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Structure and thermodynamic stability of $\beta$ -Ga $_2$ O $_3$ surfaces

This study employs first-principles calculations to comprehensively analyze the structural properties and thermodynamic stability of all low-index $\beta$ -Ga $_2$ O $_3$ surfaces, revealing that stoichiometric terminations are generally dominant while identifying specific coordination-based factors and Ga-rich reconstructions that govern surface stability under varying growth conditions.

Original authors: Konstantin Lion, Claudia Draxl

Published 2026-07-31

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CC BY 4.0

Original authors: Konstantin Lion, Claudia Draxl

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the microscopic world of materials science as a giant, bustling construction site where atoms are the bricks and the rules of physics are the foreman. In this world, some materials are like sturdy, reliable houses, while others are like fragile glass castles. One such material, Gallium Oxide ( $\text{Ga}_2\text{O}_3$ ), has recently become a superstar in the world of electronics. It's the "super-brick" that scientists are excited to use for building next-generation power devices and sensors because it can handle high voltages and temperatures better than many other materials. However, just like a house, a crystal of Gallium Oxide has a skin, or a surface. How this skin is arranged—whether the atoms are packed tight or left with gaps—determines how the crystal grows, how it reacts with the air, and how well it works in a device.

To understand this skin, scientists use a powerful mental tool called Density-Functional Theory (DFT). Think of DFT as a super-accurate video game engine that simulates how atoms behave without needing to build them in a real lab first. It calculates the energy of different atomic arrangements to see which ones are the most stable, like finding the most comfortable way to stack a pile of rocks so they don't tumble over. The big question researchers have been asking is: "If we grow a crystal of Gallium Oxide, what does its surface actually look like, and which side will be the most stable?" Knowing the answer helps engineers grow better crystals and build better gadgets.

In this study, a team of researchers took a deep dive into the surface of the most common form of Gallium Oxide, known as the $\beta$ -phase. They acted like digital architects, simulating every possible low-angle "cut" or surface orientation of the crystal to see how the atoms rearrange themselves. They found that the surface isn't just a flat, static wall; it's a dynamic landscape where atoms shift and wiggle to find the most comfortable spot. Using their super-computer simulations, they discovered that the (100) surface is the clear winner, acting as the most stable and relaxed face of the crystal, much like the smooth, flat side of a well-polished stone.

The researchers also tested how the environment changes the game. They simulated conditions ranging from very oxygen-rich (like a fresh breeze) to very oxygen-poor (like a vacuum). They found that for most of the time, the crystal prefers to keep its atoms in a perfect, balanced ratio (stoichiometric). However, if the environment becomes extremely oxygen-poor, the surface gets greedy for Gallium atoms, forming a thin, extra layer of Gallium that looks like a "skin" or an "adlayer" on top. This explains some real-world experiments where scientists saw extra Gallium appearing during growth. Interestingly, even though the atoms vibrate and jiggle due to heat (up to 1000 Kelvin), these movements don't change the ranking of which surface is the most stable; they just add a tiny bit of extra energy, like a slight shiver in a cold room.

To make sense of why some surfaces are stable and others aren't, the team created a simple rulebook based on "coordination." Imagine each atom as a person at a party who wants to hold hands with a specific number of neighbors. If an atom is cut off from its neighbors (under-coordinated), it feels lonely and unstable. The study found that if the surface leaves Oxygen atoms or certain Gallium atoms with too few neighbors, the surface becomes unstable and energetic. However, if the surface exposes a specific type of Gallium atom that is only slightly under-coordinated, it actually helps stabilize the surface. This "hand-holding" rule helps predict how the crystal will behave.

Finally, the researchers used these findings to predict the crystal's natural shape, a process called Wulff construction. They found that no matter the conditions, the (100) surface dominates the crystal's shape, covering about 42% of the total area. It's the crystal's "face" that it shows to the world. Other surfaces like (111) or (001) show up in smaller amounts, while the high-energy, unstable surfaces like (010) and (110) disappear completely from the final shape. This gives scientists a clear map for how to grow these crystals and which surfaces to use when building the powerful electronic devices of the future.

Technical Summary: Structure and Thermodynamic Stability of $\beta$ -Ga $_2$ O $_3$ Surfaces

Problem Statement
Gallium oxide ( $\beta$ -Ga $_2$ O $_3$ ) is a critical ultrawide-bandgap semiconductor for high-power electronics and gas sensing. Its performance relies heavily on controlled crystal growth and surface engineering, as surface properties dictate morphology, reactivity, and interface characteristics. While the monoclinic $\beta$ -phase is the thermodynamically stable form, its complex crystal structure (space group $C2/m$ ) features two inequivalent gallium sites (tetrahedral Ga $_2$ and octahedral Ga $_1$ ) and three inequivalent oxygen sites. This complexity allows for numerous surface terminations. Although previous studies have identified the (100) surface as the most stable low-index orientation and examined specific phenomena like reconstructions or adsorption on individual facets, a unified, comprehensive picture covering all symmetrically inequivalent low-index orientations under experimentally relevant thermodynamic conditions has been missing.

Methodology
The authors present a comprehensive first-principles investigation using Density-Functional Theory (DFT). The study covers all symmetrically inequivalent low-index surfaces: (010), (100), (001), ( $\bar{2}$ 01), (110), (111), and (11 $\bar{1}$ ).

Functionals: Calculations employ both semi-local (PBEsol) and hybrid (PBE0 with 26% exact exchange, denoted PBE0(0.26)) functionals to assess the impact of exchange-correlation treatments on surface stability.
Thermodynamics: Surface free energies ( $\gamma$ ) are calculated as a function of temperature ( $T$ ) and oxygen chemical potential ( $\mu_O$ ), referencing the bulk Gibbs free energy and experimental oxygen binding energies. The study considers a range of conditions from oxygen-poor (Ga-rich) to oxygen-rich (Ga-poor) limits.
Vibrational Contributions: Harmonic vibrational free energies are calculated using the phonopy package to evaluate their effect on relative stability up to 1000 K.
Modeling: Symmetric slabs with at least 50 Å of vacuum are used. All atoms are relaxed until forces are below $10^{-2}$ eV/Å.

Key Contributions and Results

Energetic Ordering and Functional Dependence:
The study establishes the energetic ordering of stoichiometric terminations at 1000 K: (100) < ( $\bar{2}$ 01) < (001) < (111) $\approx$ (11 $\bar{1}$ ) < (110) $\approx$ (010). The (100) surface is the most stable with a free energy of $\approx$ 0.6 J/m $^2$ , while (111) and (010) are the least stable ( $\approx$ 1.7 J/m $^2$ ). This ordering remains consistent across PBEsol and PBE0(0.26) calculations, suggesting PBEsol is sufficient for determining relative stability despite hybrid functionals being necessary for accurate electronic structure descriptions.
Vibrational Effects:
Including harmonic vibrational contributions introduces negative corrections to surface free energies that increase with temperature. At 1000 K, these contributions range from -0.05 to -0.2 J/m $^2$ . While they modify absolute values, they do not alter the relative stability ordering of the surfaces.
Coordination-Based Stability Model:
The authors develop a model correlating surface energy with the density of under-coordinated atoms. The model reveals that:
- Under-coordinated oxygen atoms (2-fold) and 3-fold tetrahedrally coordinated Ga $_2$ atoms substantially destabilize surfaces.
- 5-fold octahedrally coordinated Ga $_1$ atoms contribute negatively to surface energy, acting as a stabilizing factor.
- This explains the high stability of the (100) surface, which exposes octahedral Ga $_1$ sites without dangling bonds at tetrahedral sites. The model predicts surface energies within 10% accuracy for stoichiometric terminations.
Structural Relaxation:
- The (100) surface exhibits minimal relaxation (<0.1 Å), consistent with its low energy.
- The ( $\bar{2}$ 01) surface undergoes the most pronounced reconstruction, where under-coordinated tetrahedral Ga $_2$ atoms rearrange into edge-sharing tetrahedral units, accompanied by significant subsurface oxygen relaxation.
- High-energy surfaces like (010) show inward relaxation of surface oxygen atoms to reduce the exposure of under-coordinated species.
Thermodynamic Stability and Phase Diagrams:
Stoichiometric terminations dominate the stability diagram across nearly the entire range of chemical potentials relevant to $\beta$ -Ga $_2$ O $_3$ ( $\Delta\mu_O \approx -2.5$ to $-0.5$ eV).
- Ga-Rich Conditions: Under highly reducing conditions ( $\Delta\mu_O < -2.5$ eV), non-stoichiometric Ga-rich terminations emerge. Notably, stable Ga-rich terminations resembling Ga adlayers form on the (100) and ( $\bar{2}$ 01) surfaces.
- O-Rich Conditions: Under strongly oxidizing conditions ( $\Delta\mu_O > -0.5$ eV), oxygen-terminated structures compete for the (001), (110), and (11 $\bar{1}$ ) surfaces.
Equilibrium Morphology (Wulff Construction):
Wulff constructions predict that the (100) facet dominates the equilibrium crystal shape, comprising approximately 42% of the surface area in both O-poor and O-rich limits. The remaining area is distributed among (11 $\bar{1}$ ), (001), and ( $\bar{2}$ 01) facets. Under O-rich conditions, the (111) facet appears (10% area), while high-energy surfaces like (010) and (110) are absent from the equilibrium morphology.

Significance
The paper provides a unified framework for understanding the surface energetics of $\beta$ -Ga $_2$ O $_3$ . By demonstrating that stoichiometric terminations are stable across most growth conditions, it clarifies the baseline for crystal morphology. The prediction of stable Ga adlayers on (100) and ( $\bar{2}$ 01) surfaces under reducing conditions offers theoretical support for experimental observations of step-flow growth mechanisms in Metal-Organic Vapor Phase Epitaxy (MOVPE). Furthermore, the coordination-based model offers a qualitative tool for predicting surface stability based on atomic termination, aiding in substrate selection and processing parameter optimization for device fabrication. The work establishes a foundation for future studies on adsorbates and more complex reconstructions.

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