Enhanced complete photonic band gap for three-dimensional plasma photonic crystals in pyrochlore arrangement

Solid State Communications 190 (2014) 10–17

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Enhanced complete photonic band gap for three-dimensional plasma photonic crystals in pyrochlore arrangement Hai-Feng Zhang a,b,n, Shao-Bin Liu a,nn, Yi-Jun Tang a a Key Laboratory of Radar Imaging and Microwave Photonics, Ministry of Education, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China b Nanjing Artillery Academy, Nanjing 211132, China

art ic l e i nf o

a b s t r a c t

Article history: Received 11 February 2014 Accepted 2 April 2014 by B.-F. Zhu Available online 15 April 2014

In this paper, the properties of complete photonic band gaps (PBGs) for three-dimensional (3D) plasma photonic crystals (PPCs) composed of the isotropic positive-index materials and non-magnetized plasma with pyrochlore lattices are theoretically investigated by a modified plane wave expansion method. The eigenvalue equations for calculating the band structure for such 3D PPCs in the first irreducible Brillouin zone (spheres with the isotropic positive-index materials inserted in the non-magnetized plasma background) are theoretically deduced. Numerical simulations show that the complete PBG and one flatbands region can be achieved. It also shows that the larger PBG can be obtained in such PPCs topology structure compared to the conventional lattices as the non-magnetized plasma density is low, such as diamond, face-centered-cubic, body-centered-cubic and simple-cubic lattices. The influences of the relative dielectric constant of spheres, the filling factor of dielectric spheres, the plasma frequency and the plasma collision frequency on the properties of complete PBG are investigated in detail, and some corresponding physical explanations are also given. The numerical results also show that the PBG can be manipulated by the parameters as mentioned above except for the plasma collision frequency. Introducing the non-magnetized plasma into 3D dielectric–air photonic crystals (PCs) can obtain the larger complete PBGs as such 3D PCs in pyrochlore arrangement. Crown Copyright & 2014 Published by Elsevier Ltd. All rights reserved.

Keywords: A. Three-dimensional plasma photonic crystals B. Plane wave expansion method D. Photonic band gaps D. Flatbands

1. Introduction Since the first proposal by Yablonovitch [1] and John [2], the photonic crystals (PCs) have been obtained increasingly as an interesting material in the experiment and theory during the past 20 years. The conventional PCs are artificial materials, in which the different dielectrics are periodically arranged in spaces. The PCs can produce the spectral regions named photonic band gaps (PBGs) originating from the interface of Bragg scattering [3,4], which can control the propagation of electromagnetic wave (EM wave). Thus, PCs can be potentially used to design various applications, such as defect cavities [5], waveguide [6], defect-mode PCs lasers [7], filter [8], and omnidirectional reflector [9,10]. However, the PBGs of conventional PCs suffer from being highly sensitive to the lattices. It means that the PBGs cannot be changed as the dielectrics and topology of PCs are certain. To overcome these drawbacks, the

n Corresponding author at: Key Laboratory of Radar Imaging and Microwave Photonics, Ministry of Education, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China. nn Corresponding author. E-mail addresses: [email protected] (H.-F. Zhang), [email protected] (S.-B. Liu).

http://dx.doi.org/10.1016/j.ssc.2014.04.003 0038-1098/Crown Copyright & 2014 Published by Elsevier Ltd. All rights reserved.

researchers have to introduce the metamaterials into PCs to achieve tunable PBGs [11]. In the nature, a lot of materials can be looked as metamaterials, such as plasma [12–14], superconductors [15–17], semiconductors [18] and metals [19]. Thus, the plasma photonic crystals (PPCs) can produce the tunable PBGs, which were firstly proposed by Hojo and Mase [20]. Recently, the PPCs have become a new research focus, which have been intensively investigated in theory and experiment by many research groups. Compared to the conventional PCs, the PPCs can display many interesting properties in dispersion [20,21]. The properties of the one- and twodimensional (2D) PPCs have been studied in detail in the past few years. The properties of PBGs and defect modes were investigated by Prasad et al. [22] and Qi et al. [23], respectively. They found that the plasma density is a key to manipulate the PBGs based on the transfer matrix method (TMM). Zhang et al. [21] also investigated the transmission and defect modes of 1D PPCs as the external magnetic field is introduced by the finite difference time domain method (FDTD). They found that the external magnetic field also can affect the characters of defect modes and transmission peeks. Furthermore, Guo et al. [24] studied a more general case, and considered 1D PPCs with time-variation plasma density. They proclaimed that the PBGs can be tuned by changing time, which can modify the plasma

H.-F. Zhang et al. / Solid State Communications 190 (2014) 10–17

density. As time goes on, more and more researchers are interested in the special PPCs, in which the magnetic material and metal are introduced into the PPCs, and the magneto-optical magnetized coupled resonator PPCs [25] or plasma-magnetic PCs are proposed [26]. In experiment, the PBGs of PPCs for millimeter range and the abnormal refraction around electron plasma frequency were verified and observed by Sakai et al. [27]. Fan et al. [28] also used a dielectric barrier discharge with two liquid electrodes to obtain a tunable 1D PPCs. As mentioned above, 1D and 2D PPCs have been investigated deeply in theory and experiment. However, 1D and 2D PPCs are theoretical models. In the real applications, the PPCs always are the finite periodic structures. On the other hand, the complete PBGs hardly can be achieved in the 1D and 2D cases. Thus, the 3D PPCs are closer to the actual situation. Recently, we focused on the properties of 3D PPCs. Our research group has investigated the dispersive properties of non-magnetized or magnetized 3D PPCs [29–32]. The magneto-optical Voigt and Faraday effects in 3D PPCs also have been investigated in detail [33,34]. The propagation of the extraordinary mode and the right circular polarized wave in 3D magnetized PPCs has also been given. Especially, we study the anisotropic PBGs of 3D PPCs with different lattices as the uniaxial material is doped [35]. From these research results, we can know that the topology of 3D PPCs is a key to obtain the complete PBGs. 3D PPCs with high symmetry, such as face-centered-cubic (fcc) lattices [33,35], simple-cubic (sc) lattices [34,35], and bodycentered-cubic (bcc) lattices [35], the complete PBGs can hardly be achieved, unless the inserted dielectric constant of dielectric must be sufficiently large so that the resonant scattering of EM waves is prominent enough to open a complete band gap [36], or the plasma density must be large enough [33–35]. Unfortunately, technological difficulties can be found in fabricating high symmetry 3D PPCs to achieve the complete PBGs with the larger dielectric constant of dielectric. If the plasma density is not large enough, the complete PBGs can hardly be found in the 3D PPCs with conventional high symmetry lattices. Therefore, to solve these problems, some methods have been reported: the symmetry reduction [36], introduced anisotropy in dielectric [36], or fabricating in a new topology [37]. As reported by Garcia-Adeva [37,38], the pyrochlore lattice is a good candidate to obtain complete PBGs for the 3D PPCs, and can also be easily be realized. Thus, we can achieve the larger complete PBGs in the 3D PPCs with pyrochlore lattices as the filling factor, plasma density, and the relative dielectric constant of filling dielectric are small.

11

The aims of this paper are to explore a better geometrical structure (pyrochlore lattice) to obtain larger complete PBGs in the 3D PPCs as the plasma density is low, and the properties of PBG also are theoretically studied by a modified plane wave expansion (PWE) method. The plasma collision frequency is also considered, and the periodic structures of PCs are infinite whose spheres with isotropic dielectric are inserted into the non-magnetized plasma background periodically in pyrochlore arrangement. This paper is organized as follows. The equations for computing the band structures for such 3D PPCs are theoretically deduced in Section 2. In Section 3, the influences of the relative dielectric constant of spheres, the filling factor of dielectric spheres, the plasma frequency, and plasma collision frequency on the properties of complete PBG are investigated. Finally, conclusions are given in Section 4. An e
jωt time-dependence is implicit pffiffiffiffiffiffiffiffi throughout the paper, with t being the time, and j ¼
1. We also consider c is the speed of light in vacuum.

2. Theory and numerical method The first irreducible Brillouin zone and schematic structure of 3D PPCs with a spherical atom in pyrochlore lattice can be found in Fig. 1. The dielectric and the non-magnetized plasma are assumed to be isotropic and homogeneous, and the relative dielectric functions are εa and εp. The radii of the spheres and lattice constant are considered as R and a, respectively. In the following numerical calculations, the non-magnetized plasma is assumed to be dispersive with effective dielectric function εp, which can be written as [39]

εp ðωÞ ¼ 1


ω2p ωðω þ jνc Þ

ð1Þ

where ωp and νc are the plasma frequency and the plasma collision frequency, respectively. Plasma frequency ωp ¼ ðe2 ne =ε0 mÞ1=2 in which e, m, ne and ε0 are the electric quantity, electric mass, plasma density and dielectric constant in vacuum, respectively. In order to obtain the PBGs of the PCs, several efficient numerical methods have been reported [40–42]. Among those methods, the PWE method is the most popular method to achieve the band structures, which can also calculate successfully the PBGs for the PCs composed of the Drude-type medium [43]. As mentioned in Ref. [43], a standard linearization technique was used to compute the general nonlinear eigenvalue equation. Thus, the complete PBGs of 3D PPCs can be calculated easily by this method. In this paper, the same technique

Sphere with the dielectric

a2 a3 2 3

a1 The non-magnetized plasma background

4 1

Fig. 1. (Color online) Schematic structure of such 3D PPCs with pyrochlore lattices. (a) 3D PPCs structure, (b) rhombohedral axes with respect to the cubic unit cell and (c) the first irreducible Brillouin zone showing symmetry point used for computing the complete PBG.

12

H.-F. Zhang et al. / Solid State Communications 190 (2014) 10–17

also will be used to obtain the PBGs of 3D PPCs in pyrochlore arrangement. As we know, Maxwell's equation for the magnetic field in 3D PPCs can be expressed as   1 ω2 ∇ ∇H ¼ 2H ð2Þ εðrÞ c

Fourier coefficients κ G could be written as [33–35]

κG ¼

where k is the wave vector in the Brillouin zone of lattice, G is the _ _ reciprocal-lattice vector, and e 1 , e 2 are the orthogonal unit vectors that are both perpendicular to wave vector k þ G because of the transverse character of the magnetic field H (i.e.,∇H ¼ 0). The 3D Bravais lattice is spanned by three primitive vectors a1, a2 and a3. The dielectric structure satisfies the boundary conditions

εðr þ ai Þ ¼ εðrÞ ai dbj ¼ 2πδij

ð5Þ

where δij is the Kronecker delta symbol. The dielectric constant dyadic can also be expanded into its Fourier form as

ε


1

ðrÞ ¼ ε


1 G;G0

¼

ð4Þ

The reciprocal lattice vectors b1, b2 and b3 are defined by

¼ ∑ ηðGÞe

jGdr

ð6Þ

G

where ηðGÞ is the Fourier transform of the inverse of εðrÞ and the sum is taken over by every reciprocal lattice vector G, which is a linear combination of b1, b2 and b3 G ¼ l1 b1 þ l2 b2 þ l3 b3

ð7Þ

where l1, l2 and l3 are integers. The Fourier coefficient is expressed by Z 1 1
jGdr e ηðGÞ ¼ dr ð8Þ S1 s1 εðrÞ



ω þ jνc ω ω2 þ jνc ω
ω2p 2

ω2 c2

ns

ð9Þ

i¼1

and ηðiÞ ðGÞ is the Fourier transform of each scatter of lattice unit cell at position ri . ns is the number of scatters in a unit cell with pyrochlore lattices. It is worth noting that the pyrochlore lattice (see Fig. 1(a)) has a rhombohedral primitive unit cell with lattice vectors, a1 ¼ ð0:5; 0:5; 0Þ;

a2 ¼ ð0:5; 0; 0:5Þ;

λ; λ0 ∑ H G;G 0 hG0 ;λ0 ¼

G0 ;λ

0

ω2 c2

ð10Þ

hG;λ





4


ε1a d ∑ e
ðG U vi Þ d3f

G¼0

sin ðjGjRÞ
ðjGjRÞ cos ðjGjRÞ

i¼1

ðjGjRÞ

!

3

;

Ga0

ð14Þ

where the prime on the sum over G0 indicates that the term with _ _0 _ _ ! 2 e2 U e2
e 2 U e 10 . At this G0 ¼ G is omitted. We consider F ¼ _ _0 _ _ e1 U e1
e 1 U e 20 point we use the definition of a complex variable

μ given by

μ ¼ ω=c

ð15Þ

Eq.(13) yields 2

2

2

2

2

μ4 Ι
μ3 T
μ2 U
μ V
W ¼ 0

ð16Þ

νc

2

T ðGjG0 Þ ¼
j δGdG0 ; c )  2 ω2p  1 0 þ 4f þ ð1
4f Þ U jk þ Gjjk þG j U F δGdG0 εa c2

U ðGjG0 Þ ¼



þ 2

V ðGjG0 Þ ¼

εa

(

2

1

2
1 M

ð17bÞ

 

 2 νc 1 j 4f þ ð1
4f Þ U jk þ Gjjk þ GjU F δGdG' c εa  2 νc 1
1 M þj c εa

WðGjG0 Þ ¼

ð17cÞ

)


þ

where

ð17aÞ

(

2

a3 ¼ ð0; 0:5; 0:5Þ;

with a four atom basis at positions, v1 ¼ (0, 0, 0), v2 ¼ a2/2, v3 ¼ a3/2, and v4 ¼ a1/2, as shown in Fig. 1(b). Substituting Eqs. (3) and (6) into Eq. (2), the following linear matrix equations can be obtained:

 

AðkjGÞ

where S1 denotes the area of the unit cell. For a given lattice with a unit cell including ns scatters, ηðGÞ is given by [44]

ηðGÞ ¼ εa
1 δG;0 þ ∑ ηðiÞ ðGÞe
jGdri



ω2 þ jνc ω 4f þ ε1a ð1
4f Þ; ω2 þ jνc ω
ω2p

We can obtain the equation for the coefficients {AðkjGÞ} as ! !   2 ω2 þ jνc ω 1 ð1
4f Þ U jk þGjjk þ G0 j U F U AðkjGÞ 4f þ εa ω2 þ jνc ω
ω2p ! ! 4 ω2 þ jνc ω 1 þ∑
d ∑ e
ðG U vi Þ 2 2 εa i ¼ 1 ω þjνc ω
ωp G0 ! 2 sin ðjGjRÞ
ðjGjRÞ cos ðjGjRÞ d3f U jk þ Gjjk þ G0 j U F U AðkjGÞ 3 ðjGjRÞ

ð3Þ

G λ¼1

> > > > :



ð13Þ

Since εðrÞ is periodic, we can use Bloch's theorem to expand the H field in terms of plane wave 2 _ HðrÞ ¼ ∑ ∑ hG;λ e λ e½jðk þ GÞdr

8 > > > > <

2 ω2p 1 ð1
4f Þ U jk þ Gjjk þG0 j U F δGdG0 2 c εa

ω2p  c2

1


1

εa

2 M

2

ð17dÞ 2

M ¼ jk þ Gjjk þ G0 jd F d∑4i ¼ 1 e
ðG U vi Þ d3f ðð sin ðjGjRÞ
ðjGjRÞ 2 2

cos ðjGjRÞÞ=ðjGjRÞ3 Þ; the elements of the N  N matrices are T , U ,

2

2

V and W. This polynomial form can be transformed into a linear

where

0 _ _ e 2 U ε
1 U e 20 G;G0 0 @ λ; λ0 H G;G0 ¼ jk þ Gjjk þ G j _ _
e 1 U ε
1 U e 20 G;G0

_ _ 1
e 2 U ε
1 U e 10 G;G0 _ _ A e 1 U ε
1 U e 10 G;G0

ð11Þ

0
1 where εG;G 0 ¼ ηðG
G Þ. In order to solve Eq. (11), we use the expanded Eq. (4), and write hG;λ in the form: jðk þ GÞdr

hG;λ ¼ ∑ AðkjGÞe

ð12Þ

G

f¼ (4πR3)/(3Vm) is assumed to be the filling factor of one sphere with the isotropic dielectric and Vm is the volume of unit cell. The

problem in 4N dimension by 2 2 0 I 0 6 2 6 2 2 I 6 0 0 Q z ¼ μz; Q ¼ 6 6 0 0 0 4 2

W

2

V

2

U

2

Q that fulfills 3 0 7 7 07 27 I 7 5

ð18Þ

2

T

The complete solution of Eq. (16) is obtained by computing the eigenvalues of Eq. (18). Of course the dispersion relation can be determined by the real part of such eigenvalues.

H.-F. Zhang et al. / Solid State Communications 190 (2014) 10–17

3. Numerical results and analysis In order to study the PBGs properties of 3D PPCs with pyrochlore lattices, the band structures are calculated in the first irreducible Brillouin zone as shown in Fig. 1(c). As we know [36], the high symmetry points for the pyrochlore lattice have the coordinates as Г ¼(0, 0, 0), X ¼(2π/a, 0, 0), W ¼(2π/a, π/a, 0), K ¼ (1.5π/a, 1.5π/a, 0), L¼(π/a, π/a, π/a), and U¼ (2π/a, 0.5π/a, 0.5π/a). The convergence accuracy is better than 1% for the lower energy bands as a total number of 729 plane waves can be used [36]. Without loss of generality, we plot ωa/2πc with the normalization convention ωp0a/2πc ¼1. With this definition, we can let a take any value as long as R is shifted accordingly to achieve the same filling factors. Thus, we can define the plasma frequency as ωp ¼ ωpl ¼0.05ωp0 to make the problem scale-invariant, and we also choose the plasma collision frequency as νc ¼0.02ωpl, ma ¼1, and mp ¼1. Here, we only focus on the properties of first (1st) PBG for such 3D PPCs. In order to investigate the properties of PBG, the relative bandwidth of PBG is defined as reported in Refs. [33,34]. 3.1. The complete PBG obtained from 3D PPCs in pyrochlore arrangement In Fig. 2, we display the dispersive curves of 3D PPCs with pyrochlore lattices as εa ¼12, νc ¼0.02ωpl and R ¼0.19a but with different plasma frequency ωp. The red shaded regions indicate the PBGs. As shown in Fig. 2(a), if ωp ¼ 0, the non-magnetized plasma can be looked as air, and 3D PPCs become dielectric–air PCs. There is only one PBG in the frequency region 0
2πc/a, which presents themselves at 0.3911–0.4510 (2πc/a). As the plasma is introduced to such 3D dielectric–air PCs, the band diagram for ωp ¼0.05ωp0 is plotted in Fig. 2(b). It can be seen from Fig. 2(b) that one PBG and one flatbands region can be found. The PBG covers 0.3922–0.4526 (2πc/a), and the flatbands region runs from 0 to 0.05(2πc/a). There exists the flatbands region because of the existence of surface plasmon modes, and the plasmon resonance bands are around the cutoff frequencies of the plasma [33–35]. If the frequency of EM wave is located in the flatbands region, the real part of dielectric function of plasma is negative. However, the real part of dielectric function of dielectric sphere is positive. Therefore, the surface plasmons can be found. Compared to Fig. 2(a), the edges of PBG shift upward to higher frequencies, and the bandwidth of PBG is increased by 0.0005 (2πc/a). Thus, the PBG of 3D dielectric PCs can be enlarged by introducing the plasma. As we know, the 3D PPCs with simpler lattices including the diamond, fcc, bcc and sc

13

lattices, also can produce the PBGs. As a comparison, we plot the band structures of 3D PPCs in various lattices as εa ¼12, ωp ¼0.05ωp0, νc ¼0.02ωpl and R ¼0.19a in Fig. 3. As we know, the high-symmetry points in the Brillouin zones for fcc and diamond lattices are same as in the pyrochlore structure [37,38]. For the bcc lattices, the high-symmetry points are Г ¼(0, 0, 0), H ¼ (0, 0, 2π/a), N ¼(0, π/a, π/a) and P¼ (π/a, π/a, π/a). For the sc lattices, the high-symmetry points are Г ¼ (0, 0, 0), X ¼(π/a,0,0), M¼(π/a, π/a, 0), and R ¼(π/a, π/a, π/a). It is clearly seen that the complete PBGs cannot be found in Fig. 3(a) and (d) since band degeneracy occurs at some high-symmetry points, which are H and P points for a bcc lattice, and M and R points for a sc lattice. This can be explained by the high symmetry of those lattices and the dielectric constant of immersed dielectric sphere is not large enough to open a band gap [36]. It can also be seem from Fig. 3(b) and (c), that there are PBGs of 3D PPCs with fcc and diamond lattices, and the 1st PBGs span 0.8131–0.8161 (2πc/a) and 0.5956–0.6106 (2πc/a), respectively. Obviously, the 3D PPCs with pyrochlore lattices have a larger bandwidth of the PBG compared to such PPCs with conventional simpler lattices, such as diamond, fcc, sc and bcc lattices. As mentioned above, the introduction of plasma into the 3D dielectric PCs can enlarge the frequency range of PBG, and the larger PBG can be obtained as 3D PPCs with pyrochlore lattices compared to the conventional simpler lattices. 3.2. Effects of the filling factor of dielectric sphere on the complete PBG In Fig. 4(a), we plot the dependences of the properties of 1st PBG for such 3D PPCs on the filling factor of sphere as εa ¼12, ωp ¼0.05ωp0 and νc ¼0.02ωpl. The shaded region indicates the PBG. Fig. 4(a) reveals that the edges of 1st PBG are downward to the lower frequency region, and the bandwidth of 1st PBG increases first and then decreases with increasing the value of R/ a. If the value of R/a is less than 0.148, the PBG does not exist, and it will disappear as the value of R/a becomes higher than 0.235. As R/a is increased from 0.05 to 0.48, the maximum bandwidth of 1st PBG is 0.0674 (2πc/a), which can be found in the case of R/a ¼0.18. Compared to the case of R/a ¼0.19, the maximum frequency range of 1st PBG is increased by 0.007 (2πc/a). Thus, the 1st PBG can be tuned by the filling factor of dielectric sphere. This can also be explained in physics where increasing the radius of the dielectric sphere means that the space averaged dielectric constant of such 3D PPCs becomes larger, and the PBG can be manipulated [45].

Fig. 2. (Color online) Calculated band structures for 3D PPCs with pyrochlore lattices as εa ¼12, νc ¼ 0.02ωpl and R ¼0.19a but with different plasma frequencies: (a) ωp ¼ 0 and (b) ωp ¼0.05ωp0, respectively.

14

H.-F. Zhang et al. / Solid State Communications 190 (2014) 10–17

0.65

0.16

0.60

0.14

0.55

0.12

0.50

0.10

Δω/ω i

ωa/2πc

Fig. 3. (Color online) Calculated band structures for 3D PPCs with four different conventional lattices as εa ¼12, ωp ¼0.05ωp0, νc ¼ 0.02ωpl and R ¼0.19afor (a) sc lattices, (b) diamond lattices, (c) fcc lattices, and (d) bcc lattices, respectively.

0.45 0.40

0.06 0.04

0.35

0.02

0.30 0.14

0.08

0.16

0.18

0.20

0.22

0.24

R/a

0.00 0.14

0.16

0.18

0.20

0.22

0.24

R/a

Fig. 4. (Color online) The effects of the radius of the dielectric spheres R/a on the complete PBG and relative bandwidth for such 3D PPCs with εa ¼ 12, ωp ¼ 0.05ωp0 and νc ¼ 0.02ωpl. The shaded region indicates the complete PBG: (a) the complete PBG, and (b) the relative bandwidth.

In Fig. 4(b), we plot the relative bandwidth (Δω=Δωi ) as a function of R/a. Fig. 4(b) shows that the relative bandwidth of 1st PBG increases first and then decreases with increasing the value of R/a. The maximum relative bandwidth is 0.147, which can be found in the case of R/a¼ 0.18. Compared to the case of R/a ¼0.15, the relative bandwidth is increased by 0.124. As mentioned above, the filling factor of dielectric sphere is an important parameter which needs to be chosen. It is also noticed that if the radius of the dielectric sphere is small enough and close to null, then such 3D PPCs can be looked as a non-magnetized plasma block. The flatbands will disappear.

3.3. Effects of the relative dielectric constant of sphere on the complete PBG In Fig. 5(a), we plot the 1st PBG for such 3D PPCs as a function of the relative dielectric constant of spheres εa. The parameters of 3D PPCs are ωp ¼0.05ωp0, νc ¼0.02ωpl and R¼0.19a. The shaded region indicates the PBG. As show in Fig. 5(a), the edges of 1st PBG shift to lower frequency region, and the bandwidth increases first and then decrease with increasing εa. If εa is less than 5, then the 1st PBG will never appear. As εa is increased from 5 to 50, the 1st PBG runs from 0.1983 to 0.246 (2πc/a), and the bandwidth is

H.-F. Zhang et al. / Solid State Communications 190 (2014) 10–17

0.65

0.22

0.60

0.20

0.55

0.18

0.50

0.16

0.45

Δω/ω i

ωa/2πc

15

0.40 0.35

0.14 0.12 0.10

0.30

0.08

0.25

0.06

0.20

0.04

0.15 5

10

15

20

25

εa

30

35

40

45

5

50

10

15

20

25

εa

30

35

40

45

50

Fig. 5. (Color online) The effects of relative dielectric constant εa on the complete PBG and relative bandwidth for such 3D PPCs with ωp ¼ 0.05ωp0, νc ¼ 0.02ωpl and R¼ 0.19a. The shaded region indicates the complete PBG: (a) the complete PBG, and (b) the relative bandwidth.

0.60

0.16

0.58

0.14

0.54

0.12

0.52

0.10

Δω/ω i

ωa/2πc

0.56

0.50 0.48 0.46

0.06

0.44

0.04

0.42

0.02

0.40 0.00

0.08

0.02

0.04

0.06

0.08

0.10

0.12

ωp /ωp0

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

ωp /ωp0

Fig. 6. (Color online) The effects of the plasma frequency ωp on the complete PBG and relative bandwidth for such 3D PPCs with εa ¼ 12, νc ¼ 0.02ωpl and R¼ 0.19a. The shaded regions indicate the complete PBG: (a) the complete PBG, and (b) the relative bandwidth.

0.0477 (2πc/a). The frequency range is increased by 0.0184 (2πc/a) as compared to the case of εa ¼ 5. In Fig. 5(b), the relative bandwidth of 1st PBG is plotted. Fig. 5(b) illustrates that the general trend for 1st PBG is the increase in relative bandwidth with increasing εa, and the maximum relative bandwidth is 0.215, which can be found in the case of εa ¼50. This can be explained by the bandwidth of PBG that is governed by refractive contrast for the dielectrics which is composed of 3D PPCs, and the position of the PBG is governed by the average refractive index of such 3D PPCs. As mentioned above, the frequency range of 1st PBG for such 3D PPCs can be tuned by εa. The central frequency of 1st PBG shifts to lower frequency region, and the bandwidth can be enlarged with increasing εa.

which can be found in the case of ωp/ωp0 ¼0.1. In Fig. 6(b), the relative bandwidth as a function of plasma frequency for 1st PBG is plotted. Fig. 6(b) illustrates that the relative bandwidth of 1st PBG increases first and then decreases as the value of ωp/ωp0 is increased from 0.01 to 0.125. The maximum relative bandwidth is 0.14712, which can be found in the case of ωp/ωp0 ¼0.1. Compared to the case of ωp/ωp0 ¼0.05, the maximum relative bandwidth of 1st PBG is increased by 0.005. As mentioned above, the PBG can be tuned by the plasma frequency. This can be explained by the average refractive index of such 3D PPCs that has been changed as the plasma frequency is changed [45].

3.5. Effects of the plasma collision frequency on the complete PBG 3.4. Effects of the plasma frequency on the complete PBGs In Fig. 7(a), we plot the effects of plasma collision frequency

In Fig. 6(a), we plot the effects of the plasma frequency ωp on the 1st PBG for such 3D PPCs with εa ¼ 12, νc ¼ 0.02ωpl and R¼ 0.19a. The shaded region indicates the PBG. As shown in Fig. 6(a), the edges of PBG shift to higher frequency region with increasing the value of ωp/ωp0. The bandwidth of 1st PBG increases first and then decreases as the value of ωp/ωp0 is increased. The 1st PBG will disappear as the value of ωp/ωp0 gets higher than 0.125. As the value of ωp/ωp0 is increased from 0.01 to 0.125, the 1st PBG is located from 0.448 to 0.4575(2πc/a), and frequency range is 0.0049 (2πc/a). Compared to the case of ωp/ ωp0 ¼0.05, the bandwidth of 1st PBG is decreased by 0.0504 (2πc/ a). The maximum frequency range of 1st PBG is 0.0626 (2πc/a),

νc on the complete PBG for such 3D PPCs with εa ¼12, ωp ¼0.05ωp0 and R¼0.19a. The shaded region indicates the complete PBG. As shown in Fig. 7(a), the 1st PBG for such 3D PPCs hardly can be tuned by the plasma collision frequency. The frequency range of 1st PBG remains invariant, and the central frequency cannot be also tuned by increasing the value of νc/ωpl. The bandwidth of such PBG runs from 0.3922 to 0.4526 (2πc/a), as the value of νc/ωpl is increased from 0.002 to 0.2. The bandwidth is 0.0604 (2πc/a), and will cease to change with increasing the value of νc/ωpl. In Fig. 7(b), the relative bandwidth is also plotted. It can be seen from Fig. 7(b) that the relative bandwidth of such complete PBG remains unchanged with increasing the value of

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H.-F. Zhang et al. / Solid State Communications 190 (2014) 10–17

0.8

0.200 0.175

0.6

0.150

0.5

0.125

Δω/ω i

ωa/2πc

0.7

0.4 0.3

0.100 0.075

0.2

0.050

0.1

0.025

0.0 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200

0.000 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200

νc / ω pl

νc / ω pl

Fig. 7. (Color online) The effects of plasma collision frequency νc on the complete PBG and relative bandwidth for such 3D PPCs with εa ¼ 12, ωp ¼ 0.05ωp0 and R¼ 0.19a. The shaded regions indicate the complete PBG: (a) the complete PBG, and (b) the relative bandwidth.

νc/ωpl, which is 0.0143. As mentioned above, the plasma collision frequency has no effect on the properties of 1st PBG.

NZ2013302), in part by the Fundamental Research Funds for the Central Universities (Grant no. kfjj120214), and in part by the Foundation of Aeronautical Science (No. 20121852030).

4. Conclusions

References

In summary, the properties of PBG for the 3D PPCs with pyrochlore lattices composed of isotropic positive-index material and non-magnetized plasma are theoretically investigated by the modified PWE method, which are the dielectric spheres immersed in the uniform non-magnetized plasma background. The equations for calculating the band structures in the first irreducible Brillouin zone are theoretically deduced. Based on the numerical results, some conclusions can be drawn. Compare with the same structure composed by the isotropic dielectric and air, the complete PBG and a flatbands region can be obtained as the plasma is introduced. The flatbands caused by the existence of surface plasmon modes stem from the coupling effects between the non-magnetized plasma. Compared to the conventional topology, such as diamond, fcc, bcc and sc lattices, the 3D PPCs in pyrochlore arrangement can produce a larger complete PBG. The complete PBG of such 3D PPCs can be manipulated by the relative dielectric constant of immersed dielectric spheres, and the general trend for PBG's relative bandwidth increases with increasing εa. On the other hand, the complete PBG can also be tuned notably by the plasma frequency. Increasing the plasma frequency, the bandwidth will increase first and then decrease. The maximum relative bandwidth can be obtained at low-ωp region. With increasing the filling factor of dielectric sphere, the relative bandwidth and frequency range of complete PBG will be increased first and then decreased. It is also noticed that if filling factor is small enough and close to null, such 3D PPCs can be looked as a plasma block, and the flatbands will disappear. The plasma collision frequency has no effect on the complete PBG of such 3D PPCs. As mentioned above, we can take advantage of the non-magnetized plasma to form 3D PPCs with pyrochlore lattices for obtaining the larger complete PBG as PPCs with low filling factor and plasma density.

Acknowledgments This work was supported by the supports from the National Natural Science Foundation of China (Grant no. 61307052), the Chinese Specialized Research Fund for the Doctoral Program of Higher Education (Grant no. 20123218110017), the Jiangshu Province Science Foundation (Grant no. BK2011727), in part by the Fundamental Research Funds for the Central Universities (No.

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