Magnetic and structural dimer networks in layered ${\mathrm{K}}_2\mathrm{Ni}$(${\mathrm{MoO}}_4{)}_2$

The magnetic and thermodynamic properties of layered single-crystal ${\mathrm{K}}_2\mathrm{Ni}{\left({\mathrm{MoO}}_4\right)}_2$ having both structural and magnetic dimers have been investigated. The crystal structure of ${\mathrm{K}}_2\mathrm{Ni}{\left({\mathrm{MoO}}_4\right)}_2$ is composed of edge-sharing ${\mathrm{NiO}}_6$-octahedral pairs bridged by the ${\mathrm{MoO}}_4^{2-}$ polyatomic ion groups in a plane, and the ${\mathrm{K}}^{+}$ ions sit in the van der Waals gap between the layers. The temperature dependence of magnetic susceptibility shows a spin-singlet ground state with an activation gap of $\mathrm{\Delta }/k_B\approx 38$ K. A high-field magnetization study at $T=1.5$ K exhibits a half-magnetization plateau at ${\mu }_{0}H\sim 25$ T, corresponding to a level crossing of the singlet ground state with the lowest triplet state. Further, we have performed density functional theory calculations to determine magnetic exchange interactions. The nearest-neighbor coupling constant $J_1\sim 10$ K between the Ni spins turns out to be an order of magnitude larger than all interdimer couplings. Our experimental and theoretical results suggest that ${\mathrm{K}}_2\mathrm{Ni}{\left({\mathrm{MoO}}_4\right)}_2$ constitutes a nearly isolated two-dimensional $S=1$ dimer model.

Figure 1

(a) 2D and (b) 3D view of the crystal structure of ${\mathrm{K}}_2\mathrm{Ni}{\left({\mathrm{MoO}}_4\right)}_2$. The edge-shared ${\mathrm{NiO}}_6$ octahedral pairs form dimers connected by the polyatomic ion groups of ${\mathrm{MoO}}_4^{2-}$ in the $ac$ plane with the ${\mathrm{K}}^{+}$ ions in the van der Waals gaps.

Figure 2

Rietveld refinement of the synchrotron XRD powder pattern using crushed crystals of ${\mathrm{K}}_2\mathrm{Ni}{\left({\mathrm{MoO}}_4\right)}_2$ taken at room temperature. The observed data, Rietveld refinement fit, Bragg peaks, and difference curve are denoted by the plus sign symbols, green solid line, vertical black dashes, and purple line, respectively. The insets show an image of the as-grown single crystal with the easily cleaved $ac$ plane, and XRD pattern of a cleaved plane which has a preferred orientation of indexed (0 $K$ 0) peaks.

Figure 3

The refined lattice constants are analyzed from the neutron powder diffraction data between 2 and 200 K. (a), (b), and (c) are the temperature evolution of the lattice parameters $a$, $b$, and $c$, respectively, where the scale of $y$ axes is set to be the same for comparison. The 300 K data are derived from synchrotron XRD refinement results. (d) Change of the lattice parameters is plotted as a percentage against the base temperature lattice parameters.

Figure 4

The dc magnetic susceptibilities $\chi \left(T\right)$ of the ${\mathrm{K}}_2\mathrm{Ni}{\left({\mathrm{MoO}}_4\right)}_2$ powder sample measured in an applied field of ${\mu }_{0}H=0.01$ T. The black, red, and blue circles represent the raw $\chi \left(T\right)$, impurity and $T$-independent contribution to $\chi \left(T\right)$, and intrinsic spin dimer $\chi \left(T\right)$, respectively. The pink line is a fit to Eq. (1) and the green line to an activation expression as described in the text. The upper inset shows the $1/\chi$ plot. The red solid line is the Curie-Weiss law fit of the $T=150\text{–}300$ K data to $\chi \left(T\right)\sim C/(T-\mathrm{\Theta })$. The lower inset is a magnified view of the low-$T\chi \left(T\right)$ at ${\mu }_{0}H=0.01$ and 7 T.

Figure 5

Real ($\chi \prime$) and imaginary components ($\chi \prime \prime$) of the ac magnetic susceptibility for ${\mathrm{K}}_2\mathrm{Ni}{\left({\mathrm{MoO}}_4\right)}_2$ powder sample measured with an oscillating field of $H_{\mathrm{ac}}=1$ Oe at various frequencies of 10–1000 Hz.

Figure 6

High-field magnetization curve taken with a pulsed-field magnet at $T=1.5$ K for ${\mathrm{K}}_2\mathrm{Ni}{\left({\mathrm{MoO}}_4\right)}_2$ powder sample. The derivative $dM/dH$ of the magnetization shows two peaks at ${\mu }_0{H}_{c1}\sim 25$ T and ${\mu }_0{H}_{c2}\sim 50$ T, corresponding to magnetization plateaus.

Figure 7

The temperature dependence of specific heat $C_p\left(T\right)$ for ${\mathrm{K}}_2\mathrm{Ni}{\left({\mathrm{MoO}}_4\right)}_2$ measured at zero field. The red solid line is the fit to Eq. (2) to evaluate a phonon contribution.

Figure 8

Magnetic specific heat $C_m\left(T\right)$ vs $T$ for ${\mathrm{K}}_2\mathrm{Ni}{\left({\mathrm{MoO}}_4\right)}_2$. The green line is a fit of the low-$T{C}_m\left(T\right)$ data to Eq. (3) as discussed in the text. A change in magnetic entropy $\mathrm{\Delta }{S}_m$ from 2 to 34 K is obtained from the integration of the $C_m/T$ data.

Figure 9

Schematic representation of the spin arrangements of AFM configuration of ${\mathrm{K}}_2\mathrm{Ni}{\left({\mathrm{MoO}}_4\right)}_2$. The exchange parameters are denoted by different color arrows as discussed in Table 3.

Figure 10

(a) Electronic band structure and (b) density of states for the AFM3 configuration of ${\mathrm{K}}_2\mathrm{Ni}{\left({\mathrm{MoO}}_4\right)}_2$. The top of the valence band is set to zero.

Figure 11

The isolated dimer models for (a) ${\mathrm{K}}_2\mathrm{Ni}{\left({\mathrm{MoO}}_4\right)}_2$ and (b) ${\mathrm{Ba}}_3{\mathrm{Mn}}_2{\mathrm{O}}_8$ are compared. ${\mathrm{K}}_2\mathrm{Ni}{\left({\mathrm{MoO}}_4\right)}_2$ has the Ni dimers being arranged along the $a$ direction with weak interdimer frustration between the chains, and ${\mathrm{Ba}}_3{\mathrm{Mn}}_2{\mathrm{O}}_8$ with Mn dimers arranged in a 2D hexagonal lattice.