A theory is presented for the strong enhancement of graphene-on-substrate bandgaps by attractive interactions mediated through phonons in a polarizable superstrate. It is demonstrated that gaps of up to 1eV can be formed for experimentally achievable values of electron-phonon coupling and phonon frequency. Gap enhancements range between 1 and 4, indicating possible benefits to graphene electronics through greater bandgap control for digital applications, lasers, LEDs and photovoltaics through the relatively simple application of polarizable materials such as SiO2 and Si3N4.
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