Dark excitons in planar monolayers get a parity map: mirror symmetry splits the band and picks the phonon that lets them glow

In planar monolayers with a horizontal mirror plane, mirror parity assigns a definite value to every electronic state, phonon mode, and exciton band at every wavevector, so exciton-phonon scattering splits into two classes: even phonons connect excitons of equal parity, odd phonons connect excitons of opposite parity. Group theory had sorted exciton-phonon processes into allowed and forbidden, and a selection rule states what may happen rather than which channel a relaxing exciton actually uses. In monolayer GaN the lowest exciton band is even inside a closed contour around the center and odd throughout the rest of the Brillouin zone including K and M; the contour marks where an even and an odd band cross, and mirror symmetry forbids their mixing, so the crossing is protected and the parity switches across it.

The same rules govern emission: the dark exciton at K can only recover its momentum through one odd-parity phonon, and the light it releases is polarized in plane and circularly polarized, with the two valleys emitting opposite helicity. Phonon-assisted emission was known to be the only way out for a dark state, but which phonon carries it and what fixes the polarization had not been pinned down. Only one of the six phonon branches at K satisfies the parity requirement, the phonon-assisted emission is carried entirely by that mode with all other branches vanishing to numerical precision, and it produces a single satellite about 16 meV below the dark exciton, matching that phonon's energy at K.

Combining the parity rules with the real-time Boltzmann transport equation, a solver that follows how exciton population redistributes across the Brillouin zone in time, splits relaxation into two stages: the odd ZA phonon carries population across the parity boundary, and even phonons take over inside the odd domain. A linewidth gives the scattering rate of one state but cannot show how population crosses the parity boundary in time or which phonon branches actually move it. After in-plane optical injection at the center at 300 K, about 45% of the population reaches the K and K' valleys within 30 fs and about 90% has accumulated there by thermalization; branch-resolved scattering flux is largest for branches 4 and 6, while the polar optical branches carry exciton-phonon matrix elements an order of magnitude larger than any other mode yet contribute nothing to the linewidth of the lowest bright exciton.

The analysis targets planar monolayers that retain the horizontal mirror plane, with monolayer GaN as the test case and monolayer hBN as the contrast; only the planar phase keeps the mirror parity exact, and GaN grown so far by graphene encapsulation on SiC is buckled. The text extends the analysis to systems such as hBN and GaN, where the mirror parities of the band-edge states determine the parity of the lowest exciton-like state: even and bright in hBN, odd and dark in GaN.

The text presents linewidth and relaxation as two complementary limits that agree on the rates to within a stated tolerance under identical meshes, couplings, and delta-function integration, yet answer different questions; at the dark minimum at K only phonon absorption contributes to the linewidth, and the excitons it removes are returned by phonon emission, so the linewidth there mainly measures dephasing rather than population depletion. After thermalization the state at K acts as a bottleneck because it cannot emit a phonon, and the equilibrium flux is nearly three orders of magnitude smaller than during the initial cascade.