Finite Element Analysis of Transient Ballistic-Diffusive Heat Transfer in Two-Dimensional Structures

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014 COMSOL Conference, Boston, MA October 7-9, 014

Heat Transfer in Two-Dimensional Structures Sina Hamian

University of Utah, Salt Lake City, UT, USA Lund

1 014 COMSOL Conference, Boston, MA October 7-9, 014 Finite Element Analysis of Transient Ballistic-Diffusive Heat Transfer in Two-Dimensional Structures Sina Hamian 1, Toru Yamada, Mohammad Faghri 3, and Keunhan Park 1, 1 University of Utah, Salt Lake City, UT, USA Lund University, Lund, Sweden 3 University of Rhode Island, Kingston, RI, USA

ballistic-diffusive heat transfer Not available for public in any commercial package An SEM

2 Motivation Size of electronic devices gets smaller and smaller such as in CPUs and transistors Sub-continuum heat conduction is important Different numerical works have been done in modeling ballistic-diffusive heat transfer Not available for public in any commercial package An SEM image of an upright-type double-gate MOS transistor (Source: AIST) Singh et al. J. of Heat Transfer, 011

3 Heat transfer equations 3 Fourier equation Continuum medium T t = α Diffusive thermal transport (Parabolic equation) Cannot accurately predict sub-continuum heat transfer T Boltzmann transport equation (BTE) Based on energy carriers distribution (statistical base) Complicated scattering term Relaxation time approximation f + v g f = t f 0 τ f Ref: Joshi et al., J. of Applied Physics, 1993 f v g f 0 τ frequency dependent distribution function group velocity of energy carriers equilibrium distribution function effective relaxation time

4 Governing equation BTE for phonon energy density ωd ( 0 p ω ω ω) e (,,) rsˆ t D ( ) f d = p 1 e ˆ 0(,) rt = e (,,) td 4π rs Ω 4π vv gg e e 0 e + ( v ˆ gse ) = t τ Directional phonon energy density Equilibrium phonon energy density Phonon group velocity 4 Knudsen number Kn = Λ L For a constant phonon mean free path: Smaller domain length Larger Kn T H Phonon propagation T H T C Small Kn L >Λ T C Large Kn L <Λ Ref: Singh et al., J. of Heat Transfer, 011

5 Nondimensional -D BTE + DOM 5 1 e e e e e µ η + + = Kn t* x* y* Kn nm, nm, nm, 0 nm, n nm, µ = cosθ n ηnm, = sinθncosϕm n t* = t/ τ x* = x/ L y* = y/h Discrete Ordinate Method (DOM) e 1 ˆ 0(,) r t = (,,) 4π e rs td Ω 4π e ( t*, x*, y*) e ( t*, x*, y*) w w = 0 nm, n m 4π n m z y ϕ θ v g x

Validation (1-D thin film) 6 Nondimensional temperature, Θ 1.0 0.8 0.6 0.4 0. Fourier Kn = 0.

6 Validation (1-D thin film) 6 Nondimensional temperature, Θ Fourier Kn = 0.1 Kn = 1 Kn = 5 Kn = 10 FEA (current work) y* EPRT [1] [1] G. Chen, Nanoscale Energy Transport and Conversion,Oxford University Press, 005.

7 Results Fourier Kn = 0.03 Kn = 0.3 Kn = 1 Kn = Kn

4 directions 3 directions 56 directions Kn = 0.1 0.0 0.0 0.0 0. 0.4 0.6 0.8 1.

8 Ray effect 8 4 directions 3 directions 56 directions 4 directions 3 directions 56 directions Nondimensional temperature, Θ directions 3 directions 56 directions Kn = y* Nondimensional temperature, Θ Kn = 10 4 directions 3 directions 56 directions y* Diffusive

9 Transient solution 9 10 τ / Kn

10 Conclusion 10 COMSOL can calculate sub-continuum phonon heat transport. FEA-DOM combination is used in COMSOL for ballistic-diffusive heat transfer. Modeling nanoscale heat transfer is easily accessible. Acknowledgement This work was supported by the National Research Foundation Grant funded by the Korean Government (NRF D00014) and the National Science Foundation (CBET ). SH and KP also acknowledge the startup support at the University of Utah, including the computation at the Center for High-Performance Computing (CHPC). Recently accepted in: International Journal of Heat and Mass Transfer Doi: /j.ijheatmasstransfer

12 Introduction Application: Thermomechanical data writing/reading Thermal performance of extremely miniaturized electronic devices Thermal etching Thermal deposition Etching Deposition Pires et al., Science., 38 (010) Lee et al., Nano Lett., 10 (010) 1

13 Heat transfer equations Fourier equation Energy conservation + Fourier s heat flux approximation Used for heat conduction simulation for the last centuries Heat carriers travel with an infinite speed T t = α T Hyperbolic wave equation 1 u C t = u Hyperbolic heat equation (Cattaneo equation) Finite speed of heat carriers C = α τ Good for short time scales but not for short spatial scale T τ t T + = α t T 13

14 Fourier and hyperbolic heat equations Joshi and Majumdar, Journal of Applied Physics, 1993 Transient Fourier and Hyperbolic heat equation Steady state Fourier and Hyperbolic heat equation 14

15 Boltzmann Transport Equation (BTE) Boltzmann transport equation (BTE) BTE has a statistical base based on energy carriers distribution f t f v g f + v g f = t scattering frequency dependent distribution function group velocity of energy carriers (phonons) Relaxation time approximation f t scattering = f 0 τ f f 0 τ equilibrium Bose-Einstein distribution effective relaxation time 15

16 Eqs e ( t*, x*, y*) e ( t*, x*, y*) w w = 0 nm, n m 4π n m n m ww = n m π 4 π e ( t*, x*, y*) T t x y = = e t x y w w 0 ( *, *, *) nm, ( *, *, *) n m C C n m q ( t*, x*, y*) = v e ( t*, x*, y*) µ w w x g n, m n n m n m q ( t*, x*, y*) = v e ( t*, x*, y*) η w w y g nm, nm, n m n m e ( r,) s = e ( r ) = b 0 b CTb 4π 16

17 COMSOL model Nondimensional temperature, Θ BTE: FEA Kn = 0.1 BTE: FDM [1] BDE [1] 0. Kn = y* x y T C T C Λ Kn = d T H d 10d T C T C 5d T C 17

18 Ray effect 4 Directions 8 Directions 16 Directions 64 Directions 56 Directions Kn = 0.1 Kn = 1 Kn = 10 18

19 COMSOL model 19

International Journal of Heat and Mass Transfer

International Journal of Heat and Mass Transfer 61 (2013) 287 292 Contents lists available at SciVerse ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt