Abstract

By employing two-dimensional InGaAsP photonic band-edge lasers, we have experimentally demonstrated that a random mixture of two different photonic crystals (PCs) possesses a new band structure that is intermediate to that of the two host PCs. The photonic band-edges shift monotonically, but with a strong bowing effect, as the mixed PC system is systematically transformed from one PC to the other. The experimental observations are in excellent agreement with finite-difference time-domain simulations and model calculations based on virtual-crystal approximation with compositional disorder effect included.

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    [CrossRef]
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2009

S. Kim, Y. Park, K. Hwang, J. Lee, H. Jeon, and H. J. Kim, “High-power and large-alignment-tolerance fiber coupling of honeycomb-lattice photonic crystal Γ-point band-edge laser,” J. Opt. Soc. Am. B 26(7), 1330 (2009).
[CrossRef]

S. Kim, J. Lee, H. Jeon, and H. J. Kim, “Fiber-coupled surface-emitting photonic crystal band edge laser for biochemical sensor applications,” Appl. Phys. Lett. 94(13), 133503 (2009).
[CrossRef]

2008

H. J. Kim, D. U. Kim, Y. G. Roh, J. Yu, H. Jeon, and Q. H. Park, “Photonic crystal alloys: a new twist in controlling photonic band structure properties,” Opt. Express 16(9), 6579–6585 (2008).
[CrossRef] [PubMed]

C. Conti and A. Fratalocchi, “Dynamic light diffusion, three-dimensional Anderson localization and lasing in inverted opals,” Nat. Phys. 4(10), 794–798 (2008).
[CrossRef]

2007

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
[CrossRef] [PubMed]

2006

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys. 78(2), 455–481 (2006).
[CrossRef]

2005

H. J. Kim, Y. G. Roh, and H. Jeon, “Photonic bandgap engineering in mixed colloidal photonic crystals,” Jpn. J. Appl. Phys. 44(40), L1259–L1262 (2005).
[CrossRef]

2003

J. Mouette, C. Seassal, X. Letartre, P. Rojo-Romeo, J.-L. Leclercq, P. Regreny, P. Viktorovitch, E. Jalaguier, P. Perreau, and H. Moriceau, “Very low threshold vertical emitting laser operation in InP graphite photonic crystal slab on silicon,” Electron. Lett. 39(6), 526–528 (2003).
[CrossRef]

2002

M. Ferhat and F. Bechstedt, “First-principles calculations of gap bowing in InxGa1-xN and InxAl1-xN alloys: Relation to structural and thermodynamic properties,” Phys. Rev. B 65(7), 075213 (2002).
[CrossRef]

2001

S. Nojima, “Optical-gain enhancement in two-dimensional active photonic crystals,” J. Appl. Phys. 90(2), 545 (2001).
[CrossRef]

1990

S. J. Lee, T. S. Kwon, K. Nahm, and C. K. Kim, “Band-structure of ternary compound semiconductors beyond the virtual crystal approximation,” J. Phys. Condens. Matter 2(14), 3253–3257 (1990).
[CrossRef]

1987

F. Capasso, “Band-gap engineering - from physics and materials to new semiconductor-devices,” Science 235(4785), 172–176 (1987).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987).
[CrossRef] [PubMed]

1975

A. Baldereschi and K. Maschke, “Band structure of semiconductor alloys beyond the virtual crystal approximation. effect of compositional disorder on the energy gaps in GaPxAs1-x,” Solid State Commun. 16(1), 99–102 (1975).
[CrossRef]

1971

A. H. Lettington, D. Jones, and R. Sarginson, “Thermoreflectance studies of thin epitaxially deposited (InGa)P alloys,” J. Phys. C Solid State Phys. 4(12), 1534–1539 (1971).
[CrossRef]

1970

S. Bloom, “Bandgap Variation in Quaternary Alloys,” J. Appl. Phys. 41(4), 1864 (1970).
[CrossRef]

J. A. Van Vechten and T. K. Bergstresser, “Electronic Structures of Semiconductor Alloys,” Phys. Rev. B 1(8), 3351–3358 (1970).
[CrossRef]

1966

Z. I. Alferov and D. Z. Gabruzov, “Recombination radiation spectrum of GaAs with current excitation via p-n heterojunctions of GaP-GaAs,” Sov. Phys. Solid State 7, 1919 (1966).

1957

S. Larach, R. E. Shrader, and C. F. Stocker, “Anomalous variation of band gap with composition in zinc sulfo- and seleno-tellurides,” Phys. Rev. 108(3), 587–589 (1957).
[CrossRef]

1955

O. G. Folberth, “Mischkristallbildung Bei Aiii Bv-Verbindungen,” Z. Naturforsch. B 10a, 502 (1955).

1939

J. H. Gisolf, “The absorption spectrum of luminescent zinc-sulfide and zinc-cadmiumsulfide in connection with some optical, electrical and chemical properties,” Physica 6(1), 84 (1939).
[CrossRef]

1931

L. Nordheim, “Zur Elektronentheorie der Metalle. I and II,” Ann. Phys. 401,(5), 607–640 (1931).
[CrossRef]

1921

L. Vegard, “Die Konstitution der Mischkristalle und die Raumfüllung der Atome,” Z. Phys. 5(1), 17–26 (1921).
[CrossRef]

Alferov, Z. I.

Z. I. Alferov and D. Z. Gabruzov, “Recombination radiation spectrum of GaAs with current excitation via p-n heterojunctions of GaP-GaAs,” Sov. Phys. Solid State 7, 1919 (1966).

Baldereschi, A.

A. Baldereschi and K. Maschke, “Band structure of semiconductor alloys beyond the virtual crystal approximation. effect of compositional disorder on the energy gaps in GaPxAs1-x,” Solid State Commun. 16(1), 99–102 (1975).
[CrossRef]

Bartal, G.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
[CrossRef] [PubMed]

Bechstedt, F.

M. Ferhat and F. Bechstedt, “First-principles calculations of gap bowing in InxGa1-xN and InxAl1-xN alloys: Relation to structural and thermodynamic properties,” Phys. Rev. B 65(7), 075213 (2002).
[CrossRef]

Bergstresser, T. K.

J. A. Van Vechten and T. K. Bergstresser, “Electronic Structures of Semiconductor Alloys,” Phys. Rev. B 1(8), 3351–3358 (1970).
[CrossRef]

Bloom, S.

S. Bloom, “Bandgap Variation in Quaternary Alloys,” J. Appl. Phys. 41(4), 1864 (1970).
[CrossRef]

Capasso, F.

F. Capasso, “Band-gap engineering - from physics and materials to new semiconductor-devices,” Science 235(4785), 172–176 (1987).
[CrossRef] [PubMed]

Conti, C.

C. Conti and A. Fratalocchi, “Dynamic light diffusion, three-dimensional Anderson localization and lasing in inverted opals,” Nat. Phys. 4(10), 794–798 (2008).
[CrossRef]

Ferhat, M.

M. Ferhat and F. Bechstedt, “First-principles calculations of gap bowing in InxGa1-xN and InxAl1-xN alloys: Relation to structural and thermodynamic properties,” Phys. Rev. B 65(7), 075213 (2002).
[CrossRef]

Fishman, S.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
[CrossRef] [PubMed]

Folberth, O. G.

O. G. Folberth, “Mischkristallbildung Bei Aiii Bv-Verbindungen,” Z. Naturforsch. B 10a, 502 (1955).

Fratalocchi, A.

C. Conti and A. Fratalocchi, “Dynamic light diffusion, three-dimensional Anderson localization and lasing in inverted opals,” Nat. Phys. 4(10), 794–798 (2008).
[CrossRef]

Gabruzov, D. Z.

Z. I. Alferov and D. Z. Gabruzov, “Recombination radiation spectrum of GaAs with current excitation via p-n heterojunctions of GaP-GaAs,” Sov. Phys. Solid State 7, 1919 (1966).

Gisolf, J. H.

J. H. Gisolf, “The absorption spectrum of luminescent zinc-sulfide and zinc-cadmiumsulfide in connection with some optical, electrical and chemical properties,” Physica 6(1), 84 (1939).
[CrossRef]

Hwang, K.

Istrate, E.

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys. 78(2), 455–481 (2006).
[CrossRef]

Jalaguier, E.

J. Mouette, C. Seassal, X. Letartre, P. Rojo-Romeo, J.-L. Leclercq, P. Regreny, P. Viktorovitch, E. Jalaguier, P. Perreau, and H. Moriceau, “Very low threshold vertical emitting laser operation in InP graphite photonic crystal slab on silicon,” Electron. Lett. 39(6), 526–528 (2003).
[CrossRef]

Jeon, H.

S. Kim, J. Lee, H. Jeon, and H. J. Kim, “Fiber-coupled surface-emitting photonic crystal band edge laser for biochemical sensor applications,” Appl. Phys. Lett. 94(13), 133503 (2009).
[CrossRef]

S. Kim, Y. Park, K. Hwang, J. Lee, H. Jeon, and H. J. Kim, “High-power and large-alignment-tolerance fiber coupling of honeycomb-lattice photonic crystal Γ-point band-edge laser,” J. Opt. Soc. Am. B 26(7), 1330 (2009).
[CrossRef]

H. J. Kim, D. U. Kim, Y. G. Roh, J. Yu, H. Jeon, and Q. H. Park, “Photonic crystal alloys: a new twist in controlling photonic band structure properties,” Opt. Express 16(9), 6579–6585 (2008).
[CrossRef] [PubMed]

H. J. Kim, Y. G. Roh, and H. Jeon, “Photonic bandgap engineering in mixed colloidal photonic crystals,” Jpn. J. Appl. Phys. 44(40), L1259–L1262 (2005).
[CrossRef]

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987).
[CrossRef] [PubMed]

Jones, D.

A. H. Lettington, D. Jones, and R. Sarginson, “Thermoreflectance studies of thin epitaxially deposited (InGa)P alloys,” J. Phys. C Solid State Phys. 4(12), 1534–1539 (1971).
[CrossRef]

Kim, C. K.

S. J. Lee, T. S. Kwon, K. Nahm, and C. K. Kim, “Band-structure of ternary compound semiconductors beyond the virtual crystal approximation,” J. Phys. Condens. Matter 2(14), 3253–3257 (1990).
[CrossRef]

Kim, D. U.

Kim, H. J.

S. Kim, Y. Park, K. Hwang, J. Lee, H. Jeon, and H. J. Kim, “High-power and large-alignment-tolerance fiber coupling of honeycomb-lattice photonic crystal Γ-point band-edge laser,” J. Opt. Soc. Am. B 26(7), 1330 (2009).
[CrossRef]

S. Kim, J. Lee, H. Jeon, and H. J. Kim, “Fiber-coupled surface-emitting photonic crystal band edge laser for biochemical sensor applications,” Appl. Phys. Lett. 94(13), 133503 (2009).
[CrossRef]

H. J. Kim, D. U. Kim, Y. G. Roh, J. Yu, H. Jeon, and Q. H. Park, “Photonic crystal alloys: a new twist in controlling photonic band structure properties,” Opt. Express 16(9), 6579–6585 (2008).
[CrossRef] [PubMed]

H. J. Kim, Y. G. Roh, and H. Jeon, “Photonic bandgap engineering in mixed colloidal photonic crystals,” Jpn. J. Appl. Phys. 44(40), L1259–L1262 (2005).
[CrossRef]

Kim, S.

S. Kim, J. Lee, H. Jeon, and H. J. Kim, “Fiber-coupled surface-emitting photonic crystal band edge laser for biochemical sensor applications,” Appl. Phys. Lett. 94(13), 133503 (2009).
[CrossRef]

S. Kim, Y. Park, K. Hwang, J. Lee, H. Jeon, and H. J. Kim, “High-power and large-alignment-tolerance fiber coupling of honeycomb-lattice photonic crystal Γ-point band-edge laser,” J. Opt. Soc. Am. B 26(7), 1330 (2009).
[CrossRef]

Kwon, T. S.

S. J. Lee, T. S. Kwon, K. Nahm, and C. K. Kim, “Band-structure of ternary compound semiconductors beyond the virtual crystal approximation,” J. Phys. Condens. Matter 2(14), 3253–3257 (1990).
[CrossRef]

Larach, S.

S. Larach, R. E. Shrader, and C. F. Stocker, “Anomalous variation of band gap with composition in zinc sulfo- and seleno-tellurides,” Phys. Rev. 108(3), 587–589 (1957).
[CrossRef]

Leclercq, J.-L.

J. Mouette, C. Seassal, X. Letartre, P. Rojo-Romeo, J.-L. Leclercq, P. Regreny, P. Viktorovitch, E. Jalaguier, P. Perreau, and H. Moriceau, “Very low threshold vertical emitting laser operation in InP graphite photonic crystal slab on silicon,” Electron. Lett. 39(6), 526–528 (2003).
[CrossRef]

Lee, J.

S. Kim, Y. Park, K. Hwang, J. Lee, H. Jeon, and H. J. Kim, “High-power and large-alignment-tolerance fiber coupling of honeycomb-lattice photonic crystal Γ-point band-edge laser,” J. Opt. Soc. Am. B 26(7), 1330 (2009).
[CrossRef]

S. Kim, J. Lee, H. Jeon, and H. J. Kim, “Fiber-coupled surface-emitting photonic crystal band edge laser for biochemical sensor applications,” Appl. Phys. Lett. 94(13), 133503 (2009).
[CrossRef]

Lee, S. J.

S. J. Lee, T. S. Kwon, K. Nahm, and C. K. Kim, “Band-structure of ternary compound semiconductors beyond the virtual crystal approximation,” J. Phys. Condens. Matter 2(14), 3253–3257 (1990).
[CrossRef]

Letartre, X.

J. Mouette, C. Seassal, X. Letartre, P. Rojo-Romeo, J.-L. Leclercq, P. Regreny, P. Viktorovitch, E. Jalaguier, P. Perreau, and H. Moriceau, “Very low threshold vertical emitting laser operation in InP graphite photonic crystal slab on silicon,” Electron. Lett. 39(6), 526–528 (2003).
[CrossRef]

Lettington, A. H.

A. H. Lettington, D. Jones, and R. Sarginson, “Thermoreflectance studies of thin epitaxially deposited (InGa)P alloys,” J. Phys. C Solid State Phys. 4(12), 1534–1539 (1971).
[CrossRef]

Maschke, K.

A. Baldereschi and K. Maschke, “Band structure of semiconductor alloys beyond the virtual crystal approximation. effect of compositional disorder on the energy gaps in GaPxAs1-x,” Solid State Commun. 16(1), 99–102 (1975).
[CrossRef]

Moriceau, H.

J. Mouette, C. Seassal, X. Letartre, P. Rojo-Romeo, J.-L. Leclercq, P. Regreny, P. Viktorovitch, E. Jalaguier, P. Perreau, and H. Moriceau, “Very low threshold vertical emitting laser operation in InP graphite photonic crystal slab on silicon,” Electron. Lett. 39(6), 526–528 (2003).
[CrossRef]

Mouette, J.

J. Mouette, C. Seassal, X. Letartre, P. Rojo-Romeo, J.-L. Leclercq, P. Regreny, P. Viktorovitch, E. Jalaguier, P. Perreau, and H. Moriceau, “Very low threshold vertical emitting laser operation in InP graphite photonic crystal slab on silicon,” Electron. Lett. 39(6), 526–528 (2003).
[CrossRef]

Nahm, K.

S. J. Lee, T. S. Kwon, K. Nahm, and C. K. Kim, “Band-structure of ternary compound semiconductors beyond the virtual crystal approximation,” J. Phys. Condens. Matter 2(14), 3253–3257 (1990).
[CrossRef]

Nojima, S.

S. Nojima, “Optical-gain enhancement in two-dimensional active photonic crystals,” J. Appl. Phys. 90(2), 545 (2001).
[CrossRef]

Nordheim, L.

L. Nordheim, “Zur Elektronentheorie der Metalle. I and II,” Ann. Phys. 401,(5), 607–640 (1931).
[CrossRef]

Park, Q. H.

Park, Y.

Perreau, P.

J. Mouette, C. Seassal, X. Letartre, P. Rojo-Romeo, J.-L. Leclercq, P. Regreny, P. Viktorovitch, E. Jalaguier, P. Perreau, and H. Moriceau, “Very low threshold vertical emitting laser operation in InP graphite photonic crystal slab on silicon,” Electron. Lett. 39(6), 526–528 (2003).
[CrossRef]

Regreny, P.

J. Mouette, C. Seassal, X. Letartre, P. Rojo-Romeo, J.-L. Leclercq, P. Regreny, P. Viktorovitch, E. Jalaguier, P. Perreau, and H. Moriceau, “Very low threshold vertical emitting laser operation in InP graphite photonic crystal slab on silicon,” Electron. Lett. 39(6), 526–528 (2003).
[CrossRef]

Roh, Y. G.

H. J. Kim, D. U. Kim, Y. G. Roh, J. Yu, H. Jeon, and Q. H. Park, “Photonic crystal alloys: a new twist in controlling photonic band structure properties,” Opt. Express 16(9), 6579–6585 (2008).
[CrossRef] [PubMed]

H. J. Kim, Y. G. Roh, and H. Jeon, “Photonic bandgap engineering in mixed colloidal photonic crystals,” Jpn. J. Appl. Phys. 44(40), L1259–L1262 (2005).
[CrossRef]

Rojo-Romeo, P.

J. Mouette, C. Seassal, X. Letartre, P. Rojo-Romeo, J.-L. Leclercq, P. Regreny, P. Viktorovitch, E. Jalaguier, P. Perreau, and H. Moriceau, “Very low threshold vertical emitting laser operation in InP graphite photonic crystal slab on silicon,” Electron. Lett. 39(6), 526–528 (2003).
[CrossRef]

Sargent, E. H.

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys. 78(2), 455–481 (2006).
[CrossRef]

Sarginson, R.

A. H. Lettington, D. Jones, and R. Sarginson, “Thermoreflectance studies of thin epitaxially deposited (InGa)P alloys,” J. Phys. C Solid State Phys. 4(12), 1534–1539 (1971).
[CrossRef]

Schwartz, T.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
[CrossRef] [PubMed]

Seassal, C.

J. Mouette, C. Seassal, X. Letartre, P. Rojo-Romeo, J.-L. Leclercq, P. Regreny, P. Viktorovitch, E. Jalaguier, P. Perreau, and H. Moriceau, “Very low threshold vertical emitting laser operation in InP graphite photonic crystal slab on silicon,” Electron. Lett. 39(6), 526–528 (2003).
[CrossRef]

Segev, M.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
[CrossRef] [PubMed]

Shrader, R. E.

S. Larach, R. E. Shrader, and C. F. Stocker, “Anomalous variation of band gap with composition in zinc sulfo- and seleno-tellurides,” Phys. Rev. 108(3), 587–589 (1957).
[CrossRef]

Stocker, C. F.

S. Larach, R. E. Shrader, and C. F. Stocker, “Anomalous variation of band gap with composition in zinc sulfo- and seleno-tellurides,” Phys. Rev. 108(3), 587–589 (1957).
[CrossRef]

Van Vechten, J. A.

J. A. Van Vechten and T. K. Bergstresser, “Electronic Structures of Semiconductor Alloys,” Phys. Rev. B 1(8), 3351–3358 (1970).
[CrossRef]

Vegard, L.

L. Vegard, “Die Konstitution der Mischkristalle und die Raumfüllung der Atome,” Z. Phys. 5(1), 17–26 (1921).
[CrossRef]

Viktorovitch, P.

J. Mouette, C. Seassal, X. Letartre, P. Rojo-Romeo, J.-L. Leclercq, P. Regreny, P. Viktorovitch, E. Jalaguier, P. Perreau, and H. Moriceau, “Very low threshold vertical emitting laser operation in InP graphite photonic crystal slab on silicon,” Electron. Lett. 39(6), 526–528 (2003).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987).
[CrossRef] [PubMed]

Yu, J.

Ann. Phys.

L. Nordheim, “Zur Elektronentheorie der Metalle. I and II,” Ann. Phys. 401,(5), 607–640 (1931).
[CrossRef]

Appl. Phys. Lett.

S. Kim, J. Lee, H. Jeon, and H. J. Kim, “Fiber-coupled surface-emitting photonic crystal band edge laser for biochemical sensor applications,” Appl. Phys. Lett. 94(13), 133503 (2009).
[CrossRef]

Electron. Lett.

J. Mouette, C. Seassal, X. Letartre, P. Rojo-Romeo, J.-L. Leclercq, P. Regreny, P. Viktorovitch, E. Jalaguier, P. Perreau, and H. Moriceau, “Very low threshold vertical emitting laser operation in InP graphite photonic crystal slab on silicon,” Electron. Lett. 39(6), 526–528 (2003).
[CrossRef]

J. Appl. Phys.

S. Nojima, “Optical-gain enhancement in two-dimensional active photonic crystals,” J. Appl. Phys. 90(2), 545 (2001).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

(a) Schematic diagram of the 2D honeycomb-lattice MPC, S1– x L x , and its dielectric constant profile. The shaded regions indicate the index difference between two types of photonic atoms, S and L. (b) Photonic band diagram of the honeycomb-lattice PC for x = 0 (blue) and x = 1 (red). The insets show the magnetic field profiles for the monopole and dipole band-edge modes. (c) Scanning electron microscope images of a fabricated honeycomb-lattice MPC BEL structure. The two types of air holes with different radii are marked by the letters S and L as a guide for the eye. The radii of the air holes are r S = 0.380a and r L = 0.415a, where a = 450 nm is the distance between the closest air-holes.

Fig. 2
Fig. 2

(a) BEL lasing spectra for the dipole (left) and monopole (right) modes for a few representative composition ratios, x = 0, ⅓, ⅔, 1. (b) MPC BEL lasing spectra for the same composition ratio (x = ⅓) but different configurations.

Fig. 3
Fig. 3

(a) MPC configurations for representative composition ratios (x = 0, ⅓, ⅔, 1). (b) Modal spectra obtained by FDTD simulations on the corresponding MPC configurations given in (a). (c) Magnetic field distributions in real space, calculated at the peak frequencies observed in (b). (d) Intensity distributions in momentum space, obtained by Fourier transforming the field distributions shown in (c).

Fig. 4
Fig. 4

Experimentally determined lasing frequencies (solid squares) of the MPC BELs and theoretical Γ-point resonant frequencies (open circles) calculated by FDTD for the corresponding MPCs: (a) dipole mode and (b) monopole mode. Also shown are the band-edge frequencies calculated by PWE based on the virtual crystal models with and without the disorder effect. All frequencies are plotted as a function of the mixing composition x.

Tables (1)

Tables Icon

Table 1 Bowing parameter c for the monopole and dipole modes at the Γ-point band-edge.

Equations (3)

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ω M P C ( x ) = a x + b ( 1 x ) + c x ( 1 x ) .
ε ( r ) = ε V C ( r ) + ε d i s ( r ) ,
ε ( r ) = x ε L ( r ) + ( 1 x ) ε S ( r ) + β [ x ( 1 x ) ] 1 / 2 Δ ε ( r ) .

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