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R. Abd-Shukor and W.Y. Lim: Electron-phonon Coupling Constants of Copper Oxide-based SuperconductorsTable 1. Transition temperature, Debye temperature and electron-phonon coupling constant of the conventional BCS theory and thevarious van Hove scenarios, λ VH , λ VH1 , λ VH2 , and λ VH3 .Sample T c (K) θ D (K) λ BCS λ VH [*] λ VH1 λ VH2 λ VH3EuBa 2 Cu 3 O 6.98 90 457 0.57 0.041 0.15 0.15 0.11ErBa 2 Cu 3 O 6.9 94 375 0.66 0.046 0.19 0.17 0.12(Er 0.9 Pr 0.1 )Ba 2 Cu 3 O 6.9 89 387 0.63 0.044 0.18 0.16 0.12(Er 0.8 Pr 0.2 )Ba2Cu3O6.9 77 428 0.54 0.040 0.14 0.14 0.10ErBa 2 Cu 2.99 Zn 0.01 O 6.9 86 398 0.60 0.043 0.15 0.16 0.11ErBa 2 Cu 2.95 Zn 0.05 O 6.9 78 388 0.58 0.042 0.15 0.15 0.11GdBaSrCu 3 O 7-δ 87 385 0.62 0.044 0.16 0.16 0.12GdBaSr(Cu 2.99 Zn 0.01 )O 7-δ 84 420 0.58 0.041 0.15 0.15 0.11GdBaSr(Cu 2.97 Zn 0.03 )O 7-δ 82 449 0.55 0.040 0.14 0.14 0.10GdBaSr(Cu 2.94 Zn 0.06 )O 7-δ 73 440 0.52 0.038 0.13 0.13 0.10DyBaSrCu 3 O 7-δ 82 464 0.54 0.039 0.14 0.14 0.10(Dy 0.9 Pr 0.1 )BaSrCu 3 O 7-δ 75 400 0.56 0.040 0.14 0.14 0.11(Dy 0.8 Pr 0.2 )BaSrCu 3 O 7-δ 59 374 0.51 0.038 0.13 0.13 0.10(Dy 0.6 Pr 0.4 )BaSrCu 3 O 7-δ 28 402 0.36 0.028 0.090 0.087 0.071TlSr 2 (Ca 0.7 Y 0.3 )Cu 2 O 7-δ 71 400 0.54 0.039 0.14 0.14 0.10TlSr 2 (Ca 0.5 Y 0.5 )Cu 2 O 7-δ 73 396 0.55 0.040 0.14 0.14 0.10TlSr 2 (Sr 0.7 Y 0.3 )Cu 2 O 7-δ 81 433 0.56 0.040 0.14 0.14 0.11TlSr 2 (Sr 0.5 Y 0.5 )Cu 2 O 7-δ 87 454 0.56 0.041 0.14 0.15 0.11TlSr 2 (Ca 0.5 Pr 0.5 )Cu 2 O 7-δ 90 342 0.69 0.046 0.18 0.17 0.13Tl 2 Ba 2 Ca 2 Cu 3 O 10 123 268 1.11 0.060 0.28 0.26 0.17RuSr 2 GdCu 2 O 8 40 528 0.28 0.029 0.090 0.09 0.073Pr 2-x CexCuO 4 21 406 0.32 0.025 0.070 0.077 0.060[*] Abd-Shukor (2007)phonon modes in the plane are related to oxygen vibrationswhich are the breathing (in plane vibrations) and bucklingmodes (out of plane vibrations). For each mode therecan be a diagonal and non-diagonal electron-phononinteraction. The non-diagonal electron-phonon interactionof the breathing mode has been shown to have attractiveinteraction leading to pair formation (Piekarz et al. 1999).The three models discussed above might be somewhatsimilar and thus gave almost similar coupling constant. Thereported experimental results of electron-phonon couplingconstants for Bi based and RBa 2 Cu 3 O 7 (Devereaux et al.1998; Friedl et al. 1990; Pattnaik & Newns 1989) wascloser to the coupling constant λ VH in Abd-Shukor (2007).Hence the model by Getino et al. (1992) used in Abd-Shukor (2007) was probably more applicable to the copperoxide -based superconductors. The electron-phononcoupling constant λ VH was smaller than the three models(λ VH1 , λ VH2 and λ VH3 ) because in deriving the expressionfor Tc and λ VH , the density of states were assumed as purelogarithmic singularity. The density of states in the threemodels (λ VH1 , λ VH2 and λ VH3 ) were modified logarithmicsingularities and thus contributed to higher electronphononcoupling constant. The electron-phonon couplingconstant in this scenario was very small, which is surprisingconsidering the high transition temperature. However,from our results on a broad range of cuprates and variousreports, the small electron-phonon coupling constant mightbe sufficient and essential for the formation of Cooperpairs. The density of states diverged near the Fermi leveland even very weak electron-phonon interactions couldresult in an unexpectedly large effect.Our results also showed that the non-diagonal electronphononinteraction of the breathing mode was importantfor the pair formation in these high temperature cupratesuperconductors. This result might provide insights intothe mechanism of superconductivity in the cuprates.ACKNOWLEDGMENTSThis research has been supported by a grant from theMinistry of Higher Education, Malaysia (ERGS/1/2011/STG/UKM/01/25) and Universiti Kebangsaan Malaysia(UKM-DLP-2011-018).Date of submission: October 2012Date of acceptance: January 2013REFERENCESAbd-Shukor, R 2002, ‘Acoustic Debye temperature and therole of phonons in cuprates and related superconductors’,Supercond. Sci. and Technol., vol. 15, pp. 435–538.21