Emergent Hierarchy in Localized States of Organic Quantum Chains

TL;DR

This work addresses how emergent hierarchical states arise in Organic Quantum Chains (OQCs) and how they govern electronic transport, especially when OQCs are coupled to quantum corrals. It combines density functional theory (DFT) with molecular dynamics (MD) relaxation and a tight-binding (TB) model with exponentially decaying hopping, calibrated to DFT, to accurately reproduce band structures and transport via Green's function formalism. A robust central energy gap of $\Delta E \approx 2.0$ eV persists across unit-cell variants, and hierarchical conductance plateaus emerge from the quasi-one-dimensional geometry; compact localized states (CLS) and narrowly confined channels are identified, along with bound states in the continuum (BIC) features in chain+corral systems, characterized by a high quality factor $\mathcal{Q}=\frac{|E_{res}|}{\Delta E_{res}}$. The results, consistent with Nature2021 experimental data, point to promising carbon-based nanoelectronic and sensing devices that exploit geometry-driven localization and interference.

Abstract

Organic Quantum Chains (OQCs) represent a newly synthesized class of carbon-based nanostructures whose quasi-one-dimensional nature gives rise to unconventional electronic and transport phenomena. Here we investigate the electronic and transport properties of recently synthesized OQCs [Nature Communications, 12, 5895 (2021)]. Structural stability was first assessed through molecular dynamics relaxation combined with density functional theory (DFT). The optimized coordinates are then used in a tight-binding model with exponentially decaying hopping parameterization, which reproduces the DFT results with high accuracy. Our calculations reveal a robust and nearly constant energy gap across several OQC configurations, in agreement with experimental data. We also identify emergent hierarchical states, characterized by distinct localization behaviors within sets of localized bands. Finally, we analyze different transport responses in scenarios involving the one-dimensional OQC coupled to carbon corrals, as observed in the experimental data, highlighting their potential as promising systems for application in carbon nanodevices.

Figures (6)

• Figure 1: (a) Experimental STM image highlighting distinct motifs: ring-like structures (red arrow) and chain segments (green arrow) [Adapted Nature2021]. (b) Molecular precursor used in the synthesis. (c) Schematic representation of the pairing process that connects two precursors. (d) Resulting N$_1$-OQC, representing the smallest unit cell chain segment. (e-m) Optimized geometries of N$_i$-OQCs for $i = 2$ to $10$, illustrating the structural evolution and increasing chain complexity along the periodic direction ($L_\infty$).
• Figure 2: Electronic band structure evolution of the OQC chains for N(1-10). Red curves are the TB results, black and blue the DFT calculations adopting HSE06 and PBE, respectively. The central energy gap is $\Delta E = \text{2.0}$ eV.
• Figure 3: (a) Conductance [$G(2e^2/h)$] and DOS (a.u.) hierarchy of several OQCs for an arbitrary energy range. A pair of plateaus are separated by a gray line. The gray dashed line separates the left and right leads. Plateaus length in eV for (b) Left and (c) Right panels corresponding to the plateaus positioned at left and right of the gray axis, labeled accordingly panel (a). An exponential decaying fit is displayed at the right of panels (b) and (c). Solid black, dashed gray and black, and solid gray lines, respectively, for 1-4 labeled plateaus panel (a).
• Figure 4: (a) Localization map for the $N_4$ OQC chain, plotting charge density ($|\psi|^2$) versus site position for all energy states, with detached zoom for energies $E_a$, $E_b$, $E_c$, and $E_d$. The lower panel highlights distinct localization behaviors inside the unit cell. (b) Electronic conductance $G$ (in units of $2e^2/h$) and density of states (DOS) compared with the electronic bands.
• Figure 5: (a) Left: Experimental dI/dV spectrum of the quantum ring, showing a gap of $\Delta E= 2.35$ eV (yellow). Right: Corresponding calculated density of states (DOS). [Adapted from Ref. Nature2021]. (b) Comparison of electronic properties for the molecule with and without the precursor. The precursor introduces states (blue) inside the original ring's pseudo-gaps.