Majorana zero modes in a magnetic and superconducting hybrid vortex

We propose and investigate a new platform for the realization of Majorana zero modes in a thin-film heterostructure composed of an easy-plane ferromagnet and a superconductor with spin-orbit coupling. The system can support an energetically favorable bound state comprising a magnetic vortex and a superconducting vortex. We show that a hybrid vortex thus created can host a robust zero-energy Majorana bound state at its core over a wide range of parameters, with its partner zero mode located at the boundary of a disk-shaped topological region. We identify a novel mechanism underlying the formation of the topological phase that, remarkably, relies on the orbital effect of the magnetization field and not on the usual Zeeman effect. The in-plane components of magnetization couple to electrons as a gauge potential with nonzero curl, thus creating an emergent magnetic field responsible for the gapped topologically nontrivial region surrounding the vortex core. Our construction allows the mobility of magnetic vortices to be imposed on the Majorana zero mode at the core of the superconducting vortex. In addition, the system shows a rich interplay between magnetism and superconductivity which might aid in developing future devices and technologies.

Figure 1

System comprising a ferromagnet-superconductor heterostructure. The magnetic and superconducting vortex form an energetically favorable hybrid vortex which hosts Majorana modes.

Figure 2

Left two panels: Energetically favorable regions in parameter space for hybrid vortex formation. With the magnetic vortex texture (${\xi }_m=1$) fixed at the center of the system of size $L=21$, the superconducting vortex core (${\xi }_s=1$) is moved along the diagonal starting at the position. The yellow shaded regions in all plots correspond to the points in the parameter space where the free-energy minima roughly correspond to the limit where the superconducting and the magnetic vortex are perfectly superimposed on each other (i.e., $r_{sm}=0$) thereby forming a stable hybrid vortex. Regions of stability calculated for a magnetic vortex and superconducting vortex pair ($n_s=n_m=1$) and for a magnetic vortex and a superconducting antivortex pair ($n_s=-1,n_m=1$). Right two panels: Calculation of free energy of the hybrid vortex as a function of the distance between the magnetic and superconducting vortex. Here $F_0$ is the free energy when $r_{sm}=0$. With the magnetic vortex texture fixed at the center of the system, the superconducting vortex core is moved along the diagonal. Free energy is plotted for different magnetic decay length parameters ${\xi }_m$ at fixed $m_0=-0.35$ and for different exchange coupling $m_0$ at fixed ${\xi }_m=1$. The distance between the magnetic and the superconducting vortex is normalized with respect to ${\xi }_m$. Here $\mathrm{\Delta }=0.3, \mu =0, t=1$.

Figure 3

[(a) and (b)] System with only superconducting vortex with external Zeeman field [equivalent to hybrid vortex with ${\xi }_m\to \infty \left)\right]$. (a) The LDOS, $\rho (r_s,\epsilon )$, shows prominent zero-bias peak corresponding to Majorana zero modes. Spin-polarized LDOS reveals that the main contribution to the central Majorana zero-bias peak is from the spin-up LDOS. (b) The 2D plot shows LDOS ($\log \left[\rho \right(r,\epsilon =0\left)\right]$) plotted at zero energy shows a Majorana mode localized at the vortex core. [(c) and (d)] System with only magnetic vortex (with ${\xi }_m=1$) without any superconducting vortex. (c) LDOS at the vortex core shows that there is no isolated zero-bias peak. (d) The LDOS plotted at zero energy shows a ringlike structure ($R_{\text{mv}}$) surrounding the vortex core. The black dotted circle depicts the radius at which the local excitation gap vanishes [see Eq. (22)]. [(e)–(h)] System with hybrid vortex (${\xi }_m={\xi }_s=1$). (e) The LDOS, $\rho (r_s,\epsilon )$, shows prominent zero-bias peak corresponding to Majorana zero modes. Spin-polarized LDOS reveals that the main contribution to the central Majorana zero-bias peak is from the spin-up LDOS, as expected when $n_s=n_m$. (f) The 2D plot shows LDOS ($\log \left[\rho \right(r,\epsilon =0\left)\right]$) plotted at zero energy shows a Majorana mode localized at the vortex core. Additionally, there is a ring (radius $R_{\text{hv}}$) surrounding the core where the excitation gap vanishes. The radius, shown by the dotted black circle, is estimated where the local excitation gap in equation Eq. (B6) vanishes. [(g) and (h)] The decoupled wave-function probabilities, $|{\varphi }_e{|}^2$ and $|{\varphi }_c{|}^2$, corresponding to the two zero modes show that one Majorana mode is localized at the core while the other is present at the outer ring surrounding the vortex. Parameters: $L=75, \mathrm{\Delta }=0.3, m_0=-0.35, \phi =0, u=0.6, \mu =0, t=1$.

Figure 4

Numerical simulations of a hybrid vortex on a $75\times 75$ lattice. (a) The LDOS, $\rho (r_s,\epsilon )$, for varying magnetic vortex decay lengths ${\xi }_m$ and fixed exchange coupling strength, $m_0=-0.35$. (b) LDOS at the vortex core for varying exchange coupling, $m_0$, at fixed magnetic vortex decay length, ${\xi }_m=1$. LDOS exhibits a zero-bias peak for $|m_0|\gtrsim 0.33$. Parameters: $L=75, \mathrm{\Delta }=0.3, {\xi }_s=1, u=0.6, \mu =0$, and $t=1$. (c) LDOS at different values of magnetic vortex decay length, ${\xi }_m$ (with $m_0=-0.35$). The plot shows a diagonal cross section of the LDOS at zero energy for different values of ${\xi }_m$ (values depicted by the color bar). The central zero-bias peak suppressed in order to highlight the density at the outer ring using the envelope function $(1-{\mathrm{e}}^{-|{\delta }_{\mathbf{rs}}|})$, where $|{\delta }_{\mathbf{rs}}|=|\mathbf{r}-{\mathbf{r}}_{\mathbf{s}}|$. Outside the shaded gray region, the spin texture flattens out (up to ${10}^{-5}{m}_z^{\mathrm{center}}$). The peaks corresponding to the ring structure lie outside the shaded region and remain pinned at the same location with increasing decay length until ${\xi }_m\le 8$ and begin to diverge thereafter due to edge effects. (d) LDOS at different values of exchange coupling $m_0$ (with ${\xi }_m=1$). The plot shows a diagonal cross section of the LDOS at zero energy with the central zero-bias peak suppressed. (e) Representative 2D zero-energy LDOS plots are shown at magnetic vortex decay lengths, ${\xi }_m=(3,5,8)$ with fixed $m_0=-0.35$. (f) Representative 2D zero-energy LDOS plots are shown at magnetic vortex decay lengths, $m_0=(-0.36,-0.39,-0.42)$ with fixed ${\xi }_m=1$. Radius $R_{\text{hv}}$ calculated from Eq. (23) is shown by dotted black circles in (e) and (f). Parameters: $L=75, \mathrm{\Delta }=0.3, \phi =0, u=0.6, \mu =0$, and $t=1$.

Figure 5

Supercurrents in a system with uniform superconducting order parameter and a magnetic vortex with helicities $\phi =(0,\frac{\pi }{6},\frac{\pi }{3},\frac{\pi }{2})$. The phase of the superconducting order parameter is given by Eq. (C2). Parameters: $L=75, \mathrm{\Delta }=0.3, m_0=-0.3, {\xi }_s={\xi }_m=1, u=0.6, \mu =0$, and $t=1$.

Figure 6

[(a)–(d)] Local supercurrents plotted for different values of magnetic vortex decay lengths, ${\xi }_m=(\infty ,9,6,3)$. Case (a) corresponds to a system with superconducting vortex and uniform Zeeman field. [(e)–(h)] Local supercurrents plotted for exchange coupling strengths, $m_0=(-0.35,-0.37,-0.39,-0.41)$. The vectors (arrows) correspond to the direction of the flow of the supercurrent given by the vector ${\mathbf{J}}_{xy}$. The color bars represent the magnitude of $J_{xy}$. The insets show the orientation of the local supercurrents around the vortex core. Parameters: $L=75, \mathrm{\Delta }=0.3, {\xi }_s={\xi }_m=1, \phi =0, u=0.6, \mu =0$, and $t=1$.

Figure 7

Spatial distribution of the decoupled Majorana wave functions ${\varphi }_{e/c}$ and the ordinary CdGM wave functions (energy ${\epsilon }_1$) as a function of distance from the hybrid vortex core. The CdGM state corresponds to a dominant nonzero energy peak at ${\epsilon }_1=0.14\mathrm{\Delta }$ in the LDOS shown in Fig. 3. Here the system parameters are the same as those in Figs. 3–3 of the main text.

Figure 8

Top: Energy gap $E_g$ from Eq. (B6) for the hybrid vortex as a function of the distance from the core. Vanishing $E_g$ defines the critical radius $R_{\text{hv}}$ which is dependent on $m_0$. Bottom: $log\left(E_g\right)$ plotted for representative values of $m_0$ to illustrate the disk shaped topological region of radius $R_{\text{hv}}$. Parameters: $L=75, \mathrm{\Delta }=0.3, {\xi }_s={\xi }_m=1, \phi =0, u=0.6, \mu =0$, and $t=1$.

Figure 9

The plot shows the radius of the topological region of the magnetic vortex ($R_{\mathrm{mv}}^{\mathrm{eqn}}$) given by Eq. (22) as a function of the exchange coupling parameter and compares it to the radius of the ring obtained by simulations ($R_{\mathrm{mv}}^{\mathrm{sim}}$) for ${\mathrm{\Delta }}_0=0.3$ (top panel) and ${\mathrm{\Delta }}_0=0.4$ (bottom panel). The plot also compares the radius of the topological region around the hybrid vortex ($R_{\mathrm{hv}}^{\mathrm{eqn}}$) calculated from Eq. (B6) and the radius of the ring around the hybrid vortex obtained from the simulations ($R_{\mathrm{hv}}^{\mathrm{sim}}$).

Figure 10

Supercurrents for a pure magnetic vortex in the trivial phase (left) and the topological phase (right). The topological phase transition occurs at the critical exchange coupling $|m_0|\approx 0.33$ when $\mathrm{\Delta }=0.3$ for the chosen parameters. The phase of the superconducting order parameter is given by Eq. (C2).