- Email: [email protected]
Unified origin for superconductivity and 3D magnetism in NaxCoO2
ARTICLE IN PRESS
Physica B 403 (2008) 1151–1153 www.elsevier.com/locate/physb
Unified origin for superconductivity and 3D magnetism in NaxCoO2 K. Kurokia,, S. Okuboa, T. Nojimaa, H. Usuia, R. Aritab, S. Onaric, Y. Tanakac a
Department of Applied Physics and Chemistry, The University of Electro-Communications, Chofu, Tokyo 182-8585, Japan b RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan c Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan
Abstract We argue that the spin fluctuations that give rise to superconductivity in Nax CoO2 yH2 O and the 3D magnetism in the non-hydrated Nax CoO2 share the same origin. Due to the local minimum of the a1g band at the G point, inner and outer portions of the Fermi surface arise, where the nesting between the two results in these interesting phenomena. r 2007 Elsevier B.V. All rights reserved. PACS: 74.20.Mn; 75.30.Fv Keywords: Nax CoO2 ; Superconductivity; Magnetism; Fermi surface
1. Introduction A cobaltate Nax CoO2 has attracted much attention in that it exhibits large thermopower [1] and 3D magnetism [2] in non-hydrated, sodium rich systems, and superconductivity (SC) in the bilayer hydrated sodium poor systems [3]. A number of recent angle resolved photoemission (ARPES) experiments show that only one of the t2g bands, i.e., the a1g band crosses the Fermi level [4–7]. Here, we study the single band Hubbard model that takes into account the local minimum structure (LMS) of the a1g band at the G point (see Fig. 1), and propose that it is this peculiar band structure that results in both SC and the 3D magnetism. 2. Superconductivity We propose an electronic mechanism for SC in this cobaltate [8]. Due to the LMS of the band found in first principles calculations [9,10], an inner Fermi surface (FS) appears in addition to the large outer FS for sufficiently high Fermi level, resulting in two ‘‘disconnected FS.’’ We consider an extended Hubbard model that has on-site ðUÞ Corresponding author. Tel.: +81 424 86 9036.
E-mail address: [email protected] (K. Kuroki). 0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2007.10.352
and nearest neighbor ðV Þ repulsions on a 2D triangular lattice, taking into account the LMS by considering hopping integrals up to third nearest neighbors (t1 2t3 , where t1 is taken as the unit of the energy). We apply the fluctuation exchange (FLEX) method [11] and solve the linearized Eliashberg equation to investigate the possibility of SC. Around the band filling of n ¼ 1:5, corresponding to the cobalt valence of þ3:5 [12], we find a possibility of superconductivity with an s-wave symmetry gap, whose sign changes between the inner and the outer FS (Fig. 1). Singlet pairing is consistent with some Knight shift experiments [13,14]. Here the spin fluctuations, which mediate repulsive singlet pairing interaction, develop at wave vectors ðQs Þ that bridge the inner and the outer FS due to the nesting between the two FS, resulting in the change of the gap sign. The development of spin fluctuations that are neither purely ferromagnetic nor antiferromagnetic is consistent with NQR experiments [15–17]. Moreover, the charge fluctuations develop near the K point, namely at a wave vector ðQc Þ with a length close to the size of the outer FS. Since the charge fluctuations mediate attractive pairing interaction, this favors the gap which does not change sign within the outer FS. A recent multiorbital analysis also gives this extended s-wave pairing as far as the inner FS exists [18].
ARTICLE IN PRESS K. Kuroki et al. / Physica B 403 (2008) 1151–1153
1152
4 ΔSC>0
2 0
Qc
−2
K
Qs Γ ΔSC<0
M
−4 Γ
K
Γ
M
Fig. 1. The band for t2 ¼ 0:35, t2 ¼ 0:25 (left), FS and the sign of the SC gap (center), and the FLEX result for the SC gap for U ¼ 6, V ¼ 2, n ¼ 1:5 (right, the dashed lines are the nodes) are shown. π c/2
π c/2
1.67
kz 0
−π c/2 (−π,0)
(0,0)
π c/2
1.73
remnants of outer FS inner FS
kz 0
kz 0
−π c/2 (−π,0) (π,0)
−π c/2 (−π,0) (π,0)
(0,0) (kx ,ky)
1.82
nesting vector ~ (0,0,π)
(0,0)
(π,0)
(kx ,ky) horizontal cross section through this cut
Fig. 2. Vertical cross-section of the FS for t1 ¼ 0:35, t2 ¼ 0:07, tz ¼ 0:15 with n ¼ 1:67; 1:73, and 1.82.
3. 3D magnetism As for the magnetism in non-hydrated systems, we take an itinerant spin picture [19] as opposed to some previous approaches [20–23], and consider a 3D single band Hubbard model that takes into account the nearest neighbor hopping in the z direction [24]. In Fig. 2, we show the vertical cross-section of the FS for various band fillings (Na content x ¼ n 1). Due to the LMS, an inner FS appears again for certain band fillings, but this time it is 3D. For large enough band fillings, the FS becomes closed in the z direction and has a nesting between portions of the FS that can be considered as remnants of inner and outer FS. Since the nesting vector is close to ð0; 0; pÞ, this results in an in-plane ferromagnetic, out-of-plane antiferromagnetic spin correlation, consistent with the experiments [20,21]. We use FLEX to evaluate the ð0; 0; pÞ spin density wave transition temperature T SDW , and find that T SDW has a maximum around n ¼ 1:8, corresponding to x ¼ 0:8 [19]. Recently, Sushko et al. studied the pressure effect on this magnetism, and found that T SDW increases
with pressure [25], opposite to what is usually observed in SDW systems. Here we represent the pressure effect by multiplying the distant hopping integrals t2 , t3 , tz (which can be easily affected by small structural change) by a same factor a for simplicity. We show the a dependence of T SDW in [Fig. 3(a)]. A few percent increase of the distant hoppings results in a 10% increase of T SDW , which is about the amount of the increase observed experimentally at 9 kbar. T c increases because the increase in t2 and t3 results in a deeper LMS and a better nesting. Thus from this viewpoint, it is interesting to investigate how the pressure actually affects the hopping integrals. Finally we discuss the spin wave dispersion in the SDW ordered state. The spin wave dispersion can be determined by the condition that the real part of the eigenvalue of the "# matrix 1 Uw"# 0 ðq; oÞ equals zero, where w0 ðq; oÞ is the irreducible susceptibility matrix calculated by using the bands in the SDW state obtained within the mean field approximation [19]. The calculated spin wave dispersion shown in Fig. 3(b) strikingly resembles the experimental
ARTICLE IN PRESS K. Kuroki et al. / Physica B 403 (2008) 1151–1153
1153
0.0220 0.3
0.3
0.2
0.2
0.1
0.1
0.0212 ω/t1
Tc /t1
0.0216
0.0208 0.0204 0.02 1
1.01
1.02
1.03
α
0.0 0.0 0.20 0.15 0.10 0.05 0.0 0.1 0.2 0.3 0.4 0.5 (h,h,0) (0,0,l)
Fig. 3. (a) T SDW as a function of a (representing the effect of pressure) for U ¼ 6. (b) The spin wave dispersion with U ¼ 4. t1 ¼ 0:35, t2 ¼ 0:07, tz ¼ 0:15, and n ¼ 1:82 in both figures.
result in Ref. [20], strongly supporting the present itinerant spin picture for this magnetism. Thus in our view, the spin fluctuations that give rise to SC and the 3D magnetism share the same origin: the nesting between the inner and the outer (connected or disconnected) FS that originate from the local minimum of the a1g band. Acknowledgments Numerical calculations were performed at the facilities of the Supercomputer Center, ISSP, University of Tokyo. This study has been supported by Grants-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan, and from the Japan Society for the Promotion of Science. References [1] [2] [3] [4] [5]
I. Terasaki, et al., Phys. Rev. B 56 (1997) R12685. T. Motohashi, et al., Phys. Rev. B 67 (2003) 064406. K. Takada, et al., Nature 422 (2003) 53. H.-B. Yang, et al., Phys. Rev. Lett. 95 (2005) 146401. M.Z. Hasan, et al., Phys. Rev. Lett. 92 (2004) 246402.
[6] T. Takeuchi, et al., in: Proceedings of the 24th International Conference on Thermoelectrics IEEE, 2005, p. 435. [7] T. Shimojima, et al., Phys. Rev. Lett. 97 (2006) 267003. [8] K. Kuroki, et al., Phys. Rev. B 73 (2006) 184503. [9] D.J. Singh, Phys. Rev. B 61 (2000) 13397. [10] R. Arita, Phys. Rev. B 71 (2005) 132503. [11] N.E. Bickers, et al., Phys. Rev. Lett. 62 (1989) 961. [12] H. Sakurai, et al., Phys. Rev. B 74 (2006) 092502. [13] Y. Kobayashi, et al., J. Phys. Soc. Japan 74 (2005) 1800. [14] G.-q. Zheng, et al., Phys. Rev. B 73 (2006) 180503. [15] F.L. Ning, et al., Phys. Rev. Lett. 93 (2004) 237201; F.L. Ning, T. Imai, Phys. Rev. Lett. 94 (2005) 227004. [16] I.R. Mukhamedshin, et al., Phys. Rev. Lett. 94 (2005) 247602. [17] Y. Ihara, et al., J. Phys. Soc. Japan 74 (2005) 2177. [18] M. Mochizuki, M. Ogata, J. Phys. Soc. Japan 75 (2006) 113703; M. Mochizuki, M. Ogata, J. Phys. Soc. Japan 76 (2007) 013704; M. Mochizuki, et al., J. Phys. Soc. Japan 76 (2007) 023702. [19] K. Kuroki, et al., Phys. Rev. Lett. 98 (2007) 136401. [20] S.P. Bayrakci, et al., Phys. Rev. Lett. 94 (2005) 157205. [21] A.T. Boothroyd, et al., Phys. Rev. Lett. 92 (2004) 197201; L.M. Helme, et al., Phys. Rev. Lett. 94 (2005) 157206. [22] M.D. Johannes, et al., Phys. Rev. B 71 (2005) 214410. [23] M. Daghofer, et al., Phys. Rev. Lett. 96 (2006) 216404. [24] A related itinerant spin picture is also taken for a 2D model in Korshunov et al., JETP Lett. 84 (2007) 650; Korshunov, et al., Phys. Rev. B 75 (2007) 094511. [25] Y.V. Sushko, et al., J. Low Temp. Phys. 142 (2006) 573.
Physica B 403 (2008) 1151–1153 www.elsevier.com/locate/physb
Unified origin for superconductivity and 3D magnetism in NaxCoO2 K. Kurokia,, S. Okuboa, T. Nojimaa, H. Usuia, R. Aritab, S. Onaric, Y. Tanakac a
Department of Applied Physics and Chemistry, The University of Electro-Communications, Chofu, Tokyo 182-8585, Japan b RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan c Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan
Abstract We argue that the spin fluctuations that give rise to superconductivity in Nax CoO2 yH2 O and the 3D magnetism in the non-hydrated Nax CoO2 share the same origin. Due to the local minimum of the a1g band at the G point, inner and outer portions of the Fermi surface arise, where the nesting between the two results in these interesting phenomena. r 2007 Elsevier B.V. All rights reserved. PACS: 74.20.Mn; 75.30.Fv Keywords: Nax CoO2 ; Superconductivity; Magnetism; Fermi surface
1. Introduction A cobaltate Nax CoO2 has attracted much attention in that it exhibits large thermopower [1] and 3D magnetism [2] in non-hydrated, sodium rich systems, and superconductivity (SC) in the bilayer hydrated sodium poor systems [3]. A number of recent angle resolved photoemission (ARPES) experiments show that only one of the t2g bands, i.e., the a1g band crosses the Fermi level [4–7]. Here, we study the single band Hubbard model that takes into account the local minimum structure (LMS) of the a1g band at the G point (see Fig. 1), and propose that it is this peculiar band structure that results in both SC and the 3D magnetism. 2. Superconductivity We propose an electronic mechanism for SC in this cobaltate [8]. Due to the LMS of the band found in first principles calculations [9,10], an inner Fermi surface (FS) appears in addition to the large outer FS for sufficiently high Fermi level, resulting in two ‘‘disconnected FS.’’ We consider an extended Hubbard model that has on-site ðUÞ Corresponding author. Tel.: +81 424 86 9036.
E-mail address: [email protected] (K. Kuroki). 0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2007.10.352
and nearest neighbor ðV Þ repulsions on a 2D triangular lattice, taking into account the LMS by considering hopping integrals up to third nearest neighbors (t1 2t3 , where t1 is taken as the unit of the energy). We apply the fluctuation exchange (FLEX) method [11] and solve the linearized Eliashberg equation to investigate the possibility of SC. Around the band filling of n ¼ 1:5, corresponding to the cobalt valence of þ3:5 [12], we find a possibility of superconductivity with an s-wave symmetry gap, whose sign changes between the inner and the outer FS (Fig. 1). Singlet pairing is consistent with some Knight shift experiments [13,14]. Here the spin fluctuations, which mediate repulsive singlet pairing interaction, develop at wave vectors ðQs Þ that bridge the inner and the outer FS due to the nesting between the two FS, resulting in the change of the gap sign. The development of spin fluctuations that are neither purely ferromagnetic nor antiferromagnetic is consistent with NQR experiments [15–17]. Moreover, the charge fluctuations develop near the K point, namely at a wave vector ðQc Þ with a length close to the size of the outer FS. Since the charge fluctuations mediate attractive pairing interaction, this favors the gap which does not change sign within the outer FS. A recent multiorbital analysis also gives this extended s-wave pairing as far as the inner FS exists [18].
ARTICLE IN PRESS K. Kuroki et al. / Physica B 403 (2008) 1151–1153
1152
4 ΔSC>0
2 0
Qc
−2
K
Qs Γ ΔSC<0
M
−4 Γ
K
Γ
M
Fig. 1. The band for t2 ¼ 0:35, t2 ¼ 0:25 (left), FS and the sign of the SC gap (center), and the FLEX result for the SC gap for U ¼ 6, V ¼ 2, n ¼ 1:5 (right, the dashed lines are the nodes) are shown. π c/2
π c/2
1.67
kz 0
−π c/2 (−π,0)
(0,0)
π c/2
1.73
remnants of outer FS inner FS
kz 0
kz 0
−π c/2 (−π,0) (π,0)
−π c/2 (−π,0) (π,0)
(0,0) (kx ,ky)
1.82
nesting vector ~ (0,0,π)
(0,0)
(π,0)
(kx ,ky) horizontal cross section through this cut
Fig. 2. Vertical cross-section of the FS for t1 ¼ 0:35, t2 ¼ 0:07, tz ¼ 0:15 with n ¼ 1:67; 1:73, and 1.82.
3. 3D magnetism As for the magnetism in non-hydrated systems, we take an itinerant spin picture [19] as opposed to some previous approaches [20–23], and consider a 3D single band Hubbard model that takes into account the nearest neighbor hopping in the z direction [24]. In Fig. 2, we show the vertical cross-section of the FS for various band fillings (Na content x ¼ n 1). Due to the LMS, an inner FS appears again for certain band fillings, but this time it is 3D. For large enough band fillings, the FS becomes closed in the z direction and has a nesting between portions of the FS that can be considered as remnants of inner and outer FS. Since the nesting vector is close to ð0; 0; pÞ, this results in an in-plane ferromagnetic, out-of-plane antiferromagnetic spin correlation, consistent with the experiments [20,21]. We use FLEX to evaluate the ð0; 0; pÞ spin density wave transition temperature T SDW , and find that T SDW has a maximum around n ¼ 1:8, corresponding to x ¼ 0:8 [19]. Recently, Sushko et al. studied the pressure effect on this magnetism, and found that T SDW increases
with pressure [25], opposite to what is usually observed in SDW systems. Here we represent the pressure effect by multiplying the distant hopping integrals t2 , t3 , tz (which can be easily affected by small structural change) by a same factor a for simplicity. We show the a dependence of T SDW in [Fig. 3(a)]. A few percent increase of the distant hoppings results in a 10% increase of T SDW , which is about the amount of the increase observed experimentally at 9 kbar. T c increases because the increase in t2 and t3 results in a deeper LMS and a better nesting. Thus from this viewpoint, it is interesting to investigate how the pressure actually affects the hopping integrals. Finally we discuss the spin wave dispersion in the SDW ordered state. The spin wave dispersion can be determined by the condition that the real part of the eigenvalue of the "# matrix 1 Uw"# 0 ðq; oÞ equals zero, where w0 ðq; oÞ is the irreducible susceptibility matrix calculated by using the bands in the SDW state obtained within the mean field approximation [19]. The calculated spin wave dispersion shown in Fig. 3(b) strikingly resembles the experimental
ARTICLE IN PRESS K. Kuroki et al. / Physica B 403 (2008) 1151–1153
1153
0.0220 0.3
0.3
0.2
0.2
0.1
0.1
0.0212 ω/t1
Tc /t1
0.0216
0.0208 0.0204 0.02 1
1.01
1.02
1.03
α
0.0 0.0 0.20 0.15 0.10 0.05 0.0 0.1 0.2 0.3 0.4 0.5 (h,h,0) (0,0,l)
Fig. 3. (a) T SDW as a function of a (representing the effect of pressure) for U ¼ 6. (b) The spin wave dispersion with U ¼ 4. t1 ¼ 0:35, t2 ¼ 0:07, tz ¼ 0:15, and n ¼ 1:82 in both figures.
result in Ref. [20], strongly supporting the present itinerant spin picture for this magnetism. Thus in our view, the spin fluctuations that give rise to SC and the 3D magnetism share the same origin: the nesting between the inner and the outer (connected or disconnected) FS that originate from the local minimum of the a1g band. Acknowledgments Numerical calculations were performed at the facilities of the Supercomputer Center, ISSP, University of Tokyo. This study has been supported by Grants-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan, and from the Japan Society for the Promotion of Science. References [1] [2] [3] [4] [5]
I. Terasaki, et al., Phys. Rev. B 56 (1997) R12685. T. Motohashi, et al., Phys. Rev. B 67 (2003) 064406. K. Takada, et al., Nature 422 (2003) 53. H.-B. Yang, et al., Phys. Rev. Lett. 95 (2005) 146401. M.Z. Hasan, et al., Phys. Rev. Lett. 92 (2004) 246402.
[6] T. Takeuchi, et al., in: Proceedings of the 24th International Conference on Thermoelectrics IEEE, 2005, p. 435. [7] T. Shimojima, et al., Phys. Rev. Lett. 97 (2006) 267003. [8] K. Kuroki, et al., Phys. Rev. B 73 (2006) 184503. [9] D.J. Singh, Phys. Rev. B 61 (2000) 13397. [10] R. Arita, Phys. Rev. B 71 (2005) 132503. [11] N.E. Bickers, et al., Phys. Rev. Lett. 62 (1989) 961. [12] H. Sakurai, et al., Phys. Rev. B 74 (2006) 092502. [13] Y. Kobayashi, et al., J. Phys. Soc. Japan 74 (2005) 1800. [14] G.-q. Zheng, et al., Phys. Rev. B 73 (2006) 180503. [15] F.L. Ning, et al., Phys. Rev. Lett. 93 (2004) 237201; F.L. Ning, T. Imai, Phys. Rev. Lett. 94 (2005) 227004. [16] I.R. Mukhamedshin, et al., Phys. Rev. Lett. 94 (2005) 247602. [17] Y. Ihara, et al., J. Phys. Soc. Japan 74 (2005) 2177. [18] M. Mochizuki, M. Ogata, J. Phys. Soc. Japan 75 (2006) 113703; M. Mochizuki, M. Ogata, J. Phys. Soc. Japan 76 (2007) 013704; M. Mochizuki, et al., J. Phys. Soc. Japan 76 (2007) 023702. [19] K. Kuroki, et al., Phys. Rev. Lett. 98 (2007) 136401. [20] S.P. Bayrakci, et al., Phys. Rev. Lett. 94 (2005) 157205. [21] A.T. Boothroyd, et al., Phys. Rev. Lett. 92 (2004) 197201; L.M. Helme, et al., Phys. Rev. Lett. 94 (2005) 157206. [22] M.D. Johannes, et al., Phys. Rev. B 71 (2005) 214410. [23] M. Daghofer, et al., Phys. Rev. Lett. 96 (2006) 216404. [24] A related itinerant spin picture is also taken for a 2D model in Korshunov et al., JETP Lett. 84 (2007) 650; Korshunov, et al., Phys. Rev. B 75 (2007) 094511. [25] Y.V. Sushko, et al., J. Low Temp. Phys. 142 (2006) 573.