量子自旋霍尔体系中有限尺寸效应的理论研究
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摘要
量子自旋霍尔效应是一种新型的物质状态,它从拓扑意义上区别于传统绝缘体和半导体。其主要特征是：系统体能带处于绝缘态,但体能隙中存在无能隙的螺旋性边界态。这些螺旋性边界态具有有限的穿透深度,因而当系统尺寸足够小时,不同边界上的边界态波函数相互交叠,导致边界态色散关系在狄拉克点打开一个能隙,造成量子自旋霍尔系统中的有限尺寸效应。
本文首先按照衰减参数λ的不同组合,将螺旋性边界态分成两类：正常边界态和特殊边界态。正常边界态的穿透深度具有动量依赖性,会导致正常有限尺寸效应,其边界态能隙随系统宽度的增大而单调衰减。特殊边界态的穿透深度在动量空间保持一个恒定的最小值,会导致反常有限尺寸效应,边界态能隙随宽度变化表现出非单调振荡行为。我们将多种真实材料在参数空间进行了比较,预测了相应材料所产生的有限尺寸效应的类型,并提出了一个直观的判据来判断特殊边界态的存在性。该工作澄清了之前理论上的一些误解,并为将来的实验工作提供了理论依据。
本文还采用有限尺寸标度分析的方法,讨论了有限尺寸效应与无序导致的局域化行为的关联。HgTe量子阱体系受无序影响存在一个退局域化能量区间。该退局域化区间起源于体系对称性及非平庸Z2拓扑序,并且在小尺寸系统中表现出强烈的尺寸依赖性。有限尺寸效应所导致的边界态能隙可以通过重整局域化长度来确定。边界态能隙只存在于退局域化区间之内,随无序增强而发生位移但大小基本保持不变。随着无序强度的增加,系统发生局域-退局域转变的同时,边界态能隙表现出绝缘体-金属相变的行为。退局域化区间对边界态能隙具有保护作用,并抑制了强局域化造成的“拓扑安德森能隙”的出现。由于在真实材料中无序影响不可避免,因此本文针对中低强度无序情况的讨论为实验和应用研究提供了有益的参考。
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.
本文首先按照衰减参数λ的不同组合,将螺旋性边界态分成两类：正常边界态和特殊边界态。正常边界态的穿透深度具有动量依赖性,会导致正常有限尺寸效应,其边界态能隙随系统宽度的增大而单调衰减。特殊边界态的穿透深度在动量空间保持一个恒定的最小值,会导致反常有限尺寸效应,边界态能隙随宽度变化表现出非单调振荡行为。我们将多种真实材料在参数空间进行了比较,预测了相应材料所产生的有限尺寸效应的类型,并提出了一个直观的判据来判断特殊边界态的存在性。该工作澄清了之前理论上的一些误解,并为将来的实验工作提供了理论依据。
本文还采用有限尺寸标度分析的方法,讨论了有限尺寸效应与无序导致的局域化行为的关联。HgTe量子阱体系受无序影响存在一个退局域化能量区间。该退局域化区间起源于体系对称性及非平庸Z2拓扑序,并且在小尺寸系统中表现出强烈的尺寸依赖性。有限尺寸效应所导致的边界态能隙可以通过重整局域化长度来确定。边界态能隙只存在于退局域化区间之内,随无序增强而发生位移但大小基本保持不变。随着无序强度的增加,系统发生局域-退局域转变的同时,边界态能隙表现出绝缘体-金属相变的行为。退局域化区间对边界态能隙具有保护作用,并抑制了强局域化造成的“拓扑安德森能隙”的出现。由于在真实材料中无序影响不可避免,因此本文针对中低强度无序情况的讨论为实验和应用研究提供了有益的参考。
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.
引文
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