= Solution
A larger-gap AlGaAs barrier has a higher conduction-band edge than GaAs. Donors in AlGaAs supply <electrons> that transfer into lower-energy GaAs states, leaving a positively charged donor region. The resulting electrostatic field bends the GaAs <conduction band> into a triangular interface well. Motion normal to the interface is quantized, while motion parallel to it remains approximately free, producing a \b[<two-dimensional electron gas>].
\Image[/past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2019/paper-2-modulation-doping.png]
{title=An illustrative remotely doped heterojunction separates the electron envelope from ionized donors}
{source=paper-2-modulation-doping.py}
The confined envelope peaks just inside GaAs and decays into the AlGaAs barrier; a triangular-well treatment gives an Airy-function-like envelope with boundary matching. An undoped spacer separates the <electrons> from the donors, strongly reducing ionized-impurity scattering compared with uniformly doped 3D material and enabling high mobility. <Band offsets>, donor supply, interface roughness and electrostatics jointly determine whether a single occupied subband is obtained.
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