Josephson effect due to the long-range odd-frequency triplet superconductivity in $\mathrm{S}\mathrm{F}\mathrm{S}$ junctions with Néel domain walls

We consider a $\mathrm{S}\mathrm{F}\mathrm{S}$ Josephson junction made of two superconductors $\mathrm{S}$ and a multidomain ferromagnet $\mathrm{F}$ with an in-plane magnetization. We assume that the neighboring domains of the ferromagnet are separated by Néel domain walls. An odd-frequency triplet long-range component of superconducting correlations arises in the domain walls and spreads into the domains over a long distance of the order ${\xi }_T=\sqrt{D∕2\pi T}$, where $D$ is the diffusion coefficient (dirty limit is implied). We calculate the contribution of this component to the Josephson current in the situation when conventional short-range components exponentially decay over the thickness of the $\mathrm{F}$ layer and can be neglected. In the limit when the thickness of the $\mathrm{F}$ layer is much smaller than the penetration length of the long-range component, we find that the junction is in the $\pi$ state. We also analyze a correction to the density of states due to the long-range triplet component.

Figure 1

$\mathrm{S}\mathrm{F}\mathrm{S}$ junctions considered in the paper. (a) The domain $(y<0)$ and the region with rotating magnetization $(y>0)$ in the $\mathrm{F}$ layer are half-infinite. (b) Multidomain $\mathrm{F}$ layer. Depending on the relative orientation of rotating magnetizations in the neighboring domain walls, we distinguish the cases of positive and negative chirality ($Q$ has the same or opposite sign in the neighboring domain walls, respectively).

Figure 2

$\mathrm{S}\mathrm{F}\mathrm{S}$ junction with a half-infinite domain: absolute value of the critical current density $j_{cx}$ in the $x$ direction as a function of the $y$ coordinate at low temperatures. The current is negative, i.e., the $\pi$ state of the junction is realized. The normalization constant is $j_0=\sigma D{Q}^3∕8e{\left(d{\gamma }_b{k}_h^2\right)}^2$. Other parameters: $D{Q}^2=\Delta$ and $Qd=0.1$.

Figure 3

Multidomain $\mathrm{S}\mathrm{F}\mathrm{S}$ junction: the absolute values of the critical currents $J_{c+}$ (positive chirality) and $J_{c-}$ (negative chirality) per period of the domain structure as functions of the domain half-width $a_0$ at low temperatures. The currents are negative, i.e., the $\pi$ state of the junction is realized. The normalization constant is $J_0=\sigma D{Q}^3∕4ed{\gamma }_b^2{k}_h^4$. Other parameters: $D{Q}^2=\Delta$ and the rotation of magnetization in the domain walls corresponds to $2Q{a}_Q=\pi$.

Figure 4

Multidomain $\mathrm{F}$ layer of thickness $d$ in contact with a bulk superconductor. Depending on the relative orientation of rotating magnetizations in neighboring domain walls, we distinguish the cases of positive and negative chirality ($Q$ has the same or opposite sign in the neighboring domain walls, respectively). The proportion between the widths of domains and domain walls is chosen only for drawing purposes.

Figure 5

Addition to Fig. 2 from Ref. 28: Correction $\delta \nu \left(y\right)$ (due to the proximity effect) to the DOS at the free surface of the $\mathrm{F}$ layer in the case of positive chirality. The curves are plotted at several energies $\epsilon$. The width of the domains is $a_0=5∕Q$, while the rotation of magnetization in the domain walls corresponds to $Q{a}_Q=\pi$.

Figure 6

Correction $\delta \nu \left(\epsilon \right)$ to the DOS at the free surface of the $\mathrm{F}$ layer in the case of positive chirality. The curves are plotted at several points $y$. Other parameters are the same as in Fig. 5.