{"title":"具有隐藏拓扑相的α-锑烯中的高自旋-切尔数绝缘体","authors":"Baokai Wang, Xiaoting Zhou, Yi-Chun Hung, Yen-Chuan Lin, Hsin Lin, Arun Bansil","doi":"10.1088/2053-1583/ad3136","DOIUrl":null,"url":null,"abstract":"For a time-reversal symmetric system, the quantum spin Hall phase is assumed to be the same as the <inline-formula>\n<tex-math><?CDATA $\\mathbb{Z}_2$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"double-struck\">Z</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"tdmad3136ieqn1.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> topological insulator phase in the existing literature. The spin Chern number <inline-formula>\n<tex-math><?CDATA $\\mathcal{C}_s$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"tdmad3136ieqn2.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> is presumed to yield the same topological classification as the <inline-formula>\n<tex-math><?CDATA $\\mathbb{Z}_2$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"double-struck\">Z</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"tdmad3136ieqn3.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> invariant. Here, by investigating the electronic structures of monolayer <italic toggle=\"yes\">α</italic>-phase group V elements, we uncover the presence of a topological phase in <italic toggle=\"yes\">α</italic>-Sb, which can be characterized by a spin Chern number <inline-formula>\n<tex-math><?CDATA $\\mathcal{C}_s$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"tdmad3136ieqn4.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> = 2, even though it is <inline-formula>\n<tex-math><?CDATA $\\mathbb{Z}_2$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"double-struck\">Z</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"tdmad3136ieqn5.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> trivial. Although <italic toggle=\"yes\">α</italic>-As and Sb would thus be classified as trivial insulators within the classification schemes, we demonstrate the existence of a phase transition between <italic toggle=\"yes\">α</italic>-As and Sb, which is induced by band inversions at two generic <italic toggle=\"yes\">k</italic> points. Without spin–orbit coupling (SOC), <italic toggle=\"yes\">α</italic>-As is a trivial insulator, while <italic toggle=\"yes\">α</italic>-Sb is a Dirac semimetal with four Dirac points (DPs) located away from the high-symmetry lines. Inclusion of the SOC gaps out the DPs and induces a nontrivial Berry curvature, endowing <italic toggle=\"yes\">α</italic>-Sb with a high spin Chern number of <inline-formula>\n<tex-math><?CDATA $\\mathcal{C}_s$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"tdmad3136ieqn6.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> = 2. We further show that monolayer <italic toggle=\"yes\">α</italic>-Sb exhibits either a gapless band structure or a gapless spin spectrum on its edges, as expected from topological considerations.","PeriodicalId":6812,"journal":{"name":"2D Materials","volume":null,"pages":null},"PeriodicalIF":4.5000,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"High spin-Chern-number insulator in α-antimonene with a hidden topological phase\",\"authors\":\"Baokai Wang, Xiaoting Zhou, Yi-Chun Hung, Yen-Chuan Lin, Hsin Lin, Arun Bansil\",\"doi\":\"10.1088/2053-1583/ad3136\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For a time-reversal symmetric system, the quantum spin Hall phase is assumed to be the same as the <inline-formula>\\n<tex-math><?CDATA $\\\\mathbb{Z}_2$?></tex-math>\\n<mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\\\"double-struck\\\">Z</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math>\\n<inline-graphic xlink:href=\\\"tdmad3136ieqn1.gif\\\" xlink:type=\\\"simple\\\"></inline-graphic>\\n</inline-formula> topological insulator phase in the existing literature. The spin Chern number <inline-formula>\\n<tex-math><?CDATA $\\\\mathcal{C}_s$?></tex-math>\\n<mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math>\\n<inline-graphic xlink:href=\\\"tdmad3136ieqn2.gif\\\" xlink:type=\\\"simple\\\"></inline-graphic>\\n</inline-formula> is presumed to yield the same topological classification as the <inline-formula>\\n<tex-math><?CDATA $\\\\mathbb{Z}_2$?></tex-math>\\n<mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\\\"double-struck\\\">Z</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math>\\n<inline-graphic xlink:href=\\\"tdmad3136ieqn3.gif\\\" xlink:type=\\\"simple\\\"></inline-graphic>\\n</inline-formula> invariant. Here, by investigating the electronic structures of monolayer <italic toggle=\\\"yes\\\">α</italic>-phase group V elements, we uncover the presence of a topological phase in <italic toggle=\\\"yes\\\">α</italic>-Sb, which can be characterized by a spin Chern number <inline-formula>\\n<tex-math><?CDATA $\\\\mathcal{C}_s$?></tex-math>\\n<mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math>\\n<inline-graphic xlink:href=\\\"tdmad3136ieqn4.gif\\\" xlink:type=\\\"simple\\\"></inline-graphic>\\n</inline-formula> = 2, even though it is <inline-formula>\\n<tex-math><?CDATA $\\\\mathbb{Z}_2$?></tex-math>\\n<mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\\\"double-struck\\\">Z</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math>\\n<inline-graphic xlink:href=\\\"tdmad3136ieqn5.gif\\\" xlink:type=\\\"simple\\\"></inline-graphic>\\n</inline-formula> trivial. Although <italic toggle=\\\"yes\\\">α</italic>-As and Sb would thus be classified as trivial insulators within the classification schemes, we demonstrate the existence of a phase transition between <italic toggle=\\\"yes\\\">α</italic>-As and Sb, which is induced by band inversions at two generic <italic toggle=\\\"yes\\\">k</italic> points. Without spin–orbit coupling (SOC), <italic toggle=\\\"yes\\\">α</italic>-As is a trivial insulator, while <italic toggle=\\\"yes\\\">α</italic>-Sb is a Dirac semimetal with four Dirac points (DPs) located away from the high-symmetry lines. Inclusion of the SOC gaps out the DPs and induces a nontrivial Berry curvature, endowing <italic toggle=\\\"yes\\\">α</italic>-Sb with a high spin Chern number of <inline-formula>\\n<tex-math><?CDATA $\\\\mathcal{C}_s$?></tex-math>\\n<mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math>\\n<inline-graphic xlink:href=\\\"tdmad3136ieqn6.gif\\\" xlink:type=\\\"simple\\\"></inline-graphic>\\n</inline-formula> = 2. We further show that monolayer <italic toggle=\\\"yes\\\">α</italic>-Sb exhibits either a gapless band structure or a gapless spin spectrum on its edges, as expected from topological considerations.\",\"PeriodicalId\":6812,\"journal\":{\"name\":\"2D Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.5000,\"publicationDate\":\"2024-03-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"2D Materials\",\"FirstCategoryId\":\"88\",\"ListUrlMain\":\"https://doi.org/10.1088/2053-1583/ad3136\",\"RegionNum\":3,\"RegionCategory\":\"材料科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATERIALS SCIENCE, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"2D Materials","FirstCategoryId":"88","ListUrlMain":"https://doi.org/10.1088/2053-1583/ad3136","RegionNum":3,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, MULTIDISCIPLINARY","Score":null,"Total":0}
High spin-Chern-number insulator in α-antimonene with a hidden topological phase
For a time-reversal symmetric system, the quantum spin Hall phase is assumed to be the same as the Z2 topological insulator phase in the existing literature. The spin Chern number Cs is presumed to yield the same topological classification as the Z2 invariant. Here, by investigating the electronic structures of monolayer α-phase group V elements, we uncover the presence of a topological phase in α-Sb, which can be characterized by a spin Chern number Cs = 2, even though it is Z2 trivial. Although α-As and Sb would thus be classified as trivial insulators within the classification schemes, we demonstrate the existence of a phase transition between α-As and Sb, which is induced by band inversions at two generic k points. Without spin–orbit coupling (SOC), α-As is a trivial insulator, while α-Sb is a Dirac semimetal with four Dirac points (DPs) located away from the high-symmetry lines. Inclusion of the SOC gaps out the DPs and induces a nontrivial Berry curvature, endowing α-Sb with a high spin Chern number of Cs = 2. We further show that monolayer α-Sb exhibits either a gapless band structure or a gapless spin spectrum on its edges, as expected from topological considerations.
期刊介绍:
2D Materials is a multidisciplinary, electronic-only journal devoted to publishing fundamental and applied research of the highest quality and impact covering all aspects of graphene and related two-dimensional materials.