We inject electrons into a p-type semiconductor 5 microns...

We inject electrons into a p-type semiconductor 5 microns long such that the concentration varies linearly from $10^{20} \mathrm{~cm}^{-3}$ to 0 from left to right. If the mobility of the electrons is $500 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s}$, what is the current density if the electric fields are negligible?

Key Concepts

Carrier Mobility

Carrier mobility is a measure of how quickly electrons or holes can move through a semiconductor material when subjected to an electric field. Although the question assumes negligible electric fields, mobility is still relevant as it directly influences the diffusion coefficient via the Einstein relation, thereby affecting the diffusion current.

Einstein Relation

The Einstein relation connects the diffusion coefficient to the carrier mobility. Specifically, it expresses that the diffusion coefficient is proportional to the mobility multiplied by the thermal voltage (kT/q), which provides a useful bridge between how carriers move under thermal energy versus an applied electric field.

Fick's Law of Diffusion

Fick's Law quantifies diffusion by relating the flux of carriers to the gradient in their concentration. It indicates that the current resulting from diffusion is proportional to the steepness of the concentration gradient, making it a key principle in analyzing transport in materials when drift due to electric fields is negligible.

Diffusion Process

In semiconductors, when there is a spatial variation in carrier concentration, carriers move from regions of high concentration to regions of low concentration. This process, called diffusion, is a fundamental transport mechanism that occurs without the direct application of an external electric field.

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