Separate tuning of nematicity and spin fluctuations to unravel the origin of superconductivity in FeSe

The interplay of orbital and spin degrees of freedom is the fundamental characteristic in numerous condensed matter phenomena, including high temperature superconductivity, quantum spin liquids, and topological semimetals. In iron-based superconductors (FeSCs), this causes superconductivity to emerge in the vicinity of two other instabilities: nematic and magnetic. Unveiling the mutual relationship among nematic order, spin fluctuations, and superconductivity has been a major challenge for research in FeSCs, but it is still controversial. Here, by carrying out 77Se nuclear magnetic resonance (NMR) measurements on FeSe single crystals, doped by cobalt and sulfur that serve as control parameters, we demonstrate that the superconducting transition temperature Tc increases in proportion to the strength of spin fluctuations, while it is independent of the nematic transition temperature Tnem. Our observation therefore directly implies that superconductivity in FeSe is essentially driven by spin fluctuations in the intermediate coupling regime, while nematic fluctuations have a marginal impact on Tc.


Introduction
[4] Regardless of the origin of nematicity that is still under debate, 5,6 this raises the fundamental issue of whether superconductivity in FeSCs is closely related to nematicity 7,8 or magnetism 9,10 or both. 11To address this issue, it is much desirable to separate nematic order from magnetic one.In this respect, FeSe has been a key platform for studying the origin of nematicity and its role in superconductivity, 12 as it exhibits nematic and SC orders at well separated temperatures, T nem ∼ 90 K and T c ∼ 9 K, respectively, without involving magnetic order.Numerous recent studies in FeSe show that nematicity causes the strongly anisotropic SC gap symmetry, [13][14][15][16] and further discuss that nematic fluctuations might play an important role for the superconducting pairing mechanism. 17,180][21][22] In this spin fluctuation-mediated pairing scenario, the subsequent question arises whether weak or strong coupling approach is appropriate to establish theory of superconductivity in FeSCs.
It is quite interesting to note that recent NMR studies of FeSe under high pressure reveal the persistence of local nematicity at temperatures far above T nem , which suggests a correlation between local nematicity and magnetism. 23,24Another interesting observation by NMR is the unusual suppression of (T 1 T ) −1 at optimal pressure, 25 suggesting that the interplay of SFs and superconductivity may undergo a critical change with high pressure.
As a system undergoes a nematic transition (C 4 → C 2 ), two nematic domains are naturally formed below T nem , still preserving the C 4 symmetry on average.Accordingly, it is usually required to detwin nematic domains, for example, by an external strain to study nematicity.As a local probe in real space, on the other hand, NMR is uniquely capable of observing the two nematic domains at the same time.Indeed, it has been established that the splitting of the NMR line in FeSCs at an external field H applied along the crystallographic a axis represents the nematic order parameter and its onset temperature corresponds to the nematic transition temperature T nem (Ref.26-28, see Fig. 1c).In order to investigate whether and how nematicity is related to superconductivity, we measured the 77 Se line splitting for H a in FeSe 1−y S y and Fe 1−x Co x Se single crystals.In general, it is considered that substituting isovalent S for Se is equivalent to the application of (negative) chemical pressure, and Co substituted for Fe supplies an additional electron and also plays as a paramagnetic impurity.Therefore, a systematic NMR study on the two different doped systems may enable a full understanding of the relationship between nematicity, magnetism, and superconductivity.

Results and Discussion
Figure 1 shows how the temperature dependence of the 77 Se NMR spectrum in FeSe is modified as a function of x in Fe 1−x Co x Se (Figs. 1a and 1b) and as a function of y in FeSe 1−y S y (Figs.1d and 1e).For FeSe 1−y S y , we find that the onset temperature of the line splitting or T nem is gradually suppressed, consistent with previous studies. 17,21For Fe 1−x Co x Se, however, T nem hardly changes for x = 0.018.Upon further doping to slightly higher x = 0.025, the 77 Se line becomes significantly broad, making difficult to identify the onset of the line splitting.The much larger 77 Se line broadening for Co doping than for S doping is well understood because Co has a strong influence on Fe moments as a nonmagnetic impurity.We notice, however, that the 77 Se line broadening is not simply proportional to the concentration of Co dopants, but rather it appears to increase drastically above x ∼ 0.025.
In fact, for x = 0.036, it is not possible to observe the line splitting anymore, because the linewidth is much larger than the nematic splitting (see supplementary Fig. 1).On the other hand, the 77 Se linewidth is proportional to both S and Co dopants similarly, as long as Co-doping is smaller than 2.5%, as shown in supplementary Fig. 2.This suggests that doped Co impurities beyond ∼ 2.5% of Fe sites causes a strong disorder effect on the correlation between Fe spins, indicating the existence of a critical doping level above which the magnetic correlation length becomes sufficiently long to induce a short-range exchange.
Although T nem cannot be accurately determined for x = 0.025 due to the large line broadening in the nematic state, we clearly observed the line splitting below 80 K, as shown in Fig. 1a.While this puts a lower limit of T nem , the fitting analysis of 77 Se spectra in Interestingly, the split 77 Se lines below 80 K for x = 0.025 is notably anisotropic, i.e., the peak for the lower frequency side is broader than that for the higher frequency.The origin of the anisotropic line shape is unclear, but we note that the similar anisotropic 77 Se line shape is also observed at T < 20 K for x = 0.018 (see Fig. 1b).This implies that magnetic inhomogeneity, which otherwise appears at low temperatures, prevails at higher temperatures with higher Co doping.
Contrasting sharply with the weak dependence of nematicity on both S and Co dopants, our susceptibility measurements reveal that superconductivity is strongly dependent only on Co dopants.That is, T c is rapidly suppressed by small Co doping, whereas it is robust with regard to S doping, as shown in Figs.2a and 2b, being consistent with previous studies. 29,30The very different behavior of T nem and T c with doping indicates that nematic and superconducting orders are not directly coupled, 31,32 raising a strong question as to whether nematicity and superconductivity are closely related. 7,8,17,33ving established the lack of a coupling of the nematic and superconducting transition temperatures, we now discuss the role of SFs for superconductivity.For probing low energy SFs, we measured the spin-lattice relaxation rate, T −1 1 , as the quantity (T 1 T ) −1 is a measure of SFs at very low energy: where χ ′′ (k, ω) is the imaginary part of the dynamic susceptibility at momentum k and frequency ω, γ n is the nuclear gyromagnetic ratio, and A(k) is the structure factor of the hyperfine interaction.Figures 2c and 2d show (T 1 T ) −1 as a function of Co and S doping, respectively, at H a = 9 T. The data for the undoped FeSe crystal was taken from ref. 26.
With increasing Co doping x in Fe 1−x Co x Se, (T 1 T ) −1 or SFs above T c is rapidly suppressed, which is in exact parallel with the suppression of T c , as shown in Fig. 2a.Note that for x = 0.035 superconductivity is completely absent, and correspondingly SFs are not enhanced at all at low temperatures.On the other hand, (T 1 T ) −1 above T c is unchanged with increasing S doping y in FeSe 1−y S y up to y = 0.1, as precisely T c does (see Fig. 2b).
From the data presented in Figure 2, therefore, one sees that T c depends only on the strength of spin fluctuations, but not on T nem .(At larger S-doping near y = 0.2, it was reported that both (T 1 T ) −1 and T c are strongly suppressed in such a way that the correlation between SFs and T c persists, 21 somewhat similar to the behavior in Co-doped samples.) For further quantitative information on how SFs is related to T c , we adopt a spin fluctuation model in the Eliashberg formalism, 1 or Millis-Monien-Pines (MMP) model. 5For this, we separate out the enhancement of (T 1 T ) −1 that is solely associated with SFs from the data shown in Fig. 2c.Noting that (T 1 T ) −1 for the non-superconducting sample (x = 0.035) approaches a constant, (T 1 T ) −1 0 ≡ Γ 0 , without any enhancement at zero temperature, one may define the strength of SFs Γ for H a from the (T 1 T ) −1 (x) values just above T c : While the MMP model indicates that Γ is proportional to the square of the correlation length, 5 ξ 2 (T ), the estimation of the low energy part of the Eliashberg bosonic spectral function suggests that the coupling constant λ is proportional to ξ, i.e., √ Γ.As it was analyzed by Radtke et al. 36 and Popovich et al. 9 the direct use of the MMP-spectrum gives overestimation of T c and the gap function due to a long tail at high energies of the bosonic spectral functions ∼ 1/ω.To cure the problem it was proposed to introduce a cut-off or calculate the bosonic self-energy at high energies. 38For simplicity we use the approach of cutoff proposed in ref. 9.A detailed procedure of the calculation is described in Supplementary Note 1.
The plot of T c vs. √ Γ is shown in Fig. 3.The solid curve is a theoretical calculation (T c vs. λ ∝ √ Γ) based on the Eliashberg theory in which electron correlation effects are substantial.The good agreement of our theory with the experimental data evidences that the magnetic scenario for superconductivity in which Cooper pairing is mediated by spin fluctuations applies to FeSe, and it is likely a universal superconducting mechanism among FeSCs.
Based on our NMR finding that T c relies only on SFs, the seeming relevance of nematicity with superconductivity may be simply due to the closeness with magnetism, rather than to superconductivity itself.It should be noted that the strongly anisotropic gap structure [13][14][15][16]22 observed in FeSe may be a natural consequence of the presence of nematicity within the superconducting state. It because nematicity involves the splitting of d xz and d yz orbitals which should have an inevitable influence on the gap symmetry.However, T c itself is not necessarily affected by nematicity.39 Nevertheless, nematicity may be considered as an important barometer for superconductivity in FeSCs, as it is strongly coupled to magnetism 18 which in turn directly correlates with superconductivity.

Methods
Crystal growth and characterization.The growth of Fe was obtained by fitting the recovery of the nuclear magnetization M(t) after a saturating pulse to following fitting function, where A is a fitting parameter that is ideally unity.
Determination of T c and T nem .The superconducting transition temperature T c was determined from magnetic susceptibility (χ) measurements by comparing field-cooled and zerofield cooled data, while we obtained the nematic transition temperature T nem by measuring the temperature at which the 77 Se NMR line splits (see Fig. 1).Due to the weakness of the signal intensity, we were unable to determine T c by (T 1 T ) −1 measurements except the undoped FeSe sample.This could give an error in extracting spin fluctuations just above T c , Γ, which was reflected in an experimental error indicated in Fig. 3.The enhancement of (T 1 T ) −1 at low temperatures is progressively suppressed with increasing x (see Fig. 3).The solid lines are guides to the eyes.d, (T 1 T ) −1 as a function of temperature and S-doping y in FeSe 1−y S y .Spin fluctuations are unchanged with increasing S doping y, being consistent with T c that is nearly independent of y. denoted as χ 0 , a is the lattice constant, and Γ sf is the frequency scale characterizing the spin fluctuations.Within this model one gets the spin lattice relaxation rate in the following form at low temperatures (see also ref. 6.): b. Boson spectral function.The main input into the Eliashberg equations Eqs.( 3) and ( 4) is the spectral function of the intermediate bosons B(q, ω): where A(k, k ′ ) is the matrix element for the scattering an electron in Bloch state k to k ′ and B(q, Ω) = − 1 π N(0)Imχ(q, Ω) is the spectral function of the spin fluctuations normalized by the density of states at the Fermi level N(0).As it was pointed out in refs.7-9, that use of the MMP spectrum in the Eliashberg equation leads to overestimation of T c and superconducting gaps due to a long tail at high frequencies ∝ 1/ω.Here, we adopt the phenomenological approach proposed by Popovich et al. 9 The low energy part of the bosonic function is given by Eq. ( 6) and the function decays fast after a characteristic cut-off energy.The cut-off energy is determined by the band structure and in the leading approximation can be be taken independent on the distance to the quantum critical point.
Since we consider the leading term, we neglect the momentum dependence of A = g = const.Performing the momentum integration in Eq. ( 8) we get for small ω: It determines that λ ∼ ξ ∼ √ Γ.In Fig. 3 of the main text we show the experimental and calculated T c vs. λ and √ Γ correspondingly.The fit gives the scale of the coupling constants.

Fig. 1a also
Fig.1aalso suggests that the line splitting seems to persist even up to 100 K (see vertical 1−x Co x Se and FeSe 1−y S y single crystals was performed by using the KCl -AlCl 3 flux technique in permanent T-gradient in accordance with refs.40 and 41.All preliminary operations for the preparation of the reaction mixture were carried out in a dry box with a residual pressure of O 2 and H 2 O not higher than 0.1 ppm.At the first stage, polycrystalline samples of the composition Fe 1−x Co x Se and FeSe 1−y S y were obtained.For this, Fe, Co, S and Se powders were carefullyground in a mortar in the appropriate ratio, and then annealed in evacuated quartz ampoules at 420 • C for few days.In the second stage, 0.5 g of the prepared sample was placed on the bottom of a thick-walled ampoule, and then the mixture of AlCl 3 and KCl in a molar ratio of AlCl 3 :KCl = 2:1 is added to the ampule, after that the ampule was evacuated and sealed.The sealed ampoule with polycrystalline sample of Fe-Co-Se-S loads was placed in a horizontal 2-zone furnace and heated for five weeks in such a way that, the hot zone temperature was set to 420 • C and the cold zone temperature was set to 370 • C.After five weeks, the furnace was turned off and the ampoule was removed from the furnace.Next, the ampoule was cut and the single crystals from cold zone were separated from the flux by dissolving it in water.The single crystals obtained were thin square plates with metallic luster.The single crystals were grown with platelet like morphology and were characterized by SEM/EDX for compositional analysis.Nuclear magnetic resonance. 77Se (nuclear spin I = 1/2) NMR was carried out in undoped and doped FeSe single crystals at an external magnetic field and in the range of temperature 4.2 -160 K.The samples were oriented using a goniometer for the accurate alignment along the external field.The 77 Se NMR spectra were acquired by a standard spin-echo technique with a typical π/2 pulse length 2-3 µs.The nuclear spin-lattice relaxation rate T −1 1

FIG. 1 .FIG. 2 .
FIG. 1. 77 Se NMR spectra in undoped and doped FeSe single crystals for H a. a-b, Temperature dependence of 77 Se spectrum of Fe 1−x Co x Se.For x = 0.018 (b), the 77 Se spectrum shows a very similar behavior as the undoped one, except a moderate line broadening.For a slightly larger doping, x = 0.025 (a), the 77 Se line undergoes a considerable line broadening.While the splitting of the two 77 Se lines were clearly identified at low temperatures (vertical bars), the onset of the splitting is not well defined, being ascribed to local disorder.c, Temperature dependence of 77 Se spectrum for undoped FeSe.d-e, Temperature dependence of 77 Se spectrum of FeSe 1−y S y for y = 0.05 and 0.1, respectively.T nem is progressively suppressed with increasing S doping.

FIG. 3 .
FIG. 3. Superconducting transition temperature T c vs. √ Γ, where Γ is the strength of spin fluctuations just above T c for H a in Co-doped FeSe single crystals.The solid line represents our theory of T c vs. the coupling constant λ ∝ √ Γ (see text).The error bars represent the uncertainty in determining T c and Γ.