Theoretical Study Of Finite-Size Effects In Quantum Spin Hall Systems

The quantum spin Hall (QSH) effect belongs to a new state of matters, topologi-cally distinct from conventional insulators and semiconductors. They are characterized by the fully gapped bulk states and the non-gapped helical edge states penetrating the bulk gap. Due to the finite penetration depth of the edge states, the wave functions of the edge states in opposite edges overlap each other when the size of the quantum well is small enough. A finite-size gap subsequently opens at Dirac point. Simultaneously, the series of the properties in QSH systems changes evidently, leading to the finite-size effect in QSH systems.First, the helical edge states in QSH systems were specified into two types:the normal and special edge states, according to the decay characteristic parameter λ. The penetration depth of the normal edge state is momentum dependent, and the edge state gap decays monotonously with sample width, leading to the normal finite-size effect. In contrast, the penetration depth maintains a uniform minimal value in the special edge states, and consequently the edge state gap decays non-monotonously with sam-ple width, leading to the anomalous finite-size effect. To demonstrate their difference explicitly, we compared the real materials in phase diagrams. An intuitive way to search for the special edge states in the two-dimensional QSH system is also proposed for future applications. Some misunderstandings in the previous work had also been clarified. We hope that the present studies can provide some instructions for future experiments and applications.Next, the correlation between finite-size effect and localization induced by disor-ders in QSH systems was studied through finite-size scaling analysis. A single energy regime of delocalization showed up in the HgTe quantum well system. This delocal-ization regime originates from the symplectic symmetry and non-trivial Z2topological order, but strongly affected by the size in small systems. The edge state gap caused by the finite-size effect was distinguished from renormalized localization length. It exists only within the delocalization regime with almost unchanged magnitude of the gap, and disappeared where the localization-delocalization transition happened. The delo-calization in QSH systems protects the edge state gap and suppresses the "topological Anderson gap" which originates from the strong localization. Since the disorders are inevitable in real materials, present work provided useful information for future appli-cation.