  We study molecular beam epitaxially grown, undoped
Al$_{x}$Ga$_{1-x}$Sb/InAs/AlSb quantum wells with different buffer and barrier
designs and varying quantum well width. The highest mobilities were achieved
with Al$_{0.33}$Ga$_{0.67}$Sb buffers and lower barriers and a quantum well
width of 24 nm. These quasi-single-interface InAs/AlSb quantum well devices
reached a gate-tuned mobility of 2.4$~\times~10^{6}~$cm$^2$/Vs at a density of
1$\times 10^{12}$ cm$^{-2}$ and 1.3 K. In Hall bar devices boundary scattering
is found to strongly influence the mobility determination in this mobility
regime. Ionized background impurity scattering at low electron densities,
device boundary scattering at intermediate electron densities, and intersubband
scattering at high electron densities were identified as most likely dominant
scattering processes. Ringlike structures in the Landau fan can be explained
using a single-particle model of crossing Landau levels.
