{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:21:30Z","timestamp":1760149290350,"version":"build-2065373602"},"reference-count":75,"publisher":"MDPI AG","issue":"15","license":[{"start":{"date-parts":[[2023,7,30]],"date-time":"2023-07-30T00:00:00Z","timestamp":1690675200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Nanomaterials"],"abstract":"<jats:p>In this study, we present a theoretical study of the quantum spin Hall effect in InN\/InGaN coupled multiple quantum wells with the individual well widths equal to two atomic monolayers. We consider triple and quadruple quantum wells in which the In content in the interwell barriers is greater than or equal to the In content in the external barriers. To calculate the electronic subbands in these nanostructures, we use the eight-band k\u2219p Hamiltonian, assuming that the effective spin\u2013orbit interaction in InN is negative, which represents the worst-case scenario for achieving a two-dimensional topological insulator. For triple quantum wells, we find that when the In contents of the external and interwell barriers are the same and the widths of the internal barriers are equal to two monolayers, a topological insulator with a bulk energy gap of 0.25 meV can appear. Increasing the In content in the interwell barriers leads to a significant increase in the bulk energy gap of the topological insulator, reaching about 0.8 meV. In these structures, the topological insulator can be achieved when the In content in the external barriers is about 0.64, causing relatively low strain in quantum wells and making the epitaxial growth of these structures within the range of current technology. Using the effective 2D Hamiltonian, we study the edge states in strip structures containing topological triple quantum wells. We demonstrate that the opening of the gap in the spectrum of the edge states caused by decreasing the width of the strip has an oscillatory character regardless of whether the pseudospin-mixing elements of the effective Hamiltonian are omitted or taken into account. The strength of the finite size effect in these structures is several times smaller than that in HgTe\/HgCdTe and InAs\/GaSb\/AlSb topological insulators. Therefore, its influence on the quantum spin Hall effect is negligible in strips with a width larger than 150 nm, unless the temperature at which electron transport is measured is less than 1 mK. In the case of quadruple quantum wells, we find the topological insulator phase only when the In content in the interwell barriers is larger than in the external barriers. We show that in these structures, a topological insulator with a bulk energy gap of 0.038 meV can be achieved when the In content in the external barriers is about 0.75. Since this value of the bulk energy gap is very small, quadruple quantum wells are less useful for realizing a measurable quantum spin Hall system, but they are still attractive for achieving a topological phase transition and a nonlocal topological semimetal phase.<\/jats:p>","DOI":"10.3390\/nano13152212","type":"journal-article","created":{"date-parts":[[2023,7,31]],"date-time":"2023-07-31T01:39:32Z","timestamp":1690767572000},"page":"2212","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Quantum Spin Hall Effect in Two-Monolayer-Thick InN\/InGaN Coupled Multiple Quantum Wells"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7681-3292","authenticated-orcid":false,"given":"S\u0142awomir P.","family":"\u0141epkowski","sequence":"first","affiliation":[{"name":"Institute of High Pressure Physics\u2014Unipress, Polish Academy of Sciences, ul. Soko\u0142owska 29\/37, 01-142 Warszawa, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3045","DOI":"10.1103\/RevModPhys.82.3045","article-title":"Colloquium: Topological insulators","volume":"82","author":"Hasan","year":"2010","journal-title":"Rev. Mod. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"021004","DOI":"10.1103\/RevModPhys.88.021004","article-title":"Colloquium: Topological band theory","volume":"88","author":"Bansil","year":"2016","journal-title":"Rev. Mod. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1146\/annurev-conmatphys-062910-140538","article-title":"The quantum spin Hall effect","volume":"2","author":"Maciejko","year":"2011","journal-title":"Annu. Rev. Condens. 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