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

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.

💡 Research Summary

In this work the authors investigate how electrostatic disorder influences the electronic structure of marginally twisted bilayer MoS₂ (tb‑MoS₂) with a twist angle of approximately 0.95°. Using a tear‑and‑stack fabrication method, they assemble two monolayers of MoS₂ on a graphite/SiO₂ substrate and perform scanning tunneling microscopy (STM) and spectroscopy (STS) at cryogenic temperatures. The STM topography reveals a well‑defined moiré superlattice with a period of about 18 nm, consisting of alternating MX and XM triangular domains separated by domain walls (DW) and punctuated by MM sites where Mo atoms are directly on top of each other.

Differential conductance (dI/dV) spectra taken at the four high‑symmetry stacking regions (MM, MX, XM, DW) display three characteristic onsets: the valence‑band edge near –1.6 V (Γᵥ), the conduction‑band edge near +0.3 V (Kᶜ), and an additional valence‑band feature near –2.2 V (Kᵥ). Importantly, the MX and XM domains exhibit a systematic shift of about 15 meV in both the valence‑ and conduction‑band onsets, a splitting that is forbidden by the mirror symmetry of an ideal, disorder‑free moiré lattice. Spatial maps of these onsets over several moiré periods show the expected hexagonal pattern but also reveal long‑range, random fluctuations that extend beyond the moiré length scale.

To explain these observations the authors develop an electrostatic disorder model based on randomly distributed negatively charged sulfur vacancies (V_S) in the four sulfur layers of the bilayer. Treating the underlying graphite as a perfect metal, they employ the method of image charges to calculate the disorder potential V(x,y). At distances much larger than the bilayer thickness the potential decays as V(r)∝1/r³, characteristic of a dipole formed by the vacancy charge and its metallic image. By computing the autocorrelation function of the experimental onset maps and fitting it to the theoretical V(r) they extract a vacancy density n_V of order 10¹¹ cm⁻², consistent with previously reported defect concentrations in exfoliated and CVD‑grown MoS₂.

Next, they perform continuum‑model calculations of the moiré band structure that incorporate atomic‑scale lattice relaxation. In the presence of an out‑of‑plane electric displacement field D—originating from the disorder potential—the low‑energy moiré minibands near both the valence and conduction edges split dramatically. The calculated local density of states (LDOS) reproduces the experimentally observed MX/XM splitting only when the disorder‑induced energy broadening is added and, crucially, when lattice relaxation is included. Without relaxation the theoretical bands remain symmetric and cannot account for the measured 15 meV offset.

The combined experimental‑theoretical analysis leads to several key insights: (i) electrostatic disorder breaks the nominal MX/XM mirror symmetry, generating a measurable band‑edge offset; (ii) the disorder potential acts as a spatially varying out‑of‑plane electric field that strongly modulates the bandwidth and polarization of the moiré minibands; (iii) lattice relaxation amplifies the sensitivity of the electronic structure to the disorder, highlighting the intertwined roles of structural reconstruction and charge inhomogeneity.

These findings have broad implications for the emerging field of twisted transition‑metal dichalcogenides. Since many correlated phenomena—Mott insulating states, superconductivity, fractional quantum anomalous Hall effects, and topological phase transitions—depend sensitively on the flatness and symmetry of the moiré bands, uncontrolled electrostatic disorder can dramatically alter phase diagrams and device performance. The work therefore underscores the necessity of controlling defect densities, substrate screening, and strain during fabrication, and it provides a quantitative framework for predicting disorder effects in other marginally twisted TMD systems.