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Pseudomorphic growth on m-plane sapphire has been achieved for $$x \\ge 0.45$$<\/jats:tex-math>\n \n x<\/mml:mi>\n \u2265<\/mml:mo>\n 0.45<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. A distinct difference between the a- and m-epitaxial plane is observed in reciprocal space map measurements being in agreement with continuum elasticity theory for rhombohedral heterostructures. While pseudomorphic layers on m-plane sapphire show a pronounced shear strain $$e'_5$$<\/jats:tex-math>\n \n e<\/mml:mi>\n 5<\/mml:mn>\n \u2032<\/mml:mo>\n <\/mml:msubsup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> along the c<\/jats:italic>-axis direction, relaxed layers exhibit a global lattice tilt in the same direction. Both effects are not present on the a-epitaxial plane. Out-of-plane lattice constants as well as $$e'_5$$<\/jats:tex-math>\n \n e<\/mml:mi>\n 5<\/mml:mn>\n \u2032<\/mml:mo>\n <\/mml:msubsup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are modeled as function of x<\/jats:italic> employing elasticity theory, confirming theoretical values of the elastic stiffness tensor for $$\\alpha$$<\/jats:tex-math>\n \u03b1<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-Ga$$_2$$<\/jats:tex-math>\n \n \n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>O$$_3$$<\/jats:tex-math>\n \n \n 3<\/mml:mn>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, especially the non-zero value of the $$C_{14}$$<\/jats:tex-math>\n \n C<\/mml:mi>\n 14<\/mml:mn>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> component. Possible pyramidal slip systems for strain relaxation in c<\/jats:italic>-axis direction are examined to explain and numerically model the difference in lattice tilt for the two substrate orientations.<\/jats:p>\n <\/jats:sec>\n Graphic abstract<\/jats:title>\n \n <\/jats:sec>","DOI":"10.1557\/s43578-021-00375-3","type":"journal-article","created":{"date-parts":[[2021,9,22]],"date-time":"2021-09-22T17:17:44Z","timestamp":1632331064000},"page":"4816-4831","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":24,"title":["Strain states and relaxation for $$\\alpha$$-(Al$$_x$$Ga$$_{1-x}$$)$$_2$$O$$_3$$ thin films on prismatic planes of $$\\alpha$$-Al$$_2$$O$$_3$$ in the full composition range: Fundamental difference of a- and m-epitaxial planes in the manifestation of shear strain and lattice tilt"],"prefix":"10.1557","volume":"36","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8350-7346","authenticated-orcid":false,"given":"Max","family":"Knei\u00df","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5434-2194","authenticated-orcid":false,"given":"Daniel","family":"Splith","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3936-275X","authenticated-orcid":false,"given":"Holger","family":"von Wenckstern","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2774-6040","authenticated-orcid":false,"given":"Michael","family":"Lorenz","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0344-6302","authenticated-orcid":false,"given":"Thorsten","family":"Schultz","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6042-6447","authenticated-orcid":false,"given":"Norbert","family":"Koch","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7554-182X","authenticated-orcid":false,"given":"Marius","family":"Grundmann","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2021,9,22]]},"reference":[{"issue":"3","key":"375_CR1","doi-asserted-by":"publisher","first-page":"034001","DOI":"10.1088\/0268-1242\/31\/3\/034001","volume":"31","author":"M Higashiwaki","year":"2016","unstructured":"M. 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