Emergence of Fully-Gapped s_{++}-wave and Nodal d-wave States Mediated by Orbital- and Spin-Fluctuations in Ten-Orbital Model for KFe{}_{2}Se{}_{2}

Emergence of Fully-Gapped -wave and Nodal -wave States Mediated by
Orbital- and Spin-Fluctuations in Ten-Orbital Model for KFeSe

Tetsuro Saito, Seiichiro Onari, and Hiroshi Kontani Department of Physics, Nagoya University and JST, TRIP, Furo-cho, Nagoya 464-8602, Japan.
Department of Applied Physics, Nagoya University and JST, TRIP, Furo-cho, Nagoya 464-8602, Japan.
August 5, 2019
Abstract

We study the superconducting state in recently discovered high- superconductor KFeSe based on the ten-orbital Hubbard-Holstein model without hole-pockets. When the Coulomb interaction is large, spin-fluctuation mediated -wave state appears due to the nesting between electron-pockets. Interestingly, the symmetry of the body-centered tetragonal structure in KFeSe requires the existence of nodes in the -wave gap, although fully-gapped -wave state is realized in the case of simple tetragonal structure. In the presence of moderate electron-phonon interaction due to Fe-ion optical modes, on the other hand, orbital fluctuations give rise to the fully-gapped -wave state without sign reversal. Therefore, both superconducting states are distinguishable by careful measurements of the gap structure or the impurity effect on .

pacs:
74.20.-z, 74.20.Fg, 74.20.Rp

The pairing mechanism of high- iron-based superconductors has been significant open problem. The main characters of FeAs compounds would be (i) the nesting between electron-pockets (e-pockets) and hole-pockets (h-pockets), and (ii) the existence of orbital degree of freedom. By focusing on the intra-orbital nesting, fully-gapped sign-reversing -wave state (-wave state) had been predicted based on the spin fluctuation theories Mazin (); Kuroki (). On the other hand, existence of moderate electron-phonon (-ph) interactions due to Fe-ion optical phonons and the inter-orbital nesting can produce large orbital fluctuations Kontani (). Then, orbital-fluctuation-mediated -wave state without sign reversal (-wave state) had been predicted by using the random-phase-approximation (RPA) Kontani (); Saito () or the fluctuation-exchange (FLEX) approximation Onari (). According to the analysis in Refs. Onari-impurity, ; Onari-resonance, the -wave state is consistent with the robustness of against randomness Sato-imp (); Nakajima () as well as the “resonance-like” hump structure in the neutron inelastic scattering neutron (). Non-Fermi liquid transport phenomena in Matsuda () can be explained by the development of orbital fluctuations Onari ().

Recently, iron-selenium 122-structure compound AFeSe (A= alkaline metals) with K was discovered first (). This heavily electron-doped superconductor has been attracting great attention since both the band calculations LDA1 (); LDA2 () and angle-resolved-photoemission-spectrum (ARPES) measurements ARPES1 (); ARPES2 (); ARPES3 () indicate the absence of h-pockets. NMR measurements reports the weakness of spin fluctuations NMR (), and both ARPES ARPES1 (); ARPES2 (); ARPES3 () and specific heat measurements HHWen () indicate the isotropic SC gap. Thus, study of AFeSe will give us important information to reveal the pairing mechanism of iron pnictides.

The unit-cell of iron-based superconductors contains two Fe atoms. However, except for 122-systems, one can construct a simple “single-Fe model” from the original “two-Fe model” by applying the gauge transformation on -orbitals Miyake (). By this procedure, the original Brillouin zone (BZ) is enlarged to the “unfolded BZ”. Based on the single-Fe model, spin-fluctuation-mediated -wave state ( representation) “without nodes” had been proposed DHLee (); Graser (); Bala (), by focusing on the nesting between e-pockets. However, we cannot construct a “single-Fe model” for 122 systems since finite hybridization between e-pockets prevents the unfolding procedure Miyake (). Therefore, theoretical study based on the original two-Fe model is highly desired to conclude the gap structure.

In this paper, we study the ten-orbital (two Fe atoms) Hubbard-Holstein (HH) model for KFeSe using the RPA. When the Coulomb interaction is large, we obtain the -wave SC state due to the spin fluctuations, as predicted by the recent theoretical studies in the single-Fe Hubbard models DHLee (); Graser (); Bala (). However, the gap function on the Fermi surfaces (FSs) inevitably has “nodal structure” in the two-Fe model, due to the symmetry requirement of the body-centered tetragonal lattice. On the other hand, orbital-fluctuation-mediated -wave state is realized by small -ph coupling; the dimensionless coupling constant is just . Since the nodal SC state is fragile against randomness, study of impurity effect will be useful to distinguish these SC states.

We perform the local-density-approximation (LDA) band calculation for KFeSe using Wien2k code based on the experimental crystal structure first (). Next, we derive the ten-orbital tight-binding model that reproduces the LDA band structure and its orbital character using Wannier90 code and Wien2Wannier interface Arita (). The dispersion of the model and the primitive BZ are shown in Figs. 1 (a) and (b). Based on a similar ten-orbital model, Suzuki et al. studied the -wave gap structure for BaFeAs Suzuki ().

Figure 1: (color online) (a) Dispersion of the present ten-orbital model for KFeSe. , , and points are on the plane. (b) Primitive BZ for body-centered tetragonal lattice. (c)(d) FSs on the plane and plane. The green (light gray), red (gray), and blue (dark gray) lines correspond to , , and orbitals, respectively. The diamond-shaped shadows in (d) indicates the sign of basis function for representation. (e) Weight of each orbital on the inter FS (FS1) and the outer FS (FS2) as function of ; is shown in (d). (f) Hybridization between two e-pockets in 122 systems should create the nodal -wave gap.

In Fig. 1, we show the FSs of KFeSe for (c) and (d) planes when the electron number per Fe-ion is : On each plane, there are four large and heavy e-pockets around X and Y points, and one small and light e-pockets around Z point. For , the energy of the h-band at point from the Fermi level, , is about eV. Since the obtained FS topology and the value of are consistent with recent reports by ARPES measurements ARPES1 (); ARPES2 (); ARPES3 (), we study the case hereafter. In the present BZ in (b), and Z points and X and Y points in (c) are not equivalent, and plane is given by shifting (c) by . As for (d), T and T’ points and P and P’ points are equivalent, meaning that the reciprocal wave vector on the plane is and . The diamond-shaped shadows in the plane indicates the sign of basis function for (-type) representation, which has nodes on the P-P’ line on both FS1 (inner FS) and FS2 (outer FS).

To confirm the existence of nodes, we verify that FS1 and FS2 in KFeSe are largely hybridized. In fact, the weights of -orbitals on FS1,2 given in Fig. 1 (e) are smooth functions of , which is the strong evidence for the hybridization in wide momentum space. This hybridization disappears when inter-layer hoppings are neglected: Then, both and show cusps at , and suddenly drops to almost zero for . In (f), we explain the origin of nodal gap based on the fully-gapped -wave solution in the single-Fe model DHLee (); Graser (); Bala (): By introducing inter-layer hoppings, two elliptical e-pockets with positive and negative in the unfolded BZ are hybridized to form FS1 and 2 with four-fold symmetry. As a result, nodal lines inevitably emerge on FS1 and 2, at least near the plane.

Here, we study the ten-orbital HH model using the RPA. As for the Coulomb interaction, we consider the intra-orbital term , the inter-orbital term , Hund’s coupling or pair hopping , and assume the relation and . In addition, we consider the -ph interaction due to Fe-ion optical phonons; the phonon-mediated e-e interaction () and its matrix elements are presented in Ref. Saito, . Hereafter, we perform the RPA on the two-dimensional planes for , , and .

For and , the critical value of for the orbital-density-wave (ODW) is eV for , and the critical value of for the spin-density-wave (SDW) is eV for . These values change only for different . The obtained - phase diagram is very similar to Fig. 2 in Ref. Saito, , irrespective of the absence of h-pockets in KFeSe. The reason would be (i) the density-of-states (DOS) in KFeSe is about 1eV per Fe, which is comparable with other iron pnictides, and (ii) the nesting between e-pockets is rather strong because of their square-like shape. Figure 2 (a) shows the total spin susuceptibility at eV and for plane. is given by the intra-orbital nesting, and its peak position is , consistenlty with previous studies DHLee (); Graser (); Bala (). The obtained incommensulate spin correlation is the origin of the -wave SC gap. Figure 2 (b) shows the off-diagonal orbital susceptibility for the plane at and eV; its definition is given in Refs. Kontani, ; Saito, ; Onari, . It is derived from the inter-orbital nesting between and , and its peak position is . Note that the peak position of is . The obtained strong spin- and orbital-correlations are the origin of the -wave and -wave SC states.

Figure 2: (color online) (a) for eV and , and (b) for and eV.

In the following, we solve the linearized gap equation to obtain the gap function, by applying the Lanczos algorithm to achieve reliable results. In the actual calculation results shown below, we take -point meshes and 512 Matsubara frequencies. First, we study the spin-fluctuation-mediated SC state for by putting . Figures 3 (a)-(c) show the gap functions of the -wave solution at eV for , , and , respectively. In case of eV, the eigenvalue is for (a), for (b), and for (c); the relation corresponds to the SC state. They are relatively small since the SC condensation energy becomes small when the SC gap has complicated nodal line structure. On the (c) plane, the nodal lines are along and directions, consistently with the basis of representation in Fig. 1 (d). These nodes move to near the BZ boundary, and , on the (b) plane, and they deviate from the FSs on the (a) plane. As results, the nodal gap appears for in the whole BZ .

We also obtain the -wave state, with the sign reversal of the SC gap between e-pockets and the “hidden h-pockets below the Fermi level” given by the valence bands 5,6. The obtained solution is shown in Fig. 3 (d) for . Interestingly, the obtained eigenvalue is for eV, which is larger than for -wave state in Fig. 3 (a)-(c). Such large originates from the scattering of Cooper pairs between e-pockets and the “hidden h-pockets”, which was discussed as the “valence-band Suhl-Kondo (VBSK) effect” in the study of NaCoO in Ref. Yada, .

Figure 3: (color online) SC gap functions for outer FS (FS2) on band 7 and inner FS (FS1) on band 8: -wave gap functions on the (a) , (b) , and (c) planes. Black and green (gray) lines represent the FSs and gap nodes. (d) -wave gap function on the plane.

Here, we analyze the -dependence of based on a simple two-band model with inter-band repulsion: The set of gap equations is given by Yada () and , where is the repulsive interaction between e- and h-pockets, and is the DOS near the Fermi level. When (i) the top of the h-pocket is well above the Fermi level, , where is the cutoff energy. Thus, the eigenvalue is given as , similar to single-band BCS superconductors. On the other hand, when (ii) h-pocket is slightly below the Fermi level, , where is the energy of the top of h-band Yada (). Thus, the eigenvalue is given as . Therefore, in case (ii), the -dependence of is much moderate. In fact, as shown in Fig. 4 (a), for -wave state increases monotonically with decreasing , while for -wave state saturates at low temperatures. This result suggests that the -wave state overcomes the -wave state at K in KFeSe. Although in the -wave state is eV in Fig. 4 (a), it is greatly reduced by the self-energy correction that is absent in the RPA Onari ().

Figure 4: (color online) (a) -dependence of for - and -wave states. at eV is underestimated because of the shortage of - and Matsubara-meshes. (b) - (-) dependence of for -wave (-wave) state at eV. (c) SC gap functions for -wave state.

We discuss the VBSK effect for wave state in more detail: According to inelastic neutron scattering measurement of Ba(Fe,Co)As neutron (), the characteristic spin-fluctuation energy is K just above K. If we assume a similar in KFeSe since is close, we obtain the relation in KFeSe. Since is a monotonic decrease function of and for , we consider that -wave state overcomes the -wave state in KFeSe, as far as the spin-fluctution mediated superconductivity is considered. Although high- -wave state mighe be realized for , then the realized will be very sensitive to or the filling Yada ().

Now, we study the -wave state due to orbital fluctuations on the plane with . In Fig. 4 (b), we show the -dependence of at for the -wave state with , and the -dependence of for the -wave state with . Here, () is the charge (spin) Stoner factor introduced in Ref. Kontani, ; () corresponds to the ODW (SDW) state. In calculating the -wave state, we use rather larger phonon energy; eV, considering that the calculating temperature is about ten times larger than the real . The SC gap functions for -wave state are rather isotropic, as shown in Fig. 4 (c). However, the obtained SC gap becomes more anisotropic in case of Onari ().

We stress that the RPA is insufficient for quantitative study of since the self-energy correction is dropped: In Ref. Onari, , we have studied the present model based on ther FLEX approximation, and found that the critical region with is enlarged by the inelastic staccering . Also, the -induced suppression in for - or -wave states is more prominent than that for -wave state, since due to spin fluctuations is larger than that due to orbtial fluctuaiotns Onari ().

Recently, we found the paper by Mazin Mazin2 (), in which the dependence of the nodal -wave gap in Fig. 4 corresponds to Figs. 3 (a)-(c) in the present work.

In summary, we studied the mechanism of superconductivity in KFeSe based on the ten-orbital HH model without h-pockets. Similar to iron-pnitide superconductors, orbital-fluctuation-mediated -wave state is realized by small dimensionless -ph coupling constant . We also studied the spin-fluctuation-mediated -wave state, and confirmed that nodal lines appear on the large e-pockets, due to the hybridization between two e-pockets that is inherent in 122 systems. Therefore, careful measurements on the SC gap anisotropy is useful to distinguish these different pairing mechanisms. Study of impurity effect on is also useful since -wave (and -wave) state is fragile against impurities.

Acknowledgements.
We are grateful to D.J. Scalapino, P. Hirschfeld, A. Chubukov, Y. Matsuda, and other attandances in the international workshop “Iron-Based Superconductors” in KITP 2011 for useful and stimulating discussions. This study has been supported by Grants-in-Aid for Scientific Research from MEXT of Japan, and by JST, TRIP. Numerical calculations were performed using the facilities of the supercomputer centers in ISSP and Institute for Molecular Science.

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