Disorder-induced symmetry breaking in moiré bands of marginally twisted bilayer MoS$_2$

TL;DR

Disorder plays a fundamental role in shaping moiré bands in marginally twisted MoS2. The authors combine high-resolution STM/STS measurements with an electrostatic disorder model and a continuum moiré framework that includes lattice relaxation and interlayer bias to connect nanoscale charge puddles to band-edge onsets. They observe symmetry-breaking shifts of the valence and conduction band onsets between MX and XM domains, infer defect densities of order 10^11–10^12 cm^-2, and show that disorder-broadened, layer-polarized moiré bands reproduce the experimental data. These findings highlight the crucial influence of electrostatic disorder on moiré physics and emphasize the need to account for defects when pursuing flat-band and topological phenomena in moiré TMDs.

Abstract

Twisted transition-metal dichalcogenides host highly tunable moiré potentials, flat bands, and correlated electronic phases, yet the role of disorder in shaping these emergent properties remains largely unresolved. Using scanning tunneling spectroscopy, we investigate the impact of electrostatic disorder on the electronic structure of marginally twisted ($θ\approx 0.95^\circ$) bilayer MoS$_2$. Differences of 15 meV in the onset energies of the valence and conduction bands between MX- and XM-stacked regions are observed and are unexpected based on symmetry considerations. We further observe spatially correlated disorder in the band onset energy that is consistent with a background random charge density of a few $10^{11}\,\mathrm{cm}^{-2}$. Continuum model calculations for twisted MoS$_2$ reveal dramatic changes in the low-energy moiré bands in response to an electric displacement field, in quantitative agreement with experiment. Moreover, the calculated local density of states including disorder broadening reproduces the experimental observations only when structural relaxation is taken into account. These results highlight the critical role of electrostatic disorder in determining the electronic structure of moiré materials at the nanoscale.

Figures (4)

• Figure 1: (a) Schematic of the tb-MoS$_2$ device on a graphite/SiO$_2$ substrate. The inset shows the structure in a moiré unit cell, with MM sites, MX/XM stacking domains, and DWs. (b) Large area STM topograph of the tb-MoS$_2$ device showing a moiré lattice with a lattice constant of $\sim18$ nm corresponding to a twist angle of $\sim 0.95^\circ$. A dotted white hexagon outlines the domain walls (DW) of a few moiré unit cells. Scale bar: $40$ nm. Measurement parameters: $V_{\text{bias}}=-1.9$ V, $I_{\text{set}}=100$ pA. (c,d) Low-energy bands computed with the continuum model for (c)the conduction and (d) the valence band, along the path $\gamma\rightarrow\kappa\rightarrow\mu$, in the mBZ shown in the inset. The projection of the layer polarization onto the bands is shown, as indicated by the color scale. (e,f) Calculated spatial maps of onset energies for conduction (e) and valence (f) bands showing a honeycomb hexagonal lattice similar to one observed in (a). Given that for some locations the energy intersection with the arbitrarily-set LDOS value does not exist within the energy range, the color bars are unbounded in one direction (indicated by the arrow).
• Figure 2: (a) STS point measurements at the AA, MX, XM and DW regions of the moiré lattice. Measurement parameters are $V_{\text{bias}} = -2.5$ V , $I_{\text{set}} = 250$ pA. Each curve is the average of 10 sweeps taken using lock-in amplifier with modulation amplitude of $10$ mV and frequency of $793$ Hz. (b,c) Onset maps over a $\sim 50 \times 50$ nm$^2$ moiré region for the valence band (b) and conduction band (c) edges. Red dashed lines highlight some moiré domain walls. Scale bar is $10$ nm. (d,e) Spatially averaged LDOS for the MX/XM regions shown in (b) and (c). Vertical dotted lines show the calculated onset energies for each curve, the absolute difference between them and its uncertainty are indicated in each plot. The vertical axis of all the STS curves are in logarithmic scale.
• Figure 3: (a) Schematic of the electrostatic model used to estimate the electrostatic potential due to randomly-distributed charges in the sample; $d$ is the thickness of a MoS$_2$ monolayer and $V(x,y)$ is the total electrostatic potential. Red dots represent randomly distributed V$_{S}$ while brown dots inside the section of the graphite layer are the primary image charges. (b-e) Radially averaged disorder potential $\phi(r)$. Blue and green solid points represent the experimentally measured disorder potential $\phi(r)$, and dashed lines are power-law fits to the experimental data for $r > 10$ nm, for valence band (b,d) and conduction band (c,e) onset maps, respectively. Power-law exponents are given in the legends of d, e. Red lines in b-e are the electrostatic models that best fit the experimental profiles. In d and e we present the results for $r > 0$ nm in log-log scale with the obtained surface charge defect density, $n_{V}$ in the legends.
• Figure 4: (a,b) Moiré conduction (a) and valence (b) minibands for tb-MoS$_{2}$ with $\theta = 0.95^\circ$ and a finite interlayer electric potential difference $V_z = 20$ meV along the mBZ path shown in the inset. The color indicates the layer polarization. (c,d) LDOS of conduction (c) and valence (d) bands, obtained assuming a Lorentzian broadening $\gamma = 10$ meV to account for charged disorder and thermal effects. The horizontal axis is in logarithmic scale. (e,f) Real-space maps (coordinates in units of the moiré length) of the onset energies at which the LDOS first crosses the dashed lines in (c,d) in the conduction (e) and valence (f) bands. Similar to Figures \ref{['fig:fig1']}e, f, for some locations along the AA and DW regions, the color bars are not bounded in one direction (indicated by the arrow).