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題名 利用第一原理計算研究多鐵氧化物Cu3Mo2O9的磁性,電子態及鐵電性質
Ab Initio Studies of The Magnetic, Electronic and Ferroelectric Properties of Multiferroic Oxide Cu3Mo2O9
作者 蕭逸修
Hsiao, Yi Hsiu
貢獻者 郭光宇
Guo, Guang Yu
蕭逸修
Hsiao, Yi Hsiu
關鍵詞 第一原理
多鐵氧化物
鐵電性
幾何不穩定性
Ab Initio
multiferroic oxide
ferroelectricity
geometric frustration
日期 2012
上傳時間 30-Oct-2012 15:22:21 (UTC+8)
摘要 在此論文中，我們利用第一原理計算研究多鐵材料Cu3Mo2O9的磁性、電子態及多鐵性質。我們發現在此系統中，電子與電子間的庫倫排斥力必須被考慮，以致於導帶與價帶間能隙能夠被良好地描述。由於晶體結構所導致的幾何不穩定性，系統的磁結構尚未在實驗測量中被確定。在我們的理論計算當中得到的磁結構與Vilminot等研究人員根據實驗結果猜測出的非線性反鐵磁結構類似。交換作用與自旋軌道耦合間的爭競決定了電子自旋方向的傾斜。計算所得到的交換作用係數與實驗結果吻合良好。利用Berry’s phase計算，我們得到了系統自發電極化的理論值，其強度與實驗量測值在同一個數量級。然而，在我們計算中得到的電極化方向(平行於b軸)與實驗(平行於c軸)不符。此外，我們發現一磁結構之理論電極化方向與實驗相符，然而其磁結構之對稱性與實驗不符。目前，尚未有第一原理計算研究此氧化物，我們希望此論文能夠對同樣有興趣研究此材料的研究人員有所幫助。
In this thesis, we used the ab initio method to study a multiferroic oxide Cu3Mo2O9. The correlations of electrons must be considered in this system so that a reasonable energy gap can be obtained. Due to the geometric frustration of magnetic structure caused by crystal structure, the ground state spin configuration in this system still has not been determined experimentally. We found some spin configurations similar to the non-collinear anti-ferromagnetic spins configuration suggested by Vilminot et al.. Competition between exchange interactions and spin-orbit coupling effect determines the canting of spins on Cu atoms. The calculated exchange parameters agree with the experimental results well. By using Berry phase calculations, we obtained the theoretical value of spontaneous electric polarization. The strength of polarization in our results is in the same order of results of experiments. However, the direction of electric polarization we found (along b-axis) is different from the experimental measurements (along c-axis). We have found a spin configuration that the theoretical electric polarization of the state agrees with the experimental results. However, the symmetry of the spin configuration does not satisfy the conditions suggested by results of the neutron diffraction experiment. And, spins on neighboring Cu2 and Cu3 do not form a singlet dimer. Since there still is no ab initio calculation studying this oxide, we hope that our studies can help those who are also interested in this material.
參考文獻 [1] P. Hohenberg andW. Kohn, ”Inhomogeneous Electron Gas”, Phys. Rev. 136, B864 (1964).
[2] W. Kohn and L. J. Sham, ”Self-Consistent Equations Including Exchange and Correlation
Effects”, Phys. Rev. 140, A1133 (1965).
[3] M. Born and J. R. Oppenheimer, ”On The Quantum Theory of Molecules”, Ann. Physik
84, 457 (1927).
[4] L. H. Thomas, ”The Calculation of Atomic Fields”, Proc. Camb. Phil. Soc. 23, 542 (1927).
[5] J. P. Perdew and A. Zunger, ”Self-interaction correction to density-functional approximations
for many-electron systems”, Phys. Rev. B 23, 5048 (1981).
[6] J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and
C. Fiolhais, ”Atoms, Molecules, Solids, and Surfaces: Applications of The Generalized
Gradient Approximation for Exchange and Correlation”, Phys. Rev. B 46, 6671 (1992);
48, 4978(E)(1993).
[7] J. P. Perdew, K. Burke, and M. Ernzerhof, ”Generalized Gradient Approximation Made
Simple”, Phys. Rev. Lett. 77, 3865 (1996).
[8] J. H. de Boer and E. J. W. Verwey, ”Semi-Conductors with Partially and with Completely
Filled 3d-Lattice Bands”, Proc. Phys. Soc. 49, 59 (1937).
[9] J. Hubbard, ”Electron Correlations in Narrow Energy Bands.”, Proc. Roy. Soc. A 276, 238
(1963).
[10] J. Hubbard, ”Electron Correlations in Narrow Energy Bands. III. An Improved Solution”,
Proc. Roy. Soc. A 281, 41 (1964).
[11] A. Svane and O. Gunnarsson, ”Transition-Metal Oxides in The Self-Interaction-Corrected
Density-Functional Formalism”, Phys. Rev. Lett. 65, 1148 (1990).
[12] S. Massidda, M. Posternak and A. Baldereschi, ”Hartree-Fock LAPW Approach to The
Electronic Properties of Periodic Systems”, Phys. Rev. B 48, 5058 (1993).
[13] L. Hedin, ”New Method for Calculating The One-Particle Green’s Function with Application
to the Electron-Gas Problem”, Phys. Rev. 139, A796 (1965).
[14] A. I. Liechtenstein, V. I. Anisimov and J. Zaanen, ”Density-Functional Theory and Strong
Interactions: Orbital Ordering in Mott-Hubbard Insulators”, Phys. Rev. B 52, R5467
(1995).
[15] S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys and A. P. Sutton, ”Electron-
Energy-Loss Spectra and The Structural Stability of Nickel Oxide:An LSDA+U Study”,
Phys. Rev. B 57, 1505 (1998).
[16] H. Bethe, ”Splitting of Terms in Crystals”, Ann. Physik 3, 133 (1929).
[17] J. H. Van Vleck, ”Theory of The Variations in Paramagnetic Anisotropy Among Different
Salts of The Iron Group”, Phys. Rev. 41, 208 (1932).
[18] H. A. Jahn and E. Teller, ”Stability of polyatomic Molecules in Degenerate Electronic
States. I. Orbital Degeneracy”, Proc. Roy. Soc. A 161, 220 (1937).
[19] J. Springborg and C. E. Schaffer, ”Tetrakis (pyridine) Cobalt(III) Complexes”, Acta. Chem.
Scand. 27, 3312 (1973).
[20] L. D. Landau and E. M. Lifshitz, Electrodynamics of continuous media (Fizmatgiz,
Moscow, 1959).
[21] E. Ascher, H. Rieder, H. Schmid and H. Stossel, ”Some Properteis of Ferromagnetoelectric
Nickel-Iodine Boracite, Ni3B7O13I”, J. Appl. Phys. 37, 1404 (1966).
[22] J. Wang, J. B. Neaton, H. Zheng, V. Nagarajan, S. B. Ogale, B. Liu, D. Viehland, V.
Vaithyanathan, D. G. Schlom, U. V. Waghmare, N. A. Spaldin, K. M. Rabe, M. Wuttig
and R. Ramesh, ”Epitaxial BiFeO3 Multiferroic Thin Film Heterostructures”, Science 299,
1719 (2003).
[23] N. Hur, S. Park, P. A. Sharma, J. S. Ahn, S. Guha and S-W. Cheong, ”Electric Polarization
Reversal and Memory in A Multiferroic Material Induced by Magnetic Fields”, Nature
429, 392 (2004).
[24] T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima and Y. Tokura, ”Magnetic Control
of Ferroelectric Polarization”, Nature 426, 55 (2003).
[25] G. Toulouse, ”Theory of The Frustration Effect in Spin Glasses I”, Commun. Phys. 2, 115
(1977).
[26] J. Vannimenus and G. Toulouse, ”Theory of The Frustration Effect. II. Ising Spins on A
Square Lattice”, J. Phys. C: Solid State Phys. 10, L537 (1977).
[27] G. H. Wannier, ”Antiferromagnetism. The Triangular Ising Net”, Phys. Rev. 79, 357
(1950).
[28] L. Pauling, ”The Structure and Entropy of Ice and of Other Crystals with Some Randomness
of Atomic Arrangement”, J. Am. Chem. Soc. 57 2680 (1935).
[29] N. A. Hill, ”Why Are There so Few Magnetic Ferroelectrics?”, J. Phys. Chem. B 104, 6694
(2000).
[30] D. I. Khomskii, ”Multiferroics: Different Ways to Combine Magnetism and Ferroelectricity”,
J. Magn. Magn. Mater. 306, 1 (2006).
[31] D. V. Efremov, J. van den Brink, and D. I. Khomskii, ”Bond-Versus Site-Centred Ordering
and Possible Ferroelectricity in Manganites”, Nature Mater. 3, 853 (2004).
[32] B. B. Van Aken, T. T. M. Palstra, A. Filippetti and N. A. Spaldin, ”The Origin of Ferroelectricity
in Magnetoelectric YMnO3”, Nature Mater. 3, 164 (2004).
[33] D. Khomskii, ”Classifying Multiferroics: Mechanisms and Effects”, Physics 2, 20 (2009).
[34] H. Katsura, N. Nagaosa and A. V. Balatsky, ”Spin Current and Magnetoelectric Effect in
Noncollinear Magnets”, Phys. Rev. Lett. 95, 057205 (2005).
[35] M. V. Mostovoy, ”Ferroelectricity in Spiral Magnets”, Phys. Rev. Lett. 96, 067601 (2006).
[36] M. Fiebig, ”Revival of The Magnetoelectric Effect”, J. Phys. D 38, R123 (2005).
[37] H. Kuroe, T. Hosaka, S. Hachiuma, T. Sekine, M. Hase, K. Oka, T. Ito, H. Eisaki, M.
Fujisawa, S. Okubo and H. Ohta, ”Electric Polarization Induced by Neel Order without
Magnetic Superlattice: Experimental Study of Cu3Mo2O9 and Numerical Study of A Small
Spin Cluster”, J. Phys. Soc. Jpn. 80, 083705 (2011).
[38] S. Vilminot, G. Andre and M. Kurmoo, ”Magnetic Properties and Magnetic Structure of
Cu3Mo2O9”, Inorg. Chem, 48, 2687 (2009).
[39] T. Hamasaki, T. Ide, H. Kuroe and T. Sekine, ”Successive Phase Transitions to Antiferromagnetic
and Weak-Ferromagnetic Long-Range Order in The Quasi-One-Dimensional
Antiferromagnet Cu3Mo2O9”, Phys. Rev. B 77, 134419 (2008).
描述 碩士
國立政治大學
應用物理研究所
99755007
101
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0099755007
資料類型 thesis
dc.contributor.advisor 郭光宇zh_TW
dc.contributor.advisor Guo, Guang Yuen_US
dc.contributor.author (Authors) 蕭逸修zh_TW
dc.contributor.author (Authors) Hsiao, Yi Hsiuen_US
dc.creator (作者) 蕭逸修zh_TW
dc.creator (作者) Hsiao, Yi Hsiuen_US
dc.date (日期) 2012en_US
dc.date.accessioned 30-Oct-2012 15:22:21 (UTC+8)-
dc.date.available 30-Oct-2012 15:22:21 (UTC+8)-
dc.date.issued (上傳時間) 30-Oct-2012 15:22:21 (UTC+8)-
dc.identifier (Other Identifiers) G0099755007en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/55038-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 應用物理研究所zh_TW
dc.description (描述) 99755007zh_TW
dc.description (描述) 101zh_TW
dc.description.abstract (摘要) 在此論文中，我們利用第一原理計算研究多鐵材料Cu3Mo2O9的磁性、電子態及多鐵性質。我們發現在此系統中，電子與電子間的庫倫排斥力必須被考慮，以致於導帶與價帶間能隙能夠被良好地描述。由於晶體結構所導致的幾何不穩定性，系統的磁結構尚未在實驗測量中被確定。在我們的理論計算當中得到的磁結構與Vilminot等研究人員根據實驗結果猜測出的非線性反鐵磁結構類似。交換作用與自旋軌道耦合間的爭競決定了電子自旋方向的傾斜。計算所得到的交換作用係數與實驗結果吻合良好。利用Berry’s phase計算，我們得到了系統自發電極化的理論值，其強度與實驗量測值在同一個數量級。然而，在我們計算中得到的電極化方向(平行於b軸)與實驗(平行於c軸)不符。此外，我們發現一磁結構之理論電極化方向與實驗相符，然而其磁結構之對稱性與實驗不符。目前，尚未有第一原理計算研究此氧化物，我們希望此論文能夠對同樣有興趣研究此材料的研究人員有所幫助。zh_TW
dc.description.abstract (摘要) In this thesis, we used the ab initio method to study a multiferroic oxide Cu3Mo2O9. The correlations of electrons must be considered in this system so that a reasonable energy gap can be obtained. Due to the geometric frustration of magnetic structure caused by crystal structure, the ground state spin configuration in this system still has not been determined experimentally. We found some spin configurations similar to the non-collinear anti-ferromagnetic spins configuration suggested by Vilminot et al.. Competition between exchange interactions and spin-orbit coupling effect determines the canting of spins on Cu atoms. The calculated exchange parameters agree with the experimental results well. By using Berry phase calculations, we obtained the theoretical value of spontaneous electric polarization. The strength of polarization in our results is in the same order of results of experiments. However, the direction of electric polarization we found (along b-axis) is different from the experimental measurements (along c-axis). We have found a spin configuration that the theoretical electric polarization of the state agrees with the experimental results. However, the symmetry of the spin configuration does not satisfy the conditions suggested by results of the neutron diffraction experiment. And, spins on neighboring Cu2 and Cu3 do not form a singlet dimer. Since there still is no ab initio calculation studying this oxide, we hope that our studies can help those who are also interested in this material.en_US
dc.description.tableofcontents List of Figures . . . . . . . . . . . . . . . . . . 3
List of Tables. . . . . . . . . . . . . . . . . . . 7
1 Introduction 8
2 Density Functional Theory 9
2.1 Born-Oppenheimer approximation . . . . . . . . 9
2.2 Thomas-Fermi Theory. . . . . . . . . . . . . . 10
2.3 Density Functional Theory. . . . . . . . . . . 11
2.3.1 Hohenberg-Kohn Theorem . . . . . . . . . . . 12
2.3.2 Kohn-Sham Equation . . . . . . . . . . . . . 13
2.3.3 Exchange-Correlation Energy. . . . . . . . . 14
2.4 Mott Insulators. . . . . . . . . . . . . . . . 15
2.4.1 Hubbard Model. . . . . . . . . . . . . . . . 15
2.4.2 Beyond DFT : DFT+U . . . . . . . . . . . . . 17
3 Crystal Field Theory 18
3.1 Atomic Orbitals. . . . . . . . . . . . . . . . 18
3.2 Crystal Field Theory . . . . . . . . . . . . . 19
3.3 High Spin and Low Spin . . . . . . . . . . . . 21
3.4 Crystal Field Stabilization Energy . . . . . . 22
3.5 Jahn-Teller Theorem. . . . . . . . . . . . . . 23
3.6 Colors of Transition Metal Complexes . . . . . 23
4 Multiferroics 25
4.1 Introduction . . . . . . . . . . . . . . . . . 25
4.2 Symmetry . . . . . . . . . . . . . . . . . . . 26
4.3 Geometric Frustration. . . . . . . . . . . . . 27
4.4 Multiferroics. . . . . . . . . . . . . . . . . 27
4.4.1 Type-I Multiferroics . . . . . . . . . . . . 27
4.4.2 Type-II Multiferroics. . . . . . . . . . . . 30
5 Calculated Physical Properties of Cu3Mo2O9 . . . 33
5.1 Introduction . . . . . . . . . . . . . . . . . 33
5.2 Crystal Structure and Computational Details. . 36
5.3 Magnetic Structure . . . . . . . . . . . . . . 40
5.4 Exchange Interactions. . . . . . . . . . . . . 40
5.5 Electronic Structure . . . . . . . . . . . . . 49
5.6 Spontaneous Electric Polarization. . . . . . . 58
6 Summary and Conclusions 61
Reference. . . . . . . . . . . . . . . . . . . . . 66		zh_TW
dc.language.iso en_US-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0099755007en_US
dc.subject (關鍵詞) 第一原理zh_TW
dc.subject (關鍵詞) 多鐵氧化物zh_TW
dc.subject (關鍵詞) 鐵電性zh_TW
dc.subject (關鍵詞) 幾何不穩定性zh_TW
dc.subject (關鍵詞) Ab Initioen_US
dc.subject (關鍵詞) multiferroic oxideen_US
dc.subject (關鍵詞) ferroelectricityen_US
dc.subject (關鍵詞) geometric frustrationen_US
dc.title (題名) 利用第一原理計算研究多鐵氧化物Cu3Mo2O9的磁性,電子態及鐵電性質zh_TW
dc.title (題名) Ab Initio Studies of The Magnetic, Electronic and Ferroelectric Properties of Multiferroic Oxide Cu3Mo2O9en_US
dc.type (資料類型) thesisen
dc.relation.reference (參考文獻) [1] P. Hohenberg andW. Kohn, ”Inhomogeneous Electron Gas”, Phys. Rev. 136, B864 (1964).
[2] W. Kohn and L. J. Sham, ”Self-Consistent Equations Including Exchange and Correlation
Effects”, Phys. Rev. 140, A1133 (1965).
[3] M. Born and J. R. Oppenheimer, ”On The Quantum Theory of Molecules”, Ann. Physik
84, 457 (1927).
[4] L. H. Thomas, ”The Calculation of Atomic Fields”, Proc. Camb. Phil. Soc. 23, 542 (1927).
[5] J. P. Perdew and A. Zunger, ”Self-interaction correction to density-functional approximations
for many-electron systems”, Phys. Rev. B 23, 5048 (1981).
[6] J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and
C. Fiolhais, ”Atoms, Molecules, Solids, and Surfaces: Applications of The Generalized
Gradient Approximation for Exchange and Correlation”, Phys. Rev. B 46, 6671 (1992);
48, 4978(E)(1993).
[7] J. P. Perdew, K. Burke, and M. Ernzerhof, ”Generalized Gradient Approximation Made
Simple”, Phys. Rev. Lett. 77, 3865 (1996).
[8] J. H. de Boer and E. J. W. Verwey, ”Semi-Conductors with Partially and with Completely
Filled 3d-Lattice Bands”, Proc. Phys. Soc. 49, 59 (1937).
[9] J. Hubbard, ”Electron Correlations in Narrow Energy Bands.”, Proc. Roy. Soc. A 276, 238
(1963).
[10] J. Hubbard, ”Electron Correlations in Narrow Energy Bands. III. An Improved Solution”,
Proc. Roy. Soc. A 281, 41 (1964).
[11] A. Svane and O. Gunnarsson, ”Transition-Metal Oxides in The Self-Interaction-Corrected
Density-Functional Formalism”, Phys. Rev. Lett. 65, 1148 (1990).
[12] S. Massidda, M. Posternak and A. Baldereschi, ”Hartree-Fock LAPW Approach to The
Electronic Properties of Periodic Systems”, Phys. Rev. B 48, 5058 (1993).
[13] L. Hedin, ”New Method for Calculating The One-Particle Green’s Function with Application
to the Electron-Gas Problem”, Phys. Rev. 139, A796 (1965).
[14] A. I. Liechtenstein, V. I. Anisimov and J. Zaanen, ”Density-Functional Theory and Strong
Interactions: Orbital Ordering in Mott-Hubbard Insulators”, Phys. Rev. B 52, R5467
(1995).
[15] S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys and A. P. Sutton, ”Electron-
Energy-Loss Spectra and The Structural Stability of Nickel Oxide:An LSDA+U Study”,
Phys. Rev. B 57, 1505 (1998).
[16] H. Bethe, ”Splitting of Terms in Crystals”, Ann. Physik 3, 133 (1929).
[17] J. H. Van Vleck, ”Theory of The Variations in Paramagnetic Anisotropy Among Different
Salts of The Iron Group”, Phys. Rev. 41, 208 (1932).
[18] H. A. Jahn and E. Teller, ”Stability of polyatomic Molecules in Degenerate Electronic
States. I. Orbital Degeneracy”, Proc. Roy. Soc. A 161, 220 (1937).
[19] J. Springborg and C. E. Schaffer, ”Tetrakis (pyridine) Cobalt(III) Complexes”, Acta. Chem.
Scand. 27, 3312 (1973).
[20] L. D. Landau and E. M. Lifshitz, Electrodynamics of continuous media (Fizmatgiz,
Moscow, 1959).
[21] E. Ascher, H. Rieder, H. Schmid and H. Stossel, ”Some Properteis of Ferromagnetoelectric
Nickel-Iodine Boracite, Ni3B7O13I”, J. Appl. Phys. 37, 1404 (1966).
[22] J. Wang, J. B. Neaton, H. Zheng, V. Nagarajan, S. B. Ogale, B. Liu, D. Viehland, V.
Vaithyanathan, D. G. Schlom, U. V. Waghmare, N. A. Spaldin, K. M. Rabe, M. Wuttig
and R. Ramesh, ”Epitaxial BiFeO3 Multiferroic Thin Film Heterostructures”, Science 299,
1719 (2003).
[23] N. Hur, S. Park, P. A. Sharma, J. S. Ahn, S. Guha and S-W. Cheong, ”Electric Polarization
Reversal and Memory in A Multiferroic Material Induced by Magnetic Fields”, Nature
429, 392 (2004).
[24] T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima and Y. Tokura, ”Magnetic Control
of Ferroelectric Polarization”, Nature 426, 55 (2003).
[25] G. Toulouse, ”Theory of The Frustration Effect in Spin Glasses I”, Commun. Phys. 2, 115
(1977).
[26] J. Vannimenus and G. Toulouse, ”Theory of The Frustration Effect. II. Ising Spins on A
Square Lattice”, J. Phys. C: Solid State Phys. 10, L537 (1977).
[27] G. H. Wannier, ”Antiferromagnetism. The Triangular Ising Net”, Phys. Rev. 79, 357
(1950).
[28] L. Pauling, ”The Structure and Entropy of Ice and of Other Crystals with Some Randomness
of Atomic Arrangement”, J. Am. Chem. Soc. 57 2680 (1935).
[29] N. A. Hill, ”Why Are There so Few Magnetic Ferroelectrics?”, J. Phys. Chem. B 104, 6694
(2000).
[30] D. I. Khomskii, ”Multiferroics: Different Ways to Combine Magnetism and Ferroelectricity”,
J. Magn. Magn. Mater. 306, 1 (2006).
[31] D. V. Efremov, J. van den Brink, and D. I. Khomskii, ”Bond-Versus Site-Centred Ordering
and Possible Ferroelectricity in Manganites”, Nature Mater. 3, 853 (2004).
[32] B. B. Van Aken, T. T. M. Palstra, A. Filippetti and N. A. Spaldin, ”The Origin of Ferroelectricity
in Magnetoelectric YMnO3”, Nature Mater. 3, 164 (2004).
[33] D. Khomskii, ”Classifying Multiferroics: Mechanisms and Effects”, Physics 2, 20 (2009).
[34] H. Katsura, N. Nagaosa and A. V. Balatsky, ”Spin Current and Magnetoelectric Effect in
Noncollinear Magnets”, Phys. Rev. Lett. 95, 057205 (2005).
[35] M. V. Mostovoy, ”Ferroelectricity in Spiral Magnets”, Phys. Rev. Lett. 96, 067601 (2006).
[36] M. Fiebig, ”Revival of The Magnetoelectric Effect”, J. Phys. D 38, R123 (2005).
[37] H. Kuroe, T. Hosaka, S. Hachiuma, T. Sekine, M. Hase, K. Oka, T. Ito, H. Eisaki, M.
Fujisawa, S. Okubo and H. Ohta, ”Electric Polarization Induced by Neel Order without
Magnetic Superlattice: Experimental Study of Cu3Mo2O9 and Numerical Study of A Small
Spin Cluster”, J. Phys. Soc. Jpn. 80, 083705 (2011).
[38] S. Vilminot, G. Andre and M. Kurmoo, ”Magnetic Properties and Magnetic Structure of
Cu3Mo2O9”, Inorg. Chem, 48, 2687 (2009).
[39] T. Hamasaki, T. Ide, H. Kuroe and T. Sekine, ”Successive Phase Transitions to Antiferromagnetic
and Weak-Ferromagnetic Long-Range Order in The Quasi-One-Dimensional
Antiferromagnet Cu3Mo2O9”, Phys. Rev. B 77, 134419 (2008).		zh_TW