Lecture 2：半导体基础

What is Semiconductor ｜ 什么是半导体

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1. Conductivity lies between that of conductors (导体) and insulators (绝缘体).
2. Generally crystalline in structure for IC devices ｜ 在集成电路器件中通常呈晶体结构。

• In recent years, however, non-crystalline semiconductors (非晶半导体) have become commercially important (具有商业价值).
• Graphene ｜ 石墨烯

Typical Semiconductor ｜ 典型半导体

1. Silicon: diamond cubic structure ｜ 金刚石立方结构
2. GaAs: ZnS (Zinc Blende) structure ｜ 闪锌矿结构

How Many Silicon Atoms per cm³?

等效硅原子在一个立方晶格中是 8 个，每个立方晶格占地 $(0.543\,\text{nm})^3$，于是可求 Si 的数密度为 $5\times10^{22}\,\text{cm}^{-3}$。

Compound (or elemental) Semiconductors ｜ 化合（或元素）半导体

• A compound semiconductor is a semiconducting material formed from two or more different chemical elements, like GaN.
• This is different from an elemental semiconductor, like silicon (Si), which is made from only one element.

Energy Band Theory ｜ 能带理论

根据量子力学，在晶体中，随着相互作用原子数量 $N$ 的增加，会形成连续的能带。

Result from quantum mechanics: formation of continuous energy bands in a crystal with increasing number of interacting atoms.

Conduction (or valence) band ｜ 导（价）带

• 价带：The highest nearly-filled band is the valence band.
• 导带：The lowest nearly-empty band is the conduction band.

Energy Band Diagram ｜ 能带图

• $E_c$: Bottom edge of the conduction band
• $E_v$: Top edge of the valence band

$E_c$ and $E_v$ are separated by the band gap energy $E_G$.

Band Gap and Material Classification ｜ 带隙与材料分类

Filled bands and empty bands do not allow current flow.

• Insulators have large $E_G$ ｜ 绝缘体有很大的 band gap
• Semiconductors have small $E_G$ ｜ 半导体有较小的 band gap
• Metals have no band gap. Conduction band is partially filled ｜ 金属没有 band gap，导带是被部分填满的

How to Measure Bandgap Energy ｜ 带隙测量

$E_G$ can be determined from the minimum energy of photons (光子，注意区分声子 phonon) that are absorbed by the semiconductor.

Electronic Properties of Si ｜ 硅的电特性

1. Silicon is a semiconductor material. Pure Si has high electrical resistivity at room temperature. ｜ 纯净 Si 在室温有很高的电阻率
2. There are 2 types of mobile charge-carriers in Si:
   1. 电子：Conduction electrons are negatively charged;
   2. 空穴：Holes are positively charged.
3. The concentration of conduction electrons & holes in a semiconductor can be modulated (调整) in several ways ｜ 四种因素会影响载流子浓度：
   1. Adding impurity atoms (dopants) ｜ 掺杂
   2. Applying an electric field ｜ 施加电场
   3. Changing the temperature ｜ 改变温度
   4. Irradiation ｜ 辐照

Bond Model of Electrons and Holes ｜ 电子与空穴的键模型

• When an electron breaks loose and becomes a conduction electron, a hole is also created.
• In a pure Si crystal, conduction electrons and holes are formed in pairs.

Hole ｜ 空穴

Mobile positive charge associated with a half-filled covalent bond. （与半填充共价键相关的可移动正电荷）

• Treat as positively charged mobile particle (粒子) in the semiconductor.

Definition of Terms ｜ 术语定义

$n$ = number of electrons per $\text{cm}^3$

$p$ = number of holes per $\text{cm}^3$

$n_i$ = intrinsic carrier concentration

In a pure semiconductor, $n = p = n_i$

In a doped semiconductor, $np = n_i^2$

Band Theory: Density of States ｜ 态密度

$g(E)\,dE$ = number of states per $\text{cm}^3$ in the energy range between $E$ and $E+dE$

$m^*$ (effective mass) depends on material (材料) and crystallographic orientation (晶向).

Doping in Silicon ｜ 硅掺杂

By substituting (替代) a Si atom with a special impurity atom (Group V (Donors) or Group III (Acceptors) element), a conduction electron or hole is created.

Donors: P (Phosphorus), As (Arsenic), Sb (Antimony)

Acceptors: B (Boron), Al (Aluminum), Ga (Gallium), In (Indium)

Donor / Acceptor Levels ｜ 施主/受主能级

Charge-Carrier Concentrations ｜ 载流子浓度

$N_D$: ionized donor concentration ($\text{cm}^{-3}$)

$N_A$: ionized acceptor concentration ($\text{cm}^{-3}$)

Charge neutrality condition: $N_D + p = N_A + n$

At thermal equilibrium: $np = n_i^2$ (Law of Mass Action)

Note: Carrier concentrations depend on net dopant concentration ($N_D - N_A$)!

Summary ｜ 小结

donor: impurity atom that increases $n$, for example Group-V elements

acceptor: impurity atom that increases $p$, for example Group-III elements

n-type material: contains more electrons than holes

p-type material: contains more holes than electrons

majority carrier: the most abundant carrier

minority carrier: the least abundant carrier

intrinsic semiconductor: $n = p = n_i$

extrinsic semiconductor: doped semiconductor, $n \ne p \ne n_i$

Dopant concentrations typically range from $10^{14}\,\text{cm}^{-3}$ to $10^{19}\,\text{cm}^{-3}$

Thermal Equilibrium ｜ 热平衡

1. Condition ｜ 条件：
   1. No external forces are applied:
   2. Electric field = 0
   3. Magnetic (磁) field = 0
   4. Mechanical stress (机械应力) = 0
   5. Absolutely no light
2. Characteristic ｜ 特点
   1. Dynamic situation (动态平衡) in which every process is balanced by its inverse process.
      1. Electron-hole pair (EHP) generation rate = EHP recombination rate
   2. Thermal agitation (热运动): electrons and holes exchange energy with the crystal lattice and each other.
      1. Every energy state in the conduction band and valence band has a certain probability of being occupied by an electron.

动态平衡状态，每个过程都与其逆过程相互平衡。

电子-空穴对（EHP）的生成速率等于其复合速率。

热运动：电子与空穴在晶体晶格中相互交换能量。

导带和价带中的每个能级都有被电子占据的概率。

Fermi-Dirac Statistics ｜ 费米狄拉克统计

Probability that an available state at energy $E$ is occupied is given by the Fermi-Dirac distribution function:

There is only one Fermi level in a system at equilibrium.

Effect of Temperature on $f(E)$ ｜ 温度的影响

Boltzmann Approximation ｜ 玻尔兹曼近似

Probability that a state is filled (occupied by an electron):

Probability that a state is empty (occupied by a hole):

Remember: there is only one Fermi level in a system at equilibrium.

Distribution of Carriers ｜ 载流子分布

Obtain $n(E)$ by multiplying $g_c(E)$ and $f(E)$.

Obtain $p(E)$ by multiplying $g_v(E)$ and $1-f(E)$.

$n(E) = g_c(E)\,f(E)$，$E \ge E_c$。靠近 $E_c$ 处既有可用态 $g_c$，又有一定占据几率 $f$，因此电子主要分布在导带底附近。

$p(E) = g_v(E)\,[1-f(E)]$，$E \le E_v$。靠近 $E_v$ 处 $1-f$ 较大，因而空穴主要分布在价带顶附近。

N-type Semiconductor Distribution ｜ N 型载流子分布

Equilibrium Electron Concentration ｜ 平衡电子浓度

其中，Si 的 $N_c = 2.8 \times 10^{19}\,\text{cm}^{-3}$ (Effective DOS of conduction band, 导带有效态密度)

P-type Semiconductor Distribution ｜ P 型载流子分布

Equilibrium Hole Concentration ｜ 平衡空穴浓度

其中，Si 的 $N_v = 1.04 \times 10^{19}\,\text{cm}^{-3}$ (Effective DOS of valence band, 价带有效态密度)

Intrinsic Carrier Concentration ｜ 本征载流子浓度