能带理论Electronic band structure

从化学角度看，N 个原子的同一种原子轨道线性组合成 N 个分子轨道，N 极大时这些轨道连成能带（化学家常称“紧束缚”或分子轨道图像）。从物理角度看，电子在周期势场中运动，其波函数是布洛赫波，能量随波矢 k 变化形成 E(k) 色散关系。被电子填满的最高能带称为价带，其上的空带称为导带。金属的价带部分填充或与导带重叠；绝缘体禁带宽（>4 eV）；半导体禁带窄。

Chemically, the same orbital on N atoms combines into N molecular orbitals, which for very large N merge into a band (the tight-binding or MO picture). Physically, electrons in a periodic potential have Bloch-wave functions whose energy varies with wavevector k, giving the dispersion relation E(k). The highest filled band is the valence band and the empty band above it the conduction band. Metals have a partly filled valence band or overlapping bands; insulators have wide gaps (over 4 eV); semiconductors have narrow ones.

1928 年布洛赫在海森堡指导下证明晶体中电子的波函数为布洛赫波；1930 年佩尔斯引入能隙概念，克罗尼格-彭尼模型给出了一维能带的简单解。1931 年威尔逊用能带填充解释了金属、绝缘体和半导体的区别。此后布里渊区、费米面等概念相继建立。1960 年代起密度泛函理论使能带结构的第一性原理计算成为可能。

In 1928 Bloch, under Heisenberg, showed that electron wavefunctions in crystals are Bloch waves; in 1930 Peierls introduced energy gaps, and the Kronig–Penney model gave a simple one-dimensional solution. In 1931 Wilson explained metals, insulators and semiconductors by band filling, and concepts such as Brillouin zones and Fermi surfaces followed. From the 1960s density functional theory made first-principles band-structure calculation possible.

能带理论指导了半导体器件设计（p-n 结、带隙工程）、光伏材料选择（理想禁带约 1.1–1.5 eV）、光催化剂设计（TiO₂ 禁带 3.2 eV）和发光材料的颜色调控。拓扑绝缘体、石墨烯中的狄拉克锥（2010 年诺贝尔物理学奖）等新奇电子态也是能带理论的延伸。

Band theory guides semiconductor device design (p–n junctions, band-gap engineering), choice of photovoltaic materials (ideal gaps of about 1.1–1.5 eV), photocatalyst design (TiO₂ has a 3.2 eV gap) and tuning the colour of light emitters. Exotic electronic states such as topological insulators and the Dirac cones of graphene (Nobel Prize in Physics 2010) extend band theory further.