Dirac edge states of two-dimensional altermagnetic topological crystalline insulators

Abstract

Two-dimensional (2D) metallic altermagnets present exciting opportunities for both fundamental research and practical innovations. Their ability to enhance tunneling magnetoresistance in magnetic tunnel junctions, combined with the direct control of spin currents via electric fields, makes them highly promising for spintronic devices. Moreover, the rich electronic structure of altermagnets can host nontrivial topological phases. In particular, topological crystalline insulators are compounds where the topological states are protected by both crystalline and time-reversal symmetries. Furthermore, manipulating the state of a system between topological and trivial phases through external parameters unlocks new possibilities for quantum materials and advanced electronics. We show the edge states of a 2D altermagnetic topological crystalline insulator, using as a representative example Cr$_2$BAl, a 2D MBene metallic altermagnet with a d$_{x^2-y^2}$ altermagnetic ordering. We find that the system can host an altermagnetic phase with extremely large ``weak ferrimagnetism" which is sizeable also with light atoms, only with an in-plane component of the Néel vector. The electronic structure of Cr$_2$BAl presents multiple crossings and anti-crossings in the vicinity of the Fermi level along [100] and [010] directions. When the spin-orbit coupling interaction is included, with the Néel vector along [001] direction, energy gaps open at the band crossing points, resulting in a pronounced peak in the spin Hall conductivity. The simulated Cr-B terminated [100] edge-projected band structure reveals Dirac dispersions at the bulk crossings and anti-crossings, which are absent in Cr-Al terminations.

Figures (4)

• Figure 1: The side view (a) and the top view (b) of the 2D Cr$_2$BAl unit cell. The chromium sub-lattice with opposite spin polarization are distinguished by blue and red balls, boron atoms in green and aluminum in gray. The rectangle and arrow in blue and purple highlight the generator of opposite-spin sub-lattice rotations. Here, the rotational operation was labeled by Litvin's notation (4${_z}|$0,0,0)'.
• Figure 2: The total energy for both the AM and FM configurations as a function of the lattice parameter $a$. The AM configuration exhibits a lower energy at around 4.54 Å, indicating a more stable magnetic ground state. A biaxial strain in the a-b plane can induce an altermagnet-ferromagnet transition.
• Figure 3: (a) The non-relativistic spin-splitting (red is for spin-up and blue is for spin-down) in electronic band structure for Cr$_2$BAl monolayer with band crossings and anti-crossings (highlighted by dashed circles) along the BZ path X-$\Gamma$-Y, i.e., in [100] and [010] directions. (b) The electronic structure obtained with spin-orbit coupling (with Néel vector in [001] direction) indicates a spin-orbit interaction-induced gap (highlighted by dashed circles) along the BZ path X-$\Gamma$-Y, i.e., in [100] and [010] directions. (c) Fermi surface slice in the 2D BZ, corresponding to the spin-orbit coupling bands presented in (b). Surface-projected band structure obtained using surface projections in [100] direction: (d) The Cr-B surface termination indicates the presence of Dirac dispersions (highlighted by dashed circles) which coincide with the relativistic spin-splittings observed in (b), leading to the peak in spin Hall conductivity at $\sim$ 525 meV observed in (f). (e) The Cr-Al surface termination, on the contrary, indicates the absence of the Dirac dispersions. (f) The spin Hall conductivity indicating sharp peaks at $\sim$ 525 meV and $\sim$ 100 meV, corresponding to the multiple band crossings/anti-crossings in the non-relativistic regime and the spin-orbit induced splitting in the relativistic regime presented (a) and (b), respectively. The Fermi level is set to zero in all plots.
• Figure 4: Spin density of states of Cr$_\uparrow$ and Cr$_\downarrow$ at the low-energy for Cr$_2$BAl with Néel vector (a) along the $z$-axis and (b) along the $x$-axis. (c) Average spin-density and difference in the spin-density for the Néel vector along the $x$-axis. In the inset of panel (c), we report the magnetic spin configuration; the difference between the spins is exaggerated for graphical purposes. The difference between the spin densities of Cr$_\uparrow$ and Cr$_\downarrow$ represents the hallmark of weak ferrimagnetism.