Odd-frequency pairing in normal-metal/superconductor junctions

We theoretically study the induced odd-frequency pairing states in ballistic normal-metal/superconductor (N/S) junctions where a superconductor has even-frequency symmetry in the bulk and a normal-metal layer has an arbitrary length. Using the quasiclassical Green’s function formalism, we demonstrate that, quite generally, the pair amplitude in the junction has an admixture of an odd-frequency component due to the breakdown of translational invariance near the N/S interface where the pair potential acquires spatial dependence. If a superconductor has an even-parity pair potential (spin-singlet $s$-wave or spin-singlet $d_{xy}$-wave state), the odd-frequency pairing component with odd parity is induced near the N/S interface, while in the case of an odd-parity pair potential (spin-triplet $p_x$ wave) the odd-frequency component with even parity is generated. We show that in conventional $s$-wave junctions, the amplitude of the odd-frequency pairing state is strongest in the case of a full-transparency N/S interface and is enhanced at energies corresponding to the peaks in the local density of states (LDOS). In $p_x$- and $d_{xy}$-wave junctions, the amplitude of the odd-frequency component on the S side of the N/S interface is enhanced at zero energy where the midgap Andreev resonant state (MARS) appears due to the sign change of the pair potential. The odd-frequency component extends into the N region and exceeds the even-frequency component at energies corresponding to the LDOS peak positions, including the MARS. At the edge of the N region the odd-frequency component is nonzero while the even-frequency one vanishes. We show that the concept of the odd-frequency pairing state plays a pivotal role to interpret a number of phenomena in nonuniform superconducting systems, like McMillan-Rowell and midgap Andreev resonance states.

Figure 1

(Color online) Spatial dependence of the normalized pair potential, even-frequency pair amplitude, and odd-frequency components of the pair amplitude for $s$-wave superconductor junctions. Here, we choose $\xi =v_F∕{\Delta }_0$ in the S region $(x>0)$ and $\xi =L_0=v_F∕2\pi T_C$ in the N region. The pair amplitudes $C_0^{\left(2\right)}$, $C_1^{\left(1\right)}$, $C_3^{\left(1\right)}$, and $C_5^{\left(1\right)}$ are denoted as even $s$-wave, odd $p_x$-wave, odd $f_1$-wave, and odd $h_1$-wave pair amplitudes. (a) $Z=0$, $L=L_0$, (b) $Z=5$, $L=L_0$, (c) $Z=0$, $L=5L_0$, and (d) $Z=5$, $L=5L_0$, respectively.

Figure 2

(Color online) Energy dependence of the LDOS and the pair amplitudes in $s$-wave junctions with $L=L_0$ and $Z=0$. (a) The LDOS normalized by its value in the normal state. Solid line, LDOS on the S side of the N/S interface; dotted line, LDOS in the N region. Energy dependence of the real (solid line) and imaginary (dotted line) part of (b) even-frequency $s$-wave pair amplitude on the S side of the N/S interface, (c) odd-frequency $p_x$-wave pair amplitude on the S side of the N/S interface, and (d) even-frequency $s$-wave pair amplitude at the edge of the N region.

Figure 3

(Color online) Same as Fig. 2, but with $L=L_0$ and $Z=5$.

Figure 4

(Color online) Same as Fig. 2, but with $L=5L_0$ and $Z=0$.

Figure 5

Ratio of the pair amplitudes $f_{1+}^{\left(N\right)}(\epsilon ,\theta )∕f_{2+}^{\left(N\right)}(\epsilon ,\theta )$ on the N side of the N/S interface in $s$-wave junction as a function of energy $\epsilon$ for $\theta =0$ and $L=5L_0$.

Figure 6

(Color online) Spatial dependence of the normalized pair potential and even-frequency and odd-frequency pair amplitudes for $p_x$-wave superconductor junctions. Here, we choose $\xi =v_F∕{\Delta }_0$ in the S region $(x>0)$ and $\xi =L_0=v_F∕2\pi T_C$ in the N region. The pair amplitudes $C_1^{\left(2\right)}$, $C_0^{\left(1\right)}$, $C_2^{\left(1\right)}$, and $C_4^{\left(1\right)}$ are denoted as even $p_x$-wave, odd $s$-wave, odd $d_{x^2-y^2}$-wave, and odd $g$-wave pair amplitudes. (a) $Z=0$, $L=L_0$, (b) $Z=5$, $L=L_0$, (c) $Z=0$, $L=5L_0$, and (d) $Z=5$, $L=5L_0$, respectively.

Figure 7

(Color online) Energy dependence of the LDOS and the pair amplitudes in $p_x$-wave junctions with $L=L_0$ and $Z=0$. (a) The LDOS normalized by its value in the normal state. Solid line, LDOS on the S side of the N/S interface; dotted line, LDOS in the N region. Energy dependence of the real (solid line) and the imaginary (dotted line) part of (b) even-frequency $p_x$-wave pair amplitude on the S side of the N/S interface, (c) odd-frequency $s$-wave pair amplitude on the S side of the N/S interface, and (d) odd-frequency $s$-wave pair amplitude at the edge of the N region.

Figure 8

(Color online) Same as Fig. 7, but with $L=L_0$ and $Z=5$.

Figure 9

(Color online) Same as Fig. 8, but with $L=5L_0$ and $Z=0$.

Figure 10

Ratio of the pair amplitudes $f_{1+}^{\left(N\right)}(\epsilon ,\theta )∕f_{2+}^{\left(N\right)}(\epsilon ,\theta )$ as a function of $\epsilon$ for $L=5L_0$ at the N side of the N/S interface for $p_x$-wave junctions for $\theta =0$.

Figure 11

(Color online) Spatial dependence of the normalized pair potential and even-frequency and odd-frequency pair amplitudes for $d_{xy}$-wave superconductor junctions. Here, we choose $\xi =v_F∕{\Delta }_0$ in the S region $(x>0)$ and $\xi =L_0=v_F∕2\pi T_C$ in the N region. The pair amplitudes $S_2^{\left(2\right)}$, $S_1^{\left(1\right)}$, $S_3^{\left(1\right)}$, and $S_5^{\left(1\right)}$ are denoted as even $d_{xy}$-wave, odd $p_y$-wave, odd $f_2$-wave, and odd $h_2$-wave pair amplitudes. (a) $Z=0$, $L=L_0$, (b) $Z=5$, $L=L_0$, (c) $Z=0$, $L=5L_0$, and (d) $Z=5$, $L=5L_0$, respectively.

Figure 12

(Color online) Energy dependence of the LDOS and the pair amplitudes in $d_{xy}$-wave junctions with $L=L_0$ and $Z=0$. (a) The LDOS normalized by its value in the normal state. Solid line, LDOS on the S side of the N/S interface; dotted line, LDOS in the N region. Energy dependence of the real (solid line) and the imaginary (dotted line) part of (b) even-frequency $d_{xy}$-wave pair amplitude on the S side of the N/S interface, (c) odd-frequency $p_y$-wave pair amplitude on the S side of the N/S interface, and (d) odd-frequency $p_y$-wave pair amplitude at the edge of the N region.