# Updating Carrier Momentum after Scattering

1. Dec 1, 2007

### JasonW

1. The problem statement, all variables and given/known data
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 $$Z_{\varphi}$$.

2. Relevant 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....

3. 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 text book. 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.