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Pseudogap induced by superconducting fluctuation and anomalous metallic phase in the high-Tc superconducting materials
Physica B 284}288 (2000) 661}662
Pseudogap induced by superconducting #uctuation and anomalous metallic phase in the high-¹ superconducting materials
Akito Kobayashi *, Atsushi Tsuruta, Tamifusa Matsuura, Yoshihiro Kuroda Department of Physics, CREST, Japan Science and Technology Corporation (JST), Nagoya University, Nagoya 464-8602, Japan Department of Physics, Nagoya University, Nagoya 464-8602, Japan
Abstract Using the d}p model, we demonstrate that spin #uctuation and pseudogap induced by superconducting #uctuation in#uence each other and play key roles in the determination of the phase diagram observed in high-¹ superconducting materials. We calculate superconducting transition temperature ¹ , NMR relaxation rate 1/¹ ¹ and single particle spectra by treating both the superconducting #uctuation and spin #uctuation in a consistent fashion. As temperature decreases, 1/¹ ¹ increases at high temperatures, while it reaches a maximum followed by sharp drop at the underdoped region, which is due to evolution of the pseudogap in the single-particle spectra. The evolution is also consistent with those of ARPES experiments. 2000 Elsevier Science B.V. All rights reserved. Keywords: d}p model; Pseudogap; Superconducting #uctuations
In the high-¹ superconductors, various anomalous properties have been observed at higher temperature than the superconducting transition temperature ¹ . Key issues on the anomalous properties must be on the evolution of the pseudogap and the enhancement of the antiferromagnetic spin #uctuation (AF #uctuation) as suggested by various experiments [1}3]. We take the ;"R d}p model, where it has been shown that the in-gap state emerges inside the charge transfer gap D upon doping carriers to a Mott insulator in the leading order in the 1/N-expansion [4]. The band width of the in-gap states is proportional to the hole doping rate d in the underdoped reigon. In order to treat both the pseudogap and the AF #uctuation consistently, we extend the self-consistent t-matrix approximation, which consists of three processes
* Corresponding author. E-mail address: [email protected] (A. Kobayashi)
described as follows: (1) We treat the enhancement of AF #uctuation in the RPA of the superexchange interaction J [5]. The superconducting #uctuation (SC #uctuation) is also enhanced via the pairing interaction mediated by the AF #uctuation. (2) By treating the SC #uctuation by the self-consistent t-matrix approximation, the pseudogap in the single particle spectrum is obtained. (3) The low-energy part of the AF #uctuation is suppressed by the pseudogap because of the suppression of the density of states near the Fermi energy. We calculate ¹ , NMR relaxation rate 1/¹ ¹ and single particle spectra by using the renormalized Green's function obtained by calculating these processes (1}3) selfconsistently. Numerical results for ¹-dependence in 1/¹ ¹ are shown in Fig. 1 for d"0.06&0.19. At high temperatures, 1/¹ ¹ increases as ¹ or d decreases, which is due to development of AF #uctuation. We de"ne ¹ as a characteristic temperature where e!ects of the AF
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 2 3 2 6 - 1
662
A. Kobayashi et al. / Physica B 284}288 (2000) 661}662
Fig. 2. d}¹ phase diagram with J "0.1 and D"2.5. Fig. 1. ¹-dependences of 1/¹ ¹ obtained with J "0.1 and D"2.5 for the hole doping rate d"0.06&0.19. In the inset, 1/¹ ¹ obtained as above (solid line) is compared with that calculated by neglecting e!ects of the SC #uctuation (dashed line), where d"0.1.
#uctuation begin to be appreciable. Explicit values of ¹ is determined by the condition that [(*/*¹)(1/¹ ¹)" ]/[(1/¹ ¹)" ]"!2;10. 22 2 B At d(0.11, 1/¹ ¹ has a maximum at a certain temper ature denoted as ¹ . In the inset, the ¹-dependence of 1/¹ ¹ obtained at d"0.1 as above (solid line) is com pared with that calculated by neglecting the e!ects of the SC #uctuation (dashed line). We note that the SC #uctuation also becomes appreciable at ¹&¹ and develops as ¹ decreases inducing the pseudogap in the singleparticle spectra. The developed pseudogap kills the AF #uctuation leading to sharp drop in 1/¹ ¹ at ¹:¹ . We have calculated the e!ects of the external magnetic "eld H. It has been found that ¹ hardly depends on H in the region of H:100 T. We plot the three characteristic temperatures, ¹ , ¹ and ¹ as functions of d in Fig. 2. We note that
¹ has a maximum at d 0.11. As d decreases, the quasi particle band narrows and the nesting e!ect increases, which enhances the AF #uctuation and then pushes ¹ higher. The AF #uctuation enhances the pairing interaction and then the SC #uctuation, which develops the pseudogap and then suppresses ¹ in the underdoped region. ¹ is determined by the competition between the AF #uctuation and the SC #uctuation. The phase diagram shown in Fig. 2 accounts for essential features of the anomalous metallic phase observed in high-¹ super conducting materials [2].
References [1] T. Nishikawa, J. Takeda, M. Sato, J. Phys. Soc. Japan 63 (1994) 1441. [2] M. Sato, Physica C 263 (1996) 271. [3] M.R. Norma et al., Nature 392 (1998) 157. [4] H. Jichu, T. Matsuura, Y. Kuroda, J. Phys. Soc. Japan 61 No. 2 (1992) 649. [5] S. Fukagawa et al., J. Phys. Soc. Japan 67 (1998) 3536.
Pseudogap induced by superconducting #uctuation and anomalous metallic phase in the high-¹ superconducting materials
Akito Kobayashi *, Atsushi Tsuruta, Tamifusa Matsuura, Yoshihiro Kuroda Department of Physics, CREST, Japan Science and Technology Corporation (JST), Nagoya University, Nagoya 464-8602, Japan Department of Physics, Nagoya University, Nagoya 464-8602, Japan
Abstract Using the d}p model, we demonstrate that spin #uctuation and pseudogap induced by superconducting #uctuation in#uence each other and play key roles in the determination of the phase diagram observed in high-¹ superconducting materials. We calculate superconducting transition temperature ¹ , NMR relaxation rate 1/¹ ¹ and single particle spectra by treating both the superconducting #uctuation and spin #uctuation in a consistent fashion. As temperature decreases, 1/¹ ¹ increases at high temperatures, while it reaches a maximum followed by sharp drop at the underdoped region, which is due to evolution of the pseudogap in the single-particle spectra. The evolution is also consistent with those of ARPES experiments. 2000 Elsevier Science B.V. All rights reserved. Keywords: d}p model; Pseudogap; Superconducting #uctuations
In the high-¹ superconductors, various anomalous properties have been observed at higher temperature than the superconducting transition temperature ¹ . Key issues on the anomalous properties must be on the evolution of the pseudogap and the enhancement of the antiferromagnetic spin #uctuation (AF #uctuation) as suggested by various experiments [1}3]. We take the ;"R d}p model, where it has been shown that the in-gap state emerges inside the charge transfer gap D upon doping carriers to a Mott insulator in the leading order in the 1/N-expansion [4]. The band width of the in-gap states is proportional to the hole doping rate d in the underdoped reigon. In order to treat both the pseudogap and the AF #uctuation consistently, we extend the self-consistent t-matrix approximation, which consists of three processes
* Corresponding author. E-mail address: [email protected] (A. Kobayashi)
described as follows: (1) We treat the enhancement of AF #uctuation in the RPA of the superexchange interaction J [5]. The superconducting #uctuation (SC #uctuation) is also enhanced via the pairing interaction mediated by the AF #uctuation. (2) By treating the SC #uctuation by the self-consistent t-matrix approximation, the pseudogap in the single particle spectrum is obtained. (3) The low-energy part of the AF #uctuation is suppressed by the pseudogap because of the suppression of the density of states near the Fermi energy. We calculate ¹ , NMR relaxation rate 1/¹ ¹ and single particle spectra by using the renormalized Green's function obtained by calculating these processes (1}3) selfconsistently. Numerical results for ¹-dependence in 1/¹ ¹ are shown in Fig. 1 for d"0.06&0.19. At high temperatures, 1/¹ ¹ increases as ¹ or d decreases, which is due to development of AF #uctuation. We de"ne ¹ as a characteristic temperature where e!ects of the AF
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 2 3 2 6 - 1
662
A. Kobayashi et al. / Physica B 284}288 (2000) 661}662
Fig. 2. d}¹ phase diagram with J "0.1 and D"2.5. Fig. 1. ¹-dependences of 1/¹ ¹ obtained with J "0.1 and D"2.5 for the hole doping rate d"0.06&0.19. In the inset, 1/¹ ¹ obtained as above (solid line) is compared with that calculated by neglecting e!ects of the SC #uctuation (dashed line), where d"0.1.
#uctuation begin to be appreciable. Explicit values of ¹ is determined by the condition that [(*/*¹)(1/¹ ¹)" ]/[(1/¹ ¹)" ]"!2;10. 22 2 B At d(0.11, 1/¹ ¹ has a maximum at a certain temper ature denoted as ¹ . In the inset, the ¹-dependence of 1/¹ ¹ obtained at d"0.1 as above (solid line) is com pared with that calculated by neglecting the e!ects of the SC #uctuation (dashed line). We note that the SC #uctuation also becomes appreciable at ¹&¹ and develops as ¹ decreases inducing the pseudogap in the singleparticle spectra. The developed pseudogap kills the AF #uctuation leading to sharp drop in 1/¹ ¹ at ¹:¹ . We have calculated the e!ects of the external magnetic "eld H. It has been found that ¹ hardly depends on H in the region of H:100 T. We plot the three characteristic temperatures, ¹ , ¹ and ¹ as functions of d in Fig. 2. We note that
¹ has a maximum at d 0.11. As d decreases, the quasi particle band narrows and the nesting e!ect increases, which enhances the AF #uctuation and then pushes ¹ higher. The AF #uctuation enhances the pairing interaction and then the SC #uctuation, which develops the pseudogap and then suppresses ¹ in the underdoped region. ¹ is determined by the competition between the AF #uctuation and the SC #uctuation. The phase diagram shown in Fig. 2 accounts for essential features of the anomalous metallic phase observed in high-¹ super conducting materials [2].
References [1] T. Nishikawa, J. Takeda, M. Sato, J. Phys. Soc. Japan 63 (1994) 1441. [2] M. Sato, Physica C 263 (1996) 271. [3] M.R. Norma et al., Nature 392 (1998) 157. [4] H. Jichu, T. Matsuura, Y. Kuroda, J. Phys. Soc. Japan 61 No. 2 (1992) 649. [5] S. Fukagawa et al., J. Phys. Soc. Japan 67 (1998) 3536.