Microscopic coexistence of magnetism and superconductivity in charge-compensated ${\text{Ba}}_{1-x}$${\text{K}}_x$(${\text{Fe}}_{1-y}$${\text{Co}}_y$)${}_2$${\text{As}}_2$

We present a detailed investigation of the electronic phase diagram of effectively charge compensated ${\text{Ba}}_{1-x}$${\text{K}}_x$(${\text{Fe}}_{1-y}$${\text{Co}}_y$)${}_2$${\text{As}}_2$ with $x/2\approx y$. Our experimental study by means of x-ray diffraction, Mössbauer spectroscopy, muon spin relaxation and ac-susceptibility measurements on polycrystalline samples is complemented by density functional electronic structure calculations. For low substitution levels of $x/2\approx y\le 0.13$, the system displays an orthorhombically distorted and antiferromagnetically ordered ground state. The low-temperature structural and magnetic order parameters are successively reduced with increasing substitution level. We observe a linear relationship between the structural and the magnetic order parameter as a function of temperature and substitution level for $x/2\approx y\le 0.13$. At intermediate substitution levels in the range between 0.13 and 0.19, we find superconductivity with a maximum $T_c$ of 15 K coexisting with static magnetic order on a microscopic length scale. For higher substitution levels $x/2\approx y\ge 0.25$, a tetragonal nonmagnetic ground state is observed. Our DFT calculations yield a significant reduction of the Fe $3d$ density of states at the Fermi energy and a strong suppression of the ordered magnetic moment in excellent agreement with experimental results. The appearance of superconductivity within the antiferromagnetic state can by explained by the introduction of disorder due to nonmagnetic impurities to a system with a constant charge carrier density.

Figure 1

X-ray powder diffraction pattern (top, black), Rietveld fit (top, red). and difference curve (bottom, gray) of ${\text{Ba}}_{0.30}$${\text{K}}_{0.70}$(${\text{Fe}}_{0.84}$${\text{Co}}_{0.16}$)${}_2$${\text{As}}_2$ ($x/2\approx y=0.16$).

Figure 2

Structural changes in charge compensated ${\text{Ba}}_{1-x}$${\text{K}}_x$(${\text{Fe}}_{1-y}$${\text{Co}}_y$)${}_2$${\text{As}}_2$ (red, unfilled symbols) compared to ${\text{Ba}}_{1-x}$${\text{K}}_x$${\text{Fe}}_2$${\text{As}}_2$ (blue, filled symbols) with increasing potassium content per FeAs-layer ($x$/2). Lattice parameter $a$ (top left) and $c$ (bottom left), As-Fe-As angle $\varepsilon$ (top right) and Fe-As distance $d$ (bottom right). Data for ${\text{Ba}}_{1-x}$${\text{K}}_x$${\text{Fe}}_2$${\text{As}}_2$ are taken from Ref. [13].

Figure 3

Lattice parameters of ${\text{Ba}}_{1-x}$${\text{K}}_x$(${\text{Fe}}_{1-y}$${\text{Co}}_y$)${}_2$${\text{As}}_2$ with $\mathit{x}/2\approx y=0.00$–0.13 (in orthorhombic notation), showing the orthorhombic splitting at low temperatures. Data for ${\text{BaFe}}_2$${\text{As}}_2$ are taken from Ref. [33].

Figure 4

Representative ${}^{57}$Fe Mössbauer spectra of ${\text{Ba}}_{1-x}$${\text{K}}_x$(${\text{Fe}}_{1-y}$${\text{Co}}_y$)${}_2$${\text{As}}_2$ for $x/2\approx y=0.07$, 0.10, 0.13, and 0.16. The magnetic phase transition is evidenced by the line broadening and the concomitant transfer of spectral intensity from the central resonance to the shoulders (see text).

Figure 5

Temperature dependence of the weighted hyperfine field $B_{\mathrm{hf}}$ (full circles, left scale) and the orthorhombic splitting parameter ${\delta }_s$ (open circles, right scale). Experimental data for ${\text{BaFe}}_2$${\text{As}}_2$ ($x$/2 = $y$ = 0) are taken from Ref. [33].

Figure 6

The low temperature value of the weighted hyperfine field $B_{\mathrm{hf}}$(T = 4.2 K) and $T_N$ show a linear relationship for all magnetic samples. The experimental data for ${\text{BaFe}}_2$${\text{As}}_2$ ($y=0$) are taken from Ref. [33].

Figure 7

(Top) Zero-field $\mu$SR spectra of ${\text{Ba}}_{1-x}$${\text{K}}_x$(${\text{Fe}}_{1-y}$${\text{Co}}_y$)${}_2$${\text{As}}_2$ with $x/2\approx y=0.16$ and 0.19. (Bottom) Extracted temperature dependencies of the corresponding magnetic volume fraction and the ZF-$\mu$SR transverse relaxation rate ${\lambda }^{\mathrm{ZF}}$ (insets). Dotted lines are guides to the eyes.

Figure 8

Real part of the ac susceptibility signal for charge compensated ${\text{Ba}}_{1-x}$${\text{K}}_x$(${\text{Fe}}_{1-y}$${\text{Co}}_y$)${}_2$${\text{As}}_2$. For substitution levels $x/2\approx y$ between 0.13 and 0.19, bulk superconductivity is found.

Figure 9

Temperature dependence of the local field $\langle B\rangle$ (blue dots) and the vortex lattice relaxation rate ${\sigma }_{\mathrm{SC}}$ (black dots) within the nonmagnetic volume fraction of ${\text{Ba}}_{0.62}$${\text{K}}_{0.38}$${\text{Fe}}_{1.62}$${\text{Co}}_{0.38}$${\text{As}}_2$ ($x/2\approx y=0.19$) as determined by a transverse field experiment. The solid black line is a fit within a single $s$-wave gap scenario (see text for details).

Figure 10

(Top) Calculated magnetic momemt (per Fe) in the antiferromagnetic “stripe” phase as a function of the Co/K substitution level $x/2=y$ in ${\text{Ba}}_{1-x}$${\text{K}}_x$(${\text{Fe}}_{1-y}$${\text{Co}}_y$)${}_2$${\text{As}}_2$ together with the weighted hyperfine field $B_{\mathrm{hf}}$($T=4.2$ K). The measured hyperfine fields are scaled by a constant factor to match the calculated Fe moment for the undoped parent compound ${\text{BaFe}}_2$${\text{As}}_2$. (Bottom) Total and partial electronic density of states (DOS) at the Fermi level for ${\text{Ba}}_{1-x}$${\text{K}}_x$(${\text{Fe}}_{1-y}$${\text{Co}}_y$)${}_2$${\text{As}}_2$ for nonmagnetic calculations. With increasing substitution level the DOS decreases, and for $y\ge 0.11$ the Fe $3d$ contribution (per Fe atom and spin) drops below the Stoner criterion for a magnetic instability (marked by the dashed lines for $I$(Fe $3d$) = $1\pm 0.1$, see text).

Figure 11

The calculated hole-related Fermi surface sheets of ${\text{Ba}}_{1-x}$${\text{K}}_x$(${\text{Fe}}_{1-y}$${\text{Co}}_y$)${}_2$${\text{As}}_2$ for $x=y=0$ (left) and $x/2=y=0.25$ (right). To also indicate the change in the anisotropies, the respective Fermi velocities are indicated by the color mapping. In contrast to the hole surfaces, the shape of the electron surfaces is essentially unchanged for both compounds (within the used VCA approach).

Figure 12

Ordered magnetic moment at low temperatures in ${\text{Ba}}_{1-x}$${\text{K}}_x$(${\text{Fe}}_{1-y}$${\text{Co}}_y$)${}_2$${\text{As}}_2$ ($B_{\mathrm{hf}}$ for $T=4.2$ K from this work, bottom/left axes) and Ba(${\text{Fe}}_{1-x}$${\text{Ru}}_x$)${}_2$${\text{As}}_2$ ($M$ for $T\to 0$ from Ref. [51], top/right axes). Both systems show a linear dependence upon chemical substitution.

Figure 13

Comparison of experimental ($T_c$ from this work in black and Ref. [24] in gray) and calculated ($T/T_{c,0}$ from Ref. [56]) transition temperatures under the assumption that ${\Gamma }_0/2\pi T_{c,0}\propto n_{\mathrm{imp}}\propto y$. (a) Best possible scaling to a pure interband scattering scenario (OD, blue line). (b) Best possible scaling to a dominantly intraband scattering scenario (LRD, red line).

Figure 14

Ordered Fe magnetic moment measured by $B_{\mathrm{hf}}$ as function of the orthorhombic splitting parameter ${\delta }_s$ for ${\text{Ba}}_{1-x}$${\text{K}}_x$(${\text{Fe}}_{1-y}$${\text{Co}}_y$)${}_2$${\text{As}}_2$ with $x$/2 $\approx$ $y$ (same data and colors as in Fig. 5) together with literature data for various nominal charge equivalent 122 and 1111 systems. References are given in Table 1. In the case of Ba(${\text{Fe}}_{1-x}$${\text{Ru}}_x$)${}_2$${\text{As}}_2$, the magnetic moment was taken from neutron diffraction data and scaled identically to Fig. 12. Experimental error bars of our data have been omitted for clarity; error bars are only given for ${\mathrm{CaFe}}_2{\mathrm{As}}_2$ and PrOFeAs, where available data differ by more than 10%. Dotted lines are guides to the eyes.

Figure 15

(Left) Full phase diagram of ${\text{Ba}}_{1-x}$${\text{K}}_x$(${\text{Fe}}_{1-y}$${\text{Co}}_y$)${}_2$${\text{As}}_2$ for $x/2$ and $y\le$ 0.25: black squares refer to the corresponding superconducting transition temperatures for ${\text{Ba}}_{1-x}$${\text{K}}_x$${\text{Fe}}_2$${\text{As}}_2$ [13], Ba(${\text{Fe}}_{1-x}$${\text{Co}}_x$)${}_2$${\text{As}}_2$ [81], and ${\text{Ba}}_{1-x}$${\text{K}}_x$(${\text{Fe}}_{1-y}$${\text{Co}}_y$)${}_2$${\text{As}}_2$ from this work. Grey squares for ${\text{Ba}}_{1-x}$${\text{K}}_x$(${\text{Fe}}_{1-y}$${\text{Co}}_y$)${}_2$${\text{As}}_2$ are taken from Ref. [24]. A nonsuperconducting ground state is depicted in blue and superconductivity is shown color-coded from yellow (for $T_c\approx 10$ K) to dark red ($T_c\approx 40$ K). Dashed line: tentative border for the existence of the orthorhombic phase. (Right) Phase diagram of charge compensated ${\text{Ba}}_{1-x}$${\text{K}}_x$(${\text{Fe}}_{1-y}$${\text{Co}}_y$)${}_2$${\text{As}}_2$. Data points refer to results from x-ray diffraction (red triangles), Mössbauer spectroscopy (blue circles), and muon spin relaxation (blue squares); $T^{\mathrm{onset}}$ in full symbols, $T^{50\%}$ in half symbols. Data points for the corresponding transition temperatures from Ref. [24] were added in grey.