- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Applied

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Strain-stress relationships for coherent in-plane strain in heterostructures with monoclinic crystal systems: β-(AlxGa1−x)2O3 on (h0l)β-Ga2O3 as example

Mathias Schubert1,2,3,*, Rafał Korlacki1, Sina Khayam4, Yousra Traouli1, Preston Sorensen1, Alexis Papamichail5,6, and Vanya Darakchieva2,5,6,†

  • 1Department of Electrical and Computer Engineering, University of Nebraska-Lincoln, Lincoln, Nebraska 68588, USA
  • 2Terahertz Materials Analysis Center and Competence Center for III-N Technology, C3NiT—Janzèn, NanoLund and Solid State Physics, Lund University, 22100 Lund, Sweden
  • 3Wallenberg Initiative Materials Science for Sustainability (WISE), Department of Physics, Lund University, 22100, Lund, Sweden
  • 4Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, Nebraska 68588, USA
  • 5Competence Center for III-N Technology, C3NiT—Janzèn, Department of Physics, Chemistry and Biology (IFM), Linköping University, 58183 Linköping, Sweden
  • 6Wallenberg Initiative Materials Science for Sustainability (WISE), Department of Physics, Chemistry and Biology (IFM), Linköping University, SE-58183 Linköping, Sweden

PDF

Phys. Rev. Applied 24, 044075 – Published 23 October, 2025

DOI: https://doi.org/10.1103/5gx2-kn6m

Abstract

In this work, we derive the relationship between the states of strain and stress in pseudomorphic epitaxial layers on a substrate with differing equilibrium lattice parameters under symmetry-conserving conditions for the case of monoclinic crystal symmetry. We compare surface vectors across the template-epitaxial layer interface and impose conditions of a stress-free epitaxial layer. As a result, we demonstrate the theoretical existence of up to three possible unit cells that can establish on a given template. We demonstrate this approach for the special case of symmetry-conserving strain in a class of templates with (h0l) surface orientations. We use β-(AlxGa1−x)2O3 on (h0l)β-Ga2O3 as an example. We discuss the effects of the composition x and surface orientation on the formation of three elastically stable unit cells, their strain and stress tensors, unit-cell axes, unit-cell volumes, lattice spacings, elastic potential energies, and stress-free directions. The previous paradigm for epitaxial layer growth, where the only stress-free direction is perpendicular to the growing surface, is not generally valid for low-symmetry materials. In the example here, we find that, in addition to the stress-free growth direction for crystallographic planes with low Miller indices, two possible competing domains can exist with stress-free directions oblique to the surface of the template for almost all planes (h0l). We further demonstrate the effect of the orientation-dependent differently strained domains and calculate the strain-shifted anisotropic band-to-band transitions for β-(Al0.1Ga0.9)2O3 on (h0l)β-Ga2O3. We also provide examples of x-ray diffraction analyses, which may help guide the search for the occurrence of such differently strained unit cells in epitaxial layers grown by experiment, here taking the (2¯01) surface orientation as case for demonstration.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (58)

  1. H. Peelaers, D. Steiauf, J. B. Varley, A. Janotti, and C. G. Van de Walle, (InxGa1−x)2O3 alloys for transparent electronics, Phys. Rev. B 92, 085206 (2015).
  2. J. A. Spencer, A. L. Mock, A. G. Jacobs, M. Schubert, Y. Zhang, and M. J. Tadjer, A review of band structure and material properties of transparent conducting and semiconducting oxides: Ga2O3, Al2O3, In2O3, ZnO, SnO2, CdO, NiO, CuO, and Sc2O3, Appl. Phys. Rev. 9, 011315 (2022).
  3. P. Mazzolini, C. Wouters, M. Albrecht, A. Falkenstein, M. Martin, P. Vogt, and O. Bierwagen, Molecular beam epitaxy of β-(InxGa1−x)2O3 on β-Ga2O3 (010): Compositional control, layer quality, anisotropic strain relaxation, and prospects for two-dimensional electron gas confinement, ACS Appl. Mater. Interfaces 16, 12793 (2024).
  4. R. Korlacki, M. Hilfiker, J. Knudtson, M. Stokey, U. Kilic, A. Mauze, Y. Zhang, J. Speck, V. Darakchieva, and M. Schubert, Strain and composition dependencies of the near-band-gap optical transitions in monoclinic (AlxGa1−x)2O3 alloys with coherent biaxial in-plane strain on Ga2O3(010), Phys. Rev. Appl. 18, 064019 (2022).
  5. H. Peelaers, J. B. Varley, J. S. Speck, and C. G. Van de Walle, Structural and electronic properties of Ga2O3-Al2O3 alloys, Appl. Phys. Lett. 112, 242101 (2018).
  6. X. Liu and C.-K. Tan, Electronic properties of monoclinic (InxGa1−x)2O3 alloys by first-principle, AIP Adv. 9, 035318 (2019).
  7. J. Åhman, G. Svensson, and J. Albertsson, A reinvestigation of β-gallium oxide, Acta Crystallogr. Sect. C 52, 1336 (1996).
  8. P. Vogt and O. Bierwagen, The competing oxide and sub-oxide formation in metal-oxide molecular beam epitaxy, Appl. Phys. Lett. 106, 081910 (2015).
  9. S. J. Hao, M. Hetzl, F. Schuster, K. Danielewicz, A. Bergmaier, G. Dollinger, Q. L. Sai, C. T. Xia, T. Hoffmann, M. Wiesinger, S. Matich, W. Aigner, and M. Stutzmann, Growth and characterization of β-Ga2O3 thin films on different substrates, J. Appl. Phys. 125, 105701 (2019).
  10. D. Seo, S. Kim, H.-Y. Kim, D.-W. Jeon, J.-H. Park, and W. S. Hwang, Heteroepitaxial growth of single-crystalline β-Ga2O3 on GaN/Al2O3 using MOCVD, Cryst. Growth Des. 23, 7090 (2023).
  11. Q. Cao, L. He, H. Xiao, X. Feng, Y. Lv, and J. Ma, β-Ga2O3 epitaxial films deposited on epi-GaN/sapphire (0001) substrates by MOCVD, Mat. Sci. Semicon. Proc. 77, 58 (2018).
  12. Q. Cao, L. He, X. Feng, H. Xiao, and J. Ma, Effect of annealing on the structural and optical properties of β-Ga2O3 films prepared on gadolinium gallium garnet (110) by MOCVD, Ceram. Int. 44, 830 (2018).
  13. H. Sun, K.-H. Li, C. T. Castanedo, S. Okur, G. S. Tompa, T. Salagaj, S. Lopatin, A. Genovese, and X. Li, HCl flow-induced phase change of α-, β-, and ε-Ga2O3 films grown by MOCVD, Cryst. Growth Des. 18, 2370 (2018).
  14. W. Mi, J. Ma, C. Luan, Y. Lv, H. Xiao, and Z. Li, Characterization of β-Ga2O3 epitaxial films grown on MgO (111) substrates by metal-organic chemical vapor deposition, Mat. Lett. 87, 109 (2012).
  15. G. Pozina, C.-W. Hsu, N. Abrikossova, M. A. Kaliteevski, and C. Hemmingsson, Development of β-Ga2O3 layers growth on sapphire substrates employing modeling of precursors ratio in halide vapor phase epitaxy reactor, Sci. Rep. 10, 22261 (2020).
  16. V. Nikolaev, A. Pechnikov, S. Stepanov, I. Nikitina, A. Smirnov, A. Chikiryaka, S. Sharofidinov, V. Bougrov, and A. Romanov, Epitaxial growth of (2¯01) β-Ga2O3 on (0001) sapphire substrates by halide vapour phase epitaxy, Mat. Sci. Semicon. Proc. 47, 16 (2016).
  17. D. Guo, Z. Wu, P. Li, Y. An, H. Liu, X. Guo, H. Yan, G. Wang, C. Sun, L. Li, and W. Tang, Fabrication of β-Ga2O3 thin films and solar-blind photodetectors by laser MBE technology, Opt. Mater. Express 4, 1067 (2014).
  18. T. Itoh, A. Mauze, Y. Zhang, and J. S. Speck, Epitaxial growth of β-Ga2O3 on (110) substrate by plasma-assisted molecular beam epitaxy, Appl. Phys. Lett. 117, 152105 (2020).
  19. P. Yan, Z. Zhang, Y. Xu, H. Chen, D. Chen, Q. Feng, S. Xu, Y. Zhang, J. Zhang, C. Zhang, and Y. Hao, Heteroepitaxial growth and band alignment of β-Ga2O3 on GaN substrate grown by non-vacuum mist-CVD, Vacuum 204, 111381 (2022).
  20. R. Wakabayashi, K. Yoshimatsu, M. Hattori, and A. Ohtomo, Epitaxial structure and electronic property of β-Ga2O3 films grown on MgO (100) substrates by pulsed-laser deposition, Appl. Phys. Lett. 111, 162101 (2017).
  21. F. Egyenes-Pörsök, F. Gucmann, K. Hušeková, E. Dobročka, M. Sobota, M. Mikolášek, K. Fröhlich, and M. Ťapajna, Growth of α-and β-Ga2O3 epitaxial layers on sapphire substrates using liquid-injection MOCVD, Semicond. Sci. Technol. 35, 115002 (2020).
  22. Y. Yao, S. Okur, L. A. Lyle, G. S. Tompa, T. Salagaj, N. Sbrockey, R. F. Davis, and L. M. Porter, Growth and characterization of α-, β-, and ϵ-phases of Ga2O3 using MOCVD and HVPE techniques, Mat. Res. Lett. 6, 268 (2018).
  23. D. Wang, L. He, X. Ma, H. Xiao, Y. Le, and J. Ma, Preparation and properties of heteroepitaxial β-Ga2O3 films on KTaO3 (100) substrates by MOCVD, Mater. Charact. 165, 110391 (2020).
  24. S. Nakagomi and Y. Kokubun, Crystal orientation of monoclinic β-Ga2O3 thin films formed on cubic MgO substrates with a γ-Ga2O3 interfacial layer, J. Cryst. Growth 479, 67 (2017).
  25. X. Tang, K.-H. Li, C.-H. Liao, D. Zheng, C. Liu, R. Lin, N. Xiao, S. Krishna, J. Tauboada, and X. Li, Epitaxial growth of β-Ga2O3 (-201) thin film on four-fold symmetry CeO2 (001) substrate for heterogeneous integrations, J. Mater. Chem. C 9, 15868 (2021).
  26. A. Mock, S. Richter, A. Papamichail, V. Stanishev, M. Ghezellou, J. Ul-Hassan, A. Popp, S. Bin Anooz, D. Gogova, P. Ranga, S. Krishnamoorthy, R. Korlacki, M. Schubert, and V. Darakchieva, Effective uniaxial dielectric function tensor and optical phonons in (2¯01)-oriented β-Ga2O3 films with equally distributed sixfold-rotation domains, Phys. Rev. Appl. 22, 044003 (2024).
  27. N. Nepal, D. S. Katzer, B. P. Downey, V. D. Wheeler, L. O. Nyakiti, D. F. Storm, M. T. Hardy, J. A. Freitas, E. N. Jin, D. Vaca, L. Yates, S. Graham, S. Kumar, and D. J. Meyer, Heteroepitaxial growth of β-Ga2O3 films on SiC via molecular beam epitaxy, J. Vac. Sci. Technol. A 38, 063406 (2020).
  28. Y. Yan, Z. Zhang, D. Liu, L. Zeng, H. Chen, D. Chen, W. Zhu, Q. Feng, Y. Zhang, W. Mao, et al., High-quality heteroepitaxial growth of β-Ga2O3 with NiO buffer layer based on Mist-CVD, Vacuum 231, 113777 (2025).
  29. Y. Oshima, E. G. Víllora, and K. Shimamura, Quasi-heteroepitaxial growth of β-Ga2O3 on off-angled sapphire (0001) substrates by halide vapor phase epitaxy, J. Cryst. Growth 410, 53 (2015).
  30. Z. Wen, X. Zhai, K. Khan, O. Odabasi, M. Kim, and E. Ahmadi, (010) β-(Alx,Ga1−x)2O3 growth using tritertiarybutylaluminum as Al gas precursor via hybrid molecular beam epitaxy, Appl. Phys. Lett. 125, 162103 (2024).
  31. J. Sarker, P. Garg, A. Rauf, A. R. Nirjhar, H.-L. Huang, M. Zhu, A. F. M. A. U. Bhuiyan, L. Meng, H. Zhao, J. Hwang, E. Osei-Agyemang, S. Ahmed, and B. Mazumder, Microscopic and spectroscopic investigation of (AlxGa1−x)2O3 films: Unraveling the impact of growth orientation and aluminum content, Adv. Mater. Interfaces 12, 2301016 (2025).
  32. A. F. M. A. U. Bhuiyan, Z. Feng, L. Meng, and H. Zhao, MOCVD growth of (010) β-(AlxGa1−x)2O3 thin films, J. Mater Res. 36, 4804 (2021).
  33. A. F. M. A. Uddin Bhuiyan, L. Meng, H.-L. Huang, J. Sarker, C. Chae, B. Mazumder, J. Hwang, and H. Zhao, Metalorganic chemical vapor deposition of β-(AlxGa1−x)2O3 thin films on (001) β-Ga2O3 substrates, APL Mater. 11, 041112 (2023).
  34. F. Alema, T. Itoh, W. Brand, M. Tadjer, A. Osinsky, and J. S. Speck, N2O grown high Al composition nitrogen doped β-(AlGa)2O3/β-Ga2O3 using MOCVD, J. Vac. Sci. Technol. A 41, 042709 (2023).
  35. J. S. Lundh, K. Huynh, M. Liao, W. Olsen, K. Pan, K. Sasaki, K. Konishi, H. N. Masten, J. K. Hite, M. A. Mastro, N. A. Mahadik, M. Goorsky, A. Kuramata, K. D. Hobart, T. J. Anderson, and M. J. Tadjer, Experimental determination of critical thickness limitations of (010) β-(AlxGa1−x)2O3 heteroepitaxial films, Appl. Phys. Lett. 123, 222104 (2023).
  36. A. Mauze, T. Itoh, Y. Zhang, E. Deagueros, F. Wu, and J. S. Speck, Coherently strained (001) β-(AlxGa1−x)2O3 thin films β-Ga2O3: Growth and compositional analysis, J. Appl. Phys. 132, 115302 (2022).
  37. L. G. Bir and G. E. Pikus, Symmetry and Strain-Induced Effects in Semiconductors (John Wiley and Sons, New York, 1974).
  38. C. Zhang, X. Wu, Y. Xing, L. Zhou, H. Zhou, S. Li, and N. Xu, A first-principles study of hydrostatic strain engineering on the electronic properties of β-Ga2O3, Phys. B Condens. Matter 660, 414851 (2023).
  39. H.-L. Huang, J. M. Johnson, C. Chae, A. Senckowski, M. H. Wong, and J. Hwang, Atomic scale mechanism of β to γ phase transformation in gallium oxide, Appl. Phys. Lett. 122, 251602 (2023).
  40. S. Seacat, J. L. Lyons, and H. Peelaers, Computational design of optimal heterostructures for β−Ga2O3, Phys. Rev. Mater. 8, 014601 (2024).
  41. L. M. Barmore, J. Jesenovec, J. S. McCloy, and M. D. McCluskey, Photoluminescence of Cr3+ in β-Ga2O3 and (Al0.1Ga0.9)2O3 under pressure, J. Appl. Phys. 133, 175703 (2023).
  42. Y. Xu, J.-H. Park, Z. Yao, C. Wolverton, M. Razeghi, J. Wu, and V. P. Dravid, Strain-induced metastable phase stabilization in Ga2O3 thin films, ACS Appl. Mater. Interfaces 11, 5536 (2019).
  43. R. Zhang, M. Li, G. Wu, L. Li, Z. Zhang, K. Liang, and W. Shen, Modulating electronic properties of β-Ga2O3 by strain engineering, Results Phys. 52, 106916 (2023).
  44. Y. Hara, W. Zhu, G. Deng, E. Marin, Q. Guo, and G. Pezzotti, Spatially resolved Raman piezospectroscopy for nondestructive evaluation of residual stress in β-Ga2O3 films, J. Phys. D: Appl. Phys. 56, 125102 (2023).
  45. T. Uchida and R. Sugie, Thermal stress analysis in die-attached β-Ga2O3 using Raman spectroscopy, Jpn. J. Appl. Phys. 62, SF1003 (2023).
  46. R. Korlacki, M. Stokey, A. Mock, S. Knight, A. Papamichail, V. Darakchieva, and M. Schubert, Strain and stress relationships for optical phonon modes in monoclinic crystals with β-Ga2O3 as an example, Phys. Rev. B 102, 180101(R) (2020).
  47. N. Hasuike, I. Maeda, S. Isaji, K. Kobayashi, K. Ohira, and T. Isshiki, Micro-Raman spectroscopy of bending stresses in β-Ga2O3(001) wafer, Jpn. J. Appl. Phys. 62, SF1020 (2023).
  48. S. Mu, H. Peelaers, Y. Zhang, M. Wang, and C. G. Van de Walle, Orientation-dependent band offsets between (AlxGa1−x)2O3 and Ga2O3, Appl. Phys. Lett. 117, 252104 (2020).
  49. J.-M. Wagner and F. Bechstedt, Properties of strained wurtzite GaN and AlN: Ab initio studies, Phys. Rev. B 66, 115202 (2002).
  50. V. Darakchieva, P. P. Paskov, T. Paskova, J. Birch, S. Tungasmita, and B. Monemar, Deformation potentials of the E1(TO) mode in AlN, Appl. Phys. Lett. 80, 2302 (2002).
  51. V. Darakchieva, T. Paskova, M. Schubert, H. Arwin, P. P. Paskov, B. Monemar, D. Hommel, M. Heuken, J. Off, F. Scholz, B. A. Haskell, P. T. Fini, J. S. Speck, and S. Nakamura, Anisotropic strain and phonon deformation potentials in GaN, Phys. Rev. B 75, 195217 (2007).
  52. M.-Y. Xie, F. Tasnádi, I. A. Abrikosov, L. Hultman, and V. Darakchieva, Elastic constants, composition, and piezolectric polarization in InxAl1−xN: From ab initio calculations to experimental implications for the applicability of Vegard’s rule, Phys. Rev. B 86, 155310 (2012).
  53. M. Grundmann, Strain in pseudomorphic monoclinic Ga2O3-based heterostructures, Phys. Status Solidi B 254, 1700134 (2017).
  54. M. Schubert, R. Korlacki, S. Knight, T. Hofmann, S. Schöche, V. Darakchieva, E. Janzén, B. Monemar, D. Gogova, Q.-T. Thieu, R. Togashi, H. Murakami, Y. Kumagai, K. Goto, A. Kuramata, S. Yamakoshi, and M. Higashiwaki, Anisotropy, phonon modes, and free charge carrier parameters in monoclinic β-gallium oxide single crystals, Phys. Rev. B 93, 125209 (2016).
  55. Note that the reciprocal vectors could also be expressed by introducing the factor 2π, i.e., vivj⋆=2πδij. The difference between this and Eq. (2) lies only in a common factor applied to all lengths of the subsequent reciprocal vectors, which is irrelevant to the role they play in this derivation. The reciprocal vectors are used to express the normal to any given surface (h0l).
  56. C. Kranert, M. Jenderka, J. Lenzner, M. Lorenz, H. von Wenckstern, R. Schmidt-Grund, and M. Grundmann, Lattice parameters and Raman-active phonon modes of β-(AlxGa1−x)2O3, J. Appl. Phys. 117, 125703 (2015).
  57. This definition of the elements of ϵ here is useful because the tensor elements refer to the changes of the intrinsic lattice parameters, a, b, c, and β for a given epitaxial layer, regardless of the actual orientation of the unit cell within a given epitaxial layer relative to the substrate. In this respect, the coordinate system for the substrate is irrelevant.
  58. Note the structure of the determinant for the stress tensor, where the component σzz always factorizes out. This yields trivial solutions for all cases when the b axis is perpendicular to the surface, (010). Then the z direction is also parallel to the surface normal ghkl. Hence, for growth on (010), only one stress-free solution exists, with a stress-free direction not parallel to the surface, which is then obviously the surface normal. For almost all other crystallographic surfaces, the stress-free directions are neither parallel nor perpendicular to the surface, and all represent potential stable unit cells during growth.

Outline

Information

Phys. Rev. Applied 24, 044075– Published 23 October, 2025
  • Received 16 December 2024
  • Revised 2 April 2025
  • Accepted 16 September 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by Bibsam.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation