Interplay among Coulomb Interaction, Spin-Orbit Interaction, and ...

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Nov 24, 2010 - 1Hiroshima Synchrotron Radiation Center, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-0046, Japan. 2National Institute of ...
PRL 105, 226406 (2010)

PHYSICAL REVIEW LETTERS

week ending 26 NOVEMBER 2010

Interplay among Coulomb Interaction, Spin-Orbit Interaction, and Multiple Electron-Boson Interactions in Sr2 RuO4 H. Iwasawa,1,* Y. Yoshida,2 I. Hase,2 S. Koikegami,3 H. Hayashi,4 J. Jiang,4 K. Shimada,1 H. Namatame,1 M. Taniguchi,1,4 and Y. Aiura2,† 1

Hiroshima Synchrotron Radiation Center, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-0046, Japan 2 National Institute of Advanced Industrial Science and Technology, Tsukuba, Ibaraki 305-8568, Japan 3 Second Lab, LLC, Tsukuba, Ibaraki 305-0045, Japan 4 Graduate School of Science, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8526, Japan (Received 16 August 2010; published 24 November 2010) Using polarization- and h-dependent angle-resolved photoemission spectroscopy, we uncovered the fine details of a quasiparticle’s dynamics of a typical multiband superconductor, Sr2 RuO4 . We found strong hybridization between the in-plane and out-of-plane quasiparticles via the Coulomb and spin-orbit interactions. This effect enhances the quasiparticle mass due to the inflow of out-of-plane quasiparticles into the two-dimensional Fermi surface sheet, where the quasiparticles are further subjected to the multiple electron-boson interactions. We suggest that the spin-triplet p-wave superconductivity of Sr2 RuO4 is phonon mediated. DOI: 10.1103/PhysRevLett.105.226406

PACS numbers: 71.18.+y, 74.25.Jb, 74.70.Pq, 79.60.i

Electron correlations are interrelated with dimensionality. The single-layered ruthenate Sr2 RuO4 , which is well known as a spin-triplet p-wave superconductor [1], possesses three Ru 4dt2g orbitals: two out-of-plane dyz;zx orbitals form quasi-one-dimensional (1D)  and  Fermi surface (FS) sheets [2], while the remaining in-plane dxy orbital forms a two-dimensional (2D)  FS sheet. In general, a reduction in the dimensionality leads to narrowing of the bandwidth (Wyz;zx < Wxy ), boosting the electron correlation strength measured by U=W [3], where U is the on-site Coulomb interaction. In contrast, both de Haas– van Alphen [4] and angle-resolved photoemission spectroscopy (ARPES) [5,6] experiments indicated the significant mass enhancement not only for the 1D FS sheets but also for the 2D FS sheet. Moreover, the electron-boson (phonons [7,8] and magnons [9–11]) interaction (EBI) has been reported in Sr2 RuO4 . So far, the magnetic-fluctuations mediated pairing has been widely accepted through a number of theoretical studies [1]. However, existing ARPES data have not fully clarified the details of the bosonic modes yet, because they have been elusive due to the complex multiband electronic structure near the Fermi level (EF ). To understand the p-wave-pairing mechanism in Sr2 RuO4 , it is thus essential to clarify the bosonic modes coupled with the quasiparticles experimentally. Polarization-dependent ARPES is useful not only for identifying the wave function parity of the initial states with respect to a mirror plane of crystals [12] but also for selecting the observable electronic structure of materials. Namely, the even- (odd-)symmetry initial states are only observable with p (s) polarization following the dipole selection rule [13]. In this Letter we present the details of the quasiparticles in Sr2 RuO4 by utilizing this selectivity. Multiple kinks were found in the in-plane  band 0031-9007=10=105(22)=226406(4)

dispersion, which are not only explained well by the in-plane phonon modes but also by the out-of-plane phonon modes. Our results demonstrated that the phonon mediated p-wave pairing of Sr2 RuO4 is feasible via the out-of-plane phonon modes coupled with the 1D orbital component, which is originated from the strong hybridization between the 1D and 2D orbitals via the Coulomb interaction and spin-orbit interaction (SOI). All the data were collected at the newly constructed polarization-dependent ARPES system on the linear undulator beam line (BL-1) of the Hiroshima Synchrotron Radiation Center [14]. The details of the ARPES experiments are given in the supplemental material [13]. The topology of the FS of Sr2 RuO4 has now been established by different techniques such as de Haas– van Alphen [4], ARPES [6,15], and band-structure calculations based on the local density approximation (LDA) [16–18], and is essentially consistent with what we observed [Figs. 1(a) and 1(b)]. The spectral-weight distribution of the FS sheets strongly depends on the polarization geometry, especially along the zone horizontal (M) and zone diagonal (X) where the selection rule rigorously applies. Figures 1(c)–1(f) show large energy scale ARPES images acquired along these lines with both p and s polarizations. The electronic structure of Sr2 RuO4 consists of narrow dyz;zx bands in 1D and wide dxy band in 2D. Based on the nonrelativistic dipole selection rule, the relevant calculated bands (solid lines) are overlaid in Figs. 1(c)–1(f) and are roughly consistent with the observed ARPES dispersions. However, the bandwidths of the 1D dyz;zx bands are significantly renormalized to about half of their original widths [Figs. 1(c) and 1(f)], whereas the bandwidth of the 2D dxy is almost identical to that obtained from the LDA calculations [Figs. 1(d) and 1(e)].

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Ó 2010 The American Physical Society

PRL 105, 226406 (2010)

PHYSICAL REVIEW LETTERS

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FIG. 1 (color online). (a) and (b) FS mapping measured with 65 eV photons at 10 K using p and s polarization, respectively. The maps are obtained by integrating raw spectra over the energy window (EF  10 meV) and by symmetrizing the =4 FS. (c) and (e) The large energy scale ARPES images together with the LDA calculations at different values of kz (solid lines) along the M line using p and s polarization, respectively, where the LDA calculations are expanded by a factor of 1.1 to roughly align the O 2p position. (d) and (f) Same as panels (c) and (e) except that measured line is the X line. Considering the ARPES matrix element effects, we have chosen the excitation energy and the location of the Brillouin zone (BZ) to obtain clear band dispersions: (c) 88-eV in the 2nd BZ, (e) 88-eV in the 1st BZ, (d) 54-eV in the 2nd BZ, and (f) 38-eV in the 1st BZ. (g) and (h) Schematic electronic structure along M and X line, respectively. The solid lines represent the LDA calculation results renormalized by EEI from the LDA calculations (dashed lines). Note that the orbital assignment along the X line can be varied depending on the kz value [13].

Such a large energy scale renormalization is typically associated with the electron-electron interaction (EEI). This orbital-dependent EEI is schematically summarized in Figs. 1(g) and 1(h), where only the noninteracting 1D bands (dashed lines) are renormalized (solid lines). Focusing on the near-EF electronic structure in Figs. 2(a) and 2(b), we found the breakdown of the nonrelativistic dipole selection rule [13]. One possible explanation for this is the hybridization of the initial state due to the SOI [19]. To ensure the existence of the SOI, we performed h-dependent ARPES along the M and X planes [Figs. 2(c) and 2(d)]. We also calculated the kz dispersion with a nonrelativistic LDA calculation [Figs. 2(e) and 2(f)] and a relativistic LDA calculation including the SOI [Figs. 2(g) and 2(h)]. Along the M plane, the observed kz dispersion of the  and  FS sheets are qualitatively consistent with the calculated ones [Figs. 2(c), 2(e), and 2(g)]. Namely, the effect due to the SOI and hybridization is not appreciable along this direction, consistent with the survived dipole selection rule with certain excitation energies [Figs. 2(a) and 2(c)]. However, along the X plane, the kz dispersion of ARPES [Fig. 2(d)] and the nonrelativistic LDA calculation [Fig. 2(f)] do not agree.

Including the SOI in the LDA calculation results in a drastic narrowing of the kz dispersion and a separation of each FS [Fig. 2(h)], which reproduces ARPES [Fig. 2(d)], consistent with the previous quantum oscillation experiment [20]. Accordingly, the SOI generates strong hybridization between 1D and 2D bands along the X plane [Fig. 2(b)]. By taking advantage of the active polarization dependence along the M plane, we can selectively observe the  and  sheets [Figs. 3(a) and 3(b)]. To accurately evaluate the magnitude of EEI and EBI, we have constructed phenomenological quasiparticle dispersions [black solid lines in Fig. 3(c)] [21]. Based on the electronic structure in the wide energy range [Figs. 1(c)–1(f)], we put U 1:5 eV yz;zx [22] on the narrow 1D dyz;zx bands (WLDA 1:5 eV) but xy not on the wide 2D dxy band (WLDA  3:6 eV), and we take the SOI into account as the perturbation. Because the correlated 1D dzx band and the itinerant 2D dxy band are almost degenerate at the M point, the SOI effectively works to lift the degeneracy and the hybridization gap opens [21]. This yields the strong hybridization among 1D and 2D bands in the vicinity of the M point, as seen in the calculated orbital components [middle and bottom panels of Fig. 3(d)]. Through this hybridization, the

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PHYSICAL REVIEW LETTERS

FIG. 2 (color). (a) and (b) The momentum distribution curves (MDCs) at EF for p polarization (upper) and s polarization (lower) sliced from the FS along M and X lines, respectively [white lines in Figs. 1(a) and 1(b) ]. (c) and (d) ARPES images in the kk  kz plane acquired with 22 to 70 eV photons at 10 K using p polarization along the M and X planes, respectively. The ARPES images were obtained by integrating raw spectra over the energy window (EF  10 meV). The experimentally determined kz dispersion is indicated by grey, red, and blue marks for , , and  FS sheets, respectively. (e) and (g) Theoretical kz dispersions along the M (ZM) plane calculated by LDA and LDA with the SOI, respectively. (f) and (h) Same as (e) and (g), except that the plane is perpendicular to the X (ZX) line.

electronic mass enhancement occurs even in the 2D  FS sheet, accompanied by shrinking (expanding) of the  () FS sheet. As seen in the top panel of Fig. 3(d), the Fermi momentum (kF ) obtained from ARPES agrees well with the model dispersions, which supports the enhanced hybridization due to U and the SOI. The discrepancy between the ARPES results and those of our model in Fig. 3(c) is then assumed to originate from EBIs. Obviously, the magnitude of EBIs is highly orbital dependent: the -sheet dispersion deviates negligibly from the calculated dispersion, whereas the -sheet dispersion undergoes significant renormalization at the specific energies indicated by the three arrows despite that the group velocity agrees well with the calculation above 80 meV. Figure 3(e) shows these renormalizations, which are denoted here as the high-energy kink (HEK) at 70–80 meV, the medium-energy kink (MEK) at 50–60 meV, and the low-energy kink (LEK) at 30–40 meV. This significant renormalization is accompanied by the proximity of the van Hove singularity (vHS) to EF at the M point. Upon increasing the temperature to thermally excite the electrons above EF , we observed the vHS at 20 meV [Fig. 3(f)], consistent with previous ARPES results [23]. Figure 3(g) illustrates how the energy of vHS (EvHS ) is closer to EF : starting from EvHS  90 meV, which is the result of the

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nonrelativistic LDA calculation, EvHS is lowered to 70 meV because of the SOI, further to 40 meV including both the SOI and U, and finally to 20 meV including EBIs. How are observed EBIs and the proximity of the vHS to EF relevant to the p-wave superconductivity of Sr2 RuO4 [24]? It is unlikely that the proximity of the vHS to EF is solely responsible for the superconductivity because Ladoped Sr2 RuO4 shows no superconductivity [25] upon carrier doping [23] in spite of the vHS being closer to EF . However, EBIs in the  sheet are smeared out along with the loss of superconductivity from undoped to La-doped Sr2 RuO4 [23], suggesting that EBIs, in addition to the proximity of vHS to EF , play a key role in the superconductivity. Based on the mode energy and symmetry from our data, we can squeeze the relevant boson modes such as phonons or magnons. The energy scale of the observed kinks (30–80 meV) is obviously larger than the energy scale of the reported magnetic excitations [i.e., the incommensurate antiferromagnetic spin fluctuations ( 4 meV [9]) and the ferromagnetic spin ordering ( 8 meV [10,11])]. Note that this does not invalidate magnetic fluctuations as a potential pairing mechanism of Sr2 RuO4 , and such low-energy magnetic excitations remained to be verified in future ultra-high-resolution ARPES. On the other hand, the energy of phonon modes fits well which were observed in a recent inelastic neutron scattering experiment [8]: LEK corresponds to the oxygen buckling phonon modes, MEK to the apical oxygen bondstretching phonon modes (OBSMs), and HEK to the inplane OBSMs. The in-plane phonon modes can easily couple to the in-plane dxy orbital as manifested by LEK and HEK, though these modes do not favor the p-wave pairing and thus just enhance the density of states by tuning the energy position of the vHS. Conversely, two apical Z zone-boundary OBSMs, M 1 and Oz both at 60 meV [26], are compatible with phonon mediation of the p-wave-pairing symmetry with the 1D dzx orbitals [21]. Also, the existence of MEK in the 2D  band itself constitutes evidence of the strong hybridization between 1D dyz;zx and 2D dxy orbitals because it is indeed impossible for out-of-plane OBSMs to couple with the in-plane dxy orbital. This means that the phonon scattering process with out-of-plane OBSMs favors a wave vector connecting a vHS over a vHS because the hybridization of 1D and 2D orbitals is only significant near the M point. Our results imply that the electron-phonon interaction basically occurs as long as the phonon scattering process is allowed, yielding a high density of states in the vHS. With this high density of states, only a few phonon modes scattered via the vHS would determine the superconducting symmetry and mediate the p-wave pairing in Sr2 RuO4 . Our scenario suggests that the superconductivity of Sr2 RuO4 is incompatible with out-of-plane disorder because it seriously affects not only the apical vibration modes but also the electronic structure of the

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FIG. 3 (color). (a) and (b) ARPES images acquired along the M line (yellow line in inset) with 48-eV photons using p- and s-polarization, respectively. (c) MDC-derived energy-momentum dispersions of the  (red marks) and the  (blue marks) sheets, where  1 . The solid and dashed lines represent the model dispersions [top panel of (d)] and the Fermi the  sheet includes an offset of 0:1 A velocities in each sheet, respectively. In panels (a)–(c), the MDC dispersions are replaced by the linear function below 15 meV because of an energy-resolution effect [27], but this operation does not affect our conclusions. (d) Top panel shows the calculated phenomenological model dispersions [21]. Middle and bottom panels show the orbital components of the  and  sheets, respectively. The experimentally determined kF for the  and  sheets is indicated by the dashed lines in (d). (e) The MDC dispersions of the  sheet are closing up in the different regions, which are indicated by arrows in (c), where LEK, MEK, and HEK of the -sheet dispersion are highlighted in the top, middle, and bottom panels, respectively. (f) ARPES image measured at 100 K, s polarization, with 32-eV photons along the M line divided by the Fermi-Dirac distribution. (g) Comparison between ARPES and calculated band dispersions.

out-of-plane 1D orbitals [23], which explains the vanishing superconductivity in La-doped Sr2 RuO4 [25]. We thank N. C. Plumb for invaluable discussion. This work was supported by KAKENHI (22740233, 19340105). The synchrotron radiation experiments have been done under the approval of Hiroshima Synchrotron Radiation Center (Proposal No. 09-A-16).

*[email protected][email protected] [1] See review by A. P. Mackenzie and Y. Maeno, Rev. Mod. Phys. 75, 657 (2003). [2] We denote the dimensionality of dyz;zx orbitals as ‘‘1D’’ not ‘‘quasi-1D’’ to clarify a contrast to the ‘‘2D’’ dxy orbital. [3] A. Liebsch and A. Lichtenstein, Phys. Rev. Lett. 84, 1591 (2000). [4] A. P. Mackenzie et al., Phys. Rev. Lett. 76, 3786 (1996). [5] Y. Aiura et al., Phys. Rev. Lett. 93, 117005 (2004). [6] H. Iwasawa et al., Phys. Rev. B 72, 104514 (2005). [7] Z. Q. Mao et al., Phys. Rev. B 63, 144514 (2001). [8] M. Braden et al., Phys. Rev. B 76, 014505 (2007). [9] M. Braden et al., Phys. Rev. B 66, 064522 (2002).

[10] M. Braden et al., Phys. Rev. B 57, 1236 (1998). [11] R. Matzdorf et al., Science 289, 746 (2000). [12] W. Eberhardt and F. J. Himpsel, Phys. Rev. B 21, 5572 (1980). [13] See supplementary material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.105.226406. [14] H. Iwasawa, K. Shimada et al. (to be published). [15] A. Damascelli et al., Phys. Rev. Lett. 85, 5194 (2000). [16] T. Oguchi, Phys. Rev. B 51, 1385 (1995). [17] D. J. Singh, Phys. Rev. B 52, 1358 (1995). [18] I. Hase and Y. Nishihara, J. Phys. Soc. Jpn. 65, 3957 (1996). [19] M. W. Haverkort et al., Phys. Rev. Lett. 101, 026406 (2008). [20] C. Bergemann et al., Phys. Rev. Lett. 84, 2662 (2000). [21] Y. Aiura et al., J. Phys. Soc. Jpn. 79, 123702 (2010). [22] T. Yokoya et al., Phys. Rev. B 53, 8151 (1996). [23] K. M. Shen et al., Phys. Rev. Lett. 99, 187001 (2007). [24] S. V. Borisenko et al., Phys. Rev. Lett. 105, 067002 (2010). [25] N. Kikugawa et al., Phys. Rev. B 70, 060508 (2004). [26] The notation of the reciprocal vector for these phonon modes is from the bcc tetragonal structure, which is different from the simple tetragonal notation that we have used so far. [27] N. C. Plumb et al., Phys. Rev. Lett. 105, 046402 (2010).

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