Majorana states in prismatic core-shell nanowires

We consider core-shell nanowires with conductive shell and insulating core and with polygonal cross section. We investigate the implications of this geometry on Majorana states expected in the presence of proximity-induced superconductivity and an external magnetic field. A typical prismatic nanowire has a hexagonal profile, but square and triangular shapes can also be obtained. The low-energy states are localized at the corners of the cross section, i.e., along the prism edges, and are separated by a gap from higher energy states localized on the sides. The corner localization depends on the details of the shell geometry, i.e., thickness, diameter, and sharpness of the corners. We study systematically the low-energy spectrum of prismatic shells using numerical methods and derive the topological phase diagram as a function of magnetic field and chemical potential for triangular, square, and hexagonal geometries. A strong corner localization enhances the stability of Majorana modes to various perturbations, including the orbital effect of the magnetic field, whereas a weaker localization favorizes orbital effects and reduces the critical magnetic field. The prismatic geometry allows the Majorana zero-energy modes to be accompanied by low-energy states, which we call pseudo Majorana, and which converge to real Majoranas in the limit of small shell thickness. We include the Rashba spin-orbit coupling in a phenomenological manner, assuming a radial electric field across the shell.

Figure 1

Probability distributions of corner and side states of an electron in a polygonal ring. The upper row illustrates the corner states (including the ground state) and the lower row the side states: (a) triangle, (b) square, (c) hexagon. All polygons have radius $R=50$ nm (center-to-corners) and side thickness $t=9$ nm.

Figure 2

Low-energy spectra for the polygonal rings shown in Fig. 1. The corner states are nearly degenerated for the triangle (a) and have an increasing dispersion for the square (b) and hexagon (c). The insets show the energy span of the corner states which is denoted by ${\gamma }_{\mathrm{T},\mathrm{S},\mathrm{H}}$. The gap between corner and side states, indicated as ${\mathrm{\Gamma }}_{\mathrm{T},\mathrm{S},\mathrm{H}}$, decreases in the same order. In the numerical calculations we used $m_{\mathrm{eff}}=0.014$, as for InSb.

Figure 3

The effective Rashba electric field for a triangle and a square in the prismatic SOI model. The corner areas span angular intervals of ${40}^{\circ }$.

Figure 4

Energy dispersions for the corner states of infinite nanowires with polygonal shells including SOI within the cylindrical model. The top row shows $2N$ bands for the three polygonal shapes: (a) triangular, $N=3$, (b) square, $N=4$, (c) hexagonal, $N=6$. Due to the geometric symmetries all bands are degenerate, except one pair for the triangle, crossing at $k=0$, as shown in the zoomed inset. The bottom plots show the results with a transverse electric field of 0.22 $\mathrm{mV}/R$ which perturbs the symmetry of each polygon, lifting all spin degeneracies at $k\ne 0$. Here the radius $R=50$ nm and side thickness $t=9$ nm. The material parameters are as for InSb.

Figure 5

Triangular shell. BdG eigenenergies vs the wave vector for a triangular shell of infinite length. The superconductor gap closes and reopens depending on the values of the chemical potential $\mu$ and magnetic field $B$. The nanowire radius $R=50$ nm is fixed. The shell thickness is $t=9$ nm, i.e., $\mathrm{AR}=0.18$ for the cases (a)–(d). Other parameters: (a) $\mu =236.2$ meV, $B=0$, (b) $\mu =234.6$ meV, $B=1.32$ T, (c) $\mu =234.7$ meV, $B=1.32$ T, (d) $\mu =236.2$ meV, $B=1.32$ T. Next, $t=12.5$ nm, i.e., $\mathrm{AR}=0.25$, for (e) $\mu =126.7$ meV, $B=0.34$ T, and (f) $\mu =126.7$ meV, $B=1.32$ T.

Figure 6

Square shell. BdG energy spectra similar to those shown in Fig. 5, for a nanowire of radius $R=50$ nm. The shell thickness is $t=9$ nm, i.e., $\mathrm{AR}=0.18$ for the cases (a) $\mu =277.6$ meV, $B=0$, (b) $\mu =277.6$ meV, $B=0.33$ T, (c) $\mu =276.5$ meV, $B=0.79$ T, (d) $\mu =276.5$ meV, $B=1.32$ T. Next, $t=8$ nm, i.e., $\mathrm{AR}=0.16$ for (e) $\mu =346.4$ meV, $B=0.66$ T, and (f) $\mu =347.3$ meV, $B=0.66$ T.

Figure 7

Hexagonal shell. BdG spectra, as in Figs. 5 and 6, for a nanowire of radius $R=50$ nm and shell thickness $t=9$ nm, i.e., $\mathrm{AR}=0.18$. (a) $\mu =286.9$ meV, $B=0$, (b) $\mu =286.9$ meV, $B=0.062$ T, (c) $\mu =286.9$ meV, $B=0.26$ T, (d) $\mu =287.3$ meV, $B=0.039$ T, (e) $\mu =287.3$ meV, $B=0.17$ T, (f) $\mu =287.3$ meV, $B=0.33$ T.

Figure 8

Examples of BdG energy spectra for a nanowire of 10 000 nm length and 50 nm radius for selected values of the chemical potential and Zeeman energy and two shell geometries: triangular, the top row, with thickness $t=9$ nm and (a) $\mu =234.6$ meV, (b) $\mu =234.7$ meV, (c) $\mu =236.2$ meV, all with $B=1.32$ T; square, the bottom row, with (d) $t=9$ nm, $\mu =277.6\mathrm{meV}B=0.33\mathrm{T}$, and also $t=8$ nm where (e) $\mu =346.4\mathrm{meV}$ and (f) $\mu =347.3\mathrm{meV}$ both with $B=0.66$ T.

Figure 9

Energy dependence of the lowest eigenvalue of matrix $\mathcal{M}$ in a semi-infinite triangular wire. The zeros indicate the physically acceptable energies for states attached to $z=0$ while decaying for increasing $z$. Panels (a) to (d) correspond to a PMP configuration with increasing shell radius and fixed $t=40$ nm and $B=0.73$ T. Panels (e) to (h) correspond to a fixed $R=110$ nm and $t=37.5$ nm, and decreasing magnetic fields, with configurations PMP (e), PP (f), M (g), and trivial (h). Other parameters: $\mu =22.7\mathrm{meV}, \alpha =50\mathrm{meV}\mathrm{nm}, \mathrm{\Delta }=0.5\mathrm{meV}$.

Figure 10

Density corresponding to the M state of Fig. 9 (at zero energy). Dots show the $z$ evolution of the 1D density integrated in the transverse directions, while the inset shows in a color scale the transverse pattern for $z=150\mathrm{nm}$, near the maximum of the integrated 1D density.

Figure 11

Triangular shell with $R=50$ nm. In panels (a)–(c) the dark blue color indicates the topological Majorana phases and the yellow color the trivial phases, the frontiers being defined by the gap closed at $k=0$. In (d) the colors indicate the minimum gap at all $k$ values, on the ${log}_{10}$ scale. (a) Symmetric triangle with $t=9$ nm, with corner states almost threefold degenerated. (b) The same triangle in the presence of the weak transverse electric field which removes the degeneracy such that the phase boundary splits in three. (c),(d) A thicker shell, with $t=12.5$ nm.

Figure 12

Square shell with $R=50$ nm. The topological (Majorana) and trivial phases are shown in dark blue and yellow, respectively, with frontiers defined by the gap closed at $k=0$, whereas the smallest gap at all $k$ values is shown with continuous colors (on ${log}_{10}$ scale), like before. (a),(b) $t=9$ nm, (c), (d) $t=8$ nm.

Figure 13

Hexagonal shell with $R=50$ nm and $t=9$ nm. (a),(b) The phase diagrams are shown with the same color schemes as before (Figs. 11 and 12). (c),(d) The results after excluding the orbital Zeeman effect of the magnetic field.

Figure 14

Schematic representation of the toy model construction for a triangular wire. The shell (yellow/light gray) is coarse grained so that the vertices and the sides are represented by one-dimensional chains (red/dark gray circles). The arrows indicate the direction of the effective magnetic field ${\mathit{n}}_{\ell }$ associated with the (longitudinal) Rashba spin-orbit coupling.

Figure 15

(a) Topological phase diagram for a symmetric triangular wire, $V_{\mathrm{eff}}=0$. The white areas are topologically trivial, while the orange (gray) regions correspond to $\mathcal{M}=-1$. (b) Dependence of the minimum quasiparticle gap on the Zeeman field for $\mu =-5.4\mathrm{meV}$ [blue cut in panel (a)]. (c) Dependence of the minimum quasiparticle gap on the Zeeman field for $\mu =-4.4\mathrm{meV}$ [dark red cut in panel (a)]. The white/orange regions correspond to the phases shown in panel (a). Note the vanishing of the quasiparticle gap in certain parameter regions.

Figure 16

(a) Topological phase diagram as a function of the chemical potential and applied Zeeman field for a slightly asymmetric triangular wire, with $V_{\mathrm{eff}}=(0.67,-0.33,-0.33)$ meV at corners. The white and orange (gray) phases are topologically trivial and nontrivial, respectively. (b) Dependence of the minimum quasiparticle gap on the Zeeman field for $\mu =-5.4\mathrm{meV}$ [blue cut in panel (a)]. (c) Dependence of the minimum quasiparticle gap on the Zeeman field for $\mu =-4.4\mathrm{meV}$ [dark red cut in panel (a)]. Note that all superconducting phases are gapped.