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The c-axis optical conductivity and the superconducting fluctuation
PHYSlCA Physica C 341-348 (2000) 891-892
ELSEVIER
www.elsevier.nl/Iocate/physc
The c-axis optical conductivity and the superconducting fluctuation Hyekyung Won a * and Kazumi Maki b aDepartment of Physics, Hallym University, Chunchon 200-702, South Korea bDepartment of Physics and Astronomy, University of Southern California, Los Angeles, CA90089-0484, USA We propose that the pseudogap phenomenon is due to the standard superconducting fluctuation. The tunneling conductance and the c-axis optical conductivity measured in Bi2212 are very consistent with the present model. Also the model predicts the pseudogap behavior in a strong magnetic field but at low temperatures.
A recent review surveys the experimental aspects of pseudogap phenomenon in high-Tc cuprates very nicely [1]. However the theory for the pseudogap phenomenon appears very controversial. Here we propose one theoretical model for these pseudogap phenomenon. Since the important fluctuation in high-Tc cuprates is either spin fluctuation(antiparamagnon) or the superconducting fluctuation, we can conclude that the pseudogap should be due to the superconducting fluctuation. Such an idea has been suggested by Varlamov and collaborators [2] and their works are summarized beautifully in [3]. Here we extend Varlamov's idea in two directions; a)d-wave symmetry of superconductor is explicitly incorporated and b)the momentun transfer from the fluctualtion to the quasi-particle is explicitly included [4]. In the presence of fluctualtion the quai-particle density of states is given by N(k,w) No
_
1-
T ~
-
2
a2 cos (2¢) [[(2w) 2 + a2 ]
2wa'2 [(2w)e+ae]z/esinh-1(
)]
(1)
for
d-wave
cos~(2¢) D(q, 0) = N0(e + ~ q 2 )
(2)
*HW acknowledges the support from Hallym Academy of Science, Hallym University.
with
~
=
ductance is given by
dI
dI
T
eV
(TV)/(~)I~ = 1 + 2S~(3)EFC(~rT )
(3)
where C(x) = 1 f d ¢ Re~(2)( - ix + in sinh ¢) z J-e~ and ~ = a/4rT ~_ 0.243772V~ and ¢(2)(z) is the tetra g a m m a function. In Figure 1 we show C(eV/2rrT) versus eV/2rrTe for e = 0.1, 0.3, 0.5 and 1.0. 5 0 -5 ~9 -10 -15 -20 -25 0
where a = 1.957rTv~ , e = ln(@) and use is made of the 2D fluctuation propagator
superconductor
7¢(3)v2/2(47rT) 2 [5]. The c-axis tunneling con-
Figure 1. ~'s
0.5
1
C(eV/21rT)
1.5 2 2.5 3 eV/(2gT¢) versus eV/27rTe for several
The temperature dependence of C(x) is very comparable to the one measured by Renner et al. in Bi2212 [6].
0921-4534/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PI1 S0921-4534(00)00721-8
892
H. Won, K. Maki/Physica C 341-348 (2000) 891-892
There is a strong reason to believe that the caxis conductivity is governed by the coherent tunneling rather than usual anisotropic conduction. First of all there is no Drude tail in the optical conductivity both in YBCO and Bi2212 [1]. Therefore this is not compatible with 3D-Fermi liquid. Also there are evidence that the c-axis magnetic penetration depth has the T2-dependence rather than the T-linear dependence common to the planar penetration depth [7,8]. This T 2 dependence is most naturally interpreted in terms of Josephson tunneling [9]. Also a similar model appears to apply for a-(ET)2 salts as well [9]. Then the c-axis optical conductivity is given by T
(4)
ac/aOn = 1 + 28~(3)EF S(~-~),~Trx
OG when S (y) = ~1 f_~ ~ d¢Im¢(') ( ½+ iy + ia sinh ¢). Here ¢0)(z) is the tri gamma function. We have also a relation
c(x) = d[xs(~)]
-5
(5)
.~
......... . / - ' 7
~---o.1
-10 l-_~./"/ I" ,//
- -
E:---0.3 ....... ~o.s .........
-15
-20 -25 0
0.5
1
1.5 2 2.5 3 o)/(2~:Tc) Figure 2. S versus eV/27rTc for several e's
~ versus ~/2~rTc for In Figure 2 we show S (2--~) e -- 0.1, 0.3, 0.5 and 1.0. Indeed the behavior of S(w/2~rT) is very consistent with the recent caxis optical conductivity measurement in Bi2212 [10,11].
In a magnetic field and T/Tc << 1, we obtain the tunneling conductance
(-~)/('~-'~)]y=l + 112~(3)EF[¢(~) -x
D (~.-~1 eV
(6)
where
D(z)
'
F
v~(l + 24C2) J - - oue-U2_ I( O
+
4C(u4-3u 2 + 3))2Re¢(2)(1 - iz + z ' ~ u ) •
and ~ = v 2x/~B [12]. Again the tunnleing conductance behaves very similar as in T > Tc and B = 0. But the details of the field dependence is quite different. In summary the DOS term in the standard fluctuation theory describes the pseudogap phenomenon. The DOS term persists for large temperature and large energy (a; ~ 27rT), which sre consistent with observation. REFERENCES
1. T.Timusk and B. Statt, Rep.. Prog. Phys. 62, 61 (1999). 2. I.B. Ioffe, A.I. Larkin, A.A. Varlamov, and L. Yu, Phys. Rev. B 47, 8936 (1993). 3. A.A. Varlamov, G. Balestrino, E. Milani, and D.V. Livanov, Advance in Physics 48, 655 (1999). 4. K. Maki and H. Won, Physica C 285-287, 1839 (1997). 5. H. Won and K. Maki, ECRYS'99 proceeding. 6. Ch. Renner et al., Phys. Rev. Lett. 80, 149 (1998). 7. A. Hosseini et al., Phys. Rev. Lett. 81, 1298 (1998). 8. A.A. Tsvetkov et al., Nature 395,360 (1998). 9. M. Pinteric et al., Phys. Rev. B (in press). 10. C.C. Homes et al., Physica C 254, 265 (1995). 11. C. Bernhard et al., Phys. Rev. B 59, R6631 (1999). 12. H. Won and K. Maki, Europhys. Lett. 34,453 (1996).
ELSEVIER
www.elsevier.nl/Iocate/physc
The c-axis optical conductivity and the superconducting fluctuation Hyekyung Won a * and Kazumi Maki b aDepartment of Physics, Hallym University, Chunchon 200-702, South Korea bDepartment of Physics and Astronomy, University of Southern California, Los Angeles, CA90089-0484, USA We propose that the pseudogap phenomenon is due to the standard superconducting fluctuation. The tunneling conductance and the c-axis optical conductivity measured in Bi2212 are very consistent with the present model. Also the model predicts the pseudogap behavior in a strong magnetic field but at low temperatures.
A recent review surveys the experimental aspects of pseudogap phenomenon in high-Tc cuprates very nicely [1]. However the theory for the pseudogap phenomenon appears very controversial. Here we propose one theoretical model for these pseudogap phenomenon. Since the important fluctuation in high-Tc cuprates is either spin fluctuation(antiparamagnon) or the superconducting fluctuation, we can conclude that the pseudogap should be due to the superconducting fluctuation. Such an idea has been suggested by Varlamov and collaborators [2] and their works are summarized beautifully in [3]. Here we extend Varlamov's idea in two directions; a)d-wave symmetry of superconductor is explicitly incorporated and b)the momentun transfer from the fluctualtion to the quasi-particle is explicitly included [4]. In the presence of fluctualtion the quai-particle density of states is given by N(k,w) No
_
1-
T ~
-
2
a2 cos (2¢) [[(2w) 2 + a2 ]
2wa'2 [(2w)e+ae]z/esinh-1(
)]
(1)
for
d-wave
cos~(2¢) D(q, 0) = N0(e + ~ q 2 )
(2)
*HW acknowledges the support from Hallym Academy of Science, Hallym University.
with
~
=
ductance is given by
dI
dI
T
eV
(TV)/(~)I~ = 1 + 2S~(3)EFC(~rT )
(3)
where C(x) = 1 f d ¢ Re~(2)( - ix + in sinh ¢) z J-e~ and ~ = a/4rT ~_ 0.243772V~ and ¢(2)(z) is the tetra g a m m a function. In Figure 1 we show C(eV/2rrT) versus eV/2rrTe for e = 0.1, 0.3, 0.5 and 1.0. 5 0 -5 ~9 -10 -15 -20 -25 0
where a = 1.957rTv~ , e = ln(@) and use is made of the 2D fluctuation propagator
superconductor
7¢(3)v2/2(47rT) 2 [5]. The c-axis tunneling con-
Figure 1. ~'s
0.5
1
C(eV/21rT)
1.5 2 2.5 3 eV/(2gT¢) versus eV/27rTe for several
The temperature dependence of C(x) is very comparable to the one measured by Renner et al. in Bi2212 [6].
0921-4534/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PI1 S0921-4534(00)00721-8
892
H. Won, K. Maki/Physica C 341-348 (2000) 891-892
There is a strong reason to believe that the caxis conductivity is governed by the coherent tunneling rather than usual anisotropic conduction. First of all there is no Drude tail in the optical conductivity both in YBCO and Bi2212 [1]. Therefore this is not compatible with 3D-Fermi liquid. Also there are evidence that the c-axis magnetic penetration depth has the T2-dependence rather than the T-linear dependence common to the planar penetration depth [7,8]. This T 2 dependence is most naturally interpreted in terms of Josephson tunneling [9]. Also a similar model appears to apply for a-(ET)2 salts as well [9]. Then the c-axis optical conductivity is given by T
(4)
ac/aOn = 1 + 28~(3)EF S(~-~),~Trx
OG when S (y) = ~1 f_~ ~ d¢Im¢(') ( ½+ iy + ia sinh ¢). Here ¢0)(z) is the tri gamma function. We have also a relation
c(x) = d[xs(~)]
-5
(5)
.~
......... . / - ' 7
~---o.1
-10 l-_~./"/ I" ,//
- -
E:---0.3 ....... ~o.s .........
-15
-20 -25 0
0.5
1
1.5 2 2.5 3 o)/(2~:Tc) Figure 2. S versus eV/27rTc for several e's
~ versus ~/2~rTc for In Figure 2 we show S (2--~) e -- 0.1, 0.3, 0.5 and 1.0. Indeed the behavior of S(w/2~rT) is very consistent with the recent caxis optical conductivity measurement in Bi2212 [10,11].
In a magnetic field and T/Tc << 1, we obtain the tunneling conductance
(-~)/('~-'~)]y=l + 112~(3)EF[¢(~) -x
D (~.-~1 eV
(6)
where
D(z)
'
F
v~(l + 24C2) J - - oue-U2_ I( O
+
4C(u4-3u 2 + 3))2Re¢(2)(1 - iz + z ' ~ u ) •
and ~ = v 2x/~B [12]. Again the tunnleing conductance behaves very similar as in T > Tc and B = 0. But the details of the field dependence is quite different. In summary the DOS term in the standard fluctuation theory describes the pseudogap phenomenon. The DOS term persists for large temperature and large energy (a; ~ 27rT), which sre consistent with observation. REFERENCES
1. T.Timusk and B. Statt, Rep.. Prog. Phys. 62, 61 (1999). 2. I.B. Ioffe, A.I. Larkin, A.A. Varlamov, and L. Yu, Phys. Rev. B 47, 8936 (1993). 3. A.A. Varlamov, G. Balestrino, E. Milani, and D.V. Livanov, Advance in Physics 48, 655 (1999). 4. K. Maki and H. Won, Physica C 285-287, 1839 (1997). 5. H. Won and K. Maki, ECRYS'99 proceeding. 6. Ch. Renner et al., Phys. Rev. Lett. 80, 149 (1998). 7. A. Hosseini et al., Phys. Rev. Lett. 81, 1298 (1998). 8. A.A. Tsvetkov et al., Nature 395,360 (1998). 9. M. Pinteric et al., Phys. Rev. B (in press). 10. C.C. Homes et al., Physica C 254, 265 (1995). 11. C. Bernhard et al., Phys. Rev. B 59, R6631 (1999). 12. H. Won and K. Maki, Europhys. Lett. 34,453 (1996).