Interlayer Charge-density-wave Vector Phase Induced Structural Chirality

Abstract

Chiral charge density waves (CDWs) have attracted intense interest due to their exotic quantum properties, yet the microscopic origin of structural chirality emerging from correlated charge order remains elusive. Here, we reveal that the interlayer phases of CDW wave vectors, an overlooked degree of freedom, play a crucial role in driving chiral structural displacements in layered CDW materials. By explicitly incorporating the interlayer phases in first-principles calculations, we successfully obtained the chiral structure of the CDW phases of AV$_3$Sb$_5$ (A= K, Rb, and Cs) and 1T-TiSe$_2$. The electronic and optical properties of the predicted chiral structures are consistent with experimental measurements of these materials in their CDW phases. We further predict that 1T-NbSe$_2$ is a promising material candidate for realizing chiral CDW order. Beyond materials prediction, our theory reveals that the chiral CDW can be manipulated by electron filling. Our study opens new avenues for discovering, designing, and engineering chiral CDW materials.

Figures (3)

• Figure 1: (a) Left: Charge density modulation associated with each bulk wave vector $\bm{Q}_i$ associated with different phases $\phi_i$ishioka2010chiral. Right: Schematic of charge distribution in the "virtual" layer induced by different bulk wave vector phases ishioka2010chiral. (b) Schematic of a quasi-1D structural transition induced by in-phase and out-of-phase CDW displacements across chains. Orange and blue circles correspond to atomic positions in the original cell and the CDW cell, respectively. $u^\ell_x$ and $u^\ell_{\overline{x}}$ indicate opposite CDW displacements in real space, induced by $\varphi^\ell_{x}$=0 and $\pi$, respectively. (c) Left: Bulk CDW wave vector $\bm{Q}=\left(\pi / x_0, \pi / 2y_0\right)$ distorts chains 1 ($y= 0$) and 3 ($y= 2y_0$) oppositely while leaving chains 2 ($y= y_0$) and 4 ($y= 3y_0$) undistorted. Right: An alternative configuration enabled by independent interchain phase freedom with $\varphi_x^\ell = \{0, \pi, \pi, \pi \}$. (d) Left: Phase-marked interlayer wave vectors establish spiral patterns. Right: Illustration of the charge distribution at the atomic layer associated with different interlayer wave vector phases $\varphi_i^\ell$.
• Figure 2: (a)-(b) Inverse star-of-David patterns resulting from interlayer phases configurations $\varphi^\ell$= [0, 0, 0] and [$\pi, \pi$, 0], respectively. Black dashed lines represent the ideal kagome lattice. Schematic atomic displacements are denoted by red, blue, and green arrows, respectively. (c) Incorporation of interlayer wave vector phases in 2$\times$2$\times$4 AV$_{3}$Sb$_{5}$ (A= K, Rb, and Cs) CDWs to induce structural chirality is illustrated. Left: Spiral phase-marked interlayer wave vector. Right: Atomic structure in the chiral space group $\textit{C}$222. (d) Simulated surface charge distribution on the Sb surface.
• Figure 3: (a) Left: Schematic structure with the same interlayer phases configuration ([$\varphi_i^\ell$]) along the z direction, where only layer "A" is present. Right: Schematic structure with different [$\varphi_i^\ell$] along the z direction, where layers "A" and "B" are present. (b) Left: Schematic band for a structure with the same [$\varphi_i^\ell$]. Black solid line shows the single parabolic band along $k_z$ in the original Brillouin zone (BZ), while the black dashed line shows its folded part in the reduced BZ, where $k_z$ = [-$\pi$/2c, $\pi$/2c]. Right: A schematic band for a structure with different [$\varphi_i^\ell$]. Blue spheres denote the electron occupancy of the band. Green shadow zone represents the true BZ of the CDW structure with the same [$\varphi_i^\ell$] (left) and different [$\varphi_i^\ell$] (right). (c) Schematic band structures and the corresponding electron occupancies for structures with the same and different [$\varphi_i^\ell$] in the presence of one additional electron relative to the case of panel (b). (d) Energies for achiral-2$\times$2$\times$1 (same [$\varphi_i^\ell$]) and chiral-2$\times$2$\times$4 (different [$\varphi_i^\ell$]) CsV$_{3}$Sb$_{5}$ CDW structures at the intrinsic Fermi level (top) and with the injection of three additional electrons (bottom). For comparison, we employed the same 2$\times$2$\times$4 cell in the calculations.