Full dispersion of the surface modes of a 3D topological insulator with the transfer matrix method
Mustonen, Matias (2022)
Mustonen, Matias
2022
Teknis-luonnontieteellinen DI-ohjelma - Master's Programme in Science and Engineering
Tekniikan ja luonnontieteiden tiedekunta - Faculty of Engineering and Natural Sciences
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Hyväksymispäivämäärä
2022-10-27
Julkaisun pysyvä osoite on
https://urn.fi/URN:NBN:fi:tuni-202210117576
https://urn.fi/URN:NBN:fi:tuni-202210117576
Tiivistelmä
This thesis focuses on topological insulators and their surface states. Topology is a field in mathematics which focuses on the conservation of the relations of points within a geometric object under smooth deformations of the material. It can be used to find nontrivial properties of the electronic bands of an insulating material. At a boundary of a topologically nontrivial material and a trivial one, the collective electronic band gap must go to zero to account for the change in the topological phase. This is the defining property which separates topological insulators from regular ones. Topological insulators have a topological phase in which they have conductive channels on their edges or surfaces, which often have a spin based direction and are required to travel without resistance. This allows for interesting applications ranging from LASERs to supercomputers.
In the first part, the theoretical background of electronic band theory and topological materials are examined. The second part introduces the transfer matrix method, in which the material is divided into layers, and connected to adjacent layers by the transfer matrix. Also, topological models are introduced in a logical order where the latter is derived from the former and their edge states and their properties are solved using the transfer matrix method. In the last part the surface states of the four band 3D BHZ-model are solved using the transfer matrix method as introduced earlier. In this way the versatility and ease of the transfer matrix becomes apparent.
In the first part, the theoretical background of electronic band theory and topological materials are examined. The second part introduces the transfer matrix method, in which the material is divided into layers, and connected to adjacent layers by the transfer matrix. Also, topological models are introduced in a logical order where the latter is derived from the former and their edge states and their properties are solved using the transfer matrix method. In the last part the surface states of the four band 3D BHZ-model are solved using the transfer matrix method as introduced earlier. In this way the versatility and ease of the transfer matrix becomes apparent.