Sub-Continuum Simulations of Heat Conduction in Silicon-on-Insulator Transistors

2000 ◽  
Vol 123 (1) ◽  
pp. 130-137 ◽  
Author(s):  
Per G. Sverdrup ◽  
Y. Sungtaek Ju ◽  
Kenneth E. Goodson
Keyword(s):  
Heat Conduction ◽  
Diffusion Equation ◽  
Free Path ◽  
Temperature Rise ◽  
Mean Free Path ◽  
Channel Length ◽  
Silicon On Insulator ◽  
Heat Diffusion ◽  
Boltzmann Transport ◽  
Heat Diffusion Equation

The temperature rise in sub-micrometer silicon devices is predicted at present by solving the heat diffusion equation based on the Fourier law. The accuracy of this approach needs to be carefully examined for semiconductor devices in which the channel length is comparable with or smaller than the phonon mean free path. The phonon mean free path in silicon at room temperature is near 300 nm and exceeds the channel length of contemporary transistors. This work numerically integrates the two-dimensional phonon Boltzmann transport equation (BTE) within the silicon region of a silicon-on-insulator (SOI) transistor. The BTE is solved together with the classical heat diffusion equation in the silicon dioxide layer beneath the transistor. The predicted peak temperature rise is nearly 160 percent larger than a prediction using the heat diffusion equation for the entire domain. The disparity results both from phonon-boundary scattering and from the small dimensions of the region of strongest electron-phonon energy transfer. This work clearly shows the importance of sub-continuum heat conduction in modern transistors and will facilitate the development of simpler calculation strategies, which are appropriate for commercial device simulators.

Sub-Continuum Simulations of Heat Conduction in Silicon-on-Insulator Transistors

1999 ◽  
Author(s):  
Per G. Sverdrup ◽  
Y. Sungtaek Ju ◽  
Kenneth E. Goodson
Keyword(s):  
Diffusion Equation ◽  
Free Path ◽  
Temperature Rise ◽  
Mean Free Path ◽  
Channel Length ◽  
Silicon On Insulator ◽  
Heat Diffusion ◽  
Boltzmann Transport ◽  
Heat Diffusion Equation ◽  
Silicon Devices

Abstract The temperature rise in compact silicon devices is predicted at present by solving the heat diffusion equation based on Fourier’s law. The validity of this approach needs to be carefully examined for semiconductor devices in which the region of strongest electronphonon coupling is narrower than the phonon mean free path, Λ, and for devices in which Λ is comparable to or exceeds the dimensions of the device. Previous research estimated the effective phonon mean free path in silicon near room temperature to be near 300 nm, which is already comparable with the minimum feature size of current generation transistors. This work numerically integrates the phonon Boltzmann transport equation (BTE) within a two-dimensional Silicon-on-Insulator (SOI) transistor. The BTE is coupled with the classical heat diffusion equation, which is solved in the silicon dioxide layer beneath a transistor with a channel length of 400 nm. The sub-continuum simulations yield a peak temperature rise that is 159 percent larger than predictions using only the classical heat diffusion equation. This work will facilitate the development of simpler calculation strategies, which are appropriate for commercial device simulators.

Maximum Temperature Rise in the Reacted Zone of Electrochemical Devices for Space Applications

2003 ◽  
Vol 125 (2) ◽  
pp. 177-181 ◽  
Author(s):  
Carsie A. Hall, ◽  
Edwin P. Russo ◽  
Calvin Mackie
Keyword(s):  
Diffusion Equation ◽  
Temperature Rise ◽  
Unit Volume ◽  
Heat Diffusion ◽  
Maximum Temperature ◽  
Electrode Temperature ◽  
Discharge Current Density ◽  
Heat Diffusion Equation ◽  
Maximum Temperature Rise ◽  
Electrochemical Devices

A model to predict the temperature rise in the reacted zone of discharging electrochemical devices has been developed. The model assumes that electrode kinetics are fast and concentration gradients are negligible. In the reacted zone, a thermal boundary layer grows, and its thickness is proportional to the reacted zone thickness. In the model, the temperature rise is predicted using the one-dimensional heat diffusion equation for a porous medium. The effective heat capacity per unit volume and effective thermal conductivity are defined as a function of electrode porosity. The instantaneous power per unit area dissipated in the reacted zone is used as a source term in the heat diffusion equation. With fixed parameters such as discharge current density, charge capacity per unit volume, electrode electrical conductivity, electrode porosity, and thermophysical properties of the pore-space fluid and electrode, the transient temperature distribution in the reacted zone is derived in closed-form. Subsequently, the maximum electrode temperature is readily obtained, and the maximum electrode temperature at complete discharge is derived. A new dimensionless parameter, the electro-thermal number, emerges as one of the most important parameters controlling the discharge time and maximum temperature rise.

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.

Fast Fluid Thermodynamics Simulation by Solving Heat Diffusion Equation

2021 ◽  
Vol 11 (04) ◽  
pp. 1-11
Author(s):  
Wanwan Li
Keyword(s):  
Diffusion Equation ◽  
Real Time ◽  
Stokes Equations ◽  
Fluid Simulation ◽  
Heat Diffusion ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Heat Diffusion Equation ◽  
Acceleration Techniques ◽  
Speed Up

In mechanical engineering educations, simulating fluid thermodynamics is rather helpful for students to understand the fluid’s natural behaviors. However, rendering both high-quality and realtime simulations for fluid dynamics are rather challenging tasks due to their intensive computations. So, in order to speed up the simulations, we have taken advantage of GPU acceleration techniques to simulate interactive fluid thermodynamics in real-time. In this paper, we present an elegant, basic, but practical OpenGL/SL framework for fluid simulation with a heat map rendering. By solving Navier-Stokes equations coupled with the heat diffusion equation, we validate our framework through some real-case studies of the smoke-like fluid rendering such as their interactions with moving obstacles and their heat diffusion effects. As shown in Fig. 1, a group of experimental results demonstrates that our GPU-accelerated solver of Navier-Stokes equations with heat transfer could give the observers impressive real-time and realistic rendering results.

scholarly journals Heat-balance integral to fractional (half-time) heat diffusion sub-model

2010 ◽  
Vol 14 (2) ◽  
pp. 291-316 ◽  
Author(s):  
Jordan Hristov
Keyword(s):  
Diffusion Equation ◽  
Heat Balance ◽  
Integral Method ◽  
Time Derivative ◽  
Heat Diffusion ◽  
Half Time ◽  
Heat Diffusion Equation ◽  
Parabolic Profile ◽  
Heat Balance Integral Method ◽  
The Heat Balance

The fractional (half-time) sub-model of the heat diffusion equation, known as Dirac-like evolution diffusion equation has been solved by the heat-balance integral method and a parabolic profile with unspecified exponent. The fractional heat-balance integral method has been tested with two classic examples: fixed temperature and fixed flux at the boundary. The heat-balance technique allows easily the convolution integral of the fractional half-time derivative to be solved as a convolution of the time-independent approximating function. The fractional sub-model provides an artificial boundary condition at the boundary that closes the set of the equations required to express all parameters of the approximating profile as function of the thermal layer depth. This allows the exponent of the parabolic profile to be defined by a straightforward manner. The elegant solution performed by the fractional heat-balance integral method has been analyzed and the main efforts have been oriented towards the evaluation of fractional (half-time) derivatives by use of approximate profile across the penetration layer.

Interplay Between Thermoelectric and Thermionic Effects in Heterostructures

2000 ◽  
Author(s):  
Taofang Zeng ◽  
Gang Chen
Keyword(s):  
Film Thickness ◽  
Free Path ◽  
Thermionic Emission ◽  
Approximate Solutions ◽  
Mean Free Path ◽  
Boltzmann Transport ◽  
Thermoelectric Effects ◽  
Double Heterojunction ◽  
The Mean ◽  
Electron Mean Free Path

Abstract When electrons sweep through a double-heterojunction structure, there exist thermionic effects at the junctions and thermoelectric effects in the film. While both thermoelectric and thermionic effects have been studied for refrigeration and power generation applications separately, their interplay in heterostructures is not understood. This paper establishes a unified model including both thermionic and thermoelectric processes based on the Boltzmann transport equation for electrons, and the nonequilibrium interaction between electrons and phonons. Approximate solutions are obtained, leading to the electron temperature and Fermi level distributions inside heterostructures and discontinuities at the interfaces as a consequence of the highly nonequilibrium transport when the film thickness is much smaller than the electron mean free path. It is found that when the film thickness is smaller than the mean free path of electrons, the transport of electrons is controlled by thermionic emission. The coexistence of thermoelectric and thermionic effects may increase the power factor when the electron mean free path is comparable to the film thickness.

Modeling of Localized Heating Effect in Sub-Micron Silicon Transistors

2003 ◽  
Author(s):  
Keivan Etessam-Yazdani ◽  
Sadegh M. Sadeghipour ◽  
Mehdi Asheghi
Keyword(s):  
Transport Equation ◽  
Hot Spot ◽  
Boltzmann Transport Equation ◽  
Phonon Transport ◽  
Heat Diffusion ◽  
Normal Operation ◽  
Heating Effect ◽  
Boltzmann Transport ◽  
Heat Diffusion Equation ◽  
Localized Heating

The performance and reliability of sub-micron semiconductor transistors demands accurate modeling of electron and phonon transport at nanoscales. The continued downscaling of the critical dimensions, introduces hotspots, inside transistors, with dimensions much smaller than phonon mean free path. This phenomenon, known as localized heating effect, results in a relatively high temperature at the hotspot that cannot be predicted using heat diffusion equation. While the contribution of the localized heating effect to the total device thermal resistance is significant during the normal operation of transistors, it has even greater implications for the thermoelectrical behavior of the device during an electrostatic discharge (ESD) event. The Boltzmann transport equation (BTE) can be used to capture the ballistic phonon transport in the vicinity of a hot spot but many of the existing solutions are limited to the one-dimensional and simple geometry configurations. We report our initial progress in solving the two dimensional Boltzmann transport equation for a hot spot in an infinite media (silicon) with constant temperature boundary condition and uniform heat generation configuration.

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