Boson peak in amorphous graphene in the stable random matrix model

The effect of disorder in the distribution of force constants on optical and acoustic phonons in the scalar model of crystalline graphene is studied for both oscillations lying in the sheet plane and for flexural modes. It was shown that in the model of stable random matrices with translational symmetry, an additional to Debye vibrational density of states arises at a sufficient degree of disorder, i.e., the boson peak. The boson peak shifts to lower frequencies with increasing relative fluctuations of force constants and decreasing Young’s modulus of the system. At a weak disorder (or with no disorder), there are two peaks in the density of states $g(\omega)$, which correspond to logarithmic van-Hove singularity for acoustic and optical phonons of crystalline graphene. These peaks broaden and merge into a single boson peak with increasing disorder. Optical phonons are first destroyed due to disorder, while acoustic phonons gradually transform to the boson peak. For flexural modes there is a slightly different situation. Van-Hove singularities still spread disorder, but lead to the appearance of phonons in the system, which form the boson peak and move with it to low frequencies with increasing disorder.