We report on a systematic investigation of the dominant scattering mechanism in shallow two-dimensional electron gases (2DEGs) formed in modulation-doped GaAs/Al_{x}Ga_{1-x}As heterostructures. The power-law exponent of the electron mobility versus density, mu propto n^{alpha}, is extracted as a function of the 2DEGs depth. When shallower than 130 nm from the surface, the power-law exponent of the 2DEG, as well as the mobility, drops from alpha simeq 1.65 (130 nm deep) to alpha simeq 1.3 (60 nm deep). Our results for shallow 2DEGs are consistent with theoretical expectations for scattering by remote dopants, in contrast to the mobility-limiting background charged impurities of deeper heterostructures.
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