More Realistic Phonon Spectra: Optical Phonons, van Hove Singularities

Although Debye's theory is reasonable, it clearly oversimpli_es certain aspects of the physics. For instance, consider a crystal with a two-site basis. Half of the phonon modes will be optical modes. A crude approximation for the optical modes is an
Einstein spectrum:

In such a case, the energy will be:

With ω max chosen so that

Another feature missed by Debye's approximation is the existence of singularities
in the phonon density of states. Consider the spectrum of the linear chain:

The minimum of this spectrum is at k = 0. Here, the density of states is well described by Debye theory which, for a 1D chain predicts g(ω) ~ const:. The maximum is at k = π/a. Near the maximum, Debye theory breaks down; the density of states is singular:

In 3D, the singularity will be milder, but still present. Consider a cubic lattice. The spectrum can be expanded about a maximum as:

Then (6 maxima; 1/2 of each ellipsoid is in the B.Z.)

In 2D and 3D, there can also be saddle points, where but the eigenvalues of the second derivative matrix have di_erent signs. At a saddle point, the phonon spectrum again has a square root singularity. van Hove proved that every 3D phonon spectrum has at least one maximum and two saddle points (one with one negative eigenvalue, one with two negative eigenvalues). To see why this might be true, draw the spectrum in the full k-space, repeating the Brillouin zone. Imagine drawing lines connecting the minima of the spectrum to the nearest neighboring minima (i.e. from each copy of the B.Z. to its neighbors). Imagine doing the same with the maxima. These lines intersect; at these intersections, we expect saddle points.

Kevin M Contreras H
Electrónica del Estado Sólido
http://www.physics.ucla.edu/~nayak/solid_state.pdf