nanoHUB.orgonline simulations and more1Atomistic Green’s Function Method: IntroductionTimothy S. FisherPurdue UniversitySchool of Mechanical Engineering, andBirck Nanotechnology [email protected] on:W. Zhang, T.S. Fisher, N. Mingo, “The Atomistic Green’s Function Method: An Efficient Simulation Approach for Nanoscale Phonon Transport,” Numerical Heat Transfer: Part B (Fundamentals), Vol. 51, No. 3/4, pp. 333-349, 2007.nanoHUB.orgonline simulations and more2Overview of Phonon Simulation Tools• Boltzmann Transport Equation (BTE)♦ Requires boundary scattering models♦ Requires detailed understanding of phonon scattering and dispersion for rigorous inclusion of phonon physics• Molecular Dynamics (MD)♦ Computationally expensive♦ Not strictly applicable at low temperatures♦ Handling of boundaries requires great care for links to larger scales and simulation of functional transport processes• Atomistic Green’s Function (AGF)♦ Efficient handling of boundary and interface scattering♦ Straightforward links to larger scales♦ Inclusion of anharmonic effects is difficultnanoHUB.orgonline simulations and more3Some Background• Non-equilibrium Green’s function method initially developed to simulate electron ballistic transport (see Datta, 1995)• Very efficient in the ballistic regime but requires significant effort to implement scattering • Recently applied to phonon transport (see Mingo, 2003; Zhang et al., 2007a,b)nanoHUB.orgonline simulations and more4• Includes effects of bulk contacts by expressing their effects mathematically through Green’s functions• Suitable for ballistic transport♦ Nanoscale devices at room temperature, or♦ Low-temperature conditions, or♦ Scattering dominated by boundaries and interfaces• Required inputs♦ Equilibrium atomic positions ♦ Inter-atomic potentials♦ Contact temperaturesAtomistic Green’s FunctionnanoHUB.orgonline simulations and more5Recall Lattice Dynamics• Equation of motion for a 1D atomic chain• Plane wave assumption• Combine• Re-arrange and write in matrix form{}21122nnn ndumguuudt−+=− − −20⎡⎤ω−=⎣⎦IHuI is the identity matrix(){}()~expnut iKna t−ω{}2112nnnnguuuum−+−ω = − − −nanoHUB.orgonline simulations and more6Harmonic Matrix• Define the k matrix as• Then, define the harmonic matrix H as 2,1 ,12harmijijnnnn nnUkuukgkkg+−∂=−∂∂→=−==Here, g is the spring constantsummationindex no ,1ijMMijkHji=nanoHUB.orgonline simulations and more7Harmonic Matrixf is spring constant divided by atomic mass1. H is not the same dynamical matrix used to determine thedispersion curve (that matrix is the Fourier transform of H).2. H is symmetric.3. Sum of all elements in any row or sum of all elements in any column is zero, except in the first and last row/column.222fffffff−⎡⎤⎢⎥=−⎢⎥⎢⎥−⎣⎦HnanoHUB.orgonline simulations and more8Green’s Functions• In general, systems of equations can be written in operator form• Green’s functions are often used in such situations to determine general solutions of (usually) linear operators2[] 0⎡⎤=ω− =⎣⎦Lu I H unanoHUB.orgonline simulations and more9Green’s Functions, cont’d• The Green’s function g is the solution that results from the addition of a perturbation to the problem• In the present (matrix) problem, the uncoupledGreen’s function becomes♦ Where δ is called the broadening constant, and i is the unitary imaginary number=δL[g]()12i−⎡⎤=ω+δ −⎣⎦gIHnanoHUB.orgonline simulations and more10Uncoupled Contacts• In our context, Green’s functions first appear in the two contact regions• Imagine first that each contact is unconnected to the ‘device’ but extends to infinity in the other directionÅ to infinityContactEmpty Device RegionEmpty Device RegionnanoHUB.orgonline simulations and more11Uncoupled Contact Green’s Functions• Now, we form the uncoupled Green’s functions for each contact (g1and g2)• Where the Hi’s are the harmonic matrices for each contactnanoHUB.orgonline simulations and more12Toward Realistic Problems• So far, we have not made much progress in solving real problems • To solve most practical problems, we need to incorporate different materials and interfacesT1T2Thermal Reservoir 1Thermal Reservoir 2DevicenanoHUB.orgonline simulations and more13Assembling the System• We now need to couple the two contacts with the device• Our overall matrix equation becomesHarmonic matrixfor contact 1Harmonic matrixfor contact 2Harmonic matrixfor the deviceUncoupled displacementof contact 1Uncoupled displacementof contact 2DevicedisplacementEffects of deviceon contactdisplacementsnanoHUB.orgonline simulations and more14The Device Green’s Function• The device Green’s function G can now be expressed as♦ Note that this Green’s function does not explicitly contain a δperturbation♦ The so-called self-energy matrices (Σ1, Σ2) that involve uncoupledGreen’s functions (g’s) associated with contacts (i.e., boundaries) serve essentially as perturbations♦ τ matrices handle connections between different system elements (materials, interfaces) • Very efficient in the ballistic regime but requires significant effort to implement scatteringN21211 222TT−⎡⎤⎢⎥=ω − −τ τ−τ τ⎢⎥⎣⎦1d1ΣΣGIHg gsuperscript “T” = conjugatetransposenanoHUB.orgonline simulations and more15Toward Transport through a Devicebetween Two ContactsHot T1Cold T2Transmission function, ΞReservior 1Reservoir 2“Device”nanoHUB.orgonline simulations and more16Relation between Transmission and Green’s Functions• Some definitions of convenience• The transmission functionTjjjTjjjji⎡⎤=−⎣⎦=τ τAggΓ A12 21()TTTrace Trace⎡⎤⎡ ⎤Ξω = =⎣⎦⎣ ⎦Γ GΓ G Γ GΓ GFrom g1and τ1From g2and τ2DeviceGreen’s