Introduction

Chirality is a particularly important concept in physical and life sciences. A wide range of intriguing phenomena have been observed in chiral molecules, nanoparticles, and assembled nanostructures, such as asymmetric catalysis, chiroptical activity, and chirality-induced spin selectivity1,2,3,4,5,6. Compared with molecular systems, explorations of chiral crystals are still evolving7, such as twisted bilayers8,9 and molecule-assembled chiral materials5. Import of chirality to crystals leads to a finite set of possible symmetries, which enforce exotic transport, magnetic, and topological quantum properties, such as non-reciprocal electron transport10, skyrmions11,12, and Kramers–Weyl fermions13. In chiral crystals, the inversion, mirror, and other roto-inversion symmetries are absent, endowing nontrivial spin structures and dynamics. In particular, the electrons, spins, and symmetry are coupled via the spin–orbit coupling (SOC). The spin degeneracy of nonmagnetic chiral crystals can be lifted by SOC, creating exclusive chiral spin textures14,15. This offers an ideal playground for probing the chirality-dependent spin effects and designing solid-state chiral spintronics2,16.

The tellurium crystal and compounds generally exhibit prominent SOC effects17,18,19,20. A good example is the tellurium solid with a chiral space group (P3121 or P3221), which has been demonstrated to possess a radial spin texture. The spins point inward or outward in the right-handed or left-handed Te crystals, respectively, leading to the chirality-dependent Edelstein effect21,22,23 and thus allowing the electrical and handedness control of electron spins24. The strong SOC can also induce nontrivial quantum states, such as topological surface states in Bi2Te3 and surface superconductivity in SbTe325,26. The layered MnBi2Te4 has been confirmed as an intrinsic antiferromagnetic topological insulator, which may realize quantum anomalous Hall effect and chiral Majorana fermions27,28. The large SOC also induces strong uniaxial magnetocrystalline anisotropy in Fe3GeTe2 monolayer29, and stabilizes various types of skyrmions and antiskyrmions30. These intriguing observations shed lights in pursuing fascinating tellurides with diverse symmetries, especially with chiral structures, to explore the chirality-driven quantum effects.

Compared with conventional atomic crystals, cluster-assembly provides unlimited possibilities to create superatomic crystals with designated symmetries and electronic characteristics. A variety of cluster-assembled crystals with collective quantum phenomenon have been obtained in laboratory, such as the ionic solids formed by C60 and metal chalcogenide clusters that simultaneously host Dirac states and a flat band31,32, and the layered superatomic crystal of Au6Te12Se8 cubes carrying interweaved charge density wave (CDW) and polarized charge orders33. Moreover, the noncovalent intercube quasibonds of Au6Te12Se8 initiate a competition between CDW and superconducting phases in an ultra-narrow pressure range34,35. Featured with hierarchical structures, cluster-assembled crystals allow the tuning of both intra-cluster and inter-cluster interactions, which may induce unusual structural and topological phase transitions. It would be intriguing to explore how the coupling within a hierarchical structure interplays with the crystal symmetry and relativistic effect, and to unveil the fundamental mechanism of chirality-related charge-to-spin conversion in superatomic crystals.

Here we exploit the telluride tetrahedral clusters to assemble two-dimensional (2D) ternary compounds MXTe4 (M = transition metal; X = Ga or Ge) with strong SOC and tunable symmetry. The tetrahedral cluster is tilted in a 2D square lattice by varying the M valency, leading to the breaking of both inversion and mirror symmetries. The influence of tetrahedral tilt on the stabilization mechanism and electronic structures of the assembled 2D MXTe4 monolayers is elucidated. The chirality-dependent spin textures are discussed based on symmetry analysis and density functional theory (DFT) calculations. The spin Hall effect in these cluster-assembled materials is further explored to conceptually justify their potential for 2D chiral spintronics.

Results and discussion

2D chiral structures assembled by tetrahedral clusters

Tetrahedral clusters are the simplest motifs with high symmetry to construct many kinds of materials. For instance, white phosphorous allotrope is composed of P4 units. Among them, tetrahedral chalcogenide clusters centered by metal cations (group 12–14 or transition metal atoms) are the building blocks of supertetrahedral clusters—a large family of semiconducting nanoclusters resembling the II–VI or I–III–VI quantum dots36. These supertetrahedral clusters can be further assembled into various lattices and open frameworks37. Inspired by these experimental successes, here we exploit the group 13–14 telluride tetrahedra (XTe4) as building blocks to construct chiral crystals (Fig. 1a), where X is Ga or Ge. Tellurium is used for the assembly considering its larger SOC than the other chalcogenide elements. The tetrahedra units are patterned into 2D square lattice with a transition metal atom added in each lattice center to bridge the neighboring telluride clusters, such that both M and X atoms are coordinated by four Te atoms. Similar to the widely observed octahedral tilt in perovskites38, the telluride tetrahedra in the assembled 2D ternary compound (MXTe4) can be orientated in different angles with respect to the lattice vector, endowing either D2d (non-chiral) symmetry or S4 (chiral) symmetry shown in Fig. 1b, c.

Fig. 1: Mechanism for the formation of 2D chiral structures by cluster assembly.
figure 1

a Assembling 2D ternary compounds by tetrahedral clusters XTe4. b Top-view and side-view structures (left panels) and differential charge densities (right panels) of MXTe4 monolayers with D2d symmetry and c S4 symmetry. Electron accumulation and depletion regions are shown in blue and pink with an isosurface value of 0.005 e/Bohr3, respectively.

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It is known that the octahedral tilt in perovskites (usually denoted as ABX3) is intrinsically related to the steric effect or bonding property that mediates the interaction between A cation and BX6 octahedra39. Here different tilting angles of tetrahedra are achievable by adopting various 3d transition metals as the bridge atom, owing to their different valency that influences the bond angle in MXTe4 monolayers. Our DFT calculations show that early transition metal atoms (M = Ti, V, Cr) generally lead to non-chiral symmetry for the MXTe4 monolayers with the M–Te–X angle of 180˚, while chiral symmetry is favored by the MXTe4 monolayers composed of late transition metal atoms (M = Mn, Fe, Co, Ni, Cu). The chiral feature is strengthened as the atomic mass of 3d transition metal atom increases, i.e., the M–Te–X angle decreases from 160˚ to 145˚, associated with a larger rotation degree for the tetrahedral cluster. The energy of the chiral state relative to the non-chiral one is lowered from −0.08 eV to −0.86 eV per formula unit (see Supplementary Table 1). The energy evolution as a function of the M–Te–X angle is plotted for NiGeTe4 monolayer as a representative. As seen from Fig. 2a, transformation between the left- and right-handed enantiomers undergoes a transition state with a non-chiral structure lying at 0.60 eV above the chiral phase. In the synthetic chiral crystals, both enantiomers may coexistent separated by domain boundaries40, as well as for the present MXTe4 monolayers (see Supplementary Fig. 1 for calculation on the enantiomer domains). The non-chiral state shows negative frequencies over the entire phonon dispersion, while the imaginary phonon bands disappear in the chiral phases manifesting their outstanding lattice dynamic stability (Fig. 2b and Supplementary Fig. 2). The formation energies of these assembled monolayers are comparable to the experimentally synthetic 2D ternary compounds such as Fe3GeTe2, Fe5GeTe2, and CrGeTe3 monolayers (see Supplementary Methods for details). Furthermore, the thermal stabilities of selected MXTe4 monolayers (i.e., six semiconductors) were evaluated by performing the ab initio molecular dynamics (AIMD) simulations. As displayed in Supplementary Fig. 3, they all well maintain the chiral monolayer structures with very minor deformation at 300 K during 10 ps simulation, suggesting the outstanding stability of these superatomic crystals for experimental synthesis.

Fig. 2: The stability of chiral MXTe4 monolayers.
figure 2

a Energy evolution of NiGeTe4 monolayer from left-handed to right-handed geometry. b Phonon dispersion of NiGeTe4 monolayer with S4 symmetry. c Density of states and partial charge density of bottom conduction band (insets) for NiGeTe4 monolayer with D2d symmetry and d S4 symmetry.

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The stabilization mechanism for the assembled chiral 2D structures of MXTe4 can be understood as follows. As the formal oxidation state of Te atom is −2, formation of MXTe4 monolayer requires the transition metal atom M having a high valency of +4 (for X = Ge) or +5 (for X = Ga), such that XTe4 satisfies the closed electronic shell of a superatom (32 valence electrons for Td symmetry)41. This is feasible for early transition metals as the bridge atom. In MXTe4 monolayer containing late transition metals with lower valency, the tetrahedra tilts to form Te–Te bonds from neighboring tetrahedra units, such that each Te atom becomes three-fold coordinated to guarantee the electron saturation. The resulting Te–Te bond length of 2.77–3.11 Å is comparable to that of bulk tellurium (2.91 Å)42. In the non-chiral monolayers, the Te–Te distance is above 3.50 Å. As evidenced from the charge density distributions in Fig. 1b, c, the chiral structures present large overlap of charge density in Te–Te dimers, suggesting strong interaction between neighboring telluride tetrahedra. For the non-chiral structures, the charge overlap between the Te atoms from neighboring tetrahedral units is zero.

The tilting of telluride tetrahedra has pronounced impact on the geometry, stability, and electronic band structures of the assembled MXTe4 monolayers. The non-chiral systems with M = Ti, V and Cr are all metals, while the chiral monolayers with M = Mn, Fe and Co prefer to half metals (see band structures in Supplementary Fig. 4). Chiral monolayers of NiGeTe4 (also for M = Pd and Pt) and CuGaTe4 (also for M = Ag and Au) are semiconductors, and their transformation from chiral to non-chiral geometry leads to a semiconductor-to-metal transition (Fig. 2c, d). The electronic density of states suggest that the bottom conduction band is contributed by chalcogenide atoms, which becomes more localized and is lifted in energy as the M–Te–X angle decreases, thereby opening a band gap in the chiral phase (see Supplementary Fig. 5 for details)43. In addition, the lowering of symmetry gives rise to peculiar spin structures and transport behavior under the relativistic effect.

Spin polarization under the SOC

The chiral MXTe4 monolayers belong to S4 point group (P\(\bar{4}\) space group), which is a non-polar structure lacking both inversion symmetry and mirror symmetry. As described by Table 1, the S4 point group contains four symmetry operations: the identity operation (E), two-fold rotation about the z axis [C2: (x, y, z) → (−x, −y, z)], and fourfold improper rotation about the z axis [S4: (x, y, z) → (y, −x, −z) and S43: (x, y, z) → (−y, x, −z)]. In a crystalline solid with inversion asymmetry, the SOC interaction can be approximated by the Rashba–Dresselhaus Hamiltonian with linear expansion in k// usually taking the following forms14

$${H}_{{\rm{R}}}={\lambda }_{{\rm{R}}}({k}_{x}{\sigma }_{y}-{k}_{y}{\sigma }_{x}),{H}_{{\rm{D}}}={\lambda }_{{\rm{D}}}({k}_{x}{\sigma }_{x}-{k}_{y}{\sigma }_{y})$$
(1)

where k is wavevector; σ is the vector of Pauli spin matrices; λR and λD are the Rashba parameter and Dresselhaus parameter, respectively. The explicit forms of SOC terms in the Hamiltonian are restricted by the point group of crystal. For a 2D electron gas with S4 point group, the out-of-plane spin component Sz is not allowed under C2 operation within the linear expansion. Under S4 operation, there is transformation for the wave vector kx → ky and ky → −kx, while the spin vector does not change under the inversion operation and thus has the opposite sign as that of wave vector (i.e., σx → σy, σy → −σx). Similarly, S43 operation yields kx → −ky, ky → kx, σx → −σy, and σy → σx. It is seen that HR = kxσy − kyσx is variant under S4 and S43 operations, meaning the linear Rashba–Hamiltonian is forbidden. The Dresselhaus terms of both (kxσx − kyσy) and (kxσy + kyσx) can exist for the S4 point group. Therefore, the linear term in k vector SOC Hamiltonian for the chiral MXTe4 monolayers can be approximated as

$${H}_{{\rm{SO}}}={\lambda }_{{\rm{D}}}({k}_{x}{\sigma }_{x}-{k}_{y}{\sigma }_{y})+{\lambda }_{{\rm{D}}}\mbox{'}({k}_{x}{\sigma }_{y}+{k}_{y}{\sigma }_{x})$$
(2)
Table 1 Transformation of wave vector (k) and spin operators (σ) under symmetry operations of S4 point group.
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which yields the Dresselhaus-type spin splitting in these cluster-assembled 2D materials. The second term leads to a similar spin texture as that of the first term, with the pattern rotated by 45° counterclockwise (Supplementary Fig. 6). The ratio of two independent Dresselhaus parameters λD and λD’ determine the orientation of the spin texture.

Following the above symmetry analysis, we then examine the electronic and spin structures of MXTe4 monolayers by DFT calculations. We focused on the six semiconducting systems with dynamic stability (Supplementary Fig. 2) and clean Fermi surfaces that are desirable for spintronic applications, namely, MGeTe4 (M = Ni, Pd, and Pt) and MGaTe4 (M = Cu, Ag, and Au) monolayers. As presented in Fig. 3a and Supplementary Fig. 7, they are all indirect semiconductors and exhibit similar feature of band structures. The valence band maximum (VBM), which is dominated by the d orbital of transition metal atoms, locates at the corner (L point) of the square-shape Brillouin zone. The conduction band minimum (CBM) lies between Γ and L points or between A and B points, and is mainly contributed by the p orbital of Te atoms. Using PBE (HSE06) functional, the theoretical band gaps are 0.43–0.52 (0.71–0.95) eV for MGeTe4 (M = Ni, Pd, and Pt) monolayers and 0.92–1.08 (1.57–1.76) eV for MGaTe4 (M = Cu, Ag, and Au) monolayers, respectively (Table 2). Due to the lack of both mirror and inversion symmetries, applying SOC results in complex electronic band structures for chiral MXTe4 monolayers. They show strong spin polarization at the k points away from high-symmetry points, and the spin textures depend on the chirality of system. For the top valence band, the largest spin splitting in range of 0.09–0.18 eV occurs in the middle of L–A or L–B path. Remarkably, the CBM involves appreciable spin splitting of 0.13–0.22 eV. The SOC spin splitting in the assembled telluride compounds is considerably larger than that in selenides with the same chiral structures, and slightly depends on the transition metal element in the superatomic crystals (Supplementary Table 2). It is therefore experimentally feasible to tune the Fermi level by charge doping to access these strongly polarized spin states for transport phenomena.

Fig. 3: Spin polarization by the spin–orbit coupling in chiral MXTe4 monolayer.
figure 3

a Band structure of NiGeTe4 monolayer with (purple) and without SOC (blue). b Top valence band around L point (left panel) and bottom conduction band around Γ point (right panel) with projected expectation values of spin operator Sx (L-B, Γ-A) and Sy (L-A, Γ-B) with positive (red) and negative (blue) spin component. Bands for opposite handedness are shown in c. d Spin textures of top valence band and e bottom conduction band. f Spin textures of VBM around L point with an iso-energy of 0.27 eV below the Fermi level.

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Table 2 Tilt angle (θ), energy of chiral state relative to non-chiral one (ΔE), band gap (Eg) by PBE and HSE06 (in parentheses) functionals, SOC-induced energy splitting for top valence band (ΔVB) and bottom conduction band (ΔCB) for MXTe4 monolayers.
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To further understand the spin structure, we calculated the band structures including SOC with projected expectation values of the spin operator (Sx, Sy, and Sz) on the spinor wave functions (Fig. 3b, c). The red and blue colors of the spin-split branches represent spin-up and spin-down eigenstates at any in-plane wave vector k//, respectively44. The spin polarization in the entire Brillouin zone has a major contribution from the in-plane spin components Sx and Sy, much overwhelming the out-of-plane spin component Sz by symmetry (Supplementary Fig. 8). Taking the NiGeTe4 monolayer as a representative, its spin textures for the in-plane spin components in the kz = 0 plane are plotted in Fig. 3d, e. The spin configurations of the d orbitals of transition metal atoms and p orbitals of Te atoms are considered, as they dominate the top valence band and bottom conduction band, respectively. Noticeably, the spin textures exhibit typical linear Dresselhaus-type spin splitting. The angle between the k vector and spin depends on the direction of k. Around the Brillouin zone center, the spin is parallel to k along the kx and ky axes, and perpendicular to k along the diagonals (i.e., the A → B direction), predominantly behaving as the (kxσx − kyσy) form of spin texture. The second term of Eq. (2) has little contribution although it is allowed by symmetry. In particular, we examined the spin textures around the L point (at VBM) and Γ point (near CBM), as displayed in Fig. 3f and Supplementary Figs. 9 and 10, respectively. The iso-energy contour corresponds to a constant energy line across the top valence band at the L point with the energy of 0.28 eV below the Fermi level, and across the bottom conduction band at the Γ point with the energy of 0.29 and 0.36 eV above the Fermi level. For both cases, hyperbolic spin textures are evident. As the spin vector rotates along the iso-energy contour, the spin evolves from the radical direction to the azimuthal direction with respect to k vector in the kz = 0 plane. The spin is opposite for two energy branches, reflecting the change of spin direction with respect to the effective SOC field at a particular k point. Remarkably, the spin structures are chirality dependent. The spin points to the opposite directions for left-handed and right-handed MXTe4 monolayers, as seen from the band projection and iso-energy contour of spin-textures in Fig. 3b, c, f and Supplementary Fig. 9.

Spin Hall effect of chiral MXTe4 monolayers

In addition to the in-plane spin textures intrinsically hosted by the MXTe4 monolayers, spin current with persistent out-of-plane Sz component can be generated by applying an in-plane electric field through the spin Hall effect (SHE)45,46,47. There are two types of mechanism for SHE: the intrinsic mechanism concerns the spin-dependent variation of electron trajectories subject to the relativistic band structure, while the extrinsic mechanism is related to the electron scattering against spin–orbit-coupled impurities. Here we calculated the intrinsic spin Hall conductivity (SHC) by using the Kubo formula of linear response48:

$${\sigma }_{\alpha \beta }^{\gamma }=-\frac{{{\rm{e}}}^{2}}{{{\hbar }}}\frac{1}{V{N}_{k}^{3}}\sum _{k}{\Omega }_{\alpha \beta }^{\gamma }({\bf{k}})$$
(3)

where e is the electron charge, ħ is reduced Planck’s constant, V is volume of the primitive cell, and \({N}_{k}^{3}\) is the number of k points in the Brillouin zone. The k-resolved term of SHC is

$${\Omega }_{\alpha \beta }^{\gamma }({\bf{k}})=\sum _{n}{f}_{{nk}}{\Omega }_{n,\alpha \beta }^{\gamma }({\bf{k}})$$
(4)

where \({f}_{{nk}}\) is the Fermi–Dirac distribution function. The band-projected term \({\Omega }_{n,\alpha \beta }^{\gamma }\) (k) is given by

$${\Omega }_{n,\alpha \beta }^{\gamma }({\bf{k}})={\hslash }^{2}\mathop{\sum }\limits_{m\ne n}\frac{-2\,{\rm{Im}}\,[\langle n{\boldsymbol{k}}|\frac{1}{2}\{{\hat{\sigma }}_{\gamma },{\hat{\upsilon }}_{\alpha }\}|m{\boldsymbol{k}}\rangle \langle m{\boldsymbol{k}}|{\hat{\upsilon }}_{\beta }|n{\boldsymbol{k}}\rangle ]}{{({{\epsilon }}_{nk}-{{\epsilon }}_{mk})}^{2}}$$
(5)

where n and m are band indexes, \({\epsilon }_{n}\) and \({\epsilon }_{m}\) are the eigenvalues, \(\hat{\upsilon }\) and \(\frac{1}{2}\left\{{\hat{\sigma }}_{\gamma },{\hat{\upsilon }}_{\alpha }\right\}\) are the velocity and spin–current operators, respectively. The \({\Omega }_{n,\alpha \beta }^{\gamma }\) (k) term is sometimes called the spin Berry curvature, as it has a similar form as the Kubo-like formula for the Berry curvature49. The SHC \({\sigma }_{\alpha \beta }^{\gamma }\) is a third-order tensor that represents the spin current along the α direction generated by an electric field along the β direction, where the spin current is polarized along the γ direction.

The S4 point group of 2D MXTe4 monolayers is classified into the 4/m magnetic Laue group, which determines the shape of spin Hall conductivity tensor50. This symmetry restricts that four nonzero elements can exist and only the out-of-plane spin polarization can be generated by the in-plane charge current. The SHC in the left-handed and right-handed enantiometers are given by the form of

$${\sigma }_{L}^{z}=\left(\begin{array}{cc}{\sigma }_{xx}^{z} & {\sigma }_{xy}^{z}\\ -{\sigma }_{xy}^{z} & {\sigma }_{{\rm{xx}}}^{{\rm{z}}}\end{array}\right),{\sigma }_{R}^{z}=\left(\begin{array}{cc}-{\sigma }_{xx}^{z} & {\sigma }_{xy}^{z}\\ -{\sigma }_{xy}^{z} & -{\sigma }_{xx}^{z}\end{array}\right)$$
(6)

The two independent nonzero SHC tensor elements \({\sigma }_{{xx}}^{z}\) and \({\sigma }_{{xy}}^{z}\) mean that a longitudinal charge current can generate both transverse and longitudinal spin current51. The longitudinal spin current is reversed in the enantiometer with opposite chirality.

We calculated the intrinsic SHC of semiconducting MXTe4 monolayers using DFT, and plotted them as a function of Fermi energy shift (E − EF) in Figs. 4a and 5a. Consistent with our symmetry analysis, only \({\sigma }_{{xx}}^{z}\) and \({\sigma }_{{xy}}^{z}\) are non-zero with \({\sigma }_{{xy}}^{z}\) = 2.5–7.3 (ħ e−1) (S cm−1) and \({\sigma }_{{xx}}^{z}\) = 0.2–0.7 (ħ e−1) (S cm−1) at E = EF, comparable to the intrinsic values of typical semiconductors explored for spin Hall effect, i.e., below ~10 (ħ e−1) (S cm−1) for bulk Si, Ge, GaAs, AlAs, and 2D InSe without electronic doping52,53. Upon electronic doping, \({\sigma }_{{xy}}^{z}\) can achieve a maximal value of 28.2–35.8 (ħ e−1) (S cm−1). The mechanism for the enhanced SHC can be understood by the band structure projected by the spin Berry curvature \({\Omega }_{n,\alpha \beta }^{\gamma }\) (k) on a log scale in Fig. 4b. Large \({\Omega }_{n,\alpha \beta }^{\gamma }\) (k) occurs at certain bands near the L point indicated by the dark blue or red colors (which denotes a positive or negative contribution to the k-resolved term \({\Omega }_{\alpha \beta }^{\gamma }\), respectively). As SHC is the sum of spin Berry curvature over the occupied bands in the entire Brillouin zone, shifting the Fermi level can eliminate some negative contribution and retain the positive contribution from \({\Omega }_{\alpha \beta }^{\gamma }\) (k), thereby maximizing SHC through electronic doping (Fig. 4c, d and Supplementary Fig. 11).

Fig. 4: Spin Hall effect in chiral MXTe4 monolayer.
figure 4

a Components of SHC \({\sigma }_{{xy}}^{z}\) as a function of Fermi energy shift for MGeTe4 (M = Ni, Pd, or Pt) and MGaTe4 (M = Cu, Ag, or Au) monolayers. b Top panel: band structure of NiGeTe4 monolayer projected by spin Berry curvature \({\Omega }_{n,{xy}}^{z}\) (k); bottom panel: k-resolved \({\Omega }_{{xy}}^{z}\) (k) at E = EF (solid line) and E = EF + 0.9 eV (dashed line). c, d \({\Omega }_{{xy}}^{z}\) (k) in kz = 0 plane of Brillouin zone at E = EF and E = EF + 0.9 eV, respectively. L refers to the left-handedness.

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Fig. 5: Chirality-dependent charge-to-spin conversion.
figure 5

a Components of SHC \({\sigma }_{{xx}}^{z}\) as a function of Fermi energy shift for MGeTe4 (M = Ni, Pd, or Pt) monolayers. b Top panel: band structure of NiGeTe4 monolayer projected by spin Berry curvature \({\Omega }_{n,{xx}}^{z}\) (k); bottom panel: k-resolved \({\Omega }_{{xx}}^{z}\) (k) at E = EF (solid line) and E = EF + 0.9 eV (dashed line). c, d \({\Omega }_{{xx}}^{z}\) (k) in kz = 0 plane of Brillouin zone for left-handed and right-handed NiGeTe4 monolayers with E = EF + 0.90 eV, respectively. L and R refer to the left-handedness and right-handedness, respectively. e, f Schematic illustration of chirality-dependent spin Hall effect.

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The chirality-dependent SHC is revealed in Fig. 5. According to our DFT calculations, the SHC element \({\sigma }_{{xx}}^{z}\) has the same magnitude but different signs for the left-handed and right-handed MXTe4 enantiometers. The spin Berry curvature \({\Omega }_{{xx}}^{z}\) (k) in the Brillouin zone is antisymmetric and has opposite sign for different chirality, related by mirror reflection. Therefore, the spin current can be modulated by both electric field and geometrical handedness of MXTe4 monolayers. Indeed, recent experiments revealed chirality-dependent Edelstein effect and spin Hall effect in inorganic crystals of bulk Te and CrNb3S6, respectively21,54. Such chiral crystals are desirable for designing all-electrical solid-state devices to allow the generation and manipulation of electron spins without the need of a magnetic field. For the present MXTe4 monolayers, reversible transformation between different chirality is possible under external stimuli involving moderate energy barrier (Table 2), thus providing additional opportunity to couple multiple physical fields with geometrical handedness to control transport behavior.

In summary, we have exploited telluride tetrahedral clusters to assemble 2D ternary compounds MXTe4 (M = transition metal; X = Ga or Ge) with chiral geometry and strong SOC. With regard to atomic crystals, these superatomic crystals have the high tunability of symmetries, electronic and transport properties by changing the inter-cluster interaction within the assembled crystals. The tilting of tetrahedral clusters leads to chiral monolayer structures with high thermal and lattice dynamic stabilities. Taking six semiconductors as representatives, we demonstrate their Dresselhaus-type spin textures with dominant in-plane spin components imposed by the S4 point group, and anisotropic spin Hall effect with out-of-plane spin polarization and reversed spin current by the opposite chirality. Our results provide a general strategy for assembling hierarchical structures with tunable symmetry and designative electronic characteristics, to the access of exotic quantum effects hard to achieve by atomic crystals. Using the paradigm based on symmetry analysis and ab initio calculations, more chirality-governed physical quantities in different crystal lattices would be predicted and await to be exploited for diverse functional devices.

Methods

Theoretical calculations

DFT calculations were performed by the Vienna Ab initio Simulation Package (VASP)55, using the projector augmented-wave method, the planewave basis set with an energy cutoff of 500 eV, and the generalized gradient approximation (GGA) parameterized by Perdew, Burke and Ernzerhof (PBE) for the exchange-correlation functional56,57. The assembled 2D structures were fully relaxed for both cell and ionic degrees of freedom with the convergence criterion for energy and force of 10−5 eV and 0.01 eV/Å, respectively. A vacuum region of 20 Å was imposed in the out-of-plane direction. For structural optimization, the Brillouin zones were sampled by the Monkhorst–Pack k point grids with uniform spacing below 0.01 Å−1. Using the finite displacement method, the phonon spectra of MXTe4 monolayers were computed, as implemented in the PHONOPY package combined with VASP58. Based on PBE-optimized geometries, the Heyd–Scuseria–Ernzerhof (HSE06) hybrid functional was adopted for more accurate electronic band structure calculations59. DFT wave functions were transformed to maximally localized Wannier functions using the WANNIER90 package and the Kubo formula was employed to calculate the SHC48. A dense k mesh of 401 × 401 × 1 was employed to perform the Brillouin zone integration for the intrinsic SHC.