Abstract We show a methodology for how to constructDirac points that occur at the corners of Brillouin zone as thePhotonic counterparts of graphene. We use a triangular lat-tice with circular holes on a silicon substrate to create a Cou-pled Photonic Crystal Resonator Array (CPCRA) which itscavity resonators play the role of carbon atoms in graphene.At first we draw the band structure of our CPCRA usingthe tight-binding method. For this purpose we first designeda cavity which its resonant frequency is approximately atthe middle of the first H-polarization band gap of the ba-sis triangular lattice. Then we obtained dipole modes andmagnetic field distribution of this cavity using the Finite El-ement Method (FEM). Finally we drew the two bands thatconstruct the Dirac points together with the frequency con-tour plots for both bands and compared with the Plane WaveExpansion (PWE) and FEM results to prove the existence ofDirac point in the H-polarization band structure of latticeswith air holes.
1 Introduction
Since the discovery of graphene in 2004 by K. Novoselovand A. Geim a lot of researchers have found many inter-esting applications and specifications for this material. Themain reason for these specs is its exclusive band structurein which two bands coincide at a point in the corners of theBrillouin zone called Dirac point. These two bands are lin-ear near this point, thus the electron dispersion can be de-scribed by the Dirac equation in this region. Graphene is
M.H. Aram · R. Mohajeri · S. Khorasani (B)Department of Electrical Engineering,Sharif University of Technology, Tehran, Irane-mail: [email protected]
a good material to visualize some relativistic behavior ofelectrons such as quantum hall effect, Zitterbewegung, andKlein paradox. But because of complex interaction and lim-ited scale, some of these phenomena cannot be observed di-rectly [1].
Recently efforts have been made to construct Dirac pointsin the band structure of optical materials [2–8] to exam-ine the existence of mentioned phenomena for photons ofelectromagnetic fields. Since we have great control on thedesign of Photonic Crystals (PC), it is simpler to observethese phenomena in them than in graphene. Some of the pre-dicted or examined characteristics of PCs with Dirac pointin their band structure are inverse proportionality of photonconductance to the length of crystal [9–11], extinction of co-herent backscattering [12], Zitterbewegung phenomenon forphotons near the Dirac point [1], and one-way light trans-port [13]. Some novel applications for this kind of crystalshave been proposed as well; among them we can mentionconstruction of photonic topological insulators [14] and di-rectional optical waveguide in PCs [15].
In this paper we first explain the design of specified su-percrystal and then its band structure calculation to show theDirac point in the H-polarization mode.
2 Crystal design
In order to construct a crystal which is similar to graphenewe should use a triangular lattice as our basis. Also sincewe want to use the tight-binding method to obtain the bandstructure, we have to create cavities which in addition toform a triangular lattice should have confined electromag-netic field and be separated sufficiently. The triangular ba-sis lattice was supposed to be made of a silicon substrate
with circular air holes that their radius is 0.35 of lattice con-stant, a. The distance between cavities was set to be twotimes the lattice constant. The cavities were arranged so thatevery cavity has six cavities in its neighborhood. The de-signed CPCRA is shown in Fig. 1.
As said above the field of the cavities should be con-fined to be able to use the tight-binding method. To con-fine the field we have to design the cavity so that its reso-nance frequency falls inside a band gap of the basis lattice.We know a triangular lattice with circular air holes has awide gap between its first and second band in H-polarizationmode. By changing the cavity radius we brought its reso-nance frequency near the middle of the band gap. Of coursea cavity has different resonance frequencies depending on itsfield distribution but since we need two bands to form Diraccones it seems we should consider a field distribution thatis degenerate. So we considered only dipole field. Figure 2shows the average amplitude of z component of dipole mag-netic field inside cavities with different radii. The magentaband in the figure is the first gap of our basis lattice. Thepeaks of curves indicate the resonance frequency of eachcavity. It can be seen that the resonance frequency of thecavity with radius 0.2a is approximately in the middle ofthe gap. So we chose this radius for the cavities of our su-percrystal.
Fig. 1 Supercrystal designed in this project. Green hexagons are unitcells of this crystal. �a1 and �a2 are primitive vectors
Fig. 2 Average amplitude of z component of dipole magnetic fieldinside cavities with different radii
3 Symmetry analysis
Looking at the structure of the supercrystal in Fig. 1 it is ob-vious that it has C6v symmetry. The operators of this sym-metry group are as follows:
C6v ={
(E), (C6, C−16 ), (C3, C
−13 ),
(C2), (σx, σ′x, σ
′′x ), (σy, σ
′y, σ
′′y )
}(1)
Three highly symmetric points, the Γ point, (0,0), the Kpoint, (4π/3a,0), and the M point, (π/a,−π/
√3a) in the
reciprocal lattice have C6v , C3v , and C2v group symmetry,respectively. The T points (between Γ and K), Σ points (be-tween Γ and M), and Λ points (between K and M) have C1h
group symmetry. The compatibility relations between adja-cent points in the reciprocal lattice is summarized in Table 1.
To find modes of bands in the band structure we have tofind irreducible representations of highly symmetric pointsin reciprocal lattice. The number of a specific mode, A, ata point in the reciprocal lattice can be calculated using thefollowing formula [16]:
∑R∈G N
R.XA(R)
number of elements in G(2)
where G is the group symmetry of the point, NR
is the num-ber of invariant points of the specified type when the oper-ator R is applied to the lattice, and XA(R) is the characterof R for mode A in that point. To find N
Rwe should deter-
mine the exact location of each point in the reciprocal lattice.This is done in Fig. 3. Using this figure modes of each pointis listed in Table 2.
Table 1 Compatibility relations in the hexagonal lattice
T Λ Σ
Γ : A1 A − A
A2 B − B
B1 A − B
B2 B − A
C1,C2 A,B − A,B
K: A1 A A −A2 B B −C A,B A,B −
M: A1 − A A
A2 − B B
B1 − B A
B2 − A B
Construction of Dirac points using triangular supercrystals
Fig. 3 Position of differentpoints in the reciprocal lattice
Table 2 Modes of points of reciprocal lattice
Symmetry Point Irreducible representations
C6v Γ 1 A1
Γ 2 A1, B2,C1, C2
Γ 3 A1, B1, C1, C2
Γ 4 A1, B2, C1, C2
C3v K1 A1, C
K2 A1, C
K3 A1, A2, 2C
K4 A1, A2, 2C
C2v M1 A1, B1
M2 A1, B2
M3 A1, A2, B1, B2
M4 A1, B1
4 Band structure calculation
In this section we calculate the dispersion relation of thetwo bands which create the Dirac cone in the band structureof the designed supercrystal using the tight-binding methodand then verify these two bands with the band structure ob-tained by PWE and FE method. The results are in goodagreement with each other.
4.1 Tight-binding method
It was mentioned that a triangular lattice with air holes has awide gap between its first and second band in H-polarizationmode. So we are interested in the H-polarization bands ofthe designed CPCRA. In this mode cavities have confined
resonant field and tight binding is operative. The cavity fielddistribution which is studied here is double degenerate hencethe degenerate tight-binding method is applied. If we showeach of the electric field distribution of a cavity in its reso-nant frequency, ν, by �Eν,l (l = 1,2), then according to thefirst and the second Maxwell equations we have
∇ × ∇ × �Eν,l(�r) = εr(�r)ν2
c2�Eν,l(�r) (3)
where εr(�r) is the profile of the relative permittivity of asingle cavity and c is the speed of light in vacuum. For H-polarization mode, electric field �Eν,l , has two componentsin x and y directions. Because of the orthogonality of eigen-modes of a degenerate field distribution we can write∫
According to the Bloch theorem the electric field distribu-tion in the entire CPCRA lattice is obtained from the fol-lowing equation:
E�κ(�r) =+∞∑
n,m=−∞
2∑l=1
e−i(n�κ·�a1+m�κ·�a2)
× bl�Eν,l(r − n�a1 − m�a2) (5)
where �a1 and �a2 are primitive vectors of the CPCRA thatare shown in Fig. 1, �κ is the Bloch wave vector, and bl is aconstant. Similar to Eq. (3) we can write
∇ × ∇ × E�κ = εr(�r)ω2
c2E�κ (6)
M.H. Aram et al.
for electric field of the entire lattice, where εr(�r) is the pro-file of relative permittivity of the whole CPCRA. By replac-ing E�κ from Eq. (5) into Eq. (6) we have
+∞∑n,m=−∞
2∑l=1
(e−i(n�κ·�a1+m�κ·�a2)bl
× ∇ × ∇ × �Eν,l(�r − n�a1 − m�a2))
= εr(�r)ω2
c2
+∞∑n,m=−∞
2∑l=1
(e−i(n�κ·�a1+m�κ·�a2)
× bl�Eν,l(�r − n�a1 − m�a2)
)(7)
Now if we put the value of ∇ ×∇ × �Eν,l(r − n�a1 −m�a2)
from Eq. (3) into Eq. (7) we obtain the following equation:
ν2+∞∑
n,m=−∞
2∑l=1
(e−i(n�κ·�a1+m�κ·�a2)bl
× εr(�r − n�a1 − m�a2) �Eν,l(�r − n�a1 − m�a2))
= ω2+∞∑
n,m=−∞
2∑l=1
(e−i(n�κ·�a1+m�κ·�a2)bl
× εr(�r) �Eν,l(�r − n�a1 − m�a2))
(8)
Taking the inner product of both sides of Eq. (8) by �E∗ν,k(r)
and then integration over the x–y plane results in
ν2+∞∑
n,m=−∞
2∑l=1
{e−i(n�κ·�a1+m�κ·�a2)bl
×∫ ( �E∗
ν,k(�r) · �Eν,l(�r − n�a1 − m�a2)
× εr(�r − n�a1 − m�a2))d2r
}
= ω2+∞∑
n,m=−∞
2∑l=1
{e−i(n�κ·�a1+m�κ·�a2)bl
×∫
�E∗ν,k(�r) · �Eν,l(�r − n�a1 − m�a2)εr (�r) d2r
}(9)
If we show the overlap integrals of electric field by
αk,ln,m =
∫ ( �E∗ν,k(�r) · �Eν,l(�r − n�a1 − m�a2)
× εr(�r − n�a1 − m�a2))d2r (10)
βk,ln,m =
∫ ( �E∗ν,k(�r) · �Eν,l(�r − n�a1 − m�a2)
× εr(�r))d2r (11)
we can write Eq. (9) in an abstract form:
ν2+∞∑
n,m=−∞
2∑l=1
ble−i(n�κ·�a1+m�κ·�a2)αk,l
n,m
= ω2+∞∑
n,m=−∞
2∑l=1
ble−i(n�κ·�a1+m�κ·�a2)βk,l
n,m (12)
It is seen from Eqs. (10) and (11) that αk,ln,m = (α
k,l−n,−m)∗ and
βk,ln,m = (β
k,l−n,−m)∗ on the other hand for centrosymmetric
permittivity profiles, αk,ln,m and β
k,ln,m are real valued [17]. We
can assume
αk,ln,m βk,l
n,m 0(2 ≤ |n|, |m|) (13)
for weak coupling conditions. Thus Eq. (12) can be writtenin an approximate form:
ν22∑
l=1
bl
(α
k,l0,0 + 2Re
{α
k,l1,0e
−i(�κ·�a1)
+ αk,l−1,1e
i(�κ·�a1−�κ·�a2) + αk,l0,1e
−i(�κ·�a2)})
ω2(�κ)
2∑l=1
bl
(β
k,l0,0 + 2Re
{β
k,l1,0e
−i(�κ·�a1)
+ βk,l−1,1e
i(�κ·�a1−�κ·�a2) + βk,l0,1e
−i(�κ·�a2)})
(14)
This equation can be simplified as
ν22∑
l=1
bl
(α
k,l0,0 + 2α
k,l1,0 cos(�κ · �a1)
+ 2αk,l−1,1 cos(�κ · �a1 − �κ · �a2) + 2α
k,l0,1 cos(�κ · �a2)
)
ω2(�κ)
2∑l=1
bl
(β
k,l0,0 + 2β
k,l1,0 cos(�κ · �a1)
+ 2βk,l−1,1 cos(�κ · �a1 − �κ · �a2) + 2β
k,l0,1 cos(�κ · �a2)
)(15)
We can write Eq. (15) in a compact matrix form as
ν2[A(�κ)
][b1
b2
]= ω2(�κ)
[B(�κ)
][b1
b2
](16)
where matrices [A(�κ)] and [B(�κ)] are equal to
[A(�κ)
] =[α
1,10,0 α
1,20,0
α2,10,0 α
2,20,0
]
+ 2 cos(�κ · �a1)
[α
1,11,0 α
1,21,0
α2,11,0 α
2,21,0
]
Construction of Dirac points using triangular supercrystals
Fig. 4 ω(�κ) variations in reciprocal lattice for the first band obtainedby the tight-binding method
+ 2 cos(�κ · �a2)
[α
1,10,1 α
1,20,1
α2,10,1 α
2,20,1
]
+ 2 cos(�κ · (�a1 − �a2)
)[α
1,1−1,1 α
1,2−1,1
α2,1−1,1 α
2,2−1,1
](17)
[B(�κ)
] =[β
1,10,0 β
1,20,0
β2,10,0 β
2,20,0
]
+ 2 cos(�κ · �a1)
[β
1,11,0 β
1,21,0
β2,11,0 β
2,21,0
]
+ 2 cos(�κ · �a2)
[β
1,10,1 β
1,20,1
β2,10,1 β
2,20,1
]
+ 2 cos(�κ · (�a1 − �a2)
)[β
1,1−1,1 β
1,2−1,1
β2,1−1,1 β
2,2−1,1
](18)
Using Eq. (16) we can write the dispersion relation asfollows:
ω(�κ) = ν
√Eigen value
{[B(�κ)
]−1[A(�κ)
]}(19)
Since [A(�κ)] and [B(�κ)] are 2 × 2 matrices and have twoeigenvalues, Eq. (19) has two solutions that gives two bandsof the band structure. ω(�κ) variations obtained from thisequation is plotted in Figs. 4 and 5 as constant frequencycontours in reciprocal lattice. Each figure shows one solu-tion of equation (19). Figure 6 shows the band structure cal-culated by this method.
4.2 Plane wave expansion method
In this method the dispersion relation is obtained by solvingan eigenvalue equation too. But unlike the tight binding in
Fig. 5 ω(�κ) variations in reciprocal lattice for the second band ob-tained by the tight-binding method
Fig. 6 Band structure calculated by the tight-binding method
this method nearly all of the bands are obtained. From thefirst and the second Maxwell’ equations we have
∇ × E(�r, t) = −μ0∂
∂tH(�r, t) (20)
∇ × H(�r, t) = ε0ε(�r) ∂
∂tE(�r, t) (21)
If we take the curl of both sides of Eq. (21) and then replacethe value of ∇ × E(�r) from Eq. (20) in it we have
∇ ×(
1
ε(�r)∇ × H(�r, t))
= − 1
c2
∂2
∂t2H(�r, t) (22)
By considering sinusoidal steady state magnetic field
H(�r, t) = H(�r)e−jωt (23)
Eq. (22) becomes a phasor equation.
∇ ×(
1
ε(�r)∇ × H(�r))
= ω2
c2H(�r) (24)
It is possible to show that in the H-polarization modeEq. (24) becomes the following equation for the z compo-
M.H. Aram et al.
nent of H(�r):
∇ ·(
1
ε(�r)∇ · Hz(�r))
= −ω2
c2Hz(�r) (25)
Now we define η(�r) ≡ 1/ε(�r). Because of the periodicityof permittivity in a photonic crystal lattice we can write theFourier series expansion for it as follows:
η(�r) =∑m,n
ηm,ne−j (m�b1+n�b2).�r =
∑G
ηGe−jG.�r (26)
In this equation �b1 and �b2 are primitive vectors of recip-rocal lattice. Using the Bloch theorem we can write Hz(�r)as
Hz(�r) = e−j �κ.�r ∑G
ϕG(�κ)e−jG.�r (27)
By replacing η(�r) and Hz(�r) from equations (26) and(27) into Eq. (25) we obtain the following eigenvalue equa-tion:∑G′
∑G
(ηG′ϕG(�κ)(�κ + G) · (�κ + G + G′)
× e−j (G+G′).�r) = ω2
c2
∑G
ϕG(�κ)e−jG.�r (28)
We can simplify this equation by changing variables. Thesimplified eigenvalue equation to be solved is
∑G′
ηG′−G ϕG′(�κ)(�κ + G)2 = ω2
c2ϕG(�κ) (29)
In practice it is not possible to consider all of the Fourierseries coefficients, thus we take coefficients with indices notgreater than a specified number N . Then the final eigenvalueequation becomes
N∑p,q=−N
ηp−m,q−nϕp,q(�κ)(�κ + (m�b1 + n�b2)
)2
= ω2
c2ϕm,n(�κ) (30)
Dashed curves in Fig. 7 are the band structure obtainedby solving Eq. (30). The modes of each band are also deter-mined in this figure.
4.3 Finite element method
One of the numerical methods to calculate photonic crys-tal band structure is FEM. In general FEM is a method tosolve partial differential equations numerically. In FEM avery complicated problem is divided into small elements
Fig. 7 Band structure of designed supercrystal. Dashed and contin-uous curves are results of PWE and FE methods, respectively. Themagenta band is the first H-polarization band gap of triangular basislattice
that can be solved according to each other [18]. Continuouscurves in Fig. 7 are results of FEM for the band structure ofour supercrystal.
5 Conclusions
In this project we first designed a triangular CPCRA with airholes in the silicon substrate. Then we calculated its bandstructure in H-polarization mode using tight binding, planewave expansion, and finite element methods and proved theexistence of Dirac points at the corners of Brillouin zones.
Acknowledgements We have benefited from discussions with Dr.S.M. Etemadi. This research is supported by National Elite Foundationof Iran.
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Construction of Dirac points using triangular supercrystals
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