Real-time electron dynamics in solids under strong electromagnetic fields

Transcription

1 ISSP-CMSI Inernaonal Workshop/Symposum on Maeral Smulaon n Peaflops era (MASP01) June 7 01 ISSP Unv. Tokyo Real-me elecron dynamcs n solds under srong elecromagnec felds Kazuhro Yabana Cener for Compuaonal Scences Unversy of Tsukuba Y. Shnohara U. Tsukuba S.A. Sao U. Tsukuba T. Sugyama U. Tsukuba T. Oobe JAEA J.-I. Iwaa U. Tokyo G.F. Bersch U. Washngon

2 TDDFT: Frs-prncples ool for elecron dynamcs smulaon Lnear response regme (Perurbaon heory) +Ze ψ ( r ) = { h [ ρ( r ) ] + V ( r ) } ψ ( r ) ρ Elecronc excaon Absorpon cross secon - + σ ( ω) S - S E() + S 0 E Inense ulra-shor Laser pulses hν KS ( r ) ψ ( r ) = 3hν ex sof X ray (Hgh harmonc generaon) Laser-sold neracon Delecrc funcon Coheren phonon Elecron-hole plasma Opcal breakdown Collson of aoms and ons e- e - e - Mulple onzaon Coulomb exploson Nonlnear nonperurbave regme (Inal value problem) Conducance of nano-wre

3 CONTENTS 1. A froner n lgh - maer neracon : nense and ulrashor - Theores and compuaons requred n curren opcal scences -. Real-me and real-space calculaon for elecron dynamcs n sold - me-dependen band calculaon reamen of boundares Elecron and phonon dynamcs n sold for a gven elecrc feld - Delecrc funcon coheren opcal phonon opcal breakdown - 4. Coupled Maxwell + TDDFT mulscale smulaon - Frs-prncples smulaon for macroscopc elecromagnec feld n nense regme Dense elecron-hole plasma nduced by nense laser pulses - 3

4 Modern expermens on lgh-maer neracons Requrng new heores and large-scale compuaons beyond ordnary elecromagnesm and quanum mechancs - Nano-szed maeral oo large o calculae mcroscopcally oo small o calculae macroscopcally. large-scale mcroscopc calculaon - Near feld opcs (resoluon beyond dffracon lm) - Mea-maeral (negave refracve ndex) - Surface plasmon polaron (coupled lgh-elecron dynamcs) quanum effecs (?) - Inense and ulrashor laser pulses (exreme nonlnear elecron dynamcs) requrng me-doman descrpon coupled dynamcs of elecrons and elecromagnec felds 4

5 Froners n Opcal Scences: Inense feld QED Vacuum breakdown Nuclear reacon Parcle Acceleraon Relavsc Classcal Nonlnear elecron dynamcs Nonrelavsc Quanum G. Mourou

6 Inense laser pulse on sold W/cm Laser nensy Nonlnear opcs Coheren phonon HHG Elecron-hole plasma Opcal breakdown Laser machnng Laser acceleraon Vacuum breakdown Nonrelavsc Quamum mechancs Relavsc Classcal mechancs ee()z z Exernal elecrc feld by laser pulse Inernal elecrc feld by nucle

7 Froners n Opcal Scences: Ulra-shor dynamcs Pcosecond 10 1 s Femosecond s Aosecond s 10 3 s Tme Perod of Opcal phonon (S 64 fs) Perod of elecron orbal n Hydrogen 150as Nucleon moon n nucleus Perod of T-sapphre laser pulse Shores laser pulse 80 as (008) Real-me observaon of valence elecron moon E. Goulelmaks e.al Naure (010). 7

8 Pump-Probe Measuremen: Coheren Phonon Inense 1 s pulse exces elecrons and aoms. Weak nd pulse measure change of reflecvy. T ph : Perod of opcal phonon probe pump M. Hase e al. Naure (London) (003) ΔΔ R e Γ sn Ω ph + φ 8

9 Femo-echnology: nonhermal machnng by femosecond laser pulses R.R. Gaass E. Mazur Naure Phooncs 0 (008). Mcromachnng wavegude- Nanosurgery Opcal mcroscope mage of wavegudes wren nsde bulk glass by a 5-MHz ran of 5-nJ sub-100-fs pulses C.B. Schaffer e.al OPTICS LETTERS 6 93 (001) Ablaon of a sngle mochondron n a lvng cell. N. Shen e.al Mech. Chem. Bosysems 17 (005).

10 CONTENTS 1. A froner n lgh - maer neracon : nense and ulrashor - Theores and compuaons requred n curren opcal scences -. Real-me and real-space calculaon for elecron dynamcs n sold - me-dependen band calculaon reamen of boundares Elecron and phonon dynamcs n sold for a gven elecrc feld - Delecrc funcon coheren opcal phonon opcal breakdown - 4. Coupled Maxwell + TDDFT mulscale smulaon - Frs-prncples smulaon for macroscopc elecromagnec feld n nense regme Dense elecron-hole plasma nduced by nense laser pulses - 10

11 3D real-space grd represenaon hgh order fne dfference Tme evoluon: 4-h order Taylor expanson ( ) ( ) ( ) ( ) ( ) 4! 1 exp 0 = = + = N h k h KS N k KS ψ ψ ψ Calculaon n real-me and real-space Tme-dependen Kohn-Sham equaon ( ) ( ) [ ] ( ) ( ) ( ) = = KS r r n r r n h r ψ ψ ψ ( ) [ ] ( ) ( ) [ ] ( ) r V r n r r e dr r V m r n h ex xc on KS ' ' = µ Norm-conservng pseudopoenal

12 Treamen of opcal elecrc feld We assume a long wavelengh lm lace cons. << lgh wavelengh Elecron dynamcs n a un cell under me-dependen spaally unform feld Two key ssues Use (spaally unform) vecor poenal nsead of (spaally lnear) scalar poenal o express macroscopc elecrc feld. Boundary effec (macroscopc shape effec) needs o be consdered from ousde of he heory. 1

13 Elecron dynamcs n crysallne sold under spaally unform feld a V ( z) z ee( )z Scalar poenal ee( )z spols spaal perodcy of he Hamlonan h r + a h(r ( ) ) Employng a vecor poenal replacng he scalar poenal one may recover he perodcy of he Hamlonan Tme-dependen spaally unform elecrc feld 1 A E = φ c φ = E ( ) z A( ) = c d' E( ' )zˆ h ψ 1 m e c ( ) = p A( ) + V ( r ) ψ ( ) ( r + a) = h ( r ) ψ ka ( r + a) = e ( r ) nk ψ nk Tme-dependen Bloch funcon

14 14 ( ) ( ) ( ) ( ) ( ) ( ) 1 = + + = nk nk nk nk r u r n r u r V A c e k p m r u ( ) ( ) ( ) ( ) r u a r u r u e r nk nk nk kr nk = + = ψ Equaon for me-dependen Bloch funcon

15 Elecron dynamcs n meallc clusers by TDDFT K. Yabana G.F. Bersch Phys. Rev. B (1996). Na L Assume Icosahedral shape Dpole momen Dpole momen Tme [ev -1 ] Tme [ev -1 ]

16 Collecve elecronc excaon: plasmon Longudnal plasmon 4πe n ωp = m 1/ Me plasmon (Surface plasmon of sphercal nanoparcle) ωp ω Me = 3 n: elecron densy 16

17 Opcal response of hn flm Choce of geomery (boundary condon) - We solve elecron dynamcs n a un cell nsde a crysallne sold. - The elecrc feld n he un cell depends on macroscopc geomery of he sample. - We need o specfy he boundary condon n he calculaon. 1 e ψ nk ( ) = p A( ) + V ( r ) ψ nk ( ) m c 1 A E = A( ) = c d' E( ' )zˆ c Transverse Longudnal A( ) = A ( ) A( ) = A ( ) A ( ) ex ex + polarzaon

18 Longudnal geomery: Treamen of nduced polarzaon Bersch Iwaa Rubo Yabana Phys. Rev. B6(000)7998. parallel-plae-capacor wh delecrcs. A( ) = A ( ) exr + A polarzaon ( ) delecrcs E = E ex + E polarzaon ~ E ex ( ) Equaon for polarzaon d Apolarzaon( ) 4 = π dr j d Ω Ω( cell ) j r A ( ) ( r ) Tme-dependen Kohn-Sham equaon 1 e δexc ψ = + A ψ eφψ + ψ m c δn n = ψ j = 1 e * ψ p + A ψ c. c. m ( ) c 1 A( ) E( ) = c ( ) = c E( ' ) d' A 18

19 CONTENTS 1. A froner n lgh - maer neracon : nense and ulrashor - Theores and compuaons requred n curren opcal scences -. Real-me and real-space calculaon for elecron dynamcs n sold - me-dependen band calculaon reamen of boundares Elecron and phonon dynamcs n sold for a gven elecrc feld - Delecrc funcon coheren opcal phonon opcal breakdown - 4. Coupled Maxwell + TDDFT mulscale smulaon - Frs-prncples smulaon for macroscopc elecromagnec feld n nense regme Dense elecron-hole plasma nduced by nense laser pulses - 19

20 Lnear response: Two mehods o calculae delecrc funcon from real-me elecron dynamcs Transverse geomery J σ ε ( ) = d' σ ( ' ) ω ( ω) = de σ ( ) 4πσ ( ) ( ω) ω = 1+ ω 1 da c d Longudnal geomery A d A o o 1 ε ( ) = Aex ( ) + Apol ( ) pol ( ) = 4πJ ( ) A d 1 ( ) = d' ε ( ' ) A ( ' ) ( ω) = d e d e ω ω A A o ex ( ) ( ) ex For an mpulsve exernal feld E ( ) = kδ ( ) A( ) = kcθ ( ) J da c d o ( ) = kσ ( ) E ( ) = = kε ( ) o 1 1 0

21 Comparson of numercal resuls for delecrc funcon of S by wo mehods - Green : longudnal - Red : ransverse Re[ε] Spaal grd: 16^3 K-pons : 8^3 Dsoron srengh: A_0 = 0.01 (wh nduced) (whou) J()/A_0 Im[ε]

22

23 Delecrc funcon of S n TDDFT (ALDA) Transverse calculaon Insananeous pulse feld s appled a =0 he nduced curren as a funcon of me. Delecrc funcon by TDDFT (ALDA) s no very good. Too small bandgap (common problem wh LDA). 3

24 Dos: expermen Dash-doed: RPA Sold: Behe-Salpeer (elecron-hole neracon consdered) TDDFT (ALDA) G. Onda L. Renng A. Rubo Rev. Mod. Phys. 74(00)601. PRL (011) arxv: S. Sharma J.K. Dewhurs A. Sanna E.K.U. Gross Boosrap approx. for he exchange-correlaon kernel of me-dependen densy funconal heory

25 Pump-Probe Measuremen: Coheren Phonon Inense 1 s pulse exces elecrons and aoms. Weak nd pulse measure change of reflecvy. T ph : Perod of opcal phonon probe pump M. Hase e al. Naure (London) (003) ΔΔ R e Γ sn Ω ph + φ 5

26 The wo mechansms for coheren phonon generaon Vrual elecronc excaon (ransparen maeral) E() 1 V V = χ Ε( ) F = q E() Real elecronc excaon (opaque maeral) F() Q() Impulsve Force appears only durng laser rradaon sn Ω pp π dfference n he phase F() Q() Dsplacve Change of equlbrum 1 cos Ω pp ΔQ Y. Yan E. Gamble and K. Nelson J. Chem. Phys (1985) H.J. Zeger J. Vdal T.K. Cheng E.P. Ippen G. Dresselhaus and M.S. Dresselhaus Phys. Rev. B (199)

27 Coheren phonon measuremens n Sb - Mos popular maeral for coheren phonon generaon - Observed wo modes: A 1g E g Eg Eg A1g A1g K. Ishoka M Kajma and O. Msochko J. Appl. Phys (008)

28 Sb: crysal srucure and phonon mode - Semmeal - Rhombohedral srucure Havng C3 symmery axs parallel o vercal lne n followng fgure Bond Crysal srucure Bond Elecron densy a ground sae A 1g E g Hgh densy regon Low densy regon Each aom have hree bonds Only one bond on plane Oher wo bonds are ou of plane There are wo modes: A1g and Eg (doubly degenerae)

29 Elecron densy under laser pulse Y. Shnohara e.al submed o J. Chem. Phys. Elecron densy a ground sae Densy dfference on plane-1 Densy dfference on plane- Increasng Decreasng E Eg A1g E Includng elecrc feld Ou of elecrc feld Durng laser pulse: elecrons moves aom o aom Afer laser pulse ends: bondng elecrons are reduced

30 Densy change and force Elecron densy a ground sae Densy dfference on plane-1 Densy dfference on plane- E E g A 1g Ground sae Plane-1 01/6/5 Includng elecrc feld Ou of elecrc feld Force s domnaed by real-excaon (removal of bondng elecrons)

31 Coheren phonon measuremens n Sb - Mos popular maeral for coheren phonon generaon - Observed wo modes: A 1g E g Eg Eg A1g A1g K. Ishoka M Kajma and O. Msochko J. Appl. Phys (008) However calculaon s no conssen wh measuremen. Expermenally A 1g : dsplacve E g : mpulsve.

32 TDDFT calculaon for coheren phonon n S Y. Shnohara e al. Phys. Rev. B (010) Pulse duraon Drec bandgap:.4 ev(calc.) Vrual excaon Impulsve force.75 ev: above band gap Real excaon Dsplacve force.50 ev: around band gap.5 ev: below band gap E When laser frequency s below he drec gap mpulsve (Raman ensor) force s vsble. T g 01/6/5

33 CONTENTS 1. A froner n lgh - maer neracon : nense and ulrashor - Theores and compuaons requred n curren opcal scences -. Real-me and real-space calculaon for elecron dynamcs n sold - me-dependen band calculaon reamen of boundares Elecron and phonon dynamcs n sold for a gven elecrc feld - Delecrc funcon coheren opcal phonon opcal breakdown - 4. Coupled Maxwell + TDDFT mulscale smulaon - Frs-prncples smulaon for macroscopc elecromagnec feld n nense regme Dense elecron-hole plasma nduced by nense laser pulses - 33

34 Weak feld regme: Sandard approach for lgh-maer neracon Elecromagnesm: Maxwell equaon for macroscopc felds E D B H Consuon relaon D = [ ] = ε ( ω)e D E Quanum Mechancs: Perurbaon heory o calculae lnear suscepbles ε(ω) Inense laser pulse nduced nonlnear elecron dynamcs Quanum mechancs s useful o calculae nonlnear suscepbles as well. D = ( ) ( 3) [ ] = ε ( ω) E + χ E + χ D E E 3 + Furher ncrease of laser pulse nensy Perurbave expanson s no more useful. Dense elecron-hole plasma Opcal breakdown ec. We need o solve couple Maxwell + Schrodnger dynamcs.

35 We solve coupled Maxwell + Schroednger equaons. Maxwell equaon descrbng elecromagnec feld E ( r ) B( r ) Tme-dependen densy-funconal heory descrbng elecron dynamcs ψ polarzaon ( r ) [μm] exernal feld Dfferen spaal scales [nm] Tme-dependen Kohn-Sham eq. Maxwell eq. e δe ψ = 1 + A ψ eφψ + m c δn c A A 1 = 4π j c φ = 4π xc ψ 1 * n = ψ j = ψ + m { en } e c on en e A ψ c. c. Mulscale smulaon nroducng wo spaal grds

36 Laser pulse rradaon normal on bulk S x Maxwell eq. ( ) ε z A( z ) A( z ) = 0 c z Propagaon z-drecon polarzaon x-drecon A r = A z ˆ ( ) ( )x z A/c Laser frequency below drec bandgap λ = 800 nm ω = 1.55eV Vacuum S ε=16 [µm] Macroscopc grd Index of refracon n = Reflecance ε 1 R = 1+ n n Velocy of wave v = c n 36

37 Mulscale smulaon ( ) ( ) ( ) Z J c A Z Z A Z c 4 1 π = Macroscopc grd pons(μm) o descrbe macroscopc vecor poenal { } Z e on Z Z xc Z Z Z Z en en n E e A c e m 4 1 = + + = π φ ψ δ δ φ ψ ψ ψ ( ) A n c e m j j dr Z J Z e Z Z Z Z Z e Z e * * 4 ) ( π ψ ψ ψ ψ = = Ω A each macroscopc pons Kohn-Sham orbals are prepared and descrbed n mcroscopc grds. Exchange of nformaon by macroscopc curren and macroscopc vecor poenal. ( ) Z A ( ) Z J ψ Z A each macroscopc grd pon We consder a un cell and prepare mcroscopc grd. K. Yabana e.al Phys. Rev. B (01).

38 Effcen parallelzaon Each CPU calculaes elecron dynamcs a dfferen macroscopc poson. J Z ± 1 ( ) (We also parallelze wh repec o k-pons) ( ) A Z ± 1 CPU 1 CPU CPU 3 CPU 4 CPU- N Macroscopc vecor poenal requres A and j of nearby pons o evolve n me. (only ens of byes exchanged) ψ A 1 ( ) A ( ) A ( r ) ψ ( r 3 ( ) A ) ψ 3 ( r 4 ( ) A ) ψ 4 ( r N ( ) ) N ψ ( r ) 1 Each CPU calculaes nonlnear elecron dynamcs ndependenly. 38

39 Propagaon of weak pulse (Lnear response regme separae dynamcs of elecrons and E-M wave) I=10 10 W/cm A/c Laser frequency:1.55ev: lower han drec bandgap.4ev(lda) Vacuum S [µm] Z=0 [µm] Z=0.8 [µm] Z=1.6 [µm] 39

40 Propagaon of weak pulse (Lnear response regme separae dynamcs of elecrons and E-M wave) I=10 10 W/cm A/c Laser frequency:1.55ev: lower han drec bandgap.4ev(lda) Vacuum S [µm] Elecron excaon energy per un cell volume 40

41 Maxwell+TDDFT vs Maxwell only (consan ε) Red = Maxwell + TDDFT A/c Green = Maxwell ε c z A = ( z ) A( z ) = 0 ε 16 Vacuum S In lnear regme dsperson of delecrc funcon Induces Phase velocy vs group velocy c c v = v g = ε ω dε ε 1 + ε dω Chrp effec s also seen [µm] 41

42 More nense laser pulse Maxwell and TDKS equaons no more separae. I =5 x 10 1 W/cm Vacuum S [µm] Z=0 [µm] Z=0.8 [µm] Z=1.6 [µm] 4

43 More nense pulse (-phoon absorpon domnaes) I = 5 x 10 1 W/cm A/c Laser frequency:1.55ev: lower han drec bandgap.4ev(lda) Vacuum S [µm] Elecron excaon energy per un cell volume 43

44 Vecor poenal and elecronc excaon energy Afer he laser pulse spls no refleced and ransmed waves 44

45 Energy Conservaon Our mulscale equaons can be derved from he Lagrangan. We can derve conserved energy whch s a sum of - energy of EM feld - energy of elecrons - neracon energy beween hem.

46 Energy conservaon well sasfed n praccal calculaons. Toal energy Energy of EM feld Elecron energy W/cm 10 1 W/cm W/cm 46

47 Compuaonal aspec x z 3.5µm 56 pons z = 13nm Mcroscopc (Schroednger) spaal grd: 16 3 k-pons : 8 3 (reduced by symmery o 80) (oo small) Tme sep (commom) = Macroscopc (Maxwell) spaal grd: 56 Compuaonal coss: 104 cores (MPI only) 10 hours a SGI Alx ICE 3800EX (ISSP Unv. Tokyo) 0480 cores (MPI+openmp) 0 mn. a K-Compuer 47

48 Dense elecron-hole plasma generaon a he surface modfes delecrc properes a he surface. λ=65nm 100fs pulse Pump-probe S Srong pump-pulse exces elecrons a he surface formng dense elecron-hole plasma K. Sokoowsk-Tnen D. von der Lnde Phys. Rev. B (000) Reflecvy change Probe-pulse measures change of Delecrc properes. Drude model f Pump-pulse nensy ε ( n ) ph = ε gs 4πe n * m m*=0.18 τ=1fs ph 1 ω ω + τ 48

49 Our Pump-Probe Numercal Expermen Vacuum S I=1x10 11 W/cm hv=1.55ev We can calculae exced elecron densy a he surface by pump-pulse / reflecvy of probe pulse -phoon absorpon Reflecvy of probe pulse Exced elecron densy a he surface ε F by Drude model ( n ) ph = ε gs 4πe n * m 49 ph 1 ω ω + τ

50 We may exrac delecrc funcon of exced surface from numercal pump-probe calculaon R ( ) = A ( ) + Ar ( ) ω ( ) ( ) d e Ar ω = ω d e A ( ) A Decompose vecor poenal a he surface no ncden and refleced componens. Rao of wo Fourer componens gves us a reflecvy as a funcon of frequency. R ( ω) 1 = 1 + ε ( ω) ε ( ω) Delecrc funcon a he surface when rradaed by W/cm pulse Well descrbed by Drude model (a leas for real par). 50

51 Change of delecrc funcon as we ncrease he pump-pulse nensy. x10 1 W/cm Increase of elecron-hole densy enhances meallc response. ε ( n ) ph = ε gs 4πe n * m ph 1 ω ω + τ 5x10 1 W/cm -phoon absorpon 1x10 13 W/cm Exced elecron densy a he surface 51

52 Compuaonal Aspecs and Fuure Prospecs Large compuaonal resources requred for mul-scale calculaon A presen Normal ncdence of lnearly polarzed pulse on S crysal: 1000 cores 10 hours cores 0 mn (K-compuer Kobe) More resources o acheve More complex delecrcs(so TO ZrO ) x 10 Oblque ncdence(-dm) x 50 Self-focusng(3-dm) x 1000 Crcular polarzaon(3-dm) x 1000 K-Compuer Japan cores

53 Summary Elecron dynamcs n bulk perodc sold by real-me TDDFT - Frs-prncples descrpon of elecron dynamcs n femo- and ao-second me scale - Applcaons o - delecrc funcon (lnear response) - coheren phonon generaon - opcal breakdown Coupled Maxwell + TDDFT mul-scale smulaon - Promsng ool o nvesgae laser-maer neracon - Requres large compuaonal resources a compuaonal challenge - Furher developmens necessary Surface reamen longudnal elecromagnec wave D3D propagaon collson effecs (e-e e-v) 53