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Boltzmann transport equation-based thermal modeling approaches for hotspots in microelectronics

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Abstract

Fourier diffusion has been found to be inadequate for the prediction of heat conduction in modern microelectronics, where extreme miniaturization has led to feature sizes in the sub-micron range. Over the past decade, the phonon Boltzmann transport equation (BTE) in the relaxation time approximation has been employed to make thermal predictions in dielectrics and semiconductors at micro-scales and nano-scales. This paper presents a review of the BTE-based solution methods widely employed in the literature and recently developed by the authors. First, the solution approaches based on the gray formulation of the BTE are presented. The semi-gray approach, moments of the Boltzmann equation, the lattice Boltzmann approach, and the ballistic-diffusive approximation are also discussed. Models which incorporate greater details of phonon dispersion are also presented. Hotspot self-heating in sub-micron SOI transistors and transient electrostatic discharge in NMOS transistors are also examined. Results, which illustrate the differences between some of these models reveal the importance of developing models that incorporate substantial details of phonon physics. The impact of boundary conditions on thermal predictions is also investigated.

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Abbreviations

C :

Total volumetric heat capacity (J/m3 K)

C w :

Volumetric specific heat per unit frequency (Js/m3 K)

D(w):

Phonon density of states (m−3)

e total :

Total energy (J/m3)

f w :

Phonon distribution function

\(\hbar \) :

Reduced Planck’s constant (= h/(2π), 1.054 × 10−34 Js)

k B :

Boltzmann’s constant (1.38 × 10−23 J/K)

K :

Thermal conductivity (W/m K)

N LA , N TA :

Number of frequency bands in LA and TA branches

N bands :

Total number of frequency bands (N LA + N TA + 1)

N θ, N ϕ :

Number of θ and ϕ divisions in an octant

q vol :

Volumetric heat generation (W/m3)

\(\vec r\) :

Position vector (m)

\(\hat s\) :

Unit direction vector

t :

Time (s)

T :

Temperature (K)

v :

Phonon velocity (m/s)

w :

Phonon frequency (rad/s)

Δw :

Frequency width (rad/s)

ϕ:

Azimuthal angle

γ:

Band-averaged inverse relaxation time for interaction (s−1)

τ:

Relaxation time of a phonon (s)

θ:

Polar angle (degrees)

i:

ith frequency band

ij:

Property specific to bands i and j

L:

Lattice

O:

Optical mode

P:

Propagating mode

R:

Reservoir mode

w:

Phonon frequency

0:

Equilibrium condition

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Acknowledgements

The support of NSF grants CTS-0103082 and CTS-0219008, and the PITA program of the Pennsylvania Department of Community and Economic Development, is gratefully acknowledged.

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Authors and Affiliations

  1. National Renewable Energy Laboratory, 1617 Cole Blvd., Golden, CO, USA

    Sreekant V. J. Narumanchi

  2. School of Mechanical Engineering, Purdue University, 585 Purdue Mall, W. Lafayette, IN, USA

    Jayathi Y. Murthy

  3. Institute for Complex Engineered Systems and Department of Mechanical Engineering, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA, USA

    Cristina H. Amon

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  1. Sreekant V. J. Narumanchi
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  3. Cristina H. Amon
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Correspondence to Cristina H. Amon.

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Narumanchi, S.V.J., Murthy, J.Y. & Amon, C.H. Boltzmann transport equation-based thermal modeling approaches for hotspots in microelectronics. Heat Mass Transfer 42, 478–491 (2006). https://doi.org/10.1007/s00231-005-0645-6

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