Unsupervised machine learning to classify the confinement of waves in periodic superstructures.

We propose a rigorous method to classify the dimensionality of wave confinement by utilizing unsupervised machine learning to enhance the accuracy of our recently presented scaling method [Phys. Rev. Lett.129, 176401 (2022)10.1103/PhysRevLett.129.176401]. We apply the standard k-means++ algorithm as well as our own model-based algorithm to 3D superlattices of resonant cavities embedded in a 3D inverse woodpile photonic band gap crystal with a range of design parameters. We compare their results against each other and against the direct usage of the scaling method without clustering. Since the clustering algorithms require the set of confinement dimensionalities present in the system as an input, we investigate cluster validity indices (CVIs) as a means to find these values. We conclude that the most accurate outcome is obtained by first applying direct scaling to find the correct set of confinement dimensionalities, and subsequently utilizing our model-based clustering algorithm to refine the results.

Scaling Theory of Wave Confinement in Classical and Quantum Periodic Systems.

Functional defects in periodic media confine waves-acoustic, electromagnetic, electronic, spin, etc.-in various dimensions, depending on the structure of the defect. While defects are usually modeled by a superlattice with a typical band-structure representation of energy levels, determining the confinement associated with a given band is highly nontrivial and no analytical method is known to date. Therefore, we propose a rigorous method to classify the dimensionality of wave confinement. Starting from the confinement energy and the mode volume, we use finite-size scaling to find that ratios of these quantities raised to certain powers yield the confinement dimensionality of each band. Our classification has negligible additional computational costs compared to a band structure calculation and is valid for any type of wave, both quantum and classical, and in any dimension. In the quantum regime, we illustrate our method on electronic confinement in 2D hexagonal boron nitride (BN) with a nitrogen vacancy, in agreement with previous results. In the classical case, we study a three-dimensional photonic band gap cavity superlattice, where we identify novel acceptorlike behavior. We briefly discuss the generalization to quasiperiodic lattices.