This document discusses the optical properties of boron nitride (BN) as studied through many-body perturbation theory and electron energy loss spectroscopy (EELS) experiments. It finds that BN has an indirect bandgap of 5.79 eV according to calculations, and a sharp exciton mode is visible in EELS between 0.4-1 Å-1 momentum transfer. Many-body calculations reveal the exciton has reduced dispersion compared to the band structure, in good agreement with EELS data. Different stackings of BN layers are also considered, with ABC stacking found to have a very flat and possibly direct excitonic gap.

1. 4.5 5.0 5.5 6.0 6.5 7.0
Energy (eV)
Intensity(a.u.)
q (1/Å)
K
M K’
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
Figure 1. (Color Online) The EELS intensity in the region of the absorption onset measured parallel
K for h-BN (for clarity the curves have been shifted vertically). Note the sharp mode that is
visible in the momentum region between 0.4Å
1
and 1Å
1
. The thick line marks the momentum
transfer where this mode is strongest and at its minimum energy position. The inset shows the
hexagonal BZ illustrating that q polarized along K is also sensitive to the symmetry-equivalent
MK-direction.
5
5q0
Excitation energy (eV)
Direct and indirect excitations in boron nitride:
atomic structure and electronic correlations
1
Laboratoire d'étude des microstructures (LEM), ONERA - CNRS, Châtillon, France
2 Physics and Materials Science Research Unit, University of Luxembourg, Luxembourg, Luxembourg
3 Centre Interdisciplinaire de Nanoscience de Marseille (CINaM), Aix Marseille University - CNRS, Marseille, France
4 CEA - IRAMIS, Gif-sur-Yvette, France
KM
constructive
MK'
destructive
q=5q0=0.70Å-1
Lorenzo Sponza1
, Thomas Galvani2
, Claudio Attaccalite3
, Sylvain Latil4
, Hakim Amara1
, Ludger Wirtz2
, François Ducastelle1
Single layer of hBN
In-plane 2D honeycomb lattice
Isostructurale to graphene
Capping layer or growth substrate
...but it has also interesting properties
on its own
Boron nitride allotropes
Free-standing monolayer
AA' stacking
AB and ABC stacking
AA' (most stable)AB stacking ABC (rhombohedral)
h=3.25 Å
r=3.56 Å
Energy (eV)
0.00
0.14
0.28
0.42
0.56
0.70
0.84
0.98
1.12
1.26
1.40
1.54
1.68
Exchangedmomentum(Å-1)
Exchanged momentum (Å-1)
Energy(eV)
Gap and excitons
Indirect Gap
Momentum
Energy
photon
phonon
Exciton
Binding energy
valence
band
conduction
band
The formation of the
lowest-energy exciton
pair requires a
photon and a
phonon.
This leads to low
transition probability
and hence to
low intensity.
N. of accurate
quantities
Size of
the system
Methodes theoriques
Many Body Perturbation Theory
density, screening, gap, excitons
- ab-initio (no free parameters)
- complete and accurate
- "all" mechanisms included
- involuted analysis
Time-Dependent DFT
density, screening
Density Functional Theory
density
Tight Binding Model
ad-hoc model: exciton
- adjustable parameters
- limited descriptive power
- fundamental mechanisms
- simple analysis
Direct Gap
Momentum
valence
band
photon
Exciton
Energy
conduction
band
Binding
energy
The formation of the
lowest-energy exciton
requires a single photon.
This leads to high transition
probability and hence to
high intensity.
Gap = 1-particle excitation
Lowest energy difference between the top
valence and the bottom conduction.
Exciton = 2-particle excitation
Electron-hole bound state.
MBPT: Scissor + BSE
Exciton dspersion and Im[ε]
for momentum along ΓK.
Direct transition 5.69 eV
Indirect transition 5.79 eV
Im[ε]
MBPT: LDA + GW
Band structure calculation.
Direct gap 7.28 eV
Indirect gap 7.39 eV
Exciton formation at finite q
The exciton related to the KM directionin of the
the band structure propagates with exchanged
momentum q=k-k' parallel to the ΓK direction.
exciton 1 exciton 2
Energy(eV)
Exchanged momentum (Å-1)
exciton 1
exciton 2
Tight Binding:
Impurity model of electron
hopping between boron atoms
Exciton dspersion and exciton wave
function form momentum along ΓK.
Direct transition -1.91 eV
Indirect transition -1.88 eV
Single particle gap:
The GW band structure exhibits a
weakly dispersing conduction band with
a difference from direct and indirect gap
of approximately 0.1 eV
- T. Galvani et. al., PRB 2016
- L. Sponza et al., in preparation
- L. Sponza et. al., PRB 2018
- L. Schué et al., arXiv:1803.03766v1 2018
- R. Schuster et al., PRB 2018
Eb=0.67 eV
Eb=0.30 eV
Excitationenergy(eV)
Excitationenergy(eV)
Exchanged momentum (1/Å)
Exchanged momentum (units of q0)
Schuster et al., PRB 2018
Experimental
Loss function -Im[1/ε]
q0=0.14Å-1
Γ
2q0
4q0
6q0
8q0
10q0
K
5.0 5.5 6.0 6.5 7.0
Excitation energy (eV)
GW-BSE
Loss function
-Im[1/ε]
5.0 5.5 6.0 6.5 7.0
GW-BSE
Im[ε] and -Im[1/ε]
Excitation energy (eV)
Im[ε]
-Im[1/ε]
Origin of the exciton
- L. Sponza et. al., in preparation
The spectral intensity I(q) of the
exciton λ can be decomposed in
independent-particle (IP) transitions.
The total intensity originates from
interference of constructive and
destructive IP transitions
dipole matrix element
exciton wave function
MBPT: LDA + GW
Indirect gap
Direct gap 6.28 eV
Indirect gap 5.80 eV
Direct transition 5.61 eV
Indirect transition 5.50 eV
Energy(eV)
Energy(eV)
- The exciton is less dispersive than the non-interacting electron-hole pair.
- The quasiparticle disperison of ~0.5 eV (GW band srtucture), is reduced to ~0.1 eV in the excitonic dispersion (GW-BSE).
- The comparison with recent EELS experiments [Schuster PRB 2018] confirms the predicted exciton dispersion.
Eb=0.63 eV
Eb=0.39 eV
Eb=0.56 eV
Eb=0.38 eV
MBPT: LDA + GW
Indirect gap
Direct gap 5.80 eV
Indirect gap 5.37 eV
MBPT: LDA + GW
Indirect gap
Direct gap 6.24 eV
Indirect gap 6.09 eV
ABC stacking (rhombohedral cell)
Energy(eV)
AB stacking
Energy(eV)
Direct transition 5.12 eV
Indirect transition 4.98 eV
Direct transition 5.68 eV
Indirect transition 5.72 eV
- In the ABC, the gap is NOT in the high symmetry lines of the hexagonal cell.
- Dynamical effects of the GW self-energy lead to a smaller gap in the ABC.
- The q-dependence of the exciton binding energy Eb(q) reduces sizeably the
dispersion of the 1-particle picture (band structure), as in the AA' stacking.
- Direct and indirect gap are very close in energy.
- The exciton exhibits an extremely flat dispersion and possibly changes the
nature of the excitonic gap.
- The 2-particle excitation (exciton) is direct, while the 1-particle excitations
(band structure) display an indirect gap.