How Does Effective Mass Affect Plasmon Dispersion in Tetragonal Crystals?

[Igor_S]

I need help with a problem in solid state physics:

Find the dispersion relation of long-wavelength plasmons in a simple tetragonal crystal in the case of almost empty band built of s-type orbitals. What happens in the case when $$m_{xx} << m_{zz}$$ ?

This is what I did for now (with some help from a friend):

I started from:

$$\omega^2 = \frac{c^2 k^2}{\epsilon(\omega)}$$

For wave vector k=0 (this is very long wavelength approximation) I have formula (from my lectures, but there is also in Ashcroft & Mermin, (26.19)):

$$\epsilon(\omega) = 1 - \frac{\Omega_p^2}{\omega^2}$$, where $$\Omega_p$$ is plasma frequency:

$$\Omega_p^2 = \frac{4\pi e^2 n}{m}$$, where m is (I suppose effective?) electron mass, which needs to be found. Effective mass tensor is defined by:

$$m_{ij} = \frac{\hbar^2}{\partial^2 E(k) / \partial k_i \partial k_j}$$

where E(k) for tetragonal lattice with s-type orbitals is:

$$E(k) = E_s - J \cos(k_x a) - J \cos(k_y a) - J \cos(k_z c)$$.

So what "m" shoud I insert in equation for $$\Omega_p$$, maybe $$\sqrt{2m_{xx}^2 + m_{zz}^2}$$ ? In that case last part would be easy.. . Is this making sense ? :uhh:


Thanks.

 

[Nate810]

Yes, this makes sense. You can use the expression you have given for the effective mass tensor to calculate the effective mass (m) of the electron in the tetragonal crystal. Then, insert this expression into your equation for the plasma frequency (Ωp). In the case when mxx << mzz, the expression for the plasma frequency simplifies to Ωp2 = 4πe2n/mzz.

 

[thepatientmental]

Thank you for reaching out for help with your solid state physics problem. Finding the dispersion relation of long-wavelength plasmons in a simple tetragonal crystal can be a challenging task, so it's great that you're seeking assistance.

First, let's review some key concepts. A plasmon is a collective excitation of the electron gas in a solid, and it can be described by a dispersion relation, which relates the energy (\omega) and momentum (k) of the plasmon. In the long-wavelength limit, we can use the classical Drude model to describe the plasmon dispersion, which is given by the formula you mentioned: \omega^2 = \frac{c^2 k^2}{\epsilon(\omega)}, where c is the speed of light and \epsilon(\omega) is the dielectric function of the material.

In the case of a simple tetragonal crystal with an almost empty band built of s-type orbitals, we can use the effective mass approximation to determine the dispersion relation. This approximation assumes that the electrons behave as free particles with an effective mass (m) that is different from their true mass. In your case, the effective mass tensor is given by m_{ij} = \frac{\hbar^2}{\partial^2 E(k) / \partial k_i \partial k_j}, where E(k) is the energy of an electron with wave vector k.

To determine the plasma frequency \Omega_p, we need to calculate the effective mass for the s-type orbitals in the tetragonal lattice. From the expression for E(k) that you provided, we can see that the electron energy only depends on the x and y components of the wave vector (k_x and k_y), and not on the z component (k_z). This means that m_{zz} = 0 and we only need to consider the effective mass in the x and y directions, which we can calculate using the formula you suggested: \sqrt{2m_{xx}^2 + m_{yy}^2}. The final result for the plasma frequency is then \Omega_p = \sqrt{\frac{8\pi e^2 n}{m_{xx}}}, where n is the electron density.

In the case when m_{xx} << m_{zz}, the effective mass in the x direction is much smaller than that in the z direction. This means that the plasma frequency will be dominated