Superconducting state properties of Cu-substituted Fe_0.99Te_0.66Se_0.34 exhibiting superconductivity recovered under hydrostatic pressure

Abstract

Magnetic measurements under hydrostatic pressure (P) up to 10 kbar and x-ray diffraction measurements up to 10.7 kbar were performed for a single crystal of Fe_0.975Cu_0.025Te_0.66Se_0.34, which is non-superconducting at ambient pressure. In this compound, we have found pressure-induced recovery of the superconducting state at T_c equal to 13.0 K for P = 10 kbar. We determined the parameters characterizing the superconducting state, including the lower and the upper critical fields, coherence length, and penetration depth, and compared them with those for Fe_0.99Te_0.66Se_0.34. We found that the lower critical field for Fe_0.975Cu_0.025Te_0.66Se_0.34 at 0 K and 10 kbar is comparable to the lower critical field for Fe_0.99Te_0.66Se_0.34 at ambient pressure, while the upper critical field is significantly higher than that for Fe_0.99Te_0.66Se_0.34 at ambient pressure. The estimated increase in superconducting carrier density and effective mass under pressure can be explained if one assumes that applied pressure leads to an increase in structural disorder in the studied material. At 10 kbar, the zero-field critical current density for Fe_0.975Cu_0.025Te_0.66Se_0.34 is four times larger than that for Fe_0.99Te_0.66Se_0.34 at ambient pressure. The x-ray diffraction results indicate that under pressure crystal quality apparently degrades. Comprehensive studies of the impact of pressure on the crystal structure indicate an increasing mosaicity evolution with pressure, suggesting that the pressure-induced superconductivity of Fe_0.975Cu_0.025Te_0.66Se_0.34 originates from inhomogeneities, associated also with the superconductivity in other sulpho-iron seleno-tellurides and anti-PbO-type structures. Obviously, the pressure effect on the crystallographic structure can lead to changes in the electronic structure, which is not excluded.

Introduction

The discovery of superconductivity in the layered iron chalcogenides of general formula Fe(Se, Ch), so-called ‘11-type’ compounds with Ch = S, Te, activated intense studies of metal chalcogenides, aimed at revealing the mechanism of superconductivity, means of increasing the critical temperature T_c and improved insight into the relationship between superconductivity and magnetism^{1,2,3,4,5,6,7,8,9,10,11,12}. In the FeTe_{1− x}Se_x system, at ambient pressure, the superconducting transition temperature is the highest (~ 14 K) for FeTe_0.5Se_0.5^2,5. The application of hydrostatic pressure to this compound leads to an increase of T_c as well as of the other parameters of the superconducting state, including the upper critical field (H_c2), the lower critical field (H_c1), and the critical current density (j_c)¹³. In the case of the Fe_0.99Te_0.66Se_0.34 compound, with lower selenium content and a slightly lower critical temperature (12.8 K) at ambient pressure than that of FeTe_0.5Se_0.5, we have also noticed an enhancement of all basic parameters of the superconducting state under hydrostatic pressure¹⁴.

It was shown^15,16,17,18 that a small disorder in the magnetic sublattice due to substituting Fe atoms by Co, Ni, or Cu decreases T_c in the FeTe_{1− x}Se_x system under ambient pressure. However, for Fe_0.994Ni_0.007Te_0.66Se_0.34 subjected to hydrostatic pressure, we observed an enhancement of all superconducting properties, similar to that in Fe_0.99Te_0.66Se_0.34¹⁴. Thus, one can see that despite the weakening of superconducting state properties by the substitution of Ni at the Fe site or changes in the Se/Te ratio, the hydrostatic pressure leads to an improvement of superconducting state properties.

It was reported that for the single crystals substituted with Cu content exceeding 1.5 atomic %, superconductivity is completely destroyed¹⁵ and the restoration of superconductivity under pressure in Cu-doped polycrystalline FeSe with small doping levels of 3% and 4% has been found¹⁹. It was noticed that this result differs from earlier experiments on pressure-induced superconductivity in SrFe₂As₂ and BaFe₂As₂ (‘122’-type compounds)²⁰, because superconductivity in Cu-doped FeSe was initially suppressed due to doping and next restored with pressure, whereas in the 122 compounds, initially nonsuperconducting samples become superconducting under pressure¹⁹. Reports indicated that the enlargement of the lattice parameter a accompanies the disappearance of superconductivity for small Cu doping in FeSe, while DFT calculations attribute the vanishing of the superconducting state to the enhancement of magnetism for larger lattice parameters¹⁹. It was shown that under pressure the lattice parameter a reaches its initial value, the static susceptibility is suppressed, and the superconducting state is stabilized, consistently with theoretical predictions¹⁹. It was suggested that it is worthwhile to investigate other iron arsenide superconductors within the range of stoichiometry where they are non-superconducting. Such studies allow us to determine if their superconducting properties are also sensitive to lattice parameters, as was shown for FeSe, or whether this compound is special among iron pnictide superconductors¹⁹. Zajicek et al.²¹ have discussed the significant impact of impurity scattering on the electronic and superconducting properties of Cu-doped FeSe. The authors concluded that the Cu impurities strongly suppress nematic and superconducting states and increase residual resistivity. The authors stated that the suppression of the superconducting state caused by Cu doping is consistent with changes of sign of s_± order parameter. Moreover, compressive strain enhances superconductivity, as observed in FeSe²¹. Later on, it was shown that substitution of Cu in the Fe plane can change the balance among electronic phases competing in FeSe at high pressure²². When an external magnetic field is absent at low hydrostatic pressures, the nematic and superconducting phases are suppressed similarly. However, at pressures above 10 kbar, the superconducting state remains unchanged, in spite of the absence of any signature in transport properties related to a magnetic phase in a zero-magnetic field²², while the effects of copper doping on the electronic structure of Fe_{1− x}Cu_xSe have been studied by Huh et al. by means of ARPES²³.

Due to the best authors’ knowledge, superconducting state parameters, such as H_c1, H_c2, coherence length λ, penetration depth ξ, superconducting carrier density n_s, superconducting carrier effective mass m_s*, and j_c, of the restored Cu-doped superconductors were not determined yet. Knowledge of superconducting state properties can shed new light on the mechanisms leading to the appearance of superconductivity.

Taking the above into account, we have decided to study the influence of hydrostatic pressure on superconducting state properties of the Fe_0.975Cu_0.025Te_0.65Se_0.35 single crystal and to perform a detailed examination of crystallographic structure changes when the recovery of superconductivity under hydrostatic pressure takes place. The evolution of crystallographic changes with pressure has been presented and structural transition under pressure in the investigated pressure range was excluded. The magnetic studies were concentrated on the research performed with a maximum pressure of 10 kbar available in our experimental setup, since the main aim of our work was to investigate the possibility of the reappearance of the pressure-induced superconducting state in the compound, which is non-superconducting at ambient pressure.

Details of synthesis and experiment

Details of the synthesis of Fe_0.975Cu_0.025Te_0.65Se_0.35 single crystal were described in Ref¹⁵. The studied crystal was grown using the vertical Bridgman method. Stoichiometric amounts of iron chips (3N5), tellurium powder (4 N), selenium powder (pure) and CuSe (pure) were used to grow the crystals. These materials were weighed, mixed, and kept inside a glove box in an argon atmosphere. Evacuated sealed quartz ampoules with starting materials were placed in a furnace with an axial temperature gradient of 1.6 °C mm⁻¹. The following thermal conditions of synthesis and crystal growth were applied: heating of the ingot for 3 h at 730 °C, then increasing the temperature to 920 °C. Afterward melting, a temperature of 730 °C was kept for 3 h; next, the samples were cooled down to 500 °C at a rate of 5 °C h⁻¹, next to 200 °C at a rate of 60 °C h⁻¹, and finally cooled down to room temperature¹⁵. The obtained single crystal exhibits well-developed (001) natural planes, equivalent to a cleavage plane of the PbO-type structure.

The quantitative point analysis on the cleavage planes of the studied single crystals was performed using a field emission scanning electron microscope (FESEM) JEOL JSM-7600 F (20 kV incident energy) coupled with the Oxford INCA energy dispersive x-ray spectroscope (EDX). SEM/EDX analysis indicated that the average chemical composition of the matrices of investigated crystals is Fe_0.975Cu_0.025Te_0.66Se_0.34, with an accuracy of ± 0.02, and for the reference sample, it is Fe_0.99Te_0.66Se_0.34, also with the same accuracy¹⁵. Later in the paper we will refer to the crystals’ average composition, not to the nominal one. Our earlier papers^15,24 published all the details concerning the structure of the studied samples. One of the figures in Ref¹⁵ shows a photo of the Cu-substituted crystal with a composition similar to that studied here.

The magnetic measurements were performed, with magnetic fields up to 50 kOe, using a Quantum Design SQUID magnetometer. The magnetic field was oriented parallel to the c-axis of the crystal. Hydrostatic pressure was applied using an Almax-easyLab MCell10 cell with Daphne 7373 oil²⁵, which is considered one of the best pressure media because of the minimal decrease in pressure (in a range above 7 kbar) with a lowering temperature²⁶. As an in situ manometer, a high-purity Sn thin wire was used. The background signal of the pressure cell was subtracted, taking into account values measured under ambient pressure for the empty pressure cell. It turns out that the background contribution has a negligible impact on obtained results. Corrections for demagnetization were applied according to Ref²⁷.

X-ray diffraction measurements of Fe_0.975Cu_0.025Te_0.66Se_0.34 and Fe_0.99Te_0.66Se_0.34 crystals were completed using Xcalibur S2 Agilent diffractometers with a monochromated Mo sealed tube source (K_α radiation, λ = 0.71073 Å) and a CCD detector. Crystals were placed in a modified Merrill–Bassett Diamond-Anvil Cell (DAC)²⁸ with diamond anvils directly mounted on Inconel supports. The Fe_0.975Cu_0.025Te_0.66Se_0.34 and Fe_0.99Te_0.66Se_0.34 crystals were glued separately to the culet surface together with a small ruby chip²⁹ and placed in a 0.3 mm diameter hole of a 0.1 mm thick tungsten gasket. The chamber was filled with silicon oil (hydrostatic pressure-transmitting medium) and closed. The semi-automatic gasket-shadowing method³⁰ was applied for DAC centering along the x-ray beam. The CrysAlisPro program suite³¹ was used for collecting the data, determining the matrix orientation, and initial data reduction. The lattice parameters of Fe_0.975Cu_0.025Te_0.66Se_0.34 and Fe_0.99Te_0.66Se_0.34 under hydrostatic pressure were determined from single-crystal X-ray diffraction data collected for both compounds at 3.5, 7.2, and 10.7 kbar (Cu-doped material), and at 2.9, 7.3, and 11.3 kbar (undoped), respectively (Table 1). The analysis confirmed that both compounds maintain their characteristic tetragonal symmetry upon compression. The lattice constants a(= b) and c, as well as the unit cell volume, show a steady decrease with increasing pressure, indicating normal compressibility without any structural phase transition.

Table 1 Pressure evolution of unit-cell parameters and volume in Fe_0.99Te_0.66Se_0.34 and Fe_0.975Cu_0.025Te_0.66Se_0.34.

A slight increase in the a = b parameter observed in the Cu-doped crystal at an intermediate pressure likely reflects measurement uncertainties related to sample mosaicity and limited data points rather than evidence of negative area compressibility. With data for only three pressure values available, caution is warranted when interpreting subtle anomalies; however, the general trend of lattice contraction and symmetry retention is clear. These results provide valuable structural insight into how pressure affects the crystal framework, supporting the correlation between lattice parameter changes and the pressure-induced restoration of superconductivity discussed further in this work.

Results and discussion

Figure 1 presents the temperature dependence of the magnetic susceptibility, measured in both zero-field cooling (ZFC) and field cooling warming (FCW) mode, under ambient pressure and applied hydrostatic pressure of 10 kbar, in a dc field of 10 Oe, oriented parallel to the c-axis for Fe_0.975Cu_0.025Te_0.66Se_0.34 and at ambient pressure for Fe_0.99Te_0.66Se_0.34. The single crystal of Fe_0.975Cu_0.025Te_0.66Se_0.34 at ambient pressure does not reveal any indications of superconducting phase transition down to 5 K. Under an applied hydrostatic pressure of 10 kbar, the investigated compound is clearly superconducting, with T_c equal to 13 K. The critical temperature was determined as the temperature at which the zero-field cooled magnetization M_ZFC(T) curve deviates from a temperature-independent, constant background value. This value of the critical temperature, obtained for Fe_0.975Cu_0.025Te_0.66Se_0.34, under hydrostatic pressure, is comparable to that for Fe_0.99Te_0.66Se_0.34 at ambient pressure (12.8 K, see, Table 2) and slightly higher than that for Fe_0.994Ni_0.007Te_0.66Se_0.34 at ambient pressure (12 K)^14,15. The superconducting transition widths for Fe_0.975Cu_0.025Te_0.66Se_0.34 (P = 10 kbar) and for Fe_0.99Te_0.66Se_0.34 (at ambient pressure) are practically the same, evidenced by the nearly linear dependence of the ZFC magnetic susceptibility. The Meissner effect is not complete down to 5 K for the pressurized Fe_0.975Cu_0.025Te_0.66Se_0.34, similarly to that of Fe_0.99Te_0.66Se_0.34 at ambient pressure. The pressure-induced appearance of the superconducting transition temperature, observed for Fe_0.975Cu_0.025Te_0.66Se_0.34, confirms our earlier reports^13,14 about the possibility of an enhancement of T_c under hydrostatic pressure, previously suppressed by substitution of transition metal atoms into the Fe-site and changes in the Se/Te ratio¹⁵. The zero field-cooled magnetic susceptibility, equal to −0.88 at 5 K for Fe_0.975Cu_0.025Te_0.66Se_0.34 under a pressure of 10 kbar, indicates recovery of bulk superconductivity; however, we should note that within our experimental accuracy of a few percent, a perfect Meissner effect was not observed at the temperature range far below T_c under applied pressure. Very pronounced differences between ZFC and FCW curves imply strong pinning properties of the studied crystals.

Fig. 1

Temperature dependence of the dc magnetic susceptibility for H || c-axis, measured in a dc field of 10 Oe, for Fe_0.975Cu_0.025Te_0.66Se_0.34 under ambient pressure and applied hydrostatic pressure of 10 kbar, and for Fe_0.99Te_0.66Se_0.34 under ambient pressure. The inset shows raw magnetic data.

The critical temperature

Table 2 The impact of pressure on the thermodynamic parameters that describe the superconducting state for Fe_0.99Te_0.66Se_0.34 and Fe_0.975Cu_0.025Te_0.66Se_0.34.

In our previous studies³², we noted an excellent correlation between transition temperature determined from magnetic and transport measurements conducted on the undoped compound that is the focus of the current research.

Basic parameters describing superconducting state—the upper and the lower critical fields

We have conducted magnetic studies under pressure on the lower and the upper critical fields, as well as the critical current density, to determine whether the trend of T_c recovery can be observed in other parameters of the superconducting state. To estimate the thermodynamic parameters of a single crystal of Fe_0.975Cu_0.025Te_0.66Se_0.34, subjected to hydrostatic pressure, and to compare them with those of Fe_0.99Te_0.66Se_0.34 (see, Ref¹⁴., we have estimated the temperature dependence of the upper and the lower critical fields in H || c-axis geometry (up to 50 kOe), for Fe_0.975Cu_0.025Te_0.66Se_0.34, subjected to hydrostatic pressure of 10 kbar, and for Fe_0.99Te_0.66Se_0.34 under ambient pressure and at applied hydrostatic pressure of 9.8 kbar. Figure 2(a) presents the temperature dependence of the magnetic moment, m, measured under 10 kbar hydrostatic pressure for several magnetic fields in the geometry H || c-axis for Fe_0.975Cu_0.025Te_0.66Se_0.34. From these data, characteristic temperature T_c2(H = const.) was determined as the temperature at which m(T) deviates from linear temperature dependence, describing well normal state magnetic susceptibility. Using the T_c2(H) data from several magnetic fields, we were able to plot H_c2(T) dependence for the studied sample along the c-axis at a hydrostatic pressure of 10 kbar. From measurements of H^||c_c2, one can derive the coherence length, ξ_ab, using the following formula:

$$H_{{c2}}^{{||c}}=\frac{{{\Phi _0}}}{{2\pi \xi _{{ab}}^{2}}},$$

(1)

Fig. 2

(a) Temperature dependence of magnetic moment, measured for Fe_0.975Cu_0.025Te_0.66Se_0.34 in the vicinity and above T_c, under an applied hydrostatic pressure of 10 kbar, shown for selected magnetic fields for H || c-axis. Inset shows raw magnetic data for all magnetic fields. (b) Temperature dependence of the upper critical field, for H || c-axis, for Fe_0.975Cu_0.025Te_0.66Se_0.34 under applied hydrostatic pressure of 10 kbar, and for Fe_0.99Te_0.66Se_0.34 under ambient pressure and hydrostatic pressure of 9.8 kbar.

where Φ₀ is flux quantum. The same procedure was applied for processing experimental data for Fe_0.99Te_0.66Se_0.34 under ambient pressure and at an applied hydrostatic pressure of 9.8 kbar.

Figure 2(b) shows the temperature dependence of the upper critical field for H || c-axis (H^||c_c2) for pressurized Fe_0.975Cu_0.025Te_0.66Se_0.34 and for Fe_0.99Te_0.66Se_0.34 at ambient pressure and under a hydrostatic pressure. The upper critical field values for Fe_0.975Cu_0.025Te_0.66Se_0.34, within the range of 0–50 kOe, are slightly lower than those for Fe_0.99Te_0.66Se_0.34 at ambient pressure (Fig. 2b). A more detailed description of the temperature dependence of the upper critical field for Fe_0.99Te_0.66Se_0.34 at ambient pressure and at an applied pressure of 9.8 kOe is given in Ref¹⁴. For Fe_0.975Cu_0.025Te_0.66Se_0.34, higher magnetic fields show an apparent increase in the −dH^||c_c2/dT slope in the linear part of the H^||c_c2(0) dependence. At lower fields, in the vicinity of T_c, strong curvature is observed. In the field range 10–50 kOe, we have −dH^||c_c2/dT = 32(3) kOe/K. This value is much larger than that for Fe_0.99Te_0.66Se_0.34 determined at ambient and under hydrostatic pressure of 9.8 kbar (see, Table 2). Moreover, it is also larger than that noticed for a single crystal of FeTe_0.5Se_0.5 with a wider transition to a superconducting state¹³. Furthermore, for FeTe_0.5Se_0.5, the same shape of H^||c_c2(0) dependence was observed, which features a strong curvature near T_c and a significant increase in slope −dH^||c_c2/dT in the linear part of H^||c_c2(T); this behavior was attributed to the multiband nature of superconductivity¹³. We have determined the value of H^||c_c2 at 0 K for Fe_0.975Cu_0.025Te_0.66Se_0.34, using the Werthamer-Hefland-Hohenberg approach³³. According to that method, one can assume that the H_c2(0) value is proportional to T_c and to –dH_c2/dT, determined in a relatively wide magnetic field range above the strong curvature of H_c2(T) near T_c. This equation indicates the value of H^||c_c2(0) equal to 300(40) kOe for Fe_0.975Cu_0.025Te_0.66Se_0.34 under hydrostatic pressure of 10 kbar. Using the value of H^||c_c2(0), we estimate the zero-temperature coherence length ξ_ab(0) of 3.3 nm. The upper critical field at 0 K for Fe_0.975Cu_0.025Te_0.66Se_0.34, under P = 10 kbar, equal to 300(40) kOe (Table 2), is slightly larger than that for Fe_0.99Te_0.66Se_0.34 under P = 9.8 kbar, equal to 240(10) kOe (Table 2). Since the coherence length is proportional to the velocity at the Fermi level and inversely proportional to T_c, according to the relation³⁴:

$${\xi _{ab}}\left( 0 \right){\text{ }} = {\text{a}}\left( {\hbar {\nu _{\text{F}}}/{k_{\text{B}}}{T_{\text{c}}}} \right).$$

(2)

In this context, ν_F represents the velocity at the Fermi level, and the parameter a, which was empirically determined by Pippard, is equal to 0.15³⁴. The smaller value of ξ_ab(0) for Fe_0.975Cu_0.025Te_0.66Se_0.34, measured under P = 10 kbar and with a lower T_c (= 13.0 K) compared to Fe_0.99Te_0.66Se_0.34 under P = 9.8 kbar (T_c = 17.6 K), suggests that ν_F is smaller for Cu-substituted Fe-Se-Te than for the parent compound. Thus, one can speculate that the decrease of velocity at the Fermi level caused by Cu substitution for Fe-Se-Te is correlated with the disappearance of superconductivity.

The lower critical field H_c1 was estimated according to the procedure basing on the Bean’s critical-state model to catch the beginning of flux penetration in broad range of ZFC magnetization curves, as it was introduced in³⁵ and discussed in^13,36. The dependence of magnetic moment on field was measured at various temperatures for magnetic field parallel to the c-axis of the crystal. For a given shape of the investigated crystal, the demagnetizing factor D was calculated to correct the values of the external magnetic field. The variation of the square root of a product of magnetic induction B = 4πm/V + H_int and volume of the studied crystal V as a function of the internal magnetic field H_int = H_ext − DM (H_ext denotes the external magnetic field) for Fe_0.975Cu_0.025Te_0.66Se_0.34 under 10 kbar at 2 K is presented in the inset of Fig. 3(a). The lower critical field values H_c1(T) for H parallel to the c-axis for Fe_0.975Cu_0.025Te_0.66Se_0.34 under 10 kbar and for Fe_0.99Te_0.66Se_0.34 at ambient pressure and at an applied pressure of 9.8 kOe, presented in Fig. 3(a), were determined as the magnetic field H for which the value of (BV)^1/2 plotted as a function of H at fixed T deviates from zero.

Fig. 3

(a) Temperature dependence of H_c1 for Fe_0.975Cu_0.025Te_0.66Se_0.34 under applied hydrostatic pressure of 10 kbar and for Fe_0.99Te_0.66Se_0.34 under ambient pressure and hydrostatic pressure of 9.8 kbar for H || c-axis. Raw magnetic data for Cu-substituted sample was shown in the left inset. The right inset presents (BV)^1/2 versus the internal magnetic field, H_int, determined at 2 K at hydrostatic pressure of 10 kbar, for Fe_0.975Cu_0.025Te_0.66Se_0.34 for H || c-axis. (b) The lower critical field reduced temperature dependence at ambient pressure and under an applied hydrostatic pressure for H || c-axis.

One can noticed that a simpler and more common method to determine H_c1 is to track the point where the magnetic moment deviates from the linear diamagnetic response during the initial application of the external magnetic field in hysteresis loop measurements (see, for example³⁷). The left inset of Fig. 3(a) presents the data obtained with this method. These two methods yield essentially the same H_c1 value; however, an error bar in the method using the Bean’s critical-state model appears to be smaller.

The reduced temperature dependence of H_c1 for H || c-axis (H^||c_c1), determined for Fe_0.975Cu_0.025Te_0.66Se_0.34 under hydrostatic pressure of 10 kbar and for Fe_0.99Te_0.66Se_0.34 at ambient pressure and under hydrostatic pressure of 9.8 kbar, is presented in Fig. 3(b). The Fe_0.975Cu_0.025Te_0.66Se_0.34 sample, at non-zero temperatures, under hydrostatic pressure of 10 kbar, is characterized by smaller values of H_c1 when compared to Fe_0.99Te_0.66Se_0.34 at ambient pressure. The result indicates a smaller value of the penetration depth for Fe_0.99Te_0.66Se_0.34 and the superconducting carrier density for Fe_0.99Te_0.66Se_0.34 at ambient pressure is larger than that of Fe_0.975Cu_0.025Te_0.66Se_0.34 under hydrostatic pressure of 10 kbar. For Fe_0.975Cu_0.025Te_0.66Se_0.34, from the data presented in Fig. 3, the extrapolated zero-temperature value of H_c1 was found to be H^||c_c1(0) = 37(5) Oe for P = 10 kbar (Table 2). This value is practically the same, within the experimental error, as the value of H^||c_c1(0) for Fe_0.99Te_0.66Se_0.34, determined at ambient pressure. However, it should be noted that the zero-temperature value H^||c_c1(0) = 40(5) Oe for Fe_0.99Te_0.66Se_0.34 at ambient pressure increases to H^||c_c1(0) = 190(10) Oe under a pressure of 9.8 kbar, which is five times bigger than that of Fe_0.975Cu_0.025Te_0.66Se_0.34 for P = 10 kbar. The values of the magnetic penetration depth were extracted from the measured values of H_c1, following the basic relation³⁸:

$$H_{{c1}}^{{||c}}=\frac{{{\Phi _0}}}{{4\pi \lambda _{{ab}}^{2}}}\left[ {\ln \left( {{\kappa ^{||c}}} \right)+0.5} \right].$$

(3)

Here, λ_ab denotes the magnetic penetration depth associated with the superconducting current flowing in the ab-plane, and κ^||c = λ_ab/ξ_ab is the Ginzburg–Landau parameter. The coherence length at 0 K, ξ_ab(0), was determined by extrapolating the value of H^||c_c2 to zero temperature for H || c-axis. The zero-temperature value of the magnetic penetration depths, λ_ab(0), for pressurized Fe_0.975Cu_0.025Te_0.66Se_0.34, equals to 495(55) nm, which is comparable to that for Fe_0.99Te_0.66Se_0.34 at ambient pressure (455(15) nm, see, Table 2). Here again, λ_ab(0) for Fe_0.99Te_0.66Se_0.34 at ambient pressure decreases to 200(20) nm under a pressure of 9.8 kbar, which is about 44% of the ambient pressure value. The above similarity together with almost identical transition temperature to superconducting state indicate that an empirical relation between the zero-temperature superfluid density ρ_s(0) ∝ λ_ab⁻²(0) and T_c is here the same as the generic for various cuprate high-T_c superconductors (Uemura plot)³⁹. The obtained results, when correlated with those reported previously for FeTe_0.5Se_0.5¹³ and Fe_0.994Ni_0.007Te_0.66Se_0.34¹⁴, show that the application of hydrostatic pressure improves the superconducting properties in the Fe-Te-Se family, originally weakened due to substitutions at the Fe-site and changes in the Se/Te ratio¹⁵.

Penetration depth within the clean-limit approximation is described by the following formula⁴⁰:

$${\lambda ^{ - 2}} = 4\pi {n_{\text{s}}}{{\text{e}}^2}/\left( {{m_{\text{s}}^*}}{{\text{c}}^2} \right).$$

(4)

In this context, n_s represents the superconducting carrier density, while m_s* denotes the effective mass of the superconducting carriers. The Fermi temperature for a free electron gas in three dimensions is given by the following formula^41,42,43:

$$ {k_{\text{B}}}{T_{\text{F}}} = {E_{\text{F}}} = {\text{ }}{{{\rm{\hbar}}} ^2}/2m^*{\left( {3{\pi ^2}n} \right)^{2/3}}.$$

(5)

In this context, m* represents the carrier effective mass, while n denotes the carrier density. For the Fe-Te-Se system, which is a superconductor with low anisotropy, we can reasonably assume that the superconducting effective mass, m_s*, is well approximated by the ab-plane value of effective mass, m_ab*. Thus, the velocity at the Fermi surface ν_F in the ab-plane can be approximated by:

$${\nu _{{\text{F}}ab}} = {\text{ }}{{\rm {\hbar}}}{k_{{\text{F}}ab}}/{m_{ab}^*}{\text{ }} = {\text{ }}{{\rm{\hbar}}} /{m_{ab}^*}{\left( {3{\pi ^2}n} \right)^{1/3}}.$$

(6)

Combining Eqs. (2), (4), and (6) one can independently estimate superconducting carrier density and effective mass concentration. Obtained results are presented in Table 2. The most important achievements can be summarized as follows:

• superconducting carrier density for Fe_0.99Te_0.66Se_0.34 significantly increases with increasing T_c under applied hydrostatic pressure,

• the superconducting carrier density value for Fe_0.99Te_0.66Se_0.34 under ambient pressure with T_c = 12.8 K is very similar to that of pressurized Fe_0.975Cu_0.025Te_0.66Se_0.34, which has an almost identical T_c = 13 K,

• superconducting effective mass under applied pressure for both studied samples is higher than that at ambient pressure.

The obtained results can be explained consistently if one assumes that applied pressure leads to an increase in structural disorder in the studied material, which in turn causes an increase in the superconducting carrier’s effective mass and density. The increase of the effective mass depends on the disorder introduced to the studied material only and therefore is similar for various materials subjected to very similar applied pressure. Assuming a similar increase of superconducting carrier density under pressure, one can expect that both the actual n_s value under pressure as well as T_c depend on the starting value of superconducting carrier density at ambient pressure.

The critical current density

The critical current density was determined from M(H) dependences (see, Fig. 4), carried out for selected reduced temperatures, at ambient pressure and at non-zero hydrostatic pressure, using Bean’s model^44,45. The following formula:

$$j_{c}^{{ab}}=\frac{{20\Delta M}}{{a(1 - \frac{a}{{3b}})}}$$

(7)

was applied to determine the critical current density, j_c^ab46,47. Here, ΔM (in oersteds) is the width of the hysteresis loop for decreasing and increasing H_int (see, Fig. 4), a = 0.15 cm and b = 0.115 cm are the sample dimensions for Fe_0.975Cu_0.025Te_0.66Se_0.34 in the plane perpendicular to the external magnetic field, and the units of current densities are A/cm². Figure 5 shows how the critical current density depends on the magnetic field for both pressurized Fe_0.975Cu_0.025Te_0.66Se_0.34 and Fe_0.99Te_0.66Se_0.34 at ambient pressure, as well as under hydrostatic pressure; these values were calculated according to Eq. (7) at a reduced temperature of about 0.39T_c with the magnetic field orientated parallel to the c-axis. We note a larger value of the estimated critical current density for Fe_0.975Cu_0.025Te_0.66Se_0.34 for P = 10 kbar, as compared to that for Fe_0.99Te_0.66Se_0.34 at ambient pressure. The observed increase in j_c one can attribute to the increasing effectiveness of small-sized defects present in the crystal. This improvement occurs because of two reasons: (1) a diminution of the coherence length under pressure and (2) an enhancement of the thermodynamic critical field under pressure due to the upper critical field increase.

Fig. 4

Dependence of magnetization on magnetic field recorded at 5 K and at 6.8 K (comparable reduced temperature) for Fe_0.99Te_0.66Se_0.34 at ambient pressure and under a hydrostatic pressure of 9.8 kbar, respectively, and for Fe_0.975Cu_0.025Te_0.66Se_0.34 at ambient pressure and under a hydrostatic pressure of 10 kbar, recorded at 5 K. The inset shows raw magnetic data recorded as a function of an applied magnetic field.

Fig. 5

Magnetic field dependence of the critical current density for Fe_0.975Cu_0.025Te_0.66Se_0.34 under hydrostatic pressure of 10 kbar (T_c = 13.0 K) and for Fe_0.99Te_0.66Se_0.34 at ambient pressure (T_c = 12.8 K) and under a hydrostatic pressure of 9.8 kbar (T_c = 17.6 K), for H || c-axis, at a reduced temperature of about 0.39T_c.

X-ray diffraction studies

The crystal structure of Fe_0.975Cu_0.025Te_0.66Se_0.34 was identified as that of the anti-PbO type, but the quality of data was insufficient for determination of structure details. Confirmation of this phase is in agreement with the results of studies using x-ray diffraction^48,49 and other methods^50,51,52, confirming that the anti-PbO phase is stable in compounds with close chemical composition at pressures up to at least ∼11 kbar.

A series of high-pressure single crystal diffraction experiments were performed within DAC using an Oxford Diffraction Xcalibur Eos diffractometer on single crystals of Fe_0.975Cu_0.025Te_0.66Se_0.34 and Fe_0.99Te_0.66Se_0.34, glued to the culet surface of a diamond anvil and filled with silicon oil used as a hydrostatic medium. The DAC was centered on the diffractometer using the semiautomatic gasket-shadow procedure.

With increasing pressure one can noticed that the shape of diffraction spots changed: roughly circular spots turn into extended one along a direction agreeing with the run of Debye–Scherrer rings (see, Figs. 6 and 7), which is in our opinion caused by pressure-induced increase of mosaicity of the sample. For the diffraction-spot shape analysis the strong $\:\stackrel{-}{1}$01 (Figs. 8 and 9) and 10$\:\stackrel{-}{1}$ reflections (Figs. 10 and 11) were selected. The quoted reflection extension was calculated, with use of contour maps (see, Figs. 8, 10, 9 and 11) at the level of 50% of peak height. We found that in the radial direction there is no peak broadening, showing that pressure has insignificant impact on chemical homogeneity of the studied crystal.

Fig. 6

h0l layer of Fe_0.975Cu_0.025Te_0.66Se_0.34 crystal measured at room temperature at 3.5, 7.2, and 10.7 kbar, respectively (from left to right). All data presented in the above panel were obtained for the same crystal (with the same orientation matrix) enclosed in a modified Merrill–Bassett Diamond-Anvil Cell.

Fig. 7

h0l layer of Fe_0.99Te_0.66Se_0.34 crystal measured at room temperature at 2.9, 7.3, and 11.3 kbar, respectively (from left to right) within modified Merrill–Bassett Diamond-Anvil Cell.

Fig. 8

$\:\stackrel{-}{1}$01 reflection profile and contour maps measured at room temperature at 3.5, 7.2, and 10.7 kbar for Fe_0.975Cu_0.025Te_0.66Se_0.34 crystal. The intensity scales are shown separately for each pressure points as insets.

Fig. 9

$\:\stackrel{-}{1}$01 reflection profile and contour maps measured at room temperature at 2.9, 7.3, and 11.3 kbar for Fe_0.99Te_0.66Se_0.34 crystal. The intensity scales are shown separately for each pressure points as insets.

Fig. 10

10$\:\stackrel{-}{1}$ reflection profile and contour maps of Fe_0.975Cu_0.025Te_0.66Se_0.34 crystal measured at room temperature at 3.5, 7.2, and 10.7 kbar. The intensity scales are shown separately for each pressure points as insets. The subsequent degradation of the reflection profile and its elongation observed upon compression indicates that the crystal quality becomes inferior.

Fig. 11

10$\:\stackrel{-}{1}$ reflection profile and contour maps measured at room temperature at 2.9, 7.3, and 11.3 kbar for Fe_0.99Te_0.66Se_0.34 crystal. As previously, the intensity scales are shown separately for each pressure points as insets. The crystal quality becomes inferior between 7.3 and 11.3 kbar.

We have observed that the diffraction quality of the Fe_0.975Cu_0.025Te_0.66Se_0.34 crystal deteriorates with increasing pressure, which made reliable structure refinement and precise determination of atomic positions impossible, despite the absence of a structural phase transition under the applied pressure.

In the studied Fe_0.975Cu_0.025Te_0.66Se_0.34 crystal, structural disorder is manifested primarily as increased mosaicity and broadening of diffraction spots under pressure. These features signal enhanced local distortions, atomic site disorder, and microstrain, which have been shown in iron chalcogenide superconductors to substantially modify their electronic and magnetic ground states. Specifically, our data indicate that the application of pressure does not induce a symmetry change but increases the degree of disorder at the microscopic level. This effect is revealed by the pressure-dependent broadening and stretching of diffraction spots and the inability to achieve precise atomic coordinates at high pressure, even though the powder patterns remain characteristic of the tetragonal symmetry. Such enhanced disorder is widely considered to affect local charge carrier density, suppress long-range magnetic order, and facilitate superconducting pairing in these materials. The positive impact of these nano- and microscale inhomogeneities in restoring superconductivity is supported not only by our results where critical current density and upper critical field are also enhanced under pressure but also by literature on iron chalcogenides and analogous systems. These studies emphasize that the interplay of disorder and electronic structure can stabilize superconductivity through the enhancement of local electronic correlations and modifications to Fermi surface topology. In summary, structural disorder under pressure promotes the emergence of superconductivity in Fe_0.975Cu_0.025Te_0.66Se_0.34 by increasing electronic inhomogeneity and disrupting magnetic order, providing favorable conditions for Cooper pairing as reflected in enhanced superconducting and transport parameters.

Conclusions

In this work, we have studied single crystals of Fe_0.975Cu_0.025Te_0.66Se_0.34 and Fe_0.99Te_0.66Se_0.34 at ambient pressure and under hydrostatic pressure up to 11.3 kbar to reveal the impact of pressure on the basic parameters describing the superconducting state and on crystallographic structure changes. We have found that under hydrostatic pressure of 10 kbar, the Fe_0.975Cu_0.025Te_0.66Se_0.34 compound, which is non-superconducting at ambient pressure down to 5 K, becomes superconducting with T_c = 13 K. This value is practically the same as for Fe_0.99Te_0.66Se_0.34 at ambient pressure (12.8 K). The other investigated parameters of the superconducting state for Fe_0.975Cu_0.025Te_0.66Se_0.34, including the lower and the upper critical fields, coherence length, and penetration depth, were found to be comparable to those reported for Fe_0.99Te_0.66Se_0.34 at ambient pressure, particularly at temperatures close to 0 K. The observed increase in superconducting carrier density and effective mass under pressure can be explained if one assumes that applied pressure leads to an increase in structural disorder in the studied material. The critical current density was found to be larger for Fe_0.975Cu_0.025Te_0.66Se_0.34, for P = 10 kbar than that for Fe_0.99Te_0.66Se_0.34 at ambient pressure. Comparing the pressure effect on the superconducting properties of different substitutions at the Fe-site and changes in the Se/Te ratio, one can see the same trend based on an improvement of the superconducting state parameters under hydrostatic pressure, which were suppressed at ambient pressure by chemical modifications.

In conclusion, we observed that under pressure, the Fe_0.975Cu_0.025Te_0.66Se_0.34 crystal shows increased structural disorder, visible as broader and more diffuse diffraction spots. Despite this, the overall tetragonal symmetry remains unchanged. This increased disorder appears to weaken magnetic ordering and cause local fluctuations in the electronic environment. Similar effects in related iron-based superconductors likely contribute to the restoration of superconductivity by enabling electron pairing. Supporting this idea, we observe that superconducting properties like the critical current density improve alongside these structural changes. Therefore, rather than harming superconductivity, the disorder created under pressure seems to play an important role in reactivating it in this Cu-doped material.

Hence, the observed variation in diffraction-spot shape for Fe_0.975Cu_0.025Te_0.66Se_0.34 under pressure may correlate with the enhancement of superconductive properties, which we attribute to an increase in mosaicity. The increase in mosaicity may explain the earlier observation of superconducting property variation in related sulpho-iron seleno-tellurides of anti-PbO-type structure. After lowering pressure to ambient one, all of the crystals’ parameters recovered ambient pressure values.