  Unlike in conventional semiconductors, both the chemical potential and the
band gap in bilayer graphene (BLG) can be tuned via application of external
electric field. Among numerous device implications, this property also
designates BLG as a candidate for high-performance thermoelectric material. In
this theoretical study we have calculated the Seebeck coefficients for abrupt
interface separating weakly- and heavily-doped areas in BLG, and for a more
realistic rectangular sample of mesoscopic size, contacted by two electrodes.
For a given band gap ($\Delta$) and temperature ($T$) the maximal Seebeck
coefficient is close to the Goldsmid-Sharp value $|S|_{\rm max}^{\rm
GS}=\Delta/(2eT)$, the deviations can be approximated by the asymptotic
expression $|S|_{\rm max}^{\rm GS}-|S|_{\rm
max}=(k_B/e)\times\left[\frac{1}{2}\ln{u}+\ln{}2-\frac{1}{2}+{\cal
O}(u^{-1})\right]$, with the electron charge $-e$, the Boltzmann constant
$k_B$, and $u = \Delta/(2k_BT)\gg{}1$. Surprisingly, the effects of trigonal
warping term in the BLG low-energy Hamiltonian are clearly visible at
few-Kelvin temperatures, for all accessible values of
$\Delta\leqslant{}300\,$meV. We also show that thermoelectric figure of merit
is noticeably enhanced ($ZT>3$) when a rigid substrate suppresses out-of-plane
vibrations, reducing the contribution from $ZA$ phonons to the thermal
conductivity.
