Given the data of Prob. 5.2, plot the doping profiles Na(x)...

Given the data of Prob. $5.2$, plot the doping profiles $N_{a}(x)$ and $N_{d}(x)$ for the following double-diffused transistor: The starting wafer is $\mathrm{n}$ -type $\mathrm{Si}$ with $N_{d}=10^{16} \mathrm{~cm}^{-3} ; N_{s}=5 \times 10^{13} \mathrm{~cm}^{-2}$ boron atoms are deposited on the surface, and these atoms are diffused into the wafer at $1100^{\circ} \mathrm{C}$ for $1 \mathrm{hr}$ $\left(D=3 \times 10^{-13} \mathrm{~cm}^{2} / \mathrm{s}\right.$ for $\mathrm{B}$ in $\mathrm{Si}$ at $\left.1100^{\circ} \mathrm{C}\right) ;$ then the wafer is placed in a phosphorus diffusion furnace at $1000^{\circ} \mathrm{C}$ for $15 \mathrm{~min}\left(D=3 \times 10^{-14} \mathrm{~cm}^{2} / \mathrm{s}\right.$ for $\mathrm{P}$ in $\mathrm{Si}$ at $1000^{\circ} \mathrm{C}$ ). During the emitter diffusion, the surface concentration is held constant at $5 \times 10^{20} \mathrm{~cm}^{-3}$. You may assume that the base doping profile does not change appreciably during the emitter diffusion, which takes place at a lower temperature and for a shorter time. Find the width of the base region from plots of $N_{a}(x)$ and $N_{d}(x) .$ Hint: Use five-cycle semilog paper and let $x$ vary from zero to about $1.5 \mu \mathrm{m}$ in steps that are chosen to be simple multiples of $2 \sqrt{D t}$

Given the data of Prob. $5.2$, plot the doping profiles $N_{a}(x)$ and $N_{d}(x)$ for the following double-diffused transistor: The starting wafer is $\mathrm{n}$ -type $\mathrm{Si}$ with $N_{d}=10^{16} \mathrm{~cm}^{-3} ; N_{s}=5 \times 10^{13} \mathrm{~cm}^{-2}$ boron atoms are deposited on
the surface, and these atoms are diffused into the wafer at $1100^{\circ} \mathrm{C}$ for $1 \mathrm{hr}$ $\left(D=3 \times 10^{-13} \mathrm{~cm}^{2} / \mathrm{s}\right.$ for $\mathrm{B}$ in $\mathrm{Si}$ at $\left.1100^{\circ} \mathrm{C}\right) ;$ then the wafer is placed in a
phosphorus diffusion furnace at $1000^{\circ} \mathrm{C}$ for $15 \mathrm{~min}\left(D=3 \times 10^{-14} \mathrm{~cm}^{2} / \mathrm{s}\right.$ for $\mathrm{P}$ in $\mathrm{Si}$ at $1000^{\circ} \mathrm{C}$ ). During the emitter diffusion, the surface concentration is held constant at $5 \times 10^{20} \mathrm{~cm}^{-3}$. You may assume that the base doping profile does not change appreciably during the emitter diffusion, which takes place at a lower temperature and for a shorter time. Find the width of the base region from plots of $N_{a}(x)$ and $N_{d}(x) .$ Hint: Use five-cycle semilog paper and let $x$ vary from zero to about $1.5 \mu \mathrm{m}$ in steps that are chosen to be simple multiples of $2 \sqrt{D t}$

Key Concepts

Semi-logarithmic Plotting of Doping Profiles

Semi-log plotting is a graphical tool where one axis (usually the doping concentration) is logarithmic. This method is particularly useful in visualizing doping profiles that decay exponentially with depth, making it easier to analyze and determine critical features such as the junction depths and the transition regions between different doping concentrations.

Double-Diffused Transistor Structure

A double-diffused transistor is a device where two sequential diffusion steps are used to form overlapping regions of different dopant types, usually forming a base and an emitter. Understanding this structure involves grasping how the diffusion profiles of the two dopants interact and determine key device parameters, such as the base width and junction positions.

Diffusion Length Calculation (2?(Dt))

The diffusion length, given by 2?(Dt), is a measure of the average distance that dopant atoms travel during the diffusion process, where D is the diffusion coefficient and t is the diffusion time. It sets the scale for the spread of the dopant profile and is used both in estimating process parameters and in choosing appropriate steps for plotting concentration profiles.

Dopant Diffusion in Semiconductors

This concept refers to the process by which impurity atoms are introduced into a semiconductor material to modify its electrical properties. The process is governed by Fick’s laws of diffusion, and it is critical in forming regions with controlled carrier concentrations. Understanding dopant diffusion is fundamental to designing and fabricating semiconductor devices.

Error Function Solution to Diffusion Equations

When dopants diffuse into a semiconductor, their concentration profiles are described by solutions to Fick’s second law. Typically, the complementary error function (erfc) is used to model the diffusion from a constant source, describing how the dopant concentration decays with depth. This solution is pivotal in predicting junction depths and concentration gradients.

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