Zero-field superconducting vortices and Majorana zero modes pinned by magnetic islands in correlated Rashba systems

Abstract

We propose a route for pinning zero-field superconducting vortices in systems which are exchange-coupled to magnetic islands and feature Rashba spin-orbit coupling. We consider islands with sizes which greatly exceed those of the vortex cores and possess out-of-plane magnetic moments. A crucial ingredient of our approach is that it considers superconductors which are governed by magnetic correlations without, however, exhibiting long range magnetic order. The arising total magnetization is inhomogeneous and its gradients generate a nonzero vorticity in the superconducting phase. Vortices become energetically stable due to the energy reduction brought about from the generation of electronic magnetization. Using our developed framework, we make concrete predictions for the emergence of zero-field vortices and Majorana zero modes in superconducting topological insulator surfaces and planar Rashba superconductors. Our theory uncovers a nonstandard path for trapping composite vortex-Majorana excitations in systems which appear to be within experimental reach.

Figures (5)

• Figure 1: Illustration of an extended magnetic impurity, i.e., a magnetic island, which is embedded in a quasi-2D SC of thickness $w$, and is dictated by a Rashba-type SOC. We consider a type-II conventional spin-singlet SC which can harbor superconducting vortices. This implies that its London penetration depth $\lambda_L$ exceeds the superconducting coherence length $\xi_S$. The radius $\rho_I$ of the magnetic island is assumed to be much larger than $\xi_S$. In addition, the spin moment of the magnetic island, which is depicted with black arrows, is assumed to be polarized out of the plane of the Rashba superconducting host. By means of the Zeeman effect and the magnetoelectricity mediated by the Rashba SOC, the spin moment of the island becomes converted into a magnetic flux which induces a vortex with vorticity $\nu_\phi$, under conditions that we specify in this work. Key aspect of our approach is that we also account for possible magnetic correlations that lead to the appearance of electronic magnetization in the SC upon adding the magnetic island. The properties of the induced magnetization are controlled, among others, by the magnetic correlation length $\xi_M$ which is also much larger than $\xi_S$. The spin moment of the island is only exchange-coupled to the electrons of the substrate SC, and is considered to be sufficiently extended to not induce any Yu-Shiba-Rusinov states YSR.
• Figure 2: The heat map shows the induced vorticity $\nu_\phi$ of a superconducting vortex stabilized in the presence of a magnetic island. Here, possible magnetic correlations are not included. The above result is obtained for $\lambda_L=1500 \,\xi_S$, in which case $\nu_{\rm max}=34$. In accordance with our approximate analytical results in Eqs. \ref{['eq:Limitrholargerlambda']}-\ref{['eq:Limitrhosmallerlambda']}, the most prominent regime for the island to induce a large number of vorticity quanta appears for $\rho_I\ll\lambda_L$, while for $\rho_I\gg\lambda_L$ the vorticity $\nu_\phi$ tends to zero.
• Figure 3: Spin-to-flux conversion factor $\tilde{\zeta}$ as a function of the absolute value of the coupling strength $|{\cal G}|\in[0,|{\cal G}_{\rm max}|]$. Here, we focus on the six possible hierarchies for the lengthscales $\{\lambda_L,\xi_M,\rho_I\}$ which are obtained when these three quantities are substantially different. We find that the conversion efficiency becomes the highest (lowest) for $\lambda_L\gg\xi_M,\rho_I$ ($\lambda_L\ll\xi_M,\rho_I$), as this is reflected in panels (c)-(d) ((a)-(b)). Intermediate values of $\tilde{\zeta}$ are correspondingly obtained when $\lambda_L$ is placed in-between $\xi_M$ and $\rho_I$. See panels (e)-(f). We observe that the enhancement of the mixing between the magnetic and magnetization fields does not boost the conversion efficiency. In fact, it greatly suppresses it in all cases except for those in which $\rho_I\ll\lambda_L,\xi_M$. Approximate analytical expressions for $\tilde{\zeta}$ for weak ($|{\cal G}|=0$) and strong ($|{\cal G}|=|{\cal G}|_{\rm max}$) couplings are given in Appendix \ref{['app:AppendixB']}. The value of $\tilde{\zeta}$ for each hierarchy is obtained after multiplying the values in the vertical axis by the factor shown in each graph.
• Figure 4: Results for the parameters $\xi_M$ and ${\cal G}$ in the case of (a) disorder-free TI surface states and (b) a disordered Rashba metal. We find very similar results in both situations. The magnetic correlation length increases in terms of the pa-ra-me-ter $\eta\in[0,0.95)$, which controls the strength of the magnetic interaction. A magnetic instability appears for $\eta_c=1$, which is outside the regime of interest in the present work. We find that the correlations are required to be substantial so that $\xi_M\gg\xi_S$. We also find that the coupling ${\cal G}$ increases upon increasing $\eta$ but generally remains small in a wide range of values, unless one tunes the system extremely close to the magnetic instability. Here we multiplied ${\cal G}$ by a factor of 100, in order to conveniently plot the two quantities together.
• Figure 5: Top view of the system and vortex-MZMs for (a) a Rashba SC and (b) superconducting TI surface states. The MZMs are shown with dashed lines and extend uniformly along these. In (a) a pair of core-rim MZMs emerge due to the topological SC realized in the region enclosed. Instead, only a single domain-wall-MZM appears in (b). The arrows indicate the associated Majorana chiral edge modes of the MZMs.