Artificial relativistic molecules

The study addresses how to fabricate and stabilize artificial molecules with tailored electronic properties by exploiting relativistic effects and two-dimensional (2D) templates. Conventional atom-by-atom manipulation and self-assembly are limited by energetics and kinetics, making many target structures unfavorable. Templates such as step edges, 1D superstructures, and 2D moiré or porous networks can guide self-assembly, but the realization of true artificial molecules with designed bonding has remained elusive. The authors propose using the honeycomb charge-order superstructure at the surface of the van der Waals crystal IrTe2 to confine Pb adatoms into defined geometries, and leveraging strong spin–orbit coupling (SOC) in Pb to induce long-range interatomic bonding and relativistic (Dirac) molecular orbitals at unusually large interatomic distances. The purpose is to demonstrate the formation, spectroscopy, and theoretical understanding of such relativistic artificial molecules and to establish SOC as a key factor in both molecular orbital formation and stabilization on 2D templates.

Prior work has shown that artificial atomic chains and clusters on surfaces can host quantum confinement, topological edge modes, superlattice Dirac bands, flat bands, engineered spin interactions, and topological defects. Template-guided growth on step edges and 1D superstructures has produced anisotropic clusters, while 2D templates such as h-BN moiré, DNA-based frameworks, and metal–organic honeycomb networks demonstrated confinement of atoms and molecules. However, the formation of artificial molecules with defined molecular orbitals on such templates had not been reported. Relativistic effects in heavy-element molecules have mostly been investigated by spectroscopy and theory, with SOC known to modify bonding and electronic structure; yet direct, real-space observation of relativistic molecular orbitals is rare. Substrate-mediated long-range interactions in metal adatom systems are known but differ from the direct relativistic orbital overlap targeted here.

• Template and growth: Single-crystal IrTe2 was grown by Te flux. The surface exhibits a honeycomb charge-order superstructure below ~180 K with a period of ~2.2 nm. Samples were cleaved in UHV (~5×10^-10 torr) at room temperature. Pb atoms were deposited at room temperature on cleaved IrTe2 surfaces; coverages up to ~0.1 ML produce adatoms/clusters confined within the honeycomb cells.
• Scanning tunneling microscopy/spectroscopy (STM/STS): Measurements were performed in UHV at 4.3 K using a cryogenic STM (PtIr tips) in constant-current mode. Differential conductance dI/dV was recorded via lock-in detection at 1.17 kHz. Topographies and spatially resolved dI/dV maps were acquired at energies corresponding to spectral features to image molecular orbitals.
• Density-functional theory (DFT): Relativistic DFT calculations employed VASP with PBE-GGA and PAW potentials, including SOC. IrTe2 surfaces were modeled by (5×5) and (7×7) supercells with a single IrTe2 layer and ~23.6 Å vacuum. Calculations used a plane-wave cutoff of 211 eV and 15×15×1 k-point mesh for the 1×1 Brillouin zone. Only Pb adatoms were relaxed (forces <0.02 eV/Å) to avoid substrate distortion. To align with experiment, the Fermi level was shifted down by 0.70 eV to account for uncertainties from the unknown detailed charge-ordered structure, doping by Pb, and DFT band-energy inaccuracies. Adsorption energies and energy splittings were computed with and without SOC; charge-density visualizations and projected DOS were used to interpret molecular orbital character and substrate hybridization. Adsorption sites and energies for Pb on different Te hollow and on-top sites were evaluated.

• Confinement and cluster geometries: Pb adatoms occupy four-fold hollow sites surrounded by three Te atoms, selectively within honeycomb cells located above Ir4+ (5d4+) sites. Clusters from monomers to heptamers form within a single honeycomb, with nearest-neighbor Pb–Pb distances dictated by the substrate: either √3 a0 or 2 a0 (a0≈3.5 Å), i.e., ~6.1 Å or 7.0 Å. Benzene-like hexagonal rings (hexamers) and pentamers/heptamers composed of 2 a0 links were observed; 2 a0 dimers are more frequent than √3 a0 dimers.
• Monomer spectroscopy: STS of a Pb monomer shows three unoccupied peaks at ~1.40, 1.71, and 1.93 eV assigned to Pb 6p-derived states. DFT indicates partial ionization of Pb with 6p electrons transferred to the substrate, and SOC splits 6p into 6p1/2 and 6p3/2 (with further splitting of in-plane vs out-of-plane components), matching the three main features.
• Dimer molecular orbital splitting: For a short (2 a0) dimer, the bonding–antibonding splitting of the 6p1/2-derived state is ~0.23 eV in STS; for the longer (√3 a0) dimer it is ~0.10 eV. DFT predicts lowest molecular orbital splittings of ~0.18 eV (2 a0) and ~0.05 eV (√3 a0), slightly underestimating experiment by ~0.05 eV. Molecular orbital formation occurs at unusually large Pb–Pb separations (~7 Å), driven by direct overlap of relativistic p orbitals rather than substrate-mediated interactions.
• Dirac molecular orbital character: Using a Dirac basis, three relativistic p orbitals form g/u combinations; the lowest two peaks correspond to P1/2-derived g and u states. SOC induces significant mixing between σ and π components (e.g., g consisting of one-third σ-bond and two-thirds π-antibond character), and breaks mirror symmetry relative to the dimer axis via substrate coupling; these features are observed in dI/dV maps and reproduced in DFT charge densities.
• SOC-governed stability and optimal spacing: Calculations show that the attractive interaction between Pb adatoms and the stability of dimers and larger molecules arise only when SOC is included. The 2 a0 spacing is optimal, consistent with the higher population of 2 a0 clusters. Across trimers to heptamers, SOC stabilizes molecules with exclusively 2 a0 nearest-neighbor distances.
• Mechanism: Two SOC effects reduce ionic/dipole repulsion and enhance bonding: (i) SOC lowers Dirac orbital energies, reducing electron donation from Pb p states to the substrate; (ii) SOC enhances overlap of Pb–Te hybridized states. Substrate SOC and charge redistribution (Ir dxz to Te) further influence stability. Pb interacts substantially with Te orbitals, yielding extra energy gain in certain geometries (e.g., √3 a0 dimers’ P1/2 antibonding level).
• Benzene-like ring molecules: Hexamers show weak molecular splittings similar to 2 a0 dimers. dI/dV maps reveal (i) broken AB sublattice symmetry for the lowest state, (ii) a Kekulé-like distortion for the second state, and (iii) mirror-symmetry-broken two-fold bonding features for the third state. DFT reproduces spatial patterns though it underestimates splittings and predicts near-degenerate features around ~1.3 eV. Pentamers show Dirac molecular orbitals with edge states at truncated ends between bonding and antibonding levels, indicating circular electron delocalization. A filled-benzene-ring heptamer (hexamer with a central Pb) further enhances interatomic overlap via unique central bonding.
• Elemental suitability: Pb is ideal due to strong SOC and appropriate substrate coupling (contrasted with Sn or Tl in supplementary comparisons).

The work demonstrates that strong spin–orbit coupling, combined with a periodic 2D template on a van der Waals substrate, can drive the formation and stabilization of artificial molecules exhibiting relativistic (Dirac) molecular orbitals at unusually large interatomic separations. The observed bonding–antibonding splittings, spatial symmetry breaking, and ring-molecule characteristics directly address the hypothesis that SOC can both enhance interatomic interaction and reduce adatom–substrate charge transfer to overcome ionic repulsion. The substrate plays a crucial role: IrTe2’s charge order, Te orbital participation, and SOC-assisted charge redistribution synergize with Pb SOC to set the optimal 2 a0 spacing and stabilize larger assemblies. These findings expand the design space for artificial molecules beyond conventional covalent scales, enabling structures (e.g., benzene-like rings, open rings with edge states, filled rings) that are otherwise synthetically inaccessible. The approach offers a platform to couple relativistic molecular states to emergent substrate phenomena such as charge order and superconductivity, suggesting routes to engineered quantum systems with tailored spin and electronic textures.

The study realizes and characterizes artificial Pb molecules—from dimers to benzene-like hexamers and heptamers—templated by the honeycomb charge-order superstructure of IrTe2. SOC is essential for both molecular orbital formation and the energetic stabilization of these assemblies, yielding Dirac molecular orbitals with distinctive σ–π mixing, symmetry breaking, and long-range bonding at ~7 Å. The combination of STM/STS with relativistic DFT reveals how SOC reduces adatom–substrate charge transfer and enhances Pb–Te hybridization to favor 2 a0-spaced structures. This establishes a strategy for fabricating unprecedented relativistic molecules on 2D templates. Future directions include exploiting other 2D superstructures (e.g., twisted bilayer graphene moiré, domain-wall/twin-boundary networks in TMDs), integrating magnetic adatoms to harness proximity to novel 2D electronic states, tuning interatomic distances via atom-by-atom manipulation, and probing spin configurations of Dirac molecular orbitals.

• The DFT calculations underestimate molecular bonding interactions by roughly ~0.05 eV for dimers and predict near-degenerate spectral features for the benzene-like hexamer around ~1.3 eV, indicating quantitative discrepancies.
• The theoretical model does not fully capture the correlated, charge-ordered electronic structure of IrTe2, necessitating an empirical Fermi-level shift (−0.70 eV) to align with experiment.
• Spin textures and configurations of the Dirac molecular orbitals, while predicted, were not directly measured in the present experiments.
• The selectivity of Pb adsorption inside honeycombs above Ir4+ sites is observed but not fully understood mechanistically.
• Growth outcomes depend on deposition temperature and coverage; beyond ~0.1 ML, Pb may form continuous films or islands, potentially limiting scalability of isolated molecules.