Manifestation of the odd-frequency spin-triplet pairing state in diffusive ferromagnet/superconductor junctions

Using the quasiclassical Green’s function formalism, we study the influence of the odd-frequency spin-triplet superconductivity on the local density of states (LDOS) in a diffusive ferromagnet (DF) attached to a superconductor. Various possible symmetry classes in a superconductor are considered which are consistent with the Pauli’s principle: even-frequency spin-singlet even-parity (ESE) state, even-frequency spin-triplet odd-parity (ETO) state, odd-frequency spin-triplet even-parity (OTE) state, and odd-frequency spin-singlet odd-parity (OSO) state. For each of these states, the pairing state in the DF is studied. Particular attention is paid to the study of spin-singlet $s$-wave and spin-triplet $p$-wave superconductors as the examples of ESE and ETO superconductors. For the spin-singlet case the magnitude of the OTE component of the pair amplitude is enhanced with the increase of the exchange field in the DF. When the OTE component is dominant at low energy, the resulting LDOS in the DF has a zero-energy peak (ZEP). On the other hand, in DF/spin-triplet $p$-wave superconductor junctions the LDOS has a ZEP in the absence of the exchange field, where only the OTE pairing state exists. With the increase of the exchange field, the ESE component of the pair amplitude induced in the DF is enhanced. Then, the resulting LDOS has a ZEP splitting. We demonstrate that the appearance of the dominant OTE component of the pair amplitude is the physical reason for the emergence of the ZEP of the LDOS.

Figure 1

(Color online) Schematic illustration of DF/S junctions where the DF is connected to normal reservoirs. (a) Conventional spin-singlet $s$-wave superconductor and (b) spin-triplet $p$-wave superconductor junctions.

Figure 2

(Color online) Real (a) and imaginary (b) parts of $f_3$ and real (c) and imaginary (d) parts of $f_0$ in spin-singlet $s$-wave superconductor junctions. We choose $R_d∕R_b=1$.

Figure 3

(Color online) Normalized LDOS as a function of $\epsilon$ for $R_d∕R_b=1$ with various $h∕\Delta$ in spin-singlet $s$-wave superconductor junctions.

Figure 4

(Color online) Spatial dependence of the pair amplitudes $f_3$ and $f_0$ in the DF for $\epsilon =0$ in spin-singlet $s$-wave superconductor junctions. For $\epsilon =0$, $\mathrm{Im}{f}_3=0$ and $\mathrm{Re}{f}_0=0$ are satisfied.

Figure 5

(Color online) The pair amplitudes $f_3$ and $f_0$ as a function of $h$ in the DF for $\epsilon =0$ in spin-singlet $s$-wave superconductor junctions. (a) $x=0$, (b) $x=L∕2$, and (c) $x=L$.

Figure 6

(Color online) Pair amplitudes for DF/spin-triplet $p$-wave superconductor junctions. Real (a) and imaginary (b) parts of $f_3$. Real (c) and imaginary (d) parts of $f_0$. Here we choose $R_d∕R_b=0.1$.

Figure 7

(Color online) Normalized LDOS as a function of $\epsilon$ for $R_d∕R_b=0.1$ and various $h∕\Delta$ in $p$-wave superconductor junctions.

Figure 8

(Color online) Spatial dependence of the pair amplitudes $f_3$ and $f_0$ in the DF for $\epsilon =0$ in $p$-wave superconductor junctions. For $\epsilon =0$, $\mathrm{Im}{f}_3=0$ and $\mathrm{Re}{f}_0=0$ are satisfied.

Figure 9

(Color online) The pair amplitudes $f_0$ and $f_3$ as a function of $h$ in the DF for $\epsilon =0$ in $p$-wave superconductor junctions. (a) $x=0$, (b) $x=L∕2$, and (c) $x=L$.