Realizing square and diamond lattice $S=1/2$ Heisenberg antiferromagnet models in the $\alpha$ and $\beta$ phases of the coordination framework, $\mathrm{K}\mathrm{Ti}{\left({\mathrm{C}}_2{\mathrm{O}}_4\right)}_2·x{\mathrm{H}}_2\mathrm{O}$

We report the crystal structures and magnetic properties of two pseudopolymorphs of the $S=1/2 {\mathrm{Ti}}^{3+}$ coordination framework, $\mathrm{K}\mathrm{Ti}{\left({\mathrm{C}}_2{\mathrm{O}}_4\right)}_2·x{\mathrm{H}}_2\mathrm{O}$. Single-crystal x-ray and powder neutron diffraction measurements on $\alpha -\mathrm{K}\mathrm{Ti}{\left({\mathrm{C}}_2{\mathrm{O}}_4\right)}_2·x{\mathrm{H}}_2\mathrm{O}$ confirm its structure in the tetragonal $I4/mcm$ space group with a square planar arrangement of ${\mathrm{Ti}}^{3+}$ ions. Magnetometry and specific heat measurements reveal weak antiferromagnetic interactions, with $J_1\approx 7$ K and $J_2/J_1=0.11$ indicating a slight frustration of nearest- and next-nearest-neighbor interactions. Below 1.8 K, $\alpha -\mathrm{K}\mathrm{Ti}{\left({\mathrm{C}}_2{\mathrm{O}}_4\right)}_2·x{\mathrm{H}}_2\mathrm{O}$ undergoes a transition to G-type antiferromagnetic order with magnetic moments aligned along the $c$ axis of the tetragonal structure. The estimated ordered moment of ${\mathrm{Ti}}^{3+}$ in $\alpha -\mathrm{K}\mathrm{Ti}{\left({\mathrm{C}}_2{\mathrm{O}}_4\right)}_2·x{\mathrm{H}}_2\mathrm{O}$ is suppressed from its spin-only value to $0.62\left(3\right){\mu }_B$, thus verifying the two-dimensional nature of the magnetic interactions within the system. $\beta -\mathrm{K}\mathrm{Ti}{\left({\mathrm{C}}_2{\mathrm{O}}_4\right)}_2·2{\mathrm{H}}_2\mathrm{O}$, on the other hand, realizes a three-dimensional diamondlike magnetic network of ${\mathrm{Ti}}^{3+}$ moments within a hexagonal $P{6}_{2}22$ structure. An antiferromagnetic exchange coupling of $J\approx 54$ K—an order of magnitude larger than in $\alpha -\mathrm{K}\mathrm{Ti}{\left({\mathrm{C}}_2{\mathrm{O}}_4\right)}_2·x{\mathrm{H}}_2\mathrm{O}$—is extracted from magnetometry and specific heat data. $\beta -\mathrm{K}\mathrm{Ti}{\left({\mathrm{C}}_2{\mathrm{O}}_4\right)}_2·2{\mathrm{H}}_2\mathrm{O}$ undergoes Néel ordering at $T_N=28$ K, with the magnetic moments aligned within the $ab$ plane and a slightly reduced ordered moment of $0.79{\mu }_B$ per ${\mathrm{Ti}}^{3+}$. Through density-functional theory calculations, we address the origin of the large difference in the exchange parameters between the $\alpha$ and $\beta$ pseudopolymorphs. Given their observed magnetic behaviors, we propose $\alpha -\mathrm{K}\mathrm{Ti}{\left({\mathrm{C}}_2{\mathrm{O}}_4\right)}_2·x{\mathrm{H}}_2\mathrm{O}$ and $\beta -\mathrm{K}\mathrm{Ti}{\left({\mathrm{C}}_2{\mathrm{O}}_4\right)}_2·2{\mathrm{H}}_2\mathrm{O}$ as close to ideal model $S=1/2$ Heisenberg square and diamond lattice antiferromagnets, respectively.

Figure 1

(a) Rietveld refinement of the $I4/mcm$ model (${\chi }^2=4.80, R_p=2.83\%$) describing the structure of deuterated $\alpha -\mathrm{K}\mathrm{Ti}{\left({\mathrm{C}}_2{\mathrm{O}}_4\right)}_2·1.48\left(4\right){\mathrm{D}}_2\mathrm{O}$ using data collected on the HRPD instrument at 1.8 K. (b) Rietveld plot (${\chi }^2=1.81$ and $R_{\text{mag}}=3.16\%$) of the $I4/m^{\prime }cm$ magnetic structure to magnetic only scattering obtained by subtracting data collected on the WISH instrument at 15 K from 1.2 K data. In both plots, data points are shown in black, fitted curves in orange, difference curves in blue, Bragg reflection positions in gray and brown, and excluded data points in gray.

Figure 2

(a) The crystal structure of $\alpha -\mathrm{K}\mathrm{Ti}{\left({\mathrm{C}}_2{\mathrm{O}}_4\right)}_2·x{\mathrm{H}}_2\mathrm{O}$ as viewed along the [111] direction where the (b) oxalate-bridged square antiprismatic ${\mathrm{Ti}}^{3+}$ ions form square planar sheets within the $\mathit{ab}$ plane separated by a disordered layer of orange ${\mathrm{K}}^{+}$ ions and water molecules which are omitted for clarity. (c) G-type antiferromagnetic ordering of the magnetic moments obtained from the analysis of PND data at 1.2 K, where the orange ${\mathrm{Ti}}^{3+}$ magnetic moments lie along the crystallographic $c$ axis with nearest-neighbor ($J_1$), next-nearest-neighbor ($J_2$), and interplanar ($J_3$) exchange interactions highlighted in black, blue, and gray, respectively. Both structures were generated using the vesta [55] visualization software.

Figure 3

(a) Temperature-dependent zero-field cooled magnetic susceptibility, ${\chi }_m$ (blue), of $\alpha$ measured in an applied field of 0.1 T and its inverse, ${\chi }_m^{-1}$ (gray). A modified Curie-Weiss fit (brown) to ${\chi }_m^{-1}$ yields $C=0.233\left(1\right)$ emu K ${\mathrm{mol}}^{-1}, {\theta }_{\text{CW}}=-7.86\left(1\right)$ K, and ${\chi }_0=1.19\left(2\right)\times {10}^{-4}$ emu ${\mathrm{mol}}^{-1}$. Quantum Monte Carlo (gray) and tenth-order high-temperature series expansion (orange) fits for a Heisenberg $S=1/2$ square lattice model and a $S=1/2$ frustrated square lattice model with interplanar coupling $J_3$, respectively, to (b) ${\chi }_m$ and (c) zero-field specific heat $C_p$, simultaneously. The resulting model parameters are $J_1=6.42\left(1\right)$ for the QMC $S=1/2$ square lattice model and $J_1=6.80\left(1\right)$ K, $J_2/J_1=0.11, J_3=0$, and ${\chi }_0=1.19\times {10}^{-4}$ for the HTE $S=1/2$ FSL model.

Figure 4

(a) Rietveld plot for data collected on HRPD at 1.8 K using the $P{6}_{2}22$ structural model to describe $\beta -\mathrm{K}\mathrm{Ti}{\left({\mathrm{C}}_2{\mathrm{O}}_4\right)}_2·2{\mathrm{H}}_2\mathrm{O}$ with goodness-of-fit parameters $R_p=1.81\%$ and ${\chi }^2=3.06$. (b) Rietveld refinement ($R_{\text{mag}}=1.93\%, {\chi }^2=3.91$) of the $P_b{2}_1$ magnetic space group model to magnetic-only scattering obtained by subtracting data collected on the WISH instrument at 35 K and 1.2 K. In both plots, data points are shown in black, fitted curves in orange, difference curves in blue, Bragg reflection positions in gray and brown, and excluded data points in gray.

Figure 5

(a) The crystal structure of $\beta -\mathrm{K}\mathrm{Ti}{\left({\mathrm{C}}_2{\mathrm{O}}_4\right)}_2·2{\mathrm{H}}_2\mathrm{O}$ as viewed along the $\langle 111\rangle$ directions where (b) distorted square antiprismatic ${\mathrm{Ti}}^{3+}$ ions form a diamondlike magnetic sublattice. (c) One of the possible magnetic moment arrangements for the $P_b{2}_1$ magnetic structure describing the coplanar antiferromagnetic ordering of $\beta -\mathrm{K}\mathrm{Ti}{\left({\mathrm{C}}_2{\mathrm{O}}_4\right)}_2·2{\mathrm{H}}_2\mathrm{O}$ with the nearest-neighbor and further neighbor exchange interactions shown in blue, gray, yellow, and brown, respectively. The possible spin arrangements only differ with respect to how the moment aligns within the $\mathit{ab}$ plane. The illustrated structure in (d) is representative of one of the possible moment directions.

Figure 6

(a) Zero-field cooled magnetic susceptibility of $\beta -\mathrm{K}\mathrm{Ti}{\left({\mathrm{C}}_2{\mathrm{O}}_4\right)}_2·2{\mathrm{H}}_2\mathrm{O}, {\chi }_m$ (blue), of measured between 1.8 K and 300 K in a 0.1 T applied magnetic field and corresponding calculated effective magnetic moment (gray). A Curie-Weiss fit to ${\chi }_m$ (brown) yields ${\theta }_{\mathsf{\text{CW}}}=-109.6\left(1\right)$ K, $C=0.329\left(2\right)$ emu K ${\mathrm{mol}}^{-1}$, and ${\chi }_0=-4.88\left(3\right)\times {10}^{-5}$ emu ${\mathrm{mol}}^{-1}$. (b), (c) Simultaneous fit to ${\chi }_m$ and the zero-field specific heat $C_p{T}^{-1}$ using the diamond lattice model with a nearest-neighbor coupling $J_1$ calculated by a tenth-order high-temperature series expansion. The model yields $J=54.4\left(1\right)$ K and ${\chi }_0=-4.11\left(8\right)\times {10}^{-5}$ emu ${\mathrm{mol}}^{-1}$.

Figure 7

${\mathrm{Ti}}^{3+}{d}_{z^2}$-based Wannier functions showing the orbital overlaps for the $J_1$ superexchange pathways of (a) $\alpha -\mathrm{K}\mathrm{Ti}{\left({\mathrm{C}}_2{\mathrm{O}}_4\right)}_2·x{\mathrm{H}}_2\mathrm{O}$ and (b) $\beta -\mathrm{K}\mathrm{Ti}{\left({\mathrm{C}}_2{\mathrm{O}}_4\right)}_2·2{\mathrm{H}}_2\mathrm{O}$.