Precise Twist Angle Determination in twisted WSe2 via Optical Moiré Phonons

TL;DR

Twisted WSe2 bilayers form moiré superlattices whose local twist angle $\alpha$ governs miniband formation and potential inhomogeneity, with variations on the micron scale. The authors combine lateral force microscopy (LFM) to map the moiré lattice constant $a_\mathrm{moire}$ and micro-Raman spectroscopy to detect optical moiré phonons $E_1^t$ and $E_2^t$ relative to $E_0^{\Gamma}$, whose energies encode $\alpha$. They report twist-angle determinations with precision $\pm 0.3^\circ$ and lateral resolution $<$1 μm across $3^\circ<\alpha<12^\circ$, including fully encapsulated hBN-tWSe$_2$-hBN devices, validating a moiré backfolding model. The approach enables rapid, noninvasive precharacterization of twist-angle disorder for optical and transport experiments and is applicable to other TMDC bilayers.

Abstract

Twisted bilayers of transition metal dichalcogenides (TMDC) form moiré superlattices resulting in moiré minibands in momentum space and hosting localized excitons in real space. While moiré superlattices provide access to Mott-Hubbard physics, their energy potential landscape and electronic correlations are highly sensitive to fluctuations of the twist angle, disorder and lattice reconstructions. However, fast and non-invasive experimental access to local twist angle and its spatial variations is challenging. Here, we systematically correlate twist angle variations of twisted WSe2 bilayers across micrometer length scales using a combined lateral force microscopy (LFM) and a micro- Raman spectroscopy approach. These measurements uncover lateral variations in the twist angle by more than 1° across length scales relevant to optical and transport measurements. We demonstrate that twist angles in the range of 3° < $α$ < 12° show distinct Raman response from scattering on optical moiré phonons allowing twist angle determination with high precision and sub-micrometer spatial resolution under ambient conditions. These modes are particularly sensitive in the low-angle twist regime, predicted to host emergent quantum phases. Our results establish micro-Raman spectroscopy of optical moiré phonons as a rapid, non-invasive probe to determine twist angle and to screen local twist angle variations with a precision better than $\pm$ 0.3° and a lateral resolution below one micrometer. This methodology is also applicable to fully hBN-encapsulated heterostructures.

Figures (8)

• Figure 1: (a) Schematic showing (a) the layer sequence and (b) the moiré superlattice formed by a rigid tWSe$_2$ bilayer. (c) Top: Topographical atomic force microscopy (AFM) image of a representative tWSe$_2$ with the height information encoded in the grey scale. The tWSe$_2$ region is highlighted with a white dashed line. The dashed circles display regions at which the mean twist angle is determined from several LFM images. Bottom: Atomic-resolution LFM images collected at the marked locations showing moiré superlattice with varying $a_\mathrm{moire}$ of $5.7^\circ$ (left) and $10.7^\circ$ (right) . Insets: Corresponding 2D FFTs. (d) Selected Raman spectra from monolayer WSe$_2$ (ML), natural bilayer WSe$_2$ (nBL) and from the two twisted regions marked by the dashed circles in (c). Modes marked by an asterisk (*) appear only for tWSe$_2$ bilayers. All measurements are performed in ambient at room temperature.
• Figure 2: (a) Schematic of the first and second Brillouin zone of a tWSe$_2$ bilayer. The $\Gamma$-points of the second Brillouin zones $\Gamma_1$ and $\Gamma_2$ are displaced along the curved trajectory (green line). Their connecting vector $\vec{q}_{moir\acute{\text{e}}}$ denfines the size of the mini Brillouin zone that depends on the twist angle $\alpha$. (b) DFT calculated phonon dispersion for a monolayer WSe$_{2}$. According to (a), the momentum axis coordinate can be translated into the twist angle $\alpha$ ($x$-axis) for zero-momentum phonons in the mini-Brillouin zone. Dashed rectangle denotes the optical moiré phonons studied in this work experimentally by Raman spectroscopy.
• Figure 3: (a) Selected Raman spectra for different mean twist angles in the spectral range of the $E_g^2$-mode, showing the presence of three different modes with the energy of E$_{0}^{\Gamma}$ being independent of the twist angle and the energy of optical moiré phonons E$_{1, 2}^{t}$ highly dependent on $\alpha$. The Raman spectra are normalized to the intensity of the mode at 137 cm$^{-1}$. For clarity, spectra are shifted vertically by a constant value. (b-d) Selected representative LFM images from the areas in which the spectra in (a) are collected. The moiré cell sizes are clearly visible in real-space image as well as in 2D FFTs.
• Figure 4: Correlating twist angle $\alpha$ and energies of optical moiré phonons in tWSe$_2$ samples. Filled squares represent hBN-tWSe$_2$ and blue outlined symbols fully encapsulated hBN-tWSe$_2$-hBN samples. (a) Energies of $E_0^\Gamma$ ($E_g^2$) and the two optical moiré phonons $E_1^t$ and $E_2^t$. Solid grey lines represent DFT calculated dispersion. For the hBN-tWSe$_2$ samples, the twist angle is determined from LFM, while for the fully encapsulated hBN-tWSe$_2$-hBN samples, we used the theoretical model. (b) Energy difference $\Delta E_1^t$ and $\Delta E_2^t$ of optical moiré phonons relative to $E_0^\Gamma$ for varying twist angle.
• Figure S1: Workflow used to determine the average twist angle at different positions on the sample using LFM, and to extract averaged spectra from a Raman map at the same positions.