Nonequilibrium Greens Function Method for Thermal Transport JianSheng

Nonequilibrium Green’s Function Method for Thermal Transport Jian-Sheng Wang

OUTLINE OF THE LECTURE • Models • Definition of Green’s functions • Contour-ordered Green’s function • Calculus on the contour • Feynman diagrammatic expansion • Relation to transport (heat current) • Applications TIENCS 2010 2

MODELS Junction TIENCS 2010 Left Lead, TL Right Lead, TR 3

FORCE CONSTANT MATRIX KR TIENCS 2010 4

DEFINITIONS OF GREEN’S FUNCTIONS • Greater/lesser Green’s function • Time-ordered/anti-time ordered Green’s function • Retarded/advanced Green’s function TIENCS 2010 5

RELATIONS AMONG GREEN’S FUNCTIONS TIENCS 2010 6

STEADY STATE, FOURIER TRANSFORM TIENCS 2010 7

EQUILIBRIUM SYSTEMS, LEHMANN REPRESENTATION • The average is with respect to the density operator exp(-β H)/Z • Heisenberg operator • Write the various Green’s functions in terms of energy eigenstate H |n> = En |n> • Use the formula TIENCS 2010 8

FLUCTUATION DISSIPATION THEOREM TIENCS 2010 9

COMPUTING AVERAGE (NONEQUILIBRIUM) • Average <. . . > over an arbitrary density matrix ρ • ρ = exp(-βH)/Z in equilibrium • • Schrödinger picture: A, (t) Heisenberg picture: AH(t) = U(t 0, t)AU(t, t 0) , ρ0 , where operator U satisfies TIENCS 2010 10

CALCULATING CORRELATIONS t 0 TIENCS 2010 B A t’ t 11

CONTOUR-ORDERED GREEN’S FUNCTION Contour order: the operators earlier on the contour are to the right. τ’ t 0 TIENCS 2010 τ 12

RELATION TO OTHER GREEN’S FUNCTION τ’ t 0 TIENCS 2010 τ 13

CALCULUS ON THE CONTOUR • Integration on (Keldysh) contour • Differentiation on contour TIENCS 2010 14

THETA FUNCTION AND DELTA FUNCTION • Theta function • Delta function on contour where θ(t) and δ(t) are the ordinary theta and Dirac delta functions TIENCS 2010 15

EXPRESS CONTOUR ORDER USING THETA FUNCTION Operator A(τ) is the same as A(t) as far as commutation relation or effect on wavefunction is concerned TIENCS 2010 16

EQUATION OF MOTION FOR CONTOUR ORDERED GREEN’S FUNCTION • TIENCS 2010 Consider a harmonic system with force constant K 17

EQUATIONS FOR GREEN’S FUNCTIONS TIENCS 2010 18

SOLUTION FOR GREEN’S FUNCTIONS c and d can be fixed by initial/boundary condition. TIENCS 2010 19

HANDLING INTERACTIONS • TIENCS 2010 Transform to interaction picture, H = H 0 + Hn 20

SCATTERING OPERATOR S • Transform to interaction picture • The scattering operator satisfies: TIENCS 2010 21

CONTOUR-ORDERED GREEN’S FUNCTION τ’ t 0 TIENCS 2010 τ 22

PERTURBATIVE EXPANSION OF CONTOUR ORDERED GREEN’S FUNCTION TIENCS 2010 23

General expansion rule Single line 3 -line vertex n-double line vertex TIENCS 2010

DIAGRAMMATIC REPRESENTATION OF THE EXPANSION = + 2 i = TIENCS 2010 + 25

SELF -ENERGY EXPANSION Σn TIENCS 2010 26

EXPLICIT EXPRESSION FOR SELF-ENERGY TIENCS 2010 27

JUNCTION SYSTEM • Three types of Green’s functions: • g for isolated systems when leads and centre are decoupled • G 0 for ballistic system • G for full nonlinear system Governing Hamiltonians HL+HC+HR +V +Hn HL+HC+HR +V HL+HC+HR g t=− Equilibrium at Tα G 0 G Green’s function t=0 Nonequilibrium steady state established TIENCS 2010

THREE REGIONS TIENCS 2010

HEISENBERG EQUATIONS OF MOTION IN THREE REGIONS TIENCS 2010

RELATION BETWEEN G AND G 0 Equation of motion for GLC TIENCS 2010 31

DYSON EQUATION FOR GCC TIENCS 2010 32

THE LANGRETH THEOREM TIENCS 2010 33

DYSON EQUATIONS AND SOLUTION TIENCS 2010 34

ENERGY CURRENT TIENCS 2010 35

CAROLI FORMULA TIENCS 2010 36

BALLISTIC TRANSPORT IN A 1 D CHAIN • Force constants • Equation of motion TIENCS 2010 37

SOLUTION OF G • TIENCS 2010 Surface Green’s function 38

LEAD SELF ENERGY AND TRANSMISSION T[ω] 1 ω TIENCS 2010 39

HEAT CURRENT AND CONDUCTANCE TIENCS 2010 40

GENERAL RECURSIVE ALGORITHM FOR SURFACE GREEN’S FUNCTION TIENCS 2010 41

TIENCS 2010 Transmission Dispersion relation CARBON NANOTUBE (6, 0), FORCE FIELD FROM GAUSSIAN 42

CARBON NANOTUBE, NONLINEAR EFFECT The transmissions in a one-unit-cell carbon nanotube junction of (8, 0) at 300 K. From J-S Wang, J Wang, N Zeng, Phys. Rev. B 74, 033408 (2006). TIENCS 2010 43

1 D CHAIN, NONLINEAR EFFECT k. L=1. 56 k. C=1. 38, t=1. 8 k. R=1. 44 Three-atom junction with cubic nonlinearity (FPU- ). From J -S Wang, Zeng, PRB 74, 033408 (2006) & J -S Wang, Lü, Eur. Phys. J. B, 62, 381 (2008). Squares and pluses are from MD. TIENCS 2010 44

Molecular dynamics with quantum bath TIENCS 2010 45

AVERAGE DISPLACEMENT, THERMAL EXPANSION • TIENCS 2010 One-point Green’s function 46

THERMAL EXPANSION (a) Displacement <u> as a function of position x. (b) as a function of temperature T. Brenner potential is used. From J. W. Jiang, J. -S. Wang, and B. Li, Phys. Rev. B 80, 205429 (2009). Left edge is fixed. TIENCS 2010 47

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 J. -W. Jiang, J. -S. Wang, and B. Li, Phys. Rev. B 80, 205429 (2009). TIENCS 2010 48

TRANSIENT PROBLEMS TIENCS 2010 49

DYSON EQUATION ON CONTOUR FROM 0 TO T Contour C TIENCS 2010 50

Transient thermal current The time-dependent current when the missing spring is suddenly connected. (a) current flow out of left lead, (b) out of right lead. Dots are what predicted from Landauer formula. T=300 K, k =0. 625 e. V/(Å2 u) with a small onsite k 0=0. 1 k. From E. C. Cuansing and J. -S. Wang, Phys. Rev. B 81, 052302 (2010). See also ar. Xiv: 1005. 5014. TIENCS 2010 51

SUMMARY • The contour ordered Green’s function is the essential ingredient for NEGF • NEGF is most easily applied to ballistic systems, for both steady states and transient time-dependent problems • Nonlinear problems are still hard to work with TIENCS 2010 52

References • H. Haug & A. -P. Jauho, “Quantum Kinetics in Transport and …” • J. Rammer, “Quantum Field Theory of Non-equilibrium States” • S. Datta, “Electronic Transport in Mesoscopic Systems” • M. Di Ventra, “Electrical Transport in Nanoscale Systems” • J. -S. Wang, J. Wang, & J. Lü, Europhys B 62, 381 (2008). TIENCS 2010 53

Problems for NGS students taken credits • Work out the explicit forms of various Green’s functions (retarded, advanced, lesser, greater, time ordered, etc) for a simple harmonic oscillator, in time domain as well in frequency domain • Consider a 1 D chain with a uniform force constant k. The left lead has mass m. L, center m. C, and right lead m. R. Work out the transmission coefficient T[ω] using the Caroli formula. • Work out the detail steps leading to the Caroli formula. TIENCS 2010 54

Website • This webpage contains the review article, as well some relevant codes/thesis: http: //staff. science. nus. edu. sg/~phywjs/NEGF/negf. html TIENCS 2010 55

Thank you

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