Heat and Mass Transfer

, Volume 42, Issue 6, pp 478–491 | Cite as

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

  • Sreekant V. J. Narumanchi
  • Jayathi Y. Murthy
  • Cristina H. AmonEmail author
Special Issue


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.


Micro/nanoscale Sub-continuum thermal transport BTE Hotspot NMOS transistors SOI 



Total volumetric heat capacity (J/m3 K)


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


Phonon density of states (m−3)


Total energy (J/m3)


Phonon distribution function

\(\hbar \)

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


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


Thermal conductivity (W/m K)


Number of frequency bands in LA and TA branches


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

Nθ, Nϕ

Number of θ and ϕ divisions in an octant


Volumetric heat generation (W/m3)

\(\vec r\)

Position vector (m)

\(\hat s\)

Unit direction vector


Time (s)


Temperature (K)


Phonon velocity (m/s)


Phonon frequency (rad/s)

Greek symbols


Frequency width (rad/s)


Azimuthal angle


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


Relaxation time of a phonon (s)


Polar angle (degrees)



ith frequency band


Property specific to bands i and j




Optical mode


Propagating mode


Reservoir mode


Phonon frequency



Equilibrium condition



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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Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  • Sreekant V. J. Narumanchi
    • 1
  • Jayathi Y. Murthy
    • 2
  • Cristina H. Amon
    • 3
    Email author
  1. 1.National Renewable Energy LaboratoryGoldenUSA
  2. 2.School of Mechanical EngineeringPurdue UniversityW. LafayetteUSA
  3. 3.Institute for Complex Engineered Systems and Department of Mechanical EngineeringCarnegie Mellon UniversityPittsburghUSA

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