  Bilayers of graphitic materials have potential applications in field effect
transistors (FETs). A potential difference applied between certain ionic
bilayers made from insulating graphitic materials such as BN, ZnO and AlN could
reduce gap sizes, turning them into useful semiconductors. On the other hand,
opening of a small semiconducting gap occurs in graphene bilayers under applied
field. The aim here is to investigate to what extent substrate induced
electron-phonon interactions (EPIs) modify this gap change. We examine EPIs in
several lattice configurations, using a perturbative approach. The typical
effect of EPIs on the ionic bilayers is an undesirable gap widening. The size
of this gap change varies considerably with lattice structure and the magnitude
of the bias. When bias is larger than the non-interacting gap size, EPIs have
the smallest effect on the bandgap, especially in configurations with
$AA^{\prime}$ and $AB$ structures. Thus careful selection of substrate, lattice
configuration and bias strength to minimise the effects of EPIs could be
important for optimising the properties of electronic devices. We use
parameters related to BN in this article. In practice, the results presented
here are broadly applicable to other graphitic bilayers, and are likely to be
qualitatively similar in metal dichalcogenide bilayers such as MoS$_2$, which
are already of high interest for their use in FETs.
