# 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

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**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