Doping dependence of phase-separation morphology in (Sr,K)Fe${}_2{\mathrm{As}}_2$

The effects of the possible phase separation on macroscopic transport properties are explored in (Sr,K)Fe${}_2$As${}_2$. Three probes, the low-field shielding $4\pi {\chi }_{\mathrm{ZFC}}$, the reversible diamagnetization $M_{\mathrm{SC}}$ at 5 T, and the specific heat $C_p$, with different characteristic length scales on the order of ${10}^3$, 10, and ${10}^0$ nm, respectively, were used to examine several (Sr${}_{1-x}$K${}_x$)Fe${}_2$As${}_2$ samples. The data from different probes should be consistent if the sample can be regarded as mesoscopically homogeneous. While this seems to be the case for the optimally doped $x$ = 0.45 samples, the apparent transition temperature and the effective superfluid density strongly depend on the probes used for both the underdoped ($x$ = 0.3) and the overdoped ($x$ = 0.7) samples. Discrepancies of up to a factor of 10 are observed. The data further suggest that both the $x$ = 0.3 and $x$ = 0.7 samples are better described as Josephson-junction arrays formed over separated superconductive domains of $10-{10}^2$ nm sizes. Such a morphology evolution may drastically affect the interpretation of the observed doping effects.

Figure 1

SEM image of the $x$ = 0.3 sample. The volume-weighted average grain size is ∼3 μm. Similar grain sizes were also observed from other samples used here.

Figure 2

The morphology and symbols. The black and white areas represent the superconductive parts and the nonsuperconductive parts, respectively. An intergrain void is marked by a “V.” The thick and the thin dashed lines are the grain boundaries and intragrain weak links (nonsuperconductive layers), respectively. (a) Ceramic with spongelike morphology. (b) Ceramic with JJA-like morphology. (c) Three length scales and the associated characteristics in the JJA-like morphology. The gray areas are the parameter ranges sensitive to the marked probes. $T_{\mathit{cd}}$ and ${\lambda }_{\mathit{cd}}$ are the microscopic transition temperature and the London penetration depth, respectively. $T_{\mathit{ce}}$ and $T_{\mathit{cg}}$ are the phase-lock temperatures across the grain boundaries and the intragrain weak links, respectively. The Josephson penetration depths for these two boundaries are ${\lambda }_{\mathit{ce}}$ and ${\lambda }_{\mathit{cg}}$, respectively, which consequentially determine the overall shielding $4\pi {\chi }_{\mathrm{ZFC}}$ and the intragrain one, $4\pi {\chi }_{\mathrm{ZFC},g}$.

Figure 3

(a) The raw $C_p/T$ data (symbols) and the model fit (line) of an $x$ = 0.45 sample. (b) $C_p/T$ vs $T^2$ below 7 K for the $x$ = 0.45 (circles), 0.3 (triangles), and 0.7 (inverted triangles) samples.

Figure 4

(a) $\Delta C_p/T$ data (circles) and the two-gap $\alpha$-model fit (thin solid line) of the $x$ = 0.45 sample. Thick dashed line: The average $T_c$ based on area balance. (b) The condensation entropy $\Delta S/(\gamma -{\gamma }_0)/T_c$. Circles: data; solid line: $\alpha$-model fit.

Figure 5

(a) The 5-T magnetic moments of an $x$ = 0.45 sample under ZFC (inverted triangles) and FC (circles) conditions. (b) The assumed reversible $M_{\mathrm{SC}}$ (symbols) and the $S$-wave, two-gap, $\alpha$-model fit (thick solid line).

Figure 6

(a) The 10-Oe $4\pi {\chi }_{\mathrm{ZFC}}$ (open circles) and $4\pi {\chi }_{\mathrm{FC}}$ (inverted triangles) of the $x$ = 0.45 sample. The solid circles and the thick solid line represent the contribution from isolated grains, which was extracted from the $M$-$H$ sweep of (b) (see text). (b) $4\pi \hspace{0.5em}\mathit{dM}/\mathit{dH}$ at 10 K (circles), using the left scale. The thick dashed line is the $M$($H$) at 10 K, using the right scale.

Figure 7

The thermal and magnetic properties of an underdoped Sr${}_{0.7}$K${}_{0.3}$Fe${}_2$As${}_2$ sample. (a) The $4\pi {\chi }_{\mathrm{ZFC}}$ (open circles), $4\pi {\chi }_{\mathrm{FC}}$ (inverted triangles) at 10 Oe, and the deduced ZFC intragrain contribution $4\pi {\chi }_{\mathrm{ZFC},g}$ (solid circles, see text). Three sequential transitions at 8 K (A), 24 K (B), and 31 K (C) can be noticed. (b) $\Delta C_p/T(\gamma -{\gamma }_0)$. The average $T_c$ (gray bar) between 26 and 28.5 K is estimated from the entropy balance.

Figure 8

The thermal and magnetic properties of an underdoped Sr${}_{0.3}$K${}_{0.7}$Fe${}_2$As${}_2$ sample. (a) The $4\pi {\chi }_{\mathrm{ZFC}}$ (circles), $4\pi {\chi }_{\mathrm{FC}}$ (inverted triangles), and $4\pi {\chi }_{\mathrm{ZFC},g}$ (diamonds) at 10 Oe. The thick solid line is the linear fit of the $4\pi {\chi }_{\mathrm{ZFC},g}$. (b) $\Delta C_p/T(\gamma -{\gamma }_0)$.

Figure 9

The scaled $\Delta S/(\gamma -{\gamma }_0)/T_c$ vs $T/T_c$. (Sr${}_{1-x}$K${}_x$)Fe${}_2$As${}_2$ at $x$ = 0.3 (open circles), 0.45 (solid squares), and 0.7 (open inverted triangles).

Figure 10

The scaled $M_{\mathrm{SC}}/M_0$ vs $T/T_c$. (Sr${}_{1-x}$K${}_x$)Fe${}_2$As${}_2$ at $x$ = 0.3 (open circles), 0.45 (solid squares), and 0.7 (open inverted triangles).

Figure 11

(a) The residual ${\gamma }_0$. (b) The $T_c$ extracted from (circles) $4\pi {\chi }_{\mathrm{ZFC},g}$, (inverted triangles) specific heat, and (squares) $M_{\mathrm{SC}}$. (c) The apparent superfluid density extracted from (circles) $4\pi {\chi }_{\mathrm{ZFC},g}$ and (squares) $M_{\mathrm{SC}}$.