Gate-tunable flat bands in van der Waals patterned dielectric superlattices

We begin with a discussion of the PDS scheme using monolayer transition metal dichalogenides (TMD). The PDS that we consider here (and that can be experimentally fabricated) are of fairly long-wavelength $a \gtrsim 20 ~ {\rm nm}$. The resulting superlattice bandstructure can be captured by an effective $k\cdot p$ model expanded close to the K and $K'$ points as

$$\begin{align} \newcommand{\e}{{\rm e}} \renewcommand{\vec }[1]{{\bf #1}} \newcommand{\re}{{\rm Re}} \newcommand{\la}{\langle} \displaystyle \mathcal{H} =v\hbar (\sigma_x k_x + \xi \sigma_y k_y) + \Delta \sigma_z+V_s\sigma_0 f(\vec {r}), \label{eq:hamiltonian} \nonumber \end{align} \tag{ 1 }$$

where $v$ is the velocity, $\sigma_{x,y,z}$ are the Pauli matrices, $2\Delta$ is the bandgap of the TMD, $\newcommand{\p}{\partial} \xi = \pm 1$ for K and $K'$ valleys, $V_s$ is the amplitude of the superlattice potential applied by the PDS scheme in figure 1, and $\newcommand{\re}{{\rm Re}} \renewcommand{\vec }[1]{{\bf #1}} f(\vec {r})$ is the spatial pattern of the superlattice potential. Here we have suppressed spin indices since the bandstructure physics we discuss does not differentiate between spin/valley species (i.e. the superlattice potential is a scalar potential); spin can be included in a straightforward fashion that does not affect the physics we discuss below. Further, we note that the particularly large Ising spin splitting (∼ hundreds of meV) in the valence bands of many TMD materials, effectively lock the valley and spin in the valence bands making the model in equation (1) a good descriptor of the low-energy mini-bandstructure.

As shown in figure 1(b), dielectric inhomogeneity in the substrate material can enable electric control of a spatially inhomgeneous potential with variations on the order of the hole width (e.g. several tens of nanometers) [23]. Here gray indicates a dielectric (shown SiO₂ with $\newcommand{\e}{{\rm e}} \epsilon =4$) and empty indicates air ($\newcommand{\e}{{\rm e}} \epsilon =1$). As shown, when a gate potential is applied, the dielectric contrast yields a spatially modulated potential even in the vdW material layer (red) with potential amplitudes $V_s$ that can be switched on and off electrically [23]. Figure 1(b) was plotted using a numerical solution for a generalized Poisson equation in a spatially inhomogeneous dielectric environment (see supplementary information, SI (stacks.iop.org/TDM/7/015028/mmedia)).

To illustrate the (flat) band structure engineering in the PDS of vdW materials, we first use a triangular superlattice (see other lattices in SI) so that

$$\begin{align} \newcommand{\e}{{\rm e}} \renewcommand{\vec }[1]{{\bf #1}} \newcommand{\re}{{\rm Re}} \newcommand{\la}{\langle} \newcommand{\p}{\partial} \displaystyle f(\vec {r}) = \sum_{i=1,2,3} 2 \cos (\vec {G}_i \cdot \vec {r} + \phi_i), \label{eq:superlattice} \nonumber \end{align} \tag{ 2 }$$

where $\newcommand{\re}{{\rm Re}} \renewcommand{\vec }[1]{{\bf #1}} \vec {G}_i$ are the triangular superlattice wave vector oriented $60^\circ$ relative to each other, $\newcommand{\p}{\partial} \newcommand{\re}{{\rm Re}} \renewcommand{\vec }[1]{{\bf #1}} |\vec {G}_i| = 2\pi/a$, and $\newcommand{\p}{\partial} \phi_i$ is a relative phase. For simplicity, we will set $\newcommand{\p}{\partial} \phi_i=0$ in the following. It does not qualitatively affect the physics we discuss below. As expected, the superlattice folds the original TMD bands into a series of superlattice minibands (band folding is illustrated for $V_s=0$ as the light dashed lines in figure 2(a)). Here the bandstructure is plotted in a superlattice defined mini Brilluoin zone (MBZ), where the tilde symbol indicates the MBZ. When $V_s$ is switched on, Bragg scattering mixes the bands and produces a mini-band structure (colored lines and gray lines with $V_s = 0.5~{\rm meV}$) in both the conduction band (positive energies) and the valence band (negative energies) as shown in figure 2(a). We have colored only the first three conduction and valence minibands so as to focus our discussion on them; in solving for the mini-bandstructure a set of 162 mini-bands are employed to ensure convergence of the lower bands (we only show 12 minibands) that we focus on in the main text. Physically, the inclusion of many minibands in our numerical calculation is to capture the physics of the extreme bandstructure reconstruction regime in which the superlattice strength $V_s$ is larger than that of a kinetic energy of the electrons. Here we have used material parameters corresponding to WSe₂ with $v \hbar = 3.94~{\rm eV \mathring{\rm A}}$, $\Delta = 0.8~{\rm eV}$ [26].

Strikingly, a clear gap between the top most (valence) miniband (blue, figure 2(a)) and the rest of the minibands opens up. This is distinct from that expected from graphene PDS [23] where a $\newcommand{\p}{\partial} \pi$ berry phase prevents backscattering and gap opening; in the case of graphene PDS secondary Dirac cones form at the MBZ corners [4, 10, 11, 23, 25]. In contrast, the wavefunction (AB-sublattice pseudospinor) in TMD close to the band edge has weight mostly on a single sublattice (aligned to a pole in the Bloch sphere), allowing maximal backscattering and mini-gap formation.

Due to the large $\Delta$ in TMDs, minigap opening squeezes the top most valence miniband creating a confined energy window for it to exist with a sharp density of states (shaded blue in right panel of figure 2(a)). Indeed, as $V_s$ is increased, the top most (valence) mini-band is further flattened (figure 2(b)) with very narrow band-widths ($\ll$$1~{\rm meV}$) achievable with modest applied gate voltage creating a nearly flat band. To see this, we have plotted the maximum and minimum energy in each miniband (solid lines figure 2(b), this indicates the bandwidth) with the colors used corresponding to the colored 3 conduction and colored 3 valence minibands of figure 2(a); the shaded region between miniband maxima and minima are shaded in corresponding colors. We note that the flat top most (valence) miniband is well separated from other bands with the large $\Delta \sim {\rm eV}$ to the minibands in the conduction band, and gate tunable minigaps to the other minibands in the valence band (that can reach tens of meV); the minigaps increase with applied gate potential, see SI.

One unusual feature is that the application of superlattice PDS also renormalizes the effective bandgap between the conduction and valence band states, with changes in effective bandgap of up to $60 ~ {\rm meV}$ for the largest superlattice potential amplitude shown in the figure (see figure 2(b)). This arises from the significant miniband reconstruction. Indeed locally in real space, the electrons feel large variations in superalattice potential that range from $+6V_s$ at the peak to $-6V_s$ in the troughs. This large peak-to-trough difference enables electrons to be confined locally to produce a flatband structure (figure 2(a)).

The blue band is not the only flat band that occurs in PDS. For example, at sufficiently high superlattice potential ($V_s>0$), both green and purple minibands in the valence band flatten out. Unlike the blue band they tend to stick together. In fact, the PDS scheme yields sets of flat minibands in the valence band (some of which are nearly degenerate with each other forming bunched bundles of bands) with severely compressed bandwidths. Each of these bundles of flat bands are well separated from each other with large electrically tunable minigaps (in figure 2(b), we show three flat miniband bundles in the valence band at $V_s >0$).

We note that the PDS scheme, when applied to gapped Dirac materials such as WSe₂, naturally breaks particle-hole symmetry with (positive and negative energy) minibands exhibiting contrasting behavior (see figure 2(a)). For example, when $V_s>0$ the top (valence) miniband (blue) gets squeezed into a single non-degenerate flat miniband and is well separated from the other minibands, while the bottom (conduction) miniband (yellow) adheres closely to the next miniband (red); this displays a particle-hole asymmetric behavior. Symmetry in the miniband structure, however, is restored when both $V_s \to - V_s$ and $\newcommand{\re}{{\rm Re}} \renewcommand{\vec }[1]{{\bf #1}} \varepsilon_{\vec {k}} \to - \varepsilon_{\vec {k}}$ are interchanged (see detailed discussion in SI).

This unconventional feature allows the type of flat mini-bandstructure to be tuned by gate voltage. When $V_s>0$ the top (valence) miniband (blue) is a single non-degenerate flat band (per valley). However, when $V_s<0$ this same blue miniband, while flattening out, adheres closely to the green band (figure 2(b)); when large enough gate voltage is applied, they form a close pair of flatbands that bunch together. This flatband bundle has a width $\sim$$1~{\rm meV}$. Even though both green and blue minibands stay together, nevertheless, we find that they are separated by a very small energy gap. Indeed, close to $\tilde{K}, \tilde{K}'$ points, the blue and green miniband structure (for $V_s<0$) bands resemble Dirac cones (with an extremely small gap; we estimate the gaps to be of order several $\mu\,{\rm eV}$, see SI). We note that away from the small minigaps, the miniband spectra mimic the flattened Dirac bands in twisted bilayer graphene [15]; interestingly, the chirality of electrons at $\tilde{K}, \tilde{K}'$ points are the same (see SI) mirroring the behavior found in twisted bilayer graphene.

As a result, PDS enables to achieve multiple types of flatbands via in situ gate voltage tuning (e.g. from single non-degenerate flat-miniband for positive $V_s$ to Dirac-cone-like for negative $V_s$). This unusual asymmetric behavior (for both conduction/valence bands and opposite signs of $V_s$) arises from the gapped Dirac pseudo-spinor form of their wavefunctions (such asymmetry does not arise for a simple massive two-dimensional electron gas). Indeed, the role of the TMD pseudospinor texture is further evidenced by how PDS induced Bragg scattering also changes the winding of the gapped Dirac pseudo-spinor wavefunctions. This dramatically reconstructs the Berry curvature distribution in each of the minibands (for discussion of miniband Berry curvature, see SI).

One of the most attractive features of PDS flat bands, is that they do not occur at fine-tuned gate voltages or superlattice wavelength. The larger the superlattice wavelength, the smaller the applied superlattice potential needed to flatten the PDS minibands (see for e.g. figure 2(c) where a larger superlattice wavelength $a = 40~ {\rm nm}$ was used). Indeed, the main requirement imposed on the superlattice wavelength is that it should be smaller than the sample (disorder) mean free path to enable the electrons to Bragg scatter off the superlattice potential (see SI for an estimate of the mean free path of WSe₂). Further, even when an intrinsic conduction and valence band mass asymmetry is included in the bare TMD hamiltonian, we find that the flat minibands remain robust, with only small changes in their miniband dispersion relation, see full discussion in SI. This PDS strategy stands in stark contrast to the flat bands found in moiré superlattices where specific 'magic' twist angles between layers are required (in the case of twisted bilayer graphene) or low twist angles required in other moirè superlattice strategies.

Instead, the key to achieving flat bands in PDS is large $\Delta$ as well as low velocities. As shown in the inset of figure 2(b), when we fix $v \hbar$ and a (same parameters as panel (a)), the larger the $\Delta$, the smaller the bandwidth at a given superlattice potential $V_s$. For WSe₂, we find $\Delta = 800~{\rm meV}$ allowing very small bandwidths to be achieved even for small $V_s$ applied, e.g. the bandwidth is  ∼${\rm \mu eV}$ when $V_s = 5 ~{\rm meV}$. We note, parenthetically, that when $\Delta$ is small so that $V_s$ applied is on order $\Delta$, the minibands in the conduction and valence band can start to mix, vastly complicating the miniband structure and making the conditions for flat bands in PDS hard to achieve.

We anticipate that the PDS scheme can be applied to other van der Waals materials with large bandgaps. For example, we have computed the miniband structure for a range of TMD materials and have found that well separated sets of flat minibands generically occur—see SI for full band structure. Further, other superlattices can also be easily implemented and produce qualitatively the same results: we have computed the minibands for square as well as hexagonal superlattices as well and find similar well separated sets of flat minibands, see SI. This versatility with superlattice structure provides the ability to study flatbands and its concomittant interaction effects in other types of lattices which can have a different type of symmetry (that may be different from the underlying lattice).

Perhaps the most striking consequence of PDS flat bands is the ability to enhance correlation effects. This is because the extremely small bandwidth of single-particle mini-bandstructure of PDS ($\ll$$1~ {\rm meV}$) quenches the kinetic energy of the electrons. As a result, other energy scales such as that arising from electron-electron interactions can determine the behavior of the electronic system. For clarity, in the following, we will focus on the top-most valence miniband for $V_s>0$ which is well separated from the other superlattice minibands. Further, we note that due to the large Ising splitting in TMDs, the spin and valley degree of freedom are locked; at the non-interacting level, this band has only a two-fold degeneracy with spin up and spin down occurring in the (original) valleys at K and $K'$.

To proceed, we first estimate the strength of electron-electron interactions in the superlattice by modeling the average interaction energy of electrons confined in the troughs of the superlattice potential as [27, 28]

$$\begin{align} \newcommand{\e}{{\rm e}} \renewcommand{\vec }[1]{{\bf #1}} \newcommand{\re}{{\rm Re}} \newcommand{\la}{\langle} \newcommand{\p}{\partial} \displaystyle U = \sum_{\vec {q}} \frac{2\pi e^2}{\epsilon (q + q_s)} {\rm exp} (-q^2 a_W^2), \label{eq:interactions} \nonumber \end{align} \tag{ 3 }$$

where $q_s = r_s^{-1}$ is the effective inverse screening length, and a_W is the extent of the PDS flatband electronic wavefunction confined in the troughs of the superlattice. Here we have chosen a simple Gaussian ansatz for the extent of the wavefunction; other models do not qualitatively affect the results we discuss below. In figure 3(a), we plot the strength of U as a function of both the screening length r_s as well as a_W taking $\newcommand{\e}{{\rm e}} \epsilon =4$. This yields large interaction energies of order several tens of meV for a wide range of r_s and a_W. For WSe₂, we estimate effective r_s of a few nanometers arising either from intrinsic screening of the electron gas or from proximal gates, see discussion in SI. To estimate the extent of the electronic wavefunction a_W, we employ a variational approach on the Gaussian wavefunction ansatz. This yields typical a_W that decreases for increasing superlattice potential, reaching a fairly confined state of  ∼5–10 nm even for modest applied superlattice potential, see SI.

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Figure 3(a) indicates that U can dominate over the extremely small non-interacting PDS flatband bandwidths (see e.g. figure 2). As a result, we anticipate interaction effects can become significant. For example, adopting a Hartree–Fock approximation, we find that the exchange integral (i.e. the Fock-part of the electron self-energy for the bands) can be large and drive the spin-degenerate flatband system into a Stoner-type ferromagnetic state [28]; the exchange integral can be estimated by U in equation (3) following [27, 28]. To describe this, we use a simple mean-field model for the energy density $\mathcal{E}_{\rm MF}$ of a PDS flat miniband with spins indexed $s = \uparrow, \downarrow$ as

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\langle} \displaystyle \mathcal{E}_{\rm MF} = \sum_s \Big[\int^{\mu_s} {{\rm d}} \varepsilon \, \varepsilon Z(\varepsilon) - \frac{U}{2} n_s^2\Big], \label{eq:meanfieldenergy} \nonumber \end{align} \tag{ 4 }$$

where $\mu_s$ is the chemical potential of spin s, and $\varepsilon$ is the band energy, and U can be estimated by equation (3) (for a full description, see SI). Here we have used a simple Gaussian $\newcommand{\e}{{\rm e}} Z(\varepsilon) = A \exp [ - (\varepsilon - \varepsilon_0){}^2 / 2\Gamma^2 ]$ to model a broadened spectral weight of the flatband, and the density of each spin species is $n_s = \int^{\mu_s} {\rm d} \varepsilon Z(\varepsilon)$. The broadening, $\Gamma$, can be induced by a number of different processes for e.g. via disorder. Other forms of the spectral function do not affect the qualitative conclusions we discuss below. Here $\newcommand{\p}{\partial} A = [ \sqrt{2\pi \Gamma^2} ]^{-1}$ is a normalization constant.

For small U, the two spin species are degenerate with $\mu_\uparrow = \mu_\downarrow = \mu_0$ is the spin chemical potential. However, for large enough U this spin symmetry can become broken. To see this, we first write $\newcommand{\p}{\partial} n_s = n_0 \pm \delta n$ where $2 n_0 = n_\uparrow + n_\downarrow$. Expanding equation (4) to $\mathcal{O} (\delta n^2)$ we obtain the energy density for the flatband as

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\langle} \displaystyle \mathcal{E}_{\rm MF} = \mathcal{F}_0 + \delta n^2 [ 1/ Z (\mu_0)- U] + \mathcal{O} (\delta n^4), \label{eq:stoner} \nonumber \end{align} \tag{ 5 }$$

where $\mathcal{F}_0 = 2 \int^{\mu_0} {{\rm d}} \varepsilon\, \varepsilon Z(\varepsilon)- U n_0^2$ is the energy in the symmetry unbroken phase, and the $\delta n^2$ term describes the energy cost to imbalance the spin species. Crucially, an instability in the spin up/down population is induced when the coefficient of the second term is negative $Z(\mu_0) U > 1$; this is analogous to the Stoner criterion that has been successfully applied to understanding quantum Hall ferromagnetism [28].

The conditions for symmetry breaking depend on a competition between the broadening and U. In our model above, this is parameterized by r_s, a_W, as well as the broadening energy/width $\Gamma$. To illustrate this, we plot $Z (\mu_0 = \varepsilon_0) U$ in figure 3(b). From figure 3(b), we can see the competition clearly: for larger (smaller) $\Gamma$, we require more (less) localized electron states, or larger (smaller) U to enter the broken symmetry phase. Taking $r_s = 5 ~ {\rm nm}$ as a demonstration, we find large $Z (\mu_0 = \varepsilon_0) U$ for wide swathes of the $\Gamma$-a_W parameter space (figure 3(b)). We estimate that $a_W \sim 5$–$10~ {\rm nm}$ can be achieved by modest superlattice potentials in the PDS scheme. At low temperatures, broadening is typically dominated by disorder [29–31]. Taking values for high quality WSe₂ [24], we estimate disorder-induced broadening $\Gamma_{\rm dis} \sim$ of order several ${\rm meV}$ are available in high quality present day samples, see SI. As a result, we expect that the conditions for realizing a ferromagnetic state using PDS flatbands can be attained in WSe₂. Interestingly, this analysis is not confined to $\newcommand{\e}{{\rm e}} \mu_0=\epsilon_0$. The ferromagnetic instability can occur at a variety of chemical potentials so long as the criterion is satisfied $Z (\mu_0) U >1$, see below.

There are numerous probes of the extreme mini-bandstructure reconstruction we discuss here. For example, we anticipate a lightly hole doped WSe₂ PDS to exhibit a dramatic change in its low-frequency (THz) optical absorption characterisitcs as $V_s>0$ is switched on. While at $V_s=0$ such a sample will exhibit a Drude peak around $\omega =0$, when $V_s>0$ is switched on, the Drude peak will diminish as the topmost (valence) miniband flattens, and exhibit additional sharp THz absorption peaks corresponding to transitions between the sets of flat minibands (in the valence band).

Similarly, we expect dual-gate control to enable control of both superlattice potential as well as the filling of the minibands. Such filling control can enable to probe both the metallic as well as the insulating states induced by the superlattice potential. When pushed into the regime where flatbands exhibit ferromagnetism, the spins (locked to the valley) split; at $\mu_0 = \varepsilon_0$ the flatband would exhibit a ferromagnetic insulating state. Interestingly, ferromagnetism can also occur away from $\mu_0 = \varepsilon_0$ so long as $Z (\mu_0) U >1$. As a result, the ferromagnetic system can be metallic. Crucially, we note that the minibands exhibit non-zero Berry curvature distribution (see details in SI). As a result, when the spins/valleys are split we anticipate that a (charge) anomalous Hall effect can ensue.

In summary, PDS in gapped vdW materials provide a venue to realize flatbands without the need for sensitive twist angle alignment or stacking arrangement. Instead, extreme bandstructure engineering arise directly from gate-controlled superlattice potentials yielding flatbands when gate voltage is applied; when gate bias vanishes, the PDS system remains as an ordinary TMD system. As a result, PDS flatbands afford considerable electrical control over the character of their electronic excitations (e.g. miniband-width, miniband-gap, interaction energy), as well as an on/off switch for flatbands. While we have focussed on ferromagnetic ordering, highly localized electrons can also exhibit other types of magnetic ordering including antiferromagnetic order as well spin liquids (for a recent discussion see e.g. [17]); having two extremely flat bands separated by a small gap may provide a possible venue for unusual excitonic correlations [32]; a large local density of states may enhance pairing and unlock superconducting correlations. Lastly, we note that the PDS scheme can offer access to a variety of other mini-bandstructures. For example, in addition to a scalar superlattice potential that we focus on to illustrate PDS flatbands, in low-symmetry 2D materials (e.g. monolayer 1T'-WTe₂) spin–orbit coupling can be readily controlled by gate voltage [33, 34] enabling a spatially modulated spin texture; this spatial spin texture can be used as a further knob to control spin response. Another example arises from how the large changes in the local dielectric environment can induce gap size variations in real space [35]; these may give rise to a $\sigma_z$ superlattice potential. Given the large variety of available superlattice structures that can be patterned using PDS, the characteristic length scales for PDS (several tens of nanometers), as well as the exposed nature of the surface states, such heterostructures may provide a one-stop platform to realize and spatially probe exotic ordering.

We are grateful for useful conversations with Valla Fatemi, Ataç Imamoğlu, Brian Skinner, Javier Sanchez–Yamigishi, and Noah Yuan. This work was supported by the Singapore National Research Foundation (NRF) under NRF fellowship award NRF-NRFF2016-05, a Nanyang Technological University start-up Grant (NTU-SUG), and Singapore MOE Academic Research Fund Tier 3 Grant MOE2018-T3-1-002.