Topological Phase Transition in Layered XIn$_2$P$_2$ (X = Ca, Sr)

TL;DR

This work investigates how elastic strain and spin–orbit coupling control topology in layered XIn2P2 (X = Ca, Sr) using fully relativistic first-principles calculations. Strain in the $ab$ plane can invert the $s$- and $p_z$-derived bands, creating a line-node semimetal with a nodal ring in the $k_z=0$ plane; including SOC opens a small gap along the ring and yields a strong topological insulator with $Z_2$ index $(1;000)$. Parity analysis and surface-state computations confirm the TI phase, and a detailed orbital-resolved picture explains the strain-driven band inversion. The results demonstrate a strain-tunable pathway from a trivial semiconductor to a line-node semimetal and then to a TI, offering a new perspective on strain-engineered topological phases in layered materials.

Abstract

Based on fully relativistic first-principles calculations, we studied the topological properties of layered XIn$_2$P$_2$ (X = Ca, Sr). Band inversion can be induced by strain without SOC, forming one nodal ring in the k$_z$ = 0 plane, which is protected by the coexistence of time-reversal and mirror-reflection symmetry. Including SOC, a substantial band gap is opened along the nodal line and the line-node semimetal would evolve into a topological insulator. These results reveal a category of materials showing quantum phase transition from trivial semiconductor and topologically nontrivial insulator by the tuneable elastic strain engineering. Our investigations provide a new perspective about the formation of topological line-node semimetal under stain.

Figures (6)

• Figure 1: (Color online) (a) The Crystal structure of XIn$_2$P$_2$ with P63/mmc symmetry. (b) Brillouin zone of bulk and the projected surface Brillouin zones of (001) plane, as well as high-symmetry points. There is a line-node ring lies in the k$_z$=0 plane.
• Figure 2: (Color online) Electronic band structures of unstrained CaIn$_2$P$_2$ with SOC. (b) Electronic band structures of band-inverted CaIn$_2$P$_2$ (a/a$_0$=1.05) with SOC. The details of band-inverted CaIn$_2$P$_2$ around E$_F$ are shown in the inset.
• Figure 3: Phonon dispersion in SrIn$_2$P$_2$ along the high-symmetry directions.
• Figure 4: (Color online) The orbital characteristic band structures ( along the M - $\Gamma$ - K direction, near Fermi energy ) of SrIn$_2$P$_2$ under the different strains. (a) zero strain without SOC, (b) 5% of uniaxial strain without SOC, (c) 5% of uniaxial strain with SOC. The weight of atomic orbital In(s) and P(p) is proportional to the radius of the green (blue) circle. Band inversion could be seen clearly around the $\Gamma$ point.
• Figure 5: (Color online)(a) The band structure of unstrained SrIn$_2$P$_2$ without SOC, weighted with the In-s and P-P$_z$ characters. In-s and P-P$_z$ states are distinguished by red and green colors. (b) $\Delta$E(A), $\Delta$E($\Gamma$$_{4,6}$), $\Delta$E($\Gamma$$_{2,3}$) and $\Delta$E$_g$ as functions of the inter-slab tensile strain $\delta$=(a-a$_0$)/a$_0$ (volume is fixed), respectively.