超導銅氧化物是將電子或電洞摻雜進入莫特絕緣體中產生而成，其複雜的相圖以及各種電子結構有序性之間的相互競爭關係，造成難以被了解的超導機制。前人的實驗中被觀察到，當(La1.6-xNd0.4)SrxCuO4電洞摻雜濃度等於八分之一時，其超導現象會被異常抑制，而這個‘八分之一的問題’在電洞摻雜的超導銅氧化物中成為了一個待解的謎。(La1.6-xNd0.4)SrxCuO4 (x=1/8)化合物中，條紋相(stripe phase)的概念被提了出來，這個概念是由電荷、自旋跟晶格調變而成。非彈性中子散射的結果顯示，超導現象與條紋有序相存在競爭的關係。然而，電荷密度波的來源依舊在爭論當中，此外，電子-聲子的耦合在電荷密度波中扮演的角色也是一個重要且尚未被解開的問題。
本論文中，我們使用軟Ｘ光共振散射，在氧的Ｋ層吸收邊上量測(La1.6-xNd0.4)SrxCuO4的電子-聲子耦合以及電荷密度波。透過高解析的軟Ｘ光共振非彈性散射技術，在140 K下偵測到 (La1.6-xNd0.4)SrxCuO4的電荷密度波訊號。我們的實驗結果顯示低溫的四方結構可以穩定電荷密度波，在低溫下(25 K)電荷密度波的波向量也與前人的研究一致。隨著軟Ｘ光共振非彈性散射的能量解析度提升，我們可以分辨出三種不同聲子的振動模式。動量解析的軟Ｘ光共振非彈性散射量測結果顯示，沿著反節點方向(π,0)，某些源自於‘半呼吸’(half-breathing)振動, 頂端氧原子振動及A1g等特定聲子，具有異常能量色散現象。

The superconducting cuprates are created by doping holes or electrons into a Mott insulator. Their complex phase diagram and the competition between electronic orders hinder our understanding of the novel superconductivity of cuprates. An anomalous suppression of superconductivity was observed in (La1.6-xNd0.4)SrxCuO4 when the hole concentration is equal to 1/8. This ‘1/8 problem’ of hole-doped superconducting cuprates becomes one of long-standing mysteries. The concept of stripe phase, which consists of charge, spin, and lattice modulations, was proposed to understand the (La1.6-xNd0.4)SrxCuO4 (x=1/8) compound. Inelastic neutron scattering results suggested a competition between superconductivity and stripe order. However, the origin of charge density waves (CDW) is still under debate. In addition, the role of electron-phonon coupling (EPC) for CDW remains an important question.
Here we present a study of CDW and EPC in (La1.6-xNd0.4)SrxCuO4 by using resonant soft X-ray scattering with the incident photon energy near the O K-edge. Using high-resolution resonant inelastic X-ray scattering (RIXS), CDW signals were detected from (La1.6-xNd0.4)SrxCuO4 with the temperature up to 140 K. Our CDW results suggest that the low temperature tetragonal structure stabilizes the CDW and its wavevector (qCDW) at low temperature (25 K) is consistent with that of previous studies. With the improvement of RIXS energy resolution, three phonons features were identified. Momentum-resolved RIXS measurements revealed an anomaly in the dispersion of half-breathing, apical and A1g phonons along the anti-nodal direction (π,0), indicating that the phonons are closely coupled to CDW via EPC.

Chapter 1 Introduction 1
1.1 Overview 1
1.2 Stripe phase and superconductivity 5
Chapter 2 Experimental Techniques and Setups 7
2.1 Sample preparation and cleaving 7
2.1.1 Sample growth 7
2.1.2 Sample cleaving 7
2.2 X-ray diffraction 10
2.2.1 Diffraction condition 10
2.2.2 Four-circle X-ray diffraction station 12
2.2.3 Two-circle X-ray diffraction station 13
2.2.4 X-ray diffraction results 15
2.3 Soft X-ray absorption spectroscopy 18
2.4 Resonant soft X-ray Scattering 19
2.4.1 Introduction 19
2.4.2 Interaction of photons with electrons 21
2.4.3 RIXS process 22
2.4.4 Elementary excitations of strongly correlated systems 24
2.4.5 RIXS setups 27
Chapter 3 Charge Density Waves of Nd-LSCO 33
3.1 Charge density wave 33
3.2 Stripe phase of cuprates 34
3.3 Observation of charge density wave 36
3.4 Charge density wave of Nd-LSCO 38
Chapter 4 Phonon Excitations 45
4.1 Electron-phonon coupling & CDW 45
4.2 Detection of phonon excitations 46
4.3 Phonon excitations in Nd-LSCO 49
Chapter 5 Conclusion 58
Reference 60

[1] H. K. Onnes, Comm. Phys. Lab. Univ. Leiden 120b, 15 (1911).
[2] H. K. Onnes, Comm. Phys. Lab. Univ. Leiden, 14 (1894).
[3] J. Bardeen, L. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957).
[4] L. Cooper, Phys. Rev. 104, 1189 (1956).
[5] J. G. Bednorz, and K. A. Müller, Z. Phys.B-Condensed Matter 64, 189 (1986).
[6] B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, and J. Zaanen, Nature 518, 179 (2015).
[7] M. K. Wu, J. R. Ashburn, C. J. Torng, P. H. Hor, R. L. Meng, L. Gao, Z. J.Huang, Y.Q. Wang, and C. W. Chu, Phys. Rev. Lett. 58, 9 (1987).
[8] A. Schilling, M. Cantoni, J. D. Guo, and H. R. Ott, Nature 363, 56 (1993).
[9] D. C. Peets, J. D. F. Mottershead, B. Wu, I. S. Elfimov, R. Liang, W. N. Hardy, D. A. Bonn, M. Raudsepp, N. J. C. Ingle, and A. Damascelli, New Journal of Physics 9, 28 (2007).
[10] G. Ghiringhelli, M. Le Tacon, M. Minola, S. Blanco-Canosa, C. Mazzoli, N. B. Brookes, G. M. De Luca, A. Frano, D. G. Hawthorn, F. He, T. Loew, M. Moretti Sala, D. C. Peets, M. Salluzzo, E. Schierle, R. Sutarto, G. A. Sawatzky, E. Weschke, B. Keimer, and L. Braicovich, Science 337, 6096, 821 (2012).
[11] H. Miao, J. Lorenzana, G. Seibold, Y.Y. Peng, A. Amorese, F. Y. Harris, K. Kummer, N. B. Brookes, R. M. Konik, V. Thampy, G. D. Gu, G. Ghiringhelli, L. Braicovich, and M. P. M. Dean, P.N.A.S. 114, 12430 (2017).
[12] J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, and S. Uchida, Nature (London) 375, 561 (1995).
[13] D. F. Agterberg, J. C. Seamus Davis, S. D. Edkins, E. Fradkin, D. J. V. Harlingen, S. A. Kivelson, P. A. Lee, L. Radzihovsky, J. M. Tranquada, and Y. Wang, arXiv:1904.09687v3
[14] J. J. Wen, H. Huang, S.J. Lee, H. Jang, J. Knight, Y. S. Lee, M. Fujita, K. M. Suzuki, S. Asano, S. A. Kivelson, C. C. Kao, and J. S. Lee, Nature Communications 10, 3269 (2019).
[15] A. R. Moodenbaugh, Y. Xu, M. Suenaga, T. J. Folkerts, and R. N. Shelton, Phys. Rev. B 38, 4596 (1988).
[16] H. J. Schulz, J. Phys. France 50, 2833 (1989).
[17] C. E. Matt, C. G. Fatuzzo, Y. Sassa, M. Mansson, S. Fatale, V. Bitetta, X. Shi, S. Pailhes, M. H. Berntsen, T. Kurosawa, M. Oda, N. Momono, O. J. Lipscombe, S. M. Hayden, J. Q. Yan, J. S. Zhou, J. B. Goodenough, S. Pyon, T. Takayama, H. Takagi, L. Patthey, A. Bendounan, E. Razzoli, M. Shi, N. C. Plumb, M. Radovic, M. Grioni, J. Mesot, O. Tjernberg, and J. Chang, Phys. Rev. B 92, 134524 (2015).
[18] A. Einstein. Ann. Phys. 323, 13 (1905).
[19] S. Grenier, and Y. Joly, J. Phys.: Conf. Ser. 519, 012001 (2014).
[20] L. J. P. Ament, M. van Veenendaal, T. P. Devereaux, J. P. Hill, and J. van den Brink, Rev. Mod. Phys. 83, 705 (2011).
[21] A. Kotani, and S. Shin, Rev. Mod. Phys. 73, 203 (2001).
[22] D. Reznik, L. Pintschovius, M. Ito, S. Iikubo, M. Sato, H. Goka, M. Fujita, K. Yamada, G. D. Gu, and J. M. Tranquada, Nature 440, 1170 (2006).
[23] J. Zaanen, G. A. Sawatzky, and J. W. Allen, Phys. Rev. Lett. 55, 418 (2006).
[24] A. Higashiya, S. Imada, T. Murakawa, H. Fujiwara, S. Kasai, A. Sekiyama, S. Suga, K. Okada, M. Yabashi, and K. Tamasaku, New J. Phys. 10, 053033 (2008).
[25] NSRRC TPS 41A home page. (http://tpsbl.nsrrc.org.tw/bd_page.aspx?lang=en&pi d=1049&port=41A).
[26] C. H. Lai, H. S. Fung, W. B. Wu, H. Y. Huang, H. W. Fu, S. W. Lin, S. W. Huang, C. C. Chiu, D. J. Wang, L. J. Huang, T. C. Tseng, S. C. Chung, C. T. Chen, and D. J. Huang, J. Synchrotron Rad. 21, 325 (2014).
[27] Z. Zhang, R. Sutarto, F. He, F. C. Chou, L. Udby, S. L. Holm, Z. H. Zhu, W. A. Hines, J. I. Budnick, and B. O. Wells, Phys. Rev. Lett., 121, 067602 (2018).
[28] X. M. Chen, V. Thampy, C. Mazzoli, A. M. Barbour, H. Miao, G. D. Gu, Y. Cao, J. M. Tranquada, M. P. M. Dean, and S. B. Wilkins, Phys. Rev. Lett. 117, 167001 (2016).
[29] J. Chang, E. Blackburn, A. T. Holmes, N. B. Christensen, Jacob Larsen, J. Mesot, Ruixing Liang, D. A. Bonn, W. N. Hardy, A. Watenphul, M. v. Zimmermann, E. M. Forgan, and S. M. Hayden, Nature Physics 8, 12, 871 (2012).
[30] R.W. Cahn, and P. Haasen. editor, Physical Metallurgy. vol 1, chapter 16, North Holland, (1996).
[31] M. Hücker, M. v. Zimmermann, G. D. Gu, Z. J. Xu, J. S. Wen, Guangyong Xu, H. J. Kang, A. Zheludev, and J. M. Tranquada, Phys. Rev. B 83, 104506 (2011).
[32] A. J. Achkar, M. Zwiebler, C. McMahon, F. He, R. Sutarto, I. Djianto, Z. Hao, Michel J. P. Gingras, M. Hücker, G. D. Gu, A. Revcolevschi, H. Zhang, Y.J. Kim, J. Geck, and D. G. Hawthorn, Science 351, 6273, 576 (2016).
[33] M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 70, 1039 (1998).
[34] M. V. Zimmermann, A. Vigliante, T. Niemoller, N. Ichikawa, T. Frello, J. Madsen, P. Wochner, S. Uchida, N. H. Andersen, J. M. Tranquada, D. Gibbs, and J. R. Schneider, Europhys. Lett. 41, 6, 629 (1998).
[35] S. B. Canosa, E. Schierle, Z. W. Li, H. Guo, T. Adachi, Y. Koike, O. Sobolev, E. Weschke, A. C. Komarek, and C. S. Langeheine, Phy. Rev. B 97, 195130 (2018).
[36] C. T. Chen, L. H. Tjeng, J. Kwo, H. L. Kao, P. Rudolf, F. Sette, and R. M. Fleming, Phys. Rev. Lett. 68, 16 (1992).
[37] H. Miao, R. Fumagalli, M. Rossi, J. Lorenzana, G. Seibold, F. Yakhou-Harris, K. Kummer, N. B. Brookes, G. D. Gu, L. Braicovich, G. Ghiringhelli, and M. P. M. Dean, Phys. Rev. X 9, 031042 (2019).
[38] R. Arpaia, S. Caprara, R. Fumagalli, G. De Vecchi, Y. Y. Peng, E. Andersson, D. Betto, G. M. De Luca, N. B. Brookes, F. Lombardi, M. Salluzzo, L. Braicovich, C. Di Castro, M. Grilli, and G. Ghiringhelli, Science 365, 6456, 906 (2019).
[39] Q. Wang, M. Horio, K. V. Arx, Y. Shen, D. J. Mukkattukavil, Y. Sassa, O. Ivashko, C. E. Matt, S. Pyon, T. Takayama, H. Takagi, T. Kurosawa, N. Momono, M. Oda, T. Adachi, S. M. Haidar, Y. Koike, Y. Tseng, W. Zhang, J. Zhao, K. Kummer, M. G. Fernandez, K. J. Zhou , N. B. Christensen, H. M. Rønnow, T. Schmitt, and J. Chang, Phys. Rev. Lett. 124, 187002 (2020).
[40] M. K. Crawford, R. L. Harlow, E. M. McCarron, W. E. Farneth, J. D. Axe, H. Chou, and Q. Huang, Phys. Rev. B 44, 7749 (1991).
[41] R. E. Peierls. editor, Quantum Theory of Solids. Oxford Univ. Press, (1955).
[42] W. Kohn, Phys. Rev. Lett. 2, 393 (1959).
[43] X. Zhu, Y. Cao, J. Zhang, E. W. Plummer, and J. Guo, P.N.A.S. 24,112, 2367 (2015).
[44] H. Qin, J. Shi, Y. Cao, K. Wu, J. Zhang, E. W. Plummer, J. Wen, Z. J. Xu, G. D. Gu, and J. Guo, Phys. Rev. Lett. 105, 256402 (2010).
[45] M. D. Johannes, I. I. Mazin, and C. A. Howells, Phys. Rev. B 73, 20510 (2006).
[46] T. Fukuda, J. Mizuki, K. Ikeuchi, K. Yamada, A. Q. R. Baron, and S. Tsutsui, Phys. Rev. B 71, 060501(R) (2005).
[47] L. J. P. Ament, M. V. Veenendaal, and J. van den Brink, E.P.L. 95, 27008 (2011).
[48] G. D.Mahan, editor, Many-Particle Physics. Springer-Verlag, New York, (2000).
[49] T. P. Devereaux, A. M. Shvaika, K. Wu, K. Wohlfeld, C. J. Jia, Y. Wang, B. Moritz, L. Chaix, W. S. Lee, Z. X. Shen, G. Ghiringhelli, and L. Braicovich, Phys. Rev. X 6, 041019 (2016).
[50] D. Reznik, T. Fukuda, D. Lamago, A. Q. R. Baron, S. Tsutsui, M. Fujita, and K. Yamada. q-Dependence of the giant bond-stretching phonon anomaly in the stripe compound La1.48Nd0.4Sr0.12CuO4 measured by IXS. Journal of Physics and Chemistry of Solids 69, 12, 3103 (2008).