- Open Access
Anisotropic electron damping and energy gap in
Phys. Rev. Research 7, 023319 – Published 27 June, 2025
DOI: https://doi.org/10.1103/k32g-pknp
Abstract
The many-body electron-electron interaction in cuprates causes broadening of the electronic bands in space, leading to a deviation from the standard Fermi liquid. While a -dependent anisotropic electronic scattering (-DAES) has been assessed by photoemission, its fingerprint in space has been scarcely considered. Here, we explore the -dependent electron dynamics in optimally doped through the evolution of low-energy charge excitations as measured by resonant inelastic x-ray scattering (RIXS). In the normal state, the RIXS spectra display a continuum of excitations down to 0 meV, while the superconducting state features a spectral weight suppression below 80 meV without any enhancement at higher energies. To interpret the energy and evolution of our data, we introduce a phenomenological expression of the charge susceptibility by including the -DAES. We show that only the charge susceptibility with -DAES captures the RIXS data, highlighting the importance of -DAES when describing the dependence of charge excitations from 0 to a few eV scale. Furthermore, we also find that the inclusion of -DAES is essential when quantitative parameters such as the electronic energy gap are extracted from RIXS data.
Physics Subject Headings (PhySH)
Article Text
References (70)
- J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of superconductivity, Phys. Rev. 108, 1175 (1957).
- J. G. Bednorz and K. A. Müller, Possible high superconductivity in the Ba-La-Cu-O system, Z. Phys. B 64, 189 (1986).
- J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, and J. Akimitsu, Superconductivity at 39 K in magnesium diboride, Nature (London) 410, 63 (2001).
- B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, and J. Zaanen, From quantum matter to high-temperature superconductivity in copper oxides, Nature (London) 518, 179 (2015).
- G. A. Thomas, J. Orenstein, D. H. Rapkine, M. Capizzi, A. J. Millis, R. N. Bhatt, L. F. Schneemeyer, and J. V. Waszczak, : Electrodynamics of crystals with high reflectivity, Phys. Rev. Lett. 61, 1313 (1988).
- R. T. Collins, Z. Schlesinger, F. Holtzberg, P. Chaudhari, and C. Feild, Reflectivity and conductivity of , Phys. Rev. B 39, 6571 (1989).
- I. Božović, J. H. Kim, J. S. Harris, and W. Y. Lee, Optical study of plasmons in , Phys. Rev. B 43, 1169 (1991).
- S. L. Cooper, A. L. Kotz, M. A. Karlow, M. V. Klein, W. C. Lee, J. Giapintzakis, and D. M. Ginsberg, Development of the optical conductivity with doping in single-domain , Phys. Rev. B 45, 2549 (1992).
- E. Abrahams and C. M. Varma, What angle-resolved photoemission experiments tell about the microscopic theory for high-temperature superconductors, Proc. Natl. Acad. Sci. USA 97, 5714 (2000).
- D. A. Bonn, P. Dosanjh, R. Liang, and W. N. Hardy, Evidence for rapid suppression of quasiparticle scattering below in , Phys. Rev. Lett. 68, 2390 (1992).
- C. T. Rieck, W. A. Little, J. Ruvalds, and A. Virosztek, Infrared and microwave spectra of an energy gap in high-temperature superconductors, Phys. Rev. B 51, 3772 (1995).
- C. M. Varma, P. B. Littlewood, S. Schmitt-Rink, E. Abrahams, and A. E. Ruckenstein, Phenomenology of the normal state of Cu-O high-temperature superconductors, Phys. Rev. Lett. 63, 1996 (1989).
- M. Abdel-Jawad, M. P. Kennett, L. Balicas, A. Carrington, A. P. Mackenzie, R. H. McKenzie, and N. E. Hussey, Anisotropic scattering and anomalous normal-state transport in a high-temperature superconductor, Nat. Phys. 2, 821 (2006).
- Y. Fang, G. Grissonnanche, A. Legros, S. Verret, F. Laliberté, C. Collignon, A. Ataei, M. Dion, J. Zhou, D. Graf, M. J. Lawler, P. A. Goddard, L. Taillefer, and B. J. Ramshaw, Fermi surface transformation at the pseudogap critical point of a cuprate superconductor, Nat. Phys. 18, 558 (2022).
- T. Valla, A. V. Fedorov, P. D. Johnson, B. O. Wells, S. L. Hulbert, Q. Li, G. D. Gu, and N. Koshizuka, Evidence for quantum critical behavior in the optimally doped cuprate , Science 285, 2110 (1999).
- T. Valla, A. V. Fedorov, P. D. Johnson, Q. Li, G. D. Gu, and N. Koshizuka, Temperature dependent scattering rates at the fermi surface of optimally doped , Phys. Rev. Lett. 85, 828 (2000).
- A. Kaminski, H. M. Fretwell, M. R. Norman, M. Randeria, S. Rosenkranz, U. Chatterjee, J. C. Campuzano, J. Mesot, T. Sato, T. Takahashi, T. Terashima, M. Takano, K. Kadowaki, Z. Z. Li, and H. Raffy, Momentum anisotropy of the scattering rate in cuprate superconductors, Phys. Rev. B 71, 014517 (2005).
- J. Chang, M. Mansson, S. Pailhes, T. Claesson, O. J. Lipscombe, S. M. Hayden, L. Patthey, O. Tjernberg, and J. Mesot, Anisotropic breakdown of Fermi liquid quasiparticle excitations in overdoped , Nat. Commun. 4, 2559 (2013).
- C. M. Varma and E. Abrahams, Effective lorentz force due to small-angle impurity scattering: Magnetotransport in high- Superconductors, Phys. Rev. Lett. 86, 4652 (2001).
- M. R. Norman, Anisotropic exchange and superconductivity in , Phys. Rev. B 41, 170 (1990).
- A. J. Millis, Nearly antiferromagnetic Fermi liquids: An analytic eliashberg approach, Phys. Rev. B 45, 13047 (1992).
- T. Dahm, P. J. Hirschfeld, D. J. Scalapino, and L. Zhu, Nodal quasiparticle lifetimes in cuprate superconductors, Phys. Rev. B 72, 214512 (2005).
- M. Mitrano, A. A. Husain, S. Vig, A. Kogar, M. S. Rak, S. I. Rubeck, J. Schmalian, B. Uchoa, J. Schneeloch, R. Zhong, G. D. Gu, and P. Abbamonte, Anomalous density fluctuations in a strange metal, Proc. Natl. Acad. Sci. USA 115, 5392 (2018).
- A. A. Husain, M. Mitrano, M. S. Rak, S. Rubeck, B. Uchoa, K. March, C. Dwyer, J. Schneeloch, R. Zhong, G. D. Gu, and P. Abbamonte, Crossover of charge fluctuations across the strange metal phase diagram, Phys. Rev. X 9, 041062 (2019).
- L. J. P. Ament, M. van Veenendaal, T. P. Devereaux, J. P. Hill, and J. van den Brink, Resonant inelastic x-ray scattering studies of elementary excitations, Rev. Mod. Phys. 83, 705 (2011).
- M. Mitrano, S. Johnston, Y.-J. Kim, and M. P. M. Dean, Exploring quantum materials with resonant inelastic x-ray scattering, Phys. Rev. X 14, 040501 (2024).
- P. Marra, S. Sykora, K. Wohlfeld, and J. van den Brink, Resonant inelastic x-ray scattering as a probe of the phase and excitations of the order parameter of superconductors, Phys. Rev. Lett. 110, 117005 (2013).
- H. Suzuki, M. Minola, Y. Lu, Y. Peng, R. Fumagalli, E. Lefrançois, T. Loew, J. Porras, K. Kummer, D. Betto, S. Ishida, H. Eisaki, C. Hu, X. Zhou, M. W. Haverkort, N. B. Brookes, L. Braicovich, G. Ghiringhelli, M. Le Tacon, and B. Keimer, Probing the energy gap of high-temperature cuprate superconductors by resonant inelastic x-ray scattering, npj Quantum Mater. 3, 65 (2018).
- G. Merzoni, L. Martinelli, L. Braicovich, N. B. Brookes, F. Lombardi, F. Rosa, R. Arpaia, M. Moretti Sala, and G. Ghiringhelli, Charge response function probed by resonant inelastic x-ray scattering: Signature of electronic gaps of , Phys. Rev. B 109, 184506 (2024).
- C. Jia, K. Wohlfeld, Y. Wang, B. Moritz, and T. P. Devereaux, Using RIXS to uncover elementary charge and spin excitations, Phys. Rev. X 6, 021020 (2016).
- M. R. Norman, M. Randeria, H. Ding, and J. C. Campuzano, Phenomenological models for the gap anisotropy of as measured by angle-resolved photoemission spectroscopy, Phys. Rev. B 52, 615 (1995).
- W. S. Lee, I. M. Vishik, K. Tanaka, D. H. Lu, T. Sasagawa, N. Nagaosa, T. P. Devereaux, Z. Hussain, and Z.-X. Shen, Abrupt onset of a second energy gap at the superconducting transition of underdoped Bi2212, Nature (London) 450, 81 (2007).
- T. Kondo, R. Khasanov, T. Takeuchi, J. Schmalian, and A. Kaminski, Competition between the pseudogap and superconductivity in the high- copper oxides, Nature (London) 457, 296 (2009).
- T. Valla, I. K. Drozdov, and G. D. Gu, Disappearance of superconductivity due to vanishing coupling in the overdoped , Nat. Commun. 11, 569 (2020).
- I. M. Vishik, E. A. Nowadnick, W. S. Lee, Z. X. Shen, B. Moritz, T. P. Devereaux, K. Tanaka, T. Sasagawa, and T. Fujii, A momentum-dependent perspective on quasiparticle interference in , Nat. Phys. 5, 718 (2009).
- I. M. Vishik, M. Hashimoto, R.-H. He, W.-S. Lee, F. Schmitt, D. Lu, R. G. Moore, C. Zhang, W. Meevasana, T. Sasagawa, S. Uchida, K. Fujita, S. Ishida, M. Ishikado, Y. Yoshida, H. Eisaki, Z. Hussain, T. P. Devereaux, and Z.-X. Shen, Phase competition in trisected superconducting dome, Proc. Natl. Acad. Sci. USA 109, 18332 (2012).
- S. Onari, H. Kontani, and M. Sato, Structure of neutron-scattering peaks in both -wave and -wave states of an iron pnictide superconductor, Phys. Rev. B 81, 060504 (2010).
- M. Hashimoto, I. M. Vishik, R.-H. He, T. P. Devereaux, and Z.-X. Shen, Energy gaps in high-transition-temperature cuprate superconductors, Nat. Phys. 10, 483 (2014).
- J. Wen, Z. Xu, G. Xu, M. Hücker, J. Tranquada, and G. Gu, Large Bi-2212 single crystal growth by the floating-zone technique, J. Cryst. Growth 310, 1401 (2008).
- I. K. Drozdov, I. Pletikosić, C.-K. Kim, K. Fujita, G. D. Gu, J. C. S. Davis, P. D. Johnson, I. Božović, and T. Valla, Phase diagram of revisited, Nat. Commun. 9, 5210 (2018).
- J. Dvorak, I. Jarrige, V. Bisogni, S. Coburn, and W. Leonhardt, Towards 10 meV resolution: The design of an ultrahigh resolution soft X-ray RIXS spectrometer, Rev. Sci. Instrum. 87, 115109 (2016).
- L. Chaix, G. Ghiringhelli, Y. Y. Peng, M. Hashimoto, B. Moritz, K. Kummer, N. B. Brookes, Y. He, S. Chen, S. Ishida, Y. Yoshida, H. Eisaki, M. Salluzzo, L. Braicovich, Z.-X. Shen, T. P. Devereaux, and W.-S. Lee, Dispersive charge density wave excitations in , Nat. Phys. 13, 952 (2017).
- M. Minola, G. Dellea, H. Gretarsson, Y. Y. Peng, Y. Lu, J. Porras, T. Loew, F. Yakhou, N. B. Brookes, Y. B. Huang, J. Pelliciari, T. Schmitt, G. Ghiringhelli, B. Keimer, L. Braicovich, and M. Le Tacon, Collective nature of spin excitations in superconducting cuprates probed by resonant inelastic x-ray scattering, Phys. Rev. Lett. 114, 217003 (2015).
- M. P. M. Dean, A. J. A. James, R. S. Springell, X. Liu, C. Monney, K. J. Zhou, R. M. Konik, J. S. Wen, Z. J. Xu, G. D. Gu, V. N. Strocov, T. Schmitt, and J. P. Hill, High-energy magnetic excitations in the cuprate superconductor : Towards a unified description of its electronic and magnetic degrees of freedom, Phys. Rev. Lett. 110, 147001 (2013).
- M. P. M. Dean, A. J. A. James, A. C. Walters, V. Bisogni, I. Jarrige, M. Hücker, E. Giannini, M. Fujita, J. Pelliciari, Y. B. Huang, R. M. Konik, T. Schmitt, and J. P. Hill, Itinerant effects and enhanced magnetic interactions in bi-based multilayer cuprates, Phys. Rev. B 90, 220506 (2014).
- M. Dean, Insights into the high temperature superconducting cuprates from resonant inelastic x-ray scattering, J. Magn. Magn. Mater. 376, 3 (2015).
- J. Li, A. Nag, J. Pelliciari, H. Robarts, A. Walters, M. Garcia-Fernandez, H. Eisaki, D. Song, H. Ding, S. Johnston, R. Comin, and K.-J. Zhou, Multiorbital charge-density wave excitations and concomitant phonon anomalies in , Proc. Natl. Acad. Sci. USA 117, 16219 (2020).
- 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, Directly characterizing the relative strength and momentum dependence of electron-phonon coupling using resonant inelastic x-ray scattering, Phys. Rev. X 6, 041019 (2016).
- H. C. Robarts, M. Barthélemy, K. Kummer, M. García-Fernández, J. Li, A. Nag, A. C. Walters, K. J. Zhou, and S. M. Hayden, Anisotropic damping and wave vector dependent susceptibility of the spin fluctuations in studied by resonant inelastic x-ray scattering, Phys. Rev. B 100, 214510 (2019).
- M. Hepting, L. Chaix, E. W. Huang, R. Fumagalli, Y. Y. Peng, B. Moritz, K. Kummer, N. B. Brookes, W. C. Lee, M. Hashimoto, T. Sarkar, J.-F. He, C. R. Rotundu, Y. S. Lee, R. L. Greene, L. Braicovich, G. Ghiringhelli, Z. X. Shen, T. P. Devereaux, and W. S. Lee, Three-dimensional collective charge excitations in electron-doped copper oxide superconductors, Nature (London) 563, 374 (2018).
- A. Nag, M. Zhu, M. Bejas, J. Li, H. C. Robarts, H. Yamase, A. N. Petsch, D. Song, H. Eisaki, A. C. Walters, M. García-Fernández, A. Greco, S. M. Hayden, and K.-J. Zhou, Detection of acoustic plasmons in hole-doped lanthanum and bismuth cuprate superconductors using resonant inelastic x-ray scattering, Phys. Rev. Lett. 125, 257002 (2020).
- A. Nag, L. Zinni, J. Choi, J. Li, S. Tu, A. C. Walters, S. Agrestini, S. M. Hayden, M. Bejas, Z. Lin, H. Yamase, K. Jin, M. García-Fernández, J. Fink, A. Greco, and K.-J. Zhou, Impact of electron correlations on two-particle charge response in electron- and hole-doped cuprates, Phys. Rev. Res. 6, 043184 (2024).
- K. Scott, E. Kisiel, T. J. Boyle, R. Basak, G. Jargot, S. Das, S. Agrestini, M. Garcia-Fernandez, J. Choi, J. Pelliciari, J. Li, Y.-D. Chuang, R. Zhong, J. A. Schneeloch, G. Gu, F. Légaré, A. F. Kemper, K.-J. Zhou, V. Bisogni, S. Blanco-Canosa et al., Low-energy quasi-circular electron correlations with charge order wavelength in , Sci. Adv. 9, eadg3710 (2023).
- A. Greco, H. Yamase, and M. Bejas, Plasmon excitations in layered high- cuprates, Phys. Rev. B 94, 075139 (2016).
- A. Greco, H. Yamase, and M. Bejas, Origin of high-energy charge excitations observed by resonant inelastic X-ray scattering in cuprate superconductors, Commun. Phys. 2, 3 (2019).
- Y. Zhu, M. Liao, Q. Zhang, H.-Y. Xie, F. Meng, Y. Liu, Z. Bai, S. Ji, J. Zhang, K. Jiang, R. Zhong, J. Schneeloch, G. Gu, L. Gu, X. Ma, D. Zhang, and Q.-K. Xue, Presence of -wave pairing in josephson junctions made of twisted ultrathin Flakes, Phys. Rev. X 11, 031011 (2021).
- O. Can, T. Tummuru, R. P. Day, I. Elfimov, A. Damascelli, and M. Franz, High-temperature topological superconductivity in twisted double-layer copper oxides, Nature (London) 17, 519 (2021).
- L. Ju, T. Ren, Z. Li, Z. Liu, C. Shi, Y. Liu, S. Hong, J. Wu, H. Tian, Y. Zhou, and Y. Xie, Emergence of high-temperature superconductivity at the interface of two Mott insulators, Phys. Rev. B 105, 024516 (2022).
- S. Y. F. Zhao, X. Cui, P. A. Volkov, H. Yoo, S. Lee, J. A. Gardener, A. J. Akey, R. Engelke, Y. Ronen, R. Zhong, G. Gu, S. Plugge, T. Tummuru, M. Kim, M. Franz, J. H. Pixley, N. Poccia, and P. Kim, Time-reversal symmetry breaking superconductivity between twisted cuprate superconductors, Science 382, 1422 (2023).
- D. Li, K. Lee, B. Y. Wang, M. Osada, S. Crossley, H. R. Lee, Y. Cui, Y. Hikita, and H. Y. Hwang, Superconductivity in an infinite-layer nickelate, Nature (London) 572, 624 (2019).
- H. Lu, M. Rossi, A. Nag, M. Osada, D. F. Li, K. Lee, B. Y. Wang, M. Garcia-Fernandez, S. Agrestini, Z. X. Shen, E. M. Been, B. Moritz, T. P. Devereaux, J. Zaanen, H. Y. Hwang, K.-J. Zhou, and W. S. Lee, Magnetic excitations in infinite-layer nickelates, Science 373, 213 (2021).
- M. Hepting, M. P. M. Dean, and W.-S. Lee, Soft x-ray spectroscopy of low-valence nickelates, Front. Phys. 9, 808683 (2021).
- S. Fan, H. LaBollita, Q. Gao, N. Khan, Y. Gu, T. Kim, J. Li, V. Bhartiya, Y. Li, W. Sun, J. Yang, S. Yan, A. Barbour, X. Zhou, A. Cano, F. Bernardini, Y. Nie, Z. Zhu, V. Bisogni, C. Mazzoli et al., Capping effects on spin and charge excitations in parent and superconducting , Phys. Rev. Lett. 133, 206501 (2024).
- J. A. Sobota, Y. He, and Z.-X. Shen, Angle-resolved photoemission studies of quantum materials, Rev. Mod. Phys. 93, 025006 (2021).
- Z. Schlesinger, R. T. Collins, F. Holtzberg, C. Feild, S. H. Blanton, U. Welp, G. W. Crabtree, Y. Fang, and J. Z. Liu, Superconducting energy gap and normal-state conductivity of a single-domain crystal, Phys. Rev. Lett. 65, 801 (1990).
- P. V. Bogdanov, A. Lanzara, X. J. Zhou, W. L. Yang, H. Eisaki, Z. Hussain, and Z. X. Shen, Anomalous momentum dependence of the quasiparticle scattering rate in overdoped , Phys. Rev. Lett. 89, 167002 (2002).
- T. Yoshida, M. Hashimoto, I. M. Vishik, Z.-X. Shen, and A. Fujimori, Pseudogap, superconducting gap, and fermi arc in high- cuprates revealed by angle-resolved photoemission spectroscopy, J. Phys. Soc. Jpn. 81, 011006 (2012).
- X. He, X. Xu, X. Shi, and I. Božović, Optimization of Single crystal film growth via molecular beam epitaxy, Condens. Matter 8, 13 (2023).
- C. T. Chen, L. H. Tjeng, J. Kwo, H. L. Kao, P. Rudolf, F. Sette, and R. M. Fleming, Out-of-plane orbital characters of intrinsic and doped holes in , Phys. Rev. Lett. 68, 2543 (1992).
- Q. Li, H.-Y. Huang, T. Ren, E. Weschke, L. Ju, C. Zou, S. Zhang, Q. Qiu, J. Liu, S. Ding, A. Singh, O. Prokhnenko, D.-J. Huang, I. Esterlis, Y. Wang, Y. Xie, and Y. Peng, Prevailing charge order in overdoped beyond the superconducting dome, Phys. Rev. Lett. 131, 116002 (2023).