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V VH1 = 2k N F D o V ( E = E F ) + 2 ( T )2k T BT c FFtanh = 22k T B cln c +1 ln( 10) 2BT c2k BT ctanh E E F=1213.6 DVH1 = N o V = 2k 2BT c 1212T c = 1 2 T F exp 1NR. Abd-Shukor and W.Y. Lim: Electron-phonon Coupling Constants of Copper Oxide-based 0 V + D D , T TT Fln c2coth D+1+ ln 2 ln( T10F ) 2c = 1.36(10 D )exp 2NoV + ln k 2 2V =E F + D dE( E ENE F D( E) F ) 2 + 2 ( T )tanhB(10 D )( E E F ) 2 + 2 ( T )1tanh E 2k E BT cF=1k B 13.6 D D22T c 2TSuperconductorsc 2T F DV =E F + D2k dE( E ENE F D( E) F ) 2 + 2 BT c ( T )T tanhE( E E F ) 2 + 2 ( T ) c = 1.36(10 D 2k )exp 212BT c NoVN ( E) = N oln FE 2where, N ( EN ) o= is Na o constant ln E FEand F E F is the Fermi energy.NEEE( ) = FEN oln FModel 1. Using the BCS Eexpression EVH 2 = N 0 V =V =E F + D dEVH1 = N o V = 21T c = 1 NE ln2 2TF D( E) tanh2T c = 1.36(10 c ln 2 ( E E10) 2coth D D D , T F ) 2 + 2 T12F for energy gapT F 2T c 2T c T c = 1 D T c )exp 2+1( )Ewith the above density of states (Equation 1), and thetanh E E F= E E 2 T 1F expNFNtanh E 2k E BTFc = E 2k E oln F0 V + D D , T NoV ln( + 10) ln k 2 22 T 1F expN 0 V + D D , T F 2coth D + ln 2 T F2T c 2T c 2T F DB(10 2 D )1k13.6 B D D F 2coth D + ln 2 TVH1 = N o V = 2 F22T c 2T c 2T F Tln c D, (6)approximations tanh EBTFE F= E E 1E E Fc2k BT c2k BT cFfor |E – E F |≤ 2k B 2 T c 1c T2k BT c2k BTc 1.36T Fexp + ln D 22 T 1F exp1 N 0 V + D D , T 13.6F22coth2T c 2T ccV =E F + D dE( E ENE F D( E) F ) 2 VH1 = N o V = 2DVH 2 = N 0V = ln 2 2T c ln 2 ( 10) 2coth D D D , T 1FT + 2F 2T c 2T c 2T c( T )2( E E Eand tanh E E F ) 2 + 2 TtanhlnFE F=1 for |E – E F |≤ 2k B T c , the transition Model 3. For the third model, Newns et al. (1992)= E E tanh E 2k E VH 2 = N 0 V = ln2 2T( T ) c +1 ln( 10) 2k 2BT cc 13.6ln 2 ( 10 D 2 ) 2coth D D D , TFBTFTc =1BT derived the transition temperature:tanh E E c2k BT E F V =E F +1D dE2c N ( E) = N2k oln BT Fc F=1[ fVE Etemperature can be written FE * = e/ Z e ] 2E F D2T c 2T c 2TT ( E EE2k B as T c (Tsuei et al. 1990):D VH 2 = N 0V = ln 2 2T c ln 2 F ) 2 + 2 c= 1.36T Fexp + ln2 D 21( T ) 1E10) 2co 21N ( E) = N oln FT c = 1 1 T F, (7)T c = 1.36(10 D )exp 21NoV + ln k 2 2 2 T 2 1F expN1 2 0 V + D D , T F 2coth D + ln 2 T E E EFFB(10 D )1T c = 1.36(10 D )exp 2 k B T c = 1.36(10 D )exp 2NoV + ln k 2 2NoV + ln k 2 2 T c= 1.36T Fexp + ln2 DV =E F + D dE( E E2T c 2TNE F Dc( E) F ) 2 +2Ttanh F D( E E F ) 2 + 12 ( T )2k BT c1=1B(10 D )k BT c1tanh E E F= E E [ fVFk B D B(10 D )1= 2 T EcVH 3 ln2ln 2 D T+1 c = 1 12k BT c2k BT13.6 DE * 2 T F exp 1E * =E cTk B D by employing the conventional BCS expression togethertanh E E F= E E c= 1.36T Fexp + ln2 D N 0 V + D De/ Z e ] 2D2 2T1FF1E Fwith density of states as in (Equation 1) and using theVH1 = N o V = 2exchange of excitations with a characteristic low electronic6(10 D )exp 2, (2)VH1 = N o V = 2NoV + ln k 2 2[ fV2k BTB(10 c2k D B ) T cE * = e/ Z e ] 2EVH 2 = N 0V = ln 2 2T c ln 2 ( 10) 2coth DT D , T c = 11tanh E E EN ( E) = N Fk B D Tln cenergy scale E* given aswhere f is theVH1 = N o V = 2 +1 ln( 10) 2oln FT c= 2.72T FeD1 2 T 1F expT F N 0 V + D D , T 1F 2coth D + ln 2= 2 T c2T c 2T c2T c 2T 2T c 2VH 3 ln2ln 2 D+1F13.6=1DE *2kln 13.6T c D +1 ln( 102) 2BT E EcF[ fVE * = e/ Z e ] 2VH 2 = N 0 V = ln2 2T c lE E Ftanh E E F=1 13.6 D Tln c +1 ln( 10) 22k fraction of the Brillouin zone occupied by the van HoveBT 2k BT 13.6 F E2 T cVH 3F ln2cE c D 1where, 2 θ D is the Debye temperature = T F /10, N o V = singularity,kZ e is 13.6 ln 2 11TDF2T+1* c= 1.36T Fexp + ln2 D 2T 1c= 2.72T Fe 1the electronic D factor, D is the slave boson= N o V =λ VH1 2and T is the Fermi temperature. From (Equation propagator, V e is the electron coupling and ξ is a constant.2), the V =E F + D dE( E EN electron-phonon F( )F ) 2 + 2 ( T )T 2tanhcoupling constant can be written By using (Equation 7) and E* = 0.44D at E = 0 (Maximalas:2( E Etransition temperature), D = 1.6 eV, the electron-phononV =F + D dE( EN EE F D( )F ) 2 + 2 F ) 2 + 2 ( )2k BT cV =E F + D dE( E EN E F D( )F ) 2 ln cT c = 1.36(10+1 ln( 10 D )exp 2) 2NoV + ln k 2 2tanh E E F= E E B E1F VH 2 = N 0V = ln 2 2T c ln 2 ( 10) 2coth D DFE B(10 D )2k 1k + 2 BT c2k BT ( TB)= 2 T F T cFc13.6 tanh D1D ( T )T ( E E F ) 2 + 2 ( T )2ktanh BT c = 1.36(10 D )exp 2cNoV E Ecoupling constant could be written as:( + F ) ln k 2 2 T1B(10 D )c= 2.72T Fe 1VH 3 ln2ln 2 D+1 2 2T c13.6T c= 1.36T FexpDE * + ln2 DT E[ fV1 E * = e/ Z e ] 2F= E Fk B12+k 2 B ( T ) D 2k BT ctanh E E D2E E FF12+ dE, (3)T c = 1 12 T E1EF exp, (8)2T c = 1 N 0 V + D D , F 2coth D + ln 2 1 DVH1 (NT F 2c = 1 ( = EN o) V F ) 2 + 2 ( T )tanh= =12k2BT T2k 122T c 2T2 T F exp 1 cN 0 V + D D , T 2T F D2 T 1T F expN 0 V + D D , T F 2coth D + ln 2 TD( E E FF ) 2 2 ( T )ln c 2k BT c+1 ln( 10) 2)exp 2VH1 = N o V = 2NoV + ln k BT F= E TTFc= 2.72T Fe 1c= 1.36T Fexp + ln2 D 21[ fVcEc2 2k * 1=e/ Z e ] 2BB(10 D )1= 2 T cVH 3 ln2lnk B D13.6 D 2T 2 c 2T c 2T F F 2coth D+ ln 2 T F TTln c +1 ln( 2T ) 21 F= E 2 DD+113.6 FDE[ fV *c 2T c 2T F DModel T c = 1.36(10 2. In another D )expscenario 13.6 D2Getino et 1 al. (1993) used In previous papers the electron-phonon coupling in a2the energy gap equation:two-dimensional system where the van Hove scenario isVH 2 = N 0V = ln 2 2T c ln 2 ( 10) 2coth D Dapplicable D , T 1Fhave been reported (Abd-Shukor 2002; Abd-T F 2T c Shukor2T c 2T2007; c Getino et al. 1993; Newns et al. 1992; TsueiVH 2 = N 0V = ln 2 2T c ln 2 ( 10) 2coth DTet al. 1990). D In D the , T 1VH 2 = N 0V = ln 2 2T c ln 2 ( 10) 2coth D D D , T 1F exp 1N 0 V + D 2 D , T NoV + ln k 2 2 E * = e/ Z e ] 2k B=1B(10 D )D1= 2 T ck B DF 2coth D + ln 2 T FV =E F + D dE( E EN E2TEcF 2T D( )F ) 2 2+ 2 ( T )T c= 2.72T Fe 1VH 3 ln2ln 2 D13.6 DE1tanh2c E E 2T F DF( F ) 2 + 2 ( T )2k BTT cln c +1 2ln T F 2T c 2T c 2T c Fweak coupling limit (λ « 1), T c can beV = ( E) 21F +dE( E EN EE F D( )F ) 2 + 2 ( T ))exp 13.6 2 DtanhVH1 = N o V E E F 2T( = 2NoV + ln k 2= 2 T c2B(10 D )1Tk B D c 2T c 2TF ) 2 F= E VH 3 ln2ln 2 D+113.6 D T E *Fc= 2.72T Fe 1+ 2 ( T )2k BT2c k B c1T 12T written as where, (EF is thec= 1.36T Fexpln2 D 122 1E1T c= 1.36T Fexpln2 2N 0 V = ln 2 2T c ln 2 ( 10) 2coth 1D , T 12T c = 1 ln c +1FT F2 T F exp 1 13.6N 0 V 2T + D D , T ln( 10) 2F 2coth D + ln 2 T FTdE2 Fermi energy) Getino et al. 1992). In this paper theT c= 1.36T c2TFexp , + ln2 D 2E c 2T c 2T c2T 1F D1 (4)2T c = 1 Echaracteristic ratio of T F /θ D was given as 10 in the high T c2 T 1F expfVmaterials. In the high T c material, for example the La-Sr-Ca-[E * = e/ Z e ] 2 N 0 V + D D , T F 2coth D + ln 2 TE FF ) 2 + 2 ( T ) N ( E ( E E)F ) 2 D c= 2.72T Fe 1T F= E F+ 2 ( Tk)B2 tanh=22k BT cTln c +1 2 ln 2T c 2T c 2T F D12 Cu-O system, Tand the density [ fV of states as in (Equation 1) and showed thatF /θ D ranged from 7.4–13.9 and in the Y-Ba-E * = e/ ZD e ] 26T Cu-O system Tthe transition temperature D [ fVcould be written as:F /θ D can be as close to 11.7 (Krunacakarn etE * = e/ Z e ] Fexp + ln2V D= ( 10 E F ) 2+ D dE( E EN EE F 2 D( )F ) 2 + 2 T( T )F= E Fk13.6 D tanhB1VH E 2 = N 0 V = ln2 2T c ln 2 ( 10) 2coth D D D , T 1( E E F ) 2 + 2 ( T1 2)2k BT cF1T F 2TDal. 1998). The ratio of T F /θ D = 10 was in the correct order1 c 2T c 2T c 2 T cof magnitude. θ D values were obtained from our acousticVH 3 ln2ln 2 D 1VH 2 = N +1= 2 0 V = T ln2 2T c ln 2 ( 10) 2coth D D13.6 c1 experiments (Yahya et al. 1999).DE * VH 3 ln2ln 2 D , T 1N 0 V + D D , T F 2coth D + ln 2 T FdE 2T c 2T c 2T F D1 FfV e/ Z e ] 2T F +12T213.6 = 2 2 DTE * 1c 2T c 2TE cT c = 1 2 T 1F expN VH 3 ln2ln 2 DDT+113.6 DE * c= 1.36T Fexp 0 V + D D , T F 2coth D + ln 2 TF ) 2 + 2 ( T ) N ( E ( E E)F ) 2 + 2 ( T )tanh2k BT cF+ ln2 D2T c 2T12 c 2T F DEXPERIMENTAL DETAILSc 2.72T Fe 1 E11Tc 2.72T e 1 2ln 2c= 1.36Tcln 2 D Fexp + ln2 D 2ln 2 2T c ln 2 ( 10) 2coth D D D , T 11F21 T 1F 2T c 2T+1, (5)T c= 2.72T Fe 1 c 2TE cSamples (about 12.5 mm diameter and 2 mm thickness)13.6 DE * [ fVE * T=were prepared via solid-state reaction method using highpurityc 2TF= E e/ Z e ] 2VH 2 F1T F= E = N 0 V = ln2 2T c ln 2 ( 10) 2coth D D D , T 1N 0 V + D D , T F 2coth D + ln 2 T F2T c 2T c 2T F DFkFD T F 2T c 2T2 B(>99.9%) c metal oxides and carbonates by sinteringk[ fVin air at various temperatures followed by annealing inFrom (Equation 5), T F the = E BFT electron-phonon coupling constantFe 1E * = e/ Z e ] 2+ ln2 D 21ED1k oxygen and various gases.could be 2 written as: T B cVH 3 ln2ln 2 1ln 2 2T c ln 2 102 +1T 13.6 D * c= 1.36T Fexp + ln2 D 2( ) 2coth D D D , T 1FT F 2T c 2T c 2T 11] 2= 2 T E ccVH 3 ln2ln 2 D+113.619T c= 2.72T Fe 1DE *12 [ fVE 1* = e/ Z e ] 2+ ln2 D 21T ET c= 2.72T Fe 1c ln2 D D+1