The dispersion (\(E(k)\)) of the conduction band (CB) and valence band (VB) for a semiconductor are shown schematically in the figure. Considering the possibility of an electron making a transition from the bottom of the CB to the top of the VB, which of the following options is/are correct?
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The CB minimum and VB maximum sit at different values of \(k\) in the figure, so the transition needs a change in crystal momentum of \(\hbar q\).
A photon alone cannot supply that momentum change, so a phonon must be created alongside it, and the photon then carries less than \(E_g\).
A photon can be emitted with an energy exactly equal to \(E_g\).
A photon can be emitted with an energy less than \(E_g\).
A phonon can be created with a crystal momentum \(\hbar q\).
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The Correct Option isC, D
Solution and Explanation
Step 1: Read the band diagram.
The figure shows an indirect-gap semiconductor: the valence band maximum sits at \(k=0\) while the conduction band minimum sits at a different wavevector, \(k=q\), an energy \(E_g\) higher. The bottom of the CB and the top of the VB are not lined up on the same vertical line in \(k\).
Step 2: Recall what a photon can and cannot carry.
A photon carries energy \(\hbar\omega\) freely, but its momentum \(\hbar\omega/c\) is extremely small on the scale of a crystal's Brillouin zone, so absorbing or emitting a photon leaves the electron's crystal momentum \(k\) essentially unchanged. A transition driven by a photon alone is a vertical transition on the \(E\) vs \(k\) plot.
Step 3: Recall what a phonon can carry.
A lattice vibration (phonon) carries very little energy compared to \(E_g\) but can carry a crystal momentum comparable to \(q\), since phonon wavevectors span the same Brillouin zone as electrons. A phonon can supply or absorb the missing momentum in a transition.
Step 4: Check statement (A).
The transition from the bottom of the CB (\(k=q\)) to the top of the VB (\(k=0\)) needs a change of crystal momentum of \(\hbar q\). This is not forbidden outright, it just cannot happen through a photon alone, it needs a phonon to carry away that momentum. So the transition is allowed as a phonon-assisted process, and (A) is FALSE.
Step 5: Check statement (B).
Energy conservation across CB bottom to VB top gives exactly \(E_g\) to give away. If this energy were carried by a photon alone with no phonon involved, momentum is not conserved, since the photon cannot supply the required \(\hbar q\). A real recombination event here always needs a phonon to take some energy and momentum, so a photon carrying the full \(E_g\) alone is not the process that occurs. So (B) is FALSE.
Step 6: Check statement (C).
In the phonon-assisted process, the electron drops in energy by \(E_g\) total, splitting this between a phonon (energy \(E_{ph}\), carrying crystal momentum \(\hbar q\)) and a photon. Energy conservation gives photon energy \(= E_g - E_{ph}\), which is strictly less than \(E_g\) whenever a phonon is emitted alongside. So a photon of energy less than \(E_g\) can indeed be emitted, and (C) is TRUE.
Step 7: Check statement (D).
The electron's crystal momentum changes by \(\hbar q\) as it drops from \(k=q\) to \(k=0\). Conservation of crystal momentum requires this change to go somewhere, and a phonon with crystal momentum \(\hbar q\) is exactly what supplies it. So (D) is TRUE.
Final Answer:
The recombination is an indirect, phonon-assisted transition, so (C) and (D) hold while (A) and (B) do not.
\[ \boxed{\text{C, D}} \]
