Multistability of interstitial magnesium and its carrier recombined migration in gallium nitride

TL;DR

The paper investigates recombination-enhanced migration of Mg interstitials in GaN using hybrid-DFT ($HSE$) and a Landau–Zener–style treatment of electron capture. It maps migration pathways and barriers with CI-NEB, and builds an electronic Hamiltonian from DFT KS states to compute nonradiative capture rates during migration, including finite-size extrapolation. The key finding is that migration via the $(\mathrm{MgGa})_{ic}$ complex becomes energetically favorable when one or two electrons are captured, lowering barriers from $E_B\approx$ $2.23$ eV (2+$) to $1.65$ eV (1+$)$ and further to $1.55$ eV (neutral) for the second capture, explaining experimentally observed low barriers. Recombination rates indicate high capture probability at typical electron densities ($n \sim 10^{17}-10^{18}\ \mathrm{cm}^{-3}$), placing the timescale of recombination on or below the migration timescale and confirming a significant enhancement of Mg diffusion in GaN.

Abstract

We present density-functional-theory calculations which provide a microscopic picture of the recombination-enhanced migration of interstitial Mg in GaN. We determine stable structures and migration pathways with accurate HSE approximation to the exchange-correlation energy, and also computed recombination rates using the obtained energy spectrum and wavefunctions. It is found that the migration between the most stable octahedral sites (Mg$_{\textrm{O}}$) via newly found interstitial complex structure shows the lowest migration energy in which one or two electrons are captured during the migration, that the most stable charge state of 2+ changes to 1+ or neutral, and that by this recombination of carriers the migration barrier is significantly reduced. Starting from Mg$_{\textrm{O}}^{2+}$, Mg captures an electron becoming the 1+ charge state and overcomes the barrier of 1.65 eV, much reduced from 2.23 eV in case of the migration with the 2+ charge state kept. Moreover, further electron capture is realized accompanied by substantial structural relaxation, thus Mg becoming neutral. Detailed HSE calculations for this second capture show that the migration barrier is 1.55 eV, thus clarifying the important role of the carrier recombination for Mg migration in GaN. These findings are corroborated by the present quantitative calculations of recombination rates based on electronic Hamiltonian constructed from our DFT-obtained energy spectrum. The timescale of the recombination is clarified to be in or under the timescale of the migration with typical electron density and the enhancement is expected to be significant.

Figures (4)

• Figure 1: The total energy profile along the migration pathway from the O site (left-end point B) to the midway geometry (MgGa)ic (right-end point C, F, or I) for each charge state. The structures of points A$\sim$K are shown in Fig. \ref{['fg:path']} (c). Note that the same $\xi$ of different chare states does not refer to the same atomic structure. The solid and dashed lines show the result calculated using $2\times2\times2$ and single $\bm k$ points, respectively. Red and green dashed lines are shifted downwards by 0.05eV to match the solid lines.
• Figure 2: Schematic illustration of the atomic structure of MgO(a) and (MgGa)ic(b), respectively, viewed along the $\langle 0001 \rangle$ axis. Blue, gray and green balls represent N, Ga and Mg, respectively. (c) The trajectory of the migrating Mg atom and the nearby Ga atom, projected on the $(0001)$ plane, in the MgO$\to$ (MgGa)ic migration shown in Fig. \ref{['fg:qf']}. The energies of points A$\sim$K (for Mg, and the corresponding Ga positions are marked by A$'$$\sim$K$'$) are shown in Fig. \ref{['fg:qf']}. The projected positions B$\to$A$\to$C for the $2+$ migration, E$\to$K$\to$D$\to$F for the $1+$ migration, and H$\to$G$\to$I for the $2+$ migration are depicted by blue, red and green dots, respectively, both for the Mg and Ga positions. The projected positions of the geometry J (see text) at which the total-energy between $1+$ and neutral crosses with minimum value are shown by black squares. The inset is a zoom-out view in which gray dots denote the lattice sites (at ideal positions), corresponding to the blue and gray balls shown in Fig. \ref{['fg:path']} (a) and (b), and blue lines represent the migration trajectories of the Mg and Ga atoms.
• Figure 3: (a) The total energies of charge state $2+$ and $1+$ along the BK path. Unlike Fig. \ref{['fg:qf']}, the same $\xi$ here does correspond to the same atomic structure. (b) The same as (a), but for BJG path. (c) The KS levels calculated at $q=1+$ along the BK path that are used to evaluate the recombination rate. Only levels for up spin are shown since the down-spin electrons do not participate the capture. Electrons are filled up to the lowest level above the band gap. The KS levels exhibit an avoided crossing when total energies crosses, as marked by the circle. (d) The same as (b), but for BJG path. Here, KS levels calculated at $q=1+$ are shown for both up-spin and down-spin electrons, which participate the first and second captures, respectively.
• Figure 4: (a) and (b) The matrix elements for the electron capture on BK and BJG pathways (the second capture for the latter), respectively. Blue and red lines correspond to the total-energy difference between the two distinct charge states (see text) with its zero-value indicating the cross point between the two charge states (blue square). The matrix element ${H}_{\text{cd}}$ for the electron capture at the cross point is emphasized by the black square. (c) The overall capture probability as a function of electron density $n$ and migration speed $\mathrm{d}\xi/\mathrm{d} t$. (d) The capture probability at each $\xi$ along the BK pathway. The vertical dashed line depicts the position of $\xi_0$.