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Physica B 359–361 (2005) 591–593 www.elsevier.com/locate/physb
Superconductivity due to charge ﬂuctuation on a triangular lattice Yasuhiro Tanaka, Yoichi Yanase, Masao Ogata Department of Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
Abstract Motivated by superconductivity in the newly discovered Co-based oxide, we study a single-band extended Hubbard model on a triangular lattice within the random phase approximation. We show that f-wave triplet superconductivity is stabilized in the vicinity of charge–density–wave instability. The physical origin of the realization of the f-wave triplet state is discussed. r 2005 Elsevier B.V. All rights reserved. PACS: 71.10.Fd; 74.20.Mn; 74.20.Rp Keywords: Charge ﬂuctuation; Triplet superconductivity; Random phase approximation
The recently discovered superconductivity in Nax CoO2 yH2 O  has attracted great interest in condensed matter physics. The unique property of Nax CoO2 is that the system has a geometrical frustration since Co atoms form a two-dimensional triangular lattice, which is in sharp contrast to the high-T c cuprates or Sr2RuO4. Recent experiments have shown that Nax CoO2 has unconventional superconductivity [2–5]. The Knight shift measurements  suggest that triplet Corresponding author. Tel.: +81 3 5841 4185; fax: +81 3 5841 4185. E-mail address: [email protected]
superconductivity is realized in Nax CoO2 yH2 O; although there is still a controversy about the results . Furthermore, the relaxation rate 1=T 1 measurements in Co-NQR [4,5] suggest the presence of line nodes. There has been many mechanisms proposed for the superconductivity in Nax CoO2 so far [6–14]. One of them is spin ﬂuctuation, which has been discussed using t–J models on a triangular lattice [6–10]. However, every mean-ﬁeld-type theory has led to dx2 y2 þ idxy -wave symmetry and contradicts with a recent mSR experiment  which has shown no evidence of broken time-reversal symmetry. Another candidate is charge ﬂuctuation which is realized in an extended Hubbard model.
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.01.160
ARTICLE IN PRESS Y. Tanaka et al. / Physica B 359– 361 (2005) 591–593
It has been proposed that charge ordering is realized near the electron densities n ¼ 1 þ 13 and n ¼ 1 þ 23 (more than half-ﬁlling) . Motrunich and Lee  have discussed a strongly correlated electron system just next to charge ordering. In the present paper, we discuss superconductivity due to charge ﬂuctuation in the vicinity of charge density wave (CDW). We consider the single-band extended Hubbard model on a triangular lattice which is given by X y X H ¼ t ðcis cjs þ h:c:Þ þ U ni" ni# i
ni nj .
For the Nax CoO2 system, the electron density n is more than unity (n is approximately 1.3) and the LDA calculation  showed that there is a large hole-like Fermi surface so that the effective hopping integral is to0: In the following, we consider a case with t40 and no1; which corresponds to Nax CoO2 by electron–hole transformation. To study the effective interaction between electrons arising from the charge and spin ﬂuctuations, we use random phase approximation (RPA). The onset of the superconducting state is determined by solving the linearized E´liashberg’s equation within the weak-coupling theory. In Fig. 1, we show the effective interactions, V s for
singlet pairing and V t for triplet pairing obtained by RPA. For V ¼ 0 (i.e. ðU; V Þ ¼ ð3:64; 0Þ), V s has a positive (i.e. repulsive) peak at Q ¼ ð0; 43 pÞ ¼(K-point) which comes from the nesting condition and leads to the SDW instability of the system. V t has the same property as V s ; although its sign is opposite to that of V s and its magnitude is smaller. For moderate value of V, i.e., ðU; V Þ ¼ ð3:21; 1:2Þ; in contrast, both V s and V t show a negative (i.e. attractive) peak at q ¼ Q due to the effect of charge ﬂuctuation which induces the CDW instability. The phase diagram obtained on the ðU; V Þ plane for n ¼ 0:8 is shown in Fig. 2. The temperature is ﬁxed at T ¼ 0:01: The dashed lines indicate the boundary of the SDW or CDW state, respectively. The d-wave superconductivity is realized on the right-hand side of the solid line. It can be seen that the effect of V suppresses the d-wave pairing. This is because the repulsive peak in V s at q ¼ Q; which exists for V ¼ 0 and induces d-wave pairing, is suppressed by the effect of V as shown in Fig. 1. Near the CDW boundary, the f-wave superconductivity becomes dominant. The order parameter belongs to B2u representation of the point group D6h : It has three peaks with the same sign which are connected by wave vectors ð0; 43 pÞ; ðp2ﬃﬃ3 p; 23 pÞ and ðp2ﬃﬃ3 p; 23 pÞ in the triangular lattice Brillouin zone, as shown in Fig. 3. This property is 2
Vs (U,V)=(3.21,1.2) Vt (U,V)=(3.21,1.2) Vs (U,V)=(3.64,0.0) Vt (U,V)=(3.64,0.0)
Fig. 1. Effective interactions for singlet (V s ) and triplet (V t ) pairings at n ¼ 0:8 and T ¼ 0:01: pﬃﬃﬃ G; K and M represent ðkx ; ky Þ ¼ ð0; 0Þ; ð0; 43 pÞ and ðp= 3; pÞ in the triangular lattice Brillouin zone, respectively.
Fig. 2. Phase diagram on the ðU; VÞ plane at T ¼ 0:01: The dashed lines correspond to the CDW and SDW instabilities. On the solid and dotted lines, eigenvalues of the E´liashberg’s equation which belong to d- and f-symmetries reach unity, respectively. The d-wave pairing is dominant near the SDW and the f-wave pairing is dominant near the CDW.
ARTICLE IN PRESS Y. Tanaka et al. / Physica B 359– 361 (2005) 591–593
Note that, in general, charge ﬂuctuation equally helps triplet and singlet pairings since it comes from the charge degrees of freedom. In fact, we found that for the electron doped case, i.e. n ¼ 1:2; both of the pairings compete with each other near the CDW instability . In summary, we have shown that f-wave triplet superconductivity is realized in the vicinity of CDW instability in the triangular lattice. References
Fig. 3. Momentum dependence of the gap function for the fwave triplet pairing obtained at n ¼ 0:8; ðU; V Þ ¼ ð3:21; 1:2Þ:
favorable because V t gives large attractive interactions at these wave vectors as can be seen from Fig. 1. In fact, for the triplet pairing, spin and charge ﬂuctuations work cooperatively since they give large attractive interactions at the same wave vector, q ¼ Q: Furthermore, the stability of the fwave triplet state can be understood in a real-space picture as follows: The effect of V repels electrons from nearest-neighbor sites. Thus, it is natural that the amplitude of the order parameter in real space becomes larger at six next-nearest-neighbor sites. After Fourier transformation, this real-space order parameter results in f-wave symmetry in k-space.
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