Microscopic Magnetism of A(TiO)Cu4(PO4)4 (A = Ba, Pb, Sr): 31P and 63,65Cu NMR Study

Abstract

We report a comprehensive NMR study of the chiral square-cupola antiferromagnet Pb(TiO)Cu$_4$(PO$_4$)$_4$ and compare its microscopic hyperfine and local-field parameters with the Ba/Sr analogues in the $A$(TiO)Cu$_4$(PO$_4$)$_4$ family. Above $T_{\rm N}\simeq 6.7$ K, the $^{31}$P Knight shift tracks the bulk susceptibility and yields nearly isotropic transferred hyperfine couplings $H_{\rm hf}^{[010]}=6.77(3)$ and $H_{\rm hf}^{[001]}=6.19(3)$ kOe/$\mu_{\rm B}$. Below $T_{\rm N}$, the frequency-swept $^{31}$P spectrum splits into three lines, in contrast to the four-line pattern reported for BaTCPO. The line separation tracks the onset of the static $^{31}$P internal field with a power-law exponent $\beta\simeq 0.23$, consistent with quasi-two-dimensional criticality. Crystal-rotation $^{31}$P NMR in the ordered state resolves all eight symmetry-related P sites and their site-dependent anisotropy. In the ordered state, zero-field $^{63,65}$Cu NMR gives a Cu-site internal field $B_{\rm int}=14.50(6)$ T and a quadrupole frequency $\nu_Q=32.72(5)$ MHz, while point-charge electric-field-gradient calculations including Sternheimer corrections yield an on-site Cu hole occupancy $n_d=0.20(4)$, consistent with a ligand-hole-dominated charge-transfer character. Comparing PbTCPO with BaTCPO and SrTCPO, we find that the transferred hyperfine coupling $H_{\rm hf}$ varies across the series, reflecting changes in local Cu-O-P covalency, whereas the ordered-state $^{31}$P internal field in PbTCPO is $69.5$ mT, considerably higher than in BaTCPO ($35.6$ mT) and SrTCPO ($34.6$ mT). This enhancement is not captured by dipolar terms alone and points to the combined effects of transferred contributions and stacking-dependent cancellation.