Hybrid superconductor–semiconductor heterostructures are a promising platform for quantum devices based on mesoscopic and topological superconductivity. In these structures, a semiconductor must be in close proximity to a superconductor and form an Ohmic contact. This can be accommodated in narrow bandgap semiconductors, such as InAs, where the surface Fermi level is positioned close to the conduction band. In this work, we study the structural properties of near-surface InAs quantum wells and find that surface morphology is closely connected to low-temperature transport, where electron mobility is highly sensitive to the growth temperature of the underlying graded buffer layer. By introducing an In0.81Al0.19As capping layer, we show that we change the surface Fermi level pinning of the In0.81Al0.19As thin film as compared to the In0.81Ga0.19As, giving rise to a tuning of the Fermi level in the InAs layer. Experimental measurements show a strong agreement with Schrödinger–Poisson calculations of the electron density, suggesting the conduction band energy of the In0.81Ga0.19As and In0.81Al0.19As surface is pinned to 40 and 309 meV above the Fermi level, respectively.