Simulation of Unsteady Small Heat Source Effects in Sub-Micron Heat Conduction

2003 ◽  
Vol 125 (5) ◽  
pp. 896-903 ◽  
Author(s):  
Sreekant V. J. Narumanchi ◽  
Jayathi Y. Murthy ◽  
Cristina H. Amon
Keyword(s):  
Relaxation Time ◽  
Heat Source ◽  
Time Scales ◽  
Electric Fields ◽  
Hot Spot ◽  
Mean Free Path ◽  
Boltzmann Transport ◽  
Electron Phonon Interaction ◽  
Relaxation Time Approximation ◽  
Phonon Relaxation Time

In compact transistors, large electric fields near the drain side create hot spots whose dimensions are smaller than the phonon mean free path in the medium. In this paper, we present a study of unsteady hot spot behavior. The unsteady gray phonon Boltzmann transport equation (BTE) is solved in the relaxation time approximation using a finite volume method. Electron-phonon interaction is represented as a heat source term in the phonon BTE. The evolution of the temperature profile is governed by the interaction of four competing time scales: the phonon residence time in the hot spot and in the domain, the duration of the energy source, and the phonon relaxation time. The influence of these time scales on the temperature is investigated. Both boundary scattering and heat source localization effects are observed to have considerable impact on the thermal predictions. Comparison of BTE solutions with conventional Fourier diffusion analysis reveals significant discrepancies.

Small Heat Source Effects in Sub-Micron Heat Conduction

2001 ◽  
Author(s):  
Sreekant V. J. Narumanchi ◽  
Jayathi Y. Murthy ◽  
Cristina H. Amon
Keyword(s):  
Heat Conduction ◽  
Heat Source ◽  
Free Path ◽  
Electric Fields ◽  
Integrated Circuit ◽  
Mean Free Path ◽  
Boltzmann Transport ◽  
Source Effects ◽  
Relaxation Time Approximation ◽  
Volume Method

Abstract Decreasing dimensions of integrated circuit devices is leading to increased importance of microscale heat transfer effects and the failure of Fourier’s law in predicting sub-micron heat conduction. In compact transistors, large electric fields near the drain side create hot spots whose dimensions are smaller than the phonon mean free path in the medium. Under these conditions, the phonon Boltzmann equation (BTE) needs to be solved in order to resolve the non-local thermal conduction phenomena. In this paper, the problem of an unsteady heat source of size comparable to or smaller than the phonon mean free path is considered. The unsteady 2-D phonon Boltzmann transport equation in the relaxation time approximation is solved using a finite volume method. The interaction of the heat-up time constant with the phonon residence time in the hotspot and also its interaction with the time scales associated with scattering processes are studied. The results are useful in assessing the peak temperatures during unsteady operation in microelectronic devices.

scholarly journals Applicability of the High Field Model: An Analytical Study Via Asymptotic Parameters Defining Domain Decomposition

VLSI Design ◽  
1998 ◽  
Vol 8 (1-4) ◽  
pp. 135-141 ◽  
Author(s):  
Carlo Cercignani ◽  
Irene M. Gamba ◽  
Joseph W. Jerome ◽  
Chi-Wang Shu
Keyword(s):  
Electric Fields ◽  
High Field ◽  
Field Model ◽  
Debye Length ◽  
Mean Free Path ◽  
Field Approximation ◽  
Macroscopic Model ◽  
Boltzmann Transport ◽  
Kinetic Transition ◽  
Macroscopic States

In this paper, we present a mesoscopic-macroscopic model of self-consistent charge transport. It is based upon an asymptotic expansion of solutions of the Boltzmann Transport Equation (BTE). We identify three dimensionless parameters from the BTE. These parameters are, respectively, the quotient of reference scales for drift and thermal velocities, the scaled mean free path, and the scaled Debye length. Such parameters induce domain dependent macroscopic approximations. Particular focus is placed upon the so-called high field model, defined by the regime where drift velocity dominates thermal velocity. This model incorporates kinetic transition layers, linking mesoscopic to macroscopic states. Reference scalings are defined by the background doping levels and distinct, experimentally measured mobility expressions, as well as locally determined ranges for the electric fields. The mobilities reflect a coarse substitute for reference scales of scattering mechanisms. See [9] for elaboration.The high field approximation is a formally derived modification of the augmented drift-diffusion model originally introduced by Thornber some fifteen years ago [25]. We are able to compare our approach with the earlier kinetic approach of Baranger and Wilkins [5] and the macroscopic approach of Kan, Ravaioli and Kerkhoven [20].

RELAXATION TIME APPROXIMATION FOR ELECTRON-PHONON INTERACTION IN SEMICONDUCTORS

1995 ◽  
Vol 05 (04) ◽  
pp. 519-527 ◽  
Author(s):  
PETER A. MARKOWICH ◽  
CHRISTIAN SCHMEISER
Keyword(s):  
Relaxation Time ◽  
Hydrodynamic Limit ◽  
Asymptotic Methods ◽  
Drift Diffusion Model ◽  
Phonon Interaction ◽  
Drift Diffusion ◽  
Time Model ◽  
Electron Phonon Interaction ◽  
Relaxation Time Approximation ◽  
Equilibrium Distributions

A Boltzmann equation for semiconductors is considered. Physical assumptions include the parabolic band approximation and a new relaxation time model for electron-phonon interaction. Thermal equilibrium distributions for this scattering mechanism are products of Maxwellian distributions with periodic functions of the energy, where the period is the energy of a phonon. The hydrodynamic limit is considered and a drift-diffusion model is derived by formal asymptotic methods.

Poiseuille gas flow in the transition regime II. Application of the general theory

1967 ◽  
Vol 301 (1467) ◽  
pp. 401-409 ◽  
Keyword(s):  
Relaxation Time ◽  
Gas Flow ◽  
Capillary Tube ◽  
Collision Operator ◽  
Mean Free Path ◽  
Transition Regime ◽  
Tube Radius ◽  
Time Approximation ◽  
Relaxation Time Approximation ◽  
Flow Through

The general formulae given in the previous paper are investigated in detail using a simple relaxation-time approximation for the collision operator, and numerical results are obtained for the total gas flow through a capillary tube at various values of the ratio of tube radius to collision mean free path. For all values of this ratio, the results obtained agree with experiment to within about 2%.

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