A theoretical framework that provides a quantitative relationship between point defect formation energies and growth process parameters is presented. It enables systematic point defect reduction by chemical potential control in metalorganic chemical vapor deposition (MOCVD) of III-nitrides. Experimental corroboration is provided by a case study of C incorporation in GaN. The theoretical model is shown to be successful in providing quantitative predictions of C_{N} defect incorporation in GaN as a function of growth parameters and provides valuable insights into boundary phases and other impurity chemical reactions. The metal supersaturation is found to be the primary factor in determining the chemical potential of III/N and consequently incorporation or formation of point defects which involves exchange of III or N atoms with the reservoir. The framework is general and may be extended to other defect systems in (Al)GaN. The utility of equilibrium formalism typically employed in density functional theory in predicting defect incorporation in non-equilibrium and high temperature MOCVD growth is confirmed. Furthermore, the proposed theoretical framework may be used to determine optimal growth conditions to achieve minimum compensation within any given constraints such as growth rate, crystal quality, and other practical system limitations.

## I. INTRODUCTION

Due to its wide bandgap, (Al)GaN growth tends to incorporate native and extrinsic point defects of a compensating nature when doped. This is due to the dependence of defect formation energy on the Fermi level, which can vary across the large bandgap, thereby providing significant challenges at doping. This phenomenon results in various previously observed phenomena including mobility collapse, knee behavior, and lower internal quantum efficiency (IQE).^{1–7} The propensity to incorporate a particular defect during the growth processes may be represented by the formation energy of that defect. For a defect with charge state q, in a material at equilibrium, it is defined as

where E_{ref} is the free energy of the crystal with a single defect referenced to the free energy of the ideal crystal, n_{j} is the number of atoms of the jth type exchanged with the reservoir to form the defect, μ_{j} is the associated chemical potential, and E_{F} is the Fermi energy referenced with respect to the valence band maximum. More details can be found elsewhere.^{7,8} The energy of formation may be altered by altering the chemical potentials, the second term in Eq. (1), or by changing the Fermi level, third term in Eq. (1).

In this work, we demonstrate point defect control by controlling the energy of formation as a tool to reduce point defect incorporation and compensation. Point defect control by tuning the growth conditions, i.e., V/III ratio, temperature, pressure, etc., has been reported in the literature.^{5,9–11} Although individual studies on effects of growth conditions on incorporation of various defects exist, global quantitative theoretical models describing the dependence of point defect incorporation on growth conditions or formulating the growth parameters in terms of chemical potentials are lacking. Figure 1 illustrates the proposed theoretical model that provides a quantitative relationship between point defect formation energies, i.e., probability for incorporation, and “MOCVD knobs,” i.e., growth parameters including the V/III ratio, precursor species flows, diluent gases, and temperature. The relationship between the “MOCVD knobs” and supersaturation, i.e., input and equilibrium partial pressures, has been established earlier.^{5,12,13} Density functional theory (DFT) analyses of point defects in III-nitrides, including more accurate models in recent years, have described their formation energies in terms of chemical potentials of III-metals, nitrogen, and impurity atoms associated with the defect.^{7,14–17} The defect formation energies determined at equilibrium may be employed to describe point defect incorporation during the non-equilibrium process condition (i.e., thin film growth) since the growth of (Al)GaN in a well-controlled metalorganic chemical vapor deposition (MOCVD) reactor is typically mass transport limited and the driving force (supersaturation) or chemical potential drop is primarily across the boundary layer.^{18} Consequently, the growth surface is considered to be near equilibrium.

Carbon on the nitrogen sub-lattice is an important compensator found in MOCVD growth. The effect of the V/III ratio or diluent gases on C incorporation has been widely reported.^{5,19,20} The magnitude of the available data makes C incorporation in Si doped GaN an excellent case study. The proposed theoretical model is shown to be successful in relating growth parameters and chemical potentials of C incorporating reaction constituents, providing new insights into boundary phases and quantitatively predicting C incorporation in GaN.

## II. RELATIONSHIP BETWEEN THE SUPERSATURATION, EQUILIBRIUM PARTIAL PRESSURES, AND CHEMICAL POTENTIALS OF III-METALS AND NITROGEN

The primary requirement of developing the theoretical framework is the relationship between the change in the growth supersaturation and equilibrium partial pressures and the corresponding change in the impurity, III-metal, and nitrogen chemical potentials. The primary assumption of the proposed model is that the chemical potential of the III-atoms at the surface with respect to the standard state (metal) equals to

where $PeqIII$ is the equilibrium partial pressure of III atoms at the surface during growth and $PVIII$ is the vapor pressure of that metal at equilibrium at a given temperature. In this paper, unless specified, log is defined to base e. For any growth state, the equilibrium partial pressure of III atoms is calculated using the thermodynamic model for GaN growth by MOCVD.^{5,12,13}

Furthermore, at equilibrium, the chemical potentials of III and N atoms are related by^{7}

where $\mu IIIN$ and *ΔH _{f}* are the chemical potential and enthalpy of formation of the nitride, respectively. The partial pressure of the III-atoms and hence $\mu III$ can be calculated for any growth state using the growth thermodynamic model

^{5,12,13}and Eq. (2). Furthermore, for any change in the chemical potential of the III-atoms $\Delta \mu III$ with respect to a reference growth state, the change in the chemical potential of nitrogen can be found from

Hence, chemical potential changes of the III-metal and N may be calculated for any growth state with respect to a reference growth state. In addition, the chemical potentials of any other atoms constituting impurities during growth also need to be calculated. The impurity sources and the corresponding partial pressures in addition to boundary phases and reactions are required. To relate chemical potentials to the input metalorganic flow, Eq. (2) may be expanded as

where $PinIII$ is the input partial pressure of the III-atoms, and consequently,

where σ is the III-atom supersaturation over the III-nitride given by $\sigma =PinIII\u2212PeqIIIPeqIII$. Hence, it is clear that the supersaturation and input precursor flows relate to the III-rich or N-rich nature of the growth front and consequently quantify the incorporation of point defects. Note that the supersaturation is a function of precursor flows, pressure, and temperature, which are in turn controlled by the “MOCVD knobs.” The $kT\u2009log(PinIIIPVIII)$ term in Eq. (8) may be expressed as supersaturation for the gallium metal in the standard state, σ_{III},

Hence, the chemical potential of III-atoms (and consequently of N) may be expressed as a function of only one growth parameter, supersaturation, as

It is important to note that various combinations of growth parameters, i.e., settings of the “MOCVD knobs,” can lead to exactly the same supersaturation and will, according to the above, yield the same results, which is not obvious when one looks at the growth parameters themselves.

## III. REDUCTION OF C_{N} IN GaN BY CHEMICAL POTENTIAL CONTROL

In this section, the incorporation of compensating C_{N} in Si doped GaN is studied and the theoretical framework is applied to provide a relationship between the growth conditions and C incorporation along with insights into the boundary phases and reactions that control the chemical potential of C. In Si doped GaN, C on the nitrogen site is the point defect of interest. It is an acceptor ($CN\u22121$) with the most energetically favorable configuration of carbon that is not a complex.^{14,21} From Eq. (1), its formation energy is given by

where *μ*_{N} and *μ*_{C} are the chemical potentials of N and C, respectively. It has to be noted that net C incorporation is independent of the Si doping, and hence, the Fermi level in Eq. (11) is a constant and independent of growth conditions.^{22} The constant Fermi level is most likely due to Fermi level pinning^{23} which makes the Fermi level close to the surface mostly independent of doping, and hence, the Fermi level in the formation energy equation remains constant with doping. A similar conclusion was arrived to by Kempisty *et al.*^{24} to explain the differences in impurity incorporation between Ga-polar and N-polar films. They attribute the different incorporation to different Fermi level pinning energies (that goes into formation energy equations). The bulk Fermi level certainly changes with doping but is assumed to play no role in C incorporation at the surface. Furthermore, alternate neutral complexes of C_{N} with O or V_{N} are also a possibility whose formation energy is independent of variations in the Fermi level (above the defect energy state).^{14} In contrast to the constant Fermi level (electrochemical potential), the chemical potential is dependent on the growth conditions. We define a set of growth conditions that determine the chemical potential as a growth state. The change in the formation energy of C due to the difference in chemical potentials of N and C (i.e., Δ*μ*_{N} and Δ*μ*_{C}) between some specified experimental and reference growth states is given by

and the corresponding incorporation of C is given by

where $[CN]\u2009exp\u2009$ and $[CN]ref$ are the C incorporation for the specified experimental and reference growth states, respectively. Consequently, the changes in the chemical potentials of N and C as a function of the change in the growth state are required to determine the change in the formation energy and C incorporation. From Eq. (6), the difference in the chemical potential of N is dependent on the difference in the chemical potential of Ga by

Furthermore, for any change in the equilibrium partial pressures due to a change in the V/III ratio, diluent gases (H_{2} or N_{2}), or temperature from a reference growth state, the change in the chemical potential of gallium is calculated from Eqs. (2) and (10) as

where the subscripts “exp” and “ref” refer to specified experimental and reference growth states, respectively. It is clear from Eq. (10) that the chemical potentials of gallium may be obtained from the gallium supersaturations over GaN and Ga (standard state) and hence equilibrium and input partial pressures of gallium. The change in the chemical potential of N is then obtained from Eq. (14).

The only remaining unknown in Eq. (12) is Δ*μ _{C}* and knowledge of the boundary phase of C; impurity sources and their reactions are required to determine it. We initially follow the literature and assume a boundary phase of diamond,

^{25}which implies a chemical potential of carbon independent of chemical potentials of nitrogen or gallium. Furthermore, the influence of the C source (assumed to be ethane obtained from pyrolysis of metal organic precursor triethylgallium (TEG)) and H

_{2}on C incorporation is also expected and is incorporated by assuming the chemical reaction

which results in the chemical potential of C (*μ _{C}*) dependent on chemical potential changes (i.e., partial pressures) of ethane and hydrogen as

where the coefficients *z* and *y* are empirical coefficients to be determined from the experiment.

Three growth regimes were defined to obtain experimental data to validate the theoretical framework. Accordingly, hetero-epitaxial films of GaN were grown on c-plane sapphire via metalorganic chemical vapor deposition (MOCVD) with a silicon precursor with varying V/III ratios, diluent gases, and temperatures corresponding to growth states. The growth regimes were Regime 1—Low V/III regime (<2000) with NH_{3} flow as the V/III variable with H_{2} diluent and constant growth rate; Regime 2—Low V/III regime (<2000) with NH_{3} flow as the V/III variable with N_{2} diluent and constant growth rate; Regime 3—high V/III regime (>2000) with TEG flow as the V/III variable with H_{2} diluent and growth rate proportional to TEG flow. Details on III-nitride growth are provided elsewhere.^{5,20} The C incorporation was measured by Secondary Ion Mass Spectroscopy (SIMS). The total flow and pressure are constant at 7.5 slm and 20 Torr, respectively, in all regimes. Note that as the ammonia or TEG flow is altered to control the V/III ratio, the diluent gas is accordingly altered to maintain constant total flow. Please note that the V/III ratio is the ratio of NH_{3} flow to TEG flow (where TEG flow $fTEG\u223cfTEG+carrier\xd7PTEGPB$, where f_{TEG+carrier} is the metal organic flow including the carrier gas, P_{TEG} is the equilibrium vapor pressure of TEG at the bubbler temperature, and P_{B} is the bubbler pressure) which is equal to the ratio of input partial pressures of ammonia and Ga. Ga flow is determined with the TEG bubbler temperature and pressure being 290 K and 400 Torr, respectively. The TEG vapor pressure is ∼1.1% of bubbler pressure and represents the TEG molar fraction of the measured flow. In Regimes 1 and 2, the TEG flow is constant at 280 sccm. In Regime 3, the NH_{3} flow is constant at 5.9 slm.

As discussed earlier, in the growth regimes studied, determining the chemical potentials of Ga (and hence N) and C requires the equilibrium partial pressure of Ga and input partial pressures of the Ga precursor, hydrogen, and ammonia. Accordingly, the equilibrium partial pressures in Regimes 1 and 2 are shown in Fig. 2(a) and those in Regime 3 are shown in Fig. 2(b) at typical GaN growth temperatures of 1050 °C and 1100 °C. The partial pressures were obtained from the previously published thermodynamic model^{5,12,13} describing GaN growth. The behavior of the partial pressures shown in Fig. 2 may be understood using the growth reaction

ammonia decomposition

and the equilibrium constant

In Regime 1, in H_{2} diluent, ammonia decomposition is poor and increasing ammonia partial pressure reduces the gallium partial pressure at equilibrium. This may be understood from the equilibrium constant in Eq. (20). In Regime 2, under N_{2} diluent, the ammonia decomposition is strong, and hence, increasing the V/III ratio and ammonia partial pressure also increases hydrogen partial pressure. The dependence of ammonia decomposition on diluent gas may be understood by the decomposition reaction (19). The decomposition may be catalyzed by the growth surface^{26} and is dependent on H_{2} partial pressure, i.e., if H_{2}-rich, the equilibrium shifts to prevent decomposition and if H_{2}-poor, the equilibrium shifts towards increased decomposition. In the thermodynamic model, *α* is assumed to be 0.2 under H_{2} diluent. In contrast, it is experimentally found to be much larger under N_{2} diluent, α ∼ 0.5 at 1050 °C, and the corresponding discussion is provided later in the paper. Hence, from Eqs. (18) and (20), for a constant K, the equilibrium partial pressure of GaN is relatively unchanged in Regime 2. In Regime 3, increasing the V/III ratio by decreasing TEG flow at a constant ammonia flow has no influence on reactions (18) and (19), i.e., for a constant K, the equilibrium partial pressure of Ga does not change.^{5}

The input partial pressure of gallium may be determined from the ratio of input gallium and input total flows taking into account the ammonia decomposition [Eq. (19)] and is given by

where P_{T} is the total reactor pressure (20 Torr), and *f _{Ga}*,

*f*, and

_{NH3}*f*are the gallium, ammonia, and total input mass flows, respectively (This is a corrected input gallium partial pressure and should be considered as a correction to the thermodynamic model proposed by Mita

_{T}*et al.*in Ref. 5.). The gallium supersaturations over GaN and Ga (standard state), i.e., σ and σ

_{Ga}, can be determined from the equilibrium partial pressure shown in Fig. 2 and the input partial pressure of Ga calculated for different regimes shown in Fig. 3 and the vapor pressure of Ga (Figs. 2 and 3). The corresponding free energy change, $\Delta GGaN$, is shown in comparison to $\Delta GGa$ in Fig. 4. From Eq. (10), the difference in the two free energies determines the chemical potential of Ga (

*μ*) as shown in Fig. 4. Typically, $\Delta GGa$ is small compared to $\Delta GGaN$, and consequently, the metal supersaturation over nitride typically quantifies the chemical potentials of III-metal and N and point defect incorporation. Hence,

_{Ga}From Eq. (15), the relative change in the chemical potential of gallium (Δ*μ _{Ga}*) at any design growth state may then be determined with respect to a specified reference growth state. The reference state is defined as V/III = 100, T = 1050 °C (1323 K), H

_{2}diluent, and P = 20 Torr. The corresponding change in the chemical potential of N (Δ

*μ*) from the reference state is then calculated from Eq. (14) and is shown in Fig. 5 for different growth regimes.

_{N}A similar analysis with known input partial pressures of hydrogen and ethane (from TEG decomposition) in the studied growth regimes allows for the calculation of the chemical potential changes of hydrogen and ethane with respect to the reference growth state as shown in Fig. 5. The input partial pressure of ethane is assumed to be 3 × P_{in}^{Ga} (input partial pressure of gallium specified in Fig. 3). Furthermore, the partial pressure of ethane at the surface is assumed to be the same as the input partial pressure of ethane. The justification is as follows: The partial pressure of the C source is comparable to the input partial pressure of gallium, but C incorporation is much lower than the mass flow limited Ga incorporation (i.e., growth rate). The low rate of C incorporation ensures nearly constant ethane partial pressure across the boundary layer and consequently, the chemical potential drop for ethane is primarily at the surface and not across the boundary layer. The crystallographic surface orientation (i.e., N-polar vs. Ga-polar growth) dependence of C incorporation^{27} will be discussed later on in this paper. The partial pressure of hydrogen is obtained by including ammonia decomposition.^{5} The chemical potential differences are utilized to determine the change in the formation energy of C and hence C incorporation. In the reference growth state, the measured C concentration is ∼6 × 10^{19} cm^{−3}. We initially study C incorporation in Regime 1. From Fig. 5(a), in Regime 1, increasing the V/III ratio makes the growth condition more N-rich, i.e., Δ*μ*_{N} > 0. Furthermore,

and

Hence, the C incorporation is primarily a function of the chemical potential of nitrogen, i.e., $\Delta Ef(CN)\u223c\Delta \mu N$, and from Eq. (13), the C incorporation may be written as

From this relationship, the C_{N} incorporation is expected to decrease as the growth conditions become more N rich. Accordingly, the measured C incorporation (via SIMS) decreases with an increase in the V/III ratio, in qualitative agreement with theory as shown in Fig. 6. However, Eq. (25) is found to severely underestimate the measured reduction in C as shown in Fig. 6. Interestingly, if the reduction in C is assumed to be due to the change in the formation energy only, the experimentally observed change in the formation energy is closer to $\Delta Ef(CN)\u223c2\Delta \mu N$. Hence, the predicted change in the formation energy is about one half of the experimentally observed change.

The cause for this discrepancy is most likely the incorrect assumption of the boundary phase of C as diamond although it is commonly employed in DFT calculations. We propose a new boundary phase of C that accurately relates the chemical potentials of C and N. Accordingly, the boundary phase is assumed to be a C-N based compound given by

Hence, when the reaction (26) is at equilibrium,

and reduction in C is given by

in better agreement with the experimental results shown in Fig. 6. However, there still exists a difference between theoretical prediction and experimental C incorporation. Thus, an empirical coefficient x is employed, and the energy of formation is assumed as

where x is slightly lower than 2 possibly due to incorporation of C in configurations different from C_{N} (SIMS measures total C incorporation) and the influence of chemical potential change $\Delta \mu H2$. A study of the remaining growth regimes is required to separate the influences of the chemical potentials of nitrogen, hydrogen, and ethane.

In Regime 2, from Fig. 5, it is clear that the change in chemical potentials of nitrogen, $\Delta \mu N$, and hydrogen, $\Delta \mu H2$, due to change in diluent gas from H_{2} (Regime 1) to N_{2} (Regime 2) is significant and is responsible for the corresponding change in C incorporation between regime 1 and regime 2. Furthermore, the chemical potentials of nitrogen and hydrogen are related by the ammonia decomposition fraction, *α*, via chemical reaction (19). However, in regime 2, increasing the V/III ratio does not change the chemical potential of nitrogen for V/III > 200, i.e., Δ*μ _{N}* is a constant, and any change in C incorporation with the V/III ratio (>200) in N

_{2}diluent is not influenced by Δ

*μ*. Furthermore, similar to Regime 1, the flow of TEG is a constant in Regime 2 and $\Delta \mu C2H6$ is negligible. Hence, the change in C incorporation with the V/III ratio is primarily due to the change in the chemical potential of hydrogen $\Delta \mu H2$ only. Consequently, the effects of chemical potentials of nitrogen and hydrogen are separated and may be estimated (along with the change in

_{N}*α*due to the change in diluent gas) by studying Regimes 1 and 2. Accordingly, the net dependence of the energy of formation of carbon and its incorporation in Regimes 1 and 2 may be written as (neglecting the influence of ethane)

and

The parameters *x* and *z* are obtained by analyzing the experimentally measured C incorporation in both Regimes 1 and 2. C incorporation in growth Regimes 1 and 2 is shown in Fig. 7. At a low V/III ratio (<1000), C incorporation is lower under N_{2} diluent. At a higher V/III ratio, C incorporation is lower under H_{2} diluent. The difference in C incorporation may be understood from the equilibrium constant [Eq. (20)] where shifting to N_{2} diluent may be thought of as advantageous since reducing the partial pressure of hydrogen must reduce the partial pressure of gallium at a given partial pressure of NH_{3} (V/III ratio) and produce a N-rich environment and lower C incorporation. However, the expected C incorporation under N_{2} diluent is approximately an order of magnitude lower than experimentally observed C content if ammonia decomposition is not dependent on diluent gas. The increased (than expected) C incorporation under N_{2} diluent and the lack of agreement with experiment at a higher V/III ratio are consistent with the expected increased ammonia decomposition under N_{2} diluent, which reduces the partial pressure of ammonia and increases partial pressure of H_{2}. Hence, from Eq. (20), the partial pressure of gallium is relatively higher than expected, i.e., the growth environment is less N-rich than expected, leading to larger C incorporation. For a good agreement between the developed theoretical framework and experimental C incorporation in Regime 2, *α* is increased by 0.3, i.e., α ∼ 0.5, under N_{2} diluent at 1050 °C. The parameters *x* and *z* were found to be 1.7 and 0.1, respectively. Hence, the dependence of the energy of formation of C_{N} and C incorporation may be written as

and

respectively. The excellent agreement with the developed theoretical model and experimental results is illustrated in Fig. 7.

Finally, in Regime 3, from Fig. 5(b), increasing the V/III ratio by decreasing TEG flow at a constant ammonia flow has no influence on the chemical potentials of N and hydrogen. The chemical potential change in ethane, $\Delta \mu C2H6$, is the only influence on the formation energy of C_{N} and is given by

It is clear that as $\Delta \mu C2H6$ becomes more negative with an increase in the V/III ratio, the formation energy of C_{N} increases and the C incorporation is expected to decrease with an increase in the V/III ratio. The experimental C incorporation in the growth Regime 3 is shown in Fig. 8. The C incorporation decreases from ∼2 × 10^{16} cm^{−3} to below the detection limit of SIMS of ∼2 × 10^{15} cm^{−3}, in excellent agreement with the chemical potential model with *y* ∼ 1.7. It has to be noted that the coefficients *y* and *z* represent the shift in the equilibrium of reaction (16) with changing partial pressures of hydrogen and ethane. Therefore, in hydrogen diluent, the decrease in TEG flow and corresponding ethane partial pressure in Regime 3 does not reduce C incorporation by reducing the amount of the C source but rather shifts the equilibrium of Eq. (16) towards the left and results in an increased reduction of C by hydrogen.

Following this argument, an interesting twist happens with the choice of the metalorganic precursor. Since ethane has a slightly more negative free energy of formation than methane^{28,29} and, hence, it is more stable, changing the precursor to trimethylgallium may actually increase C incorporation as found experimentally by Saxler *et al.*^{30} Furthermore, the minor deviation of coefficient *x* from 2 may be a consequence of configurations of C alternate to C_{N} (SIMS measures net C) or influence of reaction kinetics. Similarly, *y* and *z* make for an unbalanced stoichiometry in reaction (16), which may be a consequence of a multi-step reaction or that additional reactants are required. In general, the formation energy change (with respect to the reference state) of C is given by

and it is shown in Fig. 9 for all regimes.

The role of temperature in defect incorporation was also studied. The experimental C incorporation was found to decrease by 40%–50% with an increase in temperature by 50 °C between 1000 °C to 1100 °C, as shown in Table I.

V/III ratio . | C at 1000 °C (cm^{−3})
. | C at 1050 °C (cm^{−3})
. | C at 1100 °C (cm^{−3})
. |
---|---|---|---|

2000 | … | 2 × 10^{16} | 1.3 × 10^{16} |

4000 | 1 × 10^{16} | 7 × 10^{15} | … |

V/III ratio . | C at 1000 °C (cm^{−3})
. | C at 1050 °C (cm^{−3})
. | C at 1100 °C (cm^{−3})
. |
---|---|---|---|

2000 | … | 2 × 10^{16} | 1.3 × 10^{16} |

4000 | 1 × 10^{16} | 7 × 10^{15} | … |

The partial pressures of hydrogen and ethane have negligible temperature dependence since they are dependent on input gas flows (input partial pressures) and total pressure, which are constant. Similarly, from Fig. 2, it is clear that both the equilibrium partial pressure and vapor pressure of gallium increase with an increase in temperature. The resulting change in the ratio of partial pressures is also negligible in the studied temperature range. The only significant change is in the kT component in Eq. (15), and hence, the chemical potential of gallium as a function of the temperature change from T_{1} to T_{2} is given by

Since $PeqGa\u2264PVGa$ (if $PeqGa\u2265PVGa$, gallium precipitates), for an increase in temperature (T_{2} > T_{1}), $(\Delta \mu Ga)T2\u2264(\Delta \mu Ga)T1$. Therefore, the equilibrium shifts to the more N rich (Ga poor) state with an increase in temperature. Similarly, the Fermi level E_{F} shifts towards the center of the bandgap with an increase in temperature. From Eq. (11), the chemical potential shift of ∼70 meV and the Fermi level shift of ∼50 meV result in an increased formation energy of C_{N} of ∼120 meV for a 50 °C increase in temperature in the studied temperature range of 1000 °C to 1100 °C. Since the defect concentration at any temperature T is proportional to

and assuming $Eref(CN\u22121)$ ∼ 4 to 5 eV,^{31} the increase in formation energy reduces the defect (C_{N}) incorporation in spite of increased available thermal energy kT (i.e., the ratio $Ef(CN)kT$ increases with an increase in temperature), which is in qualitative agreement with the experimental defect incorporation. Interestingly, if the incorporation was kinetically limited, the C concentration should have a strong dependence on temperature. However, a strong dependence on the chemical potentials and the relatively weak dependence on temperature strongly support the near-equilibrium assumption in this work.

The theory so far assumed C_{N} as the dominant configuration and any reduction in C_{N} resulted in an equal reduction in total C as measured by SIMS. The ability of the proposed theoretical framework to model the experimental results reasonably justifies this assumption. Interestingly, defect quasi Fermi level control based compensating defect reduction (where the formation energy of C_{N}^{−1} was increased) demonstrated in Refs. 22 and 32 produced a significant improvement in electron mobility [i.e., reduction in compensating defects (C_{N}^{−1})] with only a marginal or negligible reduction in total C incorporation. The provided explanation was the possibility of C occupying an alternative, uncharged configuration, which is not affected by the Fermi level control method, either as a neutral donor type complex or C_{Ga}.^{22,32} However, a similar improvement in electron mobility was also observed in the chemical potential control method but with a total reduction in C as measured by SIMS.^{5,20} As shown in this work, the decrease agrees well with the assumption of C_{N} being dominant, and the increase in its formation energy results in a corresponding overall decrease in C. Hence, we conclude that the dominant alternate C configuration occupied during Fermi level control is a neutral defect complex with C_{N} being one of the constituents. The justification is as follows: The Fermi level control has no effect on the neutral complex. However, chemical potential control affects charged C_{N}^{−1} and neutral C_{N}-based complex similarly. However, the decrease in *x* from 2 to 1.7 (lowering the dependence on Δ*μ _{N}*) was observed and may be due to the non-negligible presence of other alternate configurations such as C

_{Ga}with a formation energy nearly independent of Δ

*μ*given by

_{N}Furthermore, our studies have been on Ga-polar GaN. Studies in N-polar GaN show low C incorporation under similar growth conditions relative to Ga-polar GaN.^{33} Recently, Kempisty *et al.*^{24} attributed it to different Fermi level pinning and hence different formation energies from Eq. (11). Hence, the proposed model may be used to predict C or other defect incorporation on surfaces of different polarities with an additional formation energy offset equal to the change in the Fermi level pinning.

Finally, it may be concluded that the main influence on C incorporation is the metal supersaturation with additional influences by temperature and metal-organic precursor and hydrogen partial pressures depending on the growth regimes described earlier. The proposed chemical potential control framework described in this work may then be used to optimize growth conditions to obtain minimum point defect incorporation within growth constraints and may be combined with the quasi Fermi level control^{1,3,4,22,32,34} to obtain high quality epitaxy with low point defect densities.

## IV. CONCLUSION

In conclusion, a theoretical model that provides a quantitative relationship between point defect formation energies and “MOCVD knobs” is presented with experimental corroborations via a case study of C incorporation in GaN. In general, metal supersaturation primarily determines the chemical potential of III/N and consequently incorporation or formation of point defects which involves exchange of III or N atoms with the reservoir. The theoretical model was successful in providing quantitative predictions of the C_{N} defect incorporation as a function of growth conditions (V/III ratio, diluent gases, temperature, etc.) in GaN. It further provided valuable insights into boundary phases and other impurity chemical reactions, which could be extended to other defect systems in Al/GaN. The assumption of a growth surface close to equilibrium during the MOCVD growth of GaN at 1000–1100 °C is validated by the agreement between the experiment and theory. Consequently, the practical usefulness of the equilibrium formalism typically employed in 0 K DFT calculations in predicting defect incorporation in non-equilibrium and high temperature MOCVD growth is concluded. Furthermore, the proposed theoretical framework may be used to determine optimal growth conditions to achieve minimum compensation within any given constraints such as growth rate, crystal quality, and other practical/system limitations.

## ACKNOWLEDGMENTS

Partial financial support from NSF (DMR-1312582, ECCS-1508854, DMR-1508191, ECCS-1610992, and ECCS-1653383), ARO (W911NF-15-2-0068 and W911NF-16-C-0101), AFOSR (FA9550-17-1-0225), and PNNL (NA-22-WMS-#66204) is greatly appreciated.