Electronic Nematic States Tuned by Isoelectronic Substitution in Bulk FeSe_1−xS_x

Abstract

Isoelectronic substitution is an ideal tuning parameter to alter electronic states and correlations in iron-based superconductors. As this substitution takes place outside the conducting Fe planes, the electronic behaviour is less affected by the impurity scattering experimentally and relevant key electronic parameters can be accessed. In this short review, I present the experimental progress made in understanding the electronic behaviour of the nematic electronic superconductors, FeSe_1−xS_x. A direct signature of the nematic electronic state is in-plane anisotropic distortion of the Fermi surface triggered by orbital ordering effects and electronic interactions that result in multi-band shifts detected by ARPES. Upon sulphur substitution, the electronic correlations and the Fermi velocities decrease in the tetragonal phase. Quantum oscillations are observed for the whole series in ultra-high magnetic fields and show a complex spectra due to the presence of many small orbits. Effective masses associated to the largest orbit display non-divergent behaviour at the nematic end point (x ∼ 0.175(5)), as opposed to critical spin-fluctuations in other iron pnictides. Magnetotransport behaviour has a strong deviation from the Fermi liquid behaviour and linear T resistivity is detected at low temperatures inside the nematic phase, where scattering from low energy spin-fluctuations are likely to be present. The superconductivity is not enhanced in FeSe_1−xS_x and there are no divergent electronic correlations at the nematic end point. These manifestations indicate a strong coupling with the lattice in FeSe_1−xS_x and a pairing mechanism likely promoted by spin fluctuations.

1 Introduction

Nematic electronic states have been suggested to play an important role in understanding the electronic behavior of high temperature superconductors like cuprates and iron-based superconductors, quantum Hall systems, and Sr₃Ru₂O₇ [1]. Often the nematic electronic state breaks the rotational symmetry of a tetragonal plane lattice from four-fold symmetric (C₄) down to two-fold symmetric (C₂) [1]. This symmetry breaking is expected to have a number of consequences on the electronic properties leading to a series of effects involving anisotropic single-particle properties, showing a distorted Fermi surface (that can be triggered like a Pomeranchuk instability in the presence of interactions [2]), anisotropic spin-fluctuation spectra, and anisotropic transport properties that can lead to non-Fermi-liquid behavior [3]. Theoretically, in the proximity to a nematic quantum critical point, the nematic fluctuations with wave-vector q = 0 can enhance the critical temperature by pairing through the exchange of nematic fluctuations in all symmetry channels [4, 5]. In real systems, the nematic electronic phase is intimately coupled with the lattice. This coupling has significant consequences on the observed response, such as the presence of the tetragonal-to-orthorhombic structural transition at the same temperature where the nematic electronic order develops. This finite coupling of the electronic system with the lattice is expected to alter the response of the nematic critical fluctuations on superconductivity and the non-Fermi liquid power-law dependencies in transport [6–9].

Iron-based superconductors offer a unique playground to understand unconventional superconductivity and explore the normal competing electronic phases, such as nematic electronic phases and spin-density wave phases. Often the nematic and spin-density phases neighbor each other in the phase diagrams of iron-based superconductors, making it difficult to assess whether the spin or nematic fluctuations are the most relevant for stabilizing superconductivity [10]. A unique system, the chalcogenides FeSe_1−xS_x, provides an essential route to investigate the interplay between nematicity and superconductivity, in the absence of long-range magnetism. Furthermore, the isoelectronic substitution can access the experimental manifestations around a putative nematic critical point, undisturbed by the presence of a magnetic critical point, as found in other systems, like BaFe₂(As_1−xP_x)₂ [11]. The parent compound of this family, FeSe, displays a nematic electronic phase and a tetragonal-to-orthorhombic transition below 90 K [12] and no long range magnetic order was detected despite a rich spectrum of low and high-energy spin fluctuations [13–15].

The bulk superconductivity of FeSe has a relatively low critical temperature close to 9 K but it can be enhanced toward 40 K by applied external pressure [16, 17]. The nematic phase of FeSe is also suppressed at low pressures [16–19] before a new magnetic state is stabilized at high pressures [19, 20], that competes with the high-T_c phase [17]. Besides applied pressure, the bulk superconductivity of FeSe can be enhanced toward 40 K via the intercalation between the van der Waals layers of a molecular spacer [21], and by gating of thin flakes [22]. In a monolayer on FeSe, on a suitable substrate, the transition temperatures reach record values toward 65 K; a strong interfacial electron-phonon coupling and a charge transfer through the interface is proposed as a source for this two-dimensional high-T_c superconductivity [23, 24]. This effect is surprisingly absent in a monolayer of FeS [25] and in the absence of substrate in thin flakes of FeSe [26].

Isoelectronic substitution is a clean and efficient way to tune phase diagrams of iron-based superconductors, by gently suppressing the relevant electronic interactions and competing electronic phases with superconductivity, and to access quantum critical points [34]. For the FeSe family this is achieved by replacing elements with a similar number of electrons outside the Fe planes, using sulphur or tellurium ions on selenium ions sites. The availability of single crystals of these materials have allowed intense interest and study of their physical properties, summarized in recent reviews in Refs. 15, 35–37. Furthermore, by combining physical and chemical pressures, the relative position of the nematic electronic phase in relation to the spin-density wave phase can be varied and thus the influence of two competing electronic phases on superconductivity can be disentangled [38, 39]. The scope of this review is to summarize the recent experimental efforts in understanding the electronic behavior of FeSe_1−xS_x that can provide a unique insight into the role played by the nematicity, Fermi surfaces, proximity to a putative nematic critical point and electronic correlations in relation to superconductivity in the absence of any long-range magnetic order.

2 Phase Diagram of FeSe_1−xS_x

Figure 1A shows the phase diagram of FeSe_1−xS_x as a function of isoelectronic substitution with sulphur obtained from transport measurements. The isoelectronic substitution achieved by replacing selenium ions for sulfur ions outside the Fe plane causes an positive internal chemical pressure as these ions have different ionic radii (S²⁻ is 1.70 Å compared with 1.87 Å for Se²⁻) [40, 41]. The nematic electronic phase of FeSe, due to its finite coupling with the underlying lattice, triggers a structural transition from a tetragonal to an orthorhombic phase at T_s [12]. This transition gives rise to a well-defined anomaly in the transport measurements (Figure 1C) that helps to build the nematic phase diagram and to identify the expected position of each measured single crystal inside the nematic phase, as shown in Figure 1A.

FIGURE 1

The isoelectronic substitution with sulfur in FeSe leads to the efficient suppression of the nematic electronic state, similarly to the effect of applied pressure [42] (Figure 1A). In contrast to applied pressure, the nematic phase can be completely suppressed with sulfur substitution and no spin-density wave phase was detected for any available single crystals. The lowest detected value of T_s is about 25 K, followed by an abrupt drop at the nematic end point (NEP) which occurs close to x ∼ 0.175(5), [27, 29, 40, 43, 44]. Thus, FeSe_1−xS_x family is unique and permits the exploration of the nematic electronic phase transition in the vicinity of a putative nematic critical point.

Inside the nematic phase, the superconducting transition temperature displays a small dome reaching T_c ∼ 11 K close to x ∼ 0.11, varying from 8.7 (3) K for FeSe inside the nematic phase toward 6.5–5 K just outside it [29, 30, 44, 45]. For higher x values inside the tetragonal phase, the superconductivity hardly changes reaching only 4.5 (5) K toward FeS [32, 33]. The suppression of the nematic phase transition in FeSe_1−xS_x coincides with a decrease in the superconducting transition temperature T_c close to NEP. STM studies have detected a rather abrupt change in the superconducting order parameter at the nematic phase boundaries, implying that different types of pairing may be operational inside (SC1) and outside the nematic phase (SC2), as shown in Figure 1A [30, 46]. Recently, it has been suggested theoretically that a topological transition associated with the creation of a Bogoliubov Fermi surface could occur as a function of x in FeSe_1−xS_x [47, 48].

In order to understand in depth the electronic properties of FeSe_1−xS_x a good knowledge of the exact composition and the sulfur variation in each batch is required. This can be challenging for techniques, like neutron-diffraction and muon spin rotation, that require a large mass of sample made of hundreds of small single crystals [49]. At room temperature FeSe_1−xS_x crystalizes in the P4/nmm space group (No. 129), as shown in Figure 1B. The lattice parameters of FeSe are a = 3.7651 Å, c = 5.5178 Å, z_Se = 0.2672 [31] whereas FeS has a much smaller c axis (a = 3.6802 Å, c = 5.0307 Å, z_S = 0.2523) [32, 33]. The lattice parameters measured by X-ray diffraction for each crystal of FeSe_1−xS_x can be used to determine the composition of each sample. Assuming the formation of a solid solution as a function of composition, the values of the lattice parameters at room temperature, p, (that can be a,b and c or z_Se/S) for a certain composition x can be estimated using an empirical Vegard’s law p_x = xp_S + (1 − x) p_Se

2.1 Single Crystal Growth

Single crystals of FeSe_1−xS_x are normally grown by the KCl/AlCl₃ chemical vapor transport method from the FeSe end toward x ∼ 0.4 [12, 29, 40, 43, 44, 50, 51]. The growth of higher concentrations and FeS was achieved using a hydrothermal reaction of iron powder with sulfide solution, which in general is a more invasive method and can lead to single crystals with higher concentration of impurities [29, 32, 33, 52]. Epitaxial thin films of FeSe_1−xS_x with x < 0.43 were grown via pulsed laser deposition [53]. A potential anomaly was observed in the resistivity curves for films with large x, suggested to be linked to a magnetic transition [53], but these findings have not been yet confirmed in single crystals [44, 53]. The exact x composition for samples in each batch is normally checked using compositional analysis using energy-dispersive X-ray spectroscopy (EDX) or electron-probe micro-analysis (EPMA) [12, 27, 44, 50, 53]. The nominal composition, x_nom, used during the growth process is often smaller than the real composition x (by about 80%) and the higher the composition the larger degree of variation occurs within the same batch [44, 53]. For example, the phase diagrams of FeSe_1−xS_x reported in Refs. 54, 55 uses the nominal values x_nom. Thus, the linear resistivity in 35T occurs inside the nematic phase, as the measured zero resistivity shows an anomaly at T_s ∼ 51 K for a nominal composition x_nom ∼ 0.16 that would correspond around x ∼ 0.13 [28]. The residual resistivity ratio, defined as the ratio between room temperature resistivity and the resistivity at the onset of superconductivity, varies between 15 and 44 [27, 28] and it is often used as a proxy to assess the quality of each single crystal. In high magnetic field, quantum oscillations were observed for all composition of FeSe_1−xS_x reflecting their high quality with large mean free path (up to ∼350 Å) [27, 39]. For higher x composition, the mean free path decreases slightly and new hexagonal phases could be stabilized [27]. The superconductivity of Fe_1+δSe can also be destroyed by very small changes in its stoichiometry [31].

3 Electronic Structure of FeSe_1−xS_x

The main features of the electronic structure of FeSe_1−xS_x can be understood by considering the two-dimensional square lattice of Fe ions, separated by Se/S atoms residing above and below the Fe layer, as shown in Figure 1B. Due to the strong bonding between the Fe-Fe and Fe-(Se/S) sites, an Fe atom can be placed inside the center of an almost perfect tetrahedron of Fe(Se/S)₄ that determines the electronic properties of these materials. Band structure calculations show that the Fermi surface of FeSe_1−xS_x consists of well-separated hole pockets at the center of the Brillouin zone and electron pockets at the zone corners that form quasi-two dimensional Fermi surfaces, as shown in Figures 1D,E. The position of the chalcogen ion in relation to the Fe plane, z, affects significantly the predicted number and the orbital character of the hole bands, FeSe having an additional middle hole band with d_xy character which is pushed below the Fermi level in FeS (Figures 1D,E). There are two predicted cylindrical electron bands which hardly change in shape across this series, similar to the isoelectronic series BaFe₂(As_1−xP_x)₂ [56]. The positive chemical pressure in FeSe_1−xS_x results in a lattice contraction and the reduction of the c axis [41] and it would bring the Fe(Se/S) layers closer together, increasing the bandwidth and potentially leading to the suppression of the electronic correlations [29, 57]. As discussed below, DFT calculations provide essential guide to understand the origin of the observed Fermi surfaces of FeSe_1−xS_x, but the size are smaller, the number of hole bands is reduced compared with calculations and the k_z dependence is changed.

4 ARPES Studies of FeSe_1−xS_x

ARPES is highly suited for the exploration of FeSe_1−xS_x as these systems can be easily cleaved in-situ due to weak van der Waals bonds between the FeSe layers which also enable the development of devices of two-dimensional superconductors by mechanical exfoliation [22, 26]. Furthermore, ARPES studies can evaluate the role of orbital character on the nematic electronic states, as the matrix element effects affect the intensity of different bands with different orbital character. In certain conditions, ARPES spectra of iron-based superconductors does not show certain branches due to the underlying symmetry, in particular for the electron bands [59–61], as the intensity depends strongly on the polarisation of the incident beam as well as the incident photon energy. A representation of the orbital character of different pockets at high symmetry points is shown in Figure 2H.

FIGURE 2

Extensive experimental ARPES studies on FeSe found that system has many relevant electronic ingredients for a multiband system [35, 59, 62–66]. The experimental Fermi surface of FeSe is unusually small having 2 electron pockets and a single hole pocket (instead of 3), a factor 5 smaller than that predictions of the band-structure calculations (Figure 1E). Such a small Fermi surface could be sensitive to topological changes in magnetic fields or under applied strain. To bring the DFT calculated Fermi surfaces in agreement with experiments, band shifts need to be applied in opposite direction for hole and electrons of more than 200 meV for FeSe [62] and less than 100 meV for FeS [67]. Band shifts also occur at high temperatures inside the tetragonal phase of FeSe and these effects are caused by higher energy interactions [68] as well as the changes in the chemical potential [69], as found in many iron-based superconductors [70]. Furthermore, like many other iron chalcogenides, FeSe exhibits strong orbitally-dependent electronic correlations due to the larger band renormalization factor ∼7–9 of the d_xy band compared with ∼3–4 for the d_xz/yz band [62, 71]. These values are obtained by comparing the experimental band dispersion to those from DFT calculations in the tetragonal phase, as shown in Figure 1 [62, 71]. At high binding energies, ARPES spectra detected Hubbard-like bands suggesting the existence of incoherent many-body excitations originating from Fe 3d states, in addition to the renormalized quasiparticle bands near the Fermi level [63, 72]. Many high energy features of the observed ARPES data can be accounted for by considering the strong local Coulomb interactions on the spectral function via dynamical mean-field theory, including the formation of a Hubbard-like band [63, 72]. Another inherent challenge for ARPES studies inside the nematic phase is the likely presence of sample twinning (rotated by 90°), by cooling thorough the structural transition, and a lot of recent effort has been dedicated to address this issue by applying strain to FeSe [65, 66, 73].

4.1 Hole Pockets of FeSe_1−xS_x

The evolution of the hole pockets of FeSe_1−xS_x with x substitution for the two high symmetry points Z (at the top of the Brillouin zone) and (at the center of the Brillouin zone) at low temperatures is shown in Figures 2A,B, respectively. The observed energy dispersions of FeSe_1−xS_x are all renormalized and shifted, as compared to the DFT dispersions, leading to much smaller hole and electron pockets, as compared with calculations [62]. The renormalization values corresponding to the two main hole dispersions (with d_xz/d_xz orbital character) are around 3–4 and hardly change for any compositions inside the nematic phase toward x ∼ 0.180.18, but they are reduced to a factor of ∼1–2.3 for FeS, suggesting that the suppression of electronic correlations occurs from FeSe toward FeS [29, 57]. Additionally, the highly renormalized d_xy band, found at ∼50 meV below the Fermi level, remains relatively unaffected across the nematic phase transition to x ∼ 0.18 and it cannot be resolved for FeS due to the disorder effects [29]. The d_xy hole band is notoriously difficult to observe in experiments due to matrix element effects and being strongly incoherent in iron-chalcogenides [74] but its dispersion can be revealed due to band mixing caused by the spin-orbit coupling effects [29, 75]. As a function of x substitution, the Fermi velocities increase by ∼10% inside the nematic phase but more significantly outside toward FeS, reflecting the suppression of electronic correlations, as shown in Figure 4E.

FIGURE 3

FIGURE 4

The ARPES studies at the two high symmetry points using different incident energies allows for the evaluation of the k_z hole dispersion of the cylindrical Fermi surfaces of FeSe_1−xS_x and the sensitivity of the second inner hole band to any band shifts inside the nematic phase. The inner hole band, which forms a small 3D inner hole pocket around the Z point, is observed in the tetragonal phase of FeSe but it is pushed below the Fermi level inside the nematic phase [62]. With sulfur substitution, the inner hole band is shifted gradually up and it crosses the Fermi level only at the Z point from x ∼ 0.11 [40] and grows in size at x ∼ 0.18. The inner hole band does not cross the Fermi level at the point for any compositions up to x ∼ 0.18; however, it has been suggested that this pocket could grow in size and become a two-dimensional cylinder for FeS (Figures 1D,E) [29, 57, 67]. Interestingly, the Fermi surface of the tetragonal phase for x ∼ 0.18 is very similar to the Fermi surface of FeSe at high temperature. Thus, there is a direct correspondence between the temperature and sulfur substitution in FeSe of the nematic electronic structure of FeSe_1−xS_x [29, 40].

4.2 Simulations of the Effect of Nematicity on the Low-Energy Electronic Structure

To understand the effect of nematicity and the orbital effects at each high symmetry point, simulations based on a model developed in Refs. 76, 77 for a single domain sample are shown in Figure 3. The parameters for the simulations are adjusted to match the ARPES experimental data for x = 0.18 in Figures 2B,C [29] and the other variable are listed in Figure 3. In the tetragonal phase of FeSe_1−xS_x, the hole Fermi surface are expected to be circular and C₄ symmetric, originating from the d_xz/yz bands, as shown in Figure 2E. In the absence of nematicity the two hole dispersions at the center of the Brillouin zone are expected to be split only by the spin-orbit coupling [76], as shown in Figure 3A. Experimentally, the band separation gives a spin-orbit of ∆_so ∼ 13(3) meV for FeSe in the tetragonal phase at high temperatures and for the tetragonal x ∼ 0.18 at low temperatures [62, 63, 80]. As the nematic order is turned on, [76], the hole pocket is expected to become distorted and the splitting between the two hole band dispersions increases, as shown in Figure 3. In addition, the increase in the orbital ordering effects moves the inner 3D hole pocket at the Z point completely below the Fermi level, as shown in Figure 2 [40, 81]. Experimentally, at low temperature FeSe has only one quasi-two dimensional hole Fermi surface (compared with 3 predicted by the DFT calculations) with an elliptical in-plane area at the high symmetry points, as shown in Figure 2E. The signature of this nematic electronic phase can be induced by orbital-ordering effects and electronic interactions that can drive a Pomeranchuck instability of the Fermi surface [2]. Other scenarios have been addressed theoretically in detail in other works [82–86]. Since samples of FeSe_1−xS_x inside the nematic phase can form twin domains rotated by 90° below T_s often ARPES experiments visualizes two superposed ellipses, as shown in Figure 2E, but only a single ellipse may be observed in detwinned measurements on FeSe [63, 64, 81]. As the orbital ordering is reduced with S substitution, the splitting between the inner d_yz bands and outer d_yz hole band dispersion is smaller and at low temperatures from x ∼ 0.11, the inner hole band crosses the Fermi level at the Z point, leading to the formation of the small 3D hole pocket. Inside the nematic phase, the in-plane Fermi surfaces are highly anisotropic, indicated by the splitting of the two different k_F values (obtained from the MDC cuts of two different domains), as shown in Figure 3. With increasing x, in-plane Fermi surface becomes a circle for both hole pockets for the tetragonal x ∼ 0.18 but the cylindrical hole Fermi surface has a strong k_z dependence [29].

4.3 Electron Pockets of FeSe_1−xS_x

Whereas the behavior of the hole bands is well understood and consistent between different experimental reports, the behavior of the electron bands remains a highly debated subject. The P4/nmm unit cell of tetragonal FeSe includes two Fe sites which are related by a glide symmetry [76] and ARPES measurements should detect electron bands emerging from two-crossed ellipses [58, 87], similar to other systems, such as LiFeAs or NaFeAs [80]. The electron pockets suffer a significant change inside the nematic phase and the relevance of different orbital contribution is still being debated. At the corner of the Brillouin zone (M and A point) in the tetragonal phase, there are two degenerate doublet states, M1 and M3, at the zone corner protected by the space-group symmetry, even when spin-orbit coupling is taken into account [76], as shown in Figures 3C,D. Therefore, any splitting and shifts of the bands at M (or A) would reduce the crystal symmetry in the presence of the spin-orbit coupling. The nematic order can be triggered by the development of the anisotropy in the on-site energies of the d_xz and d_yz orbitals ( term) and anisotropic d_xy hopping ( term) [76]. The two-crossed ellipse, corresponding to the electron pockets (Figure 2H), are expected to have a finite splitting between the inner and outer orbits, due to the spin-orbit coupling in the tetragonal phase, as shown in Figure 3C. Inside the nematic phase, by increasing both the and , the degeneracy at the M1 and M3 points are lifted and the bands split apart; this promotes the in-plane distortion of the Fermi surface along its longest axis (Figures 3C,D). Further increasing , which is related to orbital order induced by the increase orbital polarization of the d_xz vs. d_yz bands, the inner electron band is pushed up and eventually it can disappear; thus only a single electron pocket is present, as shown in Figure 3D. Furthermore, orbitally-induced shifts could shrink the electron pocket along one direction, transforming it into two small Fermi pockets, whereas along the other direction, the electron pocket is enlarged into a peanut shape [63, 88]. Indeed, at low temperatures, the band giving rise to the inner electron pocket at the M point is very close to the Fermi level about 3 meV for FeSe (within experimental resolution of 3 meV). This proximity creates the conditions for a topological transition of the electron pocket into a peanut or Dirac-like crossing, under other perturbations, such as applied strain [65, 73], as found for thin films of FeSe under internal strain from the substrate [88].

Experimentally, in the tetragonal phase the inner electron band dispersions at M (or A) are expected to have d_xz/d_yz orbital character when probed along the diagonal of the Brillouin zone, as shown in Figure 2H. The outer electron band with d_xy orbital character is harder to observe due to matrix element effects and the incident energy used (Figures 2C,J). This behavior is detected for the tetragonal FeSe_1−xS_x with x = 0.18 shown in Figure 2C; [29] and FeSe above T_s [40, 58, 62]. In the nematic phase, the changes for the electron bands are drastic, with M1 point shifting up whereas the M3 shifts down, and additional splitting could take place around these two degenerate points, as shown in Figure 2C. The energy separation between the two intense features at the M point below T_s (EDC cuts), is defined by ∆_M which is ∼50 meV for bulk FeSe [62, 64, 89]. This splitting is much larger than what would be expected from DFT calculations simply taking into account its small orthorhombic distortion (∼5 meV) [62]. The elongated directions of the elliptical Fermi surfaces at the M point are rotated by 90° with respect to that at the point due to the momentum-dependent sign-changing orbital polarization, where the d_xz band shifts upward at the point but downward at the M point [81, 90]. Interestingly, 20 meV already separates the two doublets in the tetragonal phase at the M point, in the absence of nematicity, for the tetragonal system with x = 0.18 (see Figure 2G), and FeSe at high temperatures [63]. This implies that the energy scale of the nematic order could be smaller that 50 meV, as shown in Figure 4A [35].

A direct signature of the nematicity is the in-plane distortion of the Fermi surface. Inside the nematic phase for the electron pockets this can be related to the development of the orbital polarisation ∆n = n_xz−n_yz. The orbitally dependent band shifts cause the inner sections of the electron pockets with d_yz orbital character to contract whereas the d_xz sections to expand, but forming a cross-shape due to effect of sample twinning, as shown in Figure 1F. The degree of anisotropy of the Fermi surface can be related to (k_F−k_F0)/k_F0 [40]. where the k_F-vector is that corresponding to the inner d_yz portion of the electron pocket, and k_F0 is the Fermi k-vector in the tetragonal phase for each compound. Figure 4B shows the evolution of the Fermi surface elongation with x substitution and indicates that the nematic phase is responsible for this in-plane distortions, which is completely suppressed in the tetragonal phase.

4.4 Comparison Between ARPES and QPI

The presence of both highly elongated and isotropic Fermi surfaces of FeSe_1−xS_x is likely to significantly influence other measurements. Scanning tunneling microscopy (STM) studies shows highly anisotropic quasiparticle interference (QPI) patterns inside the nematic state, becoming isotropic in the tetragonal phase [46, 78]. The resulting QPI spectra exhibit electron-like and hole-like dispersions along different directions (q_a and q_b, respectively) corresponding to the intraband back-scatterings in the electron bands at the Brillouin zone corner and in the hole band at the zone center, respectively. Thus, the QPI spectra reflect the evolution of the scattering processes across the series FeSe_1−xS_x.

To clarify the qualitative relation between the QPI branches and the band structure, the scattering q vectors from the intraband backscattering can be compared with the Fermi wavevector extracted directly from ARPES dispersions at the Fermi level, as shown in Figure 4C. The Fermi momenta of FeSe of a distorted deformed Fermi cylinder can be estimated from the scattering vectors at zero energy (q/2∼k_F) to be ∼0.05 and 0.08 Å⁻¹ for the hole band and ∼0.04 Å⁻¹ for the electron band [37, 46]. On the other hand, the Fermi wavevector from ARPES for FeSe for the elliptical hole pocket at the point varies between ∼0.035 and 0.08 Å⁻¹ [40, 62], but is larger at the Z point (∼0.1 and 0.15 Å⁻¹), as shown in Figure 4C. These values are close to those from laser ARPES data (which are usually measured around 7 eV which corresponds to a k_z position closer to the point), varying between and  Å⁻¹. Thus, the resulting elongated hole ellipse of FeSe, with a high aspect ratio (∼3), is one of the most anisotropic Fermi surfaces among all the iron-based superconductors [91].

A direct comparison between ARPES and QPI data on FeSe_1−xS_x suggest that the scattering vectors in QPI are likely to correspond to the in-plane Fermi vectors at the center of the Brillouin zone (k_z = 0). This is in agreement with the assignment of the scattering vectors in Ref. 78. They are less sensitive to k_z dependent scattering processes outside of this plane, despite recent theoretical suggestions for FeSe [92], as the k_F vectors at Z are much larger than the scattering vectors (q/2) extracted from QPI, as shown in Figure 4C. In the case of the electrons pockets, the QPI scattering vector is close to those of the small inner electron wave vector (∼0.02(1) Å⁻¹) rather than to the long elongated axis of the ellipse (0.14 (1) Å⁻¹), found in ARPES [35, 58]. Furthermore, the estimated Fermi energies from the QPI dispersions, for the hole bands are of 10–20 meV [46], in good agreement with the top of the hole band at the point of ∼17 meV from ARPES [35, 62]. Laser ARPES data (measured away from a high-symmetry point) give slightly lower values of ∼6.7 meV or 10 meV for the hole band [91, 93]. For the electron bands the Fermi energies of 5–10 meV from QPI are close to the 3–5 meV corresponding to the inner electron band dispersion [35, 63]. With increasing x, the bottom of the inner electron bands at the M point is pushed lower below the Fermi level from about 3 meV toward 15 meV for x = 0.18 [29, 95]. The outer hole band crossing at the Z point lies around 25 (3) meV from x = 0 to x = 0.18, but decrease more significantly for the hole point at the point (Figure 4D). These shifts bring the cylindrical hole band into the regime to undergo a possible Lifshitz transition at the nematic end point, as suggested by quantum oscillations [27].

The evolution of the QPI scattering vectors along one direction performed in a single domain of FeSe_1−xS_x have been associated to two scattering vectors along short elliptical axis in different scattering planes: q_h2 at k_z = 0 ( point) and q_h1 at k_z = (Z-point) [46]. The QPI dispersion associated with the q_h1 scattering vector at the Z-point was suggested to disappear close to x ∼ 0.1, whereas q_h2 increases toward x ∼ 0.25, as shown in Figure 4C. However, there is no evidence for the disappearance of hole bands up to x = 0.18 in ARPES data at the Z point, as shown in Figures 2A–C and Figure 4C. An alternative explanation could invoke a scenario in which the two scattering vectors (q_h1 and q_h2), would merge into a single value for a isotropic Fermi surface close to x ∼ 0.17. Thus, for an isotropic system a single hole dispersion could be visible in the QPI and in that case the scattering vectors along orthogonal direction would be similar to those from ARPES at the point. As QPI is less sensitive to k_z dependent scattering processes, the small 3D hole band at the Z point in Figure 4C is not detected [46]. Further experimental QPI experiments with cuts along different directions as well as theoretical work will be needed to reconcile quantitative features obtained from ARPES and QPI for FeSe_1−xS_x.

5 Quantum Oscillations in High Magnetic Fields

A powerful technique to access directly the Fermi surface of FeSe_1−xS_x is via quantum oscillations in very high magnetic fields and at low temperatures below 1.5 K [27, 42, 62, 95]. Quantum oscillations originate from the oscillations in the density of states in the presence of the Landau quantization of a metallic system in an applied magnetic field. The quantum oscillations are periodic in 1/B and the frequency of these oscillations relates directly to extremal areas of the Fermi surface via Osanger relation ( with frequencies in Tesla ∼10⁻¹⁶ (Å⁻²) of the cross-section area of each orbit). For a slightly-warped cylindrical Fermi surface two frequencies would be observed at the center (k_z = 0) and the top of the Brillouin zone , and in the case of twinned crystals the cross section areas of different domains would coincide. Quantum oscillations are normally observed only in clean single crystals as the cyclotron energy which separates Landau levels needs to be larger than the broadening of the levels due to scattering. Quantum oscillations have been observed for all x compositions of FeSe_1−xS_x [27, 42, 67, 96]. The isoelectronic substitution result in relatively similar mean free paths (using the Dingle analysis for the maximum hole band orbit [98]), having values of ∼ 277 (35) Å for FeSe and ∼ 283 (20) Å for x ∼ 0.19 [27]. Besides impurity scattering effects, the amplitude of the quantum oscillations is significantly suppressed for heavier quasiparticle masses as a result of the smearing of the Landau levels by the Fermi-Dirac distribution and often heavier masses cannot be observed.

5.1 Comparison Between ARPES and Quantum Oscillations

As compared with ARPES, quantum oscillations are insensitive to surface states and the signal is dominated by the bulk response thus giving an unambiguous probe of the bulk Fermi surface. Furthermore, they have a much better k-space resolution of 10³ of the area of the Brillouin zone that allows very accurate determination of the cross-section orbits on the Fermi surface, for a particular magnetic field orientation. However, the location of the orbits in the k-space is not easily known for multiband systems (Figures 1D–G) making the assignment of the potential frequencies for a multi-band system difficult. In these circumstances, the angular dependence of the observed orbits is used as a guide to assign the different orbits to Fermi surfaces as the minimum and maximum orbits will have different angular dependencies and it is expected that the cyclotron effective mass for the same band is likely to have similar values [42, 62]. Even in twinned samples, the quantum oscillations are likely to be unaffected as the cross-section areas originating from different domains would be the same, however, any differences only be noticeable at very high rotation angles where the quantum oscillation amplitude disappears. The experimental Fermi surface of FeSe_1−xS_x could potentially have four different sheets, that could generate up to seven or eight extremal orbits at the high symmetry points due the strong k_z dependence, as shown in Figure 5D. In a system with many similar small orbits (with k_F values below 0.08 Å) the expected frequencies would be found below 200 T. A clear separation between individual small frequencies is hampered by the limited magnetic field window (20–45 T) caused by the presence of superconductivity and large upper critical field (<20 T) (Figure 5A. This low frequency region is also affected by extrinsic effects in a fast Fourier transform (Figure 5C, such as the 1/f noise and the peak created by a background polynomial, making any reliable assignment of the small frequencies difficult.

FIGURE 5

An interesting insight into the origin of the quantum oscillation amplitudes was provided by magnetotransport and Hall effect in ultra-high magnetic fields up to 90 T in FeSe [98]. By comparing the changes in the relative amplitudes of the quantum oscillations of the and components, and considering the positive sign of the high-field Hall signal at very low temperatures [98], the mobile carriers were assigned to the hole band (β and δ orbits) [42, 62, 96]. The frequencies of quantum oscillations assigned to the quasi-two dimensional hole cylinder of FeSe are 220 T for β orbit at the point and 660 T for the δ orbit at the Z point, when magnetic field . The cyclotron effective masses associated with these two orbits are around 4.5(5) m_e, in good agreement between different studies [35, 42, 62]. The estimated frequencies of the hole pockets using the k_F values from ARPES data (Figure 4C) are consistently smaller than those assigned in quantum oscillations (by ∼150 T or 24% smaller for the δ orbit at the Z point and  T or 50% smaller for the β orbit at the point for FeSe). One obvious difference between the two techniques is related to sensitivity to surface states in ARPES, compared with bulk, that is normally probed by quantum oscillations. ARPES resolution, k_z dependence and the energy and momentum integrations is expected to affect the precise k_F values. Variation of the values for the top of the hole band at the point of 6.7–15 meV is found between different reports for FeSe [62, 93], and this will affect the precise determination of the k_F values. On the other hand, the high frequency values from quantum oscillations have a much better agreement between different reports [42, 62, 96]. Quantum oscillations are measured below 1 K in high magnetic field above 20 T whereas ARPES is measured in zero field above 10 K (∼1 meV). As inner pockets of FeSe_1−xS_x are small, magnetic field could induce additional spin-polarization of the Fermi surfaces in very high magnetic fields (3–4 meV). Another discrepancy between quantum oscillations and ARPES is related to the orbitally averaged effective masses that are larger for the outer electron bands (∼7 m_e) compared with the hole bands (4.5 m_e) in quantum oscillations [42, 62]. In contrast, the Fermi velocities extracted from ARPES are larger at the A point (0.66 eV Å) compared with hole bands at the Z point (0.4–0.5 eV Å) [65]. However, the velocities in ARPES are extracted for only one of the highly symmetry direction and the values are not orbitally averaged (Figure 2).

Quantum oscillations in iron-based superconductors detect clearly electron Fermi surfaces with lighter effective masses in LaFePO [99], LiFe(As/P) [100] and BaFe₂As₂ [56]. These orbits originate from inner and outer quasi-two dimensional cylinders due to the finite spin-orbit coupling, as depicted for FeSe_1−xS_x in Figure 3C. In FeSe, the orbital differentiation is much more pronounced than in iron pnictides, as the d_xy band is involved in the formation of outer flower-shaped electron orbit (Figure 2H). This would lead to a much heavier orbitally averaged cyclotron mass of ∼7 m_e, associated with the outer electron orbit around the A point (γ orbit of ∼560 T) for FeSe, as shown in Figure 6A [42, 62]. Based on k_F values at the A point determined from ARPES with Fermi vector values of 0.03 (1) and 0.19 (1) Å⁻¹ [65], the area of a flower-shaped orbit would be ∼350 T (or ∼35% of γ orbit) whereas for a single ellipse pocket reaches only ∼190 T, which is or ∼66% smaller than the γ orbit from quantum oscillations. In ultra high magnetic fields, potential breakdown orbits could be generated by tunneling across the gaps created by the spin-orbit coupling, but the necessary magnetic fields are likely to be very large and the orbits would be smaller than that of the flower-shaped orbit [101]. The nematicity has a drastic effect on the electron bands and it can lead to highly elongated pockets with a very small inner electron band, as shown in Figure 3C. At 13 K, the inner band at the M point gets very close to the Fermi level within 3 meV for FeSe (within the experimental resolution). Thus, any small changes in the band positions relative to the chemical potential (1–2 meV) that could occur at low temperatures below 1.5 K and in high magnetic fields could potentially push the inner electron bands above the Fermi level at the M point and lead to single elliptical orbit or an elliptical pocket and two tiny electron pockets, as shown in Figure 3C.

FIGURE 6

Recent studies promotes the idea that FeSe would only have a single electron pocket in the corner of the Brillouin zone. For a peanut-like pocket at the A point [65] its area is almost a factor 3 smaller than the γ pocket in quantum oscillations, the change compensation of the system would be lost and the magnetotransport data of FeSe cannot be explained [98]. The proximity of the inner electron band to the Fermi level is highly sensitive to small energetic alterations within experimental resolution (∼3 meV), any differences in Fe stoichiometry, surface effects or the possible changes that can occur under applied uniaxial stress, as found for thin films of FeSe under strain from a substrate [91]). Thus, different scenarios related to the fate and the number of the electron pockets (Figure 3C) need to be considered, besides other theoretical reasons [65, 66, 102–104].

Figure 5C show the complex fast Fourier spectra of FeSe_1−xS_x due to the presence of multiple small Fermi surfaces areas. The signature associated with the inner electron band in quantum oscillations would be a peak in the Fast Fourier transform below 100 T. Previously, it was assumed that FeSe has a single cylindrical electron pocket, with areas varying between 50 T pocket for its minimum and 550 T (γ) for its maximum [42]. This variation would suggest a much more warped Fermi surface cylinder for the electron band (factor 10 between the two high symmetry areas), as compared with the hole band, for FeSe, in disagreement with the k_z dependence determined from ARPES studies [35, 62, 65]. The quantum oscillations spectra could assign the lowest frequency below 100 T to the inner quasi-two dimensional electron pocket, whereas γ and ϵ around 440 T could correspond to the outer electron band (Figure 5D) [35, 62]. A small inner electron pocket is not easy to observe using spectroscopic techniques, nor does it have a large contribution to the density of states, but it plays an important role in magnetotransport due to its high mobility [40]. With sulfur substitution for small x < 0.09, quantum oscillations show similar features to those of FeSe, as shown in Figure 5C. As the nematic effects are progressively removed, the inner electron orbits would increase in size, reaching a value of 200 T in FeS [67].

5.2 Evolution of Fermi Surface Areas of FeSe_1−xS_x

The overall evolution of the Fermi surface of FeSe_1−xS_x implies that the majority of the cross-sectional areas expand as a function of chemical pressure, in particular the maximum orbits located at the top of the Brillouin zone, as shown in Figure 6G [27]. For the outer hole band (δ orbit), the increase in areas reflects the transition from an in-plane anisotropic to isotropic Fermi surface, as the ellipse transforms into a circle, and as the in-plane areas increase due to changes in the lattice parameters [29, 35, 40]. These trends are in contrast to the small Fermi surfaces observed under applied pressure in FeSe, suggested to result from Fermi surface reconstruction inside the spin-density phase [18]. However, the Fermi surfaces of FeSe_1−xS_x are severely reduced in size compared with those predicted by DFT calculations (varying from a factor of 5 for FeSe toward a factor 3 for x ∼ 0.17). This shrinking is an important consequence of strong orbitally-dependent inter- and intra-band electronic interactions, significantly large in iron chalcogenides [62, 105], but also found in many iron-based superconductors [56, 99]. These effects are suppressed once the bandwidth increases with sulfur substitution toward FeS [29] or with phosphorus substitution in BaFe₂(As_xP_1−x)₂, as shown in Figures 6G,H [56]. The largest orbit detected in FeS is almost a factor 2 larger than for x ∼ 0.19 [67] but it is still a factor 2 smaller than that predicted by band structure, and band-shifts of 0.1 eV are required to bring experiment in agreement with DFT calculations [67]. The Fermi energies estimated from quantum oscillations of FeS have significantly increased to 27–102 meV [67], compared with 3–18 meV estimated for FeSe [42].

Transport measurements in a multi-band system like FeSe_1−xS_x are normally dominated by the pockets with the highest mobility carriers in a parallel resistor model. The magnetoresistance at low temperatures shows a prominent low frequency oscillation from x = 0.12 toward NEP (Figures 5A,B) [27]. Outside the nematic phase, the background magnetoresistance is almost quadratic in magnetic field and the dominant low-frequency oscillation has disappeared (Figure 5). This dominant low frequency is not detected at higher sulfur substitution or higher pressures beyond the nematic end point inside the tetragonal phase [27, 39]. In quantum oscillations, the disappearance of a frequency could be linked to a possible Liftshitz transition, which is a topological change of the Fermi surface in which the neck of a quasi-two dimensional is disconnected while the top of the cylinder expands such that the volume remains the same [107]. ARPES data indicate that a small inner 3D hole pocket centered at Z is expected to emerge from x ∼ 0.11, as shown in Figure 1. This small 3D pocket is supposed to grow in size with x, rather than to disappear. However, in high magnetic fields, this 3D hole pocket could become heavily spin-polarized and, therefore, one of its polarized sheet could disappear at the nematic phase boundaries. Another scenario can rely on the strong increase in the interlayer warping as a function of chemical or applied pressure, as the conducting layers come closer together when c axis decreases. DFT calculations of FeSe and FeS show indeed that the hole bands are highly sensitive to the position of the chalcogen atom above the Fe plane [29, 58, 62]. ARPES studies for the tetragonal x ∼ 0.18 suggest that the hole band at the point is smaller compared with FeSe and the top of the band is about 5 meV above the Fermi level, as shown in Figure 4D [29, 35]. Thus, the orbit associated with the hole band at the point could be a prime candidate for the observed disappearance of a significant frequency in quantum oscillations. Other scenarios could invoke magnetic field-induced Lifshitz transitions affecting the bands with very small Fermi energies, such as the inner hole and electron bands, that are comparable to the Zeeman energy (3–4 meV) [108]. Multi-band interference effects as well as oscillations of the chemical potential could be considered as other potential theoretical avenues to understand these effects in magnetic fields [109].

5.2 Electronic Correlations at a Putative Nematic Critical Point

The cyclotron-averaged effective masses of the quasiparticles for each extremal orbit can be extracted from the temperature dependence of the amplitude of the quantum oscillations [27, 110]. The quasiparticle masses associated with the largest hole orbit δ around the Z point increase slowly from 4.3(3) m_e toward a local maximum around x < 0.11 before the values continue to decrease outside the nematic phase to around 3.2 (5) m_e for x ∼ 0.19. In the end compound FeS, quantum oscillations have revealed very light effective masses ranging from 0.6–2.1 m_e [67, 95]. The overall trends shows that cyclotron masses are larger inside the nematic phase of FeSe_1−xS_x but they are getting lighter with the increasing bandwidth [57]. The reduction of the electronic correlations toward FeS is supported by the enhanced velocities from ARPES, shown in Figure 4E [29]. The effective mass of the prominent small frequency oscillation (λ) is small below 2 m_e. Due to its heavy mass and its possible proximity to the δ orbit, the γ orbit (with some orbitally averaged d_xy character) cannot be detected over the entire range but it is expected to follow similar trends to the hole bands effective mass (δ) and to the electronic contribution to the specific heat (Figure 6A) [27, 30].

The nematic state of FeSe_1−xS_x is a correlated electronic state based on the quasiparticle effective masses. Interestingly, the electronic correlations assigned to the orbits with predominant d_xz/d_yz character (outer hole band, δ) follow similar trends as T_c, as shown in Figures 6C,D, suggesting that this quasi-two dimensional hole band is likely to play a dominant role in the pairing mechanism. The trends in the effective masses are in good agreement with those from specific heat studies on FeSe_1−xS_x that show a slight increase in the Sommerfeld coefficient (7–9 mJ/mol K) inside the nematic phase before being smoothly suppressed, without any enhancement at the nematic end point [30, 111, 112]. Additionally, the Fermi liquid behavior A^1/2 coefficient extracted from the low temperature resistivity measurement has the same trends like the cyclotron mass [28], as shown in Figure 6, and it decreases slowly in the tetragonal phase [52]. Note that these values of the A^1/2 coefficient agree with those reported in Ref. 54, once adjusted for the correct value of x based on the T_s of each sample, shown by the open triangle in Figure 6B.

To asses the nematic critical behavior in FeSe_1−xS_x, it is worth emphasizing that the electronic correlations and the orbitally averaged cyclotron masses do not show any divergence close to NEP (x ∼ 0.175(5)), as shown in Figure 6C [27]. Instead, the effective mass of FeSe_1−xS_x reaches the largest value deep inside the nematic phase, where the superconductivity is the strongest and the low-energy spin-fluctuations are expected to be the largest [44]. The lack of divergent effective masses at NEP points toward a finite coupling of the electronic system with the underlying lattice that can suppressed the critical nematic fluctuations, except along certain directions in FeSe_1−xS_x [6]. Nematic susceptibility as a function of chemical pressure suggest the possibility of having a nematic critical point in FeSe_1−xS_x [43]. However, at low temperatures there are no divergent electronic correlation in any of the measured quantities in the vicinity of the nematic end point, suggesting an important role for the coupling of the electronic system with the lattice in this system [27, 28].

Signatures of quantum criticality caused by diverging spin fluctuations were detected in quantum oscillations in BaFe₂(As_1−xP_x)₂, by approaching the spin-density wave phase from the tetragonal phase. The cyclotron effective mass of the outer electron bands increases from 1.8 to 3.5 m_e over a large compositional range (x = 0.4–1) in the tetragonal phase, as shown in Figure 6E [56, 106]. This enhancement of the effective mass in BaFe₂(As_1−xP_x)₂ correlates directly with the strong increase in the superconducting transition temperature. The quantum oscillations frequencies originate from the lighter electron bands in BaFe₂(As_1−xP_x)₂ and their frequencies get smaller as the system evolves from the metallic tetragonal phase toward the spin-density wave phase. These trends are similar to those expected for FeSe_1−xS_x. It is worth emphasizing that for both systems only the effective mass is reported, not the mass enhancement in relation to the band mass, due to the complexity involved in establishing the details of the correct band structure for the mixed isoelectronic systems. Figure 6 compares the effective masses for the two isoelectronic systems and it suggests that the relevant interactions that enhance the effective masses in FeSe_1−xS_x are the same that enhance superconductivity. These pairing interactions are strongest deep inside the nematic phase not at the nematic end point. Their origin could be the spin fluctuations in both systems and they are also likely to be responsible for the linear resistivity observed inside the nematic phase for FeSe_1−xS_x [28] and for BaFe₂(As_1−xP_x)₂ for x ∼ 0.33 [114]. The superconductivity is strongly enhanced in the proximity of a magnetic critical point in BaFe₂(As_1−xP_x)₂, as opposed to the small abrupt drop in T_c at the nematic end point FeSe_1−xS_x (see Figure 6D). This suggests that a strong nematoelastic effect suppresses the critical nematic fluctuations and the superconducting mechanism has a non-nematic origin in FeSe_1−xS_x [7].

5.3 The Nematic Susceptibility of FeSe_1−xS_x

A direct measurement to test for the existence of an intrinsic nematic electronic state is the determination of the nematic susceptibility, that is the related to the in-plane resistivity anisotropy under a small amount of external strain [114]. These type of studies have established that the tetragonal-to-orthorhombic structural transition in iron pnictides is driven by the electronic instability of the system [114]. The Curie-Weiss behavior of nematic susceptibility near a nematic transition is expected to display a generic mean-field behavior. The nematic fluctuations of the nematic order parameter, which couple linearly to the orthorhombic distortion via the nematoelastic coupling, are expected to be suppressed but this may not be the case if the nematic fluctuations are driven by the spin fluctuations [116]. The nematic susceptibility of Ba(Fe_1−xCo_x)₂As₂ follows a Curie-Weiss dependence and the mean field nematic critical temperature closely tracks the actual structural transition temperature, being suppressed to zero at the optimal doping [114]. The divergence of the nematic susceptibility above T_s indicates the tendency toward an electronic nematic phase transition and the Weiss temperature indicates the strength of nematic fluctuations [114]. At a critical nematic point, the nematic susceptibility should diverge at zero temperature (in proportion to 1/T) and power law behaviors in temperature and composition are expected [116].

Nematic susceptibility measurements for FeSe and FeSe_1−xS_x [43, 62], [117] indicate a large divergence above the T_s, similar to what was previously observed in Ba(Fe_1−xCo_x)₂As₂, but in the absence of magnetic order [114]. Nematic susceptibility of FeSe has an opposite sign to other pnictides [118], but similar to other chalcogenides, like FeTe [75], as the resistance along the a (AFM) direction is larger than that along b axis (FM direction). A sign-change in the in-plane anisotropy could be induced by the different scattering rates by spin fluctuations corresponding to different Fermi velocities at the hot-spots for electron- and hole-doped pnictides [119]. In FeSe, despite the lack of long-range magnetic order, the anisotropy of the in-plane resistance below T_s follows qualitatively a model assuming that the electrons are mainly scattered by magnetic fluctuations [117, 120]. Elastoresistance measurements in FeSe_1−xS_x superconductors found that the nematic transition temperature decreases with x [43], whereas the nematic fluctuations are strongly enhanced, similar to Ba(Fe_1−xCo_x)₂As₂. The observation of strong nematic fluctuations is consistent with the presence of a nematic quantum critical point, but this observation is insufficient to determine whether these fluctuations are driven by quantum criticality. Static Raman susceptibility studies show similar trends to the nematic susceptibility, and additionally it was suggested the possible stabilization of stripe quadrupole order in FeSe_1−xS_x [121]. Future studies to establish a suitable power law of the nematic susceptibility are needed [116] in order to identify whether this point represent a critical point in FeSe_1−xS_x. However, the lack of divergent electronic correlations in quantum oscillations or enhanced superconductivity at the nematic end point suggest a strong suppression of critical fluctuations in FeSe_1−xS_x.

5.4 Magnetotransport Behaviour of FeSe_1−xS_x

In multi-band systems with different carrier mobilities, the magnetoresistivity components and have a complicated behavior in magnetic field. In the presence of a single dominant scattering time, the magnetoresistance is expected to follow Kohler’s rule and a B² dependence [122]. In the tetragonal phase of FeSe_1−xS_x, a quadratic dependence of the magnetoresistance is found up to 69 T both at high temperature above T_s or at low temperature outside the nematic phase boundaries for x ≥ 0.19 [28]. On the other hand, inside the nematic phase of FeSe_1−xS_x, Kohler’s rule is violated and the magnetoresistance of FeSe_1−xS_x, follows an unusual B^∼1.55 power law in high magnetic fields [28]. Furthermore, scaling to a modified Kohler’s rule as a function of the Hall angle was found in the vicinity of the nematic end point [123]. Another way to understand this complex behavior is to separate different components of magnetoresistance, as suggested in Ref. 55. The coefficient of one of the extracted component has the same temperature dependence as the resistivity slope in 34 T in FeSe (Figure 7), once adjusted for the correct compositions as Refs. 54, 55 uses the nominal sulfur concentration. Furthermore, other types of magnetoresistivity scaling found for BaFe₂(As_1−xP_x)₂ [125], are not found for FeSe_1−xS_x [28, 55], and these effects are likely to occur for samples with higher impurity scattering [55].

FIGURE 7

5.5 Highly Mobile Small Carriers in FeSe_1−xS_x Beyond the Two-Band Model

A compensated two-carrier model can describe the behavior of the magnetoresistance and the Hall effect of FeSe_1−xS_x in the tetragonal phase [28, 98]. For a compensated metal, the sign of the Hall coefficient depends on the difference between the hole and electron carrier mobilities [98]. At high temperatures in the tetragonal phase, the Hall effect is linear and the Hall coefficient, , is close to zero as the hole and electron pockets have rather similar mobilities [98]. Inside the nematic phase, the Hall coefficient for FeSe, extrapolated in the low-field limit (below 1 T), is negative at low temperatures with a minimum around 20 K [29, 98]. These anomalies indicate that the magnetotransport behavior of FeSe cannot be described using a two-band compensated model, and an additional higher mobility charge carrier is required (with carrier concentration of 0.7 × 10²⁰ cm⁻³, a factor 5 smaller than the size of the largest band) [98, 125, 126]. This component would correspond to a Fermi surface of small volume that could be linked to the inner electron band with frequencies below 100 T and a light effective mass (∼2 m_e) [27, 42, 62]. With increasing x, the Hall coefficient at low temperatures is positive for x ∼ 0.11, suggesting the dominance of highly mobile hole carriers, as shown in Figure 7D [28]. This sign change occurs for the same composition at which the highly mobile 3D hole pocket center is detected in ARPES studies at the Z-point around x ∼ 0.11, shown in Figure 2 [40]. High mobility carriers have been suggested to dominate the magnetotransport behavior across the whole nematic phase of FeSe_1−xS_x [126, 127], Furthermore, FeSe under pressure shows similar trends to the chemical pressure effect and the normal-state Hall resistivity changes sign from negative to positive, showing dominant highly mobile hole carriers at high pressures [79].

The orbital order significantly affects band shifts for the electron bands and it can generate very small pockets, as shown in Figure 3. The presence of the small number of highly mobile carrier was suggested to be linked to the Dirac-like dispersions in the nematic phase on some sections of the electron pockets [125, 128]. Large orbital-dependent shifts of ∆_M ∼ 70 meV in FeSe thin films on SrTiO₃ have been found to generate Dirac-like dispersion around the M point [88], but these shifts are much larger than those of ∆_M ∼ 50 meV in bulk FeSe. Magnetotransport cannot distinguish whether there are two tiny electron-like pockets or one small electron pocket (Figure 3C for large values), besides the almost compensated hole and electron bands in FeSe. It is clear that a two-band picture containing an single electron and hole pocket and assuming isotropic scattering fails to describe magnetotransport behavior of FeSe [98].

Significant changes in scattering could occur for a elongated nematic Fermi surface of FeSe_1−xS_x [35, 62]. Two scattering wave-vectors are detected by STM [46] suggesting different scattering processes along certain directions of an elongated ellipse [9]. A flower-shaped electron orbit would have a strongly varying angular velocity [129] and the scattering rate could vary strongly due to the changes of the orbital character on various sections induced by spin fluctuations [117, 130]. Hall effect in iron-based superconductors is also affected by the spin fluctuations that induce mixing of the electron and hole currents [131]. All these effects could lead to highly anisotropic scattering rates in FeSe_1−xS_x that are suppressed with x substitution. Indeed, in the tetragonal phase a single scattering process dominates the magnetotransport, as Kohler’s rule is obeyed [28]. Further theoretical work is needed to understand transport and magnetotransport data of FeSe_1−xS_x. Future models should account for anisotropic scattering and scattering of quasiparticles from the domain walls, when the nematic domain size (determined by the quenched disorder) is smaller than the normal state mean-free path [1].

FeSe_1−xS_x displays deviation from the Fermi liquid theory, expected for conventional metals, that affect the temperature and field dependencies of electron transport. The magnetoresistance of FeSe_1−xS_x increases significantly once a system enters the nematic state and shows an unusual temperature dependence that varies strongly with x, as shown in Figures 7A,B for FeSe. The temperature dependence of the resistivity slope in 34 T in FeSe changes sign at a characteristic temperature, T^* below 14 K, and the Hall coefficient R_H display a negative maximum, as shown in Figures 7B,D. Interestingly, T^* seems to be the characteristic scale for low-energy spin fluctuations in FeSe_1−xS_x [44, 132, 133]. Magnetostriction measurements in magnetic field for FeSe showed that the lattice distortion continues to increase down to T_c, different from BaFe₂As₂, where there is a intimate connection between the magnetic order and structural distortion [121]. With sulfur substitution, T^* shifts to a slightly higher temperature of ∼20 K, and eventually disappears in the tetragonal phase, as the low-energy spin fluctuations are completely suppressed [44, 134]. Changes in magnetotransport and in the resistivity slope occur from x ∼ 0.11 (Figure 7H) in the presence of the additional highly mobile 3D band, labeled as the nematic B phase [28]. It is clear that magnetic field could affect scattering inside the nematic phase that could be still dominated by spin fluctuations and it can spin polarize the multi-band small Fermi-surface of FeSe_1−xS_x. Further theoretical work will be required to explain the observed effects in magnetic fields and experimental studies in single domains crystals are needed to address the extrinsic scattering at the nematic domain boundaries.

5.6 Anomalous Transport Behavior

Linear resistivity at low temperatures is usually found near an antiferromagnetic critical point, such as in BaFe₂(As_1−xP_x) [11] and reflect scattering induced by critical spin-fluctuations [135]. FeSe_1−xS_x has a low temperature region with a linear resistivity across the whole nematic phase below T^* (using extrapolated in-plane high magnetic field data), as shown in Figures 7E,G. Low energy spin-fluctuations are present inside the nematic state in FeSe_1−xS_x [44, 132, 134], and μSR studies place FeSe near an itinerant antiferromagnetic quantum critical point at very low temperatures [136]. This region with linear T resistivity below T^* occurs over a limited temperature regime and the Fermi-liquid behavior recovers below T_FL and all compositions show quantum oscillations (Figures 7E–G) [27]. Deviations from Fermi-liquid behavior were also reported for FeSe [11] and linear T resistivity was detected in 35 T for x_nom ∼ 0.16 with T_s ∼ 50 K, which corresponds to x ∼ 0.13 inside the nematic phase [37, 54]. Thus, all existing experimental transport data for FeSe_1−xS_x in high magnetic fields up to 45 T suggest that the low-temperature linear resistivity occurs inside the nematic phase, rather than at the nematic end point, and, as in the case of the electronic correlations and T_c, it is likely a manifestation of the spin fluctuation scattering mechanism inside the nematic phase FeSe_1−xS_x.

In the tetragonal phase of FeSe_1−xS_x the resistivity exponent seem to vary with temperature [28, 52] and a resistivity with T^3/2 dependence is found over a large temperature range up to 120 K, in agreement with studies under pressure [39]. Theoretical models suggest that the exact temperature exponent of resistivity, in vicinity of nematic critical points is highly dependent on the presence of cold spots on different Fermi surfaces, due to the symmetry of the nematic order parameter [9, 137, 138] or due to the scattering from acoustic phonons [8] near the nematic end point. Near a Pomeranchuk instability the transport decay rate is linear in temperature everywhere on the Fermi surface except at cold spots on the Brillouin zone diagonal, leading to a resistivity proportional to T^3/2 for a clean 2D system or to a linear T dependence in the presence of impurities [137]. Furthermore, the scale at which the crossover to Fermi liquid behavior occurs at low temperatures depends on the strength of the coupling to the lattice [6], responsible for the lack of divergent critical fluctuation at the nematic end point in FeSe_1−xS_x [27, 28, 39].

6 Superconductivity of FeSe_1−xS_x

The normal nematic electronic phase and the anomalous scattering of FeSe_1−xS_x affects significantly its superconducting state. The gap structure of FeSe is two-fold symmetric, reaching small values on the major axis of the elliptical hole pocket and it is changing its sign between the hole and the small electron pocket [78, 139]. While nematicity is an intrinsic property of the bulk FeSe_1−xS_x, nematic fluctuations may not be the primary force driving the superconducting pairing [140], despite the fact that the relative orthorhombic distortion is reduced as the superconductivity increases in FeSe_1−xS_x [111]. Neutron scattering have detected substantial stripe spin fluctuations that are enhanced abruptly inside the nematic phase and a sharp spin resonance develops in the superconducting state with an energy of ∼4 meV, consistent with an electron-boson coupling mode [141]. By de-twinning crystals of FeSe via a BaFe₂As₂ substrate, the neutron diffraction spectra reveal that the spin fluctuations are highly anisotropic [13]. Theoretically, low-energy spin excitation were suggested to be stabilized by the frustrated magnetism among the local Fe moments [86, 142] or antiferroquadrupolar state in FeSe [143]. These low-energy spin-fluctuations are likely to provide the pairing channel in FeSe [144, 145] and this can manifest via nesting of d_yz sections of the hole and electron bands; the d_xy portions do not participate in pairing due to the orbital selective strong correlation effects [78, 146]. In this scenario, a maximum gap on the Fermi surface sections with d_yz character, and a small gap on sections with d_xz or d_xy character would occur, similar to experiments [78, 147]. Most of the thermodynamic and thermal conductivity studies of bulk FeSe in the superconducting phase have been modeled by accounting for two different nodeless superconducting gaps [148, 149]. The presence of nodes in the superconducting gap of FeSe has also been suggested by other studies [113, 150, 151]. The multi-gap superconductivity is preserved as a function of chemical pressure in FeSe_1−xS_x [112, 139], and tunneling experiments found that the vortex core anisotropy is strongly suppressed once Fermi surface becomes isotropic [151]. High-resolution thermal expansion showed a lack of coupling between the orthorhombic distortion and superconductivity in FeSe [12], whereas with increasing substitution toward x ∼ 0.15 the effect seems to be the opposite [111]. The jump in specific heat (∆C_el/γ_nT_c) for different x varies slightly around 2, which is above the weak coupling limit of the BCS theory believed to be caused by the multi-band effects [112].

For isotropic isoelectronic iron-based superconductors, the height of pnictogen acts as a switch between high-T_c nodeless and low-T_c nodal pairings [152]. FeS, like other end member compounds, displays weak correlations and nodal superconductivity, similar to other systems like LaFePO and LiFeP [153–156], as the chalcogen position is closer to the iron planes compared to their isoelectronic sister-compounds, like LiFeAs. On the other hand for FeSe, there has been suggestions both of nodal and nodeless superconductivity [113, 157, 158, 159, 160]. Abrupt changes in the superconductivity occur at the nematic end point, potentially stabilized by different pairing channels inside and outside the nematic phase [30, 46]. There is no superconductivity enhancement at the nematic end point in FeSe_1−xS_x, suggesting the presence of a non-nematic pairing mechanism and/or the lack of divergent critical fluctuations, similar to the behavior of the quasiparticle effective masses [27, 39]. The coupling to the relevant lattice strain restrict the critical behavior only along certain high symmetry directions and this can affect the nematic critical fluctuations and do not enhance superconductivity [6, 7].

6.1 BCS-BEC Crossover of the Multiband FeSe_1−xS_x

FeSe_1−xS_x are multi-band systems with relatively small Fermi energies at low temperatures. There has been a lot of interest to asses whether these systems are candidates in the crossover regime between the BCS to the BEC state, expected for ∆_SC/E_F ≤ 1 [161, 162]. These effects have been suggested to occur in Fe_1+ySe_xTe_1−x, as the hole band as the point is tuned at the Fermi level by Fe deficiency and ∆_SC/E_F varies 0.16 to 0.50 [163]. This ratio is also relevant for assessing the possibility of stabilization of a FFLO state in FeSe [164] and a good knowledge of the value of the superconducting gap and the Fermi energy of the multi-band and highly warped Fermi surfaces is needed. The amplitudes of the highly anisotropic superconducting gaps of FeSe around the hole pocket vary between ∆_SC ∼ 2.5 (or 2.3) ( point) from STM to 1.5 meV–3 meV from laser ARPES [94]. For the electron pocket (at the M point) the values of the gap vary between 3.5 and 1.5 meV [46, 78] and another potential small gap of 0.39 meV was invoked from specific heat data [79] (Figure 4F). The top of the hole band (associated to the Fermi energy E_F) is k_z dependent having a value of 16 meV at point and 25 meV at the Z point (Figure 4D) [35, 62], whereas laser ARPES reports values of 6.7–10 meV [91, 93]. Based on these values, ∆_SC/E_F ∼ 0.1–0.15 for the hole band at the point in FeSe. This ratio will decrease using the parameters at Z point and the correct values of E_F need to take into account the strong k_z dependence of the cylindrical Fermi surface and the mass anisotropy for each pocket. As a function of x, the gap associated with the hole band remains relatively constant inside the nematic phase, but it is getting smaller toward 1.5 meV in the tetragonal phase (see Figure 4F) [46]. The top of hole band and the is somewhat reduced toward 5 meV for x ∼ 0.18 but increases slightly for the Z point at 26 meV; the bottom of the inner electron band is around ∼15 meV (see Figure 4F). The variation of these parameters will affect the estimates of and one needs to consider the multiple bands and gaps of FeSe_1−xS_x together with the k_z dependence of the Fermi surface and the superconducting gap [93, 164]. Furthermore, the Fermi velocities increase with x pushing the system away from the BCS-BEC crossover regime, as not all the bands satisfy the crossover condition.

Another way to assess the proximity to the crossover is to check whether the size of the Cooper pair, given by the coherence length , is smaller than the mean inter-particle spacing 1/k_F and [15, 161]. Using the in-plane coherence length for FeSe of ξ = 4.6–5.7 nm [42, 45] and the values of k_F ∼ 0.038–0.15 for the hole bands (Figure 4), it suggests that k_F ∼ 1.75–8.55 is large and the Cooper pairs are quite extended suggesting that the superconductivity of FeSe need to be understood considering its multi-band effects. Further aspects of the pairing mechanism of FeSe and other iron-chalcogenides are discussed in detail in recent reviews [15, 37].

7 Conclusion

FeSe_1−xS_x has opened an new area of exploration of the electronic nematic state and its role in the stabilization of the unconventional superconductivity. These systems are multi-band systems which are highly sensitive to orbitally-dependent electronic interactions that affect the evolution of the electronic structure with sulfur substitution. Fermi surface of FeSe_1−xS_x are mainly quasi-two dimensional warped cylinders but an additional 3D hole pocket is present with increasing sulfur concentration from x ∼ 0.11. The Fermi energies have a broad range, that generally increases with x substitution, but they are smaller for the inner electron and hole pockets, making them prone to electronic instabilities. The development of nematic electronic phase with strong anisotropic electronic structure influences the scattering and leads to unusual magnetoresistance inside the nematic phase. Linear resistivity and anomalous magnetotransport is detected inside the nematic phase and is likely to reflect the role played by the spin fluctuations in this regime. FeSe_1−xS_x show no signatures of enhanced T_c and divergent electronic correlations at the nematic end point, which are likely to be quenched by the finite coupling with the lattice. This coupling could also be the origin of the non-Fermi liquid behavior outside the nematic phase. The superconductivity of FeSe_1−xS_x has a small enhancement inside the nematic phase and a somehow abrupt change at the nematic end point. This behavior is different from the isoelectronic family BaFe₂(As_1−xP_x)₂ where quantum critical fluctuations enhance both superconductivity and effective masses of the quasiparticles on approaching the spin-density phase and linear resistivity is found at the magnetic critical point. The study of FeSe_1−xS_x compared with other isoelectronic iron-based superconductors emphasis the important role played by the magnetic rather than nematic fluctuations for enhancing superconductivity in iron-based superconductors.

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