Possible realization of an antiferromagnetic Griffiths phase in Ba(Fe${}_{1-x}$Mn${}_x$)${}_2$As${}_2$

We investigate magnetic ordering in metallic Ba(Fe${}_{1-x}$Mn${}_x$)${}_2$As${}_2$ and discuss the unusual magnetic phase, which was recently discovered for Mn concentrations $x>10$%. We argue that it can be understood as a Griffiths-type phase that forms above the quantum critical point associated with the suppression of the stripe-antiferromagnetic spin-density-wave (SDW) order in BaFe${}_2$As${}_2$ by the randomly introduced localized Mn moments acting as strong magnetic impurities. While the SDW transition at $x=0$, 2.5$\%$, and 5$\%$ remains equally sharp, in the $x=12$$\%$ sample we observe an abrupt smearing of the antiferromagnetic transition in temperature and a considerable suppression of the spin gap in the magnetic excitation spectrum. According to our muon-spin-relaxation, nuclear magnetic resonance and neutron-scattering data, antiferromagnetically ordered rare regions start forming in the $x=12$$\%$ sample significantly above the Néel temperature of the parent compound. Upon cooling, their volume grows continuously, leading to an increase in the magnetic Bragg intensity and to the gradual opening of a partial spin gap in the magnetic excitation spectrum. Using neutron Larmor diffraction, we also demonstrate that the magnetically ordered volume is characterized by a finite orthorhombic distortion, which could not be resolved in previous diffraction studies most probably due to its coexistence with the tetragonal phase and a microstrain-induced broadening of the Bragg reflections. We argue that Ba(Fe${}_{1-x}$Mn${}_x$)${}_2$As${}_2$ could represent an interesting model spin-glass system, in which localized magnetic moments are randomly embedded into a SDW metal with Fermi surface nesting.

Figure 1

Characterization of the magnetic transitions in Ba(Fe${}_{1-x}$Mn${}_x$)${}_2$As${}_2$. (a) Temperature dependence of the normalized in-plane resistivity $\rho \left(T\right)/\rho \left(300\mathrm{K}\right)$ for all samples used in the present study. (b) In-plane resistivity (smooth curve) and its temperature derivative (noisy curve) for the $x=12$$\%$ sample, exhibiting an inflection-point anomaly at $T^{*}\approx 105$ K. (c) Temperature dependence of the elastic neutron scattering intensity at the ${\left(\frac{1}{0.8pt2}0\frac{1}{0.8pt2}\right)}_{{\mathrm{Fe}}_1}$ magnetic Bragg peak position (monotonic curve) and its temperature derivative with a minimum at $T^{*}$.

Figure 2

Temperature dependence of the main ${}^{75}$As NMR line intensity (filled symbols) for samples with different Mn concentrations, normalized to the respective high-temperature saturation values. Empty circles show the volume fraction of the tetragonal phase in the $x=12$$\%$ sample (right vertical axis), as measured by neutron Larmor diffraction (see text).

Figure 3

Neutron Larmor-diffraction measurements of the orthorhombic splitting in Ba(Fe${}_{0.88}$Mn${}_{0.12}$)${}_2$As${}_2$. (a) Experimental data for different temperatures (indicated above each curve), fitted to a model containing a mixture of the orthorhombic (O) and tetragonal (T) phases (solid lines). The dashed line shows a failed fit of the $T=4$ K data assuming a single tetragonal phase.[50] For clarity, each data set is offset vertically by an increment of 0.2 units from the one below it. (b) Modeled diffraction profiles, corresponding to every temperature in (a), as they would look like in an x-ray diffraction experiment with infinitesimally small resolution. These models account for the experimentally determined orthorhombic splitting, ratio of the tetragonal and orthorhombic phase volumes, and the peak broadening due to the finite width of the microstrain distribution, as extracted from the fits in (a). (c) Temperature dependence of the orthorhombicity parameter $\varepsilon =(a-b)/(a+b)$ extracted from the same fits. The dashed line is a temperature-independent fit. The gray dotted line is the corresponding dependence for the parent BaFe${}_2$As${}_2$, reproduced from Ref. 77 for comparison.

Figure 4

(a) Temperature dependence of the momentum width $\Delta Q/Q$ measured on the ${\left(\frac{1}{0.8pt2}0\frac{7}{0.8pt2}\right)}_{{\mathrm{Fe}}_1}$ magnetic Bragg reflection. (b) The energy width measured on the ${\left(\frac{1}{0.8pt2}0\frac{1}{0.8pt2}\right)}_{{\mathrm{Fe}}_1}$ magnetic Bragg reflection vs temperature. Measurements above 160 K were unfeasible due to the dramatically reduced intensity of the signal.

Figure 5

Temperature dependence of the $c$-axis linear thermal expansion coefficient ${\alpha }_c/T$ for Ba(Fe${}_{0.88}$Mn${}_{0.12}$)${}_2$As${}_2$ as measured by polarized-neutron Larmor diffraction (circles). The gray line shows the analogous dependence for the pure BaFe${}_2$As${}_2$, reproduced from Ref. 85 for comparison.

Figure 6

Zero-field $\mu$SR data collected on the forward-backward pair of detectors at various temperatures (as indicated in the panels) on samples with four different Mn concentrations. The solid lines represent fits described in the text.

Figure 7

Fitting parameters for the zero-field $\mu$SR data. (a) Temperature dependencies of the muon oscillation frequencies. (b) The same for the muon depolarization rate. Arrows indicate transition temperatures. For the $x=12$$\%$ sample, depolarization rates for the exponentially decaying component of the $\mu$SR signal are additionally plotted with empty symbols. The lines are guides to the eyes. The inset shows the $T\to 0$ limit of the depolarization rate, which is a measure of the degree of magnetic disorder in the ground state of the system, as a function of Mn concentration. The line is a linear fit to these data.

Figure 8

(a) Time dependence of the transverse-field $\mu$SR asymmetry at three selected temperatures for the $x=12$$\%$ sample. Fitting results are shown with solid lines. The dashed line shows the respective fit for pure BaFe${}_2$As${}_2$ at $T=200$ K for comparison. (b) Temperature dependence of the PM volume fraction extracted from the transverse-field $\mu$SR data. (c) The phase diagram of the $x=12$$\%$ sample, summarizing the zero-field (squares) and transverse-field (circles) $\mu$SR data. The $T$ dependence of the AFM Bragg peak intensity from Fig. 1, rescaled to its maximum and minimum values, is shown by the dashed line. The plot shows volume fractions of the bulk ordered AFM phase (oscillating $\mu$SR signal in zero field), the CG phase (rapid exponential muon depolarization in zero field accompanied by a magnetic Bragg peak in neutron diffraction evidencing long-range magnetic correlations), the SG phase (muon depolarization in zero field without any long-range magnetic order), and the PM phase ($\mu$SR oscillations in the transverse field).

Figure 9

INS data on the Ba(Fe${}_{0.88}$Mn${}_{0.12}$)${}_2$As${}_2$ sample at the magnetic ordering wave vector ${\mathbf{Q}}_{\mathrm{AFM}}$. (a) Several representative unprocessed momentum scans, measured at $T=1.5$ K along the ${\left(\frac{1}{0.8pt2}K\frac{1}{0.8pt2}\right)}_{{\mathrm{Fe}}_1}$ reciprocal-space direction with $k_{\mathrm{f}}=2.662$ Å${}^{-1}$, centered at ${\mathbf{Q}}_{\mathrm{AFM}}$. (b) Color map of the low-energy INS intensity in the spin-gap region, compiled out of multiple low-temperature momentum scans such as those shown in (a). (c) The background-subtracted scattering intensity $S(\mathbf{Q},\omega )$ at ${\mathbf{Q}}_{\mathrm{AFM}}={\left(\frac{1}{0.8pt2}0L\right)}_{{\mathrm{Fe}}_1}$, with $L=\frac{1}{0.8pt2}$ (gray points) or $L=\frac{3}{0.8pt2}$ (all other points). The filled symbols were obtained from fits of the full momentum scans, such as those shown in (a), whereas empty symbols result from three-point scans. The data taken with $k_{\mathrm{f}}=2.662$ and 3.837 Å${}^{-1}$ are shown with circles and squares, respectively. Data sets measured with different experimental conditions have been rescaled to match each other in the overlapping energy window. The solid curve is a guide to the eyes. The corresponding energy dependence for the BaFe${}_2$As${}_2$ parent compound from Ref. 65 is shown for comparison as a dashed curve to emphasize the suppression of the spin gap by Mn substitution. (d) Evolution of the low-energy part of $S(\mathbf{Q},\omega )$ with temperature, demonstrating a partial spin gap at intermediate temperatures with a magnitude that decreases upon heating. (e) Temperature dependence of $S(\mathbf{Q},\omega )$ at various energies within the spin-gap region. (f) The same for the dynamic spin susceptibility ${\chi }^{\prime \prime }(\mathbf{Q},\omega )$ obtained from the data in (e) after Bose-factor normalization. The lines are guides to the eyes.

Figure 10

INS data acquired on the $x=12$$\%$ sample at the magnetic zone boundary ($L=2$). (a) Three unprocessed momentum scans, measured at $T=1.5$ K along the rocking trajectory in the ${\left(H0L\right)}_{{\mathrm{Fe}}_1}$ plane with $k_{\mathrm{f}}=2.662$ Å${}^{-1}$. The inset shows the scan trajectory in the $(H,L)$ plane. (b) The background-subtracted scattering intensity $S(\mathbf{Q},\omega )$ taken at $\mathbf{Q}={\left(\frac{1}{0.8pt2}02\right)}_{{\mathrm{Fe}}_1}$. The filled symbols were obtained from fits of the full momentum scans, such as those shown in (a), whereas empty symbols result from three-point scans. The data taken with $k_{\mathrm{f}}=2.662$ and 3.837 Å${}^{-1}$ are shown with circles and squares, respectively. The solid curve is a guide to the eyes. The corresponding energy dependence for the BaFe${}_2$As${}_2$ parent compound[65] is shown for comparison with the dashed curve.

Figure 11

Comparison of the elliptical cross sections of the low-temperature INS intensity in the $Q_x{Q}_y$-plane projection for different compounds, measured at a constant energy $\hslash \omega$, which is indicated above each panel. (a) Ba(Fe${}_{0.88}$Mn${}_{0.12}$)${}_2$As${}_2$ sample from the present work. (b) Parent BaFe${}_2$As${}_2$ compound from Ref. 105. (c) Electron-doped Ba(Fe${}_{0.85}$Co${}_{0.15}$)${}_2$As${}_2$ sample from Ref. 69. (d) Hole-doped Ba${}_{0.67}$K${}_{0.33}$Fe${}_2$As${}_2$ sample from Ref. 106. The white dotted lines in all panels mark Brillouin-zone boundaries corresponding to the conventional body-centered tetragonal unit cell. Note that both the orientation and the momentum scales in all panels are identical.

Figure 12

(a) and (b) $L$ dependence of the background-subtracted intensity in Ba(Fe${}_{0.88}$Mn${}_{0.12}$)${}_2$As${}_2$ in the low-temperature AFM state ($T=4$ K) and in the PM state ($T=130$ K), measured at an energy transfer of 2 and 8 meV, respectively. (c) Comparison of the low-temperature data sets ($T=4$ K) at several energies. (d) Schematic representation of the scattering function in the normal and ordered states, the latter consisting of two broadened steplike functions corresponding to the magnetic scattering intensity with out-of-plane ($S_{zz}$) and in-plane ($S_{xy}$) polarizations. (e) The $S_{xy}/S_{zz}$ ratio extracted from the fits in (c). The expected energy dependence of this ratio is schematically shown with the dotted line.

Figure 13

Schematic phase diagram of BFMA after Refs. 50,  51, and the present work. Composition of the samples used in this study is indicated by arrows. The SDW transition temperatures ($T_{\mathrm{N}}$ or ${\tilde{T}}_{\mathrm{N}}$), below which the whole volume of the samples remains fully magnetic, as determined by transverse-field $\mu$SR spectroscopy in Sec. 3c, are marked by circles. The diamond symbol marks the onset of the elastic neutron-scattering intensity at the AFM wave vector $T_{\mathrm{CG}}$, which we define at 5$\%$ of the magnetic Bragg peak's maximal intensity. It is associated with the formation of long-range magnetic correlations in the CG phase. The star symbol stands for $T^{*}$, defined by the position of the inflection point in the temperature dependence of the resistivity (Fig. 1) or by the 50$\%$ reduction in the oscillating component of the muon asymmetry in zero field [Fig. 8c].