Phase separation and frustrated square lattice magnetism of Na${}_{1.5}$VOPO${}_4$F${}_{0.5}$

Crystal structure, electronic structure, and magnetic behavior of the spin-$\frac{1}{2}$ quantum magnet Na${}_{1.5}$VOPO${}_4$F${}_{0.5}$ are reported. The disorder of Na atoms leads to a sequence of structural phase transitions revealed by synchrotron x-ray powder diffraction and electron diffraction. The high-temperature second-order $\alpha \leftrightarrow \beta$ transition at 500 K is of the order-disorder type, whereas the low-temperature $\beta \leftrightarrow \gamma +{\gamma }^{\prime }$ transition around 250 K is of the first order and leads to a phase separation toward the polymorphs with long-range ($\gamma$) and short-range (${\gamma }^{\prime }$) order of Na. Despite the complex structural changes, the magnetic behavior of Na${}_{1.5}$VOPO${}_4$F${}_{0.5}$ probed by magnetic susceptibility, heat capacity, and electron spin resonance measurements is well described by the regular frustrated square lattice model of the high-temperature $\alpha$-polymorph. The averaged nearest-neighbor and next-nearest-neighbor couplings are ${\overline{J}}_1\simeq -3.7$ K and ${\overline{J}}_2\simeq 6.6$ K, respectively. Nuclear magnetic resonance further reveals the long-range ordering at $T_N=2.6$ K in low magnetic fields. Although the experimental data are consistent with the simplified square-lattice description, band structure calculations suggest that the ordering of Na atoms introduces a large number of inequivalent exchange couplings that split the square lattice into plaquettes. Additionally, the direct connection between the vanadium polyhedra induces an unusually strong interlayer coupling having effect on the transition entropy and the transition anomaly in the specific heat. Peculiar features of the low-temperature crystal structure and the relation to isostructural materials suggest Na${}_{1.5}$VOPO${}_4$F${}_{0.5}$ as a parent compound for the experimental study of tetramerized square lattices as well as frustrated square lattices with different values of spin.

Figure 1

Crystal structure of $\alpha$-Na${}_{1.5}$VOPO${}_4$F${}_{0.5}$: The layers in the $ab$ plane (left) and the projection along $c$ (right). The layers have an ideal FSL geometry with the couplings $J_1$ and $J_2$ (left), and connect along $c$ via the F atoms (right). Disordered Na atoms occupy the voids of the resulting framework (right).

Figure 2

ED patterns of $\beta$-Na${}_{1.5}$VOPO${}_4$F${}_{0.5}$ at RT. The patterns are indexed on a tetragonal $\sqrt{2}{a}_{\text{sub}}\times \sqrt{2}{a}_{\text{sub}}\times c$ supercell. Bright reflections correspond to the $a_{\text{sub}}\times a_{\text{sub}}\times c$ subcell, whereas weak reflections indicate the supercell.

Figure 3

Temperature dependence of the subcell volume (top) and $c/a_{\text{sub}}$ ratio (bottom) for Na${}_{1.5}$VOPO${}_4$F${}_{0.5}$. Error bars are smaller than symbols. Lines are guide for the eyes, with arrowheads showing changes upon heating/cooling in the region of the $\beta \leftrightarrow \gamma +{\gamma }^{\prime }$ transition.[36]

Figure 4

DSC data showing the first-order $\beta \leftrightarrow \gamma +{\gamma }^{\prime }$ transition at 265 K (heating, upper panel) and 235 K (cooling, bottom panel).

Figure 5

XRD pattern of Na${}_{1.5}$VOPO${}_4$F${}_{0.5}$ collected at 150 K (black dots) and the two-phase refinement (solid line). Upper and lower ticks denote the reflections of the ${\gamma }^{\prime }$- and $\gamma$-phases. The inset shows the unit cells of $\alpha$- (solid), $\beta$- (dotted), and $\gamma$- (dashed) polymorphs.

Figure 6

ED patterns of $\gamma$-Na${}_{1.5}$VOPO${}_4$F${}_{0.5}$ at 100 K. The patterns are indexed on a tetragonal $2a_{\text{sub}}\times 2a_{\text{sub}}\times c$ unit cell. Brighter reflections with even $h$, $k$, and $l$ indices arise from the subcell, whereas weaker spots evidence the formation of the superstructure.

Figure 7

ED patterns of ${\gamma }^{\prime }$-Na${}_{1.5}$VOPO${}_4$F${}_{0.5}$ at 100 K. The patterns are indexed on a body-centered tetragonal $a_{\text{sub}}\times a_{\text{sub}}\times c$ unit cell. Diffuse lines in the [110] pattern evidence short-range order.

Figure 8

Temperature dependence of the magnetic susceptibility ($\chi$) and the HTSE fit with Eq. (2): $T_{\min }=12$ K, ${\overline{J}}_1=-3.7\left(1\right)$ K, ${\overline{J}}_2=6.6\left(1\right)$ K. The inset shows the field dependence of $\chi$ at low temperatures. The kink at $2.6-2.8$ K denotes the onset of the long-range order, see Fig. 11.

Figure 9

Details of the magnetic susceptibility fit: Averaged exchange couplings ${\overline{J}}_1$ and ${\overline{J}}_2$, $g$-value, and temperature-independent contribution ${\chi }_0$ evaluated for the fitting ranges with different $T_{\min }$.

Figure 10

Zero-field specific heat ($C_p$) of Na${}_{1.5}$VOPO${}_4$F${}_{0.5}$ with a transition anomaly at $T_N=2.6$ K and the maximum of the magnetic contribution around 4 K. The inset shows the low-temperature data plotted as $C_p/T$ against $T^2$. Solid line indicates the idealized $T^3$ behavior.

Figure 11

Left: Specific heat of Na${}_{1.5}$VOPO${}_4$F${}_{0.5}$ measured in different applied fields. Arrows denote the positions of the transition anomalies. Right: The field-vs-temperature phase diagram, based on the heat capacity and NMR measurements, as well as the magnetization isotherm at 1.4 K (Ref. 19).

Figure 12

ESR spectra of Na${}_{1.5}$VOPO${}_4$F${}_{0.5}$ measured at X-band frequency for selected temperatures in the paramagnetic regime. Solid lines indicate fits with the field derivative of a Lorentzian.

Figure 13

Temperature dependence of the ESR parameters: resonance field $H_{\mathrm{res}}$ (upper frame), ESR intensity compared to the magnetic susceptibility (middle frame), and linewidth $\Delta H$ (lower frame) in Na${}_{1.5}$VOPO${}_4$F${}_{0.5}$. The solid red line indicates the fit using a mean-field BCS-like ansatz [Eq. (4)].

Figure 14

Fourier-transformed ${}^{31}$P NMR spectra at different temperatures $T$ ($T>T_N$) for Na${}_{1.5}$VOPO${}_4$F${}_{0.5}$.

Figure 15

Temperature dependence of the ${}^{31}$P NMR line shift $K$ measured at 79 MHz. The inset shows $K$ plotted against the magnetic susceptibility $\chi$ with temperature as an implicit parameter. The solid line represents the linear fit with Eq. (5).

Figure 16

Temperature dependence of the spin-lattice relaxation rate $1/T_1$ measured at two different frequencies of 79 and 8.62 MHz.

Figure 17

Temperature-dependent ${}^{31}$P NMR spectra measured at 8.62 MHz.

Figure 18

Magnetic layers in the crystal structure of $\gamma$-Na${}_{1.5}$VOPO${}_4$F${}_{0.5}$ (left) and the respective spin lattice with six inequivalent couplings (right). The $J_1-J_2^{\prime \prime }$ plaquette units are shaded.

Figure 19

LDA density of states for $\gamma$-Na${}_{1.5}$VOPO${}_4$F${}_{0.5}$. The Fermi level is at zero energy.

Figure 20

Vanadium $d_{xy}$ bands at the Fermi level (zero energy): LDA band structure (thin light lines) and the fit of the tight-binding model (thick dark lines).

Figure 21

The arrangement of Na atoms in different polymorphs of Na${}_{1.5}$VOPO${}_4$F${}_{0.5}$. Dark spheres denote filled Na positions, light spheres are partially filled Na positions, and open circles represent vacancies.

Figure 22

Zero-field specific heat at the magnetic ordering transition in Na${}_{1.5}$VOPO${}_4$F${}_{0.5}$ (powder sample), Li${}_2$VOSiO${}_4$ (powder sample), and Pb${}_2$VO(PO${}_4{)}_2$ (single crystal).[63] The data for the last two compounds are from Ref. 15.