How Is the Dynamical Matrix Used to Calculate the Phonon Spectrum of Graphene?

[nova215]

TL;DR Summary

Solving this matrix for its eigenvalues produced the phonon spectrum of graphene - what was the method for finding the eigenvalues?

Starting on page 11 of this paper on lattice dynamics, the phonon spectrum of graphene is calculated. I do not really understand how the author used the matrix they created in order to calculate the spectrum. Thanks!

 

[dRic2]

He first defines the force-constants in real space (that is, the second derivative of the potential, see eq 1) and labels them $\phi_{ij}(n, m)$. Once all the force-constants are defined, he uses the definition of the dynamical matrix (eq 4) to obtain the matrix D (which is just the "Fourier transform" of the dynamical matrix $\phi_{ij}(n,m)$).

Finally, once you know the form of D, it is just a matter of solving the eigenvalue problem:
$$D - \omega^2 I = 0$$
(where $I$ is the identity matrix). He does this final step at the end of page 12 to obtain the values in eq (26-29)

 

[nova215]

Thank you so much for the response!

Why does he vary his calculations for the different symmetry points and how is the calculation of the various frequencies changed when he uses different symmetry points? Also, what is the best way to solve that eigenvalue problem in your response?

I really appreciate the help!

 

[dRic2]

I don't understand your question. Once you solve the eigenvalue problem $D - \omega^2 I = 0$ (see the previous answer), you find four frequencies as a function of your wave-vector $\mathbf k$:
$$\omega_i = \omega_i(\mathbf k) = \omega_i(k_x, k_y)$$
since this a two-dimensional problem, i.e. $\mathbf k = (k_x, k_y)$. Now, if you are interested in the frequencies at a particular point of the Brillouin zone, you just plug in the coordinates of that point: for example, if you want to calculate the frequencies at M, which has coordinates $(2 \pi/\sqrt{3}a, 0)$, you just evaluate $\omega_i(2 \pi/\sqrt{3}a, 0)$.
If you want to plot the dispersion along the direction $\Gamma \rightarrow M$, since both $\Gamma$ and $M$ have $k_y = 0$, you simply need to calculate $\omega_i(k_x, 0)$ and, as in the paper, you'll find 4 mathematical expression as functions of $k_x$ (see eq. 26-29).