InAs quantum-well structures are grown on GaAs(001) by molecular-beam epitaxy. Although the lattice-mismatch stress is released during the heteroepitaxy, as manifested by the appearance of a cross-hatch pattern, the strain field originating from composition grading in the buffer layer prevents the extension of misfit dislocations to the quantum well. The carrier mobility in the electrical conduction is improved in such a circumstance when the irregular height fluctuations generated in the strain relaxation, in addition to the quasi-periodic modulation of the cross-hatch surface, are smoothed out prior to the growth of the quantum wells. The interface fluctuations of the quantum wells imprinted by the random roughness are thus indicated to be more crucial for causing scattering than the latter long-range meandering of the surface. The introduction of the smoothing layer makes the use of less latticemismatched but more expensive substrates, such as InP substrates, unimportant.