Inversion of specific heat oscillations with in-plane magnetic field angle in two-dimensional $d$-wave superconductors

Experiments on several novel superconducting compounds have observed oscillations of the specific heat when an applied magnetic field is rotated with respect to the crystal axes. The results are commonly interpreted as arising from the nodes of an unconventional order parameter, but the identifications of nodal directions are sometimes controversial. Here we show with a semiclassical model calculation that when the magnetic field points in the direction of the nodes, either minima or maxima can occur in the specific heat depending on the temperature $T$ and the magnetic field $H$. An inversion of the angular oscillations takes place with respect to those predicted earlier at low temperature by the nodal approximation. This result, together with the argument that the inversion takes place based on an approximation valid at moderate fields, indicates that the inversion of specific heat oscillations is an intrinsic feature of nodal superconductors.

Figure 1

(Color online) Comparison of the nodal approximation Eq. (11) for the density of states $N\left(\omega ,\mathbf{H}\right)$ versus the full density of states at $T=0.001T_c$ and $E_H=0.2T_c$. Solid line: nodal approximation for $\mathbf{H}$ along antinode. Dashed line: nodal approximation for $\mathbf{H}$ along node. Red circles: full evaluation for $\mathbf{H}$ along node. Black triangles: full evaluation for $\mathbf{H}$ along antinode. The inset magnifies the low-frequency range.

Figure 2

DOS $N\left(\theta ,\omega \right)$ vs magnetic field angle $\theta$ at $E_h=0.2T_c$ and $T=0.05T_c$ for equally spaced frequencies, $0.025{\Delta }_0$ apart, from $\omega =0$ to $0.175{\Delta }_0$. The solid line is the result of the nodal approximation for the $\omega =0$ DOS from Eq. (9).

Figure 3

Temperature dependence of the specific heat (at constant volume) normalized to the specific heat in the normal state $C_V/C_N$, where $C_N={\gamma }_{N}T$. We show both the exact (symbols) and the asymptotic low-$T$ (dotted line) behavior in the absence of the field, and compare it to the numerically determined $C_V/C_N$ at $E_H=0.4T_c$ (dashed line). There is no reduction in $T_c$ with field in our approach.

Figure 4

Left panel: $C\left(T,\mathbf{H}\right)/C\left(T,\theta =0\right)$ vs $\theta$ for a set of equally spaced temperatures, every $0.02T_c$, from 0 to $0.18T_c$, $E_h=0.2T_c$. Right panel: same for a set of equally spaced temperatures from 0 to $0.08T_c$ every $0.02T_c$, $E_h=0.05T_c$.

Figure 5

Amplitude of specific heat oscillation defined by difference of $C_V$ between node and antinode, $\left[C_V\left(0\right)-C_V\left(\frac{\pi }{4}\right)\right]/C_V\left(0\right)$ for $E_H=0.2T_c$ (solid line) and $E_H=0.05T_c$ (dashed line).

Figure 6

(Color online) Phase diagram $H/H_{c2}$ vs $T/T_c$ for specific heat oscillations of a $d$-wave superconductor in a magnetic field. The shaded regions indicate that the specific heat $C\left(T,\mathbf{H}\right)$ has a minimum when the field $\mathbf{H}$ points in the nodal direction, whereas the white regions indicate inverted oscillations, where minima correspond to field along antinode. The gray shaded regions are results from Ref. 18 using the Brandt-Pesch-Tewordt framework, whereas the blue regions represent the region of the $H\text{−}T$ plane where minima correspond to nodes within the semiclassical theory. The phase diagram comparing both approaches was determined using the assumption $H/H_{c2}={\left(E_H/{\Delta }_0\right)}^2$, valid up to a prefactor of order unity. Inset: blowup of the low $T$ and $H$ region. Note the field scale is given in terms of $E_H/{\Delta }_0$ for easier comparison with other results in this work.