Strongly entangled Quantum Spin Rings driven by Hückel rule

Abstract

Quantum spin rings represent an intriguing platform for studying unconventional magnetic order and exotic quantum phases, and they are also promising materials for emerging quantum technologies. Conventional spin systems consist of a set of weakly interacting localized spins that are well described by the Heisenberg spin models. Here, we demonstrate that strong interactions between radical centers in macrocycles of different sizes lead to fluctuations in the total number of unpaired electrons and to non-trivial antiferromagnetic order that extends beyond the Heisenberg picture. We demonstrate that the electronic structure of these spin rings is governed by the concept of 4n/4n+2 Hückel (anti)aromaticity for even-membered rings, whereas odd-membered rings possess a highly degenerate frustrated magnetic ground state. The strongly coupled spin rings are experimentally realized through the on-surface synthesis of π-magnetic carbon-based macrocycles, which consist of [2]triangulene units. The close correlation between the electronic structure and the Hückel aromaticity rule is revealed by scanning tunneling spectroscopy and multireference calculations. This work establishes a novel design principle employing the concept of Hückel aromaticity for quantum spin macrocycles.

Figures (11)

• Figure 1: Concept of Hückel spin rings.a, Spatial localization of the zero-energy radical mode $\varphi_{\mathrm{SOMO}}$ (left) and calculated spin density of $[2]$triangulene unit. b, Two connection motifs between two $[2]$triangulene units: weak coupling, where the two $\varphi_{\mathrm{SOMO}}$ do not overlap directly; and strong coupling, where the two $\varphi_{\mathrm{SOMO}}$ hybridize to form bonding and antibonding molecular orbitals. c, Antiferromagnetic Heisenberg rings: localized-spin nanorings made of $[2]$triangulene units, in which the connection through specific atomic sites results in weak hybridization. Their magnetic properties follow the Heisenberg spin model and are driven by local antiferromagnetic exchange rather than by the global $\pi$-topology. d, Hückel spin rings, in which $[2]$triangulene units are connected via polyynic C$_4$ linkers. This connectivity enforces strong hybridization, resulting in delocalized spins and magnetic properties governed by Hückel's $4n/4n+2$ rules. e, Atomic structures of Hückel spin rings with different numbers of $[2]$triangulene units and their corresponding one-electron energy spectra. Carbon atoms (highlighted in red) form an inner ring whose electron count determines the $4n$ (antiaromatic) or $4n+2$ (aromatic) character. f, Frost circles for the rings in (e), rationalizing the alternation of radical character with the number of $[2]$triangulene units.
• Figure 2: Design and synthesis of molecular Hückel spin rings.a, Schematic illustration of the synthetic pathway, involving surface-assisted dechlorination C--C coupling followed by STM tip-induced dehydrogenation. b, AFM images of molecular spin rings composed of alternating [2]triangulene units and diyne bridges. Scale bar: 1 nm. c, STS ($dI/dV$) spectra of macrocycles Hü-SR$^4$ to Hü-SR$^7$. Even-membered rings (Hü-SR$^4$ and Hü-SR$^6$) display symmetric step-like features characteristic of inelastic spin-flip excitations, whereas odd-membered rings (Hü-SR$^5$ and Hü-SR$^7$) exhibit zero-bias resonances. Scale bar: 1 nm.
• Figure 3: Magnetic properties of molecular spin rings.a, Lowest spin-excitation energies $\Delta E_{01}$ as a function of ring size $N$ obtained from experimental STS (black), CASCI calculations for Hü-SR$^N$ (red), and the Heisenberg spin ring (H-SR, blue) shown in Fig. \ref{['fig:fig1']}c. b, Total radical character $Y_d$ versus ring size $N$ for Hü-SR (red) and the Heisenberg spin ring (H-SR, blue). c, Comparison of experimental constant-current STS maps acquired at 90 meV with simulated STS maps computed from CASCI NTOs for the singlet--triplet excitation of Hü-SR$^4$. d, Comparison of experimental constant-current STS maps acquired at 147 meV with simulated STS maps computed from CASCI NTOs for the singlet--triplet excitation of Hü-SR$^6$. Scale bars: 1 nm.
• Figure 4: Frustrated magnetic order of odd-membered Hü-SR.a, Comparison of experimental constant-current STS maps acquired at $-5$ meV with simulated constant-height STS maps computed from CASCI Kondo orbitals associated with Kondo screening of the degenerate doublet ground states $\Psi^D_{0,1}$ of Hü-SR$^5$ on a metal surface. b, Comparison of experimental constant-current STS maps acquired at $-14$ meV with simulated constant-height STS maps computed from CASCI Kondo orbitals associated with Kondo screening of the degenerate doublet ground states $\Psi^D_{0,1}$ of Hü-SR$^7$ on a metal surface. c, Schematic representation of frustrated spin order in Hü-SR$^5$, showing competing local magnetic configurations with entangled singlets (dashed ovals, right). Energy-level scheme for odd-membered Hü-SR doubly degenerate doublet ground state $\Psi^D_{0,1}$ and first excited quartet states $\Psi^Q_{2,3}$, indicating magnetic frustration. d, Calculated CASCI Kondo orbitals associated with screening of the two degenerate doublet states $\Psi^D_{0,1}$ by the metallic substrate, highlighting their distinct spatial distributions. Scale bars: 1 nm.
• Figure Fig: S1 | Hückel spectra for different connections in a [2]triangulene dimer. Three distinct connection motifs between two [2]triangulene units, together with their corresponding Hückel spectra (two orbitals around zero energy) and sublattice imbalance: a, connection between non--zero-mode (non-ZM) sites, b, connection between zero-mode (ZM) sites, and c, connection between a zero-mode (ZM) site and a non-ZM site.