deviation from stoichiometry can lead to complete valence band filling and thus stabilize the structure of a covalent crystal. This explanation however raises a new question: Why was pure $\alpha$-T boron not found in bulk phases until recently? A simple answer is that there are other phases which are more stable at ambient conditions, [*i.e.*]{}, $\alpha$- and $\beta$-rhombohedral ($\alpha$-R and $\beta$-R) boron. However, the relative stability of these polymorphs can be changed at high pressures, which is a different way for stabilizing $\alpha$-T boron. To clarify the relative stability of boron phases, a detailed comparison is required, which will be addressed in another study. Here, we concentrate on $\alpha$-T boron, only and identify the best structure that pure $\alpha$-T boron can have, if it exists.

There are several theoretical studies on the structure of $\alpha$-T boron.[@Lee92a; @Morrison92; @Hayami10; @Aydin11; @Aydin12] However, these studies assumed stoichiometry. From our point of view [@Shirai12], this assumption leads to an incorrect description of the metal/insulator behavior. The important role of non-stoichiometry has been recognized only recently and it will therefore be described in Sec. \[sec:background\]. Some words are also given there concerning the terminologies related to defect states, because they are used differently in 