A material is doped such that electron concentration varies...

A material is doped such that electron concentration varies linearly across the sample, which is $0.5 \mu \mathrm{m}$ thick. Donor concentration varies from 0 (at $x=0$ ) to $10^{16} / \mathrm{cm}^{3}$ (at $\left.x=0.5 \mu \mathrm{m}\right)$. Write equations for total electron and hole concentrations as a function of distance $x .$ Determine electron and hole diffusion current densities if the diffusion coefficients are $D_{n}=30 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{sec}$ and $D_{p}=$ $12 \mathrm{~cm}^{2} / \mathrm{V}$ -sec. Find the expression for Fermi level $\left(E_{F}\right)$ as a function of $x$.

Key Concepts

Fermi Level Variation in Non-Uniformly Doped Semiconductors

The Fermi level is a measure of the chemical potential for electrons and determines the occupation probability of energy states. In a semiconductor with a non-uniform doping profile, the Fermi level is not constant but varies with position. This spatial variation is crucial for understanding carrier statistics and transport phenomena, as it sets the equilibrium condition locally. The Boltzmann approximation often provides a means to relate the local carrier concentration to the Fermi level, thereby enabling the derivation of its position-dependent expression.

Einstein Relation

The Einstein relation connects the diffusion coefficient of carriers to their mobility and the thermal voltage. This relation is fundamental in semiconductor physics because it bridges the transport properties (diffusion and drift) of carriers under thermal equilibrium. Even though the problem primarily deals with diffusion, understanding the Einstein relation helps in linking these diffusion coefficients with the corresponding mobilities, which are indirectly related to the Fermi level position in the semiconductor.

Carrier Concentrations in Semiconductors

Carrier concentrations denote the densities of free electrons and holes available for conduction in a semiconductor. In doped material, these concentrations depend on the doping profile and thermal equilibrium conditions. The total electron concentration and hole concentration can be determined using the relations that involve the intrinsic carrier concentration and the effects of doping, often applying the mass action law which relates both quantities under equilibrium conditions.

Semiconductor Doping and Doping Gradients

This concept refers to the intentional introduction of impurities into a semiconductor to modify its electrical properties. In the context of the problem, the donor concentration is varied linearly across the sample, creating a gradient. This spatial non-uniformity in dopant concentration results in corresponding variations in carrier concentrations (electrons and holes) throughout the material, influencing how the electronic properties change spatially.

Diffusion Current Density

Diffusion current density arises from the movement of charge carriers from regions of high concentration to low concentration. It is governed by Fick’s law, where the current density is proportional to the gradient (spatial derivative) of the carrier concentration. In semiconductors, both electron and hole diffusion contribute to the overall current, and the diffusion coefficients specific to each type of carrier play a key role in determining the magnitude of these currents.

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