Character of the structural and magnetic phase transitions in the parent and electron-doped BaFe${}_2$As${}_2$ compounds

We present a combined high-resolution x-ray diffraction and x-ray resonant magnetic scattering study of as-grown BaFe${}_2$As${}_2$. The structural and magnetic transitions must be described as a two-step process. At $T_S$ $=$ 134.5 K we observe the onset of a second-order structural transition from the high-temperature paramagnetic tetragonal structure to a paramagnetic orthorhombic phase, followed by a discontinuous step in the structural order parameter that is coincident with a first-order antiferromagnetic (AFM) transition at $T_N$ $=$ 133.75 K. These data, together with detailed high-resolution x-ray studies of the structural transition in lightly doped Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ and Ba(Fe${}_{1-x}$Rh${}_x$)${}_2$As${}_2$ compounds, show that the structural and AFM transitions do, in fact, occur at slightly different temperatures in the parent BaFe${}_2$As${}_2$ compound, and evolve toward split second-order transitions as the doping concentration is increased. We estimate the composition for the tricritical point for Co doping and employ a mean-field approach to show that our measurements can be explained by the inclusion of an anharmonic term in the elastic free energy and magnetoelastic coupling in the form of an emergent Ising-nematic degree of freedom.

Figure 1

(a) X-ray diffraction scans, measured using the laboratory source, along the [$\xi$ $\xi$ 0] direction through the position of the tetragonal (1 1 10)${}_T$ reflection for selected temperatures in the parent BaFe${}_2$As${}_2$ upon cooling. The lines present the fitted curves using a Lorentzian-squared line shape. The two-component fit to the broadened peak is illustrated for $T$ $=$ 134.25 K. The arrows denote the positions of peaks associated with Ort-AFM as discussed in the text. At this temperature, the integrated intensities of the Ort-AFM peaks are approximately $5\%$ of the Ort-PM diffraction peaks. (b) The orthorhombic distortion as a function of temperature upon cooling and warming determined from fits to the (1 1 10) Bragg peak.

Figure 2

(a) The measured (1 1 8)${}_T$ charge diffraction peak above the structural and magnetic transitions. (b) and (c) show the (1 1 8)${}_T$ charge peak and ($\frac{1}{2}\hspace{0.5em}\frac{1}{2}\hspace{0.5em}7$)${}_T$ magnetic peak positions at $T=130$ K, well below the transition region. (d) and (e) show the measured intensities at the (1 1 8)${}_T$ charge peak and ($\frac{1}{2}\hspace{0.5em}\frac{1}{2}\hspace{0.5em}7$)${}_T$ magnetic peak positions at $T=133.3$ K. The arrows in (c) and (e) indicate the calculated magnetic peak positions corresponding to each of the charge peaks in (b) and (d), respectively. The fitted values for the widths of the charge and magnetic peaks are the same.

Figure 3

(a) X-ray data, (b) resistance (black line) and its temperature derivative (blue line), (c) orthorhombic distortion, and (d) FWHM of the split (1 1 10)${}_T$ Bragg peaks measured for Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ with $x$ $=$ 0.018. In (c) the structural and magnetic transition temperatures are marked.

Figure 4

(a) X-ray data, (b) resistance (black line) and its temperature derivative (blue line), (c) orthorhombic distortion, and (d) FWHM of the split (1 1 10)${}_T$ Bragg peaks measured for Ba(Fe${}_{1-x}$Rh${}_x$)${}_2$As${}_2$ with $x$ $=$ 0.012. In (c) the structural and magnetic transition temperatures are marked.

Figure 5

(a) X-ray data, (b) resistance (black line) and its temperature derivative (blue line), (c) orthorhombic distortion, and (d) FWHM of the split (1 1 10)${}_T$ Bragg peaks measured for Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ with $x$ $=$ 0.047. In (c) the structural and magnetic transition temperatures are marked.

Figure 6

(a) X-ray data, (b) resistance (black line) and its temperature derivative (blue line), (c) orthorhombic distortion, and (d) FWHM of the split (1 1 10)${}_T$ Bragg peaks measured for Ba(Fe${}_{1-x}$Rh${}_x$)${}_2$As${}_2$ with $x$ $=$ 0.040. In (c) the structural and magnetic transition temperatures are marked.

Figure 7

Diagram showing the nature of the structural and magnetic phase transitions for Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ at $T_S$ and $T_N$, respectively. The thick line denotes a first-order transition whereas the thinner lines represent second-order transitions. The crosses denote values for $T_S$ and $T_N$ determined from our measurements. The open circle denotes the approximate position of a tricritical point as described in the text.

Figure 8

Jump of the orthorhombic order parameter $\Delta \delta \equiv {\delta }_{\mathrm{Ort}-\mathrm{AFM}}-{\delta }_{\mathrm{Ort}-\mathrm{PM}}$ across the first-order magnetic transition, as a function of $x$. The linear relationship $\Delta \delta \propto (x-x_{\mathrm{tri}})$ (dashed line) follows from the mean-field solution of Eq. (1).

Figure 9

The two magnetic ground states of the iron pnictides, characterized by stripes along two orthogonal directions, expressed in terms of two interpenetrating AFM sublattices with Néel vectors ${\mathbf{m}}_1$ and ${\mathbf{m}}_2$. Notice that the $(x,y)$ coordinate system used here refers to the unit cell with two Fe atoms and is rotated by ${45}^{°}$ with respect to the $(a,b)$ coordinate system relative to the single-Fe-atom unit cell.

Figure 10

Phase diagram of the system with anharmonic elastic terms. $T$ denotes temperature, $C_s$ is the bare shear modulus, and ${\chi }_0^{-1}$ is a magnetic energy scale. Thin (thick) lines refer to second-order (first-order) phase transitions, with the red (blue) lines denoting magnetic (structural) transitions; the simultaneous first-order transition line is denoted by the double line. We use the notation Ort${}^{\prime }$ to emphasize that the orthorhombic distortion jumps across the first-order magnetic transition. The orange dotted line signals the occurrence of a jump in both the magnetic and orthorhombic order parameters, without symmetry breaking. The open circle refers to the magnetic tricritical point, while the arrow indicates the value of $C_s$ for which we calculate the temperature dependence of both the magnetic and orthorhombic order parameters (see Fig. 11).

Figure 11

Magnetic ($m$, open symbols) and orthorhombic ($\delta$, filled symbols) order parameters as functions of temperature $T$ (in units of the structural transition temperature $T_S$) for the system indicated by the green arrow in the phase diagram of Fig. 10.