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Effective five band analysis on the pressure effect of FeSe
Physica C 470 (2010) S382–S384
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Effective five band analysis on the pressure effect of FeSe Hidetomo Usui a,*, Kazuhiko Kuroki a,b a b
Department of Applied Physics and Chemistry, The University of Electro-Communications, Chofu, Tokyo 182-8585, Japan JST, TRIP, Sanbancho, Chiyoda, Tokyo 102-0075, Japan
a r t i c l e
i n f o
Article history: Accepted 4 November 2009 Available online 10 November 2009 Keywords: FeSe Effective Hamiltonian Pressure Superconductivity
a b s t r a c t In the present study, we investigate the pressure dependence of the effective five band model of FeSe, where we show that the various hopping integrals are mainly anticorrelated with the Se height in the low pressure regime, while in the high pressure regime, where the reduction of the height is saturated, the hoppings are mainly correlated with the lattice constants. We show that a simple application of random phase approximation (RPA) to the model does not explain the pressure dependence of the superconducting T c . This suggests that some effects that are not taken into account properly in the RPA, such as strong correlation effects, may be stronger in 11 than in 1111 materials. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction
2. Effective five band model and the pressure dependence
In the superconductivity of iron based superconductors [1], the effect of the lattice structure has been an issue of great interest. T c ranges from 5 K in LaFePO [2] to 55 K [3] in SmFeAsO, and the importance of the Pn–Fe–Pn (Pn = pnictogen) bond angle has been pointed out [4]. In this context, the present authors have theoretically pointed out the importance of the pnictogen height measured from the iron planes in LnFePnO 1111 materials [5]. The discovery of superconductivity in the so called 11 materials, such as FeSe [6], and the succeeding studies [7–9] have provided a new avenue for the investigation of this series of materials. In particular, the extremely large enhancement by applying pressure, up to above 35 K [10–12] has attracted much attention. In the present study, we study the pressure dependence of the effective five band model of FeSe, where we show that the magnitude of the various hopping integrals are mainly anticorrelated with the Se height in the low pressure regime, while in the high pressure regime, where the reduction of the height is saturated, the hoppings are mainly correlated with the lattice constants. We show that a simple application of random phase approximation (RPA) to the model does not explain the pressure dependence of T c . This suggests that some effects that are not properly taken into account in the RPA, such as strong correlation effects, may be stronger in 11 than in 1111 materials.
In Fig. 1, we show the band structure of the five band model of FeSe at ambient pressure, obtained from the first principles band calculation [13] using maximally localized Wannier orbitals [14] as in Ref. [15]. ð0; pÞ=ðp; 0Þ nesting is seen in the Fermi surface as in the 1111 materials, which brings about the spin fluctuations around the wave vector ð0; pÞ=ðp; 0Þ and the sign reversing s-wave pairing [5,15,16]. We also obtain similarly the effective model for the case under pressure, where we use the experimentally determined lattice parameters [17]. In Fig. 2, we plot the nearest ðt 1 Þ and next nearest ðt2 Þ neighbor hopping integrals of the dX 2 Y 2 and dXZ=YZ orbitals as functions of pressure along with the Se height. (X and Y axes are those of the original unit cell, and are rotated by 45° from kx and ky axes.) We find that the hopping integrals are mainly anticorrelated with the Se height in the low to intermediate pressure regime. This is consistent with the general trend found in Ref. [5]. In the high pressure regime, where the reduction of the height saturates, the nearest neighbor dX 2 Y 2 decreases while other hoppings continue to increase gradually. The tendency at high pressures is mainly due to the shrinking of the lattice constants which continues monotonically up to high pressures. This increases most of the hoppings, but unexpectedly, the reduction of a is found to reduce the dX 2 Y 2 nearest neighbor hopping [5]. 3. Correlation between pressure and superconductivity
* Corresponding author. E-mail address: [email protected] (H. Usui). 0921-4534/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2009.11.029
We consider a many body Hamiltonian that takes into account the multiorbital electron interactions, U ¼ 1:2; U 0 ¼ 0:9; J ¼ J 0 ¼ 0:3 (eV) in standard notations, and the band filling is fixed at n ¼ 6:1. Applying random phase approximation (RPA) to our mod-
H. Usui, K. Kuroki / Physica C 470 (2010) S382–S384
10
π
E (eV)
9 8
ky
7
0
6 5 4 ( 0 ,0 ,0 )
−π −π (π,0,0)
(π,π,0)(π,π,π)
( 0 ,0 ,0 ) ( 0 ,0 ,π )
0
π
kx
Fig. 1. Left: the band structure of FeSe at ambient pressure in the unfolded Brillouin zone. Right: the Fermi surface for the band filling of n ¼ 6:1.
(b)
(a) 0.07
2
N.N. dX -Y
0.06 0
0.22
0.21
2
0.2
1 2 3 4 5 6 7 8 9
(d) 0.38
0.108
0.37 2
N.N.N. dX -Y
0.36
2
N.N.N. dXZ
0.35
0.1 0
(e)
0 1 2 3 4 5 6 7 8 9
Pressure (GPa)
Pressure (GPa)
(c) 0.112 0.104
N.N. dXZ
1 0.8 0.6
Pressure (GPa) spin SC
0.4
0 1 2 3 4 5 6 7 8 9
Pressure (GPa)
(f)
1.45
1 2 3 4 5 6 7 8 9
Pressure (GPa)
similar to what may be happening in the undoped 1111 [19] and 122 [20] materials, where applying pressure suppresses magnetism and superconductivity appears. There is, however, a large difference in FeSe, i.e., long range spin ordering is not observed in this material. Nevertheless, it is possible that the carrier doping caused by Se deficiencies prevents long range ordering of the spins, but still there is strong short range spin ordering that degrades superconductivity. In this ‘‘hidden magnetism” picture, the consistency with the experimental observations that spin fluctuations develop upon applying pressure [12,18] remain as an open problem. It may be possible that the spin fluctuations that are enhanced upon applying pressure and short range magnetism that degrades superconductivity are of different nature. The second possibility is that FeSe is a strongly correlated material, where weak coupling approaches (approaches that takes U=t as perturbation) such as RPA do not work. This may be possible since the nearest neighbor hopping integral of the dX 2 Y 2 orbital in particular is extremely small in FeSe (0.06–0.07 eV), which is less than half of that in LaFeAsO (0.16 eV). In fact, this is mainly due to the large height of Se, i.e., 1.45 Å compared to 1.32 Å in LaFeAsO and 1.38 Å in NdFeAsO. If the material is indeed in the strongly correlated regime, the magnetic interactions between neighboring Fe sites may roughly be given in the form J t 2 =U, where t is the (nearest or next nearest neighbor) hopping integral and U is the on-site interaction. If this J is the origin of the pairing interaction, superconductivity is likely to be positively correlated with t 2 , and the pressure dependence of the hopping integrals may be consistent with the experimentally observed T c variance. In either case, the explanation of the experimental observation seems to require approaches beyond RPA at the present stage. 4. Conclusion
1.44 1.43
0.2 0 0
0.34
1 2 3 4 5 6 7 8 9
S383
1.42
0 1 2 3 4 5 6 7 8 9
Pressure (GPa)
Fig. 2. The pressure dependence of the hopping integrals: (a) nearest neighbor dX 2 Y 2 , (b) dXZ , (c) next nearest neighbor dX 2 Y 2 , (d) dXZ , (e) eigenvalues of the matrix Sv0 (spin) and the Eliashberg equation (SC) as functions of pressure, (f) the Se height as a function of pressure, from Ref. [10]
el, we obtain the spin susceptibility, which is plugged into the linearized Eliashberg equation [5,15]. In Fig. 2e, we plot the eigenvalues of the matrix Sv0 and the Eliashberg equation as functions of pressure, where S is the interaction vertex matrix in the spin channel and v0 is the irreducible susceptibility matrix [5]. The spin ordering (superconductivity) occurs at the temperature where the eigenvalue of the matrix Sv0 (Eliashberg equation) reaches unity. As seen from the figure, both the spin correlation and the superconductivity anticorrelates with the pressure, especially in the low pressure regime. This is mainly because the density of states becomes smaller as the hopping integrals, and thus the band width, becomes larger. Experimentally, T c first increases upon applying pressure [10– 12,18], and decreases after taking a maximum around 6 GPa [10]. Thus, the overall anticorrelation between the pressure and the superconductivity obtained in the RPA calculation is in fact opposite to what is observed experimentally in the low to intermediate pressure regime. Here we raise two possibilities for the origin of this discrepancy. One possibility is that magnetism and superconductivity coexists in the actual material, where the former degrades the latter (this effect is not taken into account in RPA), and the pressure mainly acts to suppress the magnetism. This is
In the present study, we have studied the pressure dependence of the effective five band model of FeSe, where we show that the magnitude of the various hopping integrals are mainly anticorrelated with the Se height in the low pressure regime, while in the high pressure regime, the hoppings are mainly correlated with the lattice constants. We show that a simple application of RPA does not explain the pressure dependence of T c , i.e., superconductivity is degraded with the reduction of the height. This suggests that some effects that are not properly taken into account in the RPA, such as strong correlation effects, may be stronger in 11 than in 1111 materials due to the extremely high position of Se (or Te). If the 11 materials are indeed in the strong correlation regime, the optimum height of the pnictogen/chalcogen for the superconductivity in the iron based superconductors may be in the crossover regime from weak to strong correlation. Acknowledgments We would like to thank Kosmas Prassides and Yoshihiko Takano for valuable discussions and providing us the experimentally determined lattice parameters under pressure. We also thank Katsuhiro Suzuki for valuable discussions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Y. Kamihara et al., J. Am. Chem. Soc. 130 (2008) 3296. Y. Kamihara et al., J. Am. Chem. Soc. 128 (2006) 10012. Z.-A. Ren et al., Chin. Phys. Lett. 25 (2008) 2215. C.-H. Lee et al., J. Phys. Soc. Jpn. 77 (2008) 083704. K. Kuroki et al., Phys. Rev. B 79 (2009) 224511. F.-C. Hsu et al., Proc. Natl. Acad. Sci. 105 (2008) 14262. S. Margadonna et al., Chem. Commun. (2008) 5607. Y. Mizuguchi et al., Appl. Phys. Lett. 93 (2008) 152505. Y. Mizuguchi et al., in: Proceedings of ISS2008. arXiv:0810.5191. S. Margadonna et al. arXiv:0903.2204.
S384 [11] [12] [13] [14]
H. Usui, K. Kuroki / Physica C 470 (2010) S382–S384
S. Medvedev et al., arXiv:0903.2143. S. Masaki et al., J. Phys. Soc. Jpn. 78 (2009) 063704. S. Baroni et al.,
[16] I.I. Mazin et al., Phys. Rev. Lett. 101 (2008) 057003. [17] Kosmas Prassides, private communications. To be precise, the lattice structure is in the orthorhombic phase, but for simplicity we neglect the small difference the lattice constant in a and b directions, and adopt the tetragonal structure taking the average of a and b as the lattice constant. [18] T. Imai et al., Phys. Rev. Lett. 102 (2009) 177005. [19] H. Okada et al., J. Phys. Soc. Jpn. 77 (2009) 023701. [20] M.S. Torikachvili et al., Phys. Rev. Lett. 101 (2008) 057006.
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Effective five band analysis on the pressure effect of FeSe Hidetomo Usui a,*, Kazuhiko Kuroki a,b a b
Department of Applied Physics and Chemistry, The University of Electro-Communications, Chofu, Tokyo 182-8585, Japan JST, TRIP, Sanbancho, Chiyoda, Tokyo 102-0075, Japan
a r t i c l e
i n f o
Article history: Accepted 4 November 2009 Available online 10 November 2009 Keywords: FeSe Effective Hamiltonian Pressure Superconductivity
a b s t r a c t In the present study, we investigate the pressure dependence of the effective five band model of FeSe, where we show that the various hopping integrals are mainly anticorrelated with the Se height in the low pressure regime, while in the high pressure regime, where the reduction of the height is saturated, the hoppings are mainly correlated with the lattice constants. We show that a simple application of random phase approximation (RPA) to the model does not explain the pressure dependence of the superconducting T c . This suggests that some effects that are not taken into account properly in the RPA, such as strong correlation effects, may be stronger in 11 than in 1111 materials. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction
2. Effective five band model and the pressure dependence
In the superconductivity of iron based superconductors [1], the effect of the lattice structure has been an issue of great interest. T c ranges from 5 K in LaFePO [2] to 55 K [3] in SmFeAsO, and the importance of the Pn–Fe–Pn (Pn = pnictogen) bond angle has been pointed out [4]. In this context, the present authors have theoretically pointed out the importance of the pnictogen height measured from the iron planes in LnFePnO 1111 materials [5]. The discovery of superconductivity in the so called 11 materials, such as FeSe [6], and the succeeding studies [7–9] have provided a new avenue for the investigation of this series of materials. In particular, the extremely large enhancement by applying pressure, up to above 35 K [10–12] has attracted much attention. In the present study, we study the pressure dependence of the effective five band model of FeSe, where we show that the magnitude of the various hopping integrals are mainly anticorrelated with the Se height in the low pressure regime, while in the high pressure regime, where the reduction of the height is saturated, the hoppings are mainly correlated with the lattice constants. We show that a simple application of random phase approximation (RPA) to the model does not explain the pressure dependence of T c . This suggests that some effects that are not properly taken into account in the RPA, such as strong correlation effects, may be stronger in 11 than in 1111 materials.
In Fig. 1, we show the band structure of the five band model of FeSe at ambient pressure, obtained from the first principles band calculation [13] using maximally localized Wannier orbitals [14] as in Ref. [15]. ð0; pÞ=ðp; 0Þ nesting is seen in the Fermi surface as in the 1111 materials, which brings about the spin fluctuations around the wave vector ð0; pÞ=ðp; 0Þ and the sign reversing s-wave pairing [5,15,16]. We also obtain similarly the effective model for the case under pressure, where we use the experimentally determined lattice parameters [17]. In Fig. 2, we plot the nearest ðt 1 Þ and next nearest ðt2 Þ neighbor hopping integrals of the dX 2 Y 2 and dXZ=YZ orbitals as functions of pressure along with the Se height. (X and Y axes are those of the original unit cell, and are rotated by 45° from kx and ky axes.) We find that the hopping integrals are mainly anticorrelated with the Se height in the low to intermediate pressure regime. This is consistent with the general trend found in Ref. [5]. In the high pressure regime, where the reduction of the height saturates, the nearest neighbor dX 2 Y 2 decreases while other hoppings continue to increase gradually. The tendency at high pressures is mainly due to the shrinking of the lattice constants which continues monotonically up to high pressures. This increases most of the hoppings, but unexpectedly, the reduction of a is found to reduce the dX 2 Y 2 nearest neighbor hopping [5]. 3. Correlation between pressure and superconductivity
* Corresponding author. E-mail address: [email protected] (H. Usui). 0921-4534/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2009.11.029
We consider a many body Hamiltonian that takes into account the multiorbital electron interactions, U ¼ 1:2; U 0 ¼ 0:9; J ¼ J 0 ¼ 0:3 (eV) in standard notations, and the band filling is fixed at n ¼ 6:1. Applying random phase approximation (RPA) to our mod-
H. Usui, K. Kuroki / Physica C 470 (2010) S382–S384
10
π
E (eV)
9 8
ky
7
0
6 5 4 ( 0 ,0 ,0 )
−π −π (π,0,0)
(π,π,0)(π,π,π)
( 0 ,0 ,0 ) ( 0 ,0 ,π )
0
π
kx
Fig. 1. Left: the band structure of FeSe at ambient pressure in the unfolded Brillouin zone. Right: the Fermi surface for the band filling of n ¼ 6:1.
(b)
(a) 0.07
2
N.N. dX -Y
0.06 0
0.22
0.21
2
0.2
1 2 3 4 5 6 7 8 9
(d) 0.38
0.108
0.37 2
N.N.N. dX -Y
0.36
2
N.N.N. dXZ
0.35
0.1 0
(e)
0 1 2 3 4 5 6 7 8 9
Pressure (GPa)
Pressure (GPa)
(c) 0.112 0.104
N.N. dXZ
1 0.8 0.6
Pressure (GPa) spin SC
0.4
0 1 2 3 4 5 6 7 8 9
Pressure (GPa)
(f)
1.45
1 2 3 4 5 6 7 8 9
Pressure (GPa)
similar to what may be happening in the undoped 1111 [19] and 122 [20] materials, where applying pressure suppresses magnetism and superconductivity appears. There is, however, a large difference in FeSe, i.e., long range spin ordering is not observed in this material. Nevertheless, it is possible that the carrier doping caused by Se deficiencies prevents long range ordering of the spins, but still there is strong short range spin ordering that degrades superconductivity. In this ‘‘hidden magnetism” picture, the consistency with the experimental observations that spin fluctuations develop upon applying pressure [12,18] remain as an open problem. It may be possible that the spin fluctuations that are enhanced upon applying pressure and short range magnetism that degrades superconductivity are of different nature. The second possibility is that FeSe is a strongly correlated material, where weak coupling approaches (approaches that takes U=t as perturbation) such as RPA do not work. This may be possible since the nearest neighbor hopping integral of the dX 2 Y 2 orbital in particular is extremely small in FeSe (0.06–0.07 eV), which is less than half of that in LaFeAsO (0.16 eV). In fact, this is mainly due to the large height of Se, i.e., 1.45 Å compared to 1.32 Å in LaFeAsO and 1.38 Å in NdFeAsO. If the material is indeed in the strongly correlated regime, the magnetic interactions between neighboring Fe sites may roughly be given in the form J t 2 =U, where t is the (nearest or next nearest neighbor) hopping integral and U is the on-site interaction. If this J is the origin of the pairing interaction, superconductivity is likely to be positively correlated with t 2 , and the pressure dependence of the hopping integrals may be consistent with the experimentally observed T c variance. In either case, the explanation of the experimental observation seems to require approaches beyond RPA at the present stage. 4. Conclusion
1.44 1.43
0.2 0 0
0.34
1 2 3 4 5 6 7 8 9
S383
1.42
0 1 2 3 4 5 6 7 8 9
Pressure (GPa)
Fig. 2. The pressure dependence of the hopping integrals: (a) nearest neighbor dX 2 Y 2 , (b) dXZ , (c) next nearest neighbor dX 2 Y 2 , (d) dXZ , (e) eigenvalues of the matrix Sv0 (spin) and the Eliashberg equation (SC) as functions of pressure, (f) the Se height as a function of pressure, from Ref. [10]
el, we obtain the spin susceptibility, which is plugged into the linearized Eliashberg equation [5,15]. In Fig. 2e, we plot the eigenvalues of the matrix Sv0 and the Eliashberg equation as functions of pressure, where S is the interaction vertex matrix in the spin channel and v0 is the irreducible susceptibility matrix [5]. The spin ordering (superconductivity) occurs at the temperature where the eigenvalue of the matrix Sv0 (Eliashberg equation) reaches unity. As seen from the figure, both the spin correlation and the superconductivity anticorrelates with the pressure, especially in the low pressure regime. This is mainly because the density of states becomes smaller as the hopping integrals, and thus the band width, becomes larger. Experimentally, T c first increases upon applying pressure [10– 12,18], and decreases after taking a maximum around 6 GPa [10]. Thus, the overall anticorrelation between the pressure and the superconductivity obtained in the RPA calculation is in fact opposite to what is observed experimentally in the low to intermediate pressure regime. Here we raise two possibilities for the origin of this discrepancy. One possibility is that magnetism and superconductivity coexists in the actual material, where the former degrades the latter (this effect is not taken into account in RPA), and the pressure mainly acts to suppress the magnetism. This is
In the present study, we have studied the pressure dependence of the effective five band model of FeSe, where we show that the magnitude of the various hopping integrals are mainly anticorrelated with the Se height in the low pressure regime, while in the high pressure regime, the hoppings are mainly correlated with the lattice constants. We show that a simple application of RPA does not explain the pressure dependence of T c , i.e., superconductivity is degraded with the reduction of the height. This suggests that some effects that are not properly taken into account in the RPA, such as strong correlation effects, may be stronger in 11 than in 1111 materials due to the extremely high position of Se (or Te). If the 11 materials are indeed in the strong correlation regime, the optimum height of the pnictogen/chalcogen for the superconductivity in the iron based superconductors may be in the crossover regime from weak to strong correlation. Acknowledgments We would like to thank Kosmas Prassides and Yoshihiko Takano for valuable discussions and providing us the experimentally determined lattice parameters under pressure. We also thank Katsuhiro Suzuki for valuable discussions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Y. Kamihara et al., J. Am. Chem. Soc. 130 (2008) 3296. Y. Kamihara et al., J. Am. Chem. Soc. 128 (2006) 10012. Z.-A. Ren et al., Chin. Phys. Lett. 25 (2008) 2215. C.-H. Lee et al., J. Phys. Soc. Jpn. 77 (2008) 083704. K. Kuroki et al., Phys. Rev. B 79 (2009) 224511. F.-C. Hsu et al., Proc. Natl. Acad. Sci. 105 (2008) 14262. S. Margadonna et al., Chem. Commun. (2008) 5607. Y. Mizuguchi et al., Appl. Phys. Lett. 93 (2008) 152505. Y. Mizuguchi et al., in: Proceedings of ISS2008. arXiv:0810.5191. S. Margadonna et al. arXiv:0903.2204.
S384 [11] [12] [13] [14]
H. Usui, K. Kuroki / Physica C 470 (2010) S382–S384
S. Medvedev et al., arXiv:0903.2143. S. Masaki et al., J. Phys. Soc. Jpn. 78 (2009) 063704. S. Baroni et al.,
[16] I.I. Mazin et al., Phys. Rev. Lett. 101 (2008) 057003. [17] Kosmas Prassides, private communications. To be precise, the lattice structure is in the orthorhombic phase, but for simplicity we neglect the small difference the lattice constant in a and b directions, and adopt the tetragonal structure taking the average of a and b as the lattice constant. [18] T. Imai et al., Phys. Rev. Lett. 102 (2009) 177005. [19] H. Okada et al., J. Phys. Soc. Jpn. 77 (2009) 023701. [20] M.S. Torikachvili et al., Phys. Rev. Lett. 101 (2008) 057006.