Magnetoelectric effects and spin switching phenomena at the interface of chiral domains in spin-triplet superconductors

Spin-triplet superconductors with time-reversal symmetry breaking can naturally lead to chiral domain walls in their interior. We study the magnetic properties emerging at the interface of chiral domains with opposite winding by focusing on the effects of a superconducting phase drop across the wall and an applied electric gating. The local inversion symmetry breaking at the domain wall drives mixed singlet-triplet pairing configurations that allow a phase- and electric-controllable magnetization with resulting parallel or antiparallel orientations on the two sides of the domain wall. The magnetic switching is also generally accompanied by both spin and charge currents flowing along the edges, whose amplitudes depend on the achieved parallel or antiparallel magnetic phase. The specific magnetoelectric properties of chiral domains with zero or nonvanishing net magnetization near the wall may have several implications. They can be employed to detect the presence of unconventional pairing as well as to design information storage units based on spin-polarized states attached to topological defects.

Figure 1

Schematic view of the chiral domain wall structure made of two interfacing chiral spin-triplet superconductors with opposite windings. We depict representative $d_z$-vector configurations corresponding to a nonzero superconducting triplet order parameter and zero total spin projection along the $z$ direction in the spin space, thus implying that the parallel-spin Cooper pairs lie in the $xy$ plane, as schematically indicated by the magenta arrows. The domain wall is denoted by a black dashed line to separate the two regions with opposite windings of the superconducting order parameter for each spin direction. The arrows (green and red for up and down spins, respectively) along the domain wall stand for the spin-dependent charge currents due to the chiral symmetry of the superconductor and the presence of Andreev bound states. The $i_x=0$ position labels the interface site between the two chiral domains. Each chiral domain has a finite size of $(L_x/2)\times L_y$ sites. For our simulation we assume that $L_x=L_y=120$. Gating through $V_G$ results in electrically tuning the effective interface potential amplitude $U$ by suitably depleting or increasing the electron density near the domain wall. The two superconducting domains feel opposite phases, which is modeled by a spatially dependent factor $exp[\text{sgn}\left(i_x\right)i\varphi /2]$ which multiplies the spin-triplet order parameter.

Figure 2

Phase diagram of the most favorable magnetic configurations as a function of the applied phase difference across the chiral domain wall and of the interface potential $U$ (in units of the planar hopping amplitude in the superconductor). M1 denotes a magnetic state with a small net spin polarization pinned at the domain wall position. NM stands for a vanishing average magnetization with opposite orientations on the two sides of the chiral interface. Finally, M2 denotes a state with a magnetization profile exhibiting a nonvanishing net spin moment with a maximal amplitude away from the domain wall at a characteristic distance which is of the order of the superconducting coherence length. Big red arrows schematically depict the amplitude and orientation of the spin polarization close to the chiral interface.

Figure 3

(a) Integrated magnetization $m_{\mathrm{tot}}$ close to the chiral interface as a function of the superconducting phase difference $\varphi$ between the two domains, assuming different amplitudes of the interface barrier potential $U$ from 0 to 0.6 in units of the planar hopping value. (b) Evolution of the difference of the free energy associated with the magnetic ($E_{\mathrm{M}2}$) and nonmagnetic ($E_{\mathrm{NM}}$) configurations as a function of the superconducting phase drop $\varphi$ and by varying the strength of the interface potential $U$ towards a regime of significant electron density depletion at the chiral domain wall. (c)–(f) Spatial dependence of the magnetization (i.e., $n_{\uparrow }-n_{\downarrow }$) near the chiral domain wall (i.e., at $i_x=0$) at different values of the phase drop across the chiral interface, moving from small to large potential amplitudes. For $U=0.3$ [in (c)] a magnetization forms and is peaked at the position of the domain wall. The increase of the interface potential makes the spin polarization penetrating into the two chiral domains in an antiparallel configuration [in (d)], until a critical value of $U$ is reached, above which the spin polarization acquires the same orientation on both domains.

Figure 4

Spatial profile of (a) and (b) the amplitude and (c) phase of the superconducting order parameters in the proximity of the chiral domain wall for two representative values of the interface potential. The values $U=1$ and $U=1.6$ correspond to phases NM and M2, respectively.

Figure 5

Spatial dependence for $U=1$ of the phase of the superconducting order parameter close to the domain wall considering for the applied phase difference $\mathrm{\Delta }\varphi$ the input step-function profile ($it=0$) and the one obtained after a number $it=20$ of iterations. Two values of $\mathrm{\Delta }\varphi$ are considered, i.e., $\mathrm{\Delta }\varphi =\pi /2$ (top panel) and $\mathrm{\Delta }\varphi =\pi$ (bottom panel).