  We theoretically investigate electron-hole recollisions in high-harmonic
generation (HHG) in band-gap solids irradiated by linearly and elliptically
polarized drivers. We find that in many cases the emitted harmonics do not
originate in electron-hole pairs created at the minimum band gap, where the
tunneling probability is maximized, but rather in pairs created across an
extended region of the Brillouin zone (BZ). In these situations, the analogy to
gas-phase HHG in terms of the short- and long-trajectory categorizations is
inadequate. Our analysis methodology comprises three complementary levels of
theory: the numerical solutions to the semiconductor Bloch equations, an
extended semiclassical recollision model, and a quantum wave packet approach.
We apply this methodology to two general material types with representative
band structures: a bulk system and a hexagonal monolayer system. In the bulk,
the interband harmonics generated using elliptically-polarized drivers are
found to originate not from tunneling at the minimum band gap $\Gamma$, but
from regions away from it. In the monolayer system driven by linearly-polarized
pulses, tunneling regions near different symmetry points in the BZ lead to
distinct harmonic energies and emission profiles. We show that the imperfect
recollisions, where an electron-hole pair recollide while being spatially
separated, are important in both bulk and monolayer materials. The excellent
agreement between our three levels of theory highlights and characterizes the
complexity behind the HHG emission dynamics in solids, and expands on the
notion of interband HHG as always originating in trajectories tunnelled at the
minimum band gap. Our work furthers the fundamental understanding of HHG in
periodic systems and will benefit the future design of experiments.
