  Owing to their large effective mass, strong and tunable spin-orbit coupling,
and complex band-structure, two-dimensional hole systems (2DHSs) in GaAs
quantum wells provide rich platforms to probe exotic many-body physics, while
also offering potential applications in ballistic and spintronics devices, and
fault-tolerant topological quantum computing. We present here a systematic
study of molecular-beam-epitaxy grown, modulation-doped, GaAs (001) 2DHSs where
we explore the limits of low-temperature 2DHS mobility by optimizing two
parameters, the GaAs quantum well width and the alloy fraction ($x$) of the
flanking Al$_x$Ga$_{1-x}$As barriers. We obtain a breakthrough in 2DHS
mobility, with a peak value $\simeq 18 \times 10^6$ cm$^2$/Vs at a density of
3.8 $\times$ 10$^{10}$ /cm$^{2}$, implying a mean-free-path of $\simeq 57
\mu$m. Using transport calculations tailored to our structures, we analyze the
operating scattering mechanisms to explain the non-monotonic evolution of
mobility with density. We find it imperative to include the dependence of
effective mass on 2DHS density, well width, and $x$. We observe concomitant
improvement in quality as evinced by the appearance of delicate fractional
quantum Hall states at very low density.
