Frustrated magnetic interactions in a cyclacene crystal

R. Ortiz, J. C. Sancho-García, and J. Fernández-Rossier

Phys. Rev. Materials 6, 014406 – Published 19 January 2022

Abstract

We study the emergence of magnetism and its interplay with structural properties in a two-dimensional molecular crystal of cyclacenes, using density functional theory (DFT). Isolated cyclacenes with an even number of fused benzenes host two unpaired electrons in two topological protected zero modes, at the top and bottom carbon rings that form the molecule. We show that, in the gas phase, electron repulsion promotes an open-shell singlet with strong intramolecular antiferromagnetic exchange. We consider a closed packing triangular lattice crystal phase and we find a strong dependence of the band structure and magnetic interactions on the rotation angle of the cyclacenes with respect to the crystal lattice vectors. The orientational ground state maximizes the intermolecular hybridization, yet local moments survive. Intermolecular exchange is computed to be antiferromagnetic, and DFT predicts a broken symmetry $120^{\circ}$ spin phase reflecting the frustration of the intermolecular spin coupling. Thus, the cyclacene crystal realizes a bilayer of two antiferromagnetically coupled $S = 1 / 2$ triangular lattices. Our results provide a bottom-up route towards carbon based strongly correlated platforms in two dimensions.

Received 30 July 2021
Accepted 8 December 2021

DOI:https://doi.org/10.1103/PhysRevMaterials.6.014406

Physics Subject Headings (PhySH)

Frustrated magnetism Magnetic interactions Magnetic order

Carbon-based materials Nanostructures

Condensed Matter, Materials & Applied PhysicsInterdisciplinary Physics

Authors & Affiliations

R. Ortiz ^1,2, J. C. Sancho-García ², and J. Fernández-Rossier^3,*

¹Departamento de Física Aplicada, Universidad de Alicante, 03690 Sant Vicent del Raspeig, Spain
²Departamento de Química Física, Universidad de Alicante, 03690 Sant Vicent del Raspeig, Spain
³QuantaLab, International Iberian Nanotechnology Laboratory (INL), Av. Mestre José Veiga, 4715-330 Braga, Portugal

^*On leave from Departamento de Física Aplicada, Universidad de Alicante, Spain.

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Vol. 6, Iss. 1 — January 2022

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Images

Figure 1
(a) Chemical structure of [6]CC. (b) Molecular crystal lattice of [6]crystacene showing the rotational angle $θ$ . (c) and (d) display two cyclacenes with different relative angle $θ$ . Gray balls represent carbons and white ones represent hydrogens [22].
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Figure 2
Gas phase isolated molecule calculations. (a) Single-particle tight-binding spectra for [6]CC, calculated for the planar lattices of panels (c) and (d) with periodic boundary conditions. Note that, as we mention in the text, finite-temperature DFT calculations [27] hold the diradical character of [even]CC, so ignoring the ring curvature on this matter is justified. (b) Excitation energies computed with the CAS(6,6) approximation for the Hubbard model with $t = 2.7 eV$ . The ground state has $S = 0$ . The $S = 1$ state has $E_{S = 1} = 54$ meV for $U = t$ . We take $t = 2.7$ eV, which has been widely used in the literature [47] and is very close to the value inferred from angle-resolved photoemission experiments in graphene [48]. (c),(d) Expectation value of spin density $S_{z} (i)$ , calculated in the mean-field approximation of the Hubbard model with $U = t$ for two magnetic configurations, with total $S_{z} = 0$ (AF) and total $S_{z} = 1$ (FM). The AF is the ground state with $E_{FM} - E_{AF} = 43.3$ meV for $U = t$ and $t = 2.7 eV$ in the mean field, not far from the CAS(6,6) result.
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Figure 3
(a) Calculated local moments of the AF2 collinear phase from a side perspective when $θ = 0^{\circ}$ and (b) AF120 phase. Hydrogens were omitted for clarity. The local moment per hydrogenated carbon is $\approx 0.04 μ_{B}$ in both phases. (c), (d), and (e) are the schematic representations of the AF1, AF2, and AF120 phases (see text). (f) Ground state energy as a function of $θ$ , for the NM, AF1, and AF2 phases, referred to the value for the AF2 phase and $θ = 0^{\circ}$ . For this calculation we employed the relaxed geometries and lattice vectors at $θ = 0^{\circ}$ for each magnetic configuration. Angle steps of $Δ θ = 1^{\circ}$ were taken to calculate the curve shown.
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Figure 4
Plot of the isosurfaces with the same isovalue [22] of the magnetic moments calculated with DFT for two different angles for the magnetic phases called in the text as (a),(c) AF1 with $θ = 0^{\circ}$ and $θ = 18^{\circ}$ , respectively, and (b),(d) FM with $θ = 0^{\circ}$ and $θ = 18^{\circ}$ , respectively. The color stands for the sign.
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Figure 5
(a) Band structure for AF1 phase and $θ = 18^{\circ}$ . (b),(c),(d) wannier three-dimensional representations of the three bands in (a). Panels (b) and (c) corresponding to the bottom and middle bands match the zero-energy states from the single-molecule system. Panel (d) corresponds to the top band, which shows $σ$ character instead of $π$ .
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Figure 6
Calculated band structure for the four magnetic configurations considered in [6]crystacene: (a) NM, (b) AF1, (c) AF2, and (d) AF120. $θ = 0^{\circ}$ . The geometries are the same as those used for Fig. 3. Inset in (c) is the oblique lattice's first Brillouin zone with the high-symmetry points. For the collinear phases, the spin-up and spin-down bands are degenerate.
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