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Pressure effects on the band diagram in two-dimensional photonic crystals composed by cylindrical-shell rods
Optik - International Journal for Light and Electron Optics xxx (xxxx) xxxx
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Pressure effects on the band diagram in two-dimensional photonic crystals composed by cylindrical-shell rods Francis Segovia-Chavesa,b, Herbert Vinck-Posadaa a b
Grupo de Superconductividad y Nanotecnología, Departamento de Física, Universidad Nacional de Colombia AA 055051 Bogotá-Colombia Grupo de Física Teórica, Programa de Física, Universidad Surcolombiana AA 385 Neiva-Colombia
A R T IC LE I N F O
ABS TRA CT
Keywords: Two-dimensional photonic crystals Pressure Band diagram
In this study, we calculate the band diagram for the transverse magnetic polarization, in a twodimensional photonic crystal with a square lattice comprising cylindrical-shell rods embedded in air. Furthermore, the shells are made of GaAs with a dependence of the dielectric function on temperature and pressure. The results show that as the internal radius of the shell rods increases, the width of the first photonic band gap (PBG) decreases. Moreover, we reported a shift to larger frequencies of the band diagram by increasing the hydrostatic pressure at a fixed temperature value. Additionally, for the defective photonic crystal within the PBG, we may observe a defective mode, in which frequency increases with the pressure.
1. Introduction Photonic crystals (PC) are nanostructures constructed by alternating materials of different dielectric constant and magnetic permeability. The optical properties of PCs allow manipulation of light in a same way as the crystalline potential does with electrons [1]. Among the different properties exhibited by PCs, the photonic band gaps (PBGs) are regions in the band diagram where light cannot propagate. However, the light may be contained within the PBGs by inserting or removing impurities, such as point or line defects [2]. Materials commonly used to build PCs include dielectrics [3], semiconductors [4], superconductors [5] and nanocomposites [6]. In 1887, Lord Rayleigh first studied optical responses from the PCs and discovered that light propagation in onedimensional stratified media fully reflects depending on the angle of incidence [7]. However, PCs was not conceived for another one hundred years; then, in 1987, two different papers with independent interests were published [8,9]: Eli Yablonovitch proposed an arrangement of three-dimensional structures that could suppress spontaneous emission, while Sajeev John proposed a disordered dielectric superlattice that allowed a strong location of photons. After the PBG was effectively used to manipulate light, this field of research developed considerable interest rapidly with numerous applications in sensors for the control and detection of gases, temperature, pressure and microorganisms [10–13]. This study focuses on calculating the band diagram in a regular and defective two-dimensional (2D) PC, exhibiting a periodic pattern in the (xy) plane and remaining invariant in the direction perpendicular to the said plane (z direction). The 2D-PC dispersers are cylindrical-shell rods of circular cross-section with GaAs-shells. Herein, we assume that the GaAs dielectric function depends on both temperature and applied pressure. The document is organised as follows: Section 2 provides the theoretical framework and describes the 2D-PC model used. Section 3 denotes the main results for the band diagram. Finally, our conclusions are summarised in Section 4. 2. Theoretical Analyses Using the plane wave expansion method, we calculate the band diagram in the 2D-PC that is determined by the solution of the https://doi.org/10.1016/j.ijleo.2019.163641 Received 3 August 2019; Accepted 13 October 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Please cite this article as: Francis Segovia-Chaves and Herbert Vinck-Posada, Optik - International Journal for Light and Electron Optics, https://doi.org/10.1016/j.ijleo.2019.163641
Optik - International Journal for Light and Electron Optics xxx (xxxx) xxxx
F. Segovia-Chaves and H. Vinck-Posada
→ → → → eigenvalues equations for the electric (E ( r ) ) and magnetic (H ( r ) ) fields [14]. For dielectric and non-magnetic media, the eigenvalues equations are given by 2
ˆΘ H→ (→r ) = ωc H→ (→r )
(1)
2
2
ˆ →E (→r ) = ωc →E (→r )
(2)
2
→ 1 → ∇ × is a Hermitian operator known as the Maxwell operator. The non-Hermitian electric field operator is given where Θ = ∇ × ϵ(→ r) → 1 → by = → ∇ × ( ∇ ×) . The frequency and speed of light are given by ω and c, respectively. ϵ( r ) → → → → Using the Bloch Theorem [1], the E ( r ) and H ( r ) may be expanded in plane waves:
(
ˆ
ˆ
→→ → E k ,n ( r ) =
)
→
→
→ → →
∑ E →k ,n (G ) ei ( k + G )· r → G
→ → H→ k ,n ( r ) =
(3)
→ → → → → ∑ H →k ,n (G ) ei ( k + G )· r → G
(4) → → where the reciprocal lattice and wave vectors are represented by G and k , respectively. Similarly, the dielectric function expands in → plane waves with Fourier expansion coefficients χ (G ) , then
1 = r) ϵ(→
→
→→
∑ χ (G ) eiG · r → G
(5)
Herein, we consider a periodic pattern in the xy plane comprising cylindrical-shell rods of circular cross-section with an internal radius, Ri, and GaAs-shells with an external radius, Ro. The cylindrical-shell rods are embedded in air arranged in a square lattice of → → infinite length in the z direction. The two electromagnetic field polarizations are transverse electric (TE), wherein E ( r ) is in the → → same plane of the wave vectors, and transverse magnetic (TM), wherein the H ( r ) field is contained in the plane of the wave vectors → → and E ( r ) in the z direction [15]. For these two polarizations, the eigenvalue problems are given by
−
→
→ →
→
→
→
→
∑ χ (G∥ − G∥′)( k∥ + G∥′)·( k∥ + G∥) Hz,→k∥,n (G∥′) =
ωk2∥, n
G∥
→
→ →
→
→
∑ χ (G∥ − G∥′)| k∥ + G∥′ |2 Ez,→k∥,n (G∥′) =
ωk2∥, n c2
G∥
The
ωk2 , n ∥
c2
c2
→ Hz,→ (G∥) TEpolarization k∥, n
→ Ez,→ (G∥) TMpolarization k∥, n
(6)
(7)
→ eigenvalues are determined by the solutions to Eqs. (6)-(7), where the χ (G ) coefficients are given by
→ 1 χ (G∥) = S
→→
∫ ϵ−1(→r∥ ) e−iG · r d→r∥ ∥ ∥
(8)
→ where S the unit cell area. For the structure of interest, χ (G ) is given by 2 2 ⎧1 + π (Ro − Ri ) ⎛ 1 − 1⎞ 2 ⎪ a → ⎝ ϵGaAs ⎠ χ (G∥) = ⎨ 2π 1 − 1⎞ (Ro J1 (Ro G ) − Ri J1 (Ri G )) ⎪ 2 ⎛ ⎠ ⎩ a G ⎝ ϵGaAs ⎜
⎜
⎟
⎟
→ G∥ = 0 → G∥ ≠ 0
(9)
where a is the lattice constant, and J1(RG) is the first-class Bessel function [16,17]. In Eq. (9), the ϵGaAs dielectric function of the GaAsshells depends on both the hydrostatic pressure (P) and the temperature (T) as follows:
ϵGaAs (P , T ) = (ϵ 0 + AeT / T0 P ) e−αP
(10)
where ϵ0 = 12.446, A = 0.21125, T0=240.7 K, and α=0.00173 kbar
−1
[18].
3. Numerical results and discussion This section provides the numerical results focusing exclusively on the TM band diagram, which is generated through the diagonalization of Eq. (7). Fig. 1 (a) denotes the square lattice for cylindrical-shell rods at a temperature of 4 K, a pressure of 0 kbar and internal and external radii of Ri = 0.05a and Ro = 0.2a, respectively. The TM band diagram is calculated at the points of symmetry Γ(akx = 0, aky = 0), X(akx = π, aky = 0) and M(akx = π, aky = π), as shown in Fig. 1 (b). Herein, three PBGs (grey regions) may be observed in the following frequency intervals: 0.28 ≤ ωa/2πc ≤ 0.41, 0.54 ≤ ωa/2πc ≤ 0.56 and 0.70 ≤ ωa/2πc ≤ 0.76. Fig. 1 (c) displays the TM band diagram for three given internal radius values (0.05a, 0.1a and 0.18a) holding the external radius 2
Optik - International Journal for Light and Electron Optics xxx (xxxx) xxxx
F. Segovia-Chaves and H. Vinck-Posada
Fig. 1. a) Square lattice of cylindrical-shell rods with Ri = 0.05a. b) TM band diagram for Ri = 0.05a. c) TM band diagram for Ri = 0.05a (black line), Ri = 0.1a (blue line) and Ri = 0.18a (green line). The values used in the simulations are Ro = 0.2a, T=4 K and P=0 kbar.
3
Optik - International Journal for Light and Electron Optics xxx (xxxx) xxxx
F. Segovia-Chaves and H. Vinck-Posada
Fig. 2. TM band diagram for pressures at 0.0 kbar and 70 kbar. a) Ri = 0.05a. b) Ri = 0.1a. The values used in the simulations are Ro = 0.2a and T=4 K.
of the cylindrical-shell rods constant (Ro = 0.2a). When comparing the results from Fig. 1 (b), we determined that Ri = 0.1a indicates that the width of the first PBG decreases, while the width of the second and third PBGs increases. The 0.32 ≤ ωa/2πc ≤ 0.42, 0.61 ≤ ωa/2πc ≤ 0.57 and 0.73 ≤ ωa/2πc ≤ 0.8 intervals are located within these three PBGs (blue areas). However, when the value of the internal radius of the cylindrical-shell rods becomes close to the external radius, the TM band diagram does not display any PBG, as can be observed for Ri = 0.18a in Fig. 1 (c) Fig. 2 denotes the effects of hydrostatic pressure on the TM band diagram for two given internal radius values Ri = 0.05a (see Fig. 2 (a)) and Ri = 0.1a (see Fig. 2 (b)). Additionally, a shift towards higher frequency regions may be observed when increasing pressure in both TM band diagrams. When comparing with the results reported in Fig. 1 (b), for P=70 kbar, the width of the first PBG (located at 0.3 ≤ ωa/2πc ≤ 0.43) remains constant. However, when this pressure is increased, the width of the second and third PBG decreases, as shown in the TM band diagram of Fig. 2 (a) which is located between 0.57 ≤ ωa/2πc ≤ 0.574 and 0.72 ≤ ωa/ 2πc ≤ 0.77. Furthermore, when increasing both the internal radius of the cylindrical-shell rods (Ri = 0.1a) and the hydrostatic pressure (70 kbar), the same behaviour is shown for Ri = 0.05a. As the TM band diagram shifts towards higher frequencies, the width of the first PBG (located at 0.33 ≤ ωa/2πc ≤ 0.43) remains constant and the widths of the second (located at 0.6 ≤ ωa/2πc ≤ 0.62) and third (located at 0.75 ≤ ωa/2πc ≤ 0.81) PBGs decrease, as shown in Fig. 2 (b). When breaking the periodicity of the 2D-PC structure, defective modes originate within the PBG. Fig. 3 displays the TM band diagram of the defective crystal at the given pressure values (0.0 and 70 kbar) and different internal radius values of the cylindricalshell rods (0.05a and 0.1a) in two panels. In panel (a), the internal radius is Ri = 0.05a, where at a pressure of 0 kbar, a defective mode may be observed within the first PBG at ωa/2πc=0.37. This defective mode shifts positions at higher frequencies. When the hydrostatic pressure increases to 70 kbar, ωa/2πc=0.38. For Ri = 0.1a inside the PBG of the TM band diagram, and at P=0 kbar, the defective mode is located at ωa/2πc=0.385. However, when pressure increases to 70 kbar, the defective mode is now located at ωa/ 2πc=0.39. 4
Optik - International Journal for Light and Electron Optics xxx (xxxx) xxxx
F. Segovia-Chaves and H. Vinck-Posada
Fig. 3. TM band diagram for the defective 2D-PC at pressures of 0.0 and 70 kbar. a) Ri = 0.05a. b) Ri = 0.1a. The values used in the simulations are Ro = 0.2a and T=4 K.
4. Conclusions In this study, we calculate the TM band diagram for a 2D-PC with a square lattice comprising cylindrical-shell rods. The results show that when the internal radius of the shell increases, the width of the first PBG decreases. Additionally, the TM band diagram shifts to higher frequencies as pressure increases when holding the temperature value constant at 4 K. Further, the defective 2D-PC reports the presence of a defective mode within the PBG, which is tuned to higher frequencies as pressure increases. Acknowledgements F.S.-Ch. and H. V.-P gratefully acknowledge funding by COLCIENCIAS projects: ”Emisión en sistemas de Qubits Superconductores acoplados a la radiación. Código 110171249692, CT 293-2016, HERMES 31361” and ”Control dinámico de la emisión en sistemas de Qubits acoplados con cavidades no-estacionarias, HERMES 41611”. F.S.-Ch. also acknowledges to Vicerrectoría de Investigación, Universidad Surcolombiana Neiva-Huila. References [1] N. Aschcroft, D. Mermin, D. Wei, Solid state Physics: Revised Edition, Cengage Learning Asia, 2016. [2] J. Joannopoulos, S. Johnson, J. Winn, R. Meade, Photonic crystals: molding the flow of light, Princenton University Press, 2007. [3] F. Segovia-Chaves, H. Vinck-Posada, Dependence of the transmittance spectrum on temperature and thickness of superconducting defects coupled in dielectric one-dimensional photonic crystals, Optik 170 (2018) 384–390. [4] F. Segovia-Chaves, H. Vinck-Posada, Effects of hydrostatic pressure, temperature and angle of incidence on the transmittance spectrum of TE mode in a 1D semiconductor photonic Crystal, Optik 161 (2018) 64–69. [5] F. Segovia-Chaves, H. Vinck-Posada, Tuning of transmittance spectrum in a one-dimensional superconductor-semiconductor photonic crystal, Physica B: Condensed matter 543 (2018) 7–13.
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Optik - International Journal for Light and Electron Optics xxx (xxxx) xxxx
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[6] N.R. Ramanujam, K.S.J. Wilson, Optical properties of silver nanocomposite and photonic band gap-Pressure dependence, Optics Communications 368 (2016) 174–179. [7] L. Rayleigh, On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 24 (1887) 145–159. [8] E. Yablanovitch, Inhibited spontaneous emission in solid state physics and electronics, Phys. Rev. Lett. 58 (1987) 2059. [9] S. John, Localization of light, Phys. Rev. Lett. 58 (1987) 2486. [10] S.E.-S. Abd El-Ghany, Temperature sensors based on one dimensional photonic crystals with different double defects, Journal of Nanoelectronics and Optoelectronics 12 (2017) 1–8. [11] F. Segovia-Chaves, H. Vinck-Posada, Effects of pressure and thickness on the transmittance spectrum in a PS/PMMA photonic crystal, Optik 183 (2019) 918–923. [12] M. Li, F. He, Q. Liao, J. Liu, L. Xu, L. Jiang, Y. Song, S. Wang, Ultrasensitive DNA detection using photonic crystals, Angewandte Chemie (International ed. in English) 47 (2008) 7258–7262. [13] J. Garca-Ruprez, V. Toccafondo, M. Jos Banuls, J. Garcia Castello, A. Griol, S. Peransi-Llopis, A. Maquieira, Label-free antibody detection using band edge fringes in SOI planar photonic crystal waveguides in the slow-light regime, Opt. Express 18 (2010) 24276–24286. [14] M. Skorobogatiy, J. Yang, Fundamentals of photonic crystal guiding, Cambridge University Press, 2009. [15] F. Segovia-Chaves, H. Vinck-Posada, Dependence of photonic defect modes on hydrostatic pressure in a 2D hexagonal lattice, Physica E 104 (2018) 49–57. [16] F. Segovia-Chaves, H. Vinck-Posada, Effects of hydrostatic pressure on the band structure in two-dimensional semiconductor square photonic lattice with defect, Physica B 545 (2018) 203–209. [17] F. Segovia-Chaves, H. Vinck-Posada, Photonic band structure in square and triangular lattices of cylindrical-shell rods, Optik 181 (2019) 1013–1018. [18] G. Samara, Temperature and pressure dependences of the dielectric constants of semiconductors, Phys. Rev. B 27 (1983) 3494.
6
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Pressure effects on the band diagram in two-dimensional photonic crystals composed by cylindrical-shell rods Francis Segovia-Chavesa,b, Herbert Vinck-Posadaa a b
Grupo de Superconductividad y Nanotecnología, Departamento de Física, Universidad Nacional de Colombia AA 055051 Bogotá-Colombia Grupo de Física Teórica, Programa de Física, Universidad Surcolombiana AA 385 Neiva-Colombia
A R T IC LE I N F O
ABS TRA CT
Keywords: Two-dimensional photonic crystals Pressure Band diagram
In this study, we calculate the band diagram for the transverse magnetic polarization, in a twodimensional photonic crystal with a square lattice comprising cylindrical-shell rods embedded in air. Furthermore, the shells are made of GaAs with a dependence of the dielectric function on temperature and pressure. The results show that as the internal radius of the shell rods increases, the width of the first photonic band gap (PBG) decreases. Moreover, we reported a shift to larger frequencies of the band diagram by increasing the hydrostatic pressure at a fixed temperature value. Additionally, for the defective photonic crystal within the PBG, we may observe a defective mode, in which frequency increases with the pressure.
1. Introduction Photonic crystals (PC) are nanostructures constructed by alternating materials of different dielectric constant and magnetic permeability. The optical properties of PCs allow manipulation of light in a same way as the crystalline potential does with electrons [1]. Among the different properties exhibited by PCs, the photonic band gaps (PBGs) are regions in the band diagram where light cannot propagate. However, the light may be contained within the PBGs by inserting or removing impurities, such as point or line defects [2]. Materials commonly used to build PCs include dielectrics [3], semiconductors [4], superconductors [5] and nanocomposites [6]. In 1887, Lord Rayleigh first studied optical responses from the PCs and discovered that light propagation in onedimensional stratified media fully reflects depending on the angle of incidence [7]. However, PCs was not conceived for another one hundred years; then, in 1987, two different papers with independent interests were published [8,9]: Eli Yablonovitch proposed an arrangement of three-dimensional structures that could suppress spontaneous emission, while Sajeev John proposed a disordered dielectric superlattice that allowed a strong location of photons. After the PBG was effectively used to manipulate light, this field of research developed considerable interest rapidly with numerous applications in sensors for the control and detection of gases, temperature, pressure and microorganisms [10–13]. This study focuses on calculating the band diagram in a regular and defective two-dimensional (2D) PC, exhibiting a periodic pattern in the (xy) plane and remaining invariant in the direction perpendicular to the said plane (z direction). The 2D-PC dispersers are cylindrical-shell rods of circular cross-section with GaAs-shells. Herein, we assume that the GaAs dielectric function depends on both temperature and applied pressure. The document is organised as follows: Section 2 provides the theoretical framework and describes the 2D-PC model used. Section 3 denotes the main results for the band diagram. Finally, our conclusions are summarised in Section 4. 2. Theoretical Analyses Using the plane wave expansion method, we calculate the band diagram in the 2D-PC that is determined by the solution of the https://doi.org/10.1016/j.ijleo.2019.163641 Received 3 August 2019; Accepted 13 October 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Please cite this article as: Francis Segovia-Chaves and Herbert Vinck-Posada, Optik - International Journal for Light and Electron Optics, https://doi.org/10.1016/j.ijleo.2019.163641
Optik - International Journal for Light and Electron Optics xxx (xxxx) xxxx
F. Segovia-Chaves and H. Vinck-Posada
→ → → → eigenvalues equations for the electric (E ( r ) ) and magnetic (H ( r ) ) fields [14]. For dielectric and non-magnetic media, the eigenvalues equations are given by 2
ˆΘ H→ (→r ) = ωc H→ (→r )
(1)
2
2
ˆ →E (→r ) = ωc →E (→r )
(2)
2
→ 1 → ∇ × is a Hermitian operator known as the Maxwell operator. The non-Hermitian electric field operator is given where Θ = ∇ × ϵ(→ r) → 1 → by = → ∇ × ( ∇ ×) . The frequency and speed of light are given by ω and c, respectively. ϵ( r ) → → → → Using the Bloch Theorem [1], the E ( r ) and H ( r ) may be expanded in plane waves:
(
ˆ
ˆ
→→ → E k ,n ( r ) =
)
→
→
→ → →
∑ E →k ,n (G ) ei ( k + G )· r → G
→ → H→ k ,n ( r ) =
(3)
→ → → → → ∑ H →k ,n (G ) ei ( k + G )· r → G
(4) → → where the reciprocal lattice and wave vectors are represented by G and k , respectively. Similarly, the dielectric function expands in → plane waves with Fourier expansion coefficients χ (G ) , then
1 = r) ϵ(→
→
→→
∑ χ (G ) eiG · r → G
(5)
Herein, we consider a periodic pattern in the xy plane comprising cylindrical-shell rods of circular cross-section with an internal radius, Ri, and GaAs-shells with an external radius, Ro. The cylindrical-shell rods are embedded in air arranged in a square lattice of → → infinite length in the z direction. The two electromagnetic field polarizations are transverse electric (TE), wherein E ( r ) is in the → → same plane of the wave vectors, and transverse magnetic (TM), wherein the H ( r ) field is contained in the plane of the wave vectors → → and E ( r ) in the z direction [15]. For these two polarizations, the eigenvalue problems are given by
−
→
→ →
→
→
→
→
∑ χ (G∥ − G∥′)( k∥ + G∥′)·( k∥ + G∥) Hz,→k∥,n (G∥′) =
ωk2∥, n
G∥
→
→ →
→
→
∑ χ (G∥ − G∥′)| k∥ + G∥′ |2 Ez,→k∥,n (G∥′) =
ωk2∥, n c2
G∥
The
ωk2 , n ∥
c2
c2
→ Hz,→ (G∥) TEpolarization k∥, n
→ Ez,→ (G∥) TMpolarization k∥, n
(6)
(7)
→ eigenvalues are determined by the solutions to Eqs. (6)-(7), where the χ (G ) coefficients are given by
→ 1 χ (G∥) = S
→→
∫ ϵ−1(→r∥ ) e−iG · r d→r∥ ∥ ∥
(8)
→ where S the unit cell area. For the structure of interest, χ (G ) is given by 2 2 ⎧1 + π (Ro − Ri ) ⎛ 1 − 1⎞ 2 ⎪ a → ⎝ ϵGaAs ⎠ χ (G∥) = ⎨ 2π 1 − 1⎞ (Ro J1 (Ro G ) − Ri J1 (Ri G )) ⎪ 2 ⎛ ⎠ ⎩ a G ⎝ ϵGaAs ⎜
⎜
⎟
⎟
→ G∥ = 0 → G∥ ≠ 0
(9)
where a is the lattice constant, and J1(RG) is the first-class Bessel function [16,17]. In Eq. (9), the ϵGaAs dielectric function of the GaAsshells depends on both the hydrostatic pressure (P) and the temperature (T) as follows:
ϵGaAs (P , T ) = (ϵ 0 + AeT / T0 P ) e−αP
(10)
where ϵ0 = 12.446, A = 0.21125, T0=240.7 K, and α=0.00173 kbar
−1
[18].
3. Numerical results and discussion This section provides the numerical results focusing exclusively on the TM band diagram, which is generated through the diagonalization of Eq. (7). Fig. 1 (a) denotes the square lattice for cylindrical-shell rods at a temperature of 4 K, a pressure of 0 kbar and internal and external radii of Ri = 0.05a and Ro = 0.2a, respectively. The TM band diagram is calculated at the points of symmetry Γ(akx = 0, aky = 0), X(akx = π, aky = 0) and M(akx = π, aky = π), as shown in Fig. 1 (b). Herein, three PBGs (grey regions) may be observed in the following frequency intervals: 0.28 ≤ ωa/2πc ≤ 0.41, 0.54 ≤ ωa/2πc ≤ 0.56 and 0.70 ≤ ωa/2πc ≤ 0.76. Fig. 1 (c) displays the TM band diagram for three given internal radius values (0.05a, 0.1a and 0.18a) holding the external radius 2
Optik - International Journal for Light and Electron Optics xxx (xxxx) xxxx
F. Segovia-Chaves and H. Vinck-Posada
Fig. 1. a) Square lattice of cylindrical-shell rods with Ri = 0.05a. b) TM band diagram for Ri = 0.05a. c) TM band diagram for Ri = 0.05a (black line), Ri = 0.1a (blue line) and Ri = 0.18a (green line). The values used in the simulations are Ro = 0.2a, T=4 K and P=0 kbar.
3
Optik - International Journal for Light and Electron Optics xxx (xxxx) xxxx
F. Segovia-Chaves and H. Vinck-Posada
Fig. 2. TM band diagram for pressures at 0.0 kbar and 70 kbar. a) Ri = 0.05a. b) Ri = 0.1a. The values used in the simulations are Ro = 0.2a and T=4 K.
of the cylindrical-shell rods constant (Ro = 0.2a). When comparing the results from Fig. 1 (b), we determined that Ri = 0.1a indicates that the width of the first PBG decreases, while the width of the second and third PBGs increases. The 0.32 ≤ ωa/2πc ≤ 0.42, 0.61 ≤ ωa/2πc ≤ 0.57 and 0.73 ≤ ωa/2πc ≤ 0.8 intervals are located within these three PBGs (blue areas). However, when the value of the internal radius of the cylindrical-shell rods becomes close to the external radius, the TM band diagram does not display any PBG, as can be observed for Ri = 0.18a in Fig. 1 (c) Fig. 2 denotes the effects of hydrostatic pressure on the TM band diagram for two given internal radius values Ri = 0.05a (see Fig. 2 (a)) and Ri = 0.1a (see Fig. 2 (b)). Additionally, a shift towards higher frequency regions may be observed when increasing pressure in both TM band diagrams. When comparing with the results reported in Fig. 1 (b), for P=70 kbar, the width of the first PBG (located at 0.3 ≤ ωa/2πc ≤ 0.43) remains constant. However, when this pressure is increased, the width of the second and third PBG decreases, as shown in the TM band diagram of Fig. 2 (a) which is located between 0.57 ≤ ωa/2πc ≤ 0.574 and 0.72 ≤ ωa/ 2πc ≤ 0.77. Furthermore, when increasing both the internal radius of the cylindrical-shell rods (Ri = 0.1a) and the hydrostatic pressure (70 kbar), the same behaviour is shown for Ri = 0.05a. As the TM band diagram shifts towards higher frequencies, the width of the first PBG (located at 0.33 ≤ ωa/2πc ≤ 0.43) remains constant and the widths of the second (located at 0.6 ≤ ωa/2πc ≤ 0.62) and third (located at 0.75 ≤ ωa/2πc ≤ 0.81) PBGs decrease, as shown in Fig. 2 (b). When breaking the periodicity of the 2D-PC structure, defective modes originate within the PBG. Fig. 3 displays the TM band diagram of the defective crystal at the given pressure values (0.0 and 70 kbar) and different internal radius values of the cylindricalshell rods (0.05a and 0.1a) in two panels. In panel (a), the internal radius is Ri = 0.05a, where at a pressure of 0 kbar, a defective mode may be observed within the first PBG at ωa/2πc=0.37. This defective mode shifts positions at higher frequencies. When the hydrostatic pressure increases to 70 kbar, ωa/2πc=0.38. For Ri = 0.1a inside the PBG of the TM band diagram, and at P=0 kbar, the defective mode is located at ωa/2πc=0.385. However, when pressure increases to 70 kbar, the defective mode is now located at ωa/ 2πc=0.39. 4
Optik - International Journal for Light and Electron Optics xxx (xxxx) xxxx
F. Segovia-Chaves and H. Vinck-Posada
Fig. 3. TM band diagram for the defective 2D-PC at pressures of 0.0 and 70 kbar. a) Ri = 0.05a. b) Ri = 0.1a. The values used in the simulations are Ro = 0.2a and T=4 K.
4. Conclusions In this study, we calculate the TM band diagram for a 2D-PC with a square lattice comprising cylindrical-shell rods. The results show that when the internal radius of the shell increases, the width of the first PBG decreases. Additionally, the TM band diagram shifts to higher frequencies as pressure increases when holding the temperature value constant at 4 K. Further, the defective 2D-PC reports the presence of a defective mode within the PBG, which is tuned to higher frequencies as pressure increases. Acknowledgements F.S.-Ch. and H. V.-P gratefully acknowledge funding by COLCIENCIAS projects: ”Emisión en sistemas de Qubits Superconductores acoplados a la radiación. Código 110171249692, CT 293-2016, HERMES 31361” and ”Control dinámico de la emisión en sistemas de Qubits acoplados con cavidades no-estacionarias, HERMES 41611”. F.S.-Ch. also acknowledges to Vicerrectoría de Investigación, Universidad Surcolombiana Neiva-Huila. References [1] N. Aschcroft, D. Mermin, D. Wei, Solid state Physics: Revised Edition, Cengage Learning Asia, 2016. [2] J. Joannopoulos, S. Johnson, J. Winn, R. Meade, Photonic crystals: molding the flow of light, Princenton University Press, 2007. [3] F. Segovia-Chaves, H. Vinck-Posada, Dependence of the transmittance spectrum on temperature and thickness of superconducting defects coupled in dielectric one-dimensional photonic crystals, Optik 170 (2018) 384–390. [4] F. Segovia-Chaves, H. Vinck-Posada, Effects of hydrostatic pressure, temperature and angle of incidence on the transmittance spectrum of TE mode in a 1D semiconductor photonic Crystal, Optik 161 (2018) 64–69. [5] F. Segovia-Chaves, H. Vinck-Posada, Tuning of transmittance spectrum in a one-dimensional superconductor-semiconductor photonic crystal, Physica B: Condensed matter 543 (2018) 7–13.
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