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Microscopic magnetism of $A (TiO) {Cu}_{4} {({PO}_{4})}_{4}$ ( $A$ = Ba, Pb, Sr): ${}^{31}P$ and ${}^{63, 65}{Cu}$ NMR study

Riho Rästa ^1,2,*, Ivo Heinmaa ¹, Joosep Link ^1,2, Yusuke Kousaka ³, Tsuyoshi Kimura ⁴, Yoshihiko Ihara ⁵, Kenta Kimura ⁶, and Raivo Stern ^1,7,†

¹National Institute of Chemical Physics and Biophysics, Akadeemia tee 23, 12618 Tallinn, Estonia
²Department of Cybernetics, Tallinn University of Technology, Ehitajate tee 5, 19086 Tallinn, Estonia
³Department of Physics and Electronics, Osaka Metropolitan University, Osaka 599-8531, Japan
⁴Department of Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan
⁵Department of Physics, Faculty of Science, Hokkaido University, Sapporo 060-0810, Japan
⁶Department of Materials Science, Osaka Metropolitan University, Osaka 599-8531, Japan
⁷National High Magnetic Field Laboratory, 1800 East Paul Dirac Drive, Tallahassee, Florida 32310, USA

^*Contact author: riho.rasta@kbfi.ee
^†Contact author: raivo.stern@kbfi.ee

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Phys. Rev. B 113, 184428 – Published 7 May, 2026

DOI: https://doi.org/10.1103/fn12-sjlh

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 analogs in the $A (TiO) {Cu}_{4} {({PO}_{4})}_{4}$ family. Above $T_{N} ≃ 6.7 K$ , the ${}^{31}P$ Knight shift tracks the bulk susceptibility and yields nearly isotropic transferred hyperfine couplings $H_{hf}^{[010]} = 6.77 (3)$ and $H_{hf}^{[001]} = 6.19 (3)$ kOe/ $μ_{B}$ . Below $T_{N}$ , 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 $β ≃ 0.23$ , 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 $B_{int} = 14.50 (6) T$ and a quadrupole frequency $ν_{Q} = 32.72 (5) MHz$ , 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 $H_{hf}$ 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.

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Phys. Rev. B 113, 184428– Published 7 May, 2026

Vol. 113, Iss. 18 — 1 May 2026

Received 11 February 2026
Revised 10 April 2026
Accepted 20 April 2026

DOI: https://doi.org/10.1103/fn12-sjlh

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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Microscopic magnetism of $A (TiO) {Cu}_{4} {({PO}_{4})}_{4}$ ( $A$ = Ba, Pb, Sr): ${}^{31}P$ and ${}^{63, 65}{Cu}$ NMR study

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Microscopic magnetism of A(TiO)Cu4(PO4)4 (A = Ba, Pb, Sr): P31 and Cu63,65 NMR study

Abstract

Physics Subject Headings (PhySH)

Article Text

References (37)

Microscopic magnetism of $A (TiO) {Cu}_{4} {({PO}_{4})}_{4}$ ( $A$ = Ba, Pb, Sr): ${}^{31}P$ and ${}^{63, 65}{Cu}$ NMR study