Source of magnetic anisotropy in quasi-two-dimensional $XY$ $\{{\mathrm{Cu}}_4$(tetrenH${}_5$)W(CN)${}_8$]${}_4$$·$7.2H${}_2$O${\left)\right\}}_n$ bilayer molecular magnet

To identify the origin of the $XY$ spin dimensionality in the bilayered system $\{$Cu${}_4$(tetrenH${}_5$)[W(CN)${}_8$]${}_4·$7.2H${}_2$O${\}}_n$ (WCuT) we use a combination of single-crystal experiments (bulk magnetization, neutron flipping ratio, electron magnetic resonance, neutron diffraction) and theoretical modeling (exchange-charge model of the crystal field, dipolar energy, and density functional calculations). Our experiments show that the magnetic properties of WCuT are anisotropic and two-dimensional correlations build up below 70 K. The hard anisotropy axis is perpendicular to the layers ($b$ axis) and a small anisotropy within the $ac$ layers is present. Modeling of the crystal field validates treatment of tungsten and copper as spin $S=\frac{1}{2}$ ions with anisotropic $g$ values. The local magnetic anisotropy results from the common action of the crystal field and spin-orbit coupling and is along the $c$ axis for both ions. Density functional calculations identify the origin of the ferromagnetic exchange in different energies and symmetries of the tungsten- and copper-dominated orbitals and anticipate different exchange couplings across the apical (along the $b$ axis) and equatorial (in the $ac$ plane) Cu-CN-W bridges due to difference in the hybridization efficiency. Calculation of the dipolar energy for various spin configurations suggests that dipolar interactions play a decisive role in the $ac$-planar anisotropy in this system. We propose that the effective $XY$ spin dimensionality in WCuT is established by a combination of the axial local anisotropy of the W and Cu ions and the long-range magnetic dipolar interactions on the bilayered square lattice.

Figure 1

(Left) Arrangement of bilayers in WCuT (Ref. 13). The W atoms are shown by blue; Cu, orange; N, light blue; C, gray. The disordered cation layer is not shown. (Right) The square-grid pattern of a single layer.

Figure 2

(Left) Angular dependence of EMR (a) line shift and (b) linewidth (with respect to $B_0=\frac{h\nu }{g_0{\mu }_B}$, $\nu =239.2$ GHz) at 70 K and 45 K. Labels $cbc$ and $cac$ refer to the sense of field rotation relative to the principal crystallographic directions. (Right) Temperature dependence of EMR (c) line shift and (d) linewidth.

Figure 3

Frequency-field diagram measured at 45 K.

Figure 4

The $ab$ projection of magnetization density (gray isosurface) obtained from maximum entropy reconstruction of the FRs collected at 2 K. The color code of atoms of Fig. 1 is adopted.

Figure 5

Energy levels of the W${}^V$ and Cu${}^{II}$ ions calculated in the exchange-charge model of the CF with account of spin-orbital interaction.

Figure 6

Partial density of states (pDOS) of copper. Contribution of the $d_{y^2}$ orbitals is presented by the dark-green area, $d_{xz}$ is shown in bright blue, $d_{x^2-z^2}$ in yellow, $d_{xz}$ in blue, and $d_{yz}$ in white-red. (Insets) Magnified regions near the Fermi level with pDOS of Cu (top right) and of the equatorial nitrogen (bottom right). The orbitals of the constituting N (shown) and C (not shown) atoms contribute to the same bands. Contributions of the $s$ orbitals are shown in dark blue, $p_x$ in violet, $p_y$ in green, and $p_z$ in gray.

Figure 7

(Top) Partial density of states (pDOS) of tungsten. Magnified regions near the Fermi level with pDOS of W (top left) and of the apical nitrogen (bottom right). The color code of Fig. 6 is adopted.

Figure 8

Band structure of WCuT along the selected symmetry directions in the Brillouin zone: $\Gamma =\left(000\right)$, $X=\left(\frac{1}{2}00\right)$, $Y=\left(0\frac{1}{2}0\right)$, $Z=\left(00\frac{1}{2}\right)$, $D=\left(\frac{1}{2}0\frac{1}{2}\right)$. The majority and minority spin contributions are shown by black and red dotted lines, respectively.

Figure 9

The $ac$ projection of the single-layer $\Gamma$-point orbitals of the highest valence (left) and of the lowest conduction (right) bands. The color code of atoms of Fig. 1 is adopted. Green and brown isosurfaces correspond to two different signs of the orbitals.

Figure 10

The $ac$ (left) and $ab$ (right) projections of the DFT spin density presented as gray isosurface.

Figure 11

The energy of dipole-dipole interaction as a function of the angle $\varphi$ which shows the orientation of spins in the $ac$ plane relative to the $c$ direction in two cases: 1, with due account of the anisotropy of the $g$ factors for Cu and W ions; 2, the $g$ factors of both ions are isotropic.