Unconventional Superconductivity and Moderate Spin Fluctuations with Gap at Low Energies in Intercalated Iron Selenide Superconductor Li_x(NH₃)_yFe_2−δSe₂ Probed by ⁷⁷Se NMR

Figure 2(a) shows the temperature (T) dependence of the ⁷⁷Se NMR spectra of [Li(NH₃)]FeSe as a function of Knight shift (K), which were converted from the frequency (\(f\)) as \((f-\gamma_{n}B_{0})/\gamma_{n}B_{0}\). The spectra show a large main peak and two small peaks. The main peak, which has the highest intensity and smallest K, is indicated by hatching in the figure. It is attributed to the intrinsic Se site, which is confirmed by the sharp decreases in K and \(1/T_{1}T\) below ∼44 K. The two small peaks are presumed to be extrinsic; they may result from impurity Se sites in stacking faults, because the T dependence of their K values is similar to that of the Se site in bulk FeSe.³⁶⁾ The full width at half-maximum of the main peak is \(\Delta K\sim 300\) ppm at 140 K, which is much narrower than the value of \(\Delta K\sim 1300\) ppm in the ⁷⁷Se NMR spectra of the related compound Li_x(C₂H₈N₂)_yFe\(_{2-\delta}\)Se₂,³⁵⁾ denoted as [Li(C₂H₈N₂)]FeSe hereafter. This result suggests that the disorder in the intercalant layers is less than that in [Li(C₂H₈N₂)]FeSe,³⁵⁾ which may be attributed mainly to the smaller size and simpler shapes of the intercalant molecules. Although the presence of stacking faults in these intercalated compounds is usually unavoidable,²⁷⁾ the narrow spectra of [Li(NH₃)]FeSe enable us to detect the K and \(1/T_{1}\) at the intrinsic Se site of the intercalated FeSe layers, selectively.

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Figure 2. (a) T dependence of ⁷⁷Se NMR spectra of [Li(NH₃)]FeSe as a function of Knight shift (K). The main peak (hatched area) is attributed to the intrinsic Se site, which is narrow enough to be distinguished from extrinsic Se sites. (b) T dependence of Knight shift of [Li(NH₃)]FeSe, together with previous ⁷⁷Se NMR results for related FeSe-based compounds, [Li(C₂H₈N₂)]FeSe,³⁵⁾ K_xFe\(_{2-y}\)Se₂,³¹⁾ and (Tl_0.47Rb_0.34)Fe_1.63Se₂.³⁴⁾

Figure 2(b) shows the T dependence of K at the intrinsic Se site of [Li(NH₃)]FeSe. In the normal state, the Knight shift decreases with decreasing temperature. The Knight shift is generally represented as \(K =K_{\text{s}} + K_{\text{chem}} = A_{\text{hf}}\chi_{0} + K_{\text{chem}}\), where \(K_{\text{s}}\) is the spin part of K given by the hyperfine coupling constant \(A_{\text{hf}}\) and the local spin susceptibility \(\chi_{0}\), and \(K_{\text{chem}}\) is the T-independent chemical shift. As described below, \(K_{\text{chem}}\) is estimated to be ∼0.14%. Here \(\chi_{0}\) is proportional to the density of states (DOS) as \(\chi_{0}(T)\propto \int N(\epsilon) (\partial f(\epsilon)/\partial\epsilon)\,d\epsilon\), where \(f(\epsilon)\) and \(N(\epsilon)\) are the Fermi–Dirac distribution function and the DOS at energy ϵ, respectively. The T dependence of K in the normal state results from the characteristic band structure, namely, the large energy dependence of \(N(\epsilon)\) near \(E_{\text{F}}\). As shown in Fig. 2(b), this feature appears in many other electron-doped intercalated compounds,^31–35) which are characterized by FS components dominated by large electron pockets and a sinking small hole pocket, or an incipient hole band around \(E_{\text{F}}\). In the SC state, \(K_{\text{s}}\) decreases sharply below \(T_{\text{c}}\sim 44\) K owing to the formation of spin-singlet pairs, which is associated with line broadening, indicating that the \(T_{\text{c}}\) determined microscopically by NMR corresponds to the bulk \(T_{\text{c}}\) revealed by the macroscopic diamagnetic response (Fig. 1).

Next, we focus on the SC properties on the basis of the nuclear spin relaxation rate \(1/T_{1}\) at the intrinsic Se site of [Li(NH₃)]FeSe. As shown in Fig. 3, the T dependence of \(1/T_{1}\) decreases sharply without a coherence peak below \(T_{\text{c}}\). Assuming \(1/T_{1}\sim T^{n}\), n is close to ∼4 just below \(T_{\text{c}}\) and \(n\sim 1\) well below \(T_{\text{c}}\), indicating an unconventional SC state with the residual DOS induced at \(E_{\text{F}}\). Similar behavior has been widely observed across various Fe-based superconductor families. Regarding the FS topology, the presence of a tiny hole-like pocket was theoretically suggested for similar compounds within the framework of the random phase approximation;^45,46) however, there are no experimental reports confirming the presence or absence of the hole FS or the SC gap symmetries of this compound to date. Thus, we tried to reproduce the data by assuming possible unconventional superconductivity models based on \(s^{\pm}\)-wave, nodal d-wave, nodeless d-wave, and \(s^{++}\)-wave states for comparison, as shown in Figs. 3(a)–3(d), respectively. The energy dependence of the DOS used in each model is shown in Figs. 3(e)–3(h).

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Figure 3. (Color online) \(1/T_{1}\) for [Li(NH₃)]FeSe decreases sharply without a coherence peak below \(T_{\text{c}}\). The experimental results are well reproduced by unconventional SC states based on both (a) \(s^{\pm}\)-wave and (b) nodal d-wave models assuming the appropriate parameters,⁵⁹⁾ but not by (c) nodeless d-wave and (d) \(s^{++}\)-wave models (see text).⁵⁹⁾ For comparison, we applied these simulations of the previous results for [Li(C₂H₈N₂)]FeSe³⁵⁾ (open circles) by changing only the scattering parameter ξ within the same model. The result was well reproduced in both (a) \(s^{\pm}\)-wave and (b) nodal d-wave states (broken curves). The lower panels (e)–(h) show \(N(\epsilon)\) in the SC state corresponding to the simulated curves in upper panels (a)–(d).

For the \(s^{\pm}\)-wave state [solid curve in Fig. 3(a)], the result is well reproduced by a simple two-gap model^47,48) that assumes an FS topology composed of a small hole-like pocket and an electron-like pocket with the opposite gap function with an impurity scattering parameter ξ. Here we used the appropriate parameters of the sign-reversal gaps, \(|\Delta_{1}(0)| = 4.0k_{\text{B}}T_{\text{c}}\) and \(|\Delta_{2}(0)| = 2.0k_{\text{B}}T_{\text{c}}\), a DOS ratio \(N_{1}/N_{\text{tot}}\) of 0.4, and an impurity scattering parameter ξ of 0.0625 [Fig. 3(e)], where \(N_{\text{tot}} = N_{1} + N_{2}\). In this simulation, the amplitudes of the SC gaps are assumed to be similar to the experimental values for related intercalated FeSe compounds.^49–53) This is consistent with recent theoretical studies of similar intercalated FeSe-based high-\(T_{\text{c}}\) compounds, where the \(s^{\pm}\)-wave state is expected to be realized in the presence of a small hole FS and large electron FSs.^45,46) Even if the hole pocket is slightly below \(E_{\text{F}}\), the \(s^{\pm}\)-wave SC state is possible to be realized by the incipient hole-like pocket below \(E_{\text{F}}\) and large electron pockets at \(E_{\text{F}}\) in the sign-reversal gap function.^54–56) If the hole pocket is too far below \(E_{\text{F}}\) to contribute to the pairing interaction, the d-wave state at the large electron FSs becomes another candidate. Here we also simulated the nodal and nodeless d-wave states within the single-gap models following Ichikawa et al.^57,58) For the nodal d-wave model,⁵⁸⁾ the result was also reproduced using the appropriate parameters, a single d-wave gap \(|\Delta(0)|\) of \(3.5\,k_{\text{B}}T_{\text{c}}\) and \(\xi = 0.004\) for [Li(NH₃)]FeSe, as shown by the solid curve in Fig. 3(b). By contrast, for the nodeless d-wave model,⁵⁸⁾ it was difficult to find appropriate parameters for reproducing the experiments,⁵⁹⁾ as shown in Fig. 3(c), because an in-gap state at \(E_{\text{F}}\) is anticipated,⁵⁸⁾ as displayed in Fig. 3(g). We also tried to simulate the \(s^{++}\)-wave state using the sign-preserved version of the \(s^{\pm}\)-wave model mentioned above;^47,48) however, it was difficult to simultaneously reproduce the entire T dependence of \(1/T_{1}\) and the suppressed coherence peak of \(1/T_{1}\) just below \(T_{\text{c}}\) for any parameters,⁵⁹⁾ as shown in Fig. 3(d). If the impurity parameter ξ is assumed to be large enough to eliminate the coherence peak, the residual DOS is too large, as indicated by the solid curves in Figs. 3(d) and 3(h). If the SC gap is assumed to be extremely large to eliminate the coherence peak, the calculation results show a sharp decrease below \(T_{\text{c}}\), as shown by the dash-dotted curves in Figs. 3(d) and 3(h); however, this result is inconsistent with the experiment, even if we assume that the SC gaps are almost twice as large as those observed experimentally.^49–53)

To further compare these models, we attempted to reproduce the previous NMR results for a similar compound, [Li(C₂H₈N₂)]FeSe, with \(T_{\text{c}} = 45\) K,^23,35) where the ammonia used as an intercalant in [Li(NH₃)]FeSe is replaced by ethylenediamine. The microscopic similarity, i.e., the similar T dependence of K and \(1/T_{1}\) in the normal state, ensures that their superconductivity appears in similar electronic states [Figs. 2, 3(a), and 3(b)]. The ⁷⁷Se NMR spectra show significant differences; specifically, the linewidth is four times as large, and the \(1/T_{1}T\) values well below \(T_{\text{c}}\) are dominated by a residual DOS that is five times larger (∼50%) in the SC state than those of [Li(NH₃)]FeSe. These differences may be attributable mainly to unavoidable structural features, including the longer and larger intercalant molecule C₂H₈N₂. Here, we attempted to reproduce both data points by changing only the impurity scattering parameter ξ without changing the other parameters. As shown by the broken curves in Figs. 3(a) and 3(b), the \(1/T_{1}\) values in the SC state were well reproduced by increasing only the scattering parameter, to \(\xi = 0.3\) in the \(s^{\pm}\)-wave model and \(\xi = 0.15\) in the nodal d-wave model, where the DOS used in this simulation is shown in Figs. 3(e) and 3(f). We also note that the results cannot be reproduced in the nodeless d- and \(s^{++}\)-wave states by changing the scattering parameter.

Next, we discuss the normal-state properties on [Li(NH₃)]FeSe. As shown in Fig. 4(a), in the normal state, \(1/T_{1}T\) exhibits a monotonous decrease upon cooling, which resembles that of the Knight shift [Fig. 2(b)]. Here we extract the component of the AFM spin fluctuations following previous reports.^60,61) The observed \((1/T_{1}T)\) is expressed as \begin{equation*} (1/T_{1}T) = (1/T_{1}T)_{\text{AFM}} + (1/T_{1}T)_{0}, \end{equation*} where \((1/T_{1}T)_{0}\) is the component related to \(K_{\text{s}}^{2}\) [\(\propto N(E_{\text{F}})^{2}\)], and \((1/T_{1}T)_{\text{AFM}}\) is the component derived from spin fluctuations at a finite wave vector (\(\boldsymbol{q}\)) at low energies (\(\omega\rightarrow 0\)), as expressed by \begin{equation*} \left(\frac{1}{T_{1}T}\right)_{\text{AFM}} {}\propto \lim_{\omega \rightarrow 0} \sum_{\boldsymbol{q}} A_{\text{hf}}(\boldsymbol{q})^{2} \frac{\chi''(\boldsymbol{q},\omega)}{\omega}. \end{equation*} Figure 4(b) shows a plot of \((1/T_{1}T)^{0.5}\) vs K (\(= K_{\text{s}}+K_{\text{chem}}\)) in the normal state. If the compound is a normal metal without spin fluctuations, we expect the linear relation known as Korringa relation. The slight deviation of the experimental data in the plot indicates the presence of the non-negligible component \((1/T_{1}T)_{\text{AFM}}\), i.e., spin fluctuations at a finite wave vector \(\boldsymbol{q}\) (\(\sim Q\)). As will be discussed in Sect. 3.4, as shown in Fig. 5(b), similar non-linear deviation derived from the spin fluctuations has been observed in heavily electron doped SC compound LaFe(As_0.7Sb_0.3)(O_0.7H_0.3) with \(T_{\text{c}} = 21\) K, whereas such deviation totally disappears in non-SC compound LaFe(As_0.8P_0.2)(O_0.75H_0.25) (open diamonds).⁶²⁾ On the analogy of the case of La1111, the Korringa relation without spin fluctuations is tentatively determined as shown by the solid line of Fig. 4(b). It enable us to estimate \(K_{\text{chem}}\sim 0.14\)%, which is similar to the values evaluated in previous studies.^31,35) As a result, the T dependence of \((1/T_{1}T)_{\text{AFM}}\) can be extracted as shown in Fig. 4(c), which shows a peak near 200 K and a decrease at lower temperatures. This feature suggests the presence of a gap in the spin fluctuation spectrum \(\chi''(Q)\) at low energies. It is in contrast to the typical Fe-based compounds characterized by hole and electron FSs with similar sizes, such as bulk FeSe, where spin fluctuations are significantly enhanced at low energies toward low temperatures.^7,36,60,61)

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Figure 4. (Color online) (a) T dependence of \(1/T_{1}T\) in the normal state for [Li(NH₃)]FeSe. (b) Plot of \((1/T_{1}T)^{0.5}\) vs K in the normal state. The hatched region shows a slight deviation from linear dependence (dotted line), which corresponds to the spin fluctuation component \((1/T_{1}T)_{\text{AFM}}\). (c) T dependence of \((1/T_{1}T)_{\text{AFM}}\) shows a maximum near 200 K (see text).

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Figure 5. (Color online) T dependence of \((1/T_{1}T)_{\text{AFM}}\) extracted by the same method for the heavily electron-doped Fe-based superconductors: (a) [Li(C₂H₈N₂)]FeSe (\(T_{\text{c}} = 45\) K),³⁵⁾ K_xFe\(_{2-y}\)Se₂ (\(T_{\text{c}} = 33\) K),³¹⁾ K_0.8Fe₂Se₂ (\(T_{\text{c}} = 32\) K),³²⁾ [Tl_0.47Rb_0.34]Fe_1.63Se₂ (\(T_{\text{c}} = 32\) K),³⁴⁾ and (b) LaFe(As_0.7Sb_0.3)(O_0.7H_0.3) (⁷⁵As-NMR) (\(T_{\text{c}} = 21\) K).⁶²⁾ The inset shows the plot of \((1/T_{1}T)^{0.5}\) vs K, and the dotted straight lines exhibits Korringa relation determined from non-SC related compound LaFe(As_0.8P_0.2)(O_0.75H_0.25) (open diamonds).⁶²⁾ The hatched regions represent \((1/T_{1}T)_{\text{AFM}}\), which corresponds to the deviation from the Korringa relation. The behavior of the heavily electron-doped Fe-based superconductors^31,34,35,62) which qualitatively exhibits spin-gap behavior below \(100\,\text{K}<T< 250\,\text{K}\). This is in contrast to the case of (c) bulk FeSe (\(T_{\text{c}}\sim 8\) K),⁷⁾ where the low energy AFMSFs significantly enhance toward low temperatures.

Here we discuss the universality and diversity of the relationships between the characteristics of the spin fluctuations and superconductivity in Fe-based compounds. We note that a similar deviation from the Korringa relation in the normal state have been commonly observed in many intercalated FeSe-based compounds in the heavily electron-doped regimes, such as [Li(C₂H₈N₂)]FeSe,³⁵⁾ K_xFe\(_{2-y}\)Se₂,³¹⁾ K_0.8Fe₂Se₂,³²⁾ and [Tl_0.47Rb_0.34]Fe_1.63Se₂.³⁴⁾ It suggests that a qualitatively similar T dependence of \((1/T_{1}T)_{\text{AFM}}\) can be extracted, as summarized in Fig. 5(a), if the same analyses can be applied to the data sets on these compounds.^31,34,35) It is suggested that the spin fluctuations with a gap at low energies (\(\omega\sim 0\)) is widely seen for heavily electron-doped intercalated FeSe-based superconductors. These behaviors differ significantly from that of compounds characterized by typical nested FSs consisting of hole and electron FSs with similar sizes, where the close relationship is seen between \(T_{\text{c}}\) and the spin fluctuations critically enhanced at low energies and low temperatures.^7,36,60,61) For example, it is significantly different from the case of bulk FeSe, as shown in Fig. 5(c).^7,36–38)

In order to search the universality, we have recently investigated the reemergent high-\(T_{\text{c}}\) phase of heavily electron-doped regime in hydrogen-substituted LaFeAs(O\(_{1-y}\)H_y).⁶²⁾ As shown in Fig. 5(b), a similar nonlinear relationship was also observed in the plot of \((1/T_{1}T)^{0.5}\) vs K for heavily hydrogen-substituted LaFe(As_0.7Sb_0.3)(O_0.7H_0.3) with \(T_{\text{c}} = 21\) K, suggesting the appearance of the spin fluctuations with a gap at low energies. Remarkably, such gapped spin fluctuations totally disappear when the pnictogen height decreases with P-substitution at As site or the electron-doping level is lowered, in association with the suppression of \(T_{\text{c}}\), as will be published in detail elsewhere.⁶²⁾ These results imply the possible relationship between novel spin fluctuations and the emergence of SC phase in heavily electron-doped regime. It has been suggested theoretically that in the heavily electron-doped state, the large electron FSs and a very small hole-like pocket (or the incipient hole band) are dominated mainly by the \(d_{xy}\) orbital, because the energy levels of the \(d_{xy}\) orbital also remain high owing to the relatively high anion height.^45,46,63) Even for the sinking hole-like (incipient) band, spin fluctuations can play a key role in unconventional superconductivity in heavily electron-doped systems,^54–56) where the finite-energy component of the spin fluctuations is theoretically expected.⁶⁴⁾ To exhibit the remarkable high-\(T_{\text{c}}\) phase in heavily electron-doped state, anion height that keeps high might be important: In fact, in the case of heavily electron-doped Ba(Fe\(_{1-x}\)Co_x)₂As₂,⁶⁰⁾ both superconductivity and spin fluctuations are not observed for \(x>0.15\), since the anion height becomes low when Fe is replaced with Co. Thus, the simultaneous observation of high-\(T_{\text{c}}\) states and the presence of spin fluctuations at finite energies (even with a gap at low-energies) might be a common feature of heavily electron-doped Fe-based compounds. Theoretically it is pointed out that lack of low energy part in spin fluctuations is rather favorable in the case of heavily electron doped states.^64–68) In this context, it is suggested that the spin fluctuations play an important role even in heavily electron-doped Fe-based compounds. In this scenario, it remains unknown whether the unconventional high-\(T_{\text{c}}\) SC state with a sign-reversed gap functions is possible without the suppression of \(T_{\text{c}}\).^58,69) In this sense, the role of nematic fluctuations in the electronic states in association with the degenerated orbital states will be clarified systematically toward the wide doping range. To further examine the common features of other electron-doped intercalated FeSe- and FeAs-based compounds, it is necessary to obtain further information on spin fluctuations from the low-energy to the high-energy region with/without nematic order by combining NMR measurements, neutron scattering experiments, and other spectroscopic experiments.