Metal contacts and Schottky barriers at β-Ga₂O₃ interfaces: High-throughput-assisted first-principles calculations

The ultrawide-bandgap semiconductor Ga₂O₃ has the potential to significantly enhance power efficiency and reduce power consumption attributed to its high breakdown electric field and low on-resistance.^1–3 Ga₂O₃ has five commonly identified polymorphs (⁠ $α$⁠, $β$⁠, $γ$⁠, $ε$⁠, and $δ$⁠).⁴ Among these, monoclinic β-Ga₂O₃ is the most thermodynamically stable phase and has been the most extensively studied.^5,6 β-Ga₂O₃ has an ultra-wide bandgap of ∼4.8 eV⁷ along with a high breakdown electric field of 8 MV/cm,^1,8,9 which grants it a substantial Baliga’s figure of merit (BFOM) for power devices. In addition, since single-crystal β-Ga₂O₃ wafers can be grown using melt growth techniques, it is potentially possible to produce larger-area and uniform substrates at a lower cost.^9–11 These distinct properties make β-Ga₂O₃ a promising candidate for next-generation power devices beyond SiC and GaN.

The interfaces between metallic electrodes and β-Ga₂O₃ are crucial components of β-Ga₂O₃-based electronic and optoelectronic devices. The performance of β-Ga₂O₃ devices is significantly affected by the contact resistance, which is characterized by the interface Schottky barrier heights (SBHs).¹² The SBH can be predicted by the Schottky–Mott relation¹³ based on the relative alignment between the electron affinity of the β-Ga₂O₃ and the metal work function. However, the Schottky–Mott model has rarely been experimentally realized¹³ due to the inevitable chemical disorder and the Fermi-level pinning at typical metal–semiconductor interfaces. Moreover, the unit cell of β-Ga₂O₃ contains two crystallographically different gallium atoms [Ga(I) and Ga(II)] and three types of oxygen atoms [O(I), O(II), and O(III)]. Ga(I) and Ga(II) atoms are tetrahedral and octahedral coordinated,¹⁴ respectively. This leads to anisotropic physical, optical, and electrical properties.^15–17 

Experimentally, researchers found that the SBHs of metal/β-Ga₂O₃ interfaces are independent of the metal work functions,^18–21 while some reports believed that the SBHs of metal/β-Ga₂O₃ have a correlation with the type of the metal.^22,23 Theoretically, most studies focused on interfaces with a single facet of β-Ga₂O₃ or a few common metals.^24,25 In this study, we systematically investigated the metal contacts and SBHs of the metal/β-Ga₂O₃ interfaces with different facets using first-principles calculations.

All the calculations in this work were based on density functional theory (DFT). Specifically, the Schottky barrier heights (SBHs) were implemented using the QuantumATK²⁶ code, while all other computations were carried out with the Vienna Ab initio Simulation Package (VASP).²⁷ The Perdew–Burke–Ernzerhof (PBE)²⁸ exchange-correlation functional with the projector augmented-wave pseudopotentials²⁹ was applied to optimize the atomic structures of metal/β-Ga₂O₃ interfaces until the energy and force were below 1.0 × 10⁻⁵ eV and 0.01 eV/Å, respectively. Since the PBE method severely underestimates the bandgap of β-Ga₂O₃, the electrical properties of metal/β-Ga₂O₃ interfaces were calculated with Heyd–Scuseria–Ernzerhof (HSE)³⁰ hybrid functional to obtain more accurate results.

To study the contact characteristics of metal/β-Ga₂O₃ interfaces, nine metallic electrodes (Sc, Ag, Al, Ti, Co, Pd, Ni, Au, and Pt) with gradually increasing work functions were selected to ensure a wide enough range from 3.50 eV (Sc) to 5.65 eV (Pt).^31,32 The metal work functions applied in this paper were derived from experimental data,³³ as we did previously^34,35 The face-centered cubic (FCC) structure was adopted for all metals to minimize lattice mismatch and reduce the size of the metal/β-Ga₂O₃ interface models. Owing to the electrical anisotropy of β-Ga₂O₃, we iterated over all facets within the large cutoffs of the Miller indices “−5 to 5” of β-Ga₂O₃ to explore the high-index facets.³⁶ 

As shown in Fig. 1, our recently developed high-throughput interface prediction and generation (IPG) scheme³⁶ was adopted to predict and model the stable atomic structures of metal/β-Ga₂O₃ interfaces automatically. First, the combinations of crystal orientations for β-Ga₂O₃ and metal with the minimum mismatch and interface area were screened by the high-throughput method. The electronic properties of metal/β-Ga₂O₃ contacts are determined by Schottky barriers, not by electronic counting rules or surface charge matching. Therefore, there are only two tiers are considered for metal/β-Ga₂O₃ contacts, different from the four-tier strategy in our previous work.³⁶ The first tier screens the type of atoms exposed on the two surfaces. On the metal side, only metal atoms are exposed. Therefore, only oxygen atoms are allowed to be exposed on the β-Ga₂O₃ side. In the second tier, lattice matching is considered to screen the metal/β-Ga₂O₃ superlattices with the minimum mismatch and area tolerances. Subsequently, the atomic structures of metal/β-Ga₂O₃ interfaces were generated in batches. Ultimately, the stable interface structures were optimized through the application of the minimum hopping method with the IPG method.³⁶ 

Table I gives the parameters of the essential properties of these interfaces. It can be seen that the lattice mismatch of each metal/β-Ga₂O₃ interface is smaller than 5%, assuring the minimized interface stress. The metal/β-Ga₂O₃ interface models were composed of more than five metal atomic layers and ten oxygen/gallium atomic layers, respectively. A thick vacuum above 20 Å was added between slabs to avoid the image interactions in the z direction. The surface of β-Ga₂O₃ exposed in the vacuum was passivated by removing half of the oxygen atoms.

After geometry relaxation, the interfacial atomic structures of metal/β-Ga₂O₃ contacts are rearranged. The atomic configurations of four relaxed metal/β-Ga₂O₃ interfaces are shown as examples in Figs. 2(a)–2(d). It can be found that oxygen atoms close to the interface are attracted to the metal atoms, while gallium atoms are slightly repelled. Oxygen atoms exhibit higher reactivity with a large electronegativity, which enhances the interfacial bonding strength. Among them, negligible perturbation is observed at the Au/β-Ga₂O₃ interface, indicating the formation of weak interfacial bonding, while the atoms at the Pt/β-Ga₂O₃ contact show significant displacements.

As shown in Fig. 2(e), the interface binding energy and the interlayer distance exhibit a positive relationship. Meanwhile, the interlayer distances of the metal/β-Ga₂O₃ interfaces formed on β-Ga₂O₃ (100) and β-Ga₂O₃ (001) facets range from 1.43 to 2.91 Å, while those on the β-Ga₂O₃ $( 2 ¯01)$ facet range from 0.24 to 0.85 Å. This anisotropy in binding energy arises from the intrinsic anisotropy of the β-Ga₂O₃ structure. In our model, the $( 2 ¯01)$ surface exhibits the most dangling bonds, facilitating optimal metal bonding. As shown, Sc and Co, which bind to the $( 2 ¯01)$ surface, demonstrate the lowest binding energies. In contrast, the β-Ga₂O₃ (100) and (001) surfaces possess very few dangling bonds, resulting in higher binding energies. The β-Ga₂O₃ (001) surface, characterized by the fewest protruding oxygen atoms, forms the least number of metal–O bonds, thus exhibiting the highest interface binding energy. Additionally, for the same β-Ga₂O₃ facets, metals with smaller work functions tend to exhibit lower binding energies. For example, Sc shows a lower binding energy than Co, Al has a lower binding energy than Au, and Ti has a lower binding energy than Ni. This trend arises because metals with smaller work functions are more prone to losing electrons, which is consistent with their lower electronegativity. Lower electronegativity results in stronger, more polar metal–O bonds, which are easier to form and yield lower binding energies.

As shown in Fig. 3, the electron density difference clearly illustrates the charge transfer behavior between metals and β-Ga₂O₃. The results reveal that the electron accumulation and depletion are mainly concentrated around the metal/β-Ga₂O₃ interfaces. Charge carriers are consumed near the metal layers, while depletion and accumulation occur around β-Ga₂O₃ layers, causing the Fermi level to move away from the midgap position in β-Ga₂O₃ and resulting in SBHs. For an ideal metal–semiconductor interface, charge transfer originates from the difference in work functions between the metal and the semiconductor. Therefore, for a given work function of β-Ga₂O₃, the amount of charge transferred should show a linear relationship with the metal’s work function.

However, our calculations show that this relationship is nonlinear. For example, the Co/β-Ga₂O₃ interface exhibits the second highest peak of the planar-averaged charge density, with the second largest charge accumulation $Δ q$ of 0.73 e/Å, respectively, though the work function of Co is not the second smallest. This is primarily attributed to the high density of dangling bonds of the β-Ga₂O₃ $( 2 ¯01)$ surface, which interacts with Co, leading to greater interface charge transport. These results demonstrate that at metal–semiconductor interfaces, surface states caused by factors such as dangling bonds significantly influence charge transport and consequently affect the contact barrier height.

The p-type SBHs of metal/β-Ga₂O₃ interfaces are determined by the energy differences between the Fermi level of metals and the valence band maximum of β-Ga₂O₃ However, experimental studies on β-Ga₂O₃ Schottky contacts typically focus on n-type SBHs. To ensures consistency with experimental observations, the n-type SBHs are derived by subtracting the p-type SBHs from the bandgap of β-Ga₂O₃. To further explore the relationship between the SBH and the work function, the n-type SBHs of metal/β-Ga₂O₃ interfaces are obtained and plotted as red solid dots in Fig. 4 with different metal work functions. The experimental SBHs are represented by black hollow dots in the same figure.^{22,24,37–40} The calculated SBH (1.55 eV) of Ni is in good agreement with the previous experimental result (1.55 eV),⁴¹ while slight discrepancies are observed for the other metals. These differences may arise from the variations of the device fabrication process, the quality of the epitaxial interfaces, and the experimental conditions. In addition, the SBHs of different facets at the metal/β-Ga₂O₃ interface have slight deviations.^34,35

Then, we explore the origins of Fermi-level pinning in greater detail. For the β-Ga₂O₃ $( 2 ¯01)$ surface, surface states induced by dangling bonds introduce defect states within the β-Ga₂O₃ bandgap. Normally, the SBH depends on the difference in work functions between the metal and semiconductor, which reflects the relative positioning of the Fermi levels. However, the presence of surface states pins Fermi level, preventing the barrier height from being simply predicted based on the work function. Moreover, Fermi-level pinning also occurs on the β-Ga₂O₃ (100) and (001) surfaces, which do not contain dangling bonds, indicating that there are other sources of interface states. After the partial density of states (PDOS) of individual interface layers are calculated, we found that the presence of MIGS^44,45 also contributes to Fermi-level pinning.

Figure 5 shows the PDOS of four interfaces with the most prominent pinning strengths. Significant spikes emerge in the gap of the first layer of β-Ga₂O₃, indicating strong chemical bonding and hybridization of atomic orbitals between the metals and O atoms. The MIGS decay significantly within the β-Ga₂O₃ gap since the fourth layer, where the MIGS almost disappear. Moreover, the MIGS are very similar in shape to the continuous Bloch states on the metal side, indicating that the gap states in β-Ga₂O₃ layers mainly originate from the evanescent states of the metal's traveling wave states as they decay into β-Ga₂O₃. From this point of view, the nonlinearity between the SBH and the metal work function originates from the Fermi-level pinning, which is induced by surface states related to dangling bonds and the MIGS. These factors affect the SBH in a manner that cannot be accurately predicted based solely on the work function.

Except for the Schottky barrier, the tunneling barrier is another essential factor that determines the metal–semiconductor contact behavior,⁴⁶ which can be characterized by the tunneling barrier height (ΔV) and width (ΔW). As demonstrated in Fig. 6, the height and width of tunneling barrier can be obtained from the height and width of the green rectangular region on the electrostatic potential curves, respectively. We can see that the weak bonding Au/β-Ga₂O₃ interface has a notably high ΔV and wide ΔW, while there is no tunneling barrier at the strong bonding Co/β-Ga₂O₃ interfaces, indicating the highest electron injection efficiency, which will lead to a relatively lower resistance.

The P_TB are provided in Table II. It can be found that the tunneling probabilities of metal/β-Ga₂O₃ interfaces vary from 45.80% to 100%. Among them, Al/β-Ga₂O₃, Ti/β-Ga₂O₃, Sc/β-Ga₂O₃, Ni/β-Ga₂O₃, and Co/β-Ga₂O₃ interfaces have relatively high P_TB values ranging from 67.54% to 100%, indicating a higher electron transfer efficiency. Note that Al/β-Ga₂O₃ (100), Ti/β-Ga₂O₃ (100), Ni/β-Ga₂O₃ (100), and Co/β-Ga₂O₃ $( 2 ¯01)$ have relatively low n-type SBHs and high electron transfer efficiency, showing the promise of Al, Ti, Ni, and Co as an ohmic electrode.

To recap, we utilized the high-throughput IPG scheme to construct and systematically study the contact characteristics of nine metal/β-Ga₂O₃ interfaces with the minimum mismatch and interface area. The results indicate that Sc/β-Ga₂O₃ and Co/β-Ga₂O₃ interfaces have the smallest interlayer distance and the lowest binding energy due to the asymmetry of the β-Ga₂O₃ $( 2 ¯01)$ facet. The SBHs of metal/β-Ga₂O₃ interfaces formed by different metals are similar, demonstrating the strong pinning effect in metal/β-Ga₂O₃ interfaces. In addition, a strong linear relationship can be observed between the n-type SBHs and work function, with a pinning factor of 0.17, which is close to the available experimental result as well as the empirical result, indicating that the atomic structures of metal/β-Ga₂O₃ modeled by the IPG method are reasonable to capture the strong pinning effect. Additionally, our calculations indicate that Fermi-level pinning arises from the combined effects of β-Ga₂O₃ surface states and metal-induced gap states. Importantly, we found that Al/β-Ga₂O₃ (100), Ti/β-Ga₂O₃ (100), Ni/β-Ga₂O₃ (100), and Co/β-Ga₂O₃ $( 2 ¯01)$ have relatively low n-type SBHs and high electron transfer efficiency, embodying the promise of Al, Ti, Ni, and Co as an ohmic electrode. While our results provide important theoretical references for the rational selection of metal electrode materials of β-Ga₂O₃ power devices, we again demonstrate that the IPG scheme is a powerful and general tool for the design of high-quality semiconductor interfaces.

This work was supported by the National Natural Science Foundation of China (NNSFC) (Grant Nos. 62174122, 52302046, U2241244, 62361166628, and L2424216), the Guangdong Basic and Applied Basic Research Foundation (Grant Nos. 2024A1515011764, 2022A1515110149, and 2024A1515010383), the Knowledge Innovation Program of Wuhan-Shuguang Project (Grant No. 2023010201020262), the Natural Science Foundation of Jiangsu Province (Grant No. BK20230268), and the Open Fund of Hubei Key Laboratory of Electronic Manufacturing and Packaging Integration (Wuhan University) (Grant No. EMPI2024020). The calculations in this paper have been done on the supercomputing system in the Supercomputing Center of Wuhan University.

The authors have no conflicts to disclose.

Wei Yu: Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Jiaren Feng: Visualization (equal); Writing – review & editing (equal). Qingzhong Gui: Validation (equal); Writing – review & editing (equal). Xuhao Wan: Methodology (equal); Writing – review & editing (equal). Junjie Shi: Writing – review & editing (equal). John Robertson: Conceptualization (equal); Supervision (equal). Zhaofu Zhang: Methodology (equal); Supervision (equal); Writing – review & editing (equal). Sheng Liu: Conceptualization (equal); Funding acquisition (equal); Supervision (equal). Yuzheng Guo: Conceptualization (equal); Funding acquisition (equal); Supervision (equal); Validation (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.