Holes are injected in a p^+ n diode with an n region width similar to hole diffusion length Lp.

step 1 In this question you have to find the whole current density at x equals 0 minus 6 and minus 12 m

Holes are injected in a p^+ n diode with an n region width...

Holes are injected in a $\mathrm{p}^{+} \mathrm{n}$ diode with an $\mathrm{n}$ region width similar to hole diffusion length $L_{p}$. Excess holes at $n$ side varies linearly from $\triangle p_{n}\left(\right.$ at $\left.x_{n}=0\right)$ to zero (at $x_{n}=L_{p}$ ). Solve the diffusion equation to find the excess hole concentration.

Key Concepts

Diffusion Equation in Semiconductors

This concept refers to the differential equation that describes how excess minority carriers (in this case, holes in an n?type material) spread out due to concentration gradients while also recombining. In steady state and under low-level injection, the equation typically involves a second spatial derivative term related to diffusion and a first-order term that accounts for recombination. Solving this equation provides the spatial profile of the excess carrier concentration.

Minority Carrier Diffusion Length

The minority carrier diffusion length (L? for holes in an n-region) is the average distance that excess carriers travel before recombining. It is defined as the square root of the product of the diffusion coefficient and the carrier lifetime. In device analysis, it sets the characteristic length scale over which the carrier concentration decays in the quasi?neutral region, and it is critical for understanding device performance and designing junctions.

Boundary Conditions in Carrier Diffusion Problems

Applying appropriate boundary conditions is crucial when solving the diffusion equation because they define the physical constraints at the edges of the region of interest. Typically, one boundary is set at the edge of the depletion region where carriers are injected (with a known excess concentration) and the other boundary is chosen either far away (where the excess concentration vanishes) or at a specific distance corresponding to the physical dimensions of the device. These conditions allow the determination of constants in the general solution, often yielding a linear or exponential profile depending on the scenario.

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