Crystalline chirality and interlocked double hourglass Weyl fermion in polyhedra-intercalated transition metal dichalcogenides

Introducing crystalline chirality into transition metal dichalcogenides (TMDs) has attracted much attention due to its modulation effect on optical properties and the potential to reveal new forms of electronic states. Here, we predict a number of chiral materials by intercalating polyhedra into TMD lattices, finding a type of double hourglass Weyl fermion interlocked with crystalline chirality. The best candidate RhV3S6 (P6322) possesses the largest hourglass energy window of ~380 meV, as well as strong optical circular dichroism (CD) in the infrared regime, both of which are tunable by external strains. The chirality is originally induced by the configuration of intercalated polyhedra and then reduced by the rotational atomic displacements triggered by intercalation, as indicated by CD calculations. Our study opens the way of designing chiral materials with spin-split double hourglass Weyl fermions via structural unit intercalation in achiral crystals for future chiral-functionalized optoelectronic and spintronic devices. We realize the introduction of crystalline chirality into transition metal dichalcogenides by intercalating polyhedra into the lattice and reveal a new type of crystalline chirality interlocked double hourglass Weyl fermion. The best candidate RhV3S6 (P6322) possesses a record wide hourglass energy window of ~380 meV, as well as strong optical circular dichroism (CD) in the infrared regime, both tunable by external strains. The chirality is originally induced by the configuration of intercalated polyhedra, then reduced by the rotational atomic displacements triggered by the intercalation, as indicated by CD calculations.


Introduction
Transition metal dichalcogenides (TMDs) are the subject of intense interest due to their remarkable electronic and optical properties, such as their direct bandgap in a monolayer limit 1 , strongly bound excitons and trions 2,3 , and chiral optical selection rules 4,5 due to spin-valley locking degrees of freedom. Benefitting from these unique properties, especially the chiral selection rules, TMDs have acted as an ideal class of materials for optoelectronic and spintronic manipulation, becoming increasingly studied for valleytronic devices 6 . As the chiral optical selectivity 4,5 of the K and K′ valleys have opposite signs in TMDs without crystalline chirality, external chirality has been applied to tune the valleytronic property of TMDs [7][8][9][10] . Valley polarized photoluminescence of TMDs such as MoS 2 can be tailored through nearfield interactions with plasmonic chiral metasurfaces 8 . The coupling of valley excitons in monolayer WS 2 with chiral surface plasmons has also been demonstrated 10 , which occurs at room temperature and persists for a long lifetime. Most recently, external crystalline chirality was induced in MoS 2 nanostructures via the surface modification of chiral ligands 11,12 . The coupling of external chirality with intrinsic chiral optical selectivity significantly enriches the phenomena in the TMD material family.
Weyl fermions in TMDs [13][14][15][16][17] have also attracted much attention in light of their extremely large magnetoresistance for potential applications in magnetic sensors and memory 18,19 . The Weyl points in these achiral TMD Weyl semimetals [13][14][15] are paired with opposite chirality. Therefore, coupling external net chirality to the chirality of Weyl fermions could further advance the study of TMD-based materials and potentially open the way of finding new fermions, such as the Kramers-Weyl fermions 20 that are commonly found in nonmagnetic chiral crystals when Kramers theorem-forced band degeneracy appears near the Fermi level (E F ). The recently found new form of fermion, namely, the hourglass fermion [21][22][23][24] , is also closely related to the Weyl fermion, which, however, has not yet been found in TMDs. This class of new fermions was originally predicted on the surface of nonsymmorphic crystal KHgSb 22  , and Tl 3 PbBr 5 30 , as well as the hypothetical two-dimensional structure Bi/Cl-SiC(111) 31 . The predicted hourglass fermion on the surface of KHgSb 22 has been observed in experiments via angleresolved photoemission spectroscopy 32 . However, this type of hourglass-like electronic structure usually appears as single sets. Pairs of hourglass Weyl fermionic electronic structures have not been discussed before, which is plausible, as their splitting and annihilation could be associated with a physical quantity, such as crystalline chirality.
In this paper, we search for TMD-based chiral materials with double hourglass Weyl fermions by polyhedral intercalation in a hexagonal lattice. Based on the stability and electronic property evaluation in the framework of density functional theory (DFT) 33,34 , we found a few intercalated chiral TMDs with pairs of hourglass fermionic electronic structures around E F and identified RhV 3 S 6 (P6 3 22) as the best candidate with the largest hourglass window of~380 meV. The splitting and spin polarization of the double hourglass fermion was interlocked with the crystalline chirality. We evaluated the optical circular dichroism (CD) for RhV 3 S 6 and found that when the rotational atomic displacements around the chiral axis triggered by polyhedral intercalation were increased (decreased), the CD response was reduced (enhanced), suggesting that crystalline chirality was originally induced by polyhedral intercalation and then reduced by rotational atomic displacements. Furthermore, the energy window of the double hourglass fermion and optical CD could be effectively tuned by external strains. Our study opens the way for designing TMD-based materials with interlocked crystalline chirality and double hourglass Weyl fermions, with potential applications in future optoelectronic and spintronic devices.

Electronic structure calculation
All electronic structures were calculated by DFT 35 as implemented in the Vienna ab initio simulation package 33,36 . The projector-augmented wave pseudopotentials 33 with the exchange-correlation functional of Perdew-Burke-Ernzerhof 34 were adopted. The cutoff energy for the plane waves was selected as 520 eV, and the Brillouin zone was sampled on 10 × 10 × 4 Monkhorst-Pack 37 k-point meshes. The lattice constants and atomic positions were fully relaxed when the absolute total energy difference between two successive loops and the Hellmann-Feynman force on each atom were <10 −6 eV and 5 × 10 −3 eVÅ −1 , respectively, using the conjugate gradient algorithm. The band structure from the hybrid functional (HSE06) 38,39 was obtained for comparison with the conventional DFT result. Perturbation was added to the pseudopotential to address the spin-orbit coupling (SOC) effect. The spin state of the electron wavefunction was evaluated by projecting the calculated wavefunction on the spin and orbital basis of each atomic site, which generated spin-projected band structures with dense kmeshes in the Brillouin zone. Strain was applied for the deformation calculation by inducing lattice shrinkage and expansion for tensile and compressive situations.

Circularly polarized evaluation
The circularly polarized optical absorption was the difference between the absorption of left-and righthanded lights for all k points within the valence bands and the conduction bands, and these could be normalized and evaluated from the conventional DFT-calculated ω frequency-dependent imaginary part of the dielectric function with the random phase approximation (RPA) and Fermi golden rule [40][41][42] : where Ω is the volume of the primitive cell, e is the elementary charge, h is the Planck constant, ω represents the angular frequency, m e is the electron rest mass, c and ν indicate the conduction and valence band states, respectively, E is the eigenenergy of the wavefunction, q stands for the Bloch vector of the incident wave, u is the wavefunction, w k denotes the k-point weights that sum to 1 and the factor 2 before it accounts for the fact that degenerating spin has been considered, and ∇ is the momentum operator, in which the sum over momentum k is conducted over the full 3D space. The optical CD is obtained by decomposing the circularly polarized part from the total absorption spectrum, as implemented in the PWmat code 43,44 . The circularly polarized response for RhV 3 S 6 was determined with the RPA-GGA approach. A planewave basis set was employed at a cutoff energy of 80 Ry, and a total of 212 bands were included to ensure convergence of all computed quantities. A very dense k-point mesh of 14 × 14 × 6 over the reducible hexagonal Brillouin zone was sampled in our calculations.

Thermodynamic and kinetic stability assessment
Usually, we can construct a set of candidate structures (containing their compositions) that exist in crystal databases, e.g., ICSD 45 or Material Project 46 , compute their total energies, and then screen this list for the lowest-energy structure. To determine the thermodynamic stability of AB 3 C 6 compounds, we considered all of its possible disproportionation channels to the competing phases (elemental phases, binaries and ternaries) by solving a set of inequalities in multidimensional spaces of chemical potentials (Δμ A , Δμ B , and Δμ C ), which were presented in detail in our previous work 47 . The most stable cation intercalation position was confirmed through dividing the vdW planes of the layered TMDs with a dense mesh, globally detecting the symmetry evolution and total energy landscape of the system. To further evaluate the dynamic stability, we calculated the phonon dispersion curves using the finite-displacement approach as implemented in the Phonopy code 48 . The phonon frequency was derived from crystal forces originating from displacements of certain atoms in a 2 × 2 × 2 supercell for RhV 3 S 6 with P6 3 22 symmetry.

Searching chiral materials from cation-intercalated TMDs
Intercalated TMDs have recently attracted substantial interest due to their intriguing properties arising from intercalation applications [49][50][51] . An important factor to be considered when intercalating a structure is that the host cannot be damaged significantly, such as a large-scale rearrangement of atoms or bonding distortions, which will remarkably increase the energy of the system and make the final structure unstable. The optimal solution to introduce chirality in layered TMDs could be adding screw rotation symmetry through the pivots within the vdW region, forming a chiral axis perpendicular to the vdW plane, which can guarantee minimal damage to the initial MX 2 motifs, e.g., a triangular prism in the 2H phase and an octahedron in the 1T phase. For simplicity, we consider cation intercalation in the ffiffi ffi 3 p × ffiffi ffi 3 p × 2 supercell of the 1T phase and the ffiffi ffi 3 p × ffiffi ffi 3 p × 1 supercell of the 2H phase with AA and AB stackings of MX 2 layers. There are two types of possible intercalation patterns in each supercell with intercalated polyhedra: (i) vertically or (ii) nonvertically distributed along the c axis (Figs. 1a, S1). Regarding the AB stacked 2H TMDs with type (ii) intercalation, there are two inequivalent configurations with the same space group (SG) P6 3 22 (belonging to the 65 Sohncke groups that preserve chirality) 52 but opposite handedness in their intercalation patterns (see the circled red/blue spheres in Fig. 1a and the crystal structure in Fig.  S1), in which the chiral symmetry is endowed by breaking the inversion and mirror symmetries of the pristine AB stacked 2H phase (P6 3 /mmc). Regarding all the other intercalated situations, the products are achiral.
A series of new AB 3 C 6 compounds can be obtained from the accessible TMD library 53 with intercalated TM cations by the requirement of valence state matching (χ A þ 3χ B ¼ 6χ C , assuming chalcogen with −2 common valence states) and inequivalent species on I(II)/III sites (A ≠ B). All the 1T intercalated TMDs present CS symmetry, namely, P31m, P31c, P6 3 /m, and P6 3 /mcm. The 2H intercalated TMDs possess 1 CS (P6 3 /mcm), 1 chiral (P6 3 22), and 2 non-CS (P62m and P62c) structures (see Figs. 1b, S1). We evaluated the relative stability of the , and III(III′), which can be used to assign intercalated cations with two types of possible polyhedron motifs (octahedron and triangular prism). The gray planes represent the layered TMDs with the 2H or 1T phase, and the colored spheres (yellow, red, and blue) within the vdW region are the intercalated cations. We denote the specific chiral intercalation patterns (P6 3 22) for AB stacked 2H TMDs with red and blue dashed ellipses. b Symmetry types for the polyhedra-intercalated TMDs with centrosymmetric (CS), chiral, and (achiral) non-CS SGs (see Fig. S1 for these structures). c Symmetry evolution for the hexagonal crystal filled with nonvertically distributed ideal octahedra, which indicates that only the specific arrangement of octahedra (P6 3 22) induces chirality into the system. intercalated TMD structures and noted that (i) the intercalated cations are in octahedral (triangular prism) motifs for AA stacked 1T and AB stacked 2H (AB stacked 1T and AA stacked 2H) phases; (ii) the intercalated octahedra tend to be nonvertically, rather than vertically, distributed along the c axis, as in the low-energy P31c and P6 3 22 structures (Fig. S1). Such a nonvertical distribution of intercalated octahedra in the AA stacked 1T layers of pristine octahedra pertains to CS symmetry, whereas the nonvertical distribution of octahedra intercalated between the AB stacked 2H layers of triangular prisms leads to chiral symmetry (P6 3 22). We demonstrated the origin of how chirality is induced within the system (Figs. 1c, S2).
A number of chiral TMD materials (P6 3 22) are magnetic, including those that were synthesized in experiments, such as CoNb 3 S 6 , FeNb 3 S 6 , and MnTa 3 S 6 54-58 , and many others are nonmagnetic according to our DFT calculations (Fig. S3). We find a unique double hourglass electronic structure along the Γ − A k-path around E F for the nonmagnetic chiral TMDs (Fig. S4), as the unshelled d orbitals of their intercalated and/or host TM cations form partially occupied bands. In contrast, the double hourglass bands can be destroyed more or less in the magnetic case due to the broken time-reversal symmetry (T ) (Fig. S5). Therefore, we focus our research on the nonmagnetic chiral TMD for studying the potential chiralityassociated double hourglass fermion (Fig. 2a). Through careful examination of the electronic state and taking into account the material stability, we screened the best chiral TMD candidate, RhV 3 S 6 (P6 3 22). The most stable Rh intercalated positions are I/II and I′/II′ sites confirmed by globally detecting the total energy of the system when moving the cations in the vdW planes (Fig. S6). Further structural screening showed that RhV 3 S 6 in other known AB 3 C 6 structure types had higher energy than the P6 3 22 phase (Fig. 2b). The stability criteria in Fig. 2c, d show that RhV 3 S 6 (P6 3 22) is dynamically stable (no occurrence of virtual frequencies in its phonon spectrum) and has a relatively large stability region (0.038 eV 2 ) with respect to its competing phases. Thus, all the stability criteria prove the stability of RhV 3 S 6 , indicating the feasibility of experimental synthesis.

Double hourglass Weyl fermion in chiral TMD
The chiral TMD (RhV 3 S 6 ) crystallizes in a hexagonal structure with SG D 6 6 (P6 3 22, No. 182), which presents semimetallic properties with valence and conduction bands crossing linearly at (0, 0, ±0.447π) along Γ − A without the SOC effect (Fig. 3a, b) and forms a fourfold degenerated hourglass Dirac semimetal state ( Fig. S7 and Table S2). In the presence of SOC, the energy bands are generally nondegenerate, except at the time-reversal invariant momenta (TRIM), e.g., Γ and A, and the hourglass bands split into two nested hourglass bands (Fig. 3c, d). Here, we provide a symmetry argument for the hourglass dispersions along Γ − A. The chiral TMD (RhV 3 S 6 ) belongs to the nonsymmorphic space group P6 3 22, which contains a screw rotation operator C 2z ðx; y; zÞ ! Àx; Ày; z þ 1 2 ð2Þ where the tilde denotes a nonsymmorphic operation. The intercalated cations lead to chirality in the chiral TMDs (P6 3 22), whereas the nonsymmorphicC 2z symmetry is inherited from the parent 2H phase (P6 3 /mmc), leading to hourglass Weyl fermions. In addition, T symmetry is preserved for nonmagnetic RhV 3 S 6 . The T symmetry guarantees Kramers degeneracies at TRIM Γ and A due to T 2 = −1 in the presence of SOC. In theC 2z invariant k-path along Γ − A, we can choose each Bloch state u j i to be the eigenstate ofC 2z . It is easily obtained that which indicates that e Àik z is the eigenvalue ofC 2 2z . Thus, C 2z acts on the real-space part of the wavefunction and commutes with the time-reversal operator. Along the Γ − A k line, each k point is invariant underC 2z , and the related Bloch state u j i can be chosen as the eigenstates of C 2z , namely,C 2z u j i ¼ g z u j i. One hasC 2 2z ¼ e Àik z =2 , and the eigenvalue ofC 2z will be g z ¼ ± e Àik z =2 , i.e., g z ¼ ± 1 at the Γ point (0, 0, 0) and g z ¼ ± i at the point (0, 0, π). SinceC 2z always commutes with the time-reversal symmetry T = K (without SOC) or T = iσyK (with SOC), where K is the complex conjugate operator, the two Kramers pair u j i and T u j i must degenerate at the time-reversal invariant points Γ and A. In this case, the Kramers pair u j i and T u j i at the Γ point has the same eigenvalue, while the Kramers pair u j i and T u j i at the Α point have the opposite eigenvalue. Such partner switching between Γ and A enforces a band crossing with twofold degeneracy along Γ−A. We further evaluated the double-valued representation of the double point group for each band and evidenced the state-switching characteristics, as seen from the degenerating and separating process of irreducible representations (irreps) at arbitrary point Δ along the Γ − A line for both chiral S and R phases in Fig. 3e and Table S3. Here, Δ i is the irrep obtained by scanning each hourglass band along the Γ − A line, forming a degenerated partner with another Δ j at TRIM. The partners Δ 11 and Δ 8 (Δ 10 and Δ 9 ) form the top left (bottom) corner of one hourglass band at Γ, separate from each other while leaving from the Γ point and recombine at the A point by exchanging the degenerated pair and constituting the right corners (Fig. S8). A similar situation also holds for the other nested hourglass band formed by Δ 7 and Δ 12 (Δ 11 and Δ 12 ) degenerated partners. As a result, the occurrence of partner switching while moving from one TRIM to the other on the Γ − A line leads to the nontrivial band connectivity diagram and the nested double hourglass bands in RhV 3 S 6 . Furthermore, we note that the spin characteristics for every hourglass band are completely opposite between chiral-S/R structures for the chiral TMDs, which indicates the interlocking of spin-momentum of the double hourglass bands and crystalline chirality, Here, the r stands for spatial coordinates. It should be noted that the linear energy dispersion of the double hourglass bands extends over a relatively large energy range and thus could be experimentally observed uncomplicatedly via various spectroscopy techniques (e.g., tunneling spectroscopy, photoemission, or transport experiments) 32,59 .
Tuning the dia-chiral response and double hourglass state by strain The interlocking of crystalline chirality and double hourglass bands indicates the possibility of tuning one through another. The left-/right-handed RhV 3 S 6 configurations are determined by the placement of Rh octahedra (at I/II and I′/II′ sites) and present opposite CD responses (Fig. 4). The octahedron in pristine RhV 3 S 6 maintains a slight atomic distortion for the V − 1 and S atoms: the V − 1 atoms that are originally located on the ab plane have relative displacements along the c axis (d V−1 = 0.0015 Å); the S atoms rotate in the ab plane (0.594°/ 0.297°for α and β, respectively). In general, the local atomic distribution and bonding environment could affect the chirality of the system. The enhanced distorting operation (e.g., rotation) is usually considered the origin of chirality or the main contributor [60][61][62] . However, not all atomic bonding distortions contribute to the chirality, such as the Kekulé distortion 63 for epitaxial graphene on a copper substrate that breaks the chiral symmetry 64 . Since distortion can increase or decrease chirality, one fundamental question is whether an intrinsic chirality hidden in a specific microstructure exists and whether local atomic distortion will force it to decay. Here, we probed this issue in our predicted chiral TMD − RhV 3 S 6 , as the inherent chirality is intimated with the special configuration of the intercalated octahedra (Fig. 1c) and can be sensitively captured by the double hourglass fermion by controlling the intercalation-induced local structure distortions. This is done by applying biaxial strains (ab in-plane) on RhV 3 S 6 . We note that as the perturbation is induced via strain, the relative movement of V − 1 diminishes gradually, whereas the rotation angle of S experiences a certain degree of fluctuation (Fig. 4b). Both of these structural deformation factors reach the critical point around the +2.5% strained case, as the octahedron recovers to its pristine structure with barely any rotation. Additionally, the CD intensity presents the maximum for the pristine octahedron stacking configuration, while both the enhanced perturbation of S (α and β) and V − 1 (d V − 1 ) distortions weaken the CD signal (Fig. 4c). Even for the slightly distorted octahedra in unstrained RhV 3 S 6 , the CD response is weakened. To further demonstrate the intrinsic chirality restored within intercalated octahedra, we separately evaluated the CD response under +2.5% strain with increasing S (Rot. S; α, β) and V − 1 (Dist. V − 1; d V − 1 ) distortions, starting from the structure with quenched atomic distortion, which indicates the compensation of intrinsic chirality (for quenched distortion, see the curves with the highest CD peaks in Fig. 4d). Indeed, the intrinsic chirality of RhV 3 S 6 is introduced by the pristine octahedron placement configuration, forming the intrinsic chiral domain of the crystal. However, the domain steady state will be broken by structural perturbation due to redistributed state coupling (i.e., rebalancing of the wavefunction), leading to the opposite handedness and weakening of the intrinsic chirality, as an inevitable "dia-chiral response" that can be modulated by external strain, as discussed above.
In the "dia-chiral response" modulation, we find that the double hourglass state presents a tunable hourglass window, namely, the covered spin-splitting energy range in E-k space by hourglass bands. The splitting window for pristine RhV 3 S 6 reaches a record energy range of 380 meV (Fig. 3c), which is significantly larger than those in KHgSb (60/80 meV forZ=X ÀΓ=Ũ at 010-surface) 22 , Ag 2 BiO 3 (3 meV for X − S k line) 27 , and Bi/ Cl-SiC with a 111-plane (150/187 meV for X/Γ − M/Y direction). Strain engineering enables the wide-range modulation of the double hourglass bands (~300-440 meV) for RhV 3 S 6 (Fig. S9). In the +2.5% strained case with the maximum CD intensity, the middle band-crossing points (the normal centers of the double hourglass bands; analogous to the Weyl points in achiral TMD Weyl semimetals [13][14][15][16][17] shift to the E F (Fig. 4e). In fact, the double hourglass bands are dominated by V 3d orbitals hybridized with Rh 4d and S 3p components, forming anti-bonding states (Figs. 3c, S10, S11). When ab in-plane tensile strain is applied, the heights (h Oct. and h Tri. ) of the octahedron and triangular prism are both compressed (Fig. S11), and the Rh-S and V-S bond lengths (d RhS . and d VS ) are enlarged. As a consequence, the p-d coupling between the V 3d (or Rh 4d) and S 3p orbitals is weakened due to the decreased overlap between their wavefunctions, which tends to lower the anti-bonding double hourglass states. Therefore, the double hourglass bands continuously move down with a decreasing splitting window, presenting flexible tunability along with the modulation of crystalline chirality.

Conclusions
In summary, we predict a number of octahedral intercalated chiral TMDs with a type of spin-split double hourglass Weyl fermion, which is interlocked with crystalline chirality. The optical CD of intercalated TMDs is evaluated to monitor the chirality of the material that can be tuned by external strain. The best candidate RhV 3 S 6 is predicted to be thermodynamically stable, whose synthesis is called for as the first example of a chirality-enforced double hourglass fermion. On the other hand, this type of chiral TMD may provide a new platform for exploring tunable chiraltronics within a three-dimensional scale, e.g., a chiral plasmonic device [65][66][67] , as three-dimensional chiral plasmonic structures that exhibit a strong intrinsic chiral optical response remain an extreme challenge. Furthermore, the predicted AB 3 C 6 family members could also raise great interest in extraordinary phenomena and device applications, such as the anomalous Hall effect 54 , mesoscopic magnetic modulation 68 , three-state nematicity 69 , and tunable giant exchange bias by spin glass 70 . The stable properties and facile synthesis of these chiral TMDs guarantee the subsequent micro/nanoscale process and device integration. Our study provides an in-depth mechanism of intrinsic chirality induction through polyhedral intercalation in TMDs, which have a significant chiral optical response. More importantly, insight into introducing intrinsic chirality through chiral intercalation of polyhedra in achiral systems to realize a strong chiral response will shed light on designing/searching for new chiral structures extensively with novel characteristics. Our findings are one step forward in the exploration of chirality-related hourglass Weyl fermions in inorganic compounds and provide a new understanding of structural chirality for future chiral-functionalized optoelectronic devices. c Evolution of the CD spectrum under ab in-plane biaxial strain for RhV 3 S 6 . The strongest CD response of RhV 3 S 6 occurs at approximately +2.5% strain with the minimum local environment distortion (α, β less than 0.08°/0.04°and d V−1 < 10 −3 Å). d Evolution of the CD spectrum under +2.5% ab in-plane biaxial strain from completely quenched distortion to gradually increased S rotation (V − 1 distortion) imposed on RhV 3 S 6 . e Ideal double hourglass bands for RhV 3 S 6 (+2.5% strained case) with a spin-splitting energy window of~355 meV.