Peak position of the intersubband absorption spectrum of quantum wells with controlled electron concentrations

We investigated the influence of electron concentration $N_s$ on the peak position energy of intersubband absorption spectra in single modulation doped $\mathrm{GaAs}-{\mathrm{Al}}_{0.4}{\mathrm{Ga}}_{0.6}\mathrm{As}$ quantum well structures of different thicknesses and doping profiles. We found experimentally very small shifts of peak position as a function of $N_s.$ Also the sign of the energy shift is dependent on the well thickness. Findings were well explained by a compensation between depolarization shift and shift due to static Coulomb interaction of the electrons. The effect of the well width and the doping profile on the peak shift were also discussed.