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.

Figure 1

Illustration of supercells of size $N=5$ in a $D=3$ system with various wave confinement dimensionalities $c$ induced by defect geometries of dimensionalities $d$. Blue spheres correspond to regular unit cells, unit cells containing defects are red, and confined waves are represented by yellow. (a) Four supercells, each containing a point defect akin to a cavity [11], trap waves in all three dimensions, $d=0$, $c=3$. (b) Supercell with a linear defect, analogous to an optical waveguide or fiber, $d=1$, $c=2$. (c) Supercell with a planar defect, analogous to a 2D electron gas [64], $d=2$, $c=1$.

Figure 2

Illustration of our scaling argument in $D=1$. An $N=4$ supercell is transformed into an $N=6$ supercell by adding unit cells to its boundary. We investigate how the characteristic quantities $\tilde{V}$ and $\tilde{E}$ evolve upon this transition and obtain scaling relations for them. While scaling, we keep the total amount of energy $E_S$ within the supercell constant. (a) Confined band. (b) Extended band.

Figure 3

Flow diagram to identify bands confined in $c\ge j$ dimensions in a $D$-dimensional medium: the ratio ${\tilde{V}}^{\alpha }/\tilde{E}$ versus the system size $N$. The auxiliary power $\alpha$ is chosen so that $\kappa <0$ for all $c\ge j$ and $\kappa >0$ for all $c<j$. The sign of $\kappa$ determines if ${\tilde{V}}^{\alpha }/\tilde{E}$ increases or decreases with growing supercell size.

Figure 4

Our confinement analysis applied to a 2D quantum system: electronic waves in a hexagonal BN with nitrogen deficiency. Zero energy is set at the Fermi level. (a) Band structure of the $N=5$ supercell, flipped to have an energy abscissa. Bands are distinguished by colors. (b) Scaling analysis of confinement: every band is represented by a point. As the supercell size increases from $N_0=3$ to $N=5$, extended bands move upward, while the three point-confined bands closest to zero energy shift down.

Figure 5

Scaling analysis of wave confinement in an $N=4$ supercell of a 3D inverse woodpile photonic crystal with two proximate line defects. The horizontal axis displays the reduced frequency $\tilde{\omega }≔\omega a/(2\pi \nu )$, where $a$ is the lattice constant in the $y$ direction and $\nu$ denotes the speed of light. (a) Band structure. In the $M\text{−}A$ region degenerate bands overlap. (b) Confinement dimensionalities determined by our method for each band. Error bars denote bands assigned to $c=1$, which should not occur in our structure but appear due to the small size of the supercell.