Bloch 定理#

晶体中电子波函数：

$\psi(x) = u(x) \cdot e^{ikx}$

其中 $u(x)$ 为单胞波函数，与势函数 $U(x)$ 具有相同周期性：$u(x + d) = u(x)$

晶胞原子数#

Miller 指数晶面间距#

$d_{hkl} = a / \sqrt{h^2 + k^2 + l^2}$

电导率#

$\sigma = n e \mu_n + p e \mu_p$

漂移电流密度#

$J_{\text{drift}} = \sigma E = (n e \mu_n + p e \mu_p) E$

扩散电流密度#

$J_n = e D_n \frac{dn}{dx}$（电子扩散电流）

$J_p = -e D_p \frac{dp}{dx}$（空穴扩散电流）

Einstein 关系#

$\frac{D}{\mu} = \frac{kT}{e}$

300 K 下：$D/\mu = 0.0259$ V

漂移速度#

$v_d = \mu E$

本征载流子浓度#

$n_i^2 = N_c N_v \exp\left(-\frac{E_g}{kT}\right)$

本征 Fermi 能级#

$E_i = \frac{E_c + E_v}{2} + \frac{kT}{2} \ln\left(\frac{N_v}{N_c}\right)$

质量作用定律#

$n p = n_i^2$（热平衡下）

电中性条件#

$n + N_A^- = p + N_D^+$

N 型（$N_D \gg N_A$）：$n \approx N_D$

P 型（$N_A \gg N_D$）：$p \approx N_A$

Fermi 能级位置#

N 型：$E_F - E_i = kT \ln(N_D / n_i)$

P 型：$E_i - E_F = kT \ln(N_A / n_i)$

接触势（内建电势）#

$V_{bi} = \frac{kT}{e} \ln\left(\frac{N_A N_D}{n_i^2}\right)$

$V_{bi} = \frac{1}{e}(E_{Fn} - E_{Fp})$

耗尽层宽度#

$W = \sqrt{\frac{2\varepsilon_s}{e}\left(\frac{1}{N_A} + \frac{1}{N_D}\right)(V_{bi} - V)}$

单边突变结（$N_A \gg N_D$）：$W \approx \sqrt{\frac{2\varepsilon_s}{e N_D}(V_{bi} - V)}$，$W \approx x_n$

电场分布#

P 侧：$E(x) = -\frac{qN_A}{\varepsilon_s}(x + x_p)$

N 侧：$E(x) = -\frac{qN_D}{\varepsilon_s}(x_n - x)$

最大电场#

$|E_{\max}| = \sqrt{\frac{2e N_A N_D}{\varepsilon_s(N_A + N_D)}(V_{bi} - V)}$

耗尽层电容（势垒电容）#

$C_j = A \sqrt{\frac{e \varepsilon_s N_A N_D}{2(N_A + N_D)(V_{bi} - V)}}$

扩散电容#

$C_{\text{diff}} \propto I_F \tau$

I-V 特性#

$J = J_s (e^{eV/kT} - 1)$

$J_s = e\left(\frac{D_p n_i^2}{L_p N_D} + \frac{D_n n_i^2}{L_n N_A}\right)$

击穿电压#

$V_{BR} \propto E_g^{3/2} N_D^{-3/4}$

阈值电压#

$V_t = V_{FB} + 2\phi_F + \frac{Q_B}{C_{ox}}$

$\phi_F = \frac{kT}{e} \ln\left(\frac{N_A}{n_i}\right)$，$C_{ox} = \frac{\varepsilon_{ox}}{t_{ox}}$

DC 工作区#

沟道长度调制#

$I_D = \frac{1}{2}\mu_n C_{ox} \frac{W}{L}(V_{GS} - V_t)^2(1 + \lambda V_{DS})$

跨导#

$g_m = \frac{\partial I_D}{\partial V_{GS}}$

饱和区：$g_m = \mu_n C_{ox} \frac{W}{L}(V_{GS} - V_t)$

衬底偏置效应#

$V_t = V_{t0} + \gamma(\sqrt{2\phi_F + V_{SB}} - \sqrt{2\phi_F})$

$\gamma = \frac{\sqrt{2e \varepsilon_s N_A}}{C_{ox}}$

CMOS 对称条件#

$\mu_n (W/L)_N = \mu_p (W/L)_P$

电流关系#

$I_E = I_C + I_B$

$\beta = I_C / I_B$，$\alpha = I_C / I_E$

$\beta = \frac{\alpha}{1 - \alpha}$，$\alpha = \frac{\beta}{1 + \beta}$

集电极电流（正向有源）#

$I_C = I_S e^{V_{BE} / V_T}$，$V_T = kT/e \approx 0.0259$ V

Ebers-Moll 模型#

$I_C = \alpha_F I_E + I_{C0}$

$I_E = I_{ES}(e^{V_{BE}/V_T} - 1) - \alpha_R I_{CS}(e^{V_{BC}/V_T} - 1)$

$I_C = \alpha_F I_{ES}(e^{V_{BE}/V_T} - 1) - I_{CS}(e^{V_{BC}/V_T} - 1)$

Early 效应#

$I_C = I_S e^{V_{BE}/V_T} \left(1 + \frac{V_{CE}}{V_A}\right)$

复合寿命#

$\Delta n(t) = \Delta n(0) e^{-t/\tau}$

扩散长度#

$L_n = \sqrt{D_n \tau_n}$，$L_p = \sqrt{D_p \tau_p}$