  GaGeTe is a layered topological semimetal that has been recently found to
exist in at least two different polytypes, $\alpha$-GaGeTe ($R\bar{3}m$) and
$\beta$-GaGeTe ($P6_3 mc$). Here we report a joint experimental and theoretical
study of the structural, vibrational, and electronic properties of these two
polytypes at high pressure. Both polytypes show anisotropic compressibility and
two phase transitions, above 7 and 15 GPa, respectively, as confirmed by XRD
and Raman spectroscopy measurements. Although the nature of the high-pressure
phases is not confirmed, comparison with other chalcogenides and total-energy
calculations allow us to propose possible high-pressure phases for both
polytypes with an increase in coordination for Ga and Ge atoms from 4 to 6. In
particular, the simplification of the X-ray patterns for both polytypes above
15 GPa suggests a transition to a structure of relatively higher symmetry than
the original one. This result is consistent with the rocksalt-like
high-pressure phases observed in parent III-VI semiconductors, such as GaTe,
GaSe, and InSe. Pressure-induced amorphization is observed upon pressure
release. The electronic band structures of $\alpha$-GaGeTe and $\beta$-GaGeTe
and their pressure dependence also show similarities to III-VI semiconductors,
thus suggesting that the germanene-like sublayer induces a semimetallic
character in both GaGeTe polytypes. Above 3 GPa, both polytypes lose their
topological features, due to the opening of the direct band gap, while the
reduction of the interlayer space increases the thermal conductivity at high
pressure.
