Majorana corner modes and tunable patterns in an altermagnet heterostructure

The mutual competition and synergy of magnetism and superconductivity provide us with a very valuable opportunity to access topological superconductivity and Majorana fermions. Here, we devise a heterostructure consisting of an $s$-wave superconductor, a two-dimensional topological insulator, and an altermagnet, which is classified as the third magnet and featured by zero magnetization but spin polarization in both real and reciprocal spaces. We find that the altermagnet can induce mass terms at the edges that compete with electron pairing, and mass domains are formed at the corners of the sample, resulting in zero-energy Majorana corner modes (MCMs). The presence or absence of MCMs can be engineered by only changing the direction of the Néel vector. Moreover, uniaxial strain can effectively manipulate the patterns of the MCMs, such as moving and interchanging MCMs. Experimental realization, remarkable advantages of our proposal, and possible braiding are discussed.

Figure 1

(a) Schematic of the proposed altermagnet heterostructure. A two-dimensional topological insulator is sandwiched between an $s$-wave superconductor and an altermagnet. The emerging MZMs are localized at the corners. Altermagnets have various spin-momentum locking with the even-parity waveform, such as $d$ wave, $g$ wave, and $i$ wave. (b) The phase diagram in the plane of Néel vector polar angle $\theta$ and chemical potential $\mu$. (c) The real-space wave-function distribution with one MZM localized at each corner. (Inset) Plot of the several eigenvalues near zero energy. The common parameters are $m_0=1.0,t_x=A_x=A_y=1.0,t_y=0.5,{\mathrm{\Delta }}_0=0.5,\text{and}{J}_0=1.0$.

Figure 2

(a) Quasiparticle spectrum for a cylindrical geometry. In the presence of altermagnetism and electron pairing, the otherwise-helical edge states open a gap. The inset shows the variation of the edge gap with order parameter strength $J_0$ of altermagnets. The closure of the edge states at the critical $J_0=J_0^c$ indicates a topological phase transition at the boundary. (b) Bulk bands along the high-symmetry path at $J_0=1.0$. The inset shows the evolution of the bulk gap with $J_0$, and no bulk gap closure is found. (c) Schematic demonstrating the Dirac mass terms of adjacent edges with opposite signs, forming two domain walls. I, II, III, and IV donate four edges. (d) The real-space energy spectrum of an open-boundary square structure of size $L_x=L_y=30$. Eight MZMs emerge (red dots). (e) The distribution of the MZM wave function in real space. (f) The phase diagram in the plane of the azimuth angle $\theta$ and $J_0$. The blue dotted lines mark the phase boundary. The common parameters are $m_0=1.0$, $t_x=t_y=A_x=A_y=2.0$, and ${\mathrm{\Delta }}_0=0.5$.

Figure 3

(a) The phase diagram for the isosceles-right-triangle geometry. (b) The energy spectrum of an isosceles triangle structure with two MCMs (red dots) and the density plot of the two MCMs (c) with V representing the hypotenuse. ${\mathrm{\Delta }}_0=0.2$, $J_0=0.5$. (d) The real-space energy spectrum evolves with electron-pairing amplitude in the isosceles-right-triangle structure with MCMs marked in red existing in a large range. The common parameters are $m_0=1.0$, $t_y=A_y=t_x=A_x=2.0$, and $\mu =0.1$.

Figure 4

Results with nonzero chemical potential $\mu$. (a) The real-space wave-function distribution with one MZM localized at each corner. Inset plots the several eigenvalues near zero energy. (b) The real-space energy spectrum evolves with electron-pairing amplitude in a square structure with MCMs marked in red existing in a range. (c), (d) Tunable MCM patterns. The blue hollow (green solid) circles represent the MCMs from the $H^{+}$ ($H^{-}$) block. The arrows indicate the uniaxial stress direction. The common parameters are $m_0=1.0$, $t_y=A_y=1.0$, $t_x=A_x=2.0,J_0={\mathrm{\Delta }}_0=0.5$, and $\mu =0.1$.