  On the basis of first-principles calculations and the special displacement
method, we demonstrate the quantum confinement scaling law of the
phonon-induced gap renormalization of graphene quantum dots (GQDs). We employ
zigzag-edged GQDs with hydrogen passivation and embedded in hexagonal boron
nitride. Our calculations for GQDs in the sub-10 nm region reveal strong
quantum confinement of the zero-point renormalization ranging from 20 to 250
meV. To obtain these values we introduce a correction to the Allen-Heine theory
of temperature-dependent energy levels that arises from the phonon-induced
splitting of 2-fold degenerate edge states. This correction amounts to more
than 50% of the gap renormalization. We also present momentum-resolved spectral
functions of GQDs, which are not reported in previous contributions. Our
results lay the foundation to systematically engineer temperature-dependent
electronic structures of GQDs for applications in solar cells, electronic
transport, and quantum computing devices.
