Moiré semiconductors on the twisted bilayer dice lattice

The search for a flat-band system has become one of the new trends over the past few decades. Due to the large effective mass of the quasi particles in the flat-band system, the density of states (DOS) is high, and the kinetic energy of the carriers is strongly quenched. Therefore, the flat-band system is a good candidate for studying strongly correlated electronic states induced by strong Coulomb interaction, such as ferromagnetism JOP1991 ; PRA2010 ; PRL1992 , heavy fermions nature2020 ; np2019 , fractional Chern insulators IJMP2013 ; PRL2011 , Wigner crystals PRL2007 ; PRB2008 , and unconventional superconductivity RMP1990 ; PCS2007 ; IOS2020 .

Traditionally, the nearly-flat-band system can be achieved by invoking fine-tuned nearest-neighbor hoppings or long-ranged hoppings or by breaking time-reversal symmetry. Several lattice models have been proposed along these lines in kagome PTP1951 , Lieb PRL1989 , and dice lattices PRB1986 . The existence of the flat band is guaranteed by the destructive interference of the Wannier functions of the lattice structure, and the flat-band states are identified as compact localized states PRL2007 ; PRB2008 ; PRB1986 ; PRL2013 ; CPB2014 ; sathe1 ; sathe2 . This destructive interference protection can also be generalized to lattices with mirror symmetry chen2022 . Usually, this kind of flat band has a zero Chern number in the lattice model with nearest neighbor hopping only sathe3 . On the dice lattice, the flat band can acquire a non-zero Chern number by invoking Rashba spin-orbital coupling and exhibits an anomalous quantum Hall effect by adding onsite Hubbard interactions, which could be realized in the transition-metal oxide SrTiO3/SrIrO3/SrTiO3 trilayer heterostructure by growing in the (111) direction wang2011 . Materials with flat-band structures along these lines have also been reported in Cu(111) confined by CO molecules np2017 , optical lattices, and cold-atom systems shen2010 ; njp2014 ; prl2015a ; prl2015b ; sa2015 ; ol2016 . The topology of the dice lattice with non-Hermiticity has also been studied sarkar2023 . In three dimensions, a famous example is the Kane semimetal, where the flat band structure is associated with the triplet degenerate nodes and can be described by a three-dimensional Lieb lattice model Luo2018 . The low-energy quasi particle, the Kane fermion, can be viewed as a fermionic photon, i.e., a spin-1 fermion. The effective Hamiltonian near the triplet degenerate node is $H_{jk}=\epsilon_{ijk}p_{i}$, where $\epsilon_{ijk}$ is the totally anti-symmetric tensor, and $p_{i}$ is the momentum. By squaring the Hamiltonian, $H^{2}\sim p_{i}p_{j}-p^{2}\delta_{ij}$, which is related to the Hamiltonian of the photon. The case here is similar to squaring the Dirac Hamiltonian to obtain the Hamiltonian for the Klein-Gordon equation. Therefore, the Kane fermion can be viewed as a fermionic photon and the flat band of the Kane fermion corresponds to the longitudinal mode of the photon. Experimentally, the spinful Kane fermion has been reported in Hg_1-x Cd_xTe hgcdte and Cd₃As₂ cdas . The existence of the flat band is shown in the optical conductance by the large peaks near zero frequency hgcdte ; Luo2019 . The spinless Kane fermion is proposed to exist in materials with space groups 199 and 214, such as Ag₃Se₂Au and Pd₃Bi₂S₂ science2016 . Band structures with triple nodal points have also been proposed in ZrTe PRX2016 , LaPtBi PRL2017 , and $A$Pd₃($A$=Pb, Sn) PRB2018 . More kinds of topological materials would be found by the method of symmetry indicators and topological quantum chemistry n1 ; n2 ; n3 .

A new mechanism for generating a flat band was found in twisted bilayer graphene (TBG) at a magic twist angle $\theta\sim 1.08^{\degree}$ Bistritzer-MacDonald2011 ; mac2020 . Unlike the destructive-interference-induced flat band, the flat-band structure of TBG originates from the extremely large band folding of the moiré structure, and TBG becomes a strongly correlated electron system. Very soon thereafter, superconductivity was reported in TBG cao2018 ; bernevig2018 . The flat band in TBG has a non trivial Chern number bernevig2020 , which can be explained by the zeroth chiral Landau levels of Dirac/Weyl fermions Liu2019Pseudo . Away from the half filling, the fractional Chern insulator phase was also proposed and reported in TBG Tarnopolsky2019 ; ashvin2020 ; ashvin2021 and the twisted bilayer MoTe₂ xu2023 ; xu2023b ; shan2023 .

In this paper we consider the combination of moiré structure and destructive-interference-induced flat bands, namely, a twisted bilayer dice lattice (TBD). Inspired by Bistritzer and MacDonald’s continuum lattice model for TBG Bistritzer-MacDonald2011 , we construct a lattice model for the TBD in the reciprocal space. We find that there are flat bands with zero energy in the chiral limit at any twist angle besides the magic ones that are broadened by small perturbation away from the chiral limit. The flat bands are contributed from the ones with zero Chern number as in the Dice lattice and the non trivial ones at the magic angle; therefore, the TBD is a playground for studying the interplay between the zero-Chern-number flat bands and non trivial ones. We further confirm this scenario by considering the pseudo-Landau-level description Liu2019Pseudo and its optical conductance.

The paper is organized as follows. In Sec. II, we provide the detailed construction of the lattice model of the TBD. We also introduce the concept of chiral limit to the TBD. From the chiral limit of the TBD, we show the origin of the flat bands in the TBD. There are flat bands in the TBD in the chiral limit at all angles other than the magic ones. We also numerically calculate the Bloch band structure, and compare it with that of TBG. In Sec. III we use the pseudo-Landau-level language to describe the physics of the flat bands in the TBD, where the pseudo-magnetic field is caused by the interlayer hopping. We can directly find that, besides the topological zeroth Landau level, the higher Landau levels which are topologically trivial also contribute to the flat bands. In Sec. IV we use the degenerate perturbation method to calculate the optical conductance of the TBD and we find that the flat bands contribute a peak-splitting structure, which is conclusive evidence of the experimental prediction for the existence of the flat bands. This phenomenon also exists at all angles besides the magic ones when compared with TBG. Section V provides a brief summary and discussion of our conclusions.

The dice lattice can be viewed as two honeycomb lattices ($A-B$ and $B-C$) sharing the same sublattice site $B$. The lattice base vectors are $\bm{a}_{1}=a(\frac{1}{2},\frac{\sqrt{3}}{2})$ and $\bm{a}_{2}=a(-\frac{1}{2},\frac{\sqrt{3}}{2})$ and the corresponding reciprocal vectors $\bm{b}_{1}=\frac{4\pi}{\sqrt{3}a}(\frac{\sqrt{3}}{2}$, and $\frac{1}{2}),\bm{b}_{2}=\frac{4\pi}{\sqrt{3}a}(-\frac{\sqrt{3}}{2},\frac{1}{2})$, where $a=\sqrt{3}d$ is lattice constant, and $d$ is the distance between two nearest sites [see Fig. 1(a)]. For simplicity, we consider a spinless lattice model with nearest-neighbor hopping only. Due to this similarity, the TBD has the same moiré structure as TBG [see Fig. 2(a)]. There is a flat band $E=0$ in the lattice spectrum. The low energy behavior near the Dirac point can be captured by $H_{eff}={\bf k\cdot S}$, where ${\bf k}=(k_{1},k_{2},0)$ is the lattice momentum, and ${\bf S}$ is the spin-1 generalization of the Pauli matrix that acts on sublattice space of the order of $A$, $B$, and $C$,

Since the dice lattice can be realized in the SrTiO3/SrIrO3/SrTiO3 trilayer heterostructure by growing in the (111) direction wang2011 , it is possible to realize the TBD in this material by twisting when growing.

For TBG, Bistritzer and MacDonald proposed a low-energy effective continuum Dirac model of the moiré structure for a small twist angle $\theta<10^{\degree}$ using Bloch bands near the Dirac points. The effective model consists of two isolated graphene layers and hopping terms between them. With this model they reveal flat Bloch bands in the electric structure at magic twist angles which give rise to a high DOS Bistritzer-MacDonald2011 . Similar to the case of TBG, we follow Ref. Bistritzer-MacDonald2011 to construct a continuum model for the TBD. In principle, for the Bloch-band-based effective theory to be valid, the valley structure should be present. However, the global flat band in the single-layer dice model may negate this validity. This obstacle could be avoided by including the second-nearest-neighbor hopping in the lattice model such that the global flat bands become dispersive and acquire the valley structure. In a recent paper arv2023 Zhou $et$ $al.$ confirmed that the flat-band structure is substantiated in the TBD in the absence of the second-nearest-neighbor hopping. Therefore, we can only consider the nearest-neighbor hopping in the TBD for simplicity. Later we will show that this is the case in the chiral limit of the continuum model, and there are exact flat bands at all angles. In contrast, away from the chiral limit, the results would be less predictive if the second-nearest-neighbor hopping were zero and the valley structure were absent due to the flat bands.

In the TBD, we keep the top layer 1 fixed and rotate the bottom layer 2 by $\theta$ with respect to layer 1. The effective TBD Hamiltonian contains intralayer and interlayer parts. The low-energy intra-Hamiltonian reads

where $\psi_{1,2}$ are the annihilation operators in layers 1 and 2 respectively, and ${\bf S}_{\theta}=e^{(i\theta/2)S_{3}}(S_{1},S_{2})e^{(-i\theta/2)S_{3}}$.

For the interlayer hopping term $H_{\bot}$, we consider nearest-neighbor hopping from layer 1 in the $\alpha$ $(\alpha=A,B,C)$ sublattice to the closest sublattice $\beta$ $(\beta=A^{\prime},B^{\prime},C^{\prime})$ in layer 2 (see Fig. 1). The hopping amplitude depends on the difference between the two sites. We have,

with

where $A_{u.c}$ is unit cell area and $K_{1,2}$ represent the Dirac point for each layer which satisfies $K_{2}=R_{\theta}\cdot K_{1}$, where $R_{\theta}$ is the rotation matrix. Here $\tau_{1/2,\alpha/\beta}$ is a vector connecting the two sites in the unit cell. For the TBD, we consider the $A-B$ stacking (Bernal) configuration coordinates as $\tau_{1,A}=\tau_{2,B^{\prime}}=(0,0)$, $\tau_{1,B}=\tau_{2,C^{\prime}}=(0,d)$, and $\tau_{1,C}=\tau_{2,A^{\prime}}=(0,2d)$ [see Fig. 1(b)]. To compare the results with TBG, we choose $d=1.42$ Å, the lattice constant of graphene. Here $t({\bf k})$ is the Fourier transformation of the tunneling amplitude $t({\bf r})$ which satisfies $t^{\alpha,\beta}_{1,2}(\mathbf{r}_{1},\mathbf{r}_{2})=t^{\alpha,\beta}_{1,2}(\mathbf{r}_{1}+\tau_{1,\alpha}-\mathbf{r}_{2}-\tau_{2,\beta})$ and decays rapidly if $k$ in reciprocal space exceeds the Dirac point Bistritzer-MacDonald2011 . Considering this property we only need to choose three vectors $\mathbf{G}_{l}=\mathbf{g}_{(l),1},\mathbf{g}_{(l),2},\mathbf{g}_{(l),3}$, with $\mathbf{g}_{(l),1}=0$, $\mathbf{g}_{(l),2}=\mathbf{b}_{(l),2}$, and $\mathbf{g}_{(l),3}=-\mathbf{b}_{(l),1}$ the reciprocal lattice vectors, $(l)=(1),(2)$ the layer index, and $\mathbf{b}_{(1),i}=R_{\theta}\mathbf{b}_{(2),i}$. Substituting these three vectors into the hopping matrix (4), we have

where $\bm{q}_{b}=\frac{8\pi\sin(\theta/2)}{(3a)}(0,-1)$, $\bm{q}_{tr}=\frac{8\pi\sin(\theta/2)}{(3a)}(\frac{\sqrt{3}}{2}$, and $\frac{1}{2}),\bm{q}_{tl}=\frac{8\pi\sin(\theta/2)}{(3a)}(-\frac{\sqrt{3}}{2},\frac{1}{2})$ are the vectors connecting the nearest Dirac points of the two layers in the moiré Brillouin zone [see Fig. 2(b)], and

where $W=\frac{t(|K|)}{A_{u.c}}$ with $|K|=4\pi/(3\sqrt{3}d)$, and $\phi=2\pi/3$. Here we choose $W=110$ meV as in graphene Bistritzer-MacDonald2011 . We choose also $\tau_{0}=0$, which is the translation vector of the TBD. For later convenience, we also denote the transition amplitude between the two layers by $W_{\alpha\beta}$.

The chiral limit is an ideal model to study the flat-band structure, while in reality, the chiral limit is violated by finite $W_{AA^{\prime}}$ or $W_{AC^{\prime}}$ arising from different atomic stacking and atomic layer deformation PhysRevB.90.155451 . Fortunately, Liu $et$ $al$. showed that Bistritzer and MacDonald’s model for TBG at a small magic angle can be effectively described by pseudo-Landau levels under an $SU(2)$ gauge potential Liu2019Pseudo . After a gauge transformation on the Bloch functions and expanding the tunneling potential to the linear order of $r/L_{s}$, Liu $et$ $al.$ arrived at the pseudo-Landau-level Hamiltonian for TBG,

where $\mathbf{A}=(2\pi u^{\prime}_{0})/(L_{s}ev_{f})(y,-x)$ is the $SU(2)$ gauge potential and the Pauli matrices $\tau_{1,2}$ act on the layer index. In addition, $u^{\prime}_{0}$ denotes the hopping parameters $W_{AB^{\prime}}$ and $u_{0}$ denotes $W_{AA^{\prime}}$. The effective magnetic field ${\bf B}=\nabla\times{\bf A}\approx 120$ T for TBG at $\theta\approx 1.08^{\degree}$ Liu2019Pseudo ; San-Jose2012Non-Abelian ; ren2021 . The effective magnetic fields have opposite directions on each layer; therefore, we can define a time-reversal operator $\Theta=i(\tau_{2}\otimes\sigma_{0})\mathcal{K}$, where $\mathcal{K}$ means complex conjugation. In addition, the time-reversal symmetry is preserved. After a similar treatment, we can also derive the pseudo-Landau-level Hamiltonian for the TBD. By replacing ${\bf\sigma}$, which acts on the sublattice index of graphene, by ${\bf S}$, the result is

where $u^{\prime}_{0}$ denotes hopping parameters $W_{AB^{\prime}}$ and $W_{BC^{\prime}}$ in the TBD, and $u_{0}$ denotes $W_{AA^{\prime}}$. In the following numerical calculations, we set $u^{\prime}_{0}=0.01$ eV, and $u_{0}=0.1$ eV.

To discuss the Landau-level structure, we transform the Hamiltonian (17) into the basis that diagonalizes $\tau_{2}$,

where $\pi_{x}=\hbar v_{f}k_{x}-ev_{f}A_{x}$, $\pi_{y}=-\hbar v_{f}k_{y}+ev_{f}A_{y}$, $\pi^{\prime}_{x}=\hbar v_{f}k_{x}+ev_{f}A_{x}$, and $\pi^{\prime}_{y}=-\hbar v_{f}k_{y}-ev_{f}A_{y}$. Since $u_{0}$ is small, we can treat it as a perturbation. Then for the unperturbed Hamiltonian $H_{0}$, we define the Landau-level creation operators of the two layers, $b^{\dagger}=\sqrt{\frac{L_{s}}{8\pi u^{\prime}_{0}\hbar v_{f}}}(\pi_{x}-i\pi_{y})$ and $a^{\dagger}=\sqrt{\frac{L_{s}}{8\pi u^{\prime}_{0}\hbar v_{f}}}(\pi^{\prime}_{x}+i\pi^{\prime}_{y})$, and the nonzero commutators are $[b,b^{\dagger}]=[a,a^{\dagger}]=1$,

where $\hbar\omega=\sqrt{\frac{8\pi u^{\prime}_{0}\hbar v_{f}}{L_{s}}}$. The $H_{0}$ is block-diagonalized for each layer. We denote by $|\Psi^{(1)}\rangle$ and $|\Psi^{(2)}\rangle$ the eigen wave functions for layers 1 and 2, respectively. The components of $|\Psi^{(1)}\rangle$ and $|\Psi^{(2)}\rangle$ are made of Landau-level wave functions that satisfy $b\ket{n}=\sqrt{n}\ket{n-1}$, $b^{\dagger}\ket{n}=\sqrt{n+1}\ket{n+1}$, $a\ket{m}=\sqrt{m}\ket{m-1}$, and $a^{\dagger}\ket{m}=\sqrt{m+1}\ket{m+1}$. To be more specific,

where $A_{n}^{i}$ and $B_{m}^{j}$ are the normalizing coefficients and when $n(m)<0$, $|n(m)\rangle=0$.

For layer 1, there are three energies for $n>1$,

In the wave function for $E_{n,0}^{(1)}$, the second component $A_{n,0}^{2}=0$, which has a structure similar to that in the flat-band wave function of the dice lattice. For $n=1$, there are two eigenenergies

which are the first Landau levels, with the wave functions $\Psi_{1,\pm}^{(1)}=\frac{1}{\sqrt{2}}(0,\pm|0\rangle,|1\rangle)^{T}$. Here $n=0$ is the topological zeroth Landau level with $E_{0,0}^{(1)}=0$ and $|\Psi_{0,0}^{(1)}\rangle=(0,0,|0\rangle)^{T}$.

For layer 2, the derivation for the spectrum and wave function is similar. For $m>1$, $E_{m,\pm}^{(2)}=\pm\hbar{\omega}\sqrt{2m+1}$ and $E_{m,0}^{(2)}=0$. The second component of $|\Psi_{m,0}^{(2)}\rangle$, $B_{m,0}^{2}=0$. For $m=1$, $E_{1,\pm}^{(2)}=\pm\hbar\omega$ and $|\Psi_{1,\pm}^{(2)}\rangle=\frac{1}{\sqrt{2}}(|1\rangle,\pm|0\rangle,0)^{T}$. For $m=0$, $E_{0,0}^{(2)}=0$, and $|\Psi_{0,0}^{2}\rangle=(|0\rangle,0,0)^{T}$.

We treat small $u_{0}$ as a perturbation, which will mix the states between $|\Psi^{(1)}\rangle$ and $|\Psi^{(2)}\rangle$ and lift the degeneracy at zero energy. This is consistent with the results in Sec. II because $u_{0}$ corresponds to $W_{AA^{\prime}}$, which breaks the chiral symmetry. This behavior of small $u_{0}$ is also confirmed numerically [see Fig. 5(b)], which is similar to the band structure in Fig. 4. Here we emphasize a difference between TBG and the TBD. Because of the lack of zero-energy states $|\Psi_{n,0}\rangle$ for $n>1$ in the pseudo-Landau levels of TBG, the effect of $u_{0}$ is of the third order Liu2019Pseudo , while in the TBD, it is of the first order. Therefore, the optical conductance for finite $u_{0}$ will behave differently for TBG and the TBD (discussed below). A similarity between TBG and the TBD is that, after the perturbation of $u_{0}$, a double degeneracy remains which is related to the $S_{3}$ symmetry Liu2019Pseudo .

In reality, materials that rotate the polarization plane of linearly polarized light have many applications in various devices, which is usually achieved by magneto-optical effects, the quantum Hall effect, and the Kerr and Faraday rotations by invoking external magnetic fields oc1 ; oc2 ; oc3 ; oc4 ; oc7 , which also limits the applications in small-scale devices. Therefore, searching for materials with intrinsic properties that rotate light is urgent for recent applications. The TBG is such a candidate for advanced optical applications due to the tunable twist angles occ1 ; occ2 . The optical conductivity of the bilayer dice lattice without twisting was studied before suk1 ; suk2 . Here we study the optical conductance of the TBD, which could be another candidate for such applications.

We utilize the Kubo formula in the pseudo-Landau-level basis to calculate the optical conductance of the TBD system PhysRevB.94.125435 ,

where $l_{B}=\sqrt{(L_{s}hcv_{f})/4\pi u^{\prime}_{0}}$ is the magnetic length. Without considering the spin degree of freedom, we can simply set $g=2$. Here $f$ is the Fermi distribution and $\omega$ is the photon energy. Although we use the pseudo-Landau-level description, no external magnetic field is applied. The pseudomagnetic field is induced by the hopping of $W_{AB^{\prime}}$ and $W_{BC^{\prime}}$, which can also be induced by strain guo2022 .

We take into account the contribution of all Landau levels; however, the remaining double degeneracy of the band structure will cause divergence in conductance. For a doubly degenerate band, we set the divergent part of the Kubo formula (24) as

From the Hamiltonian (19), the current is obtained as $j_{\alpha,\beta}=\frac{\delta H_{0}(\mathbf{k})}{\delta A_{\alpha,\beta}}$.

The numerical results for $\mu=0.05$ eV and $T=300$ K of the conductance of TBG and the TBD are shown in Figs. 5(c) and 5(d). Since the chemical potential we choose is close to zero, all significant absorption peaks originate from transitions between the near-zero bands and the positive-energy bands, or between the negative-energy bands and the near-zero bands. The double-peak structure of TBG is caused by the splitting of double degeneracy, and the width of the splitting is proportional to $u^{\prime}_{0}$.

By comparing the result of TBG with that for the TBD, we see that the double-peak structure of TBG does not exist in the TBD. In the TBD, the peak splits into several small peaks. This reflects that there are many states near zero energy. In Fig. 5(c) the distance between the double peaks is $\Delta\omega\sim 0.03$ eV, which corresponds to 40 $\mu$m, while in Fig. 5(d) the typical distance between the split peaks is $\Delta\omega\sim 0.015$ eV, and the corresponding wave length is about 80 $\mu$m, which are all in the terahertz range and experimentally detectable. Therefore, this peak-splitting structure provides proof of the experimental prediction showing the existence of the large degeneracy of the flat bands in the TBD.

Although the Landau-level description for TBG fails when the twist angle is away from the magic ones, the Landau-level structure remains in the TBD [see Figs. 4(g) and 4(h)]. Therefore, in the optical conductance at angles other than the magic ones, the peak splitting structure remains in the TBD, whereas the peaks arising from the transitions that form the flat bands disappear in TBG, which is also a key experimental difference between TBG and the TBD.

The transitions between these zero-energy levels are forbidden in the TBD, which means there are no peaks at low frequency near $\omega\sim 0$, which is a key difference from the three-dimensional Kane fermion Luo2019 . The reason for this phenomenon is attributed to the structure of the current and wave function in two dimensions, that is, $\braket{\Psi}{j_{\alpha}}{\Psi^{\prime}}\braket{\Psi^{\prime}}{j_{\beta}}{\Psi}=0$ between those near-zero bands, and their contribution to the optical conductance being zero, while in three dimensions, the $k_{3}S_{3}$ part will have a nontrivial contribution, and the transition between different Landau levels has a non zero $k_{3}$, which causes the peaks near zero frequency Luo2019 .

In this paper we constructed a lattice model for the TBD. In the chiral limit, it has flat bands at all twisted angles besides the magic ones and the flat bands are broadened when chiral symmetry is broken, which could be confirmed by the peak splitting structure of the optical conductance near the magic angles. Away from the magic angles, the peak splitting remains in the TBD, whereas these peaks disappear in TBG due to the nonexistence of the flat bands. The flat bands in the TBD are composed of zero-Chern-number bands by destructive interference of the states on the dice lattice as well as the topological nontrivial bands by the moiré structure at the magic angles. In this model we have neglected the spin degrees of freedom of electrons. If the spin orbital coupling interaction is added, the bands with zero Chern number may become non-trivial wang2011 . It is possible to realize the TBD in the transition-metal oxide SrTiO3/SrIrO3/SrTiO3 trilayer heterostructure by growing and twisting in the (111) direction. As a semiconductor, due to the high DOS of the flat bands of the TBD, the TBD may have potential applications in temperature-sensitive and photosensitive manipulations. With interactions, the TBD may also be a good candidate as a fractional Chern insulator.