Quantum criticality in quasi-binary compounds of iron-based superconductors

Fig. 1a) shows the temperature dependencies of the resistance of the samples, normalized to the resistance value at 300 K. We use the normalized plots because the difference in the resistivity values of the compositions at 300 K is less than the magnitude of the possible error resulting of determining the geometric parameters of the samples. The room temperature resistivities were in the range of 0.4-1.0 m$\Omega$cm for all samples. The change in the shape of $R(T)$ curves at temperatures above the structural transitions obviously indicates the transition between good and bad-metal behavior which occurs in FeSe under the substitution of selenium to tellurium. An important difference for compositions with tellurium is the shape of the anomaly on $R(T)$ at the structural transition temperature. The change is clearly visible on the $dR/dT$ curves shown in Fig. 1b). Similar changes of $R(T)$ curves are observed in the pure FeSe under pressure Terashima et al. (2015); Kothapalli et al. (2016); Sun et al. (2016), there such a change occurs when the ground state becomes magnetic Bendele et al. (2012); Kothapalli et al. (2016). For the Fe(Se,Te) series, a change in the anomaly on $R(T)$ appears simultaneously with the change in $R(T)$ behavior at low temperatures and suggest a transformation in the ground state Ovchenkov et al. (2023).

The temperature dependence of the static magnetic susceptibility is shown in Fig. 1c). All samples demonstrate a Pauli-type dependence over a wide temperature range. Structural transitions do not cause noticeable changes in magnetic susceptibility. The increase in susceptibility at low temperatures for samples with substitution indicates the presence of a small amount of paramagnetic impurity, possibly caused by stoichiometry disorders. Measuring susceptibility using the ZFC protocol was used to determine the temperatures of the superconducting transition, which are listed in Table 1.

The samples with tellurium have noticeably higher specific heat values than pure FeSe (see Fig. 1d) ). The Debye temperatures of these compositions (Table 1) are higher by about 5-9 percent, which can be explained by the higher molar mass of Te. For FeSe, the $C(T)$ dependence is in good agreement with previously published data W. Yeh et al. (2008); Muratov et al. (2018) and shows a clear step at the structural transition point. For compositions with tellurium, there are no anomalies at the temperatures of the structural transition on the $C(T)$ curves and on their temperature derivatives. This may indicate that the temperature of the structural transition decreases due to general disorder and distortion of the local symmetry of the iron environment. Above superconducting transition temperatures, turning on a 90 kOe magnetic field has no measurable effect on the heat capacity of the samples.

The macroscopic properties show consistent change under substitution, indicating the high quality of the grown crystals. The results of transport measurements convincingly show changes in the type of anomalies at the phase transition point which suggests the presence of a new type of quantum critical point in quasi-binary IBS. For NMR experiments, we chose the S-Te3 batch because the synthesis of this compound yielded larger and higher quality crystals.

The ⁷⁷Se spectra of both powder samples show a uniformly broadened line with a shape close to Gaussian over the entire temperature range studied. Fig.2 demonstrates the temperature dependences of the Gaussian linewidth of the spectra together with its characteristic line shapes. The linewidth of S-Te3 is significantly higher than that for FeSe, which distinguishes this method of chemical pressure from hydrostatic pressure: the latter is not accompanied with such a pronounced broadening and anisotropy growth Wang et al. (2016); Wiecki et al. (2017).

The temperature dependence of the FeSe powder linewidth, which is determined by the shift anisotropy and the local ⁷⁷Se NMR linewidth, is in good agreement with the literature data. In particular, during the transition to the superconducting state, a sharp broadening is observed, just as reported in Refs. Wang et al., 2016; Wiecki et al., 2017; Li et al., 2022. Further, in the temperature range from $T_{c}$ to about 50 – 60 K, the temperature-independent linewidth is consistent with constant shift anisotropy Böhmer et al. (2015); Baek et al. (2015) at low values of the local linewidth Wiecki et al. (2021, 2017); Li et al. (2022); Wang et al. (2017). The subsequent narrowing of the line with increasing temperature is determined mainly by the decrease in anisotropy when approaching the nematic transition Baek et al. (2015), which, according to Ref. Böhmer et al., 2015, continues even with a further increase in temperature. On the other hand, in the tetragonal phase the contribution of the local linewidth increases, which, after a jump at $T_{S}$, also smoothly falls with increasing temperature (in Refs. Wang et al., 2017; Wiecki et al., 2021, 2017 this is explained by residual nematicity), experiencing a local minimum in the region of about 200 K Wiecki et al. (2017, 2021). This local minimum is in line with the observed minimum of powder linewidth in the range of 200 – 300 K.

The S-Te3 sample exhibits an almost constant linewidth in the range from 100 to 40 K, followed by a sharp broadening with further decrease in temperature. The breaking point ($\approx$ 40 K) coincides with the kink in the temperature dependence of the resistance ($T_{S}$ = 43 K, Fig. 1, Table 1.). A possible explanation of this linewidth behavior may be related to a sharp increase in line shift anisotropy below $T_{S}$, similar to the nematic transition in the parent FeSe. It is worth noting that the linewidth increase (from 24 kHz at $T_{S}$ to 41 kHz at 10 K) is noticeably higher than that for the parent FeSe. Moreover, a pronounced brake in the temperature dependence takes place exactly at $T_{S}$, in contrast to smooth dependence for FeSe. It may point to a predominance of the contribution from anisotropy to the line broadening, rather then the local linewidth. This indicates preservation of the low-temperature nematic phase at 30% substitution of selenium with a decrease in the transition temperature and a simultaneous increase in the degree of lattice distortion.

The temperature dependence of the spin-lattice relaxation rate 1/$T_{1}$ in FeSe (Fig. 3) is also consistent with literature data. In particular, in the temperature range from $T_{c}$ to $T_{S}$, a domination of the antiferromagnetic fluctuations contribution to relaxation processes is observed, which is expressed in a rather weak growth (slower than $T^{1}$) of the relaxation rate with temperature Imai et al. (2009); Baek et al. (2015, 2020); Masaki et al. (2009). In the inset of Fig. 3 the same dependence is shown in coordinates 1/($T_{1}T^{\frac{1}{2}}$) vs $T$, which clearly shows the square root dependence of the relaxation rate in this temperature range. Indeed, in the case of weakly and almost antiferromagnetic metals, the spin-lattice relaxation rate is proportional Moriya (2012) to $T/(T-T_{N})^{\frac{1}{2}}$, which at $T\gg{}T_{N}\approx 0$ gives $T^{\frac{1}{2}}$, or 1/($T_{1}T^{\frac{1}{2}}$)=const Katayama et al. (1977); Gippius et al. (2014). The corresponding approximation (inset of Fig. 3) shows good agreement with the experimental data. Below $T_{c}$ there is a sharp drop in the relaxation rate without a Hebel-Slichter peak, which is also typical for iron-containing superconductors in general and FeSe in particular Li et al. (2022); Imai et al. (2009); Masaki et al. (2009); Kotegawa et al. (2008); Molatta et al. (2021); Shi et al. (2018).

The temperature dependence of the spin-lattice relaxation rate of the S-Te3 sample with 30% substitution of tellurium for selenium (Fig. 4) resembles linear behavior in the studied temperature range from 10 to 80 K. Unlike unsubstituted FeSe, in this case it is hardly possible to talk about the influence of pronounced magnetic fluctuations, because they would give slower growth Moriya (2012). In the 1/($T_{1}T$) vs $T$ coordinates (on the inset of Fig. 4), one can clearly distinguish two horizontal regions 1/($T_{1}T$)=const, corresponding to the Korringa’s law for relaxation through conduction electrons. It is worth noting that superconductivity still takes place in this compound although partially suppressed ($T_{c}$ = 6.0 K instead of 8.9 K for parent FeSe). This result points to a rather rare case of the superconductivity preservation in iron-based superconductor without intensive magnetic fluctuations. It resembles other tellurium intermediate substitutions FeSe_1-xTe_x with $x=0.5$ Shimizu et al. (2009), 0.58 Arčon et al. (2010) and 0.6 Hara et al. (2011) following Korringa’s law and contrasts with specific low-temperature magnetic enhancement of 1/($T_{1}T$) in FeSe and Te-enriched compounds FeSe_0.33Te_0.67 Michioka et al. (2010) and FeSe_0.2Te_0.8 Hara et al. (2011). Similar situation was also observed in 122 iron selenide K_yFe_2-xSe₂ Torchetti et al. (2011); Yu et al. (2011) and some other iron-based superconductors Ma and Yu (2013). This indicates that magnetic fluctuations are not the leading, or at least the only, mechanism driving to superconductivity in iron layered systems.

The observed horizontal regions on the 1/($T_{1}T$) vs $T$ plot are separated by a jump in the vicinity of $\approx$ 30 K, close to the maximum resistivity derivative $dR/dT$ (see Fig. 1). When passing through a jump, the 1/($T_{1}T$) value increases by approximately 1.65 times with decreasing temperature. Such a change may be explained by two factors:

(i) an increase in the density of states of conducting electrons at the Fermi level $N(E_{F})$. The Korringa’s law for relaxation through conduction electrons is expressed as Slichter (2013):

where $\gamma_{e,n}$ are the electron and nuclear gyromagnetic ratios, $A_{hf}$ is the transferred hyperfine coupling. Therefore, the 1/($T_{1}T$) jump in Fig. 4 may be associated with the electron density decrease of $\sqrt{1.65}=1.28$ times above 30 K.

(ii) Switching on the weak antiferromagnetic electron-electron interaction, which causes an enhancement of the so-called Korringa ratio $K$. The latter can be introduced by the ratio Moriya (1963); Narath and Weaver (1968):

where $K_{S}$ is the spin shift of the NMR line, which does not change, since the observed total shift remains almost constant ($\approx 0.3\%$) in the vicinity of this relaxation jump. $K=1$ corresponds to a non-interacting Fermi gas, while its increase (and, accordingly, the 1/($T_{1}T$) increase in S-Te3 below 30 K) may be associated with a weak antiferromagnetic correlation. Indeed, some previous studies point to the presence of the latter factor in K_yFe_2-xSe₂ Torchetti et al. (2011); Yu et al. (2011) and FeSe_1-xS_x Rana et al. (2020, 2023) and even compare it with the Fermi liquid regime. Such a comparison looks reasonable also in our case, since the resistivity temperature dependence clearly demonstrates the prevailing $\sim{}T^{2}$ component up to 30 K (Fig. 1a), linear growth of derivative on Fig. 1b) ), which is typically considered as a Fermi liquid feature. Then the $dR/dT$ maximum can be considered as a transition to the non-Fermi liquid regime at higher temperatures, which is consistent with the NMR data.