The binding energies for the bound states of a hydrogenic impurity placed on the axis of a quantum-well wire are calculated with the use of variational solutions to the effective-mass equation. The quantum-well wire is assumed to be a cylinder of GaAs surrounded by ${\mathrm{Ga}}_{1-x}{\mathrm{Al}}_x\mathrm{As}$. In a very small wire the electrons leak out of the wire and behave as three-dimensional electrons in ${\mathrm{Ga}}_{1-x}{\mathrm{Al}}_x\mathrm{As}$. An abrupt crossover to one-dimensional behavior occurs when the wire radius becomes greater than the radial spread of the bound state. In a very large wire the bound electrons no longer interact with the wire boundary and they behave as three-dimensional electrons in GaAs. In the quasi-one-dimensional regime, the binding energies are greatly enhanced and the wave functions are squeezed radially to fit the wire. In a small wire surrounded by an infinite barrier well, the electrons always behave as quasi-one-dimensional electrons and the binding energy of the $1s$ state becomes infinite when the wire radius vanishes. For a GaAs wire surrounded by ${\mathrm{Ga}}_{0.6}$${\mathrm{Al}}_{0.4}$As, the maximum $1s$ binding energy is 6 times greater than the $1s$ binding energy in bulk GaAs and 2-3 times greater than those in comparable two-dimensional quantum wells. The implications of this enhanced binding for recent calculations of the effect of ionized impurity scattering on electron mobility in quantum-well wires are considered. Although scattering by charged impurities effectively limits the mobility in quasi-one-dimensional systems, the enhanced binding substantially reduces the number of available scattering centers.

• Received 24 October 1983

DOI:https://doi.org/10.1103/PhysRevB.29.6632