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Pseudogap behavior of high-Tc cuprates
Physica B 281&282 (2000) 800}801
Pseudogap behavior of high-¹ cuprates # To( ru Sakai*, Yoshinori Takahashi Faculty of Science, Himeji Institute of Technology, 3-2-1 Kouto, Kamigori-Cho, Ako-gun, Hyogo 678-1297, Japan
Abstract Magnetic properties of the low-doped high-¹ cuprates are investigated based on the numerical diagonalization of the # "nite cluster t}J model. We show the pseudogap behavior of the temperature dependence of the NMR relaxation rate, neutron scattering intensity and magnetic susceptibility. We also discuss on the origin of the gap-like behavior based on the calculated spin correlation functions. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: High-¹ cuprates; Pseudogap; Spin gap; t}J model #
The pseudogap behavior has attracted much of current interest in the high-¹ cuprates superconductivity. It is # characterized by a broad peak in the ¹ dependence of the relaxation rate of NMR [1], the neutron scattering intensity [2] and the magnetic susceptibility [3] etc. of the under-doped samples. There have been several theoretical explanations; the one due to the spinon pairing [4], the bi-layer coupling mechanism [5], and the one due to AF spin #uctuations [6], for instance. However, the origin is still unsolved. Since the pseudogaps are observed only in the lowdoped cuprates, it is reasonable to think that it is a property of the undoped AF. Although the 2D AF has no long-range order at "nite temperature, the short-range order (SRO) is present at low ¹. When ¹ decreases down to be of the same magnitude as the AF coupling J, SRO begins to develop. The crossover from the uncorrelated state to SRO will yield a broad peak in the ¹ dependence of the above quantities, because of the suppression of the local magnetic excitation. Since the hole motion acts to suppress SRO, the hole doping will reduce the crossover temperature. In fact the observed pseudogap temperature decreases as the hole density increases. Thus the pseudogap is consistent with the manifestation of the growth of the AF SRO.
* Corresponding author: Fax: #81-0791-58-0150. E-mail address: [email protected] (T. Sakai)
To con"rm the above picture, we perform the exact diagonalization study on 2D t}J model. Throughout the paper, all the energies are measured in units of t. In our previous study we showed the ¹ dependence of the NMR relaxation rate 1/¹ ¹ [7] evaluated by the formula, 1 1 1 J lim + Im s(q,u), ¹ ¹ u 1 u?0 q
(1)
in relation to the calculated NN spin correlation function C . In the present paper, to clarify the role of AF SRO, 1 we have also evaluated the Q,(n, n) component of the spin correlation function S(Q), as well as the dynamical magnetic susceptibilities, and the magnetic susceptibility for the comparison with experiments. We show in Fig. 1(a) the ¹ dependence of 1/¹ ¹, C , 1 1 and S(Q) for the J10]J10 cluster with 0, 1 and 2 holes, corresponding to the hole concentrations, d"0, 0.1 and 0.2, respectively, for J"0.3 estimated for cuprate superconductors. Though we see clearly a broad peak in the ¹ dependence of the 1/¹ ¹ for the undoped system 1 around ¹&J, no signi"cant suppression of 1/¹ ¹ is 1 observed in doped systems. This is, however, consistent with the absence of SRO in these cases. For small clusters the e!ect of the electron hopping tends to be overestimated. For the larger parameter J("0.6), we now see the gap-like behavior for doped systems as shown in Fig. 1(b) for d"0.1. We see from the same "gure the gap-like behavior is well correlated with the appearance of the SRO and the peak temperature rapidly decreases with increasing the hole concentration.
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 0 8 4 1 - 8
T. Sakai, Y. Takahashi / Physica B 281&282 (2000) 800}801
801
Fig. 2. Im s(Q, u), Im s(u) and s versus ¹ for J"0.6.
d"0 and 0.1. The both results are consistent with the neutron scattering experiment. [2] The result of s in Fig. 2 also exhibits a crossover behavior around the same temperature. The present results show that the pseudogap behavior of high-¹ cuprates can be well interpreted in terms of the # growth of the AF SRO.
References [1] [2] [3] [4] [5] [6] Fig. 1. 1/¹ ¹, C and S(Q) versus ¹ for (a) J"0.3 and (b) 1 1 J"0.6.
We also show in Fig. 2 Im s(Q, u), Im s(u), : dq Im s(q, u) for a small u("0.01t) and s for J"0.6. Fig. 2 shows that Im s(Q, u) increases monotonically with decreasing ¹, while Im s(u) has a broad peak around the same crossover temperature as 1/¹ ¹ for 1
A. Goto et al., J. Phys. Soc. Japan 65 (1996) 3043. B.J. Sternlieb et al., Phys. Rev. B 47 (1993) 5320. T. Nakano et al., Phys. Rev. B 49 (1994) 16000. T. Tanamoto et al., Soc. Japan 63 (1994) 2739. M.U. Ubens, P.A. Lee, Phys. Rev. B 50 (1994) 438. K. Miyake, O. Narikiyo, J. Phys. Soc. Japan 63 (1994) 3821. [7] T. Sakai, Y. Takahashi, J. Phys. Soc. Japan 67 (1998) 2630.
Pseudogap behavior of high-¹ cuprates # To( ru Sakai*, Yoshinori Takahashi Faculty of Science, Himeji Institute of Technology, 3-2-1 Kouto, Kamigori-Cho, Ako-gun, Hyogo 678-1297, Japan
Abstract Magnetic properties of the low-doped high-¹ cuprates are investigated based on the numerical diagonalization of the # "nite cluster t}J model. We show the pseudogap behavior of the temperature dependence of the NMR relaxation rate, neutron scattering intensity and magnetic susceptibility. We also discuss on the origin of the gap-like behavior based on the calculated spin correlation functions. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: High-¹ cuprates; Pseudogap; Spin gap; t}J model #
The pseudogap behavior has attracted much of current interest in the high-¹ cuprates superconductivity. It is # characterized by a broad peak in the ¹ dependence of the relaxation rate of NMR [1], the neutron scattering intensity [2] and the magnetic susceptibility [3] etc. of the under-doped samples. There have been several theoretical explanations; the one due to the spinon pairing [4], the bi-layer coupling mechanism [5], and the one due to AF spin #uctuations [6], for instance. However, the origin is still unsolved. Since the pseudogaps are observed only in the lowdoped cuprates, it is reasonable to think that it is a property of the undoped AF. Although the 2D AF has no long-range order at "nite temperature, the short-range order (SRO) is present at low ¹. When ¹ decreases down to be of the same magnitude as the AF coupling J, SRO begins to develop. The crossover from the uncorrelated state to SRO will yield a broad peak in the ¹ dependence of the above quantities, because of the suppression of the local magnetic excitation. Since the hole motion acts to suppress SRO, the hole doping will reduce the crossover temperature. In fact the observed pseudogap temperature decreases as the hole density increases. Thus the pseudogap is consistent with the manifestation of the growth of the AF SRO.
* Corresponding author: Fax: #81-0791-58-0150. E-mail address: [email protected] (T. Sakai)
To con"rm the above picture, we perform the exact diagonalization study on 2D t}J model. Throughout the paper, all the energies are measured in units of t. In our previous study we showed the ¹ dependence of the NMR relaxation rate 1/¹ ¹ [7] evaluated by the formula, 1 1 1 J lim + Im s(q,u), ¹ ¹ u 1 u?0 q
(1)
in relation to the calculated NN spin correlation function C . In the present paper, to clarify the role of AF SRO, 1 we have also evaluated the Q,(n, n) component of the spin correlation function S(Q), as well as the dynamical magnetic susceptibilities, and the magnetic susceptibility for the comparison with experiments. We show in Fig. 1(a) the ¹ dependence of 1/¹ ¹, C , 1 1 and S(Q) for the J10]J10 cluster with 0, 1 and 2 holes, corresponding to the hole concentrations, d"0, 0.1 and 0.2, respectively, for J"0.3 estimated for cuprate superconductors. Though we see clearly a broad peak in the ¹ dependence of the 1/¹ ¹ for the undoped system 1 around ¹&J, no signi"cant suppression of 1/¹ ¹ is 1 observed in doped systems. This is, however, consistent with the absence of SRO in these cases. For small clusters the e!ect of the electron hopping tends to be overestimated. For the larger parameter J("0.6), we now see the gap-like behavior for doped systems as shown in Fig. 1(b) for d"0.1. We see from the same "gure the gap-like behavior is well correlated with the appearance of the SRO and the peak temperature rapidly decreases with increasing the hole concentration.
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 0 8 4 1 - 8
T. Sakai, Y. Takahashi / Physica B 281&282 (2000) 800}801
801
Fig. 2. Im s(Q, u), Im s(u) and s versus ¹ for J"0.6.
d"0 and 0.1. The both results are consistent with the neutron scattering experiment. [2] The result of s in Fig. 2 also exhibits a crossover behavior around the same temperature. The present results show that the pseudogap behavior of high-¹ cuprates can be well interpreted in terms of the # growth of the AF SRO.
References [1] [2] [3] [4] [5] [6] Fig. 1. 1/¹ ¹, C and S(Q) versus ¹ for (a) J"0.3 and (b) 1 1 J"0.6.
We also show in Fig. 2 Im s(Q, u), Im s(u), : dq Im s(q, u) for a small u("0.01t) and s for J"0.6. Fig. 2 shows that Im s(Q, u) increases monotonically with decreasing ¹, while Im s(u) has a broad peak around the same crossover temperature as 1/¹ ¹ for 1
A. Goto et al., J. Phys. Soc. Japan 65 (1996) 3043. B.J. Sternlieb et al., Phys. Rev. B 47 (1993) 5320. T. Nakano et al., Phys. Rev. B 49 (1994) 16000. T. Tanamoto et al., Soc. Japan 63 (1994) 2739. M.U. Ubens, P.A. Lee, Phys. Rev. B 50 (1994) 438. K. Miyake, O. Narikiyo, J. Phys. Soc. Japan 63 (1994) 3821. [7] T. Sakai, Y. Takahashi, J. Phys. Soc. Japan 67 (1998) 2630.