Antiferromagnetic dimers of Ni(II) in the $S=1$ spin-ladder ${\mathrm{Na}}_2{\mathrm{Ni}}_2{\left({\mathrm{C}}_2{\mathrm{O}}_4\right)}_3{\left({\mathrm{H}}_2\mathrm{O}\right)}_2$

We report the synthesis, crystal structure, and magnetic properties of the $S=1$ 2-leg spin-ladder compound ${\mathrm{Na}}_2{\mathrm{Ni}}_2{\left({\mathrm{C}}_2{\mathrm{O}}_4\right)}_3{\left({\mathrm{H}}_2\mathrm{O}\right)}_2$. The magnetic properties were examined by magnetic susceptibility and pulsed high field magnetization measurements. The magnetic excitations have been measured in high field high frequency ESR. Although the Ni(II) ions form structurally a 2-leg ladder, an isolated dimer model consistently describes the observations very well. The analysis of the temperature dependent magnetization data leads to a magnetic exchange constant of $J=43\mathrm{K}$ along the rungs of the ladder and an average value of the $g$-factor of 2.25. From the ESR measurements, we determined the single ion anisotropy to $D=11.5\mathrm{K}$. The validity of the isolated dimer model is supported by quantum Monte Carlo calculations, performed for several ratios of interdimer versus intradimer magnetic exchange and taking into account the experimentally determined single ion anisotropy. The results can be understood in terms of the different coordination and superexchange angles of the oxalate ligands along the rungs and legs of the 2-leg spin ladder.

Figure 1

Crystal structure of ${\mathrm{Na}}_2{\mathrm{Ni}}_2{\left({\mathrm{C}}_2{\mathrm{O}}_4\right)}_3{\left({\mathrm{H}}_2\mathrm{O}\right)}_2$. For clarity, Na and H atoms are not shown. The numbers indicate the distance (in Å) between opposing oxygen atoms on the distorted $\mathrm{Ni}{\mathrm{O}}_6$ octahedra which are tilted by $17\text{degrees}$ with respect to the $a$ axis.

Figure 2

Temperature dependence of the magnetization of a SNOX single crystal $\left(2.04\mathrm{mg}\right)$ at an external field of $B_{\mathit{ex}}=2\mathrm{T}$ along the $a$ (circles), $b$ (diamonds), and $c$ axes (triangles).

Figure 3

Relative energies of the spin states, calculated for the magnetic field applied perpendicular to the $z$ axis of the uniaxial anisotropy tensor: the triplet $S=1$ and the quintet $S=2$ are well separated from the $S=0$ ground state by an activation energy of roughly $\mid J\mid =43\mathrm{K}$ and $3\mid J\mid$, respectively. Both multiplets exhibit a zero field splitting due to crystal field anisotropy. Note the ground state level crossings around 30 and $53\mathrm{T}$ (for details see text).

Figure 4

Main panel: Relationship $\nu$ versus $B_i^{\mathrm{res}}$ for the ESR modes $i=1,\dots ,4$. Solid line is a theoretical fit of the data. Dashed lines are a linear approximation of branches 2, 3, and 4 to high fields. Gray error bar indicates a field region where a broad absorption mode is observed in the pulsed field ESR measurement. Inset, ESR spectrum in a static (bottom curve) and pulsed magnetic field ESR experiment (top curve). Sharp lines are due to the DPPH field marker. Resonance modes associated with a bulk ESR spectrum of Ni-oxalate are indicated by black arrows and numbered. The anticipated positions of modes 2, 3, and 4 at high magnetic fields are marked by gray dashed arrows (for details see text).

Figure 5

Intensity of the strongest ESR mode 1 as a function of temperature. Note an activating behavior similar to the static magnetization $M\left(T\right)$ (Fig. 2).

Figure 6

High field magnetization of a SNOX powder sample. The figure shows measurements at $1.47\mathrm{K}$ and $4.2\mathrm{K}$, the inset measurements at $10\mathrm{K}$ and $26\mathrm{K}$. The solid line represents the simulation as described in the text.

Figure 7

Differential magnetization curves $dM∕dB$ for temperatures $T=1.47\mathrm{K}$ and $T=4.2\mathrm{K}$ (inset). The open circles are calculated from measured data, the solid lines represent simulations as described in the text.

Figure 8

Quantum Monte Carlo simulations for $R=1$ (open circles), 0.95 (open triangles), and 0.9 (open squares). We plotted the reduced susceptibility versus reduced temperature. The solid circles show the measured data for $B=2\mathrm{T}$ along the $a$ axis corrected by a Curie tail and a temperature independent contribution. The inset shows the high temperature results.

Figure 9

QMC calculations of the field dependent magnetization for different ratios of $R$ with $D∕J=0.27$ and $B$ parallel to the anisotropy axis. Open circles, $R=1$; open triangles, $R=0.95$; open squares, $R=0.9$. The calculations were done for a reduced temperature $T∕J=0.034$, corresponding to $J+K=43\mathrm{K}$ and $T=1.47\mathrm{K}$. The solid squares are the normalized measurements scaled with $J+K=43\mathrm{K}$.

Figure 10

Ni-(ox)-Ni coordination and proposed magnetic exchange pathways on the rungs (left-hand panel) and on the legs of the spin ladder (right-hand panel). The involved Ni $3d$ and oxygen $2p$ orbitals are shown in black.