Deformation of the triangular spin-$\frac{1}{2}$ lattice in ${\mathrm{Na}}_2{\mathrm{SrCo}(\mathrm{PO}}_4{)}_2$

Crystal structure and thermodynamic properties of ${\mathrm{Na}}_2\mathrm{SrCo}{\left({\mathrm{PO}}_4\right)}_2$, the chemical sibling of the triangular quantum spin-liquid candidate ${\mathrm{Na}}_2\mathrm{BaCo}{\left({\mathrm{PO}}_4\right)}_2$, are reported. From single crystal x-ray diffraction and high-resolution synchrotron x-ray powder diffraction, the compound was found to crystallize in the monoclinic space group $P{2}_1/a$ at room temperature, in contrast to the trigonal ${\mathrm{Na}}_2\mathrm{BaCo}{\left({\mathrm{PO}}_4\right)}_2$. Above 650 K, the symmetry of ${\mathrm{Na}}_2\mathrm{SrCo}{\left({\mathrm{PO}}_4\right)}_2$ changes to $C2/m$, while around 1025 K a further transformation toward trigonal symmetry is observed. The monoclinic symmetry leads to a small deformation of the ${\mathrm{CoO}}_6$ octahedra beyond the trigonal distortion ubiquitous in this structure type and results in the stronger $g$-tensor anisotropy ($g_{\text{z}}/g_{\text{xy}}=1.6$) as well as the increased XXZ anisotropy ($J_{\text{z}}/J_{\text{xy}}=2.1$) compared to the Ba compound ($g_{\text{z}}/g_{\text{xy}}=1.1, J_{\text{z}}/J_{\text{xy}}=1.5$), while the average coupling strength, $J_{\text{av}}/k_{\text{B}}=(2J_{\text{xy}}+J_{\text{z}})/3k_{\text{B}}\simeq 1.3\text{K}$, remains unchanged. The Néel temperature increases from 140 mK (Ba) to 600 mK (Sr), and an uncompensated in-plane moment of $0.066\left(4\right){\mu }_{\text{B}}/\text{f.u.}$ appears. We show that the ordering temperature of a triangular antiferromagnet is capably controlled by its structural distortions.

Figure 1

Crystal structure of ${\mathrm{Na}}_2\mathrm{SrCo}{\left({\mathrm{PO}}_4\right)}_2$. (a) Unit cell viewed perpendicular to the $c$-axis showing the layered structure. The tilting of the ${\mathrm{CoO}}_6$ octahedra is indicated with the black arrow. (b) Triangular layer formed by the ${\mathrm{CoO}}_6$ octahedra in the low-temperature monoclinic phase $P{2}_1/a$ viewed along the $c$ axis. The Co-O-P bonds referred to in Sec. 3b are drawn with thick black lines. (c) Same triangular layer in the intermediate monoclinic phase $C2/m$. d) Triangular layer in the high-temperature trigonal phase $P\overline{3}m1$. In the top view perspective, the ${\mathrm{Na}}^{+}$ and ${\mathrm{Sr}}^{2+}$ ions were omitted for clarity.

Figure 2

Pink single crystals of ${\mathrm{Na}}_2\mathrm{SrCo}{\left({\mathrm{PO}}_4\right)}_2$.

Figure 3

Rietveld refinement for the synchrotron XRD data collected at 300 K in the trigonal space group for ${\mathrm{Na}}_2\mathrm{BaCo}{\left({\mathrm{PO}}_4\right)}_2$ [31] and in the monoclinic space group $P{2}_1/a$ for ${\mathrm{Na}}_2\mathrm{SrCo}{\left({\mathrm{PO}}_4\right)}_2$. The illustrated peak splitting is indicative of the monoclinic symmetry of the Sr compound. The scattering angles for ${\mathrm{Na}}_2\mathrm{BaCo}{\left({\mathrm{PO}}_4\right)}_2$ were renormalized for an easier comparison with the Sr compound.

Figure 4

Crystal structure of ${\mathrm{Na}}_2\mathrm{SrCo}{\left({\mathrm{PO}}_4\right)}_2$ as a function of temperature. (a) Temperature dependence of the monoclinic angle $\beta$. The weak anomaly at 200 K is likely a measurement artifact, because thermodynamic measurements, including Fisher's heat capacity $d\left(\chi T\right)/dT$ (inset), reveal no anomaly at this temperature. The second anomaly at around 650 K marks the structural transition from the space group $P{2}_1/a$ to $C2/m$. The color gradient marks the gradual transition to a trigonal space group which is not fully completed at 1025 K. (b) Synchrotron XRD data for selected temperatures illustrating the symmetry changes. The reflections labeled in black vanish for $C2/m$ and the red labels indicate the remaining reflections for $P\overline{3}m1$. The scattering angles were rescaled with respect to the data collected at 500 K. (c) Red are the refined $a/\sqrt{(}3)$ and blue are the refined $b$ lattice parameters. Visible is the kink at the phase transition from $P{2}_1/a$ to $C2/m$ at around 650 K in both lattice parameters. At higher temperatures $a/\sqrt{(}3)$ and $b$ converge due to the transition into a trigonal structure. The inset shows the continuous change of the unit cell volume with temperature. (d) Rietveld refinement for the synchrotron XRD data collected at 1025 K in the trigonal space group $P\overline{3}m1$. The insets show the merging of the split peaks as the temperature increases.

Figure 5

Deviation from the trigonal structure. (a) The ratio of short ($d^{\prime }$) to long ($d$) Co-Co bonds gauges deformation of the triangular lattice. The error bars are smaller than the symbol size. [(b) and (d)] Tilting angles of the $\mathrm{Co}1{\mathrm{O}}_6$ and $\mathrm{Co}2{\mathrm{O}}_6$ octahedra, respectively. [(c) and (e)] Co1-O-P and Co2-O-P angles, respectively. The angles were normalized in respect to the angle in the trigonal case.

Figure 6

Temperature dependence of the magnetic heat capacity at different magnetic fields parallel to $c$. The broad Schottky-like peak shifts to higher temperatures with increasing magnetic field. In low magnetic fields the broad peak due to short-range order is superimposed on the $\lambda$-type anomaly that indicates the magnetic ordering transition at 600 mK. The fit of the zero-field data is depicted as red line. The inset shows the zero-field magnetic entropy, which approaches $\text{R}ln\left(2\right)$.

Figure 7

Inverse magnetic susceptibility data at 0.1 T. The red line depicts the fit with the Griffith model as described in Ref. [43].

Figure 8

(Top) Magnetic field perpendicular to $c$. (Bottom) Magnetic field parallel to $c$. [(a) and (d)] Magnetic susceptibility for different magnetic fields below 3 K. The antiferromagnetic transition manifests itself by the drop of the susceptibility and remains visible up to at least 0.1 T. The inset contrasts the susceptibilities for both field directions at 0.05 T. In low fields up to 0.05 T, the increase of the susceptibility below $T_N$ for $H\perp c$ originates from small uncompensated moments in the $ab$ plane. [(b) and (e)] Curie-Weiss fits of the susceptibility between 5 and 20 K. [(c) and (f)] The TLAF fit of the magnetic susceptibility describes the data accurately. The coefficients are listed in Ref. [44].

Figure 9

Comparison of the magnetization versus magnetic field for ${\mathrm{Na}}_2\mathrm{SrCo}{\left({\mathrm{PO}}_4\right)}_2$ and ${\mathrm{Na}}_2\mathrm{BaCo}{\left({\mathrm{PO}}_4\right)}_2$ [31] for both field directions at 400 mK. The Sr compound shows a three times larger magnetic anisotropy. The inset shows the small hysteresis that indicates the uncompensated spin component in the $ab$ plane.

Figure 10

Local distortions of the ${\mathrm{CoO}}_6$ octahedra, crystal-field splitting of $3d$ orbitals, and the corresponding density of states for ${\mathrm{Na}}_2\mathrm{SrCo}{\left({\mathrm{PO}}_4\right)}_2$ and ${\mathrm{Na}}_2\mathrm{BaCo}{\left({\mathrm{PO}}_4\right)}_2$. The Fermi level is at zero energy. Note that calculations are performed on the PBE level, resulting in the metallic band structures.

Figure 11

Comparison of the rescaled susceptibilities of ${\mathrm{Na}}_2\mathrm{SrCo}{\left({\mathrm{PO}}_4\right)}_2$ and ${\mathrm{Na}}_2\mathrm{BaCo}{\left({\mathrm{PO}}_4\right)}_2$ [31] for both field directions at 0.1 T. Deviations are prominent at low temperatures.