Magnetic structure in square cupola compound Ba(TiO)Cu$_4$(PO$_4$)$_4$: a $^{31}$P NMR Study

The magnetic structure of the antiferromagnetic square cupola compound Ba(TiO)Cu$_4$(PO$_4$)$_4$ with the tetragonal structure is studied with 31P nuclear magnetic resonance techniques. The magnetic hyperfine shift K shows a clear splitting at the N\'eel temperature T$_N$ = 9.5 K, where the resonance splits 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(T) follows temperature dependence of the magnetic susceptibility $\chi(T)$. From K vs $\chi$ plot we determined nearly isotropic hyperfine field values Ha_hf = 765 mT/$\mu_B$ and Hc_hf = 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 K-tensor in the paramagnetic region. In the antiferromagnetic state at T = 6 K we found that the local field at phosphorus is mainly due to dipolar field of coppers. Here the rotation of the single crystal shows eight different orientations of the local field Bint = 35.6 mT. The orientations correspond to the calculation of dipolar fields at phosphorus assuming magnetic quadrupolar configuration of magnetic moments $\varGamma_3(1)$ described previously [Nat. Commun. 7, 13039 (2016); Phys. Rev. B 96, 214436 (2017)].

The magnetic structure of the antiferromagnetic square cupola compound Ba(TiO)Cu4(PO4)4 with the tetragonal structure is studied with 31 P nuclear magnetic resonance techniques. The magnetic hyperfine shift K shows a clear splitting at the Néel temperature TN = 9.5 K, where the resonance splits 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(T ) follows temperature dependence of the magnetic susceptibility χ(T ). From K vs χ plot we determined nearly isotropic hyperfine field values H a hf = 765 mT/µB and H c hf = 740 mT/µB 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 K-tensor in the paramagnetic region. In the antiferromagnetic state at T = 6 K we found that the local field at phosphorus is mainly due to dipolar field of coppers. Here the rotation of the single crystal shows eight different orientations of the local field Bint = 35.6 mT. The orientations correspond to the calculation of dipolar fields at phosphorus assuming magnetic quadrupolar configuration of magnetic moments Γ3 (1)

I. INTRODUCTION
Recently, an interesting class of compounds,AE(TiO)Cu 4 (PO 4 ) 4 (AETCPO; AE = Ba, Sr, or Pb), was newly synthesized 1,2 . The structure of AETCPO is tetragonal with the space group P 42 1 2 and consists of the layers of up and down square cupola Cu 4 O 12 (FIG. 1). Each square cupola is made of four corner sharing CuO 4 plaquets. The layers are separated by AE ions and TiO 5 pyramids. These compounds undergo an antiferromagnetic (AF) transition at low temperatures. For example, Ba(TiO)Cu 4 (PO 4 ) 4 (BaTCPO) exhibits an AF ordered state at temperatures below the Néel temperature T N = 9.5 K. Below T N the arrangement of magnetic moments of coppers shows an antiferroic quadrupolar order. Because of the antiferroic order, BaTCPO does not exhibit a magnetoelectric effect where an electric (magnetic) field causes magnetization (electric polarization). Instead, this compound exhibits a remarkable magnetodielectric effect. Sr(TiO)Cu 4 (PO 4 ) 4 (SrTCPO) also shows similar magnetic and magnetodielectric properties below T N = 6.2 K 3 , while Pb(TiO)Cu 4 (PO 4 ) 4 (PbTCPO) exhibits a ferroic quadrupolar order resulting in a linear magnetoelectric effect below T N = 6.5 K 4,5 .
Magnetic structure of BaTCPO was carefully studied by neutron scattering experiments 1 . These studies yielded two possible arrangements of the spins: one, denoted Γ 3 (1), where the moments are confined approximately in the CuO 4 planes; and the other, Γ 3 (2), where the moments are approximately perpendicular to the CuO 4 planes forming two-in-two-out-type structure (FIG. 1). The latter arrangement had smaller R-factor, thus being the most probable arrangement. Later Babkevich et al. 6 confirmed the two possible structures using spherical neutron polarimetry, inclining by a discernible, albeit small advantage towards the Γ 3 (2) spin structure.
In the present report we will use 31 P NMR techniques to study the local magnetic fields in a single crystal of BaTCPO. Phosphorus ions in PO 4 tetrahedra see four coppers in two cupolas as next-nearest-neighbours. Therefore, the local field at phosphorus is definitely influenced by the magnetic arrangement of coppers. Previous NMR studies of SrTPCO 7 and BaTPCO 8 have used powder samples where detailed information about the magnetic structure is difficult to get.

II. EXPERIMENTAL DETAILS
Single crystals of Ba(TiO)Cu 4 (PO 4 ) 4 were grown with the flux method by Kimura et al. 2,9 . The sample crystal used in the experiments sized 1.9 x 2.0 x 3.9 mm and weighed 61.84 mg. NMR measurements were conducted using the spectrometer MAGRes2000 attached to a B = 4.7 T superconducting magnet, 31 P resonance frequency 80.97 MHz. He-flow cryostat (JANIS Research Inc.) allowed measurements in the temperature range of 4.5 to 300 K. The NMR probe was our own made and had a single-axis goniometer. LakeShore CERNOX calibrated sensors and Model 332 temperature controller were used for temperature regulation. Spin-lattice relaxation T 1 was measured using inversion-recovery method. The magic angle spinning (MAS) room temperature spectrum was recorded using a home-built probe with the 1.8 mm O.D. rotor spinning 25 kHz. The magnetic shifts are given relative to the resonance of H 3 PO 4 .  At high temperature χ (T ) followed the Curie-Weiss law: Here χ 0 is temperature-independent susceptibility. At T > 100 K the fitting gives the following parameters: for the magnetic field along the [001] direction: χ 0 = −7.3(4) × 10 −5 cm 3 /mole per Cu, C = 0.460(1) cm 3 K per mole Cu, θ CW = −29.0(4) K; and for the field along the [100] direction: χ 0 = −3.5(2) × 10 −5 cm 3 /mole-Cu, C = 0.454(7) × 10 −4 cm 3 K /mole-Cu, and θ CW = −29.3(2) K. From the Curie constant one gets the effective copper magnetic moment as µ eff = 3k B C/N A , where k B is the Boltzmann factor, and N A is Avogadro's number. We get µ eff = 1.920 µ B and 1.911 µ B for [001] and [100] directions, respectively. Here, µ B is the Bohr magneton. Using g = S(S + 1)µ eff we get almost equal Lande g-factor values as g = 2.22 and g = 2.20 for [001] and [100], respectively, which are common for Cu 2+ ions.
Here we note, that in regular, collinear antiferromagnets the χ vanishes in the direction where the local magnetic moments are aligned parallel or antiparallel to an external magnetic field and stays constant in the directions where the external field is perpendicular to the local magnetic moments 10 . Thus the local moments here may prefer the [001] direction, but the non-vanishing χ could also just be a signature of the non-collinear order. Above the Néel temperature T N = 9.5 K, the Knight shift K(T) follows perfectly the magnetic susceptibility curve χ (T), as given 11 by Eq. (2): Here, K 0 is the temperature-independent shift, the chemical shift, and H hf is the hyperfine coupling constant. From the Kvs χ plots (insets of FIG. 3) we found the hyperfine field values as H hf = 7.65(5) kOe/µ B for a B direction and slightly smaller value, H hf = 7.40(5) kOe/µ B for direction c B. Below T N , due to the onset of local magnetic fields, the resonance line splits into four lines in the orientation a B and into two lines when the crystal is oriented with c B. Following the approach given in Ref. 7 we will estimate the critical order parameter β of the phase transition from the growth of the internal field B int in the vicinity of the ordering temperature. As a measure of the internal field we use the frequency difference between the most shifted resonance lines instead of the internal ones, which is proportional to the internal field by gyromagnetic ratio 31 γ/2π = 17.237 MHz/T. The temperature dependences of the frequency differences in the two orientations of the single crystal are given in FIG. 4.
The temperature dependence of the frequency differ- ence is fitted by the formula The best fit was obtained with T N = 9.48 K, β = 0.26 for the orientation a B, and T N = 8.82 K, β = 0.20 for c B. The critical exponent β values, if compared to some theoretical values (as given e. g. in Ref. 12 ), indicate that the ordering scheme is the closest to the 2D XY case.
C. 31 P NMR of powder sample Before starting 31 P analysis of single crystal we recorded the spectrum of a powder sample. FIG. 5(a) shows 31 P NMR spectrum recorded with magic angle spinning 13 of the sample. The spectrum shows a single sharp line at isotropic magnetic shift. The spectrum of a static sample (FIG. 5(b)) shows a typical powder line shape with singularities at the principal values of the Knight shift tensor: Determination of tensor orientation in a single crystal is not very easy task 14 . For that, in general case, one needs to record resonance frequencies rotating the sample around three different axes. Depending on symmetry or having some principal values of the shift tensor predetermined, the number of necessary rotation patterns may be smaller. After obtaining those rotation patterns, one needs to find unitary transformation that will transform the principal axis system (PAS) into the crystal frame. As usual, the Hamiltonian of spin-1 2 nucleus consists of the Zeeman and the Knight shift interaction where I = (I x , I y , I z ) T , H 0 = (0, 0, H 0 ), and I x , I y , I z are Pauli matrices: Three successive rotations, each characterized by three Euler angles, transform the Hamiltonian from the principal axes frame (PAS) into the laboratory frame: There are four frames of reference: PAS with diagonal tensor (K P AS ), crystal frame (K * ), goniometer frame (K g ), and laboratory frame (K). The transformations between them are R p , R i , R g with the corresponding Euler angles. We used the conventional Euler ZYZ rotation of the Hamiltonian 15 . In BaTCPO there are eight different positions of the phosphorus ions in the unit cell, each giving the resonance corresponding to different transformation R p . The rotation patterns around the c and a axis at temperatures T = 295 K and T = 18 K are shown in FIG. 6. The  angles b 1 , b 2 , b 3 and c 1 , c 2 , c 3 are unique for each experiment; a 1 , a 2 , a 3 are desired eight sets of the Euler angles transforming the tensor in PAS to the crystal frame for each phosporus site. It turns out that the principal axis K 33 of the Knight shift tensor is tilted by 45 degrees from the crystal c axis. Schematics of the Knight tensor's eight orientations are presented in Fig. 7.

E. Spin-lattice relaxation results
Spin-lattice relaxation T 1 was measured with inversionrecovery pulse sequence at magnetic fields along [001] and [100] directions (Fig. 8). The magnetization recovery was exponential throughout all the measurements: where M (τ ) is the magnetization at delay τ after inversion, M 0 is the equilibrium magnetization, and A ≤ 2 is a constant depending on the accuracy of the inversion. The relaxation rate at T > 60 K is almost constant which is typical for a paramagnetic material, where the relaxation is caused by the fluctuation of the magnetic moments. Before the phase transition at T N , a sharp spike occurs in the relaxation speed which is connected to the rapid slowing of the fluctuations. Below T N , relaxation speed decreases sharply proportional to T 7 . In case of c B, the relaxation rate seems to have a discontinuity in close vicinity of T N . Here we note, that 1/T 1 ∝ T 7 was also observed for SrTCPO 7 .
In paramagnetic temperature region we can use the Moriya's theory of relaxation 16 , where: Here γ N is the nuclear gyromagnetic ratio, S is the nuclear spin, H hf is the hyperfine field as in Eq. 2, z = 2 is the number of nearest Cu 2+ neighbors for the nucleus and ω E = (|J|k B / ) 2zS(S + 1)/3 is the Heisenberg exchange frequency (in units rad −1 ), where z = 2 is the Figure 6. 31 P NMR resonance frequencies by rotating BaTCPO single crystal around the a (a) and c axis (b) at T=295 K (upper panels) and T=18 K (lower panels). The full lines correspond to the angle dependencies according to the parameters given in Table I. Due to symmetry, eight different sites in the unit cell contribute only to two different rotation patterns. The red lines in panel (a) correspond to the sites 1, 3, 6 and 7 (see Table I), and to the sites 2, 3, 6 and 8 in panel (b), the rest of the sites are given by blue lines. number of Cu 2+ ions as nearest neighbors to 31 P, S = 1/2 is the electronic spin, and J is the exchange interaction.
With the hyperfine field value of H hf = 7.650 kOe/µ B and the relaxation rate for the paramagnetic region 1/T 1 = 1410 s −1 we get an estimate for the exchange interaction inside the square cupola J/k B = 35 K with an exchange frequency of ω E = 4.5·10 12 rad/s. This value is in coherence with previous results of J = 3.0 meV = 34.8 K 1 .

F. Local magnetic structure in the ordered state
The resonance frequencies by rotation of the single crystal around the [001] and [100] at T = 6 K are given in Fig. 9. The results are in coherence with the Knight shift temperature dependence (FIG. 3). Once the magnetic field is turned around the c axis, one can see that when a B, there are four different magnetic field projections in the AFM region as given in FIG. 3(a). When the crystal is oriented c B, there are two different local field projections to the external field direction as given in FIG. 3(b). Rotating the sample around c and a gives eight different rotation patterns for eight phosphorus ions in the unit cell. Each rotation pattern can be described  The inset shows that relaxation rate 1/T1 is proportional to T 7 below TN . by the equation: (9) where the constant term K is the Larmor frequency plus average chemical shift; the second term describes the angle dependence of the local field projection to the external field direction. The third term describes the angle dependence of the resonance frequency due to turning of the chemical shift tensor.
The rotation patterns around the [001] axis show two sets of lines (blue and red). The phase shift within the set is 90 degrees. The red lines are shifted from b B by +16 degrees, while the blue lines are shifted -16 degrees. We can assign the blue lines to the phosphorus ions in "up" cupola and the red lines to the ions in "down" cupola. The rotation patterns of the crystal around the [100] (FIG. 9(b), Table IIb) are not so well resolved. Here, the approximation of the frequencies by Eq. 9 is not particularly good. A possible reason might be that the magnetic structure in 4.7 T magnetic field at c B direction is not yet well settled at temperature 6 K. We found above (see FIG. 4) that the ordering temperature in the direction c B was T N = 8.8 K, while in case of b B we had T N = 9.5 K.
Despite of that, one can clearly see the rotation patterns with two different amplitudes as expected for two different local field projections along the b axis. The assignment of the resonances to "up" and "down" cupola is not unique. For example, we cannot distinguish the cases, where all the local field directions of one cupola have positive projection to the c axis and the moments of the other cupola have negative projection, from the case, where the ions of one cupola have two positive and two negative projections and the field on ions of the other cupola have, respectively, two negative and two positive projections to the c axis. The assignment in FIG. 9(b) corresponds to the local field configuration as given below in FIG. 11.
The analysis of the data given in Table II can be carried out using the scheme of the local field direction as given in FIG. 10. Three cosine amplitudes L in Table II correspond  As noted above, unique assignment of the resonances to certain phosphorus in the unit cell is not possible. One possible local field configuration is given in FIG. 11.
The internal field at phosphorus ions consists of two components: where B hf is transferred hyperfine field, and B dip is the dipolar field from the magnetic moments of Cu 2+ ions.   . 9(a)), lower table corresponds to rotation around the a axis (FIG. 9(b)) at temperature T = 6 K.  Figure 10. Scheme of local field direction on a phosphorus ion in BaTCPO crystal. Bint is the local field, Ba, B b , and Bc are the projections of Bint to the crystal a, b, and c axis, respectively. B1, B2, and B3 are the projections of Bint to the bc, ac, and bc planes, respectively. The latter amplitudes can be found from the rotation pattern parameters given in Table II. In the following we assume that B hf is very well cancelled in AF ordered state, and the local magnetic field at phosphorus is due to dipolar field of Cu 2+ ions. We did calculate the dipolar field at each phosphorus ion of the unit cell assuming the two magnetic structures proposed in Refs. 1,6 . In the calculation we did sum the dipolar field from every Cu 2+ inside a sphere of 50 Å around given phosphorus. At that we took into account that the unit cell of magnetic structure is doubled along c direction, i. e. the magnetic moments of every other layer along the c axis were reversed. As a result, we obtain the following dipolar field projections along the a, b, and c axis Comparison of the experimental field pattern to the calculated dipolar fields gives remarkable similarity to the case calculated for Γ 3 (1) structure -the calculated dipolar field directions are close to the experimental values in FIG. 11, although the calculated B c value is relatively small and the calculated local field is 2.7 times larger than the experimental value. Three times larger value of the calculated dipolar field compared to the experimentally determined local field was reported earlier for dipolar field at Ba site in antiferromagnetic YBa 2 Cu 3 O 6.05 Ref. 17 . The authors ascribed this controversy to the possible effect of delocalization of the copper d -electron. Dipolar field calculated for Γ 3 (2) structure is quite different. It is almost confined to the ab plane, with nearly equal B a and B b components. Therefore, we can conclude that the NMR data are better consistent with the magnetic structure Γ 3 (1).

IV. CONCLUSION
We performed 31 P NMR of the antiferromagnetic square cupola compound Ba(TiO)Cu 4 (PO 4 ) 4 in the applied magnetic field B = 4.7 T and provided an in-depth overview of the local magnetic environment around the Cu 2+ cupolas. From the 31 P NMR frequency dependence of the single crystal orientation we successfully determined the principal values of the 31 P magnetic shift tensor and the orientation of eight magnetic tensors in the unit-cell at room temperature and at temperature T = 6 K. The Knight shift temperature dependence in comparison of that of the bulk magnetic susceptibility enabled to determine the hyperfine field on 31 P nuclei H hf = 7.65(5) kOe/µ B for a B and H hf = 7.40(5) kOe/µ B for c B. The temperature dependence of 31 P spin-lattice relaxation resulted in an approximation for the exchange interaction constant between Cu 2+ ions J = 35K. 31 P NMR frequency dependence on the single crystal orientation in the antiferromagnetic state gave a clear picture of local magnetic fields at 31 P ions. The static magnetic field at every phosphorus was determined as B int = 35.6 mT. Experimental configuration of the local field was compared to the calculated dipolar field for several magnetic arrangement of the copper magnetic moments. We found that the magnetic structure Γ 3 (1) determined by the previous neutron diffraction studies 1,6 is most consistent with the NMR data.