Maximizing Tc by tuning nematicity and magnetism in FeSe1−xSx superconductors

A fundamental issue concerning iron-based superconductivity is the roles of electronic nematicity and magnetism in realising high transition temperature (T c). To address this issue, FeSe is a key material, as it exhibits a unique pressure phase diagram involving non-magnetic nematic and pressure-induced antiferromagnetic ordered phases. However, as these two phases in FeSe have considerable overlap, how each order affects superconductivity remains perplexing. Here we construct the three-dimensional electronic phase diagram, temperature (T) against pressure (P) and isovalent S-substitution (x), for FeSe1−xSx. By simultaneously tuning chemical and physical pressures, against which the chalcogen height shows a contrasting variation, we achieve a complete separation of nematic and antiferromagnetic phases. In between, an extended non-magnetic tetragonal phase emerges, where T c shows a striking enhancement. The completed phase diagram uncovers that high-T c superconductivity lies near both ends of the dome-shaped antiferromagnetic phase, whereas T c remains low near the nematic critical point.

O ne of the common aspects among unconventional superconductors, including high-T c cuprates, heavyfermion and organic materials, is the appearance of a superconducting dome in the vicinity of magnetic order. This has naturally led to the notion of superconducting pairing mechanism driven by magnetic fluctuations 1,2 . In iron pnictides, high-T c superconductivity also appears near the antiferromagnetic phase 3 , which however is accompanied by the tetragonal-toorthorhombic structural transition with significant electronic anisotropy (nematicity). This gives rise to new theoretical proposals involving the fluctuations of this electronic nematicity as a glue for the electron pairing [4][5][6] . Although enhanced nematic fluctuations of ferro-type (q = 0) are observed experimentally 7 , the antiferromagnetic fluctuations are also enhanced 8 , and thus it is difficult to pinpoint the impact of nematic fluctuations on the superconductivity in iron pnictides.
From this viewpoint, the FeSe-based superconductors are a suitable system for addressing the importance of nematic fluctuations, as it has a unique phase diagram 9 . At ambient pressure, FeSe shows a nematic transition at T s = 90 K without magnetic order down to the lowest temperature 10 . Under pressure, antiferromagnetic order is induced [11][12][13][14][15][16] , and the superconducting T c is enhanced by more than a factor of 4 9,17 . Recently, it has been shown that the nematic transition can be tuned to a quantum critical point by isovalent substitution of Se with S, but without inducing magnetic order 18 . These results indicate the nonequivalence of physical and chemical pressure in this system. This implies that one can control the magnetism and nematicity independently by these tuning knobs, isovalent substitution and physical pressure, which offers the possibility to disentangle intertwined effects of nematic and magnetic fluctuations on high-T c superconductivity.
Here we present our systematic study of temperature-pressuresubstitution (T-P-x) phase diagrams of FeSe 1−x S x in wide ranges of pressure (up to P~8 GPa) and sulphur content (0 ≤ x ≤ 0.17). In pure FeSe, it has been shown by several groups that the nematic transition temperature T s is suppressed by pressure (P < 2 GPa) but before the complete suppression of T s , antiferromagnetic or spin density wave (SDW) order is induced, resulting in an overlap region of these two phases 9,[12][13][14][15] . With increasing x, the nematic transition temperature T s is lowered and correspondingly the nematic phase is rapidly suppressed by pressure. However, an opposite trend is found for pressureinduced magnetism: the SDW onset pressure is shifted to higher pressure. These lead to the emergence of the tetragonal nonmagnetic phase in between, which becomes wider with increasing x. Most importantly, a new high-T c superconducting dome emerges in the tetragonal phase. Based on the obtained threedimensional phase diagram, we discuss the relationship between the two orders and superconductivity in this system.

Results
Phase diagram. In FeSe 1−x S x at ambient pressure, the temperature dependence of resistivity ρ(T) for x < 0.17 exhibits a slight upturn upon cooling at T s due to tetragonal-to-orthorhombic structural transitions, and then it goes to zero below the superconducting transition temperature T zero c À Á 18 . By measuring ρ(T), we determine the structural transition temperature (T s ) and  The structural (T s , blue squares), magnetic (T m , green triangles) and superconducting transition temperatures (T zero c , red circles) are plotted against hydrostatic pressure P and S content x. Following the procedure reported for x = 0 by Sun et al. 9 , T s , T m and T zero suppression of T s as reported previously 18 . In Fig. 2a-d, we show the evolution of ρ(T) under pressure measured using a cubic anvil cell (CAC) which can generate pressure with a good hydrostatic condition and maintain constant pressure upon cooling 19 . With applying pressure, the T s anomaly observed at ambient pressure in x = 0.04, 0.08 and 0.12 disappears completely at P ≲ 1 GPa. This is a natural consequence of the fact that both S substitution and applying pressure suppress the structural transition in FeSe.
In x = 0.04, the ρ(T) curve at 2.0 GPa exhibits a clear upturn around 40 K. The temperature of the upturn increases with pressure, and then it turns to a kink above 4.0 GPa. This evolution of resistive transition is reminiscent of the magnetic transition seen in FeSe under pressure 9 . Therefore, we follow the procedure of ref. 9 to determine the magnetic transition temperatures (T m ) by using a dip or peak in dρ/dT (Supplementary Note 2 and Supplementary Figs. [3][4][5][6], and the pressure-evolution of T m is shown in Fig. 1 and Supplementary Fig. 1b. With increasing pressure, T m is enhanced monotonically up to 6.0 GPa, while T zero c determined by the zero resistivity is slightly suppressed just after the emergence of magnetism. Inside the magnetic phase, the superconducting transition in ρ(T) becomes broad. When we define T peak c as the peak temperature in dρ(T)/dT ( Supplementary  Fig. 3), the difference between T zero c and T peak c is significant in the magnetic phase ( Supplementary Fig. 1). Above 7.0 GPa, the kink anomaly due to the magnetic transition disappears, and concomitantly T zero c increases gradually up to 32 K, resembling the evolution of the electronic phases in FeSe at high pressure 9 .
In x = 0.08 and 0.12, we observe remarkable features at moderate pressures. As shown in Fig. 2b, c, there is no discernible upturn anomaly in ρ(T) between 1.0 and 3.0 GPa for x = 0.08 and 0.12. At 3.0 GPa, a clear T-linear behaviour in the normal-state resistivity is observed (Fig. 3b), which is accompanied by a sharp superconducting transition with enhanced T zero c of~32 K. We checked for x = 0.12 that T c determined by ac susceptibility is consistent with that determined by the zero resistivity (Supplementary Note 3, Supplementary Figs. 7, 8). Further increase of pressure leads to the emergence of magnetism seen as the kink anomaly around 40 K, then it persists up to 6.0 (7.0) GPa for x = 0.08 (0.12). The marked difference compared with FeSe under pressure is the strong enhancement of T c in the lower pressure side of the magnetic phase, forming a peak in T zero c around 3.0 GPa for both x = 0.08 and 0.12. With increasing pressure above 7.0 (8.0) GPa in x = 0.08 (0.12), the kink anomaly due to the magnetic transition disappears and T zero c exhibits another gradual enhancement up to~32 K after the disappearance of T m , resulting in the double-dome structure in T zero c having two maxima with almost identical magnitudes.
In x = 0.17, where there is no T s at ambient pressure as shown in Fig. 2d, its initial T zero c of~4 K gradually increases up to~35 K with pressure, and turns to decrease above 6.0 GPa, forming a broad superconducting dome as a function of pressure as illustrated in Fig. 1. We observe T m only at 5.0 GPa for this S content, implying that the system is approaching the verge of the pressure-induced SDW phase ( Supplementary Fig. 2f).
Emergent tetragonal phase. As x is increased, the pressureinduced SDW dome shifts to higher pressure and shrinks, while low-pressure non-magnetic nematic phase shifts to lower pressure and disappears at x~0. 17. We stress that the nematic phase is completely separated from the SDW phase at x ≥ 0.04. To confirm the separation between two distinct phases under pressure, we performed synchrotron X-ray diffraction measurements under pressure for x = 0.08 (Fig. 3a, and Supplementary Note 4). In Fig. 3b, c, we show (331) Bragg intensity as a function of temperature at 3.0 and 4.9 GPa together with the ρ(T) and dρ/dT data. At 3.0 GPa, no discernible change of the Bragg-peak is observed down to the lowest temperature of 10 K (Fig. 3b and Supplementary Fig. 9a). At 4.9 GPa, on the other hand, the splitting of the Bragg peak is clearly resolved around T s 41 K, evidencing the presence of the tetragonal-to-orthorhombic structural transition (Fig. 3c and Supplementary Fig. 9b). This structural transition is located very close to the SDW transition at T m at 5.0 GPa as indicated by the sharp peak in dρ/dT curve in Fig. 3b. Thus it is natural to consider that the magnetic phase has an orthorhombic structure, similar to the case of the highpressure SDW phase of FeSe 13,14 .
These results demonstrate that the high-T c superconductivity in FeSe 1−x S x is realised in the tetragonal phase newly emerged between the orthorhombic nematic and magnetic phases. In the non-magnetic tetragonal phase (1 GPa ≲ P ≲ 3 GPa), T c shows a strong increase with P (Figs. 1, 3a), indicating that the enhancement of superconductivity is most pronounced near the  verge of the magnetic phase, not the nematic phase. It is also likely that the T-linear resistivity observed near the SDW boundary (Fig. 3b) is a consequence of enhanced antiferromagnetic fluctuations, as reported in other high-T c cases 8,20,21 .

Discussion
Why the effects of two tuning parameters, physical pressure and isovalent substitution, are so different? In general, applying pressure reduces lattice constants, and it leads to an increase of bandwidth as well as a change in the Coulomb interactions 22 , often affecting the ground state of the system. The chemical substitution by smaller ions also leads to a decrease of lattice constants, which results in similar effect on the system as the pressure effect. Indeed, in BaFe 2 As 2 system, the physical and chemical pressure effects on superconductivity are essentially similar 23 . To address the origin of the difference between chemical and physical pressure effects in FeSe, we determine the structure parameters of FeSe 1−x S x at room temperature by singlecrystal X-ray diffraction, which are compared with the published data under pressure 24 (Fig. 4a-c and Supplementary Table 1). As expected, both a-axis and c-axis lattice constants decrease with S content x, which follow the trends under physical pressure. The quantitative comparison suggests that 10% substitution corresponds to~0.3 GPa (Fig. 4a, b). This can be compared with effects of chemical and physical pressure on the phase diagrams of BaFe 2 As 2 , where the 30% substitution of P for As 21 and application of~0.55 GPa 25 both lead to the maximum T c 30 K.
In sharp contrast to the a-axis and c-axis lattice constants, there is a significant difference in the trends of the chalcogen height h Ch from the iron plane (Fig. 4c). It has been pointed out in iron pnictides that the height of the anion atoms from the Fe plane plays an important role on the existence of hole-like Fermi surface around the zone corner of the unfolded Brillouin zone, which has significant influence on the nesting properties between the Fermi surfaces 26 . It has also been shown that the chalcogen height in FeSe 1−x Te x is an important factor for the magnetic interactions 27 . We find that the isovalent substitution reduces h Ch monotonically which is opposite to the observed increasing trend due to physical pressure effect. Thus, this contrasting variation of the chalcogen height against chemical and physical pressures is likely responsible for the shift of the magnetic dome to the higher pressure side with S-substitution, because for samples with a larger S content a higher pressure is required for obtaining a large enough h Ch to induce magnetic order. Indeed, recent theoretical calculation investigating the pressure effect in FeSe points out that the increase of h Ch results in the appearance of Fermi surface in the Brillouin zone corner, which explains the emergence of magnetism under pressure 28 .
Another intriguing observation is that inside the SDW phase the superconducting transition becomes broad with a relatively high onset temperature ( Fig. 2 and Supplementary Figs. 3-6). This is demonstrated in Supplementary Fig. 1, which shows that T peak c defined by the peak temperature in dρ(T)/dT continues to rise with pressure upon entering the SDW phase, whereas the zero resistivity T zero c is suppressed. This implies the competing nature of the magnetic and superconducting orders, which may disturb the development of high-T c superconductivity. Indeed, a recent detailed study of ac susceptibility measurements under pressure shows that the volume fraction in the diamagnetic shielding is suppressed in the pressure-induced SDW phase, pointing to non-bulk superconductivity inside the magnetic phase 29 .
We have shown that by combining physical and chemical pressure effects, which can change the chalcogen height in different manners, nematicity and magnetism can be tuned to control T c in FeSe-based superconductors. The most notable feature is that the high-T c superconductivity in the tetragonal phase emerges at the verge of both side of the SDW dome, while T c is little influenced by the non-magnetic nematic phase. Magnetic order appears to exert two effects on superconductivity: on one hand, their competing nature suppresses the formation of bulk high-T c superconductivity inside the magnetic phase, and on the other hand, the enhanced fluctuations near the end points of the magnetic phase may help increase T c , much more significant than the non-magnetic fluctuations near the nematic quantum critical point at x 0:17 and P 0 GPa. In view of the orthorhombicity found in the pressure-induced SDW phase, an intriguing issue that deserves further studies is whether the nematic and magnetic fluctuations cooperatively promote the superconducting pairing, as recently suggested theoretically 28 .

Methods
Single crystals. High-quality single crystals of FeSe 1−x S x (x = 0, 0.04, 0.08, 0.12 and 0.17) have been grown by the chemical vapour transport technique 18 . The x values are determined by the energy dispersive X-ray spectroscopy. In the crystals obtained under identical conditions, quantum oscillations have been observed in a wide range of x(≤0.19) 30 , indicating superior crystal quality.
High-pressure measurements. High-pressure resistivity ρ(T, P) measurements have been performed under hydrostatic pressures up to 8 GPa with a constantloading type cubic anvil apparatus which can maintain a nearly constant pressure over the whole temperature range from 300 to 2 K 9,19 . For all these high-pressure resistivity measurements, we employed glycerol as the pressure-transmitting medium, and used the conventional four-terminal method with current applied within the ab plane. High-pressure ac susceptibility measurements have been done by using a mutual inductance technique in a moissanite anvil cell with glycerol as the pressure-transmitting medium 23 . The pressure achieved was determined by measuring the wavelength of the R 1 peak of ruby fluorescence. Synchrotron X-ray diffraction measurements under pressure have been performed at BL22XU in SPring-8 by using diamond anvil cell diffractometer equipped with a gas membrane for maintaining constant pressure on cooling 31 . Helium is used as the pressure-transmitting medium. The pressure value in the sample space is monitored by tracking the ruby fluorescence wavelength for the whole temperature range.
Data availability. The data that support the findings of this study are available from the corresponding author upon reasonable request.