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7 The pn Junction

7.1 Basic Structure of the pn Junction

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7.2 Zero Applied Bias

假设：

  • The first assumption is that the Boltzmann approximation is valid, which means that each semiconductor region is nondegenerately doped.
  • The second assumption is that complete ionization exists, which means that the temperature of the pn junction is not “too low.”

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\[ \begin{align*} n_0&=n_i\exp\left[ \frac{E_F-E_{Fi}}{kT} \right]\\ &=n_i\exp\left[ \frac{-e\phi_{Fn}}{kT} \right] \end{align*} \]
\[ \phi_{Fn}=\frac{-kT}{e}\ln(\frac{N_d}{n_i}) \]

同理：

\[ \phi_{F_p}=\frac{kT}{e}\ln(\frac{N_a}{n_i}) \]
\[ V_{bi}=\frac{kT}{e}\ln(\frac{N_aN_d}{n_i^2})=V_t\ln(\frac{N_aN_d}{n_i^2}) \]
  • \(V_t=\frac{kT}{e}\) 热电压
电场强度

假设：

We will assume that the space charge region abruptly ends in the n region at \(x=+x_n\) and abruptly ends in the p region at \(x=-x_p\) (\(x_p\) is a positive quantity).

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\[ \rho(x)= \begin{cases} -eN_a\quad &-x_p<x<0\\ eN_d\quad &0<x<x_n \end{cases} \]
\[ \frac{d^2\phi(x)}{dx^2}=\frac{-\rho(x)}{\epsilon_s}=-\frac{dE(x)}{dx} \]
\[ E= \begin{cases} \frac{-eN_a}{\epsilon_s}(x+x_p)&-x_p\le x\le 0\\ \frac{-eN_d}{\epsilon_s}(x_n-x)&0\le x\le x_n \end{cases} \]
\[ N_ax_p=N_dx_n \]

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The potential in the junction
\[ \begin{align*} \phi(x)&=-\int E(x)dx=\int \frac{eN_a}{\epsilon_s}(x+x_p)dx\\ &=\frac{eN_a}{\epsilon_s}(\frac{x^2}2+x_p\cdot x)+C'_1\\ \end{align*} \]

如果设定\(x_p\)处为0电势点，则

\[ C'_1=\frac{eN_a}{2\epsilon_s}x_p^2 \]
\[ \therefore \phi(x)=\frac{eN_a}{\epsilon_s}(x+x_p)^2 \]

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Space Charge Width
\[ \begin{align*} x_n&=\left\{ \frac{2\epsilon_sV_{bi}}{e}\left[ \frac{N_a}{N_d} \right]\left[ \frac1{N_a+N_d} \right] \right\}^{1/2}\\ x_p&=\left\{ \frac{2\epsilon_sV_{bi}}{e}\left[ \frac{N_a}{N_d} \right]\left[ \frac1{N_a+N_d} \right] \right\}^{1/2}\\ W&=\left\{ \frac{2\epsilon_sV_{bi}}{e}\left[ \frac{N_a+N_d}{N_aN_d} \right] \right\}^{1/2} \end{align*} \]

7.3 Reverse Applied Bias

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\[ V_{total}=V_{bi}+V_R \]
\[ W=\left\{ \frac{2\epsilon_s(V_{bi}+\bold{V_R})}{e}\left[ \frac{N_a+N_d}{N_aN_d} \right] \right\}^{1/2} \]
Junction Capacitance
\[ C'=\frac{dQ'}{dV_R}=\frac{eN_ddx_n}{dV_R}=\frac{eN_adx_p}{dV_R} \]
\[ C'=\left[\frac{e\epsilon_sN_aN_d}{2(V_{bi+V_R})(N_a+N_d)}\right]^{1/2} \]
  • \(C'\) 单位面积电容

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One-Sided Junctions

\(N_a>>N_d\), \(p^+n\)结

  • \(x_p<<x_n\)
  • \(W\approx x_n\)
  • \(C'=\left[\frac{e\epsilon_sN_d}{2(V_{bi+V_R})}\right]^{1/2}\Leftrightarrow \frac1{C'^2}=\frac{2(V_{bi}+V_R)}{e\epsilon_sN_d}\)

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7.4 Junction Breakdown

Breakdown Voltage

the Zener effect and the avalanche effect

Zener breakdown occurs in highly doped pn junctions through a tunneling mechanism. In a highly doped junction, the conduction and valence bands on opposite sides of the junction are sufficiently close during reverse bias that electrons may tunnel directly from the valence band on the p side into the conduction band on the n side. This tunneling process is schematically shown in Figure 7.12a.

The avalanche breakdown process occurs when electrons and/or holes, moving across the space charge region, acquire sufficient energy from the electric field to create electron–hole pairs by colliding with atomic electrons within the depletion region. The avalanche process is schematically shown in Figure 7.12b. The newly created electrons and holes move in opposite directions due to the electric field and thereby create a reverse-biased current. In addition, the newly generated electrons and/or holes may acquire sufficient energy to ionize other atoms, leading to the avalanche process. For most pn junctions, the predominant breakdown mechanism will be the avalanche effect.

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\[ I_n(W)=M_nI_{n0} \]
  • \(M_n\) multiplication factor
\[ dI_n(x)=I_n(x)\alpha_ndx+I_p(x)\alpha_pdx \]
  • \(\alpha_n, \alpha_p\) ionization rate
\[ I=I_n(x)+I_p(x) \]

assumption: $$ \alpha_n=\alpha_p=\alpha $$

\[ \begin{align*} I_n(W)-I_n(0)&=I\int_0^W\alpha dx\\ \frac{M_nI_{n0}-I_n(0)}{I}&=\int_0^W\alpha dx\\ \therefore 1-\frac1{M_n}&=\int_0^W \alpha dx \end{align*} \]

The avalanche breakdown voltage is defined to be the voltage at which \(M_n\) approaches infinity.

\[ \int_0^W\alpha dx=1 \]

\(\alpha\) is strong function of electric field.