Orbital Origin of Extremely Anisotropic Superconducting Gap in Nematic Phase of FeSe Superconductor

The iron-based superconductors are characterized by multiple-orbital physics where all the five Fe 3$d$ orbitals get involved. The multiple-orbital nature gives rise to various novel phenomena like orbital-selective Mott transition, nematicity and orbital fluctuation that provide a new route for realizing superconductivity. The complexity of multiple-orbital also asks to disentangle the relationship between orbital, spin and nematicity, and to identify dominant orbital ingredients that dictate superconductivity. The bulk FeSe superconductor provides an ideal platform to address these issues because of its simple crystal structure and unique coexistence of superconductivity and nematicity. However, the orbital nature of the low energy electronic excitations and its relation to the superconducting gap remain controversial. Here we report direct observation of highly anisotropic Fermi surface and extremely anisotropic superconducting gap in the nematic state of FeSe superconductor by high resolution laser-based angle-resolved photoemission measurements. We find that the low energy excitations of the entire hole pocket at the Brillouin zone center are dominated by the single $d_{xz}$ orbital. The superconducting gap exhibits an anti-correlation relation with the $d_{xz}$ spectral weight near the Fermi level, i.e., the gap size minimum (maximum) corresponds to the maximum (minimum) of the $d_{xz}$ spectral weight along the Fermi surface. These observations provide new insights in understanding the orbital origin of the extremely anisotropic superconducting gap in FeSe superconductor and the relation between nematicity and superconductivity in the iron-based superconductors.

The iron-based superconductors are characterized by multiple-orbital physics where all the five Fe 3d orbitals get involved. The multiple-orbital nature gives rise to various novel phenomena like orbital-selective Mott transition, nematicity and orbital fluctuation that provide a new route for realizing superconductivity. The complexity of multiple-orbital also asks to disentangle the relationship between orbital, spin and nematicity, and to identify dominant orbital ingredients that dictate superconductivity. The bulk FeSe superconductor provides an ideal platform to address these issues because of its simple crystal structure and unique coexistence of superconductivity and nematicity. However, the orbital nature of the low energy electronic excitations and its relation to the superconducting gap remain controversial. Here we report direct observation In the iron-based superconductors, all the five Fe 3d orbitals (d xz , d yz , d xy , d x 2 −y 2 and d z 2 ) are involved in the low energy electronic excitations [1,2]. The multiple-orbital character provides a new degree of freedom which, when combined with charge and spin, brings new phenomena like orbital-selective Mott transition [3,4], orbital ordering [5,6] and nematicity [7].
Orbital fluctuation may provide a new channel for realizing superconductivity [8,9]. On the other hand, such multiple-orbital nature also brings complexity in finding the key ingredients of superconductivity in the iron-based superconductors. FeSe is unique in the iron-based superconductors because it has the simplest crystal structure [10]. It shows a nematic transition at ∼90 K without being accompanied by a magnetic transition [11]. In particular, FeSe superconductor provides an ideal case for studying the relationship between nematicity and 2 superconductivity [12][13][14][15] because superconductivity occurs in the nematic state. However the experimental results are controversial regarding the superconducting gap of FeSe on whether it is nodeless [16][17][18][19][20][21][22] or nodal [23][24][25]. It is also under debate on the orbital nature of the low energy excitations that dictates superconductivity [22,26,27]. Direct determination of the correspondence between the orbital nature of the low energy electronic states and the superconducting gap is crucial to understand the superconductivity mechanism of the iron-based superconductors.
In this paper, we performed high resolution laser-based angle-resolved photoemission (ARPES) measurements on the electronic structure and superconducting gap of bulk FeSe superconductor (T c =8.0 K) in the nematic state. Highly anisotropic Fermi surface around the Brillouin zone (BZ) center is observed with the aspect ratio of ∼3 between the long axis (along k y ) and short axis (along k x ). The superconducting gap along the Fermi pocket is extremely anisotropic, varying between ∼3 meV along the short axis of Fermi surface to zero along the long axis within our experimental precision (±0.2 meV). Detailed band structure analysis, combined with band structure calculations, indicates that the Fermi surface is dominated by a single d xz orbital. Moreover, we find that the superconducting gap size shows an anti-correlation with the d xz spectral weight near the Fermi level; the gap minimum (maximum) corresponds to the d xz spectral intensity maximum (minimum) along the Fermi surface. These observations provide key insights on the orbital origin of the anisotropic electronic structure and superconducting gap in FeSe and the interplay between nematicity and superconductivity in the iron-based superconductors.
The electronic structure and superconducting gap of FeSe superconductor (T c =8.0 K) (see Methods and Fig. S1 for sample details) were measured by high resolution laser-based angle-resolved photoemission system based on the time-of-flight electron energy analyzer (see Methods and Fig. S2 for experimental details). This new ARPES system has an advantage of covering two-dimensional momentum space at the same time with high energy resolution (∼1 meV) and momentum resolution. It is also equipped with an ultra-low temperature cryostat which can cool down the sample to 1.6 K. The laser polarization can be tuned to identify the orbital character of the observed band structure by taking the advantage of the photoemission matrix element effects [28]. This system is particularly suitable for bulk FeSe because of its low T c , tiny Fermi pockets and very small superconducting gap.
The FeSe sample we measured has a structural phase transition T s around 90 K (Fig.  S1b). It assumes a tetragonal crystal structure above T s and becomes orthorhombic below T s where the distance between adjacent Fe is slightly different (Fig. 1a, where a F e (b F e ) is defined as the long x (short y) axis). Dramatic anisotropy of physical properties between a F e and b F e axes occurs below the nematic transition temperature [29][30][31][32] although the lattice constant difference between a F e and b F e is only ∼0.5% [33]. Fig. 1b shows the measured Fermi surface of FeSe around the BZ center which occupies a small portion of the entire BZ.
Our laser-based ARPES system made it possible to cover the whole Fermi pocket around the Γ point simultaneously under the same condition with very dense momentum points. K. The measured Fermi surface (leftmost panel of Fig. 1c) is highly anisotropic: the Fermi momentum along the k x direction is ∼0.036π/a while it is ∼0.11π/a along the k y direction resulting in an elongated ellipse with a high aspect ratio of ∼3. This is the Fermi surface that exhibits the strongest anisotropy observed among all the iron-based superconductors [34]. For the photon energy we used (6.994 eV), the measured Fermi surface corresponds to a k z close to zero [35]. With increasing binding energy, the constant energy contours increase in area ( Fig. 1c), consistent with the Fermi pocket being hole-like. In the meantime, the spectral weight is gradually concentrated onto the two vertex areas along the long axis while it gets strongly suppressed in the central region. We note that for the FeSe single crystal sample we measured here it has nearly pure single domain even though we did not detwin the sample in advance. This may be due to accidental internal stress exerted in the sample during the preparation process because in most cases we can see signals from coexisting two domains   Fig. 1e. When the sample cools down from the normal state to the superconducting state, for both momentum points A and B, sharp superconducting coherence peaks develop which get stronger with decreasing temperature (Fig. 1d). The sharp coherence peak with a width of ∼6 meV is observed at 1.6 K. The EDC symmetrization is a standard procedure to remove the Fermi-Dirac function in order to extract the superconducting gap [36]. The gap size can be determined by the distance between the two peaks in the symmetrized EDCs which can be fitted by a phenomenological gap formula [36]. For point A, the symmetrized EDCs always show a peak at the Fermi level (left panel in Fig. 1e  two-fold symmetry, together with two near-zero points, is rather reminiscent of a p-wave gap form. The measured gap is fitted by the p-wave form ∆ p =|∆ 0 cos(θ)| and the fitted curve is shown as a green line in Fig. 2d. An alternative gap form is s+d type which can also assume two-fold symmetry [40,42]. In Fig. 2d, we also fitted the measured data with two types of s+d form. We first tried ∆ s,d =|∆ 0 +∆ 1 cos(2θ)| which contains a simple s-wave form ∆ 0 and a d-wave form ∆ 1 cos(2θ). The fitted curve (blue line in Fig. 2d) exhibits an obvious deviation from the measured data, particularly near the minimal and maximal gap regions. Then we tried ∆ es,d =|∆ 0 +∆ 1 cos(2θ)+∆ 2 cos(4θ)| which contains an anisotropic s-wave ∆ 0 +∆ 2 cos(4θ) and a d-wave form ∆ 1 cos(2θ). The fitted curve is marked as a red line in Fig. 2d. We note that within our experimental precision (±0.2 meV), we can not differentiate between the cases of zero node, two nodes and four nodes along the Fermi surface for the es+d gap form (Fig. S6). The observation of two nodes on the Fermi surface in terms of es+d gap form is quite accidental because three fitting parameters are needed and a constraint has to be imposed between these parameters, i.e., (∆ 0 +∆ 2 )-∆ 1 =0 (Fig.   S6e). Our precise measurement on the momentum dependence of the superconducting gap puts a strong constraint on the possible gap form in FeSe superconductor.  Fig. 3b). The top of the β band stays at ∼20 meV below the Fermi level (dashed green line in the leftmost panel in Fig. 3b). This band gets flatter when the momentum cuts change from vertical (θ=90 • ) to horizontal (θ=0 • ) directions. Based on the analysis of the photoemission matrix elements (see Fig. S2), the α and β bands can be attributed to the d xz and d yz orbitals, respectively. This assignment is further supported by our band structure calculations ( Fig. 3c and 3d) which capture the momentum dependence of the two bands well (see Methods and Supplementary materials). Above the nematic transition, d xz and d yz bands are split because of the spin orbital coupling (Fig. S3a and S3b) [35]. In the nematic state, the d xz band is pushed up while the d yz band is pushed down below the Fermi level ( Fig. S3c and S3d) [26], resulting in only one hole-like Fermi pocket around Γ which is dominated by the d xz orbital. In particular, the electronic states at the Fermi level along the k x and k y axes are purely from the d xz orbitals, as demonstrated by the matrix elements analysis (Fig. S2) and the band structure calculations (Fig. 3c and 3d).
We note that there is a slight band hybridization of d xz and d yz orbitals on some portions of the Fermi surface with an Fermi surface angle between θ=15 • ∼45 • (Fig. 3c and 3d). But 6 even in this case, the d xz orbital still dominates the electronic states at the Fermi level.
There have been a controversy regarding the orbital nature of the electronic states near the Fermi level [22,26,35,37]. Our results demonstrate unambiguously that it is the d xz orbital that dominates the low energy electronic structure of FeSe around Γ point in the nematic state.
After resolving the dominant d xz orbital nature that dictates the Fermi surface and superconducting gap of FeSe superconductor, now we come to investigate some key factors that are closely related to its superconductivity. Fig. 4a shows the momentum distribution curves (MDCs) at the Fermi level along various Fermi surface angles θ (schematically shown in top-left inset in Fig. 4a) measured in normal state at 9.8 K. The spectral weight (red region in Fig. 4a) is strongest near the θ=90 • and gets weaker when moving to θ=0 • and 180 • . Quantitative analysis of the spectral weight (MDC area) as a function of the Fermi surface angle is plotted in Fig. 4d; it is peaked near θ=90 • and 270 • . Fig. 4b shows photoemission spectra (EDCs) along the Fermi surface measured at 9.8 K in normal state and the spectral weight of the d xz orbital (red region in Fig. 4b) at different Fermi surface angles is plotted in Fig. 4e. It shows similar variation with the MDC weight. The analysis in the superconducting state gives similar results (Fig. S4). For comparison, the superconducting gap of FeSe as a function of the Fermi surface angle measured at 1.6 K is re-plotted in Fig. 4c. Overall the superconducting gap exhibits an anti-correlation relation with the spectral weight of the d xz orbital, i.e., the gap minimum corresponds to the spectral weight maximum. We also analyzed the effective mass of the α band as the function of the Fermi surface angle, as shown in Fig. 4f. The effective mass displays a dramatic variation along the Fermi surface; its value along the k x axis is nearly ten times that along the k y axis.
It also shows a perfect anti-correlation with the superconducting gap: the gap maximum (minimum) corresponds to the minimum (maximum) of the effective mass.
The present observations provide key information on understanding the relationship between nematicity, electronic structure and superconductivity in FeSe. In the nematic state, the splitting of the d xz and d yz orbitals gives rise to an anisotropic Fermi surface around the Γ and X/Y points (corresponding to BZ of 1-Fe unit cell) [35,37]. In the extreme case when the low energy electronic states are dominated by a single d xz orbital, the electron hopping along the x direction is much more enhanced than that along the y direction (Fig.   4h), resulting in highly anisotropic Fermi surface topology (Fig. 1c) and effective mass (Fig.  4f) in the nematic state of FeSe. Since nearly isotropic superconducting gap with four-fold symmetry is observed in tetragonal Fe(Se,Te) superconductors [38,39], the observed extremely anisotropic superconducting gap with two-fold symmetry in FeSe is expected to be associated with its nematicity. The question comes to how nematicity can generate such an anisotropic superconducting gap. One scenario proposed involves the splitting of the d xz and d yz orbitals [40,41]. The orbital order in the nematic state introduces a d-wave component on top of the existing s-wave component, leading to a highly anisotropic superconducting gap with even accidental nodes [40]. Our measured superconducting gap (Fig. 2d) is consistent with the expected es+d type gap form in this picture [40]. However, it is expected that the superconducting gap is positively correlated with the orbital occupation [41], which is opposite to our observation that the gap and the spectral weight exhibit an anti-correlation ( Fig. 4c-4e). This picture involves two hole-pockets around the Γ point, each of which is composed of both d xz and d yz orbitals [40,41]. It needs to be modified in order to make a direct comparison with our results of FeSe where there is only one Fermi pocket around the Γ point which is dominated by a single d xz orbital.
To understand the anisotropic superconducting gap of FeSe, another scenario proposed considers orbital-selective Cooper pairing in a modified spin-fluctuation theory via suppression of pair scattering processes involving those less coherent states [22]. This picture can produce a superconducting gap structure that is consistent with our results (Fig. 2d) and previous experimental observations [22,42]. It was proposed that the pairing is mainly based on electrons from the d yz orbital of the iron atoms [22,27]. Our observations provide direct evidence that, in the nematic state of FeSe, the extremely anisotropic superconducting gap opens on the highly anisotropic Fermi surface (Fig. 4g) that is dominated by the d xz orbital. In FeSe, it was also proposed that the superconducting gap shows a positive correlation with the spectral weight of the d yz orbital that is considered to be responsible for superconductivity [22]. From our direct ARPES results, although we find that the spectral weight of the d yz orbital exhibits a good correspondence to the superconducting gap ( Fig. S5), the β band with d yz orbital character lies ∼20 meV below the Fermi level, thus contributing little to superconductivity of FeSe. Our results indicate that superconductivity in FeSe is dictated by the d xz orbital and its spectral weight shows an anti-correlation with the superconducting gap: the gap minimum corresponds to the spectral weight maximum of the d xz orbital (Fig. 4c-4e). Our results ask for a reexamination on the picture of the orbital-selective superconductivity in FeSe [22].
The fact that the measured gap function can be fitted to a simple p-wave gap function also leads us to ask whether the spin-triplet pairing is possible in FeSe. In the nematic state, because of the dominant single d xz orbital feature of the Fermi surface around the Γ and the strong Hund's rule coupling, the interaction of the d xz band with other bands near X/Y is expected to become rather weak. For the d xz orbital, the dominant intra-orbital coupling is the nearest neighbor magnetic exchange couplings along the x axis, namely the J 1x -interaction shown in Fig. 4h, which is ferromagnetic in iron-chalcogenides from neutron scattering measurements [43][44][45][46]. If this is the strongest pairing interaction, the superconducting pairing on the Fermi surface is possible to have a triplet p-wave symmetry with a gap function proportional to cos(θ), which is consistent with our observations. No drop of the Knight shift across T c is observed in NMR measurements of bulk FeSe, which is also compatible with a possible triplet pairing [31,32]. In addition to the coexisting Néel and stripe spin fluctuations observed by neutron scattering [47,48], recent observation of charge ordering in FeSe suggests a presence of an additional magnetic fluctuation with a rather small wavevector [49] that is related to intra-pocket scattering around Γ point [50].
Our present work raises an interesting possibility of p-wave pairing symmetry in FeSe. We note that ARPES can only measure the magnitude of the superconducting order parameter.
Also because of the limitation of the laser photon energy (6.994 eV), it is not possible for us to measure the superconducting gap on the Fermi surface around the X/Y point (Fig.   1b). Considering that the measured superconducting gap can be fitted quite well by both a simple p-wave form (green line in Fig. 2d)
Electrical resistance measurement and magnetic measurement (Fig. S1b, c) show a T c of 8.0 K with a sharp transition width of ∼0.4 K.
High-resolution ARPES measurements were performed at our new laser-based system equipped with the 6.994 eV vacuum ultraviolet laser and the time-of-flight electron energy analyser (ARToF 10k by Scienta Omicron) [53]. This latest-generation ARPES system is capable of measuring photoelectrons covering two-dimensional momentum space (k x , k y ) simultaneously. The system is equipped with an ultra-low temperature cryostat which can cool the sample to a low temperature of 1.6 K. Measurements were performed using both LH and LV polarization geometries (see Fig. S2). The energy resolution is ∼1 meV and the angular resolution is ∼0.1 degree. All the samples were measured in ultrahigh vacuum with a base pressure better than 5.0×10 −11 mbar. The samples were cleaved in situ and measured at different temperatures. The Fermi level is referenced by measuring polycrystalline gold which is in good electrical contact with the sample, as well as the normal state measurement of the sample above T c .
To simulate the band structure of FeSe, we adopted 5-orbital tight-binding model including onsite spin-orbital coupling (λ). In the nematic state, we further consider s-wave (∆ s ) and d-wave (∆ d ) orbital order which break C 4 rotational symmetry. The Hamiltonian of these two orders are given by, where n α,k = n α,k↑ + n α,k↓ is the density for α orbital. To match the data in ARPES experiments in the nematic state, the adopted parameters in the calculations are λ=10 meV, ∆ s =17.5 meV and ∆ d =-10 meV. In normal state, we set λ=10 meV, ∆ s =0 meV and ∆ d =0 meV.