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Rigid Unit Modes in $sp$-$sp^2$ Hybridized Carbon Systems: Origin of Negative Thermal Expansion

Using density functional theory combined with quasi-harmonic approximation, we investigate the thermal expansion behaviors of three different types (α, β, and γ) of graphyne which is a two-dimensional carbon allotrope composed of sp and $sp^2$ bonds. For each type of graphyne, we obtain the temperature dependent area variation by minimizing its free energy as a function of temperature, which is calculated by considering all the phonon modes in the whole Brillouin zone. We find that all three types of graphyne exhibit negative in-plane thermal expansion up to $T\lesssim1000$ K. The observed in-plane thermal contraction can be attributed partially to the ripple effect, similarly in graphene. The ripple effect itself, however, is not sufficient to explain anomalously larger thermal contraction found in graphyne than in graphene. Our deliberate analysis on the phonon modes observed in graphyne enables us to discover another source causing such thermal expansion anomaly. We find that there are particular phonon modes with frequencies around a few hundreds of cm$^{-1}$ existing exclusively in graphyne that may fill empty spaces resulting in area reduction. These modes are identified as "rigid unit modes" corresponding to the libration of each rigid unit composed of $sp^2$ bonds.