Frustrated magnetic interactions in a cyclacene crystal

Magnetism and strongly correlated phases have been traditionally alien to the realm of graphitic crystals and polycyclic aromatic hydrocarbons (PAH). Thus, graphite, graphene and carbon nanotubes feature wide bands with extended states (small or vanishing density of states at the Fermi energy) so that the effect of Coulomb interactions in these systems are mostly negligible. In spite of this, strong correlations and magnetism were predicted in graphene zigzag edgesnakada1996 ; fujita1996 ; JFR07 and radical PAHmorita11 , but in both cases their strong reactivity is expected to compromise their chemical stability.

However, recent experimental breakthroughs have significantly changed this scenario. First, the discovery of strongly correlated electronic phases in twisted bilayer graphenecao2018correlated ; cao2018unconventional has shown that narrow bands are indeed possible in suitably designed carbon based 2D materials. Second, the on-surface assisted synthesis of multiradical PAHs, that remain stable in ultrahigh vacuum, and the study of their electronic excitations using scanning tunneling microcope spectrocopy (STS) have significantly paved the way towards the fine control of their synthesis. This includes triangulenespavlivcek2017synthesis ; mishra2019synthesis ; mishra2020 ; mishra2021 and the Clar’s Gobletmishra19b , among others mishra19b ; mishra2020 . Therefore, using STSortiz2020probing to probe the fluctuating local moments, from PAH with open-shell singletsmishra19b ; mishra2020 ; mishra2021 , is a firmly stablished tool for this kind of systems.

An ingredient common to both twisted bilayer and radical PAHs is the presence of localized electronic states close to the Fermi energy that host the strongly correlated electrons. This leads us to propose here a bottom-up route to engineer strongly correlated carbon based 2D crystals. Our approach is to use as building blocks radical molecules that can assemble in such a way that intermolecular interactions lead to a weak hybridization of those molecular states hosting the unpaired electrons. In-plane assembly of radical PAH leads, with some exceptionsmishra2020 , to strong covalent bonding of the molecular states, compromising the ultimate goal of preserving the localized molecular levels.

Here we consider cyclacenes with diradical character as building blocks for a molecular crystal (see Figure 1) with a weak intermolecular interaction, so that the diradical character can be preserved in the crystal phase. These aromatic hydrocarbon nanobeltsguo2021 are very appealing molecules subject of strong interest, both from theory and experiments alike. In the gas phase, the local moments are hosted by molecular orbitals formed by $\pi$ atomic orbitals, that are topological symmetry protected zero modeskoshino2014topological . A natural configuration for their self-assembly is a closed packing crystal, where the separation between molecules is in the range of the interlayer distance in graphite. We find this preserves the local moments of the isolated molecules in the crystal phase. Importantly, the resulting organization in this idealized cyclacene crystal yields a network of states that does not live in a bipartite lattice. This opens the door, as we will show next, to frustrated antiferromagnetic interactions that are known to promote quantum spin liquid statesANDERSON1973153 ; Balents2010spin , and thus different from the widely studied broken symmetry molecular magnetismgatteschi2006 and from the Lieb theorem paradigm Lieb89 ; JFR07 ; ortiz19 ; mishra2021 .

Cyclacenes, hereafter [$n$]CC’s, with $n$ denoting the number of fused benzene rings, can be thought of as the shortest possible carbon nanotubes with zigzag edges. The electronic properties of [$n$]CC’s depend on the parity of $n$, showing an odd-even patternchoi1999structures ; chen2007open ; wu2016electronic . This is confirmed at several levels of theory, where a simple one-orbital per atom tight-binding model yields two zero energy states for [even]CC’s and four states split in two pairs for [odd]CC’sperez2019cyclic . This provides a simple picture to understand the di/tetraradical character obtained by more sophisticated calculationsperez2018role . On the experimental side, besides the multiple efforts to synthesize carbon nanobeltspovie2017synthesis ; povie2018synthesis ; chisynthesis ; cyclaceneDiego ; jacsZhang2020 ; shudo2020synthesis ; jacsShi2020 , the obtention of pristine cyclacenes is still missing.

In figure 1a we show the structure of [6]CC. We can think of this molecule as two carbon rings with a strong covalent coupling through the non-hydrogenated carbons (NHC). Their tight-binding energy spectra presents two zero energy states localized on the hydrogenated carbon (HC) sites. In this work, we consider a two dimensional crystal of these [6]CC molecules, as if they had gone through a process of self-assembly over a surfacexu1998situ ; bain1988molecular ; fukui2017control (hereafter [n]crystacene, see figure 1b). We assume that the molecules assemble in a triangular lattice, with the axis of the tubes perpendicular to the plane of the crystal. This idealized (monolayer) self-assembly agrees with that recently found for closely related systems, as cycloparaphenylene nanoringsperez2019structure . In order to characterize the orientational order of the crystal, we define the relative angle $\theta$ formed between the line joining the geometric center of the molecule and the projection of the hydrogenated carbon atoms on the plane containing the center. This angle is determined by minimization of the ground state energy obtained using DFT for different magnetic phases.

In the following we address the study of the electronic structure of the [6]crystacene, focusing on the emergence of molecular local moments, the nature of intermolecular exchange, and their interplay with the rotational angle $\theta$.

We describe the electronic structure of cyclacenes with three different levels of theory. Individual molecules are described with a Hubbard model with a single orbital per carbon site. The model is treated both in the mean-field approximationJFR07 and using multi-configurational methods in a complete active space (CAS) with 6 single particle states and 6 electronsortiz19 ; ortiz2020probing . The molecular crystal is treated using density functional theory(DFT) calculations, carried out with Quantum Espresso (QE) codegiannozzi2009quantum ; giannozzi2017advanced . We use a Perdew-Burke-Ernzerhof (PBE)perdew1996generalized ultrasoft non-relativistic density functional for both carbon and hydrogen atoms. The van der Waals interactions, when considered, were implemented with the ’vdw-DF’ flagthonhauser2015spin ; thonhauser2007prb ; berland2015van . The kinetic energy cut-off considered for wavefunctions was 30Ry, for the charge density and potential it was 700Ry and we employed a k-grid of 10x10x1.

We consider 5 magnetic states for the crystal: a spin-unpolarized non-magnetic state (NM), ferromagnetic (FM), and three states with intramolecular antiferromagnetic correlations that differ by the intermolecular spin order: AF1, with ferromagnetic intermolecular correlations; AF2, a stripe phase with two antiferromagnetically coupled molecules per unit cell; and AF120, with non-collinear magnetic order where first neighbour spins are misaligned by $120^{\circ}$, and 3 molecules per unit cell (see figure 3a,b,c,d,e).

In the case of NM, AF1 and AF2, we have explored the $\theta$ depedence of the ground state energy, using the atomic coordinates obtained upon relaxation of the unit cell at $\theta=0^{\circ}$ for these three magnetic configurations; this geometry was further employed for any angle $\theta$ and for the band structures. We have also computed the FM phase for two different angles $\theta=0^{\circ}$ and $\theta=18^{\circ}$ using the same relaxed coordinates as those employed for the AF120 phase, as we will explain in the following. The results for this ferromagnetic phases are in table 1.

In the case of the AF120 phase, the unit cell has 3 molecules. This makes the computational cost of the structural relaxation prohibitively large. We use instead atomic configurations obtained with smaller unit cells and we only consider $\theta=0^{\circ}$. We have tried two different structures and, in both cases, the magnetic ground state with 3 molecules in the unit cell converges to the 120 phase. Below we show results obtained using the atomic coordinates obtained from the relaxation of the AF1 phase with $\theta=-20^{\circ}$, with the lattice constants $|\vec{a}|,|\vec{b}|=14.29$Å (which corresponds to a center-to-center intermolecular distance of 8.25Å ) : the approximate position of the energy minimum in the E vs $|\vec{a}|,|\vec{b}|$ curve for the frustrated AF2 phase, see suppl. mat.). No vdW interactions were considered for the AF120 phase. For the NM and AF2 phases we employed a Marzari-Vanderbilt smearing with a gaussian spreading of $10^{-4}$Ry, whilst no smearing at all was used for the AF1 or AF120 phases. ²²2We have verified that AF1 phase’s energies do not depend on the smearing, that was introduced to improve the convergence of the calculations in the neighbourhood of $\theta=0^{\circ}$.

In order to characterize the magnetic state of the crystals, we use two indicators:

$$M_{\rm ring}^{tot}=\sum_{i\in{\rm ring}}m_{i}$$

(1)

and

$$M_{\rm ring}^{abs}=\sum_{i\in{\rm ring}}|m_{i}|$$

(2)

where $m_{i}$ is the length of the atomic magnetic moments, as obtained from the output of our DFT calculations. The use of these two indicators is motivated by the fact that, in a given ring, there is a small residual magnetization in the non-hydrogenated carbon atoms, typical of the broken symmetry solutions in bipartite systemsJFR07 . As a reference, the absolute magnetic moment per ring in the single molecule mean-field calculation is $M_{\rm ring}^{abs}\sim 1.6\mu_{B}$ for the FM solution, to be compared with $M_{\rm ring}^{tot}=1\mu_{B}$.

The calculation of the atomic magnetic moments, carried out by Quantum Espresso by integrating the magnetic density over a sphere centered around the atom, turns out to be non-reliable in this system. For instance, let us consider the FM phase in a strongly insulating case where the total moment per CC can be determined by counting the number of occupied bands per spin channel, and is unambiguously determined to be 2$\mu_{B}$. We can now obtain a lower limit for the magnetic moment per hydrogenated carbon atom using the following argument. The calculation of the magnetic density, shown in figure 4d, shows that non hydrogenated carbon atoms have a residual magnetization opposite to the dominant contribution of the hydrogenated carbon. We thus have $6(|m_{\rm HC}|-|m_{\rm NHC}|)=1\mu_{B}$, the magnetization per ring. However, the values given by QE for this system is $6(|m_{\rm HC}|-|m_{\rm NHC}|)=0.43\mu_{B}$ which gives a good account of the unreliability of the integration method. We attribute the source of this discrepancy to the poor matching between a sphere and the $\pi$ shape electronic cloud that host the local moments (see figure 4).

For the phases with vanishing total spin per unit cell, we cannot rely on the use of the total moment to assess the error of the integration method. For this reason, in table 1 we choose to normalize the values of the atomic moments obtained by QE for all phases to the ones obtained for the FM phase with $\theta=18^{\circ}$. This local moments can be considered as an upper limit for the magnetization per ring. We thus define:

$$\tilde{M}_{\rm ring}^{tot}=\frac{1}{M_{\rm ring}^{tot}(\rm{FM,\theta=18^{\circ}})}\sum_{i\in{\rm ring}}m_{i}$$

(3)

and analogously for the absolute magnetization.

We have studied different magnetic configurations from a molecular crystal consisting in a two-dimensional triangular array of cyclacenes. In the gas phase, we have explored the diradical nature of the molecules using both multiconfigurational and mean-field calculations for the Hubbard model. We have shown that the diradical nature of the molecules has a topological origin, that leads to an open-shell $S=0$ singlet formed by two antiferomagnetically coupled electrons localized at the top and bottom ring of the molecule. We have studied the fate of these local moments in the crystal phase, using density functional theory. We have explored both the relative orientation of the molecules with respect to the crystal lattice vectors, characterized by the angle $\theta$, and we have considered four different magnetic states: NM and four types of magnetic order, FM, AF1, AF2, AF120. We find a strong dependence of the electronic structure with $\theta$. For all angles, the AF2 magnetic solution has lower energy than the NM case, showing that the antiferromagnetically correlated local moments of the gas phase survive in the crystal. We find that the ground state occurs for the AF120 phase, for $\theta=0^{\circ}$, which indicates antiferromagnetic intermolecular interactions. Our results portray the crystacene crystal as a versatile platform to promote strongly correlated phases, akin to the twisted bilayer graphene.

We acknowledge J. L. Lado and A. Perez Guardiola for fruitful discussions. We ackowledge for funding support to Ministry of Science and Innovation of Spain (grant numbers PID2019-106114GB-I00 and PID2019-109539GB), Generalitat Valenciana and Fondo Social Europeo (grant number ACIF/2018/175) and FEDER/Junta de Andalucía Consejería de Transformación Económica, Industria, Conocimiento y Universidades, grant PY18-4834.