On compensation in Si-doped AlN

Aluminum nitride (AlN) and aluminum gallium nitride (AlGaN) are important next-generation materials for high-power electronics and deep UV light sources (light emitting and laser diodes).^1–4 Realization of these applications centers on the ability to controllably dope over a wide range of doping regimes. For example, drift regions in power electronics require low doping (⁠$∼1015$ cm⁻³), while opto-electronics typically require higher dopant concentrations (⁠$∼1019$ cm⁻³). Silicon is often preferred for donor doping applications due to its hydrogenic nature in gallium nitride (GaN) and Ga-rich AlGaN, enabling well controlled electron populations over a very wide doping regime.^5–8 Similar behavior might be expected for Si doping in Al-rich AlGaN and AlN. Nevertheless, two issues arise in translating Si to these systems.

First, while Si still prefers the cation site (i.e., $SiAl/Ga$⁠), it is no longer a shallow dopant, due to the emergence of a DX transition near the conduction band (CB) edge with the increasing Al content. Experimentally, the DX transition has been assigned a wide range of activation energies ranging between 78 and 345 meV.^9–12 First principles methods using hybrid exchange-correlation functionals predict the transition with activation energies of 150 meV (Gordon⁸) and 200 meV (this work) below the CB edge in AlN. In principle, it should be possible to overcome the higher DX activation energy by doping with more Si, at the cost of lower mobility.

The second issue, however, is that additional doping only yields more carriers in the low doping regime. The carrier concentration actually decreases with additional Si in the high doping regime. The increase and subsequent decrease in the carrier concentration as a function of Si content is often referred to as a compensation knee. Several studies have observed a compensation knee in AlGaN with the Al content ranging from 60% to 100%.^13–19 The mechanism of this knee has been attributed to many factors, which include the DX transition,¹⁴ cation vacancies,¹⁶ $SiN$⁠,¹⁶ and cation vacancy-silicon complexes.^13,19

The true knee mechanism, whatever its origin, is deeply tied to how the system achieves charge neutrality. There are many routes to charge neutrality in a wide bandgap material like AlN, and the dominant mechanism depends on the position of the Fermi level and the availability of each species in the growth environment. Many possibilities for defect-related compensation have been explored computationally, including native defects (vacancies, interstitials, etc.), common impurities (C, O, and Si-DX), and first-nearest neighbor complexes between Si and C and between O and cation vacancies.^8,20–23 However, the second-nearest neighbor complexes between Si donors and cation vacancies proposed by Chichibu¹³ and more recently by Bryan¹⁹ have not been explored computationally, and the roles played by any of these defects in the compensation knee have not been definitively determined.

We have used state-of-the-art density functional theory (DFT) calculations utilizing hybrid exchange-correlation functionals to develop a model which predicts that $VAl+nSiAl$ complexes are central to forming the compensation knee and to a mechanistic understanding of compensation in the high doping limit of Si in AlN. These defects are consistent with previous experiments that measured high concentrations of cation vacancies observed in n-type AlN and AlGaN^23–25 as well as the observation by Taniyasu that the calculated donor concentration is lower than the actual Si concentration in highly doped samples.¹⁶ Furthermore, our DFT calculations predict that the main compensating defect should change in the low and high Si-doping limits and that this should lead to an observable shift in the optical emissions. Detailed photoluminescence (PL) measurements on metalorganic chemical vapor deposition (MOCVD) grown AlN samples confirm such a luminescence shift for low and highly Si-doped samples, and the DFT predictions are in good agreement with these measurements.

All DFT calculations were carried out in the Vienna Ab initio Simulation Package (VASP 5.3.4).^26–29 All calculations used the hybrid exchange-correlation functional of Heyd, Scuseria, and Ernzerhof (HSE), with an exact exchange amount of $α=0.32$⁠, yielding a bandgap of 6.1 eV.^30,31 All defect calculations were carried out in 96-atom supercells with a 2 × 2 × 2 k-point mesh and a 500 eV kinetic energy cutoff.

Formation energies were calculated using the grand canonical formalism.^32,33 In this formalism, the formation energy of a point defect D^q, where q is the charge state, is given by

In this expression, $EDqtot$ and $Ebulktot$ are the total energies, respectively, of a supercell containing defect D^q and of the corresponding bulk supercell. μ_i is the chemical potential of species i, and n_i is the number of atoms of species i exchanged between the bulk and a chemical reservoir to create the defect. μ_e is the Fermi level relative to the valence band maximum (E_v). $EDqcorr$ is a post-hoc finite-size correction to the energy of the charged defect supercell based on the method of Kumagai,³⁴ using an experimental relative permittivity of 9.14.³⁵ Convergence of the formation energies, including those of neutral vacancy complexes, was found to be on the order of 0.2 eV as compared to supercells containing up to 360 atoms. The formation energies and optical transitions were calculated for native vacancies; on-site and DX configurations of Si, O, and C impurities; first nearest neighbor (1-NN) complexes for all pairs of defects on their dominant sites; $VAl+nON$ 1-NN complexes with $n=1−4$ O atoms surrounding an Al vacancy; $VN+CN$ 2-NN complexes; and $VAl+nSiAl$ 2-NN complexes with $n=1−3$⁠.

Figure 1 shows a plot of formation energy versus Fermi level for $VAl, SiAl$⁠, and $VAl+nSiAl$ complexes. For each defect at each Fermi level, only the most stable charge state is plotted. The slope in the formation energy plot gives the charge state of the defect. A change in the slope in Fig. 1 corresponds to the Fermi level at which two charge states of a defect are equally stable (i.e., a thermodynamic transition level).

In the silicon-vacancy complexes, the Si impurities sit on Al second nearest neighbor (2-NN) sites to the Al vacancy; there are 12 such 2-NN sites. Thus, the $VAl+1SiAl$ complex has 12 total configurations, 3 of which are unique due to symmetry; $VAl+2SiAl$ has 66 total configurations, 13 of which are unique; and $VAl+3SiAl$ has 220 configurations, 42 of which are unique. All symmetrically inequivalent configurations of these complexes were explicitly modeled in all relevant charge states. It is clear from Fig. 1 that there are widespread formation energies for different configurations of $VAl+nSiAl$ especially as the number of Si atoms in the complex increases. This dispersion in energy is due to electrostatic interactions between the Si impurities in each complex. For instance, the formation energy of neutral $VAl+3SiAl$ increases approximately linearly as the average Si-Si distance decreases in the configuration. The binding energy of each $VAl+nSiAl$ complex at relevant Fermi levels (relative to $VAl−3$ and $SiAl+1$⁠) is greater than 1.5 eV. Regardless of binding energy, the concentration of all species is governed by the formation energy.^32,33

Formation energies provide a route to determine the concentration of impurities, native defects, and complexes for various growth conditions (Al-rich to N-rich). In Fig. 1, the Fermi level is taken as a free parameter, but its exact position is determined by conditions of charge neutrality (i.e., $n−p=∑i,qq[Diq]$⁠) and is influenced by all defects present in the system. Based on this principle, we utilize a numerical charge balance solver^22,36,37 to simulate doping at an elevated temperature followed by a quench to room temperature. After the quench, high temperature defect concentrations are frozen in, but each defect is allowed to change its charge state in order to achieve charge balance. Electron and hole concentrations are calculated using an effective density of states from experiment.

Figure 2 shows a representative plot of $log n$ versus $log [Si]$⁠, where [C] = 3 × 10¹⁶ cm⁻³, [O] = 4 × 10¹⁷ cm⁻³, the doping temperature is representative of MOCVD growth (1400 K), and the native chemical potential is 0.2 eV from the Al-rich limit. Impurity chemical potentials are determined by imposing fixed concentrations on each impurity. Thus, $μO$ and $μC$ vary with the Si concentration in order to maintain constant levels of O and C. Chemical potentials are compared with stability-limiting phases in post-processing. The knee maximum n in this plot is consistent with carrier concentrations common in Si-doped AlN (order of $1015 cm−3$⁠), but the Si concentration where it occurs is predicted to be too low (experimentally, the peak is close to $1019 cm−3$⁠).^18,19 We therefore examined how the predicted knee responds to changes in processing conditions and background impurities, as well as the effect of defect vibrational energies. In all cases, we obtained a compensation knee from the results of charge balance solutions when vacancy-silicon complexes were included in the defect set.

The shape of the knee and the location of the knee maximum depend on the model parameters. Both the predicted Si concentration and the carrier concentration at the knee maximum increase as the growth conditions vary from N-rich to Al-rich. Common background impurities (C and O) typically compensate the free carriers. As a result, increasing their concentration reduces the maximum carrier concentration and shifts the knee maximum to higher Si concentrations. Quantitative predictions of the knee are highly sensitive to the Fermi level, and so, even small contributions to the formation energies, including vibrational energies, may be relevant. Due to the use of hybrid functionals and the large number of defects and configurations, it is a substantial undertaking to compute vibrational contributions. We have explored the sensitivity to vibrational energies in Si-containing defects by assuming a constant vibrational contribution per Si atom in each defect, which alters the relative energies of complexes with different numbers of constituent defects. In this approximation, vibrational energies shift the knee maximum to higher Si concentrations, more consistent with experiment. This does not contradict the convention that vibrational energies tend to be similar for single-site defects. Rather, it points to the potential importance of including vibrational energies when complexes are considered.

We now turn to understanding the mechanism of compensation in Si-doped AlN. Si is added to the system by increasing its availability in the growth environment, which is equivalent to increasing the Si chemical potential. Figure 3 plots the formation energy of the most energetically favorable configuration of each $VAl+nSiAl$ complex for low (dotted) and high (solid) Si concentrations, corresponding to either side of the knee. As can be seen, the change in formation energy is equal to $nΔμSi$⁠, where n is the number of Si atoms in the complex. That is, higher-n $VAl+nSiAl$ complexes are more strongly affected by changes in Si availability than low-n $VAl+nSiAl$ complexes and Si_Al.

Thus, the formation of the knee can be understood in terms of equilibrium concepts. In the low [Si] regime, $SiAl$ and $VAl$ dominate and thus control the equilibrium Fermi level. In this regime, the Fermi level increases with Si doping. But in the high [Si] regime, the favorability of higher-n complexes is enhanced, and the Fermi level is controlled by the intersection of $ESiAlf$ and $E(VAl+nSiAl)f$⁠. Because $E(VAl+nSiAl)f$ shifts down more rapidly than $ESiAlf$ for n = 2 and n = 3, the equilibrium Fermi level decreases. That is, the Fermi level (carrier concentration) decreases with Si doping in the high [Si] regime.

Our predictions point to the following observations. The second nearest neighbor defects in AlN are possible and stable in certain conditions. These defects are central to the formation of the compensation knee and are expected to be in their highest concentrations when Si is most available in the growth environment. This change in compensating defects between the high and low doping regimes motivates the exploration of predicted optical signatures to see if these predictions can be further tested by experimental measurements. We focus this analysis on midgap emissions (2.5–4.5 eV) of compensating defects in high concentration.

Two points that have roughly the same carrier concentration on opposite sides of the compensation knee were selected for this and are indicated in Fig. 2. Table I summarizes the concentrations of all optically relevant defects above $1015 cm−3$⁠, as well as their absorption and emission energies, which were calculated using the Franck-Condon approximation.^32,33 Optical absorptions and emissions of $VAl$ and $(VAl+nON)$ complexes have recently been studied in detail by Yan et al.,²¹ and the energies in Table I are in line with their results.

At low Si concentrations, the dominant defects with emissions in the midgap range are $(VAl+3ON)0$ and $CN−1$⁠, which are predicted to emit between 3.6 and 3.7 eV. The 3.6 eV emission prediction for $CN−1$ underpredicts the experimental measurement of 3.9 eV as it may emit before it fully relaxes in the excited state.³⁸ There is also a small concentration of vacancy-oxygen complexes which are expected to emit at 3.0 and 3.3 eV in the low Si doping limit.

By contrast, the dominant defects in the high [Si] regime are vacancy-silicon complexes, which emit in the range of 2.9 to 3.4 eV, depending on the configuration. Oxygen now forms $(VAl+1ON)−2$ and $(VAl+2ON)−1$⁠, and so, oxygen-related emissions shift to lower values around 3.0 and 3.3 eV. Although its concentration is small relative to other defects, $VAl−3$⁠, with an emission at 2.7 eV, may also contribute to the PL peak at high [Si]. The intensities of the emission peaks due to $(VAl+nSiAl)$ in this doping regime are expected to increase relative to the $(VAl+nON)$ and $CN−1$ peaks since [O] and [C] are fixed, while [Si] is higher relative to the low doping regime.

Thus, in the low [Si] regime, we predict an optical emission peak around 3.6 or 3.7 eV due to $(VAl+3ON)0$⁠, perhaps with a shoulder near 3.3 eV due to $(VAl+2ON)−1$⁠. $CN−1$ should also contribute with an emission peak at or above 3.9 eV based on experimental assignment. In the high [Si] regime, we predict a peak centered closer to 3.2 eV, with some broadening due to the ensemble of defects emitting at various energies in the range of the peak.

To test these predictions, AlN films with low and high Si concentrations (⁠$2×1018$ and $2×1019 cm−3$⁠, respectively) were grown on single crystal AlN substrates using an rf-heated low-pressure MOCVD reactor equipped with an open showerhead.³⁹ The growth temperature was 1373 K. PL spectra were acquired using a pulsed ArF excimer laser (λ = 193 nm) together with a Princeton Instruments Acton SP2750 0.75 m high-resolution monochromator and a PIXIS: 2KBUV cooled charge-coupled device camera. Room temperature PL spectra of the low and high Si content samples are shown in Fig. 4. The sample with a low Si content corresponds to a doping level below the onset of the compensation knee, while the sample with a high Si content corresponds to a doping level above the compensation knee. A clear shift is observed in the peak PL value from about 3.7 eV in the low [Si] doping regime to about 3.2 eV in the high [Si] doping regime. In the low [Si] regime, we also observe a shoulder around 3.2 eV. These observations are consistent with the theoretical model proposed above and provide further support for our defect model. In a recent study of AlGaN grown on different substrates, a corresponding PL peak transition was observed at high Si doping levels.¹⁹ The peak shift was found to depend only on the Si concentration, despite the observation that samples grown on different substrates had different dislocation densities and hence different strain states. This rules out the possibility that the PL peak shift could be due to differences in strain associated with low and high Si doping conditions. Further experimental details, including deconvolution of the PL, can be found in the study by Bryan et al.^18,19

In conclusion, $VAl+nSiAl$ complexes are crucial to explaining the compensation knee observed in Si doping experiments in AlN. The basic mechanism behind the compensation knee can be explained using simple arguments with a formation energy diagram, together with the fact that the formation energies of $VAl+nSiAl$ vary with Si chemical potential at different rates depending on n. Finally, the predicted optical emission peaks for relevant defects on either side of the compensation knee agree well with experimental PL measurements, suggesting that our defect model appropriately represents the physics of Si doping in AlN.

See supplementary material for additional information, including a list of all defects and charge states simulated.

The authors would like to acknowledge partial financial support from NSF (DMR-1151568, ECCS-1508854, ECCS-1610992, and ECCS-1653383), DOE (DE-SC0011883), ARO (W911NF-15-2-0068 and W911NF-16-C-0101), and AFOSR (FA9550-14-1-0264 and FA9550-17-1-0225). Computer time was provided by NERSC and DoD HPCMP.