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Functional defects in periodic media confine waves - acoustic,electromagnetic, electronic, spin, etc. - in various dimensions, depending onthe structure of the defect. While defects are usually modelled by asuperlattice with a typical band-structure representation of energy levels,determining the confinement associated with a given band is highly non-trivialand no analytical method is known to date. Therefore, we propose a rigorousmethod to classify the dimensionality of the confinement. Starting from theconfinement energy and the mode volume, we use finite-size scaling to find thatratios of these quantities to certain powers yield the confinementdimensionality of each band. This classification has negligible additionalcomputational costs compared to a band structure calculation and is valid forany type of wave in both quantum and classical regimes, and any dimension. Inthe quantum case, we illustrate our method on electronic confinement in 2Dhexagonal BN with a nitrogen vacancy, which confirms the previous results. Inthe classical case, we study a three-dimensional photonic band gap cavitysuperlattice, where we identify novel acceptor-like behavior.

Scaling theory of wave confinement in classical and quantum periodic systems