Pseudodiffusive transmission of nodal Dirac fermions through a clean $d$-wave superconductor

We calculate the transmission of electrons and holes between two normal-metal (N) electrodes, separated over a distance $L$ by an impurity-free superconductor (S) with $d$-wave symmetry of the order parameter. Nodal lines of vanishing excitation gap form ballistic conduction channels for coupled electron-hole excitations, described by an anisotropic two-dimensional Dirac equation. We find that the transmitted electrical and thermal currents both have the pseudodiffusive $1/L$ scaling characteristic of massless Dirac fermions—regardless of the presence of tunnel barriers at the NS interfaces. Tunnel barriers reduce the slope of the $1/L$ scaling in the case of the electrical current while leaving the thermal current unaffected.

Figure 1

(Color online) Geometry to measure the transmission of nodal fermions through a $d$-wave superconductor. A current $I_1$ is injected into the superconductor from metal contact ${\text{N}}_1$ (at a voltage $V$) and drained to ground via the superconductor (current $I_{\text{S}}$) or via a second metal contact ${\text{N}}_2$ (current $I_2$). If the separation $L$ of the metal contacts is large compared to the superconducting coherence length ${\xi }_0$, the current $I_2$ is predominantly due to transmission parallel to the nodal lines $x=0$ or $y=0$ of vanishing excitation gap.

Figure 2

Ellipsoidal equal-energy contours of low-energy excitations in the Brillouin zone of a superconductor with $d_{xy}$ symmetry. Long and short axes have ratio $v_F/v_{\Delta }$. The contours are centered at the four nodal points (solid dots), where the order parameter vanishes on the Fermi surface. The normal $\hat{\mathit{n}}$ to the NS interfaces is indicated. The dashed line, displaced from the nearest nodal point by $q$, indicates points of constant wave-vector component parallel to the interface.

Figure 3

Dependence on the separation $L$ of the NS interfaces of the current $I_2$ into contact ${\text{N}}_2$, for the interface orientation $\alpha =0$ of aligned nodes $A$ and $C$. Calculated from Eqs. (C3, C4, C5) for parameters ${\Gamma }_1={\Gamma }_2=0.3$ and $v_F/v_{\Delta }=10$.

Figure 4

Same as Fig. 2 but now for an angle $\alpha =\pi /4$ between the normal $\hat{\mathit{n}}$ to the NS interface and the lines $x=0$ and $y=0$ of vanishing order parameter. For this orientation the nodal points $A\text{−}D$ and $B\text{−}C$ are pairwise aligned with $\hat{\mathit{n}}$ (dashed lines) so that they are pairwise coupled by a tunnel barrier at the interfaces.

Figure 5

Same as Fig. 3 but now for an interface orientation $\alpha =\pi /4$ of pairwise aligned nodes $A\text{−}D$ and $B\text{−}C$, calculated from Eq. (C10) for parameters ${\Gamma }_1={\Gamma }_2=0.3$ and $v_F/v_{\Delta }=10$.