Advances in Nonequilibrium Green's Function Method for Energy Transfer Mediated by Photons

Research discusses the application of the Nonequilibrium Green's Function method for analyzing energy, momentum, and angular momentum transfers mediated by photons. Various topics covered include near-field heat transfer, Coulomb interactions, electron-photon transitions, local equilibrium assumptions, and heat transfer between graphene sheets, nanotubes, and triangles.

• Greens Function Method
• Energy Transfer
• Photons
• Near-field Heat Transfer
• Coulomb Interaction

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1. Nonequilibrium Greens function method for energy, momentum, and angular momentum transfers mediated by photons Jian-Sheng Wang Department of Physics, National University of Singapore 1

2. Near-field heat transfer by Coulomb force of electrons, Meir-Wingreen formulas NEGF theory of energy, momentum, and angular momentum transfer N + 1 objects, bath at infinity Meir-Wingreen/Landauer formula Applications Angular momentum emission from current-driven benzene molecule Nonreciprocity, graphene edge effect Outline 2

3. Coulomb problem, Meir- Wingreen formula dH dt = I L L + dEE ( ) = Tr G G L L 2 = L H c H c L L L 1 2 1 r = + + i c c c c H H H L R j j i r 4 3 , i j 0 i j

4. From electrons to photons dH dt = I L L + dEE ( ) = Tr G G L L 2 + d ( ) ( ) = = + + + + G D g v v D GD g G Tr D D L R n L L 4 , GG n 4

5. Assuming local equilibrium dH dt = I L L ( ) = + r a D D D L R + dEE ( ) ( ) = Tr G G = a r D D L L 2 ( ) = r a ( ) , N + d ( ) Tr D D ( ) L L 4 = r a i + d = ( )( , L R ) = r a Tr D D N N L R L R 2 0 5

6. Heat transfer between two graphene sheets First principles QE/BerkeleyGW calculation for the ratio of energy transfer to blackbody value between two graphene sheets at temperatures 300 K and 1000 K, = 0.05 eV, electron chemical potential at Dirac point. Zhu & Wang, PRB 104, L121409 (2021). Heat transfer ratio based on electron tight-binding model with nearest neighbor hopping t = 2.8 eV, between 300 K and 1000 K sheets at chemical potential = 0.1 eV. Slope 2.2. Jiang & Wang, PRB 96, 155437 (2017). 6

7. Kloppstech, et al, Nature Comm (2017) Geometry effects Distance dependent hopping, ?exp( 4 ? ? ? ?1= 350 ?,?2= 300 ?. ),? = 0.288 nm. From Wang et al, PRE (2018) 7

8. Heat transfer between zigzag nanotubes or triangles Heat transfer from 400K to 300K objects. (a), (b) zigzag carbon nanotubes. (c), (d) nano-triangles. d: gap distance, M: nanotube circumference, L: triangle length. : dielectric constant. From Tang, Yap, Ren, and Wang, Phys. Rev. Appl. 11, 031004 (2019). 8

9. Heat transfer driven by current Bottom layer fixed at ?1= 300 K, with a drift velocity ?1. Top layer at local equilibrium ?2. Heat flux as a function of temperature ?2. From Peng arXiv:1805.09493. 1.5x106 v1=1.0 106 m/s Current density (W/m2) 1.0x106 = r a ( ) D f G G neq 5.0x105 1 k T = N 0.0 eq / 1 e B -5.0x105 -1.0x106 1 = -1.5x106 N 300 305 310 315 320 325 330 335 340 T2 (K) neq ( )/( ) v q k T 1 e 1 1 x B 9

10. Nonequilibrium Green s function, general theory https://phyweb.physics.nus.edu.sg/~phywjs/NEGF/review-2022.pdf 10

11. System setup N+1 or infinity ? 1 Question: what are the energy emitted, force and torque applied to, for each of the object 1 to N+1. N 2 11

12. NEGF preliminaries contour time ? + ? t A D D D D 1 i ( , ; ', ') r r = = = ( , ) r ( ', ') r E B A , D T A A t t = ( , ) t = A , , , x y z A = r t D D D + = + = = = + t t K r a t t r a , , D D D D D D D D D D D G G Keldysh 1965, Haug & Jauho 2008 12

13. NEGF definitions A 1 i = = = ( , ; ' ') t r r ( , ) r ( ', ') t r E B A , D t A t A t = A , , , A x y z 1 i = ( , ; ' ') t r r ( ', ') r ( , ) t r D t A t A ( ) = Tr ( ) = r ( ') D t t D D + = = = K r a , D D D D D D D iA 13

14. Fluctuation-dissipation theorem in thermal equilibrium ( ) ( ) = = + r a r a , ( 1) D N D D D N D D ( ) = a r D D Callen and Welton 1951, Eckhardt 1984 1 = N 1 e ( ; , ') r r D D ( ) = D D 14

15. Dyson equations ( ) D = + r a D D D 2 v v D = + = + 1 2 D v U c 0 2 = + r r r r r D v v 2 + = ') ( + 2 r ( , '; r r r r ') ( ') U c D t t U t t 0 2 t 3 r r r ( , '', r r ( '', ', '' r r '' '' '') ') d dt t t D t t 1 0 0 0 1 0 0 0 1 r = = j A U 15

16. Keldysh equation ( ) D = + r a D D D 2 v v D = + = + 1 2 D v U c 0 2 t = + r r r r r D v v = + ( + + ( + r r r a a a D v D v D v D D v D D ) + r a r r a a = +( ) ( ) D I v I ) + + 1 1 r r a a = ( ) ( ) D N v v D = r a environment: = ( ) v N v v 1 r r ( ) v 16

17. = 0 gauge, fundamental equation for vector potential A 2 = + = 1 2 A A j v c 0 2 t A = = = A j E G , dyadic Green's function / v v 0 t Quantization: i ( ) = ( ), r ( ') r r r ' A E 0 17

18. Poynting theorem, steady state average 1 2 1 u t + = S = + 2 2 E j E B u 0 0 A u t = = = E 0 u t t = = S E j so or d dV S E j 18

19. Momentum and angular momentum conservation S 1 c + = + j B = T E f 2 t 1 = + T EE BB U u 0 0 angular momentum density / c = = l r S ( ) 2 r E B 0 l ( ) = r f T r t 19

20. From surface integral to volume integral A 1 = = = ( ) I d dV dV E B E j j t 0 f ( ) = = = F d dV dV A j T ( ) = = = r f + j A N r T d dV dV A j r = + j B f E 1 1 2 1 = + = + 2 2 T EE BB U E B , u u 0 0 0 0 20

21. A-j correlation function 1 i ; ' ') r r = ( , ) r ( ', ') r ( F T A j = ; '' '') r r ( '' ''; ' ') r 3 r r '' '' ( F D d d D N = Assuming additivity: , then F D = 1 In frequency domain, using Langreth rule, we have: = + = = K )K r K K a ( F F F D D D 21

22. Self energy Aslamazov-Larkin diagram (superconductivity, Coulomb drag) RPA 2 1 e = + = H H H 0 int 4 137 c 0 = = l jk j A j ( ) r H dV c M c A int k l jkl ( ) v v D = + = = 2 r = 1 D 0 i ( ) ' ( ', ) l l ( , ') Tr ( , ') i M G M G , ' l l e 22

23. Meir-Wingreen formula Kr ger, et al (2012); Strekha, et al, poster p = I F N d ( ) , = + K J ReTr 1,2, , , 1 F N N 2 0 = + K r K K a ( , ', ) r r F D D F J = = + r p S = p , , ( ) S i i 23

24. Bath at infinity Eckhardt, PRA 29, 1991 (1984) = 1 r r ( ) v Kr ger, et al, PRB 86, 115423 (2012) ( ) RR = i c r U 0 R = R R 24

25. Recover blackbody Planck result ( ) = i c r U RR 0 R = R R R 2 B 1 2 = + 2 E u 0 0 1 d = = r 2 r 0 ( ) Tr u i D D r 0 ' 0 2 0 0 = = r r 0 ' 2 = ( ) d N 2 3 c 0 ( ) = = ( ) r a r a , D D D N i R ( ) e c RR r U D 2 4 c R 0 25

26. From Meir-Wingreen to Landauer: local equilibrium approximation = + K r K K a F D D ( ) = + K 2 1 i N ( ) = r a i No Landauer form for force and torque! + 1 N = K r K a D D D = 1 + 1 N d ( ) ( ) = r a Tr I N N D D 2 = 1 0 26

27. Emission to infinity p = I F N d r a a J ReTr D D 2 0 ( ) RR = a U i c 0 J = = + r p S = p , , ( ) S i i 27

28. Far field approximations Ignore screening or multiple reflection: ?? ?? Multi-pole expansion: ???,? = ???,0 ? ????,0 + Integrating over solid angle analytically, eigenmode representation for ? 4 2 ( ) 2 = ( ) | | ' n (1 ) I n V f f ' ' ' n n n n n n 2 3 c , , ' n n 2 Fermi golden rule 1 e = 4 137 c 0 28

29. Torque and force on object = = , , , x y z N d 2 3 6 c , 0 0 3 ( ) ( ) ( ) = ' ' ' l l l l l l 4 F d r r r r r r , ' l , ' l , ' l l l l 2 5 60 c , ', l l 0 0 Force is zero if system is reciprocal, i.e., if < ?= < 29

30. Applications 30

31. Angular momentum emission from a benzene molecule 2 = ( ) Im P d , ' l l 2 3 6 c , ', l l 0 0 z dL dt ( ) = = ( ) ( ) z N d , ' , ' lx l y ly l x 2 3 6 c , ' l l 0 0 + dE ( ) = ' l l v Tr ( ) ( ) i M G E M G E , ' l l 2 = l jk j ( ) r H c M c A int k l , , , l j k Far field monopole approximation (all atoms are at the origin), ignore screening/multiple scatterings. 31

32. Angular momentum emission resonance effect E ? = ?, ? = 2 ? = 2?, ? = 0 Largest angular momentum emission when one of the chemical potential meets the ? = +? energy level. From Zhang, L , and Wang, PRB 101, 161406(R) (2020). 32

33. Force and torque from the nonequilibrium edge ? ? ?? ?? 33

34. Emission from graphene edge (a) Emission power as a function of temperature, (b) P, N, F as a function of ?? fixing ?? ?= 0.8. Y.-M. Zhang, etc. 34

35. I, N, F scan ??vs ??. System size width 700 unit cells, k-point 401. From Zhang, et al, PRB 105, 205421 (2022). 35

36. NEGF advantage Fully quantum-mechanical Local equilibrium or not, reciprocal or not, the theory is the same Strong coupling regime? E.g., cavity QED (different free Green s function ?), or need self- consistent ?(??,??)? First-principle based materials properties, i.e., . Challenges: Large system sizes? Moving objects? 36

37. Acknowledgements left to right: Dr. Zhang Yong-Mei, Dr. Zhu Tao, Dr. Gao Zhibin, Prof. Wang Jian-Sheng, Mr. Sun Kangtai, and Dr. Zhang Zuquan. 37