencies are close. So, accidental degeneracy does not take place in this case. Nevertheless, this case is included in Table 1 and other tables in the following sections to show that their dispersion curves are quadratic for small k.

4. Square lattice of C_4v symmetry

Next, we examine the two-dimensional square lattice of C_4v symmetry. An example of such structures was previously analyzed by tight-binding approximation in Ref. [10]. There are four one-dimensional representations (A₁, A₂, B₁, B₂) and one two-dimensional representation (E) [16]. We can verify easily that vector k, which should be regarded as a two-dimensional vector for this case, has the E symmetry. The nature of $C_{i j}^{(k)}$ is summarized in Table 2. It is non-zero for the combinations of an A₁, A₂, B₁, or B₂ mode and an E mode.

Table 2. Types of dispersion curves generated by accidental degeneracy of two modes (Mode 1 and Mode 2) for square lattices of the C₄_v symmetry. Symmetries of the magnetic field of the two modes on the Γ point are given in the left column. The middle column shows whether { $C_{i j}^{(k)}$ } vanishes by symmetry (×) or it can be non-zero (○). Shapes of the dispersion curves are given in the right column, where D and Q denote Dirac cone and quadratic dispersion surface, respectively.

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Let us first examine the case of the combination of an E mode and an A₁ mode. From group theory [16], we can assume without loss of generality that one of the two eigen functions of the E mode is transformed like the x coordinate and the other like the y coordinate by the symmetry operation ℛ ∈ C₄_v. We denote the first and second functions by u₀₁ and u₀₂, respectively. The eigen function of the A₁ mode is denoted by u₀₃. By examining all transformation ℛ ∈ C_4v, we can prove that matrix C_k has the following form:

C_{k} = (\begin{matrix} 0 & 0 & b k_{x} \\ 0 & 0 & b k_{y} \\ b^{*} k_{x} & b^{*} k_{y} & 0 \end{matrix}) .

The secular equation for the first-order correction is given by

| \begin{matrix} - Δ λ, & 0, & b k_{x} \\ 0, & - Δ λ, & b k_{y} \\ b^{*} k_{x}, & b^{*} k_{y}, & - Δ λ, \end{matrix} | = 0,

whose solutions are

Δ λ = 0, \pm | b | k,

where

k = \sqrt{k_{x}^{2} + k_{y}^{2}}

. Therefore, two of the three dispersion curves compose an isotropic Dirac cone. For the third mode, the linear term is absent, so it gives a quadratic dispersion. This result agrees with Ref. [10], in which the presence of the Dirac cone was shown by the tight-binding approximation. When we compare the calculation given in Ref. [10] with the present one, the matrix that we have to diagonalize is much simpler in the present study due to the perturbative nature of the calculation, so the diagonalization is much easier.

For the combinations of (E, A₂), (E, B₁), and (E, B₂) modes, we can prove that C_k has the following forms, respectively:

\begin{array}{l} (E, A_{2}) & : & C_{k} = (\begin{matrix} 0 & 0 & b k_{y} \\ 0 & 0 & - b k_{x} \\ b^{*} k_{y} & - b^{*} k_{x} & 0 \end{matrix}), \\ (E, B_{1}) & : & C_{k} = (\begin{matrix} 0 & 0 & b k_{x} \\ 0 & 0 & - b k_{y} \\ b^{*} k_{x} & - b^{*} k_{y} & 0 \end{matrix}), \\ (E, B_{2}) & : & C_{k} = (\begin{matrix} 0 & 0 & b k_{y} \\ 0 & 0 & b k_{x} \\ b^{*} k_{y} & b^{*} k_{x} & 0 \end{matrix}) . \end{array}

For all these cases, the solutions are given by the same equation as Eq. (31). So, there is an isotropic Dirac cone and a quadratic dispersion surface. These conclusions are consistent with Ref. [12] in which the shapes of dispersion curves were analyzed by tight-binding approximation for metamaterials and by numerical photonic-band calculation for photonic crystals.

5. Triangular lattice of C_6v symmetry

In this section, we examine the two-dimensional triangular lattice of C_6v symmetry. There are four one-dimensional representations (A₁, A₂, B₁, B₂) and two two-dimensional representations (E₁, E₂) [16]. We can verify easily that vector k has the E₁ symmetry. The nature of $C_{i j}^{(k)}$ is summarized in Table 3.

Table 3. Types of dispersion curves generated by accidental degeneracy of two modes for the triangular lattice of the C_6v symmetry. Shapes of the dispersion curves are given in the right column, where D, DD and Q denote Dirac cone, double Dirac cones, and quadratic dispersion surface, respectively.

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Let us examine the shapes of the dispersion curves for the cases of non-zero { $C_{i j}^{(k)}$ }. For the combination of an E₁ mode and an E₂ mode, we can prove by examining all transformations ℛ ∈ C_6v that matrix C_k has the following form:

C_{k} = (\begin{matrix} 0 & 0 & - b k_{y} & - b k_{x} \\ 0 & 0 & - b k_{x} & b k_{y} \\ - b^{*} k_{y} & - b^{*} k_{x} & 0 & 0 \\ - b^{*} k_{x} & b^{*} k_{y} & 0 & 0 \end{matrix}),

where we assumed without loss of generality that the two eigen functions of the E₁ mode (u₀₁, u₀₂) are transformed like x and y and those of the E₂ mode (u₀₃, u₀₄) are transformed like 2xy and x² − y² [16]. The secular equation for the first-order correction can be solved easily and its solutions are given by

Δ λ = \pm | b | k (double roots) .

So, we have isotropic double Dirac cones with the same slope. This conclusion agrees with the result of the tight-binding calculation given in Ref. [11]. The matrix that has to be diagonalized is again much simpler in the present study due to the perturbative nature of the calculation, so the diagonalization is much easier compared with that in Ref. [11].

For the (E₁, A₁) and (E₂, B₂) combinations, matrix C_k has the following form:

C_{k} = (\begin{matrix} 0 & 0 & b k_{x} \\ 0 & 0 & b k_{y} \\ b^{*} k_{x} & b^{*} k_{y} & 0 \end{matrix}) .

On the other hand, for the (E₁, A₂) and (E₂, B₁) combinations,

C_{k} = (\begin{matrix} 0 & 0 & b k_{y} \\ 0 & 0 & - b k_{x} \\ b^{*} k_{y} & - b^{*} k_{x} & 0 \end{matrix}) .

For these cases, the solutions of the secular equation are the same and given by

Δ λ = 0, \pm | b | k .

So, there are an isotropic Dirac cone and a quadratic dispersion surface. These conclusions are consistent with Ref. [12] in which dispersion curves were analyzed by tight-binding approximation and numerical photonic-band calculations.

6. Simple-cubic lattice of O_h symmetry

Finally, we examine the simple-cubic lattice of the O_h symmetry. There are four one-dimensional representations (A_1g, A_1u, A_2g, A_2u), two two-dimensional representations (E_g, E_u), and four three-dimensional representations (T_1g, T_1u, T_2g, T_2u) [16]. We can verify that vector k has the T_1u symmetry. The nature of $C_{i j}^{(k)}$ is summarized in Table 4.

Table 4. Types of dispersion curves generated by accidental degeneracy of two modes for the simple-cubic lattice of the O_h symmetry. Symmetries of the magnetic field of the two modes on the Γ point are given in the left column. The middle column shows whether { $C_{i j}^{(k)}$ } vanishes by symmetry (×) or it can be non-zero (○). Shapes of the dispersion curves are given in the right column, where D, DD, and Q denote Dirac cone, double Dirac cones, and quadratic dispersion surface, respectively.

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Let us start by examining the shapes of the dispersion curves for the combination of a T_1u mode and an A_1g mode [10]. From group theory, we can assume without loss of generality that three eigen functions of the T_1u mode are transformed like the x, y, and z coordinates [16], which we denote by u₀₁, u₀₂, and u₀₃, respectively. The eigen function of the A_1g mode is denoted by u₀₄. We can prove by examining all transformations ℛ ∈ O_h that matrix C_k has the following form:

C_{k} = (\begin{matrix} 0 & 0 & 0 & b k_{x} \\ 0 & 0 & 0 & b k_{y} \\ 0 & 0 & 0 & b k_{z} \\ b^{*} k_{x} & b^{*} k_{y} & b^{*} k_{z} & 0 \end{matrix}) .

Because matrix C_k is sufficiently sparse, we can easily diagonalize it to obtain the first-order correction to the eigenvalues. The result is

Δ λ = {\begin{matrix} 0 & (double roots), \\ \pm | b | k, \end{matrix}

where

k = \sqrt{k_{x}^{2} + k_{y}^{2} + k_{z}^{2}}

. So, there are an isotropic Dirac cone and two quadratic dispersion surfaces. This result agrees with our previous calculation on periodic metamaterials by tight-binding approximation [10]. For the combinations of (T_1g,A_1u), (T_2u,A_2g), and (T_2g,A_2u), we can also prove that C_k has the same form as in Eq. (38). So, all these combinations yield an isotropic Dirac cone and two quadratic dispersion surfaces.

Next, we examine the combination of a T_1u mode (u₀₁, u₀₂, u₀₃) and an E_g mode (u₀₄, u₀₅). We can assume that two eigen functions of the E_g mode are transformed like 2z² − x² − y² and $\sqrt{3} (x^{2} - y^{2})$ [16]. We can prove easily that matrix C_k has the following form:

C_{k} = (\begin{matrix} 0 & 0 & 0 & b k_{x} & - \sqrt{3} b k_{x} \\ 0 & 0 & 0 & b k_{y} & \sqrt{3} b k_{y} \\ 0 & 0 & 0 & - 2 b k_{z} & 0 \\ b^{*} k_{x} & b^{*} k_{y} & - 2 b^{*} k_{z} & 0 & 0 \\ - \sqrt{3} b^{*} k_{x} & \sqrt{3} b^{*} k_{y} & 0 & 0 & 0 \end{matrix})

Then, the secular equation is

Δ λ [{(Δ λ)}^{4} - 4 {| b |}^{2} k^{2} {(Δ λ)}^{2} + 12 (k_{x}^{2} k_{y}^{2} + k_{y}^{2} k_{z}^{2} + k_{z}^{2} k_{x}^{2}) {| b |}^{4}] = 0 .

When we express the solutions by the spherical coordinates, (k, θ, ϕ),

Δ λ = {\begin{array}{l} \begin{array}{l} 0, \\ \pm \sqrt{2} | b | k \sqrt{1 \pm \sqrt{1 - 3 F (θ, ϕ)}}, \end{array} \end{array}

where

F (θ, ϕ) = \frac{{sin}^{4} θ {sin}^{2} 2 ϕ + {sin}^{2} 2 θ}{4} .

Thus, there are two anisotropic Dirac cones and a quadratic dispersion surface unless F(θ, ϕ) ≠ 0. When F(θ, ϕ) = 0, which happens for k in the (1,0,0) direction and its equivalent directions, two of the four k-linear eigenvalues vanish, so the corresponding dispersion curves are quadratic in k. For the combination of (T_1g,E_u), we can prove that C_k has the same structure as Eq. (40), so it also yields two anisotropic Dirac cones and a quadratic dispersion surface.

For the combinations of (T_2u,E_g) and (T_2g,E_u), C_k has the following form:

C_{k} = (\begin{matrix} 0 & 0 & 0 & \sqrt{3} b k_{x} & b k_{x} \\ 0 & 0 & 0 & - \sqrt{3} b k_{y} & b k_{y} \\ 0 & 0 & 0 & 0 & - 2 b k_{z} \\ \sqrt{3} b^{*} k_{x} & - \sqrt{3} b^{*} k_{y} & 0 & 0 & 0 \\ b^{*} k_{x} & b^{*} k_{y} & - 2 b^{*} k_{z} & 0 & 0 \end{matrix})

This C_k leads to exactly the same secular equation as Eq. (41). So, the combinations of (T_2u,E_g) and (T_2g,E_u) also yield two anisotropic Dirac cones and a quadratic dispersion surface.

Finally, let us examine the case of two triply degenerate modes. For the combination of a T_1g mode (u₀₁, u₀₂, u₀₃) and a T_1u mode (u₀₄, u₀₅, u₀₆), C_k has the following structure:

C_{k} = (\begin{matrix} 0 & 0 & 0 & 0 & b k_{z} & - b k_{y} \\ 0 & 0 & 0 & - b k_{z} & 0 & b k_{x} \\ 0 & 0 & 0 & b k_{y} & - b k_{x} & 0 \\ 0 & - b^{*} k_{z} & b^{*} k_{y} & 0 & 0 & 0 \\ b^{*} k_{z} & 0 & - b^{*} k_{x} & 0 & 0 & 0 \\ - b^{*} k_{y} & b^{*} k_{x} & 0 & 0 & 0 & 0 \end{matrix})

which is sufficiently sparse so that we can easily obtain its analytical solutions:

Δ λ = {\begin{matrix} 0 & (double roots), \\ \pm | b | k & (double roots) . \end{matrix}

So, there are isotropic double Dirac cones with the same slope and two quadratic dispersion surfaces. We can prove that the combination of T_2g and T_2u modes has C_k of the same structure, so this combination also yields double Dirac cones and two quadratic dispersion surfaces.

For the combinations of (T_1g, T_2u) and (T_2g, T_1u), C_k has the following form:

C_{k} = (\begin{matrix} 0 & 0 & 0 & 0 & b k_{z} & b k_{y} \\ 0 & 0 & 0 & b k_{z} & 0 & b k_{x} \\ 0 & 0 & 0 & b k_{y} & b k_{x} & 0 \\ 0 & b^{*} k_{z} & b^{*} k_{y} & 0 & 0 & 0 \\ b^{*} k_{z} & 0 & b^{*} k_{x} & 0 & 0 & 0 \\ b^{*} k_{y} & b^{*} k_{x} & 0 & 0 & 0 & 0 \end{matrix}),

which leads to the following secular equation for the first-order correction.

{(Δ λ)}^{3} - {| b |}^{2} k^{2} Δ λ \pm 2 {| b |}^{3} k_{x} k_{y} k_{z} = 0 .

In general, three solutions of

ξ^{3} + p ξ + q = 0

are given by

ξ_{l} = e^{2 π i l / 3} \sqrt[3]{- \frac{q}{2} + \sqrt{{(\frac{q}{2})}^{2} + {(\frac{p}{3})}^{3}}} + e^{- 2 π i l / 3} \sqrt[3]{- \frac{q}{2} - \sqrt{{(\frac{q}{2})}^{2} + {(\frac{p}{3})}^{3}}},

where l = 1, 2, 3. So, six solutions of Eq. (48) are

Δ λ_{l \pm} = | b | k (e^{2 π i l / 3} F_{1 \pm} + e^{- 2 π i l / 3} F_{2 \pm}),

where

F_{1 \pm} = {[\pm \frac{sin θ sin 2 θ sin 2 ϕ}{4} + \sqrt{\frac{{sin}^{2} θ {sin}^{2} 2 θ {sin}^{2} 2 ϕ}{16} - \frac{1}{27}}]}^{\frac{1}{3}},

F_{2 \pm} = {[\pm \frac{sin θ sin 2 θ sin 2 ϕ}{4} - \sqrt{\frac{{sin}^{2} θ {sin}^{2} 2 θ {sin}^{2} 2 ϕ}{16} - \frac{1}{27}}]}^{\frac{1}{3}} .

So, there are three anisotropic Dirac cones.

To check these analytical results, we performed photonic-band calculations by the plane-wave expansion method [15]. We assumed a regular simple-cubic lattice of dielectric spheres with a dielectric constant of 12.6 (GaAs). By changing the radius of the spheres, we found several cases of accidental degeneracy among the lowest 20 dispersion curves, three of which are shown in Fig. 1, which agree with the results of analytical calculations listed in Table 4. The rest of the cases also agreed with Table 4.

Fig. 1 Dispersion curves of a photonic crystal composed of the simple-cubic lattice of dielectric spheres with a dielectric constant of 12.6. The vertical axis is the normalized frequency (ωa/2πc), and the horizontal axis is the wave vector in the R (= (π/a, π/a, π/a)), and X (= (π/a, 0, 0)) directions, where R/6, for example, means that the horizontal axis is magnified by six times. The radius of the spheres is 0.160a for (a), 0.210a for (b), and 0.416a for (c). The number associated with each dispersion curve is the multiplicity of the mode.

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Let us make six remarks. First, in this paper, we formulated a degenerate perturbation theory for the vector electromagnetic field of periodic structures and applied it to the problem of the creation of Dirac cones by accidental degeneracy of two modes. We obtained an efficient necessary condition (non-zero C_k) by which we can select candidates of mode combinations that enable the creation of single and double Dirac cones. In addition, by examining the transformation of C_k by symmetry operations for four periodic systems of different spatial symmetries, we could determine the structure of C_k and obtain the shapes of dispersion curves in the vicinity of the zone center by purely analytical calculations. All these findings clearly show that the presence or absence of the Dirac cone by accidental degeneracy does not depend on the details of the periodic structure but is solely determined by the spatial symmetry of the two modes.

Second, the present theory can of course be applied to periodic structures of other spatial symmetries. We can examine C_k and obtain the shapes of dispersion curves in quite a similar manner. We should also note that if the periodic structure does not have a spatial symmetry, any two modes are mixed and repel each other when their eigen frequencies are close, so they do not create Dirac cones.

Third, the secular equations obtained by the present theory are generally much simpler than those obtained by tight-binding approximation in Refs. [8], [10], [11], and [12] due to the perturbative nature of the calculations. As a result, we could easily solve equations up to the sixth degree to obtain the first-order corrections to the eigen values.

Fourth, we assumed that the dielectric constant of the periodic structure does not depend on frequency in order to make our problem well-defined. When we have to deal with dispersive materials with frequency-dependent dielectric constants, we can use the value at the degenerate frequency. Thus, we can obtain qualitatively correct solutions in the vicinity of the degenerate frequency.

Fifth, although the numerical confirmation was performed for photonic crystals because of the ease of calculation, the present theory is equally applicable to periodic metamaterials with well-defined resonant states localized in unit cells.

Finally, the present theory may be regarded as an extension of the k · p theory in semiconductor physics [17] to the vector electromagnetic field. In the usual k · p theory, the first-order term is absent and an effective mass is derived by the second-order perturbation, so the energy bands are quadratic in the Brillouin-zone center. However, in the present theory, the first-order term can be non-zero because of the accidental degeneracy of two modes, which resulted in the creation of Dirac cones.

7. Conclusion

We formulated a degenerate perturbation theory for the vector electromagnetic field of periodic structures like photonic crystals and metamaterials and applied it to the problem of the creation of Dirac cones in the center of the Brillouin zone by accidental degeneracy of two modes. We derived an efficient necessary condition for the Dirac cone by which we can easily select candidates of mode combinations that enable the creation of the Dirac cones. We can analyze the structure of a matrix (C_k) that determines the first-order correction to eigen frequencies by examining its transformation by symmetry operations. Thus, we can obtain the analytical solutions of the dispersion curves in the vicinity of the zone center and judge the presence of the Dirac cone.

This method was applied to four periodic systems, that is, the one-dimensional lattice of C_2v symmetry, the square lattice of C_4v symmetry, the triangular lattice of C_6v symmetry, and the simple-cubic lattice of O_h symmetry. We succeeded in examining all possible combinations of mode symmetries and obtaining the analytical solution of dispersion curves for each case. Thus, we fully clarified the conditions required to obtain Dirac cones and double Dirac cones in these four systems. These results are consistent with our previous calculations for periodic metamaterials by tight-binding approximation and for photonic crystals by plane-wave expansion.

All these findings clearly show that the presence or absence of the Dirac cone in the zone center due to accidental degeneracy of two modes does not depend on the details of the periodic structure that we analyze but is solely determined by the spatial symmetry of the two modes. Thus, we succeeded in proving the universality of mode symmetries in creating photonic Dirac cones, which was anticipated in our recent study [12].

Acknowledgments

This study was supported by a Grant-in-Aid for Scientific Research on Innovative Areas from the Japanese Ministry of Education, Culture, Sports and Technology (Grant number 22109007).

References and links

1. F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008). [CrossRef] [PubMed]

2. S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008). [CrossRef]

3. T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80, 155103 (2009). [CrossRef]

4. X. Zhang, “Observing zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100, 113903 (2008). [CrossRef] [PubMed]

5. R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75, 063813 (2007). [CrossRef]

6. M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405, 2990–2995 (2010). [CrossRef]

7. X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nature Mater. 10, 582–586 (2011). [CrossRef]

8. K. Sakoda and H-F. Zhou, “Role of structural electromagnetic resonances in a steerable left-handed antenna,” Opt. Express 18, 27371–27386 (2010). [CrossRef]

9. K. Sakoda and H.-F. Zhou, “Analytical study of two-dimensional degenerate metamaterial antennas,” Opt. Express 19, 13899–13921 (2011). [CrossRef] [PubMed]

10. K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express 20, 3898–3917 (2012). [CrossRef] [PubMed]

11. K. Sakoda, “Double Dirac cones in triangular-lattice metamaterials,” Opt. Express 20, 9925–9939 (2012). [CrossRef] [PubMed]

12. K. Sakoda, “Universality of mode symmetries in creating photonic Dirac cones,” J. Opt. Soc. Am. B 29, 2770–2778 (2012). [CrossRef]

13. M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. 97, 157403 (2006). [CrossRef] [PubMed]

14. A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007). [CrossRef]

15. K. Sakoda, Optical Properties of Photonic Crystals, 2nd Ed. (Springer-Verlag, Berlin, 2004).

16. T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer, Berlin, 1990). [CrossRef]

17. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 5th Edition (World Scientific, Singapore, 2009). [CrossRef]

Proof of the universality of mode symmetries in creating photonic Dirac cones

Abstract

1. Introduction

2. Theory

3. One-dimensional lattice of C_2v symmetry

4. Square lattice of C_4v symmetry

5. Triangular lattice of C_6v symmetry

6. Simple-cubic lattice of O_h symmetry

7. Conclusion

Acknowledgments

References and links

Cited By

Figures (1)

Tables (4)

Equations (53)

Optics Express

Mode 1	Mode 2	$C_{i j}^{(k)}$	Dispersion

A₁	B₁	○	D
A₂	B₂	○	D
A₁	A₁, A₂, B₂	×	2Q
A₂	A₂, B₁	×	2Q
B₁	B₁, B₂	×	2Q
A₂	A₂	×	2Q

Mode 1	Mode 2	$C_{i j}^{(k)}$	Dispersion

E	A₁, A₂, B₁, B₂	○	D + Q
A₁, A₂, B₁, B₂	A₁, A₂, B₁, B₂	×	2Q
E	E	×	4Q

Mode 1	Mode 2	$C_{i j}^{(k)}$	Dispersion

E₁	E₂	○	DD
E₁	E₁	×	4Q
E₂	E₂	×	4Q
E₁	A₁, A₂	○	D + Q
E₂	B₁, B₂	○	D + Q
E₂	A₁, A₂	×	3Q
E₁	B₁, B₂	×	3Q
A₁, A₂, B₁, B₂	A₁, A₂, B₁, B₂	×	2Q

Mode 1	Mode 2	$C_{i j}^{(k)}$	Dispersion

T_1g	T_1u	○	DD + 2Q
T_2g	T_2u	○	DD + 2Q
T_1g	T_2u	○	3D
T_2g	T_1u	○	3D
T_1g	T_1g, T_2g	×	6Q
T_2g	T_2g	×	6Q
T_1u	T_1u, T_2u	×	6Q
T_2u	T_2u	×	6Q
T_1g, T_2g	E_u	○	2D + Q
T_1u, T_2u	E_g	○	2D + Q
T_1g, T_2g	E_g	×	5Q
T_1u, T_2u	E_u	×	5Q
E_g	E_u	×	4Q
T_1u	A_1g	○	D + 2Q
T_1g	A_1u	○	D + 2Q
T_2u	A_2g	○	D + 2Q
T_2g	A_2u	○	D + 2Q
T_1u	A_1u, A_2g, A_2u	×	4Q
T_1g	A_1g, A_2g, A_2u	×	4Q
T_2u	A_1g, A_1u, A_2u	×	4Q
T_2g	A_1g, A_1u, A_2g	×	4Q
E_g, E_u	A_1g, A_1u, A_2g, A_2u	×	3Q
A_1g, A_1u, A_2g, A_2u	A_1g, A_1u, A_2g, A_2u	×	2Q

Abstract

1. Introduction

2. Theory

3. One-dimensional lattice of C2v symmetry

4. Square lattice of C4v symmetry

5. Triangular lattice of C6v symmetry

6. Simple-cubic lattice of Oh symmetry

7. Conclusion

Acknowledgments

References and links

Cited By

Figures (1)

Tables (4)

Equations (53)

Optics Express

3. One-dimensional lattice of C_2v symmetry

4. Square lattice of C_4v symmetry

5. Triangular lattice of C_6v symmetry

6. Simple-cubic lattice of O_h symmetry