Introduction

Recently, quasi-two-dimensional kagome metals AV3Sb5 (A = K, Rb, Cs)1 have attracted tremendous attention due to their ability to exhibit exotic quantum phenomena such as unconventional superconductivity2,3,4,5,6, charge density wave (CDW)7,8,9,10,11,12, nontrivial Z2 topological surface2,13, anomalous Hall effect14,15, anomalous Nernst effect16 and time-reversal symmetry breaking (TRSB)17,18. The interplay between electron correlation effects, topology and geometric frustration AV3Sb5 has motivated a rapid exploration of this group of materials19,20,21. In addition, the electronic instability22,23 provided by van Hove singularities23,24,25,26,27 and the structural instability revealed by phonon dispersion28 make it possible to manipulate various properties of these materials. Include high pressure-induced Tc enhancement29, superconductivity reproduction30,31, dimensional crossover, topological phase transitions32, magnetic field-excited superconducting two-fold symmetry33, rotational symmetry breaking34, and magnetic breakdown35.

Importantly, as emphasized in earlier works36,37,38,39,40, the structural deformation and symmetry breaking play a significant role in manipulating the properties of AV3Sb5. Since the CDW features lattice distortions, studies of the lattice degree of freedom can offer insight into the mechanism. The instabilities of van Hove singularities near the Fermi surface are also considered to arise from structural information. Although many studies on tuned quantum phenomena have been reported, they require harsh and tough experimental conditions, such as high pressures and even strong magnetic fields. Optical control has provided a more adjustable and flexible avenue to achieve symmetry breaking without changing the volume and exceeding the material’s elastic limit. Moreover, the light-induced symmetry breaking, electronic interactions, and the dynamic response of light in AV3Sb5, potentially benefiting emerging functional devices and superconductive devices, have remained elusive until now.

Driving certain lattice vibrations by high-intensity Terahertz frequency optical pulses is an emerging technique to control the properties of materials41,42,43,44. Such optical manipulation can rapidly direct photoinduced coherent motion of atoms to metastable states, away from their equilibrium counterparts, resulting in remarkable phenomena such as light-induced ferroelectricity45, transient superconductivity46,47,48 and ultrafast control of magnetic interactions49. This ultrafast structural control of materials starts from the nonlinear coupling between Raman QR and infrared QIR phonon modes that can cause a distortion along the QR coordinate when the QIR mode is externally pumped41. The excitation of an infrared-active phonon serves as the intermediate state for ionic Raman scattering (IRS), a process that relies on lattice anharmonicities rather than electron-phonon interactions. Compared to the conventional Raman effect, IRS has proven to be highly selective and potentially less dissipative. The approach currently focuses on symmetry breaking and transient distortion of the lattice by using an external pump to prompt nonlinear coupling between resonantly driven IR phonons of the same frequency and the Raman phonon. In particular, the nonlinear coupling of multiple infrared to Raman phonons at different frequencies requires further study. Since the electric field of the combination of multi-frequency laser pulses is asymmetric, the rapid oscillation may enhance the acceleration process of electrons, leading to significant quantum phenomena in materials.

The two-dimensional (2D) system can be gradually tuned to a 3D system due to the rapid reduction of interlayer distance, while the superconductivity is tied to the dimensional crossover in AV3Sb5, the abnormal evolution of superconductivity can be attributed to the formation of interlayer Sb2-Sb2 bonding32. In particular, Raman-active phonon amplitude mode A1g in CsV3Sb5, which are collective excitations typically thought to emerge out of frozen soft phonons, exhibits the character of Sb2 motion. Hence, it could also be induced through the displacements of certain optical phonons alone, with the advantage that such displacements preserve the unit-cell volume. Moreover, the A1g vibration is presumably the key to the formation of the CDW, and A1g mode hardening is observed in both bulk samples and thin flakes as the material undergoes a CDW transition50,51,52,53. Thus, the Raman phonon mode must be considered to comprehensively understand the CDW state, dimensional crossover and superconductivity in CsV3Sb5.

Here, we describe an innovative two-frequencies laser control mechanism for ultrafast manipulation of transient structural distortions. By employing CsV3Sb5 as a model system, we demonstrate that the three optical phonons’ coupling of two different infrared-active phonons A2u (3. 28, 7.547) with a Raman-active phonon A1g (4.06) can generate photoinduced dimensional crossover and a topological phase transition. This is achieved by exciting the two infrared-active modes using tailored two laser pulses. Since these two IR modes A2u (3. 28, 7.547) coupled with the target Raman mode A1g (4.06) anharmonically, the Raman mode is triggered. This approach offers excellent tunability and flexibility to achieve ultrafast transitions of topological phases without disrupting the material structure. In contrast to the case of applied strain, this dynamical response exhibits disparate topological conversions, providing a vital path for nonlinear phononic switching of transient structural distortions and offering an excellent prospect for topological and superconductive device applications.

Results and discussion

Nonlinear phonon dynamics

The vanadium sublattice of CsV3Sb5 is a structurally perfect kagome lattice. The Sb1 sublattice is a simple hexagonal periodic system, centered on each kagome hexagon. Between different kagome V3Sb layers, the Sb2 sublattice forms two honeycomb-type Sb sheets embedded by the Cs atom, as shown in Fig. 1a. We calculated the lowest-energy structure of CsV3Sb5 to obtain a force-free DFT reference structure by employing the lattice constants a = b = 5.45Ǻ, c = 9.35Ǻ, α = β = 90°, γ = 120° with space group P6/mmm. All the parameters are in good agreement with the experimental values2. Phonon frequencies and eigenvectors of the optimized structure were computed by using density functional perturbation theory. Since light radiation only excites phonons close to the center of the Brillouin zone, q = (0,0,0), we only calculated modes that span the irreducible representatives of the P6/mmm point group. The corresponding phonon frequencies for both Raman and infrared (IR) active phonon modes are listed in Table 1. Parenthetically, (a) and (b) represent two degenerate phonon modes at the same frequency, with the same symmetry but different vibrational directions.

Fig. 1: Structure and polarization configurations in CsV3Sb5.
figure 1

a Crystal structure of the kagome superconductor CsV3Sb5. b Sketch of the atomic displacements corresponding to the IR-active mode A2u (3.28 THz) (left), Raman-active mode A1g (4.06 THz) (middle) and IR-active mode A2u (7.547 THz) (right); the blue arrow represents the direction of phonon displacement.

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Table 1 Calculated phonon frequencies for Raman and infrared-active phonon modes of CsV3Sb5.
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In the attempt to investigate the ultrafast structure switching through nonlinear phononics of CsV3Sb5, we focused on the fundamental properties of these lattice rearrangements (induced by anharmonic mode coupling), which allow for the transformation of an oscillatory motion into a quasistatic deformation. We find that the rectification of the Raman-active mode A1g (4.06) in CsV3Sb5 would produce the bond displacements of the Sb2 atoms, with consequent rapid motion along the c-axis, among the V3Sb layers. This phenomenon may presumably affect the dimension, topological properties and superconductivity of the analyzed system. Since the ionic Raman scattering we employed here is a process that relies on lattice anharmonicities rather than electron- interactions. This important effect, which indirectly generates the optical response, can be triggered by a laser pump. Moreover, despite the fact that oscillating vibrations of A2u IR-active modes (3.28 and 7.547) via corresponding lights can induce the motion of Sb2 atoms, additional transitions within the V3Sb1 layers are also triggered. The phenomenon of symmetry breaking induced by anharmonic lattice distortion is shown in Fig. 1b. Thus, by employing the nonlinear phonons between IR-active phonons A2u (3.28, 7.547) and Raman phonon A1g (4.06), it is possible to identify a candidate pathway to deviate the motions of Sb2 from equilibrium while keeping the V3Sb layers atoms immobile, resulting in the formation of photoinduced metastable states.

We first consider the previously proposed quadratic coupling approach, with particular emphasis on the interplay between the infrared-active mode A2u (7.547) and the Raman-active mode A1g (4.06). Noticeably, the IR-active mode A2u (7.547) can be excited resonantly with the aid of a co-frequency laser pump, while the oscillation of Raman-active mode A1g (4.06) is not driven effectively due to the weak interaction between the two phonons, which is susceptible to the atomic displacements of V-Sb1 sheets.

In order to stifle the movements of V-Sb1 atoms and consolidate the displacements of Sb2, we focused on the three phonons nonlinear coupling approach, in which two IR-active modes A2u with similar symmetries but different eigenfrequencies (3.28, 7.547) would couple to a single Raman mode A1g (4.06). According to the established approach of nonlinear phononics43, the relevant anharmonic potential energy reads

$$V\left(Q\right)=\frac{{\omega }_{1}^{2}}{2}{Q}_{I{R}_{1}}^{2}+\frac{{\omega }_{2}^{2}}{2}{Q}_{I{R}_{2}}^{2}+\frac{{\omega }_{R}^{2}}{2}{Q}_{R}^{2}+O({Q}_{{IR}1},{Q}_{{IR}2},{Q}_{R})$$
(1)

where QIR1 and QIR2 represent the real-space eigen-displacements of IR-active modes A2u (3.28) and A2u (7.547) and QR is the eigen-displacement of Raman-active mode A1g (4.06). The atomic displacements due to an amplitude QIR, R is given by \({U}_{j}=\frac{{Q}_{{IR},R}}{\sqrt{{m}_{j}}}{\omega }_{j}^{{IR},R}\), where Uj is the displacement of the jth atom, mj is the mass of the atom, and \({\omega }_{j}^{{IR},R}\) represents the corresponding component of the IR and Raman mode vector. Here we use the atomic unit of u for masses and unit of u1/2Å for the mode amplitude QIR, R. While O represents the higher order term, ωα (α = 1,2, R) denote the corresponding eigenfrequencies of the phonon modes. The calculated energy surface as a function of the atom displacements (along the phonon modes of different QIR1, QIR2 and QR values) is displayed in Fig. 2a. It is noticeable that the three phonons coupling, that is, two IR-active phonons (with different frequencies) coupled with a Raman-active phonon, causes the anharmonic potential energy to shift from the equilibrium position. The calculated anharmonic constants are listed in the Supplementary Table 1. The result obtained demonstrates that the two IR-active modes, A2u (3.28) and A2u (7.547), exhibit a strong coupling to the desired single Raman mode A1g (4.06). This effect is observable when the two IR modes are excited simultaneously and coherently.

Fig. 2: Nonlinear phononic coupling in CsV3Sb5.
figure 2

a Total energy as a function of the two QIR and the single QR mode coordinates of CsV3Sb5, the numerical value represented the amplitude of QIR1 = A2u (3.28 THz), QIR2 = A2u (7.547 THz) and QR = A1g (4.06 THz), QR vary from −2 u1/2Å to 2 u1/2Å. Here, we plot V (QR, QIR1, QIR2)-V (0, QIR1, QIR2) for visual clarity. b The maximum value of QR corresponds to various field intensities E1 and E2. c Influence of the varied delay time Δt on the maximum value of QR.

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To manipulate the symmetry breaking of CsV3Sb5 via light and to investigate the evolution of the anharmonic system, we further discuss the dynamic process when the two IR modes are pumped externally. We solve the dynamical equation of motion to clarify the response of each IR and Raman phonon mode42:

$$\ddot{Q}+{{\gamma }}\dot{Q}+{\nabla }_{Q}\left[V\left(Q\right)-F(t,\theta ){Q}_{{IR}}\right]=0$$
(2)

Where Q = QIR, QR describes the excited IR modes and the Raman modes coupled to them. γ is the damping constant of each mode. Here we set γ = 0.1ω. F (t, θ) denotes the specific intense THz excitation pulse.

Since the V and Sb1 atoms of CsV3Sb5 have opposite vibrational directions in the selected two infrared phonon modes A2u (3.28, 7.547), we might effectively induce a transient structural distortion along the eigenvectors of a coupled A1g Raman distortion, with only Sb2 atoms in motion. However, external field modulation methods based on anharmonic phonon coupling have not been used to investigate the coherent excitation of IR phonons with different eigenfrequencies. In this circumstance, we propose a two-laser model with suitable time delay and frequency to achieve diverse infrared mode excitation:

$$\begin{array}{l}{\boldsymbol{F}}\left(t,\theta \right)={E}_{1}{e}^{-\frac{{t}^{2}}{2{\left(\frac{\alpha }{2\sqrt{2\mathrm{ln}\left(2\right)}}\right)}^{2}}}\cos \left(\theta \right){\int }_{\!-{{\infty }}}^{{{\infty }}}d\omega \cos \left({\omega }_{{IR}1}t\right){e}^{-\frac{{\omega }^{2}}{2{\left(\frac{\alpha }{2\sqrt{2\mathrm{ln}\left(2\right)}}\right)}^{2}}}\\\qquad\qquad +\,{E}_{2}{e}^{-\frac{{\left(t-{t}_{0}\right)}^{2}}{2{{\rm{\alpha }}}^{2}}}\cos (\theta ){\int }_{\!-{{\infty }}}^{{{\infty }}}d\omega \cos ({{\rm{\omega }}}_{{IR}2}(t-{t}_{0}))e^{-\frac{{\left({\rm{\omega }}-{{\rm{\omega }}}_{0}\right)}^{2}}{2{{\rm{\alpha }}}^{2}}}\end{array}$$
(3)

Where ωIR1 = 3.28 THz, ωIR2 = 7.547 THz, α = 2 ps. θ is the polarization angle of the linearly polarized light from the laser. We set θ = 0° corresponding to polarization along the a-axis of CsV3Sb5. E1 and E2 are the maximum intensity of two lights, respectively, and t0 indicates the time delay between two lasers.

Figure 2b shows the values for the amplitudes of two pulses used to pump the QIR1 and QIR2 coordinates at their eigenfrequencies that induced rectification of the QR coordinate. We sampled the two electric fields from 0 to 2000 mV/Å with a step of 50 mV/Å to identify the light-phonon coupling. It is limited to the value of pump amplitude that induced rectification of Raman phonon displacement without possible breakdown of the system. We can see that when the electric field strength is less than 1000 mV/Å, almost no oscillation of the QR is induced. It is noticeable that stronger fields appear to trigger the enhancement of Raman phonon modes; however, the map of max(QR) is irregular and disordered. Consequently, the maximum values of QR are not linearly dependent on E1 and E2, which may originate from the competition, periodic oscillation and damping effect of two resonantly excited infrared phonon modes. In addition, since the frequencies of two tailored lasers were consistent with the eigenfrequencies of the material, the excitation of phonons is closely related to the delay time Δt. In particular, the same conclusion can be drawn in most instances of our calculation. Therefore, for a clearer illustration, we only exhibit three coupling cases in Fig. 2c. We can see that the trends of max(QR) appear to be similar, and all reach a maximum when Δt = 3 ps. This implies that the optimal delay time is related to the FWHM and frequency of the laser, which is susceptible to phonon coupling but is almost independent of the field strength.

By using the determined field strengths and delay time, we then validated the advantages and benefits of this two-pulse model for inducing targeted Raman phonon modes. Figure 3 shows the evolution of IR modes A2u (3.28, 7.547) and Raman mode A1g (4.06) when pumped with laser along the a-axis of the crystal. Numerical integration of Eq. 2 reveals that the two IR modes can be excited via infrared absorption by exciting CsV3Sb5 via a laser pump with a center frequency of 3.28 THz and 7.547 THz.

Fig. 3: Response of the infrared and Raman-active phonons under THz light.
figure 3

Dynamics of the IR-active phonon mode QIR1 (red), IR-active phonon mode QIR2 (green), and Raman-active phonon mode QR (blue) for a single-frequency pump pulse with amplitude E1 = 200 mV/Å and frequency ω1 = 3.28 THz (yellow) (a–c), for a single-frequency pump pulse with amplitude E2 = 1400 mV/Å and frequency ω2 = 7.547 THz (purple) (d–f) and for the two-frequency pump pulse with delay time Δt = 3 ps (g–i).

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When the light strength is set to 200 mV/Å with frequency ω1 = 3.28 THz, the weak excitation for the lower pump frequency, the IR-active mode QIR1 (red) and QIR2 (green) exhibit antisymmetric oscillations and eventually converge to a fixed value. This result indicates that the weak excitation can effectively change the position of Sb and V atoms and shift them away from their equilibrium position, leading to a sub-stable state. However, this does little to motivate the rectification of QR coordinates.

When the field strength increases to 1400 mV/Å and frequency ω2 up to 7.547 THz, the IR-active phonon modes show similar vibrational trends, with QIR1 (red) and QIR2 (green) being resonantly excited at higher pumping frequencies and returning to the equilibrium value of zero at 10 ps. Though the field strength and frequency enhanced, the amplitude of the oscillations of the IR modes get reduced and the displacement of Raman mode is almost unaffected in this case.

When we consider two laser pumps to excite phonons of different frequencies, the amplified oscillations of the pumped IR mode demonstrate other process dynamics. The vibration of QIR1 lasts for a longer duration, and the QIR2 mode makes several cycles while being displaced from the equilibrium position. Remarkably, the Raman mode, which can not be excited directly by laser, is highly amplified and oscillates with an amplitude of 1.0 u1/2Å. A straightforward interpretation of this simulation is that in order to reach the excitation of the two IR-active phonon modes simultaneously, two optical pulses of different frequencies are needed. If there is a phase difference between the two pulses, a larger displacement of the desired Raman mode can be achieved due to the nonlinear phononic coupling. Here we only list the case of two-frequencies pump pulses of E1 = 200 mV/Å and E2 = 1400 mV/Å with the delay time Δt = 3 ps for general illustration. However, using our method, the amplitude of Raman mode A1g (4.06) can be adjusted to 2.0 u1/2Å (Sb2 ion displacement of 0.18125 Å) sufficiently when the intensity of the two-frequency pump pulse is E1 = 1900 mV/Å and E2 = 100 mV/Å. Hence, breaking symmetry by anharmonic coupling of the IR with Raman modes can efficiently modify the Sb-Sb distance.

Intriguingly, we also find that the two Raman-active modes E2g (4.005), which are orthogonal and degenerate, represent sources of intraformational vibration which then consequently trigger the motion of the Sb2 atoms along the x-y plane direction. We interpret the oscillation of those two Raman modes as being complementary to each other. In an attempt to explore the perturbative regime of quadrilinear coupling for four different phonons, we then focused on the phonon-phonon coupling interaction between the two IR modes E1u (5.717) and the two degenerate Raman-active modes E2g (4.005). Being these IR and Raman modes coupled quadratically to each other, the Raman E2g modes can be driven in the context of electron-phonon and spin-phonon coupled phenomena.

Topological phase transition

Further, when focusing on the impact of the modulated structure on the superconductive ordering and topological band structure evolution, we calculated the electronic properties and the evolution of topological properties (of the system) with various Raman modes while the optimized lattice parameters remained unchanged. We calculated the parity of the wave function at the time-reversal invariant momentum (TRIM) points54 and the Z2 topological invariant of various QR between the bands crossing EF, as shown in Eq. 4,

$$(-1)^{v}=\mathop{\prod }\limits_{i=1}^{4}{\delta }_{i}$$
(4)

where ν is the Z2 topological invariant, and δi is the parity product of bands at the TRIM point i (Γ, A, L, or M). Since the IR phonons served as the intermediate state for ionic Raman scattering, it is necessary to take their effects into account before investigating the evolution with Raman phonons. The results of Z2 topological invariants for the three bands across the Fermi level (band #47, #49, #51) with QIR varying from −2.0 to 2.0 u1/2Å reveal that the Z2 topological invariants remained constant at 1, indicating that the oscillations of the infrared phonons here have a negligible effect on the material’s topological properties. The calculated results are listed in the Supplementary Table 2. Next, we computed Z2 topological invariants of the three bands across the Fermi level with QR varying from −2.0 to 2.0, a range that avoids secondary effects above the elasticity limits of CsV3Sb5. Figure 4c shows schematic representations of Z2 when the Raman mode is A1g (4.06). It is clearly shown that the three bands across the Fermi level remain topological nontrivial characters with QR varying from 0.0 to 1.6, and topological phase transitions occur when the structural distortion of QR = −0.2, −0.6 and −0.8 u1/2Å, corresponding to the Sb2 ions displacements of −0.018125, −0.054375 and −0.0725 Å. Combined with the calculated energy band diagram in Fig. 4b, the transition at QR = -0.2 can be attributed to the band gap at the L point for the #47 band and the band shift at the M point for the #51 band. Prominently, owing to the parity change at the A point, the topological invariant of the #51 band returns to 1 with QR = −0.6. The conversion at QR = −0.8 possibly originates from the band split at point L. The invariants then remain unchanged till QR = −2.0.

Fig. 4: Topological phase transitions corresponding to the Raman-active mode A1g (4.06 THz).
figure 4

a Depiction of the set-up with two THz pumps. b The band structures of CsV3Sb5 within QR = −0.2, −0.6, −0.8, 2.0 u1/2Å, respectively. The three bands across the Fermi level are labeled yellow (#51), red (#49), and blue (#47), respectively. c Visual image for the evolution of Z2 topological invariants of the three bands.

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The results confirm that light-induced nonlinear phonons coupling in CsV3Sb5 can efficiently modify the parity of bands near EF, leading to the topological phase transition, which is inextricably related to superconducting properties. Raman mode A1g (4.06) can well alter the Sb2-Sb2 bond between different V3Sb layers, leading to a structural evolution from 2D to 3D. Furthermore, the formation of Sb2-Sb2 bonds can enormously affect the van Hove singularities and electric properties, driving topological phase transition in CsV3Sb5.

We further investigate another Raman mode E2g. In contrast to Raman mode A1g (4.06), Raman mode E2g (4.005) exhibits plenty of significant differences. This vibration mode has no significant effects on the distance between the V3Sb layers, as shown in Fig. 5a. By utilizing Eq. 4, we investigated the evolution of the Z2 topological invariant of the system with Raman mode E2g (4.005) varying from −2.0 to 2.0 in Fig. 5b, c. When compared with the Raman mode A1g (4.06), the E2g (4.005) vibrational mode exhibits a higher tendency to accomplish the transition into a topological phase. In the interval of QR = 0.2–2.0, the topological properties change almost continuously. Additionally, when exploring the evolution of the projected band structures (PBAND) of Sb1 and Sb2 p orbitals upon distortion of the system with the QR = E2g mode, we find an almost invariable contribution of Sb1 to the electronic properties, while the p orbital of Sb2 dominates near the Fermi surface, as shown in Fig. 5d. Being consistent with our calculation for the parity product of bands at the TRIM point, the p orbital of Sb2 moves drastically at the high symmetry points M and L, with consequential effects on the density of states near EF (topological transitions).

Fig. 5: Topological phase transitions corresponding to the Raman-active mode E2g (4.005 THz).
figure 5

a Sketch of the atomic displacements corresponding to the Raman-active mode E2g (4.005 THz). b Visual image for the evolution of Z2 topological invariants of the three bands across the Fermi level with different QR amplitude, where QR denotes the Raman mode vibration at 4.005 THz. c Z2 topological invariants of the three bands. d The projected band structure (PBAND) of Sb1 p (blue) and Sb2 p (red) orbitals of CsV3Sb5 within QR = 0.2, 0.6, 1.0, 2.0 u1/2Å, respectively.

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We have extended the nonlinear lattice rectification mechanism to difference-frequency generation between pairs of non-degenerate excitations, resulting in ultrafast topological transition of quantum materials with light. We predict that this optical engineering will substantially affect the superconductivity of CsV3Sb5. We strongly recommend further experimental verification.

In conclusion, in this manuscript, we investigated the transient structural distortion along the eigenvectors of Raman phonon mode A1g (4.06) with two excited IR phonon modes A2u (3. 28, 7.547) of similar symmetries in the superconductor CsV3Sb5. By employing two pump pulses, we extensively promoted symmetry breaking, resulting in the dramatic amplitude of Raman mode A1g. Furthermore, we found that the Raman modes A1g (4.06) can significantly affect the Sb2-Sb2 distance between different V3Sb kagome layers, leading to the transition of the topological phase in CsV3Sb5. Noticeably, despite the fact that the oscillation of the Raman phonon modes E2g (4.005) barely impacts the interlayer interaction, the topological properties are also sensitive to this vibrational mode. The presented work not only highlights a mechanism for nonlinear phononics control and switching of structural distortions by changing the field intensities and the delay time between two light pulses but also provides a feasible pathway for manipulating the light-induced topological phase transition in CsV3Sb5. We hope the mechanisms and strategies for photoinduced topological phase transition shown here will stimulate further experimental research and open new avenues for optical devices in which the topological properties, superconductivity and optical responses of materials can be modulated or switched by optical pulses.

Methods

DFT calculations

The structural optimization and electronic structure calculations were performed by using density functional theory with the projected augmented wave method55 as implemented in the Vienna ab initio simulation package (VASP) package56,57,58. We employed the generalized gradient approximation of the Perdew-Burke-Ernzerh59 of functional with DFT-D3 van der Waals correction60 for exchange correlation. We increased the k-mesh to 6 × 6 × 3 points to sample the Brillouin zone of the unit cell and used a plane-wave energy cutoff of 600 eV with an energy precision of 10–6 eV. Both lattice parameters and atomic positions were fully optimized until the Hellmann-Feynman forces on each ion were less than 10–3 eV/Å. The spin-orbit coupling is taken into account in all calculations. Phonon calculations were performed using the frozen phonon method implemented in the PHONOPY code61. Z2 topological invariant calculations were performed using the Irvsp program62 in conjunction with VASP.