Rare-earth defect pairs in GaN: $\text{LDA}+U$ calculations

The structural and electronic properties of rare-earth (RE) (Eu, Er, and Tm) related defect pairs in GaN have been investigated theoretically. Based on $\text{LDA}+U$ total-energy calculations, their possible role in the luminescence process is discussed. In all charge states, the lanthanides show a strong preference for the Ga-lattice site, either as isolated substitutional or complexed with intrinsic defects. With respect to the electronic valence, a proper description of correlation effects of the strongly localized $4f$ electrons is shown to be crucial, especially if the ${\text{RE}}_{\text{Ga}}$ is paired with donors like the Ga interstitial or the N vacancy. The pairs formed by ${\text{RE}}_{\text{Ga}}$ substitutionals and vacancies or interstitials lower the symmetry and are found to locally distort the environment. By this, they are quite effective in relaxing the selection rules for the luminescent intra-$4f$-shell transitions. While for $n$-type GaN, the next-nearest-neighbor pair ${\text{RE}}_{\text{Ga}}{V}_{\text{Ga}}$ pair is energetically favored, for $p$-type GaN, the ${\text{RE}}_{\text{Ga}}{V}_{\text{N}}$ pair provides the most stable configuration and introduces shallow levels close to the conduction band, which can act as assistant levels in the luminescence process.

Figure 1

(Color online) The ${\text{RE}}_{\text{Ga}}$ substitutionals in hexagonal GaN. Ga atoms are white, nitrogen atoms light gray. Eu (left-hand side), Er (center), and Tm (right-hand side) are the darker atoms at the center of each picture. The numbers show the bond lengths in $\text{Å}$.

Figure 2

(Color online) The RE interstitials in hexagonal GaN. Rare-earth ions will not stay in the interstitial configuration (left-hand side) but will rather take the place of a neighboring Ga atom, which is shifted in the interstitial site (right-hand side). N atoms are black, Ga white, and Er green.

Figure 3

(Color online) Complexes formed by RE substitutionals and N interstitials in hexagonal GaN result in the so-called N split-interstitial configuration. Dark gray in the picture: The Tm impurity occupying the Ga place. The arrows indicate the two N atoms sharing a lattice site.

Figure 4

(Color online) Defect pairs formed by ${\text{Er}}_{\text{Ga}}$ substitutionals and $V_{\text{N}}$ in hexagonal GaN. Because of the inequivalent N sites in hexagonal GaN (the Er-N bond along the $c$ axis is longer than the other three), the complexes can be formed in two configurations. The basal configuration (left-hand side) has $C_{1h}$ symmetry, whereas the ideal axial configuration (right-hand side) has the $C_{3v}$ symmetry, or the $C_{1h}$ if a Jahn-Teller distortion occurs.

Figure 5

(Color online) Formation energy and binding energy of the ${\text{Er}}_{\text{Ga}}{V}_{\text{N}}$ pair as a function of the Fermi energy in hexagonal GaN. Dotted lines are the calculated formation energies of the isolated ${\text{Er}}_{\text{Ga}}$ and $V_{\text{N}}$, the solid lines are their sum (blue), the formation energy of the ${\text{Er}}_{\text{Ga}}{V}_{\text{N}}$ pair (red) and its binding energy (black), given as difference of the two curves.

Figure 6

(Color online) The lanthanide in ${\text{RE}}_{\text{N}}{V}_{\text{Ga}}$ pairs in hexagonal GaN (left-hand side) leaves the N site to occupy the Ga site (${\text{RE}}_{\text{Ga}}{V}_{\text{N}}$, right-hand side).

Figure 7

(Color online) ${\text{Er}}_{\text{Ga}}{V}_{\text{Ga}}$ pairs in hexagonal GaN. Because of the inequivalent Ga sites in hexagonal GaN, the complexes can have different symmetry. The configurations (a) and (b) have the $C_{1h}$ symmetry, while the configuration (c) has the $C_1$, i.e., has no symmetry.

Figure 8

(Color online) Formation energy and binding energy of the ${\text{Er}}_{\text{Ga}}{V}_{\text{Ga}}$ pair as a function of the Fermi energy in hexagonal GaN. Dotted lines are the formation energies of the isolated ${\text{Er}}_{\text{Ga}}$ and $V_{\text{Ga}}$, the solid lines are their sum (blue), the formation energy of the ${\text{Er}}_{\text{Ga}}{V}_{\text{N}}$ pair (red) and its binding energy (black), given as difference of the two curves. The formation energy of the ${\text{Er}}_{\text{Ga}}{V}_{\text{N}}$ pair (dashed line) is reported for comparison (see also Fig. 5).