Consider the ideal long silicon pn junction shown in Figure P 8.17 . T=300 K. The n region is d

step 1 In this question we have to find the excess whole concentration as a function of x and find the

Consider the ideal long silicon pn junction shown in Figure...

Consider the ideal long silicon pn junction shown in Figure $\mathrm{P} 8.17 . T=300 \mathrm{~K}$. The n region is doped with $10^{16}$ donor atoms per $\mathrm{cm}^{3}$ and the p region is doped with $5 \times 10^{16}$ acceptor atoms per $\mathrm{cm}^{3} .$ The minority carrier lifetimes are $\tau_{n 0}=0.05 \mu \mathrm{s}$ and $\tau_{p 0}=0.01 \mu \mathrm{s}$. The minority carrier diffusion coefficients are $D_{n}=23 \mathrm{~cm}^{2} / \mathrm{s}$ and $D_{p}=8 \mathrm{~cm}^{2} / \mathrm{s}$. The forward-bias voltage is $V_{a}=0.610 \mathrm{~V} .$ Calculate $(a)$ the excess hole concentration as a function of $x$ for $x \geq 0,(b)$ the hole diffusion current density at $x=3 \times 10^{-4} \mathrm{~cm}$, and $(c)$ the electron current density at $x=3 \times 10^{-4} \mathrm{~cm} .$

Key Concepts

PN Junction Fundamentals

A pn junction is formed when p-type and n-type semiconductor materials are joined, creating a depletion region due to the diffusion of carriers and the formation of an internal electric field. Under forward bias, the barrier is lowered, allowing for the injection of minority carriers from one side of the junction into the other, which is essential for device operation in diodes and other semiconductor devices.

Minority Carrier Injection and Diffusion

In a forward-biased pn junction, carriers from the majority side are forced into the opposite region where they become minority carriers. These injected carriers then diffuse away from the junction, establishing a spatially varying excess carrier concentration that typically decays exponentially with distance. This diffusion process is fundamental to understanding carrier transport in semiconductor devices.

Carrier Lifetime and Recombination

The minority carrier lifetime represents the average time a carrier exists before recombining with a majority carrier. Recombination processes limit the diffusion of excess minority carriers, effectively defining the diffusion length and influencing the spatial profile of injected carriers in a device. This concept is vital in quantifying how far carriers can travel from the junction before recombination occurs.

Diffusion Equation and Boundary Conditions

The diffusion equation models the spatial distribution of minority carriers in a semiconductor, incorporating both diffusion due to concentration gradients and carrier recombination. Solving this equation requires appropriate boundary conditions, such as the excess carrier concentration at the edge of the depletion region and conditions imposed at distant points, which yield solutions that usually express the concentration as an exponential function of distance.

Diffusion Current Density

Diffusion current density is the flow of charge resulting from the gradient in excess minority carrier concentration, as described by Fick’s law. It is calculated by multiplying the diffusion coefficient with the spatial derivative of the carrier concentration. This current density is an essential parameter for analyzing the performance of semiconductor devices, particularly in determining how carrier gradients contribute to the total current flow.

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