Robustness of bound states in the continuum in metasurface based on Ge$_2$Sb$_2$Te$_5$ versus structural imperfections

Technical Summary: Robustness of Bound States in the Continuum in Ge2Sb2Te5 Metasurfaces

Problem Statement
Bound states in the continuum (BICs) are localized states within a continuous spectrum that, in ideal periodic structures, possess infinite quality (Q) factors due to zero radiative loss. However, in practical photonic devices, ideal BICs are transformed into quasi-BICs (q-BICs) with finite Q factors due to material absorption, finite-size effects, and fabrication-induced geometric imperfections. Phase-change materials (PCMs), specifically Ge2Sb2Te5 (GST), offer a unique platform for reconfigurable BICs because their optical properties change reversibly between amorphous (a-GST) and crystalline (c-GST) phases. A critical challenge arises in GST-based metasurfaces: the interplay between strong material dispersion/absorption and inevitable lithographic imperfections. Specifically, etching processes often result in trapezoidal cross-sections rather than the ideal rectangular designs. The extent to which these geometric deviations, combined with the intrinsic material losses and dispersion of GST, degrade the Q factor of symmetry-protected BICs remains a subject requiring quantification.

Methodology
The authors employ a combined analytical and numerical approach to investigate a one-dimensional metasurface composed of GST bars on a SiO2 substrate.

• Structure: The system consists of a periodic array of GST bars with a period of 700 nm, height of 250 nm, and width of 500 nm. The study considers both amorphous and crystalline phases.
• Geometric Modeling: The authors model deviations from the ideal rectangular cross-section by introducing trapezoidal shapes. They analyze two scenarios:
  1. Uniform Trapezoids: All bars share an identical inclination angle $\alpha$, preserving the $C_2$ symmetry of the unit cell.
  2. Random Disorder: Each bar possesses independent, random sidewall inclination angles ($\alpha_\ell$) drawn from a uniform distribution, breaking the $C_2$ symmetry.
• Analytical Framework: To address the disorder, the authors utilize a supercell approach. They expand the permittivity profile into a Fourier series to derive the perturbation caused by random angle fluctuations. This allows for the derivation of an analytical expression for the disorder-induced Q factor ($Q_{dis}$) based on the mean square of the permittivity perturbation.
• Numerical Simulation: Finite element method (FEM) simulations are performed on supercells comprising 20 periods. Statistical ensembles of 100 structures for each disorder amplitude are averaged to determine the mean Q factor. The total Q factor is modeled as the inverse sum of disorder-induced and absorption-induced contributions ($Q_{tot}^{-1} = Q_{dis}^{-1} + Q_{abs}^{-1}$).

Key Contributions and Results

1. Impact of Uniform Trapezoidal Deformations:

   • Crystalline Phase (c-GST): The Q factor remains robust against uniform trapezoidal deformations. Since the permittivity of c-GST is nearly constant in the relevant wavelength range, the shift in resonance wavelength caused by the angle $\alpha$ does not significantly alter absorption losses. The preservation of $C_2$ symmetry ensures the topological protection of the BIC.
   • Amorphous Phase (a-GST): In contrast, the Q factor decreases as the trapezoidal angle increases. This degradation is driven by the strong dispersion of a-GST near the BIC wavelength. As the geometry changes, the resonance shifts to a spectral region where the material absorption (loss tangent, $\tan \delta$) is higher. The authors establish a scaling law where $Q \propto 1/\tan \delta(\alpha)$, identifying absorption changes accompanying the resonance shift as the dominant mechanism for Q-factor reduction, even when symmetry is preserved.

2. Impact of Random Geometric Disorder:

   • Lossless Case: In the absence of material loss, the authors analytically derive and numerically verify that the Q factor follows an inverse-quadratic scaling with the disorder amplitude ($\sigma$) and the mean inclination angle ($\alpha_0$). Specifically, $Q_{dis} \propto \sigma^{-2}$ and $Q_{dis} \propto \alpha_0^{-2}$. This scaling is attributed to the breaking of $C_2$ symmetry, which transforms the BIC into a finite-Q resonance.
   • With Material Loss: When intrinsic material absorption is included, the total Q factor becomes dominated by absorption ($Q_{abs}$). Consequently, the Q factor becomes largely insensitive to the disorder amplitude $\sigma$. The fluctuations in geometry cause only weak shifts in resonance frequency, resulting in negligible changes to the absorption loss. Thus, the Q factor remains stable across both GST phases despite random element-to-element variations.

Significance and Claims
The paper claims to quantify the robustness of BICs in GST-based metasurfaces against lithographic imperfections, specifically trapezoidal cross-sections. The primary significance lies in demonstrating that while geometric disorder theoretically scales the Q factor inversely quadratically in lossless systems, in realistic GST structures, intrinsic material absorption dominates the total loss. This dominance renders the Q factor tolerant to geometric imperfections.

The authors conclude that this combination of disorder resilience and phase-switching capability makes GST-based metasurfaces suitable for applications requiring stable, moderate Q factors rather than extreme Q values. They explicitly identify potential applications in tunable filters, optical switches, modulators, and reconfigurable sensing platforms, where the ability to switch between the telecom C-band (amorphous phase, ~1.55 $\mu$m) and the emerging 2-$\mu$m communication band (crystalline phase) via phase change is essential. The work suggests that for such devices, achieving the highest possible Q factor is less critical than maintaining a stable resonant response that is robust against fabrication tolerances.