Unconventional superconductors under a rotating magnetic field. II. Thermal transport

We present a microscopic approach to the calculations of thermal conductivity in unconventional superconductors for a wide range of temperatures and magnetic fields. Our work employs the nonequilibrium Keldysh formulation of the quasiclassical theory. We solve the transport equations using a variation of the Brandt-Pesch-Tewordt method that accounts for the quasiparticle scattering on vortices. We focus on the dependence of the thermal conductivity on the direction of the field with the respect to the nodes of the order parameter, and discuss it in the context of experiments aiming to determine the shape of the gap from such anisotropy measurements. We consider quasi-two-dimensional Fermi surfaces with vertical line nodes and use our analysis to establish the location of gap nodes in heavy-fermion $\mathrm{Ce}\mathrm{Co}{\mathrm{In}}_5$ and the organic superconductor $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2\mathrm{Cu}{\left(\mathrm{NCS}\right)}_2$.

Figure 1

(Color online) The model considered in this paper. Calculation of the thermal conductivity is done for a quasicylindrical Fermi surface, when the heat current or temperature gradient and (rotated) magnetic field are in the $ab$ plane. The order parameter is assumed to have a $d$-wave symmetry.

Figure 2

(Color online) Two distinct experimentally relevant setups for the thermal conductivity. Left panel: $d_{x^2-y^2}$ gap symmetry; right panel: $d_{xy}$ gap symmetry. Response of the $d_{xy}$ superconductor to the thermal gradient along the [100] $\left(\mathbf{x}\right)$ direction with the field at an angle ${\varphi }_0^{\prime }$ to this axis is equivalent to the response of a $d_{x^2-y^2}$ system to the thermal gradient along the [110] $\left({\mathbf{x}}^{\prime }\right)$ direction and the field at the angle ${\varphi }_0={\varphi }^{\prime }+\pi ∕4$ to the $\mathbf{x}$ axis. Note that the experiment is done with the heat current ${\mathbf{j}}_h$ along [100], while the calculations are for the thermal gradient along this direction (see text for details).

Figure 3

(Color online) $T\text{−}H$ diagram of the longitudinal heat conductivity anisotropy ${\kappa }_{xx}\left({\varphi }_0\right)$. The thermal gradient is along the $x$ axis (maximal gap). Regions of different anisotropy are marked as $[{\varphi }_0^a,{\varphi }_0^b,{\varphi }_0^c]$, which denotes ${\kappa }_{xx}\left({\varphi }_0^a\right)>{\kappa }_{xx}\left({\varphi }_0^b\right)>{\kappa }_{xx}\left({\varphi }_0^c\right)$. For points marked by circles and squares the profiles of the angle-dependent ${\kappa }_{xx}\left({\varphi }_0\right)$ are shown in Fig. 4.

Figure 4

(Color online) Left panel: evolution of the heat conductivity with temperature for $H∕H_{c2}\approx 0.14$. Right panel: heat conductivity as a function of the field angle for different fields at $T∕T_c=0.25$. The curves are shifted vertically for clarity. The framed numbers give true values of normalized ${\kappa }_{xx}({\varphi }_0=0)$ for several of them. Corresponding temperatures and fields are shown in each panel in the same order as the curves.

Figure 5

(Color online) Longitudinal heat conductivity for $\mathbf{H}\parallel \mathbf{\nabla }T\parallel \widehat{x}$ antinodal direction as a function of temperature for several fields (left); as a function of the field for temperatures {0.05, 0.2, 0.4, 0.6} (right).

Figure 6

(Color online) Transverse heat conductivity for $\mathbf{\nabla }T\Vert \widehat{x}$ antinodal direction: $\mathbf{H}\Vert \text{node}$ as a function of temperature (left) and field (right).

Figure 7

(Color online) Heat conductivity profiles for $T$ scans at two different fields and an $H$ scan at $T∕T_c=0.25$. Different symbols correspond to points in Fig. 10 (left). The curves are shifted vertically to fit on the same plot, and the value for ${\varphi }_0=0$ for each of the shifted curves is given in the corresponding boxes.

Figure 8

(Color online) Anisotropy profile of the transverse heat conductivity. This component gives the heat current along the $y$ direction, when $\mathbf{\nabla }T\Vert \mathbf{x}$ and the field makes an angle ${\varphi }_0$ with the antinodal $x$ axis.

Figure 9

(Color online) Longitudinal heat conductivity anisotropy for $d_{xy}$ symmetry. The curves are shifted vertically, so the scale on the vertical axis shows only the absolute anisotropy. The relative anisotropy can be deduced from the true values of ${\kappa }_{xx}\left(0\right)$ shown in boxes. On the left is shown a temperature scan for $H∕H_{c2}=0.065$, and on the right is a field scan for $T∕T_c=0.25$, shown correspondingly by filled circles and squares in Fig. 10 (right).

Figure 10

(Color online) Anisotropy phase diagram of the longitudinal heat conductivity ${\kappa }_{xx}\left({\varphi }_0\right)$. Notation $[{\varphi }_0^a,{\varphi }_0^b,{\varphi }_0^c]$ is short for ${\kappa }_{xx}\left({\varphi }_0^a\right)>{\kappa }_{xx}\left({\varphi }_0^b\right)>{\kappa }_{xx}\left({\varphi }_0^c\right)$. The solid blue line corresponds to the change of sign in the ${\kappa }_2$, which is positive at high $T$ and negative at low $T$. Its position on the phase diagram is almost the same for the heat current along antinode and node. Left panel: temperature gradient is along the antinodal direction. The symbols (circles, squares, and triangles) correspond to anisotropy curves in Fig. 7. The variation of the transverse heat conductivity anisotropy for this gap orientation is shown in Fig. 8. Right panel: temperature gradient is along a nodal direction.

Figure 11

(Color online) Fourfold symmetry coefficient ${\kappa }_4$ of heat conductivity for the scans shown above in Fig. 7.

Figure 12

(Color online) $T\text{−}H$ phase diagram for the fourfold component ${\kappa }_4$ of the thermal conductivity under rotated magnetic field. Differently shaded regions correspond to different signs of the fourfold term, and its connection to the nodal directions. The dashed line marks the crossover from ${\kappa }_{xx}\left(0\right)<{\kappa }_{xx}(90°)$ at low $T$ to ${\kappa }_{xx}\left(0\right)>{\kappa }_{xx}(90°)$ at high $T$.

Figure 13

(Color online) The $T\text{−}H$ phase diagram for anisotropy of the heat capacity under rotated magnetic field. We choose for our model a quasicylindrical Fermi surface and a $d$-wave order parameter. For rotations of the field in the $ab$ plane the oscillations of the heat conductivity are fourfold, and a different feature (maximum or minimum) of this dependence determines the node position (arrows), depending on the location on this diagram.