Lecture 2：半导体基础

What is Semiconductor | 什么是半导体

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

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$(.543nm)^3$，于是可求Si 的数密度为 $5\times10^{22}cm^{-3}$

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 ($E_c$)
• $E_v$: Top edge of the valence band ($E_v$)

$E_c$ 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 $E_G$ ｜ 绝缘体有很高的band gap
• Semiconductors have small $E_G$ ｜ 半导体有很小的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
2. There are 2 types of mobile charge-carriers in Si:
   • 电子：Conduction electrons are negatively charged;
   • 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：
   • Adding impurity atoms (dopants) 掺杂
   • Applying to an electrical field 施加电场
   • Changing the temperature 改变温度
   • Irradiation 辐照

Bond Model of Electrons and Holes

• When an electron breaks loose and becomes a conduction electron, a hole is also created.
• In a pure Si crystal, conduction electronics 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

n = number of electrons/cm3

p = number of holes/cm3

$n_i$ = intrinsic carrier concentration

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

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

Bond theory: Density of state 状态密度

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

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

Doping in Sillicon 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 (cm-3)

$N_A$: ionized acceptor concentration (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 ($N_D$ - $N_A$) !

Thermal Equilibrium ｜ 热平衡

1. Condition 条件：No external forces are applied:
   • Electric field = 0
   • Magnetic(磁) field = 0
   • Mechanical stress(机械应力) = 0
   • Absolutely no light
2. Characteristic 特点
   1. Dynamic situation（动态平衡）in which every process is balanced by its inverse process.
      • 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.
      • 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 Fermi-Dirac distribution function:

$f(E) = \frac{1}{1 + e^{(E - E_F) / kT}}$ 费米狄拉克函数

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

Effect of Temperature on f(E)

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)

$n(E)=g_c(E)f(E)(E≥Ec)$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

$p(E)=g_v(E)[1−f(E)] (E≤Ev)$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$N_c = 2.8 * 10^{19} cm^{-3}$ (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$N_v = 1.04 * 10^{19} cm^{-3}$ (Effective DOS of valence band, 价带有效态密度)

Intrinsic Carrier Concentration（本征载流子浓度）