1) A tight-binding model for excitons in 2D and layered hexagonal boron nitride provides insight into its optical properties in the bulk and different polymorphs. 2) Ab initio calculations show the flattening of the exciton dispersion in bulk hBN leads to a mixing of direct and indirect transitions, consistently explaining conflicting data on band structure and optical spectra. 3) The model predicts an even stronger direct gap behavior for Bernal stacked hBN, despite it having an indirect fundamental gap.

1. Des questions ouvertes depuis quinze ans !
Direct/indirect band gap
and exciton dispersion:
Monolayer and bulk hexagonal boron nitride
L. Sponza, L. Schué, H. Amara, C. Attaccalite, F. Ducastelle, A. Loiseau, J. Barjon
Graphene 2018 26-29/06 Dresden

2. - The curious spectra of hexagonal boron nitride (hBN)
contradiction between electronic and optical properties
modeling free carriers and excitons
- Tight-binding model of the exciton
validation of the model in the monolayer
insight from the monolayer to the bulk
- Ab initio exciton dispersion in bulk hBN
conciliation of the contraddictions (theory)
experimental evidences
- Predictions about the Bernal stacking
Outlook

3. Hexagonal boron nitride: electronic structure
Monolayer
Bulk: AA' stacking
Energy(eV)Energy(eV)
quasiparticle gap
GGA + 2.75 eV
direct K
indirect KM
7.25 eV
7.35 eV
T
L. Sponza et al., arXiv:1806.06201 (2018)
direct T
direct M
indirect TM
6.45 eV
6.28 eV
5.80 eV
quasiparticle gap
LDA + G0W0

4. Direct gap
coupling only
to photons (q=0)
- High probability
- High efficiency
- Low probability
- Low efficiency
Expectations from direct and indirect gaps
Indirect gap
Momentum
Energy
AbsorptionEmission
coupling
to photons (q=0)
and phonons (q≠0)
Momentum
Energy
AbsorptionEmission

5. Diamant
indirect
∼0.1%
hBN indirect
∼15%
ZnO direct >50%
5.8 6.0 6.2
5.3 5.4 5.5
L. Schué et. al., arXiv:1803.03766 (2018)
High luminescence efficiency No abs/lum mirror symmetry
Experimental luminescence of bulk hBN
energy (eV)
energy (eV)
Lum. PLE ∼ Abs.
AbsorptionLuminescenceParadigmatic indirect gap material
(diamond) has low luminescence
efficiency and abs/lum mirror
symmetry holds for spectra.
Absorption
Luminescence

6. Diamant
indirect
∼0.1%
hBN indirect
∼15%
ZnO direct >50%
5.8 6.0 6.2
5.3 5.4 5.5
Absorption
Luminescence
High luminescence efficiency
Experimental luminescence of bulk hBN
energy (eV)
energy (eV)
Lum.
AbsorptionLuminescenceParadigmatic indirect gap material
(diamond) has low luminescence
efficiency and abs/lum mirror
symmetry holds for spectra.
Despite the indirect gap,
luminescence efficiency is very high
abs/lum specularity does not hold
in bulk hBN
L. Schué et. al., arXiv:1803.03766 (2018)
No abs/lum mirror symmetry
PLE ∼ Abs.

7. Exciton: electron-hole pairFree carriers: electron + hole
Excitonic Hamiltonian
on a basis of free carriers.
Excitonic levels.
Quasiparticle: scissor, GW...
two 1-particle propagators
Dipole-allowed transitions
between points of the
band structure.
1-particle and 2-particle excitations
Bethe-Salpeter equation
single 2-particle propagator

8. 1-particle
triangular lattice
attractive source
1-particle and 2-particle excitations
L. Sponza et al., arXiv:1806.06201 (2018)
T. Galvani, PRB 94 (2016)
Excitonic model
eh
N B
Arbitraryunits
1-particle model
h
e
energy(eV)
Density
of states
q
Brillouin
zone
q q
ab initio tight-binding

9. t⊥ = -2.33 eV t|| = 0.5 eV t2⊥ = -0.4 eV t2|| = -0.1 eV
Γ K
-0.6
-0.3
0.0
Energy(eV)
From monolayer to bulk AA'
L. Sponza et al., arXiv:1806.06201 (2018)
Γ KΓ KΓ K
-0.6
-0.3
0.0
Energy(eV)
TB exciton TB exciton
TB excitonTB exciton

10. 5.69 eV5.80 eV
6.28 eV
Variations of the binding energy
lead to a flattening of the
excitonic dispersion.
Direct (bright) and indirect
excitons lie within 0.2 eV.
ab-initio band structure
Exciton dispersion in bulk AA'
L. Sponza et al., arXiv:1806.06201 (2018)
Free carriers
Excitons
bright
ab initio e-h dispersion
5.50 eV
Energy(eV)
q
5.4
6.2
6.4
6.0
Energy(eV)
dark

11. L. Sponza et al., arXiv:1806.06201 (2018)
5.69 eV
Free carriers
Excitons
dark
bright
ab initio e-h dispersion
q
5.4
6.2
6.4
6.0
Energy(eV)
Consistent explanation of the optical properties
The energy difference is
comparable to phonon energies.
Thus final states are strongly
coupled by phonons, and
effectively mixed.
5.8 6.0
PLE ∼ Abs.
Lum.
energy (eV)
2) No abs/lum mirror
because absorption
passes through a
resonant state,
while luminescence
does not.
1) high luminescence
efficiency due to
coupling between the
two states. 5.50 eV

12. L. Sponza et al., Phys. Rev. B 97, 075121 (2018)
Confirmation from experimental evidences
Loss function at finite q
R. Schuster et. al.,
Phys. Rev. B (2018)
experiment
K
0.7 Å-1
1.1 Å-1
0.7 Å-1
1.1 Å-1
theory
GW+BSE+0.47 eV
experiment
GW+BSE+0.47 eV
5.8 6.1 6.4 6.7
Energy (eV)
0.7 Å-1
1.1 Å-1
KK/2
6.0
6.1
6.2
6.3
Energy(eV)
6.4
5.5 6.0 6.5 7.0
Energy (eV)

13. Intriguing prediction about Bernal stacking
L. Sponza et al., arXiv:1806.06201 (2018)
Bernal stacking TB exciton
Γ KΓ K
Energy(eV) -0.6
-0.3
-0.1
-0.2
-0.4
-0.5
-0.7
TB exciton
Standard stacking for comparison
-0.6
-0.3
-0.2
-0.4
-0.5
-0.7
Energy(eV)
Γ KK/2Γ K K' ΓM
T
5.5
6.0
6.5
7.0
7.5
8.0
-2.0
-1.5
-1.0
-0.5
0.0
Energy(eV)
Energy(eV)
5.6
6.2
5.7
5.8
5.9
6.0
6.1
ab initio band structure ab initio exciton dispersion
6.13 eV6.01 eV
5.63 eV
Free carriers
Excitons
5.64 eV

14. Intriguing prediction about Bernal stacking
L. Sponza et al., arXiv:1806.06201 (2018)
Bernal stacking TB exciton
Γ KΓ K
Energy(eV) -0.6
-0.3
-0.1
-0.2
-0.4
-0.5
-0.7
TB exciton
Standard stacking for comparison
-0.6
-0.3
-0.2
-0.4
-0.5
-0.7
Energy(eV)
Γ KK/2Γ K K' ΓM
T
5.5
6.0
6.5
7.0
7.5
8.0
-2.0
-1.5
-1.0
-0.5
0.0
Energy(eV)
Energy(eV)
5.6
6.2
5.7
5.8
5.9
6.0
6.1
ab initio band structure ab initio exciton dispersion
6.13 eV6.01 eV
5.63 eV
Free carriers
Excitons
5.64 eV
Even though the gap is indirect,
the Bernal stacking is predicted to
behave as a direct-gap material
from the spectroscopic point of view.

15. 1) Tight-bidning model for excitons in 2D and layered hBN.
Insight when applied to bulk and to different polymorphs.
2) Flattening of the exciton dispersion in hBN leads to a
mixed direct/indirect transition.
Consistent explanation of conflicting data coming from
experimental spectra and theoretical band structure.
Experimental evidence of the theoretical explanation.
3) Prediction of even stronger effect in Bernal hBN.
Conclusions
Reasoned bibliography:
- K. Watanabe et al., Nat. Materials 3, 404 (2004)
- L. Schué et al., Nanoscale 8, 6986 (2016)
- G. Cassabois et al., Nat. Photonics 10, 262 (2016)
- L. Schué, L. Sponza et. al., arXiv:1803.03766 (2018)
- T. Galvani et al., Phys. Rev. B 94, 125303 (2016)
- L. Sponza et. al., Phys. Rev. B 97, 075121 (2018)
- L. Sponza et al., arXiv:1806.06201 (2018)
mostly experimental
on luminescence
mostly theoretical
TB & ab initio