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sapphire photonic crystals
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Physica E 17 (2003) 423 – 425 www.elsevier.com/locate/physe
Equifrequency surfaces in GaN/sapphire photonic crystals D. Peyradea , J. Torresa , D. Coquillata;∗ , R. Legrosa , J.P. Lascaraya , Y. Chenb , L. Manin-Ferlazzob , S. Ru4enacha , O. Briota , M. Le Vassor d’Yervillea , E. Centenoa , D. Cassagnea , J.P. Alberta a GES,
UMR 5650, CNRS-Universite Montpellier II, Pl. E. Bataillon, Montpellier 34095, France b LPN-CNRS, Route de Nozay, Marcoussis 91460, France
Abstract Photonic crystals are a new class of materials where photonic band gaps, large dispersion and anisotropy occur. By exploiting these properties GaN photonic crystals should have important potential for optoelectronic applications, principally in the areas of high-e;ciency light emitters and second-harmonic generators. We present measurements of the equifrequency surfaces of the radiative Bloch modes for a photonic crystal etched in a GaN/sapphire =lm. The photonic band structure is calculated by using a scattering matrix method that reproduces well the anisotropy of the equifrequency surfaces exhibited by the photonic crystal. ? 2002 Elsevier Science B.V. All rights reserved. PACS: 42.70.Qs; 42.65.Ky Keywords: Photonic crystal; GaN; Dispersion anisotropy; Equifrequency surface
Photonic crystals (PCs) are arti=cial materials in which the dielectric constant is periodically modulated at length scale comparable to optical wavelengths. The photonic band structure is characterised by photonic band gaps, large dispersions and anisotropy. By exploiting these properties, a number of improved optoelectronic devices such as high-e;ciency light emitters and second-harmonic generators can be realised. On the other hand, the III–V nitride semiconductors have enormous potential for UV and visible light-emitting diodes. PC structures fabricated in light-emitting diodes based on III–V nitrides have the potential to enhance the spontaneous emission ∗ Corresponding author. Tel.: +33-(0)4-6714-3239; fax: +33(0)4-6714-3760. E-mail address: [email protected] (D. Coquillat).
by changing the density of electromagnetic states at the emission frequency and by increasing vertical out-coupling [1]. Moreover, because of their large second-order optical nonlinearity, thin =lms of III–V nitrides might be useful as non-linear optical materials for optical wavelength conversion. A PC etched in a waveguide =lm can also be used for e;cient second-harmonic generation [2]. These phenomena, e;cient extraction and enhanced second-harmonic generation from an III–V nitride structure modi=ed by a PC, are sensitive to the detailed photonic band structure of the radiative modes. Therefore, it appears that the rigorous topology of the photonic band structure in the overall 2-k space is important, because bands display very strong anisotropy. This work is devoted to the determination of the equifrequency surfaces (EFS) of radiative Bloch modes, obtained by
1386-9477/03/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S1386-9477(02)00903-7
424
D. Peyrade et al. / Physica E 17 (2003) 423 – 425
1
The structure we have fabricated is a monomode GaN on sapphire waveguide in the visible range, of thickness h = 260 nm, patterned with a 2D triangular lattice of holes with a periodicity a = 500 nm and a 0.22 =lling factor [3]. Systematic measurements of the transmission spectra were carried out for collimated white light incident on the surface of the 2D PC over a range the two angles and ’, where is the polar angle of incidence relative to the surface normal, and ’ denotes the azimuthal angle between the plane of incidence and one of the axes of symmetry of the photonic crystal. For certain angles of incidence , depending both on the frequency and on ’, light can couple to the radiative Bloch modes of the PC [4–7]. Fig. 1 shows experimental transmission spectra when the sample was rotated around its surface normal in the azimuthal direction (0◦ 6 ’ 6 30◦ ) at =xed value of , for s-polarised light. The in-plane wave vectors of the resonances at constant frequencies are displayed for all ’ in order to construct the detailed EFS. Resulting EFS for the =rst, the second and the sixth radiative Bloch modes are displayed in Figs. 2a–c for di4erent normalised frequencies (!a=2c). For each EFS, the in-plane wave vector k = (!=c) sin of the excited mode has its end point on the contour of the EFS, and the incident angle varies with the azimuthal angle ’. The EFS
θ
= 21˚ Pol s
2 3 45
7
Transmission (arb. unit)
6
ϕ =28.7˚ ϕ =24.5˚ ϕ =20.1˚ ϕ =16.2˚ ϕ =11.8˚ ϕ =8.2˚ ϕ =5.8˚ ϕ =2.0˚ ϕ =0˚
1.2
1.4
1.6
1.8
2.0
2.2
2.4
Photon Energy (eV) Fig. 1. s-polarised transmission spectra for GaN photonic crystal with triangular lattice of holes. The incident angle is kept constant, =21◦ , whereas the azimuthal angle ’ varies from 0◦ to 30◦ .
cutting the dispersion surfaces at constant frequencies in the case of a GaN PC. 60
40000
90
(a)
-1
120
30 M
150 20000
Γ 0 180
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(b)
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M K
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Fig. 2. EFS as determined from the experimental transmission spectra (dots) and from calculation (solid) for the =rst (!a=2c = 0:455, 0.48, 0.50, 0.52, 0.55 from outer to inner) (a), second (!a=2c = 0:48, 0.50, 0.52, 0.55, 0.57 from outer to inner) (b), and sixth (!a=2c = 0:74, 0.72, 0.69, 0.66 from outer to inner) (c) lowest s-polarised radiative modes.
D. Peyrade et al. / Physica E 17 (2003) 423 – 425
exhibit a variety of remarkable shapes in contrast to the circular surface obtained for classical optical materials. The most notable feature of the plots for the =rst lowest radiative mode (Fig. 2a) are the almost perfect hexagonal shape reNecting the symmetry of the PC and the very Nat sides of the EFS. The EFS get smaller in size as the frequency increases, the EFS shape becomes rounded when the in-plane wave vectors approach the symmetry point. This e4ect can be explained by the fact that the mixing among di4erent reciprocal vectors becomes more pronounced close to the band edge around the symmetry points [8]. For the second mode (Fig. 2b), the greater the EFS size, the more noticeable is the distortion. Note that the almost hexagonal shape at 0.52 is facing in di4erent direction to those in Fig. 2a. Fig. 2c shows EFS corresponding to the sixth lowest mode. The dispersion surface (data no shown) of this mode possesses positive curvature in the 2-k space and is separated from the higher surfaces by a small band gap. In this frequency range, all of the EFS have quasicircular shape centred on point. Fig. 2a–c also show the simulated EFS obtained by using a scattering matrix method.
425
The complexity of the EFS is again apparent and the calculated EFS present a very good agreement with the measured ones. Small quantitative discrepancies could be related to the experimental uncertainty on angles and ’, and to the energy position estimation of the experimental Fano-shaped resonances. In conclusion, the design of GaN PC will enable to select di4erent resonant frequencies and di4erent directions of emitted light to create e;cient structures that emit UV and visible light. The conditions of quasi-phase matching for second-harmonic generation in these structures can be directly given by the construction of the EFS at ! and at 2!. References [1] [2] [3] [4] [5] [6] [7] [8]
A.A. Erchak, et al., Appl. Phys. Lett. 78 (2001) 563. A.R. Cowan, et al., Phys. Rev. B 65 (2002) 85 106. D. Peyrade, et al., Microelect. Engin. 57/58 (2001) 843. V. Astratov, et al., Appl. Phys. Lett. 77 (2000) 178. V. Pacradouni, et al., Phys. Rev. B 62 (2000) 4204. D. Coquillat, et al., Appl. Phys. B 73 (2001) 1. M. Galli, et al., Eur. Phys. J. B 27 (2002) 79. N. Notomi, Phys. Rev. B 62 (2000) 10 696.
Physica E 17 (2003) 423 – 425 www.elsevier.com/locate/physe
Equifrequency surfaces in GaN/sapphire photonic crystals D. Peyradea , J. Torresa , D. Coquillata;∗ , R. Legrosa , J.P. Lascaraya , Y. Chenb , L. Manin-Ferlazzob , S. Ru4enacha , O. Briota , M. Le Vassor d’Yervillea , E. Centenoa , D. Cassagnea , J.P. Alberta a GES,
UMR 5650, CNRS-Universite Montpellier II, Pl. E. Bataillon, Montpellier 34095, France b LPN-CNRS, Route de Nozay, Marcoussis 91460, France
Abstract Photonic crystals are a new class of materials where photonic band gaps, large dispersion and anisotropy occur. By exploiting these properties GaN photonic crystals should have important potential for optoelectronic applications, principally in the areas of high-e;ciency light emitters and second-harmonic generators. We present measurements of the equifrequency surfaces of the radiative Bloch modes for a photonic crystal etched in a GaN/sapphire =lm. The photonic band structure is calculated by using a scattering matrix method that reproduces well the anisotropy of the equifrequency surfaces exhibited by the photonic crystal. ? 2002 Elsevier Science B.V. All rights reserved. PACS: 42.70.Qs; 42.65.Ky Keywords: Photonic crystal; GaN; Dispersion anisotropy; Equifrequency surface
Photonic crystals (PCs) are arti=cial materials in which the dielectric constant is periodically modulated at length scale comparable to optical wavelengths. The photonic band structure is characterised by photonic band gaps, large dispersions and anisotropy. By exploiting these properties, a number of improved optoelectronic devices such as high-e;ciency light emitters and second-harmonic generators can be realised. On the other hand, the III–V nitride semiconductors have enormous potential for UV and visible light-emitting diodes. PC structures fabricated in light-emitting diodes based on III–V nitrides have the potential to enhance the spontaneous emission ∗ Corresponding author. Tel.: +33-(0)4-6714-3239; fax: +33(0)4-6714-3760. E-mail address: [email protected] (D. Coquillat).
by changing the density of electromagnetic states at the emission frequency and by increasing vertical out-coupling [1]. Moreover, because of their large second-order optical nonlinearity, thin =lms of III–V nitrides might be useful as non-linear optical materials for optical wavelength conversion. A PC etched in a waveguide =lm can also be used for e;cient second-harmonic generation [2]. These phenomena, e;cient extraction and enhanced second-harmonic generation from an III–V nitride structure modi=ed by a PC, are sensitive to the detailed photonic band structure of the radiative modes. Therefore, it appears that the rigorous topology of the photonic band structure in the overall 2-k space is important, because bands display very strong anisotropy. This work is devoted to the determination of the equifrequency surfaces (EFS) of radiative Bloch modes, obtained by
1386-9477/03/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S1386-9477(02)00903-7
424
D. Peyrade et al. / Physica E 17 (2003) 423 – 425
1
The structure we have fabricated is a monomode GaN on sapphire waveguide in the visible range, of thickness h = 260 nm, patterned with a 2D triangular lattice of holes with a periodicity a = 500 nm and a 0.22 =lling factor [3]. Systematic measurements of the transmission spectra were carried out for collimated white light incident on the surface of the 2D PC over a range the two angles and ’, where is the polar angle of incidence relative to the surface normal, and ’ denotes the azimuthal angle between the plane of incidence and one of the axes of symmetry of the photonic crystal. For certain angles of incidence , depending both on the frequency and on ’, light can couple to the radiative Bloch modes of the PC [4–7]. Fig. 1 shows experimental transmission spectra when the sample was rotated around its surface normal in the azimuthal direction (0◦ 6 ’ 6 30◦ ) at =xed value of , for s-polarised light. The in-plane wave vectors of the resonances at constant frequencies are displayed for all ’ in order to construct the detailed EFS. Resulting EFS for the =rst, the second and the sixth radiative Bloch modes are displayed in Figs. 2a–c for di4erent normalised frequencies (!a=2c). For each EFS, the in-plane wave vector k = (!=c) sin of the excited mode has its end point on the contour of the EFS, and the incident angle varies with the azimuthal angle ’. The EFS
θ
= 21˚ Pol s
2 3 45
7
Transmission (arb. unit)
6
ϕ =28.7˚ ϕ =24.5˚ ϕ =20.1˚ ϕ =16.2˚ ϕ =11.8˚ ϕ =8.2˚ ϕ =5.8˚ ϕ =2.0˚ ϕ =0˚
1.2
1.4
1.6
1.8
2.0
2.2
2.4
Photon Energy (eV) Fig. 1. s-polarised transmission spectra for GaN photonic crystal with triangular lattice of holes. The incident angle is kept constant, =21◦ , whereas the azimuthal angle ’ varies from 0◦ to 30◦ .
cutting the dispersion surfaces at constant frequencies in the case of a GaN PC. 60
40000
90
(a)
-1
120
30 M
150 20000
Γ 0 180
0
K
20000 210
330
40000 240
60000
In-Plane Wavevector (cm )
-1
In-Plane Wavevector (cm )
90 60000
300
60000
120
(b)
60
40000 150
30
M
20000
Γ
0 180
0
K
20000 210
330
40000 60000
240
300
270
270
120
-1
In-Plane Wavevector (cm )
90 60000
60
(c)
40000 150
30
ky
M
20000
Γ
0 180
0
K
Γ
M K
20000 210
kx
330
40000 60000
240
300 270
Fig. 2. EFS as determined from the experimental transmission spectra (dots) and from calculation (solid) for the =rst (!a=2c = 0:455, 0.48, 0.50, 0.52, 0.55 from outer to inner) (a), second (!a=2c = 0:48, 0.50, 0.52, 0.55, 0.57 from outer to inner) (b), and sixth (!a=2c = 0:74, 0.72, 0.69, 0.66 from outer to inner) (c) lowest s-polarised radiative modes.
D. Peyrade et al. / Physica E 17 (2003) 423 – 425
exhibit a variety of remarkable shapes in contrast to the circular surface obtained for classical optical materials. The most notable feature of the plots for the =rst lowest radiative mode (Fig. 2a) are the almost perfect hexagonal shape reNecting the symmetry of the PC and the very Nat sides of the EFS. The EFS get smaller in size as the frequency increases, the EFS shape becomes rounded when the in-plane wave vectors approach the symmetry point. This e4ect can be explained by the fact that the mixing among di4erent reciprocal vectors becomes more pronounced close to the band edge around the symmetry points [8]. For the second mode (Fig. 2b), the greater the EFS size, the more noticeable is the distortion. Note that the almost hexagonal shape at 0.52 is facing in di4erent direction to those in Fig. 2a. Fig. 2c shows EFS corresponding to the sixth lowest mode. The dispersion surface (data no shown) of this mode possesses positive curvature in the 2-k space and is separated from the higher surfaces by a small band gap. In this frequency range, all of the EFS have quasicircular shape centred on point. Fig. 2a–c also show the simulated EFS obtained by using a scattering matrix method.
425
The complexity of the EFS is again apparent and the calculated EFS present a very good agreement with the measured ones. Small quantitative discrepancies could be related to the experimental uncertainty on angles and ’, and to the energy position estimation of the experimental Fano-shaped resonances. In conclusion, the design of GaN PC will enable to select di4erent resonant frequencies and di4erent directions of emitted light to create e;cient structures that emit UV and visible light. The conditions of quasi-phase matching for second-harmonic generation in these structures can be directly given by the construction of the EFS at ! and at 2!. References [1] [2] [3] [4] [5] [6] [7] [8]
A.A. Erchak, et al., Appl. Phys. Lett. 78 (2001) 563. A.R. Cowan, et al., Phys. Rev. B 65 (2002) 85 106. D. Peyrade, et al., Microelect. Engin. 57/58 (2001) 843. V. Astratov, et al., Appl. Phys. Lett. 77 (2000) 178. V. Pacradouni, et al., Phys. Rev. B 62 (2000) 4204. D. Coquillat, et al., Appl. Phys. B 73 (2001) 1. M. Galli, et al., Eur. Phys. J. B 27 (2002) 79. N. Notomi, Phys. Rev. B 62 (2000) 10 696.