Negative thermal expansion: Mechanisms and materials

Abstract

Negative thermal expansion (NTE) of materials is an intriguing phenomenon challenging the concept of traditional lattice dynamics and of importance for a variety of applications. Progresses in this field develop markedly and update continuously our knowledge on the NTE behavior of materials. In this article, we review the most recent understandings on the underlying mechanisms (anharmonic phonon vibration, magnetovolume effect, ferroelectrorestriction and charge transfer) of thermal shrinkage and the development of NTE materials under each mechanism from both the theoretical and experimental aspects. Besides the low frequency optical phonons which are usually accepted as the origins of NTE in framework structures, NTE driven by acoustic phonons and the interplay between anisotropic elasticity and phonons are stressed. Based on the data documented, some problems affecting applications of NTE materials are discussed and strategies for discovering and design novel framework structured NET materials are also presented.

Keywords

negative thermal expansion / mechanisms of thermal contraction / negative thermal expansion materials / lattice thermal dynamics / magnetovolume effect / ferroelectrostriction / charge transfer / anisotropic elasticity

References

[1] L. P. Prisco, P. I. Ponton, M. V. Guaman, R. R. Avillez, C. P. Romao, M. B. Johnson, M. A. White, and B. A. Marinkovic, Assessment of the thermal shock resistance figures of merit of Al₂W₃O₁₂, a low thermal expansion ceramic, J. Am. Ceram. Soc. 99(5), 1742 (2016)

[2] T. A. Mary, J. S. O. Evans, T. Vogt, and A. W. Sleight, Negative thermal expansion from 0.3 to 1050 Kelvin in ZrW₂O₈, Science 272(5258), 90 (1996)

[3] N. Zhang, L. Li, M. Wu, Y. Li, D. Feng, C. Liu, Y. Mao, J. Guo, M. Chao, and E. Liang, Negative thermal expansion and electrical properties of α-Cu₂V₂O₇, J. Eur. Ceram. Soc. 36(11), 2761 (2016)

[4] X. H. Ge, Y. C. Mao, X. S. Liu, Y. G. Cheng, B. H. Yuan, M. J. Chao, and E. J. Liang, Negative thermal expansion and broad band photo luminescence in a novel material of ZrScMo₂VO₁₂, Sci. Rep. 6(1), 24832 (2016)

[5] N. K. Shi, A. Sanson, Q. Gao, Q. Sun, Y. Ren, Q. Z. Huang, D. O. deSouza, X. R. Xing, and J. Chen, Strong negative thermal expansion in a low-cost and facile oxide of Cu₂P₂O₇, J. Am. Chem. Soc. 142(6), 3088 (2020)

[6] M. K. Gupta, R. Mittal, and S. L. Chaplot, Negative thermal expansion behavior in orthorhombic Sc₂(MoO₄)₃ andSc₂(WO₄)₃, J. Appl. Phys. 126(12), 125114 (2019)

[7] A. L. Goodwin and C. J. Kepert, Negative thermal expansion and low-frequency modes in cyanide-bridged framework materials, Phys. Rev. B 71, 140301(R) (2005)

[8] K. W. Chapman, P. J. Chupas, and C. J. Kepert, Compositional dependence of negative thermal expansion in the Prussian blue analogues MI^I Pt^IV (CN)₆(M)Mn, Fe, Co, Ni, Cu, Zn, Cd), J. Am. Chem. Soc. 128(21), 7009 (2006)

[9] S. d’Ambrumenil, M. Zbiri, A. M. Chippindale, S. J. Hibble, E. Marelli, and A. C. Hannon, Lattice dynamics and negative thermal expansion in the framework compound ZnNi(CN)₄ with two-dimensional and three-dimensional local environments, Phys. Rev. B 99(2), 024309 (2019)

[10] Q. Gao, J. Wang, A. Sanson, Q. Sun, E. Liang, X. Xing, and J. Chen, Discovering large isotropic negative thermal expansion in framework compound AgB(CN)₄ via the concept of average atomic volume, J. Am. Chem. Soc. 142(15), 6935 (2020)

[11] B. K. Greve, K. L. Martin, P. L. Lee, P. J. Chupas, K. W. Chapman, and A. P. Wilkinson, Pronounced negative thermal expansion from a simple structure: Cubic ScF₃, J. Am. Chem. Soc. 132(44), 15496 (2010)

[12] J. C. Hancock, K. W. Chapman, G. J. Halder, C. R. Morelock, B. S. Kaplan, L. C. Gallington, A. Bongiorno, C. Han, S. Zhou, and A. P. Wilkinson, Large negative thermal expansion and anomalous behavior on compression in cubic ReO₃-type AI^I B^IV F₆: CaZrF₆ and CaHfF6, Chem. Mater. 27(11), 3912 (2015)

[13] B. R. Hester, J. C. Hancock, S. H. Lapidus, and A. P. Wilkinson, Composition, response to pressure, and negative thermal expansion in MI^I B^IV F₆ (M= Ca, Mg; B= Zr, Nb), Chem. Mater. 29(2), 823 (2017)

[14] D. Yoon, Y. W. Son, and H. Cheong, Negative thermal expansion coefficient of graphene measured by Raman spectroscopy, Nano Lett. 11(8), 3227 (2011)

[15] G. Liu and J. Zhou, First-principles study of thermal expansion and thermomechanics of group-V monolayers: blue phosphorene, arsenene, and antimonene, J. Phys.: Condens. Matter 31(6), 065302 (2019)

[16] K. Takenaka and H. Takagi, Giant negative thermal expansion in Ge-doped anti-perovskite manganese nitrides, Appl. Phys. Lett. 87(23), 261902 (2005)

[17] Y. Sun, C. Wang, Y. C. Wen, K. G. Zhu, and J. T. Zhao, Lattice contraction and magnetic and electronic transport properties of Mn₃Zn_1−xGe_xN, Appl. Phys. Lett. 91(23), 231913 (2007)

[18] X. G. Guo, P. Tong, J. C. Lina, C. Yang, K. Zhang, M. Wang, Y. Wu, S. Lin, W. H. Song, and Y. P. Sun, Large negative thermal expansion in(Ga_0.7Cu_0.3)_1−xMnxNMn₃ (x≤0.4), compensating for the thermal expansion of cryogenic materials, Scr. Mater. 128, 74 (2017)

[19] R. J. Huang, Y. Y. Liu, W. Fan, J. Tan, F. R. Xiao, L. H. Qian, and L. F. Li, Giant negative thermal expansion in NaZn₁₃-type La(Fe, Si, Co)₁₃ compounds, J. Am. Chem. Soc. 135(28), 11469 (2013)

[20] W. T. Sun, H. Zhang, W. Li, R. J. Huang, Y. Q. Zhao, W. Wang, and L. F. Li, Controllable negative thermal expansion in NaZn₁₃-type La(Fe, Co, Al)₁₃ compounds, AIP Adv. 10(7), 075123 (2020)

[21] X. R. Xing, J. X. Deng, J. Chen, and G. R. Liu, Novel thermal expansion of lead titanate, Rare Met. 22(4), 294 (2003)

[22] H. Fang, Y. Wang, S. Shang, and Z. K. Liu, Nature of ferroelectric–paraelectric phase transition and origin of negative thermal expansion in PbTiO₃, Phys. Rev. B 91(2), 024104 (2015)

[23] E. T. Ritz and N. A. Benedek, Interplay between phonons and anisotropic elasticity drives negative thermal expansion in PbTiO₃, Phys. Rev. Lett. 121(22), 255901 (2018)

[24] Y. W. Long, N. Hayashi, T. Saito, M. Azuma, S. Muranaka, and Y. Shimakawa, Temperature-induced A–B intersite charge transfer in an A-site-ordered LaCu₃Fe₄O₁₂ perovskite, Nature 458(7234), 60 (2009)

[25] M. Azuma, W. T. Chen, H. Seki, M. Czapski, S. Olga, K. Oka, M. Mizumaki, T. Watanuki, N. Ishimatsu, N. Kawamura, S. Ishiwata, M. G. Tucker, Y. Shimakawa, and J. P. Attfield, Colossal negative thermal expansion in BiNiO₃ induced by intermetallic charge transfer, Nat. Commun. 2(1), 347 (2011)

[26] A. P. Giddy, M. T. Dove, G. S. Pawley, and V. Heine, The determination of rigid-unit modes as potential soft modes for displacive phase transitions in framework crystal structures, Acta Crystallogr Sec. A: Found. Crystallogr. A49, 697 (1993)

[27] L. Goodwin, Rigid unit modes and intrinsic flexibility in linearly bridged framework structures, Phys. Rev. B 74(13), 134302 (2006)

[28] M. T. Dove, Flexibility of network materials and the Rigid Unit Mode model: A personal perspective, Phil. Trans. R. Soc. A 377(2149), 20180222 (2019)

[29] W. Kim, S. H. Kang, and Y. K. Kwon, Rigid unit modes in sp-sp(2) hybridized carbon systems: Origin of negative thermal expansion, Phys. Rev. B 92(24), 245434 (2015)

[30] J. Z. Tao and A. W. Sleight, The role of rigid unit modes in negative thermal expansion, J. Solid State Chem. 173(2), 442 (2003)

[31] K. D. Hammonds, A. Bosenick, M. T. Dove, and V. Heine, Rigid unit modes in crystal structures with octahedrally coordinated atoms, Am. Mineral. 83(5–6), 476 (1998)

[32] A. K. A. Pryde, K. D. Hammonds, M. T. Dove, V. Heine, J. D. Gale, and M. C. Warren, Origin of the negative thermal expansion in ZrW₂O₈ and ZrV₂O₇, J. Phys.: Condens. Matter 8(50), 10973 (1999)

[33] L. H. N. Rimmer and M. T. Dove, Simulation study of negative thermal expansion in yttrium tungstate Y₂W₃O₁₂, J. Phys.: Condens. Matter 27(18), 185401 (2015)

[34] A. Sanson, F. Rocca, G. Dalba, P. Fornasini, R. Grisenti, M. Dapiaggi, and G. Artioli, Negative thermal expansion and local dynamics in Cu₂O and Ag₂O, Phys. Rev. B 73(21), 214305 (2006)

[35] L. H. N. Rimmer, M. T. Dove, B. Winkler, D. J. Wilson, K. Refson, and A. L. Goodwin, Framework flexibility and the negative thermal expansion mechanism of copper (I) oxide Cu₂O, Phys. Rev. B 89(21), 214115 (2014)

[36] C. W. Li, X. Tang, J. A. Munöz, J. B. Keith, S. J. Tracy, D. L. Abernathy, and B. Fultz, Structural relationship between negative thermal expansion and quartic anharmonicity of cubic ScF₃, Phys. Rev. Lett. 107(19), 195504 (2011)

[37] C. P. Romao, Anisotropic thermal expansion in flexible materials, Phys. Rev. B 96(13), 134113 (2017)

[38] E. Grüneisen, Theory of the solid state of monoatomic elements, Ann. Phys. 344(12), 257 (1912)

[39] N. Mounet and N. Marzari, First-principles determination of the structural, vibrational and thermodynamical properties of diamond, graphite, and derivatives, Phys. Rev. B 71(20), 205214 (2005)

[40] Z. Y. Wang, Y. L. Zhou, X. Q. Wang, F. Wang, Q. Sun, Z. X. Guo, and Y. Jia, Effects of in-plane stiffness and charge transfer on thermal expansion of monolayer transition metal dichalcogenide, Chin. Phys. B 24(2), 026501 (2015)

[41] R. M. Hazen and A. Y. Au, High-pressure crystal chemistry of phenakite (Be₂SiO₄) and bertrandite (Be₄Si₂O₇(OH)₂), Phys. Chem. Miner. 13(2), 69 (1986)

[42] R. Stevens, B. F. Woodfield, J. Boerio-Goates, and M. K. Crawford, Heat capacities, third-law entropies and thermodynamic functions of the negative thermal expansion material Zn₂GeO₄ from T=(0 to 400) K, J. Chem. Thermodyn. 36(5), 349 (2004)

[43] H. L. Yuan, Q. L. Gao, P. Xu, J. Guo, L. H. He, A. Sanson, M. J. Chao, and E. J. Liang, Understanding negative thermal expansion of Zn₂GeO₄ through local structure and vibrational dynamics, Inorg. Chem. 60(3), 1499 (2021)

[44] J. Q. Wang, P. Xu, H. L. Yuan, Q. L. Gao, Q. Sun, and E. J. Liang, Negative thermal expansion driven by acoustic phonon modes in rhombohedral Zn₂GeO₄, Results Phys. 19, 103531 (2020)

[45] J. N. Plendl and L. C. Mansur, Anomalous thermal expansion with infrared spectroscopy, Appl. Opt. 11(5), 1194 (1972)

[46] M. Vaccari, R. Grisenti, P. Fornasini, F. Rocca, and A. Sanson, Negative thermal expansion in CuCl: An extended X-ray absorption fine structure study, Phys. Rev. B 75(18), 184307 (2007)

[47] A. M. Gopakumar, M. K. Gupta, R. Mittal, S. Rolsd, and S. L. Chaplot, Investigating anomalous thermal expansion of copper halides by inelastic neutron scattering and ab initio phonon calculations, Phys. Chem. Chem. Phys. 19(19), 12107 (2017)

[48] T. H. K. Barron and R. W. Munn, Analysis of the thermal expansion of anisotropic solids: Application to zinc, Philos. Mag. 15(133), 85 (1967)

[49] C. Ablitt, S. Craddock, M. S. Senn, A. A. Mostofi, and N. C. Bristowe, The origin of uniaxial negative thermal expansion in layered perovskites, npj Comput. Mater. 3, 44 (2017)

[50] N. Mounet and N. Marzari, First-principles determination of the structural, vibrational and thermodynamical properties of diamond, graphite, and derivatives, Phys. Rev. B 71(20), 205214 (2005)

[51] G. Lucazeau, Effect of pressure and temperature on Raman spectra of solids: Anharmonicity, J. Raman Spectrosc. 34(7–8), 478 (2003)

[52] L. C. Gallington, K. W. Chapman, C. R. Morelock, P. J. Chupas, and A. P. Wilkinson, Dramatic softening of the negative thermal expansion material HfW₂O₈ upon heating through its WO₄ orientational order–disorder phase transition, J. Appl. Phys. 115(5), 053512 (2014)

[53] S. Allen and J. S. O. Evans, Negative thermal expansion and oxygen disorder in cubic ZrMo₂O₈, Phys. Rev. B 68(13), 134101 (2003)

[54] A. Kennedy, M. A. White, A. P. Wilkinson, and T. Varga, Low thermal conductivity of the negative thermal expansion material, HfMo₂O₈, Appl. Phys. Lett. 90(15), 151906 (2007)

[55] J. S. O. Evans, T. A. Mary, and A. W. Sleight, Negative thermal expansion materials, Physica B 241, 311 (1998)

[56] M. G. Tucker, A. L. Goodwin, M. T. Dove, D. A. Keen, S. A. Wells, and J. S. O. Evans, Negative thermal expansion in ZrW₂O₈: Mechanisms, rigid unit modes, and neutron total scattering, PRL 95(22), 255501 (2005)

[57] D. Cao, F. Bridges, G. R. Kowach, and A. P. Ramirez, Frustrated softmodes and negative thermal expansion in ZrW₂O₈, Phys. Rev. Lett. 89(21), 215902 (2002)

[58] F. Cao, F. Bridges, G. R. Kowach, and A. P. Ramirez, Correlated atomicmotions in the negative thermal expansion material ZrW₂O₈: A local structure study, Phys. Rev. B 68(1), 014303 (2003)

[59] F. Bridges, T. Keiber, P. Juhas, S. J. L. Billinge, L. Sutton, J. Wilde, and G. R. Kowach, Local vibrations and negative thermal expansion in ZrW₂O₈, Phys. Rev. Lett. 112(4), 045505 (2014)

[60] T. R. Ravindran, A. K. Arora, and T. A. Mary, High-pressure Raman spectroscopic study of zirconium tungstate, J. Phys.: Condens. Matter 13(50), 11573 (2001)

[61] E. J. Liang, Y. Liang, Y. Zhao, J. Liu, and Y. J. Jiang, Low frequency phonon modes and negative thermal expansion in A(MO₄)₂ (A=Zr, Hf and M=W, Mo) by Raman and terahertz time-domain spectroscopy, J. Phys. Chem. A 112(49), 12582 (2008)

[62] J. N. Hancock, C. Turpen, Z. Schlesinger, G. R. Kowach, and A. P. Ramirez, Unusual low-energy phonon dynamics in the negative thermal expansion compound ZrW₂O₈, Phys. Rev. Lett. 93(22), 225501 (2004)

[63] V. Gava, A. L. Martinotto, and C. A. Perottoni, Firstprinciples mode Gruneisen parameters and negative thermal expansion in α-ZrW₂O₈, Phys. Rev. Lett. 109(19), 195503 (2012)

[64] M. K. Gupta, R. Mittal, and S. L. Chaplot, Negative thermal expansion in cubic ZrW₂O₈: Role of phonons in the entire Brillouin zone from ab initio calculations, Phys. Rev. B 88(1), 014303 (2013)

[65] N. Khosrovani, A. W. Sleight, and T. Vogt, Structure of ZrV₂O₇ from –263 to 470 °C, J. Solid State Chem. 132(2), 355 (1997)

[66] C. Turquat, C. Muller, E. Nigrelli, C. Leroux, J. L. Soubeyroux, and G. Nihoul, Structural investigation of temperature-induced phase transitions in HfV₂O₇, Eur. Phys. J. Appl. Phys. 10(1), 15 (2000)

[67] R. L. Withers, J. S. O. Evans, J. Hanson, and A. W. Sleight, An in situ temperature-dependent electron and X-ray diffraction study of structural phase transitions in ZrV2O7, J. Solid State Chem. 137(1), 161 (1998)

[68] L. C. Gallington, B. R. Hester, B. S. Kaplan, and A. P. Wilkinson, Pressure-dependence of the phase transitions and thermal expansion in zirconium and hafnium pyrovanadate, J. Solid State Chem. 249, 46 (2017)

[69] V. Korthuis, N. Khosrovani, A. W. Sleight, N. Roberts, R. Dupree, and W. W. Warren, Negative thermal expansion and phase transitions in the ZrV_2−xP_xO₇ series, Chem. Mater. 7(2), 412 (1995)

[70] P. P. Sahoo, S. Sumithra, G. Madras, and T. N. G. Row, Synthesis, structure, negative thermal expansion, and photocatalytic property of Mo doped ZrV₂O₇, Inorg. Chem. 50(18), 8774 (2011)

[71] H. L. Yuan, B. H. Yuan, F. Li, and E. J. Liang, Phase transition and thermal expansion properties of ZrV_2−xP_xO₇, Acta Physica Sinica 61(22), 226502 (2012)

[72] Q. Q. Liu, J. Yang, X. J. Sun, X. N. Cheng, H. Tang, and H. H. Li, Influence of W doped ZrV2O7 on structure, negative thermal expansion property and photocatalytic performance, Appl. Surf. Sci. 313, 41 (2014)

[73] H. Yanase, H. Chida, and H. Kobayashi, Fabrication and negative thermal expansion properties of P-substituted ZrV₂O₇ sintered bodies, J. Eur. Ceram. Soc. 38(1), 221 (2018)

[74] T. Hisashige, T. Yamaguchi, T. Tsuji, and Y. Yamamura, Phase transition of Zr_1−xHf_xV₂O₇ solid solutions having negative thermal expansion, J. Ceram. Soc. Jpn. 114(1331), 607 (2006)

[75] T. Yanase, T. Kojima, and H. Kobayashi, Effects of Nb and Y substitution on negative thermal expansion of ZrV_2−xP_xO₇ (0≤x≤0.8), Solid State Commun. 151(8), 595 (2011)

[76] B. H. Yuan, H. L. Yuan, W. B. Song, X. S. Liu, Y. G. Cheng, M. J. Chao, and E. J. Liang, High solubility of hetero-valence ion (Cu²⁺) for reducing phase transition and thermal expansion of ZrV1.6P0.4O7, Chin. Phys. Lett. 31(7), 076501 (2014)

[77] B. H. Yuan, X. S. Liu, W. B. Song, Y. G. Cheng, M. J. Chao, and E. J. Liang, High substitution of Fe³⁺ for Zr⁴⁺ in ZrV_1.6P_0.4O₇ with small amount of FeV0.8P0.2O4 for low thermal expansion, J. Phys. Lett. A 378(45), 3397 (2014)

[78] P. Wang, Q. D. Chen, S. L. Li, Y. J. Ji, W. Y. Mu, W. W. Feng, G. J. Zeng, Y. W. Liu, and E. J. Liang, Phase transition and near-zero thermal expansion of Zr_0.5Hf_0.5VPO₇, Chin. Phys. B 27(6), 066501 (2018)

[79] P. Wang, Q. D. Chen, S. L. Li, Y. J. Ji, G. J. Zeng, Y. W. Liu, and E. J. Liang, Phase transition and nearzero thermal expansion properties of Zr_0.5Hf_0.5V_2−xP_xO₇ (0≤ x≤1.2), Solid State Commun. 287, 7 (2019)

[80] B. H. Yuan, W. S. Cao, X. H. Ge, Y. G. Cheng, X. S. Liu, and E. J. Liang, Substitutions of dual-ion Al³⁺/Mo⁶⁺ for Zr⁴⁺/V⁵⁺in ZrV₂O₇ for realizing near-zero thermal expansion, Acta Physica Sinica 66(7), 076501 (2017)

[81] B. H. Yuan, X. S. Liu, Y. C. Mao, J. Q. Wang, J. Guo, Y. G. Cheng, W. B. Song, E. J. Liang, and M. J. Chao, Low thermal expansion over a wide temperature range of Zr_1−xFexV_2−xMoxO₇ (0≤x≤0.9), Mater. Chem. Phys. 170, 162 (2016)

[82] W. Wei, Q. L. Gao, J. Guo, M. J. Chao, L. H. He, J. Chen, and E. J. Liang, Realizing isotropic negative thermal expansion covering room temperature by breaking the superstructure of ZrV₂O₇, Appl. Phys. Lett. 116(18), 181902 (2020)

[83] R. Mittal and S. L. Chaplot, Lattice dynamical calculation of negative thermal expansion in ZrV₂O₇ and HfV₂O₇, Phys. Rev. B 78(17), 174303 (2008)

[84] M. Maczka, W. Paraguassu, A. G. S. Filho, P. T. C. Freire, J. M. Filho, F. E. A. Melo, and J. Hanuza, High-pressure Raman study of Al2(WO4)3, J. Solid State Chem. 177(6), 2002 (2004)

[85] G. J. Zeng, H. L. Yuan, J. Guo, Q. Sun, Q. L. Gao, M. J. Chao, X. Ren, and E. J. Liang, Hydrate formation and its effects on the thermal expansion properties of HfMgW3O12, Phys. Chem. Chem. Phys. 22(22), 12605 (2020)

[86] J. Pommer, V. Kataev, K. Y. Choi, P. Lemmens, A. Ionescu, Y. Pashkevich, A. Freimuth, and G. Güntherodt, Interplay between structure and magnetism in the spinchain compound (Cu, Zn)₂V₂O₇, Phys. Rev. B 67(21), 214410 (2003)

[87] M. Schindler and F. C. Hawthorne, Structural characterization of the β-Cu₂V₂O₇–α-Zn₂V₂O₇ solid solution, J. Solid State Chem. 146(1), 271 (1999)

[88] B. V. Slobodin and R. F. Samigullina, Thermoanalytical study of the polymorphism and melting behavior of Cu₂V₂O₇, Inorg. Mater. 46(2), 196 (2010)

[89] K. Katayama, K. Otsuka, M. Mitamura, Y. Yokoyama, Y. Okamoto, and K. Takenaka, Microstructural effects on negative thermal expansion extending over a wide temperature range in β-Cu_1.8Zn_0.2V₂O₇, Appl. Phys. Lett. 113(18), 181902 (2018)

[90] L. Wang, J. Werner, A. Ottmann, R. Weis, M. Abdel-Hafiez, J. Sannigrahi, S. Majumdar, C. Koo, and R. Klingeler, Magnetoelastic coupling and ferromagnetic-type ingap spin excitations in multiferroic α-Cu₂V₂O₇, New J. Phys. 20(6), 063045 (2018)

[91] K. Shi, A. Sanson, A. Venier, L. L. Fan, C. L. Sun, X. R. Xing, and J. Chen, Negative and zero thermal expansion in α-(Cu_2−xZn_x)V₂O₇ solid solutions, Chem. Commun. (Camb.) 56(73), 10666 (2020)

[92] M. Sato, V. Warne-Lang, Y. Kadowaki, N. Katayama, Y. Okamoto, and K. Takenaka, Sol–gel synthesis of doped Cu₂V₂O₇ fine particles showing giant negative thermal expansion, AIP Adv. 10(7), 075207 (2020)

[93] Y. W. Lee, T. H. Jang, S. E. Dissanayake, S. Lee, and Y. H. Jeong, Magnetism and magnetoelectricity in the polar oxide α-Cu₂V₂O₇,Europhys. Lett. 113(2), 27007 (2016)

[94] H. Wang, M. Yang, M. Chao, J. Guo, Q. Gao, Y. Jiao, X. Tang, and E. Liang, Negative thermal expansion property of β-Cu₂V₂O₇, Solid State Ion. 343, 115086 (2019)

[95] H. Wang, M. Yang, M. Chao, J. Guo, X. Tang, Y. Jiao, and E. Liang, Negative thermal expansion properties of Cu_1.5Mg_0.5V₂O₇, Ceram. Int. 45(8), 9814 (2019)

[96] K. Takenaka, Y. Okamoto, T. Shinoda, N. Katayama, and Y. Sakai, Colossal negative thermal expansion in reduced layered ruthenate, Nat. Commun. 8(1), 14102 (2017)

[97] K. Takenaka, N. Inoue, Y. Mizuno, Y. Okamoto, N. Katayama, Y. Sakai, T. Nishikubo, and M. Azuma, Extended operating temperature window of giant negative thermal expansion in Sn-doped Ca₂RuO₄, Appl. Phys. Lett. 113(7), 071902 (2018)

[98] S. Xu, Y. M. Hu, Y. Liang, C. F. Shi, Y. L. Su, J. Guo, Q. L. Gao, M. J. Chao, and E. J. Liang, Negative thermal expansion of Ca₂RuO₄ with oxygen vacancies, Chin. Phys. B 29(8), 086501 (2020)

[99] A. K. Tyagi, S. N. Achary, and M. D. Mathews, Phase transition and negative thermal expansion in A₂(MoO₄)₃ system (A= Fe³⁺, Cr³⁺ and Al³⁺), J. Alloys Compd. 339(1–2), 207 (2002)

[100] T. Varga, J. L. Moats, S. V. Ushakov, and A. Navrotsky, Thermochemistry of A₂M₃O₁₂ negative thermal expansion materials, J. Mater. Res. 22(9), 2512 (2007)

[101] X. S. Liu, B. H. Yuan, Y. G. Cheng, E. J. Liang, and W. F. Zhang, Combined influences of A³⁺ and Mo⁶⁺ on monoclinic–orthorhombic phase transition of negativethermal- expansion A₂Mo₃O₁₂, J. Alloys Compd. 776, 236 (2019)

[102] G. Yang, X. S. Liu, X. W. Sun, E. J. Liang, and W. F. Zhang, Synthesis process control of low-thermalexpansion Fe₂W₃O₁₂ by suppressing the intermediate phase Fe₂WO₆, Ceram. Int. 44(17), 22032 (2018)

[103] Z. P. Zhang, W. K. Sun, H. F. Liu, B. T. Xiao, X. H. Zeng, and X. B. Chen, Densification and negative thermal expansion property of In_0.5Sc_1.5(MoO₄)₃ ceramics, J. Alloys Compd. 783, 77 (2019)

[104] T. Suzuki and A. Omote, Negative thermal expansion in (HfMg)(WO₄)₃, J. Am. Ceram. Soc. 87(7), 1365 (2004)

[105] B. A. Marinkovic, P. M. Jardim, M. Ari, R. R. de Avillez,F. Rizzo, and F. F. Ferreira, Low positive thermal expansion in HfMgMo₃O₁₂, Phys. Status Solidi. (b) 245(11), 2514 (2008)

[106] W. B. Song, E. J. Liang, X. S. Liu, Z. Y. Li, B. H. Yuan, and J. Q. Wang, A negative thermal expansion material of ZrMgMo₃O₁₂, Chin. Phys. Lett. 30(12), 126502 (2013)

[107] F. Li, X. S. Liu, W. B. Song, B. Yuan, Y. G. Cheng, H. L. Yuan, F. X. Cheng, M. J. Chao, and E. liang, Phase transition, crystal water and low thermal expansion behavior of Al_2−2x(ZrMg)_xW₃O₁₂·n(H₂O), Solid State Chem. 218, 15 (2014)

[108] X. H. Ge, Y. C. Mao, L. Li, L. P. Li, N. Yuan, Y. G. Cheng, J. Guo, M. J. Chao, and E. J. Liang, Phase transition and negative thermal expansion property of ZrMnMo₃O₁₂, Chin. Phys. Lett. 33(4), 046503 (2016)

[109] Y. Y. Liu, B. H. Yuan, Y. G. Cheng, E. J. Liang, X. H. Ge, H. L. Yuan, Y. Zhang, J.Guo, and M. J. Chao, Phase transition and negative thermal expansion of HfMnMo₃O₁₂,Mater. Res. Bull. 99, 255 (2018)

[110] X. H. Ge, X. S. Liu, Y. G. Cheng, B. H. Yuan, D. X. Chen, M. J. Chao, J. Guo, J. Q. Wang, and E. J. Liang, Negative thermal expansion and photoluminescence properties in a novel material ZrScW₂PO₁₂, J. Appl. Phys. 120(20), 205101 (2016)

[111] Y. G. Cheng, Y. Liang, X. H. Ge, X. S. Liu, B. H. Yuan, J. Guo, M. J. Chao, and E. J. Liang, A novel material of HfScMo₂VO₁₂ with negative thermal expansion and intense white-light emission, RSC Advances 6(59), 53657 (2016)

[112] Y. G. Cheng, Y. Liang, Y. C. Mao, X. H. Ge, B. H. Yuan, J. Guo, M. J. Chao, and E. J. Liang, A novel material of HfScW₂PO₁₂ with negative thermal expansion from 140 K to 1469 K and intense blue photoluminescence, Mater. Res. Bull. 85, 176 (2017)

[113] X. Chen, B. H. Yuan, Y. G. Cheng, X. H. Ge, Y. Jia, E. J. Liang, and M. J. Chao, Phase transition and nearzero thermal expansion in ZrFeMo₂VO₁₂, Phys. Lett. A 380(48), 4070 (2016)

[114] D. X. Chen, Y. Zhang, X. H. Ge, Y. G. Cheng, Y. Y. Liu, H. L. Yuan, J. Guo, M. J. Chao, and E. J. Liang, Structural, vibrational and thermal expansion properties of Sc₂W₄O₁₅, Phys. Chem. Chem. Phys. 20(27), 20160 (2018)

[115] Y. Liang, Y. G. Cheng, X. H. Ge, B. H. Yuan, J. Guo, Q. Sun, and E. J. Liang, Negative thermal expansion and photoluminescence in solid solution (HfSc)_0.83W_2.25P_0.83O_12−δ, Chin. Phys. B 26(10), 106501 (2017)

[116] L. Wang, F. Wang, P. F. Yuan, Q. Sun, E. J. Liang, Y. Jia, and Z. X. Guo, Negative thermal expansion correlated with polyhedral movements and distortions in orthorhombic Y₂Mo₃O₁₂, Mater. Res. Bull. 48(7), 2724 (2013)

[117] H. L. Yuan, C. Y. Wang, Q. L. Gao, X. H. Ge, H. Sun, S. H. Lapidus, J. Guo, M. J. Chao, Y. Jia, and E. J. Liang, Structure and negative thermal expansion in Zr_0.3Sc_1.7Mo_2.7V_0.3O₁₂, Inorg. Chem. 59(6), 4090 (2020)

[118] T. G. Amos, A. Yokochi, and A. W. Sleight, Phase transition and negative thermal expansion in tetragonal NbOPO₄, J. Solid State Chem. 141(1), 303 (1998)

[119] T. G. Amos and A. W. Sleight, Negative thermal expansion in orthorhombic NbOPO₄, J. Solid State Chem. 160(1), 230 (2001)

[120] X. Wang, Q. Huang, J. Deng, R. Yu, J. Chen, and X. Xing, Phase transformation and negative thermal expansion in TaVO₅, Inorg. Chem. 50(6), 2685 (2011)

[121] J. Wang, J. Deng, R. Yu, J. Chen, and X. Xing, Coprecipitation synthesis and negative thermal expansion of NbVO₅, Dalton Trans. 40(13), 3394 (2011)

[122] N. P. Salke, M. K. Gupta, R. Rao, R. Mittal, J. Deng, and X. Xing, Raman and ab initio investigation of negative thermal expansion material TaVO₅: Insights into phase stability and anharmonicity, J. Appl. Phys. 117(23), 235902 (2015)

[123] N. P. Salke, R. Rao, S. N. Achary, C. Nayak, A. B. Garg, P. S. R. Krishna, A. B. Shinde, S. N. Jha, D. Bhattacharyya, Jagannath, and A. K. Tyagi, High pressure phases and amorphization of a negative thermal expansion compound TaVO5, Inorg. Chem. 57(12), 6973 (2018)

[124] M. Ivanda, D. Waasmaier, A. Endriss, J. Ihringer, A. Kirfel, and W. Kiefer, Low-temperature anomalies of cuprite observed by Raman spectroscopy and X-ray powder diffraction, J. Raman Spectrosc. 28(7), 487 (1997)

[125] A. Sanson, G. Dalba, P. Fornasini, R. Grisenti, F. Rocca, G. Artioli, M. Dapiaggi, and W. Tiano, EXAFS and XRD study of local dynamics in Cu₂O and Ag₂O, Phys. Scr. T 115, 271 (2005)

[126] S. A. Beccara, G. Dalba, P. Fornasini, R. Grisenti, A. Sanson, and F. Rocca, Local thermal expansion in a cuprite structure: The case of Ag₂O, Phys. Rev. Lett. 89(2), 025503 (2002)

[127] M. K. Gupta, R. Mittal, S. Rols, and S. L. Chaplot, Inelastic neutron scattering an ab–initio calculation of negative thermal expansion in Ag₂O, Physica B 407(12), 2146 (2012)

[128] T. Lan, C. W. Li, J. L. Niedziela, H. Smith, D. L. Abernathy, G. R. Rossman, and B. Fultz, Anharmonic lattice dynamics of Ag₂O studied by inelastic neutron scattering and first-principles molecular dynamics simulations, Phys. Rev. B 89(5), 054306 (2014)

[129] G. Wallez, N. Clavier, N. Dacheux, and D. Bregiroux, Negative thermal expansion in Th₂O(PO₄)₂, Mater. Res. Bull. 46(11), 1777 (2011)

[130] J. Xu, L. Hu, Y. Song, F. Han, Y. Qiao, J. Deng, J. Chen, and X. Xing, Zero thermal expansion in cubic MgZrF₆, J. Am. Ceram. Soc. 100(12), 5385 (2017)

[131] B. R. Hester and A. P. Wilkinson, Negative thermal expansion, response to pressure and phase transitions in CaTiF₆, Inorg. Chem. 57(17), 11275 (2018)

[132] M. T. Dove, J. Du, A. E. Phillips, D. A. Keen, and M. G. Tucker, A real-space experimental model for negative thermal expansion in scandium trifluoride, arXiv: 1905.09250 (2019)

[133] Y. Oba, T. Tadano, R. Akashi, and S. Tsuneyuki, Firstprinciples study of phonon anharmonicity and negative thermal expansion in ScF₃, Phys. Rev. Mater. 3(3), 033601 (2019)

[134] D. Bocharov, M. Krack, Y. Rafalskij, A. Kuzmin, and J. Purans, Ab initio molecular dynamics simulations of negative thermal expansion in ScF₃: The effect of the supercell size, Comput. Mater. Sci. 171, 109198 (2020)

[135] M. K. Gupta, B. Singh, R. Mittal, and S. L. Chaplot, Negative thermal expansion behavior in MZrF₆ (M= Ca, Mg, Sr): Ab initio lattice dynamical studies, Phys. Rev. B 98(1), 014301 (2018)

[136] T. A. Bird, J. Woodland-Scott, L. Hu, M. T. Wharmby, J. Chen, A. L. Goodwin, and M. S. Senn, Anharmonicity and scissoring modes in the negative thermal expansion materials ScF3 and CaZrF₆, Phys. Rev. B 101(6), 064306 (2020)

[137] A. Sanson, M. Giarola, G. Mariotto, L. Hu, J. Chen, and X. R. Xing, Lattice dynamics and anharmonicity of CaZrF₆ from Raman spectroscopy and ab initio calculations, Mater. Chem. Phys. 180, 213 (2016)

[138] T. Chatterji, T. C. Hansen, M. Brunelli, and P. F. Henry, Negative thermal expansion of ReO3 in the extended temperature range, Appl. Phys. Lett. 94(24), 241902 (2009)

[139] Y. M. Liu, Z. H. Wang, M. Wu, Q. Sun, M. J. Chao, and Y. Jia, Negative thermal expansion in isostructural cubic ReO₃ and ScF₃: A comparative study, Comput. Mater. Sci. 107, 157 (2015)

[140] A. L. Goodwin, M. Calleja, M. J. Conterio, M. T. Dove, J. S. O. Evans, D. A. Keen, L. Peters, and M. G. Tucker, Colossal positive and negative thermal expansion in the framework material Ag₃[Co(CN)₆], Science 319(5864), 794 (2008)

[141] L. Gao, N. K. Shi, Q. Sun, A. Sanson, R. Milazzo, A. Carnera, H. Zhu, S. H. Lapidus, Y. Ren, Q. Huang, J. Chen, and X. Xing, Low-frequency phonon driven negative thermal expansion in cubic GaFe(CN)₆ Prussian blue analogues, Inorg. Chem. 57(17), 10918 (2018)

[142] L. Gao, Y. Sun, N. Shi, R. Milazzo, S. Pollastri, L. Olivi, Q. Huang, H. Liu, A. Sanson, Q. Sun, E. Liang, X. Xing, and J. Chen, Large isotropic negative thermal expansion in water-free Prussian blue analogues of ScCo(CN)₆, Scr. Mater. 187, 119 (2020)

[143] P. Ding, E. J. Liang, Y. Jia, and Z. Y. Du, Electronic structure, bonding and phonon modes in the negative thermal expansion materials of Cd(CN)₂ and Zn(CN)₂, J. Phys.: Condens. Matter 20(24), 275224 (2008)

[144] Z. Y. Du, E. J. Liang, P. Ding, J. P. Wang, and E. M. Xu, Lattice vibrational analysis of the negative thermal expansion materials of Zn(CN)₂ and Zn(CN)₂, Chin. J. Light Scatt. 20(2),145 (2008)

[145] J. Hibble, A. M. Chippindale, E. Marelli, S. Kroeker, V. K. Michaelis, B. J. Greer, P. M. Aguiar, E. J. Bilbe, E. R. Barney, and A. C. Hannon, Local and average structure in zinc cyanide: Toward an understanding of the atomistic origin of negative thermal expansion, J. Am. Chem. Soc. 135(44), 16478 (2013)

[146] M. K. Gupta, B. Singh, R. Mittal, M. Zbiri, A. B. Cairns, A. L. Goodwin, H. Schober, and S. L. Chaplot, Anomalous thermal expansion, negative linear compressibility, and high-pressure phase transition in ZnAu₂(CN)₄: Neutron inelastic scattering and lattice dynamics studies, Phys. Rev. B 96(21), 214303 (2017)

[147] M. Mittal, H. Zbiri, E. Schober, S. J. Marelli, A. M. Hibble, A. M. Chippindale, and S. L. Chaplot, Relationship between phonons and thermal expansion in Zn(CN)₂ and Ni(CN)₂ from inelastic neutron scattering and ab initio calculations, Phys. Rev. B 83(2), 024301 (2011)

[148] H. Fang, M. T. Dove, L. H. N. Rimmer, and A. J. Misquitta, Simulation study of pressure and temperature dependence of the negative thermal expansion in Zn(CN)₂, Phys. Rev. B 88(10), 104306 (2013)

[149] J. Hibble, A. M. Chippindale, A. H. Pohl, and A. C. Hannon, Surprises from a simple material — The structure and properties of nickel cyanide, Angew. Chem. Int. Ed. 46(37), 7116 (2007)

[150] A. M. Chippindale, S. J. Hibble, E. Marelli, E. J. Bilbé, A. C. Hannon, and M. Zbiri, Chemistry and structure by design: Ordered CuNi(CN)₄ sheets with copper(ii) in a square-planar environment, Dalton Trans. 44(25), 12502 (2015)

[151] S. d’Ambrumenil,M. Zbiri, A. M. Chippindale, and S. J. Hibble, Phonon dynamics in the layered negative thermal expansion compounds CuxNi_2−x(CN)₄, Phys. Rev. B 100(9), 094312 (2019)

[152] M. Li, Y. Li, C. Y. Wang, and Q. Sun, Negative thermal expansion of GaFe(CN)₆ and effect of Na insertion by first-principles calculations, Chin. Phys. Lett. 36(6), 066301 (2019)

[153] Y. Li, Q. L. Gao, D. H. Chang, P. J. Sun, J. Z. Liu, Y. Jia, E. J. Liang, and Q. Sun, Effect of bond on negative thermal expansion of Prussian blue analogues MCo(CN)₆ (M= Fe, Ti and Sc): A first-principles study, J. Phys.: Condens. Matter 32(45), 455703 (2020)

[154] P. Kroll, M. Andrade, X. Yan, E. Ionescu, G. Miehe, and R. Riedel, Isotropic negative thermal expansion in β-Si(NCN)₂ and itsorigin, J. Phys. Chem. C 116(1), 526 (2012)

[155] L. Li, K. Refson, and M. T. Dove, Negative thermal expansion of cubic silicon dicarbodiimide, Si(NCN)₂, studied by ab initio lattice dynamics, J. Phys.: Condens. Matter 32(46), 465402 (2020)

[156] D. Dubbeldam, K. S. Walton, D. Ellis, and R. Snurr, Exceptional negative thermal expansion in isoreticular metal–organic frameworks, Angew. Chem. Int. Ed. 119(24), 4580 (2007)

[157] S. S. Han, and W. A. Goddard, Metal-organic frameworks provide large negative thermal expansion behavior, J. Phys. Chem. C 111(42), 15185 (2007)

[158] W. Zhou, H. Wu, T. Yildirim, J. R. Simpson, and A. R. H. Walker, Originof the exceptional negative thermal expansion in metal–organic framework-5 Zn₄O(1, 4-benzenedicarboxylate)₃, Phys. Rev. B 78(5), 054114 (2008)

[159] N. Lock, M. Christensen, Y. Wu, V. K. Peterson, M. K. Thomsen, R. O. Piltz, A. J. Ramirez-Cuesta,G. J. McIntyre, K. Norén,R. Kutteh, C. J. Kepert, G. J. Kearley, and B. B. Iversen, Scrutinizing negative thermal expansion in MOF-5 by scattering techniques and ab initio calculations, Dalton Trans. 42(6), 1996 (2013)

[160] L. H. N. Rimmer, M. T. Dove, A. L. Goodwin, and D. C. Palmer, Acoustic phonons and negative thermal expansion in MOF-5, Phys. Chem. Chem. Phys. 16(39), 21144 (2014)

[161] Y. Wu, A. Kobayashi, G. Halder, V. Peterson, K. Chapman, N. Lock, P. Southon, and C. Kepert, Negative thermal expansion in the metal–organic framework material Cu₃(1, 3, 5-benzenetricarboxylate)₂, Angew. Chem. Int. Ed. 47(46), 8929 (2008)

[162] W. L. Queen, C. M. Brown, D. K. Britt, P. Zajdel, M. R. Hudson, and O. M. Yaghi, Site specific CO₂ adsorption and zero thermal expansion in ananisotropic pore network, J. Phys. Chem. C 115(50), 24915 (2011)

[163] S. Henke, A. Schneemann, and R. A. Fischer, Massive anisotropic thermal expansion and thermo-responsive breathing in metal–organic frameworks modulated by linker functionalization, Adv. Funct. Mater. 23(48), 5990 (2013)

[164] Z. N. Liu, Q. Li, H. Zhu, K. Lin, J. X. Deng, J. Chen, and X. Xing, 3D negative thermal expansion in orthorhombic MIL-68(In), Chem. Commun. (Camb.) 54(45), 5712 (2018)

[165] F. X. Coudert and J. D. Evans, Nanoscale metamaterials: Meta-MOFs and framework materials with anomalous behavior, Coord. Chem. Rev. 388, 48 (2019)

[166] Z. Y. Wang, Y. L. Zhou, X. Q. Wang, F. Wang, Q. Sun, Z. X. Guo, and Y. Jia, Effects of in-plane stiffness and charge transfer on thermal expansion of monolayer transition metal dichalcogenide, Chin. Phys. B 24(2), 026501 (2015)

[167] D. Kumar, B. Singh, P. Kumar, V. Balakrishnan, and P. Kumar, Thermal expansion coefficient and phonon dynamics in coexisting allotropes of monolayer WS₂ probed by Raman scattering, J. Phys.: Condens. Matter 31(50), 505403 (2019)

[168] Y. Aierken, D. Cakir, C. Sevik, and F. M. Peeters, Thermal properties of black and blue phosphorenes from a first-principles quasiharmonic approach, Phys. Rev. B 92(8), 081408 (2015)

[169] L. Wang, C. Wang, and Y. Chen, Black phosphorene exhibiting negative thermal expansion and negative linear compressibility, J. Phys.: Condens. Matter 31(46), 465003 (2019)

[170] H. Sun, G. Liu, Q. Li, and X. G. Wan, First-principles study of thermal expansion and thermomechanics of single-layer black and blue phosphorus, Phys. Lett. A 380(24), 2098 (2016)

[171] X. J. Ge, K. L. Yao, and J. T. Lü, Comparative study of phonon spectrum and thermal expansion of graphene, silicene, germanene, and blue phosphorene, Phys. Rev. B 94(16), 165433 (2016)

[172] P. R. Shaina, L. George, V. Yadav, and M. Jaiswal, Estimating the thermal expansion coefficient of graphene: The role of graphene-substrate interactions, J. Phys.: Condens. Matter 28(8), 085301 (2016)

[173] C. W. Kim, S. H. Kang, and Y. K. Kwon, Rigid unit modes in sp-sp2 hybridized carbon systems: Origin of negative thermal expansion, Phys. Rev. B 92(24), 245434 (2015)

[174] A. I. Lebedev, Negative thermal expansion in CdSe quasitwo- dimensional nanoplatelets, Phys. Rev. B 100(3), 035432 (2019)

[175] Y. Hu, J. Chen, and B. Wang, On the intrinsic ripples and negative thermal expansion of graphene, Carbon 95, 239 (2015)

[176] H. Sun, G. Liu, Q. Li, and X. G. Wan, First-principles study of thermal expansion and thermomechanics of single-layer black and blue phosphorus, Phys. Lett. A 380(24), 2098 (2016)

[177] C. W. Kim, S. H. Kang, and Y. K. Kwon, Rigid unit modes in sp-sp2 hybridized carbon systems: Origin of negative thermal expansion, Phys. Rev. B 92(24), 245434 (2015)

[178] Y. X. Gao, C. Y. Wang, Q. L. Gao, J. Guo, M. J. Chao, Y. Jia, and E. J. Liang, Zero thermal expansion in Ta₂Mo₂O₁₁ by compensation effects, Inorg. Chem. 59(24), 18427 (2020)

[179] C. Guillaume, Recherches sur les aciers au nickel, Dilatations auxtemperatures elevees, resistance electrique, CR Acad. Sci. 125(235), 18 (1897)

[180] R. J. Weiss, The origin of the “Invar” effect, Proc. Phys. Soc. 82(2), 281 (1963)

[181] T. Moriya and K. Usami, Magneto-volume effect and invar phenomena in ferromagnetic metals, Solid State Commun. 34(2), 95 (1980)

[182] H. Yamada, Metamagnetic transition and susceptibility maximum in an itinerant-electron system, Phys. Rev. B 47(17), 11211 (1993)

[183] H. Akai and P. H. Dederichs, Local moment disorder in ferromagnetic alloys, Phys. Rev. B 47(14), 8739 (1993)

[184] V. Crisan, P. Entel, H. Ebert, H. Akai, D. D. Johnson, and J. B. Staunton, Magnetochemical origin for Invar anomalies in iron–nickel alloys, Phys. Rev. B 66(1), 014416 (2002)

[185] Y. Wang, G. M. Stocks, D. M. C. Nicholson, W. A. Shelton, V. P. Antropov, and B. N. Harmon, Noncollinear magnetic structure in Ni_0.35Fe_0.65, J. Appl. Phys. 81(8), 3873 (1997)

[186] M. Schilfgaarde, I. A. Abrikosov, and B. Johansson, Origin of the Invar effect in iron–nickel alloys, Nature 400(6739), 46 (1999)

[187] H. Yamada, Pressure effect in an itinerant-electron metamagnet at finite temperature, J. Magn. Magn. Mater. 139(1–2), 162 (1995)

[188] G. G. Lonzarich and L. TaiUefer, Effect of spin fluctuations on the magnetic equation of state of ferromagnetic or nearly ferromagnetic metals, J. Phys. (Paris) 18, 4339 (1985)

[189] T. Moriya, Spin Fluctuations in Itinerant Electron Magnetism, Springer-Verlag, Berlin, 1985

[190] A. V. Andreev, F. R. de Boer, T. H. Jacobs, and K. H. J. Buschow, Thermal expansion anomalies and spontaneous magnetostriction in R₂Fe₁₇C_x intermetallic compounds, Physica B 175(4), 361 (1991)

[191] Y. M. Hao, X. M. Zhang, B. W. Wang, Y. Z. Yuang, and F. Wang, Anomalous thermal expansion and magnetic properties of Tm₂Fe_17−xCr_x compounds, J. Appl. Phys. 108(2), 023915 (2010)

[192] P. Álvarez-Alonso, P. Gorria, J. A. Blanco, J. Sánchez- Marcos, G. J. Cuello, I. Puente-Orench, J. A. Rodríguez- Velamazán, G. Garbarino, I. dePedro, J. R. Fernández, and J. L. S. Llamazares, Magnetovolume and magnetocaloric effects in Er₂Fe₁₇, Phys. Rev. B 86, 184411 (2012)

[193] D. Givord and R. Lemaire, Magnetic transition and anomalous thermal expansion in R₂Fe₁₇ compounds, IEEE Trans. Magn. 10(2), 109 (1974)

[194] R. J. Huang, Y. Liu, W. Fan, J. Tan, F. Xiao, L. Qian, and L. Li, Giant negative thermal expansion in NaZn₁₃- type La(Fe, Si, Co)₁₃ compounds, J. Am. Chem. Soc. 135(28), 11469 (2013)

[195] B. Li, X. H. Luo, H. Wang, W. J. Ren, S. Yano, C. W. Wang, J. S. Gardner, K. D. Liss, P. Miao, S. H. Lee, T. Kamiyama, R. Q. Wu, Y. Kawakita, and Z. D. Zhang, Colossal negative thermal expansion induced by magnetic phase competition on frustrated lattices in Laves phase compound (Hf, Ta)Fe₂, Phys. Rev. B 93(22), 224405 (2016)

[196] Y. Song, J. Chen, X. Liu, C. Wang, Q. Gao, Q. Li, L. Hu, J. Zhang, S. Zhang, and X. Xing, Structure, magnetism, and tunable negative thermal expansion in (Hf, Nb)Fe₂ alloys, Chem. Mater. 29(17), 7078 (2017)

[197] L. F. Li, P. Tong, Y. M. Zou, W. Tong, W. B. Jiang, Y. Jiang, X. K. Zhang, J. C. Lin, M. Wang, C. Yang, X. B. Zhu, W. H. Song, and Y. P. Sun, Good comprehensive performance of Laves phase Hf_1−xTa_xFe₂ as negative thermal expansion materials, Acta Mater. 161, 258 (2018)

[198] H. Yibole, A. K. Pathak, Y. Mudryk, F. Guillou, N. Zarkevich, S. Gupta, V. Balema, and V. K. Pecharsky, Manipulating the stability of crystallographic and magnetic sub-lattices: A first-order magnetoelastic transformation in transition metal based Laves phase, Acta Mater. 154, 365 (2018)

[199] K. Takenaka and H. Takagi, Giant negative thermal expansion in Ge-doped anti-perovskite manganese nitrides, Appl. Phys. Lett. 87(23), 261902 (2005)

[200] K. Takenaka, K. Asano, M. Misawa, and H. Takagi, Negative thermal expansion in Ge-free antiperovskite manganese nitrides: Tin-doping effect, Appl. Phys. Lett. 92(1), 011927 (2008)

[201] C. Wang, Y. Sun, Y. C. Wen, L. H. Chu, and M. Nie, Investigation of lattice contraction in Mn₃XN (X=Zn, Cu, Sn), Mater. Sci. Forum 638– 642, 2195 (2010)

[202] R. J. Huang, L. F. Li, F. S. Cai, X. D. Xu, and L. H. Qian, Low-temperature negative thermal expansion of the antiperovskite manganese nitride Mn₃CuN codoped with Ge and Si, Appl. Phys. Lett. 93(8), 081902 (2008)

[203] T. Hamada and K. Takenaka, Giant negative thermal expansion in antiperovskite manganese nitrides, J. Appl. Phys. 109(7), 07E309 (2011)

[204] K. Takenaka and H. Takagi, Zero thermal expansion in a pure-form antiperovskite manganese nitride, Appl. Phys. Lett. 94(13), 131904 (2009)

[205] C. Wang, L. H. Chu, Q. R. Yao, Y. Sun, M. M. Wu, L. Ding, J. Yan, Y. Y. Na, W. H. Tang, G. N. Li, Q. Huang, and J. W. Lynn, Tuning the range, magnitude, and sign of the thermal expansion in intermetallic Mn₃(Zn, M)_xN (M= Ag, Ge), Phys. Rev. B 85(22), 220103 (2012)

[206] E. F. Wasserman, in: Ferromagnetic Materials, ed. K. H. J. Buschow and E. P. Wohlfartht, Elsevier Science Publishers B. V., Ch. 3, Vol. 5, pp 238–322, 1990

[207] E. F. Wassermann, The Invar problem, J. Magn. Magn. Mater. 100(1–3), 346 (1991)

[208] M. Shiga, Invar alloys, Curr. Opin. Solid State Mater. Sci. 1(3), 340 (1996)

[209] J. Chen, L. Hu, J. Deng, and X. Xing, Negative thermal expansion in functional materials: Controllable thermal expansion by chemical modifications, Chem. Soc. Rev. 44(11), 3522 (2015)

[210] S. Khmelevskyi, I. Turek, and P. Mohn, Large negative magnetic contribution to the thermal expansion in ironplatinum alloys: Quantitative theory of the Invar effect, Phys. Rev. Lett. 91(3), 037201 (2003)

[211] D. D. Johnson and W. A. Shelton, in: The Invar effect: A centennial symposium, Ed. J. Wittenauer, The Minerals, Metals and Materials Society, Warrendale, PA, pp 63–74, 1997

[212] M. Shiga and Y. Nakamura, Magnetovolume effects and Invar characters of (Zr_1−xNb_x)Fe₂, J. Phys. Soc. Jpn. 47(5), 1446 (1979)

[213] H. Wada and M. Shigaand, Thermal expansion anomaly and Invar effect of Mn_1−xCo_xB, J. Magn. Magn. Mater. 104– 107, 1925 (1992)

[214] A. G. Kuchin, I. V. Medvedeva, V. S. Gaviko, and V. A. Kazantsev, Magnetovolume properties of Y₂Fe_17−xM_x alloys (M= Si orAl), J. Alloys Compd. 289(1–2), 18 (1999)

[215] P. Álvare, P. Gorria, J. S. Marcos, I. P. Orench, J. A. R. Velamazán, G. Cuello, J. L. S. Llamazares, and J. A. Blanco, Magnetic structure and magneto-volume anomalies in Er₂Fe₁₇ compound, J. Phys.: Conf. Ser. 325, 01 (2011)

[216] A. Shuitcev, R. N. Vasin, A. M. Balagurov, L. Li, I. A. Bobrikov, and Y. X. Tong, Thermal expansion of martensite in Ti_29.7Ni_50.3Hf₂₀ shape memory alloy, Intermetalllics 125, 106889 (2020)

[217] I. S. Winter, J. Montoya, K. A. Persson, and D. C. Chrzan, Ab initio calculation of thermal expansion with application to understanding Invar behavior in gum metal, Phys. Rev. Mater. 2(7), 073601 (2018)

[218] R. Li, R. Huang, W. Wang, H. Liu, Y. Han, C. Huang, and L. Li, Low-temperature negative thermal expansion property of Mn doped La(Fe, Si)₁₃ compounds, J. Alloys Compd. 628, 308 (2015)

[219] F. Shen, H. Zhou, F. Hu, J. T. Wang, S. Deng, B. Wang, H. Wu, Q. Huang, J. Wang, J. Chen, L. He, J. Hao, Z. Yu, F. Liang, T. Liang, J. Sun, and B. Shen, Cone-spiral magnetic ordering dominated lattice distortion and giantnegative thermal expansion in Fe-doped MnNiGe compounds, Mater. Horiz. 7(3), 804 (2020)

[220] J. Xu, X. Zheng, S. Yang, L. Xi, S. Wang, L. Zhang, W. Yang, J. Yang, X. Ma, D. Chen, L. He, S. Deng, J. Zhang, Y. Wu, and B. Shen, Large linear negative thermal expansion in NiAs-type magnetic intermetallic Cr−Te−Se compounds, Inorg. Chem. 59(12), 8603 (2020)

[221] L. Y. Hao, T. Yang, and M. Tan, Negative thermal expansion and spontaneous magnetostriction of Nd₂Fe_16.5Cr_0.5 compound, Chin. Phys. Lett. 37(1), 016501 (2020)

[222] Y. Yan, J. Yang, N. Zhao, G. Yao, X. Fu, Y. Zhang, and W. Cui, Large negative thermal expansion and magnetoelastic coupling in metamagnetic tetragonal (Mn, T)2Sb (T= Cr, V) alloys, J. Supercond. Nov. Magn. 33(9), 2551 (2020)

[223] J. H. Belo, A. L. Pires, I. T. Gomes, V. Andrade, J. B. Sousa, R. L. Hadimani, and C. David, Giant negative thermal expansion at the nanoscale in the multifunctional material Gd₅(Si, Ge)₄, Phys. Rev. B 100(13), 134303 (2019)

[224] Y. Z. Song, Q. Sun, T. Yokoyama, H. H. Zhu, Q. Li, R. J. Huang, Y. Ren, Q. Z. Huang, X. R. Xing, and J. Chen, Transforming thermal expansion from positive to negative: The case of cubic magnetic compounds of (Zr, Nb)Fe₂, J. Phys. Chem. Lett. 11(5), 1954 (2020)

[225] Y. Z. Song, Q. Sun, M. Xu, J. Zhang, Y. Q. Hao, Y. Q. Qiao, S. T. Zhang, Q. Z. Huang, X. R. Xing, and J. Chen, Negative thermal expansion in (Sc, Ti)Fe₂ induced by an unconventional magnetovolume effect, Mater. Horiz. 7(1), 275 (2020)

[226] T. Yokoyama, A. Koide, and Y. Uemura, Local thermal expansions and lattice strains in Elinvar and stainless steel alloys, Phys. Rev. Mater. 2(2), 023601 (2018)

[227] X. Y. Song, Z. G. Sun, Q. Z. Huang, M. Rettenmayr, X. M. Liu, M. Seyring, G. N. Li, G. G. Rao, and F. X. Yin, Adjustable zero thermal expansion in antiperovskite manganese nitride, Adv. Mater. 23(40), 4690 (2011)

[228] M. Kobayashi and M. Mochizuki, Theory of magnetismdriven negative thermal expansion in inverse perovskite antiferromagnets, Phys. Rev. Mater. 3(2), 024407(2019)

[229] K. Takenaka, M. Ichigo, T. Hamada, A. Ozawa, T. Shibayama, T. Inagaki, and K. Asano, Magnetovolume effects in manganese nitrides with antiperovskite structure, Sci. Technol. Adv. Mater. 15(1), 015009 (2014)

[230] H. Lu, Y. Sun, S. Deng, K. Shi, L. Wang, W. Zhao, H. Han, S. Deng, and C. Wang, Tunable negative thermal expansion and structural evolution in antiperovskite Mn₃Ga_1−xGe_xN (0≤x≤1.0), J. Am. Ceram. Soc. 100(12), 5739 (2017)

[231] S. Tan, C. Gao, C. Wang, T. Zhou, G. Yin, M. Sun, F. Xing, R. Cao, and Y. Sun, The tunable negative thermal expansion covering a wide temperature range around room temperature in Sn, Mn co-substituted Mn3ZnN, Dalton Trans. 49(27), 10407 (2020)

[232] S. Iikubo, K. Kodama, K. Takenaka, H. Takagi, M. Takigawa, and S. Shamoto, Local lattice distortion in the giant negative thermal expansion material Mn₃Cu_1−xGe_xN, Phys. Rev. Lett. 101(20), 205901 (2008)

[233] P. Tong, D. Louca, G. King, A. Llobet, J. C. Lin, and Y. P. Sun, Magnetic transition broadening and local lattice distortion in the negative thermal expansion antiperovskite Cu_1−xSn_xNMn₃, Appl. Phys. Lett. 102(4), 041908 (2013)

[234] J. C. Lin, P. Tong, X. J. Zhou, H. Lin, Y. W. Ding, Y. X. Bai, L. Chen, X. G. Guo, C. Yang, B. Song, Y. Wu, S. Lin, W. H. Song, and Y. P. Sun, Giant negative thermal expansion covering room temperature in nanocrystalline- GaN_xMn₃,App, Phys. Lett. 107, 131902 (2015)

[235] J. Tan, R. Huang, W. Wang, W. Li, Y. Zhao, S. Li, Y. Han, C. Huang, and L. Li, Broad negative thermal expansion operation-temperature window in antiperovskite manganese nitride with small crystallites, Nano Res. 8(7), 2302 (2015)

[236] B. Paul, S. Chatterjee, A. Roy, A. Midya, P. Mandal, V. Grover, and A. K. Tyagi, Geometrically frustrated GdInO₃: An exotic system to study negative thermal expansion and spin-lattice coupling, Phys. Rev. B 95(5), 054103 (2017)

[237] P. Miao, X. Lin, A. Koda, S. Lee, Y. Ishikawa, S. Torii, M. Yonemura, T. Mochiku, H. Sagayama, S. Itoh, K. Ikeda, T. Otomo, Y. Wang, R. Kadono, and T. Kamiyama, Large magnetovolume effect induced by embedding ferromagnetic clusters into antiferromagnetic matrix of cobaltite perovskite, Adv. Mater. 29(24), 1605991 (2017)

[238] B. Ranjbar and B. J. Kennedy, Unusual thermal expansion of Sr₂IrO₄: A variable temperature synchrotron Xray diffraction study, J. Solid State Chem. 232, 178 (2015)

[239] M. S. Senn, A. Bombardi, C. A. Murray, C. Vecchini, A. Scherillo, X. Luo, and S. W. Cheong, Negative thermal expansion in hybrid improper ferroelectric Ruddlesden– Popper perovskites by symmetry trapping, Phys. Rev. Lett. 114(3), 035701 (2015)

[240] S. C. Abrahams, S. K. Kurtz, and P. B. Jamieson, Atomic displacement relationship to Curie temperature and spontaneous polarization in displacive ferroelectrics, Phys. Rev. 172(2), 551 (1968)

[241] J. Chen, F. Wang, Q. Huang, L. Hu, X. Song, J. Deng, R. Yu, and X. Xing, Effectively control negative thermal expansion of single-phase ferroelectrics of PbTiO₃- (Bi, La)FeO₃ over a giant range, Sci. Rep. 3(1), 2458 (2013)

[242] J. Chen, K. Nittala, J. S. Forrester, J. L. Jones, J. X. Deng, R. Yu, and X. R. Xing, The role of spontaneous polarization in the negative thermal expansion of tetragonal PbTiO₃-based compounds, J. Am. Chem. Soc. 133(26), 11114 (2011)

[243] P. E. Janolin, P. Bouvier, J. Kreisel, P. A. Thomas, I. A. Kornev, L. Bellaiche, W. Crichton, M. Hanfland, and B. Dkhil, High-pressure effect on PbTiO₃: An investigation by Raman and X-ray scattering up to 63 GPa, Phys. Rev. Lett. 101(23), 237601 (2008)

[244] L. Wang, P. Yuan, F. Wang, E. Liang, Q. Sun, Z. Guo, and Y. Jia, First-principles study of tetragonal PbTiO₃: Phonon and thermal expansion, Mater. Res. Bull. 49, 509 (2014)

[245] H. Fang, Y. Wang, S. Shang, and Z. K. Liu, Nature of ferroelectric–paraelectric phase transition and origin of negative thermal expansion in PbTiO₃, Phys. Rev. B 91(2), 024104 (2015)

[246] D. Zhou, W. Tan, W. Xiao, M. Song, M. Chen, X. Xiong, and J. Xu, Structural properties of PbVO₃ perovskites under hydrostatic pressure conditions up to 10.6 GPa, J. Phys.: Condens. Matter 24(43), 435403 (2012)

[247] K. Oka, T. Yamauchi, S. Kanungo, T. Shimazu, K. Ohishi, Y. Uwatoko, M. Azuma, and T. Saha-Dasgupta, Experimental and theoretical studies of the metallic conductivity in cubic PbVO₃ under high pressure, J. Phys. Soc. Jpn. 87(2), 024801 (2018)

[248] T. Ogata, K. Oka, and M. Azuma, Negative thermal expansion in electron doped PbVO_3−xF_x, Appl. Phys. Express 12(2), 023005 (2019)

[249] T. Ogata, Y. Sakai, H. Yamamoto, S. Patel, P. Keil, J. Koruza, S. Kawaguchi, Z. Pan, T. Nishikubo, J. Rödel, and M. Azuma, Melting of d_xy orbital ordering accompanied by suppression of giant tetragonal distortion and insulator-to-metal transition in Cr-substituted PbVO₃, Chem. Mater. 31(4), 1352 (2019)

[250] H. Ishizaki, Y. Sakai, T. Nishikubo, Z. Pan, K. Oka, H. Yamamoto, and M. Azuma, Negative thermal expansion in lead-free La-substituted Bi_0.5Na_0.5VO₃, Chem. Mater. 32(11), 4832 (2020)

[251] M. Rong, M. Li, J. Chen, M. Zhou, K. Lin, L. Hu, W. Yuan, W. Duan, J. Deng, and X. Xing, Large negative thermal expansion in non-perovskitelead-free ferroelectric Sn₂P₂S₆, Phys. Chem. Chem. Phys. 18(8), 6247 (2016)

[252] T. Chattopadhyay, J. X. Boucherle, and H. G. von Schnering, Neutron diffraction study on the structural phase transition in GeTe, J. Phys. C: Solid State Phys. 20(10), 1431 (1987)

[253] T. Chatterji, C. M. N. Kumar, and U. D. Wdowik, Anomalous temperature-induced volume contraction in GeTe, Phys. Rev. B 91(5), 054110 (2015)

[254] Đ. Dangić, A. R. Murphy, É. D. Murray, S. Fahy, and I. Savić, Coupling between acoustic and soft transverse optical phonons leads to negative thermal expansion of GeTe near the ferroelectric phase transition, Phys. Rev. B 97(22), 224106 (2018)

[255] R. D. Shannon, Revised effective ionic radii and systematic studies of interatomic distances in halides and chaleogenides, Acta Crystallogr. A 32(5), 751 (1976)

[256] J. Arvanitidis, K. Papagelis, S. Margadonna, K. Prassides, and A. N. Fitch, Temperature-induced valence transition and associated lattice collapse in samarium fulleride, Nature 425(6958), 599 (2003)

[257] S. Margadonna, J. Arvanitidis, K. Papagelis, and K. Prassides, Negative thermal expansion in the mixed valence ytterbium fulleride, Yb2.75C60, Chem. Mater. 17(17), 4474 (2005)

[258] A. Jayaraman, P. Dernier, and L. D. Longinotti, Study of valence transition in SmS induced by alloying, temperature, and pressure,Phys. Rev. B 11(8), 2783 (1975)

[259] K. Takenaka, D. Asai, R. Kaizu, Y. Mizuno, Y. Yokoyama, Y. Okamoto, N. Katayama, H. S. Suzuki, and Y. Imanaka, Giant isotropic negative thermal expansion in Y-doped samarium monosulfides by intra-atomic charge transfer, Sci. Rep. 9(1), 122 (2019)

[260] D. Asai, Y. Mizuno, H. Hasegawa, Y. Yokoyama, Y. Okamoto, N. Katayama, H. S. Suzuki, Y. Imanaka, and K. Takenaka, Valence fluctuations and giant isotropic negative thermal expansion in Sm_1−xR_xS (R= Y, La, Ce, Pr, Nd), Appl. Phys. Lett. 114(14), 141902 (2019)

[261] D. G. Mazzone, M. Dzero, A. M. M. Abeykoon, H. Yamaoka, H. Ishii, N. Hiraoka, J. P. Rueff, J. M. Ablett, K. Imura, H. S. Suzuki, J. N. Hancock, and I. Jarrige, Kondo-induced giant isotropic negative thermal expansion, Phys. Rev. Lett. 124(12), 125701 (2020)

[262] H. Li, S. Lv, Z. Wang, Y. Xia, Y. Bai, X. Liu, and J. Meng, Machanism of A-B intersite charge transfer and negative thermal expansion in A-site-ordered perovskite LaCu₃Fe₄O₁₂, J. Appl. Phys. 111(10), 103718 (2012)

[263] K. Yamada, K. Shiro, H. Etani, S. Marukawa, N. Hayashi, M. Mizumaki, Y. Kusano, S. Ueda, H. Abe, and T. Irifune, Valence transitions in negative thermal expansion material SrCu₃Fe₄O₁₂, Inorg. Chem. 53(19), 10563 (2014)

[264] M. W. Shimakawa, M. W. Lufaso, and P. M. Woodward, Negative and positive thermal expansion-like volume changes due to intermetallic charge transfer based on an ionic crystal model of transition-metal oxides, APL Mater. 6(8), 086106 (2018)

[265] Z. Liu, Z. Wang, D. Chang, Q. Sun, M. Chao, and Y. Jia, Charge transfer induced negative thermal expansion in perovskite BiNiO3, Comput. Mater. Sci. 113, 198 (2016)

[266] M. Naka, H. Seo, and Y. Motome, Theory of valence transition in BiNiO3, Phys. Rev. Lett. 116(5), 056402 (2016)

[267] E. Pachoud, J. Cumby, J. Wright, B. Raguž, R. Glaum, and J. P. Attfield, Charge order and negative thermal expansion in V₂OPO₄, J. Am. Chem. Soc. 140(2), 636 (2018)

[268] E. Pachoud, J. Cumby, J. Wright, B. Raguz, R. Glaumc, and J. P. Attfield, Electronic origin of negative thermal expansion in V₂OPO₄, Chem. Commun. (Camb.) 56(48), 6523 (2020)

[269] K. Nabetani, Y. Muramatsu, K. Oka, K. Nakano, H. Hojo, M. Mizumaki, A. Agui, Y. Higo, N. Hayashi, M. Takano, and M. Azuma, Suppression of temperature hysteresis in negative thermal expansion compound BiNi_xFe_xO₃ and zero-thermal expansion composite, Appl. Phys. Lett. 106(6), 061912 (2015)

[270] S. Yamada, S. Marukawa, M. Murakami, and S. Mori, “True” negative thermal expansion in Mndoped LaCu₃Fe₄O₁₂ perovskite oxides, Appl. Phys. Lett. 105(23), 231906 (2014)

[271] E. E. McCabe, E. Bousquet, C. P. J. Stockdale, C. A. Deacon, T. T. Tran, P. S. Halasyamani, M. C. Stennett, and N. C. Hyatt, Proper ferroelectricity in the Dion– Jacobson material CsBi₂Ti₂NbO₁₀: Experiment and theory, Chem. Mater. 27(24), 8298 (2015)

[272] M. J. Pitcher, P. Mandal, M. S. Dyer, J. Alaria, P. Borisov, H. Niu, J. B. Claridge, and M. J. Rosseinsky, Tilt engineering of spontaneous polarization and magnetization above 300 K in a bulk layered perovskite, Science 347(6220), 420 (2015)

[273] J. Mangeri, K. C. Pitike, S. P. Alpay, and S. Nakhmanson, Amplitudon and phason modes of electrocaloric energy interconversion, npj Comput. Mater. 2(1), 16020 (2016)

[274] Y. Kimura, Y. Tomioka, H. Kuwahara, A. Asamitsu, M. Tamura, and Y. Tokura, Interplane tunneling magnetoresistance in a layered manganite crystal, Science 274(5293), 1698 (1996)

[275] L. F. Huang, X. Z. Lu, and J. M. Rondinelli, Tunable negative thermal expansion in layered perovskites from quasi-two-dimensional vibrations, Phys. Rev. Lett. 117(11), 115901 (2016)

[276] T. Vogt and D. Buttrey, Temperature dependent structural behavior of Sr₂RhO₄, J. Solid State Chem. 123(1), 186 (1996)

[277] B. Ranjbar and B. J. Kennedy, Anisotropic thermal expansion in Sr₂RhO₄ — A variable temperature synchrotron X-ray diffraction study, Solid State Sci. 49, 43 (2015)

[278] J. Takahashi and N. Kamegashira, X-ray structural study of calcium manganese oxide by Rietveld analysis at high temperatures, MRS Bull. 28(6), 565 (1993)

[279] M. S. Senn, A. Bombardi, C. A. Murray, C. Vecchini, A. Scherillo, X. Luo, and S. W. Cheong, Negative thermal expansion in hybrid improper ferroelectric Ruddlesden–Popper perovskites by symmetry trapping, Phys. Rev. Lett. 114(3), 035701 (2015)

[280] K. J. Cordrey, M. Stanczyk, C. A. L. Dixon, K. S. Knight, J. Gardner, F. D. Morrison, and P. Lightfoot, Structural and dielectric studies of the phase behaviour of the topological ferroelectric La_1−xNd_xTaO₄, Dalton Trans. 44(23), 10673 (2015)

[281] J. W. Zhang, A. S. McLeod, Q. Han, X. Chen, H. A. Bechtel, Z. Yao, S. N. Gilbert Corder, T. Ciavatti, T. H. Tao, M. Aronson, G. L. Carr, M. C. Martin, C. Sow, S. Yonezawa, F. Nakamura, I. Terasaki, D. N. Basov, A. J. Millis, Y. Maeno, and M. Liu, Nano-resolved currentinduced insulator–metal transition in the Mott insulator Ca₂RuO₄, Phys. Rev. X 9(1), 011032 (2019)

[282] G. Zhang and E. Pavarini, Mott transition, spin–orbit effects, and magnetism in Ca₂RuO₄, Phys. Rev. B 95(7), 075145 (2017)

[283] Q. Han and A. Millis, Lattice energetics and correlationdriven metal–insulator transitions: The case of Ca2RuO4, Phys. Rev. Lett. 121(6), 067601 (2018)

[284] M. Braden, G. Andre, S. Nakatsuji, and Y. Maeno, Crystal and magnetic structure of Ca₂RuO₄: Magnetoelastic coupling and the metal–insulator transition, Phys. Rev. B 58(2), 847 (1998)

[285] O. Friedt, M. Braden, G. Andre, P. Adelmann, S. Nakatsuji, and Y. Maeno, Structural and magnetic aspects of the metal–insulator transition in Ca_2−xSr_xRuO₄, Phys. Rev. B 63(17), 174432 (2001)

[286] A. S. Alexander, G. Cao, V. Dobrosavljevic, S. McCall, J. E. Crow, E. Lochner, and R. P. Guertin, Destruction of the Mott insulating ground state of Ca₂RuO₄ by a structural transition, Phys. Rev. B 60(12), R8422 (1999)

[287] T. F. Qi, O. B. Korneta, S. Parkin, L. E. De Long, P. Schlottmann, and G. Cao, Negative volume thermal expansion via orbital and magnetic orders in Ca2Ru_1−xCr_xO₄ (0<x<0.13), Phys. Rev. Lett. 105(17), 177203 (2010)

[288] T. F. Qi, O. B. Korneta, S. Parkin, J. P. Hu, and G. Cao, Magnetic and orbital orders coupled to negative thermal expansion in Mott insulators Ca₂Ru_1−xM_xO4 (M= Mn and Fe), Phys. Rev. B 85(16), 165143 (2012)

[289] K. Takenaka, T. Shinoda, N. Inoue, Y. Okamoto, N. Katayama, Y. Sakai, T. Nishikubo, and M. Azuma, Giant negative thermal expansion in Fe-doped layered ruthenate ceramics, Appl. Phys. Express 10(11), 115501 (2017)

[290] Y. G. Cheng, Y. C. Mao, B. H. Yuan, X. H. Ge, J. Guo, M. J. Chao, and E. J. Liang, Enhanced negative thermal expansion and optical absorption of In_0.6(HfMg)_0.7Mo₃O₁₂ with oxygen vacancies, Phys. Lett. A 381(24), 2195 (2017)

[291] J. Zegkinoglou, J. Strempfer, C. S. Nelson, J. P. Hill, J. Chakhalian, C. Bernhard, J. C. Lang, G. Srajer, H. Fukazawa, S. Nakatsuji, Y. Maeno, and B. Keimer, Orbital ordering transition in Ca₂RuO₄ observed with resonant X-ray diffraction, Phys. Rev. Lett. 95(13), 136401 (2005)

[292] M. C. Lee, C. H. Kim, I. Kwak, J. Kim, S. Yoon, B. C. Park, B. Lee, F. Nakamura, C. Sow, Y. Maeno, T. W. Noh, and K. W. Kim, Abnormal phase flip in the coherent phonon oscillations of Ca₂RuO₄, Phys. Rev. B 98(16), 161115 (2018)

[293] M. C. Lee, C. H. Kim, I. Kwak, C. W. Seo, C. H. Sohn, F. Nakamura, C. Sow, Y. Maeno, E. A. Kim, T. W. Noh, and K. W. Kim, Strong spin–phonon coupling unveiled by coherent phonon oscillations in Ca₂RuO₄, Phys. Rev. B 99(14), 144306 (2019)

[294] V. Sivasubramanian, T. R. Ravindran, R. Nithya, and A. K. Arora, Structural phase transition in indium tungstate, J. Appl. Phys. 96(1), 387 (2004)

[295] B. A. Marinkovic, M. Ari, P. M. Jardim, R. R. de Avillez, F. Rizzo, and F. F. Ferreira, In₂Mo₃O₁₂: A low negative thermal expansion compound, Thermochim. Acta 499(1– 2), 48 (2010)

[296] J. S. O. Evans, T. A. Mary, and A. W. Sleight, Negative thermal expansion in Sc₂(WO₄)₃, J. Solid State Chem. 137(1), 148 (1998)

[297] J. S. O. Evans and T. A. Mary, Structural phase transitions and negative thermal expansion in Sc₂(MoO₄)₃, Inter. J. Inorg. Mater. 2(1), 143 (2000)

[298] H. Liu, W. Sun, Z. Zhang, X. Zhang, Y. Zhou, J. Zhua, and X. Zeng, Tailored phase transition temperature and negative thermal expansion of Sc-substituted Al₂Mo₃O₁₂ synthesized by a co-precipitation method, Inorg. Chem. Front. 6(7), 1842 (2019)

[299] H. F. Liu, Z. P. Zhang, J. Ma, J. Zhu, and X. H. Zeng, Effect of isovalent substitution on phase transition and negative thermal expansion of In_2−xSc_xW₃O₁₂ ceramics, Ceram. Int. 41(8), 9873 (2015)

[300] Y. G. Cheng, Y. C. Mao, X. S. Liu, B. H. Yuan, M. J. Chao, and E. J. Liang, Near-zero thermal expansion of In₂(_1−x)(HfMg)_xMo₃O₁₂ with tailored phase transition, Chin. Phys. B 25(8), 086501 (2016)

[301] W. B. Song, B. H. Yuan, X. S. Liu, Z. Y. Li, J. Q. Wang, and E. J. Liang, Tuning the monoclinic-to-orthorhombic phase transition temperature of Fe₂Mo₃O₁₂ by substitutional coincorporation of Zr⁴⁺ and Mg²⁺, Mater. Res. 29(7), 849 (2014)

[302] W. B. Song, J. Q. Wang, Z. Y. Li, X. S. Liu, B. H. Yuan, and E. J. Liang, Phase transition and thermal expansion property of Cr_2−xZr_0.5xMg_0.5xMo₃O₁₂ solid solution, Chin. Phys. B 23(6), 066501 (2014)

[303] K. J. Miller, C. P. Romao, M. Bieringer, B. A. Marinkovic, L. Prisco, and M. A. White, Near-zero thermal expansion in In(HfMg)_0.5Mo₃O₁₂, J. Am. Ceram. Soc. 96(2), 561 (2013)

[304] S. L. Li, X. H. Ge, H. L. Yuan, D. X. Chen, J. Guo, R. F. Shen, M. J. Chao, and E. J. Liang, Near-zero thermal expansion and phase transitions in HfMg_1−xZn_xMo₃O₁₂, Front. Chem. 6, 115 (2018)

[305] A. Madrid, P. I. Ponton, F. Garcia, M. B. Johnson, M. A. White, and B. A. Marinkovic, Solubility limit of Zn²⁺ in low thermal expansion ZrMgMo₃O₁₂ and its influence on phase transition temperature, Ceram. Int. 46(3), 3979 (2020)

[306] R. F. Shen, B. H. Yuan, S. L. Li, X. H. Ge, J. Guo, and E. J. Liang, Near-zero thermal expansion of ZrxHf_1−xMgMo₃O₁₂ in a larger temperature range, Optik (Stuttg.) 165, 1 (2018)

[307] S. Sumithra and A. M. Umarji, Role of crystal structure on the thermal expansion of Ln₂W₃O₁₂ (Ln=La, Nd, Dy, Y, Er and Yb), Solid State Sci. 6(12), 1313 (2004)

[308] X. L. Xiao, Y. Z. Cheng, J. Peng, M. M. Wu, D. F. Chen, Z. B. Hu, R. Kiyanagi, J. S. Fieramosca, S. Short, and J. Jorgensen, Thermal expansion properties of A₂(MO₄)₃ (A= Ho and Tm; M=W and Mo) , Solid State Sci. 10(3), 321 (2008)

[309] E. J. Liang, H. Huo, J. Wang, and M. J. Chao, Effect of water species on the phonon modes in orthorhombic Y₂(MoO₄)₃ revealed by Raman spectroscopy, J. Phys. Chem. C 112(16), 6577 (2008)

[310] Z. Y. Li, W. B. Song, and E. J. Liang, Structures, phase transition, and crystal water of Fe_2−xY_xMo₃O₁₂, J. Phys. Chem. C 115(36), 17806 (2011)

[311] N. A. Banek, H. I. Baiz, A. Latigo, and C. Lind, Autohydration of nanosized cubic zirconium tungstate, J. Am. Chem. Soc. 132(24), 8278 (2010)

[312] N. Duan, U. Kameswari, and A. W. Sleight, Further contraction of ZrW₂O₈, J. Am. Chem. Soc. 121(44), 10432 (1999)

[313] Q. L. Gao, J. Chen, Q. Sun, D. H. Chang, Q. Z. Huang, H. Wu, A. Sanson, R. Milazzo, H. Zhu, Q. Li, Z. N. Liu, J. X. Deng, and X. R. Xing, Switching between giant positive and negative thermal expansions of a YFe(CN)6- based Prussian blue analogue induced by guest species, Angew. Chem. Int. Ed. 56(28), 9023 (2017)

[314] A. L. Goodwin, K. W. Chapman, and C. J. Kepert, Guest-dependent negative thermal expansion in nanoporous Prussian blue analogues M^II Pt^IV (CN)_{6 · x}(H₂O) (0≤ x≤2; M=Zn, Cd), J. Am. Chem. Soc. 127(51), 17980 (2005)

[315] T. Pretsch, K. W. Chapman, G. J. Halder, and C. J. Kepert, Dehydration of the nanoporous coordination framework Er^III [Co^III (CN)₆]·4(H₂O): Single crystal tosingle crystal transformation and negative thermal expansion in Er^III [CoIII (CN)₆], Chem. Commun. (Camb.) 17(17), 1857 (2006)

[316] M. K. Gupta, R. Mittal, B. Singh, and S. L. Chaplot, Effect of hydration and ammonization on the thermal expansion behavior of ZrW₂O₈: Ab initio lattice dynamical perspective, Phys. Rev. B 98(22), 224303 (2018)

[317] H. Liu, Z. Zhang, W. Zhang, X. Zeng, and X. Chen, Synthesis and negative thermal expansion property of Y_2−xLa_xW₃O₁₂ (0≤x≤2), Ceram. Int. 39(3), 2781 (2013)

[318] Q. J. Li, B. H. Yuan, W. B. Song, E. J. Liang, and B. Yuan, Phase transition, hygroscopicity and thermal expansion properties of Yb_2−xAl_xMo₃O₁₂, Chin. Phys. B 21(4), 046501 (2012)

[319] X. S. Liu, Y. G. Cheng, E. J. Liang, and M. J. Chao, Interaction of crystal water with the building block in Y₂Mo₃O₁₂ and the effect of Ce³⁺ doping, Phys. Chem. Chem. Phys. 16(22), 12848 (2014)

[320] X. S. Liu, B. H. Yuan, Y. G. Cheng, X. H. Ge, E. J. Liang, and W. F. Zhang, Avoiding the invasion of H₂O into Y₂Mo₃O₁₂ by coating with C₃N₄ to improve negative thermal expansion properties, Phys. Chem. Chem. Phys. 19(21), 13443 (2017)

[321] A. Sanson, On the switching between negative and positive thermal expansion in framework materials, Mater. Res. Lett. 7(10), 412 (2019)

[322] M. Cetinkol and A. P. Wilkinson, Pressure dependence of negative thermal expansion in Zr₂(WO₄)(PO₄)₂, Solid State Commun. 149(11–12), 421 (2009)

[323] M. Cetinkol, A. P. Wilkinson, and P. L. Lee, Structural changes accompanying negative thermal expansion in Zr₂(MoO₄)(PO₄)₂, J. Solid State Chem. 182(6), 1304 (2009)

[324] M. Y. Wu, L. Wang, Y. Jia, Z. X. Guo, and Q. Sun, Theoretical study of hydration in Y2Mo3O12: Effects on structure and negative thermal expansion, AIP Adv. 5(2), 027126 (2015)

[325] M. Y. Wu, Y. Jia, and Q. Sun, Effects of A³⁺ cations on hydration in A₂M₃O₁₂ family materials: A firstprinciples study, Comput. Mater. Sci. 111, 28 (2016)

[326] G. Jefferson, T. A. Parthasarathy, and R. J. Kerans, Tailorable thermal expansion hybrid structures, Int. J. Solids Struct. 46(11–12), 2372 (2009)

[327] L. Wu, B. Li, and J. Zhou, Isotropic negative thermal expansion metamaterials, ACS Appl. Mater. Interfaces 8(24), 17721 (2016)

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