Nodal gap detection through polar angle-resolved density of states measurements in uniaxial superconductors

We propose a spectroscopic method to identify the nodal gap structure in unconventional superconductors. This method is best suited for locating the horizontal line node and for pinpointing the isolated point nodes by measuring polar angle $\left(\theta \right)$ resolved zero-energy density of states $N\left(\theta \right)$. This is measured by specific heat or thermal conductivity at low temperatures under a magnetic field. We examine a variety of uniaxially symmetric nodal structures, including point and/or line nodes with linear and quadratic dispersions, by solving the Eilenberger equation in vortex states. It is found that (a) the maxima of $N\left(\theta \right)$ continuously shift from the antinodal to the nodal direction $\left({\theta }_{\mathrm{n}}\right)$ as a field increases accompanying the oscillation pattern reversal at low and high fields. Furthermore, (b) local minima emerge next to ${\theta }_{\mathrm{n}}$ on both sides, except for the case of the linear point node. These features are robust and detectable experimentally. Experimental results of $N\left(\theta \right)$ performed on several superconductors, ${\mathrm{UPd}}_2{\mathrm{Al}}_3,{\mathrm{URu}}_2{\mathrm{Si}}_2,{\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$, and ${\mathrm{UPt}}_3$, are examined and commented on in light of the present theory.

Figure 1

Schematic figure of ZDOS as a function of $B$ for two directions. CR indicates the crossing field of two ZDOS curves. The oscillation pattern of $N\left(\theta \right)$ at a fixed field is reversed at around $B_{\mathrm{CR}}$. (a) A line node at the equator and (b) point nodes at two poles in momentum space.

Figure 2

Field evolutions of polar angle-resolved ZDOS normalized by the normal DOS $N_{\mathrm{F}}$ for (a) $\varphi \left(\mathit{k}\right)\propto k_z^2$, (b) $k_z^2(k_x^2+k_y^2)$, and (c) $k_z(k_x+i{k}_y)$. The magnetic field is scaled by the Eilenberger unit $B_0={\varphi }_0/2\pi {\xi }^2$.

Figure 3

Momentum-resolved ZDOS $N_{\mathit{k}}(E=0)$ on the Fermi sphere for $\varphi \left(\mathit{k}\right)\propto k_z^2$. The arrows indicate the field direction. (a) $B=0.01,\theta =0^{\circ }$, (b) $B=0.01,\theta ={90}^{\circ }$, (c) $B=0.1,\theta =0^{\circ }$, and (d) $B=0.1,\theta ={90}^{\circ }$.

Figure 4

Weighted ZDOS $sin{\theta }_{\mathit{k}}{N}_{{\theta }_{\mathit{k}}}(E=0)$ for selected field angles for $\varphi \left(\mathit{k}\right)\propto k_z^2$ at $B=0.03$. Inset shows schematic configuration of the line node on the Fermi sphere and two field directions, ${75}^{\circ }$ (black arrow) and ${90}^{\circ }$ (white arrow), with two integration paths over ${\varphi }_{\mathit{k}}$ for ${\theta }_{\mathit{k}}=5^{\circ }$ (small circles) and ${45}^{\circ }$ (large circles).

Figure 5

Field evolutions of $N\left(\theta \right)$ within the KPA for three Fermi surfaces: (a) cigar, (b) sphere, and (c) spheroid, corresponding to the anisotropy parameters ${\mathrm{\Gamma }}_0=1.2$, 1, and 0.71, respectively. The linear line node is on the equator.

Figure 6

Field evolutions of $N\left(\theta \right)$ within the KPA for the (a) linear point nodes $\varphi \left(\mathit{k}\right)\propto \sqrt{k_x^2+k_y^2}$ and (b) quadratic point nodes $\varphi \left(\mathit{k}\right)\propto k_x^2+k_y^2$.

Figure 7

Field evolutions of $N\left(\theta \right)$ within the KPA for the (a) linear point and line nodes $\varphi \left(\mathit{k}\right)\propto k_z\sqrt{k_x^2+k_y^2}$ and (b) quadratic point and line nodes $\varphi \left(\mathit{k}\right)\propto k_z^2(k_x^2+k_y^2)$.

Figure 8

Field evolution of $N\left(\theta \right)$ within the KPA for the off-symmetric line nodes $\varphi \left(\mathit{k}\right)\propto (5k_z^2-1)$.