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Owing to their large effective mass, strong and tunable spin-orbit coupling,and complex band-structure, two-dimensional hole systems (2DHSs) in GaAsquantum wells provide rich platforms to probe exotic many-body physics, whilealso offering potential applications in ballistic and spintronics devices, andfault-tolerant topological quantum computing. We present here a systematicstudy of molecular-beam-epitaxy grown, modulation-doped, GaAs (001) 2DHSs wherewe explore the limits of low-temperature 2DHS mobility by optimizing twoparameters, the GaAs quantum well width and the alloy fraction ($x$) of theflanking Al$_x$Ga$_{1-x}$As barriers. We obtain a breakthrough in 2DHSmobility, with a peak value $\simeq 18 \times 10^6$ cm$^2$/Vs at a density of3.8 $\times$ 10$^{10}$ /cm$^{2}$, implying a mean-free-path of $\simeq 57\mu$m. Using transport calculations tailored to our structures, we analyze theoperating scattering mechanisms to explain the non-monotonic evolution ofmobility with density. We find it imperative to include the dependence ofeffective mass on 2DHS density, well width, and $x$. We observe concomitantimprovement in quality as evinced by the appearance of delicate fractionalquantum Hall states at very low density.

Ultraclean two-dimensional hole systems with mobilities exceeding 10$^7$ cm$^2$/Vs