ZrTe$_5$ and HfTe$_5$ have attracted increasingly attention recently sincethe theoretical prediction of being topological insulators (TIs). However,subsequent works show many contradictions about their topological nature.Threepossible phases, i.e. strong TI, weak TI, and Dirac semi-metal, have beenobserved in different experiments until now. Essentially whether ZrTe$_5$ orHfTe$_5$ has a band gap or not is still a question. Here, we present detailedfirst-principles calculations on the electronic and topological properties ofZrTe$_5$ and HfTe$_5$ on variant volumes and clearly demonstrate thetopological phase transition from a strong TI, going through an intermediateDirac semi-metal phase, then to a weak TI when the crystal expands.Our workmight give a unified explain about the divergent experimental results andpropose the crucial clue to further experiments to elucidate the topologicalnature of these materials.
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