Phase diagram of Eu magnetic ordering in Sn-flux-grown $\mathbf{Eu}{\left({\mathbf{Fe}}_{1-x}{\mathbf{Co}}_x\right)}_2{\mathbf{As}}_2$ single crystals

The magnetic ground state of the ${\mathrm{Eu}}^{2+}$ moments in a series of $\mathrm{Eu}{\left({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\right)}_2{\mathrm{As}}_2$ single crystals grown from the Sn flux has been investigated in detail by neutron diffraction measurements. Combined with the results from the macroscopic properties (resistivity, magnetic susceptibility and specific heat) measurements, a phase diagram describing how the Eu magnetic order evolves with Co doping in $\mathrm{Eu}{\left({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\right)}_2{\mathrm{As}}_2$ is established. The ground-state magnetic structure of the ${\mathrm{Eu}}^{2+}$ spins is found to develop from the A-type antiferromagnetic (AFM) order in the parent compound, via the A-type canted AFM structure with some net ferromagnetic (FM) moment component along the crystallographic $\mathit{c}$ direction at intermediate Co doping levels, finally to the pure FM order at relatively high Co doping levels. The ordering temperature of Eu declines linearly at first, reaches the minimum value of 16.5(2) K around $\mathit{x}=0.100\left(4\right)$, and then reverses upwards with further Co doping. The doping-induced modification of the indirect Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between the ${\mathrm{Eu}}^{2+}$ moments, which is mediated by the conduction $\mathit{d}$ electrons on the (Fe,Co)As layers, as well as the change of the strength of the direct interaction between the ${\mathrm{Eu}}^{2+}$ and ${\mathrm{Fe}}^{2+}$ moments, might be responsible for the change of the magnetic ground state and the ordering temperature of the Eu sublattice. In addition, for $\mathrm{Eu}{\left({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\right)}_2{\mathrm{As}}_2$ single crystals with 0.$10\le \mathit{x}\le 0.18$, strong ferromagnetism from the Eu sublattice is well developed in the superconducting state, where a spontaneous vortex state is expected to account for the compromise between the two competing phenomena.

Figure 1

Temperature dependencies of the normalized in-plane resistivity $\left({\rho }_{ab}\right)$ of $\mathrm{Eu}{\left({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\right)}_2{\mathrm{As}}_2$ single crystals with $\mathit{x}=0.014\left(2\right)$, 0.027(2), 0.053(2), 0.075(2), 0.100(4), and 0.180(5). The inset is an enlarged illustration of the resistivity below 20 K. ${\mathit{T}}_{\mathit{s}}$ (downward arrows) and ${\mathit{T}}_{\mathit{SC}}$ (upward arrows) mark the structural phase transition and the superconducting transition, respectively. The right inset shows the absolute resistivity values of different crystals at 300 K, in which the error bars are given by the uncertainties in the estimation of the geometric factors of the resistivity measurements.

Figure 2

Temperature dependencies of the dc magnetic susceptibility $\left({\chi }_c\right)$ of $\mathrm{Eu}{\left({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\right)}_2{\mathrm{As}}_2$ single crystals below 25 K with $\mathit{x}=0.014\left(2\right)$ (a), 0.027(2) (b), 0.053(2) (c), 0.075(2) (d), 0.100(4) (e), and 0.180(5) (f) [22], measured in an applied field of 10 Oe [(a) to (e)] or 30 Oe (f) parallel to the $\mathit{c}$ direction in ZFC and FC processes, respectively. The dotted lines denote the ordering temperature of the ${\mathrm{Eu}}^{2+}$ moments $\left({\mathit{T}}_{\mathit{Eu}}\right)$. The upward arrow in (e) and (f) marks the superconducting transition temperature $\left({\mathit{T}}_{\mathit{SC}}\right)$ of the samples with $\mathit{x}=0.100\left(4\right)$ and 0.180(5), respectively.

Figure 3

The low-temperature molar specific heat of $\mathrm{Eu}{\left({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\right)}_2{\mathrm{As}}_2$ single crystals as a function of the Co doping level. The inset shows the first derivative of the specific heat to the temperature $\left(-\mathit{dC}/\mathit{dT}\right)$ around the ordering temperature of the ${\mathrm{Eu}}^{2+}$ moments $\left({\mathit{T}}_{\mathit{Eu}}\right)$.

Figure 4

The ground-state magnetic structure of $\mathrm{Eu}{\left({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\right)}_2{\mathrm{As}}_2$ with $\mathit{x}\le 0.053$ (a), $\mathit{x}=0.075\left(2\right)$ (b), $\mathit{x}=0.100\left(4\right)$ (c), and $\mathit{x}=0.180\left(5\right)$ (d) [22], as determined by single-crystal neutron diffraction measurements. The ${\mathrm{Eu}}^{2+}$ spins in (b) and (c) are canted out of the $\mathit{ab}$ plane with angles of $23.8{\left(6\right)}^{\circ }$ (at 2 K) and $\sim {65}^{\circ }$ (at 4 K), respectively.

Figure 5

The summary of neutron diffraction measurements on $\mathrm{Eu}{\left({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\right)}_2{\mathrm{As}}_2$ single crystals with $\mathit{x}=0.014\left(2\right)$ (a)–(c), 0.027(2) (d)–(f), and 0.053(2) (g)–(h), suggesting the A-type AFM structure as the magnetic ground state of the ${\mathrm{Eu}}^{2+}$ spins. (a),(d),(g) The rocking-curve scans of the $(0,0,3)$ magnetic reflection performed on diffractometer HEiDi [(a) and (d)] or TriCS (g) at different temperatures for $\mathit{x}=0.014\left(2\right)$ (a), $\mathit{x}=0.027\left(2\right)$ (d), and $\mathit{x}=0.053\left(2\right)$ (g). The solid curves represent the fits using the Gaussian profiles. (b),(e) The rocking-curve scans of the $(2,0,0)$ nuclear reflection performed on diffractometer HEiDi at the base temperature and 25 K, for $\mathit{x}=0.014\left(2\right)$ (b) and $\mathit{x}=0.027\left(2\right)$ (e). (h) The contour map in the $(H,0,L)$ reciprocal plane for $\mathit{x}=0.053\left(2\right)$ at $T=3.5$ K obtained via polarized neutron diffraction at spectrometer DNS with the neutron polarization parallel to the scattering vector $\mathit{Q}$ $(\mathit{x}$ polarization). For $\mathit{x}$ polarization, the intensity in the spin-flip (SF) channel solely arises from the magnetic reflections [36]. (c),(f),(i) The temperature dependencies of the integrated intensity (at HEiDi and TriCS) or peak intensity (at DNS) of the $(0,0,3)$ magnetic reflection. The vertical dashed lines mark the antiferromagnetic ordering temperature $\left({\mathit{T}}_{\mathit{N}}\right)$ of the ${\mathrm{Eu}}^{2+}$ moments for each composition.

Figure 6

(a),(b) The rocking-curve scans of the $(0,0,3)$ and $(2,0,0)$ reflections, respectively, performed at diffractometer D23 at different temperatures for the $\mathrm{Eu}{\left({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\right)}_2{\mathrm{As}}_2$ single crystal with $\mathit{x}=0.075\left(2\right)$. The solid curves represent the fits using the Gaussian profiles. (c) The temperature dependencies of the integrated intensity of the $(0,0,3)$ and $(2,0,0)$ reflection, for $\mathit{x}=0.075\left(2\right)$. The vertical dashed lines mark the same onset temperature for the antiferromagnetic and ferromagnetic contribution $\left({\mathit{T}}_{\mathit{N}}={\mathit{T}}_{\mathit{C}}=17.0\left(2\right)\mathrm{K}\right)$. (d) The temperature dependencies of the ratio of the net magnetic scattering intensities between the $(2,0,0)$ and $(0,0,3)$ reflections, $\mathit{R}={\mathit{I}}_{\mathit{M}}(2,0,0)/I_M(0,0,3)$, and the canting angle of the ${\mathrm{Eu}}^{2+}$ moments. (e) Comparison between the observed intensities of the collected magnetic reflections at 2 K (black spheres), the fitted intensities using the A-type canted AFM structure (red squares), and the calculated intensities using the A-type AFM structure (blue diamonds) or the pure FM structure along the $\mathit{c}$ axis (green triangles). The observed intensities for $(1,1,1),(2,0,0)$, and $(3,1,1)$ reflections are the net magnetic contributions at 2 K after subtracting the nuclear contributions at 25 K.

Figure 7

(a) The contour map in the $(H,0,L)$ reciprocal plane for $\mathrm{Eu}{\left({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\right)}_2{\mathrm{As}}_2$ $\mathrm{[}\mathit{x}=0.100\left(4\right)]$ obtained at $T=3.5$ K via polarized neutron diffraction on spectrometer DNS with $\mathit{z}$ polarization. (b) The temperature dependencies of the SF channel (orange circles), NSF channel (green), and the total (blue) integrated intensities of the $(0,0,2)$ reflection. (c) The temperature dependence of the integrated intensity of the $(0,0,3)$ and $(2,0,0)$ reflections. The vertical dashed lines mark the same onset temperature for the antiferromagnetic and ferromagnetic contributions $\mathrm{[}{\mathit{T}}_{\mathit{N}}={\mathit{T}}_{\mathit{C}}=16.5\left(2\right)$ K]. (d) The temperature dependence of the ratio of the magnetic scattering intensities in the SF channel between the $(2,0,0)$ and $(0,0,3)$ reflections, $\mathit{R}={\mathit{I}}_{\mathit{ZSF}}(2,0,0)/I_{ZSF}(0,0,3)$. (e),(f) The rocking-curve scans of the $(2,0,2)$ reflection at 4.1 and 25 K in the SF and NSF channels, respectively. The solid curves represent the fits using the Gaussian profiles. (g) The temperature dependencies of the SF channel (orange circles), NSF channel (green), and the total (blue) integrated intensities of the $(2,0,2)$ reflection. (h) The ratio between the ferromagnetic contribution and the nuclear scattering for the $(2,0,2)$ reflection within the A-type canted AFM model (solid line) calculated as a function of the canting angle, from which the canting angle for $\mathit{x}=0.100\left(4\right)$ is determined to be $\sim {65}^{\circ }$ off the $\mathit{ab}$ plane.

Figure 8

The temperature dependencies of the order parameter associated with the SDW transition of Fe in $\mathrm{Eu}{\left({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\right)}_2{\mathrm{As}}_2$ single crystals with $\mathit{x}=0.014\left(2\right)$, 0.027(2), 0.075(2), and 0.180(5) [22]. The peak intensity of the $(1,0,3)$ or $(1,2,1)$ magnetic reflection, measured on DNS or TriCS, is plotted for $\mathit{x}=0.014\left(2\right)$ and 0.180(5), respectively. For $\mathit{x}=0.027\left(2\right)$ and 0.075(2), the integrated intensity of the $(1,0,3)$ reflection measured on HEiDi or D23 is shown instead [40]. The solid lines are guides to the eye.

Figure 9

The phase diagram of Sn-flux-grown $\mathrm{Eu}{\left({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\right)}_2{\mathrm{As}}_2$ single crystals, illustrating the evolution of the magnetic ground state of the ${\mathrm{Eu}}^{2+}$ spins with Co doping. ${\mathit{T}}_{\mathit{C}}\left(x\right)$ is assumed to vary linearly as ${\mathit{T}}_{\mathit{N}}\left(x\right)$ does. The ${\mathit{T}}_{\mathit{N}}$ value for the parent compound $\left(\mathit{x}=0\right)$ is obtained from the result of a previous neutron diffraction measurement on Sn-flux-grown ${\mathrm{EuFe}}_2{\mathrm{As}}_2$ single crystals [8]. ${\mathit{T}}_{\mathit{S}}$ and ${\mathit{T}}_{\mathit{SDW}}$ denote the structural phase transition and the SDW transtion of Fe, respectively. The dotted lines are linear fittings to ${\mathit{T}}_{\mathit{S}}$ and ${\mathit{T}}_{\mathit{SDW}}$. The right end part is faded because no neutron data are available for that regime.