On the Nanoscale Structure of K_xFe_2−yCh₂ (Ch = S, Se): A Neutron Pair Distribution Function View

Abstract

Comparative exploration of the nanometer-scale atomic structure of K_xFe_2−yCh₂ (Ch = S, Se) was performed using neutron total scattering-based atomic pair distribution function (PDF) analysis of 5 K powder diffraction data in relation to physical properties. Whereas K_xFe_2−ySe₂ is a superconductor with a transition temperature of about 32 K, the isostructural sulphide analogue is not, which instead displays a spin glass semiconducting behavior at low temperatures. The PDF analysis explores phase separated and disordered structural models as candidate descriptors of the low temperature data. For both materials, the nanoscale structure is well described by the iron (Fe)-vacancy-disordered K₂Fe_5−yCh₅ (I4/m) model containing excess Fe. An equally good description of the data is achieved by using a phase separated model comprised of I4/m vacancy-ordered and I4/mmm components. The I4/mmm component appears as a minority phase in the structure of both K_xFe_2−ySe₂ and K_xFe_2−yS₂, and with similar contribution, implying that the phase ratio is not a decisive factor influencing the lack of superconductivity in the latter. Comparison of structural parameters of the Fe-vacancy-disordered model indicates that the replacement of selenium (Se) by sulphur (S) results in an appreciable reduction in the Fe-Ch interatomic distances and anion heights, while simultaneously increasing the irregularity of FeCh₄ tetrahedra, suggesting the more significant influence of these factors. Structural features are also compared to the non-intercalated FeSe and FeS parent phases, providing further information for the discussion about the influence of the lattice degrees of freedom on the observed properties in layered iron chalcogenides.

1. Introduction

Binary Fe_1+ySe [1] has attracted considerable attention [2] due to its critical superconducting transition temperature (T_C) up to 8 K at ambient pressure. Externally applied pressure increases the T_C to 37 K at 7 GPa [3], whereas single FeSe layers grown on SrTiO₃ show strikingly higher T_Cs (ca. 65 K) [4,5] than many other Fe-based superconductors. Their crystal structure is quasi two-dimensional, composed of slabs of FeSe edge-sharing tetrahedra held together by van der Waals interactions. Isovalent substitution with sulphur or tellurium for selenium does not change the carrier density [6,7], but substantially modifies the regularity of FeCh₄ (Ch = Se, S, Te) tetrahedron, which is believed to be a crucial factor governing electronic and magnetic properties, having particular influence on electron pairing. Consequently, the chemistry and physics of these simple iron-based materials result in complex relationships in the Fe_1+yTe_1−xSe_x series [8].

The transition between itinerant antiferromagnetism and superconductivity can be controlled at elevated pressures, but also at ambient pressure by altering the crystal structure through substitutions either at the iron or chalcogen sites and favorably by intercalation of electron donating spacers (guests) between the FeCh sheets. Typically, alkali metals inserted between the FeCh layers expand the lattice and result in a ternary family of iron-superconductors, A_xFe₂Se₂ (A = K, Rb, Cs or Tl), with T_Cs in the order of 30 K. The first example of these so-called 122-iron selenides, the K_0.8Fe₂Se₂ derivative [9,10], was shown to adopt a ThCr₂Si₂-type structure. Much like the iron-pnicitides, with Fe tetrahedrally coordinated, superconductivity remains at the boundary with antiferromagnetism. However, whereas pnictides display superconductivity in the fully occupied, stoichiometric ThCr₂Si₂ structure, divalent Se²⁻ (instead of trivalent As³⁻) leads to the presence of alkali and iron vacancies and issues of chemical instability that is promoted by the high-temperature solid-state synthesis of the K_xFe_2−ySe₂ compounds. This entails microscale phase separation, with heterogeneity creating less than 20% of I4/mmm K_xFe₂Se₂, adopting the iron-pnictide ThCr₂Si₂ superconducting crystal structure, and the majority I4/m K₂Fe₄Se₅ Fe-vacancy-ordered antiferromagnetic phase (Figure 1) [11,12]. The origin of superconductivity and the precise stoichiometry of iron and potassium content that leads to the superconducting phase are still under debate owing to the intrinsic phase separation and inhomogeneity in this type of chemically complex material [13,14]. Interestingly, the substitution of Se for S in the K_xFe_2−ySe_2−zS_z (0 ≤ z ≤ 2) series suppresses the superconducting state (z = 1.6) [15] and eventually the sulphide end-member, K_xFe_2−yS₂, is not superconducting but exhibits a spin-glass behavior at temperatures below 32 K [16]. Therefore, experimental efforts to understand superconductivity in iron chalcogenides must not only tackle the sensitive stoichiometry required to observe a sizeable T_C, but also decipher the subtle differences in the atomic structure configuration that lead to such markedly different physical properties.

Here, we use neutron total scattering experiments combined with atomic pair distribution function (PDF) analysis [17] to explore the nanometer scale aspects of phase separation and inhomogeneity for the two end-members of K_xFe_2−ySe_2−zS_z (z = 0, 2) series, at 5 K. We probe their structural evolution based on the regularity of the FeCh₄ tetrahedra and the Fe anion heights, which have been considered before as empirical parameters that mediate the evolution of the superconducting critical temperature in these intercalated materials.

2. Materials and Methods

Powdered samples of K_xFe_2−ySe₂ and K_xFe_2−yS₂ were prepared by pulverizing single crystals stored in a glove box. The single crystals were synthesized and characterized according to protocols reported previously [15]. Laboratory X-ray powder diffraction data of the samples used in this study were fully indexed within the I4/m space group. Low temperature magnetic susceptibility and electrical resistivity measurements of the materials were previously reported [15]. The selenium-containing sample exhibited bulk superconductivity at T_C = 33 K, whereas the sulfur-containing sample was not superconducting and displayed magnetic semi-metallic behavior at low temperatures.

The neutron total scattering experiment was performed at the Nanoscale-Ordered Materials Diffractometer (NOMAD) [18] at the Spallation Neutron Source at Oak Ridge National Laboratory. Two pulverized samples, approximately 0.5 g each, were sealed in helium atmosphere in extruded cylindrical vanadium containers 6 mm in diameter, and mounted in the diffractometer equipped with an Orange cryostat sample environment. The instrument was calibrated using diamond standard. The data for each sample were collected at 5 K for 60 min. Data reduction was performed following standard protocols described elsewhere [19]. Pair distribution functions were obtained by Sine Fourier transformation of the measured reduced total scattering functions F(Q), where Q is the momentum transfer, over a range of Q_MIN = 0.5 Å⁻¹ to Q_MAX = 26 Å⁻¹. Refinement of the structural models against the experimental PDF data was completed over a 1 nm range via the small-box approach used within the PDFGUI software suite [20].

3. Results and Discussion

3.1. Qualitative Data Comparison

We began by observing and comparing the raw neutron powder diffraction data for K_xFe_2−ySe₂ and K_xFe_2−yS₂ shown in Figure 2, as seen by the forward scattering bank of detectors. Notably, this detector bank provided rather poor resolution featuring broad peaks and inherently lower statistics. This is a consequence of the construction of the NOMAD instrument, which is optimized for total scattering type measurements, where broad Q-range coverage and high resolution in backscattering banks are far more important, at the expense of the low angle banks. Nevertheless, for both, the data clearly identify the presence of, albeit weak, intensity associated with the (110) reflection that is a hallmark of I4/m and is absent in I4/mmm symmetry. This qualitative assessment revealed that both systems have, at least in part, I4/m structural character, in agreement with previous reports [15,16].

The neutron total scattering data collected over a broad range of momentum transfer Q were examined next. The findings are illustrated in Figure 3a,b for K_xFe_2−ySe₂ and K_xFe_2−yS₂, respectively, in the form of reduced total scattering structure function F(Q). Observable are appreciable arcs of diffuse scattering, particularly at higher momentum transfer values, indicative of appreciable disorder, as expected for this class of materials. Fourier transforms of F(Q) resulted in PDF and G(r), that are presented in Figure 3c,d, respectively. Some of the PDF peaks appear more intense in the K_xFe_2−ySe₂ pattern than in that of K_xFe_2−yS₂. In principle, this could be due to the differences in the underlying bond-length distributions reflecting the nanoscale structures and the complexity of underlying disorder. However, the coherent neutron scattering lengths, which scale the individual PDF contributions of any given pair of atoms in the structure, are nearly three times larger for selenium than for sulfur (b(Se) = 7.970 fm, whereas b(S) = 2.847 fm). The change in relative PDF peak intensity also originates from the renormalization due to different scattering properties of the samples.

The PDFs of the two samples studied were directly compared in Figure 3. Notably, the PDF pattern of the sulphide is shifted considerably to lower interatomic distances, r, as indicated by the block arrow in Figure 3d, compared to the selenide counterpart. This lattice contraction occurred due to the lower ionic size of S²⁻ (ionic radius 1.84 Å) compared to that of Se²⁻ (ionic radius 1.98 Å) [21]. By examining the very local scale (Figure 3c), observably narrower bond-length distribution pertaining to the FeCh₄ tetrahedral environments was apparent in K_xFe_2−ySe₂ compared with K_xFe_2−yS₂. However, considering the appreciably different neutron scattering lengths of Se and S, as well as their effective radii, this assertion requires further confirmation from explicit structural modeling.

3.2. Model Dependent PDF Analysis

Structural information from experimental PDFs was extracted using the small box modeling approach based on small symmetrized unit cells (typically ~10 to 100 atoms) [17]. This approach provides a simple framework for crystal structure interpretation at the expense of being limited to the symmetry constraints of the underlying models. The approach was previously reported in detail [20].

Our modeling was based on the two phases described above and shown in Figure 1. The phase with I4/mmm symmetry, often referred to as metallic, features stoichiometric slabs of FeCh₄ tetrahedra, whereas the phase with I4/m symmetry, considered insulating and antiferromagnetic, contains two symmetrically distinct Fe crystallographic sites, one of which features Fe-vacancies and is completely empty in the case of full vacancy order. Here, we refer to the latter as the Fe-vacancy-ordered (I4/m VO) model. In this model, the Fe and K occupancies at the low multiplicity sites, 4d and 2b, respectively, were fixed to zero under the assumption that all Fe and K atoms reside only at the high multiplicity sites, 16i and 8h, consistent with our room temperature Rietveld refinements reported earlier [22]. In addition, we considered a modified version of the I4/m model, in which the occupancy of the Fe-vacancy 4d site was allowed to vary, which we refer to as the vacancy-disordered (I4/m VD) model. We used these atomic configurations both individually in single phase fits and combined in two-phase refinements. All fractional coordinates were constrained to follow the respective space group symmetries. All atomic displacement parameters (ADP) were set to be isotropic and identical for all atomic species of the same type. The occupancy and ADP constraints were introduced to minimize the number of refined parameters and to enable the sensible semiquantitative comparison of the two systems within the realm of these models. Although various model combinations were tested, the two that showed the best agreement with the data were (1) two-phase mixture composed of I4/mmm and I4/m VO ingredient phases, and (2) single phase I4/m VD model. We limit our discussion to these two models.

The refinements of the structural models that result in the lowest fit residual (R_W) are summarized in Figure 4a for selenide and in Figure 4b for sulphide. The refinements results revealed that, at the nanoscale, the structures of both K_xFe_2−ySe₂ and K_xFe_2−yS₂ within the sensitivity of our data, were consistent with both a two-phase mixture (metallic I4/mmm and insulating I4/m VO) model and with a single phase disordered I4/m VD model. Previous reports confirmed the coexistence of these two phases in both selenide [22] and sulphide analogue [23]. Notably, the single phase I4/m VD model is effectively equivalent to a phase separated model, albeit disordered. In the explicit two-phase modeling of the selenide and sulphide data, the majority phase was found to be the I4/m VO with a percentage of ca. 89(5)% and ca. 73(8)% by weight, respectively. Notably, according to this step in the analysis, the minority phase I4/mmm in the case of K_xFe_2−yS₂ exhibited a higher percentage in the phase mixture than K_xFe_2−ySe₂. Whereas the exact phase percentage is difficult to acquire and is somewhat dependent on the type of constraints used, the robust observation in all our fits was that the metallic I4/mmm phase contribution in non-superconducting K_xFe_2−yS₂ was at least comparable or probably slightly larger than in superconducting K_xFe_2−ySe_2. This implied that the presence or absence of this phase is not the sole factor that affects the superconductivity in this system. Furthermore, the phase separation picture on the nanoscale likely represents an oversimplification of the true complexity of the system, and may possibly reflect the limitations of the small box modeling approach.

We next explored the vacancy-disordered K₂Fe_5−yCh₅ I4/m VD model in which the Fe occupancy at the 4d site is explicitly refined. The Fe occupancy at the high multiplicity 16i site was fixed to one, as an attempt to explicitly refine the marginal deviations, in agreement with previous reports. Although attempts to refine K-content were made, this resulted in fit instabilities and the correlation of refined parameters was indicative of over-parametrization, so the occupancy of K sites was adjusted in all models to reflect nominal 0.8 potassium content. In I4/m-type models, the K 2b site was empty. Importantly, the refinement results were qualitatively very similar to those of the phase mixture model. The refinements show finite and similar Fe occupancies at the 4d site in both materials, which increase the overall Fe content to K₂Fe_4.27(1)Se₅ and K₂Fe_4.27(1)S_5, indicating no substantial difference across the two systems in terms of phase-separation sensitive parameters and within the limitations of the applied modeling methodology. We interpreted this as an indicator that the phase separation aspect is not of fundamental importance for the lack of superconductivity in the sulphide, in the sense that non-superconducting sulphide does not account for a smaller fraction of the metallic phase.

In

Table 1. Lattice parameters, interatomic Fe-Ch distances (d) and anion heights obtained from PDF refinements of I4/m VD model to the K₂Fe_5−yCh₅ 5 K data (Ch = Se, S). Neutron powder diffraction results for FeSe at 10 K [24] and x-ray powder diffraction results for FeS at 300 K [25] are also shown as a reference.

I4/m	a = b (Å)	c (Å)	d1 (Å) 1	d2 (Å) 2	d3 (Å) 2	d4 (Å) 2	hCh (Å)
K2Fe4.27(1)Se5	8.683(1)	14.001(3)	2.337(8)	2.446(2)	2.451(3)	2.512(5)	1.396(9)
K2Fe4.27(1)S5	8.395(2)	13.437(2)	2.283(5)	2.321(8)	2.332(5)	2.37(1)	1.282(8)
Cmma	a (Å)	b (Å)	c (Å)	d (Å) × 4	-	-	hCh (Å)
FeSe	5.3147(5)	5.3367(5)	5.4855(3)	2.383(1)	-	-	1.461(2)
P4/nmm	a = b (Å)	c (Å)	d (Å) × 4	-	-	-	hCh (Å)
FeS	3.6802(5)	5.0307(7)	2.235(1)	-	-	-	1.270(1)

¹ Fe-Ch(1) distance, where Ch(1) belongs to 4e site; ² Fe-Ch(2) distances, where Ch(2) belongs to 16i site.

and

Table 2. Interatomic Fe-Fe distances and isotropic ADPs from the PDF refinements of I4/m VD model to the K₂Fe_5−yCh₅ data (Ch = Se, S) at 5 K and neutron powder diffraction refinement for FeSe at 10 K.

I4/m	Intra-Cluster Fe-Fe (Å)	Inter-Cluster Fe-Fe (Å)	K, Uiso (Å2)	Fe, Uiso (Å2)	Ch, Uiso (Å2)
K2Fe4.27(1)Se5	2.649(4)	2.897(5)	0.0231(7)	0.0087(1)	0.0067(1)
K2Fe4.27(1)S5	2.672(1)	2.746(1)	0.0236(2)	0.0093(1)	0.0032(1)
Cmma	Fe-Fe(1) (Å)	Fe-Fe(2) (Å)
FeSe	2.657(1)	2.667(1)	-	0.0043(7)	0.0033(9)

, we quantitatively compare various structural parameters derived from the single-phase I4/m VD model at 5 K, such as lattice parameters, interatomic Fe-Ch and Fe-Fe distances, anion heights, and ADP values. Most values are compared in reference to respective quantities obtained for non-intercalated FeSe [24] and FeS [25] parent materials. As expected, the lattice parameters and the interatomic distances (all Fe-Ch and the inter-cluster Fe-Fe lengths) are longer in the case of selenide, resulting in a large increase in the unit cell volume of ca. 11%. In the K-intercalated systems, two types of Fe-Fe distances are associated with the square Fe clusters: intracluster and intercluster, as defined within the I4/m model. In the FeSe reference, there are two distinct Fe-Fe distances; however, these are along the two distinct crystallographic directions, as defined by the orthorhombic Cmma model.

One prominent structural parameter known to be correlated with the superconducting transition temperature T_C in iron-based superconductors is the anion height, defined as the c-axis normal distance between the chalcogen/pnictogen and the Fe-sheet. According to this empirical observation, the critical temperature, T_C increases as the anion height approaches a value of ~1.38 Å, inferring the existence of an optimal bonding environment for maximizing the superconducting response [26]. The anion height values obtained for the samples studied here were generally consistent with the empirical rule established by Mizuguchi et al. [26]. For our superconducting K₂Fe_4.27(1)Se₅ sample, with T_C ≈ 32 K, the anion height was indeed relatively close to this ideal value, whereas for the insulating K₂Fe_4.27(1)S₅ material, the anion height was appreciably further away from the optimal 1.38 Å. This is illustrated schematically in the inset of Figure 5. The replacement of Se by smaller S results in reduction of all interatomic cation-anion distances, which in turn gives rise to a decrease in the anion height to about ~1.28 Å. The anion height in the potassium-intercalated selenide can also be compared to the parent FeSe, which crystallizes in the Cmma structure at low temperatures. For the latter, the neutron powder diffraction data collected at 10 K on the same instrument were used [24]. Intercalation of K in between the FeSe layers induced an effective reduction in the anion height, shifting the value closer to the optimum, while simultaneously increasing the Fe-interlayer spacing from 5.316(1) Å to 7.001(1) Å, with a concomitant increase of T_C from about 8 K to about 32 K. In the sulphide analogue, the K intercalation also increased the Fe interlayer spacing from 5.031(1) Å to 6.747(4) Å, but the anion height increased only slightly, and at 1.28 Å, is still substantially far from the optimum value for superconductivity. Whereas FeS is a superconductor with a T_C ~5 K [25], the K-intercalated compound exhibited a spin-glass behavior.

Lastly, we considered the regularity of the FeCh₄ tetrahedron, which represents yet another structural aspect of relevance to superconductivity [12,15,27]. Indeed, the superconducting critical temperature T_C in some iron-based superconductors was maximized for structures featuring regular tetrahedral units, defined by the Ch-Fe-Ch angles all being equal to 109.47°, thus resulting in perfect tetrahedral geometry. Given this, the mean and standard deviation of the underlying bond angle distributions were estimated from the neutron scattering data analyzed for the systems of interest here. To facilitate the comparison, the data are schematically presented as Gaussian distributions in Figure 5, whose centroids reflect the average tetrahedral angle, whereas the widths represent the standard deviation of the angular distribution. For the samples studied, these values were obtained for the six tetrahedral angles of K₂Fe_5−yCh₅ (Ch = Se, S) based on the I4/m VD model for 16i Fe site, and the three angles of FeSe (with multiplicity two) assuming an orthorhombic Cmma crystal type. In the case of FeS, the distribution was the narrowest (not shown), implying that the tetrahedral regularity is not a required condition for optimizing superconductivity. Nevertheless, PDF results evidenced a larger degree of tetrahedral irregularity in the case of K₂Fe_5−yS₅ as compared with K₂Fe_5−ySe₅. Replacement of Se by S substantially increased the irregularity of the FeCh₄ tetrahedron on average, which, together with the substantial departure of the anion height from the optimum value, coincide with the observed decrease in superconductivity and the appearance of the spin glass state in the sulphide. These modeling results confirm the narrower distribution of interatomic distances in the selenide, determined from the direct comparison of the experimental PDFs. Importantly, the parent FeSe showed a slightly wider distribution compared with the K-intercalated counterpart, and consistently lower value of T_C. As the anion heights of the selenide and sulfide samples studied here were positioned on the opposite sides of the optimal value of 1.38 Å, where the observed T_C peaks in iron chalcogenides and pnictides, tracking this subtle structural parameter on the basis of the K_xFe_2−ySe_2−zS_z solid-solution testing ground would be valuable. In particular, systematically and quantitatively characterizing the evolution of the disorder across the phase diagram of K_xFe_2−ySe_2−zS_z at low temperatures would help correlate how the key building blocks relay the competition of magnetism with superconductivity (and vice versa), a concept of fundamental importance in Fe-based correlated electron systems.

4. Conclusions

In summary, we performed a neutron PDF analysis to explore the nanoscale atomic structures of two K_xFe_2−yCh₂ materials (Ch = S, Se) at 5 K. The study was motivated by the phase separation concept in these compositions, which enabled the comparison of their structural features to those observed in their non-intercalated counterparts. Interestingly, the diffraction patterns of both systems displayed the (110) reflection, a hallmark of I4/m crystal symmetry. In the respective reduced total scattering structure functions F(Q), appreciable arcs of diffuse scattering, particularly at high momentum transfers Q, were indicative of the presence of substantial disorder in the system. The diffuse intensity appeared to be more pronounced for K_xFe_2−ySe₂ than for K_xFe_2−yS₂. The PDF analysis revealed that the data for both systems could be explained by either a mixture of majority I4/m vacancy-ordered and minority I4/mmm components, or by a single phase I4/m vacancy-disordered model. Both descriptions are effectively consistent with the underlying phase separation and complex disorder on the nanoscale. Quantitatively, in the two-phase description, the I4/mmm minority phase content did not decrease from K_xFe_2−ySe₂ to K_xFe_2−yS₂ as would be expected on the grounds that this phase is commonly associated with superconductivity. Similarly, the single-phase vacancy disordered model description resulted in comparable values of the occupancy of the Fe 4d site for the two systems. This indicates that the potassium intercalated FeCh is more complex than outlined by the simplistic phase separation scenario, and implies that the phase separation does not play a key role in the collapse of superconductivity in the intercalated sulphide system. An explanation might be provided by the fine details of disorder that are yet to be explored. Comparison of relevant structural parameters derived from the Fe-vacancy-disordered model demonstrated that replacement of larger Se by smaller S induced a dramatic reduction in the unit cell volume and interatomic Fe-Ch distances. Corresponding anion heights conformed to the empirical rule of Mizuguchi et al. [26]. Furthermore, the FeCh₄ tetrahedra were more irregular in the non-superconducting sulphide than in the superconducting selenide. The same parameters obtained for the non-intercalated parent materials appear to also follow the same empirical trend. Overall, this neutron total scattering study highlights that deciphering the subtle differences in the atomic structure configuration of the K-intercalated end-members and their markedly different physical properties is a complex physical crystallography problem. Seeking additional insights from K_xFe_2−ySe_2−zS_z solid-solutions may be crucial for uncovering the mediating role of key structural building blocks in this puzzle.