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

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.