The Hume-Rothery electron concentration rule studied through interference phenomena in a series of cP12 compounds with either a true gap or pseudogap across the Fermi level

ABSTRACT

Among binary MX₂-type cP12 compounds with space group $Pa\overline{3}$, there are many compounds with either a true gap or pseudogap at the Fermi level. They are grouped into two with respect to the total valence electrons per atom (VEC), the first being at 14/3, while the second at 20/3. The ratio of the number of itinerant valence electrons per atom e/a over VEC is used as a measure to evaluate a drastic change in the electronic structure from the nearly free electron-like to the tight binding-like. The Full-potential Linearized Augmented Plane Wave (FLAPW) band calculations were performed to study to what extent the value of e/a can be well defined. We claim that the Hume-Rothery electron concentration rule holds for As₂Si, P₂Si and MgTe₂, where the ratio above is higher than 90%. For compounds with M = Fe, Ru and Os and X = S, Se and Te, where the ratio is between 60 and 70%, the discussion based on a large Brillouin zone is still meaningful but the validity of Hume-Rothery electron concentration rule is judged to be marginal. The value of e/a is lowered below 20% and the Hume-Rothery electron concentration rule fails in MgO₂ and O₂Os.

KEYWORDS:

• Hume-Rothery electron concentration rule
• interference phenomena
• MX₂-type cP12 compounds
• transition metal elements
• NFE-like versus TB-like
• large Brillouin zone

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 In simple metals like Na and Al, the electrons are mostly free and so we can treat the single particle states in terms of plane waves. This is the NFE model. In contrast, the TB model assumes the states in a crystal as combinations of wave functions of isolated atoms. The FLAPW method can treat both models in a single formalism. It is essentially reduced to the former, provided that $|C_{{\mathit{k}}_i+{\mathit{G}}_p}^j{|}_{max}^2$defined in equation (1)(1) ${\psi }^j\left(\mathit{r},{\mathit{k}}_i\right)=\frac{1}{\sqrt{V}}\sum _p{C}_{{\mathit{k}}_i+{\mathit{G}}_p}^{j}exp\left\{i\left({\mathit{k}}_i+{\mathit{G}}_p\right)\cdot \mathit{r}\right\},$(1) in the text is close to unity or higher than 0.2 [1-4], while it is reduced to the latter, provided that the FLAPW wave functions are almost localised in the atomic spheres and that $|C_{{\mathit{k}}_i+{\mathit{G}}_p}^j{|}_{max}^2$is significantly low or lower than the level of 0.1.

2 The threshold factor 0.2 is rather arbitrarily chosen without any theoretical basis. The factor 0.15 may be still acceptable but 0.10 is probably too marginal to justify it as its lower limit (see Sections 3.2 and 3.4).

3 A careful inspection into Figs.3 (a) and (b) reveals the existence of very small amounts of states (yellow dots) beyond $[2\left|{\mathit{k}}_i+{\mathit{G}}_p\right|{]}^2$=16 and 17 even below the Fermi level. This means that these electrons may fall outside the complex LBZ bounded by a combination of {400}- and {410}+{322}-zone planes.

4 In the evaluation of (e/a)_av for the MX₂ compounds with X=S, Se and Te in Group 16 in the periodic table, (e/a)_X is assumed to be six. The e/a determination for individual elements over Group 1 up to 15 can be reliably made with the assistance of the centre-of-gravity method [1-3]. However, it fails for solid N in Group 15 and all elements in Group 16 and 17. It has been confirmed that the e/a values close to six and seven are retrieved for metallic S (hR1) and metallic Cl (oI2) under pressures of 206.5 and 83 GPa, respectively [1, 3]. This might be taken as an indication that (e/a)_X=6.0 and 7.0 are assigned to elements in Group 16 and 17, respectively, when they are dissolved in a matrix.

 

Additional information

Funding

This work was supported by the Japan Society for the Promotion of Science [Grant-in-Aid for Scientific Research (Contract 17K06780)].