量子自旋霍爾效應(quantum spin Hall effect)是在反轉能帶(inverted band)的材料下,產生的成對且行進方向與自旋方向皆相反的對掌性邊界電流(chiral edge states),即螺旋邊界態(helix edge states)。理論上螺旋邊界態在量子自旋霍爾絕緣體中是被時間反演對稱所保護,而磁場會破壞時間反演對稱,但即使在有磁場的情況下,螺旋邊界態還是有可能存在,此現象被稱為偽量子自旋霍爾效應(pseudo quantum spin Hall effect)。此論文探討在不同磁場條件下造成半導體量子井的拓墣相變,在不同的費米能階下,垂直磁場越大,會發生從量子自旋霍爾相位轉變至絕緣體或是先轉變到量子霍爾相位再轉變到絕緣體的變化,而高水平磁場會使偽量子自旋霍爾態轉變為金屬態,但是目前的研究都是只有單獨調控垂直磁場或水平磁場的研究。在此論文中,討論在混合的磁場中對於反轉半導體量子井造成的相變,另外也有發現只有在混合磁場中才會產生的返回量子自旋霍爾效應(reentrant quantum spin Hall effect),就是指從量子自旋霍爾相位轉變至絕緣體後再轉變為量子自旋霍爾相位的現象,造成此現象發生的原因是因為藍道能階(Landau levels)在垂直磁場下並非單調而是有個彎曲的曲線以及在水平磁場下造成藍道能階的混合所共同引響造成的。最後由計算邊緣態的機率分布以及陳數值(Chern numbers)證明返回量子自旋霍爾是拓樸的,並為此態的穩定提供了一些證明。
Quantum spin Hall effect is an effect which has two counter conduction currents and spin order chiral edge states in inverted band materials, in other word, helix edge state. The helical edge states are presumably protected by time reversal symmetry in a quantum spin Hall insulator. However, even in the presence of magnetic field which breaks time-reversal symmetry, the helical edge conduction can still exist, dubbed as pseudo quantum spin Hall effect. The effects of the magnetic fields on the pseudo quantum spin Hall effect and the phase transitions are studied, when out-of-plane magnetic field becomes bigger, it makes quantum spin Hall phase change into normal insulator, or change into quantum Hall phase first and then change into normal insulator. We also illustrate that an in-plane magnetic field drives a pseudo quantum spin Hall state to metallic state at a high field, but never study what happen in tilted magnetic field. In this master thesis, at a fixed in-plane magnetic field, an increasing out-of-plane magnetic field leads to a reentrance of pseudo quantum spin Hall state in an inverted InAs/GaSb quantum well. The origin of the reentrant behavior is attributed to the nonmonotonic bending of Landau levels and the Landau level mixing caused by the orbital effect induced by the in-plane magnetic field. The edge state probability distribution and Chern numbers are calculated to verify that the reentrant states are topologically nontrivial and provide an evidence to the robustness of the reentrant quantum spin Hall effect.
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