Field-angle-resolved anisotropy in superconducting CeCoIn${}_5$ using realistic Fermi surfaces

We compute the field-angle-resolved specific heat and thermal conductivity using realistic model band structures for the heavy-fermion superconductor CeCoIn${}_5$ to identify the gap structure and location of nodes. We use a two-band tight-binding parametrization of the band dispersion as input for the self-consistent calculations in the quasiclassical formulation of the superconductivity. Systematic analysis shows that modest in-plane anisotropy in the density of states and Fermi velocity in tetragonal crystals significantly affects the fourfold oscillations in thermal quantities, when the magnetic field is rotated in the basal plane. The Fermi-surface anisotropy substantially shifts the location of the lines in the $H$-$T$ plane, where the oscillations change sign compared to quasicylindrical model calculations. In particular, at high fields, the anisotropy and sign reversal are found even for isotropic gaps. Our findings imply that a simultaneous analysis of the specific heat and thermal conductivity, with an emphasis on the low-energy sector, is needed to restrict potential pairing scenarios in multiband superconductors. We discuss the impact of our results on recent measurements of the Ce-115 family, namely, Ce$T$In${}_5$ with $T=$ Co, Rh, Ir.

Figure 1

Electronic structure of CeCoIn${}_5$: (a) tight-binding fits of the two most relevant bands $\alpha$ (red) and $\beta$ (green) to the electronic dispersions of CeCoIn${}_5$ calculated in the local density approximation by Maehira et al. (Ref. 14); (b) $\alpha$ and $\beta$ FSs at three representative $k_z$ values, colored by corresponding Fermi velocity from low (green) to high (yellow); (c) three-dimensional rendering of the computed FSs for $\alpha$ and $\beta$ bands (bottom panel) compared with the dHvA experiments (Ref. 17) (top panel). The $\alpha$ FS has a narrow waist, while the $\beta$ FS has a belly. The color map of the calculated FSs gives the anisotropy of the magnitude of the Fermi velocities ranging from low (blue) to high (red).

Figure 2

Fermi surfaces (FSs) and Fermi velocities at $k_z=0$ (left panel) and $k_z=\pi /c$ (right panel). The relative magnitude of the Fermi velocities (in arbitrary units) is given by the length of the (red and green) arrows in the top panels. The bottom panels show the Fermi velocities along the Fermi lines of the corresponding $k_z$ slice.

Figure 3

Fermi-surface anisotropy of the normal-state DOS and SC gaps contrasted with the field-angle anisotropy of the Sommerfeld coefficient and the SC DOS. (a) The calculated FS anisotropy of the normal-state DOS juxtaposed with gap functions of three pairing symmetries. All the SC gaps are computed at the FS and all curves are shifted by their corresponding minimum value, except for the $s$-wave gap. (b) Specific-heat coefficient $\gamma \left(\alpha \right)=C\left(\alpha \right)/T$, normalized to its value at $T_c$, calculated at $T/T_{c0}=0.1$ and $H/H_{c2}=0.1$ for $d$-wave gaps and $H/H_{c2}=0.5$ for the $s$-wave gap. (c)–(e) Field-induced total SC DOS at $T=0$ vs energy at two representative field angles $\alpha =0^{\circ }$ and ${45}^{\circ }$ for all three pairing symmetries. Here, we used $H/H_{c2}=0.5$ for $s$ wave and 0.1 for both $d$ waves. Note the low- and high-energy crossings in the SC DOS (arrows) are related to the low- and high-$T$ sign reversals in the oscillations of $\gamma$ and $\kappa$ in Fig. 4.

Figure 4

Calculated oscillations of the heat capacity and thermal conductivity as a function of the field direction relative to the $x$ axis. (a1) Sommerfeld coefficient $\gamma =C/T$ normalized to its normal-state value $C_N/T_{c0}$ at fixed field $H/H_{c2}=0.5$ for $s$ wave, plotted from low to high $T$ in units of $T/T_{c0}$ (bottom to top curves). Each curve is colored by the sign of the fourfold oscillation; a uniform color map is used for values below $-0.025$ and above $0.025$. (b1) Same as in (a1) but for the normalized thermal conductivity coefficient $\kappa /T$. (c1) The fourfold amplitudes of $\gamma$ (dashed line) and $\kappa /T$ (solid line) are plotted as a function of $T$. For direct comparison, the results for nodal $d_{xy}$ ($H/H_{c2}=0.1$) and $d_{x^2-y^2}$ ($H/H_{c2}=0.33$) symmetries are plotted in panels (a2) and (a3), (b2) and (b3), and (c2) and (c3), respectively. Note that the nonvanishing of ${\kappa }_{4\alpha }$ as temperature approaches the phase transition line in panels (c1) and (c3) is a consequence of the in-plane anisotropy of $H_{c2}$.

Figure 5

Contour maps of fourfold amplitude oscillations of normalized specific heat $C_{4\alpha }$ (row $a$) and normalized thermal conductivity ${\kappa }_{4\alpha }$ (row $b$) in the $H$-$T$ phase diagram. Each column denotes a different gap symmetry studied (isotropic nodeless $s$, nodal $d_{xy}$, and $d_{x^2-y^2}$ gaps). All plots use the same color map scale from minimum (red) to maximum (blue). Note that the fourfold amplitude is given with respect to field $\mathit{H}\parallel \left(100\right)$, i.e., a negative value corresponds to a minimum at $\alpha =0^{\circ }$. Here, $T_c\left(\mathit{H}\right)$ is defined by the vanishing of both gaps for given symmetry, which determines the line of the (minimum) upper critical field. Since the BPT approximation for isotropic $s$-wave pairing is not valid at low $H$, we shaded the corresponding area where our approach is not applicable.

Figure 6

Theoretical contour maps of fourfold amplitude oscillations for $d_{x^2-y^2}$ gap for specific heat (upper panel) and thermal conductivity (lower panel) of CeCoIn${}_5$, reproduced from Figs. 5(a3)–5(b3). The same theoretical data are repeated in three columns, but compared with three different experimental data for isostructural superconductors within the Ce-115 family. (a1) Specific-heat data of CeCoIn${}_5$ by An et al. (Ref. 6) (circles) and Aoki et al. (Ref. 13) (squares). (b1) Thermal conductivity data of CeCoIn${}_5$ by Izawa et al. (Ref. 12). (a2) Specific-heat data for CeRhIn${}_5$ by Park et al. (Refs. 43 and 44). (a3) Specific-heat data for CeIrIn${}_5$ by Lu et al. (Ref. 41) (circles) and Kittaka et al. (Ref. 7) (squares). (b3) Thermal conductivity data of CeIrIn${}_5$ by Kashara et al. (Ref. 45). The symbol size gives the corresponding reported amplitude of the oscillation, whereas the filled color depicts its sign. We find reasonable agreement between theory and experiment in both sign and amplitude of oscillations.