Phase diagram of Ba${}_{1-x}$K${}_x$Fe${}_2$As${}_2$

We report the results of a systematic investigation of the phase diagram of the iron-based superconductor Ba${}_{1-x}$K${}_x$Fe${}_2$As${}_2$ from $x$ $=$ 0 to $x$ $=$ 1.0 using high-resolution neutron and x-ray diffraction and magnetization measurements. The polycrystalline samples were prepared with an estimated compositional variation of $\Delta x$ $\lesssim$ 0.01, allowing a more precise estimate of the phase boundaries than reported so far. At room temperature, Ba${}_{1-x}$K${}_x$Fe${}_2$As${}_2$ crystallizes in a tetragonal structure with the space group symmetry of $I$4/mmm, but at low doping, the samples undergo a coincident first-order structural and magnetic phase transition to an orthorhombic (O) structure with the space group Fmmm and a striped antiferromagnet (AF) with the space group $F_{\mathrm{c}}mm$′$m$′. The transition temperature falls from a maximum of 139 K in the undoped compound to 0 K at $x$ $=$ 0.252, with a critical exponent as a function of doping of 0.25(2) and 0.12(1) for the structural and magnetic order parameters, respectively. The onset of superconductivity occurs at a critical concentration of $x$ $=$ 0.130(3), and the superconducting transition temperature grows linearly with $x$ until it crosses the AF/O phase boundary. Below this concentration, there is microscopic phase coexistence of the AF/O and superconducting order parameters, although a slight suppression of the AF/O order is evidence that the phases are competing. At higher doping, superconductivity has a maximum $T_{\mathrm{c}}$ of 38 K at $x$ $=$ 0.4 that falls to 3 K at $x$ $=$ 1.0. We discuss reasons for the suppression of the spin density wave order and the electron–hole asymmetry in the phase diagram.

Figure 1

Structure of BaFe${}_2$As${}_2$, which crystallizes in a tetragonal ThCr${}_2$Si${}_2$-type structure with the space group symmetry of $I$4/mmm. Potassium substitutes onto the barium sites.

Figure 2

Variation of lattice constants $a$–$c$ with temperature for $x$ $=$ 0, 0.1, 0.21, and 0.3. The lattice constants, $a$ and $b$, in the orthorhombic phase are divided by $\sqrt{2}$.

Figure 3

Temperature dependence of the (110) peak in the vicinity of structural transition temperature $T_{\mathrm{s}}$. The bold curve shows the peak at the estimated $T_{\mathrm{s}}$. For 0 ≤ $x$ ≤ 0.24, the other curves show the peak at intervals of 2 K around $T_{\mathrm{s}}$. For $x$ $=$ 0.3, the peak is shown in 20 K intervals between 1.7 and 120 K.

Figure 4

Variation of lattice constants and volume in Ba${}_{1-x}$K${}_x$Fe${}_2$As${}_2$ with $x$ at 1.7 K. Solid lines are guides for the eye. The lattice constants $a$ and $b$ and the volume in the orthorhombic phase are divided by $\sqrt{2}$ and 2, respectively, from those for the Fmmm space group.

Figure 5

Variation of Fe-Fe and Fe-As bond lengths with $x$ at 1.7 K and with temperature for different K substitutions. Blue triangles represent the Fe-As bonds. Black squares and red circles represent the Fe-Fe bond lengths merging at $T_{\mathrm{s}}$. Solid lines are guides for the eye.

Figure 6

Variation of As-Fe-As bond angles with $x$ at 1.7 K and with temperature for different K substitutions. The top panel shows ${\alpha }_1$, ${\alpha }_2^{\prime }$, and ${\alpha }_2^{\prime \prime }$ in an orthorhombic setting. Blue circles represent ${\alpha }_{1;}$ red triangles and black squares represent ${\alpha }_2^{\prime }$ and ${\alpha }_2^{\prime \prime }$ merging into one ${\alpha }_2$ at $T_{\mathrm{s}}$. Solid lines are guides for the eye.

Figure 7

Neutron diffraction at 1.7 K with the magnetic Bragg peaks at a $d$-spacing of 2.45 and 3.43 $\AA$ indicated by the arrows. They are absent above $T_{\mathrm{N}}$ for $x$ $=$ 0, 0.1, 0.21, 0.22, and 0.25. At $x$ $=$ 0.3, no magnetic peaks are observed.

Figure 8

SQUID magnetization measurements for $x$ $=$ 0, 0.125, 0.15, 0.175, 0.2, and 0.24 in a 2-kG applied magnetic field. Insets are the first derivatives of the magnetization curves d$M$/d$T$ (10${}^{-5}$) used to determine the Néel temperatures given by the arrows. For $x$ > 0.2, the magnetization anomaly at $T_{\mathrm{N}}$ is too weak to be detected.

Figure 9

Dependence of the neutron diffraction intensity for Ba${}_{1-x}$K${}_x$Fe${}_2$As${}_2$ at 1.7 K with $x$. The magnetic Bragg peaks are shown by the arrows. The solid lines represent the calculated intensity of the Rietveld refinement.

Figure 10

Dependence of the square of the magnetic moment ${\mu }^2$ (red stars) and the orthorhombic order parameter $\delta$ $=$ ($a$ $-$ $b$)/($a$ $+$ $b$) (blue circles) at 1.7 K on the potassium concentration $x$. The inset shows a comparison of $T_{\mathrm{N}}$ and ${\mu }^2$ vs $x$, showing that $T_{\mathrm{N}}$ ∝ ${\mu }^2$. Solid lines in the main panel and the inset are a fit to (1 $-$ $x$/$x_{\mathrm{c}}$)${}^{2\beta }$, with $\beta$ $=$ 0.125.

Figure 11

SQUID magnetization measurements (zero field cooled) in the 0.1-G magnetic field for (a) $x$ $=$ 0.15 (solid squares), 0.175 (open squares), 0.21 (solid circles), 0.22 (open circles), 0.25 (solid triangles), and 0.3 (open triangles) and (b) $x$ $=$ 0.5 (solid triangles), 0.7 (solid circles), and 0.9 (solid squares), showing well-defined superconducting transitions. Magnetization values are normalized to the mass of the samples. (c) Superconducting transition temperatures (onset $T_{\mathrm{c}}$) of the underdoped compounds. Solid line represents the linear regression showing that the critical concentration for superconductivity is 0.130(3).

Figure 12

Phase diagram of Ba${}_{1-x}$K${}_x$Fe${}_2$As${}_2$ with the superconducting critical temperatures $T_{\mathrm{c}}$ (circles), the Néel temperatures $T_{\mathrm{N}}$ (stars), and the structural transition temperatures $T_{\mathrm{s}}$ (squares).

Figure 13

(a) Magnetic and structural phase diagram of electron-doped Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ and hole-doped Ba${}_{1-x}$K${}_x$Fe${}_2$As${}_2$ with $T_{\mathrm{c}}$ (squares), $T_{\mathrm{N}}$ (stars), and $T_{\mathrm{s}}$ (circles). The $x$-axis is normalized to the charge carrier per iron atom. Data for the electron-doped side where the transition temperatures are represented by open symbols are taken from Ref. 50. The error bars for $T_{\mathrm{N}}$ and $T_{\mathrm{s}}$ values in the hole-doped side are within the symbols. The dashed line enveloping the superconducting dome represents the Lindhard function taken from Ref. 33. (b) Charge carrier dependence of the As-Fe-As bond angles for both electron and hole doping. Solid triangles represent the results of our neutron diffraction study at 1.7 K for the hole-doped Ba${}_{1-x}$K${}_x$Fe${}_2$As${}_2$. At this temperature, one of the As-Fe-As angles splits due to orthorhombic distortion below $x$ $=$ 0.3. Therefore, we took the average of these two splitting angles. The As-Fe-As bond angle data for the electron doped side are taken from Ref. 51. Solid lines are visual guides.