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[Takayoshi Oshima](https://orcid.org/0000-0001-8550-9735)

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[Mapping primary crystallographic planes in                    <i>β</i>                    -Ga                    <sub>2</sub>                    O                    <sub>3</sub>                    based on a pseudo-cubic oxygen sublattice](https://mdr.nims.go.jp/datasets/2e2f0b9f-5c04-4352-ae0f-d1644852db4f)

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Mapping primary crystallographic planes in β-Ga2O3 based on a pseudo-cubic oxygen sublatticeaaaMapping primary crystallographic planes in β-Ga2O3 based on a pseudo-cubicoxygen sublatticeTakayoshi Oshima*Research Center for Electronic and Optical Materials, National Institute for Materials Science, Tsukuba, Ibaraki 305-0044, Japan*E-mail: OSHIMA.Takayoshi@nims.go.jpReceived January 18, 2026; revised February 3, 2026; accepted February 4, 2026; published online February 16, 2026Although β-Ga2O3 crystallizes in the monoclinic system, its structural complexity can be simplified by focusing on the oxygen framework. Theoxygen sublattice forms a cubic close-packed arrangement, corresponding to a distorted face-centered cubic (fcc) structure defined by latticevectors: afcc = b + 1/2 c, bfcc = −b + 1/2 c, cfcc = 1/3 a + 1/6 c, giving lattice parameters of afcc = bfcc = 4.20 Å, cfcc = 3.95 Å, αfcc = βfcc= 90.1°, γfcc = 92.7°. This pseudo-cubic representation provides a straightforward understanding of the crystal shape, thereby facilitatingsystematic exploration of its crystal orientations. © 2026 The Author(s). Published on behalf of The Japan Society of Applied Physics by IOPPublishing LtdSupplementary material for this article is available onlineRecently, β-Ga2O3 has attracted increasing attention as apromising semiconductor for ultraviolet optoelectronicsand power electronics,1,2) owing to its large bandgap(⩾4.43 eV)3) and the availability of scalable, melt-grownsingle crystals.4) In particular, the availability of low-defectnative substrates4) provides a reproducible platform for theoptimization of homoepitaxy5) and device processing,6)thereby enabling rapid device demonstration.1,2) However,establishing clear guidelines for substrate orientation selec-tion and the associated process design remains challengingowing to its low-symmetry monoclinic structure,7) in con-trast to conventional semiconductors with high-symmetrycrystal structures (e.g. diamond, zinc blende, and wurtzitestructures).β-Ga2O3 crystallizes in the monoclinic system (spacegroup C2/m) and has lattice parameters of a = 12.214 Å,b = 3.0371 Å, c = 5.7981 Å, α = 90°, β = 103.83°, and γ =90°.7) This monoclinic structure, unique to β-Ga2O3 andθ-Al2O3,8) hampers an intuitive understanding of crystalorientations. For example, the planes that are exactly andnearly perpendicular to (100) are {010} and {112}/{102},respectively; however, such orientation relationships are notreadily apparent from the monoclinic unit cell. Therefore, analternative and more straightforward representation is re-quired to facilitate understanding of this complex crystalstructure.A useful way to obtain a more intuitive picture of the low-symmetry monoclinic structure of β-Ga2O3 is to view it fromthe oxygen sublattice. The oxygen sublattice governs thecrystal framework of β-Ga2O3 because oxygen ions havelarger ionic radii (1.36 and 1.38 Å for threefold and fourfoldcoordination, respectively) than gallium ions (0.47 and 0.62 Åfor fourfold and sixfold coordination, respectively).9) Theoxygen ions in β-Ga2O3 are known to form a cubic close-packed (ccp) arrangement and can be regarded as a slightlydistorted face-centered cubic (fcc) structure.7,10)This oxygen-framework-based viewpoint has often beenemployed to facilitate the understanding of various phe-nomena in β-Ga2O3 and other Ga2O3 polymorphs. Forexample, slip systems in β-Ga2O3 have been discussed inrelation to oxygen ccp planes, accompanied by experimentalobservations of dislocation formation.11,12) In addition, thecrystal structures of Ga2O3 polymorphs (α, β, γ, ε, and κ)can be broadly understood in terms of whether the oxygensublattice adopts hexagonal close packing or ccp and howGa-site occupancy is arranged within the framework,13)which also helps rationalize structural changes during phasetransitions (e.g. α→β, γ→β, ε→β, and κ→β).14–17) Notably,because β- and γ-Ga2O3 share the same ccp oxygen frame-work, radiation-induced β→γ transformations can preservethe oxygen sublattice, thereby accounting for the highradiation tolerance.18) Furthermore, the fcc-based frameworkprovides a convenient basis for describing heteroepitaxialorientation relationships between β-Ga2O3 and other cubicmaterials.19,20)Despite these demonstrated utilities, detailed and conso-lidated information on the oxygen-sublattice-based pseudo-cubic framework of β-Ga2O3—such as the averaged fcclattice parameters and the correspondence between primaryfcc planes and monoclinic β-Ga2O3 planes—has not yet beensummarized in a single, systematic form. Therefore, in thisBrief Note, we provide a unified and readily accessiblesummary of this pseudo-cubic representation.Here, we first define the averaged fcc reference unit cellfor the oxygen sublattice. The averaged fcc structure can bedefined by lattice vectors: afcc = b + 1/2 c, bfcc = −b + 1/2c, and cfcc = 1/3 a + 1/6 c, where a, b, and c are latticevectors of monoclinic β-Ga2O3, as shown in Fig. 1. Here,afcc, bfcc, and cfcc are not fcc primitive lattice vectors butrather define a non-primitive fcc-based reference unit cell forthe averaged oxygen sublattice (see Fig. 1). Note that theassignment of the directions of afcc, bfcc, and cfcc is notunique and may be permuted. In the present definition, afccand bfcc are crystallographically equivalent, reflecting thesymmetric equivalence of the b and −b directions in themonoclinic lattice.From the fcc lattice vectors, the corresponding latticeparameters are calculated to be afcc = bfcc = 4.20 Å, cfcc= 3.95 Å, αfcc = βfcc = 90.1°, and γfcc = 92.7°. Thedeviation from an ideal cubic lattice is relatively small,Content from this work may be used under the terms of the Creative Commons Attribution 4.0 license. Any further distribution ofthis work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.038003-1© 2026 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdJapanese Journal of Applied Physics 65, 038003 (2026) BRIEF NOTEhttps://doi.org/10.35848/1347-4065/ae42achttps://crossmark.crossref.org/dialog/?doi=10.35848/1347-4065/ae42ac&domain=pdf&date_stamp=2026-02-16https://orcid.org/0000-0001-8550-9735mailto:OSHIMA.Takayoshi@nims.go.jphttps://doi.org/10.35848/1347-4065/ae42achttps://creativecommons.org/licenses/by/4.0/https://doi.org/10.35848/1347-4065/ae42acjustifying the use of pseudo-cubic approximation, whichgreatly facilitates the understanding of the crystal structure.In the cubic fcc oxygen structure, the principal crystal-lographic planes are {100}fcc (6 planes), {110}fcc (12planes), and {111}fcc (8 planes). All of these planes havecorresponding planes in monoclinic β-Ga2O3, as summarizedin Table I. Note that (hkl), (hk l ), (hk l), and (h kl ) planes arecrystallographically equivalent. Figure 2 shows the crystalshape constructed using all the planes listed in Table I. Thisthree-dimensional shape was visualized using the crystal-lographic visualization software VESTA,21,22) assuming thateach crystal facet is located at an equal distance from theorigin. We also prepared a printable papercraft template tobuild a physical model of this crystal shape, enabling hands-on inspection of its three-dimensional geometry (see thesupplementary file).Table I presents the primary crystallographic planes of theoxygen fcc sublattice. The {100} and {112} planes areamong the most fundamental, as they correspond to the six{100}fcc faces of the pseudo-cubic structure. The {201},{101}, and {310} planes exhibit oxygen close-packedconfigurations corresponding to the {111}fcc planes, whereasthe {010}, {102}, {512}, and {712} planes correspond to thelow-index {110}fcc planes. Several of these planes have notyet been selected as substrate orientations; therefore, theirinvestigation may be worthwhile for exploring new substrateorientations. Note that the (001) plane—the most commonlyused substrate orientation for vertical power devices—is notincluded in Table I; instead, the (102) plane, which iscrystallographically close to (001),23–25) is included.Similarly, the (011) plane—another substrate orientationthat has recently attracted attention for vertical powerdevices26–29)—is also excluded from the Table I, whereasthe (112) plane, which is close to (011) is included.Figure 2 clearly illustrates the orientation relationships ofthese planes, particularly the orthogonal and nearly ortho-gonal relationships resulting from the pseudo-cubic approx-imation. For example, the (100) plane is perpendicular to the{010} and almost perpendicular to the {112} and {102}planes with interplanar angles of 89.9° and 89.9°, respec-tively; the (010) plane is perpendicular to the {100}, {102},{101}, and {201} planes; the (102) plane is perpendicular to{010} and almost perpendicular to {100} and {310} withinterplanar angles of 89.9° and 89.9°, respectively; and the (201) plane is perpendicular to {010} and nearly perpendi-cular to {512} with an interplanar angle of 91.7°. Thisinformation is particularly useful when considering combi-nations of substrate orientations and sidewall planes formedby etching processes. In plasma-free gas etching, the (100)surface becomes exceptionally flat as it exhibits theminimum surface energy density.30) Therefore, trenchesand fins with (100)-faceted vertical and near-vertical side-walls should be fabricated using gas etching by employingsubstrates oriented along {010} and {112}/{102}, respec-tively. This approach has been experimentally verified with(010) and (102) substrates.24,31–35)Furthermore, the correspondence between the fcc andmonoclinic planes summarized in Table I allows directestimation of epitaxial relationships between cubic crystalsand β-Ga2O3. Since β-Ga2O3 is a unipolar n-type semicon-ductor, the realization of p–n junctions relies on theFig. 1. Averaged pseudo-cubic unit cell of the distorted face-centeredcubic (fcc) oxygen sublattice in β-Ga2O3. The unit cell is defined by thelattice vectors afcc, bfcc, and cfcc, giving lattice parameters of afcc, bfcc,cfcc, αfcc, βfcc, and γfcc. Oxygen atoms are placed at the fcc sites and areshown in the unit cell as open circles.Table I. Correspondence between crystallographic planes of the slightly distorted face-centered cubic (fcc) oxygen sublattice and monoclinic β-Ga2O3.fcc oxygen sublattice planes Corresponding planes in monoclinic β-Ga2O3{100}fcc (100), (100), (112), (11 2), (1 12), (112){110}fcc (010), (010), (102), (102), (512), (5 1 2), (512), (512), (712), (71 2), (7 12), (712){111}fcc (Close packing planes) (201), (201), (101), (101), (310), (3 10), (310), (310)Fig. 2. Crystal shape faceted with the representative oxygen sublatticeplanes of β-Ga2O3. The Miller indices of the hidden (hk l ) facets can beidentified because they are crystallographically equivalent to their opposite(hkl) facets that are already labeled.038003-2© 2026 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdJpn. J. Appl. Phys. 65, 038003 (2026) T. Oshimaheteroepitaxial growth of p-type oxides, particularly rock-salt-structured NiO (a= 4.1684 Å).36,37) Therefore, deter-mining the epitaxial relationships between cubic NiO andmonoclinic β-Ga2O3 is essential for understanding p–nheterojunctions, and such relationships can be readilyestimated using Table I. For example, (100)-oriented NiOshould be obtained on {100} and {112} β-Ga2O3 substrates.The validity of this correspondence has been experimentallydemonstrated in several studies. To date, (100)-, (110)-,(110)- and (111)-oriented NiO films have been epitaxiallygrown on (100), (010), (102), and (201) β-Ga2O3 substrates,respectively.25,38,39) Similar oxygen-sublattice-aligned epitaxyhas also been reported for other cubic oxides grown on β-Ga2O3substrates, further supporting the validity of this approach. Forp-type materials other than NiO, reported examples include(100) and (110) NiGa2O4 (spinel structure, a = 8.258 95 Å)40)on (100) and (010) β-Ga2O3, respectively;41) (111) Cr2MnO4(spinel structure, a = 8.41 Å)42) on (201) β-Ga2O3;43) and(110) LiGa5O8 (spinel structure, a = 8.203 Å)44) on (010)β-Ga2O3.45) For dielectric materials, reported examples include(110) γ-Al2O3 (defective spinel structure, a = 7.911 Å)46) on(010) β-Ga2O3;47) and (110) and (100) γ-(AlxGa1−x)2O3(defective spinel structure) on (010) and (100) β-Ga2O3,respectively.48,49)In summary, we systematically investigated a pseudo-cubic description of monoclinic β-Ga2O3 based on itsdistorted fcc oxygen sublattice. By defining an averagedfcc unit cell and establishing the correspondence between fccand monoclinic planes, the complex orientation relationshipsof β-Ga2O3 can be understood in an intuitive and straight-forward manner. This approach identifies the primaryplanes constituting the oxygen sublattice, clarifies theorthogonal relationships among these planes, enables rea-sonable selection of substrate orientations suitable forvertical etching, and allows estimation of epitaxial relation-ships with cubic oxides such as NiO. The pseudo-cubicrepresentation therefore provides a practical crystallo-graphic basis for β-Ga2O3 and is useful for screeningunexplored substrate orientations.Acknowledgments This work was financially supported by a Grant-in-Aid for Scientific Research (B) from the Japan Society for the Promotion ofScience (JSPS), MEXT, Japan (No. JP24K01368).ORCID iDs Takayoshi Oshima https://orcid.org/0000-0001-8550-97351) S. J. Pearton, F. Ren, A. Y. Polyakov, A. Haque, M. Labed, and Y. S. Rim,Appl. Phys. Rev. 12, 031336 (2025).2) K. Sasaki, Appl. Phys. Express 17, 090101 (2024).3) T. Onuma, S. Saito, K. Sasaki, T. Masui, T. Yamaguchi, T. Honda, andM. Higashiwaki, Jpn. J. Appl. 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