Vortex state in $d$-wave superconductors with strong paramagnetism: Transport and specific heat anisotropy

We analyze the combined effect of orbital and Pauli depairing on the superconducting state, and apply the results to the heavy fermion ${\text{CeCoIn}}_5$. We find that (a) standard extrapolation based on the slope of $H_{c2}\left(T\right)$ in the vicinity of the transition temperature does not always give accurate values of the orbital upper critical field; (b) critical value of the Maki parameter, $\alpha$, which determines onset of the first order transition, depends on the Fermi surface shape and the symmetry of the gap and is ${\alpha }^{\ast }\approx 3$ for ${\text{CeCoIn}}_5$; and (c) the anisotropy of the thermodynamic and transport coefficients in the low-temperature, low-field part of the phase diagram is essentially insensitive to the Zeeman field and can be used to determine the nodal directions in Pauli-limited superconductors. The latter result confirms the finding of the $d_{x^2-y^2}$ order parameter ${\text{CeCoIn}}_5$.

Figure 1

(Color online) Difference between estimated and computed orbital upper critical field, $H_{c2}^{orb}$, in strongly paramagnetic superconductors for (a) in-plane field and (b) field normal to the planes. Lower solid lines denote $H_{c2}$ for the paramagnetically limiting case $Z=0.8$ and are continued as broken lines in regions of the first order transition. The orbital field is computed by ignoring Zeeman splitting $\left(Z=0\right)$, the upper solid lines. It is in reasonable agreement with the values based on the HW result (Ref. 70), thick dot-dashed lines. If, however, the HW plot is based on the slope $d{H}_{c2}/dT$ at $T_c$ determined from the points $T/T_c=\left(0.9,1\right)$ for $Z=0.8$, thin dot-dashed lines, the extracted orbital limit, $H_{c2}^{ex}$, is significantly different from $H_{c2}^{orb}$ as indicated by vertical arrows.

Figure 2

(Color online) Upper critical field of a pure $d_{x^2-y^2}$ superconductor with quasicylindrical FS for three orientations of the field. The critical value of the Maki parameter is about ${\alpha }^{\ast }\sim 3$ for all directions (see text). The dotted lines indicate first order transition below $T^{\ast }$. Inset of the right-hand panel: $T^{\ast }$ as a function of the relative strength of the Pauli limiting. For purely Pauli-limited case $T_P^{\ast }\approx 0.56T_c$.

Figure 3

(Color online) A fit of the experimental $H_{c2}\parallel \left[100\right]$ in ${\text{CeCoIn}}_5$. The data are taken from measurements of magnetization by Tayama et al. (Ref. 26) and specific heat by Bianchi et al. (Ref. 65). We find that $Z=0.5$ gives the best fit of second order transition line and the onset of the first order transition $T^{\ast }$.

Figure 4

(Color online) The anisotropy of heat capacity phase diagram for three Zeeman couplings (a) $Z=0.0$, (b) $Z=0.2$, and (c) $Z=0.4$. Shaded areas correspond to minimum of the heat capacity for $\mathbf{H}\parallel \left(\text{node}\right)$. Panels (c1) and (c2) demonstrate how anisotropy of heat capacity, $C\left({\varphi }_0\right)$, varies with temperature and magnetic field for $Z=0.4$. The curves for different $T/T_c$ are shifted vertically for clarity.

Figure 5

(Color online) Anisotropy of the specific heat (left) and heat conductivity (right) for $Z=0.0$ and 0.4 at $T/T_c=0.3$ and various fields. The field values are shown on the left in terms of $Z$-dependent $H_{c2}$. The curves were shifted vertically for clarity. The absolute values of $C$ and $\kappa$ and their anisotropies are presented in Figs. 6, 7, 8, 9 below. The range of fields in $H/B_0$ for convenience indicated on the right.

Figure 6

(Color online) $C_{0\varphi }$ and $C_{4\varphi }$ coefficients in heat capacity expansion at two different temperatures, $T=0.05T_c$ and $T=0.3T_c$. Positive fourfold coefficient means that the specific heat has minimum for the field along a node in the gap while negative $C_{4\varphi }$ corresponds to the maximum for the field along a node.

Figure 7

(Color online) The anisotropic component of the specific heat at temperatures $T=0.05T_c$ (left panel) and $T=0.3T_c$ (right panel) for $Z=0$ and 0.4.

Figure 8

(Color online) Field dependence of the coefficients ${\kappa }_0$ and ${\kappa }_2$ in the expansion of the thermal conductivity, Eq. (18) at $T=0.05T_c$ (top two panels), and $T=0.3T_c$ (bottom panels) for $Z=0$ and 0.4.

Figure 9

(Color online) The anisotropic fourfold term of the thermal conductivity in Eq. (19) is shown for $T=0.05T_c$ in panels (a) and (c), and for $T=0.3T_c$ in panels (b) and (d) as function of the field, panels (a) and (b) and the reduced field, $H/H_{c2}$, panels (c) and (d).

Figure 10

(Color online) The phase diagram for the anisotropy of the specific heat under rotated magnetic field for different strength of Pauli pairbreaking term. Shaded areas correspond to minimum of the heat capacity for $\mathbf{H}\parallel \left(\text{node}\right)$. Notice that the shaded region at low $T$ and $H$ is identical in all three panels.