Understanding chiral charge-density wave by frozen chiral phonon

Fermi surface nesting (FSN) and strong coupling theory are two accessible theories to understand and explain the CDW transitions in solids [52]. FSN is a weak coupling theory generalized from the Peierls idea for the one-dimensional atomic chain, while it has been ruled out in a few typical CDW materials such as NbSe₂ by both the first-principles calculations and experiments, and suffering from explaining the CDW supercell size with nesting vectors in many materials [53, 54]. The strong coupling theory highlights the role of EPC in CDW transitions. Strong EPC usually results in a strong screening of lattice dynamics, giving rise to soft phonon modes when the temperature is lower than $T_{\text{CDW}}$, as shown in Fig. 1 (a). This softening, in turn, precipitates lattice instability and contributes to the CDW distorted structure.

The soft modes $\mathbf{u}(\mathbf{R},t)=\mathbf{e}\cdot e^{i(\mathbf{q}\cdot\mathbf{R}-\omega t)}$ can be further cataloged into two cases based on the atomic motion behaviors, i.e., the phonon polarization vector $\mathbf{e}$. We take the system shown in Fig. 1 (b) as an example, in which, there are four atoms located at (1,0), (0,1), (0,0), and (1,1) in the primitive cell (which are also denoted by $s_{1...4}$. For specifying a certain mode of atomic motion in a primitive cell, we write the phonon polarization vector on the basis of $\{u_{\kappa,\alpha}\}(\kappa=s_{1,...,4};\alpha=x,y)$. In the case of $\mathbf{e}=(1,0,0,1,0,0,0,0)$ under the Cartesian coordinates, which corresponds to a phonon mode with two of the mirror ($M_{xy}$)-related atoms vibrate linearly and the other two atoms keep still, an achiral mode preserving mirror symmetry is obtained, as shown in Figs. 1 (b) and (d). This to an achiral displacement, which contributed to achiral CDW at an arbitrary time $\tau$. In other cases where $\mathbf{e}=(1,0,0,e^{-i\pi/2},0,0,0,0)$, those two non-related linearly atomic vibrations will contribute to a chiral phonon mode for the mass center, which breaks the mirror symmetry, leading to a CCDW as shown in Figs. 1 (c) and (e).

With the relationship between the atomic displacements of the soft modes and the CDW, one can further explore the mechanism of CCDW from the chiral phonon point of view. To uncover the rarity of CCDW candidates in nature and offer an effective way to engineer CCDW in experiments, in the following, a careful calculation on monolayer 1T-TiSe₂ is used as a case study to show the mechanism for the CDW, the absence of CCDW, the way to obtain and engineer CCDW in experiments. We will utilize TiSe₂ to illustrate our theory.

Layered 1T-TiSe₂ (space group P$\bar{3}$m1) is the first system reported to host CCDW [19, 8], and it undergoes a lattice rearrangement to a $2\times 2\times 2$ supercell near $200$ K. One of the possible explanations of the chirality is that the locking of relative $\frac{\pi}{2}$ phases between degenerate CDWs along three orthogonal directions, and the chiral feature survives even when the system is reduced to three layers. For monolayer 1T-TiSe₂, the $2\times 2$ CDW phase has been proposed with $T_{\text{CDW}}=232K$ [55], yet debate remains. This is not immediately evident by claiming that chirality is only well-defined in 3D because CCDW in other monolayer materials has been observed, such as 1T-NbSe₂ has a CCDW structure identified by STM [56]. This anomaly inspires us to revisit CDW transition in monolayer 1T-TiSe₂.

Figure. 2 (a) shows the phonon spectra of monolayer 1T-TiSe₂, where the soft modes show the instability of the structure. To study the CDW mechanism of the system, the Fermi nesting function $\chi_{0}^{\prime\prime}({\bf q})$ is calculated and shown in the supplementary. Excluding the artificial singularity at $\Gamma$ coming from the double-delta approximation, strong peaks of $\chi_{0}^{\prime\prime}({\bf q})$ are widely distributed around $M$, indicating that the FSN cannot quantitatively explain the selective softening of phonon modes in the vicinity of $M$, as has also been established in a few other TMD materials [53, 54].

Hereafter, we turn to study the CDW mechanism under the strong coupling scenario. We calculate phonon linewidth to estimate the EPC of each mode at momentum ${\bf q}$, i.e.

as plotted by yellow dots in Fig.2 (a), where $g_{mn\nu}({\bf k},{\bf q})$ is the EPC vertex and $f_{T}(\varepsilon)$ is the Fermi-Dirac distribution function. Dominant EPCs are contributed from three in-plane acoustic phonon modes at $M$, indicating that the CDW transition may arise from anomalous strong couplings of electrons to atomic displacements in line with these phonon modes. Next, we move to study the CDW structure in terms of total energy and atomic displacements under the EPC scenario. In principle, identifying the structure distortion from the normal state to $2\times 2$ CDW state needs a specification of 36 degrees of freedom (12 atoms $\times$ 3 directions). We decompose the atomic displacements $\Delta\tau_{\kappa\alpha p}$ to $36$ phonon normal coordinates $z_{{\bf q}\nu}$ involved in the lattice reconstruction from $1\times 1$ to $2\times 2$, which shows three soft modes are the key degrees of freedom that drive CDW transition (see supplementary for details). All the other $30$ phonon modes only slightly modify the shape of the double well and form a parabolic potential energy surface, as shown in Fig. 2 (b). The amplitude of the soft modes and the corresponding PES are qualitatively consistent across the different exchange-dependent (XC) functionals, as shown in the supplementary.

Three coupled order parameters that dominate the CDW state are

where $|z_{i}|e^{i\phi_{i}}$ is the complex normal coordinate, ${\bf R}_{p}$ is the lattice vector of unit cell $p$ in the Born von-Karman supercell, $M_{\kappa}$ is the mass of $\kappa$-th atom in a unit cell, $M_{0}$ is the proton mass as a mass reference to follow the convention, and ${\bf e}_{\kappa\alpha}({\bf M}_{i})$ are polarization vectors of soft mode at ${\bf M}_{i}$. A linear combination of those three coupled order parameters forms the actual pattern of a CDW state, i.e., $\Delta\tau_{\kappa\alpha p}=\sum_{i=1}^{3}\Delta\tau_{\kappa\alpha p}^{(i)}$, where the amplitude $|z_{i}|$ and phase $\phi_{i}$ specifying the contribution of each order. In 3D systems, relative phases ($\phi_{j}-\phi_{i}$) between degenerate transverse phonon modes could share a common propagation direction, thus they can define the chirality. However, due to the absence of a common propagation direction in 2D systems, the relative phases simply relate to a global shift of a fixed point and are irrelevant to the chirality. Thereafter, we set $\phi_{i}=0$ for simplicity, and the chirality defined by the lack of mirror symmetry we are interested in must come from the amplitude $|z_{i}|$, and $|z_{1}|=|z_{2}|=|z_{3}|$ restricted by $C_{3}$ and mirror symmetry. Crystal structure induced by this mode preserves $C_{3}$ rotation symmetry while breaking the mirror symmetry, as shown in Fig. 2 (c), yet there is no decisive experimental evidence about the chirality of the CDW phase of the monolayer TiSe₂, such as XRD, STM and other experimental means that can directly observe the crystal structure.

Monolayer TiSe₂ is composed of 3 atomic layers, Se-Ti-Se from bottom to top, and each layer has mirror symmetry, thus it is difficult for STM (which can only detect the configuration of the surface atoms) to capture mirror symmetry breaking. Furthermore, we performed $ab$ $initio$ calculations of diffraction pattern corresponding to XRD/electric diffraction observations [57, 58], as shown by the mirror-symmetric scattering pattern in Fig. 2 (d) under 230 K. The results for the other different temperatures are shown in supplementary materials. Therefore, no mirror symmetry breaking is observed for both elastic and inelastic scattering patterns, which is also consistent with the recent experiment result [59].

Since neither of the two main methods can detect the chirality of the CDW phase for TiSe₂, a way to obtain and modulate the CCDW phase is needed. In the recent studies concerning extant CCDW phases, there are at least four distinct external stimuli that have the potential to switch the chirality of CDW, ascertained through experimental observations [7, 8, 9, 51, 60, 61, 62], including magnetic field, circularly polarized light, electric field with polarization, and local electric field pulse delivered through STM tips. However, altering the chirality of CDW is mostly done randomly, such as the electric pulse modulation in NbSe₂ [56]. Here, we provide a possible method to induce the shear strain and hence the CCDW by the external electric/magnetic field, through the nonlinear electro- or magnetostrictive effects [63, 64, 65] ( see supplementary for details), as shown in Fig. 3 (a), which is a trigger to break the degeneracy of left-handed CCDW (L-CDW) and right-handed CCDW (R-CDW) phases. Furthermore, the chirality of CCDW can be modulated by changing the direction of the magnetic/electric field.

Due to the breaking of vertical mirror symmetry under the perturbation, chiral phonons are expected to be obtained. Though $M$ is the time-reversal-invariant momenta with linearly vibrating modes, the contribution for each $M$ mode will be no longer equal due to the breaking of mirror symmetry and $C_{3}$ rotation symmetry, as shown in Fig. 3 (b), resulting in a soften chiral phonon mode with circular motion for the mass center, as shown in Fig. 3 (c). The chiral phonon of the mass center contributes to the CCDW, as the case in monolayer 1T-TiSe₂. That is, by breaking the vertical mirror symmetry, the chiral soft mode induces a CCDW at $\tau$. By changing the direction of the shear strain, the chirality of CCDW can be selectively switched. In contrast to alternative methods for inducing CCDW, shear strain provides a clear advantage by a continuous modulation on the strength of the CCDW with a customizable strain strength, represented as shear strain angle $\theta^{0}$ in our case.

XRD/Electron diffraction are powerful method to identify the crystal structure, and hence the chirality of CDW. However, only the location or the intensity of the spots is used for identifying the crystal structures in previous studies. We propose that the anisotropy of the Bragg peak profile can provide us with more information about the CCDW structure. To get a more intuitive picture, we calculated the diffraction pattern of 1T-TiSe₂ under 230 K, with left-hand chiral stimuli $\theta^{0}=$ 0.05^∘, as shown in Fig. 4 (a). Moreover, we have also calculated the diffuse scattering pattern under lower temperatures, which shows no qualitative difference (see supplementary for details). Though the shape of the scattering pattern at $(220)$ (or $(420)$) is ellipse for both of the structures of CDW (Fig. 2 (d)) and CCDW (Fig. 4 (a)), the latter one breaks the mirror symmetry and shows a chiral pattern.

Here we define a quantity named “chirality angle” ($\theta^{c}$) to identify the angle between the major axis of elliptical scattering spot in the CDW and CCDW phase, i.e., the degree of destruction of the mirror symmetry. The chirality angle $\theta^{c}\approx 15^{\circ}$, which is much larger than the shear angle $\theta^{0}$. A further calculation shows that the 2$nd$ mode contributes $\theta^{c}$ most, resulting in a mirror-breaking structure, as shown in Fig. 4 (b). Other patterns for the acoustic modes are in the supplementary. Based on the L-CDW structure with $\theta^{c}=0.05^{\circ}$, we also calculated the dynamic structure factor (DSF, which could be seen as the diffraction with only the one phonon process), passing through $(220)$ in two paths (green and blue lines). The first one is from $(110)$ to $(330)$, as shown in in Fig. 4 (c), while the second one is from $(400)$ to $(040)$, as shown in Fig. 4 (d). Both (b) and the blue cut in (d) show that the 2$nd$ mode contributes the most, thus the change of intensity along the major axis is mainly contributed by the 2$nd$ phonon mode. The chirality angle is found to be qualitatively robust by using different XC functionals in the first-principles simulations. Details for these features and their dependencies are shown in supplementary materials.

Ab initio calculations were carried out for 1T-TiSe₂ (space group $P\bar{3}m1$), using the relaxed in-plane lattice parameter $a=3.442$ $\mathring{\text{A}}$, and interlayer distance $c=3.762a$. The Titanium atom is located at $1a$ position (0,0,0), and two Senium atoms are located at $2d$ $(1/3,2/3,0.443a)$. We used DFT within the generalized gradient approximation of PW91 LDA functionals [66]. The core–valence interaction was described using norm-conserving pseudopotentials. Electron wavefunctions were expanded in a plane waves basis set with a kinetic energy cutoff of 100 Ry, and the Brillouin zone was sampled using a 24×24×1 $\Gamma$-centered Monkhorst–Pack mesh. Lattice-dynamical properties were calculated using density functional perturbation theory (DFPT) [67]. All DFT and density functional perturbation theory calculations were performed using the Quantum ESPRESSO package [68, 69]. We adopt a degauss $\sigma=$1.9 mRy of Fermi-Dirac smearing function in total energy calculations. Note that, the soft modes can be stabilized by increasing degauss to 4 mRy.

Calculations of electron-phonon couplings were performed using the EPW code [70]. To evaluate phonon self-energy and linewidth, we computed electronic and vibrational states as well as the scattering matrix elements on a 24×24×1 and 4×4×1 Brillouin-zone grid, respectively. These quantities were interpolated with ab initio accuracy onto a fine grid with 200×200×1 $k$-points and a high-symmetry path of $q$-points using EPW. Temperature effects were accounted for by including the Fermi–Dirac occupation in the spectral function, corresponding to 50K below the CDW transition temperature.

The multi-phonon diffused scattering method [57, 58], which is implemented in the EPW package suite [70]. For both the CDW and CCDW states, we have chosen a $40\times 40\times 1$ supercell for getting a converged scattering pattern. For the $2\times 2$ CDW and CCDW structures, the dynamic matrix is calculated on a $2\times 2\times 1$ $q$ mesh, with the electronic self-consistent field calculation implemented on a $12\times 12\times 1$ mesh.

Mechanism for CDW and CCDW. (a) Phonon mode softening in the CDW process as temperature goes down. (b) Atomic vibrations for an achiral phonon mode, which corresponds to linear atomic vibrations of the mass center. Grey dots are for the atoms in the unit cell at different times, $\tau_{0}$ (atoms located at the equilibrium positions), $\tau_{1}$, $\tau_{2}$, and $\tau_{3}$, while blue dots represent the motion of the mass center. (c) Atomic vibration for a chiral phonon mode, which corresponds to the circular motion of the mass center. (d) An achiral CDW induced by the mirror-symmetric atomic displacement. (e) A chiral CDW structure induced by a chiral displacement.

First-principles calculations of 2D 1T-TiSe₂. (a) Phonon dispersion and linewidth (yellow dots) based on the primitive cell, showing significant electron-couplings of soft modes at three $M$. The inset shows the BZ of the primitive cell and its high-symmetry points. (b) Potential energy surface (PES) of 1T-TiSe₂ by linearly displacing atoms from normal state to $2\times 2$ CDW state. Two degenerate CDW states shown in the insets are connected by mirror symmetries. The dashed double-well potential in red is formed by three frozen soft modes, while the inclusion of the other 30 normal phonon modes only slightly modifies the shape of the energy surface (solid line). (c) Atomic displacements (marked by arrows) for the $2\times 2$ CDW structure. The blue, pink, and green ones represent the Ti (locating at $z=0$), Se ($z>0$), and Se ($z<0$) atoms, respectively. The main displacement of the CDW mode is contributed by the Ti atom, and the minor part is contributed by the Se atoms (the displacement vectors of Se atoms are enlarged for clearness). Each layer does not break the mirror symmetry, yet the CDW structure breaks mirror symmetry. (d) Calculated diffraction pattern of the CDW structure.

Mirror symmetry breaking external stimuli induced CCDW. (a) The left-handed CDW (L-CDW, green one) induced by mirror symmetry-breaking stimuli, such as a time-dependent electronic/magnetic field. The right-handed CDW (R-CDW) is likewise. (b) Phonon dispersion and linewidth of 1T-TiSe₂ based on the left-handed strained primitive cell. The degeneracy of three soft modes at $M$ is lifted due to the breaking of the mirror and $C_{3}$ rotation symmetries, while their coupling strengths with electrons are still the strongest among others. (c) CCDW induced by the chiral soft modes in TiSe₂. The blue dots denote the equilibrium positions of Ti atoms for the $2\times 2$ supercell (Here we admit all the Se atoms for clearness). The light-gray, gray, and dark-gray solid circles represent the positions at $\tau_{1}$, $\tau_{2}$ and $\tau_{3}$. The hollow circles represent the mass center of Ti atoms at these $\tau$, resulting in a chiral elliptical motion for the mass center (dashed blue line).

The diffuse scattering factors are calculated based on the left-handed CCDW structure with $\theta=0.05^{\circ}$. (a) The scattering pattern with all phonon modes. The scattering pattern located at (420) is left rotated (represented by the rotation angle $\theta^{c}=15^{\circ}$), compared to the pattern in the no-sheared structure, which indicates the mirror symmetry breaking and left-chirality of the crystal structure. (b) The scattering pattern with $2nd$ phonon mode shows that the left rotated pattern at (420) is mainly contributed by the $2nd$ mode. (c) and (d) is the dynamic structure factors with the same structure along two paths marked in (a), passing the rotated pattern at (220). (c) DSF from (110) to (330), donated by the green line. (d) The DSF from (400) to (040), donated by the blue line. The black lines in (c) and (d) represent the phonon spectra from DFT.