The short answer is yes. For details, consult Sec. 2-4 (Emission and Absorption of Photons by Atoms) of the book Advanced Quantum Mechanics, by JJ Sakurai. The equation most relevant to your question is Eq. (2.95).

Dear

Behnam Fraid

Thank you very much for your reply. I have read in some conversation in research gate like that "........ Spin is directional and can be said to have odd parity. It follows that transitions in which the spin "direction" changes are forbidden. In formal terms, only states with the same total spin quantum number are "spin-allowed"."

Please help me to understand spin allowed transition from spin polarized band structure.

Dear Rana, in your main question you asked about the possibility of optical spin flip transitions, to which I responded by indicating that it is possible, since spin couples to the magnetic component of the electromagnetic radiation. Your latest question has to do with whether first-order transition is possible from any arbitrary initial state to any arbitrary final state, the answer to which is no. There are selection rules which have to be satisfied in order for a transition to be possible.

Selection rules are best understood by the group-theoretic symmetry considerations, under the rubric of Wigner-Eckart theorem, referred to in the book by Sakurai that I cited earlier. In particular, in the long-wavelength limit (that is, in the limit where the wave-vector k of light approaches 0) where the dipole approximation becomes applicable (approximating the exponential function corresponding to the propagating electromagnetic mode by its first two leading terms -- see Eq. (2.121) in the book by Sakurai; the coefficient of the spin-flip term in the Hamiltonian in the dipole approximation is given in the paragraph following Eq. (2.121)), optical (spin-flip) selection rules are easily deduced -- since x, or r, has odd parity under inversion (it is a vector), optical transitions in the long-wavelength limit are between states with different parities, from odd-parity states to even-parity states and vice versa. Consult any other good book on quantum mechanics, such as that by Merzbacher.

In dealing with the states in crystals, the symmetry considerations of the atomic physics do not apply, because of the crystal field that breaks the continuous rotation symmetry of the atomic physics. Here one will have to consider the irreducible representations of the space group of the underlying crystal corresponding to the relevant k point in the first Brillouin zone. Essentially, if you consider one state as the initial and another as the final state, you will have to decompose them in atomic orbitals (s, p, d, ...), and then apply atomic selection rules to these. For instance, if the initial state has a non-vanishing s component, and the final state a non-vanishing p component, then first-order optical transition is possible. Etc.

Incidentally, Chapter 15 of the book Quantum Processes in Semiconductors, 5th edition (Oxford University Press, 2013), by BK Ridley, is wholly devoted to spin processes. For group-theoretical considerations, consult the excellent book Group Theory: Applications to the Physics of Condensed Matter (Springer, Berlin, 2008), by MS Dresselhaus, G Dresselhaus, and A Jorio.

Very impresssive explanation Dr. Behnam Fraid. Thank You.

You are welcome Rana.

Dear

Dr. Fraid

I have get confused about difference between 'electronic band gap' and optical band gap. In most DFT  literatures I have read they report optical band gap as just the difference between VBM and CBM. I have also found some paper where there indroduce 'Symmetry Forbidden Transition' term to show the difference between fundamental gap and optical gap. Kindly make me clear about the difference. 

The electronic bandgap is the gap in the spectral function of the one-particle Green function, the optical bandgap is that in the spectral function of the density-density correlation function. The latter correlation function, which determines the dielectric response function, is related to the two-particle Green function. The two gaps are identical when the density-density correlation function is calculated in the random-phase approximation (RPA) -- which neglects the vertex corrections to the density-density response function -- using the one-particle Green function at hand.

As regards the Green functions, consult the book Quantum Theory of Many-Particle Systems, by AL Fetter abd JD Walecka, or Many-Particle Physics, 3rd edition, by GD Mahan. You may also wish to consult the book Fundamentals of Semiconductors: Physics and Materials Properties, 4th edition (Springer, Berlin, 2010), by PY Yu, and M Cardona.

Dear

Dr. Fraid

Greetings. I hope you are well.

I want to know about Majority and Minority Band gap. For spin polarized DFT calculation I have got Spin Up (Majority) and Spin Down (Minority) Band structure. Now to calculate band gap what should I use?

1. Spin up band?

2. spin down band?

3. both?

4. Spin up- spin down gap?

I my calculation I have found that CBM is for spin up and VBM for spin down.

My experimental gap is 2.10 eV. DFT result is much lower that experimental although I have use = 3 to 9 eV with interval 0.5eV. What i am noticing is that direct band gap at L point between spin down channels are matching very well with experiment.

Is it logical to explain band gap with such spin down band? If yes, Why?

Please check the attached figures.