论文标题
从球形到周期性对称性：半导体晶体的轨道角动量的类似物
From spherical to periodic symmetry: the analog of orbital angular momentum for semiconductor crystals
论文作者
论文摘要
角动量形式主义提供了一种对原子状态进行分类的有力方法。然而，即使第一行需要球形对称性，这种形式主义也不能用于周期性系统，即使三次半导体状态通常根据原子符号进行分类。尽管从未注意到，但只能通过使用电子触感来定义轨道角动量的类似物。然后，晶体的自旋轨道交互采用$ \ mathbfcal {\ hat {l}} \ cdot \ hat {\ vs} $形式，带有$ \ mathbfcal {\ hat {\ hat {l}} $降低到$ \ hat {$ \ hat {\ vl} = \ vr} = \ vr \ vp \ vp for for $ \ hat。 This provides the long-missed support for using the eigenvalues of $\mathbfcal{\hat{L}}$ and $\mathbfcal{\hat{J}}=\mathbfcal{\hat{L}}+\hat{\vS}$, as quantum indices to label cubic semiconductor states.重要的是，这些量子指数还控制着将价电子与孔算子与粒子与反粒子相同的相关的相因，尽管孔绝对不是价电子反体反粒子。与更广泛的定义相关联，（$ \ MathBfcal {\ hat {l}}，\ Mathbfcal {\ hat {\ hat {j}} $）$（\ hat {\ vl}，\ hat hat \ hat \ hat \ hat {\ vj}）$ angular momenta by name for for formemang： $ \ mathbfcal {\ hat {l}} $，在真实空间中作用，而对于$ \ mathbfcal {\ hat {j}} $，也对旋转起作用的$ \ mathbfcal {\ hat {j}} $，潜在的对称性被指定为“立方空间动量”。这会将$ \ hat {\ vj} $作为“球形混合动力”，这对这个概念有些尴尬。
The angular momentum formalism provides a powerful way to classify atomic states. Yet, requiring a spherical symmetry from the very first line, this formalism cannot be used for periodic systems, even though cubic semiconductor states are commonly classified according to atomic notations. Although never noted, it is possible to define the analog of the orbital angular momentum, by only using the potential felt by the electrons. The spin-orbit interaction for crystals then takes the $\mathbfcal{\hat{L}}\cdot \hat{\vS}$ form, with $\mathbfcal{\hat{L}}$ reducing to $\hat{\vL}=\vr\times\hat{\vp}$ for spherical symmetry. This provides the long-missed support for using the eigenvalues of $\mathbfcal{\hat{L}}$ and $\mathbfcal{\hat{J}}=\mathbfcal{\hat{L}}+\hat{\vS}$, as quantum indices to label cubic semiconductor states. Importantly, these quantum indices also control the phase factor that relates valence electron to hole operators, in the same way as particle to antiparticle, in spite of the fact that the hole is definitely not the valence-electron antiparticle. Being associated with a broader definition, the ($\mathbfcal{\hat{L}},\mathbfcal{\hat{J}}$) analogs of the $(\hat{\vL},\hat{\vJ})$ angular momenta, must be distinguished by names: we suggest "spatial momentum" for $\mathbfcal{\hat{L}}$ that acts in the real space, and "hybrid momentum" for $\mathbfcal{\hat{J}}$ that also acts on spin, the potential symmetry being specified as "cubic spatial momentum". This would cast $\hat{\vJ}$ as a "spherical hybrid momentum", a bit awkward for the concept is novel.
