Long-range crossed Andreev reflections in high-temperature superconductors

We analyze the nonlocal transport properties of a $d$-wave superconductor coupled to metallic electrodes at nanoscale distances. We show that the nonlocal conductance exhibits an algebraical decay with distance rather than the exponential behavior which is found in conventional superconductors. Crossed Andreev processes, associated with electronic entanglement, are favored for certain orientations of the symmetry axes of the superconductor with respect to the leads. These properties would allow its experimental detection using present technologies.

Figure 1

Two leads on a $d$-wave superconductor. The distance between the leads is $d$; $\alpha$ is the angle between the crystallographic axes of the superconductor and the direction normal to the surface. An incoming electron from lead $a$ can be reflected in lead $b$ as (b) an electron or (c) a hole while a Cooper pair is created in the superconductor.

Figure 2

(Color online) The anomalous propagator ${\left|g_{ba,12}\right|}^2$ for $d_{x^2-y^2}$ symmetry at zero energy as a function of the separation between the leads. The propagator has been normalized to its value in the normal state at $d=0$. The corresponding electron propagator $g_{ba,11}$ is negligible within this scale. The envelope curve decaying as $1/d^2$ is indicated by a full line. The corresponding curve for $s$ symmetry with the same choice of parameters, exhibiting an exponential decay, is also represented for comparison. The inset shows the exponents $r_{1,2}$ in the decay laws of Eq. (5) as a function of energy.

Figure 3

(Color online) Plots of the CAR and EC contributions to the nonlocal conductance for the $d_{x^2-y^2}$ symmetry as one of the contacts moves inside the superconductor while the other remains fixed at $x=y=0$. The contacts are in the tunnel limit and the voltage is set to zero. Both contributions are plotted in a logarithmic scale normalized to their maximum value.

Figure 4

(Color online) Spatial averaged nonlocal resistance at $d=10{\xi }_0$ for $d_{x^2-y^2}$ symmetry as a function of the voltage for different values of the transmission $T_N$. The inset shows the nonlocal conductance at $eV=0$ as a function of $d$ and for different values of $T_N$. For $T_N\approx 0.9$ there is a sign change in ${\sigma }_{ba}$.

Figure 5

(Color online) Nonlocal conductance at $eV=0$ for $d_{xy}$ symmetry as a function of the distance between the leads for $T_N=0.04$,0.15, and 0.3. The inset shows the spatial-averaged nonlocal conductance at $d=10{\xi }_0$ and $T_N=0.04$ as a function of the voltage.