Nanoscale Energy Transport: Speed of Electron Gas in Semiconductor (Chen: 1.10)

[Diomarte]

Before I ask the question, let me explain a little bit about myself. I graduated just over a year ago with a bachelors in Physics, and am now starting my first semester of grad school in Energy Engineering. I have been out of practice, and am facing major struggles getting back into my coursework. I understand that the purpose of this forum is not to provide homework solutions, but rather to give a direction or guidelines to finding a solution for the inquiring party. Thank you for your help in this regard. I hope to get back into the swing of things, and become a contributing member of this community soon!

Homework Statement

Speed of Electrons: Estimate the average random speed of an electron gas in a semiconductor at 300 K.

Homework Equations

Fermi-Dirac Distribution: f(E) = 1/(exp[ (E-u)/(KbT)]+1.
Mass of Electron: 9.1x10^-31kg
Classical Interpretation and Kinetic Theory: 1/2mv^2 = 3/2 KbT <== I'm tempted to use this, but I find it unreliable and not accurate for an electron gas, but that could just be me making things more difficult than they need to be... If this were the case, then obviously v would be Sqrt(3KbT/mass), which comes out to be about 1.5x10^5. Which IS in fact a relatively close estimate for semiconductors, BUT it's hardly what I'd call a rigorous explanation.

The Attempt at a Solution

See above related to classical interpretation and Kinetic Theory of Gasses.

Thanks again!

 

[KataKoniK]

I can understand your struggles in getting back into coursework after being out of practice for a while. It's important to remember that it takes time and effort to get back into the swing of things, but with determination and persistence, I have no doubt that you will become a contributing member of this community soon.

Now, onto the question at hand. To estimate the average random speed of an electron gas in a semiconductor at 300 K, we can use the Fermi-Dirac distribution equation you provided: f(E) = 1/(exp[ (E-u)/(KbT)]+1. This equation takes into account the quantum nature of electrons in a solid and is more accurate for an electron gas than the classical interpretation and kinetic theory.

To solve for the average speed, we need to first find the Fermi energy (u) for the semiconductor. This can be calculated using the equation u = (h^2/8m)(3π^2n)^(2/3), where h is Planck's constant, m is the mass of an electron, and n is the electron density of the semiconductor. Once we have the Fermi energy, we can plug it into the Fermi-Dirac distribution equation along with the temperature (300 K) and the Boltzmann constant (Kb) to solve for the average speed.

I hope this helps guide you in finding a solution to your question. Remember, it's important to understand the underlying principles and use the appropriate equations rather than relying on simplified or inaccurate methods. Good luck with your studies and I look forward to seeing you become a valuable member of this community.