# Tunable Photonic Band Gaps in Two-Dimensional Bravais–Moiré Photonic Crystal Composed of High-Tc Superconductors

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## Abstract

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## 1. Introduction

## 2. Description of the System

#### 2.1. Theoretical Framework

#### 2.2. Simulation Settings

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(Color online) (

**a**) Bravais–Moiré unit cell placed in regular position with commensurable rotation $\varphi =53.{13}^{\circ}$ with $r=3,s=1$ (BM-R3S1). (

**b**) Schematic diagram of the proposed waveguide. The unit cell utilized for the analysis is represented by the yellow rectangle with orange edges. The dashed rectangle indicates the direction along which the wave propagates inside the guide. $\beta $ is the propagation constant and ${k}_{x}$ is the x-component of the wavevector. (

**c**) Shows the Brillouin zone and the k-path in waveguide case. (

**d**) (in red) the cores removed from the unit cell in order to build the waveguide in (

**b**). In our model, the cylindrical active cores are assumed to be made of a high critical temperature $B{i}_{1.85}P{b}_{0.35}Sr{}_{2}C{a}_{2}C{u}_{3.1}{O}_{y}$ superconductor compound, and the background is air.

**Figure 2.**(Color online) (

**a**) Photonic gap mapping for TM modes in the BM-R3S1 structure with a lattice constant a, as depicted in Figure 1, for $T=15\phantom{\rule{0.166667em}{0ex}}$K. We choose this value of temperature for the sake of exemplifying. The radii of both rotated and unrotated cores are the same. (

**b**) The photonic gap mapping as a function of temperature for the value $r/a=0.0334$, at which the maximum gap is obtained in (

**a**). In this case, the maximum ($\Delta \omega =0.37$) appears for $T/{T}_{c}=1$. The lowest blue stripes represent the evolution of the cut-off frequency in each case. (

**c**) Dispersion relation for the same TM modes, corresponding to the filling fraction $r/a=0.0334$—associated to the widest photonic gap depicted in (

**a**), for $T=15\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}K$, $\Delta \omega =0.25$. Here, the cutoff frequency is ${\omega}_{c}a/2\pi c=0.67$. (

**d**) The plasma frequency of superconducting electrons ${\omega}_{sp}$ as a function of temperature.

**Figure 3.**(Color online) (

**a**) The gap mapping of TM modes for the photonic crystal with a BM-R3S1 unit cell (Figure 1a) a fixed value of the radius of unrotated cylinder, ${r}_{nr}/a=0.0334$, is kept and the radius of rotated cores, ${r}_{r}$, changes (see explanation in text). (

**b**) Gap mapping as a function of temperature for the configuration of the maximum gap in (

**a**): ${r}_{nr}/a=0.0334$ and ${r}_{r}/a=0.01$. The maximum PBG ($\Delta \omega =0.55$) is obtained for $T/{T}_{c}=1$. (

**c**) Dispersion relation for the same TM modes, corresponding to the filling fraction ${r}_{nr}/a=0.0334$ and ${r}_{r}/a=0.01$—associated to the widest photonic gap depicted in (

**a**)—with $T=15\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}K$, $\Delta \omega =0.39$, is located between $\omega a/2\pi c=1.68$ and $\omega a/2\pi c=2.07$, and a cutoff frequency of ${\omega}_{c}a/2\pi c=0.58$. The bottom stripes in the plots associate with the cut-off frequency.

**Figure 4.**(Color online) (

**a**) The same as in Figure 3 but for the fixed value of rotated cores, ${r}_{r}/a=0.0334$, and fo the changing of unrotated ones. (

**b**) The gap mapping as a function of temperature, for the geometry at which the widest gap in (

**a**) appears: ${r}_{r}/a=0.0334$ and ${r}_{nr}/a=0.01$. Again, when $t={T}_{c}$ (i.e., when the maximum phonic gap is obtained $\Delta \omega =0.33$), then $T/{T}_{c}=1$ is obtained. (

**c**) Dispersion relation for the same TM modes, corresponding to the same filling fraction with ${r}_{r}/a=0.0334$ and ${r}_{nr}/a=0.01$, for $T=15\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}K$. There is a double gap with almost the same amplitudes: $\Delta {\omega}_{1}=0.26$, located between $\omega a/2\pi c=1.83$ and $\omega a/2\pi c=2.09$ and $\Delta {\omega}_{2}=0.24$, located between $\omega a/2\pi c=1.56$ and $\omega a/2\pi c=1.80$, as well as a cutoff frequency of ${\omega}_{c}a/2\pi c=0.55$. The bottom stripes correspond to the cut-off frequency.

**Figure 5.**Light dispersion relations for the coupled resonator optical waveguide shown in Figure 1b (see text for the description). Red lines represent guided modes. $\beta a/\pi $ is the normalized propagation constant. (

**a**) Photonic unit cell parameters, taken to be the same used to generate Figure 2c. (

**b**) Produced with the same cell parameters used to generate Figure 3c. (

**c**) Cell parameters, in this case, are the same used to generate Figure 4c. Blue stripes correspond to continuum states, and (

**d**–

**f**) represent the enhanced views of PBG regions where guided modes of cases (

**a**–

**c**) respectively appear.

**Figure 6.**(

**a**) Evolution of the guided modes depicted in Figure 5a,d as a result of an increase in temperature. In addition, their calculated group velocities as functions of the propagation constant (

**b**).

**Table 1.**Main results from the analysis of gap mapping in the Bravais–Moiré photonic crystals with an R3S1 structure, showing different configurations of rotated (r) and unrotated ($nr$) cylindrical dielectric core radii.

System | ${\mathit{r}}_{\mathit{r}}$ (mm) | ${\mathit{r}}_{\mathbf{nr}}$ (mm) | $\frac{\mathit{\omega}\mathit{a}}{2\mathit{\pi}\mathit{c}}$ | The Mid-Gap Frequency of PBG $\left({\mathit{\omega}}_{\mathit{m}}\right)$ | PBG Width $\Delta \mathit{\omega}$ | Gap Mid-Gap Ratio $\left(\frac{\Delta \mathit{\omega}}{{\mathit{\omega}}_{\mathit{m}}}\right)$ | PBG Percentatge (%) |
---|---|---|---|---|---|---|---|

1 | $0.0334$ | $0.0334$ | 2.01–2.26 | 2.14 | 0.25 | 0.1168 | 11.68 |

2 | $0.01$ | $0.0334$ | 1.68–2.07 | 1.875 | 0.39 | 0.208 | 20.8 |

3 | $0.0334$ | $0.01$ | 1.83–2.09 | 1.96 | 0.26 | 0.132 | 13.2 |

**Table 2.**Maximum normalized group velocities corresponding to the guided modes analyzed in Figure 6.

T (K) | Mode | Max $|{\mathit{v}}_{\mathit{g}}|/\mathit{c}$ |
---|---|---|

15 | ${A}_{0}{B}_{0}$ | 0.0069 |

45 | ${A}_{0}{B}_{0}$ | 0.0063 |

75 | ${A}_{0}{B}_{0}$ | 0.0058 |

107 | ${A}_{0}{B}_{0}$ | 0.0054 |

**Table 3.**Maximum normalized group velocities corresponding to the guided modes analyzed in Figure 7.

T (K) | Mode | Max $|{\mathit{v}}_{\mathit{g}}|/\mathit{c}$ |
---|---|---|

15 | ${A}_{0}{B}_{0}$ | 0.0092 |

15 | ${A}_{1}{B}_{1}$ | 0.0058 |

45 | ${A}_{0}{B}_{0}$ | 0.0096 |

45 | ${A}_{1}{B}_{1}$ | 0.0043 |

75 | ${A}_{0}{B}_{0}$ | 0.0098 |

75 | ${A}_{1}{B}_{1}$ | 0.0027 |

107 | ${A}_{0}{B}_{0}$ | 0.0098 |

107 | ${A}_{1}{B}_{1}$ | 0.0013 |

**Table 4.**Maximum normalized group velocities corresponding to the guided modes analyzed in Figure 8.

T (K) | Mode | Max $|{\mathit{v}}_{\mathit{g}}|/\mathit{c}$ |
---|---|---|

15 | ${A}_{0}{B}_{0}$ | 0.0668 |

15 | ${A}_{1}{B}_{1}$ | 0.0733 |

45 | ${A}_{0}{B}_{0}$ | 0.0628 |

45 | ${A}_{1}{B}_{1}$ | 0.0643 |

75 | ${A}_{0}{B}_{0}$ | 0.0579 |

75 | ${A}_{1}{B}_{1}$ | 0.0571 |

107 | ${A}_{0}{B}_{0}$ | 0.0321 |

107 | ${A}_{1}{B}_{1}$ | 0.0242 |

**Table 5.**Maximum normalized group velocities corresponding to the guided modes analyzed in Figure 9.

T (K) | Mode | Max $|{\mathit{v}}_{\mathit{g}}|/\mathit{c}$ |
---|---|---|

15 | ${A}_{2}{B}_{2}$ | 0.0248 |

15 | ${A}_{3}{B}_{3}$ | 0.0686 |

45 | ${A}_{2}{B}_{2}$ | 0.0167 |

45 | ${A}_{3}{B}_{3}$ | 0.0842 |

75 | ${A}_{2}{B}_{2}$ | 0.0150 |

75 | ${A}_{3}{B}_{3}$ | 0.0917 |

107 | ${A}_{2}{B}_{2}$ | 0.0094 |

107 | ${A}_{3}{B}_{3}$ | 0.0958 |

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**MDPI and ACS Style**

Gómez-Urrea, H.A.; Cardona, J.G.; Mora-Ramos, M.E.; Duque, C.A.
Tunable Photonic Band Gaps in Two-Dimensional Bravais–Moiré Photonic Crystal Composed of High-Tc Superconductors. *Condens. Matter* **2023**, *8*, 51.
https://doi.org/10.3390/condmat8020051

**AMA Style**

Gómez-Urrea HA, Cardona JG, Mora-Ramos ME, Duque CA.
Tunable Photonic Band Gaps in Two-Dimensional Bravais–Moiré Photonic Crystal Composed of High-Tc Superconductors. *Condensed Matter*. 2023; 8(2):51.
https://doi.org/10.3390/condmat8020051

**Chicago/Turabian Style**

Gómez-Urrea, Hernán A., José G. Cardona, Miguel E. Mora-Ramos, and Carlos A. Duque.
2023. "Tunable Photonic Band Gaps in Two-Dimensional Bravais–Moiré Photonic Crystal Composed of High-Tc Superconductors" *Condensed Matter* 8, no. 2: 51.
https://doi.org/10.3390/condmat8020051