  We analyze fluctuation of the layer thicknesses and its influence on the
strain state of (In,Ga)As/(Al,Ga)As micro-tubes containing quantum well
structures. In those structures a curved high-mobility two-dimensional electron
gas (HM2DEG) is established. The layer thickness fluctuation studied by atomic
force microscopy, x-ray scattering, and spatially resolved cathodoluminescence
spectroscopy occurs on two different lateral length scales. On the shorter
length scale of about 0.01~$\mu$m, we found from atomic force micrographs and
the broadening of the satellite maxima in x-ray diffraction curves a very small
value of the mean square roughness of 0.1~nm. However, on a longer length scale
of about 1.0~$\mu$m, step bunching during epitaxial growth resulted in layer
thickness inhomogeneities of up to 2~nm. The resulting fluctuation of the
strain in the micro-tubes leads to a local variation of the chemical potential,
which results in the fluctuation of the carrier density as well. This leads to
a phase cancelation of the Shubnikov-de-Haas oscillations in the curved HM2DEG
and a reduction of the single-electron scattering time, while the electron
mobility in the structures remains high. The estimated fluctuation of the
carrier density agrees well with the energy fluctuation measured in the
cathodoluminescence spectra of the free-electron transition of the quantum
well.
