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CH_{4} Adsorption Probability on GaN(0001) and (000−1) during Metalorganic Vapor Phase Epitaxy and Its Relationship to Carbon Contamination in the Films

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## Abstract

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_{4}produced by the decomposition of the Ga source, Ga(CH

_{3})

_{3}, and its incorporation into the GaN sub-surface layers are investigated. In this sequential analysis, the dataset of the adsorption probability of CH

_{4}on reconstructed surfaces is indispensable, as is the energy of the C impurity in the GaN sub-surface layers. The C adsorption probability is obtained based on steepest-entropy-ascent quantum thermodynamics (SEAQT). SEAQT is a thermodynamic ensemble-based, non-phenomenological framework that can predict the behavior of non-equilibrium processes, even those far from equilibrium. This framework is suitable especially when one studies the adsorption behavior of an impurity molecule because the conventional approach, the chemical potential control method, cannot be applied to a quantitative analysis for such a system. The proposed sequential model successfully explains the influence of the growth orientation, GaN(0001) and (000−1), on the incorporation of C into the film. This model can contribute to the suppression of the C contamination in GaN MOVPE.

## 1. Introduction

^{16}cm

^{−3}, is suitable for the control of the main motor of a hybrid vehicle (HV) or an electric vehicle (EV) [2]. The drift layer (i.e., epitaxial GaN film) for the device is grown using metalorganic vapor phase epitaxy (MOVPE). In the GaN MOVPE system, Ga(CH

_{3})

_{3}(trimethylgallium, TMG) is used as the source gas of a group-III element (Ga) and is also the source of an unintentional C impurity. To reduce the manufacturing cost of the GaN power device with the thick drift layer, the high growth rate MOVPE technology is needed. However, the increase in TMG input has a negative impact on the C impurity contamination. For this reason, the theoretical prediction of the C concentration is important. According to thermodynamic analyses, activation energy calculations [3], and time-of-flight (TOF) high-resolution and high-sensitivity mass spectrometry measurements [4], TMG decomposes via reacting with the H

_{2}carrier gas and/or nitrogen source (NH

_{3}), producing CH

_{4}. Therefore, the interaction between the CH

_{4}molecule and the GaN surface during the MOVPE growth process must be understood clearly from the standpoint of reducing the unintentional C impurity contamination. It is worth noting that the chemical potential control method is not able to quantitatively explain the differences in the incorporation of impurities at different crystallographic orientations. This is due to the fact that the gaseous chemical potential is a macroscopic property, whose mean value is calculated based on the distribution of gaseous microscopic states. For example, for the case of a Maxwell distribution, most of molecules are near the peak of the distribution, which means that the molecules near the peak are reflected strongly in the mean value (i.e., the translational energy of the molecule) and the molecules far from the peak are not. This is true in a similar vein for the chemical potential. Thus, the chemical potential control method can explain whether or not most of the molecules are deposited on the surface and is, thus, adequate for describing the deposition of material molecules (e.g., Ga monoatomic molecule after TMG decomposition). However, it is unable to account for the minor part of the CH

_{4}molecules, which have values far from the mean. This small number of CH

_{4}molecules can influence the GaN power device performance via C impurities. In fact, References [5,6] indicate that the electron mobility of GaN films is strongly influenced by C impurities. For a quantitative analysis of the adsorption behavior of such an impurity, steepest-entropy-ascent quantum thermodynamics (SEAQT) is suitable because the expression of a state in SEAQT is based on the distribution itself and has the information of the minor part. In addition, the surface structure, on which the CH

_{4}is adsorbed, and the stability of the C impurity in the sub-surface layers should be considered for the quantitative prediction of the C concentration in GaN films. In this study, this sequential analysis (surface reconstruction, CH

_{4}adsorption, and C incorporation) was carried out to understand the influence of growth orientation, GaN(0001) and (000−1), on the C concentration in GaN films. First, the GaN surface reconstructions during MOVPE are determined using an approach that compares surface formation energies. (Section 2.1). For these reconstructed surfaces, CH

_{4}adsorption structures are then studied using a density functional theory (DFT) total energies comparison (Section 2.2). In addition, the stabilization of the adsorption structures is discussed based on an adsorption free energies comparison. Then, based on SEAQT, Section 3 describes the non-equilibrium adsorption process and adsorption probability for the adsorption structure revealed in Section 2.2. Finally, Section 4 provides the model for the growth orientation dependence of the C concentration using the adsorption probability obtained in Section 3.

## 2. Adsorption Structure

#### 2.1. Reconstructed Surface During the MOVPE Process

_{2}partial pressure on surface phase transition can be provided. The chemical potential of a molecule $\mu $ is given by:

_{2}molecule, and H

_{2}molecule, respectively; and ${n}_{\mathrm{Ga}}$, ${n}_{\mathrm{N}}$, and ${n}_{\mathrm{H}}$ are the numbers of excess atoms of the reconstructed surface. The stable surface, which has the minimum formation energy among the possible candidates, is shown on the surface phase diagram. Figure 1 shows the dependence of the H

_{2}partial pressure on surface reconstruction. The growth conditions are set as follows:$\text{}{p}_{\mathrm{Ga}}=2.5\times {10}^{-4}\text{}\mathrm{atm}$ and ${p}_{\mathrm{carrier}}={p}_{\mathrm{N}2}+{p}_{\mathrm{H}2}=0.7\text{}\mathrm{atm}$, which means that the V/III ratio, ${p}_{\mathrm{NH}3}/{p}_{\mathrm{Ga}}$, is set at 1200. The details of the structures of candidate reconstructed surfaces are provided in Reference [19]. In the case of GaN(0001) growth, a hydrogen adsorbed surface, 3Ga-H (2 × 2), appears at a H

_{2}partial pressure greater than about 0.5 atm and around a typical growth temperature of 1050 °C. In the case of GaN(000−1) growth, a hydrogen adsorbed surface, 3N-H (2 × 2), appears in the H

_{2}partial pressure range of about 0.0 atm to 0.7 atm above 800 °C . Thus, these growth surfaces (i.e., surface reconstructions: 3Ga-H and 3N-H surfaces) are assumed hereafter.

#### 2.2. Adsorption Structure and Its Stabilization

_{4}molecule and the desorption of one H

_{2}molecule onto/from 3Ga-H (3N-H) are designated as Stage-1, while those after the desorption of an additional one H

_{2}molecule and two H

_{2}molecules are designated as Stage-2 and Stage-3, respectively. The candidate structures are constructed on the basis of the electron counting (EC) rule [20]. Figure 2 shows the stable structures determined by DFT energetics for each stage. All electron calculations are made using the DMol

^{3}software package [21,22] with the Perdew–Burke–Ernzerhof (PBE) functional [23] and the double numerical plus polarization (DNP) basis set. The calculated 2 × 2 slab models comprise a vacuum layer of more than 20 Å and five GaN bilayers whose bottom layer is fixed and passivated with fictitious hydrogen atoms [24]. A basis set cutoff of 4.8 Å and a 3 × 3 × 1 Monkhorst-Pack (MP) k-point mesh [25] are used. The geometry optimization convergence thresholds are 2.0 × 10

^{−5}Ha, 0.0005 Ha/Å, and 0.005 Å for the energy change, maximum force, and maximum displacement, respectively. Ga-CH

_{3}and N-CH

_{3}are the first adsorption structures of CH

_{4}and these structures are constructed by the reaction between the H atom of CH

_{4}and the adsorbed H atom of 3Ga-H (3N-H) producing H

_{2}. Comparing Stage-2 structures, the stable structure in (000−1) is not a bridge structure like (0001). This implies that the strained bridge structure in (000−1) is unstable because of the short N–C bond length. In Stage-3, the (000−1) structure is different from the (0001) structure because of a similar reason.

_{4}onto the 3Ga-H and 3N-H (2 × 2) surfaces is given by:

_{4}adsorbed surface, the 3Ga-H (3N-H) surface, and the CH

_{4}molecule, respectively. ${n}_{\mathrm{H}2}$ is the number of desorbed H

_{2}molecules, and is equal to 1 for Stage-1, 2 for Stage-2, and 3 for Stage-3. Figure 3 shows the adsorption free energy for the case of the H

_{2}carrier gas (${p}_{\mathrm{H}2}=0.7\text{}\mathrm{atm}$, solid lines) and the N

_{2}carrier gas (${p}_{\mathrm{H}2}=0.01\text{}\mathrm{atm}$, dashed lines). In (0001), it is suggested that thermodynamically, the stabilization of the CH

_{4}adsorbed structure at typical growth temperatures, ≈1000 °C, proceeds as follows: S[Ga-CH

_{3}+ 2Ga-H] + H

_{2}(g) → S[C

_{br}H2 + Ga-H] + 2H

_{2}(g) → S[C

_{ad}H(H3)] + 3H

_{2}(g) (see Stage-1, Stage-2, and Stage-3 in Figure 2, respectively). At temperatures less than about 900 °C, this stabilization would not occur in the case of the H

_{2}carrier gas. In (000−1), the structures of Stage-2 and Stage-3 are much more unstable than the structure of Stage-1. Thus, the adsorption structure would not change from Stage-1 regardless of the H

_{2}partial pressure.

## 3. Adsorption Probability

#### 3.1. Steepest-Entropy-Ascent Quantum Thermodynamics

^{1}reduces to:

^{1}Note that the basis for the formulation of the SEAQT equation of motion is one, which uses the quantum mechanical operator format, so that the number and types of generators of the motion, i.e., quantum mechanical operators, used to derive this equation vary according to the type of system and types of phenomena being modeled. Thus, Equation (11) as written is specific to the case at hand.

#### 3.2. System Definition and Calculation Condition

_{4}adsorption (i.e., 3Ga-H (3N-H) → Stage-1) is discussed quantitatively using the SEAQT framework to theoretically predict the concentration of C in a GaN film. Doing so requires knowing the growth orientation dependence of the adsorption probability. The reaction mechanism for GaN(0001) and (000−1) considered is as follows:

_{4}molecule and a 2 × 2 S1 surface (i.e., three H adatoms), subsystem 2 (i.e., the products) is comprised of one H

_{2}molecule and a 2 × 2 S2 surface (i.e., one CH

_{3}admolecule and two H adatoms). In 2017, Kusaba et al. proposed the SEAQT chemical adsorption model at the semiconductor surface coupled with DFT calculations [50]. In the paper, the excited states of the vibrational modes of the adsorbates were neglected for simplification. In the present study, these excited states were taken into account for a more accurate analysis. Therefore, the energy eigenlevels of the eigenstructures for subsystems 1 and 2 (i.e., $\left\{{\u03f5}_{i}^{\mathrm{sub}1}\right\}$ and $\left\{{\u03f5}_{i}^{\mathrm{sub}2}\right\}$) are given by:

_{4}and H

_{2}molecules and the surface slab models of S1 and S2 determined using DFT calculations, respectively; ${E}_{\mathrm{ZPV}}^{\mathrm{CH}4}$, ${E}_{\mathrm{ZPV}}^{\mathrm{H}2}$, ${E}_{\mathrm{ZPV}}^{\mathrm{ad}1}$, and ${E}_{\mathrm{ZPV}}^{\mathrm{ad}2}$ are the zero-point energies of the CH

_{4}and H

_{2}molecules and the adsorbates of S1 and S2, respectively; the $\left\{{\u03f5}_{i}^{\mathrm{CH}4+\mathrm{ad}1}\right\}$ are the joint energy eigenlevels of the CH

_{4}molecule and the adsorbates of S1; and the $\left\{{\u03f5}_{i}^{\mathrm{H}2+\mathrm{ad}2}\right\}$ are those of the H

_{2}molecule and the adsorbates of S2. Note that adsorbates only have a vibrational mode, while gaseous molecules have translational, rotational, and vibrational modes. These joint energy eigenlevels are constructed from the translational density of states (DOS), the rotational DOS, and the vibrational discrete levels, given as:

^{3}), $m$ is the particle mass, $I$ is the moment of inertia, $\sigma $ is the symmetry factor (2 for H

_{2}, 12 for CH

_{4}), $L$ is the quantum number (0, 1, 2, …), and ${\nu}_{\mathrm{M}}$ is the vibrational frequency of the M-th mode. The translational and rotational DOS is constructed as follows [41]:

_{4}and H

_{2}molecules and the adsorbates of S1 and S2 have 9, 1, 9, and 18 vibrational modes, respectively. Thus, M = 1, 2, …, 18 for subsystem 1 and M = 1, 2, …, 19 for subsystem 2. Finally, the $\left\{{\u03f5}_{i}^{\mathrm{CH}4+\mathrm{ad}1}\right\}$ and $\left\{{n}_{i}^{\mathrm{CH}4+\mathrm{ad}1}\right\}$, and $\left\{{\u03f5}_{i}^{\mathrm{H}2+\mathrm{ad}2}\right\}$ and $\left\{{n}_{i}^{\mathrm{H}2+\mathrm{ad}2}\right\}$ are obtained. Using Equations (15) and (16), the pseudo-eigenstructure for the whole system is obtained as $\left\{{\u03f5}_{i}^{\mathrm{sub}1},\text{}{\u03f5}_{i}^{\mathrm{sub}2}\right\}$ and $\left\{{n}_{i}^{\mathrm{sub}1},\text{}{n}_{i}^{\mathrm{sub}2}\right\}$. The initial state is chosen to be a second-order hypoequilibrium state [41], for which the probability distribution in each subsystem $\left\{{p}_{i}^{\mathrm{sub}1}\right\}$, $\left\{{p}_{i}^{\mathrm{sub}2}\right\}$ takes a canonical form, namely:

#### 3.3. Non-Equilibrium State Evolution

_{4}adsorbed structure) because the chemical potential does not have detailed information about the small number of molecules different from the average ones. Only a few CH

_{4}molecules are adsorbed onto the surface in the present case. Figure 5 shows the evolution of the total probability of each subsystem, $\sum {p}_{i}^{\mathrm{sub}1}\left(t\right)$ and $\sum {p}_{i}^{\mathrm{sub}2}\left(t\right)$. The adsorption probability, i.e., ${P}_{ad}\equiv \sum {p}_{i}^{\mathrm{sub}2}\left(t\right)$, (red line) reached 7.06 × 10

^{−3}in (0001), and 3.32 × 10

^{−6}in (000−1) at equilibrium. This difference in probabilities is remarkable and should influence the orientation dependence of the C impurity concentration, which will be discussed in Section 4.

## 4. Impurity Concentration

_{4}obtained in Section 3.3, the growth orientation dependence of the C impurity concentration is discussed here. In 2017, Kempisty et al. reported the depth profiles of the C impurity energy in (0001) and (000−1), i.e., the comparison of the total energies of two-surface slab models, where the one nitrogen atom at the selected layer near the surface (i.e., the 1st, 2nd, …, 8th, or 10th atomic layers from the surface) is substituted by a carbon atom [49]. Figure 7 shows the result of Reference [51] for 3Ga-H and 3N-H surfaces and the Boltzmann factors, $\mathrm{exp}\left(-E/{k}_{\mathrm{B}}T\right)$, at 1000 °C calculated from these energies. As can be seen, the C atoms at the fourth layer onwards from the surface were almost in the same situation as those in the bulk (i.e., layers deep enough). However, in (000−1), the C atoms at the second and third layers were somewhat different and the one at the first layer was greatly different from those in the bulk, which should influence the C impurity incorporation. Thus, here, only the difference in the first layer is considered for the C impurity incorporation model.

_{4}adsorption probability. Note that there are two kinds of sites with different stability in the (2 × 2) area: three triple sites and one single site (see Reference [51]). Subsequently, these C atoms are incorporated into deeper layers via a non-equilibrium process, which means that the ${c}_{\mathrm{C}}$ at the first layer almost decides the ${c}_{\mathrm{C}}$ at the bulk. Therefore, the growth orientation dependence of the C impurity concentration at 1000 °C is estimated as follows:

_{4}adsorption probability and the higher stability of the C impurity in the sub-surface than the (0001) growth orientation. As the result of our theoretical model, the C impurity concentration of GaN film grown in (000−1) was lower than that in (0001) by one order of magnitude. In experiments [52], the C impurity in (000−1) was also lower than that in (0001) by one order of magnitude or more, although it depended on the growth conditions. The model-estimated value, thus, agrees well quantitatively with the experimental one.

## 5. Conclusions

_{4}adsorption structures are found to be those with the CH

_{3}admolecule substituting the H adatom. In (0001), the stabilization of the structure proceeded to the CH

_{2}(bridge-site) admolecule structure and to the CH (3H-site) admolecule structure. In addition, the SEAQT adsorption model is essential for a detailed discussion of the minor structure (i.e., the adsorption probability of the impurity atom) although the major structure (i.e., the stable surface reconstruction) can be discussed via a surface formation energies comparison, in which the macroscopic property as the expected value from the microscopic picture is used. In other words, the expected value (i.e., macroscopic property) can not be representative of the minor molecules, which play an important role in the impurity adsorption process. From the SEAQT analysis, a non-equilibrium state evolution of the adsorption process was obtained, and the adsorption probability was found to be 7.06 × 10

^{−3}in (0001) and 3.32 × 10

^{−6}in (000−1) at 1000 °C. A C impurity incorporation model was proposed here in which it was assumed that the C impurity concentration was proportional to the CH

_{4}adsorption probability and to the Boltzmann factor calculated from the energy of the C impurity at the first surface layer [51]. As a result of this model, the ratio of the C impurity concentration in (0001) to that in (000−1), ${c}_{\mathrm{C}}^{\left(0001\right)}/{c}_{\mathrm{C}}^{\left(000-1\right)}$, was $1.25\times {10}^{1}$. This estimate agrees well with experimental results [52]. Therefore, the feasibility of estimating the growth orientation dependence of the C impurity concentration using our sequential analysis approach was confirmed. That is, both the stability in crystal and the adsorption amount of the impurity, which depend on surface reconstruction, have to be considered for the quantitative estimation. For future work, the prediction could be improved by considering other possible carbon source molecules (e.g., Ga(CH

_{3}), Ga(CH

_{3})

_{3}, C

_{2}H

_{4}, etc. [3,53]) or the mixed ratio of the most stable and metastable reconstructions.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Surface phase diagram of GaN(0001) and (000−1) during the MOVPE growth process. The growth conditions are p

_{Ga}= 2.5 × 10

^{−4}atm and p

_{carrier}= p

_{N2}+ p

_{H2}= 0.7 atm.

**Figure 2.**Adsorption structures of CH

_{4}onto the 3Ga-H and 3N-H (2 × 2) surfaces. Red atoms represent Ga, blue atoms N, yellow atoms C, and white atoms H. Stage-1, Stage-2, and Stage-3 show the structures after the desorption of one, two, and three H

_{2}molecules, respectively.

**Figure 3.**Adsorption free energy of CH

_{4}onto the 3Ga-H and 3N-H (2 × 2) surfaces. The red, green, and blue lines correspond to Stage-1, Stage-2, and Stage-3 in Figure 2, respectively. The growth conditions are p

_{CH4}= 3p

_{Ga}= 7.5 × 10

^{−4}atm, p

_{H2}= 0.7 atm (solid lines), and p

_{H2}= 0.01 atm (dashed lines).

**Figure 4.**Probability distribution among the energy eigenlevels for the adsorption reaction in (0001) and (000−1). The bold solid lines correspond to the stable equilibrium state, while the narrow dashed lines correspond to intermediate states during the relaxation.

**Figure 5.**Evolution of the total probability of each subsystem for the adsorption reaction in (0001) and (000−1); that of subsystem 2 corresponds to the adsorption probability of CH

_{4}.

**Figure 6.**Evolution of the specific energy of each subsystem for the adsorption reaction in (0001) and (000−1). The green and blue horizontal lines correspond (going from top to bottom) to the specific energies at 1000, 999, 998, 997, and 996 °C.

**Figure 7.**Depth profiles of the C impurity energy in (0001) and (000−1) as reported in Reference [51]. Weighted sum of Boltzmann factors (i.e., the sum of the triple one for the triple-site and the one for the single-site) at 1000 °C calculated from the energies.

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**MDPI and ACS Style**

Kusaba, A.; Li, G.; Kempisty, P.; von Spakovsky, M.R.; Kangawa, Y.
CH_{4} Adsorption Probability on GaN(0001) and (000−1) during Metalorganic Vapor Phase Epitaxy and Its Relationship to Carbon Contamination in the Films. *Materials* **2019**, *12*, 972.
https://doi.org/10.3390/ma12060972

**AMA Style**

Kusaba A, Li G, Kempisty P, von Spakovsky MR, Kangawa Y.
CH_{4} Adsorption Probability on GaN(0001) and (000−1) during Metalorganic Vapor Phase Epitaxy and Its Relationship to Carbon Contamination in the Films. *Materials*. 2019; 12(6):972.
https://doi.org/10.3390/ma12060972

**Chicago/Turabian Style**

Kusaba, Akira, Guanchen Li, Pawel Kempisty, Michael R. von Spakovsky, and Yoshihiro Kangawa.
2019. "CH_{4} Adsorption Probability on GaN(0001) and (000−1) during Metalorganic Vapor Phase Epitaxy and Its Relationship to Carbon Contamination in the Films" *Materials* 12, no. 6: 972.
https://doi.org/10.3390/ma12060972