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A follow up question for this question. What are the different properties that can be deduced from a phonon dispersion curve and Phonon DOS? Also if possible explain how to deduce them?

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    $\begingroup$ 1: Identify the dynamical stability of your structure: No imaginary frequencies means your structure is the dynamical stability. 2: Obtain the frequencies of the Raman active modes. 3: Reference: aip.scitation.org/doi/10.1063/1.5010881 $\endgroup$ – Jack Oct 1 at 2:00
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The phonon dispersion relates the phonon frequencies $\omega_{\mathbf{q}\nu}$ for each branch $\nu$ with the phonon wave vector $\mathbf{q}$, typically along a path in the Brillouin zone joining high-symmetry points. The phonon density of states compresses this information by integrating over $\mathbf{q}$ and summing over $\nu$:

$$ \tag{1} g(\omega)=\sum_{\nu}\int\frac{d\mathbf{q}}{(2\pi)^3}\delta(\omega-\omega_{\mathbf{q}\nu}), $$

and some information is thus lost in the density of states. Quantities that can be extracted from a phonon dispersion include (happy for others to add more to the list):

  1. Speed of sound. The linear slope of the three acoustic branches as $\mathbf{q}\to0$ provide the speed of sound propagation in the material.
  2. Raman mode frequencies. The Raman mode frequencies are given by some of the optical mode frequencies as $\mathbf{q}\to0$. However, to determine which modes are Raman active you need to complement the dispersion relation with a symmetry analysis of the optical modes, and to determine the intensity of the Raman signal you need to complement the dispersion with a calculation of the Raman activity tensor.
  3. Infrared mode frequencies. Similar to Raman frequencies, but with different symmetry rules and intensities.
  4. LO-TO splitting. Longitudinal optical modes set up oscillating dipoles that lead to long-range electric fields that split the degeneracy between transverse and optical modes as $\mathbf{q}\to0$. The magnitude of LO-TO splitting depends on the dielectric permitivity and the Born effective charges, so you can assess how polar the material is through the LO-TO splitting.
  5. Helmholtz free energy. The phonon contribution to the free energy can be deduced from the phonon dispersion by occupying each phonon mode according to the Bose-Einstein distribution at the corresponding temperature. As this quantity only depends on the phonon energy, this is something that you can also directly extract from the phonon density of states.
  6. Dynamical stability. The abscence of imaginary phonon frequencies indicates that the system is dynamically stable (at a local minimum of the potential energy surface). Conversly, if there are imaginary phonon frequencies the system is dynamically unstable. Complementing this information with the atomic displacements of the imaginary modes provides information about the direction in which the structure needs to be distorted to lower its energy.
  7. Topological phonons. Although less well-known than their electronic counterparts, phonon dispersions also support a topological classification. For example, you may encounter topologically protected Weyl phonons or nodal line phonons. The dispersion itself will provide the degeneracy points/lines, but for a full classification you need to complement it with a symmetry/topology analysis.

Phonon dispersions also serve as starting points to study phonon-related properties. For example, the inclusion of anharmonic terms (phonon-phonon interactions) can help explore thermal transport or temperature-driven structural phase transitions. The coupling of phonons with electrons (electron-phonon interactions) can help explore electronic transport, superconductivity, carrier relaxation in semiconductors, etc.

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