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Exciton interference in hexagonal boron nitride

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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.

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Exciton interference in hexagonal boron nitride

  1. 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

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