Floppy modes in network glasses

We discuss how concepts from rigidity percolation can be used to understand the low-frequency excitations and elastic properties of network glasses like ${\mathrm{Ge}}_{\mathrm{x}}$ ${\mathrm{As}}_{\mathrm{y}}$ ${\mathrm{Se}}_{1\mathrm{-}\mathrm{x}\mathrm{-}\mathrm{y}}$. When the mean coordination 〈r〉=2+2x+y is low, these materials are soft and their properties are strongly influenced by low-frequency phonons. We use a bond-depleted diamond lattice to mimic the coordination properties of the glass. We show that a model with only covalent forces is unstable for 〈r〉<2.4 but can be stabilized by small additional forces. We calculate the elastic constants, the density of states, and the Debye-Waller factor as a function of 〈r〉. Despite the simplicity of the model, the rounded phase transition at 〈r〉=2.4 is consistent with recent experimental results involving ultrasonics, inelastic neutron scattering, and Mössbauer experiments on ${\mathrm{Ge}}_{\mathrm{x}}$ ${\mathrm{Se}}_{1\mathrm{-}\mathrm{x}}$ glasses. The floppy modes, observed in inelastic neutron scattering, disappear as 〈r〉 increases from 2 up to around 2.4.