Motion and Recombination of Electrons and Holes

\begin{align*} \newcommand{\dif}{\mathop{}\!\mathrm{d}}\\ \newcommand{\p}{\partial}\\ \newcommand{\bd}{\boldsymbol}\\ \end{align*}

2.1 Thermal Motion

$\frac{\int f(E)D(E)(E-E_c) \dif E}{\int f(E)D(E)\dif E} = \frac{3}{2} kT$

$v_\text{th} = \sqrt{\frac{3kT}{m}}\\ m 取 m_n 或 m_p$

2.2 Drift

Drift （漂移运动）指载流子在电场下的运动

2.2.1 Electron and Hole Mobilities

$m_p v = q\mathscr{E} \tau_\text{mp}$

$\begin{cases} v_p =\mu_p \mathscr{E}\\ \mu_p = \dfrac{q \tau_\text{mp}}{m_p} \end{cases} \tag{2.2.3a}$ $\begin{cases} v_n = - \mu_n \mathscr{E}\\ \mu_n = \dfrac{q \tau_\text{mn}}{m_n} \end{cases} \tag{2.2.3b}$

Si Ge GaAs InAs
$\mu_n$ 1400 3900 8500 30,000
$\mu_p$ 470 1900 400 500

2.2.2 Mechanisms of Carrier Scattering

1. phonon scattering 声子散射

Phonons are the particle representation of the vibration of the atoms in the crystal

2. ionized impurity scattering 电离杂质散射

$\frac{1}{\tau} = \frac{1}{\tau_\text{phonon}} + \frac{1}{\tau_\text{impurity}}\\ \frac{1}{\mu} = \frac{1}{\mu_\text{phonon}} + \frac{1}{\mu_\text{impurity}}$

$\tau_\text{phonon}$ 和 $\tau_\text{impurity}$ 与温度的关系如下：

$\mu_\text{phonon} \propto \tau_\text{phonon} \propto T^{-3/2}\\ \mu_\text{impurity} \propto \tau_\text{impurity} \propto \frac{T^{3/2}}{N_a+N_d}$

2.2.3 Drift Current and Conductivity

$J_{p,\text{drift}} = qpv = qp \mu_p \mathscr{E} \tag{2.2.11}$ $J_{n,\text{drift}} = qnv = qn \mu_n \mathscr{E} \tag{2.2.12}$

$J_\text{drift} = J_{p,\text{drift}} + J_{n,\text{drift}}= (qp \mu_p + qn \mu_n )\mathscr{E}$

2.3 Diffusion Current

Diffusion is the result of particles undergoing thermal motion，一般来说热运动总是有使各处浓度一致的倾向，当浓度不一致时，热运动就会导致高浓度向低浓度扩散。因此，扩散电流与浓度梯度有关：

$J_{n,\text{diffusion}} = q D_n \frac{\dif n}{\dif x} \tag{2.3.2}\\$ $J_{p,\text{diffusion}} =- q D_p \frac{\dif p}{\dif x} \tag{2.3.3}$

$D_n,D_p$ 称为 diffusion constant. 比较令人困惑的就是上两式中，空穴反而需要加负号？！这是因为梯度本身有一个负号，所以空穴反而要加负号才是正电荷电流。

$J_n = J_{n,\text{drift}} + J_{n,\text{diffusion}} = qn \mu_n \mathscr{E} + q D_n \frac{\dif n}{\dif x}\\ J_p = J_{p,\text{drift}} + J_{p,\text{diffusion}} = qp \mu_p \mathscr{E} - q D_p \frac{\dif p}{\dif x}\\ J=J_n+J_p$

2.4 Relation between the Energy Diagram and V,E

The electrons roll downhill like stones in the energy band diagram and the holes float up like bubbles.

$E_c(x)=\text{constant} - qV(x)$

constant 的值由参考电位来决定，而由于 $E_c$ 的单位是 eV，所以 $V(x)$ 要乘上 $q$

$\mathscr{E}(x)=-\frac{\dif V}{\dif x}=\frac{1}{q} \frac{\dif E_c}{\dif x}=\frac{1}{q} \frac{\dif E_v}{\dif x}$

2.5 Einstein Relationship between D and μ

$J_n = qn\mu_n \mathscr{E} +q D_n \frac{\dif n}{\dif x} = 0$

$0 = q n \mu_n \mathscr{E} - qn\frac{qD_n}{kT} \mathscr{E}\\ \Rightarrow D_n = \frac{kT}{q} \mu_n \tag{2.5.6a}$

$D_p = \frac{kT}{q}\mu_p \tag{2.5.6b}$

2.6 Electron-Hole Recombination

$n'\equiv p' \tag{2.6.2}\\ n \equiv n_0 + n'\\ p \equiv p_0 + p'$

$\frac{\dif n'}{\dif t} = -\frac{n'}{\tau}=-\frac{p'}{\tau}$

recombination rate 结合律 为 $\dfrac{n’}{\tau}=\dfrac{p’}{\tau}$（单位：$/(\text{cm}^3\cdot s)$ ）

1. 直接复合 direct recombination, or radiative recombination：导带电子与价带空穴直接结合
• direct gap semiconductors: efficient. used for light emission.
• indirect gap semiconductors: inefficient. the electrons and holes at the edges of the band gap do not have the same wave vectors
2. 间接复合：导带电子与价带电子在 recombination centers 处复合
3. 俄歇复合？？？

2.7 Thermal Generation

The reverse process of recombination is called thermal generation 热激发，显然，过剩载流子产生的速度等于 rate of generation minus rate of recombination

2.8 Quasi-Equilibrium and Quasi-Fermin Levels

$n = N_c e^{-(E_c-E_{F_n})/kT} \tag{2.8.1}\\$ $p = N_v e^{-(E_{F_p}-E_v)/kT} \tag{2.8.2}$