Advanced Methods in Transport Phenomena and Quantum Field Theory

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Explore cutting-edge research in transport processes mediated by photons, tight-binding models for electron behavior, electron-field interactions, NEGF definitions, fluctuation-dissipation theorem, Dyson equations, and Keldysh equations in the context of quantum field theory and super-Planckian near-field radiation.

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NEGF method for transport mediated by photons and the super-Planckian problem Jian-Sheng Wang Department of Physics, National University of Singapore 1

Tight-binding model for electrons NEGF theory of energy, momentum, and angular momentum transfer, Meir-Wingreen formulas Applications van der Waals force vs Biot-Savart law Super-Planckian near-field radiation in 2D metals Outline 2

Electron tight-binding model with bath(s) 1 ??? = ? ? ?? system ?<? = ??? <? ??? <? = ?(?) ?? ?? ?? = ?? bath 3

Electron-field interaction, Peierls substitution system ? ?????exp ??/ ?? ? ?? ? ?:vector field bath 4

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 5

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 6

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 7

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 8

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 9

Self energy diamagnetic 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 10

Meir-Wingreen formula Kr ger, et al (2012); Strekha, et al, PRA 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 11

van der Waals force between current-carrying graphene nanostrips nearest neighbor hopping t = 2.7 eV Chemical potential bias, ??= ??= 1eV, electric current 1.2 10 4A. Temperature at 300 K. d ?? ?? ?? ?? 12

Force between two 13 8 armchair graphene with chemical potential biases of 1 eV. Black: fluctuational force; red: additional force due to current (Biot- Savart law). J.-S. Wang and M. Antezza, in preparation. 13

Heat transfer in on 2D metals, super-Planckian problem 1000 K 300 K x y z d 14

Drude model approximation to photon self-energy ? ? ?2?2? ? ??,? ? ? = ??,? ??,? 2?? + ? ?: electron surface density ?: electron effective mass 2? = 1/?: inverse relation time ?: lattice constant (4 a.u.) a 15

Fig. 1. Heat flux between coplanar 2D metals as a function of gap sizes. The materials properties are calculated by the Drude model. The red line is the total heat flux; the blue- and green- dotted lines are results from evanescent waves and propagating waves, respectively. The black-dashed line is the Black-body result given by the Stefan-Boltzmann law with a geometrical view factor F = a/2d, where a = 4 a.u. is the lattice constant and d is the gap size. Some parameters: Lx = 160, Lz = 640, T1=1000 K, T2=300 K w_max = 10*Kb*(T1+T2) q_max = 5/d for total and evanescent q_max = w/c for propagating W_n = 512 a_lat = 4 a.u. Rcut = 1.6

Fig. 2. Spectrum of transmission function as a function of frequency and wavevector with the gap size (a) d = 10 a.u. and (b) d = 10000 a.u. The materials properties are calculated by the Drude model. The red lines in (b) are given by ?/c, which represents the boundary of propagating and evanescent waves. T. Zhu, Y.-M. Zhang, and J.-S. Wang, in preparation.