Disentangling superconducting and magnetic orders in ${\mathrm{NaFe}}_{1-x}{\mathrm{Ni}}_x\mathrm{As}$ using muon spin rotation

Muon spin rotation and relaxation studies have been performed on a “111” family of iron-based superconductors, ${\mathrm{NaFe}}_{1-x}{\mathrm{Ni}}_x\mathrm{As}$, using single crystalline samples with Ni concentrations $x=0$, 0.4, 0.6, 1.0, 1.3, and 1.5%. Static magnetic order was characterized by obtaining the temperature and doping dependences of the local ordered magnetic moment size and the volume fraction of the magnetically ordered regions. For $x=0$ and 0.4%, a transition to a nearly-homogeneous long range magnetically ordered state is observed, while for $x\gtrsim 0.4\%$ magnetic order becomes more disordered and is completely suppressed for $x=1.5\%$. The magnetic volume fraction continuously decreases with increasing $x$. Development of superconductivity in the full volume is inferred from Meissner shielding results for $x\gtrsim 0.4\%$. The combination of magnetic and superconducting volumes implies that a spatially-overlapping coexistence of magnetism and superconductivity spans a large region of the $T-x$ phase diagram for ${\mathrm{NaFe}}_{1-x}{\mathrm{Ni}}_x\mathrm{As}$. A strong reduction of both the ordered moment size and the volume fraction is observed below the superconducting $T_{\mathrm{C}}$ for $x=0.6$, 1.0, and 1.3%, in contrast to other iron pnictides in which one of these two parameters exhibits a reduction below $T_{\mathrm{C}}$, but not both. The suppression of magnetic order is further enhanced with increased Ni doping, leading to a reentrant nonmagnetic state below $T_{\mathrm{C}}$ for $x=1.3\%$. The reentrant behavior indicates an interplay between antiferromagnetism and superconductivity involving competition for the same electrons. These observations are consistent with the sign-changing $s^{\pm }$ superconducting state, which is expected to appear on the verge of microscopic coexistence and phase separation with magnetism. We also present a universal linear relationship between the local ordered moment size and the antiferromagnetic ordering temperature $T_{\mathrm{N}}$ across a variety of iron-based superconductors. We argue that this linear relationship is consistent with an itinerant-electron approach, in which Fermi surface nesting drives antiferromagnetic ordering. In studies of superconducting properties, we find that the $T=0$ limit of superfluid density follows the linear trend observed in underdoped cuprates when plotted against $T_{\mathrm{C}}$. This paper also includes a detailed theoretical prediction of the muon stopping sites and provides comparisons with experimental results.

Figure 1

Magnetic characterization of ${\mathrm{NaFe}}_{1-x}{\mathrm{Ni}}_x\mathrm{As}$. (a) Temperature-dependent DC susceptibility measurements in a magnetic field of 5 Oe applied in the $ab$ plane. Full SC volume is obtained for $x\ge 0.4\%$. Susceptibility spectra for $x\ge 0.8\%$ are vertically offset for visual clarity. Black solid lines show how $T_{\mathrm{C}}$ (indicated by black arrows) was determined for each doping. (b) Phase diagram of ${\mathrm{NaFe}}_{1-x}{\mathrm{Ni}}_x\mathrm{As}$ illustrating temperature and doping dependences of various orders, with structural and magnetic transitions obtained from Ref. [18] and displayed as fully-colored symbols. Superconducting transition temperatures $T_{\mathrm{C}}$ were determined from magnetic susceptibility measurements shown in (a). Black arrows indicate the doping concentrations measured by $\mu \mathrm{SR}$ in our present investigation. (c) Collinear AFM spin structure of the undoped compound NaFeAs [19, 20]. Only Fe atoms (green) are shown for visual clarity. Dashed lines indicate the boundaries of a single unit cell of the crystal structure.

Figure 2

ZF-$\mu \mathrm{SR}$ spectra on ${\mathrm{NaFe}}_{1-x}{\mathrm{Ni}}_x\mathrm{As}$. (a),(b) Muon spin polarization in zero field for ${\mathrm{NaFe}}_{1-x}{\mathrm{Ni}}_x\mathrm{As}$ for $x=0$ and 0.4%, respectively. (c),(d) Zoomed-in view of the first 1.5 microseconds of the spectra shown in (a) and (b). (e)–(h) Time spectra for the $x=0.6$, 1.0, 1.3, and 1.5% compound in zero field, respectively. Solid lines in all panels are fits of the data to the ZF-$\mu \mathrm{SR}$ model in (1) for each temperature. Additional details on the $\mu \mathrm{SR}$ time spectra and experimental geometry can be found in Refs. [25, 26, 27].

Figure 3

Muon precession frequencies in ${\mathrm{NaFe}}_{1-x}{\mathrm{Ni}}_x\mathrm{As}$. (a),(b) Precession frequencies ${\nu }_j$ from the model used in (1) on the $x=0$ and 0.4% compounds, respectively. Solid lines are power-law fits to the data. (c) Simulation results from dipolar field calculations on lightly-doped ${\mathrm{NaFe}}_{1-x}{\mathrm{Ni}}_x\mathrm{As}$ using the three muon stopping sites obtained from DFT calculations.

Figure 4

Summary of ZF-$\mu \mathrm{SR}$ fit results on ${\mathrm{NaFe}}_{1-x}{\mathrm{Ni}}_x\mathrm{As}$. (a) Magnetically ordered volume fraction $\left(F\right)$ as a function of temperature and Ni concentration. (b) Static magnetic order parameter $\left(\mathcal{M}\right)$ as a function of temperature and Ni concentration. Arrows in the figures denote $T_{\mathrm{C}}$ for the various samples.

Figure 5

Magnetic and SC volume fractions in ${\mathrm{NaFe}}_{1-x}{\mathrm{Ni}}_x\mathrm{As}$. (a) Doping and temperature evolution of the magnetic fraction $V_{\mathrm{Mag}}=F$ from ZF-$\mu \mathrm{SR}$. Orange diamonds indicate $T_{\mathrm{N}}$ and the dashed curve is a guide to the eye. Observe the bend in the curve near $x=0.013$ indicating a reentrant non-AFM phase. (b) Doping and temperature evolution of the SC volume fraction from magnetic susceptibility measurements presented in Fig. 1. Green circles indicate $T_{\mathrm{C}}$ and the dashed curve is a guide to the eye. (c) Summary of magnetic and SC volume fractions. $V_{SC}$ is the SC volume fraction, $V_{\mathrm{Mag},\mathrm{Max}}$ is the maximum value of $F$ for each doping, and $V_{\mathrm{Mag}}(T\ll T_{\mathrm{C}})$ is $F$ at temperatures much less than $T_{\mathrm{C}}$.

Figure 6

Comparison of $\mu \mathrm{SR}$ and elastic neutron scattering measurements [18] for the (a) $x=1.0\%$ and (b) $x=1.3\%$ compounds. The magnetic strength ${\mathcal{M}}^{2}F$ from $\mu \mathrm{SR}$ measurements is plotted in red and the Bragg peak intensity from neutron scattering is shown in blue.

Figure 7

Correlation between low-temperature muon precession frequency $\nu (T\to 0)$ and the magnetic ordering temperature $T_{\mathrm{N}}$ for various Fe-HTS. The black dashed line is a linear least-squares model of the data. For systems with more than one precession frequency, the maximum frequency is taken. Circle symbols indicate the “111” family of Fe-HTS: NaFeAs [68], ${\mathrm{NaFe}}_{1-x}{\mathrm{Co}}_x\mathrm{As}$ [68], and ${\mathrm{NaFe}}_{0.996}{\mathrm{Ni}}_{0.004}\mathrm{As}$. Diamond symbols indicate the “122” family of Fe-HTS: ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ [65], ${\mathrm{Ba}}_{0.5}{\mathrm{K}}_{0.5}{\mathrm{Fe}}_2{\mathrm{As}}_2$ [69], ${\mathrm{BaFe}}_{2-x}{\mathrm{Co}}_x{\mathrm{As}}_2$ [70], ${\mathrm{CaFe}}_2{\mathrm{As}}_2$ [16], ${\mathrm{EuFe}}_2{\mathrm{As}}_{2-x}{\mathrm{P}}_x$ [71], ${\mathrm{SrFe}}_2{\mathrm{As}}_2$ [16], ${\mathrm{Sr}}_{0.5}{\mathrm{Na}}_{0.5}{\mathrm{Fe}}_2{\mathrm{As}}_2$ [16]. Upwards-pointing triangle symbols indicate the “1111” family of Fe-HTS: ${\mathrm{CaO}}_{0.94}{\mathrm{F}}_{0.06}\mathrm{FeAs}$ [65], LaOFeAs [17], ${\mathrm{LaO}}_{0.97}{\mathrm{F}}_{0.03}\mathrm{FeAs}$ [72]. Downwards-pointing triangle symbols indicate the “11” family of Fe-HTS: ${\mathrm{FeSe}}_{1-x}$ [32] under pressure.

Figure 8

Correlation between the low-temperature ordered moment size on the Fe atom ${\mu }_{\mathrm{Fe}}$ from $\mu \mathrm{SR}$ measurements and the magnetic ordering temperature $T_{\mathrm{N}}$ for various Fe-HTS. The black dashed line is a linear least-squares model of the data. The “111,” “122,” “1111,” and “11” families of Fe-HTS are represented by circle, diamond, upwards-pointing triangle, and downwards-pointing triangle symbols, respectively. See the caption for Fig. 7 for references to data points.

Figure 9

Theoretical calculations of the ordered moment $M$ at $T=0$ versus the AFM critical temperature $T_{\mathrm{N}}$. The ellipticity of the electron band, given by ${\delta }_2$, is fixed for each curve, whereas the parameter ${\delta }_0$, corresponding to doping, is varied continuously. $M_0$ is the AFM moment $M$ at $T=0$ when the hole and electron Fermi pockets are perfectly nested.

Figure 10

TF-$\mu \mathrm{SR}$ polarization on field-cooled $x=1.3\%$ in an applied field of 300 G. Time spectra in the $x=1.3\%$ system with an applied transverse field at $T=40$ K (PM) and $T=2$ K (SC).

Figure 11

TF-$\mu \mathrm{SR}$ results on field cooled $x=1.3\%$ in an applied field of 300 G. (a) Temperature dependence of the field shift from 40 K (normal state). (b),(c) Temperature dependences of the relaxation rates ascribed to the SC and magnetic orders, respectively. The solid line in (b) represents an isotropic two-band SC model fit to the temperature evolution of ${\sigma }_{\mathrm{SC}}$. The peak in ${\lambda }_{\mathrm{Mag}}$ in (c) is close to $T_{\mathrm{C}}$, indicating that the onset of SC order affects the dilute electronic moment distribution.

Figure 12

Uemura plot for hole and electron doped Fe-HTS (see Ref. [99] and references therein). The linear relation observed for underdoped cuprates is shown as a blue colored solid line for hole doping [100, 101] and as a red colored dashed line for electron doped systems [102]. The points for conventional BCS superconductors are also shown. The orange star marker shows the data point for ${\mathrm{NaFe}}_{1-x}{\mathrm{Ni}}_x\mathrm{As}$ obtained in this work.

Figure 13

A toy model potential $V(0,0,z)$ (solid line) together with the ground state energy, $E_0=0.17$ eV from solving the Schrödinger equation for a muon in a potential of the form $V(x,y,z)=\frac{1}{2}a(x^2+y^2)+\frac{1}{2}(b{z}^4-c{z}^2+dz)+f$ with $a=2.44\times {10}^{-3}, b=5.04\times {10}^{-4}, c=3\times {10}^{-3}, d=2.85\times {10}^{-3}$, and $f=4.79\times {10}^{-3}$, all in Hartree atomic units. The green dots show the minimum energy profile map from the DBO for the symmetric site D to site E. These simulations imply that the muon is likely delocalized over the two sites in cluster II (sites D and E).

Figure 14

Summary of numerical results for the ordered moment $M$ and $T_{\mathrm{N}}$. (a) Ordered moment $M$ at $T=0$ and (b) $T_{\mathrm{N}}$ as functions of the area difference parameter ${\delta }_0$ for fixed values of the ellipticity parameter ${\delta }_2$. Results are shown in the regime where the transition is second order. Colors represent different values of ${\delta }_2/M_0$ listed in Fig. 9.