Spin-orbit effects in heavy-atom organic radical ferromagnets

We discuss the effects of the spin-orbit interaction on heavy-atom organic magnets with specific reference to a series of isostructural sulfur- and selenium-based radical ferromagnets of tetragonal space group $P\overline{4}{2}_{1}m$. By using a perturbative approach, we show the spin-orbit effects lead to a pairwise anisotropic exchange interaction between neighboring radicals that provides an easy magnetic axis running parallel to the $c$-axis. Estimates of the magnitude of this magnetic anisotropy explain the significant increase in the coercive fields by virtue of selenium incorporation. Complementing this theoretical discussion are the results of ferromagnetic resonance studies, which provide an experimental verification of both the magnitude and symmetry of the spin-orbit terms. Taken as a whole, the results underscore the importance of heavy atoms and crystal symmetry in the design of molecular ferromagnets with large magnetic anisotropy and high ordering temperatures.

Figure 1

Crystal packing of 1 viewed (a) parallel and (b) perpendicular to the $\pi$-stacking direction, the crystallographic $c$-axis. Radicals 1–4 are isostructural. The positions of the crystallographic mirror planes are indicated in blue.

Figure 2

Splitting of the three singlet and triplet states arising from the two-site, two-orbital treatment of a lattice of interacting $S=\frac{1}{2}$ radicals. $U$ represents the onsite Coulomb repulsion, $t$ is the charge transfer or hopping integral, and $K$ is the electronic exchange integral. The singlet-triplet energy gap is defined such that the exchange coupling constant $\mathcal{J}=\Delta E_{\text{ST}}$. The notation $|a\overline{b}\rangle$ denotes a state with an $m_s=+\frac{1}{2}$ in orbital $a$, and $m_s=-\frac{1}{2}$ in orbital $b$.

Figure 3

(a) UB3LYP/6-311G(d,p) Kohn-Sham isosurface for the $A^{\prime \prime }$ SOMO of 1. (b) Schematic view of overlay of SOMOs on adjacent radicals along slipped $\pi$-stacks. (c) Near orthogonal overlap of neighboring SOMOs along the $\pi$-stack, leading to a small $t_{00}^{ij}$ and thus a positive ferromagnetic exchange interaction ${\mathcal{J}}^{\pi }$.

Figure 4

Spin-orbit splitting, for $C_s$ symmetry, of the triplet manifold $\left\{\mathcal{T}\right\}$ in the two-site two-orbital case (continued from Fig. 2). The product of the spin and orbital symmetry is $A^{\prime }$ for ${\mathcal{T}}_x$ and ${\mathcal{T}}_y$ and $A^{\prime \prime }$ for ${\mathcal{T}}_z$. The splitting stabilizes the triplet with respect to the open-shell singlet ${\mathcal{S}}^{\left(0\right)}$ and provides the source of magnetic anisotropy.

Figure 5

(a) Definition of local coordinate system of a single radical, viewed parallel to the [$\overline{1}$10] and [110] directions. (b) Relative orientation of local coordinates of four radicals in the unit cell, viewed parallel to the [110] direction. (c) Anisotropic exchange terms viewed parallel to the [001] direction for three unit cells. Dashed blue lines indicate both the crystallographic mirror planes and the local easy $xy$-planes (see Fig. 1). The relative orientation of local ${\mathbf{D}}_{ij}$ vectors is indicated by an arrow along the local $z$-axis.

Figure 6

Transmission spectra for 2 as a function of polar angle ${\theta }_H$ for $\omega =$ 73 GHz and $T=2$ K. The needle axis and crystallographic $c$-axis are approximately equivalent. The resonance for ${\mathbf{H}}_{\text{ext}}\left|\right|$ $c$-axis (${\theta }_H=0^{\circ }$) is sharp and consists of a single peak, whereas the resonance for ${\mathbf{H}}_{\text{ext}}\perp$ $c$-axis (${\theta }_H={90}^{\circ }$) is broad and multipeaked.

Figure 7

Transmission spectra for 2 as a function of azimuthal angle $\phi$ for ${\mathbf{H}}_{\text{ext}}\perp$ $c$-axis (${\theta }_H={90}^{\circ }$) and $\omega =127$ GHz. The average position of the resonance is constant, suggesting little anisotropy in the $ab$-plane, but a splitting is observed with period of $\Delta {\phi }_H={90}^{\circ }$. The relationship between the experimental ${\phi }_H$ angle and the crystal axes is unknown, although the results imply ${\phi }_H=0^{\circ }$ corresponds to ${\mathbf{H}}_{\text{ext}}\left|\right|$ $a$- or $b$-axis (see discussion).

Figure 8

Angular dependence of the resonance field $H_{\text{ext}}$ for 1 and 2 at high frequency. Where splitting of the resonance is observed, the average position of the two peaks is displayed. The black curves are a fit of Eq. (14). Data for 1 reproduced from Ref. 11.

Figure 9

Temperature dependence of the anisotropy field $H_A$ for 1 and 2. The ferromagnetic ordering temperatures $T_C$ $=$ 12.8, 17 K for 1 and 2, respectively, are indicated by an arrow. Data for 1 reproduced from Ref. 11.