Electron Energies in Solids | 固体中的电子能量

📚 Electron Energies in Solids | 固体中的电子能量

In isolated atoms, electrons exist in sharply defined energy levels. When atoms join together to form a crystalline solid, these discrete levels interact and spread into bands of allowed energies separated by forbidden gaps. Understanding how electrons occupy these bands is the key to explaining the electrical behaviour of conductors, insulators and semiconductors.

在孤立的原子中，电子存在于能量分明的分立能级上。当原子结合形成晶体固体时，这些分立的能级会发生相互作用，并扩展成由禁带隔开的允带。理解电子如何占据这些能带，是解释导体、绝缘体和半导体导电行为的关键。

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1. From Atomic Orbitals to Energy Bands | 从原子轨道到能带

In a single atom, the allowed electron energies are represented by sharp lines. When a large number of atoms are brought close together in a crystal lattice, the outer electron orbitals overlap. Because of the Pauli exclusion principle, no two electrons can have the same set of quantum numbers, so each atomic energy level splits into a huge number of closely spaced sub-levels, forming a quasi-continuous energy band.

在单个原子中，允许的电子能量由尖锐的谱线表示。当大量原子在晶格中靠近时，外层的电子轨道发生重叠。由于泡利不相容原理，两个电子不可能具有完全相同的一组量子数，因此每个原子能级分裂成大量间距极小的子能级，形成一个准连续的能量带。

Inner electrons are tightly bound to their nuclei and their energy levels hardly broaden. It is the outermost, or valence, electrons that give rise to the broad bands that determine the solid’s electrical properties.

内层电子被原子核紧紧束缚，其能级几乎不展宽。正是最外层的价电子形成了决定固体电学性质的宽能带。

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2. Valence and Conduction Bands | 价带和导带

At the absolute zero of temperature, the highest energy band that is completely filled with electrons is called the valence band. Above it lies an empty or partially filled band known as the conduction band. Electrons in the conduction band are free to move through the crystal and contribute to electrical conduction.

在绝对零度下，被电子完全占据的最高能量带称为价带。在其上方是一个空的或部分填充的能带，称为导带。导带中的电子可以在晶体中自由运动，从而对导电作出贡献。

The energy separation between the top of the valence band and the bottom of the conduction band is the forbidden energy gap, or simply the band gap, denoted by E_g. The size of this band gap is the primary factor that distinguishes conductors, semiconductors and insulators.

价带顶部与导带底部之间的能量间隔即为禁带宽度，简称带隙，用 E_g 表示。带隙的大小是区分导体、半导体和绝缘体的首要因素。

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3. Band Gap and Conductivity Classification | 带隙与导电性分类

The width of the energy band gap allows us to classify solids into three broad categories. A simple comparison is shown in the table below.

根据能带隙的宽度，可以将固体分为三大类。下表给出了一个简单的比较。

Material Type	Band Gap Eg (eV)	Example
Conductor	0 (overlapping bands)	Copper, Aluminium
Semiconductor	0.2 – 3.0	Silicon (1.1 eV), Germanium (0.7 eV)
Insulator	> 5	Diamond (5.5 eV), Silicon dioxide

For a current to flow, electrons must gain enough energy to cross the band gap and reach the conduction band. The ease with which this happens determines the observed conductivity.

要形成电流，电子必须获得足够能量以跨越带隙并到达导带。这一过程的难易程度决定了人们观察到的电导率。

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4. Conductors: Overlapping Bands | 导体：能带重叠

In a metallic conductor, the valence band and the conduction band overlap, or the conduction band is already partially filled even at 0 K. There is effectively no forbidden gap. A very large number of electrons are always available for conduction, and they can be easily accelerated by an applied electric field.

在金属导体中，价带和导带重叠，或者即使在 0 K 时导带也已被部分填充，实际上不存在禁带。始终有大量电子可用于导电，它们很容易被外加电场加速。

This explains why metals such as copper and silver have very high electrical conductivity. The application of a tiny potential difference instantly sets a large number of free electrons into motion, producing a current.

这就解释了铜和银等金属为何具有极高的电导率。施加一个微小的电势差就能立即使大量自由电子运动起来，形成电流。

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5. Insulators: Large Energy Gap | 绝缘体：大能量间隙

In an insulator, the band gap E_g is very large, typically exceeding 5 eV. At ordinary temperatures, thermal energy k_BT is only around 0.025 eV, far too small to excite a significant number of electrons across the gap. The conduction band therefore remains almost completely empty, and the material cannot conduct electricity.

在绝缘体中，带隙 E_g 非常大，通常超过 5 eV。在常温下，热能 k_BT 仅约为 0.025 eV，远不足以激发大量电子跨越带隙。因此，导带几乎完全空着，材料无法导电。

If an extremely strong electric field or very high temperature is applied, breakdown can occur, but under normal conditions insulators block current flow entirely.

如果施加极强的电场或极高的温度，可能会发生击穿，但在正常条件下，绝缘体完全阻断电流。

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6. Semiconductors: Small Energy Gap | 半导体：小能量间隙

Semiconductors possess a band gap intermediate in size, typically between 0.2 eV and 3.0 eV. Silicon, the most widely used semiconductor, has a band gap of about 1.1 eV. At room temperature, a small but noticeable fraction of valence electrons gain enough thermal energy to jump into the conduction band, leaving behind positively charged holes in the valence band.

半导体的带隙大小介于中间，通常在 0.2 eV 至 3.0 eV 之间。硅是应用最广泛的半导体，其带隙约为 1.1 eV。在室温下，一小部分但不容忽视的价电子获得了足够的热能，跃迁进入导带，同时在价带中留下带正电的空穴。

Both the liberated electrons in the conduction band and the holes in the valence band contribute to electrical conduction. The conductivity of a pure semiconductor rises rapidly with increasing temperature because more electron–hole pairs are generated.

导带中释放出来的电子和价带中的空穴都对导电有贡献。纯半导体的电导率随温度升高而急剧上升，因为产生了更多的电子–空穴对。

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7. The Fermi Level | 费米能级

The Fermi level, E_F, is the highest occupied energy level at absolute zero temperature. In a conductor, the Fermi level lies within an allowed band, so there are many electrons available for conduction even at 0 K. In an intrinsic semiconductor, E_F is located approximately halfway between the valence and conduction bands, inside the forbidden gap.

费米能级 E_F 是绝对零度下被电子占据的最高能级。在导体中，费米能级位于允带之内，因此即使在 0 K 时也有大量电子可用于导电。在本征半导体中，E_F 大致位于价带和导带的中间，处于禁带内部。

At any temperature, the probability that an energy level E is occupied by an electron is given by the Fermi–Dirac distribution. A useful form of this distribution is shown below.

在任何温度下，一个能量为 E 的能级被电子占据的概率由费米–狄拉克分布给出。下面展示了该分布的一种有用形式。

f(E) = 1 / (1 + e^{(E – E_F) / (k_B T)})

When E = E_F, the exponential term becomes e⁰ = 1, so the occupation probability f(E_F e step-like function becomes slightly smeared, allowing a few electrons to be excited above E_F.

当 E = E_F 时，指数项为 e⁰ = 1，因此占据概率 f(E_F) = 0.5。随着温度升高，阶梯状函数会略微变模糊，使得少量电子被激发到 E_F 以上。

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8. Intrinsic Semiconductors and Thermal Excitation | 本征半导体与热激发

A pure semiconductor without any significant impurity is called an intrinsic semiconductor. In such a material, the number of conduction electrons n_i exactly equals the number of holes p_i, because each electron promoted to the conduction band leaves behind one hole.

不含明显杂质的纯半导体称为本征半导体。在这类材料中，导带电子数 n_i 恰等于空穴数 p_i，因为每一个被激发到导带的电子都会留下一个空穴。

The intrinsic carrier concentration depends exponentially on temperature and band gap, following an approximate relationship:

本征载流子浓度随温度和带隙呈指数变化，近似关系为：

n_i ∝ e^{–E_g / (2 k_B T)}

Because the exponential factor contains E_g/(2k_BT), even a small change in temperature can produce a large change in conductivity. This is the physical basis of thermistors.

由于指数因子中含有 E_g/(2k_BT)，温度的微小变化也能引起电导率的显著变化。这就是热敏电阻的物理基础。

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9. Doping: n-type and p-type | 掺杂：N 型和 P 型

The conductivity of a semiconductor can be dramatically increased by the deliberate addition of impurity atoms – a process called doping. If a pentavalent atom such as phosphorus (five valence electrons) is introduced into silicon, four of its electrons form covalent bonds with neighbouring silicon atoms, while the fifth electron becomes almost free. This type of doping produces an n-type semiconductor, where the majority charge carriers are negative electrons.

通过有目的地加入杂质原子——这一过程称为掺杂——可以显著提高半导体的电导率。如果在硅中引入五价原子（如磷），它的四个电子与相邻硅原子形成共价键，而第五个电子几乎成为自由电子。这种掺杂产生 N 型半导体，其中多数载流子是带负电的电子。

In contrast, adding a trivalent impurity such as boron (three valence electrons) leaves one bond short of an electron, creating a hole. The material becomes a p-type semiconductor, with positive holes as the majority carriers.

相反，加入三价杂质（如硼，三个价电子）会使一个键缺少一个电子，从而产生一个空穴。该材料成为 P 型半导体，以带正电的空穴为多数载流子。

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10. Donor and Acceptor Levels | 施主与受主能级

In an n-type semiconductor, the extra electron provided by the donor impurity occupies a discrete energy level, called the donor level E_D, located just below the bottom of the conduction band, typically within 0.05 eV. At room temperature, most of these donor electrons are easily excited into the conduction band, greatly enhancing the electron concentration.

在 N 型半导体中，施主杂质提供的多余电子占据一个分立的能级，称为施主能级 E_D，它刚好位于导带底部下方，通常在 0.05 eV 以内。在室温下，这些施主电子中的大多数很容易被激发到导带，极大提高了电子浓度。

In a p-type semiconductor, the trivalent impurity introduces an acceptor level E_A located just above the top of the valence band. Electrons from the valence band readily jump into this level, leaving mobile holes behind. The acceptor level is the energy destination for these captured electrons.

在 P 型半导体中，三价杂质引入一个受主能级 E_A，位于价带顶部上方很近的位置。价带中的电子很轻易地跃迁到这个能级上，在价带中留下可移动的空穴。受主能级就是这些被俘获电子的能量归宿。

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11. Charge Carriers and Conductivity | 载流子与导电性

oncentration n is much greater than the hole concentration p, so the conductivity can be approximated by σ ≈ e n μ_n, where μ_n is the electron mobility. For a p-type sample, σ ≈ e p μ_p.

半导体的总电导率是电子和空穴贡献的总和。在 N 型样品中，电子浓度 n 远大于空穴浓度 p，因此电导率可近似为 σ ≈ e n μ_n，其中 μ_n 为电子迁移率。对于 P 型样品，σ ≈ e p μ_p。

The mobility describes how quickly a charge carrier drifts in response to an electric field. Although holes have lower mobility than electrons, in a heavily doped p-type semiconductor the large number of holes ensures high conductivity.

迁移率描述了载流子在电场作用下漂移的快慢。尽管空穴的迁移率低于电子，但在重掺杂的 P 型半导体中，大量的空穴仍能保证高电导率。

oduct of electron and hole concentrations remains constant at a given temperature: n p = n_i². This law of mass action is fundamental to understanding how doping shifts the Fermi level towards the conduction band (n-type) or towards the valence band (p-type).

一个重要的事实是，在给定温度下，电子和空穴浓度的乘积保持不变：n p = n_i²。这一质量作用定律对于理解掺杂如何使费米能级靠近导带（N 型）或靠近价带（P 型）至关重要。

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12. Summary of Electron Energies in Solids | 固体中电子能量总结

The energy band model provides a unified framework for understanding why some materials conduct, some insulate, and some can be switched between insulating and conducting states. Conductors have overlapping bands, giving free electrons at any temperature. Insulators have a gap too wide for thermal excitation to overcome. Semiconductors, with their moderate band gaps, allow controlled conduction that can be finely tuned by doping and by changes in temperature.

能带模型为理解为何有些材料导电、有些绝缘、有些可以在绝缘态和导电态之间切换提供了统一的框架。导体具有重叠的能带，在任何温度下都有自由电子。绝缘体的带隙太宽，热激发无法克服。半导体具有适中的带隙，允许可控的电导，并且可通过掺杂和温度变化进行精细调节。

The concept of the Fermi level, together with the Fermi-Dirac distribution, quantitatively explains electron occupancy across the bands. Donor and acceptor levels created by doping introduce energy steps within the forbidden gap, drastically increasing the number of mobile charge carriers and enabling the enormous variety of solid-state electronic devices.

费米能级的概念以及费米–狄拉克分布定量地解释了电子在各能带中的占据情况。掺杂产生的施主和受主能级在禁带中引入了能量台阶，极大地增加了可移动载流子的数量，使种类繁多的固态电子器件成为可能。

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