ARTICLE IN PRESS Journal of Physics and Chemistry of Solids 69 (2008) 2265– 2267
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Fermi-arc superconductivity, pseudogap and charge order in Bi2Sr2CaCu2O8+d T. Kurosawa a, M. Oda a,, Y.H. Liu a, K. Takeyama a, N. Momono b, M. Ido a a b
Department of Physics, Hokkaido University, Sapporo 060-0810, Japan Department of Materials Science and Engineering, Muroran Institute of Technology, Muroran 050-8585, Japan
abstract We report STM/STS observations on the 4a 4a charge order in the pseudogap (PG) and superconducting (SC) states of Bi2Sr2CaCu2O8+d, and suggest that the charge order is associated with incoherent quasiparticle or pair states around the antinodal fermi surface (FS), which are also responsible for the PG, and it can coexist with the superconductivity caused by the pairing of coherent quasiparticles on the nodal fermi arc. We also suggest that the nanometer-scale gap inhomogeneity in the SC state, reported previously [Pan, et al., Nature 413 (2001) 282; Mommo, et al., J. Phys. Soc. Jpn. 74 (2005) 2400; Hashimoto, et al., Phys. Rev. B74 (2006) 064508] arises from that in the PG state, which occurs around the antinodal FS. & 2008 Elsevier Ltd. All rights reserved.
1. Introduction One of the interesting features in high-temperature cuprate superconductors is the relation between their superconducting (SC) transition temperature Tc and low-temperature (T5Tc) energy gap amplitude D0. As is well known, in the BCS mean ﬁeld theory, Tc scales with D0. In high-Tc cuprates, however, Tc does not scale with D0; even though the hole-doping level p is lowered down to the underdoped (UD) region where Tc turns to decrease, D0 continues to increase. It was demonstrated in our previous works that Tc is nearly proportional to the product of p and D0, pD0, in many high-Tc cuprates such as Bi2Sr2CaCu2O8+d (Bi2212) and La2xSrxCuO4 (La214) . Interestingly, in the UD region, where Tc is largely reduced from the mean-ﬁeld value Tco, deﬁned as Tco2D0/4.3kB, a PG develops below Tco at incoherent states around the antinodal FS near (p/a, 0) and (0, p/a), where a is the lattice constant or the Cu–O–Cu distance; the SC gap seems to develop mainly on the coherent part of the FS, including the node point near (p/2a, p/2a), which is called the nodal fermi-arc or fermi-arc (Fig. 1) [2–5]. This fact tempts us to suppose that the PG will be responsible for the reduction of Tc in the UD region and the energy gap on the fermi-arc will function as an effective SC gap (Deff p pD0) in determining Tc. Recently, in the PG state of Bi2212, Vershinin et al. found a charge order (CO) in two-dimensional (2-D) maps of the local density of states (LDOS) at speciﬁed energies, which were obtained by STM/STS, and suggested that the CO is a candidate for the hidden order of the PG state . Furthermore, such a CO was also observed in the SC state [7–9]. In this paper, we take up some interesting results in STM/STS
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experiments on the CO, PG and SC gap of Bi2212, and discuss how the CO, which develops in the PG state and persists to the SC state, coexists with the superconductivity.
2. Experiments Single crystals of Bi2212 were grown by traveling solvent ﬂoating-zone method. The grown crystals are slightly UD. To obtain clean sample surface, which are indispensable for STM, the crystals were cleaved in ultrahigh vacuum, better than 109 Torr, before being inserted in situ into an STM unit whose temperature is 7 K. STM experiments in the PG state were performed after ﬁnishing those at T5Tc and warming the sample gradually up to a temperature above Tc.
3. Results and discussions Fig. 2(a) is an example of STM images observed on Bi2212 cleaved surface at 85 K, higher than Tc ¼ 77 K, and a low bias voltage Vs of 20 mV. The cleavage in Bi2212 usually occurs between the semiconducting Bi–O planes with an energy gap Eg of the order of 0.1 eV, forming a bilayer. In STM experiments on the cleaved surface, the topmost atomic plane closest to the STM tip is the semiconducting Bi–O plane, the second the insulating Sr–O plane, and the third the metallic or SC Cu–O plane, and the Bi–O and Cu–O planes can be observed selectively, as reported in our previous works; STM images measured at |Vs|4Eg/e100 meV reﬂect the Bi–O plane, while STM images measured at |Vs|o100 meV reﬂect the Cu–O plane [8–10]. Therefore, the image (Vs ¼ 20 mV) in Fig. 2(a) corresponds to the Cu–O plane. One can see in this image that a 2-D CO develops in the PG state above
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Tc. From the fourier transform (Fig. 2(b)), furthermore, it is found that the period of the 2-D CO is four times the lattice constant (4a) along the Cu–O bond directions, i.e., 4a 4a. Fig. 2(c) shows some of the STS spectra measured along the solid line in the STM image in Fig. 2(a). Similar PG structures are seen in the STS spectra, regardless of the position. The spectrum,
Fig. 1. Schematic illustration of the PG and the effective SC gap. The PG develops below T*–Tco (4Tc) on the antinodal fermi surface near (p/a, 0) and (0, p/a). On the other hand, the effective SC gap opens below Tc on the fermi arc.
which tends to increase gradually with the lowering of Vs, is largely reduced in the range around Vs ¼ 0, corresponding to EF; thus, it exhibits a broad peak around the positive voltage Vps , while a broad bend appears around Vps in the negative Vs region. We deﬁne the energy size of PG, Dpg, from the peak position, Vps , DPGeVps . One can see in Fig. 2(c) that the DPG value is largely modulated or inhomogeneous in the nanometer scale for the present sample, which exhibited the strong 4a 4a CO. Furthermore, it was found in our recent works that the PG structure is homogeneous in samples with a very weak CO. To understand such a correlation between the electronic CO and the gap inhomogeneity, it has been proposed that the CO is dynamic in itself and, if Bi2212 samples involve strong scattering centers for quasiparticles, leading to the gap inhomogeneity, such as crystallographic imperfections, the scattering centers will function as effective pinning centers for the dynamically ﬂuctuating CO and make it static . On the basis of this pinning picture, the dynamically ﬂuctuating CO would be a candidate for the hidden order of the homogeneous PG state, and the degree of development of the static CO can be explained in terms of the density and/or strength of pinning centers. From the Vs-dependence of STM image in other Bi2212 samples, it was found that the period of CO was energy independent, i.e. nondispersive, while its amplitude decreased rapidly with increasing energy and became comparable to the background level above
Fig. 2. (a) STM image for an UD Bi2212 sample (Tc ¼ 77 K) obtained at 85 K and Vs ¼ 20 mV. (b) Line cuts taken along the qx and qy directions on the fourier map (inset). (c) dI/dVVs curves (STS spectra) measured along the white line in the image.
ARTICLE IN PRESS T. Kurosawa et al. / Journal of Physics and Chemistry of Solids 69 (2008) 2265–2267
(bias-voltage) range as the PG, incoherent, antinodal quasiparticle or pair states outside the FA will also be responsible for the CO. In fact, as has been demonstrated in low-temperature (T5Tc) STM/STS experiments, at low energies around EF, reﬂecting the quasiparticle states inside the FA, the gap structure is characterized by a spatially homogeneous d-wave gap and the CO tends to fade out, while at high energies around the gap edge, reﬂecting the quasiparticle states outside the FA, the gap structure is strongly inhomogeneous and the CO becomes marked [9,12]. Thus, it is suggested that if the 4a 4a CO, which is considered to be dynamic in itself and associated with antinodal quasiparticle or pair states outside the FA, is pinned down and static in the inhomogeneous PG state above Tc, it will remain below Tc, together with the inhomogeneous gap structure in the antinodal region, and coexist with the superconductivity caused by the pairing of coherent quasiparticles on the FA, i.e., the so-called ‘‘FA superconductivity.’’
Fig. 3. (a) Spatial average of dI/dVVs curves at 88 K for an UD Bi2212 sample (Tc ¼ 81 K), which exhibited a strong 4a 4a CO in the PG and SC states. (b) Biasvoltage dependence of the peak intensity (PI) at (2p/4a, 0), corresponding to the 4a 4a CO, in the fourier map at 88 K for the same sample. The peak intensity at (2p/4a, 0) is normalized with that at the Bragg point (2p/a, 0).
the PG energy, DPG (Fig. 3). This, consistent with the result observed by Vershinin et al., indicates that the characteristic energy of the 4a 4a CO in the PG state above Tc is the corresponding energy gap, as in the SC state below Tc [6,11]. As mentioned in Section 1, ARPES experiments on UD Bi2212 demonstrated that the PG starts to develop around the temperature T, well above Tc, on a part of the FS near (p, 0) and (0, p) it evolves gradually toward the node point of the d-wave gap near (p/2a, p/2a) with the lowering of T, but an ungapped region still remains around the node point just above Tc, leading to an arcshaped FS, i.e. fermi-arc (FA) [2,3,5]. The FS parts, inside and outside the FA, have been considered to consist of coherent and incoherent electronic states, respectively. In light of these facts, we have argued that even if incoherent quasiparticles outside the FA form pairs in the PG state below T, they cannot establish longrange phase coherence in collective motion, which will be done by the pairing of coherent quasiparticles on the FA, and the energy gap which opens up on the FA below Tc will function as an effective SC gap in determining Tc . It should be remembered here that the PG, which is formed on the incoherent part of the FS, is spatially inhomogeneous in samples exhibiting the strong, pinned 4a 4a CO, and vice versa. This fact is naturally understandable, because incoherent electronic states are easily modiﬁed by external perturbation, which is due to the randomness associated with pinning potentials for the CO. Furthermore, since the 4a 4a CO can be seen in almost the same energy
In the present STM/STS experiments on Bi2212, we demonstrated that a strong correlation between the static 4a 4a CO and the gap inhomogeneity held in the PG state as in the SC state; the static CO develops markedly in the inhomogeneous PG state, while it is very weak in the homogeneous PG state. The static CO can be understood in terms of a pinning picture where the dynamically ﬂuctuating CO, which seems to be intrinsic in the homogeneous PG and SC states, is pinned down by scattering potentials, leading to the gap inhomogeneity. We also suggested that the 4a 4a CO is associated with incoherent quasiparticle or pair states on the antinodal FS, which are responsible for the PG, and can coexist with the FA superconductivity below Tc.
Acknowledgments The authors thank Prof. F.J. Ohkawa for valuable discussions. This work was supported by the 21st century COE program ‘‘Topological Science and Technology’’ and Grants-in-Aid for Scientiﬁc Research from the Ministry of Education, Culture, Sports, Science and Technology, Japan.
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