Magnetism and its coexistence with superconductivity in $\mathrm{Ca}\mathrm{K}{\left({\mathrm{Fe}}_{0.949}{\mathrm{Ni}}_{0.051}\right)}_4{\mathrm{As}}_4$: Muon spin rotation/relaxation studies

The magnetic response of $\mathrm{Ca}\mathrm{K}{\left({\mathrm{Fe}}_{0.949}{\mathrm{Ni}}_{0.051}\right)}_4{\mathrm{As}}_4$ was investigated by means of the muon spin rotation/relaxation. The long-range commensurate magnetic order sets in below the Néel temperature $T_{\mathrm{N}}=50.0\left(5\right)$ K. The density-functional theory calculations have identified three possible muon stopping sites. The experimental data were found to be consistent with only one type of magnetic structure, namely, the long-range magnetic spin-vortex-crystal order with the hedgehog motif within the $ab$ plane and the antiferromagnetic stacking along the $c$ direction. The value of the ordered magnetic moment at $T\approx 3$ K was estimated to be $m_{\mathrm{Fe}}=0.38\left(11\right){\mu }_{\mathrm{B}}$ (${\mu }_{\mathrm{B}}$ is the Bohr magneton). A microscopic coexistence of magnetic and superconducting phases accompanied by a reduction of the magnetic order parameter below the superconducting transition temperature $T_{\mathrm{c}}\simeq 9$ K is observed. Comparison with 11, 122, and 1144 families of Fe-based pnictides points to existence of correlation between the reduction of the magnetic order parameter at $T\to 0$ and the ratio of the transition temperatures $T_{\mathrm{c}}/T_{\mathrm{N}}$. Such correlations were found to be described by Machida's model for coexistence of itinerant spin-density-wave magnetism and superconductivity [K. Machida, J. Phys. Soc. Jpn. 50, 2195 (1981); S. L. Bud'ko et al., Phys. Rev. B 98, 144520 (2018)].

Figure 1

The crystal structure of $\mathrm{Ca}\mathrm{K}{\mathrm{Fe}}_4{\mathrm{As}}_4$ within the $P4/mmm$ group representation. Three muon stopping sites (denoted by diamonds) are $\mu 1$, the Wyckoff position is (0.5,0.5,0.301), the local symmetry is $2h$; $\mu 2$, the Wyckoff position is (0,0,0.186), the local symmetry is $2g$; and $\mu 3$, the Wyckoff position is (0,0,0.5), the local symmetry is $1b$. The crystal structure is visualized by using the vesta package [18].

Figure 2

(a) The schematic representation of an arrangement of positron counters at the GPS (General Purpose Surface) spectrometer at the Paul Scherrer Institute, Switzerland [19]. The letters F, B, U, and D denote the forward, backward, up, and down detectors, respectively. The single-crystalline sample (blue rectangle) has its $c$ axis aligned along the incoming muon beam (red arrow). The initial muon spin polarization $P_{\mu }$ was rotated by ${45}^{\mathrm{o}}$ within the vertical ($y-z$) plane. (b) $\mathrm{Ca}\mathrm{K}{\left({\mathrm{Fe}}_{0.949}{\mathrm{Ni}}_{0.051}\right)}_4{\mathrm{As}}_4$ sample sandwiched between Ag (degrader and stopper) sheets.

Figure 3

(a) WTF-$\mu \mathrm{SR}$ time spectra ($B_{\mathrm{ex}}=3$ mT) of $\mathrm{Ca}\mathrm{K}{\left({\mathrm{Fe}}_{0.949}{\mathrm{Ni}}_{0.051}\right)}_4{\mathrm{As}}_4$ measured on up/down set of positron counters below ($T\simeq 1.5$ K) and above ($T=70$ K) the magnetic transition temperature ($T_N\simeq 50$ K). The solid lines are fits of Eq. (3) to the data (see text for details). (b) The same as (a) but for backward/forward set of detectors.

Figure 4

Temperature dependencies of the initial asymmetry $A\left(0\right)$ for the up/down and backward/forward set of positron counters. The solid lines are fits of Eq. (4) to $A\left(0\right)$ vs $T$ data. See text for details.

Figure 5

(a) ZF-$\mu \mathrm{SR}$ time spectra of $\mathrm{Ca}\mathrm{K}{\left({\mathrm{Fe}}_{0.949}{\mathrm{Ni}}_{0.051}\right)}_4{\mathrm{As}}_4$ measured on up/down set of positron counters. The experiment probes the time evolution of $P^{\perp c}$ component of the muon spin polarization. In order to improve statistics, the data sets collected at temperatures ranging from 1.5 up to 5 K are combined together. The solid line is the fit of Eq. (1) with the sample part described by Eq. (5) to the data. The dashed line is the time evolution of the background component. (b)–(d) Contributions of components Nos. 1, 2, and 3, respectively. (e) The same as in (a), but for backward/forward set of positron counters. In this experiment the time evolution of $P^{\parallel c}$ component of the muon spin polarization is probed. (f)–(h) Represent contributions of the first, second, and third components, respectively.

Figure 6

Temperature dependence of the internal field $B_{\mathrm{int},1}$. The solid line is the fit of Eq. (6) to the data with the parameters $T_{\mathrm{N}}=50.2\left(3\right)$ K, $\alpha =2.52\left(9\right), \beta =0.35\left(2\right)$, and $B_{\mathrm{int}}\left(0\right)=179\left(1\right)$ mT.

Figure 7

The magnetic structure of $\mathrm{Ca}\mathrm{K}{\left({\mathrm{Fe}}_{1-x}{\mathrm{Ni}}_x\right)}_4{\mathrm{As}}_4$ which is consistent with the results of ZF-$\mu \mathrm{SR}$ measurements. Only the magnetic (Fe) ions are shown. Arrows represent the ordered Fe moments. Diamond symbols correspond to different muon stopping sites. The magnetic unit cell ($P4/mbm$ group representation) is doubled and ${45}^{\mathrm{o}}$ rotated in the $ab$ plane with respect to the crystallographic one ($P4/mmm$ group representation). See Appendix pp1 for details. The structure is visualized by using vesta package [18].

Figure 8

Temperature dependence of the magnetic order parameter $m_{\mathrm{Fe}}$ of $\mathrm{Ca}\mathrm{K}{\left({\mathrm{Fe}}_{0.949}{\mathrm{Ni}}_{0.051}\right)}_4{\mathrm{As}}_4$. The solid line is the analysis by using the approach of Machida [14, 17]. The inset shows extension of the low-temperature part. The position of the superconducting transition temperature $T_{\mathrm{c}}$ is marked by the arrow.

Figure 9

Dependence of the normalized magnetic $m=M\left(0\right)/M_0$ vs the superconducting order parameter $\delta =\mathrm{\Delta }\left(0\right)/M_0$ obtained within the framework of Machida approach [17] [$M_0$ is the zero-temperature value of the magnetic order parameter in absence of superconductivity, while $M\left(0\right)=M(T=0)$ and $\mathrm{\Delta }\left(0\right)=\mathrm{\Delta }(T=0)$ are $T=0$ values of the magnetic and the superconducting order parameters, respectively]. The closed and open stars are experimental points obtained within present studies and by means of Mössbauer spectroscopy and neutron scattering experiments in $\mathrm{Ca}\mathrm{K}{\left({\mathrm{Fe}}_{1-x}{\mathrm{Ni}}_x\right)}_4{\mathrm{As}}_4$ (1144 family of Fe-SC's) [14, 15]. The open squares and closed circles are the experimental data for samples belonging to 11 [42, 43] and 122 families of Fe-SC's [30, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57], respectively.

Figure 10

The crystal structure of $\mathrm{Ca}\mathrm{K}{\mathrm{Fe}}_4{\mathrm{As}}_4$ and the muon stopping sites $\mu 1, \mu 2$, and $\mu 3$ within the tetragonal subgroup $P4/mbm$ (N127) of the space group $P4/mmm$ (N123). The $P4/mbm$ subgroup has the same origin as the parent group $P4/mmm$ and the basis $(a,b,c)$ rotated in the $ab$ plane on ${45}^{\mathrm{o}}$ compared to the basis $(a^{\prime },b^{\prime },c^{\prime })$ of the $P4/mmm$. Atoms and muon stopping sites are at the following positions: Ca, $2a$ (0,0,0); K, $2c$ (0,0.5,0.5); As1, $4e$ (0,0,0.35408); As2, $4f$ (0,0.5,0.12310); Fe, $8k$ (0.25, 0.75, 0.76800); $\mu 1$–$4f$ (0,0.5, 0.301); $\mu 2$–$4e$ (0,0, 0.186); and $\mu 3$–$2b$ (0,0,0.5). The crystal structure is visualized by using the vesta package [18].

Figure 11

(a) $m$ and (b) $\delta$ as a function of the parameter $R=T_{\mathrm{c}0}/T_{\mathrm{s}0}={\mathrm{\Delta }}_0/M_0$ corresponding to solutions of the system of Eqs. (B3) and (B5) for $n_1=0.3$. The blue (red) branches represent solutions for $m$ increasing (decreasing) as a function of $R$ (see text for details). (c) The $R$ dependence of the quantity ${\delta }^2+n_1{m}^2$ entering the condensation energy at $T=0$ [Eq. (B9)].

Figure 12

The same as in Fig. 11 but for $n_1=0.05$.

Figure 13

Solutions of Eqs. (B3) and (B5). Points ($m,\delta$) on the red/blue curve are solutions of (B3). The black curves are contours of constant $R$'s calculated with the help of Eq. (B5) [or Eq. (B6)] for $n_1=0.3$. The points, where two curves cross, correspond to solutions of the system of Eqs. (B3) and (B5).

Figure 14

The same as in Fig. 13, but for $n_1=0.05$.