# Modeling the Non-Equilibrium Process of the Chemical Adsorption of Ammonia on GaN(0001) Reconstructed Surfaces Based on Steepest-Entropy-Ascent Quantum Thermodynamics

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## Abstract

**:**

_{ad}-H + Ga-H on a 2 × 2 unit cell) is investigated using steepest-entropy-ascent quantum thermodynamics (SEAQT). SEAQT is a thermodynamic-ensemble based, first-principles framework that can predict the behavior of non-equilibrium processes, even those far from equilibrium where the state evolution is a combination of reversible and irreversible dynamics. SEAQT is an ideal choice to handle this problem on a first-principles basis since the chemical adsorption process starts from a highly non-equilibrium state. A result of the analysis shows that the probability of adsorption on 3Ga-H is significantly higher than that on N

_{ad}-H + Ga-H. Additionally, the growth temperature dependence of these adsorption probabilities and the temperature increase due to the heat of reaction is determined. The non-equilibrium thermodynamic modeling applied can lead to better control of the MOVPE process through the selection of preferable reconstructed surfaces. The modeling also demonstrates the efficacy of DFT-SEAQT coupling for determining detailed non-equilibrium process characteristics with a much smaller computational burden than would be entailed with mechanics-based, microscopic-mesoscopic approaches.

## 1. Introduction

_{ad}-H + Ga-H structures can appear in (0001) at ordinary conditions according to the literature [20].

## 2. Theory and Model

#### 2.1. SEAQT Equation of Motion

#### 2.2. System and Energy Eigenstructure

_{ad}-H + Ga-H structures is modeled. The corresponding reaction mechanisms are

_{3}(g) + S[3Ga-H] → H

_{2}(g) + S[NH

_{2}(br) + 2Ga-H],

_{3}(g) + S[N

_{ad}-H + Ga-H] → H

_{2}(g) + S[N

_{ad}-H + Ga-NH

_{2}].

_{3}molecule and the 2 × 2 surface S[3Ga-H], while subsystem 2 (i.e., the products) is comprised of one H

_{2}molecule and the 2 × 2 surface S[NH

_{2}(br) + 2Ga-H]. In a like manner, for the system subject to reaction mechanism (8), subsystem 1 (i.e., the reactants) is comprised of one NH

_{3}molecule and the 2 × 2 surface S[N

_{ad}-H + Ga-H], while subsystem 2 (i.e., the products) is comprised of one H

_{2}molecule and the 2 × 2 surface S[N

_{ad}-H + Ga-NH

_{2}]. The energy eigenlevels of the eigenstructures for subsystems 1 and 2 are then given by

_{3}and H

_{2}molecules, respectively; ${E}_{\mathrm{ZPV}}^{\mathrm{ad}1}$ and ${E}_{\mathrm{ZPV}}^{\mathrm{ad}2}$ are the zero-point energies of the subsystem adsorbates calculated from the vibrational frequencies of the adsorbates. The $\left\{{\u03f5}_{i}^{\mathrm{NH}3}\right\}$ and $\left\{{\u03f5}_{i}^{\mathrm{H}2}\right\}$ are the energy eigenlevels of the NH

_{3}and H

_{2}molecules, respectively, and are constructed from the energy eigenlevels of each degree of freedom of the molecules, i.e., translation, rotation and vibration, which are determined using the infinite potential well, the rigid motor, and the harmonic oscillator models, i.e.,

_{2}) and the non-linear molecules (i.e., NH

_{3}), respectively. $I$ in these equations is the moment of inertia, while $A,B,C$ are the rotational constants, ${B}_{\mathrm{av}}$ is the geometrical mean of the rotational constants, and $\sigma $ is the symmetry factor. When $A=B=C$ (i.e., ${B}_{\mathrm{av}}=B$), Equation (13) corresponds to the expression for a spherical top. The use of this expression with ${B}_{\mathrm{av}}$ for the NH

_{3}molecule is an approximation. In Equation (14), the ${\u03f5}_{\mathrm{vib}}$ are the discrete eigenenergies for vibrational motion, $n$ is the quantum number, and $\nu $ is the vibrational frequency. The procedure for developing each subsystem energy eigenstructure using Equations (11) and (12) can be found in Reference [31]. In a similar way, that for the non-linear molecules is developed. The final energy eigenstructure for each reactive system is then given by $\left\{{\u03f5}_{i}\right\}=\{{\u03f5}_{i}^{\mathrm{sub}1},{\u03f5}_{i}^{\mathrm{sub}2}$}. In order to closely approximate the system’s non-equilibrium state evolution in infinite-dimensional state space with an effective finite-dimensional one, the SEAQT equation of motion, Equation (4), is numerically solved using the density of states method developed by Li and von Spakovsky [31].

^{3}software package [44,45] with the Perdew-Burke-Ernzerhof (PBE) functional [46] and the double numerical plus polarization (DNP) basis set for the isolated molecule and the 2 × 2 surface slab model. The slab model comprises a vacuum layer of more than 20 Å and five GaN bilayers whose bottom layer is fixed and passivated with fictitious hydrogen atoms [47]. A basis set cutoff of 4.8 Å and a 3 × 3 × 1 Monkhorst-Pack (MP) k-point mesh [48] 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. For the frequency of the adsorbates, partial Hessian calculations are performed.

#### 2.3. Initial State and Model Parameters

^{3}. The relaxation time $\tau $ in the equation of motion is fixed at 1 so that the unique state evolution predicted for a given initial state describes the kinetics of the state trajectory only and not its dynamics, i.e., the real time required to traverse the trajectory of intermediate non-equilibrium states through which the system passes. To capture the latter, $\tau $ can be determined via experiment [27,28,29] or a microscopic/mesoscopic model (e.g., one from kinetic theory) [28,29,32,33,38,39] or in a completely ab initio fashion as is done in [41].

## 3. Results and Discussion

#### 3.1. Probability Distribution Among Energy Eigenlevels

_{ad}-H + Ga-H. This is the principal difference between the two adsorption systems and results in more ammonia adsorption on 3Ga-H than N

_{ad}-H + Ga-H. This is not because the probability flows towards lower energy eigenlevels but because the probability scatters to increase the entropy of the whole system.

#### 3.2. Adsorption Probability

_{3}on 3Ga-H and on N

_{ad}-H + Ga-H, respectively. In other words, ammonia is adsorbed on 3Ga-H approximately 7.5 times as much as on N

_{ad}-H + Ga-H. The sticking coefficient of ammonia on a GaN surface is reported in the literature to be 0.04 [49]; and it is this figure, which is used in GaN MOVPE models [50,51]. The value of ${P}^{\mathrm{sub}2}$ in the present study (i.e., 0.0120) is the same order of magnitude as the coefficient value found in the literature, although an exact comparison between these two properties cannot be made because the reconstructed surfaces in this paper are different from those in the literature.

#### 3.3. Temperature Increase by Adsorption

_{ad}-H + Ga-H is estimated to be approximately 1000 °C because of the position of the black (or red) curve relative to the first green (or blue) line from below. For adsorption onto N

_{ad}-H + Ga-H, the temperature increase is insignificant because the adsorption probability is quite small. However, for adsorption onto 3Ga-H, the temperature increase is much more important.

## 4. Conclusions

_{ad}-H + Ga-H on a 2 × 2 unit cell) is performed using the first-principle, non-equilibrium thermodynamic-ensemble based framework SEAQT. Results show that the adsorption probability on 3Ga-H is approximately 7.5 times higher than that on N

_{ad}-H + Ga-H for the case when the initial temperature is 1000 °C. This difference should affect the MOVPE process significantly. In addition, it is demonstrated that the difference in adsorption probability at equilibrium between the two reconstructed surfaces becomes much more significant the lower the initial temperature is.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Akasaki, I. Nobel Lecture: Fascinated journeys into blue light. Rev. Mod. Phys.
**2015**, 87, 1119–1131. [Google Scholar] [CrossRef] - Amano, H. Nobel Lecture: Growth of GaN on sapphire via low-temperature deposited buffer layer and realization of p-type GaN by Mg doping followed by low-energy electron beam irradiation. Rev. Mod. Phys.
**2015**, 87, 1133–1138. [Google Scholar] [CrossRef] - Nakamura, S. Nobel Lecture: Background story of the invention of efficient blue InGaN light emitting diodes. Rev. Mod. Phys.
**2015**, 87, 1139–1151. [Google Scholar] [CrossRef] - Kushimoto, M.; Tanikawa, T.; Honda, Y.; Amano, H. Optically pumped lasing properties of (1$\overline{1}$01) InGaN/GaN stripe multiquantum wells with ridge cavity structure on patterned (001) Si substrates. Appl. Phys. Express
**2015**, 8, 022702. [Google Scholar] [CrossRef] - Shojiki, K.; Tanikawa, T.; Choi, J.H.; Kuboya, S.; Hanada, T.; Katayama, R.; Matsuoka, T. Red to blue wavelength emission of N-polar (000$\overline{1}$) InGaN light-emitting diodes grown by metalorganic vapor phase epitaxy. Appl. Phys. Express
**2015**, 8, 061005. [Google Scholar] [CrossRef] - Ichikawa, S.; Iwata, Y.; Funato, M.; Nagata, S.; Kawakami, Y. High quality semipolar (1$\overline{1}$02) AlGaN/AlN quantum wells with remarkably enhanced optical transition probabilities. Appl. Phys. Lett.
**2014**, 104, 252102. [Google Scholar] [CrossRef] - Okumura, H. Present Status and Future Prospect of Widegap Semiconductor High-Power Devices. Jpn. J. Appl. Phys.
**2006**, 45, 7565–7586. [Google Scholar] [CrossRef] - Kachi, T. Recent progress of GaN power devices for automotive applications. Jpn. J. Appl. Phys.
**2014**, 53, 100210. [Google Scholar] [CrossRef] - Amano, H. Progress and Prospect of the Growth of Wide-Band-Gap Group III Nitrides: Development of the Growth Method for Single-Crystal Bulk GaN. Jpn. J. Appl. Phys.
**2013**, 52, 050001. [Google Scholar] [CrossRef] - Imade, M.; Imanishi, M.; Todoroki, Y.; Imabayashi, H.; Matsuo, D.; Murakami, K.; Takazawa, H.; Kitamoto, A.; Maruyama, M.; Yoshimura, M.; et al. Fabrication of low-curvature 2 in. GaN wafers by Na-flux coalescence growth technique. Appl. Phys. Express
**2014**, 7, 035503. [Google Scholar] [CrossRef] - Däweritz, L.; Hey, R. Reconstruction and defect structure of vicinal GaAs(001) and Al
_{x}Ga_{1−x}As(001) surfaces during MBE growth. Surf. Sci.**1990**, 236, 15–22. [Google Scholar] [CrossRef] - Kangawa, Y.; Ito, T.; Taguchi, A.; Shiraishi, K.; Ohachi, T. A new theoretical approach to adsorption-desorption behavior of Ga on GaAs surfaces. Surf. Sci.
**2001**, 493, 178–181. [Google Scholar] [CrossRef] - Kangawa, Y.; Ito, T.; Hiraoka, Y.S.; Taguchi, A.; Shiraishi, K.; Ohachi, T. Theoretical approach to influence of As
_{2}pressure on GaAs growth kinetics. Surf. Sci.**2002**, 507, 285–289. [Google Scholar] [CrossRef] - Kangawa, Y.; Akiyama, T.; Ito, T.; Shiraishi, K.; Nakayama, T. Surface Stability and Growth Kinetics of Compound Semiconductors: An Ab Initio-Based Approach. Materials
**2013**, 6, 3309–3360. [Google Scholar] [CrossRef] - Northrup, J.E.; Di Felice, R.; Neugebauer, J. Energetics of H and NH
_{2}on GaN (10$\overline{1}$0) and implications for the origin of nanopipe defects. Phys. Rev. B**1997**, 56, R4325–R4328. [Google Scholar] [CrossRef] - Northrup, J.E.; Neugebauer, J. Strong affinity of hydrogen for the GaN (000-1) surface: Implications for molecular beam epitaxy and metalorganic chemical vapor deposition. Appl. Phys. Lett.
**2004**, 85, 3429–3431. [Google Scholar] [CrossRef] - Van de Walle, C.G.; Neugebauer, J. First-Principles Surface Phase Diagram for Hydrogen on GaN Surfaces. Phys. Rev. Lett.
**2002**, 88, 066103. [Google Scholar] [CrossRef] [PubMed] - Van de Walle, C.G.; Neugebauer, J. Structure and energetics of nitride surfaces under MOCVD growth conditions. J. Cryst. Growth
**2003**, 248, 8–13. [Google Scholar] [CrossRef] - Akiyama, T.; Ammi, D.; Nakamura, K.; Ito, T. Surface reconstruction and magnesium incorporation on semipolar GaN (1$\overline{1}$01) surfaces. Phys. Rev. B
**2010**, 81, 245317. [Google Scholar] [CrossRef] - Akiyama, T.; Yamashita, T.; Nakamura, K.; Ito, T. Stability of hydrogen on nonpolar and semipolar nitride surfaces: Role of surface orientation. J. Cryst. Growth
**2011**, 318, 79–83. [Google Scholar] [CrossRef] - Beretta, G.P.; Gyftopoulos, E.P.; Park, J.L.; Hatsopoulos, G.N. Quantum Thermodynamics. A New Equation of Motion for a Single Constituent of Matter. Nuovo Cimento B
**1984**, 82, 169–191. [Google Scholar] [CrossRef] - Beretta, G.P.; Gyftopoulos, E.P.; Park, J.L. Quantum Thermodynamics. A New Equation of Motion for a General Quantum System. Nuovo Cimento B
**1985**, 87, 77–97. [Google Scholar] [CrossRef] - Beretta, G.P. Nonlinear model dynamics for closed-system, constrained, maximal-entropy-generation relaxation by energy redistribution. Phys. Rev. E
**2006**, 73, 026113. [Google Scholar] [CrossRef] [PubMed] - Beretta, G.P. Nonlinear quantum evolution equations to model irreversible adiabatic relaxation with maximal entropy production and other nonunitary processes. Rep. Math. Phys.
**2009**, 64, 139–168. [Google Scholar] [CrossRef] - Beretta, G.P. Steepest entropy ascent model for far-nonequilibrium thermodynamics: Unified implementation of the maximum entropy production principle. Phys. Rev. E
**2014**, 90, 042113. [Google Scholar] [CrossRef] [PubMed] - Montefusco, A.; Consonni, F.; Beretta, G.P. Essential equivalence of the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) and steepest-entropy-ascent models of dissipation for nonequilibrium thermodynamics. Phys. Rev. E
**2015**, 91, 042138. [Google Scholar] [CrossRef] [PubMed] - Smith, C.E.; von Spakovsky, M.R. Comparison of the non-equilibrium predictions of Intrinsic Quantum Thermodynamics at the atomistic level with experimental evidence. J. Phys. Conf. Ser.
**2012**, 380, 012015. [Google Scholar] [CrossRef] - Cano-Andrade, S.; Beretta, G.P.; von Spakovsky, M.R. Steepest-entropy-ascent quantum thermodynamic modeling of decoherence in two different microscopic composite systems. Phys. Rev. A
**2015**, 91, 013848. [Google Scholar] [CrossRef] - Cano-Andrade, S.; von Spakovsky, M.R.; Beretta, G.P. Steepest-Entropy-Ascent Quantum Thermodynamic Non-Equilibrium Modeling of Decoherence of a Composite System of Two Interacting Spin-1/2 Systems. In Heat Transfer and Thermal Engineering, Proceedings of ASME 2013 International Mechanical Engineering Congress and Exposition, San Diego, CA, USA, 15 November 2013; Paper No. IMECE2013-63596; ASME: New York, NY, USA, 2013; pp. V08BT09A043:1–V08BT09A043:8. [Google Scholar] [CrossRef]
- Beretta, G.P.; Al-Abbasi, O.; von Spakovsky, M.R. Steepest-entropy-ascent nonequilibrium quantum thermodynamic framework to model chemical reaction rates at an atomistic level. Phys. Rev. E
**2017**, 95, 042139. [Google Scholar] [CrossRef] [PubMed] - Li, G.; von Spakovsky, M.R. Steepest-entropy-ascent quantum thermodynamic modeling of the relaxation process of isolated chemically reactive systems using density of states and the concept of hypoequilibrium state. Phys. Rev. E
**2016**, 93, 012137. [Google Scholar] [CrossRef] [PubMed] - Li, G.; von Spakovsky, M.R. Generalized thermodynamic relations for a system experiencing heat and mass diffusion in the far-from-equilibrium realm based on steepest entropy ascent. Phys. Rev. E
**2016**, 94, 032117. [Google Scholar] [CrossRef] [PubMed] - Li, G.; von Spakovsky, M.R. Modeling the nonequilibrium effects in a nonquasi-equilibrium thermodynamic cycle based on steepest entropy ascent and an isothermal-isobaric ensemble. Energy
**2016**, 115, 498–512. [Google Scholar] [CrossRef] - Li, G.; von Spakovsky, M.R. Steepest-Entropy-Ascent Quantum Thermodynamic Modeling of the Far-From-Equilibrium Interactions between Nonequilibrium Systems of Indistinguishable Particle Ensembles. to be submitted for publication. arXiv
**2016**, arXiv:1601.02703. [Google Scholar] - Li, G.; Al-Abbasi, O.; von Spakovsky, M.R. Atomistic-level non-equilibrium model for chemically reactive systems based on steepest-entropy-ascent quantum thermodynamics. J. Phys. Conf. Ser.
**2014**, 538, 012013. [Google Scholar] [CrossRef] - Li, G.; von Spakovsky, M.R. Study of Nonequilibrium Size and Concentration Effects on the Heat and Mass Diffusion of Indistinguishable Particles using Steepest-Entropy-Ascent Quantum Thermodynamics. J. Heat Transf.
**2017**, 139, 122003. [Google Scholar] [CrossRef] - Li, G.; von Spakovsky, M.R. Study of the Transient Behavior and Microstructure Degradation of a SOFC Cathode Using an Oxygen Reduction Model Based on Steepest-Entropy-Ascent Quantum Thermodynamics. In Energy, Proceedings of ASME 2015 International Mechanical Engineering Congress and Exposition, Houston, TX, USA, 13 November 2015; Paper No. IMECE2015-53726; ASME: New York, NY, USA, 2015; pp. V06BT07A016:1–V06BT07A016:12. [Google Scholar] [CrossRef]
- Li, G.; von Spakovsky, M.R.; Shen, C.; Lu, C. Multiscale Transient and Steady State Study of the Influence of Microstructure Degradation and Chromium Oxide Poisoning on Solid Oxide Fuel Cell Cathode Performance. J. Non-Equilib. Thermodyn. under review.
- Li, G.; von Spakovsky, M.R. Application of Steepest-Entropy-Ascent Quantum Thermodynamics to Predicting Heat and Mass Diffusion From the Atomistic Up to the Macroscopic Level. In Energy, Proceedings of ASME 2015 International Mechanical Engineering Congress and Exposition, Houston, TX, USA, 13 November 2015; Paper No. IMECE2015-53581; ASME: New York, NY, USA, 2015; pp. V06BT07A015:1–V06BT07A015:10. [Google Scholar] [CrossRef]
- Von Spakovsky, M.R.; Gemmer, J. Some Trends in Quantum Thermodynamics. Entropy
**2014**, 16, 3434–3470. [Google Scholar] [CrossRef] - Kim, I.; von Spakovsky, M.R. Ab initio relaxation times and time-dependent Hamiltonians within the steepest-entropy-ascent quantum thermodynamic framework. Phys. Rev. E
**2017**, 96, 022129. [Google Scholar] [CrossRef] - Gyftopoulos, E.P.; Ҫubukҫu, E. Entropy: Thermodynamic definition and quantum expression. Phys. Rev. E
**1997**, 55, 3851–3858. [Google Scholar] [CrossRef] - Zanchini, E.; Beretta, G.P. Recent Progress in the Definition of Thermodynamic Entropy. Entropy
**2014**, 16, 1547–1570. [Google Scholar] [CrossRef] - Delley, B. An all-electron numerical method for solving the local density functional for polyatomic molecules. J. Chem. Phys.
**1990**, 92, 508–517. [Google Scholar] [CrossRef] - Delley, B. From molecules to solids with the DMol
^{3}approach. J. Chem. Phys.**2000**, 113, 7756–7764. [Google Scholar] [CrossRef] - Perdew, J.P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett.
**1996**, 77, 3865–3868. [Google Scholar] [CrossRef] [PubMed] - Shiraishi, K. A New Slab Model Approach for Electronic Structure Calculation of Polar Semiconductor Surface. J. Phys. Soc. Jpn
**1990**, 59, 3455–3458. [Google Scholar] [CrossRef] - Monkhorst, H.J.; Pack, J.D. Special points for Brillouin-zone integrations. Phys. Rev. B
**1976**, 13, 5188–5192. [Google Scholar] [CrossRef] - Mesrine, M.; Grandjean, N.; Massies, J. Efficiency of NH
_{3}as nitrogen source for GaN molecular beam epitaxy. Appl. Phys. Lett.**1998**, 72, 350–352. [Google Scholar] [CrossRef] - Karpov, S.Y.; Prokofyev, V.G.; Yakovlev, E.V.; Talalaev, R.A.; Makarov, Y.N. Novel approach to simulation of group-III nitrides growth by MOVPE. MRS Internet J. Nitride Semicond. Res.
**1999**, 4, e4. [Google Scholar] [CrossRef] - Karpov, S.Y.; Bord, O.V.; Talalaev, R.A.; Makarov, Y.N. Gallium droplet formation during MOVPE and thermal annealing of GaN. Mater. Sci. Eng. B
**2001**, 82, 22–24. [Google Scholar] [CrossRef]

**Figure 1.**Surface structures before and after the chemical adsorption reactions: (upper row) NH

_{3}(g) + S[3Ga-H] → H

_{2}(g) + S[NH

_{2}(br) + 2Ga-H], (lower row) NH

_{3}(g) + S[N

_{ad}-H + Ga-H] → H

_{2}(g) + S[N

_{ad}-H + Ga-NH

_{2}]. Brown, blue, and white atoms are gallium, nitrogen, and hydrogen, respectively.

**Figure 2.**Probability distribution among the energy eigenlevels for the adsorption reactions on (

**a**) S[3Ga-H] (and (

**c**) zoomed-in) and (

**b**) S[N

_{ad}-H + Ga-H] (and (

**d**) zoomed-in). The narrow solid, dashed, and bold solid lines correspond to the initial state, a number of intermediate states during relaxation, and the stable equilibrium state, respectively.

**Figure 3.**Evolution of the total probability of each subsystem as a function of the dimensionless time for the adsorption reactions on (

**a**) S[3Ga-H] and (

**b**) S[N

_{ad}-H + Ga-H]. This probability corresponds to the sum of the probabilities in Figure 2 over the energy eigenlevels.

**Figure 5.**Evolution of the specific energy of each subsystem as a function of the dimensionless time for the adsorption reactions on (

**a**) S[3Ga-H] and (

**b**) S[N

_{ad}-H + Ga-H]. Green and blue horizontal lines correspond going from bottom to top to the specific energies at 1000, 1005, 1010, 1015, 1020 °C.

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**MDPI and ACS Style**

Kusaba, A.; Li, G.; Von Spakovsky, M.R.; Kangawa, Y.; Kakimoto, K. Modeling the Non-Equilibrium Process of the Chemical Adsorption of Ammonia on GaN(0001) Reconstructed Surfaces Based on Steepest-Entropy-Ascent Quantum Thermodynamics. *Materials* **2017**, *10*, 948.
https://doi.org/10.3390/ma10080948

**AMA Style**

Kusaba A, Li G, Von Spakovsky MR, Kangawa Y, Kakimoto K. Modeling the Non-Equilibrium Process of the Chemical Adsorption of Ammonia on GaN(0001) Reconstructed Surfaces Based on Steepest-Entropy-Ascent Quantum Thermodynamics. *Materials*. 2017; 10(8):948.
https://doi.org/10.3390/ma10080948

**Chicago/Turabian Style**

Kusaba, Akira, Guanchen Li, Michael R. Von Spakovsky, Yoshihiro Kangawa, and Koichi Kakimoto. 2017. "Modeling the Non-Equilibrium Process of the Chemical Adsorption of Ammonia on GaN(0001) Reconstructed Surfaces Based on Steepest-Entropy-Ascent Quantum Thermodynamics" *Materials* 10, no. 8: 948.
https://doi.org/10.3390/ma10080948