Native Point Defects in GaN: A Hybrid-Functional Study

We present a systematic study of properties of common native point defects in GaN based on hybrid density-functional calculations. These defects include vacancies, interstitials, antisites, and common complexes. Using configuration coordinate diagrams, we estimate the likelihood of defects to be radiative or nonradiative. Our results show that gallium vacancies exhibit a large magnetic moment in the neutral charge state and are most likely nonradiative. This is in contrast to nitrogen vacancies, which are probable sources of the experimentally observed green-luminescence band peaking at 2.35 eV in undoped GaN. We also show that infrared photoluminescence (PL) bands that are created by 2.5-MeV electron irradiation in GaN can be explained by the formation of native defects. Namely, the interstitial gallium is likely to be responsible for the narrow infrared PL band centered around 0.85 eV, with a phonon fine structure at 0.88 eV; the gallium-nitrogen divacancies are possible sources of the broad PL band with a peak at 0.95 eV.

Figure 1

Formation energy of $V_{\mathrm{Ga}}$ as a function of the Fermi energy in (a) Ga-rich and (b) N-rich growth conditions. The zero and maximum of the Fermi-level scale correspond to the top of the valence band and the bottom of the conduction band, respectively. The solid lines correspond to the formation energies for the most stable charge states of the defects. The points where each line changes slope (charge state) mark thermodynamic transition levels in the GaN band gap.

Figure 2

Schematic configuration coordinate diagram displaying possible radiative and nonradiative transitions between the excited and ground states of a defect. The potential curve of the excited state is vertically displaced from that of the ground state according to their formation energies and assuming the presence of an electron in the conduction band. The ZPL describes the transition between the zero-point vibrational states in excited-state and ground-state configurations. ${\epsilon }_{\text{exc}}$ is the resonant excitation energy (absorption energy), ${\mathrm{PL}}_{\max }$ corresponds to the peak of the PL band. $\mathrm{\Delta }{\epsilon }_e$ and FC (Franck-Condon shift) describe the relaxation energies of the excited state and the ground state, respectively. $\mathrm{\Delta }{\epsilon }_b$ (energy barrier) is the energy between the vibrational ground state of the upper curve and the crossover between the two curves. The dashed arrow labeled NR represents a nonradiative transition.

Figure 3

Configuration coordinate diagram of $V_{\mathrm{Ga}}$ obtained from the harmonic approximation fitting of total energies at relaxed defect lattices only (solid black lines) and direct HSE calculations (filled circles). The filled circles correspond to the total energies of ten intermediate defect geometries between two minima in the $3-$ and $2-$ charge states. An average difference in energy of 4 meV is found between the CCD based on the harmonic approximation and the CCD obtained from direct HSE calculations. A calculated emission of 0.60 eV, a FC shift of 0.54 eV, and a ZPL of 1.14 eV are obtained. The energy barrier for a nonradiative transition is 0.13 eV, making the $V_{\mathrm{Ga}}$ likely nonradiative.

Figure 4

Charge-density isosurfaces of the four defect states of $V_{\mathrm{Ga}}$. The wave functions are calculated in the $3-$ charge state of the defect. For clarity, two different orientations are used for states (a),(b) and (c),(d) indicated by the lattice vectors. The (a)–(d) charge densities correspond to the eigenvalues shown in Fig. 5 (right panel, $3-$ charge state of $V_{\mathrm{Ga}}$) from lowest (a) to highest (d) energy. The small (gray) and large (green) spheres indicate the nitrogen and gallium atoms, respectively. The isosurface values are set at 5% of the maximum.

Figure 5

Single-electron energy levels of $V_{\mathrm{Ga}}$ for all the possible charge states $q$ with their respective magnetic moments $m$. Zero energy corresponds to the VBM.

Figure 6

Magnetization density isosurfaces of $V_{\mathrm{Ga}}$ in the singly positive charge state in the (a) FM spin configuration and (b) AFM spin configuration. The positive (yellow) and negative (light blue) isosurface values are plotted at 10% of the maximum. AFM spin alignment has lower energy than FM alignment by 75 meV.

Figure 7

Formation energies of the $V_{\mathrm{N}}$ defect in GaN grown under (a) Ga-rich and (b) N-rich conditions. The dashed lines are used to emphasize the presence of negative-$U$ behavior where $U=-0.13\mathrm{eV}$. The insets show the regions with the $2+/+$ and $3+/2+$ transition levels located at 0.47 and 0.61 eV above the VBM, respectively.

Figure 8

Charge density of the localized defect state of $V_{\mathrm{N}}$ calculated in the $3+$ charge state. The isosurfaces with the value 6% of the maximum are shown. There is a strongly localized charge density at the vacancy site, which is of $s$ character, while the $s$- and $p$-hybridized parts of the defect state are formed at the neighboring N sites.

Figure 9

Configuration coordinate diagram of optical transitions for $V_{\mathrm{N}}$. The PL maximum is 2.24 eV. Here the vibrational ground state of ${V_{\mathrm{N}}}^{2+}$ is 1.53 eV lower than the energy of the crossover of the potential curves, which makes transitions via the $+/2+$ level of $V_{\mathrm{N}}$ most likely radiative. The Franck-Condon shift of ${V_{\mathrm{N}}}^{+}$ is computed to be 0.78 eV, and the ZPL is 3.02 eV. These parameters are in close agreement with the experimentally observed GL2 band.

Figure 10

(a) Formation energy of the most stable charge states of the $V_{\mathrm{Ga}}\text{−}{V}_{\mathrm{N}}$ defect (solid black line) in GaN grown under either Ga- or N-rich conditions. The dashed lines display the instability of the $2+$ charge state or the negative-$U$ behavior ($U=-0.13\mathrm{eV}$). The insets show the $2+/+$ and $3+/2+$ transition levels occurring at 0.68 and 0.81 eV above the VBM, respectively. (b) Binding energy ($B$) of $V_{\mathrm{Ga}}\text{−}{V}_{\mathrm{N}}$ as a function of the Fermi level across the band gap. In $n$-type GaN, the binding energy is calculated to be 3.04 eV.

Figure 11

Configuration coordinate diagram for the optical transitions via the $V_{\mathrm{Ga}}\text{−}{V}_{\mathrm{N}}$ defect in GaN. The emission is predicted to occur at 0.99 eV, while the FC shift and ZPL are calculated to be 0.54 and 1.53 eV, respectively. Here the vibrational ground state of $V_{\mathrm{Ga}}\text{−}{V_{\mathrm{N}}}^{-}$ is 0.35 eV below the energy of the crossover of the potential curves, suggesting that this defect is radiative only at low temperatures.

Figure 12

(a) Atomic configuration of the tetrahedral (labeled $T$) and octahedral (labeled $O$) interstitial sites in the wurtzite GaN. Large green spheres represent Ga atoms and small gray spheres represent N atoms. (b) Relaxed atomic structure of ${\mathrm{Ga}}_i$ in the $+$ charge state. The Ga atom occupies a slightly distorted octahedral site, where the distances between ${\mathrm{Ga}}_i$ and nearest N atoms decrease by 4.28%, when compared to ideal octahedral configuration.

Figure 13

Formation energies of the Ga native defects as a function of the Fermi energy calculated for (a) Ga-rich and (b) N-rich growth conditions. In the $p$-type GaN and Ga-rich environment, both interstitial and antisite Ga possess the lowest formation energies among the investigated substitutional and interstitial native defects, while displaying very high formation energies in $n$-type GaN in both growth conditions.

Figure 14

Configuration coordinate diagram for the isolated interstitial Ga displaying calculated optical transitions via the $+/2+$ transition level. The peak of the PL band is at 0.72 eV. The two potential curves never intersect, making ${\mathrm{Ga}}_i$ likely a radiative defect. The ZPL is found to be 0.84 eV and the Franck-Condon shift (relaxation energy) is 0.12 eV.

Figure 15

(a) Atomic structure of the section of ideal wurtzite GaN; (b) equivalent section of the relaxed N split interstitial (${\mathrm{N}}_i─{\mathrm{N}}_i$) in the singly negative ($-$) charge state and (c) in the $3+$ charge state. Large green spheres represent Ga atoms and small gray spheres represent N atoms. In the $-$ charge state, the ${\mathrm{N}}_i─{\mathrm{N}}_i$ bond is 1.41 Å; in the $3+$ charge state, the ${\mathrm{N}}_i─{\mathrm{N}}_i$ bond is reduced to 1.11 Å.

Figure 16

Formation energies of native nitrogen defects as a function of the Fermi energy for GaN grown in (a) Ga-rich and (b) N-rich environments. The dashed lines are used to show the instability of the $+$ charge state for the N antisite, displaying a negative-$U$ character ($U=-0.73\mathrm{eV}$).

Figure 17

Thermodynamic transition levels $ϵ(q_1/q_2)$ of all investigated native defects in GaN, with the reference to the VBM. The solid lines denote the positions of the deep-defect transition levels. The $+/0$ transition levels of ${\mathrm{Ga}}_i$ and $V_{\mathrm{N}}$ are calculated to be resonant with the conduction band, which suggests that experimentally, shallow donor levels (dashed lines) of these defects should be observed.