Cyclotron Resonance (Solid State Physics)

[cepheid]

Homework Statement

Cyclotron resonance for a spheroidal energy surface. Consider the energy surface

$$\epsilon(\mathbf{k}) = \hbar^2 \left( \frac{k_x^2 + k_y^2}{2m_t} + \frac{k_z^2}{2m_l} \right)$$​

where m_t is the transverse mass parameter and m_l is the longitudinal mass parameter. A surface on which $\epsilon(\mathbf{k})$ is constant will be a spheroid. Use the equation of motion with $\mathbf{v} = \hbar^{-1} \nabla_{\mathbf{k}} \epsilon$ to show that $\omega_c = eB/(m_l m_t)^{1/2}c$ when the static magnetic field B lies in the xy plane.

Homework Equations

Dynamics of Bloch Electrons

The equation of motion for an electron subject to the periodic potential of a crystal lattice is

$$\hbar \frac{d\mathbf{k}}{dt} = -\frac{e}{c}\mathbf{v} \times \mathbf{B} \ \ \ \textrm{cgs}$$$$\hbar \frac{d\mathbf{k}}{dt} = -e\mathbf{v} \times \mathbf{B} \ \ \ \textrm{SI}$$
​

The Attempt at a Solution

$$\mathbf{v} = \hbar^{-1} \nabla_{\mathbf{k}} \epsilon(\mathbf{k}) = \hbar^{-1} \left( \hat{k}_x \frac{\partial}{\partial k_x} + \hat{k}_y \frac{\partial}{\partial k_y} + \hat{k}_z \frac{\partial}{\partial k_z} \right) \hbar^2 \left( \frac{k_x^2 + k_y^2}{2m_t} + \frac{k_z^2}{2m_l} \right)$$​

$$= \hbar \left( \hat{k}_x \frac{k_x}{m_t} + \hat{k}_y \frac{k_y}{m_t} + \hat{k}_z \frac{k_z}{m_l} \right)$$​

Apply the equation of motion with

$$\mathbf{B} = B_x \hat{x} + B_y \hat{y}$$

$$\frac{d\mathbf{k}}{dt} = -\frac{e}{c}\left( \hat{k}_x \frac{k_x}{m_t} + \hat{k}_y \frac{k_y}{m_t} + \hat{k}_z \frac{k_z}{m_l} \right) \times (B_x \hat{x} + B_y \hat{y})$$
​

Right so, um how am I supposed to proceed to compute such a cross product?

 

[mjsd]

so, you have problem with mixing coordinate space vectors with momentum space vectors. you can still do this as long as you know what is
$$\hat k_i \times \hat x_j$$
at the moment I can't recall what this is... but i think $$\hat k_i \perp \hat x_i$$
should try the momentum space representation of the position operator as a guide to convert them, perhaps?

 

[Aaronse_r]

Did you ever figure this out?

 

[Aaronse_r]

I just had this on a test, but didn't get it done so as you can imagine i was curious about it. Also, just realized you put this on here a year ago so nevermind.