Application of Nonequilibrium Greens Function Method
The application of Nonequilibrium Greens Function (NEGF) method to thermal transport and thermal expansion is explored in this study by Wang Jian-Sheng. The content covers an introduction to NEGF, heat transport, counting statistics, and the problem of thermal expansion. Detailed insights into the NEGF method, including evolution operators on contour, contour-ordered Greens function, relation to other Greens functions, Heisenberg equation on contour, thermal conduction at a junction, three regions analysis, and Dyson equations and solutions are provided.
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Nonequilibrium Greens Function Method: application to thermal transport and thermal expansion Wang Jian-Sheng 1
Outline An introduction to nonequilibrium Green s function (NEGF) method Heat transport, counting statistics Problem of thermal expansion 2
NEGF Our review: Wang, Wang, and L , Eur. Phys. J. B 62, 381 (2008); Wang, Agarwalla, Li, and Thingna, Front. Phys. (2013), DOI:10.1007/s11467-013-0340-x 3
Evolution Operator on Contour i 2 ( , ) = exp , U T H d 2 1 2 1 c 1 ( , ) ( , ) U = = ( , ), U U 3 2 2 ( , ) , 1 3 1 3 2 1 1 ( , ) U U 1 2 2 1 1 2 + + ( ) = ( , ) ( , ) U t O OU t 0 0 4
Contour-ordered Greens function i ( , ') = ( ) ( ') u T G T u C i ( ) H d = ( ) t T u u e T Tr C 0 ' C Contour order: the operators earlier on the contour are to the right. See, e.g., H. Haug & A.-P. Jauho. t0 5
Relation to other Greens functions ( , ), t = = or , t ++ + G G G G ( , ') = ' ( , ') t t or G G G + ++ + = = = = t , G G G G + t , G G G G t0 6
Heisenberg Equation on Contour i 2 ( , ) = exp , U T H d 2 1 2 1 c 1 + + ( ) = ( , ) ( , ) U t O OU t 0 0 ( ) dO = [ ( ), O ] i H d 7
Thermal conduction at a junction Left Lead, TL Right Lead, TR semi-infinite Junction Part 8
Three regions 1 u u L L 2 = = = , , u u u u u C L L C u R i = = T ( , ) ' ( ) ( ) ' , , , , G T u u L C R C 9
Dyson equations and solution = + , G g g G 0 0 C C = + G G G G 0 0 n ( G ) 1 + = + 2 r 0 C r ( ) , 0 G i I K = r 0 a 0 G G 0 ( G ) I 1 = 1 r r 0 r n ( G ( ) , G = + + + r a r r n a n a G ) ( ) G G G I G 0 n = + r a ( G ) G n 10
Energy current dH dt = = T L LC I u V u L L C t a L ( ', ) t t t ( ', ) t t t = + r CC ( , ') t t ( , ') t t ' i G G dt L CC t 0 + 1 ( ) = [ ] LC Tr V G d CL 2 + 1 ( ) = [ ] [ ] + [ ] [ ] r CC a L Tr G G d L CC 2 11
Landauer/Caroli formula + 1 dH dt ( )( ) = = r CC a CC Tr I G G f f d L L L R L R 2 0 ( ) = r a i I I , I L R L 2 = = = + r a , G G G i f f L L R R + a r r a ( ) G G iG G L R 12
Ballistic transport in a 1D chain Force constants k 0 k + k 2 0 k k k 0 = + k K 2 k k k 0 + 0 2 k k 0 0 0 k Equation of motion u ku = + + = , 1,0,1,2, (2 ) , k k u ku j 1 0 1 j j j j 13
Lead self energy and transmission 0 0 0 0 0 0 k 0 0 0 0 0 T[ ] = L 1 0 = 2 1 r C ( ) , G K L R j k | | = r jk G ( ) 1 k + 2 1, 0, 4 k k k ( ) [ ] = = r a 0 0 Tr T G G L R otherwise 14
Heat current and conductance + d ( ) = [ ] I T f f L L R 2 0 1 I f d T max = = = lim L T , f L 2 1 T T e T R L R min 2 2 B k T h = , 0, 0 T k 0 3 15
Arbitrary time, transient result 1 ln 0 G Z L = A Tr ln( 1 ) 2 = + + A L ( , ) ' ( ( ), ' ( ' )) ( , ) ' x x L L n ln i Z n = = n Q x ( ) 2 = 0 A L ln i 1 Z = = = Tr Q Q G 0 ( ) 2 ( ) i = 0 2 ln i Z 2 = = 2 2 Q Q Q 2 ( ) in ( long time) Q t I 16 M
Numerical results, 1D chain 1D chain with a single site as the center. k= 1eV/(u 2), k0=0.1k, TL=310K, TC=300K, TR=290K. Red line right lead; black, left lead. From Agarwalla, Li, and Wang, PRE 85, 051142, 2012. 17
Thermal Expansion Gr neisen theory ln ln 1 BV = = , c 3 V NEGF compute the displacement of each atom <uj>. It is obtained by the standard Feynman-diagrammatic expansion with respect to nonlinear interactions. 18
Average displacement, thermal expansion One-point Green s function i G T u = ( ) ( ) j C j = ( ', '', ''') ( ', '') ( ''', ) 0 lm 0 nj ' '' ''' d d d T G G lmn lmn = = = r nj ( 0) [ 0] G T G t G j lmn lm lmn dG dT 1 x i M = j j j 20
Connection, see Jiang, Wang, Wang, Park, arXiv:1408.1450. c 1 L ( ) = NEGF N * m 1 T S S K lmn l nN 2 lmn ln ln L = = * m T S S lmn l n 2 3 V lmn = / R L n n FL B = = = 1 NEGF N , u K F n nN n 21
Thermal expansion (a) Displacement <u> as a function of position x. (b) as a function of temperature T. Brenner potential is used. From Jiang, Wang, and Li, Phys. Rev. B 80, 205429 (2009). Left edge is fixed. 22
Graphene Thermal expansion coefficient The coefficient of thermal expansion v.s. temperature for graphene sheet with periodic boundary condition in y direction and fixed boundary condition at the x=0 edge. is onsite strength. From Jiang, Wang, and Li, Phys. Rev. B 80, 205429 (2009).
Phonon Life-Time t + 1 ( ) i t q q 2 = r q r q , ( ) G G t e q ( ) 2 + i [ ] 2 q r n q , r n q Re [ ] 2 , , q q q For calculations based on this, see, Xu, Wang, Duan, Gu, and Li, Phys. Rev. B 78, 224303 (2008). q r n q Im [ ] , q q 1 3 = 2 q q c v q q 24
Summary NEGF: powerful tool for steady state and transient, best for ballistic system, difficult for interaction systems Thermal expansion problem: NEGF does not need to assume uniform expansion, suited for any nanostructure or bulk 25
Acknowledgements NEGF, transport: Wang Jian, L Jingtao, Eduardo C Cuansing, Zhang Lifa, Bijay Kumar Agarwalla, Li Huanan Thermal expansion: Jiang Jinwu (now at Shanghai Univ) 26
