# PN and Metal-Semiconductor Junctions

\begin{align*} \newcommand{\dif}{\mathop{}\!\mathrm{d}} \newcommand{\belowarrow}[1]{\mathop{#1}\limits_{\uparrow}} \newcommand{\bd}{\boldsymbol} \newcommand{\L}{\mathscr{L}} \end{align*}

PN junction 就是 P-type 和 N-type 靠在一起，其主要特性是 rectifying current–voltage（见 fig 4-2），所以也叫 rectifier or diode

# 4.1 Building Blocks of the PN Junction Theory

## 4.1.1 Energy Band Diagram and Depletion Layer of a PN Junction

• 左边：the neutral N layer
• 右边：the neutral P layer
• 中间：depletion layer 费米能级在中间，表示 $n\approx p \approx 0$

## 4.1.2 Built-In Potential

$\text{N-region} \quad n=N_d=N_c e^{-qA/kT} \Rightarrow A = \frac{kT}{q} \ln \frac{N_c}{N_d}\\ \text{P-region} \quad n=\frac{n_i^2}{N_a}=N_c e^{-qA/kT} \Rightarrow B=\frac{kT}{q}\ln \frac{N_cN_a}{n_i^2}$ $\phi_\text{bi} = B-A = \frac{kT}{q} \ln \frac{N_dN_a}{n_i^2}\\ 其中，n_i=\sqrt{N_c N_v} e^{-E_g/2kT}\\ 所以，\phi_\text{bi} = \frac{kT}{q} \ln \frac{N_dN_a}{N_c N_v}+\frac{E_g}{q}$

## 4.1.3 Poisson’s Equation

\begin{aligned} \mathscr{E}(x+\Delta x) \cdot A- \mathscr{E}(x) \cdot A &= \frac{\rho \Delta x A}{\varepsilon_s}\\ \frac{\mathscr{E}(x+\Delta x)- \mathscr{E}(x)}{\Delta x} &= \frac{\rho}{\varepsilon_s}\\ \frac{\dif \mathscr{E}}{\dif x} &= \frac{\rho}{\varepsilon_s}\\ \frac{\dif^2 V}{\dif x^2} = - \frac{\dif \mathscr{E}}{\dif x} &= -\frac{\rho}{\varepsilon_s} \end{aligned}

# 4.2 Depletion-Layer Model

## 4.2.1 Field and Potential in the Depletion Layer

$\frac{\dif \mathscr{E}}{\dif x} = \frac{\rho}{\varepsilon_s} = \begin{cases} \dfrac{-q N_a}{\varepsilon_s}\\ \dfrac{q N_d}{\varepsilon_s} \end{cases}\\ \Rightarrow \mathscr{E}(x) = \begin{cases} -\frac{q N_a}{\varepsilon_s} x +C_1 & x>0\\ \frac{q N_d}{\varepsilon_s} x +C_2 & x<0 \end{cases}$

$\mathscr{E}(x) = \begin{cases} \frac{q N_a}{\varepsilon_s} (x_P-x) & x>0\\ \frac{q N_d}{\varepsilon_s} (x-x_N) & x<0 \end{cases}$

$N_a |x_P| = N_d |x_N| \tag{4.2.5}$

$V(x)= \begin{cases} \frac{qN_a}{2\varepsilon_s}(x_P-x)^2 & 0\leq x \leq x_P\\ \phi_\text{bi} - \frac{qN_d}{2 \varepsilon_2}(x-x_N)^2 & x_N\leq x\leq 0 \end{cases} \tag{4.2.6}$

## 4.2.2 Depletion-Layer Width

$\begin{cases} N_a |x_P| = N_d |x_N| &电场连续\\ V(x_N)-V(x_P)=\phi_\text{bi} &电势差一定 \end{cases}$

$\frac{qN_a}{2\varepsilon_s}x_P^2 = \phi_\text{bi} - \frac{qN_d}{2 \varepsilon_2} x_N^2$

$|x_N|=\sqrt{\phi_\text{bi}\bigg/\frac{q}{2\varepsilon_s}\frac{N_d}{N_a}\left( N_a + N_d \right)}\\ |x_P|=\sqrt{\phi_\text{bi}\bigg/\frac{q}{2\varepsilon_s}\frac{N_a}{N_d}\left( N_a + N_d \right)}$ \begin{aligned} |x_P| + |x_N| &= \sqrt{\frac{2 \varepsilon_s \phi_\text{bi}}{q(N_a+N_d)}} \cdot \left( \sqrt{\frac{N_d}{N_a}}+ \sqrt{\frac{N_a}{N_d}}\right) \\ &= \sqrt{\frac{2 \varepsilon_s \phi_\text{bi}}{q(N_a+N_d)}} \cdot \left( \frac{N_d}{\sqrt{N_a N_d}}+ \frac{N_a}{\sqrt{N_a N_d}}\right) \\ &= \sqrt{\frac{2 \varepsilon_s \phi_\text{bi}}{q}\left( \frac{1}{N_a}+\frac{1}{N_d} \right)} \end{aligned}\tag{4.2.8}

• If $N_a \gg N_d$, as in a ${\rm P^+N}$ junction, $W_\text{dep} \approx \sqrt{\frac{2 \varepsilon_s \phi_\text{bi}}{qN_d}} \approx |x_N| \tag{4.2.9}$
• If $N_d \gg N_a$, as in an $\rm{ N^+P}$ junction, $W_\text{dep} \approx \sqrt{\frac{2 \varepsilon_s \phi_\text{bi}}{qN_a}} \approx |x_P| \tag{4.2.10}$

$W_\text{dep} = \sqrt{\frac{2 \varepsilon_s \phi_\text{bi}}{qN}}$

# 4.3 Reverse-Biased PN Junction

$\phi_\text{bi} \rightarrow \phi_\text{bi} +V_r$

$W_\text{dep} = \sqrt{\frac{2 \varepsilon_s (\phi_\text{bi} +V_r)}{qN}}$

# 4.4 Capacitance-Voltage Charateristics

$C_\text{dep} = A\frac{\varepsilon_s}{W_\text{dep}} \tag{4.4.1}$

$\frac{1}{C_\text{dep}^2} = \frac{W_\text{dep}^2}{A^2 \varepsilon_s^2}=\frac{2(\phi_\text{bi}+V_r)}{q N \varepsilon_s A^2}$

# 4.5 Junction Breakdown

Junction Breakdown 指的是当反偏电压大于某个值时，电流急剧增大的情况。

## 4.5.1 Peak Electric Field

$\mathscr{E}_p = \mathscr{E}(0)=\frac{q N_a}{\varepsilon_s} x_P = \sqrt{\frac{2 qN}{\varepsilon_s}(\phi_\text{bi}+V_r)}$

$V_B = \frac{\varepsilon_s \mathscr{E}_\text{crit}^2}{2qN}-\phi_\text{bi}$

## 4.5.2 Tunneling Breakdown

tunneling current density：$J = G e^{-H/\mathscr{E}_p}$

# 4.6 Carrier Injection Under Forward Bias

\begin{aligned} n(x_P)&=N_c e^{-(E_c-E_{Fn})/kT}\\ &= N_c e^{-(E_c-E_{Fp})/kT} \cdot e^{(E_{Fn}-E_{Fp})/kT}\\ &=n_{P0} e^{(E_{Fn}-E_{Fp})/kT}\\ &= n_{P0} e^{qV/kT} \end{aligned}

\begin{aligned} n(x_P)&= n_{P0} e^{qV/kT}=\frac{n_i^2}{N_a} e^{qV/kT} \\ n(x_N)&= n_{N0} e^{qV/kT}=\frac{n_i^2}{N_d} e^{qV/kT} \end{aligned} \tag{4.6.2}

\begin{aligned} n'(x_P)&= n(x_P) - n_{P0}=n_{P0} (e^{qV/kT}-1) \\ n'(x_N)&= n(x_N) - n_{N0}=n_{N0} (e^{qV/kT}-1) \end{aligned} \tag{4.6.3}

# 4.7 Current Continuity Equation

$A \cdot \frac{J_p(x)}{q} = A \cdot \frac{J_p(x+\Delta x)}{q} + A \cdot \Delta x \cdot \frac{p'}{\tau}\\ -\frac{J_p(x+\Delta x)-J_p(x)}{q} = q\frac{p'}{\tau}\\ -\frac{\dif J_p}{\dif x} = q\frac{p'}{\tau}$

$q D_p \frac{\dif^2 p}{\dif x^2} = q\frac{p'}{\tau_p}$

$D_p$ 是空穴的扩散系数 diffusion constant. 由于 $p=p_0+p’$，$p_0$ 取决于 $N_a$，是不变的，所以微分部分可以减去常数 $p_0$，并进行变形：

$\frac{\dif^2 p'}{\dif x^2} = \frac{p'}{D_p\tau_p}=\frac{p'}{L_p^2} \tag{4.7.5}\\ 其中，L_p \equiv \sqrt{D_p \tau_p}$

$\frac{\dif^2 n'}{\dif x^2} = \frac{n'}{L_n^2} \tag{4.7.7}\\ 其中，L_n \equiv \sqrt{D_n \tau_n}$

$L_n$ 和 $L_p$ 称为电子和空穴的 diffusion lengths

# 4.8 Excess Carriers in Forward-Biased PN Juntion

$\begin{cases} p'(\infty)=0\\ p'(x_N)=p_{N0} (e^{qV/kT}-1) \end{cases}$

$p'(x)=p_{N0} (e^{qV/kT}-1)e^{-(x-x_N)/L_p}, x>x_N \tag{4.8.2}$

$n'(x)=n_{P0} (e^{qV/kT}-1)e^{(x-x_P)/L_n}, x<x_P \tag{4.8.3}$

$x_N,x_P$ 和 $L_n,L_p$ 带来的影响比较小（一般都是同一数量级），主要看 $p_{N0}$ 和 $n_{P0}$，由于掺杂越少，少子就越多，因此可以得出：轻掺杂一侧的少子注入更多

# 4.9 PN Diode I-V Characteristics

$J_{pN}= - q D_p \frac{\dif^2 p'(x)}{\dif x^2}=q \frac{D_p}{L_p} p_{N0} (e^{qV/kT}-1)e^{-(x-x_N)/L_p} \tag{4.9.1}$ $J_{nP}= - q D_n \frac{\dif^2 n'(x)}{\dif x^2}=q \frac{D_n}{L_n} n_{P0} (e^{qV/kT}-1)e^{-(x-x_P)/L_n} \tag{4.9.2}$

\begin{aligned} J &= J_{pN}+J_{nP}\\ &= \left( q \frac{D_p}{L_p} p_{N0}+q \frac{D_n}{L_n} n_{P0} \right) (e^{qV/kT}-1)\\ \end{aligned} \tag{4.9.3} $I=I_0 (e^{qV/kT}-1) \tag{4.9.4}$ $I_0=A \cdot q n_i^2 \left( \frac{D_p}{L_p N_d}+\frac{D_n}{L_n N_a} \right) \tag{4.9.5}$

## 4.9.1 Contributions from the Depletion Region

\begin{aligned} pn &= N_c N_v e^{-(E_c-E_v)/kT} e^{-(E_{F_p}-E_{F_n})/kT}\\ &=n_i^2 e^{qV/kT} \end{aligned}

$\text{Net recombination rate per unit volume} = \frac{n_i}{\tau_\text{dep}}\left( e^{qV/2kT}-1 \right)$

$\tau_\text{dep}$ is the generation/recombination lifetime in the depletion layer.（不太清楚这是啥，总之题目会给吧），后面减 1 是为了在没有偏压时，净复合率为 0.

$I=I_0 (e^{qV/kT}-1) + A \frac{ q n_i W_\text{dep}}{\tau_\text{dep}}\left( e^{qV/2kT}-1 \right) \tag{4.9.4}$

$I_\text{leakage} = I_0+A \frac{q n_i W_\text{dep}}{\tau_\text{dep}} \tag{4.9.10}$

# 4.10 Charge Storage

$I=Q/\tau_s \tag{4.10.1}\\ Q=I\tau_s$

$\tau_s$ is called the charge-storage time. In a one-sided junction, $\tau_s$ is the recombination lifetime on the lighter-doping side, where charge injection and recombination take place.

# 4.11 Small-Signal Model of the Diode

diode 在高频小信号电路中等效为 RC 并联：

$G \equiv \frac{1}{R} = \frac{\dif I}{\dif V} = \frac{q}{kT} I_0 e^{qV/kT} = I_\text{DC} / \frac{kT}{q} \tag{4.11.1}$ $C = \frac{\dif Q}{\dif V}=\tau_s \frac{\dif I}{\dif V} = \tau_s I_\text{DC} / \frac{kT}{q} \tag{4.11.2}$

# 4.16 Schottky Barriers

two kinds of metal–semiconductor junction

• Schottky diodes： metal and lightly doped semiconductors
• ohmic contacts：metal and heavily doped semiconductors

$\phi_B$ 称为 Schottky barrier height，常见金属的肖特基势垒如下表：

Metal Mg Ti Cr W Mo Pd Au Pt
$\phi_{Bn}$（V） 0.4 0.5 0.61 0.67 0.68 0.77 0.8 0.9
$\phi_{Bp}$（V）   0.61 0.50   0.42   0.3
$\psi_M$（V） 3.7 4.3 4.5 4.6 4.6 5.1 5.1 5.7

$\phi_{Bn}=\phi_M -\chi_\text{Si}$

$\phi_{Bn}=0.7 \text{V} + 0.2 (\phi_M - 4.75)$

$q \phi_\text{bi}=q\phi_{Bn} - (E_c-E_F)=q \phi_{Bn} - kT \ln \frac{N_c}{N_d}$

$\frac{1}{C^2}=\frac{2(\phi_\text{bi}+V)}{q N_d \varepsilon_s A^2}$

# 4.17 Thermionic Emission Theory

$n = N_c e^{-q(\phi_B-V)/kT}= 2 \left[ \frac{2 \pi m_n kT}{h^2} \right]^{3/2} e^{-q(\phi_B - V)/kT} \tag{4.17.1}$

$v_\text{thx} = -\sqrt{2kT/\pi m_n}$

$J_{\rm S\rightarrow M}=-\frac{1}{2} qnv_\text{thx} = \frac{4\pi q m_n k^2}{h^3} T^2 e^{-q \phi_B /kT} e^{qV/kT}\\ \equiv J_0 e^{qV/kT} \tag{4.17.3}$

$I_0 = AKT^2 e^{-q \phi_B /kT}\\ K=\frac{4\pi q m_n k^2}{h^3}$

$K\approx 100 {\rm A/(cm^2/K)}$ 称为 Richardson constant

# 4.18 Schottky Diodes

$I = J_{\rm S\rightarrow M} + J_{\rm M\rightarrow S} = I_0 e^{qV/kT} - I_0=I_0 (e^{qV/kT}-1) \tag{4.18.4}$

# 4.19 Applications of Schottky Diodes

$I=I_0 (e^{qV/kT}-1) \tag{4.19.1}$

# 4.20 Quantum Mechanical Tunneling

$P \approx \exp\left( -2 T \sqrt{\frac{8 \pi ^2 m}{h^2}}(V_H-E) \right)$

$m$ is the effective mass and $h$ is the Planck’s constant.

# 4.21 Ohmic Contacts

$T \approx W_\text{dep}/2=\sqrt{\varepsilon_s \phi_{Bn}/(2qN_d)}\\ P\approx e^{-H\phi_{Bn}/\sqrt{N_d}}\\ 其中，H\equiv \frac{4\pi}{h} \sqrt{(\varepsilon_s m_n)/q}$

$J_{\rm S\rightarrow M}=-J_{\rm M\rightarrow S}\approx \frac{1}{2} q N_d v_\text{thx} P\\ v_\text{thx} = -\sqrt{2kT/\pi m_n} \quad和 4.17 中的一样$

$J_{\rm S\rightarrow M}=\frac{1}{2} q N_d v_\text{thx} e^{-H(\phi_{Bn} - V)/\sqrt{N_d}}$

\begin{aligned} J &= \left. \frac{\dif J_{\rm S\rightarrow M}}{\dif V} \right|_{V=0}\cdot V\\ &=V \cdot \frac{1}{2} q v_\text{thx} H \sqrt{N_d} e^{-H \phi_{Bn}/\sqrt{N_d}} \end{aligned}\tag{4.21.6}

$R_c \equiv \frac{V}{J} = \frac{2 \cdot e^{H \phi_{Bn}/\sqrt{N_d}}}{q v_\text{thx} H \sqrt{N_d}}\\ \propto e^{H \phi_{Bn}/\sqrt{N_d}} \tag{4.21.7}$