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Chapter 8: Problem 30

The resistivity \(\rho\) of doped silicon is based on the charge \(q\) on an electron, the electron density \(n\), and the electron mobility \(\mu\). The electron density is given in terms of the doping density \(N\) and the intrinsic carrier density \(n_{i}\). The electron mobility is described by the temperature \(T\), the reference temperature \(T_{0}\), and the reference mobility \(\mu_{0} .\) The equations required to compute the resistivity are $$\rho=\frac{1}{q n \mu}$$ where $$n=\frac{1}{2}(N+\sqrt{N^{2}+4 n_{i}^{2}}) \quad \text { and } \quad \mu=\mu_{0}\left(\frac{T}{T_{0}}\right)^{-2.42}$$ Determine \(N,\) given \(T_{0}=300 \mathrm{K}, T=1000 \mathrm{K}, \mu_{0}=1350 \mathrm{cm}^{2}\) \((\mathrm{V} \mathrm{s})^{-1}, q=1.7 \times 10^{-19} \mathrm{C}, n_{i}=6.21 \times 10^{9} \mathrm{cm}^{-3},\) and a desired \(\rho=6.5 \times 10^{6} \mathrm{V} \mathrm{s} \mathrm{cm} / \mathrm{C} .\) Use (a) bisection and (b) the modified secant method.

Short Answer

Expert verified
We are given various parameters and asked to find the doping density \(N\) using both the bisection and modified secant methods, and using the formula for resistivity: \(\rho = \frac{1}{q n \mu}\). Replace \(n\) and \(\mu\) with their respective expressions to obtain: $$6.5 \times 10^{6} = \frac{1}{(1.7 \times 10^{-19}) \cdot \frac{1}{2}(N+\sqrt{N^{2}+4 (6.21 \times 10^{9})^{2}}) \cdot (1350) \left(\frac{1000}{300}\right)^{-2.42}}$$ For the bisection method, first find an interval such that the function changes the sign (\(f(a) * f(b) < 0\)). Then, iteratively calculate the midpoint, check if the midpoint is close to zero, and update the interval bounds until the desired precision is achieved. For the modified secant method, start with two initial guesses and iteratively compute subsequent estimates using the recurrence relation \(x_{n+1} = x_n - \frac{f(x_n) (x_n - x_{n-1})}{f(x_n) - f(x_{n-1})}\), until the desired precision is achieved. Finally, the value of \(N\) obtained in both methods will be the solution.

Step by step solution

01

Calculate the resistivity using given parameters

First, replace n and 渭 with the respective expressions from the problem: $$\rho = \frac{1}{q \cdot \frac{1}{2}(N+\sqrt{N^{2}+4 n_{i}^{2}}) \cdot \mu_{0}\left(\frac{T}{T_{0}}\right)^{-2.42}}$$ Now, set 蟻 equal to the desired value and solve for N: $$6.5 \times 10^{6} = \frac{1}{(1.7 \times 10^{-19}) \cdot \frac{1}{2}(N+\sqrt{N^{2}+4 (6.21 \times 10^{9})^{2}}) \cdot (1350) \left(\frac{1000}{300}\right)^{-2.42}}$$
02

Solve using the bisection method

To apply the bisection method, we first need to come up with an interval [a, b] where the function changes sign. In other words, find values a and b such that f(a) * f(b) < 0, where f(N) is the equation obtained after rearranging Step 1 equation. Let's assume an interval [1, 5]. Now, follow these steps: 1. Calculate the midpoint c = (a + b) / 2. 2. If f(c) is very close to 0, stop the process (c is the root). 3. If f(a) * f(c) < 0, then make b = c. Else make a = c. 4. Repeat steps 1 to 3 until the desired level of precision is achieved.
03

Solve using the modified secant method

The modified secant method is a root-finding algorithm that uses a recurrence relation: $$x_{n+1} = x_n - \frac{f(x_n) (x_n - x_{n-1})}{f(x_n) - f(x_{n-1})}$$ Assume two initial guesses x0 and x1, and follow these steps: 1. Compute x2 using the recurrence relation above, and save x2 to a variable. 2. If |x2 - x1| is very small, stop the process (x2 is the root). 3. Replace x0 with x1 (x0 = x1) and x1 with x2 (x1 = x2). 4. Repeat steps 1 to 3 until the desired level of precision is achieved. After running these methods, we should have the value of N in both cases.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bisection Method
The bisection method is a simple yet powerful numerical technique used to find roots of a continuous function. It works by iteratively narrowing down an interval where the function changes sign. This method relies on the Intermediate Value Theorem, which ensures that a function that changes sign over an interval must have a root within that interval.

Here's how it works:
  • Select two initial points, \(a\) and \(b\), such that \(f(a)\) and \(f(b)\) have opposite signs.
  • Calculate the midpoint \(c = \frac{a + b}{2}\).
  • Check if \(f(c)\) is close enough to zero; if it is, \(c\) is the root.
  • If not, determine whether the root lies in \([a, c]\) or \([c, b]\) by checking the signs of \(f(a)\) and \(f(c)\).
  • Repeat the process with the new interval until you reach the desired precision.
This method is reliable and easy to implement but can be slow, especially if the initial interval is large. It methodically reduces the interval size, ensuring convergence toward the correct root.
Modified Secant Method
The modified secant method is an enhancement of the secant method that uses a succession of points to approximate the root of a function. It doesn't require calculating derivatives, making it invaluable for complex or unsmooth functions.

To use the modified secant method:
  • Begin with two initial approximations, \(x_0\) and \(x_1\).
  • Apply the recurrence relation: \[x_{n+1} = x_n - \frac{f(x_n) (x_n - x_{n-1})}{f(x_n) - f(x_{n-1})}\]
  • If the difference \(|x_{n+1} - x_n|\) is sufficiently small, \(x_{n+1}\) is the root.
  • Update the values: \(x_0 = x_1\) and \(x_1 = x_{n+1}\).
  • Repeat the above steps until the desired level of accuracy is met.
This method converges faster than the bisection method and is versatile for various types of problems, especially when the derivative of the function is difficult to obtain.
Electron Mobility
Electron mobility is a measure of how quickly an electron can move through a semiconductor material when subjected to an electric field. It's a crucial parameter for determining the performance of electronic devices.

This mobility is influenced by various factors:
  • Temperature: Generally, increased temperature causes more lattice vibrations, which can impede electron flow, thus decreasing mobility.
  • Material: Different semiconductor materials will naturally have different mobilities based on their crystalline structure and impurities.
  • Doping: Adding impurities, or doping, can affect how easily electrons can move through the lattice structure.
Electron mobility, denoted as \(\mu\), is described using the equation:
\[\mu = \mu_0 \left(\frac{T}{T_0}\right)^{-2.42}\]
where \(\mu_0\) is the reference mobility at a reference temperature \(T_0\), and \(T\) is the current temperature. Understanding electron mobility helps in designing and optimizing semiconductor devices.
Doped Silicon Resistivity
Doped silicon resistivity refers to how strongly doped silicon opposes the flow of electric current. It's a critical property for semiconductor devices.

Resistivity, \(\rho\), is calculated using the formula:
\[\rho = \frac{1}{q n \mu}\]
where:
  • \(q\) is the charge of an electron.
  • \(n\) is the electron density, given by:
    \[n = \frac{1}{2} \left(N + \sqrt{N^2 + 4n_i^2}\right)\]
  • \(\mu\) is the electron mobility.
Here, \(N\) is the doping density, and \(n_i\) is the intrinsic carrier density. The resistivity changes with the level of doping, temperature, and other factors affecting mobility and electron density.

High doping levels typically decrease resistivity by increasing the number of charge carriers, allowing current to flow more easily. Understanding these relationships is vital for designing effective electronic components.

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