Nonequilibrium Green s function (NEGF) method in thermal transport and some applications

Transcription

1 Nonequlbrum reen s funcon NEF mehod n hermal ranspor and some applcaons Jan-Sheng Wang Naonal Unversy of Sngapore

2 Oulne of he al Inroducon Mehod of nonequlbrum reen s funcons Applcaons Thermal currens n1d chan and nanoubes Transen problems Full counng sascs ICTP worshop 1

3 Fourer s law for hea conducon J κ T f [ ω] f e ω d Fourer Jean Bapse Joseph Baron

4 Thermal conducance I κ σ I SJ S where I : hermal curren J : curren densy T σ : T T T σ : emperaure of lef and rgh lead conducance κ : conducvy S : cross secon area 4

5 Thermal ranspor of a juncon Juncon ef ead T gh ead T sem-nfne leads ICTP worshop 1 5

6 Models C C C n 1 T 1 T α α pα pα uαk uα u mx α C ac T αc uv u u u α C α j 1 1 Tuuu T uuuu 3 4 C C C C C C C n j j jl j l j jl α Juncon ef ead T gh ead T ICTP worshop 1 6

7 ICTP worshop 1 7 Force consan marx C C C C C V V K V V K

8 Defnons of reen s funcons reaer/lesser reen s funcon > < j ' uj u ' j ' Tme-ordered/an-me ordered reen s funcon ' θ ' ' θ ' ' θ ' ' θ earded/advanced reen s funcon r a ' ' θ > > ' θ ' ' > < > < < < ' ' See ex boos e.g.. aug and A.-P. Jauho or J. ammer or S. Daa or D Venra ICTP worshop 1 8

9 Conour-ordered reen s funcon j ' Tu C j u ' d C Tr ρ Tu C j u ' e Conour order: he operaors earler on he conour are o he rgh. ICTP worshop 1 9

10 elaon o he real-me reen s funcons σ σ or σ ± σσ ' ' ' or < > > < ICTP worshop 1 1

11 ICTP worshop 1 11 Equaons for reen s funcons n ballsc sysems ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' < > < > K I K I K I K I K a r a r δ δ δ σδ δ σσ σσ σσ

12 Soluon for reen s funcons ' ' δ ' usng Fourer ransform: ra ra K I ra ra ω [ ω] K [ ω] I ra 1 ω ωi K cδω K dδω K [ ] 1 r a [ ω] [ ω] ω η I K η r a βω f e < > < r < c and d can be fxed by nal/boundary condon. ICTP worshop 1 1

13 Transform o neracon pcure j ' uj u ' '' d'' C Tr ρ Tu C j u ' e I n '' d'' I I C Tr ρitu C j u ' e ICTP worshop 1 13

14 Perurbave expanson of conour ordered reen s funcon j ' Tu C j u ' e 3 n '' d'' 1 TCuj u ' 1 n 1 d1 n 1 d1 n d! 1 1 Tu C j u ' Tu C j u ' Tlmn 1 3 ul 1 um un 3 d1dd3 3lmn 1 3 opq T u u u d d d opq o 4 p 5 q '... Tu u ' u u u u u u j C j l 1 m n 3 o 4 p 5 q 6 Wc's heorem Tu u Tu u Tu u Tu u ' C j l 1 C m p 5 C n 3 q 6 C o 4 See e.g. A.. Feer & J. D. Waleca. ICTP worshop 1 14

15 Dagrammac represenaon of he expanson ħ ħ ħ ' ' Σ ' dd conn 1 n 1 conn 1 ICTP worshop 1 15

16 Dagrammac represenaon of he nonlnear self-energy ICTP worshop 1 16

17 Explc expresson for self-energy σσ ' ' ' n j TjlmT σσ σσ rs lr ms lmrs dω ' Σ [ ω] [ ω'] [ ωω'] π OT 4 j σσ '' σ '' σ '' σσ ' jl mrs lm rs lmrs σ '' dω ' σδ σ '' T T [] [ ω'] π ICTP worshop 1 17

18 eneral expanson rule Sngle lne 3-lne verex n-double lne verex T ' j j jj 1 j n n Tu C j uj uj n n ICTP worshop 1

19 Juncon sysems Three ypes of reen s funcons: g for solaed sysems when leads and cenre are decoupled for ballsc sysem for full nonlnear sysem overnng amlonans C V n C g α Equlbrum a T α C V reen s funcon Nonequlbrum seady sae esablshed ICTP worshop 1 19

20 ICTP worshop 1 Three regons C u u T u u u u u u u u T C C C ' ' 1 β α β α αβ

21 Dyson equaons and soluons g g Σ Σ V gv V gv C C C C C C Σ n ω η I K Σ η r C r Σ < r < a 1 n 1 Σ r r r ICTP worshop Keldysh equaon Σ I Σ I Σ < r < a r r < a a n n n r < < Σ Σ n a

22 Energy curren Mer-Wngreen d I uv u d T C C 1 π C < C ω Tr V [ ] ωdω 1 r < < a Tr [ ω] Σ [ ] [ ] [ ] ω ω Σ ω ωdω π ICTP worshop 1

23 andauer/carol formula d ω Tr CCΓ CCΓ d r a r a I f f Γ Σ Σ α α α dω π I I I < r < a < Σ Σ f Γ f Γ Γ Γ a r r a ICTP worshop 1 3

24 ICTP worshop 1 4 Ballsc ranspor n a 1D chan Force consan marx Equaon of moon K j j j j u u u u j

25 Soluon of g Surface reen s funcon ω η K g I η K g j j λ j 1 / < 1 1 λ ω η λ λ ICTP worshop 1 5

26 ead self energy and ransmsson λ T[ω] 1 Σ r r j ω K Σ Σ C 1 j λ 1 λ λ ω r a 1 < ω < 4 T[ ω] Tr Γ Γ oherwse ICTP worshop 1 6

27 ea curren and unversal hermal conducance I ωt[ ω] f f mn dω π max I f dω 1 σ lm ω f T T βω T T T π e 1 π T B σ T 3h ego and Krczenow ω ω ICTP worshop 1 7

28 1D nonlnear model Three-aom juncon wh cubc nonlneary FPU-α. From JSW Wang Zeng PB Cross from QMD: JSW Wang ü Eur. Phys. J. B ICTP worshop 1 8

29 Carbon nanoube nonlnear effec The ransmssons n a one-un-cell carbon nanoube juncon of 8 a 3K. From J-S Wang J Wang N Zeng Phys. ev. B ICTP worshop 1 9

30 Transen problems T uv u > I Im < 1 ICTP worshop 1 3 1

31 Dyson equaon on conour from o Conour C ' dg V g ' C d dg V g V ' C C ICTP worshop 1 31

32 Transen hermal curren The me-dependen curren when he couplng s suddenly swched on. a Curren flow ou of lef lead b ou of rgh lead. Dos are wha predced from andauer formula. T3K.65 ev/å u wh a small onse.1. From E. C. Cuansng and J.-S. Wang Phys. ev. B See also PE ICTP worshop 1 3

33 Fne lead problem Top : no onse a N N 1 b NN5 emperaure T1 1 3 K blac red blue. T1.1T T.9T. gh : wh onse.1 and smlarly for oher parameers. From E. C. Cuansng. and J.-S. Wang Phys. ev. E ICTP worshop 1 33

34 Full counng sascs FCS Wha s he amoun of energy hea Q ransferred n a gven me? Ths s no a fxed number bu gven by a probably dsrbuon PQ enerang funcon ξq Z ξ e P Q dq All momens of Q can be compued from he dervaves of Z. The objecve of full counng sascs s o compue Zξ. ICTP worshop 1 34

35 A bref hsory on full counng sascs. S. evov and. B. esov proposed he concep for elecrons n 1993; rederved for nonneracng elecron problems by I. Klch K. Schönhammer and ohers K. Sao and A. Dhar obaned he frs resul for phonon ranspor n 7 J.-S. Wang B. K. Agarwalla and. PB 11; B. K. Agarwalla B. and J.-S. Wang PE 1. ICTP worshop 1 35

36 Defnon of generang funcon based on wo-me measuremen Z Tr ρ' P a [ ] ρ' e ξ e ξ P ρ a a P a P a a >< a Conssen hsory quanum mechancs. rffhs. a > a a > U e.g. U e U / ICTP worshop 1 36

37 Approaches o compue Z Express Z as expecaon value of some effecve evoluon operaor over a conour Evaluae he expresson usng Feynman pah negral/nfluence funconal Feynman dagrammac expanson ICTP worshop 1 37

38 Produc nal sae β ρ ρ e α α [ P ] ρ' ρ α C Z Tr ρe e ξ ξ ξ ξ Tr ρe U e U Tr ρe U e U e ξ / ξ ξ / Tr ρuξ/ U ξ/ x U ' e U ' e x x ICTP worshop 1 38

39 ICTP worshop 1 39 Ux ' ' ' ' x u e u e u u V u e e e e e e Te e Te e U x x x x I C C T x C x C x C C x C C C x x x x d x d x x x

40 ICTP worshop 1 4 Schrödnger esenberg and neracon pcures I d I I I I Ae e A Te S S S S e A A AU U A U U w U I ' ' ' ' ' ' ' ] [ ' ' ' ' '' '' > > Ψ > Ψ > Ψ > Ψ > Ψ >< > Ψ > Ψ ρ Schrödnger pcure esenberg pcure Ineracon pcure

41 ICTP worshop 1 41 Compue Z n neracon pcure x A C C A A C x C C x C C C C x I x I C x I C d C g g g g Z V g V g d d d T e T U U Z C x I Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ ] Tr ln[1 1 ] 1 ln de[1 1 ] Tr ln[1 1 ln ] Tr[ 1 1 ' ' 1 1 Tr ] Tr[ ] Tr[ / / ρ ρ ρ ξ ξ -ξ/ ξ/

42 ICTP worshop 1 4 Imporan resul [ ] oherwse f / f / ' ' ' ' ln1 Tr 1 ln M M A x A C C A x x x x x g g Z < < < < Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ ξ ξ

43 ICTP worshop 1 43 ong-me resul evov-esov formula [ ] [ ] { } 1/ 1 1/ 1 ] [ 1 ] [ where 1 1 ln de 4 1 ln Tr 1 ln1 Tr 1 ln α α ω β α α α ω ξ ω ξ ω ξ ω ξ ω ξ ω ξ β α ω ω π ω α T e f e b e a f f e e f e f e I d Z B a a r a r M a K r A b a b a b a b a Σ Σ Γ Σ Σ Γ Γ Σ < >

44 ICTP worshop 1 44 Arbrary me ransen resul n long me ln Tr 1 ln ln Tr ln1 1 ln I Q Z Q Q Q Z Q Q Z Q Z M A n n n A Σ Σ ξ ξ ξ ξ ξ ξ

45 Numercal resuls 1D chan 1D chan wh a sngle se as he cener. 1eV/uÅ.1 T 31K T C 3K T 9K. ed lne rgh lead; blac lef lead. B. K. Agarwalla B. and J.-S. Wang PE ICTP worshop 1 45

46 FCS for coupled leads C K V V C C K V KC V g gv C V V K dω ln Z M ln de I e 1 f1 f e 1 f1 f Tg[ ] 4π { ξω ξω ω } r a a r α α α 1 1 Tg [ ω] ΓΓ Γ g g α If cener s absen V C C V r a T [ ω] T [ ω] Γ Γ g I hen From. B. K. Agarwalla and J.-S. Wang Phys. ev. E ; and arxv: ICTP worshop 1 46

47 FCS n nonlnear sysems Can we do nonlnear? Yes we can. Formal expresson generalzes Mer- Wngreen Ineracon pcure defned on conour Some example calculaons ICTP worshop 1 47

48 Vacuum dagrams Dyson equaon neracon pcure ransformaon on conour ln Z 1 Σ Tr j ξ ξ Σ Σ A n I I n d I I T C 1 Tr ρ Tu C 1 u e Z ZZ ρ ρv V / Z I n M M I O V O V C[ ] V Te C x T C T C u V uc uv C u d 1 Z n ICTP worshop 1 48

49 Nonlnear sysem self-conssen resuls Σ n ' 3 λ ' δ ' Sngle-se 1/4λu 4 model cumulans. 1 ev/uå.1 K c 1.1 V C -1V C 1-.5 T 66 K T 41K. From. B. K. Agarwalla B. and J.-S. Wang arxv:: ICTP worshop 1 49

50 Oher resuls We can also compue he cumulans for he projeced seady sae ρ Enropy producon Flucuaon heorem Zξ Z-ξ β -β The heory s appled equally well o elecron number of elecron energy ranspor ICTP worshop 1 5

51 Summary remars NEF s a powerful ool o handle hermal ranspor problems n nanosrucures Seady sae curren s obaned from andauer and Carol formula New resuls for ransen and full counng sascs. A ey quany s he self-energy Σ A ICTP worshop 1 51

52 roup members/acnowledgemens Fron lef o rgh Mr. Thngna Juzar Yahya Mr. Zhou angbo Mr. Bjay Kumar Agarwalla Dr. ee Meng ee Dr/Ms N Xaox Mr. Tan Kuo ang; Bac: Dr. Jose us arca Palacos Prof. Wang Jan Dr. Jang Jnwu Prof. Wang Jan-Sheng Dr. Zhang fa Mr. uanan Dr. Eduardo C. Cuansng. ICTP worshop 1 5

53 Than you see hp://saff.scence.nus.edu.sg/~phywjs/ for more nformaon