Fig. 1: Schematic illustration of the pathway for the formation of nonstoichiometric M_1+θX₂ from stoichiometric MX₂.

a Pristine atomic structure of three layers stoichiometric MX₂ with 1 T phase. This structure is layered and centrosymmetric without piezo/ferroelectricity. The yellow ball represents chalcogen atoms, while transition metal atom balls labeled in blue and red indicate primary and extra-introduced metal atoms, respectively. The metal and chalcogen atom vacancies are indicated by a ball surrounded by a dotted circle line with light yellow and light blue colors, respectively. b Chalcogen atoms intercalated MX₂ atomic structure. This structure is unstable as lack of charge compensation to chalcogen atoms. c Atomic structure illustration of transition metal atom intercalated MX₂ (left) and stoichiometric Cr₂Se₃ compound (right). Due to the formation of covalent bonds between transition metal atoms and chalcogen atoms, this pathway is thermodynamically predisposed to occur. Stoichiometric Cr₂Se₃ is stable and centrosymmetric with a quasi-layered structure, in which sandwich layers are linked by Cr atoms. Both interlayer and intralayer Cr atoms are octahedrally coordinated by Se atoms. d Typical case of several random metal vacancies in layered MX₂. Note that a metal atom symbol in the atomic structure is overlapped by many metal atoms in the side view. The atoms are removed at this site to clearly present vacancy defects, rather than all the atoms below this site are missing. e A chalcogen vacancy in MX₂ (left) and then transfer into a metal vacancy/antisite (right). The red dash box is for eye guidance. This is just a simple possible transformation of chalcogenide vacancies. The real situation may be more complex to involve long-range atoms.