  Exciton level structure and interwell relaxation are studied in
Cd(Mn,Mg)Te-based asymmetric double quantum wells (ADQWs) in normal to plane
magnetic fields up to B = 10 T by a steady-state optical spectroscopy. As grown
structures with nonmagnetic CdTe quantum wells (QWs) were subjected to quick
temperature annealing to introduce Mn and Mg atoms from the barriers inside the
wells resulting in formation of magnetic (MW) and nonmagnetic (NMW) QWs,
respectively, with the concentration of diffused atoms of about 3--5%. A
significant change of exciton energies occurs with magnetic field: at low
fields exciton, localized in the MW, is higher in energy than that localized in
the NMW and efficient exciton relaxation from the MW to the NMW takes place for
all ADQWs. In all structures the giant Zeeman effect in the MW changes the
energy order of $\sigma^+$-polarized heavy hole (hh) excitons, localized in
different wells, at high B. Levels' crossing is accompanied by a reverse of
tunneling direction without anticrossing. Calculations of single-particle
states and their change with B indicate that the interwell exciton transfer is
forbidden in the single-particle picture at B $\gtrsim$ 1 T for all studied
structures. Experimentally, nevertheless, a very efficient interwell relaxation
of excitons is found in the whole magnetic field range regardless of tunneling
direction which evidences about importance of electron-hole Coulomb
correlations in tunneling process. Different charge-transfer mechanism are
analysed in details and elastic scattering due to strong disorder is suggested
as the main tunneling mechanism of excitons with underlying influence of the
valence band-mixing effects on the hh-exciton transfer in ADQWs with relaxed
stress.
