The study and investigation of various material properties, as well as the invariance of these properties, have always been primary ob- jectives in the field of condensed matter physics. The Ginzburg-Landau theory of phase transitions, which is rooted in symmetry breaking, known as as one of the most successful mod- els available. However, the discovery of the quantum Hall effect in 1980 pro‎mp‎ted a reeva‎luation of physicists’ perspectives. This dis- covery led to the recognition of a unique form of invariances known as topological invariance, which, in turn, unveiled a class of mate- rials known as topological materials. The first prediction of three- dimensional (3D) topological materials emerged in 2005 and was experimentally confirmed just a year later. To elucidate this class of materials, numerous theoretical models were proposed. Notably that, the Bernevig, Hughes and Zhang (BHZ) model, grounded in the Kane-Melle theory, successfully predicted the quantum spin Hall effect in quantum wells composed of HgTe-CdTe, offering a practical model. Despite the quantum spin Hall effect’s advantages over the quantum Hall effect, which relies on spin-orbit interactions, mercury telluride quantum wells remain a subject of scientificaly fascination. This is primarily due to the continued application of the BHZ model to multiple wells, often considering additional bands. Nevertheless, the theoretical foundation of these studies still leans on Kane’s multi- band perturbation model. As a result, within a device featuring an HgTe quantum well sand- wiched between CdTe barriers, an inversion in the energy bands occurs, allowing for tunability in the overall band structure of the system. The main motivation behind our current research lies in the poten- tial applications of this work and the further development of Kane’s perturbation theory. We have explained the energy bands of a CdTe-HgTe-CdTe quantum well and compared it to a CdTe-HgTe-CdTe-HgTe-CdTe double quantum well. Our investigation delved into the role of orbitals in energy band inversion, then we estimated the topological phase transitions independently, for the single quantum well. Finally, to investigate the role of combined orbitals in the energy band inversion, we examine the 8-band Kane model for the HgTe quantum well and calculate the possible states in the band structure of a single well

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بهروز ميرزا , جواد هاشمي فر