Ideal Topological Flat Bands in Two-dimensional Moiré Heterostructures with Type-II Band Alignment

TL;DR

This work presents a design framework for realizing topological flat bands with ideal quantum geometry in two-dimensional moiré heterostructures exhibiting type-II band alignment. By formulating a moiré Chern-band model and mapping it to a topological heavy fermion (THF) model, the authors show that gate-tunable energy offsets can yield exact flat bands with trace(Fubini-Study) equal to Berry curvature, independent of twist angle. They provide analytical and numerical analyses in first-shell and δ-function moiré-potential limits, establish the condition ΔE = β^2/α for ideal geometry, and demonstrate an interacting fractional Chern insulator (FCI) phase in the projected regime. Material realizations are proposed in semiconductor heterostructures and Γ-valley 2D-material heterostructures, with Tl$_2$Se$_2$/Zn$_2$Te$_2$ as a concrete example, highlighting the practical potential for exploring correlated topological states and related phenomena. Gate-control of the atomic gap enables systematic exploration of THF physics and quantum geometry in realistic platforms.

Abstract

Topological flat bands play an essential role in inducing exotic interacting physics, ranging from fractional Chern insulators to superconductivity, in moiré materials. In this work, we propose a design principle for realizing topological flat bands with "ideal quantum geometry", namely the trace of Fubini-Study metric equals to the Berry curvature, in a class of two-dimensional moiré heterostructures with type-II band alignment. We first introduce a moiré Chern-band model to describe this system and show that topological flat bands can be realized in this model when the moiré superlattice potential is stronger than the type-II atomic band gap of the heterostructure. Next, we map this model into a topological heavy fermion model that consists of a localized orbital for "f-electron" and a conducting band for "c-electron". We find that both the flatness and quantum geometry of the flat band in the topological heavy fermion model depend on the energy gap between c-electron and f-electron bands at $Γ$ which is experimentally controllable via external gate voltages. This tunability will allow us to realize an ideal topological flat band with zero band-width and ideal quantum geometry. Our design strategy of topological flat bands is insensitive of twist angle. We also discuss possible material candidates for moiré heterostructures with type-II band alignment.

Figures (5)

• Figure 1: (a) Illustration of the setup, which consists of two parts, A and B. In part A, electrons with a small effective mass behave as itinerant “c-electrons”, depicted by green spheres. In part B, the electron wavefunctions are periodic and localized due to moiré potential, and can be regarded as “f electrons”, depicted by pink spheres. An electric field can be applied to tuning band alignment between the parts A and B. (b) Part B can be formed by either twisted homobilayers or heterobilayers. The moiré potential arises from twisting or lattice mismatch. (c) Illustration of type-II band alignment. The conduction and valence bands originate from the parts A and B, respectively. The atomic band gap of the heterostructure is labeled as $M$. Due to the moiré potential, the moiré mini-band from the top valence band can become flat and shift upward in energy. When the strength of the moiré potential $\Delta_0$ is larger than $M$, band inversion can occur, leading to a topological flat band. (d) Schematic figure of localized wavefunctions at the peak positions of moiré potential. (e) The formation of topological flat band after band inversion in the THF model. The sizes of orange and blue dots represent the contribution from “c-electrons” and “f-electrons”. Inset shows the concentration of Berry curvature around $\mathbf{\Gamma}$. (f) Phase diagram for $\Delta_0>0$. $E_g$ and $E_w$ denote the band gap between VB1 and the other minibands and the bandwidth of VB1, respectively. The background color shows $E_g$/$E_w$. White dashed lines separate different phases and the red line guides the peak for $E_g/E_w$ of VB1.
• Figure 2: (a)-(c) show the band dispersion for the parameters $M=0.135$eV and $\Delta_0 = 0.054$ eV, $0.089$ eV and $0.094$ eV, respectively, as denoted by the points (a)-(c) in the phase diagram of Fig.\ref{['fig;1']}(f). The band irreps of VB1 and CB1 are labeled in (a)-(c). In (c), the size of red dots in the VB1 and CB1 represents the overlap between the Bloch states and the Wannier orbital that is localized at the Wyckoff positions $1a$, as shown in (d). Three different Wyckoff positions are labeled by green dots in (d). (e) shows the Berry curvature distribution of VB1 in (c).
• Figure 3: (a) and (b) show the energy spectra before and after band inversion for moiré Chern-band model with periodic $\delta$-function-like potential. The potential strength $\Delta_0$ is $3.85$meV and $4.05$meV, respectively. In panel (b), the white dashed line represents the dispersion obtained from the THF model. (c) $E_w/E_g$ and $\chi=\frac{\int \Omega(\mathbf{k}) d\mathbf{k}}{\int Tr (g(\mathbf{k}))d\mathbf{k} }$ as a function of $\lambda$. When $\lambda=\frac{\Delta E\alpha}{\beta^2}=1$, $\frac{\Omega_{xy}(\mathbf{k})}{Tr (g(\mathbf{k}))}=1$ for every $\mathbf{k}$ and $E_w/E_g=0$. For the parameters of the THF fitting to the band dispersion in (b), we find $\lambda=2.01$ indicated by the red dashed line. The insert shows the Berry curvature distribution of topological flat band.
• Figure 4: (a) The single particle spectrum. Red lines give the band structure of the moiré Chern band model with the parameter $M=0.02$eV and $\Delta_0=0.046$eV lying at the peak line of $E_g/E_w$. The blue dashed line is the band structure calculated from the THF model with the parameters $\alpha=1.238$nm$^2\cdot$eV, $\Delta_E=0.043$eV, $\beta=0.27$nm$\cdot$eV. (b) The Berry curvature distribution of the band VB1 in the moiré BZ indicated by the dashed white lines. (c) The many-body spectrum by the ED calculation for the hole filling $-2/3$ of the flat band VB1. The total crystal momentum is $k_1 \mathbf b_1^\text{M}/N_1+k_2 \mathbf{b}_2^\text{M}/N_2$ with $N_1=6,N_2=4$. The blue region includes the nearly degenerate three-fold ground states of FCI. (d) The flow of the many-body spectrum under the flux insertion in the $\mathbf b_1^\text{M}$ direction.
• Figure 5: (a) and (b) show band dispersions near Fermi energy for monolayer Tl$_{2}$Se$_{2}$ and Zn$_{2}$Te$_{2}$, individually. The irreps of conduction band bottom and valence band top at $\mathbf{\Gamma}$ are labeled in the plot for both compounds. (c) The band dispersion of Tl$_2$Se$_2$/Zn$_2$Te$_2$ heterostructure without moiré forms a type-II band alignment. The sizes of red and blue dots represent the projections to the Tl$_2$Se$_2$ and Zn$_2$Te$_2$ layers. The inset of (c) presents a zoomed-in view of the band dispersion near $\mathbf{\Gamma}$ point from DFT calculation (black curves) and the corresponding effective model fitting (orange curves). (d) Schematics for Tl$_2$Se$_2$/Zn$_2$Te$_2$ heterostructure with moiré potential applied to Tl$_2$Se$_2$ layer. (e) Schematics of type-II band alignment. The conduction and valence bands originate from Tl$_2$Se$_2$ and Zn$_2$Te$_2$, respectively. (f) The band dispersion of Tl$_2$Se$_2$/Zn$_2$Te$_2$ heterostructure with moiré potential on Tl$_2$Se$_2$ layer. Topological flat bands, labelled by CB1, are highlighted in red color. (g) The Wannier centers flow of CB1 with a winding feature demonstrate non-trivial $Z_2$ number of CB1.