nanoHUB.orgonline simulations and more1Atomistic Green’s Function Method: Energy TransportTimothy 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 more21D Atomic Chain• Can be visualized as an atomic chain between two isothermal contacts (Note: contacts are still atomic chains in this example)nanoHUB.orgonline simulations and more3An Idealized Solution for Heat Flow• Consider a perfect, free-standing atomic chain overwhich a small temperature difference (ΔT) is applied• In the absence of scattering, the net heat flow through the chain would beT+ΔT/2 T-ΔT/2JQ,netNNN22,phonon phononphonon phonon# phonons per length # phonons per lengthenergy energygroup group in left contact in right contactvelocity veloc(,) (,)TTKp Kp Kp KpQ net Kp KpNT d NT dJLdKLdKΔΔ+ω ω −ω ω=ω+ω==N(<0)00itypK K><⎧⎫⎪⎪⎪⎪⎪⎪⎨⎬⎪⎪⎪⎪⎪⎪⎩⎭∑∑ ∑nanoHUB.orgonline simulations and more4Conversion of Sum to Integral• First, use the usual conversion of a sum to integral in K-space by recognizing that each allowed state in K-space occupies a 1D ‘volume’ of 2π/L• Now, take advantage of the symmetry of K-space to re-write the net heat flow as 11 1() () ()22KLFKFKdKFKdKLL==ππ∑∫∫,max,min/,220/01(,)(,)2(, ) (, )22ppaKpTTQ net Kp Kp KppaKp Kp pKp p pppdJNTNT dKdKNT d NTTTdK dTdK TπΔΔωπω⎧⎫ω⎡⎤=+ω−−ωω⎨⎬⎣⎦π⎩⎭⎧⎫⎧⎫∂ω ω ∂ωΔΔ⎪⎪≈ω= ωω⎨⎬⎨⎬π∂ π ∂⎩⎭⎪⎪⎩⎭∑∫∑∑∫∫===nanoHUB.orgonline simulations and more5Thermal Conductance and Conductivity• The thermal conductance (σ = JQ/ΔT) is independent of the wire length L…• …and the thermal conductivity is therefore length-dependent♦ Where s is the effective cross-sectional areaLsσκ=,max,min,(, )12ppQnet ppppJNTdTTωω⎧⎫∂ω⎪⎪σ= = ω ω⎨⎬Δπ ∂⎪⎪⎩⎭∑∫=nanoHUB.orgonline simulations and more6Effects of Interfaces and Scattering• When scattering at an interface or in the bulk is included, the heat flux integral must include a transmission function (Ξ) because some phonons will be ‘blocked’• Consequently, the essence of the solution involves finding Ξ• Note, in the AGF, the transmission function itself includes all polarizations p and therefore the summation over p is unnecessary()max,01(,)2QnetJNTdTTω∂ωσ= = Ξ ω ω ωΔπ ∂∫=nanoHUB.orgonline simulations and more7Lattice Energy• The ithdegree of freedom (i.e., atom) possesses both potential and kinetic energy♦ Where uiis the time-dependent displacement of atom i and • The time derivative of energy is• Using Newton’s 2ndLaw, this simplifies tonanoHUB.orgonline simulations and more8Energy Flow• Now, assume a form of the displacements as• The time derivative of energy becomes•Where Jpqtakes the form of an energy flux, where has no time dependenceitii iieuMω=φ φnanoHUB.orgonline simulations and more9Final Form of Energy Flux• After normalizing the displacement amplitude φand relating φ and H to other AGF matrices (see Zhang et al.), we find the following expression for the transmission functionnanoHUB.orgonline simulations and more10Transmission and Heat Flux0()()2JNdωωωωπ∞=ΔΞ∫=(units = W)Phonon occupationdifferenceTransmissionPhonon energyWe need to evaluate transmission in order to calculate heat flux()/22/1BBkTkTBeNN N TkTe+−Δ= − ≈ Δ−===ωωωnanoHUB.orgonline simulations and more11The AGF AlgorithmAssemble harmonic matrices (H)Calculate the Green’s function (g) of uncoupled contactsCalculate device G and phonon transmission (Ξ)Integrate (Ξ) over phonon frequencies and k||to obtain the thermal conductanceUses decimation algorithmU and xiEstablish atomic positions and potential parametersnanoHUB.orgonline simulations and more12nanoHUB Tool: Atomistic Green’s Function 1D Atomic Chain SimulationnanoHUB.orgonline simulations and more13Link to nanoHUB• https://www.nanohub.org/index.php?option=com_mw&invoke=greentherm&appcaption=The%20Atomistic%20Green's%20Function%201D%20Atomic%20Chain%20SimulationnanoHUB.orgonline simulations and more14Results for Simple Atomic ChainsHomogeneous chain density of statesContact atomic masses = 4.6x10-26kgHeavy device masses = 9.2x10-26kgLight device masses =