Updating Carrier Momentum after Scattering

[JasonW]

Homework Statement

Updating the carrier momentum after scattering is most easily accomplished in the rotated coordinate system. The rotated x-axis is related to the original x-axis by $$x_{r} = Y_{\theta}Z_{\varphi}x$$ where $$Y_{\theta}$$ describes a rotation of $${\theta}$$ about the y-axis, and $$Z_{\varphi}$$ describes a rotation of $${\varphi}$$ about the z-axis. The angles $${\theta}$$ and $${\varphi}$$ represent the polar and the azimuthal angles of the carrier momentum in the original coordinate system before the scattering event.

(a) Calculate the rotation matrices $$Y_{\theta}$$ and $$Z_{\varphi}$$.

Homework Equations

For Polar Optical Phonon Scattering
$$cos{\theta} = \frac{1+f-(1+2f)^r}{f}$$ where $$f = \frac{2\sqrt{E_{k}E_{k'}}}{(\sqrt{E_k}-\sqrt{E_{k'}})^2}$$

For isotropic scattering:
$$\varphi = 2\pi r_3$$
$$cos\theta = 1-2r_4$$
where r3 and r4 are uniform random numbers lying between 0 and 1

Other scattering mechanisms...

The Attempt at a Solution

To start with, this is my first class dealing with semiconductor physics and device modeling so I'm very hazy with a lot of the terminology used in this class which where many of my problems arize from.

I have two issues with this question, first is I don't know what a rotation matrix is. There is no mention of it in the class notes or our textbook. Anyone able to describe what a rotation matrix is?

2nd, from reading Numerical Simulation of Submicron Semiconductor Devices, I find several equations dealing with angles after scattering depending on what the scattering mechanism is, yet the problem does not state the scattering mechanism so I'm not sure how to tackle this problem.

Any advice?

 

[Jhero]

Thank you for your question. A rotation matrix is a mathematical tool used to describe the transformation of coordinates in a three-dimensional space. It is a square matrix that represents a rotation in a specific direction and by a specific angle. In this case, the rotation matrices Y_{\theta} and Z_{\varphi} are used to describe the rotation of the coordinate system in the y and z directions, respectively.

To calculate these matrices, you can use the following equations:

Y_{\theta} = \begin{bmatrix}
cos{\theta} & 0 & sin{\theta} \\
0 & 1 & 0 \\
-sin{\theta} & 0 & cos{\theta}
\end{bmatrix}

Z_{\varphi} = \begin{bmatrix}
cos{\varphi} & -sin{\varphi} & 0 \\
sin{\varphi} & cos{\varphi} & 0 \\
0 & 0 & 1
\end{bmatrix}

As for the scattering mechanism, the equations you have provided are for polar optical phonon scattering and isotropic scattering. Depending on the specific scattering mechanism, different equations may be used to calculate the angles after scattering. It would be helpful to clarify with your professor or refer to your textbook for the specific equations and parameters needed for the scattering mechanism in question.

I hope this helps clarify your understanding of the problem. Good luck with your studies!