Theory of the effective Hamiltonian for degenerate bands in an electric field
Theory of the effective Hamiltonian for degenerate bands in an electric field
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电场中简并能带的有效哈密顿量理论
DOI：
10.1088/0953-8984/12/34/201
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发表时间：
2000
期刊：
影响因子：
--
通讯作者：
Bradley A Foreman
中科院分区：
文献类型：
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作者：
Bradley A Foreman
最近的实验重新引起了人们对均匀外电场作用下半导体简并价态性质的兴趣。作为回应，一些作者提出，标准的Luttinger-Kohn有效质量哈密顿量应该被修改，以包括电场和相关区域中心布洛赫函数的偶极矩阵元素之间的相互作用的能量。本文通过比较提出的偶极相互作用与严格推导的布洛赫和Luttinger-Kohn表示中的场依赖哈密顿量来研究这些建议。它表明，偶极矩阵元的单位细胞是不是一个合适的基础的哈密顿量，因为它取决于选择的单位细胞，这是任意的。此外，正确的Luttinger-Kohn哈密顿量没有波矢无关的矩阵元素，这些矩阵元素在施加的电场中是线性的，除了相对论起源的小项。因此，对Luttinger-Kohn理论提出的修改是无效的。在这里，通过阶数k3、k1和k2的项，沿着k和k2的一阶动量矩阵，导出了Luttinger-Kohn哈密顿量的正确形式。此外，最近发现的场诱导混合在布里渊区边界的Luttinger-Kohn表示进行了详细的研究。
Recent experiments have generated a renewed interest in the properties of the degenerate valence states in semiconductors under the influence of a uniform external electric field. In response, a number of authors have proposed that the standard Luttinger-Kohn effective-mass Hamiltonian should be modified to include the energy of interaction between the electric field and the dipole matrix elements of the relevant zone-centre Bloch functions. This article examines these proposals by comparing the proposed dipole interaction with rigorous derivations of the field-dependent Hamiltonian in the Bloch and Luttinger-Kohn representations. It is shown that the dipole matrix element of a unit cell is not a suitable foundation for a Hamiltonian because it depends on the choice of unit cell, which is arbitrary. Moreover, the correct Luttinger-Kohn Hamiltonian has no wave-vector-independent matrix elements that are linear in the applied field ℰ, except for small terms of relativistic origin. Therefore, the proposed modifications to the Luttinger-Kohn theory are not valid. The correct form of the Luttinger-Kohn Hamiltonian is derived here through terms of order k3, kℰ, and ℰ2, along with the momentum matrix to first order in k and ℰ. In addition, the recent discovery of field-induced mixing at the Brillouin zone boundary in the Luttinger-Kohn representation is studied in detail.