GW+BSE Robert Laskowski

Transcription

1 GW+BSE Robert Laskowski star.edu.sg Institute of High Performance Computing Singapore

2 outline Thanks to Hong Jiang from College of Chemistry, Peking University, contributing GW package, and portion of the slides BSE package contributed by Robert Laskowski

3 Simple picture, independent particles 2 6 π ℑ(ϵij )= 2 Ωω vk pi ck ck p j vk δ(εkc εvk ω) vck Energy levels calculated within DFT are not true addition and removal energies absorption or emission is more complex process than just electrons jumping between energy levels

4 Energy levels calculated within DFT

5 Beyond standard DFT Effective functionals (mbj, F. Tran) F. Tran, P. Blaha PRL 02, (2009) scissor shift QP LDA QP vk LDA vk εck =εck Δ scissor ε =ε E ck E vk vk p ck = ε ε vk p ck ck vk QP ℑ ε(ω)=ℑ ε(ω Δ) non-locality of the self energy operator or scissor shift

6 Beyond standard DFT Hybrid DFT (thanks to F. Tran in wien2k) H-F Exchange energy included into KS theory F. Tran, P. Blaha PHYSICAL REVIEW B 83, 2358 (20) GW method (available for wien2k) i r, r ', = d 'G r,r, ' W r,r ', Self-energy 2 LDA QP LDA QP = nk V nk nk nk xc nk M. S. Hybertsen and S. G. Louie, Phys. Rev. Lett. 55, 48 (985) M. S. Hybertsen and S. G. Louie, Phys. Rev. B 34, 5390 (986) R. Gómez-Abal, et al, PRL 0, (2008).

7 Quasi particles Can mean field approach do the good job? Quasi-particle approach

8 GW approximation Hedin s self-consistent equations L. Hedin, Phys. Rev. 39, A769 (965) F. Aryasetiawan, O.Gunnarsson, Rep. Prog. Phys. (998)

9 G0W0 Approximation Hybertsen and Louie(985); Godby, Schlüter and Sham (986)

10 FHI-gap: a LAPW GW code Based on the FP-LAPW basis set Mixed basis set to expand the GW -related quantities Interfaced with WIEN2k G0 W0, H. Jiang et al., Comput. Phys. Comput. 84, 348 (203)

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12 FHI-gap: workflow Run a WIEN2k SCF calculation prepare the input files for FHI-gap in gwdir: gap_init -d <gwdir> -nkp <nkp> -s 0//2 -orb -emax <emax> modify gwdir.ingw Execute gap.x or gap-mpi.x in gwdir Analyse the results from: Look at gwdir.outgw the plot of the DOS/band structure generated by gap_analy

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15 Independent particles absorption or emission is more complex process than electrons jumping between energy levels e C Ψ c (r) C ℏν Ψ v (r) e V ρ(r)=ψ c (r) Ψ c (r) h V ρ(r)=ψ v (r) Ψ v (r) ρc (r) ρv (r) H initial H final

16 When independent particles picture fails LiF absorption spectra

17 Bethe-Salpeter Equation excitation is a two-particle process (electronhole pair is created) L 2; 2 =L0 2; 2 d 3456 L0 4 ; 3 K 35 ; 46 L 62; 52 hν equation of motion of two particle Green's function BSE is simplified into a two particles eigenvalue equation (in a basis of valence (vk) and conduction (ck) states) S,, vc ( E c E v ) A + K vc, v c ( E λ ) A =E λ A λ vc band energies, v, c, x K =V W d interaction kernel, excitation energies S vc e-h coupling coef. G. Strinati, Phys. Rev. Lett. 49, 59 (982). M. Rohlfing and S. G. Louie, Phys. Rev. B 62, 4927(2000) P Puschnig and C. Ambrosch-Draxl PRB 66, 6505 (2002)

18 BSE, kernel and dielectric function Exchange: vc V x ( E λ ) v, c, = dr dr, ψ c (r) ψc (r, ) υ(r, r, )ψv (r, ) ψ v (r),, Direct term: i iω 0 vc W ( E λ ) v c = dr dr ψ (r) ψ c (r) ψ v (r ) ψ (r ) d ω e W (r, r, ω) 2π E λ ω ( E c E v )+i O E λ +ω ( E c E v )+i O d c + v [ ] screened e-h interaction Usual approximation valid for: ( E c E v ) E λ d vc W ( E λ ) v c = dr dr ψ (r )ψc (r) ψv (r ) ψ (r )W (r, r,ω=0) c v

19 BSE, kernel and dielectric function macroscopic dielectric function from BSE ℑ ϵ M (ω)= λ E k 2 A vck λ vck vk p ck δ( E λ ℏ ω) (εck εvk ) oscillator strength are proportional to coherent sum of the momentum matrix elements macroscopic dielectric function from DFT E ℑ ϵ M (ω)= λ=vck k 2 vk p ck δ( E λ ℏ ω) (εck εvk )

20 LiF absorption spectra

21 Exciton envelope function in AlN s - dipole active 2 n oscil. strength~ 3 n E n =E g E B 8 4 p - dipole inactive d - dipole inactive A in BZ plotted for s, p and d excitons vck

22 X-ray absorption in IPA X-ray absorption is proportional to the projected DOS of the conduction band 2 ℑ M = vk p ck E ℏ ck vk branching ratio (L2/L3) is :2 (proportional to occupation of 2p/2, 2p3/2) L2,3 edge of Ca in CaF2 L3 L2 plain DFT core hole DFT (supercell with core hole on excited atom, wrong branching ratio)

23 Measured L edges of 3d metals L3 L2 BSE gives correct L2/L3branching rations

24 L2,3 edge for Ca in CaF2 transitions from 2p/2 and 2p3/2 are included at the same time into the BSE Hamiltonian experiment Only coherent mixing of transitions from 2p/2 and 2p3/2 results in proper branching ratio

25 BSE in wien2k Run a WIEN2k SCF calculation prepare the input files for BSEwien2k : init_bse -??? (it creates new bsedir) Inside bsedir optionally, modify input files (master is called input) Inside bsedir execute bse.job or optionally modify it before executing Analyze results: Look at exciton_singlet/triplet, epsilon_singlet/tripled There are some programs usefult for detailed analysis of excitonic envelope function

26 init_bse init_bse -h (get help) ~/SRC/BSE/BSE_SCRIPTS/init_bse h rolask@rlws:~/data/lif/lif_bse> ~/SRC/BSE/BSE_SCRIPTS/init_bse h This prepares the input for BSE This prepares BSE You need to run the thisinput in a for converged SCF calculation directory You need to run this in a converged SCF calculation directory Choose your options: hchoose your options: for this message h for this message p [path/to/bse] specify path to BSE [def: ${BSEROOT}] specify path to BSE [def: ${BSEROOT}] s p [path/to/bse] will do spin orbit will from do spin orbit f s start hybrid calc [experimental] f start from hybrid calc n [cpu] number of cores [def: 6][experimental] [cpu] number the of cores [def: d n[name] specify name of the 6] new directory [def: *_BSE] d [name] specify the name of the new directory [def: 4,0] *_BSE] b [VB,CB] number of valence and conduction bands [def: b [VB,CB] number of valence and conduction bands [def: 4,0] k [DMx:DMy:DMx,WDx:WDy:WDz,iWDx:iWDy:iWDz] k meshes for DM, WDVX and (optional) interpolation q k [DMx:DMy:DMx,WDx:WDy:WDz,iWDx:iWDy:iWDz] shift q mesh from Gamma k meshes for DM, WDVX and (optional) interpolation shiftfor q mesh from g q[dm,wd,vx,dm_max] G max WD, VX andgamma DM [DM,WD,VX,DM_max] for WD, VX DM specified G max for interpolation) i g[g max] dog max interpolated runand (with [G max] do energy interpolated e i[emin,emax] in ranges run [Ry](with (def:specified from SCF)G max for interpolation) e [emin,emax] in energy ranges [Ry] (def: from in SCF) r [emin,emax,de,broad] energy ranges, step and broadening BSE [ev] [def: 0.0,26.0,0.00] [emin,emax,de,broad] energy ranges, step(in andev) broadening c r[shift (ev)] apply scissor shift [def: 0] in BSE [ev] [def: 0.0,26.0,0.00] [shift (ev)] shift (in ev)and [def: 0] the spheres [def: 3,3] l c[pw,sphere] l apply value scissor for expansion of PW inside [PW,sphere] l value for expansion of PW inside /eps_{gg}(q) the spheres [def: 3,3] v l[ 4] screening level ( full, 2 and diagonal v [ 4] ( full, diagonal /eps_{gg}(q) 3 screening diagonal level eps_{gg}(q), 4 2long range (optional constant after colon)) 3 diagonal eps_{gg}(q), 4 long range (optional constant after colon)) Prepares input files based on provided parameters Copy vsp, vns, defines k-meshes, in for lapw,...

27 Bse.job cat bse.job cat bse.job!/bin/bash ex!/bin/bash ex echo $BSEROOT echo $BSEROOT $BSEROOT/execBSEkgen $BSEROOT/execBSEkgen $BSEROOT/execlapwoptic DM $BSEROOT/execlapwoptic DM mpirun machinefile.bse_machines envall np 2 $BSEROOT/DM > outputdm mpirun machinefile.bse_machines envall np 2 $BSEROOT/DM > outputdm $BSEROOT/execlapwoptic WDVX $BSEROOT/execlapwoptic WDVX mpirun machinefile.bse_machines envall np 2 $BSEROOT/WD > outputwd mpirun machinefile.bse_machines envall np 2 $BSEROOT/WD > outputwd mpirun machinefile.bse_machines envall np 2 $BSEROOT/VX > outputvx mpirun machinefile.bse_machines envall np 2 $BSEROOT/VX > outputvx mpirun machinefile.bse_machines envall np 2 $BSEROOT/BSE_diag > outputbse_diag mpirun machinefile.bse_machines envall np 2 $BSEROOT/BSE_diag > outputbse_diag execbsekgen creates k-meshes for DM, WD,VX execlapwoptic executes lapw, optic, generates eigenvectores, and momentum matric elements DM computes dielectric matrix WD computes direct part of the BSE Hamiltonian VX computes exchange part of the BSE Hamiltonian BSE_diag solves BSE equation

28 Input file (input) scissors scissors scalebands scalebands eminmaxdm eminmaxdm gmax_wd gmax_wd gmax_vx gmax_vx gmax_dm gmax_dm gmax_dm_max gmax_dm_max lmax_besel lmax_besel lmax_wave lmax_wave offesdm offesdm broaddm broaddm omegadm omegadm formatvxwd formatvxwd nbandsvxwd nbandsvxwd nbandsint nbandsint nbandsbse nbandsbse noccbands noccbands noccbandsdm noccbandsdm scrlevel scrlevel diroffbse diroffbse nexoutbse nexoutbse eminemaxbse eminemaxbse broadbse broadbse polbse polbse spinbse spinbse 0.0 scissors shift [ev] scissors shiftfor [ev].0 scaling factor (valence, conduction) bands.0.0 scaling factor energy cutoff forfor DM (valence, conduction) bands 5.0 energy cutoff for maximum magnitude of GDMvector for exp(igr) 4.0 maximummagnitude magnitudeofofg Gvector vectorfor forexp(igr) exp(igr) 4.0 maximum 4.0 maximum magnitude of G vector for exp(igr) maximum magnitude of G vector for exp(igr) maximum of G of vector for exp(igr) max of lmagnitude in expansion exp( i(q+g)r) 3 max of l in expansion of exp( i(q+g)r) 3 max of l in expansion of wave function (sphere) max of l off diag in expansion of wave function (sphere) 3 calculate elements of eps_ggp(q) calculate off diag elements of eps_ggp(q) VX, WD files 0 ascii, binary VX, WDoffiles 0 ascii, binarybands in VX, WD 4 4 number (valence,conduction) 4 4 number of (valence,conduction) bandsused in VX, WD 4 4 number of (valence,conduction) bands in BSE 4 4 number of (valence,conduction) bands used in BSE 4 4 number of (valence,conduction) bands used in BSE 4 4 number of (valence,conduction) bands used in BSE 5 number of occupied bands 5 number of occupied bands 5 number of occupied bands 5 number of level occupied bands screening in WD: screening level in the full {g, g'}wd: dependence is used full {g, g'} dependence is used 2 a the diagonal approximation is used (g = g') using 2 a diagonal approximation is (g = g') using the inverse dielectric matrix used eps^{ }_{gg}(q) the inverse dielectric matrix eps^{ }_{gg}(q) 3 a diagonal approximation is used (g = g') using 3 the a diagonal approximation is used (g = g') using dielectric matrix eps_{gg}(q) the dielectric matrix eps_{gg}(q) 4 a long range screening is used (constant tensor) 4 a long range screening is used (constant tensor) 0 : local field effects only (direct interaction = zero) 0 : local field writen effectsinto onlyfiles (direct interaction = zero) 000 number of states numberemax, of states emin, de in writen [ev] into files emin, emax, de in [ev][ev], lineshape 0. 0 broadending of spectrum 0. 0 broadending of spectrum [ev], lineshape polarization polarization singlet, 2 triplet singlet, 2 triplet

29 GW, BSE in wien2k GW calculations require large computational resources BSE is computationally very expensive