Tuneable topological domain wall states in engineered atomic chains

The study investigates how topological domain wall states can be engineered and tuned in one-dimensional chains built from localized electronic states. Tight-binding models capture essential physics via hopping between localized sites, and atomic manipulation with an STM enables assembling artificial materials with designed geometries and couplings. While the SSH dimer chain hosts a symmetry-protected mid-gap state at a domain wall between displaced unit cells, more complex unit cells (trimer chains) or chains with different on-site energies allow domain wall states whose energies are not fixed at mid-gap. The research question is whether one can realize atomically precise trimer and coupled dimer chains where domain wall states are deterministically created and their energies tuned by modifying local couplings. Using chlorine vacancies in the c(2 × 2)-Cl structure on Cu(100) as building blocks, the work aims to demonstrate tuneable, localized in-gap states at designed domain walls, with implications for realizing fractional charges and controllable topological interface modes in solid-state architectures.

Prior work has shown atomic manipulation can create designer quantum materials, quantum corrals, and artificial lattices (e.g., honeycomb, Lieb). The SSH dimer chain is the prototypical 1D topological model with protected zero-energy domain wall (edge) states, realized in solid-state atomic lattices and cold atom systems, and analogs in graphene nanoribbons. For more complex 1D systems (e.g., trimer lattices or chains with inequivalent on-site energies), theory predicts domain wall states at tunable, nonzero energies. Coupled dimer (double Peierls) chains self-assembled from indium on Si(111) exhibit four distinct topological bulk phases and interface states understood as chiral solitons; however, such systems have stochastic defects and domain walls with limited control. Atomic manipulation overcomes these limitations by enabling defect-free, predesigned domain walls, allowing explicit tests of tunability and topology in trimer and coupled dimer chains.

Artificial 1D chains are constructed from chlorine vacancies in the c(2 × 2)-Cl layer on Cu(100) using low-temperature STM atom manipulation. Sample preparation: Cu(100) cleaned by Ne+ sputtering (1.5 kV) and annealing (600 °C); Cl layer formed by depositing anhydrous CuCl2 from an effusion cell at 300 °C onto the crystal held at 150–200 °C for 180 s, followed by 10 min at the same temperature. Experiments: Performed at T = 4.2 K in a Unisoku USM-1300 STM. STM images acquired in constant-current mode; dI/dV spectroscopy by lock-in (20 mV p–p modulation at 709 Hz) in open-feedback. Line spectra acquired in constant height. Vacancy manipulation: Tip positioned above a Cl atom adjacent to a vacancy at 0.5 V, current increased to 1–2 µA with feedback on; tip dragged toward the vacancy up to 250 pm s−1 until a sharp z-jump, swapping the Cl atom and vacancy reliably. Tight-binding (TB) modeling: Finite chains with parameters matched to the vacancy system. Typical intratrimer/strong hopping t1 = 0.14 eV, intertrimer/weak hopping t2 = 0.04 eV. For trimer domain walls, the coupling to the domain wall site, t3, is tuned experimentally by adjusting nearest-neighbor distances, achieving t3 = 0.04, 0.07, and 0.14 eV. On-site energy and spectral broadening are extracted from single vacancy spectra: on-site energy 3.49 ± 0.01 V, Lorentzian half-width Δ = 0.18 ± 0.01 eV; spatial extent modeled by a Gaussian with FWHM σ = 0.71a (a is the c(2 × 2)-Cl lattice constant). Simulated local density of states (LDOS) maps are generated from the TB model and compared directly to experimental constant-height dI/dV maps across energy. For coupled dimer chains, domain walls between AA, AB, BA, and BB phases are constructed; TB band structures and wavefunctions are computed with several unit cells on each side of the interface to identify in-gap states and their spatial character.

- Trimer chains with domain walls: TB calculations for finite chains (300 unit cells per side) show localized in-gap states at trimer domain walls, with energies tunable by the coupling t3 to the domain wall site. In contrast to dimer (SSH) chains where the domain wall state is pinned to mid-gap, trimer domain wall states move within the gaps as t3 varies, remaining in-gap provided t3 is not much smaller than t2. - Experimental realization: STM-fabricated trimer chains exhibit localized domain wall states visible in dI/dV spectra at biases around 3.5 V. Spatially resolved constant-height dI/dV maps reveal domain wall localization consistent with TB predictions. Energy-resolved LDOS contrasts within unit cells evolve with bias toward the on-site energy, matching simulations. - Tunability via hopping: By adjusting nearest-neighbor distances, t3 is tuned experimentally between approximately 0.04, 0.07, and 0.14 eV (with t1 ≈ 0.14 eV, t2 ≈ 0.04 eV). Corresponding dI/dV maps near the in-gap state biases (e.g., ~3.45–3.48 V for weaker/intermediate couplings and ~3.38 V for strongest coupling) show excellent agreement with parameter-free simulations using the independently determined on-site energy and broadening (3.49 ± 0.01 V; Δ = 0.18 ± 0.01 eV; σ = 0.71a). - Fractional charge context: Trimer domain walls between chains shifted by one-third (two-thirds) of a unit cell support e/3 (2e/3) charge per spin when the lowest band is filled; these charges are topologically protected, while the energies of the associated states are movable within the gaps. - Coupled dimer chains: All four bulk phases (AA, AB, BA, BB) and multiple domain walls (e.g., AA → AB, AA → BA, AA → BB) are constructed. TB calculations show domain wall in-gap states whose number and energies depend on domain wall geometry and interchain coupling. For AA → BA, three domain wall–segment states are expected; with the experimental interchain coupling, the highest overlaps and hybridizes with a bulk band, leaving two in-gap states. For AA → BB, three in-gap states are found: two from bonding/antibonding combinations on the two-site domain wall and a third hybrid state with the middle band. Experimental dI/dV maps for AA → BA and AA → BB domain walls corroborate these predictions. - Overall, domain wall states are robust (cannot be removed by small perturbations) yet their energies traverse the gap depending on geometry and couplings, enabling deterministic, atomic-scale tuning.

The work addresses whether topological domain wall states in 1D chains with complex unit cells can be deterministically engineered and energetically tuned in a solid-state platform. By using chlorine vacancies on Cu(100) assembled with atomic precision, the authors realize trimer and coupled dimer chains with designed domain walls, demonstrating that the associated in-gap states are localized, robust, and their energies can be tuned via local hopping strengths (notably t3). The strong agreement between STM dI/dV maps and tight-binding simulations without free fitting parameters validates the model and shows controllable coupling between the domain wall segment and the bulk determines the energy placement within the gap. In coupled dimer chains, distinct domain walls that alter dimerization in one or both chains yield different numbers and energies of in-gap states, consistent with a hidden topological origin linked to pumping between configurations. These results establish an atomically precise, versatile platform for exploring topological interface physics beyond the SSH model, including the preparation and manipulation of fractional charges and the design of domain wall states with additional chirality degrees of freedom.

The study demonstrates atomically engineered trimer and coupled dimer chains with controllable domain wall states using chlorine vacancy lattices on Cu(100). In trimer chains, domain wall state energies are tunable within the gaps by adjusting the local coupling t3 between the domain wall segment and the bulk (with t1 ≈ 0.14 eV, t2 ≈ 0.04 eV, t3 varied from 0.04 to 0.14 eV). In coupled dimer chains, different domain wall geometries between AA, AB, BA, and BB phases produce distinct in-gap state counts and energies, reflecting a hidden topological origin and enabling additional control. The experimental STM spectroscopy and mapping are in excellent agreement with tight-binding predictions, enabling rational design of more complex structures. Future directions include automated atomic assembly to scale device complexity, direct exploration of fractional charges at domain walls as a function of chemical potential, realization of topological charge pumping, and integration into exotic quantum device architectures.

- Spectroscopic constraints: Higher-energy in-gap states in trimer chains are difficult to access because they lie near the conduction band of the chlorine layer, and spectral broadening (Lorentzian HWHM ~0.18 eV) hampers precise extraction of state energies directly from dI/dV. - Indirect validation: Energetic tunability is demonstrated primarily via qualitative and spatial correspondence between experimental dI/dV maps and tight-binding LDOS simulations rather than precise peak energy tracking for all states. - Coupled chains: Strong interchain coupling complicates simple mapping to decoupled dimer models; some domain wall states hybridize with bulk bands, reducing observability. - Scope: Fractional charge states and topological charge pumping are discussed as prospective properties; direct experimental detection/manipulation of fractional charge and pumping was not performed here.