Superconducting gap evolution in overdoped BaFe$_{2}$(As$_{1-x}$P$_x$)$_2$ single crystals through nanocalorimetry

We report on specific heat measurements on clean overdoped $\mathrm{BaFe_{2}(As_{1-x}P_x)_2}$ single crystals performed with a high resolution membrane-based nanocalorimeter. A nonzero residual electronic specific heat coefficient at zero temperature $\gamma_\mathrm{r}={C/T}|_{T \to 0}$ is seen for all doping compositions, indicating a considerable fraction of the Fermi surface ungapped or having very deep minima. The remaining superconducting electronic specific heat is analyzed through a two-band s-wave $\alpha$ model in order to investigate the gap structure. Close to optimal doping we detect a single zero-temperature gap of $\Delta_0 \sim 5.3\,\mathrm{meV}$, corresponding to $\Delta_0 / k_\mathrm{B} T_\mathrm{c} \sim 2.2$. Increasing the phosphorus concentration $x$, the main gap reduces till a value of $\Delta_0 \sim 1.9\,\mathrm{meV}$ for $x = 0.55$ and a second weaker gap becomes evident. From the magnetic field effect on $\gamma_\mathrm{r}$, all samples however show similar behavior [$\gamma_\mathrm{r}(H) - \gamma_\mathrm{r}(H=0) \propto H^n$, with $n$ between 0.6 and 0.7]. This indicates that, despite a considerable redistribution of the gap weights, the total degree of gap anisotropy does not change drastically with doping.