Identifying the odd-frequency pairing state of superconductors by a field-induced Josephson effect

Superconducting order parameters that are odd under exchange of time coordinates of the electrons constituting a Cooper pair are potentially of great importance both conceptually and technologically. Recent experiments report that such an odd-frequency superconducting bulk state may be realized in certain heavy-fermion compounds. While the Josephson current normally only flows between superconductors with the same symmetries with respect to frequency, we demonstrate that an exchange field may induce a current between diffusive even- and odd-frequency superconductors. This suggests a way to identify the possible existence of bulk odd-frequency superconductors.

Figure 1

(Color online) A diffusive metal, which may be either normal or ferromagnetic (with exchange field $h$) of width $d$, separates two superconductors with a phase difference $2\chi$, thus constituting a Josephson current. The individual superconductors have either an even-frequency or odd-frequency bulk symmetry. The exchange field may have any orientation in the $yz$ plane, which is characterized by the angle $\varphi$.

Figure 2

(Color online) Plot of the singlet $f_s$ (thick, blue lines) and triplet $f_t$ (thin, red lines) anomalous Green’s function induced in a $N$ (upper panels) and $F$ (lower panel) in proximity with an even-frequency and/or odd-frequency superconductors by using $\chi =\pi /4$ and $\epsilon /{\Delta }_0=0.1$. The solid lines are the real parts of the anomalous Green’s functions, while the dashed lines are the imaginary parts. To ensure that $f_{s,t}⪡1$, we take $d/\xi =50$ in the $N$ case and $d/\xi =2$, $h/{\Delta }_0=50$ in the $F$ case. Here, $\xi =\sqrt{D/\left(2\pi T_c\right)}$ is the superconducting coherence length. While a Josephson current flows when the symmetries of the superconductors are equal in the $N$ case, it vanishes when the superconductors have different symmetries. This is because the triplet and singlet components are absent at the interface of the even- and odd-frequency superconductors, respectively. Replacing the $N$ with a $F$ establishes a Josephson coupling between superconductors with different symmetries.

Figure 3

(Color online) (a) The critical current $\left(\chi =\pi /4\right)$ as a function of $d/\xi$ for $T/T_{\text{c}}=0.001$. (b) Deviation from the LDOS at $x=d/2$ for an even-frequency/$F$/odd-frequency junction with $\chi =\pi /4$ and $h/{\Delta }_0=50$. The zero-energy peak is clearly discernible, which originates with the odd-frequency pairs.

Figure 4

(Color online) Field dependence of the critical current for several values of $d/\xi$ with $T/T_c=0.001$.

Figure 5

(Color online) Plot of the deviation from the normal-state zero-energy density of states in the middle of the ferromagnetic region $\left(x/d=0.5\right)$. The symbols denote approximately the region where the linearized treatment becomes less accurate, i.e., the anomalous Green’s functions $f$ no longer satisfy $\left|f\right|⪡1$.

Figure 6

(Color online) Plot of the modulation of the Josephson current with orientation $\varphi$ of the exchange field in the ferromagnet. The symbols correspond to an even-frequency/$F$/odd-frequency junction, while the dashed line denotes an even-frequency/$F$/even-frequency junction with $h/{\Delta }_0=30$ and $d/\xi =2.0$.

Figure 7

(Color online) Plot of the current-phase relationship for several values of $\varphi$. We have chosen $h/{\Delta }_0=30$ and $d/\xi =2.0$.