Eliminating imaginary phonon frequencies in the phonon dispersion of coved graphene nanoribbons (GNRs) is a complex task that involves understanding the underlying physics and potentially modifying the material's structure or synthesis process. Here are some general strategies that could be considered:

1. Theoretical Analysis: Use first-principles calculations to understand the electronic and vibrational properties of the GNRs. Identify the regions where the imaginary phonon frequencies occur and analyze the atomic displacements associated with these modes.

2. Structural Optimization: Modify the atomic structure of the GNRs to eliminate the conditions that give rise to imaginary frequencies. This could involve changing the edge structure, adjusting the ribbon width, or introducing defects or dopants that alter the phonon behavior.

3. Strain Engineering: Apply strain to the GNRs to modify their lattice parameters and potentially eliminate the imaginary modes. Strain can significantly affect the phonon dispersion relations.

4. Temperature Control: Since phonon frequencies can be temperature-dependent, controlling the temperature of the system might help in reducing or eliminating the imaginary modes.

5. Chemical Functionalization: Introduce chemical groups or functionalize the edges of the GNRs to change the electronic and vibrational properties, which could affect the phonon dispersion.

6. Hybrid Structures: Create hybrid structures by combining GNRs with other materials that have different phonon properties, which might help in stabilizing the phonon modes.

7. External Fields: Apply external fields such as electric or magnetic fields to influence the electronic structure and potentially the phonon properties.

8. Defect Engineering: Introduce controlled defects in the GNRs that can modify the phonon dispersion without significantly degrading the material's properties.

9. Synthesis Process: Optimize the synthesis process to achieve a higher quality of GNRs with fewer defects that might give rise to imaginary phonon frequencies.

10. Experimental Validation: Use experimental techniques such as inelastic neutron scattering or Raman spectroscopy to validate the theoretical predictions and guide the optimization process.

It's important to note that the presence of imaginary phonon frequencies might be an indication of instabilities in the material or the calculation method. Therefore, a thorough investigation is necessary to determine the most appropriate approach to eliminate or mitigate these frequencies.

Dear Guohua zhang Zhang,

Thank you very much for providing this valuable information. It is a pleasure to have these insights, as they may help solve my problem.

I am currently focusing on point 2, structure optimization, which includes changing the pseudo potential, energy cut-off, and other related parameters.

Once again, thank you for your assistance.