Thermodynamic phase diagram, phase competition, and uniaxial pressure effects in BaFe${}_2$(As${}_{1-x}$P${}_x$)${}_2$ studied by thermal expansion

High-resolution thermal-expansion and specific-heat data of isovalently substituted single-crystalline BaFe${}_2$(As${}_{1-x}$P${}_x$)${}_2$ ($0\le x\le 0.33$, $x=1$) are presented. We show that crystals can be detwinned in situ in the capacitance dilatometer, allowing a study of all three independent crystallographic directions. From the thermal-expansion data, we determine the phase diagram via a thermodynamic probe, study the coupling of the spin-density wave (SDW) and superconducting order parameters, and determine various pressure dependencies of the normal and superconducting states. Our results show that in the underdoped region, superconductivity and SDW order coexist and compete with each other. The resulting phase diagram, however, exhibits a smaller coexistence region of SDW and superconductivity with a steeper rise of $T_c$ on the underdoped side than in, e.g., Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$. On the overdoped side, where there is no sign of SDW order, the lattice parameters respond to superconductivity in much the same way as to the SDW on the underdoped side, which demonstrates the intimate connection between both kinds of order. Using thermodynamic relations, the uniaxial pressure derivatives of the superconducting critical temperature and the electronic Sommerfeld coefficient are determined from our thermal-expansion data together with the specific-heat data. We find that uniaxial pressure is proportional to P substitution and that the electronic density of states has a maximum at optimal doping. Overall, the coupling of the SDW and superconducting order to the lattice parameters of BaFe${}_2$(As${}_{1-x}$P${}_x$)${}_2$ is found to be qualitatively very similar to that of the well-studied, supposedly electron-doped Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ system.

Figure 1

Uniaxial thermal-expansion coefficients ${\alpha }_i$ as a function of temperature $T$ for (a) twinned in-plane and (b) detwinned measurements (along the $b$ axis) and (c) along the $c$ axis for various substitution levels as indicated in panel (a). Arrows indicate examples of the temperature of the magnetostructural transition $T_{\mathrm{sm}}$ and $T_c^{\mathrm{on}}$. (d) Magnified view of the low-temperature region of (a). (e) and (f) show a comparison of twinned, detwinned, and $c$-axis data around the magnetostructural transition temperature for $x=0.18$ and $0.25$, respectively.

Figure 2

The superconducting transition of samples with $x=0.27$ (underdoped, left column) and $x=0.33$ (overdoped, right column) seen by different probes. Magnetization [panels (a) and (b)], specific heat [panels (c) and (d)], and thermal expansion [panels (e) and (f)] all show a broad superconducting transition with a sharp onset for the underdoped sample and more standard, sharp anomalies for the overdoped sample. The lines in panels (c)–(f) represent idealized second-order transitions.

Figure 3

Thermodynamic phase diagrams of P-Ba122 (lower axis) compiled from the present thermal-expansion data and of Co-Ba122 (upper axis) scaled so that the maximum $T_c$'s coincide. Data for Co-Ba122 were obtained from specific-heat (Ref. 29) and thermal-expansion (Ref. 9) measurements. Also added are previously published resistivity data (Ref. 5) on the P-Ba122 system (onset of superconducting transition $T_c^{\mathrm{on}}$ and temperature where zero resistivity is achieved $T_c^{\mathrm{zero}}$).

Figure 4

Relative change of sample length of the BaFe${}_2$(As${}_{0.75}$P${}_{0.25}$)${}_2$ sample, measured in the twinned (in-plane average, black line) and detwinned ($b$ axis, red line) configuration and along the $c$ axis (blue line). The $a$-axis length (broken purple line) was estimated from the twinned and detwinned in-plane measurements and the volume change (green broken line) from the twinned in-plane and the $c$-axis measurements. The inset shows the difference of $a$ axis and $b$ axis. For this sample, $T_{\mathrm{sm}}=48.5$ K and $T_c^{\mathrm{mid}}=16$ K.

Figure 5

Relative change of the $b$-axis length (in-plane length for the overdoped samples) for various doping levels. Its shortening signals the magnetostructural transition, and its increase below $T_c$ for the superconducting samples demonstrates the competition between the orthorhombic phase and superconductivity. The upper inset shows the data for $x=0.33$ on an expanded vertical scale. The lower inset shows the doping evolution of $\Delta b_{\mathrm{tot}}$ as defined in the main panel. The reference lines for the determination of $\Delta b_{\mathrm{tot}}$ (dashed black lines) are a smooth extrapolation. The upper line is provided by the data for $x=1$ scaled with a constant factor.

Figure 6

(a) Relative change of the $c$-axis length for various doping levels. Panel (b) shows the data after subtraction of a background (the data for $x=1$ times an individual factor close to 1, see text and Ref. 34).

Figure 7

The electronic contribution to the uniaxial thermal-expansion coefficients divided by $T$, ${\alpha }_{\mathrm{el}}/T$, obtained from subtracting the data of the $x=1$ sample as a phonon background (a) in the twinned in-plane measurements, (b) along the orthorhombic $b$ axis (detwinned in-plane measurements), and (c) along the $c$ axis. Arrows and dashed lines indicate how values of the normal state ${\alpha }_i^{\mathrm{el}}/T(T=0)$, which are reported in Fig. 8, are extracted.

Figure 8

(a) Values of the electronic thermal expansivity ${\alpha }_{\mathrm{el},i}/T$ extrapolated to zero temperature. The right-hand scale shows the corresponding values of $d\gamma /d{p}_i$. Lines are a guide to the eye. (b) The Sommerfeld coefficient $\gamma$ obtained by integrating the smooth lines from (a) (black line) (see text for details). Full squares indicate directly measured values of $\gamma$ and open squares indicate values estimated from the measured specific-heat jump (Ref. 36) (see also Ref. 38).