Our new paper on exciton interference in hexagonal boron nitride is online. In this paper we show that the excitonic peak at finite momentum is formed by the superposition of two groups of transitions that we call KM and MK′ from the k-points involved in the transitions. These two groups contribute to the peak intensity with opposite signs, each damping the contributions of the other. The variations in number and amplitude of these transitions determine the changes in intensity of the peak. Our results unveil the non-trivial relation between valley physics and excitonic dispersion in h-BN.

1. Lorenzo Sponza1
, Hakim Amara1
, François Ducastelle1
, Claudio Attaccalite2
, Annick Loiseau1
Exciton interference in hexagonal boron nitride
1
Laboratoire d'étude des microstructures (LEM), ONERA - CNRS, Châtillon, France
2
Centre Interdisciplinaire de Nanoscience de Marseille, Aix Marseille University - CNRS, Marseille, France
High-accuracy electron energy loss (1)
High-accuracy electron
energy loss
spectroscopy (EELS)
has been carried out
on bulk-hBN along the
ΓK direction (1).
Dispersive peak of
excitonic nature.
q=0.7Å-1
: minimum
energy & maximum
intensity
q=1.1Å-1
almost
disappears4.5 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
Image
taken from (1)
(1) Scuster et al, arXiv:1706.04806v1 (2017)
Single-particle band structure: G0W0 at LDA
Band structure of bulk hBN (AA' stacking)
has indirect gap between
a point close to K and M.
Established theoretical result.
Computational details:
- contour deformation
- nbands=600
- cutoff wfn.=816 eV
- cutoff mat.dim.=816 eV
- Γ-centered 6x6x4
-2.0
0.0
2.0
4.0
6.0
8.0
K M K’
Energy-EF(eV)
6.285.79
Analysis of q=5q0 = 0.7Å-1
Analysis of the peak intensity: Decomposition of IP-transitions
KM
MK'constructive
destructive
5.0 6.0 7.0 8.0 9.0 10.0
0.0
0.2
0.4
0.6
0.8
1.0
λ=1
IntensityofIm[ε(q,ω)](arb.units)
NormalisedCumulant
Energy (eV)
J1(q,E)
BSE
low energy high energy
steepincrease
slow increase
Normalized cumulant weight
Information abaout the
building-up of the excitonic
peak.
Analysis of q=8q0 = 1.12Å-1
KM
MK'constructive
destructive
5.0 6.0 7.0 8.0 9.0 10.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
λ=3
IntensityofIm[ε(q,ω)](arb.units)
NormalisedCumulant
Energy (eV)
J3(q,E)
BSE
low energy high energy
steepincrease
trend
inversion
slow
decrease
Conclusions
- The GW-BSE dispersion of the
peak of the loss function
perfectly reproduces
experimental data (1).
- Transitions can be classified
in groups depending on the
phase of Mλ
t
(q).
- Competition between two
groups of transitions (KM
and MK' groups). KM
dominates at q=0.7Å-1; the
two groups interfere
destructively at q=1.1Å-1.
- Intriguing valleys physics.
- General framework and
analysis tools, applicable to
any study of excitonic
properties.
- Possible way to redefine an
optimized basis for excitons.
GW-BSE Excitonic spectra and dispersion
GW-BSE spectra along ΓK
An additional shift of 0.4 eV to align to experiments
for the lack of self-consistency in GW.
q=8q0=1.12Å-1
q0=0.14Å-1
Computational details:
screening
nbands = 350
mat.dim. = 120 eV
cutoff wfn.=200 eV
Bethe-Salpeter
nbands = 6+3
mat.dim. = 80 eV
cutoff wfn.=110 eV
Γ-centered 36x36x4
q=5q0=0.70Å-1
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
5.0 5.5 6.0 6.5 7.0
0.00
0.05
0.10
0.15
0.20
LossFunction
DielectricFunction
Energy (eV)
Im[ε]
-Im[1/ε]
Γ
2q0
4q0
6q0
8q0
10q0
K
5.0 5.5 6.0 6.5 7.0
Loss
Function
-Im[1/ε]
Excitation energy (eV)
LossFunction
DielectricFunction
Energy (eV)
Im[ε]
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
5.0 5.5 6.0 6.5 7.0
0.00
0.20
0.40
0.60
0.80
1.00
-Im[1/ε] peak intensity IP-transition matrix element
independent-particle transition index
The GW-BSE dispersion of the loss function perfectly
reproduces the experimental dispersion of the peak.
The intensity of the loss function is maximum at q=5q0
and mnimum at q=8q0
Intensity and dispersion of loss function follow the peaks of
the dielectric function.
What is the origin of these intensity variations?
Answer by decomposing the spectral intensity in
contributions from the IP-transitions.
Preprint available:
-10.0
-5.0
0.0
5.0
10.0
15.0
Γ K H A Γ M L A
Energy-EF(eV)
K M H L
-10.0
-5.0
0.0
5.0
10.0
15.0
Energy-EF(eV)
6.28
5.79
8.43
Excitationenergy(eV)
Excitationenergy(eV)
5.9
6.0
6.1
6.2
6.3
6.4
6.5
6.6
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
5.9
6.0
6.1
6.2
6.3
6.4
6.5
6.6
Γ 2q0 4q0 6q0 8q0 10q0 K
Loss Function
Exchanged momentum (Å)−1
Exchanged momentum (units of q0)
GW−BSE (+0.4 eV)
q0=0.14Å-1
GW-BSE (+0.4 eV)
Schuster et al. (ref. 1)
Loss function dispersion
Positive phase; group KM
Band-contracted map of
the IP-transitions
Negative phase; group MK'
Band-contracted map of
the IP-transitions
Sketch of IP-transitions
At 5q0 KM dominates: higher number
and intensity.
At 8q0 the two groups are competing.
lorenzo.sponza@onera.fr