Presentation on theme: "Nonequilibrium Green’s Function Method in Thermal Transport"— Presentation transcript:
1Nonequilibrium Green’s Function Method in Thermal Transport Wang Jian-Sheng
2Lecture 1: NEGF basics – a brief history, definition of Green’s functions, properties, interpretation.Lecture 2: Calculation machinery – equation of motion method, Feynman diagramatics, Dyson equations, Landauer/Caroli formula.Lecture 3: Applications – transmission, transient problem, thermal expansion, phonon life-time, full counting statistics, reduced density matrices and connection to quantum master equation.
3ReferencesJ.-S. Wang, J. Wang, and J. T. Lü, “Quantum thermal transport in nanostructures,” Eur. Phys. J. B 62, 381 (2008).J.-S. Wang, B. K. Agarwalla, H. Li, and J. Thingna, “Nonequilibrium Green’s function method for quantum thermal transport,” Front. Phys. (2013).See alsoSee also
4History, definitions, properties of NEGF Lecture OneHistory, definitions, properties of NEGF
5A Brief History of NEGF Schwinger 1961 Kadanoff and Baym 1962 Keldysh 1965Caroli, Combescot, Nozieres, and Saint-James 1971Meir and Wingreen 1992
6Equilibrium Green’s functions using a harmonic oscillator as an example Single mode harmonic oscillator is a very important example to illustrate the concept of Green’s functions as any phononic system (vibrational degrees of freedom in a collection of atoms) at ballistic (linear) level can be thought of as a collection of independent oscillators in eigenmodes. Equilibrium means that system is distributed according to the Gibbs canonical distribution.
7Harmonic Oscillator m k du/dt = sqrt(hbar/(2Omega)) ( - a + a\dagger) (i Omega)
16Nonequilibrium Green’s Functions By “nonequilibrium”, we mean, either the Hamiltonian is explicitly time-dependent after t0, or the initial density matrix ρ is not a canonical distribution.We’ll show how to build nonequilibrium Green’s function from the equilibrium ones through product initial state or through the Dyson equation.
17Definitions of General Green’s functions Other types such as Matsubara Green’s function or canonical (Kubo) correlation are also used. i = sqrt(-1). The average < … > can be equilibrium or with respect to arbitrary density matrix. For classical systems, G< = G>.17
18Relations among Green’s functions The relation implies two (three?) linearly independent ones, and one independent one only if nonlinear dependence is considered.18
19Steady state, Fourier transform The inverse Fourier transform has a factor of 2 . We use square brackets [ ] to denote Fourier transformed function.19
20Equilibrium Green’s Function, Lehmann Representation Lehmann representation is to expand everything in the eigenstates.
21Kramers-Kronig Relation G^a is when z -> w – I eta.
22Eigen-mode Decomposition Only for finite system, in equilibrium? See a nice summaryin appendix of Phys Rev B (2009).
23Pictures in Quantum Mechanics Schrödinger picture: O, (t) =U(t,t0)(t0)Heisenberg picture: O(t) = U(t0,t)OU(t,t0) , ρ0, where the evolution operator U satisfies< . > = Tr(ρ . ). T here denotes time order, assuming t > t’. Anti-time order if t < t’.See, e.g., Fetter & Walecka, “Quantum Theory of Many-Particle Systems.”23
24Calculating correlation t0 is the synchronization time where the Schrödinger picture and Heisenberg picture are the same.BAt0t’t24
26Contour-ordered Green’s function Contour order: the operators earlier on the contour are to the right. See, e.g., H. Haug & A.-P. Jauho.τ’τt026
27Relation to other Green’s function σ = + for the upper branch, and – for the returning lower branch.τ’τt027
28An InterpretationThis is originally due to Schwinger.G is defined with respect to Hamiltonian H and density matrix ρ and assuming Wick’s theorem.
29Calculus on the contour Integration on (Keldysh) contourDifferentiation on contourdτ is σdt for integration, but just dt in differentiation. σ is either just the sign + or -, or +1, -1.29
30Theta function and delta function Delta function on contourwhere θ(t) and δ(t) are the ordinary theta and Dirac delta functions30
31Transformation/Keldysh Rotation Also properly known as Larkin-Ovchinikov transform, JETP 41, 960 (1975).
32Convolution, Langreth Rule The last two equations are known as Keldysh equations.
33Equation of motion method, Feynman diagrams, etc Lecture twoEquation of motion method, Feynman diagrams, etc
34Equation of Motion Method The advantage of equation of motion method is that we don’t need to know or pay attention to the distribution (density operator) ρ. The equations can be derived quickly.The disadvantage is that we have a hard time justified the initial/boundary condition in solving the equations.
35Heisenberg Equation on Contour The integral on the exponential is from a point tau_1 specified on the contour to tau_2 at a later point on the contour. The integral is along the contour.
36Express contour order using theta function Operator A(τ) is the same as A(t) as far as commutation relation or effect on wavefunction is concernedu uT is a square matrix, and [u, vT] is a matrix formed by (j,k) element [uj, vk].36
37Equation of motion for contour ordered Green’s function We use the equation of motion d2u/dt2 = -Ku.37
39Solution for Green’s functions Note that if ax = b, x = b/a, but not if a = 0, in that case, x = infinity, thus, the Dirac delta function.c and d can be fixed by initial/boundary condition.39
40Feynman diagrammatic method The Wick theoremCluster decomposition theorem, factor theoremDyson equationVertex functionVacuum diagrams and Green’s function
41Handling interaction Transform to interaction picture, H = H0 + Hn For the concept of pictures and relations among them, see A. L. Fetter and J. D. Walecka, “Quantum theory of many-body systems”, McGraw-Hill, The meaning of t0 and t’ is quite different. t0 is such that three pictures are equivalent, t’ is an arbitrary time. Note that ρ behaves like |><|.41
42Scattering operator SA(t) and B(t’) are in interaction picture (i.e. AI(t) etc).42
43Contour-ordered Green’s function Another possible representation of the result is using Feynman path integral formulation.τ’τt043
44Perturbative expansion of contour ordered Green’s function Every operator is in interaction picture, dropped superscript I for simplicity. The Wick theorem is valid if the expectation < … > is with respect to a quadratic weight. For a proof, see Rammer.44
45General expansion rule Single line3-line vertexn-double line vertexSee our review article or Rammer’s book.
46Diagrammatic representation of the expansion =+ 2iħ+ 2iħ+ 2iħ=+Due to the last term, G should be read as connected part G_c only in the Dyson equation.46
47Self -energy expansion The self-energy diagrams are obtained by including all doubly or more connected diagrams removing the two external legs from the Green’s function diagrams. For a more systematic exposition of the Feynman-diagrammatic expansion, see Fetter and Welecka. hbar was set to 1 in the diagrams.47
48Explicit expression for self-energy See also figure 3 in J.-S. Wang, X. Ni, and J.-W. Jiang, Phys. Rev. B 80, (2009).48
49Junction system, adiabatic switch-on gα for isolated systems when leads and centre are decoupledG0 for ballistic systemG for full nonlinear systemHL+HC+HR +V +HnHL+HC+HR +VGTwo adiabatic switch-on’s.HL+HC+HRG0gt = − Equilibrium at Tαt = 0Nonequilibrium steady state established4949
50Sudden Switch-on g t = − t = ∞ t =t0 50 HL+HC+HR +V +Hn Green’s function GTwo adiabatic switch-on’s.HL+HC+HRgt = − Equilibrium at Tαt = ∞t =t0Nonequilibrium steady state established5050
54Relation between g and G0 Equation of motion for GLCIn obtaining GLC, an arbitrary function satisfying (d/dtau^2 + KL) f = 0 is omitted. Absence of this term is consistent with adiabatic switch-on from Feynman diagrammatic expansion.54
55Dyson equation for GCCIn this derivation, we assume Hn = 0.55
58Diagrammatic WayFeynman diagrams for the nonequilibrium transport problem with quartic nonlinearity. (a) Building blocks of the diagrams. The solid line is for gC, wavy line for gL, and dash line for gR; (b) first few diagrams for ln Z; (c) Green’s function G0CC; (d) Full Green’s function GCC; and (e) re-sumed ln Z where the ballistic result is ln Z0 = (1/2) Tr ln (1 –gCΣ). The number in front of the diagrams represents extra combinatorial factor.
59The Langreth theoremNeed to use the relation between G++, G- -, G+-, and G+- with Gr and G<. See Haug & Jauho, page 66.59
60Dyson equations and solution In deriving the last line, we use the fact (gr)-1 g< = 0.60
61Energy currentIL is energy flowing out of the left lead into the center. This expression can be proving to be real, no need to take Re. Above equation works for general nonlinear case.61
62Landauer/Caroli formula The symmetrization appears necessary to get the Caroli formula. Caroli formula is valid only for ballistic case where Hn = 0.62
631D calculationIn the following we give a complete calculation for a simple 1D chain (the baths and the center are identical) with on-site coupling and nearest neighbor couplings. This example shows the general steps needed for more general junction systems, such as the need to calculate the “surface” Green’s functions.
64Ballistic transport in a 1D chain Force constantsEquation of motionThe matrix K is infinite in both directions.64
65Solution of gKR is semi-finite, V is nonzero only at the corner.65
66Lead self energy and transmission 1ωΣR is nonzero only for the lower right corner.66
67Heat current and conductance The last formula σ is called universal conductance.67
68General recursive algorithm for g The algorithm is due to Lopez Sancho et al (1985). Typically, 10 to 20 iterations should be already converged.68
70Carbon nanotube (6,0), force field from Gaussian Dispersion relationTransmissionThe force constants were symmetrized using the method of Mingo et al (2008). Calculation was performed by Lü Jingtao.Calculation was performed by J.-T. Lü.70
71Carbon nanotube, nonlinear effect The transmissions in a one-unit-cell carbon nanotube junction of (8,0) at 300K. From J-S Wang, J Wang, N Zeng, Phys. Rev. B 74, (2006).The nonlinear result may be not correct at the low frequency end.71
72u4 Nonlinear modelOne degree of freedom (a) and two degrees freedom (b) (1/4) Σ Tijkl ui uj uk ul nonlinear model. Symbols from quantum master equation, lines from self-consistent NEGF. For parameters used, see Fig.4 caption in J.-S. Wang, et al, Front. Phys. Calculated by Juzar Thingna.1510
73Molecular dynamics with quantum bath For more detail, see Phys. Rev. B 80, (2009). Any idea how to generalize this so that quantum mechanics is reproduced exactly?See J.-S. Wang, et al, Phys. Rev. Lett. 99, (2007);Phys. Rev. B 80, (2009).73
74Average displacement, thermal expansion One-point Green’s functionThe idea of calculating the thermal expansion coefficient this way due to Jiang Jinwu.74
75Thermal expansion Left edge is fixed. Displacement <u> as a function of position x.as a function of temperature T. Brenner potential is used. From J.-W. Jiang, J.-S. Wang, and B. Li, Phys. Rev. B 80, (2009).Left edge is fixed.75
76Graphene 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, (2009).76
77Phonon Life-TimeFor calculations based on this, see, Y. Xu, J.-S. Wang, W. Duan, B.-L. Gu, and B. Li, Phys. Rev. B 78, (2008).Xu Yong’s formula may have a factor 2 wrong.
78Transient problemsThe IL is valid for 1D chain with spring constant k. Eduardo will present a talk on this topic Friday morning.78
79Dyson equation on contour from 0 to t Contour CIn steady state we use Keldysh contour which is from - to +. But here for transient study, the contour is a finite segment.79
80Transient 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=300K, k =0.625 eV/(Å2u) with a small onsite k0=0.1k. From E. C. Cuansing and J.-S. Wang, Phys. Rev. B 81, (2010). See also arXiv:8080
81Finite lead problemTop ↥: no onsite (k0=0), (a) NL = NR = 100, (b) NL=NR=50, temperature T=10, 100, 300 K (black, red, blue). TL=1.1T, TR = 0.9T.Right ↦: with onsite k0=0.1k and similarly for other parameters. From E. C. Cuansing, H. Li, and J.-S. Wang, Phys. Rev. E 86, (2012).8181
82Full counting statistics (FCS) What is the amount of energy (heat) Q transferred in a given time t ?This is not a fixed number but given by a probability distribution P(Q)Generating functionAll moments of Q can be computed from the derivatives of Z.The objective of full counting statistics is to compute Z(ξ).8282
83A brief history on full counting statistics L. S. Levitov and G. B. Lesovik proposed the concept for electrons in 1993; rederived for noninteracting electron problems by I. Klich, K. Schönhammer, and othersK. Saito and A. Dhar obtained the first result for phonon transport in 2007J.-S. Wang, B. K. Agarwalla, and H. Li, PRB 2011; B. K. Agarwalla, B. Li, and J.-S. Wang, PRE 2012.83
84Definition of generating function based on two-time measurement Consistent history quantum mechanics (R. Griffiths).84
86For electron number measurement, the effect to phase a phase, see B. K For electron number measurement, the effect to phase a phase, see B. K. Agarwalla, B. Li, and J.-S. Wang, “Full-counting statistics of heat transport in harmonic junctions: transient, steady states, and fluctuation theorems,” Phys. Rev. E 85, (2012).86
92Numerical results, 1D chain 1D chain with a single site as the center.k= 1eV/(uÅ2), k0=0.1k,TL=310K, TC=300K, TR=290K. Red line right lead; black, left lead. B. K. Agarwalla, B. Li, and J.-S. Wang, PRE 85, , 2012.92
93Cumulants in Finite System Plot of the culumants of heat <<Qn>> for n=1,2,3, and 4 for one-atom center connected with two finite lead (one-dimensional chain) as a function of measurement time tM. The black and red line correspond to NL =NR =20 and 30, respectively. The initial temperatures of the left, center, right parts are 310K, 360K, and 290K, respective. We choose k=1 eV/(uÅ2) and k0=0.1 eV/(uÅ2) for all particles. From J.-S. Wang, et al, Front. Phys
94FCS for coupled leadsFrom H. Li, B. K. Agarwalla, and J.-S. Wang, Phys. Rev. E 86, (2012); H. Li, B. K. Agarwalla, and J.-S. Wang, Phys. Rev. B 86, (2012). For multiple leads, see B. K. Agarwalla, et al, arXiv:94
95FCS in nonlinear systems Can we do nonlinear?Yes we can. Formal expression generalizes Meir-WingreenInteraction picture defined on contourSome example calculations95
96Vacuum diagrams, Dyson equation, interaction picture transformation on contour 96
97Nonlinear system self-consistent results Single-site (1/4)λu4 model cumulants.k=1 eV/(uÅ2), k0=0.1k, Kc=1.1k, V LC-1,0=V CR0,1=-0.25k, TL=660 K, TR=410K.From H. Li, B. K. Agarwalla, B. Li, and J.-S. Wang, arxiv::97
98Quantum Master Equations When we tried to calculation the reduced density matrix from Dyson expansion, we found that the 2nd order (of the system-bath coupling) diagonal terms do not agree with master equation results, and there are also divergences in higher orders. Similarly, the currents also diverge at higher order. We were quite puzzled. The following is our effort to solve this puzzle and found an expression of the high order current which is finite.
99Quantum Master Equations Some detail is in the last section of J.-S. Wang, B. K. Agarwalla, H. Li, and J. Thingna, “Nonequilibrium Green's function method for quantum thermal transport,” Front. Phys. (2013). rho_0 is in fact rho_0^I, i.e., interaction picture one.
101Group & Acknowledgements Group members: front (left to right) Mr. Zhou Hangbo, Prof. Wang Jian-Sheng, Dr./Ms Lan Jinghua, Mr. Wong Songhan; Back: Mr. Li Huanan, Dr. Zhang Lifa, Dr. Bijay Kumar Agarwalla, Mr. Kwong Chang Jian, Dr. Thingna Juzar Yahya. Picture taking 17 April Others not in the picture: Qiu Hongfei. Thanks also go to Jian Wang, Jingtao Lü, Jin-Wu Jiang, Edardo Cuansing, Meng Lee Leek, Ni Xiaoxi, Baowen Li, Peter Hänggi, Pawel Keblinski.