Properties of $\mathrm{GaN}∕\mathrm{ScN}$ and $\mathrm{InN}∕\mathrm{ScN}$ superlattices from first principles

First-principles calculations have been carried out to reveal structural, electromechanical, electronic, and lattice dynamical properties of $\mathrm{GaN}∕\mathrm{ScN}$ and $\mathrm{InN}∕\mathrm{ScN}$ superlattices—made by alternating hexagonal layers of GaN, or InN, with hexagonal layers of ScN—for different periods and overall compositions. These nitride systems belong to two different structural classes (having different coordination number), depending on the overall composition. For Sc compositions larger than 50%, each atom has nearly five nearest neighbors. On the other hand, Sc-deficient superlattices adopt a ground state that is nearly fourfold coordinated. This change of structure, and the change in composition or period within the same structure, considerably affect the piezoelectric response, the electronic band gap in magnitude as well as in character (indirect versus direct), and the phonon spectra. We also discussed the relevance of some of these predictions for designing future technological applications.

Figure 1

Variation of the equilibrium in-plane lattice constant (a), the axial ratio (b), the averaged internal parameters (c), and the unit-cell volume (d) as a function of Ga $\left(x\right)$ concentration in ${\left(\mathrm{GaN}\right)}_n∕{\left(\mathrm{ScN}\right)}_m$ (e)–(h): same as (a)–(d), but for ${\left(\mathrm{InN}\right)}_n∕{\left(\mathrm{ScN}\right)}_m$. The $x$ composition is given by $x=n∕(m+n)$, and the axial ratio shown here corresponds to four atoms per cell. Properties of the parent compounds are also given.

Figure 2

Piezoelectric coefficient $e_{33}$ as a function of Ga (a) concentration in ${\left(\mathrm{GaN}\right)}_n∕{\left(\mathrm{ScN}\right)}_m$ systems. The clamped-ion $\left(e_{33,c}\right)$ and internal-strain $\left(e_{33,i}\right)$ contributions to the the piezoelectric coefficient $\left(e_{33}\right)$ are depicted in (b) of the figure. (c) and (d) depict the same as (a) and (b), but for ${\left(\mathrm{InN}\right)}_n∕{\left(\mathrm{ScN}\right)}_m$.

Figure 3

Electronic band structure of the ${\left(\mathrm{GaN}\right)}_n∕{\left(\mathrm{ScN}\right)}_m$ systems. (a) corresponds to pure ScN, whereas (b)–(g) show the electronic band structure of $1\times 3$ and $4\times 6$, $1\times 1$, $2\times 2$, $3\times 1$, and GaN systems, respectively. The dashed horizontal line shows the zero energy level chosen for the valence band maximum.

Figure 4

Electronic band-structure of the ${\left(\mathrm{InN}\right)}_n∕{\left(\mathrm{ScN}\right)}_m$ systems: (a) pure ScN, (b) $1\times 3$, (c) $2\times 2$, (d) $1\times 1$, (e) $3\times 1$, and (f) InN. The trends are described in the text. The dashed horizontal line shows the zero energy level chosen for the valence-band maximum.

Figure 5

Phonon spectrum for ${\left(\mathrm{GaN}\right)}_n∕{\left(\mathrm{ScN}\right)}_m$: (a) ScN, (b) $1\times 3$, (c) $1\times 1$, (d) $2\times 2$, (e) $3\times 1$, and (f) GaN. Anisotropy can be observed via the splitting of the highest optical modes along $\Gamma \to A$ and $\Gamma \to M$ directions.

Figure 6

Phonon spectrum for ${\left(\mathrm{InN}\right)}_n∕{\left(\mathrm{ScN}\right)}_m$: (a) ScN, (b) $1\times 3$, (c) $2\times 2$, (d) $1\times 1$, (e) $3\times 1$, and (f) InN. Anisotropy can be observed the way the highest optical mode splits along $\Gamma \to A$ and $\Gamma \to M$ directions.