Nonlinear pressure dependence of the direct band gap in adamantine ordered-vacancy compounds

A strong nonlinear pressure dependence of the optical absorption edge has been measured in defect chalcopyrites ${\text{CdGa}}_2{\text{Se}}_4$ and ${\text{HgGa}}_2{\text{Se}}_4$. The behavior is due to the nonlinear pressure dependence of the direct band-gap energy in these compounds as confirmed by ab initio calculations. Our calculations for ${\text{CdGa}}_2{\text{Se}}_4$, ${\text{HgGa}}_2{\text{Se}}_4$ and monoclinic $\beta {\text{-Ga}}_2{\text{Se}}_3$ provide evidence that the nonlinear pressure dependence of the direct band-gap energy is a general feature of adamantine ordered-vacancy compounds irrespective of their composition and crystalline structure. The nonlinear behavior is due to a conduction band anticrossing at the $\Gamma$ point of the Brillouin zone caused by the presence of ordered vacancies in the unit cell of these tetrahedrally coordinated compounds.

Figure 1

(Color online) Structure of the defect chalcopyrite (a) (DC) ${\text{CdGa}}_2{\text{Se}}_4$, (b) pseudocubic (PS) ${\text{CdGa}}_2{\text{Se}}_4$, and (c) $\beta {\text{-Ga}}_2{\text{Se}}_3$. Big light atoms are Cd, medium dark atoms are Ga, and small dark atoms are Se. To distinguish between nonequivalent atoms in the DC structure the Wyckoff sites relevant for the discussion of Fig. 6 are given in parenthesis. The DC- and ${\text{PS-HgGa}}_2{\text{Se}}_4$ structures are obtained by substituting Cd atoms by Hg atoms in the (a) and (b) pictures.

Figure 2

Absorption edge of defect chalcopyrite ${\text{CdGa}}_2{\text{Se}}_4$ on increasing pressure up to (a) 7.2 GPa and from 7.2 up to (b) 16 GPa.

Figure 3

Absorption edge of defect chalcopyrite ${\text{HgGa}}_2{\text{Se}}_4$ on increasing pressure up to (a) 10.4 GPa and (b) from 10.4 up to 15.4 GPa.

Figure 4

(a) Pressure dependence of the band-gap energy in ${\text{CdGa}}_2{\text{Se}}_4$. Symbols correspond to experimental values for $\text{I}\overline{4}$ defect chalcopyrite (DC) phase. Theoretical calculations: direct band gap for DC phase (solid line), indirect $\Gamma \text{-Z}$ band gap for DC phase (dashed line), direct band gap for $\text{P}\overline{4}2\text{m}$ pseudocubic (PS) phase (dotted line). (b) Same but for ${\text{HgGa}}_2{\text{Se}}_4$. (c) Pressure dependence of the band-gap energy in $\beta {\text{-Ga}}_2{\text{Se}}_3$. Symbols correspond to experimental data of the direct band gap taken from Ref. 7. Solid and dashed lines correspond to theoretical values of the direct and $\Gamma \text{-L}$ indirect band gaps of the $\beta$ phase. All calculated energies are upshifted to match the experimental direct band-gap energy at room pressure.

Figure 5

(Color online) (a) Symbols: calculated pressure dependence of the energies of the topmost valence band and the lowest conduction bands at the $\Gamma$ point for ${\text{DC-CdGa}}_2{\text{Se}}_4$. Solid (dashed) lines: expected pressure dependence of the first (second) conduction band energy in absence of band anticrossing. (b) Same for ${\text{DC-HgGa}}_2{\text{Se}}_4$. (c) Same for $\beta {\text{-Ga}}_2{\text{Se}}_3$.

Figure 6

(Color online) Atomic character of the first (b) and second (a) conduction band at the $\Gamma$ point as a function of pressure for ${\text{DC-CdGa}}_2{\text{Se}}_4$. Wyckoff positions of the different atoms according to Fig. 1a are noted in parenthesis.