Correlated disorder-to-order crossover in the local structure of ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_{2-z}{\mathrm{S}}_z$

A detailed account of the local atomic structure and disorder at 5 K across the phase diagram of the high-temperature superconductor ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_{2-z}{\mathrm{S}}_z$ $(0\le z\le 2)$ is obtained from neutron total scattering and associated atomic pair distribution function (PDF) approaches. Various model-independent and model-dependent aspects of the analysis reveal a high level of structural complexity on the nanometer length scale. Evidence is found for considerable disorder in the $c$-axis stacking of the ${\mathrm{FeSe}}_{1-x}{\mathrm{S}}_x$ slabs without observable signs of turbostratic character of the disorder. In contrast to the related FeCh (Ch = S, Se)-type superconductors, substantial Fe-vacancies are present in ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_{2-z}{\mathrm{S}}_z$, deemed detrimental for superconductivity when ordered. Our study suggests that the distribution of vacancies significantly modifies the iron-chalcogen bond-length distribution, in agreement with observed evolution of the PDF signal. A crossoverlike transition is observed at a composition of $z\approx 1$, from a correlated disorder state at the selenium end to a more vacancy-ordered (VO) state closer to the sulfur end of the phase diagram. The S-content-dependent measures of the local structure are found to exhibit distinct behavior on either side of this crossover, correlating well with the evolution of the superconducting state to that of a magnetic semiconductor toward the $z\approx 2$ end. The behavior reinforces the idea of the intimate relationship of correlated Fe-vacancy order in the local structure and the emergent electronic properties.

Figure 1

Two possible crystallographic models of the atomic structure of ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_{2-z}{\mathrm{S}}_z$. (a) A perspective view of a single unit cell in the $I4/mmm$ Fe-vacancy-free mode. (b) A single $\mathrm{Fe}{Ch}_4$ tetrahedron and other structural elements discussed in the text. $h_A$ denotes the anion height. (c) The Fe sublattice viewed both along the $a-b$ plane (top) and down the $c$ axis (bottom) in the $I4/mmm$ model. (d) A perspective view of a single unit cell in the $I4/m$ VO model. (e) The Fe sub-lattice in the $I4/m$ VO model as viewed both along the $a-b$ plane (top) and down the $c$ axis (bottom). Symmetry-distinct Fe-sites ($4d$ and $16i$), Fe vacancies, as well as various Fe-Fe interatomic distances discussed in text are indicated by arrows.

Figure 2

Vacancy ordering in ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_{2-z}{\mathrm{S}}_z$. Neutron powder diffraction data (sulphur content, $z$, as indicated) in the vicinity of the (110) and (002) reflections, positions of which for $z=0$ sample are represented by vertical red and green dashed lines, respectively. Patterns are offset vertically by 0.07 $y$-axis units for clarity, and a background curve is plotted for the $z=0$ data set to highlight the diffuse (110) feature. The presence of a sharp (110) reflection would indicate perfect vacancy ordering (the $I4/m$ model). Observation of broad and diffuse intensity of this reflection would imply correlated vacancy disorder. A broad shoulder at the low-$Q$ side of the (002) reflection is noticeable for each composition. The evolution with $z$ of its normalized integrated intensity is shown in the inset. The origin of this feature is subsequently addressed in Fig. 3 and in the text.

Figure 3

Observation of stacking disorder in ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_{2-z}{\mathrm{S}}_z$ system. (a) A side view of an atomistic model depicting two distinct interlayer distances along the $c$ axis, represented by either a purple or a green double-sided arrow. Green spheres represent chalcogen, brown spheres iron, and purple spheres potassium atoms. The red vertical dashed line highlights the lack of turbostratic disorder, as discussed in the text. (b) The simulated neutron diffraction pattern (blue profile) of ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{S}}_2$ for the case depicted in (a), where two distinct interlayer distances are present. Also shown is the experimental diffraction pattern for ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{S}}_2$ sample at 5 K with the associated background (orange profile and dashed line, respectively) featuring (002) basal reflection at around $0.95{\AA }^{-1}$. The feature at $Q\approx 0.8{\AA }^{-1}$ is an irrational basal reflection implying the presence of a second, longer interlayer spacing. This feature is observed for all as-prepared samples studied here (Fig. 2). Inset in (b) features data for the ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_2$ sample at 5 K, for both as-prepared (red line) and thermally quenched (green line) variants.

Figure 4

Observation of stacking disorder in ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_{2-z}{\mathrm{S}}_z$ system. (a) A side view of an atomistic model depicting turbostratic disorder or uniform random layer displacements along the $a-b$ lattice directions, highlighted by the vertical red dashed line. Atoms are represented as in Fig. 3, with the $c$ axis running vertically. (b) The neutron diffraction pattern simulated for ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{S}}_2$ (blue profile) in the case of turbostratic disorder depicted in (a). This type of disorder reduces the number of observed Bragg reflections, and produces characteristic sawtoothlike features, as discussed in the text. The experimental data for ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{S}}_2$ (orange profile) collected at 5 K do not display the features that would imply turbostratic character of the disorder.

Figure 5

Comparison of neutron total-scattering-derived data for several samples of interest. Reduced total-scattering structure functions, $F\left(Q\right)$, are shown in the left panels ($1.5\le Q\le 25{\AA }^{-1}$), while corresponding PDFs, $G\left(r\right)$, are displayed in the right panels ($1.5\le r\le 25\AA$). The panels feature (a), (b) ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_2$, (c), (d) ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{S}}_2$, and (e), (f) FeSe. Data for potassium containing samples were collected at 5 K, while for FeSe at 10 K.

Figure 6

Analysis of the diffuse scattering signal in $F\left(Q\right)$ at high momentum transfers $Q$ in ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_{2-z}{\mathrm{S}}_z$. (a) The measured (lighter line) and fitted (heavier line) $F\left(Q\right)$ signal. The curves are sequentially offset vertically by $2{\AA }^{-2}$ for clarity. (b) The model $F\left(Q\right)$ signal over a regime of concentrations where the oscillation amplitude and frequency decrease. (c) The model $F\left(Q\right)$ signal over a regime of concentrations where oscillation amplitude and frequency increase. (d) The amplitude of $F\left(Q\right)$ diffuse oscillations as a function of sulfur content, $z$. (e) The frequency of diffuse oscillations as a function of $z$. Colors represent sulfur content, $z$, as indicated, and in (b) and (c) are consistent with those in (a).

Figure 7

5 K PDF data of the ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_{2-z}{\mathrm{S}}_z$ series. (a) The experimental PDFs in the local structure range $1.5\le r\le 8\AA$. The sulfur and selenium end-members are shown with a blue and red line, respectively, while intermediate compositions are shown in gray. A black arrow is included to emphasize the continuous evolution of features at high $r$. Inset is a false color map representation of the same PDF data plotted vs $r$ and $z$, highlighting the lack of apparent change in the position of the first PDF peak at $\sim 2.48\AA$ as a function of sulfur content until approximately $z=1$. The peak starts evolving for higher $z$ values. (b) A plot of the peak position of the feature marked by an orange arrow in (a) as a function of sulfur content $z$. A dashed line is a guide to the eye, included to highlight the nearly linear behavior of this peak position with $z$. (c) A plot of the Fe-Ch peak position, marked by a blue arrow in (a), as a function of $z$. Two dashed lines are guides to the eye, highlighting the two-regime behavior.

Figure 8

Fits of the structural models to the PDF data. The fit to the observed ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_2$ PDF data using either (a) a single-phase $I4/m$ vacancy-disordered (VD) model or (b) a two-phase model with $I4/m$ VO and $I4/mmm$ vacancy-free components. Similar one- and two-phase fits for ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{S}}_2$ are shown in (c) and (d), respectively, for the ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{S}}_2$ PDF data. In both compositions, single- and two-phase models yielded fits of similar quality: $R_w=0.12$ and $R_w=0.15$ for ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_2$ and ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{S}}_2$, respectively. In various panels, arrows point to intracluster and intercluster Fe-Fe PDF peaks, as indicated. See text and Fig. 1 for details. In all cases, fits were over the displayed $r$-range ($1.5\le r\le 10\AA$). PDF data and fits are shown with open circles and solid red lines, respectively, and the difference curves (solid green lines) are shown offset vertically for clarity. The inset in (b) summarizes the single-phase PDF refinement-derived evolution with sulfur content of Fe occupancy of the two distinct Fe sites ($4d$ and $16i$) in the $I4/m$ VD model. Inset in (d) shows evolution of chalcogen anion height $h_A$ with sulfur content as obtained from the $I4/mmm$ vacancy-free component of the two-phase PDF model. A horizontal dashed line represents the reported optimal anion height for superconductivity [55].

Figure 9

The effect of vacancy ordering on the local ${\mathrm{FeCh}}_4$ structural unit. (a) An atomistic model (viewed down the $c$ axis) featuring uniform vacancy (white dots) disorder in the iron (brown dots) sublattice. (b) A histogram enumerating the total number of chalcogen species (expressed as a fraction) coordinated by either 0, 1, 2, 3, or 4 Fe in the atomic configuration represented in (a). (c) Similar to (a), except the vacancies show correlated disorder, with an increased likelihood of like neighbors in the first coordination shell and unlike neighbors in the second coordination shell. (d) Similar to (b) except pertaining to the atomic configuration in (c). (e) Similar to (a), except the vacancies are fully ordered over the Fe sublattice. (f) Similar to (b) except pertaining to the atomic configuration in (e). (g) The distribution of Fe-Ch pair distances following energy minimization in the presence of a Lennard-Jones (LJ) potential between Fe-Ch nearest neighbors, shown for the vacancy distributions represented by the atomistic model in (a), (c), and (e). LJ potential parameters were identical between the each atomic configuration, the differences in the pair distances distributions are a result of the vacancy configuration, only.

Figure 10

Evolution of the local structure with sulfur content. (a) ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_2$ radial distribution function showing Fe-Ch and distinct Fe-Fe pair distances (see text). (b) Similar to (a) except with ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{S}}_2$, highlighting the relative increase in prevalence of intra-cluster Fe-Fe pairs. (c) Quantification of the prevalence of different inter- and intracluster Fe-Fe pairs as a function of sulfur content $z$. (d) Quantification of the inter- and intracluster Fe-Fe pair distances as a function of sulfur content $z$.

Figure 11

Qualitative comparison of PDF derived local structural parameters and electronic properties of ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_{2-z}{\mathrm{S}}_z$. (a) Reduced observables, representing electronic and structural quantities rescaled to a fraction of their values at the end-members. By definition (see text) these reduced observables are one at $z=0$ and zero at $z=2$. (b) The previously reported [18] electronic phase diagram of ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_{2-z}{\mathrm{S}}_z$ $(0\le z\le 2)$ as a function of temperature $T$ and composition $z$, obtained by resisitivity and DC magnetic susceptibility measurements using a Quantum design PPMS-9 and MPMS-XL5 on single-crystal samples prior to grinding.