Emergent momentum scale, localization, and van Hove singularities in the graphene twist bilayer
Emergent momentum scale, localization, and van Hove singularities in the graphene twist bilayer
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DOI：
10.1103/physrevb.87.245403
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发表时间：
2013-06-03
影响因子：
3.7
通讯作者：
Pankratov, O.
中科院分区：
文献类型：
--
作者：
Shallcross, S.;Sharma, S.;Pankratov, O.
我们确定了一个角度依赖的动量尺度作为相互旋转石墨烯层组成的双层的基本性质。在这些特征动量值处的层间散射过程定义了一个有效的布里渊区（琼斯区），通常不同于由实空间晶格产生的布里渊区，这在物理上是无关的。由此，我们开发了一种数值方法，该方法在不损失精度的情况下，将标准紧密结合方法的效率提高了大约10(3)倍。该方法的效率基于(i)扭曲哈密顿量在单层石墨烯（SLG）状态的基础上异常稀疏的事实，（ii）琐碎的Diophantine问题（Bezout的身份）的解决方案允许人们提前知道哪些矩阵元素取非零值，以及（iii）在Dirac点附近的几个电子伏特内访问电子结构，截断的SLG基只需要在更大的能量窗口中包含状态。从而大大减小了哈密顿函数的大小。这允许对系统进行完整的调查，揭示(i)与角度相关的van Hove奇点系列，（ii）随着扭转角的减小，SLG状态的混合增加，导致在小角度极限下所有能量的扭转双层波函数的局域化，以及（iii）在近似自相似的小角度区域中，状态密度的零能量峰值。
We identify an angle-dependent momentum scale as the fundamental property of a bilayer composed of mutually rotated graphene layers. The interlayer scattering processes at these characteristic momentum values define an effective Brillouin zone (a Jones zone) which, in general, differs from the Brillouin zone generated by the real-space lattice, which is physically irrelevant. From this we develop a numerical method that increases, for the twist bilayer, the efficiency of the standard tight-binding method by a factor of approximate to 10(3) at no loss of accuracy. The efficiency of the method is based on (i) the fact that the twist Hamiltonian is exceptionally sparse in a basis of single-layer graphene (SLG) states, (ii) a solution of trivial Diophantine problem (Bezout's identity) allows one to know in advance which matrix elements take nonzero values, and (iii) to access the electronic structure in a few electron volts about the Dirac point a truncated SLG basis consisting only of states in a somewhat larger energy window are required, leading to a much reduced size of the Hamiltonian. This allows a complete survey of the system which reveals (i) an angle-dependent series of van Hove singularities, (ii) an increasing mixing of SLG states as the twist angle is reduced leading to the appearance of localization of the twist bilayer wave functions at all energies in the small-angle limit, and (iii) a zero-energy peak in the density of states in an approximately self-similar small-angle regime.