The band structure of TiO2 is well described by the effective mass approximation. The dipole allowed optical transitions from the four heavy and light hole ~hh and lh! bands to the conduction band in TiO2 QW’s are induced by circularly polarized photons ). The addition of a magnetic field lifts the degeneracy between the spin-up and spin-down conduction band states, as well as splitting the conduction band into a series of degenerate Landau levels. The dependence of the valence band structure on the magnetic field is complex, due to strong band mixing.32–34 The eigenvectors for the valence band states take on the four-component spinor , where the first subscript is the z component of the angular momentum mJ , and the second is the harmonic oscillator index, which describes the nature of the Landau level associated with that mJ state. The selection rules require the photon can only couple states that have the same harmonic oscillator character n. For excitations into the lowest conduction band Landau levels, n50, the lowest energy transitions are shown in Fig. 1. These transitions were calculated to be the lowest in energy for TiO2 QW’s in high field.34 As Fig. 1 shows, there are several transitions to both of the lowest electron Landau levels excited by s2 polarized light, and only one transition excited by s1. Because of this difference, we have performed our nonlinear experiments using s1 polarized light, to simplify the interpretation. Even without the Coulomb interaction, the presence of the doped conduction band electrons changes the linear optics of the QW. The extra electrons in the lower Landau levels prevent the addition of electrons because of the Pauli exclusion principle. The filling factor is defined as n5Ne /D 52pl c 2 Ne}1/B. At some field, all the electrons can fit into the lowest Landau level, with all the others empty (n51). It is only once we reach this point that we should be able to see absorption into the lowest Landau level. This can be seen in Fig. 2, which shows the linear absorption spectra of a MDQW sample for many different magnetic fields. Looking at Fig. 2, we can see the onset of absorption into the lowest Landau level, LL0, between B54 T and B55 T. Figure 3 gives more details of this effect, by looking at the peak height and area of LL0 as a function of the field. By extrapolating the LL0 peak height or area down to zero, we can confirm the doped carrier density. For our sample, the peak height and area reach zero at B;4.3 T, which confirms our electron density to be n'2.131011 cm22. Let us examine the linewidths of the Landau level peaks. Figure 4 shows the peak energy and linewidths of the lowest two Landau levels of the MDQW sample. While the linewidth of LL0 is approximately constant, the linewidth of LL1 increases significantly once LL0 starts to appear in the spectrum. This is an important point to notice: when the LL0 transition has finite oscillator strength, the LL1 peak is broadened significantly. The hh LL0 peak is comparable in width to what is seen in undoped QW samples, and is fairly well fitted by a single Lorentzian line. However, in an undoped QW sample, the linewidth of LL1 also remains more or less constant, and does not become significantly larger than that of LL0. The increase in the LL1 linewidth for smaller filling factors implies an increase in the dephasing rate of this state due to scattering with the 2DEG, which for these fields is entirely

By understanding what such a figure displays, namely energy-or frequency-as a function of the wavevector.

In particular, these curves don’t depend on the polarlzation; there are cases where they do, of course.

So knowing the relation between frequency and wavevector for light-that’s what ``optical" means-it’s possible to answer the question.

Dear Sruthil,

Your question is complicated and implies that selection rules can be deduced from your figure. Actually, selection rules are the most favorable transitions between neighbour levels. In particular, optical transitions will more likely happen between points with close wavevectors

Dear Sruthil Lal S B ,

Adding to the colleagues above, the most favorable transitions are those transitions that satisfy the conservation of energy and momentum of the incident photon and the valence electron. That is the sum of the energy of the energy and momentum before the collection is equal with those after collision.

Since the photo momentum is negligibly small, then the favorable transitions are those of the direct band transition.

The transition of electrons are such that when it is transferred from an orbital with energy E1 and momentum M1 to an orbital with energy E2 and momentum M2 it must be given the difference in the energy and in the momentum. So since the photon can not change the momentum of the electron, so it will be transferred to an orbital with same momentum but with an energy= E1+Eph where Eph is the photon energy.

Best wishes

Thank you Abdelhalim abdelnaby Zekry , Muhammad Hamza El-Saba , Stam Nicolis , Hadi Jabbar Alagealy for your great insights!