Heisenberg and Dzyaloshinskii-Moriya interactions controlled by molecular packing in trinuclear organometallic clusters

Motivated by recent synthetic and theoretical progress we consider magnetism in crystals of multinuclear organometallic complexes. We calculate the Heisenberg symmetric exchange and the Dzyaloshinskii-Moriya antisymmetric exchange. We show how, in the absence of spin-orbit coupling, the interplay of electronic correlations and quantum interference leads to a quasi-one-dimensional effective spin model in a typical trinuclear complex, ${\mathrm{Mo}}_3{\mathrm{S}}_7$(dmit)${}_3$, despite its underlying three-dimensional band structure. We show that both intra- and intermolecular spin-orbit coupling can cause an effective Dzyaloshinskii-Moriya interaction. Furthermore, we show that even for an isolated pair of molecules the relative orientation of the molecules controls the nature of the Dzyaloshinskii-Moriya coupling. We show that interference effects also play a crucial role in determining the Dzyaloshinskii-Moriya interaction. Thus, we argue that multinuclear organometallic complexes represent an ideal platform to investigate the effects of Dzyaloshinskii-Moriya interactions on quantum magnets.

Figure 1

A Wannier orbital on a single ${\mathrm{Mo}}_3{\mathrm{S}}_7$(dmit)${}_3$ complex. The other two Wannier orbitals are related by the $C_3$ rotational symmetry of the molecule. Data from Ref. [26].

Figure 2

Sketches of (the tight-binding terms in) the effective Hamiltonians discussed in this paper. (a) A single molecule. The local $x$ and $y$ axes, defined by the phase convention for the SMOC [Eq. (3)], are shown. (b) Interlayer coupling model discussed in Sec. 3b; see particularly Eq. (21). (c) Inversion-symmetric interlayer coupling discussed in Sec. 3a; cf. Eq. (5). The inversion center is marked by the X. (d) $C_2^z$-symmetric interlayer coupling discussed in Sec. 3a3. The rotation axis is marked by the oval. Note that in panel (c) the numbering on both molecules runs counterclockwise, whereas in panel (d) the numbering is clockwise on the molecule labeled $\nu$.

Figure 3

Crystal structure of ${\mathrm{Mo}}_3{\mathrm{S}}_7$(dmit)${}_3$, after [14]. (a), (b) Triangular tubes perpendicular to the plane (along the $c$ axis). (c), (d) Truncated hexagonal net within a plane. (a), (c) The full molecular structure: with carbons black, sulphurs yellow, and molybdenums cyan. (b), (d) Simplified model showing only the Mo atoms, which are in a one-to-one correspondence to the Wannier orbitals; cf. Fig. 1. A unit cell is marked in all panels.

Figure 4

Comparison of the exact solution for a single molecule with the energy to first order in the SMOC. For small $J_c$ the spectrum is not strongly affected by the value of $J_c$; here we set $J_c=0.3t_c$ in all panels. The perturbation theory is most accurate for ${\lambda }_{xy}=0$. This is unsurprising because, at first order, there are corrections to the energies due to ${\lambda }_z$ but not ${\lambda }_{xy}$. This is natural because in the absence of SMOC Bloch's theorem requires that the single-molecule-energy eigenstates also be eigenstates of ${\widehat{L}}_{\mu }^z$.

Figure 5

Molecules packing with a rotation relative to one another leads to a DM coupling. (a) Initial arrangement with inversion symmetry and hence no DM interaction. The local axes of the $\nu \mathrm{th}$ molecule are marked (the $z$ axis is perpendicular to the page). (b) The $\nu \mathrm{th}$ molecule is rotated by an angle $\varphi$ about the $z$ axis, defining a new local coordinate system, $(x^{\prime },y^{\prime },z^{\prime })$. (c) The $\nu \mathrm{th}$ molecule is rotated by an angle $\theta$ about the $x^{\prime }$ axis, defining a new local coordinate system, $(x^{\prime \prime },y^{\prime \prime },z^{\prime \prime })$. To produce an arbitrary rotation $\mathbf{\vartheta }$, one must also complete another rotation about the $z^{\prime \prime }$, not shown here. Translations have a less complicated effect on the effective Hamiltonian (only via changes of the parameters of the microscopic Hamiltonian) and are therefore not shown in this figure.

Figure 6

Relative strengths of the effective DM and exchange interactions as a function of the interaction strength. Here we have set $J_c=4t_c^2/U, J_g=4t_g^2/U$, and $J_z=4t_z^2/U$ to reduce the parameter space. The ratio of the in-plane DM and Heisenberg exchange constants saturates to $D_0/{\mathcal{J}}_{\parallel }=\sqrt{2}({\lambda }_{xy}/t_c)[(19/33)+(92/121)(t_c/U)+O({(t_c/U)}^2)]$ in the strongly correlated regime ($U\to \infty$), whereas ${\mathcal{J}}_{\perp }/{\mathcal{J}}_{\parallel }={(t_z/t_g)}^2[(3/22)(U/t_c)+(324/121)+(702/1331)(t_c/U)+O({(t_c/U)}^2)]$ grows linearly.

Figure 7

Classical cartoon of a process that contributes to ${\mathcal{J}}_{\perp }$ in the strongly correlated limit, $J_c,J_z\to 0$ ($U\to \infty$). On the far left we sketch one of the states in the low-energy subspace, $|{\mathcal{S}}_{\text{upper}}^z=1,{\mathcal{S}}_{\text{lower}}^z=-1\rangle$, with holes marked by (yellow) arrows; the actual single-molecule eigenstates are linear superpositions of cyclic permutations of the states sketched here. One hole hops from the upper molecule to the lower molecule along the right-hand bond with an amplitude proportional to $t_z$. This leaves an intermediate state that is higher in energy than the initial state by $\mathrm{\Delta }E\equiv E_5^{2{\mathrm{\Gamma }}^{\prime }}(k_1,\sigma )+E_3^{(2S+1)\mathrm{\Gamma }}-2E_4$. For ${\lambda }_z={\lambda }_{xy}=0$ and $J_c\to 0$ we have $E_3^{(2S+1)\mathrm{\Gamma }}=0$ for all $S,\mathrm{\Gamma }$ and $E_5^{2{\mathrm{\Gamma }}^{\prime }}(k_1,\sigma )=2t_{c}cos(2\pi k_1/3)$ for all ${\mathrm{\Gamma }}^{\prime },\sigma$; hence $\mathrm{\Delta }E=2t_c[1+cos(2\pi k_1/3)]$. This intermediate excited state has an amplitude proportional to $t_z$ for the hole on the left-hand site to hop back to the top molecule giving the state $|0,0\rangle$, which is again part of the low-energy manifold. From this and similar processes one expects ${\mathcal{J}}_{\perp }\propto t_z^2/t_c$ in the strongly correlated limit, as we find explicitly; cf. Eq. (22). However, this classical picture is somewhat oversimplified because the holes are not localized in the true single-molecule eigenstates. Therefore a more complete treatment must include quantum mechanical interference, as described in the main text (Sec. 3b).

Figure 8

Classical cartoon illustrating the suppression of ${\mathcal{J}}_{\parallel }\to 0$ in the strongly correlated limit, $J_c,J_g\to 0$ ($U\to \infty$). Electrons can still hop between the two molecules, but no processes at second order can change the net spin on either molecule. Thus, there is a constant offset in the energy at second order, but the effective exchange coupling between molecules vanishes.

Figure 9

Effective interlayer DM and exchange interactions in ${\mathrm{Mo}}_3{\mathrm{S}}_7$(dmit)${}_3$ as a function of the interaction strength. Here we have set $J_c=4t_c^2/U$ and taken all other parameters from first-principles calculations [17, 26]. Note that the perturbation theory breaks down as $U\to 2t_c$.