Superconductivity and magnetism in Rb${}_x$Fe${}_{2-y}$Se${}_2$: Impact of thermal treatment on mesoscopic phase separation

An extended study of the superconducting and normal-state properties of various as-grown and post-annealed Rb${}_x$Fe${}_{2-y}$Se${}_2$ single crystals is presented. Magnetization experiments evidence that annealing of Rb${}_x$Fe${}_{2-y}$Se${}_2$ at 413 K, well below the onset of phase separation $T_{\mathrm{p}}\simeq 489$ K, neither changes the magnetic nor the superconducting properties of the crystals. In addition, annealing at 563 K, well above $T_{\mathrm{p}}$, suppresses the superconducting transition temperature $T_{\mathrm{c}}$ and leads to an increase of the antiferromagnetic susceptibility accompanied by the creation of ferromagnetic impurity phases, which are developing with annealing time. However, annealing at $T=488\mathrm{K}\simeq T_{\mathrm{p}}$ increases $T_{\mathrm{c}}$ up to 33.3 K, sharpens the superconducting transition, increases the lower critical field, and strengthens the screening efficiency of the applied magnetic field. Resistivity measurements of the as-grown and optimally annealed samples reveal an increase of the upper critical field along both crystallographic directions as well as its anisotropy. Muon spin rotation and scanning transmission electron microscopy experiments suggest the coexistence of two phases below $T_{\mathrm{p}}$: a magnetic majority phase of Rb${}_2$Fe${}_4$Se${}_5$ and a nonmagnetic minority phase of Rb${}_{0.5}$Fe${}_2$Se${}_2$. Both microscopic techniques indicate that annealing the specimens just at $T_{\mathrm{p}}$ does not affect the volume fraction of the two phases, although the magnetic field distribution in the samples changes substantially. This suggests that the microstructure of the sample, caused by mesoscopic phase separation, is modified by annealing just at $T_{\mathrm{p}}$, leading to an improvement of the superconducting properties of Rb${}_x$Fe${}_{2-y}$Se${}_2$ and an enhancement of $T_{\mathrm{c}}$.

Figure 1

Differential heat $\Delta Q$ for a Rb${}_x$Fe${}_{2-y}$Se${}_2$ single crystal recorded between 400 and 600 K with a constant heating rate of 20 K/min. Three distinct peaks are observed, related to the three onset temperatures $T_{\mathrm{p}}\simeq 489$ K, $T_{\mathrm{N}}\simeq 517$ K, and $T_{\mathrm{s}}\simeq 540$ K (see text). The three annealing temperatures 413, 488, and 563 K were chosen to post anneal the as-grown Rb${}_x$Fe${}_{2-y}$Se${}_2$ crystals for the subsequent experiments.

Figure 2

STEM images of as-grown Rb${}_x$Fe${}_{2-y}$Se${}_2$ single crystal taken with the direction of the electron beam perpendicular to the tetragonal $c$ axis. Picture (a) was taken on a square of $\sim 1.5\times 1.5\mu$m${}^2$, (b) on a square of $\sim 250\times 250$ nm${}^2$, (c) on a square of $\sim 50\times 50$ nm${}^2$. The atomic composition of the darker regions was found to correspond to Rb${}_{0.5}$Fe${}_2$Se${}_2$, whereas in the brighter regions, the composition is Fe- and Rb-deficient Rb${}_{0.4}$Fe${}_{1.6}$Se${}_2$.

Figure 3

Normalized ZFC magnetization $M\left(T\right)/M\left(0\right)$ for the Rb${}_x$Fe${}_{2-y}$Se${}_2$ single crystals A${}_{413}\left[t_{\mathrm{ann}}\right]$, A${}_{488}\left[t_{\mathrm{ann}}\right]$, A${}_{563}\left[t_{\mathrm{ann}}\right]$, and A${}_{488}^{*}\left[t_{\mathrm{ann}}\right]$ in a magnetic field ${\mu }_{0}H=0.3$ mT applied along the $c$ axis. The panels present the data for the as-grown samples with $t_{\mathrm{ann}}=0$ h (a), annealed samples for $t_{\mathrm{ann}}=3$ h (b), and for $t_{\mathrm{ann}}=36$ h (c). The respective insets show closeups of the onset of diamagnetism.

Figure 4

Zero-field-cooled (ZFC) magnetization curves for the Rb${}_x$Fe${}_{2-y}$Se${}_2$ single crystals A${}_{413}\left[t_{\mathrm{ann}}\right]$, A${}_{488}\left[t_{\mathrm{ann}}\right]$, A${}_{563}\left[t_{\mathrm{ann}}\right]$, and A${}_{488}^{*}\left[t_{\mathrm{ann}}\right]$ measured at 2.0 K as a function of $H_{\mathrm{int}}$ along the $c$ axis. The corresponding $t_{\mathrm{ann}}$ of the different panels are $t_{\mathrm{ann}}=0$ h (a), $t_{\mathrm{ann}}=3$ h (b), and $t_{\mathrm{ann}}=36$ h (c).

Figure 5

In-plane resistivity $\rho$ of the Rb${}_x$Fe${}_{2-y}$Se${}_2$ samples B${}_{488}\left[0\mathrm{h}\right]$ and B${}_{488}\left[3\mathrm{h}\right]$. The pronounced hump in the normal-state resistivity of the as-grown sample B${}_{488}\left[0\mathrm{h}\right]$ decreases dramatically after annealing and the superconducting $T_{\mathrm{c}}$ increases from 31.5 to 33.1 K.

Figure 6

Resistivity of the samples B${}_{488}\left[t_{\mathrm{ann}}\right]$ for magnetic fields between 0 and 9 T, varied by 0.5-T steps, for fields in the $ab$ plane and along the $c$ axis. The measurements were performed for the as-grown sample B${}_{488}\left[0\mathrm{h}\right]$ [panels (a) and (b)] and for the annealed sample B${}_{488}\left[3\mathrm{h}\right]$ [panels (c) and (d)], with $H$ applied along the $c$ axis and in the $ab$ plane. The dashed lines denotes 50$\%$ of the extrapolated normal-state resistivity, which was used as a criterion to determine $H_{\mathrm{c}2}\left(T\right)$, shown in panel (e). The transition temperature $T_{\mathrm{c}}$ increases by 1.6 K as a result of annealing. The solid lines are guides to the linear part of the $H_{\mathrm{c}2}\left(T\right)$ curves, used in the WHH approximation [Eq. (6)].

Figure 7

Measured magnetic moment $m\left(T\right)$ of pristine and annealed Rb${}_x$Fe${}_{2-y}$Se${}_2$ single crystals in the temperature range between 50 and 370 K for magnetic fields of 1 T (a) and 3 T (b), applied along the $c$ axis.

Figure 8

(a) Antiferromagnetic susceptibility ${\chi }_{\mathrm{AFM}}\left(T\right)$ in the normal state of pristine and annealed Rb${}_x$Fe${}_{2-y}$Se${}_2$ single crystals determined from the data shown in Fig. 7 using Eq. (8). For clarity, the curves representing the four different annealing sets are vertically shifted to each other. Whereas no change of ${\chi }_{\mathrm{AFM}}\left(T\right)$ is found by annealing for sample A${}_{413}\left[t_{\mathrm{ann}}\right]$, for all other samples, ${\chi }_{\mathrm{AFM}}\left(T\right)$ increases with increasing $t_{\mathrm{ann}}$. (b) Ferromagnetic component $M_{\mathrm{FM}}\left(T\right)$, being constant for sample A${}_{413}\left[t_{\mathrm{ann}}\right]$ as a function of $t_{\mathrm{ann}}$. For all other samples, $M_{\mathrm{FM}}\left(50\mathrm{K}\right)$ increases substantially with increasing $t_{\mathrm{ann}}$.

Figure 9

Results of the TF $\mu$SR investigation of as-grown and 60-h annealed Rb${}_x$Fe${}_{2-y}$Se${}_2$ single crystals, C${}_{488}\left[0\mathrm{h}\right]$ and C${}_{488}\left[60\mathrm{h}\right]$, in a magnetic field of 70 mT applied along the $c$ axis. (a) $P\left(B\right)$ for both samples at 5 K. The line shape for C${}_{488}\left[60\mathrm{h}\right]$ is more asymmetric compared to that for the as-grown sample C${}_{488}\left[0\mathrm{h}\right]$. (b) and (c) $\mu$SR time spectra at 40 K for sample C${}_{488}\left[0\mathrm{h}\right]$ and C${}_{488}\left[60\mathrm{h}\right]$. The thin solid line is a fit to the data assuming a single relaxation rate $\sigma$. The thick solid line is the envelope of the oscillating function. The data for the annealed sample C${}_{488}\left[60\mathrm{h}\right]$ exhibit a significantly faster damping.

Figure 10

Results of the ZF $\mu$SR investigation of as-grown and 60-h annealed Rb${}_x$Fe${}_{2-y}$Se${}_2$ single crystals, C${}_{488}\left[0\mathrm{h}\right]$ and C${}_{488}\left[60\mathrm{h}\right]$. All data were modeled assuming two internal magnetic fields $B_{\mathrm{int},1}\approx 1$ T and $B_{\mathrm{int},2}\approx 3$ T.

Figure 11

STEM images of as-grown and annealed Rb${}_x$Fe${}_{2-y}$Se${}_2$ single crystals D${}_{488}\left[0\mathrm{h}\right]$ and D${}_{488}\left[3\mathrm{h}\right]$. The microstructure caused by mesoscopic phase separation in the annealed sample D${}_{488}\left[3\mathrm{h}\right]$, shown in panel (b), is modified compared to the one of the as-grown sample D${}_{488}\left[0\mathrm{h}\right]$, shown in panel (a).

Figure 12

Series of temperature-dependent ZFC magnetization measurements on a Rb${}_x$Fe${}_{2-y}$Se${}_2$ single crystal in 0.3 mT. The curves obtained after the various post annealings of the sample are labeled by the respective number.