Atomistic Green s Function Method: Energy Transport

Transcription

1 Atomistic Green s Function Method: Energy Transport Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center tsfisher@purdue.edu Based 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 ,

2 1D Atomic Chain Can be visualized as an atomic chain between two isothermal contacts (Note: contacts are still atomic chains in this example) 2

3 An Idealized Solution for Heat Flow Consider a perfect, free-standing atomic chain over which a small temperature difference (ΔT) is applied In the absence of scattering, the net heat flow through the chain would be J T+ΔT/2 J Q,net T-ΔT/2 ΔT ΔT NT ( + 2, ωkp) dωkp NT ( 2, ωkp ) dωkp = ω + ω L dk L p K> 0 K< 0 dk phonon phonon # phonons per length energy phonon # phonons per length energy phonon in left contact group in right contact group velocity velocity (<0) Q, net Kp Kp 3

4 Conversion 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 1 1 L 1 F( K) F( K) dk F( K) dk L = L = 2π 2π K Now, take advantage of the symmetry of K-space to re-write the net heat flow as π / a 1 dωkp ΔT ΔT JQ, net = N( T + 2, ωkp ) N( T 2, ωkp ) ωkp dk 2π p dk 0 π / a ωp,max ΔT NT (, ωkp) dωkp ΔT NT (, ωp) ω Kp dk = ωpdωp 2π p T dk 2π 0 p T ωp,min 4

5 Thermal Conductance and Conductivity The thermal conductance (σ = J Q /ΔT) is independent of the wire length L ωp,max J, 1 Qnet N( T, ωp) σ= = ωpdω ΔT 2π p T ωp,min and the thermal conductivity is therefore lengthdependent Lσ κ= s Where s is the effective cross-sectional area p 5

6 Effects 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 J ωmax Qnet, 1 NT (, ω) σ= = Ξ( ω) ωdω ΔT 2π T 0 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 6

7 Lattice Energy The i th degree of freedom (i.e., atom) possesses both potential and kinetic energy Where u i is the time-dependent displacement of atom i and The time derivative of energy is Using Newton s 2 nd Law, this simplifies to 7

8 Energy Flow Now, assume a form of the displacements as iωt e ui =φi, where φi has no time dependence M i The time derivative of energy becomes Where J pq takes the form of an energy flux 8

9 Final Form of Energy Flux After normalizing the displacement amplitude φ and relating φ and H to other AGF matrices (see Zhang et al.), we find the following expression for the transmission function 9

10 Transmission and Heat Flux ω J = ΔN( ω) Ξ( ω) dω (units = W) 2π 0 Phonon energy Phonon occupation difference B ω / kt B + ω e Δ N = N N ΔT 2 2 k T ( ω / kt e 1) Transmission We need to evaluate transmission in order to calculate heat flux B 10

11 The AGF Algorithm Establish atomic positions and potential parameters U and x i Assemble harmonic matrices (H) Calculate the Green s function (g) of uncoupled contacts Uses decimation algorithm Calculate device G and phonon transmission (Ξ) Integrate (Ξ) over phonon frequencies and k to obtain the thermal conductance 11

12 nanohub Tool: Atomistic Green s Function 1D Atomic Chain Simulation 12

13 Link to nanohub mw&invoke=greentherm&appcaption=the%20ato mistic%20green's%20function%201d%20atomic %20Chain%20Simulation 13

14 Results for Simple Atomic Chains light device heavy device Homogeneous chain density of states Contact atomic masses = 4.6x10-26 kg Heavy device masses = 9.2x10-26 kg Light device masses = 2.3x10-26 kg 14