Duality between atomic configurations and Bloch states in twistronic materials

The relative orientation (twist) of successive layers of stacked two-dimensional (2D) materials creates variations in the interlayer atomic registry. The variations often form a superlattice, called a moiré pattern, which can alter electronic properties. In this work we introduce a classification of the single-particle electronic structures that can occur in twisted stacks of 2D layers by characterizing them as “moiré molecules” or “moiré crystals.” The molecules generate localized electronic states and moiré flat bands, while the crystals are sometimes unconventional and produce electronic banding in the configuration basis. The underpinning of this classification is the duality between interlayer configuration and monolayer Bloch momentum in moiré Hamiltonians. We apply this understanding to diagrams of local electron density in untwisted geometries to produce intuitive and quantitative predictions of twistronic properties. We provide a conceptual introduction to this framework through a one-dimensional model, and then apply it to 2D twisted bilayers of the semimetal graphene, and of ${\mathrm{MoS}}_2$, a representative material of the transition metal dichalcogenide family of semiconductors. This level of thorough understanding of twistronic phenomena is vital in the search for new material platforms for localized moiré electrons.

Figure 1

(a) Geometry of a twisted bilayer formed by two identical honeycomb lattices. The bottom (top) layer is red (blue). The moiré supercell is shown in gray and the moiré Wigner-Seitz cell is outlined in black, with colored boxes highlighting the high-symmetry stacking locations shown in the insets, corresponding to $AB$, $AA$, and $BA$ stackings. (b) Configuration-dependent local density of states (LDOS) for $p_z$ orbitals in untwisted bilayer ${\mathrm{MoS}}_2$ in an energy region 4 eV below the valence band maximum. (c) Configuration-dependent LDOS for a $4^{\circ }$ twisted bilayer of ${\mathrm{MoS}}_2$. (d), (e) The same as (b) and (c), but for twisted bilayer graphene at $0.0^{\circ }$ and $1.0^{\circ }$. (f), (g) Top-down views of the real-space variation in the LDOS at the energy values identified by the dashed colored lines. The LDOS in (g) appears in both tBLG and ${\mathrm{tBLMoS}}_2$, and includes the moiré Wigner-Seitz cell outlined in white.

Figure 2

(a) The 1D two-chain model, with the top (bottom) chain in red (blue) with its lattice parameter in the same color. The relative interlayer configuration between the top and bottom layer ($\mathbf{d}$) and its interlayer coupling $\left[T\right(\mathbf{d}\left)\right]$ are labeled for a pair of atoms. (b) The interlayer coupling function $T$, with parameters $\xi =0.3\left(0.6\right)$ and $T_1=3\left(1\right)$ in black (red). (c) Configuration-dependent LDOS of the top atom in the two-chain model. Each of the four panels consists of an “untwisted” geometry with $\mathrm{\Theta }=0$, and two “twisted” geometries with $\mathrm{\Theta }=0.05$ and 0.1, for two different values of $\xi$, $\xi =0.3$ (left set), and $\xi =0.6$ (right set), and two different values of $T_1$, $T_1=1$ (top set), and $T_1=3$ (bottom set).

Figure 3

(a) Band structure of a single 1D chain, with a harmonic approximation of a band extremum in red. (b) Configuration dependence of the interlayer coupling function for the bilayer 1D chain model, with a harmonic approximation of its maximum in red. (c) Total density of states (DOS) for commensurate bilayer chains for $\mathrm{\Theta }$ in $[0,0.4]$. A fractal butterfly spectrum is evident, with a region highlighted in the top left corner. (d) Expanded view of the region highlighted in (c), that shows a number of isolated $\mathrm{\Theta }$-dependent states in the DOS. The dashed red lines correspond to the energy levels $\omega (n+1/2)$ of a real-space harmonic oscillator based on the red fits in (a) and (b). The red numbers indicate the $n$ for that level. (e) Schematic of the fitted real-space harmonic well, with the first three energy levels of the oscillator indicated by the red lines and numbers. The frequency $\omega ={\omega }_{\mathrm{\Theta }}\mathrm{\Theta }$ is displayed between the first two energy levels.

Figure 4

Electronic structure calculations for bilayer chains with $T_1=3$ and $\xi =0.3$, with varying $\mathrm{\Theta }$. Top: LDOS for atoms of the top layer with specific configurations $\mathbf{d}$. The harmonic approximation of $T\left(\mathbf{d}\right)$ is highlighted in red. Bottom: LDOS for Bloch waves of the top layer with specific momenta $\mathbf{k}$. The harmonic approximation of the bands $E\left(\mathbf{k}\right)$ is highlighted in red.

Figure 5

(a) Diagram of the bilayer chain Hamiltonian when the monolayer couplings ($H_i$) are much larger than the interlayer coupling $\left(T\right)$. (b) Configuration-dependent LDOS for the atoms of the top layer in this limiting case with $\mathrm{\Theta }=0$. (c) Momentum-dependent LDOS for Bloch waves of the top layer in this limiting case with $\mathrm{\Theta }=0$. (d)–(f) Same as (a)–(c), but the interlayer coupling is now much larger than the monolayer couplings. (g)–(i) Configuration and momentum LDOS for the bilayer chain model ($\mathrm{\Theta }=0.1$, $T_1=3$, $\xi =0.3$) with the central panel describing the twistronic regime at that energy.

Figure 6

(a)–(i) Configuration-dependent LDOS for twisted bilayer graphene (tBLG) for selected energy regions and various twist angles. The configurations correspond to a diagonal traversal of the moiré supercell. The results were calculated for finite flakes of diameter 200 nm with a kernel polynomial method. (j) Band structure for a simple nearest-neighbor tight-binding model of graphene. The energy regions associated with the selected LDOS energy regions of the twisted bilayer are highlighted in red. (k) Top-down view of (d). The Bloch states that are within the interlayer coupling strength (0.3 eV) of each selected energy are highlighted in red.

Figure 7

Configuration-dependent LDOS for the $p_z$ orbitals of bilayer ${\mathrm{MoS}}_2$ in selected energy regions of the valence bands. The configurations go across the diagonal of the moiré cell, starting and ending at aligned $AA$-type stacking regions. (j)–(m) Show a top view of the configuration-dependent LDOS at four energy values for the $4^{\circ }$ twisted bilayer, interpolated from a $20\times 20$ sampling of configuration space. All results were calculated for finite flakes of radius 40 nm with the kernel polynomial method. The energy spectrum resolution is limited by roughly 10 meV ($p=4000$).

Figure 8

Interpretation of the moiré flat bands in unrelaxed twisted bilayers of ${\mathrm{MoS}}_2$ as a quantum harmonic oscillator model. (a) Band structure comparison between a monolayer (orange) and $AA$-stacked bilayer (black) of ${\mathrm{MoS}}_2$. The effective interlayer coupling can be easily interpreted at any band and momentum by looking at the strength of the band splitting. The dashed red line indicates the bilayer's valence band maximum, which is the zero-energy reference for the following panels. (b) Band structure of the twisted bilayer at $\theta =2.{28}^{\circ }$, showing the characteristic splitting and flattening of the bands, as captured by the QHO model. (c) Summary of band structures for twisted bilayer ${\mathrm{MoS}}_2$ near the valence band maximum. The average band energy for each band is plotted as a function of the twist angle $\theta$, with the bandwidth $b_w$ given in color. The red lines indicate the expected energy levels of a harmonic oscillator defined by the monolayer's effective mass $m^{*}$, and the shape of the moiré potential $T\left(\mathit{r}\right)$, both expanded around $\mathrm{\Gamma }$. The dashed lines are a guide to the eye for $\theta$ dependence of the first (singly degenerate) level and the second (double degenerate) level.

Figure 9

(a) LDOS for the 1D bilayer model as a function of local configuration $\mathbf{d}$ at $\mathrm{\Theta }=0.1$. For a commensurate geometry, only a finite number of configurations are sampled. The sampled configurations for two different choices of ${\mathbf{d}}_0$ are indicated by the red “moiré combs.” (b), (c) Real-space LDOS for the two commensurate geometries (C) introduced in (a). The configuration banding is lost due to the finite sampling of configurations. The moiré length $\lambda$ is given at the bottom of $\left(\mathbf{c}\right)$. (d)–(f) Real-space LDOS for three selected incommensurate geometries (I), parametrized by $\mathrm{\Theta }=0.1\times (1+\mathrm{\Delta })$. As these geometries are not periodic on the moiré length, the configuration bands are visible in real space.