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.
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