  Near-surface InAs two-dimensional electron gas (2DEG) systems have great
potential for realizing networks of multiple Majorana zero modes towards a
scalable topological quantum computer. Improving mobility in the near-surface
2DEGs is beneficial for stable topological superconducting states as well as
for correlation of multiple Majorana zero modes in a complex network. Here, we
investigate near-surface InAs 2DEGs (13 nm away from the surface) grown on
GaSb(001) substrates, whose lattice constant is closely matched to InAs, by
molecular beam epitaxy. The effect of 10-nm-thick top barrier to the mobility
is studied by comparing Al$_{0.9}$Ga$_{0.1}$Sb and In$_{0.75}$Ga$_{0.25}$As as
a top barrier on otherwise identical InAs quantum wells grown with identical
bottom barrier and buffer layers. A 3-nm-thick capping layer on
Al$_{0.9}$Ga$_{0.1}$Sb top barrier also affects the 2DEG electronic transport
properties by modifying scattering from 2D remote ionized impurities at the
surface. The highest transport mobility of 650,000 cm$^2$/Vs with an electron
density of 3.81 $\times$ 10$^{11}$ cm$^{-2}$ was observed in an InAs 2DEG with
an Al$_{0.9}$Ga$_{0.1}$Sb top barrier and an In$_{0.75}$Ga$_{0.25}$As capping
layer. Analysis of Shubnikov-de Haas oscillations in the high mobility sample
suggests that long-range scattering, such as remote ionized impurity
scattering, is the dominant scattering mechanism in the InAs 2DEGs grown on
GaSb(001) substrates. In comparison to InAs quantum wells grown on
lattice-mismatched InP, the ones grown on GaSb show smoother surface morphology
and higher quantum mobility. However, In$_{0.75}$Ga$_{0.25}$As top barrier in
InAs quantum well grown on GaSb limits the transport mobility by charged
dislocations formed in it, in addition to the major contribution to scattering
from the alloy scattering.
