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\subsubsection{\texorpdfstring{Thermal expansion of graphyne}{Thermal expansion of graphyne}}
\label{sec:thermal_expansion_of_graphyne}


We have investigated the thermal expansion behaviors of three different types (α, β, and γ) of graphyne
which is a two-dimensional carbon allotrope composed of \emph{sp} and \emph{sp\textsuperscript{2}} bonds.
We have found that all three types of graphyne exhibit negative in-plane thermal expansion.
The observed in-plane thermal contraction can be attributed to the ripple effect, similarly in graphene. However, 
the ripple effect itself 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 amplified contraction behavior.
We found that there are particular phonon modes with frequencies around a few hundreds of cm\textsuperscript{-1} existing exclusively in
graphyne that may fill empty spaces resulting in area reduction.
These modes are attributed to the libration of ,,rigid units" composed of triple bonds

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