Intrinsic emergence of Majorana modes in Luttinger $j=\frac{3}{2}$ systems

We analyze theoretically two different setups for $s$-wave superconductivity proximitized $j=\frac{3}{2}$ particles in Luttinger materials that are able to host Majorana bound states (MBSs). First, we consider a one-dimensional superconductor (SC) wire with intrinsic bulk inversion asymmetry (BIA). In contrast to wires, modeled by a quadratic dispersion with Rashba spin-orbit coupling, there are two topological phase transitions in our systems at finite magnetic fields. Second, we analyze a two-dimensional Josephson junction on the Luttinger model finding a topological region even in the absence of BIA and Rashba spin-orbit couplings. This originates from the hybridization of the light and heavy hole bands of the $j=\frac{3}{2}$ states in combination with the SC pairing. As a consequence, both systems can be driven into a topological phase hosting MBSs. Hence, we predict that MBSs form in any SC proximitized Josephson junction on two-dimensional Luttinger materials by the application of magnetic field alone. This opens an avenue for the search of topological SC.

Figure 1

Sketch of the setups: (a) A one-dimensional Luttinger wire of length $L$ in contact with an $s$-wave superconductor and in the presence of a Zeeman field $B_x$ or $B_y$. (b) A two-dimensional Josephson junction of length $L$ with a Zeeman field applied in a normal region of width $W$. Two $s$-wave superconductors of width $W_S$ and phases ${\varphi }_{S_1}=-\varphi /2$ and ${\varphi }_{S_2}=\varphi /2$, respectively, separated by a normal regime N on top of the Luttinger material.

Figure 2

(a)–(d) Bulk band-structure calculations of an infinite wire with proximitized superconductivity and Zeeman field $B_y$ [see Fig. 1], using the parameters of HgTe $({\alpha }_0,{\alpha }_z,{\alpha }_{\square })=(16.58,9.36,-1.6)\frac{{\hslash }^2}{2m}$ and $\beta =-4.31\mathrm{meV}\mathrm{nm}$ [47, 48]. We set the chemical potential at $\mu =0.25\mathrm{meV}$ and $\mathrm{\Delta }=0.2\mathrm{meV}$. The color code indicates the character of the bands being light-hole (red) or heavy-hole (blue) like. The insets in (d) specify the topological invariant $\mathcal{Q}$. The critical magnetic fields $B_{\mathrm{HH}}$ and $B_{\mathrm{LH}}$ are given in Eq. (4). (e) Size of the minimum gap for all momenta as a function of the superconducting gap $\mathrm{\Delta }$ and the strength of the bulk inversion asymmetry $\beta$ in the topological phase at $B_y=(B_{\text{HH}}+B_{\text{LH}})/2$. (f) Band structure of a wire with finite length $L$ as a function of $B_y$ with a system size much larger than the localization length of the MBS ($L\gg \lambda$). States localized at the boundaries of the wire are indicated in red, showing the Majorana bound states in the topologically nontrivial region.

Figure 3

(a) Band structure of the nonsuperconducting semi-infinite 2D semimetal regime with the bulk continuum (blue shaded) and the edge states (red). The dashed line indicates the chemical potential $\mu$. (b) Andreev bound states as a function of the superconducting phase mismatch $\varphi$ for the Zeeman field $B_x=0.5\mathrm{meV}$ and $k_y=0$. The dashed lines indicate the topological phase transitions characterized by the topological invariant $\mathcal{Q}=\pm 1$ (framed insets). (c) Energy gap $E_g$ of the Andreev spectrum of the 2D Josephson junction at $k_y=0$ as a function of $\varphi$ and $B_x$. The topological regions in the phase diagram are separated by the gap closings (black lines). (d) Topological gap of the Andreev spectrum for all momenta $[E_g^{\text{top}}={min}_{k_y}{E}_g\left(k_y\right)]$. (e)–(h) Density of the lowest energy wave functions in real space for $\varphi =1.2\pi$. (e) Nonlocalized Andreev bound state in the trivial region ($B_x=0.05\mathrm{meV}$), (f) localized zero-energy Majorana bound state in the topological region ($B_x=2.0\mathrm{meV}$), and (g) and (h) two localized zero-energy states in the trivial region ($B_x=4.0\mathrm{meV}$). (i) Conductance calculated at the edge of the $N$ region of the Josephson junction as a function of energy $E$ and $\varphi$ at $B_x=0\mathrm{meV}$ and (j) $B_x=0.5\mathrm{meV}$ with a zero-bias peak in the topological region. Here, we used the parameters for Josephson junction based on $\alpha$-Sn $({\alpha }_0,{\alpha }_z,{\alpha }_{\square })=(18.62,11.88,0)\frac{{\hslash }^2}{2m}$ [49], $\mu =-1.0\mathrm{meV}$, while the $s$-wave induced SC gap is taken for $\beta$-Sn with $\mathrm{\Delta }=0.56\mathrm{meV}$ [50]. The dimensions of the junction are $W=20\mathrm{nm}$, $W_S=150\mathrm{nm}$, and $L=1000\mathrm{nm}$.