Magnetic structure of the square cupola compound Ba(TiO)${\mathrm{Cu}}_4({\mathrm{PO}}_4{)}_4$

The magnetic structure of the antiferromagnetic square cupola compound ${\mathrm{Ba}\left(\mathrm{TiO}\right)\mathrm{Cu}}_4{(\mathrm{PO}}_4{)}_4$ with the tetragonal structure is studied with ${}^{31}\mathrm{P}$ and ${}^{63,65}\mathrm{Cu}$ nuclear magnetic resonance techniques. The ${}^{31}\mathrm{P}$ magnetic hyperfine shift $K$ shows a clear splitting at the Néel temperature $T_N=9.5$ K, where the resonance breaks down into two lines when an external magnetic field is oriented along the $c$ axis and into four lines when the field is along the $a$ axis. In the paramagnetic region $K\left(T\right)$ follows the temperature dependence of the magnetic susceptibility $\chi \left(T\right)$. From the $K$ vs $\chi$ plot we determined nearly equal hyperfine field values $H{}_{\text{hf}}^a=765$ mT/${\mu }_B$ and $H{}_{\text{hf}}^c=740$ mT/${\mu }_B$ for the magnetic field oriented along $a$ and $c$, respectively. From the rotation of the single crystal in the external magnetic field we determined eight different orientations of the $K$ tensor in the paramagnetic region. In the antiferromagnetic state at $T=6$ K, the rotation of the single crystal shows at phosphorus eight different orientations of the local field $B_{\text{int}}=35.6$ mT. The NMR spectrum of copper at zero external field shows local field at copper nucleus $B_{\text{loc}}=14.77$ T, directed perpendicular to the plane of the ${\mathrm{CuO}}_4$ plaquette, confirming a previously determined quadrupolar configuration of magnetic moments ${\mathrm{\Gamma }}_3\left(2\right)$ [K. Kimura et al., Nat. Commun. 7, 13039 (2016); P. Babkevich et al., Phys. Rev. B 96, 214436 (2017)].

Figure 1

Description of the BTCPO unit cell with up and down ${\mathrm{Cu}}_4{\mathrm{O}}_{12}$ square cupolas (blue), ${\mathrm{PO}}_4$ tetrahedrons (green), and ${\mathrm{TiO}}_5$ pyramids (brown), with Ba ions shown as large gray spheres along the $c$ (left panel) and $a$ (middle panel) axes. Two possible arrangements of spins ${\mathrm{\Gamma }}_3\left(1\right)$ and ${\mathrm{\Gamma }}_3\left(2\right)$ (right panel).

Figure 2

Temperature dependence of magnetic susceptibility $\chi \left(T\right)$ in an applied magnetic field $B=4.7$ T, measured in two different directions: $B\parallel \left[100\right]$ (black solid circles) and $B\parallel \left[001\right]$ (red open circles). The cyan line represents the Curie-Weiss fit. The inset shows the detailed behavior of susceptibilities below $T_N=9.5$ K.

Figure 3

Temperature dependence of the ${}^{31}\mathrm{P}$ Knight shift as measured when the single crystal is oriented (a) with the $a$ axis parallel to the external field $B$ (blue dots) and (b) with the $c$ axis parallel to $B$ (open circles). The insets show the Clogston-Jaccarino plots, giving values quite close to the hyperfine field in both the directions.

Figure 4

Temperature development of the splitting of the resonance lines below the ordering temperature in the orientation $a\parallel B$ (top panel) and $c\parallel B$ (bottom panel). The solid lines are fits by Eq. (3).

Figure 5

(a) ${}^{31}\mathrm{P}$ MAS NMR of the BaTCPO powder sample at $T=305$ K with a spinning frequency $\nu =25$ kHz. The NMR spectrum of the rotating sample consists of a center band at isotropic Knight shift $K_{\text{iso}}=0.1643\%$ and of a number of spinning sidebands denoted by asterisks at multiples of the spinning frequency from the center band. (b) ${}^{31}\mathrm{P}$ NMR of a static powder sample at slightly lower temperature $T=295$ K. The red line is a fit to the powder line with the eigenvalues of the nearly axially symmetric Knight shift tensor: $K_{11}=0.1952\%,K_{22}=0.1832\%,K_{33}=0.1298\%$.

Figure 6

${}^{31}\mathrm{P}$ NMR resonance frequencies by rotating the BaTCPO single crystal around the (a) $a$ and (b) $c$ axes at $T=295$ K (top panels) and $T=18$ K (bottom panels). The solid lines correspond to the angle dependences according to the parameters given in Table 1. Due to symmetry, eight different sites in the unit cell contribute to only two different rotation patterns. The red lines correspond to sites 1, 3, 6, and 7 (see Table 1) in (a) and to sites 2, 3, 6, and 8 in (b); the rest of the sites are given by the blue lines.

Figure 7

Orientations of the Knight shift tensor at eight phosphorus locations of the unit cell at temperature $T=295$ K. The tensors have three principal vectors, green, orange, and dark violet, which correspond to $K_{PAS}$ values $K_{11},K_{22}$, and $K_{33}$, respectively.

Figure 8

Temperature dependence of ${}^{31}\mathrm{P}$ spin-lattice relaxation rate in directions [100] (red dots) and [001] (blue dots). The inset shows that relaxation rate $1/T_1$ is proportional to $T^7$ below $T_N$.

Figure 9

Rotation patterns of ${}^{31}\mathrm{P}$ NMR frequencies of the BaTCPO single crystal at temperature $T=6$ K rotating around (a) [001] and (b) [100]. The lines are approximations with Eq. (9) using data in Table 2. The main directions of a crystal, $a\parallel \mathrm{B},b\parallel \mathrm{B}$, and $c\parallel \mathrm{B}$, and $b\parallel \mathrm{B}$, are noted on the respective $x$ axes. The blue symbols are assigned to the sites in the up cupola, and the red symbols are assigned to the sites in the down cupola.

Figure 10

Scheme of the local field direction on a phosphorus ion in the BaTCPO crystal. $B_{\text{int}}$ is the local field; $B_a,B_b$, and $B_c$ are the projections of $B_{\text{int}}$ to the crystal $a,b$, and $c$ axes, respectively. $B_1,B_2$, and $B_3$ are the projections of $B_{\text{int}}$ to the $bc,ac$, and $bc$ planes, respectively. The latter amplitudes can be found from the rotation pattern parameters given in Table 2.

Figure 11

Possible configuration of induced static magnetic fields at phosphorus ions in the BaTCPO unit cell. Red arrows represent the directions and sizes of the static magnetic fields; plus and minus signs indicate the field direction to the front or to the back of the figure plane. Blue squares are ${\mathrm{Cu}}^{2+}$ ions, and the purple tetrahedrons show P ions.

Figure 12

Calculated dipolar field direction at phosphorus ions of the BaTCPO unit cell (a) assuming the structure of ${\mathrm{Cu}}^{2+}$ magnetic moments ${\mathrm{\Gamma }}_3\left(1\right)$ (given in Ref. [1]) and (b) assuming the structure ${\mathrm{\Gamma }}_3\left(2\right)$. The red arrows represent the dipole fields. The length of the arrows corresponds to the relative value of dipolar field in both cases and that in Fig. 11. Blue arrows represent directions of the ${\mathrm{Cu}}^{2+}$ magnetic moments and their sizes in the two cases.

Figure 13

${}^{63,65}\mathrm{Cu}$ ZFNMR spectrum of BaTCPO at 4.2 K. The blue lines correspond to the resonances of ${}^{63}\mathrm{Cu}$; the red lines belong to ${}^{65}\mathrm{Cu}$ resonances. Red dots are experimental echo amplitudes, and the solid lines are computer fits by Lorentzian lines.