Nature of acceptor states in magnesium-doped gallium nitride

The most effective method of producing $p$-type gallium nitride is currently through incorporation of magnesium. However, such doping leads not only to the desired shallow acceptors but also to the creation of much deeper energy levels. Magnesium-related acceptors have been observed in many optically detected magnetic resonance experiments and their magnetic properties, as characterized by the $g$ values of the holes which they trap, have been found to vary significantly according to the growth conditions and doping levels. The purpose of the present paper is to present a model that accounts for these observations. The model assumes that, in the deep acceptors, the hole is located in an atomic orbital of $p$ character, presumed to be on a nitrogen atom. The orbital degeneracy is partly removed by the wurtzite crystal field and finally by a reduction in local symmetry associated with the relative positions of the magnesium dopant and the nitrogen atom upon which the hole is localized. Further changes in the local crystal field are caused by the presence of nearby defects or by strain. These changes in the crystal field are accompanied by changes in acceptor depth. The approach leads to the correct $g$ values and the correct correlation between the $g$ values and acceptor depth for reasonable choices of the parameters. In the limit that the low symmetry fields become small, the model evolves to one that is consistent with the correct forms of the ground and near-ground Kramers doublets that are observed by other workers in studies of shallow acceptors in material that is not doped with magnesium. Finally, the model is shown to be entirely consistent with a range of acceptor states of different depths being formed by simple substitution of a magnesium ion at a gallium site, rather than by the creation of more complicated defects. The conclusion also highlights the need for the GaN to be of high crystalline quality if effective $p$-type doping is to be achieved.

Figure 1

(a) The ODMR spectra for specimen 623 with detection at different PL energy ranges: (i) $2.75\text{to}3.1\mathrm{eV}$; (ii) $2.48\text{to}2.75\mathrm{eV}$; (iii) $2.25\text{to}2.48\mathrm{eV}$. The microwave frequency was $13.76\mathrm{GHz}$ and the temperature was $2\mathrm{K}$. The magnetic field was along the crystal $c$ axis. (b) The values of $g_{\Vert }$ for deep acceptors in specimen 623, measured at three different detection windows, centered at the energies shown. (c) The PL spectrum at $10\mathrm{K}$ for specimen 623 under UV excitation from a $\mathrm{He}\text{−}\mathrm{Cd}$ laser.

Figure 2

The energy levels for holes in $p$-type orbits. Each level represents a Kramers doublet, which is split when a magnetic field is applied. In each case the hole occupies one of the states of the highest energy doublet. Case (a) represents a $c$-axis axial field which causes the $\mid p_z,1∕2⟩$), $\mid p_z,-1∕2⟩$ doublet to lie highest. In case (b), the sign of the $c$-axis field is reversed and the orbitally degenerate $\mid p_x⟩$, $\mid p_y⟩$ doublet is split by spin-orbit coupling into the doublets ${\varphi }_1^{\pm }=\mid 1,1∕2⟩$, $\mid -1,-1∕2⟩$ and ${\varphi }_2^{\pm }=\mid 1,-1∕2⟩$, $\mid -1,1∕2⟩$. The diagram is drawn for the notional case that $\mid {\Delta }_{cfz}\mid ⪢\mid \lambda \mid$ and where ${\varphi }_3^{\pm }$ denotes $\mid p_z,\pm 1∕2⟩$. Increasing spin-orbit coupling increasingly mixes ${\varphi }_3^{\pm }$ with ${\varphi }_2^{\mp }$. In case (c), an additional crystal field is taken to be present which removes the orbital degeneracy, the states now being of the form $\mid p_x,1∕2⟩$, $\mid p_x,-1∕2⟩$ and $\mid p_y,1∕2⟩$, $\mid p_y,-1∕2⟩$; these are mixed by spin-orbit coupling, resulting in the $g$ shifts observed for the deep acceptors.

Figure 3

The $g$ values for holes at deep acceptors as a function of an axial field in the direction of the $c$ axis $(U=\mid {\Delta }_{cfz}\mid )$. For large values of the crystal field (extreme right side of the diagram), the $g$ values are approximated by the perturbation expressions given in the text [Eqs. (3, 4)]. The broken lines show calculated values according to the exact parts of Eqs. (5, 6).

Figure 4

The $g$ values for holes at deep acceptors as a function of an additional axial crystal field in the $x$ direction. The $c$-axis field is fixed (${\Delta }_{cfz}=-1800\mathrm{meV}$, $\lambda =-15\mathrm{meV}$). If the $c$-axis field is strong, as assumed here, the components $g_x$ and $g_y$ are identical. For large values of the crystal fields, the behavior is shown in expanded form in the inset, for the case where $g_l=1.0$. In this range (extreme right of the diagram), the behavior is approximated by the perturbation expressions given in the text [Eqs. (7, 8, 9)]. The broken lines show the calculations for reduced values of the orbital $g$ value.

Figure 5

The recombination processes leading to the different parts of the broad PL band. As the perturbation of the acceptor by nearby defects is increased, the $p_x$ and $p_y$ states move further into the gap and their splitting increases.