Symmetrical Analysis of Complex Two-Dimensional Hexagonal Photonic Crystals

PHYSICAL REVIEW B 67, 125203 ͑2003͒

Symmetrical analysis of complex two-dimensional hexagonal photonic crystals

N. Malkova, S. Kim, T. DiLazaro, and V. Gopalan Materials Research Institute and Department of Materials Science and Engineering, Pennsylvania State University, University Park, Pennsylvania 16802 ͑Received 29 August 2002; revised manuscript received 6 January 2003; published 12 March 2003͒ We study complex hexagonal photonic crystals with unit cells that include different dielectric cylinders. A general symmetrical perturbation approach for a hexagonal lattice with up to three basis rods is presented that systematically develops other structural derivatives including comblike structures. We show how the band spectrum of these complex structures evolves from the most symmetrical prophase. The results are in agreement with the plane-wave calculations of the band spectrum.

DOI: 10.1103/PhysRevB.67.125203 PACS number͑s͒: 71.15.Dx, 42.70.Qs, 71.20.Nr

I. INTRODUCTION semiconductors in Refs. 11 and 12. In Ref. 10 the model has been applied to complex square photonic crystals in which, In recent years, experimental and theoretical studies of perpendicular to the selected diagonal, the layers with differ- artificially manufactured dielectric media, the so-called pho- ent rods alternated. When studying the band spectrum of the tonic band-gap materials or photonic crystals, have attracted complex crystals, we start from the band spectrum of the considerable attention.1 The photonic crystals may be di- prophase, suggested by simple symmetry analysis. Introduc- vided by their application either as photonic insulators or as tion and identification of the appropriate prophase is a crucial photonic conductors. In the first case, the most important step in this model, which differentiates this approach from property of the photonic structures is a band gap, where the nearly free electron model for the electronic band propagating modes for any magnitude and direction of the structure13 or with the symmetrical model of the photonic wave vector are forbidden for either specific or all crystals developed in Ref. 4 in which the starting point is the polarizations.2 The most important feature of the photonic band spectrum of the free electrons or free photons. In the conductors, such as those exhibiting the superprism effect, is present approach, the plane-wave functions for the prophase the possibility of tuning the opening of the band gap at some states are used as the basis. The band spectrum of the pertur- low-symmetry points of the Brillouin zone.3 Thus, under- bative phase is then obtained as a perturbation of the plane- standing the behavior of defect-free photonic crystals and wave spectrum of the prophase. designing devices based on them require a knowledge of the Two important classes of periodic lattice symmetries are wave propagation properties in the crystals.4 In the same way studied in photonic structures: the class of square lattices and as electron properties of electronic crystals are governed by their complex derivatives ͑containing fourfold or twofold ro- the solid’s band structure, the information about the photon tational symmetry axes͒,14,15 and the class of hexagonal lat- propagation properties is contained in the band structure and tices and their complex derivatives ͑containing sixfold- or eigenmodes of the dielectric periodic structure. threefold rotational symmetry axes͒.16,17 Reference 10 dealt A variety of methods have been used to calculate the pho- with the first class of square lattices. This paper deals with tonic band structure. All these approaches may be divided the second important class of hexagonal lattices and their into two groups. The first group comprises the so-called ab complex derivatives. The motivation of this work is two- initio models, implementation of which does not demand any step. First is that since the model depends critically on the empirical parameters. They are the plane-wave technique,5 presence and selection of a suitable prophase lattice, demon- transfer-matrix methods,6 and different types of numerical strating the presence of a suitable prophase for the complex schemes to solve Maxwell’s equations.7 A disadvantage of square lattices ͑as in Ref. 10͒ does not automatically imply the ab initio calculations is that they are very time and or naturally suggest a similar suitable prophase for the entire memory consuming. Another group of the photonic band- class of complex hexagonal lattices. In this paper, we show structure models consists of empirical models. For periodic that indeed such a prophase exists. The second goals is to dielectric structures, this group includes the tight-binding understand the common scheme of the evolution of the band model, first adopted for photonic crystals in Ref. 8. This spectrum for the whole class of the complex crystals based model contains some empirical parameters, coupled matrix on hexagonal and square symmetries within the framework elements, which have to be fitted to ab initio or experimental of one model. We compare and contrast these two important results. Tight-binding calculations of the band structure are crystal classes. very simple in their numerical implementation, giving an In particular, we consider a hexagonal lattice with alter- analytical solution. Recently, this scheme was also success- nating layers of the dielectric rods perpendicular to a selected fully used for various photonic structures.9 hexagon diagonal. The selection of this one direction in the In our previous paper10 a symmetrical model for the crystal will be shown to be of utmost importance in all band analysis of the band structure of the complex photonic struc- properties of these structures. We start from a simple hexa- tures has been developed. A similar theoretical approach has gon lattice with one rod in the basis, when all the layers are been suggested for electronic band structures of the complex equivalent. Then we simulate complex lattices by introduc-

0163-1829/2003/67͑12͒/125203͑9͒/$20.0067 125203-1 ©2003 The American Physical Society N. MALKOVA, S. KIM, T. DiLAZARO, AND V. GOPALAN PHYSICAL REVIEW B 67, 125203 ͑2003͒ ing two additional rods on the selected diagonal of the initial lattice. As a limit case when the radius of one of the additional rods is equal to zero, we can derive the hexagonal comblike lattice considered ﬁrst in Refs. 16 and 17. Thus we show that in the framework of the developed model we can systematically study the band structure of the entire class of complex lattices, including square, hexagonal, and their complex derivatives such as comblike lattices. From the symmetrical analysis, the opening of the band gap is shown to be predicted for the class of the layered photonic structures. We state that these layered photonic crystals present a wide class of tunable photonic structures. The outline of this paper is as follows. A theoretical development of the model for the case of the complex hexagonal lattices is presented in Sec. II. In Sec. III we compare our theoretical predictions with the plane-wave calculations of the photonic crystals in question. In Sec. IV we give conclu- sions, where discussion of the evolution of the band spectrum for complex crystals with square and hexagonal symmetries is presented.

II. SYMMETRICAL MODEL FOR COMPLEX HEXAGONAL LATTICES We consider a two-dimensional dielectric system periodical in the x-y plane and homogeneous along the z axis. The dielectric constant for the periodical system is a position de- pendent and periodic function of the vector r in the x-y ͑ ͒ plane, satisfying the relation ⑀(rϩR )ϭ⑀(r), where for any FIG. 1. Hexagonal lattice with two rods in basis a and its l Brillouin zone ͑b͒. The unit cells of the simple hexagonal prophase integers l , R ϭl a ϩl a defines a two-dimensional Bra- 1,2 l 1 1 2 2 and perturbative phase are shown by dashed and bold lines, respec- vais lattice with the unit cell constructed on the primitive ϭ͉ ϫ ͉ tively. translation vectors a1 and a2 . The magnitude A a1 a2 gives the area of a primitive unit cell of this lattice. In the framework of the plane-wave method, the band- We study the complex hexagonal lattice, shown in Fig. structure problem for the two-dimensional lattice is reduced 1͑a͒. This lattice is obtained from the simple hexagonal lat- to the eigenvalue problem4 tice by placing equidistantly, two different rods on one of the diagonals of hexagon. We first note that this is a layer lattice ␻2 with alternating layers of the different rods perpendicular to detͩ HϪ ͪ ϭ0, ͑1͒ 2 the selected diagonal. The unit cell is defined by the vectors c ϭ ϭ ͱ ͑ ͒ a1 a(1,0) and a2 a(1/2, 3/2), shown in Fig. 1 a by bold where lines. The point group symmetry of this lattice is C3 . The most important point is that this lattice can be represented as H͑G, GЈ͒ϭ͉kϩG͉͉kϩGЈ͉␩͑GϪGЈ͒ three embedded identical hexagonal sublattices, determined by the set of same primitive vectors a and a , but which are for the E polarization ͑electric field is parallel to the z axis͒ 1 2 shifted with respect to each other by the vector ␶ and ϭa/2(1,1/ͱ3). These sublattices are shown in Fig. 1͑a͒ by H͑G,GЈ͒ϭ͑kϩG͒ ͑kϩGЈ͒␩͑GϪGЈ͒ dashed-dotted lines. The inverse dielectric constant of such a • lattice can be expressed as for the H polarization ͑magnetic field is parallel to the z axis͒. The function ␩(G) is defined as Fourier transforms of the inverse dielectric constant 1 ϩ␶ ϭ͚ ͓␩1͑G͒eiG•rϩ␩2͑G͒eiG•(r ) ⑀͑r͒ G 1 1 Ϫ ␩͑ ͒ϭ ͵ iG•r ͑ ͒ G e dr, 2 3 iG (rϩ2␶) A uc⑀͑r͒ ϩ␩ ͑G͒e • ͔. ͑3͒ where the integral is taken over the unit cell with an area A. ϭ ϩ ͒ Here k is the two-dimensional reciprocal vector lying inside Here G n1b1 n2b2 (n1,2 are integers are the vectors of the Brillouin zone and the translation vectors of the recipro- the corresponding reciprocal lattice, with the primitive vec- ϭ ϩ ϭ ␲ Ϫ ͱ ϭ ␲ ͱ cal lattice Gn n1b1 n2b2 , where n1,2 are integers, and b1,2 tors b1 (2 /a)(1, 1/ 3) and b2 (2 /a)(0,2/ 3); are the primitive reciprocal vectors. ␩1,2,3(G) are the Fourier transforms of the inverse dielectric

125203-2 SYMMETRICAL ANALYSIS OF THE COMPLEX TWO-... PHYSICAL REVIEW B 67, 125203 ͑2003͒

constants for the three selected sublattices. The Brillouin point of the perturbed phase, all the points J0 of the prophase zone of this lattice is a hexagon shown as a shaded area in being combined into one point of the prophase. The points ͑ ͒ Fig. 1 b . X0 of the prophase become the points X in the perturbative ␶ϭ ␲ ϩ We note that G• 2 (n1 n2)/3. So the inverse dielec- phase. But the vector Q picks up one of the diagonals of the tric constant can be expressed as a sum of the three potentials hexagonal Brillouin zone of the prophase in such a way that two of the X points of the perturbative phase are obtained 1 ⌫ ϭ ͒ϩ ͒ϩ ͒ ͑ ͒ from the the middle points of the J0 branches of the V0͑r V1͑r V2͑r . 4 ⑀͑r͒ prophase and four others—from the X0 points of the prophase. The potential V0 is represented as The periodicity of the V0 potential implies that it can mix only the states differing by the reciprocal vectors G0 of the ͒ϭ ␩1 ͒ϩ␩2 ͒ϩ␩3 ͒ iG0 r ͑ ͒ V0͑r ͚ ͓ ͑G0 ͑G0 ͑G0 ͔e • . 5 simple hexagon lattice while the translation properties of the G 0 V1,2 potentials result in mixing of the prophase states sepa- Ϯ Here the sum runs over the reciprocal vectors of the simple rated by the vectors Q. Perturbative analysis similar to that H ϭ ␲ ͓ ϩ ͱ Ϫ ͔ presented in Ref. 10 shows that the matrix Hamiltonian hexagonal lattice G0 (2 /a) n1 n2 , 3(n1 n2) , with ͑ ͒ ϫ 0 ϭ ␲ Ϯͱ 1 is 3 3 block matrix: the primitive vectors b1,2 2 /a(1, 3). The corresponding real space unit cell is defined by the primitive vectors Hk Vk,kϩQ Vk,kϪQ a0 ϭ(a/2)(1,Ϯ1/ͱ3). The potential V may be considered 1,2 0 H˜ ϭ ͑ ͒ G ,GЈ ͩ VkϩQ,k HkϩQ VkϩQ,kϪQ ͪ . 7 as a potential of the simple hexagonal prophase obtained 0 0 from the lattice in question if all the cylinders were identical. VkϪQ,k VkϪQ,kϩQ HkϪQ It has to be characterized by C6v symmetry of the simple hexagonal lattice, satisfying the translation relation Here the diagonal blocks give the band spectrum of the prophase in the points k, kϩQ and kϪQ. They interact with ϩ␶͒ϭ ͒ V0͑r V0͑r . each other through the block matrices Vk,kЈ characterized by the matrix elements of the potentials V1 and V2 . In the The unit cell and Brillouin zone of the simple hexagonal plane-wave basis these matrix elements, for the case of the E prophase are shown in Fig. 1 by dashed lines. polarization, are defined as10 Two potentials V1,2 have the form ϭ͉ ͉͉ Ј͉␩Ϫ ͑ ͒ vk,kЈ k k kϪkЈ , 8 ͑ ͒ϭ ͓␩1͑ ϯ ͒Ϫ␩2͑ ϯ ͒ ϯi␲/3 V1,2 r ͚ G0 Q G0 Q e ␩Ϫϭ␩1Ϫ ␩2ϩ␩3 ␲ Ϫ ␩2Ϫ␩3 ␲ G0 where ( )cos( /3) i( )sin( /3). Ϯ ␲ ϯ We can interpret the effect of including the V1,2 potentials as Ϫ␩3͑ ϯ ͒ i /3͔ i(G0 Q) r ͑ ͒ G0 Q e e • . 6 triple folding of the simple hexagon Brillouin zone. So that Here summing is over the reciprocal vectors of the simple the spectrum of the perturbative phase along any direction hexagon lattice G again, a new reciprocal vector Q M 1M 2 will be a sum of the three prophase branches: M 1M 2 , 0 ϩ ϩ Ϫ Ϫ ϭ ␲ͱ (M 1 Q)(M 2 Q) and (M 1 Q)(M 2 Q). Here we define (4 3/3a)(0,1) being selected. Ϯ We note that the potentials V increase the translation (M i Q) as a point inside the Brillouin zone with the coor- 1,2 ␲ Ϯ ␲ ͱ scaling of the lattice, satisfying the translation relation: dinate (2 /3a)(xi ,y i) (2 /3a)(0,2 3), assuming that the ␲ coordinates of the point M are (2 /3a)(xi ,y i). We conclude ϩ␶͒ϭϪ ͒ ϯi␲/3 that the folding of the Brillouin zone of the simple hexagonal V1,2͑r V1,2͑r e . lattice is governed by the vector Q, resulting in a strong In general case, when all the rods in the basis are different mixing of the states separated by this vector. ͓this is the case in Fig. 1͑a͔͒, the symmetry of the lattice is lowered to C3v . But in the particular case, when two rods in ϭ III. PLANE-WAVE CALCULATION the basis are equivalent, that is, V1 V2 , the lattice keeps the symmetry C6v of the prophase. As seen from Fig. 1, the area We apply this symmetrical analysis to study the plane- of the unit cell for the complex hexagon lattice is greater wave band spectrum of the complex hexagonal lattice, than three times the area of the simple hexagon lattice of the shown in Fig. 1. It is obvious10 that in the region of the light prophase, while the areas of the reciprocal unit cells, namely, wavelength much larger than the radii of the rods, the effect the Brillouin zones, are characterized by the inverse relation- of varying the radii of the rods versus varying their dielectric ship. constants on the band spectrum should be similar in the pho- The potentials V1,2 can be treated as perturbations charac- tonic crystals studied. We have performed numerical calcu- terized by the difference in properties of the dielectric rods in lations for wavelengths larger than the lattice period. There- the basis. Inclusion of the perturbation potentials results in fore, in our numerical analysis, since the radius of the tripling the area of the direct lattice and in complete nesting dielectric rods is a more flexible parameter, we have mostly of the Brillouin zone of the simple hexagonal prophase into paid attention to the case of varying the radii of the rods, and the Brillouin zone of perturbative phase. This is governed by keeping the dielectric constants of the rods unchanged. But ͑ ͒ ⌫ the vector Q shown in Fig. 1 b . So that the points and J0 the results presented can be easily extended for the case of of the prophase will be combined by the vector Q into one ⌫ varying the dielectric constants of the rods.

125203-3 N. MALKOVA, S. KIM, T. DiLAZARO, AND V. GOPALAN PHYSICAL REVIEW B 67, 125203 ͑2003͒

ϭ FIG. 2. The band spectrum for E polarization of the simple hexagonal lattice with one rod (r1 0.2a) in the basis constructed in the Brillouin zone of the prophase ͑a͒ and in the Brillouin zone of the perturbative phase ͑b͒; and the band spectrum of the complex hexagonal ϭ ϭ ϭ ͑ ͒ ϭ ͑ ͒ lattices with r1 r2 0.2a, r3 0.19a c ; r3 0.17a d . Discussed points are labeled by circles.

A system of dielectric rods in the air has been considered, other, and as follows from Eq. ͑6͒, the perturbative potentials ⑀ ϭ ͑ ͒ with the dielectric constants of the rods a 11.9 Si and air V1,2 are complex conjugate. ⑀ ϭ b 1. To calculate the band spectrum for the photonic crys- A key point in our symmetrical analysis is demonstrated tals with complex basis we used the technique developed in by comparing Fig. 2͑a͒ and Fig. 2͑b͒. By doing this, we can Ref. 18. The results that follow were obtained using 569 follow the folding of the band spectrum of the prophase into plane waves for the simple hexagonal lattice with one rod in the Brillouin zone of the perturbative phase. From the previ- basis and 961 plane waves for the complex lattices. The nu- ous analysis, we know that in the perturbative phase, the merical data were tested using 1221 plane waves showing band spectrum along ⌫X direction will be a sum of the three that the accuracy of the results is better than 1%. prophase branches ⌫X, ⌫XϮQ. Simple symmetrical analy- At first we study the generation of the band spectrum of sis shows that the band spectrum along ⌫X branch in the the complex crystals from the band spectrum of the prophase perturbative phase must be composed from the three differ- ⌫ ϭ ⌫ prϩ pr under small perturbation. Figure 2 shows the spectrum for ent branches of the prophase: X ( X) (J0X) ϩ pr E-polarization of the simple hexagonal lattice with one rod (J0X0) , where index pr means the states of the prophase. ϭ ͑ ͒ (r1 0.2a) in the basis constructed in the Brillouin zone of The dashed line in Fig. 2 a separates the first two branches. the prophase ͑a͒ and in the Brillouin zone of the perturbative In order to get the band spectrum of the perturbative phase phase ͑b͒; and the band spectrum of the complex lattices ͓Fig. 2͑b͔͒, we have to fold the band spectrum of the ϭ ϭ ϭ ͑ ͒ ϭ ͑ ͒ with r1 r2 0.2a, r3 0.19a c and r3 0.17a d . This is prophase first along J0 line, and then along the dashed line so the simplest case of the hexagonal photonic lattice with two as to overlap the states separated by the vector Q shown in rods in the basis when the radius of one of the rods is kept Fig. 2͑a͒. The branches involved in this mapping are marked unchanged and only the radius of the second rod changes. In by one dash in Fig. 1͑b͒. All these branches are different in this case, the matrix elements ␩1 and ␩2 are equal to each the prophase. They have to show splitting at the points of the

125203-4 SYMMETRICAL ANALYSIS OF THE COMPLEX TWO-... PHYSICAL REVIEW B 67, 125203 ͑2003͒ degeneracy when perturbation is included. This is a case for the degeneracy point between seventh and eighth bands marked by the circle in Figs. 2͑b, c, d͒. The band spectrum along ⌫J direction of the perturbative phase is composed from the three branches of the prophase ⌫ ϭ ⌫ prϩ prϩ pr again: J ( J) (J0J)1 (J0J)2 . They are mapped in Fig. 1͑b͒ as cuts with double dashes marker. To get the band spectrum of the perturbative phase ͓Fig. 2͑b͔͒ from the prophase spectrum ͓Fig. 2͑a͔͒ we have to take the states from the point ⌫ up to the dash-dotted line in Fig. 2͑a͒, which marks the point J of the perturbative phase, and then overlap pr the (J0J) branches. But these latter branches do not lie on the symmetrical directions of the hexagonal lattice ͓Fig. 1͑b͔͒, and so they are not present in the spectrum of the prophase ͓Fig. 2͑a͔͒. From Fig. 1͑b͒ we note that the pr branches (J0J)1,2 are equivalent in the simple hexagonal lattice, and therefore, they completely overlap in the prophase ͓the second and third bands in Fig. 2͑b͒, for example͔. When including the perturbations V1,2 , the scaling of the lattice increases ͓Fig. 1͑a͔͒ and these branches split. This process is clearly shown in Figs. 2͑c, d͒. We can also follow the opening of the band gaps at the points of the degeneracy between the seventh and the eighth bands on the ⌫J branch, marked by the circle in Figs. 2͑b, c͒. Figure 2͑d͒ shows that the splitting of the seventh and eighth bands at the degeneracy points on the ⌫J and ⌫X directions results in opening the complete band gap for E bands, bordered by two dashed lines. An obvious reason for the opening of the band gaps at the band intersection points created by the Brillouin zone folding described above, is that these bands are related to the two different rods in the basis of the lattice. In support of this statement, we present in Fig. 3, the spatial distribution of the Poynting vector for the seventh ͓␻˜ ϭ␻a/(2␲c)ϭ0.6056͔ ˜ ͑a͒ and the eighth ͓␻ϭ0.6196͔ ͑b͒ bands at the labeled point FIG. 3. The space distribution of the Poynting vector for the ⌫ ϭ on the J direction of the photonic crystal with r1 r2 seventh (␻˜ ϭ0.6056) ͑a͒ and the eighth (␻˜ ϭ0.6196) ͑b͒ bands at ϭ ϭ ͓ ͑ ͔͒ 0.2a and r3 0.19a Fig. 2 c . Figure 3 shows that the the labeled point on the ⌫J direction for the photonic crystal with maximum intensity of the energy for these states is located ϭ ϭ ϭ ͓ ͑ ͔͒ r1 r2 0.2a; r3 0.19a Fig. 2 c . The unit cell is superimposed. on different rods in the basis. In the prophase, all the rods are identical and the two bands have the same energy. However, 0 2 2 ͑E ͒ Ϫ␻˜ v ϩ v Ϫ in the perturbative phase, the greater the difference between k k,k Q k,k Q dielectric rods, the greater the energy splitting between these ͑ 0 ͒2Ϫ␻˜ 2 ϭ ͑ ͒ ͯ vkϩQ,k EkϩQ vkϩQ,kϪQ ͯ 0. 9 two bands ͓Figs. 2͑c, d͔͒. ͑ 0 ͒2Ϫ␻˜ 2 As was noted above, this lattice has the same group sym- vkϪQ,k vkϪQ,kϩQ EkϪQ metry C as a hexagonal lattice of the prophase. That is why 6v Here E0 is a normalized frequency for the treated bands at the symmetry of the bands at the symmetrical points of the k Brillouin zone should not change. Data in Figs. 2͑c, d͒ show the point k of the prophase, vk,kЈ are the corresponding matrix element of the block matrices V ͑10͒, and ␻˜ that the double degeneracy, permitted by the C6v , C3v , and k,kЈ ⌫ ϭ␻a/(2␲c) is an unknown normalized frequency. C2v point groups at the , J, and X points, respectively, survives. We perform the perturbative calculations for the second The above analysis provides evidence for a qualitative and third split bands at the ⌫ point for the photonic crystal agreement between symmetrical model and exact plane- shown in Fig. 2͑c͒. This is the lowest labeled circle on the ⌫ wave calculation of the band spectrum of the complex hex- axis in Fig. 2͑c͒. We note that in this case the problem can be agonal crystals. In the case of the small difference between reduced to a 2ϫ2 matrix, because for these bands kϭ0 and ͑ ͒ the rods, the perturbative approach can be used to quantita- all the matrix elements vk,kϩQ 8 are equal to zero. The tively solve the eigenvalue problem with Hamiltonian ͑7͒. nonzero matrix element is

For the points of degeneracy, the problem can be reduced to Ϫ 10 ϭ͉ ͉2␩ ͑ ͒ the three-band approximation vkϩQ,kϪQ Q 2Q , 10

125203-5 N. MALKOVA, S. KIM, T. DiLAZARO, AND V. GOPALAN PHYSICAL REVIEW B 67, 125203 ͑2003͒ where

1 1 2␲J ͑2QR͒ ␩Ϫ ϭ͑ Ϫ ͒ͩ Ϫ ͪ 1 2Q r1 r2 ⑀ ⑀ . a b 2AQ ϭ 2 ͱ Here J1 is the ﬁrst-order Bessel function and A a / 3is the area of the prophase unit cell. In this approximation we immediately arrive at

v ϩ Ϫ ⌬ ϳ k Q,k Q ͑ ͒ Ek 0 . 11 Ek 0ϭ ͓ Taking into account that for these bands Ek 0.37 Fig. ͑ ͔͒ ⌬ ϭ 2 b ,weget E0 0.0093. The exact plane-wave solution gives the splitting ⌬Eϭ0.3727Ϫ0.3680ϭ0.0047, which is ⌬ the same order of magnitude as E0 . We apply the same approach for the fourth and ﬁfth split ⌫ bands marked by the second circle on ⌫ axis in Fig. 2͑c͒. In this case as well, the problem allows the two-band approximation. The nonzero matrix element is

ϭ͉ ϩ Ϫ ͉͉ ϩ Јϩ ͉␩Ϫ vkϩG ϪQ,kϩGЈϩQ k G0 Q k G0 Q G ϪGЈϪ2Q , 0 0 0 0 ͑12͒ ϭ ␲ ͱ Јϭ ␲ Ϫͱ where G0 2 /a(1, 3) and G0 2 /a(1, 3). Using the ͑ ͒ 0ϭ approximation formula 11 with Ek 0.53, we have got ⌬Eϭ0.0189. Within the limits of the accuracy of the plane- wave calculations, this is in good agreement with the exact solution for these bands ⌬Eϭ0.5405Ϫ0.5265ϭ0.014. We note that the above estimations have been performed within the very rough two-band approximation in compari- son with the more accurate plane-wave calculations using ϳ1000 plane waves ͑bands͒. A reasonable agreement between perturbative and ab initio data, without any ﬁtting procedure, demonstrates the power of even this simple two-band model for the prediction of the band splitting for small perturbations. However, for large perturbations compared with the distance from far bands in the prophase, one needs to consider the many-band model of the perturbative Hamil- tonian in order to get a good agreement with the ab initio calculations. In this case, the perturbative approach loses its simplicity and the standard plane-wave calculations are more reasonable. Next we study the band spectrum of the complex hexagonal lattice when perturbation potential is increased, but ␩1 ϭ␩2. Figure 4 presents the plane-wave band spectrum for ϭ ϭ ϭ ͑ ͒ the E and H bands in the cases r1 r2 0.2a, r3 0.1a a , ϭ ͑ ͒ ϭ ͑ ͒ r3 0.05a b and r3 0a c . First we note that the comblike lattices are included in the family of these lattices as a lim- iting case when the radius of one of the rods tends to zero ͓Fig. 4͑c͔͒. These lattices have been carefully studied in Refs. 16 and 17. Here, we shall only emphasize that the comblike lattices can be considered within the framework of our model. However, in this case, the structures do not allow FIG. 4. The plane-wave band spectrum for the E and H bands in the perturbative approach, because both the perturbative po- the cases r ϭr ϭ0.2a, r ϭ0.1a ͑a͒, r ϭ0.05a ͑b͒, and r ϭ0.0a → 1 2 3 3 3 tentials V1,2 do not have limit transition V1,2 0. ͑c͒. The E and H bands are shown by dots and stars, respectively. Figure 4 shows that the common character of the changes The E-band gaps are shown by dashed lines, the complete band of the H bands is similar to the E bands discussed above. The gaps are shown by the dashed stripes. Discussed points are labeled perturbation potential is the reason for splitting of the second by circles.

125203-6 SYMMETRICAL ANALYSIS OF THE COMPLEX TWO-... PHYSICAL REVIEW B 67, 125203 ͑2003͒ and the third H bands, which overlap in the prophase ͑not shown͒. We also note that the splitting of the second and third E bands that had appeared with a small perturbation in Figs. 2͑c, d͒ results, in the case of a large perturbation, in the first band gap, shown by two dashed lines in Fig. 4. This band gap increases with decreasing the radius of the second ϭ rod, reaching the maximum for a comblike lattice at r3 0. It is worth mentioning that in the prophase at the point ⌫, the fourth and fifth E bands ͓Fig. 2͑b͔͒ and the second and third H bands are degenerate. Moreover, accidentally for the crystal studied, the energy of the E bands overlaps with the energy of the H bands. The degeneracy is removed in the perturbative phase. It is remarkable here that the splitting of the E and H bands at the ⌫ point, labeled by circles in Fig. 4, has approximately the same magnitude for all the crystals shown in Fig. 4. The explanation of this fact is the following. First, the relatively large distance of the split H bands from the far bands results in a small effect of the far bands and the possibility of using a two-band approximation even in the case of a large perturbation. The splitting of the second and third H-bands is determined by the matrix element ͑10͒.A direct estimation by formula ͑11͒, for the crystal shown in Fig. 4͑a͒, gives the magnitude ⌬Eϭ0.05 that is close to the exact value 0.06. As seen from Fig. 4, the two-band approximation for the E bands is hardly applicable. But in a rough approximation we may still use formula ͑12͒ to estimate the splitting of the fourth and fifth E bands. Since the energies in 0 the unperturbative phase Ek for the E and H bands are equal each other, Eq. ͑11͒ estimates that the splitting of the H and E bands is approximately equal. The binding of the E and H bands explains the fact that the lowest complete band gap in these structures may be opened only between the sixth ͑sev- enth͒ E band and the fourth H band. The complete band gap is shown by the shaded stripes in Fig. 4. We note the dra- matic increase in the first complete band gap for the comblike crystal ͓Fig. 4͑c͔͒. As the last step, we study the band spectrum of the complex hexagonal crystals when all the rods are different. Fig- ure 5 shows the spectrum of the E bands and the H bands of ϭ ϭ ϭ the hexagonal lattice with r1 0.2a,r2 0.21a, r3 0.05a ͑ ͒ ϭ ϭ ϭ ͑ ͒ ϭ a ; r1 0.2a,r2 0.24a,r3 0.05a b ; and r1 0.2a,r2 ϭ ϭ ͑ ͒ 0.3a,r3 0.05a c . First, we note that in this case both the potentials V1 and ͑ ͒ V2 6 are different, and the group symmetry of the lattice is lowered to C3v . The different magnitudes of the potentials V1,2 differentiate the two rods in the basis. This results in the separation of the states at the point J into two groups related to two different rods. This is the reason for the removing of the degeneracy of the first and second as well as of the fourth and fifth E bands at the point J and for the opening of these band gaps between these E bands. These states are marked by circles in Figs. 5͑a, b͒. We can follow, in Fig. 5, the ͑ opening and increasing of these band gaps bordered by two FIG. 5. The plane-wave spectrum of the E and H bands of the dashed lines͒ when the difference between two rods in the ϭ ϭ ϭ ͑ ͒ hexagonal lattice with r1 0.2a,r2 0.21a,r3 0.05a a ; r1 basis increases. The splitting of the second and third as well ϭ ϭ ϭ ͑ ͒ ϭ ϭ 0.2a,r2 0.24a,r3 0.05a b ; and r1 0.2a,r2 0.3a,r3 as fourth and fifth H bands ͑marked by dashed circles͒ is ϭ0.05a ͑c͒. The E and H bands are shown by dots and stars, re- again caused by the difference in the V1 and V2 potentials. It spectively. The E band gaps are shown by dashed lines, the com- is also important to note that the C3v point group symmetry plete band gaps are shown by the dashed stripes. Discussed points of the points ⌫ and J allows the existence of the doubly are labeled by circles.

125203-7 N. MALKOVA, S. KIM, T. DiLAZARO, AND V. GOPALAN PHYSICAL REVIEW B 67, 125203 ͑2003͒ degenerate ⌫ and J bands, while all the X bands are has been shown to open along the direction parallel to the nondegenerate.4,19 This explains the fact that the band gap vector Q. This theoretical prediction of the symmetrical between first and second as well as fourth and fifth H bands model is supported by the numerical calculations. An impor- cannot be opened. Figure 5 also shows that the complete tant difference in the symmetrical analysis of the both sym- band gaps for these crystals, marked by shaded stripes, de- metries is the very low C2 symmetry of the perturbative potential of square lattice with three alternating layers, while crease with increasing the difference between two rods, and in the case of the layered hexagonal lattice, the symmetry of at last disappear for the last crystal ͓Fig. 5͑c͔͒. The overlap the perturbative potential is C3v . It is generally easier to find of the E- and H band gaps and, as a result, the appearing of complete band gaps in hexagonal lattices due to its high the complete band gap, are described by the relative move- symmetry, C6v . The perturbation of this lattice with a C3v ment of the bands related to photon states with different po- potential still maintains a high enough symmetry to maintain larizations. This problem is beyond the developed model. an overlapping of the band gaps for both polarizations, this resulting in a complete band gap. This fact is an advantage IV. CONCLUSION for the complex hexagonal structures to be used as photonic We have developed in this paper the symmetrical model10 isolators. In contrast, the low symmetry of the complex for analyzing the band spectrum of the complex hexagonal square lattices results in a complete splitting of the degener- photonic crystals. We have shown that this symmetrical ate states at all symmetrical points, which is not the case for the complex hexagonal crystals. Therefore, by varying the model allows one to analyze the band spectrum and to pre- ͑ ͒ dict the opening of the band gaps for the whole class of parameters radii or dielectric constants of the rods of the complex hexagonal photonic crystals. complex square structures it is possible to create a band gap Within the framework of the developed model, a qualita- for a specific polarization of light. However, complete band tive understanding of the evolution of the band spectrum of gaps for such low-symmetry structures are difficult to the complex crystals can be obtained, starting from the band achieve. This fact is of great interest in possibly designing a spectrum of the prophase and symmetry of the perturbative complex square lattice based photonic insulator based on potential only, without any numerical calculations. The com- materials with low dielectric constant contrast. mon scheme for the analysis of the band spectrum of the The theoretical background of perturbative approach of complex crystals is to select the simplest prophase with the this model was based on the plane-wave expansion for the most symmetrical spectrum, and then to follow the genera- wave function of the prophase. We would like to point out tion of the band spectrum of the complex photonic crystals that further theoretical development of this approach holds a when including the perturbative potentials. The generation of possibility for the tight-binding formulation, based on using the Mai resonances as a basis for the prophase wave the band spectrum of the complex crystals goes in two steps. 8 In the first step, the band spectrum of the prophase, governed functions. In this case the problem may be reduced to an by the corresponding vector Q, is folded into the Brillouin analytical solution, having a simple numerical implementa- zone of the perturbative phase. In the second step, the degen- tion. eracy of the band spectrum is lifted by the perturbative po- To conclude, the developed symmetrical model has been tential. We emphasize that, without any fitting procedure, the shown to be useful for understanding and for the prediction results of the perturbative analysis have shown a reasonable of the points in the Brillouin zone where the band gap can be agreement with the plane-wave calculations of the complex opened by some lattice perturbation. The hexagonal complex photonic crystals. crystals considered here present an entire class of photonic We now compare the complex hexagonal photonic crys- structures with predictable and tunable properties that can be tals with complex square photonic lattices. It is important to applicable both for photonic insulator and photonic conduc- note that the evolution of the band spectrum of the complex tor devices. crystals with hexagonal and square10 symmetries is similar ACKNOWLEDGMENTS within the the theoretical framework presented. In both cases, the qualitative understanding of the changes in the We would like to acknowledge the support from the Cen- band spectrum can be obtained from the band spectrum of ter for Collective Phenomena in Restricted Geometries ͑Penn the prophase and the symmetry of the prophase potentials. State MRSEC͒ under NSF Grant No. DMR-00800190, and For both symmetries, the band gap inside the Brillouin zone the National Science Foundation Grant No. ECS-9988685.

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