Spatially Inhomogeneous Triplet Pairing Order and Josephson Diode Effect Induced by Frustrated Spin Textures

Abstract

We demonstrate that frustrated spin textures can generate anisotropic Josephson couplings between $d$-vectors that can stabilize spatially varying pairing orders in spin triplet superconductors. These couplings depend on the relative orientation of $d$-vectors, analogous to Dzyaloshinskii-Moriya and $Γ$-type interactions in magnetism, leading to an effective "pliability" of the pairing order that competes with superfluid stiffness. Such couplings cannot originate from spin-orbit coupling; rather, they can arise, for example, when itinerant electrons are coupled to a local exchange field composed of frustrated spin moments. Using a $T$-matrix expansion, we show that coupling to a local exchange field leads to an effective tunneling of itinerant electrons that is dependent on the underlying spin configurations at the barrier between superconducting grains. Furthermore, Josephson tunneling through frustrated spin textures can produce a Josephson diode effect. The diode effect originates either from nonvanishing spin chirality in the barrier, or from antisymmetric Josephson coupling between noncollinear $d$-vectors, both of which break inversion and time-reversal symmetries.

Figures (8)

• Figure 1: Discretized Josephson free energy describing the contribution from a spatially varying order. We consider a system consisting of multiple weakly-linked superconducting grains. For an $s$-wave superconductor, the pairing correlation defined locally for the $n\mathrm{th}$ superconducting grain is given by $\psi_n$. The discretized superfluid stiffness $J_{nm}$ describes the Josephson coupling between grains $n$ and $m$.
• Figure 2: Josephson couplings between superconducting grains $n$ and $m$ described by $d$-vectors $\hat{\mathbf{d}}_n$ and $\hat{\mathbf{d}}_m$. (a) Heisenberg-like coupling, corresponding to the discretized superfluid stiffness, which leads to collinear $d$-vector configurations. (b) DM-like Josephson coupling, which can lead to a noncollinear $d$-vector textures, leading to an effective pliability in the superconducting order.
• Figure 3: Local spin moments and effective tunneling for the three-sublattice systems for the (a) kagome and (b) triangular lattices. Nearest neighbor hopping between sites $i$ and $j$ on the boundaries of superconducting grains $n$ and $m$ consist of nearest neighbor hopping and effective tunneling from the underlying spin texture. In addition, there are higher order tunneling processes mediated by third spin $\mathbf{s}_k$.
• Figure 4: (a) Fermi surfaces for the $s$-$d$ model on the kagome lattice, shown in the first Brillouin zone. States are plotted according to the azimuthal angle of their in-plane spin, $\arctan(\langle s_y \rangle/ \langle s_x \rangle)$. Parameters are $\mu = -1.6 |t_0|$ and $J_{sd} = 0.2 |t_0|$. (b) Fermi surfaces $s$-$d$ model on the triangular lattice, shown in the first Brillouin zone. States are colored on a continuum according to $z$-component of spin, $\langle s_z \rangle$. Parameters are $\mu = -0.1 |t_0|$, $J_{sd} = 0.1 |t_0|$, and $\lambda_\mathrm{SO} = 0.3 |t_0|$. For both the kagome and triangular lattices, the local spins $\mathbf{s}_a$, $\mathbf{s}_b$, and $\mathbf{s}_c$ form a coplanar $120^\circ$ ordered antiferromagnetic configuration corresponding to $\theta_0 = 0$, $\varphi_0 = 0$, and $\nu = -1$ in Eq. \ref{['three_spins']}.
• Figure 5: BdG excitation spectra and spin triplet pairing correlations for systems on the (a) kagome lattice and (b) triangular lattice. The spectrum is colored on a continuum according to the magnitude of the total spin triplet pairing correlations, including inter- and intrasublattice pairing. Parameters for the system are the same as those in Fig. \ref{['fig:FS_spin_texture']}, and the magnitude of the pairing gap function is $\Delta_0 = 0.05 |t_0|$.