- Letter
- Open Access
Pressure-invariant isotope effect as evidence for electronically driven intertwined order in
Phys. Rev. B 114, L080508 – Published 24 August, 2026
DOI: https://doi.org/10.1103/wzcz-z62k
Abstract
We report muon-spin rotation measurements of the pressure dependence of the oxygen-isotope effect on the spin-density wave (SDW) transition in the trilayer Ruddlesden-Popper nickelate . At ambient pressure, the SDW transition shows a finite isotope shift, with and . Under hydrostatic pressure, decreases linearly at rates that are equal within uncertainty for the two isotope compositions, K/GPa and K/GPa, such that the isotope shift remains constant within uncertainty under compression. The absence of pressure enhancement of the isotope effect points to a predominantly electronic origin of the SDW transition and is consistent with recent inelastic x-ray scattering results, suggesting a regime of intertwined order in trilayer RP nickelates, which is stabilized by strong spin interactions.
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References (116)
- E. Maxwell, Isotope effect in the superconductivity of mercury, Phys. Rev. 78, 477 (1950).
- C. A. Reynolds, B. Serin, and L. B. Nesbitt, The isotope effect in superconductivity. I. Mercury, Phys. Rev. 84, 691 (1951).
- J. L. Olsen, Superconductivity of cadmium isotopes, Cryogenics 2, 356 (1962).
- R. W. Shaw, D. E. Mapother, and D. C. Hopkins, Isotope effect in superconducting lead, Phys. Rev. 121, 86 (1961).
- B. T. Matthias, T. H. Geballe, E. Corenzwit, and G. W. Hull Jr, Intermediate isotope effect of molybdenum and , Phys. Rev. 129, 1025 (1963).
- E. Bucher and C. Palmy, Superconductivity and isotope effect in compounds and molybdenum, Phys. Lett. A 24, 340 (1967).
- R. E. Fassnacht and J. R. Dillinger, Isotope effect in superconducting zinc, Phys. Rev. Lett. 17, 255 (1966).
- S. L. Bud'ko, G. Lapertot, C. Petrovic, C. E. Cunningham, N. Anderson, and P. C. Canfield, Boron isotope effect in superconducting , Phys. Rev. Lett. 86, 1877 (2001).
- D. G. Hinks, H. Clauss, and J. D. Jorgensen, The complex nature of superconductivity in as revealed by the reduced total isotope effect, Nature (London) 411, 457 (2001).
- R. A. Hein and J. W. Gibson, Isotope effect in superconducting osmium, Phys. Rev. 131, 1105 (1963).
- T. H. Geballe, B. T. Matthias, G. W. Hull Jr., and E. Corenzwit, Absence of an isotope effect in superconducting ruthenium, Phys. Rev. Lett. 6, 275 (1961).
- D. K. Finnemore and D. E. Mapother, Absence of an isotope effect in superconducting ruthenium, Phys. Rev. Lett. 9, 288 (1962).
- J. W. Gibson and R. A. Hein, Critical magnetic fields of superconducting ruthenium isotopes and search for an isotope effect, Phys. Rev. 141, 407 (1966).
- T. H. Geballe and B. T. Matthias, Isotope effects in low temperature superconductors, IBM J. Res. Dev. 6, 256 (1962).
- R. D. Fowler, J. D. G. Lindsay, R. W. White, H. H. Hill, and B. T. Matthias, Positive isotope effect on the superconducting transition temperature of α-, Phys. Rev. Lett. 19, 892 (1967).
- B. Stritzker and W. Buckel, Superconductivity in the palladium-hydrogen and the palladium-deuterium systems, Z. Phys. 257, 1 (1972).
- B. Batlogg, G. Kourouklis, W. Weber, R. J. Cava, A. Jayaraman, A. E. White, K. T. Short, L. W. Rupp, and E. A. Rietman, Nonzero isotope effect in , Phys. Rev. Lett. 59, 912 (1987).
- Tanya A. Faltens, W. K. Ham, S. W. Keller, K. J. Leary, J. N. Michaels, A. M. Stacy, Hans-Conrad zur Loye, D. E. Morris, T. W. Barbee III, L. C. Bourne, M. L. Cohen, S. Hoen, and A. Zettl, Observation of an oxygen isotope shift in the superconducting transition temperature of , Phys. Rev. Lett. 59, 915 (1987).
- B. Batlogg, R. J. Cava, A. Jayaraman, R. B. van Dover, G. A. Kourouklis, S. Sunshine, D. W. Murphy, L. W. Rupp, H. S. Chen, A. White, K. T. Short, A. M. Mujsce, and E. A. Rietman, Isotope effect in the superconductors and , Phys. Rev. Lett. 58, 2333 (1987).
- L. C. Bourne, M. F. Crommie, A. Zettl, H.-C. zur Loye, S. W. Keller, K. L. Leary, A. M. Stacy, K. J. Chang, M. L. Cohen, and D. E. Morris, Search for isotope effect in superconducting Y-Ba-Cu-O, Phys. Rev. Lett. 58, 2337 (1987).
- J. P. Franck, J. Jung, M.-K. Mohamed, S. Gygax, and G. I. Sproule, Observation of an oxygen isotope effect in superconducting (), Phys. Rev. B 44, 5318 (1991).
- N. Babushkina, A. Inyushkin, V. Ozhogin, A. Taldenkov, I. Kobrin, T. Vorobeva, L. Molchanova, L. Damyanets, T. Uvarova, and A. Kuzakov, Oxygen isotope effect in LaSrCuO system with modified by Ni and Zn doping, Physica C 185-189, 901 (1991).
- Guo-meng Zhao, M. B. Hunt, H. Keller, and K. A. Müller, Evidence for polaronic supercarriers in the copper oxide superconductors , Nature (London) 385, 236 (1997).
- D. Zech, H. Keller, K. Conder, E. Kaldis, E. Liarokapis, N. Poulakis, and K. A. Müller, Site-selective oxygen isotope effect in optimally doped , Nature (London) 371, 681 (1994).
- D. Zech, K. Conder, H. Keller, E. Kaldis, and K. A. Müller, Doping dependence of the oxygen isotope effect in , Physica B 219-220, 136 (1996).
- R. Khasanov, A. Shengelaya, K. Conder, E. Morenzoni, I. M. Savić, and H. Keller, The oxygen-isotope effect on the in-plane penetration depth in underdoped as revealed by muon-spin rotation, J. Phys.: Condens. Matter 15, L17 (2003).
- R. Khasanov, A. Shengelaya, E. Morenzoni, M. Angst, K. Conder, I. M. Savić, D. Lampakis, E. Liarokapis, A. Tatsi, and H. Keller, Site-selective oxygen isotope effect on the magnetic-field penetration depth in underdoped , Phys. Rev. B 68, 220506(R) (2003).
- J. L. Tallon, R. S. Islam, J. Storey, G. V. M. Williams, and J. R. Cooper, Isotope effect in the superfluid density of high-temperature superconducting cuprates: Stripes, pseudogap, and impurities, Phys. Rev. Lett. 94, 237002 (2005).
- H. Keller, A. Bussmann-Holder, and K. A. Müller, Jahn–Teller physics and high- superconductivity Mater. Today 11, 38 (2008).
- R. Khasanov, D. G. Eshchenko, H. Luetkens, E. Morenzoni, T. Prokscha, A. Suter, N. Garifianov, M. Mali, J. Roos, K. Conder, and H. Keller, Direct observation of the oxygen isotope effect on the in-plane magnetic field penetration depth in optimally doped , Phys. Rev. Lett. 92, 057602 (2004).
- Z. Q. Mao, Y. Maeno, Y. Mori, S. Sakita, S. Nimori, and M. Udagawa, Sign reversal of the oxygen isotope effect on in , Phys. Rev. B 63, 144514 (2001).
- R. Khasanov, A. Shengelaya, K. Conder, E. Morenzoni, I. M. Savić, J. Karpinski, and H. Keller, Correlation between oxygen isotope effects on transition temperature and magnetic penetration depth in high-temperature superconductors close to optimal doping, Phys. Rev. B 74, 064504 (2006).
- R. Khasanov, A. Shengelaya, D. Di Castro, E. Morenzoni, A. Maisuradze, I. M. Savić, K. Conder, E. Pomjakushina, A. Bussmann-Holder, and H. Keller, Oxygen isotope effects on the superconducting transition and magnetic states within the phase diagram of , Phys. Rev. Lett. 101, 077001 (2008).
- C. W. Rischau, D. Pulmannová, G. W. Scheerer, A. Stucky, E. Giannini, and D. van der Marel, Isotope tuning of the superconducting dome of strontium titanate, Phys. Rev. Res. 4, 013019 (2022).
- J. A. Schlueter, A. M. Kini, B. H. Ward, U. Geiser, H. H. Wang, J. Mohtasham, R. W. Winter, and G. L. Gard, Universal inverse deuterium isotope effect on the of BEDT-TTF-based molecular superconductors, Physica C 351, 261 (2001).
- R. H. Liu, T. Wu, G. Wu, H. Chen, X. F. Wang, Y. L. Xie, J. J. Yin, Y. J. Yan, Q. J. Li, B. C. Shi, W. S. Chu, Z. Y. Wu, and X. H. Chen, A large iron isotope effect in and , Nature (London) 459, 64 (2009).
- P. M. Shirage, K. Kihou, K. Miyazawa, C.-H. Lee, H. Kito, H. Eisaki, Y. Tanaka, and A. Iyo, Inverse iron isotope effect on the transition temperature of the superconductor, Phys. Rev. Lett. 103, 257003 (2009).
- R. Khasanov, M. Bendele, K. Conder, H. Keller, E. Pomjakushina, and V. Pomjakushin, Iron isotope effect on the superconducting transition temperature and the crystal structure of , New J. Phys. 12, 073024 (2010).
- R. Khasanov, M. Bendele, A. Bussmann-Holder, and H. Keller, Intrinsic and structural isotope effects in iron-based superconductors, Phys. Rev. B 82, 212505 (2010).
- G. M. Zhao, K. Conder, H. Keller, and K. A. Müller, Giant oxygen isotope shift in the magnetoresistive perovskite , Nature (London) 381, 676 (1996).
- Z. Guguchia, R. Khasanov, M. Bendele, E. Pomjakushina, K. Conder, A. Shengelaya, and H. Keller, Negative oxygen isotope effect on the static spin stripe order in superconducting () observed by muon-spin rotation, Phys. Rev. Lett. 113, 057002 (2014).
- M. Medarde, P. Lacorre, K. Conder, F. Fauth, and A. Furrer, Giant isotope effect on the metal-insulator transition of perovskites ( earth), Phys. Rev. Lett. 80, 2397 (1998).
- H. Luetkens, M. Stingaciu, Y. G. Pashkevich, P. Lemmens, E. Pomjakushina, K. Conder, and H.-H. Klauss, Oxygen isotope effect on the AFM–FM phase transition of the layered cobaltite , J. Magn. Magn. Mater. 310, 1566 (2007).
- E. Amit, A. Keren, J. S. Lord, and P. A. King, Precise measurement of the oxygen isotope effect on the Néel temperature in cuprates, Adv. Condens. Matter Phys. 2011, 178190 (2011).
- R. Khasanov, V. Sazgari, I. Plokhikh, L. Shi, K. Y. Ma, M. Medarde, E. Pomjakushina, T. Klimczuk, T. J. Hicken, H. Luetkens, C. W. Schneider, Z. Guguchia, S. Medvedev, and D. J. Gawryluk, Oxygen-isotope effect on the density wave transitions in , Phys. Rev. Res. 8, L012055 (2026).
- R. Khasanov, V. Sazgari, T. J. Hicken, I. Plokhikh, M. Medarde, E. Pomjakushina, L. Keller, V. Pomjakushin, M. Bartkowiak, S. Królak, M. J. Winiarski, A. Steppke, J. A. Krieger, H. Luetkens, T. Klimczuk, C. W. Schneider, D. J. Gawryluk, and Z. Guguchia, Pressure and oxygen-isotope substitution on density-wave transitions in , Phys. Rev. Res. 8, 013249 (2026).
- A. Shengelaya, G.-M. Zhao, C. M. Aegerter, K. Conder, I. M. Savić, and H. Keller, Giant oxygen isotope effect on the spin glass transition in as revealed by muon spin rotation, Phys. Rev. Lett. 83, 5142 (1999).
- G.-M. Zhao, K. K. Singh, and D. E. Morris, Oxygen isotope effect on Néel temperature in various antiferromagnetic cuprates, Phys. Rev. B 50, 4112 (1994).
- A. Lanzara, G.-M. Zhao, N. L. Saini, A. Bianconi, K. Conder, H. Keller, and K. A. Müller, Oxygen-isotope shift of the charge-stripe ordering temperature in from x-ray absorption spectroscopy, J. Phys.: Condens. Matter 11, L541 (1999).
- Z. Guguchia, D. Sheptyakov, E. Pomjakushina, K. Conder, R. Khasanov, A. Shengelaya, A. Simon, A. Bussmann-Holder, and H. Keller, Oxygen isotope effects on lattice properties of (), Phys. Rev. B 92, 024508 (2015).
- M. Bendele, F. von Rohr, Z. Guguchia, E. Pomjakushina, K. Conder, A. Bianconi, A. Simon, A. Bussmann-Holder, and H. Keller, Evidence for strong lattice effects as revealed from huge unconventional oxygen isotope effects on the pseudogap temperature in , Phys. Rev. B 95, 014514 (2017).
- H. Sun, M. Huo, X. Hu, J. Li, Z. Liu, Y. Han, L. Tang, Z. Mao, P. Yang, B. Wang, J. Cheng, D.-X. Yao, G.-M. Zhang, and M. Wang, Signatures of superconductivity near 80 K in a nickelate under high pressure, Nature (London) 621, 493 (2023).
- Y. Zhang, D. Su, Y. Huang, Z. Shan, H. Sun, M. Huo, K. Ye, J. Zhang, Z. Yang, Y. Xu, Y. Su, R. Li, M. Smidman, M. Wang, L. Jiao, and H. Yuan, High-temperature superconductivity with zero resistance and strange-metal behaviour in , Nat. Phys. 20, 1269 (2024).
- N. Wang, G. Wang, X. Shen, J. Hou, J. Luo, X. Ma, H. Yang, L. Shi, J. Dou, J. Feng, J. Yang, Y. Shi, Z. Ren, H. Ma, P. Yang, Z. Liu, Y. Liu, H. Zhang, X. Dong, Y. Wang, et al., Bulk high-temperature superconductivity in pressurized tetragonal , Nature (London) 634, 579 (2024).
- Z. Liu, M. Huo, J. Li, Q. Li, Y. Liu, Y. Dai, X. Zhou, J. Hao, Y. Lu, M. Wang, and H.-H. Wen, Electronic correlations and energy gap in the bilayer nickelate , Nat. Commun. 15, 7570 (2024).
- M. Zhang, C. Pei, Q. Wang, Y. Zhao, C. Li, W. Cao, S. Zhu, J. Wu, and Y. Qi, Effects of pressure and doping on Ruddlesden-Popper phases , J. Mater. Sci. Technol. 185, 147 (2024).
- Y. Li, Y. Cao, L. Liu, P. Peng, H. Lin, C. Pei, M. Zhang, H. Wu, X. Du, W. Zhao, K. Zhai, X. Zhang, J. Zhao, M. Lin, P. Tan, Y. Qi, G. Li, H. Guo, L. Yang, and L. Yang, Distinct ultrafast dynamics of bilayer and trilayer nickelate superconductors regarding the density-wave-like transitions, Sci. Bull. 70, 180 (2025).
- Y. Li, X. Du, Y. Cao, C. Pei, M. Zhang, W. Zhao, K. Zhai, R. Xu, Z. Liu, Z. Li, J. Zhao, G. Li, Y. Qi, H. Guo, Y. Chen, and L. Yang, Electronic correlation and pseudogap-like behavior of high-temperature superconductor , Chin. Phys. Lett. 41, 087402 (2024).
- Y. Zhou, J. Guo, S. Cai, H. Sun, P. Wang, J. Zhao, J. Han, X. Chen, Y. Chen, Q. Wu, Y. Ding, T. Xiang, H.-K. Mao, and L. Sun, Investigations of key issues on the reproducibility of high Tc superconductivity emerging from compressed , Matter Radiat. Extremes 10, 027801 (2025).
- X. Zhou, W. He, Z. Zhou, K. Ni, M. Huo, D. Hu, Y. Zhu, E. Zhang, Z. Jiang, S. Zhang, S. Su, J. Jiang, Y. Yan, Y. Wang, D. Shen, X. Liu, J. Zhao, M. Wang, M. Liu, Z. Du, et al., Revealing nanoscale structural phase separation in single crystal via scanning near field optical microscopy, arXiv:2410.06602.
- H. Sakakibara, M. Ochi, H. Nagata, Y. Ueki, H. Sakurai, R. Matsumoto, K. Terashima, K. Hirose, H. Ohta, M. Kato, Y. Takano, and K. Kuroki, Theoretical analysis on the possibility of superconductivity in the trilayer Ruddlesden-Popper nickelate under pressure and its experimental examination: Comparison with , Phys. Rev. B 109, 144511 (2024).
- M. Wang, H.-H. Wen, T. Wu, D.-X. Yao, and T. Xiang, Normal and superconducting properties of , Chin. Phys. Lett. 41, 077402 (2024).
- C. Pei, M. Zhang, D. Peng, S. Huangfu, S. Zhu, Q. Wang, J. Wu, Z. Xing, L. Zhang, Y. Chen, J. Zhao, W. Yang, H. Suo, H. Guo, Q. Zeng, and Y. Qi, Pressure-induced superconductivity in single crystals, Sci. China Phys. Mech. Astron. 69, 237011 (2026).
- G. Wang, N. N. Wang, X. L. Shen, J. Hou, L. Ma, L. F. Shi, Z. A. Ren, Y. D. Gu, H. M. Ma, P. T. Yang, Z. Y. Liu, H. Z. Guo, J. P. Sun, G. M. Zhang, S. Calder, J.-Q. Yan, B. S. Wang, Y. Uwatoko, and J.-G. Cheng, Pressure-induced superconductivity in polycrystalline , Phys. Rev. X 14, 011040 (2024).
- F. Li, Y. Hao, N. Guo, D. Jia, J. Dou, D. Peng, G. Zhang, L. Xiao, J. Zhang, W. Zhou, Y. Huang, X. Wang, Z. Guo, M. Mezouar, J. E. F. S. Rodrigues, J. Luo, J. Yang, Q. Zeng, R. Zhou, H. Gou, Q. Zheng, G. Liu, and J. Zhang, Single-crystal structure determination of superconducting under high pressure, Adv. Mater. 37, e07365 (2025).
- P. Puphal, P. Reiss, N. Enderlein, Y.-M. Wu, G. Khaliullin, V. Sundaramurthy, T. Priessnitz, M. Knauft, A. Suthar, L. Richter, M. Isobe, P. A. van Aken, H. Takagi, B. Keimer, Y. E. Suyolcu, B. Wehinger, P. Hansmann, and M. Hepting, Unconventional crystal structure of the high-pressure superconductor , Phys. Rev. Lett. 133, 146002 (2024).
- Y. Zhu, D. Peng, E. Zhang, B. Pan, X. Chen, L. Chen, H. Ren, F. Liu, Y. Hao, N. Li, Z. Xing, F. Lan, J. Han, J. Wang, D. Jia, H. Wo, Y. Gu, Y. Gu, L. Ji, W. Wang, et al., Superconductivity in pressurized trilayer single crystals, Nature (London) 631, 531 (2024).
- E. Zhang, D. Peng, Y. Zhu, L. Chen, B. Cui, X. Wang, W. Wang, Q. Zeng, and J. Zhao, Bulk superconductivity in pressurized trilayer nickelate single crystals, Phys. Rev. X 15, 021008 (2025).
- R. Khasanov, T. J. Hicken, D. J. Gawryluk, V. Sazgari, I. Plokhikh, Loïc Pierre Sorel, M. Bartkowiak, S. Bötzel, F. Lechermann, I. M. Eremin, H. Luetkens, and Z. Guguchia, Pressure-enhanced splitting of density wave transitions in , Nat. Phys. 21, 430 (2025).
- C. Huang, J. Li, X. Huang, H. Zhang, D. Hu, M. Huo, X. Chen, Z. Chen, H. Sun, and M. Wang, Superconductivity in monolayer-trilayer phase of under high pressure, arXiv:2510.12250.
- R. Khasanov, I. Plokhikh, T. J. Hicken, H. Luetkens, D. J. Gawryluk, and Z. Guguchia, Pressure effect on the spin density wave transition in , Phys. Rev. Res. 7, L022046 (2025).
- R. Khasanov, T. J. Hicken, H. Luetkens, Z. Guguchia, D. J. Gawryluk, V. Sundaramurthy, A. Suthar, M. Isobe, B. Keimer, G. Khaliullin, M. Hepting, and P. Puphal, Magnetism of the alternating monolayer-trilayer phase of , arXiv:2512.22371.
- K. Chen, X. Liu, J. Jiao, M. Zou, C. Jiang, X. Li, Y. Luo, Q. Wu, N. Zhang, Y. Guo, and L. Shu, Evidence of spin density waves in , Phys. Rev. Lett. 132, 256503 (2024).
- K. Chen, X. Liu, Y. Wang, J. Han, Z. Zhu, J. Jiao, C. Jiang, Y. Guo, C. Zheng, and L. Shu, Effect of Pr-doping and oxygen vacancies on spin density wave in : A investigation, Phys. Rev. Res. 7, L032014 (2025).
- D. Zhao, Y. Zhou, M. Huo, Y. Wang, L. Nie, Y. Yang, J. Ying, M. Wang, T. Wu, and X. Chen, Pressure-enhanced spin-density-wave transition in double-layer nickelate , Sci. Bull. 70, 1239 (2025).
- M. Kakoi, T. Oi, Y. Ohshita, M. Yashima, K. Kuroki, T. Kato, H. Takahashi, S. Ishiwata, Y. Adachi, N. Hatada, T. Uda, and H. Mukuda, Multiband metallic ground state in multilayered nickelates and probed by -NMR at ambient pressure, J. Phys. Soc. Jpn. 93, 053702 (2024).
- X. Chen, J. Choi, Z. Jiang, J. Mei, K. Jiang, J. Li, S. Agrestini, M. Garcia-Fernandez, X. Huang, H. Sun, D. Shen, M. Wang, J. Hu, Y. Lu, K.-J. Zhou, and D. Feng, Electronic and magnetic excitations in , Nat. Commun. 15, 9597 (2024).
- X. Ren, R. Sutarto, X. Wu, J. Zhang, H. Huang, T. Xiang, J. Hu, R. Comin, X. J. Zhou, and Z. Zhu, Resolving the electronic ground state of films, Commun. Phys. 8, 52 (2025).
- N. K. Gupta, R. Gong, Y. Wu, M. Kang, C. T. Parzyck, B. Z. Gregory, N. Costa, R. Sutarto, S. Sarker, A. Singer, D. G. Schlom, K. M. Shen, and D. G. Hawthorn, Anisotropic spin stripe domains in bilayer , Nat. Commun. 16, 6560 (2025).
- J. Luo, J. Feng, G. Wang, N. N. Wang, J. Dou, A. F. Fang, J. Yang, J. G. Cheng, G.-Q. Zheng, and R. Zhou, Microscopic evidence of charge- and spin-density waves in revealed by -NQR, Chin. Phys. Lett. 42, 067402 (2025).
- H. Wang, L. Chen, A. Rutherford, H. Zhou, and W. Xie, Long-range structural order in a hidden phase of Ruddlesden–Popper bilayer nickelate , Inorg. Chem. 63, 5020 (2024).
- G. Wu, J. J. Neumeier, and M. F. Hundley, Magnetic susceptibility, heat capacity, and pressure dependence of the electrical resistivity of and , Phys. Rev. B 63, 245120 (2001).
- Z. Liu, H. Sun, M. Huo, X. Ma, Y. Ji, E. Yi, L. Li, H. Liu, J. Yu, Z. Zhang, Z. Chen, F. Liang, H. Dong, H. Guo, D. Zhong, B. Shen, S. Li, and M. Wang, Evidence for charge and spin density waves in single crystals of and , Sci. China Phys. Mech. Astron. 66, 217411 (2023).
- D.-K. Seo, W. Liang, M.-H. Whangbo, Z. Zhang, and M. Greenblatt, Electronic band structure and Madelung potential study of the nickelates , , and , Inorg. Chem. 35, 6396 (1996).
- J. Zhang, D. Phelan, A. S. Botana, Y.-S. Chen, H. Zheng, M. Krogstad, S. G. Wang, Y. Qiu, J. A. Rodriguez-Rivera, R. Osborn, S. Rosenkranz, M. R. Norman, and J. F. Mitchell, Intertwined density waves in a metallic nickelate, Nat. Commun. 11, 6003 (2020).
- A. M. Samarakoon, J. Strempfer, J. Zhang, F. Ye, Y. Qiu, J.-W. Kim, H. Zheng, S. Rosenkranz, M. R. Norman, J. F. Mitchell, and D. Phelan, Bootstrapped dimensional crossover of a spin density wave, Phys. Rev. X 13, 041018 (2023).
- M. R. Norman, Landau theory of the density wave transition in trilayer Ruddlesden-Popper nickelates, Phys. Rev. B 112, 075149 (2025).
- X. Jia, Y. Shen, H. LaBollita, X. Chen, J. Zhang, Y. Li, H. Zhao, M. G. Kanatzidis, M. Krogstad, H. Zheng, A. H. Said, A. Alatas, S. Rosenkranz, D. Phelan, M. P. M. Dean, M. R. Norman, J. F. Mitchell, A. S. Botana, and Y. Cao, Lattice-charge coupling in a trilayer nickelate with intertwined density wave order, Phys. Rev. X 16, 011013 (2026).
- See Supplemental Material at http://link.aps.org/supplemental/10.1103/wzcz-z62k for details of the sample preparation, room-temperature x-ray diffraction, thermogravimetric analysis, and the experimental methods employed in this work, including muon-spin rotation/relaxation () and Raman spectroscopy, which includes Refs. [105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116].
- K. Conder, Oxygen diffusion in the superconductors of the YBaCuO family: Isotope exchange measurements and models, Mater. Sci. Eng. R Rep. 32, 41 (2001).
- K. Conder, Material aspects of oxygen isotope effect studies in high-temperature superconductors, Physica C 614, 1354376 (2023).
- M. Cardona and M. L. W. Thewalt, Isotope effects on the optical spectra of semiconductors, Rev. Mod. Phys. 77, 1173 (2005).
- J. Menéndez, J. B. Page, and S. Guha, The isotope effect on the Raman spectrum of molecular , Philos. Mag. B 70, 651 (1994).
- J. M. Zhang, T. Ruf, M. Cardona, O. Ambacher, M. Stutzmann, J.-M. Wagner, and F. Bechstedt, Raman spectra of isotopic GaN, Phys. Rev. B 56, 14399 (1997).
- R. Khasanov, T. J. Hicken, Z. Guguchia, S. Huangfu, H. Luetkens, E. Pomjakushina, V. Pomjakushin, A. Schilling, I. Plokhikh, and D. J. Gawryluk, Multiple magnetic transitions in the trilayer nickelate revealed by muon-spin rotation, Supercond. Sci. Technol. 39, 085020 (2026).
- R. Khasanov, Z. Guguchia, A. Maisuradze, D. Andreica, M. Elender, A. Raselli, Z. Shermadini, T. Goko, F. Knecht, E. Morenzoni, and A. Amato, High pressure research using muons at the Paul Scherrer Institute, High Pressure Res. 36, 140 (2016).
- Z. Shermadini, R. Khasanov, M. Elender, G. Simutis, Z. Guguchia, K. V. Kamenev, and A. Amato, A low-background piston–cylinder-type hybrid high pressure cell for muon-spin rotation/relaxation experiments, High Press. Res. 37, 449 (2017).
- R. Khasanov, Perspective on muon-spin rotation/relaxation under hydrostatic pressure, J. Appl. Phys. 132, 190903 (2022).
- R. Khasanov, R. Urquhart, M. Elender, and K. Kamenev, Three-wall piston-cylinder type pressure cell for muon-spin rotation/relaxation experiments, High Pressure Res. 42, 29 (2022).
- R. J. McQueeney, Y. Petrov, T. Egami, M. Yethiraj, G. Shirane, and Y. Endoh, Anomalous dispersion of LO phonons in at low temperatures, Phys. Rev. Lett. 82, 628 (1999).
- D. Reznik, L. Pintschovius, M. Ito, S. Iikubo, M. Sato, H. Goka, M. Fujita, K. Yamada, G. Gu, and J. Tranquada, Electron–phonon coupling reflecting dynamic charge inhomogeneity in copper oxide superconductors, Nature (London) 440, 1170 (2006).
- A. Q. Baron, J. P. Sutter, S. Tsutsui, H. Uchiyama, T. Masui, S. Tajima, R. Heid, and K.-P. Bohnen, First study of the buckling phonon mode in optimally doped, de-twinned, by inelastic x-ray scattering, J. Phys. Chem. Solids 69, 3100 (2008).
- M. Le Tacon, A. Bosak, S. Souliou, G. Dellea, T. Loew, R. Heid, K. Bohnen, G. Ghiringhelli, M. Krisch, and B. Keimer, Inelastic x-ray scattering in reveals giant phonon anomalies and elastic central peak due to charge-density-wave formation, Nat. Phys. 10, 52 (2014).
- H. Miao, D. Ishikawa, R. Heid, M. Le Tacon, G. Fabbris, D. Meyers, G. D. Gu, A. Q. R. Baron, and M. P. M. Dean, Incommensurate phonon anomaly and the nature of charge density waves in cuprates, Phys. Rev. X 8, 011008 (2018).
- V. V. Poltavets, K. A. Lokshin, T. Egami, and M. Greenblatt, The oxygen deficient Ruddlesden–Popper () phase: Structure and properties, Mater. Res. Bull. 41, 955 (2006).
- I. Plokhikh, T. J. Hicken, L. Keller, V. Pomjakushin, S. H. Moody, P. Foury-Leylekian, J. J. Krieger, H. Luetkens, Z. Guguchia, R. Khasanov, and D. J. Gawryluk, Unraveling spin density wave order in layered nickelates and via neutron diffraction, Phys. Rev. Res. 8, 033151 (2026).
- P. Atkins, J. de Paula, and J. Keeler, Atkins' Physical Chemistry, 12th ed. (Oxford University Press, Oxford, 2023).
- J. Rodríguez-Carvajal, Recent advances in magnetic structure determination by neutron powder diffraction, Physica B 192, 55 (1993).
- J. Rodriguez-Carvajal, J. Gonzalez-Platas, and N. A. Katcho, Magnetic structure determination and refinement using FullProf, Acta Cryst. B 81, 302 (2025).
- T. Roisnel and J. Rodríguez-Carvajal, WinPLOTR: A Windows tool for powder diffraction pattern analysis, Mater. Sci. Forum 378-381, 118 (2001).
- O. Arcelus, J. Rodríguez-Carvajal, N. A. Katcho, M. Reynaud, A. P. Black, D. Chatzogiannakis, C. Frontera, J. Serrano-Sevillano, M. Ismail, J. Carrasco, F. Fauth, M. R. Palacin, and M. Casas-Cabanas, FullProfAPP: a graphical user interface for the streamlined automation of powder diffraction data analysis, J. Appl. Cryst. 57, 1676 (2024).
- A. Amato, H. Luetkens, K. Sedlak, A. Stoykov, R. Scheuermann, M. Elender, A. Raselli, and D. Graf, The new versatile general purpose surface-muon instrument (GPS) based on silicon photomultipliers for measurements on a continuous-wave beam, Rev. Sci. Instrum. 88, 093301 (2017).
- A. Suter and B. Wojek, Musrfit: A free platform-independent framework for data analysis, Phys. Procedia 30, 69 (2012).
- A. Amato and E. Morenzoni, Introduction to Muon Spin Spectroscopy: Applications to Solid State and Material Sciences, Lecture Notes in Physics Vol. 961 (Springer Nature Switzerland, Cham, 2024).
- A. Yaouanc and P. Dalmas de Reotier, Muon Spin Rotation, Relaxation, and Resonance: Applications to Condensed Matter (Oxford University Press, Oxford, 2011).
- Muon Spectroscopy. An Introduction, edited by S. J. Blundell, R. De Renzi, T. Lancaster, and F. L. Pratt (Oxford University Press, Oxford, 2022).