An unstructured finite volume scheme is applied to the solution of sub-micron heat conduction problems. The phonon Boltzmann transport equation (BTE) in the relaxation time approximation is considered. The similarity between the radiative transfer equation (RTE) and the BTE is exploited in developing a finite volume scheme for the BTE. The spatial domain is divided into arbitrary unstructured polyhedra, the angular domain into control angles, and the frequency domain into frequency bands, and conservation equations for phonon energy are written. The unsteady wave propagation term, not usually present in thermal radiation problems, is differentiated using a fully implicit scheme. A sequential multigrid scheme is applied to solve the nominally linear set. Isotropic scattering due to a variety of mechanisms such as impurity and Umklapp scattering is considered. The numerical scheme is applied to a variety of sub-micron conduction problems, both unsteady and steady. Favorable comparison is found with the published literature and with exact solutions.

1.
Tien
,
C.
, and
Chen
,
G.
,
1994
, “
Challenges in Microscale Radiative and Conductive Heat Transfer
,”
ASME J. Heat Transfer
,
116
, pp.
799
807
.
2.
Asheghi
,
M.
,
Touzelbaev
,
M.
, and
Goodson
,
K.
,
1997
, “
Phonon-Boundary Scattering in Thin Silicon Layers
,”
Appl. Phys. Lett.
,
71
, pp.
1798
1800
.
3.
Sverdrup
,
P.
,
Ju
,
Y.
, and
Goodson
,
K.
,
1998
, “
Sub-Continuum Simulations of Heat Conduction in Silicon-on-Insulator Devices
,”
ASME J. Heat Transfer
,
120
, pp.
30
36
.
4.
Sverdrup, P., Banerjee, K., Dai, K., Shih, W., Dutton, R., and Goodson, K., 2000, “Sub-Continuum Simulations of Deep Sub-Micron Devices under ESD Conditions,” Proceedings of the International Conference on Simulation of Semiconductor Processes and Devices, IEEE, pp. 54–57.
5.
Chen
,
G.
,
1997
, “
Size and Interface Effects on Thermal Conductivity of Superlattices and Periodic Thin Film Structures
,”
ASME J. Heat Transfer
,
119
, pp.
220
229
.
6.
Majumdar
,
A.
,
1993
, “
Microscale Heat Conduction in Dielectric Thin Films
,”
ASME J. Heat Transfer
,
115
, pp.
7
16
.
7.
Joshi
,
A.
, and
Majumdar
,
A.
,
1993
, “
Transient Ballistic and Diffusive Phonon Heat Transport in Thin Films
,”
J. Appl. Phys.
,
74
, pp.
31
39
.
8.
Chen
,
G.
,
1996
, “
Nonlocal and Nonequilibrium Heat Conduction in the Vicinity of Nanoparticles
,”
ASME J. Heat Transfer
,
118
, pp.
539
545
.
9.
Kumar
,
S.
,
Majumdar
,
A.
, and
Tien
,
C.
,
1990
, “
The Differential Discrete-Ordinates Method for Solutions of the Equation of Radiative Transfer
,”
ASME J. Heat Transfer
,
112
, pp.
424
429
.
10.
Modest, M. F., 1993, Radiative Heat Transfer, Series in Mechanical Engineering, McGraw Hill, New York, NY.
11.
Murthy
,
J.
, and
Mathur
,
S.
,
1998
, “
Finite Volume Method for Radiative Heat Transfer Using Unstructured Meshes
,”
J. Thermophys. Heat Transfer
,
12
(
3
), pp.
313
321
.
12.
Kittel, C., 1996, Introduciton to Solid State Physics, John Wiley & Sons, New York.
13.
Mathur
,
S.
, and
Murthy
,
J.
,
1997
, “
A Pressure Based Method for Unstructured Meshes
,”
Numer. Heat Transfer Part B
,
31
(
2
), pp.
195
216
.
14.
Murthy
,
J.
, and
Mathur
,
S.
,
1998
, “
A Conservative Numerical Scheme for the Energy Equation
,”
ASME J. Heat Transfer
,
120
, pp.
1081
1085
.
15.
Murthy
,
J. Y.
, and
Mathur
,
S. R.
,
1998
, “
Radiative Heat Transfer in Axisymmetric Geometries Using an Unstructured Finite Volume Method
,”
Numer. Heat Transfer, Part B
,
33
(
4
), pp.
397
416
.
16.
Mathur
,
S.
, and
Murthy
,
J.
,
1998
, “
Radiative Heat Transfer in Periodic Geometries Using a Finite Volume Scheme
,”
ASME J. Heat Transfer
,
121
, pp.
357
364
.
You do not currently have access to this content.