Abstract
We combine first principles calculations with a group theory analysis to investigate topological phase transitions in the stacking of SnTe monolayers. We show that distinct finite stacking yields different symmetry-imposed degeneracy, which dictates the hybridization properties of opposite surface states. For SnTe aligned along the [001] direction, an (even) odd number of monolayers yields a (non)symmorphic space group. For the symmorphic case, the hybridization of surface states lead to band inversions and topological phase transitions as the sample height is reduced. In contrast, for a nonsymmorphic stacking, an extra degeneracy is guaranteed by symmetry, thus avoiding the hybridization and topological phase transitions, even in the limit of a few monolayers. Our group theory analysis provide a clear picture for this phenomenology and matches well the first principles calculations.
Highlights
The recently discovered topological insulators (TIs) are classified to the symmetries of its crystal lattice and/or time-reversal symmetry, which yields its topologically protected edge or surface states[1,2]
The spinful double group of both bulk and monolayer admits doubly degenerate irreducible representations (IRREPs) with odd and even inversion characters[38], which can be characterized by the orbital projections Φ− = [φPSn + φSTe] and Φ+ = [φPTe + φSSn], respectively[39]
We have shown that the band structure of the symmorphic and nonsymmorphic stackings evolve differently as a function of N
Summary
The recently discovered topological insulators (TIs) are classified to the symmetries of its crystal lattice and/or time-reversal symmetry, which yields its topologically protected edge or surface states[1,2]. The class of topological crystalline insulators (TCIs) have Dirac-like bands protected by space group symmetries[3,4]. The monolayers of IV-VI materials have been predicted to be two-dimensional (2D) TCIs11,12 Both the bulk and monolayers of SnTe are classified by their mirror Chern number |nM| = 2, yielding an even number of Dirac cones in the Brillouin zone. In between these two limits, the stacking of monolayers show an intriguing nonmonotonic evolution of the band gap for stackings oriented along the [001]12–15, [111]16–20 and [110]21,22 directions. The nonsymmorphism of the even stackings enforces an extra degeneracy, avoiding a band inversion This protects the system from a topological phase transition. We focus here on the [001] stacking of SnTe monolayers, equivalent considerations shall hold for other directions, and as well for any material that may present distinct space groups for different stacking heights
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