Ab-initio calculations of (non)linear optical properties of solids

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1 Ab-initio calculations of (non)linear optical properties of solids Myrta Grüning, Queen's University Belfast Absorption Experimental measurements Energy(eV) Absorption Calculated spectra Energy (ev) CCP9 workshop

2 Phenomenological absorption coefficient related to electronic structure/microscopical picture: Phenomenological/macroscopic: Beer-Lambert law: incident radiation transmitted radiation Quantum mechanical/microscopic: Absorbed energy/time is proportional to: E F Ef Ei Perturbation (first order)

3 Absorption coefficient is related to the density response: Unperturbed system Perturbation Perturbed system described by its electronic density: change in the electronic density: Density-density response

4 How to describe optical properties ab-initio? Time-dependent density functional theory Green's function theory How to translate theory in computational tools? GW approximation and Bethe-Salpeter equation How to go beyond the linear regime? Nonlinear optics Real-time spectroscopy

5 Optical absorption can be treated within Time Dependent-Density Functional Theory: Key quantity density response Unperturbed system Perturbation Perturbed system described by its electronic density: change in the electronic density: From Time-dependent density functional theory (TD-DFT):

6 } } For extended systems it is important to model correctly the response of e-e interaction : corrections due to response of e-e GOOD APPROXIMATION FOR EXCITATIONS IN MOLECULES IMPORTANT IN EXTENDED SYSTEMS!!!

7 Standard approximations describe poorly response of e-e interaction: example: Absorption spectrum of LiF: Exp Theory Energy [ev] Energy [ev]

8 Green's function theory is an alternative to describe excited states: Key quantity Green's function: Addition/removal of electron (charged excitations): r, t r, t r', t' Fetter, Walecka "Quantum Theory of Many-Particle Systems"; Mattuck "A Guide to Feynman Diagrams in the Many-body Problem"

9 Green's function theory is an alternative to describe excited states: Key quantity Green's function: Addition/removal of electron (charged excitations): Lehmann representation

10 Electron correlation accounted by using Many-Body perturbation theory: Self-energy (sum of infinite number of diagrams): + =

11 Optical absorption cannot be described simply as removal+addition of electron: Ef e- Ei e- E F Ef hv Ei =/ Ei Ek hv Evac E F Ef Ek hv OPTICAL ABSORPTION =/ SUM OF INVERSE/DIRECT PHOTOEMISSION PROCESSES OPTICAL EXCITATION ENERGY =/ DIFFERENCE OF QUASIPARTICLE ENERGIES

12 Optical absorption cannot be described simply as removal+addition of electron: = + electron-hole interaction = + electron-hole interaction analogy with 1-particle case:

13 Implementation of two-particle Dyson-equation not trivial: 1. : How to calculate G? 2. : Viable physical approximations? 3. Invert 4-points quantities?!

14 Green's function calculated using the GW approximation: a. b. a. Independent particle GF from Kohn-Sham: universal, parameter free approach b. GW approximation with RPA screening (calculated from KS quantities) c. First order perturbation to Kohn-Sham energies

15 Finally, we obtain the Bethe-Salpeter equation: approximated QP: KS wfs + GW energies Hartree (RPA) + static screened exchange (COHSEX) 3. solve in the electron-hole space of phases (two-particles effective Hamiltonian) see e.g. Onida et al Mod. Phys. Rev. 74, 601 (2002); G. Strinati. Rivista del Nuovo Cimento 11, (12)1 (1988)

16 GW+BSE contains the physics we need to describe optical absorption iin solids: independent particles: Ground state DFT: LiF quasiparticles: GW corrections: electron-hole attraction: Bethe-Salpeter: Exp Theory Energy [ev] Energy [ev] Energy [ev] From where poles and pole-strengths are obtained from eigensolutions of: see e.g. Onida et al Mod. Phys. Rev. 74, 601 (2002); G. Strinati. Rivista del Nuovo Cimento 11, (12)1 (1988)

17 Some more examples on GW+BSE performance: Application to optical absorption in periodic solid (alpha-alumina): Band structure: LDA 6.70 ev Energy (ev) + Quasiparticle corrrections: GW ev QP corrections (ev) LDA eigenvalues (ev) Optical absorption + Bethe-Salpeter Energy (ev) L L -20 Z A D A Band gap ~9.4 ev (experimental estimates ev) Exciton position: ~9 ev (within ev from exp) Exciton binding energy: ~0.4 ev (exp ~ ev) GW+BSE calculations: yambo ( A. Marinopolous, M.G, PRB 83, (2011) Exp: French et al J. Am. Cer. Soc. 81, 2549 (1998); Tomiki, et al., J Phys Soc Jap 62, 573 (1993).

18 Some more examples on GW+BSE performance: Application to optical absorption in layered TMDs (spin-orbit coupling): Excitonic wave-function Optical absorption GW+BSE calculations: yambo ( A. Molina-Sanchez et al PRB 88, (2013)

19 Can we use those methods for nonlinear optics and time-resolved spectroscopy?!? Optical pulse duration Intensity (W/cm 2 ) rotational motion of molecules 10 ps 1 ps photonuclear fission vibrational motion of nuclei 100 fs relativistic nonlinear optics of vacuum electrons fs electron motion 1 fs electrostatic tunneling of electrons from atoms M. Wegener, Extreme nonlinear optics, an introduction

20 Can we use those methods for nonlinear optics and time-resolved spectroscopy?!? CORRELATION: TD-DFT GW+BSE NONLINEAR EFFECTS: response theory real-time Depends on XC approximation. Linear/second/third order response Depends on XC approximation. Extreme nonlinear out-of-equilibrium state-of-the-art: photoemission optical properties linear/shg TO EXPLORE!

software: Theory&approximations Many-Body Perturbation Theory Time-dependent density functional theory Interfaces GPL: Quasiparticles Optical absorption EELS magnetic properties electron-phonon

21 software: Theory&approximations Many-Body Perturbation Theory Time-dependent density functional theory Interfaces GPL: Quasiparticles Optical absorption EELS magnetic properties electron-phonon effects surface spectroscopy Properties Develop/pre-GPL: non-equilibrium dynamics non-linear optics total energies super-parallelism for HPC Planewave- Pseudopotential codes: Support & reach-out Dedicated users forum Online documentation/tutorials Schools & workshops Community Publications about 200 publications about 200 users (mostly from EU, USA, China) Andrea Marini, Conor Hogan, Myrta Grüning, Daniele Varsano Comp. Phys. Comm. 180, 1392 (2009)

22 How to describe optical properties ab-initio? Green's function theory: correlation effects systematically described by MBPT How to translate theory in computational tools? GW approximation + BSE offer a viable though accurate approach to describe optical absorption How to go beyond the linear regime? Within this framework, approaches to nonlinear optics and real-time spectroscopy not yet developed: a lot of work to be done!

Introduction to Green s functions

Introduction to Green s functions Matteo Gatti ETSF Users Meeting and Training Day Ecole Polytechnique - 22 October 2010 Outline 1 Motivation 2 Green s functions 3 The GW Approximation 4 The Bethe-Salpeter