Photonic moiré crystals

How a two-scale description of twisted crystals led to Blaze2D.

A photonic crystal controls light through a repeating pattern of dielectric material. Twist two such patterns against each other and their local alignment changes slowly across space. That extra structure can reshape the optical bands, reduce propagation speeds and confine modes. The question behind this thesis was how to calculate those effects directly from Maxwell’s equations without resolving every microscopic cell of a large twisted structure.

The answer developed in Photonic Band Theory of Moire Crystals: A Two-Scale Approach is an envelope theory built from local Bloch modes. It replaces one large calculation with many small crystal calculations and a separate problem for the slow modulation. Blaze2D was built to supply those local calculations, including the operator data needed to connect them.

Why direct calculations become expensive

A periodic band solver needs a repeating cell. A generic twist has no exact repeating bilayer cell. Special, commensurate angles do, but those cells grow rapidly as the angle shrinks. At a fixed microscopic resolution, the number of grid points grows roughly as $\theta^{-2}$. Memory, factorization and repeated eigensolves then become the bottleneck.

This was concrete in the revised square-bilayer study. The FDFD reference at $2.01^\circ$ used about 3.3 million unknowns and took 3,221 seconds. The next cell at $1.01^\circ$ would need about 13 million unknowns and did not fit in the workstation’s memory. An angle sweep therefore needed a description whose grid follows the slow physics.

Two scales from one twist

Each monolayer has a lattice spacing $a$. Their changing alignment introduces the much larger moiré period $L$. For identical lattices near alignment,

Here the small-angle approximation uses radians. A $2^\circ$ twist gives $L\approx29a$. The dots below show the microscopic scale; the white arrows and cell show the slow stacking pattern. Change the angle to see the two scales separate. The enlarged monolayer cells keep their size while the moiré cell changes.

The moiré cell describes a repeating pattern of local alignment. It is not generally an exact supercell of all the dielectric objects. That distinction is what lets the theory address angles at which a periodic full-structure calculation is unavailable. At large twists the geometric construction still works, but the two scales are no longer well separated and the small-angle theory loses its basis.

From Maxwell to a slow envelope

For a lossless, non-magnetic medium with scalar permittivity $\varepsilon(x,y)$, the strictly two-dimensional problem separates into two polarizations:

Locally, a small-angle bilayer resembles an ordinary periodic crystal with a relative shift between its two constituents. This shift is the registry, $\boldsymbol\delta$. Moving through the moiré pattern changes the registry slowly, so the local Bloch modes can serve as a basis for the full field:

The $u_n$ resolve the fast crystal oscillations near a chosen carrier momentum $\mathbf k_0$. The amplitudes $f_n$ describe the slow envelope. Only a selected group of bands, $P$, is retained. Its separation from the other bands matters: a small twist alone cannot make a poorly isolated band group into a valid reduced model.

What the effective Hamiltonian contains

Projecting Maxwell’s operator onto these local modes and expanding through second order in $\eta$ gives the following compact TE form:

It acts on the envelope via $\mathcal H_{\mathrm{eff}}f=\lambda f$, with the dimensionless $\lambda=(\omega a/c)^2$. Repeated spatial indices are summed.

• Local band energies $\Lambda$ describe how the spectrum changes with registry.
• Velocity and inverse mass describe the local band slopes and curvatures. The mass includes virtual coupling to bands outside $P$ through Löwdin downfolding.
• Covariant momentum $\Pi_i=-i\mathcal D_i$ includes the Berry connection, which tracks how the retained Bloch basis changes from one registry to another.
• Remainder operators contain the remaining scalar and derivative terms, including the Born-Huang contribution from the basis changing into the discarded band space.

These coefficients are tabulated on the registry domain for a fixed material and band choice, then reused at different twists. For TM, transforming the field with $\widetilde E_z=\varepsilon^{1/2}E_z$ makes the eigenproblem Hermitian in the ordinary inner product. Derivatives of the factor $\rho=\varepsilon^{-1/2}$ then contribute additional terms. The complete expressions and definitions are collected in these two annotated posters.

Why this required Blaze2D

The reduced problem is smaller, but assembling it needs more than band frequencies. Each registry point requires Bloch eigenvectors, velocity and mass matrix elements, momentum and registry derivatives, and couplings between retained and remote bands. Keeping four bands can require solving sixteen or more, repeated at thousands of slightly different crystal configurations.

MPB provided a valuable reference. Its Python interface also exposes fields, raw eigenvectors and group velocities. The missing part for this thesis was direct access to the specific projected operators and interband couplings needed by the envelope construction. Those quantities were not available as the operator dataset the pipeline required. Extracting them, with consistent normalization and gauge conventions across a registry sweep, was a separate problem from computing a band diagram.

Blaze2D combines that access with a Rust plane-wave solver designed for repeated local solves. Iterative eigensolves, warm starts, mixed precision and parallel parameter studies address throughput; explicit Bloch and operator outputs address the data bottleneck. Blaze supplies the microscopic inputs. The MSL research pipeline assembles and solves the moiré envelope problem from them.

What the second edition establishes

The central test is whether the small envelope problem reproduces the full crystal’s frequencies. EA labels the envelope approximation, assembled by MSL from Blaze’s local Bloch data. The independent references solve Maxwell’s equations on the full structure: PWE uses plane waves; FDFD uses a spatial grid. MPB adds a check for the square bilayer. Every comparison uses the same dielectric and declared band domain.

Eigenvalues determine the crystal’s allowed frequencies. Similar-looking bands can hide missing, extra or misplaced modes. Matching every state in a range chosen in advance tests both the frequencies and their count. Agreement near the reference’s own numerical error shows how much the small model preserves. Frequency differences below use $f=\omega a/(2\pi c)$.

Convergence under controlled conditions

A frame is the set of local Bloch field patterns used to build the envelope solution. Recall how the blue and orange dots change their local alignment across the explorer’s white cell. That change in registry also changes the local modes.

• A frozen frame reuses the modes from one registry and carrier momentum everywhere. The envelope can vary, but its microscopic field patterns stay fixed.
• A registry-adapted frame recalculates those patterns as the local alignment changes. The envelope is built on modes that follow the crystal underneath it.

The first plot asks whether a smaller twist makes the prediction more accurate. On this smooth, weakly coupled crystal, adaptation lowers the error about fifteenfold. At fixed material, however, both models reach an error floor: the registry potential becomes stronger relative to the slow kinetic energy as the angle shrinks.

A controlled test reduces the second-layer amplitude as $a_2\propto\eta^2$, keeping that balance fixed. Its adapted model converges approximately as $\eta^{3.8}$ and reaches agreement with both full-Maxwell references of about $8\times10^{-7}$ in frequency. This rate belongs to that scaled family.

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Matching every state in a declared domain

A frame also depends on momentum, which sets the wave’s direction and wavelength. The “exact frames” below use a Bloch solution at each retained momentum, summing beyond the low-order expansion. This separate weak-coupling test uses modes of the registry-averaged crystal.

At $0.607^\circ$, a supercell contains 8,911 microscopic cells. The chosen domain covers states near one minimum of the local band, called a valley. The exact-frame model matches all 51 states in that domain, with frequency differences of $4.7\times10^{-8}$ to $8.1\times10^{-8}$. These are comparable to the measured numerical error of the full-Maxwell reference.

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Bands outside the retained set also influence how its modes propagate. The second-order Löwdin correction includes this influence and recovers two to three orders of accuracy on the affected states. Feshbach downfolding treats the same coupling without that second-order truncation. Exact momentum-dependent frames improve it further. The discarded bands contribute corrections, while the envelope state set stays fixed.

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What remains unresolved at strong coupling

The square rod bilayer is harder: its changing registry creates deep wells that confine the envelopes. The revised model brings the lowest frequency close to the full-wave references and reaches $0.50^\circ$, where the full calculation did not fit in memory. Higher in the spectrum, however, symmetry should make groups of four modes share a frequency. The model splits these groups incorrectly. Its adapted basis still needs a consistent phase and mixing convention, or gauge, that respects the crystal’s symmetry.

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What the operator data reveal

Figure 61 collects the local band data and Hamiltonian coefficients for a honeycomb TM crystal. Its retained Dirac pair consists of two bands that meet at a cone-shaped crossing when aligned. Each sheet maps a different quantity over the same local alignments, like moving around the explorer’s white cell.

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Read from the bottom up

Local spectrum. Above the crystal sit the two band energies and their gap. Their variation supplies the landscape the envelope moves through.

Propagation. The velocity and kinetic mass describe the slopes and curvatures of the local bands, including the influence of remote bands.

Changing modes. The Berry fields track changes within the retained pair. The Born-Huang term at the top records how its basis changes into discarded bands.

The same registry data can be mapped onto different moiré cell sizes. Changing the twist changes that spatial scale, as in the explorer above.

The maps below unpack the stack’s geometric part. The upper row shows the local gap and two measures of how strongly the retained basis changes into other bands. Bright lines mark the strongest changes.

A computed mode has an arbitrary phase; a nearly degenerate pair can also be mixed. A gauge fixes these choices between neighbouring registries. The two Berry-connection maps show how inconsistent choices can dominate the apparent variation. The Berry curvature at bottom right is independent of that choice.

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Does a smaller twist make flatter bands?

Flat bands and magic angles are of particular interest in physics. A study of band flattening was therefore a natural application of the thesis’s effective model.

A moiré band traces an envelope mode’s frequency as its momentum changes. Its slope sets the wave packet’s propagation speed, so flat bands are candidates for slow light. The bandwidth measures the frequency spread along the sampled momentum path.

The pipeline scanned 237 angles from $0.2^\circ$ to $10^\circ$. Above $1.5^\circ$, the typical bandwidth follows roughly $\theta^{1.6}$. At smaller angles the fixed-grid calculation plateaus. No magic angle was established: the shallow minimum does not demonstrate a collapse to a flat band. Resolving that question would require finer, angle-dependent grids and a direct check of the propagation velocity.

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Blaze exposes the local modes and operator matrix elements so a smaller model can be assembled, tested and understood term by term. The second edition establishes that approach in controlled regimes. Strong-coupling symmetry and an extension to photonic slabs remain open.