Question
Derive the expression for the excess minority hole concentration by solving the diffusion equation in a uniformly doped p-n-p bipolar transistor in forward active region having base width $W_{B}$.
Step 1
In the forward active region of a p-n-p bipolar transistor, the emitter-base junction is forward biased, and the collector-base junction is reverse biased. Show more…
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Derive the expression for the excess minority carrier hole concentration in the base region of a uniformly doped pnp bipolar transistor operating in the forward-active region.
Consider a uniformly doped silicon pnp bipolar transistor biased in the forward-active mode at low injection. The excess minority carrier hole concentration at $x=0$ is $\delta p_{E}(0)=10^{15} \mathrm{~cm}^{-3}$ and the excess minority carrier hole concentration at $x=x_{B}$ is $\delta p_{\bar{B}}\left(x_{B}\right)=-5 \times 10^{3} \mathrm{~cm}^{-3} \cdot(a)$ What is the majority carrier electron concentration in the base region and what is the $\mathrm{E}-\mathrm{B}$ voltage? $(b)$ Assuming $x_{\bar{n}}=0.80 \mu \mathrm{m}$ and $\mathrm{D}_{B}=10 \mathrm{~cm}^{2} / \mathrm{s}$, calculate the magnitude of diffusion current density at $(i) x=0$ and $(i i) x=x_{B}$ for the case when $x_{B} \ll L_{\mathrm{B}} .$ (See Equation (12.15b).) ( $c$ ) Repeat part (b) for the case when $x_{B}=L_{s}=12 \mu \mathrm{m}$. (See Equation (12.15a).) (d) Determine the ratio $J\left(x=x_{B}\right) / J(x=0)$ for parts $(b)$ and $(c)$.
(a) A uniformly doped npn bipolar transistor at $T=300 \mathrm{~K}$ is biased in saturation. Starting with the continuity equation for minority carriers, show that the excess electron concentration in the base region can be expressed as $$ \delta n_{B}(x)=n_{B 0}\left\{\left[\exp \left(\frac{e V_{B E}}{k T}\right)-1\right]\left(1-\frac{x}{x_{B}}\right)+\left[\exp \left(\frac{e V_{B C}}{k T}\right)-1\right]\left(\frac{x}{x_{B}}\right)\right\} $$ for $x_{B} / L_{B} \ll 1$ where $x_{B}$ is the neutral base width. (b) Show that the minority carrier diffusion current in the base is then given by $$ J_{n}=-\frac{e D_{i} n_{B D}}{x_{B}}\left[\exp \left(\frac{e V_{B E}}{k T}\right)-\exp \left(\frac{e V_{B C}}{k T}\right)\right] $$ (c) Show that the total excess minority carrier charge (C/cm $^{2}$ ) in the base region is given by $$ \delta Q_{n B}=\frac{-e n_{\mathrm{Bn}} x_{\mathrm{F}}}{2}\left\{\left[\exp \left(\frac{e V_{\mathrm{BE}}}{k T}\right)-1\right]+\left[\exp \left(\frac{e V_{B C}}{k T}\right)-1\right]\right\} $$
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