  This study examines the spin-lattice relaxation rate of Mn$^{2+}$ ions in
strained diluted magnetic semiconductor (Cd,Mn)Te/(Cd,Mg)Te quantum wells using
the optically detected magnetic resonance (ODMR) technique. By adjusting the
magnesium (Mg) content in the buffer layer, we created samples with different
strain levels. Our time-resolved ODMR results show that the spin-lattice
relaxation time becomes faster as strain increases. We also found that the
relaxation rate increases with both magnetic field and temperature, showing a
power-law behavior. To understand these observations, we used a theoretical
model based on six-level rate equations with non-equal level separations. This
model suggests that the main factor affecting relaxation in our samples is a
"direct" mechanism. The model's predictions match well with our experimental
data. Overall, our findings give insights into spin-lattice relaxation in
strained quantum wells and could be important for the development of future
quantum and spintronic devices.
