Moiré Superlattices

A moiré superlattice is a long-wavelength modulation produced when two or more periodic structures are combined with a small relative rotation, lattice mismatch, strain, or displacement texture. The atomic lattices supply rapidly varying phases; their differences produce a much longer geometric scale. Interlayer tunneling, local band-edge shifts, dielectric response, or lattice relaxation can then turn that geometric beat into a periodic quantum Hamiltonian.

A visible moiré pattern establishes a geometric length scale. To understand its electronic consequences, we must next determine how local stacking changes the Hamiltonian. We will derive the period, construct a mini Brillouin zone, and compare the energy scales that make narrow bands and strong interactions possible.

Required background. van der Waals Heterostructures supplies stacking, alignment and interlayer coupling. Reciprocal Lattice supplies the dual vectors used to form a reciprocal mismatch; Brillouin Zones supplies reduced-zone geometry. Bloch’s Theorem supplies the translation-symmetry conditions under which a periodic moiré Hamiltonian has minibands.

Moiré periodicity and electronic reconstruction

Two nearby spatial frequencies produce a slow beat. The same idea applies to two crystal layers: nearly matching reciprocal vectors differ by a small vector, so the local alignment repeats over a long distance. The geometry is our starting point; Twisted Bilayer Graphene and Transition-Metal Dichalcogenides explain how particular electronic structures respond to it.

Registry is a slowly varying field

Consider two nearly matched two-dimensional Bravais lattices. Let $\mathbf R$ label a point in the undeformed reference lattice and let the rigid direct-space deformation of layer 2 be $F=R(\theta)(I+\varepsilon)$. Its physical position is $\mathbf r=F\mathbf R$. A relative displacement written in the material coordinate is

$$\mathbf d_{\mathrm{mat}}(\mathbf R) =(F-I)\mathbf R+\mathbf d_0.$$

Here $R(\theta)$ is a relative rotation, $\varepsilon$ is heterostrain or lattice mismatch, and $\mathbf d_0$ fixes the global translation. Registry is defined modulo a reference atomic Bravais vector. At a fixed physical position, the same rigid registry is

$$\mathbf d_{\mathrm{lab}}(\mathbf r) =\mathbf d_{\mathrm{mat}}(F^{-1}\mathbf r) =(I-F^{-1})\mathbf r+\mathbf d_0.$$

These are different coordinate arguments for the same field, not two interchangeable formulas at the same $\mathbf r$. More general relaxed layers require inverse maps $\mathbf R_\ell(\mathbf r)$ from physical space to a common reference lattice; their local registry is $\mathbf R_1(\mathbf r)-\mathbf R_2(\mathbf r)+\mathbf d_0$. For small displacements this gives the familiar relative relaxation contribution, but an exact rigid rotation or dilation must retain the coordinate transformation. Balents’s continuum construction explains why this distinction matters for a local laboratory-frame Hamiltonian.

Any registry-dependent scalar or matrix $\mathcal F(\mathbf d)$ is periodic on the reference atomic displacement space, so

$$\mathcal F(\mathbf d) =\sum_{\mathbf G}\mathcal F_{\mathbf G} e^{i\mathbf G\cdot\mathbf d}.$$

For a rigid deformation, its material and physical wavevectors are related as covectors:

$$\begin{aligned} \mathbf g_{\mathrm{mat}}&=(F-I)^T\mathbf G, \\ \mathbf g_{\mathrm{lab}}&=F^{-T}\mathbf g_{\mathrm{mat}} =(I-F^{-T})\mathbf G. \end{aligned}$$

The second line is the physical reciprocal mismatch. Small rotation or mismatch makes its magnitude much smaller than the atomic reciprocal scale, so the real-space modulation is much longer than the lattice period. This is the two-dimensional version of beating between nearby frequencies.

Reciprocal differences define the emergent lattice

Choose paired reciprocal bases $\mathbf b_i^{(1)}$ and $\mathbf b_i^{(2)}$ for the two layers. A convenient moiré reciprocal basis is

$$\mathbf g_i = \mathbf b_i^{(1)} - \mathbf b_i^{(2)}.$$

The associated direct basis $\mathbf A_i$ is defined by

$$\mathbf A_i\cdot\mathbf g_j = 2\pi\delta_{ij}.$$

Use corresponding primitive vectors in the two nearly aligned lattices. Applying the same unimodular integer basis change to both pairs relabels the same moiré lattice; independently re-pairing unrelated vectors need not do so. Reversing every difference changes the sign convention, not the lattice or its spectra.

Exact periodicity is not automatic

Only special relative rotations and lattice ratios produce an exact common atomic supercell. A generic twisted bilayer is incommensurate: no finite translation maps every atom in both layers onto an equivalent atom. Nevertheless, smooth registry-dependent terms can be extremely well represented by a periodic continuum model over a moiré cell.

That approximation must be named. Three descriptions are common:

1. an exact commensurate supercell at a permitted angle;
2. a nearby commensurate approximant to an incommensurate structure;
3. a continuum model periodic in the slow registry coordinate.

They need not agree when atomic relaxation, quasiperiodicity, disorder, or boundaries matter on the target scale.

Moiré Length Scale

Twist and mismatch enter together

For identical triangular lattices with lattice constant $a$ and relative twist $\theta$, the rigid moiré period is

$$L_M = \frac{a} {2\sin\!\left(|\theta|/2\right)} \simeq \frac{a}{|\theta|}, \qquad |\theta|\ll1,$$

where the small-angle expression requires $\theta$ in radians.

Now let $a_1=a$ and $a_2=(1+\delta)a$. For isotropic mismatch $\delta$ and relative twist $\theta$, subtract two reciprocal vectors with lengths $b$ and $b/(1+\delta)$. The cosine rule gives

$$\begin{aligned} g_M^2&=b^2\left[1+\frac{1}{(1+\delta)^2} -\frac{2\cos\theta}{1+\delta}\right], \\ b&=\frac{4\pi}{\sqrt3\,a}. \end{aligned}$$

Using $1-\cos\theta=2\sin^2(\theta/2)$ then yields

$$\begin{aligned} g_M &= \frac{4\pi}{\sqrt3\,a} \frac{ \sqrt{ \delta^2 + 4(1+\delta)\sin^2(\theta/2) } }{1+\delta}, \\ L_M &= \frac{4\pi}{\sqrt3\,g_M} \\ &= \frac{ a(1+\delta) }{ \sqrt{ \delta^2 + 4(1+\delta)\sin^2(\theta/2) } }. \end{aligned}$$

When $|\delta|,|\theta|\ll1$,

$$L_M \simeq \frac{a} {\sqrt{\delta^2+\theta^2}}.$$

Twist and mismatch therefore do not define independent long periods in this isotropic limit; they combine as orthogonal components of the reciprocal mismatch. Heterostrain is tensorial and generally makes the three shortest moiré vectors unequal, distorting a hexagonal cell into an oblique one.

Top: the arrow from the tip of $\mathbf b^{(2)}$ to the tip of $\mathbf b^{(1)}$ is their difference $\mathbf g$. The angle is exaggerated for visibility. Bottom: exact rigid triangular-lattice periods for $a=0.330\,\mathrm{nm}$. Identical layers have $L_M\propto1/|\theta|$ at small twist; a $4\%$ lattice mismatch instead gives the finite zero-twist limit $a(1+\delta)/|\delta|=8.58\,\mathrm{nm}$. Small twist alone therefore does not guarantee a very large cell.

Cell area converts density into filling

For a triangular moiré lattice,

$$A_M = \frac{\sqrt3}{2}L_M^2, \qquad n_M = \frac{1}{A_M}.$$

If $\nu$ denotes carriers per moiré cell relative to a stated reference,

$$n-n_{\mathrm{ref}} = \frac{\nu}{A_M}.$$

The conversion is purely geometric. Spin, valley, layer, and orbital degeneracies determine how many carriers fill a chosen miniband manifold, but they do not alter $n_M$. Authors use incompatible $\nu$ origins and signs, so a filling label is incomplete unless the reference state and degeneracy are specified.

For identical graphene lattices with $a=0.246\,\mathrm{nm}$ and $\theta=1.10^\circ$,

$$\begin{aligned} L_M &\simeq 12.8\,\mathrm{nm}, \\ A_M &\simeq 142\,\mathrm{nm}^2, \\ n_M &\simeq 7.0\times10^{11}\,\mathrm{cm}^{-2}. \end{aligned}$$

A fourfold manifold therefore changes occupancy by four carriers per cell over a density interval near $2.8\times10^{12}\,\mathrm{cm}^{-2}$. The numerical value is a geometry check, not proof that the fourfold manifold is isolated or symmetry-degenerate.

Angle uncertainty grows at small twist

For a twist-dominated pattern, $L_M\propto1/|\theta|$, so

$$\frac{\Delta L_M}{L_M} \simeq \frac{\Delta\theta}{|\theta|}.$$

An absolute angular uncertainty that looks small can therefore produce a substantial spread in cell area, filling density, and bandwidth. Local heterostrain and twist gradients add spatial variation. Reporting one nominal angle without a distribution can conceal the dominant uncertainty in a phase diagram.

Relaxation reconstructs the rigid pattern

The rigid geometry supplies a registry field, not the final atomic structure. A schematic elastic-adhesion functional is

$$\begin{aligned} E[\mathbf u] &= \int d^2r \left[ \mathcal E_{\mathrm{el}}(\mathbf r) + V_{\mathrm{stack}}\!\left(\mathbf d(\mathbf r)\right) \right], \\ \mathcal E_{\mathrm{el}}(\mathbf r) &= \frac12 C_{ijkl} \epsilon_{ij}(\mathbf r) \epsilon_{kl}(\mathbf r). \end{aligned}$$

Elasticity penalizes rapid deformation, while the stacking energy favors selected registries. At sufficiently long moiré periods, the compromise can enlarge low-energy domains and concentrate mismatch into narrow soliton-like walls. Out-of-plane corrugation may accompany in-plane reconstruction. The electronic potential must then be built from the relaxed registry map, not merely from two rigid lattices.

Carr et al. (2018) formulate this elastic and stacking-energy competition in configuration space and calculate relaxation and domain formation for incommensurate graphene and MoS$_2$ bilayers. Their construction separates a smooth continuum description from an exact atomic supercell.

Mini Brillouin Zones

The moiré cell has its own reciprocal fundamental domain

Once a periodic moiré Hamiltonian is justified, the reciprocal vectors

$$\mathbf G_M = m_1\mathbf g_1 + m_2\mathbf g_2, \qquad m_i\in\mathbb Z,$$

define a moiré reciprocal lattice. Its first Brillouin zone has area

$$\mathcal A_{\mathrm{mBZ}} = \frac{(2\pi)^2}{A_M}.$$

The mini Brillouin zone is not an additional set of states. It represents the same states using momenta in the smaller reciprocal cell. Atomic-layer momenta separated by a moiré reciprocal vector are represented at the same mini-zone momentum and can hybridize when the Hamiltonian supplies the corresponding Fourier component.

Continuum Hamiltonians retain the slow couplings

A generic bilayer continuum model has the block form

$$\begin{gathered} \mathcal H(\mathbf r) = \begin{pmatrix} \widetilde h_1(\mathbf r) & T(\mathbf r) \\ T^\dagger(\mathbf r) & \widetilde h_2(\mathbf r) \end{pmatrix}, \\ \widetilde h_\ell(\mathbf r) = h_\ell(-i\boldsymbol\nabla) + V_\ell(\mathbf r). \end{gathered}$$

in a basis chosen so that the coefficients are periodic:

$$\begin{aligned} V_\ell(\mathbf r+\mathbf A_i) &= V_\ell(\mathbf r), \\ T(\mathbf r+\mathbf A_i) &= T(\mathbf r). \end{aligned}$$

A valley-centered basis can instead make the tunneling acquire a common phase under a moiré translation. A layer-dependent gauge transformation transfers that phase into a momentum shift of $h_\ell$; it must be carried through consistently when applying Bloch’s theorem.

$h_\ell$ describes the isolated-layer bands in a chosen valley and orbital basis. $V_\ell$ contains registry-dependent intralayer terms, and $T$ contains interlayer tunneling. Their matrix structure can act on sublattice, orbital, spin, valley, layer, and Nambu indices. A scalar sinusoidal potential is only one special case.

For graphene, Bistritzer and MacDonald (2011) show how continuum interlayer coupling produces moiré Bloch bands and strong velocity renormalization. Balents (2019) derives the continuum Hamiltonian for arbitrary smooth displacement fields with small gradients. These are specific realizations of the model hierarchy, not a universal parameter set for every bilayer.

Expand the slow terms as

$$T(\mathbf r) = \sum_{\mathbf G_M} T_{\mathbf G_M} e^{i\mathbf G_M\cdot\mathbf r},$$

and similarly for $V_\ell$. Bloch states of the continuum model obey

$$\Psi_{n\mathbf k}(\mathbf r) = e^{i\mathbf k\cdot\mathbf r} u_{n\mathbf k}(\mathbf r), \qquad \mathbf k\in\mathrm{mBZ},$$

where $u_{n\mathbf k}$ is moiré-periodic. In a plane-wave calculation, its components carry momenta $\mathbf k+\mathbf G_M$. Truncating that reciprocal set is a numerical approximation whose convergence must be checked for energies, wavefunctions, symmetry eigenvalues, and Berry quantities.

Folding is not flattening

If the layer coupling were set to zero, writing the original dispersions in the mini zone would merely fold them into many branches. Fourier components of $V_\ell$ and $T$ then open avoided crossings, exchange layer and orbital character, and can isolate groups of bands. The relevant local two-level structure is

$$\mathcal H_{\mathrm{cross}} = \begin{pmatrix} \epsilon_1(\mathbf k) & t_{\mathbf g} \\ t_{\mathbf g}^{*} & \epsilon_2(\mathbf k+\mathbf g) \end{pmatrix}.$$

At degeneracy, the direct minigap is $2|t_{\mathbf g}|$. Away from degeneracy, hybridization falls with detuning. Observing a replica band without a controlled anticrossing can reflect matrix elements, final-state diffraction, domains, or an analysis convention rather than a coherent miniband.

Use a model hierarchy

Description	Uses and limitations
Atomistic model	Resolves local chemistry, relaxation and orbital-scale tunneling. Cell size, fitted parameters and numerical convergence limit accuracy.
Continuum model	Resolves slow coupling and minibands at small twist. Results depend on the basis, harmonics, relaxed structure and momentum cutoff.
Wannier or lattice model	Supports many-body calculations with localized orbitals. Band isolation, symmetry, topology and interaction range constrain the reduction.

Agreement at one band edge does not validate the whole hierarchy. A controlled reduction should reproduce the target dispersion, wavefunction character, symmetries, and response over a stated energy and parameter window.

Flat Minibands

Flatness requires a scale comparison

For an isolated band $n$, define

$$W_n = \max_{\mathbf k}E_{n\mathbf k} - \min_{\mathbf k}E_{n\mathbf k}.$$

Global gaps to adjacent bands may be defined as

$$\begin{aligned} \Delta_+ &= \min_{\mathbf k}E_{n+1,\mathbf k} - \max_{\mathbf k}E_{n,\mathbf k}, \\ \Delta_- &= \min_{\mathbf k}E_{n,\mathbf k} - \max_{\mathbf k}E_{n-1,\mathbf k}. \end{aligned}$$

Positive $\Delta_\pm$ establish energetic isolation. A ratio such as

$$\mathcal F = \frac{\min(\Delta_+,\Delta_-)}{W_n}$$

is a useful flatness indicator, but it is not universal. For a connected multiband manifold, $W$ and the remote-band gaps must be defined for the whole manifold. Disorder broadening $\Gamma$, thermal energy $k_BT$, interaction scales, and measurement resolution also belong in the comparison.

Several mechanisms narrow moiré bands

1. Large-cell kinetic suppression. A long period reduces the momentum scale over which a miniband disperses.
2. Hybridization interference. Coupling among momentum-space copies can cancel velocities or hopping amplitudes at special parameters.
3. Registry localization. A deep moiré potential can trap wavefunctions near selected stackings, making intercell tunneling exponentially small.
4. Structural reconstruction. Domains, corrugation, and walls reshape both onsite energies and tunneling networks.

Zone folding alone does not reduce the physical velocity of an uncoupled band. Conversely, a band can have a small total width yet remain unsuitable for a one-orbital model because it overlaps remote bands or carries a Wannier obstruction.

Kinetic scales depend on the parent dispersion

For a parabolic band with effective mass $m^*$, a geometric kinetic estimate is

$$E_M^{\mathrm{par}} \sim \frac{\hbar^2}{2m^*L_M^2}.$$

For a Dirac cone with velocity $v$,

$$E_M^{\mathrm{Dirac}} \sim \frac{\hbar v}{L_M}.$$

These estimates use the characteristic momentum $1/L_M$ and omit geometric factors. Using the shortest reciprocal vector instead gives $q=4\pi/(\sqrt3 L_M)$ for a triangular lattice, so the corresponding parabolic energy is larger by $16\pi^2/3$. Neither convention alone predicts the actual bandwidth. These estimates give the scaling before detailed hybridization. They explain why increasing $L_M$ suppresses parabolic kinetic energy faster than the bare Coulomb scale. In a Dirac system, strong flattening generally requires more than a large unit cell: interlayer interference and velocity renormalization are central.

Tunable Interactions

Coulomb and kinetic scales transform differently

A rough unscreened interaction scale across one moiré cell is

$$E_C \sim \frac{e^2} {4\pi\epsilon_0\epsilon_{\mathrm{eff}}L_M}.$$

Combining this with the parabolic estimate gives

$$\frac{E_C}{E_M^{\mathrm{par}}} \sim \frac{2L_M}{a_B^*}, \qquad a_B^* = \frac{ 4\pi\epsilon_0\epsilon_{\mathrm{eff}}\hbar^2 }{ m^*e^2 }.$$

Thus a longer moiré period can parametrically strengthen interactions relative to parabolic kinetic energy. For an unrenormalized Dirac cone,

$$\frac{E_C}{E_M^{\mathrm{Dirac}}} \sim \frac{e^2} {4\pi\epsilon_0\epsilon_{\mathrm{eff}}\hbar v},$$

which is independent of $L_M$ at this level. Moiré hybridization can nevertheless reduce the operative velocity or bandwidth and thereby enhance correlations.

These are scaling estimates, not Hubbard parameters. The actual matrix elements depend on wavefunction extent, layer polarization, dielectric nonlocality, metallic gates, form factors, and remote-band screening.

Gates reshape the interaction, not just the density

For a two-dimensional charge layer centered between metallic gates a distance $d$ above and below, the ideal homogeneous-dielectric interaction, with perfectly conducting grounded gates, is

$$V(q) = \frac{e^2} {2\epsilon_0\epsilon_{\mathrm{eff}}q} \tanh(qd).$$

At $qd\gg1$, the gates are ineffective at that wavelength. At $qd\ll1$, long-range components are strongly reduced. Real stacks can have unequal gate distances, anisotropic dielectrics, multiple conducting layers, and density-dependent internal screening, so the electrostatic geometry belongs in every quoted interaction parameter.

Projection produces an extended model

Projecting isolated minibands into localized orbitals gives a schematic Hamiltonian

$$\begin{aligned} H_{\mathrm{eff}} &= H_t+H_V+H_{\mathrm{res}}, \\ H_t &= \sum_{ij,ab} t_{ij}^{ab} c_{ia}^{\dagger}c_{jb} \\ H_V &= \frac12 \sum_{ij,ab} V_{ij}^{ab} :n_{ia}n_{jb}: , \\ H_{\mathrm{res}} &= H_{\mathrm{exchange}} + H_{\mathrm{pair}} + \cdots . \end{aligned}$$

Here $n_{ia}=c_{ia}^\dagger c_{ia}$ and normal ordering removes a particle’s interaction with itself; equivalently omit identical fermionic single-state indices from the density-pair sum. The labels $a,b$ may encode spin, valley, layer, orbital, or sublattice. Long moiré orbitals often make intersite repulsion appreciable; intervalley exchange, assisted hopping, phonons, and remote bands can also matter. If the active manifold is topological or symmetry-obstructed, a single localized orbital per apparent potential minimum may not exist without adding bands or sacrificing a symmetry.

Hubbard Physics in Materials explains how to construct and test such an effective model. A moiré pattern makes Hubbard-family descriptions tunable; it does not make a nearest-neighbor Hubbard model automatic.

An explicit theoretical example is the generalized triangular-lattice Hubbard description derived for selected TMD heterobilayers by Wu et al. (2018). The orbital construction and interaction estimates are part of that model assignment; triangular registry alone is insufficient.

Twist changes the period and orbital overlap; a displacement field changes layer polarization and band offsets; density changes both filling and screening. These effects are coupled, so tuning one control rarely changes only one Hamiltonian parameter. Twist is usually fixed during fabrication, whereas gate voltages can be varied in an operating device.

Why Moiré Systems Are Quantum Simulators in Solids

Connecting experimental controls to a model

An analog quantum simulator realizes a target Hamiltonian well enough that controlled changes of physical inputs map to known changes of model parameters and measured observables test the model’s predictions. In moiré matter, the simulator degrees of freedom are real electrons, holes, or excitons inside a solid. The platform is tunable, but it is not a universal digital quantum computer and is not automatically programmable term by term.

Moiré systems are attractive because geometry can create large, accessible unit cells; gates tune filling without chemical substitution; displacement fields and pressure reshape wavefunctions; and transport, capacitance, optics, tunneling, and local probes interrogate complementary observables. Different registry patterns can produce triangular, honeycomb, rectangular, or multiorbital effective lattices.

For example, one dominant potential minimum per triangular moiré cell can support a triangular-lattice description, while two active registries can require a two-orbital model. Exciton lattices introduce bosonic degrees of freedom, but finite lifetime and optical pumping can prevent an equilibrium description.

The analogy to Optical Lattices is useful: geometry and external fields tune a many-body Hamiltonian. In a solid, however, long-range Coulomb interactions, disorder, phonons and contacts are part of the physical system. A useful comparison therefore uses one set of parameters to explain several observables, such as compressibility, excitation energies and transport, rather than fitting a single resistance curve. Kennes and collaborators (2021) discuss these opportunities and limitations.

Reading measurements across scales

Real-space imaging or diffraction measures the period and its spatial variation. Spectroscopy can reveal miniband widths and avoided crossings. Capacitance relates gate charge to filling and can identify incompressibility; transport also depends on scattering and contacts. These measurements answer different questions. In particular, a resistance maximum at integer filling does not distinguish a Mott gap from a one-body gap, charge order or localization. See Correlated Insulators in Moiré Systems for that distinction.

The large moiré area makes one flux quantum per cell reachable at laboratory magnetic fields:

$$B_{\Phi_0} = \frac{\Phi_0}{A_M}, \qquad \Phi_0 = \frac{h}{e}.$$

For $L_M=10\,\mathrm{nm}$, this field is about $47.8\,\mathrm T$; larger cells reduce it quadratically. Hofstadter features also occur at rational fractions of a flux quantum, so a full quantum need not be reached. Dean and collaborators (2013) observed magnetic-superlattice structure in aligned graphene/hBN. Magnetic Translations explains the translation algebra behind the magnetic subbands.

Exercises

Exercise 1: twist period and angular sensitivity

Two identical hexagonal layers have $a=0.330\,\mathrm{nm}$ and twist angle $\theta=1.50^\circ\pm0.05^\circ$. Find the rigid moiré period and its leading uncertainty.

Solution

Convert the angle to radians:

$$\theta = 1.50\frac{\pi}{180} \simeq 0.02618.$$

The exact period is

$$\begin{aligned} L_M &= \frac{0.330\,\mathrm{nm}} {2\sin(0.75^\circ)} \\ &\simeq 12.6\,\mathrm{nm}. \end{aligned}$$

For a twist-dominated pattern,

$$\frac{\Delta L_M}{L_M} \simeq \frac{0.05}{1.50} = 0.0333.$$

Thus

$$\Delta L_M \simeq 0.42\,\mathrm{nm},$$

so the result is $L_M\simeq(12.6\pm0.4)\,\mathrm{nm}$. This uncertainty excludes local twist gradients and heterostrain.

Exercise 2: twist and mismatch

Two triangular lattices have $a=0.330\,\mathrm{nm}$, isotropic mismatch $\delta=0.040$, and twist $\theta=1.0^\circ$. Estimate $L_M$ using the small-parameter formula. Which contribution dominates?

Solution

In radians, $\theta\simeq0.01745$. Therefore

$$\begin{aligned} L_M &\simeq \frac{0.330\,\mathrm{nm}} {\sqrt{0.040^2+0.01745^2}} \\ &\simeq 7.56\,\mathrm{nm}. \end{aligned}$$

Because $\delta^2=1.60\times10^{-3}$ while $\theta^2\simeq3.05\times10^{-4}$, mismatch supplies about $84\%$ of the squared reciprocal offset. Calling this a twist-controlled period would therefore be misleading.

Exercise 3: density per moiré cell

A triangular moiré lattice has $L_M=10.0\,\mathrm{nm}$. Compute the density change corresponding to one carrier per cell and to four carriers per cell.

Solution

The cell area is

$$A_M = \frac{\sqrt3}{2} (10.0\,\mathrm{nm})^2 \simeq 86.6\,\mathrm{nm}^2.$$

Since $1\,\mathrm{nm}^{-2}=10^{14}\,\mathrm{cm}^{-2}$,

$$\begin{aligned} n_M &= \frac{1}{A_M} \simeq 1.15\times10^{12}\,\mathrm{cm}^{-2}, \\ 4n_M &\simeq 4.62\times10^{12}\,\mathrm{cm}^{-2}. \end{aligned}$$

The second number is the density span of a four-state-per-cell manifold only if those four states form the active isolated manifold.

Exercise 4: kinetic and Coulomb estimates

Take a parabolic moiré band with $m^*=0.50m_e$, $L_M=8.0\,\mathrm{nm}$, and $\epsilon_{\mathrm{eff}}=10$. Estimate $E_M^{\mathrm{par}}$, $E_C$, and their ratio using $\hbar^2/(2m_e)=0.0381\,\mathrm{eV\,nm^2}$ and $e^2/(4\pi\epsilon_0)=1.44\,\mathrm{eV\,nm}$.

Solution

The kinetic estimate is

$$\begin{aligned} E_M^{\mathrm{par}} &= \frac{0.0381\,\mathrm{eV\,nm^2}} {(0.50)(8.0\,\mathrm{nm})^2} \\ &\simeq 1.19\,\mathrm{meV}. \end{aligned}$$

The Coulomb estimate is

$$\begin{aligned} E_C &= \frac{1.44\,\mathrm{eV\,nm}} {(10)(8.0\,\mathrm{nm})} \\ &= 18.0\,\mathrm{meV}. \end{aligned}$$

Hence $E_C/E_M^{\mathrm{par}}\simeq15$. This signals a potentially interaction-dominated regime, but it is not a prediction of an onsite $U=18\,\mathrm{meV}$: gates, orbital form factors, intersite terms, and internal screening still have to be included.

Exercise 5: gate screening at a moiré wavevector

For symmetric metallic gates at $d=5.0\,\mathrm{nm}$, compare the interaction factor $\tanh(qd)$ at $q=1/L_M$ for $L_M=10\,\mathrm{nm}$ with the ungated value.

Solution

Here

$$qd = \frac{d}{L_M} = 0.50.$$

Therefore

$$\tanh(qd) = \tanh(0.50) \simeq 0.462.$$

At this representative wavevector, the symmetric gates reduce the bare Fourier-space interaction to about $46\%$ of its ungated value. Components with smaller $q$ are suppressed more strongly, while short-wavelength components with $qd\gg1$ are nearly unchanged.

Exercise 6: what does an insulating peak establish?

A device has a resistance peak at a gate voltage assigned to one carrier per moiré cell. Does this establish a Mott insulator? Identify two distinct missing pieces of information and a useful measurement for each.

Solution

First, the filling assignment may be wrong: trapped charge, quantum capacitance and a nonuniform moiré area can change the relation between gate voltage and carriers per cell. Combine local structural imaging with a charge-density or capacitance measurement to test it.

Second, even a correct integer filling does not identify the insulating mechanism. A thermodynamic chemical-potential jump helps establish a charge gap, but a band gap or charge order can also produce it. Compare spectroscopy and symmetry-sensitive measurements with the one-body band structure and an interacting model. Spectral-weight transfer or resolved charge order can distinguish possibilities that resistance alone cannot.

References

• Balents, Leon. “General continuum model for twisted bilayer graphene and arbitrary smooth deformations.” SciPost Physics 7, 048 (2019). doi:10.21468/SciPostPhys.7.4.048; author preprint.
• Bistritzer, R., and A. H. MacDonald. “Moiré bands in twisted double-layer graphene.” Proceedings of the National Academy of Sciences 108, 12233–12237 (2011). doi:10.1073/pnas.1108174108.
• Carr, Stephen, Daniel Massatt, Steven B. Torrisi, Paul Cazeaux, Mitchell Luskin, and Efthimios Kaxiras. “Relaxation and domain formation in incommensurate two-dimensional heterostructures.” Physical Review B 98, 224102 (2018). doi:10.1103/PhysRevB.98.224102.
• C. R. Dean et al., “Hofstadter’s Butterfly and the Fractal Quantum Hall Effect in Moiré Superlattices,” Nature 497, 598–602 (2013), doi:10.1038/nature12186.
• D. M. Kennes et al., “Moiré Heterostructures as a Condensed-Matter Quantum Simulator,” Nature Physics 17, 155–163 (2021), doi:10.1038/s41567-020-01154-3.
• Wu, F., T. Lovorn, E. Tutuc, and A. H. MacDonald. “Hubbard Model Physics in Transition Metal Dichalcogenide Moiré Bands.” Physical Review Letters 121, 026402 (2018). doi:10.1103/PhysRevLett.121.026402.