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Analysis of the electron-phonon coupling in the superconductor Ba0.7K0.3BiO3
PHYSICA EL~EVIFR
Physica C 282-287 (1997) 1825-1826
Analysis of the electron-phonon coupling in the superconductor
Bao.7Ko.3Bi03. O. Navarro and E. Chavira Instituto de Investigaciones en Materiales, UNAM, Apartado Postal 70-360, 04510, Mdxico D.F., MEXICO. Within the Ehashberg theory we present an analysis of the eleetron-phonon coupling in the compound Bao.TKo.3BiOa. This analysis has been carried out evaluating the isotope coefficient (c~), using the Rainer and Culetto approach and the electron-phonon spectra a2(w)F(w) as calculated by Shirai et 02. for the oxide superconductor Bao.TKo.sBi03. With numerical calculations of some important BCS-ratios and the analysis of the isotope coefficient in this bismuthate, we outlined the importance of the electron-phonon coupling in this oxide superconductor.
The discover3, in 1988 [1] of superconductivity in the strictly 3-dimensional perovskite Ba=KI_=Bi03 with a highest critical temperature of approximately 30 K at x = 0.7, invalidated thc up-to-then prevalent belief that quasi two-dimensional oxide layers containing Cu are a necessary condition, for high-T~ superconductors. Some of the important particularities about this potassium-based compound are: the absence of magnetic effects and the very low density of states N(EF), which is considerably lower than that of A15 alloys such as Nb3Ge [2] with a lower critical temperature (23 K). In this paper, we make an analysis of the electron-phonon coupling in the Bao.TKo.3Bi03 compound evaluating the isotope coefficient (c~), using the Ralner and Culetto approach [3] and the electron-phonon spectra c~2(w)F(w) as calculated by Shiral et al. [4] for this oxide superconductor. Rainer and Culetto uses the functional derivative of T~ with respect to az(w)F(w) to find the isotope coefficient. This method is very revealbag and was first derived for a multimass system Mi with corresponding isotopic coefficient ai -- - d l n T J d l n M ~ . It can be show that the total isotope effect, ~tot -- - d In Tc / d In M, is just the stun of all the partial ~ ' s (atot = ~-'~,c~,) [3], where M is a reference mass given by the equation Mi = f ~ M with fli a constant that depends on i. The functional derivative tec.hnlque [5] is quite 0921-4534/97/$17.00 © Elsevier Science B.V. All rights reserved. PII S0921-4534(97)01041-1
helpful to know how the changes in the electronphonon interaction will affect the thermodynamic properties. For example, the functional derivative of the critical temperature, 6Tc/6(a2(w)F(oJ)), tells one how favorable a certain frequency is for an increase of To. That is, if a2(w)F(w) is changed by an infinitesimal amount 6(a2(w)F(w)), the resulting change of the transition temperature is given by 6T~ = f0 ~ dw6(a2(w)F(w)) 6To df(a2 (w)F(w)), which can be rewritten as 6inTo =-
where
Using the above equation for R(w), P~iner and Culetto introduced the partial isotope effect coefficient according to
The total isotope effect coefficient is then (~tot =
dw(~(w).
(I)
In Figure 1 we show the weighting function
R(w) vs w for Bao.TKo.3BiOa. This curve is for
O. Navarrc E. Chavira/Physica C 282-287 (1997) 1825-1826
1826
0.07
for Bao.7Ko.3BiO3, we have shown that not all phonons contribute the same amount to the isotope effect coefficient. We also showed that this oxide superconductor can be analyzed as a normal strong coupled superconductor, where the Cooper pairing is phonon-mediated. However, more calculations on the bases of other pairing mechanisms is necessary before one can make definite statements and draw strong conclusions. This work was partially supported by DGAPAUNAM, Grant IN102196 and by CONACyT 2661P-A9507.
i
0.06
0.05 0.04 s 0.03
0.02 0.01
REFERENCES
0.00 I
0
i
I
10 20
i
I
3040
I
50
60
70
(meV)
Figure 1. Weighting function R(w) vs. w for /z* = 0.12. This function is based on the spectral function a2(w)F(w) for Bao.TKo.3BiO3.
It* = 0.12 and looks much like the functional derivative of To. We note that R(w) goes to zero at w = 0 and w --* co, it means, that very low and very high phonon energy has a small contribution to the isotope effect. 'The shape for R(w) also shows a maximum at approximately 1/5 of the maximum phonon energy of the system. In other words, the fi,equencies which are more important for enhancing the isotope coefficient are precisely those in the maximum of the curve. Our calculated value of the total isotope effect for #* = 0.12 using Eq. (1) is atot = 0.49, which is close to the classical BCS result. We have also calculated, some important BCSratios for the Bao.7Ko.3Bi03 compound, such as 2Ao/kBTc = 4.15, 7[To~He(O)]2 = 0.145, and AC(T¢)/TTc ---- 1.88, given a clear indication of a strong coupling behavior in this potassium system [5]. In conclusion, based on the Rainer and Culetto approach and the electron-phonon spectral density az(w)F(w) as calculated by Shirai et al.
1. R.J. Cava et al., Nature 332 (1988) 814; L.F. Mattheiss et al., Phys. Rev B 37 (1988) 3745. 2. O. Navarro and R. Escudero, Physica C 170 (1990) 4 0 5 . 3. J.P. Carbotte, Rev. Mod. Phys. 62 (1990) 1027. 4. M. Shirai et al., J. Phys.: Condensed Matter 2 (1990) 3553. 5. O. Navarro, Physica C 265 (1996) 73.
Physica C 282-287 (1997) 1825-1826
Analysis of the electron-phonon coupling in the superconductor
Bao.7Ko.3Bi03. O. Navarro and E. Chavira Instituto de Investigaciones en Materiales, UNAM, Apartado Postal 70-360, 04510, Mdxico D.F., MEXICO. Within the Ehashberg theory we present an analysis of the eleetron-phonon coupling in the compound Bao.TKo.3BiOa. This analysis has been carried out evaluating the isotope coefficient (c~), using the Rainer and Culetto approach and the electron-phonon spectra a2(w)F(w) as calculated by Shirai et 02. for the oxide superconductor Bao.TKo.sBi03. With numerical calculations of some important BCS-ratios and the analysis of the isotope coefficient in this bismuthate, we outlined the importance of the electron-phonon coupling in this oxide superconductor.
The discover3, in 1988 [1] of superconductivity in the strictly 3-dimensional perovskite Ba=KI_=Bi03 with a highest critical temperature of approximately 30 K at x = 0.7, invalidated thc up-to-then prevalent belief that quasi two-dimensional oxide layers containing Cu are a necessary condition, for high-T~ superconductors. Some of the important particularities about this potassium-based compound are: the absence of magnetic effects and the very low density of states N(EF), which is considerably lower than that of A15 alloys such as Nb3Ge [2] with a lower critical temperature (23 K). In this paper, we make an analysis of the electron-phonon coupling in the Bao.TKo.3Bi03 compound evaluating the isotope coefficient (c~), using the Ralner and Culetto approach [3] and the electron-phonon spectra c~2(w)F(w) as calculated by Shiral et al. [4] for this oxide superconductor. Rainer and Culetto uses the functional derivative of T~ with respect to az(w)F(w) to find the isotope coefficient. This method is very revealbag and was first derived for a multimass system Mi with corresponding isotopic coefficient ai -- - d l n T J d l n M ~ . It can be show that the total isotope effect, ~tot -- - d In Tc / d In M, is just the stun of all the partial ~ ' s (atot = ~-'~,c~,) [3], where M is a reference mass given by the equation Mi = f ~ M with fli a constant that depends on i. The functional derivative tec.hnlque [5] is quite 0921-4534/97/$17.00 © Elsevier Science B.V. All rights reserved. PII S0921-4534(97)01041-1
helpful to know how the changes in the electronphonon interaction will affect the thermodynamic properties. For example, the functional derivative of the critical temperature, 6Tc/6(a2(w)F(oJ)), tells one how favorable a certain frequency is for an increase of To. That is, if a2(w)F(w) is changed by an infinitesimal amount 6(a2(w)F(w)), the resulting change of the transition temperature is given by 6T~ = f0 ~ dw6(a2(w)F(w)) 6To df(a2 (w)F(w)), which can be rewritten as 6inTo =-
where
Using the above equation for R(w), P~iner and Culetto introduced the partial isotope effect coefficient according to
The total isotope effect coefficient is then (~tot =
dw(~(w).
(I)
In Figure 1 we show the weighting function
R(w) vs w for Bao.TKo.3BiOa. This curve is for
O. Navarrc E. Chavira/Physica C 282-287 (1997) 1825-1826
1826
0.07
for Bao.7Ko.3BiO3, we have shown that not all phonons contribute the same amount to the isotope effect coefficient. We also showed that this oxide superconductor can be analyzed as a normal strong coupled superconductor, where the Cooper pairing is phonon-mediated. However, more calculations on the bases of other pairing mechanisms is necessary before one can make definite statements and draw strong conclusions. This work was partially supported by DGAPAUNAM, Grant IN102196 and by CONACyT 2661P-A9507.
i
0.06
0.05 0.04 s 0.03
0.02 0.01
REFERENCES
0.00 I
0
i
I
10 20
i
I
3040
I
50
60
70
(meV)
Figure 1. Weighting function R(w) vs. w for /z* = 0.12. This function is based on the spectral function a2(w)F(w) for Bao.TKo.3BiO3.
It* = 0.12 and looks much like the functional derivative of To. We note that R(w) goes to zero at w = 0 and w --* co, it means, that very low and very high phonon energy has a small contribution to the isotope effect. 'The shape for R(w) also shows a maximum at approximately 1/5 of the maximum phonon energy of the system. In other words, the fi,equencies which are more important for enhancing the isotope coefficient are precisely those in the maximum of the curve. Our calculated value of the total isotope effect for #* = 0.12 using Eq. (1) is atot = 0.49, which is close to the classical BCS result. We have also calculated, some important BCSratios for the Bao.7Ko.3Bi03 compound, such as 2Ao/kBTc = 4.15, 7[To~He(O)]2 = 0.145, and AC(T¢)/TTc ---- 1.88, given a clear indication of a strong coupling behavior in this potassium system [5]. In conclusion, based on the Rainer and Culetto approach and the electron-phonon spectral density az(w)F(w) as calculated by Shirai et al.
1. R.J. Cava et al., Nature 332 (1988) 814; L.F. Mattheiss et al., Phys. Rev B 37 (1988) 3745. 2. O. Navarro and R. Escudero, Physica C 170 (1990) 4 0 5 . 3. J.P. Carbotte, Rev. Mod. Phys. 62 (1990) 1027. 4. M. Shirai et al., J. Phys.: Condensed Matter 2 (1990) 3553. 5. O. Navarro, Physica C 265 (1996) 73.