A-Level Physics: Electron Energies and Energy Bands in Solids | A-Level 物理：固体中的电子能量与能带

📚 A-Level Physics: Electron Energies and Energy Bands in Solids | A-Level 物理：固体中的电子能量与能带

In a solid, the behaviour of electrons determines whether a material is a metal, a semiconductor or an insulator. This article explains how discrete atomic energy levels broaden into continuous energy bands, how the band gap controls electrical conductivity, and how doping creates n-type and p-type semiconductors. The ideas presented here are central to the CIE A-Level Physics topic on solid-state electronics and semiconductor devices.

在固体中，电子的行为决定了材料是金属、半导体还是绝缘体。本文将解释分立原子能级如何扩展为连续的能带，带隙如何控制电导率，以及掺杂如何形成 n 型和 p 型半导体。这里所阐述的概念是 CIE A-Level 物理中固态电子学与半导体器件专题的核心内容。

1. Energy Levels in Isolated Atoms | 孤立原子中的能级

In a single isolated atom, such as an atom in a gas, electrons occupy discrete energy levels. For a hydrogen atom, the allowed energies are E₁, E₂, E₃, and so on. These levels are determined by quantum mechanics and are unique to each element. When an electron moves from one level to another, the atom emits or absorbs a photon of a specific frequency, producing sharp spectral lines.

在单个孤立原子中，例如气体中的一个原子，电子占据分立的能级。以氢原子为例，其允许能量为 E₁、E₂、E₃ 等等。这些能级由量子力学决定，并且每种元素都有自己独特的能级。当电子从一个能级跃迁到另一个能级时，原子会发射或吸收特定频率的光子，从而产生锐利的光谱线。

2. From Discrete Levels to Continuous Bands | 从分立能级到连续能带

When a large number of atoms are brought together to form a solid, the outer valence electrons interact with one another and with the lattice of positive ions. Because of the Pauli exclusion principle, no two electrons in the solid can occupy exactly the same quantum state. Consequently, each original atomic energy level splits into N very closely spaced levels, where N is the number of atoms in the crystal.

当大量原子聚集形成固体时，外层价电子之间以及它们与正离子晶格之间会发生相互作用。由于泡利不相容原理，固体中不能有两个电子占据完全相同的量子态。因此，原有的每一个原子能级都会分裂成 N 个间距极小的能级，其中 N 是晶体中的原子数。

A typical crystal contains about 10²³ atoms, so these split levels are so close together that they effectively form a continuous band. The lower-energy core electrons remain close to their own nuclei and do not take part in the formation of bands; it is mainly the valence electrons whose energy levels broaden significantly.

典型晶体约含 10²³ 个原子，因此这些分裂能级彼此靠得极近，实际上形成连续的能带。能量较低的内层电子仍紧靠自身原子核，不参与能带的形成；主要是价电子的能级发生显著展宽。

3. Allowed Bands and Forbidden Gap | 允许能带与禁带

The allowed energies in a solid therefore lie inside energy bands separated by forbidden gaps. At absolute zero, the highest band that is completely filled with electrons is called the valence band. The next allowed band above it, which is normally empty, is called the conduction band. The energy difference between the top of the valence band E_v and the bottom of the conduction band E_c is called the band gap E_g.

因此，固体中的允许能量位于能带内，能带之间由禁带隔开。在绝对零度时，完全被电子填满的最高能带称为价带。其上方通常为空的下一个允许能带称为导带。价带顶部 E_v 与导带底部 E_c 之间的能量差称为带隙 E_g。

E_g = E_c − E_v

Electrons in the valence band are still localised in covalent or metallic bonds; only electrons in the conduction band are free to move through the crystal when an electric field is applied.

价带中的电子仍局域在共价键或金属键中；只有导带中的电子才能在施加电场时自由地穿过晶体移动。

4. Metals: Partially Filled Bands | 金属：部分填充的能带

In metals, the conduction band is either partially filled with electrons, or it overlaps with the valence band. This means that there are accessible empty energy states immediately above the occupied states. There is no large band gap that electrons must jump across.

在金属中，导带要么被电子部分填充，要么与价带重叠。这意味着在已占能级上方不远处就存在可以进入的空态，不存在电子必须跨越的大带隙。

Because empty states are so close in energy, a small electric field can accelerate conduction electrons into nearby empty states, producing a large electric current. This is why metals have very high electrical conductivity. Typical examples include copper, aluminium and silver.

由于空态在能量上非常接近，很小的电场就能将传导电子加速到邻近的空态，从而产生很大的电流。这就是金属具有很高电导率的原因。典型例子包括铜、铝和银。

5. Insulators: Large Band Gap | 绝缘体：大带隙

In an insulator, the valence band is completely full and the conduction band is empty. The band gap is large, typically greater than about 3 eV. At ordinary temperatures, the thermal energy available to an electron is far too small to excite it across this gap. As a result, almost no electrons reach the conduction band and the material conducts electricity extremely poorly.

在绝缘体中，价带完全填满，导带为空。带隙通常很大，大于约 3 eV。在常温下，电子能够获得的平均热能远不足以使其越过这个带隙。因此，几乎没有电子能到达导带，材料的导电性能极差。

Diamond is a good example of an insulator, with a band gap of about 5.5 eV. Even under a strong electric field, very few electrons can become mobile, so the current remains extremely small.

金刚石是绝缘体的好例子，其带隙约为 5.5 eV。即使施加很强的电场，能成为可移动载流子的电子也极少，因此电流极小。

6. Semiconductors: Small Band Gap | 半导体：小带隙

Semiconductors such as silicon and germanium have band gaps that are much smaller than those of insulators. For silicon, E_g ≈ 1.1 eV; for germanium, E_g ≈ 0.7 eV. At room temperature, a small number of valence-band electrons receive enough thermal energy to jump into the conduction band.

硅和锗等半导体的带隙比绝缘体小得多。硅的 E_g ≈ 1.1 eV，锗的 E_g ≈ 0.7 eV。在室温下，少量价带电子能获得足够的热能跃迁到导带。

Every electron that jumps into the conduction band leaves behind a missing electron in the valence band, called a hole. A hole behaves as a positively charged carrier. Because both electrons and holes can carry charge, the conductivity of

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