arXiv:1711.06686v2 [cond-mat.supr-con] 28 Feb 2018

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Feb 28, 2018 - 3Istituto Officina dei Materiali (IOM)-CNR, Laboratorio TASC,. Area Science Park, S.S.14, Km 163.5, I-34149 Trieste, Italy. 4International Centre ...
Influence of Spin Orbit Coupling in Iron-Based Superconductors R.P. Day,1, 2 G. Levy,1, 2 M. Michiardi,1, 2 M. Zonno,1, 2 F. J,1, 2 E. Razzoli,1, 2 F. Boschini,1, 2 S. Chi,1, 2 R. Liang,1, 2 P.K. Das,3, 4 I. Vobornik,3 J. Fujii,3 D.A. Bonn,1, 2 W.N. Hardy,1, 2 I.S. Elfimov,1, 2 and A. Damascelli1, 2

arXiv:1711.06686v2 [cond-mat.supr-con] 28 Feb 2018

1

Department of Physics and Astronomy, University of British Columbia, Vancouver, BC V6T 1Z1, Canada 2 Quantum Matter Institute, University of British Columbia, Vancouver, BC V6T 1Z4, Canada 3 Istituto Officina dei Materiali (IOM)-CNR, Laboratorio TASC, Area Science Park, S.S.14, Km 163.5, I-34149 Trieste, Italy 4 International Centre for Theoretical Physics (ICTP), Strada Costiera 11, I-34100 Trieste, Italy We report on the influence of spin-orbit coupling (SOC) in the Fe-based superconductors (FeSCs) via application of circularly-polarized spin and angle-resolved photoemission spectroscopy. We combine this technique in representative members of both the Fe-pnictides and Fe-chalcogenides with ab initio density functional theory and tight-binding calculations to establish an ubiquitous modification of the electronic structure in these materials imbued by SOC. The influence of SOC is found to be concentrated on the hole pockets where the superconducting gap is generally found to be largest. This result contests descriptions of superconductivity in these materials in terms of pure spin-singlet eigenstates, raising questions regarding the possible pairing mechanisms and role of SOC therein.

In order to establish the fundamental criteria for superconductivity in the iron-based superconductors (FeSCs), there is considerable debate at present as to the significance and ubiquity of various interactions in these materials with regards to the superconducting pairing [1]. The weak crystal field and 3d6 configuration of the Fe2+ ions result in several closely spaced shallow or incipient Fermi surface pockets which are highly susceptible to various perturbations [1–4]. Regarding superconductivity, the effects of Coulomb’s and Hunds’ coupling, as well as nematic ordering have been of primary focus. While the relativistic effects of spin-orbit coupling (SOC) are indeed universal to all materials, it remains to be confirmed whether this energy scale is relevant to the low temperature phase diagram of the FeSCs, where superconductivity and magnetic order evolve. Recently, SOC has been put forth to explain deviations in ARPES from the non-relativistic band structure [5–10]. To attribute definitively the gaps observed in ARPES to SOC and to establish the influence of SOC on not only the dispersion but the electronic eigenstates as well, explicit evidence for a coupling of the nominative spin and orbital degrees of freedom is necessary. Alternatively, at the Brillouin zone centre, other perturbations such as nematicity can lead to nearly indistinguishable changes to the electronic dispersion from those induced by SOC [11]. For example, ferro-orbital order has been suggested to account for these splittings [12]. By clarifying this situation, we hope to verify the correct starting point for pairing theories, taking into account the relationship between spin and orbital degrees of freedom on those states relevant to superconductivity. Circularly Polarized Spin and Angle-Resolved Photoemission Spectroscopy (CPS-ARPES) is an experimental technique capable of making this distinction, providing evidence that the spin-orbit interaction binds these degrees of freedom in a strongly momentum-dependent way.

By employing CPS-ARPES to study archetypal members of both Fe-pnictides (LiFeAs) and Fe-chalcogenides (FeSe), we demonstrate here evidence for entanglement in the spin and orbital angular momentum in the FeSCs. We combine these experimental results with ab initio density functional theory (DFT) and tight-binding calculations of LiFeAs to simulate photoemission from this material and further explore the influence of SOC on the electronic structure of the FeSCs. FeSe crystals were grown via vapour transport technique [13] and LiFeAs by a self-flux method [14]. Samples were cleaved and measured in the non-superconducting phase at 20 K at pressure of 10−10 mbar at the APELE endstation at ELETTRA using a Scienta DA30 analyzer equipped with a VLEED-based spin-detector [15]. The electronic structure has been calculated using the QUANTUM ESPRESSO [16] package, and the output post-processed using Wannier90 [17], projecting the lowenergy electronic states near EF onto a reduced basis of maximally localized wavefunctions of Fe 3d and As 4p character. To demonstrate the capacity for CPS-ARPES to elucidate the nature of spin-orbit coupling, it is instructive to consider the form of the photoemission intensity in an ARPES experiment. Following Fermi’s Golden Rule, in the dipole approximation I(~k, ω) ∝ A(~k, ω)| hψf |ˆ  · ~r|ψi i |2

(1)

where |ψi,f i describe the initial and final one-electron states of the emitted photoelectron, and ˆ  the polarization of light used in the experiment. A(~k, ω) is the one-electron removal spectral function [18]. From Equation 1, we see that while ARPES is defined broadly as a probe of the one electron-removal spectral function, the photocurrent is modulated by dipole matrix element effects which encode information regarding both the experimental configuration such as light polarization as well as

2 the nature of the wavefunction being probed [19, 20]. In CPS-ARPES, circularly polarized light photoemits preferentially from states of specific angular momentum ml , and by spin-filtering the resulting photocurrent one can measure directly the correlation between the spin and orbital degrees of freedom [21–23]. With the aim then of ~ and S ~ in the FeSCs, exploring the relationship between L we report here on the study of archetypal members of the Fe-SCs via CPS-ARPES. To illustrate the application of the technique of CPSARPES to the FeSCs, we begin by considering the two hole pockets common to all FeSCs near the Brillouin zone center. Each band is Kramers’ (two-fold) degenerate and composed primarily of Fe dxz and dyz orbitals. At the zone centre (k|| = 0 ˚ A−1 ) these bands meet at a four-fold degenerate point. In the presence of SOC, this four-fold degeneracy is split and two Kramers’ doublets separated by 10 - 20 meV remain (Figure (1a)): one doublet defined with the ˆ z projection of the orbital and spin angular momentum aligned parallel (d↑z +1z = Y21 |+χi ↓z and d−1z = Y2,−1 |−χi) and the other doublet antipar↓z allel (d↑z −1z = Y2,−1 |+χi and d+1z = Y2,1 |−χi). These pairs are illustrated for the case of LiFeAs in Figure (1). Here we have used the basis of the direct product space over spherical harmonics Yl,m (θ, φ) and spin- 21 spinors |±χi with the quantization axis ˆ z defined out of the Fepnictogen/chalcogen plane. Under the dipole approxi-

FIG. 1. Electronic structure for LiFeAs with colour-scale indicating the expectation value of hLi Si i. λSOC taken as 40 ~ ·S ~ promeV before renormalization. (a) Component of L jected along the ˆ z direction (i.e. along the ΓZ direction), and in (b), that along the x ˆ direction (i.e. ΓM ). In (c) and (d), the hLz Sz i and hLx Sx i as in (a) and (b), but around constant energy surfaces in momentum space at an energy of E = EF ~ and L ~ show largest (anti)collinear alignment - 5 meV. The S near the zone centre (ΓZ) for states near EF on the two inner hole bands. This is directed primarily out of plane, and varies for constant energy contours along the kz direction.

mation ∆l = ±1 during photoemission, and for circularly polarized light selection rules are further restricted allowing only ∆ml = 1 for C+ , and ∆ml = −1 for C− light. Here and throughout, the photoelectron final state is ap-

proximated by a free-electron plane wave [19]. For C+ light as an example, photoemission from the Y2,−1 state is dominant, and here this would restrict photoemission ↓z from only d↑z −1z and d−1z [24]. Photoelectrons from these states are then spin-filtered by the VLEED detector [15]. In this way one has control over both the orbital from which photoemission occurs, and the spin channel which is measured. To apply this technique towards elucidating the binding of spin and orbital degrees of freedom, one measures the spin polarization asymmetry defined as [21] q q ↑ ↓ ↑ ↓ I− I+ − I+ I− q Pi = q ↑ ↓ ↑ ↓ I− I+ + I+ I−

(2)

↑(↓)

where I+(−) indicates the photocurrent intensity for C+ (C− ) light incident on the ↑(↓) spin detector oriented along the i = x ˆ, y ˆ, ˆ z direction. Under consideration of ↑ ↓ I+ is a measure of the dipole selection rules above, I− ↑ ↓ states with orbital and spin aligned parallel, and I+ I− a measure of those aligned antiparallel. Consequently, CPS-ARPES is the most direct measure of the effects of SOC, with Pi offering an energy and momentum-resolved measure of the spin-orbit coupling polarization. In the absence of SOC, where another interaction such as nematicity has induced a splitting of bands, Pi would vanish everywhere, as C± would photoemit from the Kramers’ degenerate states of |+χi and |−χi indiscriminately. By measuring Pi throughout the Brillouin zone and along different axes of spin projection, we may study the impact of spin-orbit coupling throughout the electronic structure of the FeSCs. In connection to the experiment, we plot the evolution of SOC in LiFeAs along high symmetry directions in Figure (1). We emphasize hLz Sz i and hLx Sx i in Figure (1a) and (1b) since experimentally, we measure Pi along the out of plane and ΓM (i.e. Fe-Fe) directions which we designate as ˆ z and x ˆ respectively. To achieve agreement between the singleparticle DFT model and the ARPES dispersion, atomic SOC of strength λSOC = 40 meV has been added to the non-relativistic Wannier-Hamiltonian, and the electronic band structure then renormalized by a factor of 2.19 to account for correlation effects in ARPES, as in Figure (2a). This renormalization of the spectral function is consistent with those observed throughout the FeSCs [1, 4, 25]. The influence of SOC is most pronounced on the hole pockets at the zone centre near EF . As illustrated in Figure (2c), an experimental splitting of 13 ± 3 meV at the zone centre is extracted from a fit to energy distribution curves (EDCs), which is similar to the splitting measured elsewhere [6]. D E ~ ·S ~ , experimenMotivated by the calculations of L tal emphasis was placed on emission angles near k|| = 0, where we collected spin-resolved EDCs from LiFeAs with

3 ficiently small that the Kramers’ doublets reduce to the usual |dxz /dyz i ⊗ |±χi used to describe these bands.

FIG. 2. (a) ARPES near normal emission for LiFeAs at hν = 26 eV, corresponding to 0.1ΓZ. Fits to MDCs are shown in blue, with the Wannier band structure underneath in orange. (b) Calculated ARPES intensity using this model at T =20 K for C+ polarization. (c) EDC from panel (a) at k|| = 0 ˚ A−1 , fit to the sum of background (dashed line) and two Lorentzians (blue and orange) split by 13 ± 3 meV, consistent with other reports [6]. The total fit is in purple with data points in grey.

C+ and C− light and spin projected along both ˆ z and x ˆ. For each emission angle we computed Pz as in Equation 2 and the result is plotted in Figure (3a). The two spectral peaks denoted in Figure (2c) can be ascribed to the ↑↓z d↑↓z ±1z and d∓1z states defined earlier with spin and orbital angular momentum antiparallel at high binding energy, and parallel closer to the Fermi level. In a plot of the polarization asymmetry (Figure (3)), this is reflected by a switch in the sign of Pz between the two peaks, reflective of the switch in sign of hLz Sz i between the two doublets. Moving to larger k|| , the switch in Pz moves with the dispersion to higher binding energies, and at large enough k|| , only the d↑↓z ±1z states contributes to the ARPES intensity, resulting in Pz > 0. While the inherent collection inefficiency of S-ARPES limits the resolution of our experimental evaluation of Pz in both energy and momentum, simulated CPS-ARPES intensity across the entire region of momenta and energy near the zone centre not only agrees well with the measured curves, but allows for interpolation of Pz throughout this volume of the Brillouin zone. In order to do this, simulated ARPES spectra, as in Figure (2b) were generated based on the experimental configuration (See Supplementary). The full simulated Pz spectra is plotted in Figure (3c). For more direct comparison with the data, three simulated Pz curves for the k|| values probed experimentally are reproduced in Figure (3b). From these results we infer that the spin and orbital degrees of freedom in LiFeAs are coupled primarily near the zone centre where these bands approach the Fermi level. Only for larger in plane momenta k|| as the bands move away from EF do the effects of SOC become suf-

In addition to the spin-projection out of plane, we measured spin along the ΓM direction and found that away from k|| = 0, the spin vector tilts away from (anti)parallel ~ for the two hole bands. The results, alignment with L plotted in Figure (S4) show this small, but larger than anticipated, in-plane spin-orbit polarization Px . This implicitly requires hybridization with orbitals beyond dxz and dyz , as SOC only introduces Lz Sz terms between these states. Similar measurements (both in and out of plane) on the electron pockets at the zone corner produced no observable spin-orbital polarization (Figure (S5)), suggesting the effects of SOC on the independence ~ and S ~ to be more relevant to the hole pockets at of L the zone centre. This has important implications for spin and orbital fluctuation pairing mechanisms which involve intra- and inter-band exchange in these channels [2] which are evidently co-dependent in regions of k-space relevant to superconductivity. Specifically SOC has been ~ connectshown to suppress the spin-susceptibility for Q ing hole and electron pockets in FeSCs, a point of particular relevance to the viability of spin-fluctuation mediated superconductivity [26]. Due to the conservation of energy in the photoemission process, variation of the photon energy hν allows for exploration of the full three-dimensional Brillouin zone [18]. In consideration of Figure (1b) and (1d), the hLz Sz i is anticipated to vary substantively along the kz direction between Γ and Z. While at Γ the tops of the two hole bands are below EF and characterized by parallel and antiparallel SOC projection, at Z these bands have both moved above EF , and an additional band drops below EF around kz = 0.55Z. Near to Z then, we anticipate the polarization asymmetry to be dominated by states with orbital and spin angular momentum aligned antiparallel, resulting in a strictly negative Pz curve. Experimentally, we have explored the evolution of Pz across several values of kz by varying photon energy between 26 eV and 46 eV (Figure 3d). This range allows for us to tune kz from Γ to Z and on towards the next Γ point. The characteristic switch of Pz is observed for kz near Γ, and the anticipated Pz < 0 curves were found near kz = Z. The variation of SOC along kz demonstrates that the occupied states have co-aligned and counter-aligned spin and orbital angular momentum states for kz near Γ evolving to a situation with the occupied states having solely counter-aligned spin and orbital vectors closer to Z. In contrast to LiFeAs, many of the FeSCs undergo a nematic transition as temperature is reduced towards Tc . The observation of a nematic phase near Tc in many FeSCs has motivated theoretical interest in its role in Febased superconductivity [27]. Recently, it was suggested that SOC plays an important role in the phenomenology of the nematic phase [9]: the influence and connection

4

˚−1 (red) and FIG. 3. (a) Measurement of out of plane Spin Polarization Asymmetry at normal emission (orange), k|| = -0.06A −1 k|| = -0.14 ˚ A (purple). Inset shows the Fermi surface for this region of ~k, with symbols indicating the momenta for the three curves. (b) Calculation of Pz for the same momenta as in (a). (c) The calculated map of Pz (red maximum, blue minimum) near normal emission, vertical lines correspond to curves in (a,b). (d) Spin Polarization Asymmetry along ΓZ. Spin polarization asymmetry was measured at hν = 26 (black), 31(red), 36(yellow), 41 (orange), and 46 (purple) eV, corresponding to kz = 0.1Z, 0.5Z, 0.9Z, 0.7Z, and 0.35Z respectively. Inset: ARPES at normal emission as a function of photon energy–vertical lines indicate the photon energies (and kz values) for the spin measurements.

of these different interactions may be more complicated than has been assumed. We may ask if SOC is still of importance, or if the system is dominated by the energy scales associated with orbital ordering. In FeSe, there is a well-known structural distortion from tetragonal to orthorhombic around 90 K, associated with the onset of orbital-ordering [13]. To explore SOC in the presence

FIG. 4. a) Out of plane Spin Polarization Asymmetry (Pz ) for normal emission, with inset showing cursors on the FeSe Fermi surface. Inset indicates the relevant momenta points on the Fermi surface. (b) ARPES near normal emission for orthorhombic FeSe at hν = 37 eV (kz = Γ). Vertical lines indicate the direction for spin-resolved measurements.

of nematicity, CPS-ARPES was performed on FeSe near both the zone centre and corners, well within the orthorhombic phase at 20 K. As evidenced by the ARPES spectra in Figure (4b) and the Fermi surface in the inset of Figure (4a), we probe primarily one orthorhombic domain [28]. The polarization asymmetry near zone centre for FeSe displays a close similarity to that of LiFeAs, over the entire region of k|| explored in Figure (4b). The enhanced splitting between dxz and dyz levels imbued by the orthorhombic distortion [11] results in a larger amplitude in the measured Pz for FeSe. This combined influence of orbital order and SOC near Γ emphasizes

the need for CPS-ARPES to address SOC directly. Similar to LiFeAs, CPS-ARPES at the zone corner recovered no evidence for substantive coupling (Figure S6). Through study of orthorhombic FeSe, we have demonstrated clearly that the orbital and spin degrees of freedom are coupled by relativistic effects in a way which is not suppressed by the introduction of nematicity. Furthermore, this provides evidence that the effects of SOC are not limited to the pnictides, but rather SOC appears to be of generic importance to the low-energy electronic structure of both principal families of the FeSCs. We have demonstrated here via CPS-ARPES that the modest spin-orbit coupling in FeSCs results in a substantive modification of the electronic states and dispersion near the Fermi-level relevant to superconductivity and other low-temperature phases. While many interactions influence the low energy electronic dispersion, we have distinguished the influence of SOC from other perturbations such as orbital ordering. In the context of unconventional superconductivity, this further distinguishes the FeSCs from the strongly correlated cuprates, and rather is suggestive of comparison with the relativistic superconducting ruthenates [21]. The FeSCs however, illustrate the possibility for supporting high temperature superconductivity in the presence of both correlations and relativistic effects. The principal influence of SOC on the low energy electronic structure has been shown here to be the collapse of the independence of spin and orbital degrees of freedom on the hole pockets near the Brillouin zone centre. Interestingly, these hole pockets are associated in general with the largest reported superconducting gaps [6]. Beyond the material-specific prospect of suppressing spin-susceptibility [26], one consequence of SOC on any theory of superconductivity is the need to incorporate both spin singlet and triplet terms in the pairing equations [29]. These effects can be subtle, and in the case

5 of strongly hole-doped FeSCs, this can even be shown to be necessary for attractive pairing to occur in the swave channel [30]. Discussions to date have primarily disregarded triplet terms and emphasized either orbital or spin-based fluctuation mechanisms supporting some type of s-wave superconductivity. However, the results here suggests that further consideration of the pairing mechanisms put forth thus far and their possible interplay will be needed for a more complete understanding of superconductivity in the Fe-pnictides and chalcogenides. We thank O. Vafek for helpful discussions on the topic. This research was undertaken thanks in part to funding from the Max Planck UBC Centre for Quantum Materials and the Canada First Research Excellence Fund, Quantum Materials and Future Technologies Program. The work at UBC was supported by the Killam, Alfred P. Sloan, and Natural Sciences and Engineering Research Council of Canada (NSERC) Steacie Memorial Fellowships (A.D.), the Alexander von Humboldt Fellowship (A.D.), the Canada Research Chairs Program (A.D.), NSERC, Canada Foundation for Innovation (CFI) and CIFAR Quantum Materials Program. E.R. acknowledges support from the Swiss National Science Foundation (SNSF) grant no. P300P2-164649. This work has been partly performed in the framework of the nanoscience foundry and fine analysis (NFFA-MIUR Italy Progetti Internazionali) facility.

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