Tailoring spontaneous symmetry breaking in engineered van der Waals superlattices

Abstract

Superlattice engineering in van der Waals heterostructures (e.\,g.\ by moiré engineering) provides a powerful platform for designing electronic bands and realising correlated and topological quantum phenomena. Here, we pioneer a scheme to tailor superpotentials based on intrinsic substrate electronic orders. We show that this establishes a robust, self-aligned, and highly versatile route to band-structure control as we demonstrate in graphene by engineering two distinct, nearly commensurate superlattices using the charge density waves of 1T-NbSe$_2$. In these superlattices the graphene's Dirac cones are folded either to the $Γ$-point or to the K-points of the mini-Brillouin zone. Using scanning tunnelling microscopy, we observe that the $Γ$-folded system preserves C$_3$ symmetry, while the K-folded system exhibits spontaneous symmetry breaking. Combining density functional theory with an interlayer interaction model, we reveal that this difference is not electronically driven but originates from a structural instability. Our work establishes superlattice engineering for designer quantum states and unveils a structural mechanism for controlled emergent symmetry breaking.

Figures (4)

• Figure 1: Superlattice Engineering.a, Scheme of the graphene on 1T-NbSe2/2H-NbSe2 van der Waals heterostructure. The side view illustrates the charge density wave (CDW) maxima. b, Twist-angle $\theta$ dependent mismatch between the graphene lattice and the $2\times\!2$ (blue) and $\sqrt{3}\times\!\sqrt{3}\ \mathrm{R}30^{\circ}$ (orange) superlattice of the 1T-CDW in units of the graphene lattice constant. Here, we use the Wood's notation $\sqrt{3}\times\!\sqrt{3}\ \mathrm{R}30^{\circ}$ to denote a superlattice with unit vectors $\sqrt{3}$ times larger than those of the CDW lattice, rotated by $30^{\circ}$wood1964vocabulary-bfd. c, Scheme of the stacking configuration of a $2\times\!2$ commensurate superlattice realised by compressing the graphene lattice by a heterostrain of $\varepsilon=4.1\%$. d, Zoom-in of the supercell. e, f, Same as (c, d) but for a $\sqrt{3}\times\!\sqrt{3}\ \mathrm{R}30^{\circ}$ commensurate superlattice with $\varepsilon=1.8\%$. g, h, Atomically resolved STM topography and its FFT with $\theta=9.0°$ and $\varepsilon=0.3\%$ ($V =0.5V$, $I=200pA$). i, j, Same as (g, h) for the sample with $\theta=26.8^{\circ}$ and $\varepsilon=0.4\%$ ($V =0.3V$, $I=60pA$). Colored hexagons mark the 3 different CDW maxima which repeat periodically, building up the $\sqrt{3}\times\!\sqrt{3}\ \mathrm{R}30^{\circ}$ supercell.
• Figure 2: Electronic structure of the $2\times 2$ superlattice.a, Spectra measured between the A- and B-sites, marked in panel c, show three peaks at energies $E_{\rm L1}= 0.66(1)eV$, $E_{\rm L2}= 0.92(1)eV$, and $E_{\rm L3}= 1.36(1)eV$ and a dip at $E_{\rm v1} = 1.10(1)eV$. b, On graphene/2H-NbSe2 and 1T-NbSe2/2H-NbSe2 these features are absent. c-f,$\mathrm{d}I/\mathrm{d}V$ maps measured at biases corresponding to $E_{\rm L1-L3}$, and $E_{\rm v1}$. g, Evolution of the FFT amplitudes $A_{1-3}$ (top) and phases $\varphi_{1-3}$ (bottom) extracted from the $\mathrm{d}I/\mathrm{d}V$ maps. h, Schematic Model used for the DFT calculations. i, Calculated band structure of the superlattice mBZ and zoom-ins to the bands near the $L_{1-3}$ states. The hybridised states are coded in red. j, Calculated LDOS at the A- and B-sites. k-n, Calculated LDOS maps at energies corresponding to the $L_{1-3}$, and $v_1$ features.
• Figure 3: Electronic structure of the $\sqrt{3}\times\!\sqrt{3}\ \mathrm{R}30^\circ$ superlatticea, Spectrum taken at the A-site (solid line) show peaks at energies $E_{\rm L1}=0.71\pm0.01$ eV, $E_{\rm L2}=0.90\pm0.01$ eV, and $E_{\rm L3}=1.27\pm0.01$ eV and a dip at $E_{\rm v1}=1.10\pm0.01$ eV. The spectrum measured in the $2\times 2$ superlattice (dashed line, $\times25$) is shown for comparison. b, Spectra recorded at the CDW minima B$_1$ (blue), B$_2$ (orange) and B$_3$ (green), which are rotated around the CDW maximum by 120° with respect to each other. At B$_1$ an additional peak $L_1'$ appears at $E_{\rm L1'}=0.62\pm0.01$ eV. c, Phase and amplitude from the FFT analysis of the $\mathrm{d}I/\mathrm{d}V$ data as a function of bias. d-h,$\mathrm{d}I/\mathrm{d}V$ maps at the indicated energies. i, Calculated band structure of the $\sqrt{3}\times\!\sqrt{3}\ \mathrm{R}30^{\circ}$ superlattice mini-Brillouin zone. j-n, DFT calculated LDOS maps, at the corresponding states. o, Similar phase analysis using the DFT-calculated LDOS data.
• Figure 4: Geometric origin of the symmetry breaking.a, Schematic of the interlayer interaction model. b, 2D resolved sliding map of the calculated hybridisation magnitude $|d_{pp}|$ for the $2\times\!2$ superlattice with sliding directions defined in a. c, 2D resolved sliding map of the $|d_{pp}|$ for the $\sqrt{3}\times\!\sqrt{3}\ \mathrm{R}30^{\circ}$ superlattice. d, Comparison of the $|d_{pp}|$ as a function of sliding between these two commensurate configurations, along the sliding paths indicated in b and c.