Theory of enhanced proximity effect by midgap Andreev resonant state in diffusive normal-metal/triplet superconductor junctions

Enhanced proximity effect by the midgap Andreev resonant state (MARS) in a diffusive normal-metal/insulator/triplet superconductor (DN/TS) junction is studied based on the Keldysh-Nambu quasiclassical Green’s-function formalism. By choosing a $p$-wave superconductor as a typical example of the TS, conductance of the junction and the spatial variation of the quasiparticle local density of states (LDOS) in the DN are calculated as the function of the magnitudes of the resistance $R_d$, Thouless energy in the DN, and the transparency of the insulating barrier. The resulting conductance spectrum has a zero-bias conductance peak (ZBCP) and the LDOS has a zero energy peak (ZEP) except for $\alpha =\pi ∕2$ $(0⩽\alpha ⩽\pi ∕2)$, where $\alpha$ denotes the angle between the lobe direction of the $p$-wave pair potential and the normal to the interface. The widths of the ZBCP and the ZEP are reduced with the increase of $R_d$ while their heights are drastically enhanced. These peaks are revealed to be suppressed by applying a magnetic field. When the magnitude of $R_d∕R_0$ is sufficiently large, the total zero voltage resistance of the junction is almost independent of the $R_d$ for $\alpha \ne \pi ∕2$. The extreme case is $\alpha =0$, where total zero voltage resistance is always $R_0∕2$. We also studied the charge transport in $p_x+i{p}_y$-wave junctions, where only the quasiparticles with perpendicular injection feel the MARS. Even in this case, the resulting LDOS in the DN has a ZEP. Thus the existence of the ZEP in the LDOS of the DN region is a remarkable feature for DN/TS junctions which have never been expected for the DN/singlet superconductor junctions where the MARS and proximity effect compete with each other. Based on these results, a crucial test to identify triplet pairing superconductors based on tunneling experiments is proposed.

Figure 1

Schematic illustration of the model of the constriction area.

Figure 2

Normalized conductance ${\sigma }_T\left(eV\right)$ for $Z=3$, and $\alpha =0$. $E_{\mathit{Th}}={\Delta }_0$ for the left panel and $E_{\mathit{Th}}=0.02{\Delta }_0$ for the right panel, respectively. (a) $R_d∕R_b=0$; (b) $R_d∕R_b=0.1$; and (c) $R_d∕R_b=1$.

Figure 3

Normalized conductance ${\sigma }_T\left(eV\right)$ for $Z=0$, and $\alpha =0$. $E_{\mathit{Th}}={\Delta }_0$ for the left panel and $E_{\mathit{Th}}=0.02{\Delta }_0$ for the right panel, respectively. (a) $R_d∕R_b=0$; (b) $R_d∕R_b=0.1$; and (c) $R_d∕R_b=1$.

Figure 4

Normalized conductance ${\sigma }_T\left(eV\right)$ for DN/USS junctions with $d_{xy}$-wave superconductor. $Z=3$ for the left panel and $Z=0$ for the right panel. (a) $R_d∕R_b=0$; (b) $R_d∕R_b=0.1$; and (c) $R_d∕R_b=1$.

Figure 5

Normalized conductance ${\sigma }_T\left(eV\right)$ for $Z=3$, and $R_d∕R_b=1$. $E_{\mathit{Th}}={\Delta }_0$ for the left panel and $E_{\mathit{Th}}=0.02{\Delta }_0$ for the right panel, respectively. (a) $\alpha =0$; (b) $\alpha =\pi ∕4$; and (c) $\alpha =\pi ∕2$.

Figure 6

Normalized local density of states $\rho \left(\epsilon \right)$ in the DN for $Z=3$, $R_d∕R_b=1$, and $\alpha =0$. $E_{\mathit{Th}}={\Delta }_0$ (left panel) and $E_{\mathit{Th}}=0.02{\Delta }_0$ (right panel). (a) $x=-L∕4$; (b) $x=-L∕2$; and (c) $x=-L$.

Figure 7

Normalized local density of states $\rho \left(\epsilon \right)$ in the DN for $Z=3$, $R_d∕R_b=1$, and $E_{\mathit{Th}}={\Delta }_0$. $\alpha =\pi ∕4$ (left panel), $\alpha =3\pi ∕8$ (middle panel), and $\alpha =\pi ∕2$ (right panel). (a) $x=-L∕4$; (b) $x=-L∕2$; and (c) $x=-L$.

Figure 8

Normalized local density of states $\rho \left(\epsilon \right)$ in the DN for the DN/USS junction with $Z=3$, $R_d∕R_b=1$, and $E_{\mathit{Th}}={\Delta }_0$. We have chosen a $d$-wave superconductor with ${\Delta }_{\pm }={\Delta }_0\cos \left[2(\theta \mp \beta )\right]$. $\beta =0$ (left panel), $\beta =\pi ∕8$ (middle panel), and $\beta =\pi ∕4$ (right panel). (a) $x=-L∕4$; (b) $x=-L∕2$; and (c) $x=-L$.

Figure 9

Real (upper panels) and imaginary (lower panels) parts of ${\theta }_0$ are plotted as a function of $\epsilon$ for $\alpha =0$ and $Z=0$ with $E_{\mathit{Th}}={\Delta }_0$ (left panels) and $E_{\mathit{Th}}=0.02{\Delta }_0$ (right panels). (a) $R_d∕R_b=0.1$; (b) $R_d∕R_b=0.5$; and (c) $R_d∕R_b=1$.

Figure 10

Real (upper panels) and imaginary (lower panels) parts of ${\theta }_0$ are plotted as a function of $\epsilon$ for $\alpha =0$ and $Z=3$. $E_{\mathit{Th}}={\Delta }_0$ (left panels) and $E_{\mathit{Th}}=0.02{\Delta }_0$ (right panels). (a) $R_d∕R_b=0.1$; (b) $R_d∕R_b=0.5$; and (c) $R_d∕R_b=1$.

Figure 11

Real (upper panels) and imaginary (lower panels) parts of ${\theta }_0$ are plotted as a function of $\epsilon$ for $R_d∕R_b=1$ and $Z=3$. $E_{\mathit{Th}}={\Delta }_0$ (left panels) and $E_{\mathit{Th}}=0.02{\Delta }_0$ (right panels). (a) $\alpha =0$; (b) $\alpha =0.25\pi$; and (c) $\alpha =0.375\pi$.

Figure 12

$E_C$ is plotted as a function of $R_d∕R_b$ for $E_{\mathit{Th}}={\Delta }_0$ (left panel) and $E_{\mathit{Th}}=0.02{\Delta }_0$ (right panel). (a) $Z=1$; (b) $Z=2$; and (c) $Z=3$.

Figure 13

$E_{\rho }$ is plotted as a function of $R_d∕R_b$ for $E_{\mathit{Th}}={\Delta }_0$ (left panel) and $E_{\mathit{Th}}=0.02{\Delta }_0$ (right panel). (a) $Z=1$; (b) $Z=2$; and (c) $Z=3$.

Figure 14

Total zero voltage resistance of the junctions $R$ is plotted as a function of $R_d∕R_b$ for $Z=1.5$ with (a) $\alpha =0.25\pi$; and (b) $\alpha =0.4\pi$. Curves c and d present the same dependence for the DN/CSS junctions with the $s$-wave superconductor and DN/USS junctions with the $d_{xy}$-wave superconductor, respectively.

Figure 15

Total zero voltage resistance of the junctions $R$ is plotted as a function of $R_d∕R_b$ for $Z=0$ (left panel) and $Z=3$ (right panel). (a) $\alpha =0$; (b) $\alpha =\pi ∕2$; and (c) $\alpha =0.4\pi$.

Figure 16

Zero voltage electrostatic potential distribution at zero temperature ${\varphi }_0\left(x\right)$ is plotted as a function of $x$ for $Z=3$. The DN/USS junction with $d_{xy}$-wave superconductor (left panel) and the DN/TS junction with $p_x$-wave superconductor (left panel). (a) $R_d∕R_b=0$; (b) $R_d∕R_b=0.1$; (c) $R_d∕R_b=1$; and (d) $R_d∕R_b=2$.

Figure 17

Normalized conductance ${\sigma }_T\left(\mathit{eV}\right)$ for DN/TS junctions with $Z=1.5$ and $R_d∕R_b=1$ where $p$-wave superconductor with $\alpha =0$ is chosen. We choose $E_{\mathit{Th}}={\Delta }_0$ for the left panel and $E_{\mathit{Th}}=0.1{\Delta }_0$ for the right panel, respectively. (a) $\hslash ∕{\tau }_H=0$; (b) $\hslash ∕{\tau }_H=E_{\mathit{Th}}$; and (c) $\hslash ∕{\tau }_H=10E_{\mathit{Th}}$.

Figure 18

Normalized local density of states of quasiparticle $\rho \left(\mathit{eV}\right)$ for DN/TS junctions with $Z=1.5$ and $R_d∕R_b=1$ where the $p$-wave superconductor with $\alpha =0$ is chosen. We choose $x=-L∕4$. $E_{\mathit{Th}}={\Delta }_0$ for the left panel and $E_{\mathit{Th}}=0.1{\Delta }_0$ for the right panel, respectively. (a) $\hslash ∕{\tau }_H=0$; (b) $\hslash ∕{\tau }_H=E_{\mathit{Th}}$; and (c) $\hslash ∕{\tau }_H=10E_{\mathrm{Th}}$.

Figure 19

Normalized conductance ${\sigma }_T\left(eV\right)$ for DN/TS junctions with the $p_x+i{p}_y$ superconductor for $Z=3$ is plotted. We choose $E_{\mathit{Th}}={\Delta }_0$ for the left panel and $E_{\mathit{Th}}=0.02{\Delta }_0$ for the right panel, respectively. (a) $R_d∕R_b=0$; (b) $R_d∕R_b=0.1$; and (c) $R_d∕R_b=1$.

Figure 20

Normalized local density of states $\rho \left(\epsilon \right)$ for DN/TS junctions with the $p_x+i{p}_y$-wave superconductor is plotted for $Z=3$. We choose $E_{\mathit{Th}}={\Delta }_0$ for the left panel and $E_{\mathit{Th}}=0.02{\Delta }_0$ for the right panel, respectively. (a) $x=-L∕4$; (b) $x=-L∕2$; and (c) $x=-L$.

Figure 21

Schematic illustration of the experimental setup to discriminate the triplet superconducting state from the singlet one. The LDOS measured by STS only has a ZEP for DN/TS junctions.

Figure 22

Spatial dependence of the local density of state in the DN region at zero energy is plotted for $p_x$-wave (left panel) and $p_x+i{p}_y$-wave (right panel) superconductors with $Z=1$. For the $p_x$-wave case (left panel), $R_d∕R_b=1$ (curve a), and $R_d∕R_b=0.5$ (curve b). For the $p_x+i{p}_y$-wave case (left panel), $R_d∕R_b=2$ (curve a), and $R_d∕R_b=1$ (curve b). Since we are concentrating on the $\epsilon =0$ case, $\rho \left(0\right)$ is independent of $E_{\mathit{Th}}$.