The electronic and structural properties of the Bi-covered $\mathrm{Si}\ensuremath{-}\mathrm{Ge}$ two-dimensional superlattices (SLs) have been investigated employing ab initio total energy calculations. The $\mathrm{Si}\text{\ensuremath{-}}\mathrm{Ge}$ SLs are formed by Ge stripes with a height of one atomic layer, $\ensuremath{\sim}0.3\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$, separated by Si layers on the Si(111) surface covered with a monolayer of Bi trimers. At the equilibrium geometry we find that the Bi trimers on $\mathrm{Ge}∕\mathrm{Si}(111)$ are higher by $0.10\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$ compared with the Bi trimers on the $\mathrm{Si}∕\mathrm{Si}(111)$ region. In both regions the Bi trimers lie on the $T4$ sites. Our simulated scanning tunneling microscopy images exhibit bias-dependent ``electronic steps,'' which are in accordance with recent experimental findings [Phys. Rev. Lett. 91, 096102 (2003)]. These bias-dependent electronic steps, which can reach up to $0.8\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$, are due to the different electronic density of states of the Bi trimers adsorbed on the Ge stripes compared with the Bi trimers on the Si stripes. Since the Si and Ge regions of the SL are electronically distinct, we compared the electronic structures of the isolated systems, $\mathrm{Bi}∕\mathrm{Si}∕\mathrm{Si}(111)$ and $\mathrm{Bi}∕\mathrm{Ge}∕\mathrm{Si}(111)$, with the one of the $\mathrm{Si}\text{\ensuremath{-}}\mathrm{Ge}$ SL. We find a valence band offset of $0.21\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ for the $\mathrm{Si}\text{\ensuremath{-}}\mathrm{Ge}$ SL, with the highest occupied states lying on the Ge layers. In contrast, for the lowest unoccupied states we find an almost negligible conduction band offset.