{"title":"Vacancy-induced localized modes and impurity band formation in the Haldane model: A quantum dot analogy","authors":"Hussein Al-Shuwaili , Zahra Noorinejad , Mohsen Amini , Morteza Soltani , Ebrahim Ghanbari-Adivi","doi":"10.1016/j.ssc.2025.115923","DOIUrl":null,"url":null,"abstract":"<div><div>In this study, the Haldane model’s edge states are utilized to illustrate that a zero-energy localized state forms around a single vacancy in the model. In order to complete this task, the conventional unit cell associated to the Haldane hexagonal structure is transferred onto a two-leg ladder in momentum space, effectively forming an extended Su–Schrieffer–Heeger (SSH) lattice through a one-dimensional Fourier transform. Through the application of a suitable unitary transformation, the two-leg SSH ladder in momentum space is converted into an equivalent lattice with two distinct on-site states with different momentum that are suitable for the calculations. Ultimately, the desired zero-energy localized mode formed around the vacant-site is represented by a combination of the armchair edge states. Furthermore, the scenario involving two vacant sites is investigated and it is revealed that an effective hopping interaction exists between the localized states formed around the on-site vacancies created along a zigzag chain in the lattice. This structure can be likened to the structure of a quantum dot with two none-degenerate energy levels. Such a hopping interaction is absent for the same vacancies created on the armchair chains. Finally, it is shown that introducing vacancies periodically on the sites of a zigzag row along a finite-width ribbon with the Haldane structure leads to the emergence of an impurity band within the energy gap.</div></div>","PeriodicalId":430,"journal":{"name":"Solid State Communications","volume":"401 ","pages":"Article 115923"},"PeriodicalIF":2.1000,"publicationDate":"2025-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Solid State Communications","FirstCategoryId":"101","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0038109825000985","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, CONDENSED MATTER","Score":null,"Total":0}
引用次数: 0
Abstract
In this study, the Haldane model’s edge states are utilized to illustrate that a zero-energy localized state forms around a single vacancy in the model. In order to complete this task, the conventional unit cell associated to the Haldane hexagonal structure is transferred onto a two-leg ladder in momentum space, effectively forming an extended Su–Schrieffer–Heeger (SSH) lattice through a one-dimensional Fourier transform. Through the application of a suitable unitary transformation, the two-leg SSH ladder in momentum space is converted into an equivalent lattice with two distinct on-site states with different momentum that are suitable for the calculations. Ultimately, the desired zero-energy localized mode formed around the vacant-site is represented by a combination of the armchair edge states. Furthermore, the scenario involving two vacant sites is investigated and it is revealed that an effective hopping interaction exists between the localized states formed around the on-site vacancies created along a zigzag chain in the lattice. This structure can be likened to the structure of a quantum dot with two none-degenerate energy levels. Such a hopping interaction is absent for the same vacancies created on the armchair chains. Finally, it is shown that introducing vacancies periodically on the sites of a zigzag row along a finite-width ribbon with the Haldane structure leads to the emergence of an impurity band within the energy gap.
期刊介绍:
Solid State Communications is an international medium for the publication of short communications and original research articles on significant developments in condensed matter science, giving scientists immediate access to important, recently completed work. The journal publishes original experimental and theoretical research on the physical and chemical properties of solids and other condensed systems and also on their preparation. The submission of manuscripts reporting research on the basic physics of materials science and devices, as well as of state-of-the-art microstructures and nanostructures, is encouraged.
A coherent quantitative treatment emphasizing new physics is expected rather than a simple accumulation of experimental data. Consistent with these aims, the short communications should be kept concise and short, usually not longer than six printed pages. The number of figures and tables should also be kept to a minimum. Solid State Communications now also welcomes original research articles without length restrictions.
The Fast-Track section of Solid State Communications is the venue for very rapid publication of short communications on significant developments in condensed matter science. The goal is to offer the broad condensed matter community quick and immediate access to publish recently completed papers in research areas that are rapidly evolving and in which there are developments with great potential impact.