A theoretical framework for a general approach to reduce point defect density in materials via control of defect quasi Fermi level (dQFL) is presented. The control of dQFL is achieved via excess minority carrier generation. General guidelines for controlling dQFL that lead to a significant reduction in compensating point defects in any doped material is proposed. The framework introduces and incorporates the effects of various factors that control the efficacy of the defect reduction process such as defect level, defect formation energy, bandgap, and excess minority carrier density. Modified formation energy diagrams are proposed, which illustrate the effect of the quasi Fermi level control on the defect formation energies. These formation energy diagrams provide powerful tools to determine the feasibility and requirements to produce the desired reduction in specified point defects. An experimental study of the effect of excess minority carriers on point defect incorporation in GaN and AlGaN shows an excellent quantitative agreement with the theoretical predictions. Illumination at energies larger than the bandgap is employed as a means to generate excess minority carriers. The case studies with C_{N} in Si doped GaN, H and V_{N} in Mg doped GaN and V_{M}-2O_{N} in Si doped Al_{0.65}Ga_{0.35}N revealed a significant reduction in impurities in agreement with the proposed theory. Since compensating point defects control the material performance (this is particularly challenging in wide and ultra wide bandgap materials), dQFL control is a highly promising technique with wide scope and may be utilized to improve the properties of various materials systems and performance of devices based upon them.

## I. INTRODUCTION

Fundamental semiconductor structures and devices including bipolar junction transistors (BJT), field-effect transistors (FET), Schottky diodes, light-emitting diodes (LED), and laser diodes require highly controlled impurity (dopant) incorporation. The dopant incorporation is required to be controlled from very low densities (e.g., drift region in Schottky diodes) to very high densities (e.g., Esaki diodes). However, incorporation of dopant atoms is accompanied by the formation of native defects and incorporation of impurity atoms/ions. These additional point defects can limit the performance of devices through compensation. Primarily compensating point defects reduce free carriers and mobility and significantly reduce the device performance.^{1} Point defects are also reported to lower the internal quantum efficiencies in LED and laser diode active regions.^{2} In general, point defect control is found to be a more serious issue in wide bandgap semiconductors due to the dependence of defect formation energies on the Fermi level.^{3} The compensating defects in wide bandgap semiconductors include impurity ions, vacancies (self-compensation), and complexes.^{3} Consideration of the energy of formation of a point defect is an important tool in the reduction of point defect incorporation. Conventional methods to increase the energy of formation typically utilize its dependence on the chemical potential of the constituents, i.e., by choice of growth environment, dopants, and dopant sources.^{3,4} However, the choice of growth environment may be influenced by other requirements including growth rate, chemical stability of the product, reaction kinetics, etc., and altering it to reduce point defects may not be feasible. Although the dependence of defect formation energy on equilibrium Fermi level is well known,^{5} it has not been utilized in point defect reduction since the Fermi level primarily depends on the type and concentration of doping and cannot be changed without changing the doping profile. However managing the quasi Fermi level (QFL) of point defects presents an interesting possibility in point defect reduction. Among the recent efforts in this direction, QFL control technique via illumination with photons with energies greater than the bandgap was reported previously by us.^{1,6,7} The advantage of such a technique is that its use does not alter the growth conditions.

In this work, we present the general theory of compensating defect density reduction via an increase of the formation energy of defects through the modification of defect quasi Fermi level (dQFL) achieved by introduction of excess minority carriers into the system. The proposed theoretical model is shown to be successful in predicting the existence of a defect energy dependent process efficiency with excellent qualitative and quantitative agreements with experimental data. The proposed theory is also material independent and may be applied to the doping process of any semiconductor. We present certain general guidelines for and consequences of controlling the point defect formation and incorporation via excess minority carrier injection. Since III-nitrides form an important class of semiconductors with excellent optical and electronic properties and present issues with point defect control, Al/GaN is employed in this work as a model material system as an additional test to the theory. Excess minority carriers are in this case generated via the above bandgap illumination.

## II. EXPERIMENTAL

In addition to the previously published results related to a qualitative experimental validation of this theory, we include in this article a quantitative validation based on Al/GaN. To test C_{N} incorporation in Si doped GaN, GaN films doped with Si were grown heteroepitaxially on (0001)-sapphire substrates using a metal organic chemical vapor deposition (MOCVD) reactor at a growth temperature of 1040 °C, at a total reactor pressure of 20 Torr and under a V/III ratio of 100 (ensures sufficient C_{N} to test its reduction). This V/III ratio was reached by flowing 67 *μ*mol/min of Trimethylgallium and 0.3 slm of ammonia, under a total flow rate of 7.4 slm using nitrogen as the diluent gas. Silane was used as a n-type Si precursor. A 20 nm low temperature (650 °C) AlN nucleation layer was deposited on sapphire prior to GaN growth, providing for a Ga-polar film.^{8} Si doped 700 nm thick GaN:Si layers were deposited on 1.3 *μ*m undoped GaN films grown onto the AlN nucleation layer. The n-type doping was varied from 1 × 10^{18} cm^{−3} to 7 × 10^{19} cm^{−3}. Growth of the studied AlGaN and Mg doped GaN is described elsewhere.^{1,6,7} The UV-illumination during growth for the samples was achieved by using a mercury-xenon arc lamp uniformly irradiating the wafer surface at a power of ∼1 W/cm^{2}. The illumination was only used during the growth of the intentionally doped Al/GaN.^{6} Secondary Ion Mass Spectrometry (SIMS) analysis was obtained using a CAMECA IMS-6f magnetic sector analyzer instrument. The analysis for Si and C was achieved using Cs^{+} primary 11 nA beam rastered over a 120 × 120 *μ*m^{2} area and detection of negative secondary ions from a 30 *μ*m diameter region at the center of the raster. Electrical characterization was performed on an Ecopia HMS-3000 Hall effect measurement system. Room temperature Hall conductivity measurements on all films were performed using the van der Pauw method. Photoluminescence (PL) measurements were performed using a 325 nm (56 mW) HeCd laser at room temperature or at 3 K. The PL setup consists of Janis (SHI-RDK-415D) closed cycle cryostat using helium and a Princeton Instruments (SP2750 0.75 m) spectrometer with an attached PIXIS 2 K charge-coupled-device (CCD) camera with a resolution better than 0.01 nm. Dislocation density was estimated by XRD using a Philips X'Pert Research X-ray Diffractometer.

## III. THEORY OF POINT DEFECT REDUCTION VIA DEFECT QUASI FERMI LEVEL CONTROL

The tendency to incorporate a particular point defect during the growth processes may be represented by the formation energy of the defect. For a defect with charge state q, in a material at equilibrium, it is defined as^{3,9}

where E_{ref} is the free energy of a crystal with a single defect referenced to the free energy of an ideal crystal, n_{j} is the number of atoms of jth-type exchanged with the reservoir to form the defect, and μ_{j} is the associated chemical potential. E_{F} + E_{V} is the Fermi energy referenced with respect to the valence band maximum. More details can be found elsewhere.^{3,9} As suggested by Van de Walle *et al*., the equilibrium formalism used to describe the defect incorporation during film growth, a non-equilibrium process, may be employed for processes such as MOCVD due to high temperatures, growth conditions close to equilibrium and defects with high mobility.^{3} Further, MOCVD growth of GaN is a mass transport limited process at growth temperatures 1000–1100 °C and supersaturation chemical potential drop is primarily across the boundary layer.^{10} Consequently growth surface is expected to be near equilibrium. To understand the increased tendency to incorporation of compensating defects in doped materials, let us consider a case where the material is n-doped. In this case, the Fermi level shifts towards the conduction band relative to the intrinsic material. The shift is approximately equal to half the bandgap. Consequently, from Equation (1), for a compensating acceptor-type (negatively charged (−1)) defect, the formation energy decreases by half the bandgap. This results in an increased incorporation of such defects. This problem is exacerbated in wide bandgap materials due to larger swings of the Fermi level and associated electron exchange energy. As a consequence, a decrease in carrier concentration beyond a critical doping level is expected and has been reported in Si doped AlGaN at high Al compositions.^{1,11} While the formation energy is defined for each charge state, the defect energy level is defined for each charge state transition. Consider the charge balance reaction where a defect changes its charge state by 1

The ratio of defect densities in different charge states can be calculated using the Fermi Dirac statistics and is given by

where E_{x} is the defect energy corresponding to the transition $Xq\u22121\u2194Xq$. Here, we extend the concept of a defect energy term to a general charge transition $Xm\u2194Xn$. Consider a charge balance equation

where $t=n\u2212m$. From Equation (2)

where E^{i}_{x} is the defect energy for the ith charge state transition $Xm+i\u22121\u2194Xm+i$ with $1\u2264i\u2264t$. Consequently, it may be shown that the ratio of defect densities at charge states n and m is then given by

where E_{X} is defined as the average defect level and is given by

The ratio of charge state densities may also be expressed in terms of defect formation energies for the corresponding charge states. Assuming E^{f}(X^{m}) and E^{f}(X^{n}) as the formation energies of defect X in charge states m and n, respectively, one can obtain

From Equations (4) and (6), the difference in the formation energies of the defect in different charge states can be written as

Similarly, the difference in the formation energy between a charge state n and neutral state (m = 0) is

Equation (7) expresses the formation energy of a defect at a given charge state in terms of defect level, Fermi level, and formation energy of defect in its neutral state. If minority carriers are injected into the system at a constant rate such that a steady state is achieved, the charge carrier populations are described by QFLs. At typical doping levels, the change in the majority carrier density is negligible and the QFL of the majority carrier can be assumed to be the same as the equilibrium Fermi level. If a defect is present, it will interact with the electrons and holes by absorbing and emitting them. Here, we have to define a defect QFL E^{i}_{Fx} for the ith transition $Xm+i\u22121\u2194Xm+i$ with $1\u2264i\u2264t$, as discussed previously, in order to describe the probability of occupation of each of the defect energy levels. From the definition of QFL and Equation (4), the ratio of defect densities at different charge states under steady state is written as

where E_{FX} is the average QFL defined similar to E_{X} in Equation (5) as

The difference in the formation energies between different charge states at steady state is therefore given by

If one of the states is neutral (m = 0), the difference in the formation energy at steady state between a charge state n and neutral state is

The formation energy of a neutral defect is independent of the Fermi level and does not change between the steady state and equilibrium, i.e., $Ef(X0)=Essf(X0)$. Hence from Equations (7) and (9), the change in the formation energy of a defect at charge state n between equilibrium and steady state is

Hence, the change in the formation energy of the charged defect is proportional to the deviation of the defect QFL from the equilibrium Fermi level. Consequently Equation (10) provides us with a means to alter the formation energy of any defect. Consider an n-type material with an acceptor-type compensator as the defect of interest. The charge on the acceptor takes negative values. From Equation (10), for n < 0, the formation energy of this acceptor can be increased by decreasing E_{FX} below the equilibrium Fermi level. Therefore the material must appear more p-type for the acceptor.

The dQFL is a function of electron and hole QFLs. The analysis of defect QFL in a reverse biased p-n junction was done by Lutz.^{12} A similar analysis in any material with excess steady state carrier density results in the defect QFL for the transition $Xq\u22121\u2194Xq$ to be defined by

where E_{x} and E_{Fx} are the defect transition energy for $Xq\u22121\u2194Xq$ and the defect quasi Fermi level for the corresponding transition respectively, E_{i} is the Fermi energy of intrinsic semiconductor, E_{Fn} and E_{Fp} are electron and hole QFLs, respectively, and c_{n} and c_{p} are the capture coefficients of electrons and holes, respectively. Hence, for all defect transitions $Exi$, the defect quasi Fermi level $EFxi$ may be calculated and E_{FX} and E_{X} are obtained from Equations (8) and (5). The net change in the formation energy is then calculated from Equation (10). The intrinsic Fermi energy is dependent on the bandgap (E_{g}) and effective densities of states of the valence (N_{V}) and conduction (N_{C}) bands and is given by

with 2E_{i} ∼ E_{g}. It is important to recognize that E_{Fn} − E_{Fp} represents a photo-voltage, eV_{p}, where ‘e’ is the electron charge. More details are found elsewhere.^{12} The average dQFL defined for a general charge state n is then given by

where E^{i}_{Fx} is the QFL defined for the ith charge state transition $Xi\u22121\u2194Xi$ with $1\u2264i\u2264n$. It is clear that the defect QFL is a function of the carrier QFLs and the defect energy level. The effect of differences in capture coefficients on defect QFL is minor, and c_{n} is assumed to be equal to c_{p} unless otherwise specified. QFLs require an assumption that the carriers are in equilibrium with the lattice since Fermi-Dirac distribution is used to describe the carrier population. Similarly, defects are also assumed to be in thermal equilibrium with the lattice. The formation energy of the defect in steady state may hence be written as

The dependence of compensating dQFL in an n-type material (E_{Fn} = E_{F} = E_{g} = 6 eV) as a function of defect energy (E_{X} − E_{V}) calculated for different photo-voltages (eV_{p} = E_{Fn} − E_{Fp}) from Equation (11) is shown in Figure 1(a). Here, the minority carrier band is the valence band. The energy difference between the defect level and the minority carrier band is termed as defect depth in this work. It is clear that there exists a maximum defect energy depth, equal to the generated photo-voltage, beyond which deviation of dQFL from the equilibrium Fermi level is negligible. Hence, a threshold photo-voltage can be defined as the minimum photo-voltage equal to the defect depth beyond which dQFL varies linearly with a slope = 1 for any additionally generated photo-voltage. Figure 1(b) shows the presence of the threshold photo-voltage (V_{th}) in the dependence of dQFL on the hole QFL. The threshold voltage represents a sharp (gradual) transition in the behavior of the dQFL near V_{th} at low (high) temperatures, as shown in Figure 2. Further, it can be concluded that for a shallow acceptor-type defect transition (E_{x} ≈ 0) and a shallow donor-type defect (E_{x} ≈ E_{g}), the dQFL is equal to hole QFL and electron QFL respectively with V_{th} ≈ 0. This corroborates with the analysis on the possible role of defects in lasing by Bernard *et al*.^{13} At lower donor concentrations, i.e., E_{Fn} = E_{F} < E_{g}, the threshold is lower than E_{x}, i.e., if E_{Fn} > E_{g} − E_{x}, the threshold photo-voltage V_{th} = E_{Fn} − (E_{g} − E_{x}) and if E_{Fn} < E_{g} − E_{x}, V_{th} = 0. Therefore, for a zero threshold, the defect needs to be sufficiently shallow so as to satisfy E_{x} ≤ E_{g} − E_{Fn}. Similarly, for a compensating donor defect in p-type material, zero threshold requires E_{x} ≥ E_{g} − E_{Fp}. This is illustrated in Figure 3 where E_{Fx} is determined only by E_{Fp} for E_{x} < E_{g} − E_{Fn} and by E_{Fn} for E_{x} > E_{g} − E_{Fp}. A similar analysis reveals a symmetric behavior in p-type (E_{Fp} = E_{v} = 0) materials where the threshold is determined by the defect level referenced with respect to the conduction band, i.e., defect depth = E_{g} − E_{x}. The zero threshold (for E_{Fp} > 0) corresponds to E_{x} ≥ E_{g} − E_{Fp}. The summary of the general behavior of the defect QFL is illustrated in Figures 3 and 4.

It may hence be concluded that the compensating defect QFL always decreases (increases) in n-type (p-type) materials in the presence of excess minority carriers. This results in a corresponding increase in the defect formation energy and consequently a decrease in defect incorporation. Hence, in general, minority carrier injection/generation always decreases the density of compensating defects, and there exists a minimum required carrier density corresponding to a threshold photo-voltage determined by the defect depth to initiate the decrease in defect density. This is illustrated in Figure 5. Further, since dopant atoms can be considered as extremely deep impurities that are close to the opposite band with a threshold photo-voltage close to the bandgap of the material, their incorporation remains practically unaffected.

The effect of QFLs on the formation energy may also be described in terms of excess free energy (driving force) for reaction (3). The free energy change for a reaction is

where K is the ratio of concentrations of products and reactants and the standard free energy change $\Delta G0=\u2212kTln(Keq)$, where K_{eq} is the equilibrium constant. The equilibrium constant for the reaction (3) is given by

where N_{C} and N_{V} are effective densities of states in the conduction and valence bands, respectively. In steady state, the ratio of the concentrations of the products and reactants can be obtained using QFLs. The ratio is given by

At steady state, $\Delta G\u22600$ and the excess free energy or the driving force can be written as

The excess free energy depends on the generated photo-voltage, eV_{p} = E_{Fn} − E_{Fp}. Figure 6 shows the excess free energy in the system and the defect QFL of a particular transition with defect level E_{x} as a function of the induced photo-voltage. It is clear that the threshold photo-voltage corresponds to a transition from an increasing free energy to a decreasing free energy.

The ratio of the change in dQFL from the equilibrium Fermi level (or the corresponding increase in defect formation energy) and the input energy can be interpreted as an efficiency of the process. It is clear that this efficiency is determined by the defect energy. The energy supplied to the system through light can be described using the work done by the generated photo-voltage given by

The useful work done against defect formation is the increase in the formation energy given by

The efficiency is then given by

Figure 7 illustrates the approximate dependence of the efficiency of the process on the applied photo-voltage (a measure of illumination intensity) and defect depth, assuming the majority carrier QFL is close to the corresponding energy band. The dependence can be illustrated with a right angled triangle ABC with the hypotenuse AC representing the increasing threshold energy. The illumination has an effect with finite efficiency inside the triangle, with BC representing the maximum achievable efficiency

The defect density is hence reduced by introducing a driving force (similar to supersaturation in growth) against defect incorporation itself (by increasing the energy of formation) or in favor of other defect removal mechanisms such as out-diffusion.

The discussion so far contributes to the understanding of dQFL control in doped materials with the Fermi level close to the conduction or valence bands. Typically, defect formation energies are presented as a function of the equilibrium Fermi level at assumed chemical potentials of the elements involved. In the following, we study the effect of excess charge carriers on the formation energy as a general function of the equilibrium Fermi level, i.e., we present modified formation energy diagrams. Since the above bandgap illumination presents a highly promising approach to conveniently provide excess minority carriers, the modified formation energy diagrams are presented assuming generation of electron-hole pairs resulting in excess free carriers. We assume a constant illumination and band to band recombination, i.e., the recombination rate R = Bn_{maj}n_{min} and minority carrier lifetime τ = 1/(Bn_{maj}), where B is the radiative constant, and photo-voltage (eV_{p} = E_{Fn} − E_{Fp}) is determined by illumination intensity independent of doping concentration. Accordingly, the minority carrier lifetime increases as the Fermi level is pushed away (decrease in n_{maj}) from the conduction/valence band. The modified formation energy diagram is shown in Figure 8 assuming a bandgap of 3 eV, T = 1300 K, I = 1 Wcm^{−2} at 300 nm being absorbed within 100 nm, and B = 10^{−8} cm^{3}s^{−1} (reported for GaN^{14}). The formation energies at equilibrium and the defect transition energies are assumed for illustrative purposes. In Figure 8(a), the dashed lines represent the formation energies without illumination and the solid lines represent the formation energies during illumination. The green plot represents a deep acceptor with charge transition at the middle of the bandgap, and the blue plot represents a relatively shallow donor-defect. Figure 8(b) shows the deviations of corresponding dQFL from the Fermi level, which determines the change in defect formation energy as discussed earlier. The following observations may be made:

The difference between steady state and equilibrium defect formation energy is a function of the Fermi level such that the efficiency (change in the formation energy for a given photo-voltage) increases as Fermi level shifts away from the valence/conduction bands i.e., $\eta max(EX)\u22481\u2212EthEg$ with a decreased threshold energy of $Eth=EX\u2212(Eg\u2212EFn)$ for n-type and $Eth=Eg\u2212EX\u2212(EFp)$ for p-type materials.

The formation energy can only be increased up to that of the uncharged state.

As the Fermi level shifts away from the band edges, the excess majority carrier concentration may exceed the equilibrium majority carrier concentration, i.e., the majority carrier quasi Fermi level is different from the Fermi level and the dopant formation energy consequently increases.

## IV. RULES OF POINT DEFECT CONTROL

High doping of GaN leads to the incorporation of several native vacancy defects in addition to impurities like C, H, and O. The concentration of such point defects varies significantly depending on the applied doping concentration, type of doping, and growth conditions.^{15,16} High concentrations of such point defects lead to compensation of the dopant and reduce free carrier concentrations and mobility or even lead to a completely insulating crystal film. In the case of n-type GaN:Si, C acceptor impurities can compensate Si donors,^{17} which leads to a strongly reduced mobility (mobility collapse), reduced free carrier densities, and even highly resistive films.^{8} A deliberately chosen low V/III ratio, as described in the experimental section, results in a relatively high incorporation of C acceptors (10^{18 }cm^{−3}) and improves the sensitivity of testing the above theoretical concepts.^{18} The lack of dependence of C incorporation on Si concentration (n ∼ carbon concentration to n > carbon concentration) is clearly illustrated via the secondary ion mass spectroscopy (SIMS) measurements (Figure 9) on the following “ladder” structure: a 1.3 *μ*m undoped GaN template followed by intercalated 200 nm thick Si-doped layers with four different Si doping concentrations separated by 200 nm undoped layers. At high Si concentrations, the shift in the Fermi level and corresponding change in formation energy is negligibly small. Other impurities like H (below 7 × 10^{17} cm^{−3}) and O (below 2 × 10^{17} cm^{−3}) are found by SIMS in low concentrations. Hence C is the main compensating defect in GaN:Si when doping in the low 10^{18 }cm^{−3} range.^{18}

Certain conclusions may be formulated for illumination-controlled dQFL-based point defect control scheme. We assume the growth conditions remain unaltered, i.e., the moderate light intensity or charge injection/generation have no significant effect on process conditions, including V/III-ratio and temperature, gas-phase reactions, or surface kinetics. The following principles are applicable for reducing compensating point defects in any n- or p-doped semiconductor:

The dopant incorporation remains unchanged as the above bandgap illumination at moderate intensity has no effect on its formation energy; the threshold photovoltage is large and is equal to bandgap as discussed previously.

The UV-illumination changes the formation energy of compensators only and leads to a decreased incorporation of impurities and reduced self-compensation by native defects; this is true for n- and p-type semiconductors;

For a given minority carrier concentration and defect depth, the reduction in compensating defect concentration increases with bandgap;

The reduction in the defect incorporation/formation for a given photo-voltage is only a function of defect depth and is independent of bandgap or type of doping;

Relatively low minority carrier concentrations generated at moderate light intensities are sufficient to significantly decrease the relatively deep point defect densities;

The change in the charge state of the compensating acceptor/donor state in n-/p-type material due to minority carrier generation/injection is typically +/−1; hence typically E

_{X}= E_{x}and E_{FX}= E_{Fx}.The excess formation energy is zero for a photo-voltage around two times the defect energy depth. This was assumed in previous studies.

^{1}

It has to be noted that the energy of formation of the defect is assumed to be positive in calculating the reduction in defect density. A negative formation energy (spontaneous defect formation) introduces an additive threshold energy equal to the negative formation energy to the previously discussed threshold energy. Any binding energies due to defect complex formation not already incorporated into the formation energy calculations can further increase the threshold, i.e., the increase in the formation energy of the defect has to exceed the sum of the threshold voltage, negative formation energy, and defect complex binding energies.

To support the enumerated principles, Figure 10 shows the carrier concentration as a function of silane flow rate. It is clear that the carrier concentration does not change with illumination at higher silane flow rates, where Si incorporation is much larger than C. This strongly suggests that Si incorporation is independent of illumination (1st principle). Further, studies in Si-doped AlGaN and Mg-doped GaN strongly support the first rule.^{1,6}

Compensation of Si donors has a major effect on carrier mobility at low concentrations as Si doping levels become comparable to the C acceptor background concentration. This so-called mobility collapse is also shown in Figure 10. For Si concentrations below 5 × 10^{18} cm^{−3}, the mobility starts decreasing significantly due to decreased shielding^{19} and drops to an extremely low 115 cm^{2} V^{−1} s^{−1} at a Si concentration of 2 × 10^{18} cm^{−3}. The introduction of the above bandgap illumination results in a significant increase in mobility at low Si concentrations. The increase in the mobility is a consequence of the decreased ionized carbon acceptor scattering, i.e., the mobility increases with a decrease in doping as expected in materials without compensating impurities. There is a corresponding increase in the free carrier concentration at low silane flow rates due to reduced compensation by C, as seen in Figure 10. Based on the carrier concentrations and known incorporation (Figure 9), the C_{N} is expected to decrease by ∼4 times from >10^{18 }cm^{−3} to low 10^{17 }cm^{−3}. Further low temperature (3 K) photoluminescence (PL) experiments on Si doped GaN grown with and without illumination are shown in Figure 11. The spectra were normalized to the donor bound exciton (DBX) transition at 3.48 eV due to either O or Si.^{16} In addition to the DBX transition, a strong (weak) transition at 2.2 eV is observed in the sample without (with) illumination. This yellow luminescence at 2.2 eV is typically attributed to C_{N}.^{20,21} Consequently the significant decrease in the yellow luminescence (by ∼5 times) can be attributed to significant reduction in the C acceptor concentration, which is in strong agreement with the mobility studies. Note that the decrease is in C_{N} configuration, i.e., acceptor (compensating) configuration and total C content may or may not change depending on other favorable non-compensating (uncharged) configurations. Accordingly, SIMS of C in GaN in UV illuminated region and “dark” region shows a marginal decrease in C by ∼25% as shown in Figure 12. An earlier study on Mg doped GaN^{7} revealed a reduced incorporation of compensating H under UV illumination as measured by SIMS and electrical measurements.^{7} The same study expectedly showed a reduced incorporation of acceptor Mg in the presence of excess donor oxygen ([O] > [Mg]). These observations strongly support the 2nd principle.

While considering principles 3 and 4, one needs to realize that the defect depth typically increases with increase in bandgap (like when increasing the Al content in AlGaN). When the change in the charge state of the compensating defect is ±1 (principle 5), E_{X} = E_{x} and E_{FX} = E_{Fx} are typically true. However, there are cases, such as V_{N}^{3+}/V_{N}^{1+} transition in p-type GaN, where V_{N}^{2+} is not stable.^{3,22} Hence, the general formalisms described by Equations (5) and (13) are required. Specifically, in case of V_{N}^{3+} to V_{N}^{1+} transition, due to doubly degenerated energy states interacting with electrons resulting in V_{N}^{3+} ↔ V_{N}^{1+}, the average defect energy state corresponds to 0.59 eV (Ref. 3) or 1.8 eV (Ref. 22) (the Fermi energy corresponding to transition from stable 3+ to 1+ states of nitrogen vacancy). The average dQFL in this case is merely the dQFL calculated for a defect state at 0.59 eV or 1.8 eV due to degeneracy.

A quantitative comparison of the experimental data with theoretical estimations requires the knowledge of defect depth and the relative reduction in defect concentration. DFT-based estimates made at T = 0 K with formation energies varying between Al/Ga-rich and N-rich environments make accurate comparison with experimental results difficult since growth temperatures are ∼1300 to 1500 K. Further differences between the experimental and theoretical predictions are expected within the order of magnitude due to deviations of assumed defect energies and carrier concentrations from their actual values. Surface pinning^{23–25} may introduce some additional deviations. However, with reasonable assumptions, valuable quantitative conclusions can be made. An experimental reduction of C_{N} by ∼4 to 5 times is suggested by photoluminescence and electrical studies as discussed earlier. SIMS suggests a marginal decrease in net C. Recent theoretical formation energy estimates of carbon as an acceptor (C_{N}) places the defect level at 0.9 eV above the valence band maximum.^{26} Figure 13(a) shows modified formation energy diagrams showing the increased formation energies (solid lines) with respect to equilibrium formation energy (dashed lines) for various defects, and Figure 13(b) shows the corresponding decrease in steady state defect concentrations relative to their equilibrium densities in GaN and Al_{0.65}Ga_{0.35 }N. These calculations utilize the DFT calculated formation energy diagram and assume a band to band recombination with a radiative constant B = 1.1 × 10^{−8} cm^{3}s^{−1}.^{14} Accordingly, the formation energy of C_{N} is expected to increase by ∼110 meV at the Fermi level calculated for known Si doping, and C_{N} incorporation is expected to reduce by ∼70%, under the experimental growth and illumination conditions as shown by filled squares in Figure 13(b). Hence, considering the expected difficulties in directly correlating DFT results with experiment, there is an excellent agreement between the predictions of the proposed theoretical model and experimental results (hollow squares in Figure 13(b)).SIMS suggests a conversion of C from C_{N} configuration into a donor type configuration since the net decrease in C is marginal and much lower than the decrease in C_{N}. Further, we have previously reported UV illumination studies with Si doped AlGaN and Mg doped GaN under identical illumination intensities.^{1,6} Studies on Si doped Al_{0.65}Ga_{0.35 }N with metal vacancy related compensation indicated a 1/3rd reduction of compensation under moderate illumination.^{1} Photoluminescence indicates the defect to be around 2 eV, assuming a Frank Condon shift of ∼0.5 eV. A good theoretical fit requires the −1/0 acceptor state transition at 2.1 eV for a fractional reduction of ∼30% with the formation energy increasing by ∼50 meV, which is in excellent agreement with the photoluminescence (PL) studies. A good candidate for the defect complex is (V_{M}–2O_{N})^{−1} (−1/0 at ∼2.35 eV in AlN)^{27} and the associated DFT calculated formation energy is used to represent the defect in Figure 13(a) and the reduction in Figure 13(b) in agreement with PL studies. PL studies on UV illuminated Mg doped GaN growth show a decrease in compensating H by roughly 5 times (∼80%), showing activated samples without the requirement of thermal annealing.^{6} PL studies were corroborated by SIMS analysis showing a clear decrease in H incorporation in UV illuminated samples.^{28} The decrease as measured by SIMS is lower at ∼50%. As discussed for C, it has to be noted that a direct quantitative correlation with SIMS is not appropriate since we reduce the likelihood of only specific configurations of defects. Other configurations may not be effected and SIMS characterizing atomic density, i.e., defects in all configurations may underestimate the efficacy of compensating defect reduction. The reduction (according to photoluminescence) is in good agreement (Figure 13(b)) with predicted increase in the formation energy (Figure 13(a)) for H (+1/0) considering the spontaneity of H incorporation in hydrogen diluent growth conditions. At the calculated Fermi level (0.3 eV at growth temperature and 2 × 10^{19} cm^{−3} of Mg (Ref. 6)), the formation energy increases to ∼150 meV and the reduction in H incorporation is ∼80%, which is in agreement with the experimental results. Further, the effect of UV illumination on V_{N} self-compensation is studied. Since the donor +3/+1 transition occurs at 0.59 eV with an activation energy of E_{g}–0.59 eV, V_{N} is extremely deep with no expected reduction with UV (Figure 13). Experimental results agree with no observed change in the self-compensation regime. These results strongly indicate that the theoretical model is capable of valuable quantitative predictions and giving direction to experiments. Due to the excellent agreement with experiment, preventing defect incorporation is likely the dominant defect reducing mechanism during growth, i.e., defect incorporation is reduced by shifting the equilibrium of impurity incorporating surface reaction by changing the defect quasi Fermi level. Further the ability to change the formation energy enables the generated photo-voltage to be a convenient tunable growth parameter that may be employed to control the energetics of point defects. Although, in this work, we assume growth conditions close to equilibrium, the technique of defect reduction by introducing an opposing driving force by dQFL control may be employed in growth conditions away from the equilibrium as well (e.g., molecular beam epitaxy). However, quantitative predictions in such cases may deviate from the experiment.

## V. CONCLUSION

We have developed a quantitative theoretical framework that describes a general technique of point defect reduction in semiconductors via the defect quasi Fermi level control. Modified formation energy diagrams are proposed, which provide powerful tools to determine the feasibility of defect reduction and process conditions such as the excess carrier generation rate (illumination intensity) that can produce the desired reduction in point defect density. The general rules that describe the effects of defect level, defect formation energy, and bandgap on efficacy of dQFL control are given. Studies on GaN and AlGaN corroborate the theoretical predictions with good quantitative agreement. The feasibility of the above bandgap illumination as a means of generating minority carriers controlling dQFL is confirmed. The significant observed reduction in point defect density supports the conclusion that dQFL control is a highly promising and general means of improving electronic and optical properties of any semiconductor or even dielectric.

## ACKNOWLEDGMENTS

The authors thank Fred Stevie and his co-workers from the Analytical Instrumentation Facility at the North Carolina State University for their great contribution by the SIMS analysis of our samples. Partial financial support from NSF (DMR-1108071, DMR-1508191, DMR-1312582 and ECCS-1508854), ARO (W911NF-04-D-0003 and W911NF-14-C-0008), ARPA-E (DE-AR0000299), GAANN Fellowship, and NDSEG Fellowship was greatly appreciated. Part of this research was performed while the second author held a National Research Council Research Associateship Award.