Degeneracy of Landau Levels in Crystals
Degeneracy of Landau Levels in Crystals
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DOI：
10.1002/pssb.2220650262
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发表时间：
1974-10
期刊：
影响因子：
--
通讯作者：
A. Rauh
中科院分区：
文献类型：
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作者：
A. Rauh
对于恒定磁场中晶体电子对应能级的简并程度，人们做出了不同的预测。在基于Onsager假设(2)的半经典描述(1)中，简并假设是连续的。另一方面，根据磁平移群理论，我们发现有理磁场(3)的简并是有限的，无理磁场(4到6)的简并是无穷的。此外，在有理场的情况下，简并度不规律地依赖于这个场集合(3)。最近，通过考虑能量矩阵，对于一类特殊的无理性场，人们发现在周期势的存在下，能级(朗道能级)是连续简并的。我们将在下面应用与参考文献(6)中类似的方法，并将表明，在某些条件下，对于所有磁场，晶体中的朗道能级是连续简并的。我们的方法本质上在于考虑哈密顿量X=Ac+V(R)在Landau函数的空间@[$i]中的矩阵。这里Z是自由电子在恒定磁场fi中的哈密顿量，V(7)=is，v(G)exp(I*T)是晶格势，为方便起见，用具有倒数晶格矢量G的傅里叶级数来表示。I是由
Different predictions have been made on the degeneracy of the energy levels corresponding to a crystal electron in a constant magnetic field. Within the semiclassical description (1) which is based on the Onsager hypothesis (2) the degeneracy is assumed to be continuous. On the other hand, by the theory of the magnetic translation group one finds a finite degeneracy for rational magnetic fields (3) and a denumerably infinite degeneracy for irrational fields (4 to 6). Moreover, in the case of rational fields, the degeneracy depends erratically on this field set (3). Recently, by considering the energy matrix and for a special class of irrational fields, it has been found that the energy levels (Landau levels) are continuously degenerate in the presence of a periodic potential. We shall apply in the following a similar method as in reference (6) and shall show that, under certain conditions, the Landau levels in a crystal are continuously degenerate for all magnetic fields. Our method consists essentially in considering the matrix of the Hamiltonian X= aC+ V (r) in the space@[$ I of Landau functions. Here Z is the Hamiltonian of a free electron in a constant magnetic field fi, V (7)= Is, v (G) exp (i* T) is the crystal potential which, for convenience, is represented by a Fourier series with reciprocal lattice vectors G. A typical representative of@[? I is given by the