Heat capacity jump at $T_c$ and pressure derivatives of superconducting transition temperature in the Ba${}_{1-x}$Na${}_x$Fe${}_2$As${}_2$ ($0.1\le x\le 0.9$) series

We present the evolution of the initial (up to $\sim 10$ kbar) hydrostatic pressure dependencies of $T_c$ and of the ambient pressure, and the jump in the heat capacity associated with the superconducting transition as a function of Na doping in the Ba${}_{1-x}$Na${}_x$Fe${}_2$As${}_2$ family of iron-based superconductors. For Na concentrations $0.15\le x\le 0.9$, the jump in specific heat at $T_c$, $\Delta C_p{|}_{T_c}$, follows the $\Delta C_p\propto T^3$ (the so-called BNC scaling) found for most BaFe${}_2$As${}_2$-based superconductors. This finding suggests that, unlike the related Ba${}_{1-x}$K${}_x$Fe${}_2$As${}_2$ series, there is no significant modification of the superconducting state (e.g., change in superconducting gap symmetry) in the Ba${}_{1-x}$Na${}_x$Fe${}_2$As${}_2$ series over the whole studied Na concentration range. Pressure dependencies are nonmonotonic for $x=0.2$ and $0.24$. For other Na concentrations, $T_c$ decreases under pressure in an almost linear fashion. The anomalous behavior of the $x=0.2$ and $0.24$ samples under pressure is possibly due to the crossing of the phase boundaries of the narrow antiferromagnetic tetragonal phase, unique for the Ba${}_{1-x}$Na${}_x$Fe${}_2$As${}_2$ series, with the application of pressure. The negative sign of the pressure derivatives of $T_c$ across the whole superconducting dome (except for $x=0.2$) is a clear indication of the nonequivalence of substitution and pressure for the Ba${}_{1-x}$Na${}_x$Fe${}_2$As${}_2$ series.

Figure 1

Temperature-dependent heat capacity of Ba${}_{0.2}$Na${}_{0.8}$Fe${}_2$As${}_2$ near the superconducting transition plotted as $C_p/T$ vs $T$. Criteria for $T_c$ and $\Delta C_p{|}_{T_c}$ (isoentropic construct) are shown. Inset: $C_p\left(T\right)$ in a wider temperature range.

Figure 2

$\Delta C_p$ at the superconducting transition vs $T_c$ for the Ba${}_{1-x}$Na${}_x$Fe${}_2$As${}_2$ series, plotted together with literature data for various FeAs-based superconducting materials. The plot from [27] was updated to include published data for K${}_{1-x}$Na${}_x$Fe${}_2$As${}_2$ ($0\le x\le 0.31$), Ca${}_{1-x}$Na${}_x$Fe${}_2$As${}_2$, Ba${}_{1-x}$Na${}_x$Fe${}_2$As${}_2$ ($x=0.35,0.4$), and LaFeAs${}_{0.9}$F${}_{0.1}$ [17, 45, 46, 47, 48]. The line corresponds to $\Delta C_p\propto T_c^3$. Numbers near the symbols are Na concentrations $x$.

Figure 3

Example of temperature-dependent magnetization (zero-field-cooled, taken in 25 Oe applied magnetic field) of Ba${}_{0.7}$Na${}_{0.3}$Fe${}_2$As${}_2$ measured at 1.1, 2.9, 5.9, and 10.7 kbar. The arrow indicates increasing pressure. The onset criterion for $T_c$ used in this work is shown for $P=10.7$ kbar curve as an example. Superconducting transitions in Pb used as a pressure gauge are seen near 7 K.

Figure 4

Summary plot of the pressure dependence of $T_c$ for the Ba${}_{1-x}$Na${}_x$Fe${}_2$As${}_2$ ($0.15\le x\le 0.9$) series studied in this work. Dashed lines are linear fits to the data except for $x=0.2$ and $0.24$, where they are a guide for the eye. The linear fits are extended to $P=0$.

Figure 5

Pressure dependence of the superconducting transition temperature for (a) Ba${}_{0.8}$Na${}_{0.2}$Fe${}_2$As${}_2$ and (b)Ba${}_{0.76}$Na${}_{0.24}$Fe${}_2$As${}_2$. Different symbols in panel (a) correspond to two different pressure runs. Numbers near the symbols in panels (a) and (b) correspond to the order in which the pressure was changed. Panel (c) shows both data sets with the one for $x=0.24$ shifted by $+8$ kbar along the $X$ axis and by $-4$ K along the $Y$ axis.

Figure 6

$d(ln{T}_c)/dP$, $d{T}_c/dP$, and $T_{c0}$ (top to bottom) as a function of Na concentration in Ba${}_{1-x}$Na${}_x$Fe${}_2$As${}_2$. Stars: $T_c$ values from heat capacity measurements. For Ba${}_{0.8}$Na${}_{0.2}$Fe${}_2$As${}_2$ and Ba${}_{0.76}$Na${}_{0.24}$Fe${}_2$As${}_2$, initial, low-pressure, pressure derivative values are used (half-filled hexagons).

Figure 7

Na-concentration dependence of the normalized concentration derivatives, $d(ln{T}_c)/dx=\frac{1}{T_{c0}}d{T}_c/dx$ (left axis, circles), and the normalized pressure derivatives, $d(ln{T}_c)/dP=\frac{1}{T_{c0}}d{T}_c/dP$ (right axis, triangles), of the superconducting transition temperatures. For Ba${}_{0.8}$Na${}_{0.2}$Fe${}_2$As${}_2$ and Ba${}_{0.76}$Na${}_{0.24}$Fe${}_2$As${}_2$, initial, low-pressure, normalized pressure derivative values are used (half-filled hexagons). Inset: schematic exhibiting different signs of the change in $T_c$ with substitution and with pressure for the underdoped samples, and similar effects of substitution and pressure for the overdoped samples.