Nodal gap structure of superconducting BaFe${}_2$(As${}_{1-x}$P${}_x$)${}_2$ from angle-resolved thermal conductivity in a magnetic field

The structure of the superconducting order parameter in the iron-pnictide superconductor BaFe${}_2$(As${}_{0.67}$P${}_{0.33}$)${}_2$ ($T_c=31$ K) with line nodes is studied by the angle-resolved thermal conductivity measurements in a magnetic field rotated within the basal plane. We find that the thermal conductivity displays distinct fourfold oscillations with minima when the field is directed at $\pm {45}^{\circ }$ with respect to the tetragonal $a$ axis. We discuss possible gap structures that can account for the data, and conclude that the observed results are most consistent with the closed nodal loops located at the flat parts of the electron Fermi surface with high Fermi velocity.

Figure 1

(a) Angular variation of $\kappa \left(\phi \right)$ in BaFe${}_2$(As${}_{0.67}$P${}_{0.33}$)${}_2$ with a fit to $\kappa \left(\phi \right)={\kappa }_0+{\kappa }_{2\phi }+{\kappa }_{4\phi }$ (solid line). The sinusoidal curves with one minimum (two minima) represent the twofold (fourfold) term obtained by the fitting. The inset illustrates the measurement configuration. (b) The magnitudes of ${\kappa }_{2\phi }$ and ${\kappa }_{4\phi }$ at 2 K are plotted against the field magnitude $H$. (c)–(e) The fourfold term ${\kappa }_{4\phi }$ at ${\mu }_{0}H=2$ T. (f)–(h) ${\kappa }_{4\phi }$ at $T=2$ K.

Figure 2

(a) Fermi surface of BaFe${}_2$(As${}_{0.67}$P${}_{0.33}$)${}_2$. The shading represents the in-plane Fermi velocity $v_F^{ab}$. Note that the flat parts of the outer electron sheets (near point A in Figs. 3a, 3b, 3c) have high $v_F^{ab}$ values, which correspond to the gray shading in Fig. 2b and 2c. (b), (c) The DOS, $N\left({\varepsilon }_F\right)(\propto$$C/T)$, and $N\left({\varepsilon }_F\right){\left(v_F^{ab}\right)}^2(\propto$$\kappa /T)$ of each band as a function of $v_F^{ab}$.

Figure 3

(a)–(c) Possible node positions on the electron Fermi surfaces. For $k_z=0$ and $\pm \frac{2\pi }{c}$, electron surface cross sections are approximately elliptical, (d)–(g). (d), (e) When the nodes are on the flat part, the field-induced states near all nodes (filled red circles) have a comparable contribution to the heat current for $\mathit{H}\parallel \mathit{q}$ (d) and for $\mathit{H}\perp \mathit{q}$ (e), so that the twofold component of $\kappa$ is nearly absent. (f), (g) When the nodes are on the sharp ends, the field-induced DOS is small for nodes with ${\mathit{v}}_F$ nearly parallel to $\mathit{H}$ (crosses). The nodes with more quasiparticles (filled circles) have a different angle between ${\mathit{v}}_F$ and $\mathit{q}$ for the two field orientations, leading to a dominant twofold anisotropy. (h) Comparison of $\kappa \left(\phi \right)$ for the two cases with the scattering rate $\gamma =\Gamma /2\pi T_c=0.01$, phase shift of scattering $\delta ={60}^{\circ }$, and the ratio of interband to intraband scattering $\delta v=0.9$.

Figure 4

Results of the calculations for the 3D FSs with nodal loops. (a) The superfluid density remains linear down to low temperatures for different values of the scattering rate $\gamma$. (b) At the same time a large residual term in the thermal conductivity in zero field is observed. (c), (d) Anisotropy of thermal conductivity at low $T$ for several values of $H$. A large fourfold term with a minimum at 45${}^{\circ }$ is observed for cleaner material with sizable interband scattering (as well as for the dirtier case with small interband scattering).