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Hofstadter's butterfly and the fractal quantum Hall effect in moire superlattices

机译:莫尔超晶格中的霍夫施塔特的蝴蝶和分形量子霍尔效应

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摘要

Electrons moving through a spatially periodic lattice potential develop a quantized energy spectrum consisting of discrete Bloch bands. In two dimensions, electrons moving through a magnetic field also develop a quantized energy spectrum, consisting of highly degenerate Landau energy levels. When subject to both a magnetic field and a periodic electrostatic potential, two-dimensional systems of electrons exhibit a self-similar recursive energy spectrum1. Known as Hofstadter's butterfly, this complex spectrum results from an interplay between the characteristic lengths associated with the two quantizing fields, and is one of the first quantum fractals discovered in physics. In the decades since its prediction, experimental attempts to study this effect have been limited by difficulties in reconciling the two length scales. Typical atomic lattices (with periodicities of less than one nanometre) require unfeasihly large magnetic fields to reach the commensurability condition, and in artificially engineered structures (with periodicities greater than about 100 nanometres) the corresponding fields are too small to overcome disorder completely. Here we demonstrate that moire superlattices arising in bilayer graphene coupled to hexagonal boron nitride provide a periodic modulation with ideal length scales of the order of ten nanometres, enabling unprecedented experimental access to the fractal spectrum. We confirm that quantum Hall features associated with the fractal gaps are described by two integer topological quantum numbers, and report evidence of their recursive structure. Observation of a Hofstadter spectrum in bilayer graphene means that it is possible to investigate emergent behaviour within a fractal energy landscape in a system with tunable internal degrees of freedom.
机译:移动通过空间周期性晶格电势的电子会形成由离散的Bloch带组成的量化能谱。在二维中,通过磁场移动的电子还形成了量子化的能谱,该谱由高度简并的朗道能级组成。当同时受到磁场和周期性静电势的影响时,二维电子系统会表现出自相似的递归能谱。这种复杂的光谱被称为霍夫施塔特的蝴蝶,是由于与两个量化场相关的特征长度之间的相互作用而产生的,并且是物理学中最早发现的量子分形之一。自从预测以来的几十年中,研究这种效果的实验尝试一直受到协调两个长度范围的困难的限制。典型的原子晶格(周期小于1纳米)需要不容易的大磁场才能达到可比性条件,而在人工工程结构(周期大于约100纳米)中,相应的磁场太小而无法完全克服混乱。在这里,我们证明了在与六角形氮化硼偶联的双层石墨烯中出现的莫尔超晶格提供了具有十纳米量级的理想长度尺度的周期性调制,从而实现了对分形谱的空前实验。我们确认与分形间隙相关的量子霍尔特征由两个整数拓扑量子数描述,并报告了其递归结构的证据。在双层石墨烯中观察Hofstadter光谱意味着,有可能研究内部自由度可调的系统中分形能量景观内的紧急行为。

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  • 来源
    《Nature》 |2013年第7451期|598-602|共5页
  • 作者单位

    Department of Physics, The City College of New York, New York, New York 10031, USA;

    Department of Mechanical Engineering, Columbia University, New York, New York 10027, USA;

    Department of Physics, Columbia University, New York, New York 10027, USA;

    Department of Physics, Columbia University, New York, New York 10027, USA;

    Department of Physics, Columbia University, New York, New York 10027, USA;

    Department of Mechanical Engineering, Columbia University, New York, New York 10027, USA;

    Department of Physics and Nanoscience Technology Center, University of Central Florida, Orlando, Florida 32816-2385, USA;

    Department of Physics and Nanoscience Technology Center, University of Central Florida, Orlando, Florida 32816-2385, USA;

    department of Physics, Tohoku University, Sendai 980-8578, Japan;

    department of Physics, Tohoku University, Sendai 980-8578, Japan;

    National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan;

    National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan;

    Department of Electrical Engineering, Columbia University, New York, New York 10027, USA;

    Department of Mechanical Engineering, Columbia University, New York, New York 10027, USA;

    Department of Physics, Columbia University, New York, New York 10027, USA;

  • 收录信息 美国《科学引文索引》(SCI);美国《工程索引》(EI);美国《生物学医学文摘》(MEDLINE);美国《化学文摘》(CA);
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