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Direct and indirect excitations in boron nitride: atomic structure and electronic correlations


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Direct and indirect excitations in boron nitride:
atomic structure and electronic correlations

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Direct and indirect excitations in boron nitride: atomic structure and electronic correlations

  1. 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.