TDCDFT: Nonlinear regime

Transcription

1 Lecue 3 TDCDFT: Noliea egime Case A. Ullich Uivesiy of Missoui Beasque Sepembe 2008

2 Oveview Lecue I: Basic fomalism of TDCDFT Lecue II: Applicaios of TDCDFT i liea espose Lecue III: TDCDFT i he oliea egime Time-depede Koh-Sham wih memoy Eegy dissipaio TDDFT i he Lagagia fame Aohe way o ea memoy i TDDFT: ime-depede OEP TDOEP

3 TDKS equaio i TDCDFT 1 2 i + 1 c A ex + 1 c A 2 + V ex + V H i 0 A igoous exesio of he LDA io he oliea dyamical egime has ecely bee fomulaed Lagagia TDDFT see lae Howeve he viscoelasic expessio of liea-espose TDCDFT ca be easily bu somewha ad hoc exeded io he dyamical egime: 1 c A V ALDA + G. Vigale C.A.U. ad S. Coi PRL Valid up o secod ode i he spaial deivaives The gadies eed o be small bu he velociies hemselves ca be lage

4 Noliea VK-TDCDFT: sess eso ime-depede velociy field: / v i i i i i d d δ ζ δ η v v 3 2 v v + + whee he viscosiy coefficies ae defied as Fouie asfoms: ~ 2 i e d ω ω η π ω η

5 Noliea TDCDFT: 1D sysems Coside a 3D sysem which is uifom alog wo diecios ca asfom veco poeial io scala poeial: z ALDA V z V z + V M z wih he memoy-depede poeial V M z z dz z zz z z

6 The memoy keel Assumig ha he sysem has bee i he goud sae wih zeo velociy fo <0 he zz compoe of he sess eso is z v z z zz d Y z' z' whee he memoy keel is give by 0 4 Y η + ζ 3 Usig he defiiio of he viscosiy coefficies oe fids explicily Y 4 3 S 0 2 dω L Im f cos[ ω ω ] π ω saic shea modulus

7 The memoy keel H.O. Wiewadae ad C.A.Ullich PRL apid memoy loss dissipaive s S Y0 has zeo slope puely elasic i high-fequecy limi log-age compoe elasic 4 3 S 0 ~ 0.25 T plasma

8 poeial wih memoy: simple model + L z A L z L N z s π ω π si si 1 cos 2 2 ALDA z V z V z V z QV M GK M XC Memoy Phase Lag Readaio Foce

9 poeial wih memoy: full TDKS calculaio 40 m GaAs/AlGaAs Weak eiaio iiial field 0.01 Sog eiaio iiial field 0.5 ALDA ALDA+M H.O. Wiewadae ad C.A. Ullich PRL

10 ...bu whee does he eegy go? The sysem is o dive by exeal fields so he eegy should be coseved. I liea espose calculaios of aomic eiaio eegies he VK fucioal gives a fiie liewidh which is uphysical. R. D Agosa ad G. Vigale PRL collecive moio alog z is coupled o he i-plae degees of feedom he x-y degees of feedom ac like a esevoi decay io muliple paicle-hole eiaios This is he siuaio fo ifiie sysems. Bu wha abou fiie sysems?

11 Example: wo elecos o a 2D quaum sip had walls z x sadig waves peiodic boudaies avelig waves Chage-desiy oscillaios Δ L C.A. Ullich JCP

12 Example: wo elecos o a 2D quaum sip Δ10 L50 F0.02 iiial-sae desiy exac LDA Compae exac calculaio ime-depede CI wih TDKS Iiial sae: cosa elecic field which is suddely swiched off Afe swich-off fee popagaio of he chage-desiy oscillaios

13 2D quaum sip: ime-depede dipole mome Δ10 L50 F0.02 Δ10 L100 F0.02 exac ALDA Exac calculaios give a beaig pae of d due o a supeposiio of dipole oscillaios ivolvig sigle ad double eiaios Recuece ime iceases wih legh of he sip To modulae d he exac V aleaely damps ad dives he sysem ALDA misses he beaig pae sice i has o muliple eiaios

14 2D quaum sip: ALDA+M d is expoeially damped L50 L100 Ulike he exac V he VK fucioal oly damps bu does o dive back oly accous fo eadaio The VK fucioal cao ell ha he sysem is fiie. I eas he sysem locally like a homogeeous eleco gas. ifiie ecuece ime emeges i he hemodyamic limi of he sysem dampig of d is due o decoheece ivolvig may eiaios wih a coiuous specum

15 Summay fis pa I he oliea eal-ime domai he fequecy-depedece of he XC sess eso aslaes io memoy depedece We solved TDKS equaios wih memoy fo chage-desiy oscillaios i quaum well The VK fucioal causes dissipaio whee eegy ges asfeed io icohee muliple paicle-hole eiaios Model calculaios fo 2D quaum sip show how he exac TDKS poeial causes muliple eiaios by is oadiabaic behavio divig ad dampig. The VK fucioal misses his behavio bu becomes coec i he hemodyamic limi ifiie sysem size ad paicle umbe.

16 TDDFT i he Lagagia fame L-TDDFT I.V. Tokaly PRB ad ad TDDFT book Ch. 8 C.A.U. ad I.V. Tokaly PRB ; I.V. Tokaly PRB g i ξ v ξ ξ0 ξ ξk i ξk use a efeece fame ha moves wih he fluid. basic vaiables: posiios of fluid elemes ad hei defomaios oliea coodiae asfomaio ξ Lagagia coodiae Cauchy s defomaio eso i he laboaoy fame a fucioal of he velociy ξ g 0

17 TDDFT i he Lagagia fame: sess eso A i + A A P [ g ] c v i i i i whee P i P i T KS i sess eso of ieacig mius kieic sess eso of KS sysem This is a fomally exac ime-depede may-body heoy. The ieacig sess eso is of couse oly appoximaely kow. Fo small gadies of he sess eso is a spaially local fucioal of bu a olocal fucioal i ime. g i g i This is he exac exesio of LDA io he dyamical egime. I geeal i coais boh elasic ad dissipaive effecs.

18 The small defomaio appoximaio P i P ALDA δ i + 0 δ d' 2 i K δg kk + μ δ 1 3 i δg kk δg i u u i + ad u i v i he egime of small defomaios we ecove he oliea fom of VK-TDCDFT i.e. ALDA+M whee μ iωη~ K ~ iωζ This pus oliea VK-TDCDFT o fim gouds. Remembe he defomaios ae small bu he velociies ca be lage.

19 Noliea elasic appoximaio If we eglec dissipaio a oliea local appoximaio fo he sess eso ca be igoously deived: P whee i E 2 3 ki g i g E ki g + L i g kl E po uif uif 7 /3 e po 8/3 e 3 ad E 3 4/3 5/ 3 ad L i is a kow fucio. g Exac dyamical LDA i he high-fequecy limi fo ay defomaio Fo small defomaios his educes o he puely elasic high-fequecy limi of VK-TDCDFT. deviaios of he defomaio eso g fom δ i ca be viewed as a measue of oadiabaiciy.

20 L-TDDFT vesus VK-TDCDFT: simple 1D models C.A.Ullich ad I.V. Tokaly PRB ξ x g x 0 ξ x ad g x le 2N 0 ξ cos L 2 πξ L ad choose aalyical expessios fo which ca easily be iveed. v ξ ad x ξ 2 sloshig mode beahig mode

21 L-TDDFT vesus TDCDFT: simple 1D models desiy velociy defomaio sloshig mode: o oo sogly defomed cousi of Koh s mode beahig mode: sogly defomed eveywhee vey u-hydodyamic

22 L-TDDFT vesus TDCDFT: high-fequecy limi powe i he high-fequecy limi he elasic appoximaio fo L-TDDFT becomes he exac dyamical exesio of he LDA fo all defomaios fo small defomaios TDCDFT becomes exac fo all fequecies powe fo lages ampliudes TDCDFT deviaes: <2.5% fo sloshig mode ~100% fo beahig mode VK-TDCDFT L-TDDFT exac The oliea TDCDFT emais good fo modeae defomaios!

23 Beakdow of he ALDA L-TDDFT i he high-fequecy puely elasic limi ω>>ω p ALDA L TDDFT V V exac Sloshig mode: small defomaio mio coecios o ALDA Beahig mode: lage defomaio ALDA beaks dow

24 Summay secod pa A igoous fomulaio of local ime-depede effecs is esablished by TDDFT i he Lagagia fame VK-TDCDFT emeges as small-defomaio appoximaio. Noadiabaic effecs ae boh elasic ad dissipaive. I depeds o he fequecy which effec is moe impoa. The ALDA beaks dow whe he elecoic desiy apidly udegoes lage defomaios. A moe geeal fomulaio of Lagagia TDDFT has ecely become available: TDDefFT TD defomaio fucioal heoy icludig veco poeials Tokaly 2007.

25 Time-depede opimized effecive poeial C.A.U. U.J. Gossma E.K.U. Goss PRL [ ] * * 1 3 c c u V d d i k k k N + { } [ ] 1 * A u i δ δ exac ehage: N k k k x d u 1 * * 3 * ' 1 whee

26 Applicaios of TDOEP i he liea egime Opical speca of solids Kim ad Goelig PRL Molecula eiaio eegies ad dyamic polaizabiliies Hiaa e al. PRA Shigea Hiao Hiaa PRA saic ω0 keel ω-depede keel Kim ad Goelig Silico Shigea e al. Neo

27 Time-depede KLI { }] + 3 u V d u V KLI KLI Applicaios of TDKLI i he oliea egime: Aoms i sog fields C.A.U. ad E.K.U. Goss Commes A. Mol. Phys M. Mud ad S. Kuemmel PRL Meallic cluses C.A.U. P.-G. Reihad E. Suaud PRA H.S. Nguye A.D. Badauk C.A.U PRA No zeo-foce heoem fo TDKLI Mud Kuemmel va Leeuwe Reihad PRA Slae poeial

28 Adiabaic appoximaio o TDOEP AOEP: saic OEP which poduces as selfcosise goud-sae desiy V [ ] V + V [ ] + V KS ex H AOEP saic KS equaio { ε } selfcosise ukow! saic OEP evaluaed wih { ε } Sep 1: ive saic KS equaio: AOEP { } Sep 2: cosuc fom ε V { V ε } KS see also M. Thiele E.K.U. Goss S. Kümmel PRL

29 Numeical soluio of TDOEP: sep-by-sep? Mud ad Kümmel PRA : umeical isabiliies?? [ ] * * 1 3 c c u V d d i k k k N + Iegad vaishes a uppe limi oly deemied fo < Sep-by-sep ime popagaio fails V

30 Numeical soluio of TDOEP: global ieaio 0 T 0 T global selfcosisecy +1 s ieaio: V i d d d d d T T 0 ew 0 ew old G Covegece idex: 2 h ieaio: give ] [0 1 2 T V i + { } V TDOEP ] 0 [ T

31 Fee chage-desiy oscillaios sudde swichig ε ε0 iiially TDOEP AOEP TDKLI ime a.u. TDOEP slighly blueshifed: memoy gives ise o elasic coibuio x-oly

32 Dive oscillaios ω11 mev ω20 mev close o ω 12 esoace close o ω 23 esoace XC powe: P [ ] 0 dz z V z V z z x x

33 Dive oscillaios ω40 mev ω50 mev close o ω 14 esoace bewee ω 14 ad ω 25 away fom esoaces

34 Summay hid pa Sable algoihm fo full TDOEP i quaum wells AOEP is vey simila o TDKLI No poblems wih zeo-foce heoem i TDKLI Memoy effecs i TDOEP become sigifica fo high fequecies ad close o esoaces Iisically oadiabaic pheomea muliple eiaios dissipaio: eed o go beyod exac-ehage H.O. Wiewadae ad C.A. Ullich PRL

35 ... ad fially... Thaks o: D. Hashai Wiewadae ow a CUNY D. Volodymy Tukowski ow a UCF Yoghui Li gad sude a MU Posdoc opeig TDDFT fo eioic effecs i maeials