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I'm currently trying to understand this paper explaining the origin of band inversion in the topological insulator Bi2Se3. In Fig. 2 (see below) they explain by means of 4 steps how certain atomic (and later bonding) bismuth and selenium p-orbitals behave under consideration of hybridization (I), the formation of chemical bonds (II), crystal field splitting and (III) and finally spin-orbit coupling (IV). enter image description here I'm wondering why the energy of the $P2_z^-$ state is increased while the one of $P1_z^+$ is lowered. Maybe someone could explain to me why these states are affected by spin-orbit coupling in opposite ways?

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  • $\begingroup$ +1 for a very interesting question! Welcome to our new community and we hope to see much more of you in the future!!! $\endgroup$ Commented Feb 18, 2023 at 15:35

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Welcome to MMSE! That's an interesting question! At first the "crossing" of energy levels seemed peculiar to me, but when I look at the energy levels for Se and for Bi separately, everything seems to follow a consistent pattern and the cross-over just happens as a matter of coincidence because of how close the Bi and Se levels are to each other (at least in this figure's depiction).

As for why the various perturbations seem to be affecting the Bi and Se levels in opposite ways, all I had to do is to notice that this was Figure 2, and then find where "Fig. 2" was mentioned in the text, and the answer was given in the same sentence:

"Since all the Se layers are separated by Bi layers,the strongest coupling in this system is the coupling between Bi layers and Se layers. Such coupling causes level repulsion, so that the Bi energy levels are pushed up and form new hybridized states |B> and |B'> while the Se energy levels are pushed down and yield three states |S>,|S'>, and |S0>, as shown in Fig.2(I)."

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  • $\begingroup$ Thanks for your answer. Unfortunately, the quote you referred to explains Step (I) which is still quite clear to me. I actually was wondering about the process occurring in the right green rectangle. The Spin-Orbit coupling supposedly increases the Selenium state energy while decreasing the energy of the Bismuth state. My question was why these two states are affected oppositely by the same effect (SOC). It seems to be due to the opposite parities (+ in Bi and - in Se respectively) but I haven't found a source yet explaining the connection between parity and SOC $\endgroup$
    – Mika R.
    Commented Feb 18, 2023 at 17:13
  • $\begingroup$ Was the choice of +/- labeling done based on the effect of spin-orbit coupling? $\endgroup$ Commented Feb 18, 2023 at 17:18
  • $\begingroup$ The +/- refers to an even/odd parity eigenstate if I understood correctly $\endgroup$
    – Mika R.
    Commented Feb 18, 2023 at 17:21
  • $\begingroup$ Let's talk in the spectroscopy chat room! $\endgroup$ Commented Feb 18, 2023 at 17:24

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