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Quasiparticle density of states in high-Tc superconductors in the superconducting state
Physica C 463–465 (2007) 123–125 www.elsevier.com/locate/physc
Quasiparticle density of states in high-Tc superconductors in the superconducting state T. Hata, S. Onari *, S. Honda, H. Itoh, Y. Tanaka, J. Inoue Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan Available online 18 May 2007
Abstract In order to clarify the tunneling spectroscopy in hole-doped high-Tc cuprates, we study the strongly correlated electron systems with surface using the 2D Hubbard model with fluctuation exchange (FLEX) approximation from the side of weak coupling theory. We have found that (i) LDOS near the (1, 1, 0) surface has a zero energy peak, and (ii) LDOS is asymmetric, reflecting the asymmetry of band structure and electron correlation in the hole-doped high-Tc cuprates. Ó 2007 Elsevier B.V. All rights reserved. PACS: 74.20.Mn Keywords: Superconductivity; High-Tc cuprates; Tunneling spectroscopy
Tunneling spectroscopy is a powerful probe to analyze the electronic state of superconductors [1]. Several experimental results for scanning tunneling spectroscopy (STS) on high-Tc cuprates have shown so-called the zero-bias conductance peak (ZBCP) for (1, 1, 0)-oriented junctions [2,3]. It was clarified that ZBCP is a direct consequence of dx2 y 2 symmetry of pair potential [4–6]. Most theories have dealt with tunneling spectroscopy based on non-interacting model and explained the behavior of the line shape of the tunneling conductance of normal metal–insulatordx2 y 2 wave superconductor junctions. However, the electron–electron interaction is important for strongly correlated electron systems such as high-Tc cuprates to discuss detailed structure of the line shapes of tunneling spectroscopy. In the previous work using t–J model, the tunneling surface density of states [6] and the tunneling conductance is calculated within the Gutzwiller approximation with Bogoliubov de Gennes equation [7]. However, in the previous papers [6,7], the self-energy originating from the many
*
Corresponding author. Tel.: +81 52 789 3701; fax: +81 52 789 3298. E-mail address: [email protected] (S. Onari).
0921-4534/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.05.019
body effect, especially the energy dependence of the life time of the quasiparticles is not taken into account. To take into account the energy dependence of the self-energy, we study this problem from the weak coupling limit. The purpose of this paper is to calculate microscopically the detailed structure of the line shape of the tunneling spectroscopy including the electron–electron interaction on hole-doped high-Tc cuprate YBCO. It is known that a tunneling spectroscopy corresponds to a local density of states (LDOS) at the surfaces of the superconductors in the limit of low transparency. Thus, we calculate the local density of states for the surface of high-Tc cuprates using the method developed by Matsumoto and Shiba [8]. In the actual calculation, we first obtain the Green’s function of the bulk system using the fluctuation exchange (FLEX) approximation [9,10], where normal Green’s function (G) and anomalous Green’s function (F ) are calculated selfconsistently. We have found that (i) LDOS has a peak at zero energy for (1, 1, 0)-oriented surface, which corresponds to the ZBCP, and (ii) LDOS is asymmetric for energy axis, because the band structure and electron correlation are asymmetric in hole-doped high-Tc cuprate YBCO.
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T. Hata et al. / Physica C 463–465 (2007) 123–125
Our calculation is based on the Hubbard model on a 2D square lattice with the next-nearest hopping integral t 0 and the third-nearest hopping integral t00 in addition to the nearest-hopping integral t. We take t = 1 as a unit of energy. We have also taken into account the on-site Coulomb interaction U. We set the lattice constant a = 1 as a unit of length and denote the number of electron per site by n, the temperature by T. For the calculation described below we have taken t 0 /t = 1/6, t00 /t = 0.2, n = 0.88, T/t = 0.01, U/t = 6.0. In order to take account of the effect of the many body interactions, we have employed the FLEX approximation. It is known that the plausible results are obtained by using the FLEX except for the boarder of the metal insulator transition. First, we calculate the Green’s function in the superconducting state. The Green’s function can be written as [9] GðkÞ ¼ F ðkÞ ¼
ixn Z k þ ek þ vðkÞ 2
2
2
ð1Þ
2
2
2
ð2Þ
ðixn Z k Þ ðek þ vðkÞÞ /ðkÞ /ðkÞ ðixn Z k Þ ðek þ vðkÞÞ /ðkÞ
where Zk is the renormalization parameter, ixn(1 Zk) and v are the antisymmetric and symmetric part of the normal self-energy with respect to ixn, respectively, / is the anomalous self-energy, k (k, ixn), and ixn is the Matsubara frequency. Next, in order to describe the effect of the surface, we put a scattering potential U0 which is perpendicular to x axis and is located at x = 0. In the limit U0 ! 1, the Green’s function matrix at position x is obtained according to the formula given in Ref. [8], 1 k y Þ Gðx; ky Þ 1 ðx; k y Þ ¼ Gð0; G k y Þ Gðx; k y Þ Gð0;
Fig. 1. The momentum dependence of D(k, ipT) = /(k, ipT)/Z(k, ipT). Dashed lines indicate nodes.
Fig. 2. The local density of states (LDOS) near the (1, 0, 0) surface of the dx2 y 2 superconductor at three different positions (x/a = 7, 11, 15, where x is the distance from the surface). Dashed line denotes the LDOS of the bulk dx2 y 2 superconductor.
ð3Þ
1 denotes the where x is the distance from the surface and G 1 ðx; k y Þ can be matrix Green’s function at the position x. G calculated by Fourier transform along the kx direction of 1 ðk x ; k y Þ. Finally, using Pade´ approximation, we have perG formed the analytic continuation of the temperature Green’s 1 ðx; k y Þ and obtained the retarded Green’s funcfunction G tion GR ðx; k y ; wÞ. In terms of the obtained retarded Green’s 1 function, the LDOS D(x,w) at P the position x can be expressed as Dðx; wÞ ¼ 1=ðpN Þ ky Im½GR 1 ðx; k y ; wÞ. In Fig. 1, we show the momentum-dependence of the gap function D(k, ipT) = /(k, ipT)/Z(k, ipT) for the lowest Matsubara frequency. The dashed line in Fig. 1 indicates nodes. We see that the gap function has dx2 y 2 symmetry as expected. In Fig. 2, we show the bulk density of states (dashed line) and LDOS near the flat (1, 0, 0) surface for three different positions (x/a = 7, 11, 15). The results show that in the case of (1, 0, 0) surface, the zero energy state (ZES) does not appear since there is no sign change of the pair potential felt by the quasiparticle. Fig. 3 shows our results in the case of a flat (1, 1, 0) surface. We show LDOS near the (1, 1, 0) surface for three dif-
Fig. 3. LDOS in the dx2 y 2 -wave superconductor and that in the dx2 y2 pair potential near the (1, 1, 0) surface at three different positions.
pffiffiffi ferent positions (x= 2a ¼ 7; 11; 15) and the bulk density of states (dashed line) in comparison. In this case, zero energy peak (ZEP) appears due to the sign change of the pair potential felt by the quasiparticle. ZEP is suppressed at large x and the resulting LDOS converges to the bulk density of states. LDOS in Figs. 2 and 3 show asymmetric structure reflecting the asymmetry of the band structure and the electron correlations induced by the second and the third nearest hopping integrals.
T. Hata et al. / Physica C 463–465 (2007) 123–125
In summary, we have studied the LDOS near the surfaces of dx2 y 2 wave superconductors treating the many body interaction within the FLEX approximation. We have found that the LDOS near the (1, 0, 0) surface is the same as the bulk one. On the other hand, the LDOS near the (1, 1, 0) surface has a ZEP and the peak value decreases as distance from the surface increases. These behaviors are consistent with those obtained in the previous studies [6–8]. Acknowledgements This work was supported by a Grant-in-Aid for 21st Century COE ‘‘Frontiers of Computational Science’’. Numerical calculations were performed at the supercomputer center, ISSP.
125
References [1] E.L. Wolf, Principles of Electron Tunneling Spectroscopy, Oxford University Press, New York, 1985. [2] J. Geerk, X.X. Xi, G. Linker, Z. Phys. B 73 (1988) 329. [3] T. Walsh, Int. J. Mod. Phys. 6 (1992) 125. [4] Y. Tanaka, S. Kashiwaya, Phys. Rev. Lett. 74 (1995) 3451. [5] S. Kashiwaya, Y. Tanaka, Rep. Mod. Phys. 63 (2000) 1641. [6] Y. Tanuma, Y. Tanaka, M. Ogata, S. Kashiwaya, Phys. Rev. B 60 (1999) 9817; Y. Tanuma, Y. Tanaka, M. Ogata, S. Kashiwaya, J. Phys. Soc. Jpn. 67 (1998) 1118. [7] Y. Tanaka, H. Itoh, Y. Tanuma, H. Tsuchiura, J. Inoue, S. Kashiwaya, J. Phys. Soc. Jpn. 71 (2002) 2005. [8] M. Matsumoto, H. Shiba, J. Phys. Soc. Jpn. 64 (1995) 1703. [9] P. Monthoux, D.J. Scalapino, Phys. Rev. Lett. 72 (1994) 1874. [10] N.E. Bickers, D.J. Scalapino, S.R. White, Phys. Rev. Lett. 62 (1989) 961.
Quasiparticle density of states in high-Tc superconductors in the superconducting state T. Hata, S. Onari *, S. Honda, H. Itoh, Y. Tanaka, J. Inoue Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan Available online 18 May 2007
Abstract In order to clarify the tunneling spectroscopy in hole-doped high-Tc cuprates, we study the strongly correlated electron systems with surface using the 2D Hubbard model with fluctuation exchange (FLEX) approximation from the side of weak coupling theory. We have found that (i) LDOS near the (1, 1, 0) surface has a zero energy peak, and (ii) LDOS is asymmetric, reflecting the asymmetry of band structure and electron correlation in the hole-doped high-Tc cuprates. Ó 2007 Elsevier B.V. All rights reserved. PACS: 74.20.Mn Keywords: Superconductivity; High-Tc cuprates; Tunneling spectroscopy
Tunneling spectroscopy is a powerful probe to analyze the electronic state of superconductors [1]. Several experimental results for scanning tunneling spectroscopy (STS) on high-Tc cuprates have shown so-called the zero-bias conductance peak (ZBCP) for (1, 1, 0)-oriented junctions [2,3]. It was clarified that ZBCP is a direct consequence of dx2 y 2 symmetry of pair potential [4–6]. Most theories have dealt with tunneling spectroscopy based on non-interacting model and explained the behavior of the line shape of the tunneling conductance of normal metal–insulatordx2 y 2 wave superconductor junctions. However, the electron–electron interaction is important for strongly correlated electron systems such as high-Tc cuprates to discuss detailed structure of the line shapes of tunneling spectroscopy. In the previous work using t–J model, the tunneling surface density of states [6] and the tunneling conductance is calculated within the Gutzwiller approximation with Bogoliubov de Gennes equation [7]. However, in the previous papers [6,7], the self-energy originating from the many
*
Corresponding author. Tel.: +81 52 789 3701; fax: +81 52 789 3298. E-mail address: [email protected] (S. Onari).
0921-4534/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.05.019
body effect, especially the energy dependence of the life time of the quasiparticles is not taken into account. To take into account the energy dependence of the self-energy, we study this problem from the weak coupling limit. The purpose of this paper is to calculate microscopically the detailed structure of the line shape of the tunneling spectroscopy including the electron–electron interaction on hole-doped high-Tc cuprate YBCO. It is known that a tunneling spectroscopy corresponds to a local density of states (LDOS) at the surfaces of the superconductors in the limit of low transparency. Thus, we calculate the local density of states for the surface of high-Tc cuprates using the method developed by Matsumoto and Shiba [8]. In the actual calculation, we first obtain the Green’s function of the bulk system using the fluctuation exchange (FLEX) approximation [9,10], where normal Green’s function (G) and anomalous Green’s function (F ) are calculated selfconsistently. We have found that (i) LDOS has a peak at zero energy for (1, 1, 0)-oriented surface, which corresponds to the ZBCP, and (ii) LDOS is asymmetric for energy axis, because the band structure and electron correlation are asymmetric in hole-doped high-Tc cuprate YBCO.
124
T. Hata et al. / Physica C 463–465 (2007) 123–125
Our calculation is based on the Hubbard model on a 2D square lattice with the next-nearest hopping integral t 0 and the third-nearest hopping integral t00 in addition to the nearest-hopping integral t. We take t = 1 as a unit of energy. We have also taken into account the on-site Coulomb interaction U. We set the lattice constant a = 1 as a unit of length and denote the number of electron per site by n, the temperature by T. For the calculation described below we have taken t 0 /t = 1/6, t00 /t = 0.2, n = 0.88, T/t = 0.01, U/t = 6.0. In order to take account of the effect of the many body interactions, we have employed the FLEX approximation. It is known that the plausible results are obtained by using the FLEX except for the boarder of the metal insulator transition. First, we calculate the Green’s function in the superconducting state. The Green’s function can be written as [9] GðkÞ ¼ F ðkÞ ¼
ixn Z k þ ek þ vðkÞ 2
2
2
ð1Þ
2
2
2
ð2Þ
ðixn Z k Þ ðek þ vðkÞÞ /ðkÞ /ðkÞ ðixn Z k Þ ðek þ vðkÞÞ /ðkÞ
where Zk is the renormalization parameter, ixn(1 Zk) and v are the antisymmetric and symmetric part of the normal self-energy with respect to ixn, respectively, / is the anomalous self-energy, k (k, ixn), and ixn is the Matsubara frequency. Next, in order to describe the effect of the surface, we put a scattering potential U0 which is perpendicular to x axis and is located at x = 0. In the limit U0 ! 1, the Green’s function matrix at position x is obtained according to the formula given in Ref. [8], 1 k y Þ Gðx; ky Þ 1 ðx; k y Þ ¼ Gð0; G k y Þ Gðx; k y Þ Gð0;
Fig. 1. The momentum dependence of D(k, ipT) = /(k, ipT)/Z(k, ipT). Dashed lines indicate nodes.
Fig. 2. The local density of states (LDOS) near the (1, 0, 0) surface of the dx2 y 2 superconductor at three different positions (x/a = 7, 11, 15, where x is the distance from the surface). Dashed line denotes the LDOS of the bulk dx2 y 2 superconductor.
ð3Þ
1 denotes the where x is the distance from the surface and G 1 ðx; k y Þ can be matrix Green’s function at the position x. G calculated by Fourier transform along the kx direction of 1 ðk x ; k y Þ. Finally, using Pade´ approximation, we have perG formed the analytic continuation of the temperature Green’s 1 ðx; k y Þ and obtained the retarded Green’s funcfunction G tion GR ðx; k y ; wÞ. In terms of the obtained retarded Green’s 1 function, the LDOS D(x,w) at P the position x can be expressed as Dðx; wÞ ¼ 1=ðpN Þ ky Im½GR 1 ðx; k y ; wÞ. In Fig. 1, we show the momentum-dependence of the gap function D(k, ipT) = /(k, ipT)/Z(k, ipT) for the lowest Matsubara frequency. The dashed line in Fig. 1 indicates nodes. We see that the gap function has dx2 y 2 symmetry as expected. In Fig. 2, we show the bulk density of states (dashed line) and LDOS near the flat (1, 0, 0) surface for three different positions (x/a = 7, 11, 15). The results show that in the case of (1, 0, 0) surface, the zero energy state (ZES) does not appear since there is no sign change of the pair potential felt by the quasiparticle. Fig. 3 shows our results in the case of a flat (1, 1, 0) surface. We show LDOS near the (1, 1, 0) surface for three dif-
Fig. 3. LDOS in the dx2 y 2 -wave superconductor and that in the dx2 y2 pair potential near the (1, 1, 0) surface at three different positions.
pffiffiffi ferent positions (x= 2a ¼ 7; 11; 15) and the bulk density of states (dashed line) in comparison. In this case, zero energy peak (ZEP) appears due to the sign change of the pair potential felt by the quasiparticle. ZEP is suppressed at large x and the resulting LDOS converges to the bulk density of states. LDOS in Figs. 2 and 3 show asymmetric structure reflecting the asymmetry of the band structure and the electron correlations induced by the second and the third nearest hopping integrals.
T. Hata et al. / Physica C 463–465 (2007) 123–125
In summary, we have studied the LDOS near the surfaces of dx2 y 2 wave superconductors treating the many body interaction within the FLEX approximation. We have found that the LDOS near the (1, 0, 0) surface is the same as the bulk one. On the other hand, the LDOS near the (1, 1, 0) surface has a ZEP and the peak value decreases as distance from the surface increases. These behaviors are consistent with those obtained in the previous studies [6–8]. Acknowledgements This work was supported by a Grant-in-Aid for 21st Century COE ‘‘Frontiers of Computational Science’’. Numerical calculations were performed at the supercomputer center, ISSP.
125
References [1] E.L. Wolf, Principles of Electron Tunneling Spectroscopy, Oxford University Press, New York, 1985. [2] J. Geerk, X.X. Xi, G. Linker, Z. Phys. B 73 (1988) 329. [3] T. Walsh, Int. J. Mod. Phys. 6 (1992) 125. [4] Y. Tanaka, S. Kashiwaya, Phys. Rev. Lett. 74 (1995) 3451. [5] S. Kashiwaya, Y. Tanaka, Rep. Mod. Phys. 63 (2000) 1641. [6] Y. Tanuma, Y. Tanaka, M. Ogata, S. Kashiwaya, Phys. Rev. B 60 (1999) 9817; Y. Tanuma, Y. Tanaka, M. Ogata, S. Kashiwaya, J. Phys. Soc. Jpn. 67 (1998) 1118. [7] Y. Tanaka, H. Itoh, Y. Tanuma, H. Tsuchiura, J. Inoue, S. Kashiwaya, J. Phys. Soc. Jpn. 71 (2002) 2005. [8] M. Matsumoto, H. Shiba, J. Phys. Soc. Jpn. 64 (1995) 1703. [9] P. Monthoux, D.J. Scalapino, Phys. Rev. Lett. 72 (1994) 1874. [10] N.E. Bickers, D.J. Scalapino, S.R. White, Phys. Rev. Lett. 62 (1989) 961.