Effects of phase coherence on local density of states in superconducting proximity structures

We theoretically study the local density of states in superconducting proximity structures where two superconducting terminals are attached to a side surface of a normal-metal wire. Using the quasiclassical Green's function method, the energy spectrum is obtained for both spin-singlet $s$-wave and spin-triplet $p$-wave junctions. In both cases, the decay length of the proximity effect at zero temperature is limited by a depairing effect due to inelastic scattering. In addition to the depairing effect, in $p$-wave junctions the decay length depends sensitively on the transparency at the junction interfaces, which is a unique property of odd-parity superconductors where the anomalous proximity effect occurs.

Figure 1

Schematics of (a) T-shaped and (b) Volkov-Takayanagi (VT) junctions. The N'/DN interfaces are located at $x=\pm L$, where N' indicates an N lead. The barrier potential is present only at the S/DN interfaces. The widths and the thickness of the wires are assumed much narrower and thinner than the coherence length. The superconductor(s) is attached to the DN at $x=0$ in (a) and at $x=\pm L_1$ (b). The superconductors have the phase difference $\delta \mathrm{\Phi }$ in (b). Schematics of the (c) $s$-wave and (d) $p$-wave pair potentials in momentum space. The inner circles indicate the Fermi surface. The sign means the phase of the pair potential.

Figure 2

Deviations of the densities of states $\delta \nu (x,\varepsilon )$ in the T-shaped junction with an $s$-wave superconducting wire. The results are obtained at $x=0,{\xi }_0,2{\xi }_0,3{\xi }_0,4{\xi }_0$. The barrier parameter is set to ${\gamma }_B=1$ in (a), (b), and (c), ${\gamma }_B=3.33$ in (d). The interface-potential parameter is set to $z_0=0.1$ in (a), $1.0$ in (b) and (d), and $3.0$ in (c). The length of the DN is set to $L=6{\xi }_0$. A superconductor with a width $w=0.3{\xi }_0$ is attached to the DN at $x=0$. The depairing ratio is set to $\gamma =0.01{\mathrm{\Delta }}_0$. The structures such as coherence peak and low-energy dip become sharper with increasing $z_0$. The amplitude becomes smaller with increasing ${\gamma }_B$.

Figure 3

Deviations of the densities of states $\delta \nu (x,\varepsilon )$ in the T-shaped junction with a $p$-wave SC. The parameters are set to $L=10{\xi }_0,w=0.3{\xi }_0$, and $\gamma =0.01{\mathrm{\Delta }}_0$. The zero-energy peak appears because of the $p$-wave nature. The zero-energy peak becomes narrower and higher with increasing the interface potential $z_0$.

Figure 4

Correction of the local density of states (LDOS) in the VT junction with $s$-wave superconducting arms. The phase difference is set to (a) $\delta \mathrm{\Phi }=0$ and (b) $\delta \mathrm{\Phi }=\pi$. The results are obtained between the center of the junction (i.e., $x=0$) and the point where a superconducting arm is attached (i.e., $x=L_1$). The parameters are set to $L=6{\xi }_0,L_1=5{\xi }_0,w=0.3{\xi }_0,\gamma =0.01{\mathrm{\Delta }}_0,{\gamma }_B=1$, and $z_0=1$. The LDOS at the center of the junction is modified when $\delta \mathrm{\Phi }=0$, whereas the correction vanishes when $\delta \mathrm{\Phi }=\pi$. The results mean Cooper pairs from each superconductor interfere in the DN.

Figure 5

LDOS in the VT junction with $p$-wave superconducting arms. The parameters are set to $L=6{\xi }_0$, $L_1=5{\xi }_0$, $w=0.3{\xi }_0$, $\gamma =0.01{\mathrm{\Delta }}_0$, ${\gamma }_B=1$, and $z_0=1$. The parameters are set to the same values as those used in Fig. 4. The zero-energy peak spreads spatially between the two superconducting arms when $\delta \mathrm{\Phi }=0$, whereas it vanishes when $\delta \mathrm{\Phi }=\pi$ because of the long-range phase coherence.

Figure 6

Junction-length dependence of the LDOS correction at the center of VT junctions with (a) $s$-wave and (b) $p$-wave superconducting arms. The junction length is changed from $L=2{\xi }_0$ to $10{\xi }_0$, where $\delta \mathrm{\Phi }=0$ and the interval between the superconducting wire and the normal lead is fixed at $L-L_1={\xi }_0$. The other parameters are set to $w=0.3{\xi }_0,\gamma =0.01{\mathrm{\Delta }}_0,{\gamma }_B=1$, and $z_0=1$.

Figure 7

Junction-length dependence of the LDOS at $x=0$ and $\varepsilon =0$. The results for the $s$-wave case are shown in (a) and (c), where as those for $p$-wave are in (b) and (d). The depairing ratio is fixed at $\gamma =0.01{\mathrm{\Delta }}_0$ in (a) and (b), and $\gamma =0.01{\mathrm{\Delta }}_0$ in (c) and (d). The other parameters are set to ${\gamma }_B=1$ and $L-L_1={\xi }_0$. In $s$-wave cases, the correction becomes small with increasing the barrier potential $z_0$, whereas it becomes large with increasing $z_0$. The correction decreases more rapidly with increasing $L_1$ when $\gamma$ is large.

Figure 8

Depairing-ratio dependence of the LDOS correction at $x=0$ and $\varepsilon =0$ for (a) an $s$-wave junction and (b) an $p$-wave junction. The interface barrier and the junction length are set to $z_0=1.0$ or 0.1 and $L/{\xi }_0=4$ or 8. The length $L_1$ is fixed to $L_1=L-{\xi }_0$. The correction of the LDOS converges at a certain value regardless of the junction length, meaning that the decay length of $\delta \nu$ is determined by $\gamma$.

Figure 9

Depairing-ratio dependence of the normalized correction. The normalized correction $\overline{\delta \nu }\left(\gamma \right)$ is given in Eq. (28). The parameters are set to ${\gamma }_B=1,L=8{\xi }_0$, and $L-L_1={\xi }_0$. For $p$-wave junctions, the correction at the zero-energy depends strongly on the interface barrier $z_0$ because the large barrier potential results in the high zero-energy peak.