Spatially anisotropic $S=1$ square-lattice antiferromagnet with single-ion anisotropy realized in a Ni(II) pyrazine-$n,n^{\prime }$-dioxide coordination polymer

The $\mathrm{Ni}{\left(\mathrm{NCS}\right)}_2{\left(\mathrm{pyzdo}\right)}_2$ coordination polymer is found to be an $S=1$ spatially anisotropic square lattice with easy-axis single-ion anisotropy. This conclusion is based upon considering in concert the experimental probes x-ray diffraction, magnetic susceptibility, magnetic-field-dependent heat capacity, muon-spin relaxation, neutron diffraction, neutron spectroscopy, and pulsed-field magnetization. Long-range antiferromagnetic (AFM) order develops at $T_N=18.5\mathrm{K}$. Although the samples are polycrystalline, there is an observable spin-flop transition and saturation of the magnetization at $\approx 80\mathrm{T}$. Linear spin-wave theory yields spatially anisotropic exchanges within an AFM square lattice, $J_x=0.235\mathrm{meV}$, $J_y=2.014\mathrm{meV}$, and an easy-axis single-ion anisotropy $D=-1.622\mathrm{meV}$ (after renormalization). The anisotropy of the exchanges is supported by density functional theory.

Figure 1

Crystal structure of $\mathrm{Ni}{\left(\mathrm{NCS}\right)}_2{\left(\mathrm{pyzdo}\right)}_2$. (a) A section of one layer of nickel ions bridged by pyzdo ligands. (b) Stacking of the layers. Molecular units of (c) ${\mathrm{NiO}}_4{\mathrm{N}}_2$ octahedra, (d) NCS, and (e) pyzdo.

Figure 2

Magnetic susceptibility of $\mathrm{Ni}{\left(\mathrm{NCS}\right)}_2{\left(\mathrm{pyzdo}\right)}_2$ measured in 0.1 T. (a) Data and Lines [29] model fit described in the text. (b) Inverse susceptibility data and Curie-Weiss law fit described in the text. (c) Product of susceptibility and temperature with derivative inset.

Figure 3

$\mathrm{Ni}{\left(\mathrm{NCS}\right)}_2{\left(\mathrm{pyzdo}\right)}_2$ heat capacity. Heat capacity divided by temperature with the magnetic field dependence inset.

Figure 4

Muon-spin relaxation of $\mathrm{Ni}{\left(\mathrm{NCS}\right)}_2{\left(\mathrm{pyzdo}\right)}_2$. (a) Example asymmetry spectra, $A$($t$), measured at 9 K and 19 K. The 9 K data are shown in percent; the 19 K data have been displaced upward by 4% for clarity. (b) Temperature dependence of the muon precession frequency. (c) Relaxation rate as a function of temperature. In (c), the line is a guide to the eye.

Figure 5

Neutron diffraction of $\mathrm{Ni}{\left(\mathrm{NCS}\right)}_2{\left(\mathrm{pyzdo}\right)}_2$ at $T=25\mathrm{K}$ in the paramagnetic state. Data are red circles, the model is a black line, vertical marks are Bragg peak positions, and residuals are an offset blue line.

Figure 6

Neutron diffraction of $\mathrm{Ni}{\left(\mathrm{NCS}\right)}_2{\left(\mathrm{pyzdo}\right)}_2$ at $T=1.5\mathrm{K}$ in the magnetically ordered state. Data are red circles, the model is a black line, vertical marks are Bragg peak positions, and residuals are an offset blue line. Data are shown over (a) a region of interest that highlights the magnetic Bragg peaks and vertical marks for magnetic Bragg peak locations and (b) a large range of $d$ spacing with upper vertical marks for structural Bragg peaks and lower vertical marks for magnetic Bragg peaks.

Figure 7

Visualization of magnetic structure from neutron diffraction of $\mathrm{Ni}{\left(\mathrm{NCS}\right)}_2{\left(\mathrm{pyzdo}\right)}_2$. This image shows a section of one layer of nickel ions bridged by pyzdo ligands overlayed with magnetic moment vectors.

Figure 8

Ordered magnetic moment vs temperature from neutron diffraction for $\mathrm{Ni}{\left(\mathrm{NCS}\right)}_2{\left(\mathrm{pyzdo}\right)}_2$. Uncertainties are from counting statistics.

Figure 9

Superexchange pathways in $\mathrm{Ni}{\left(\mathrm{NCS}\right)}_2{\left(\mathrm{pyzdo}\right)}_2$. (a) The $2\times 2\times 2$ unit cell containing 8 nickel atoms as white circles with numbered labels for the density functional theory (DFT) magnetic structure calculations. Views along (b) $a$ axis, (c) $b$ axis, and (d) $c$ axis are shown with only nickel atoms shown as white circles but with all atom-atom bonds shown. The thick blue solid line pyzdo connections are along the $J_5$ superexchange, the medium thickness green solid line pyzdo connections are along the $J_3$ superexchange, and the NCS bonds are thin gray solid lines. For (b)–(d), individual unit cells within the $2\times 2\times 2$ unit cell are shown.

Figure 10

Neutron spectroscopy of $\mathrm{Ni}{\left(\mathrm{NCS}\right)}_2{\left(\mathrm{pyzdo}\right)}_2$. These polycrystalline data are averaged from momentum transfers of 1 to $3{\AA }^{-1}$. Models are as described in the text. The model 1 dashed red line is barely visible $>5.7\mathrm{meV}$, as it is nearly identical to models 2 and 4 in that region.

Figure 11

Comparison of linear spin-wave theory (LSWT) models for $\mathrm{Ni}{\left(\mathrm{NCS}\right)}_2{\left(\mathrm{pyzdo}\right)}_2$. The momentum dependence of (a) energy transfer and (b) intensity are shown for the models described in the text.

Figure 12

Mean-field model of magnetization vs field for $T=0.5\mathrm{K}$ using neutron spectroscopy and magnetic susceptibility derived parameters for $\mathrm{Ni}{\left(\mathrm{NCS}\right)}_2{\left(\mathrm{pyzdo}\right)}_2$. Ten different orientations of the applied field with respect to the anisotropy axis are shown. (inset) Directions of magnetic fields for magnetization calculations are shown. The anisotropy axis is along the $z$ direction, and the dash spacings of the unit vectors correspond to the dash spacings of the lines for the expectation value of the magnetization along the field axis, $\langle S_{\mathrm{Ni}}\parallel \mathbf{B}\rangle$, shown in the main plot. The solid line is nearly $\mathbf{B}\left|\right|\mathbf{D}$, while the dashed line with the largest spacing is nearly $\mathbf{B}\perp \mathbf{D}$.

Figure 13

Isothermal pulsed-field magnetization of $\mathrm{Ni}{\left(\mathrm{NCS}\right)}_2{\left(\mathrm{pyzdo}\right)}_2$ at $T=0.5\mathrm{K}$ and models. Models are as described in the text. The (a) derivative of magnetization with respect to field and (b) the magnetization normalized to the saturation magnetization are shown.

Figure 14

Dimer models of $\mathrm{Ni}{\left(\mathrm{NCS}\right)}_2{\left(\mathrm{pyzdo}\right)}_2$. (a) Along the $J_{5}a$, $-b$ direction. (b) Along the $J_{3}c$ direction. (c) and (d) Views of the spin density distribution for the broken symmetry state of the dimer models. Isosurface value $\pm 0.001$.

Figure 15

Comparison of different models for the $\mathrm{Ni}{\left(\mathrm{NCS}\right)}_2{\left(\mathrm{pyzdo}\right)}_2$ thermal phase transition, details described in the text. (inset) Temperature dependence of the linear spin-wave theory (LSWT) correction to the ordered moment.

Figure 16

$\mathrm{Ni}{\left(\mathrm{NCS}\right)}_2{\left(\mathrm{pyzdo}\right)}_2$ phase diagram.

Figure 17

Intensity maps of $\mathrm{Ni}{\left(\mathrm{NCS}\right)}_2{\left(\mathrm{pyzdo}\right)}_2$ experimental neutron spectroscopy data. (a) $Ei=12\mathrm{meV}$, $T=2\mathrm{K}-T=22\mathrm{K}$ data. (b) $Ei=6.59\mathrm{meV}$, $T=2\mathrm{K}-T=22\mathrm{K}$ data. (c) $Ei=3.32\mathrm{meV}$, $T=2\mathrm{K}$ data. (d) $Ei=1.55\mathrm{meV}$, $T=2\mathrm{K}$ data. White regions are where the scattering condition is not satisfied by the spectrometer.

Figure 18

Spin-spin correlations from density matrix renormalization group (DMRG) for (a)–(c) isolated chains and (g)–(i) chains interacting via a mean-field as described in the text. For (a)–(c), the region $>4J$ was not calculated. Fits of the DMRG to linear spin-wave theory (LSWT) are shown to the right of the respective spectra, with the LSWT parameters plotted in Fig. 20.

Figure 19

Comparison of the dominant transverse density matrix renormalization group (DMRG) mode at (a) the Brillouin zone (BZ) center, (b) the BZ edge, and (c) their ratio as a function of $D/J_y$.

Figure 20

Comparison of density matrix renormalization group (DMRG) and linear spin-wave theory (LSWT) parameterizations of a spin-chain mode.