Table Of Contenti
Thermal
Conductivity
Theory, Properties,
and Applications
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ii
PHYSICS OF SOLIDS AND LIQUIDS
Editorial Board: Jozef T. Devreese ● University of Antwerp, Belgium
Roger P. Evrary ● University of Lie`ge, Belgium
Stig Lundqvist ● Chalmers University of Technology, Sweden
Gerald D. Mahan ● University of Tennessee, USA
Norman H. March ● University of Oxford, England
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EXCITATION ENERGY TRANSFER PROCESSES IN CONDENSED MATTER: Theory
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QUANTUM TRANSPORT IN SEMICONDUCTORS
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THERMAL CONDUCTIVITY: Theory, Properties, and Applications
Edited by Terry M. Tritt
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iii
Thermal
Conductivity
Theory, Properties,
and Applications
Edited by
Terry M. Tritt
Clemson University
Clemson, South Carolina
Kluwer Academic/Plenum Publishers
New York, Boston, Dordrecht, London, Moscow
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iv
Library of Congress Cataloging-in-Publication Data
Thermal conductivity: theory, properties, and applications/[edited by] Terry Tritt.
p. cm. — (Physics of solids and liquids)
Includes bibliographical references and index.
ISBN 0-306-48327-0
1. Thermal conductivity. I. Tritt, Terry M. II. Series.
QC176.8.T39T48 2004
536(cid:1).2012—dc22
2003064141
ISBN 0-306-48327-0
(cid:1)2004, Kluwer Academic/Plenum Publishers, New York
233 Spring Street, New York, New York 10013
http://www.wkap.nl
10 9 8 7 6 5 4 3 2 1
A C.I.P. record for this book is available from the Library of Congress
All rights reserved
No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any
means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written
permission from the Publisher, with the exception of any material supplied specifically for the purpose of
being entered and executed on a computer system, for exclusive use by the purchaser of the work.
Permissions for books published in Europe: [email protected]
Permissions for books published in the United States of America: [email protected]
Printed in the United States of America
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PREFACE
It has been almost thirty years since a book was published that was entirely dedi-
cated to the theory, description, characterization and measurement of the thermal
conductivity of solids. Forexample, theexcellent texts byauthors suchas Berman1,
Tye2 and Carlslaw & Jaeger3 remain as the standards in the ¢eld of thermal con-
ductivity. Tremendous e¡orts were expended in the late 1950’s and 1960’s in rela-
tion to the measurement and characterization of the thermal conductivity of solid-
state materials. These e¡orts were made by a generation of scientists, who for the
most part are no longer active, and this expertise would be lost to us unless we are
aware of the great strides they made during their time.
I became interested in the ¢eld of thermal conductivity in the mid 1990’s in relation
to my own work on the next generation thermoelectric materials, of which the
measurement and understanding of thermal conductivity is an integral part.4 In
my search for information, I found that most of the books on the subject of
thermal conductivity were out of print. Much of the theory previously formulated
by researchers such as Klemens5 and Slack6 contain considerable theoretical insight
into understanding thermal conductivity in solids. However, the discovery of new
materials over the last several years which possess more complicated crystal struc-
tures and thus more complicated phonon scattering mechanisms have brought new
challenges to the theory and experimental understanding of these new materials.
These include: cage structure materials such as skutterudites and clathrates, metal-
lic glasses, quasicrystals as well as many of new nano-materials which exist today.
In addition the development of new materials in the form of thin ¢lm and super-
lattice structures are of great theoretical and technological interest. Subsequently,
new measurement techniques (such as the 3-! technique) and analytical models to
characterize the thermal conductivity in these novel structures were developed.
Thus, with the development of many new and novel solid materials and new mea-
surement techniques, it appeared to be time to produce a more current and readily
available reference book on the subject of thermal conductivity. Hopefully, this
book, Thermal Conductivity-2004: Theory, Properties and Applications, will serve
not only as a testament to those researchers of past generations whose great care in
experimental design and thought still stands today but it will also describe many of
the new developments over the last several years. In addition, this book will serve
as an extensive resource to the next generation researchers in the ¢eld of thermal
conductivity.
First and foremost, I express great thanks to the authors who contributed to this
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VI PREFACE
book for their hard work and dedication in producing such an excellent collection
ofchapters.Theywereveryresponsivetothemanydeadlines andrequirementsand
they were a great group of people to work with. I want to personally acknowledge
my many conversations with Glen Slack, Julian Goldsmid, George Nolas and
Ctirad Uher, as well as many other colleagues in the ¢eld, as I grasped for the
knowledge necessary to personally advance in this ¢eld of research. I also want to
acknowledge the support and encouragement of my own institution, Clemson Uni-
versity (especially my Chairman: Peter Barnes), during this editorial and manu-
scriptpreparationprocess.Iamtrulyindebtedtoallmygraduatestudents,fortheir
contributionstothesevolumes,theirhardwork,forthepatienceandunderstanding
they exempli¢ed during the editorial and writing process. A special thanks goes to
the publishers and editors at Kluwar Press for their encouragement and patience in
allstages of development ofthesemanuscripts forpublication. Iamindebted to my
assistant at Clemson University, Lori McGowan, whose attention to detail and
hard work (copying, reading, ¢ling, corresponding with authors, etc.) helped
make this book possible. I especially wish to acknowledge my wonderful wife,
Penny, and my great kids for their patience and understanding for the many hours
that I spent on this work.
Terry M. Tritt
September 18, 2003
Clemson University
Clemson, SC
REFERENCES
1. ThermalConductionin Solids, R. Berman, Clarendon Press, Oxford,1976.
2. ThermalConductivityVol. I and Vol. II, editedby R. P.Tye, Academic Press,New York, 1969.
3. H.S.CarslawandJ.C.Jaeger,‘‘ConductionofHeatinSolids’’(OxfordUniversityPress,Oxford,
1959).
4. ‘‘Recent Ternds in Thermoelectric Materials Research’’ Terry M. Tritt, editor, (Academic Press,
Volumes69^71, Semiconductors andSemimetals, 2002).
5. SeeforexampleP.G.Klemens,Chapter1,p1ofThermalConductivityVol.I,editedbyR.P.Tye,
Academic Press,New York, 1969.
6. SolidState Physics, G. A. Slack, 34, 1, AcademicPress, New York,1979.
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LIST OF SYMBOLS
Ch. 1.1 Yang
(cid:1) ^ total thermal conductivity
(cid:2) ^ excitation
Q ^ heat £ow rate/ heat £ux vector
T ^ absolute temperature
c ^ heat capacity
n ^ concentration of particles
rT ^ temperature gradient
v ^ velocity
@E=@t^ Time rate of change of energy
(cid:3) ^ relaxation time
C ^ total heat capacity
l ^ particle mean free path
k ^ electron wave vector
E ^ electric ¢eld
f0k ^ distribution function
EF ^ Fermi energy constant
kB ^ Boltzmann’s constant
vk ^ electron velocity
e ^ electron charge
Eeff ^ e¡ective ¢eld
J ^ elecrical current density
Kn ^ general integral
(cid:4) ^ electrical conductivity
Lo ^ standard Lorentz number
(cid:5)D ^ Debye temperature for phonons
kf ^ electron wave number at Fermi surface
qD ^ phonon Debye wave number
G’ ^ constant representing strength of electron-phonon interaction
(cid:1)h ^ Planck constant
m* ^ electron e¡ective mass
nc ^ number unit cells per unit volume
vF ^ electron velocity at Fermi surface
(cid:6) ^ resistivity
We ^ electronic thermal resistivity
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VIII LIST OF SYMBOLS
Wi ^ constant
(cid:1)o ^ residual electrical resistivity
EG ^ energy gap
f0e ^ equilibrium distribution for electron
f0h ^ equilibrium distribution for holes
Ee ^ distances of Fermi level below bottom of conduction band
F
Eh ^ distances of Fermi level above top of valence band
F
Kn ^ integral for electrons
K0 ^ integral for holes
n
(cid:2) ^ constant
h ^ holes
(cid:3) ^ mobility
!~qq ^ phonon frequencey
@!
~qq ^ group velocity
@
q
q ^ wave vector
N~qq ^ phonon distribution function
~(cid:4)(cid:4)g ^ phonon group velocity
(cid:5)q ^ phonon scattering relaxation time
(cid:6) ^ angle between ~(cid:4)(cid:4)g and rT
(cid:7)L ^ lattice thermal conductivity
!D ^ Debye frequency (resonant frequency ?)
(cid:8)(cid:8)~ ^ constant vector-determine s anisotropy of phonon distribution
(cid:9) ^ parameter de¢ned in the text (units of time)
B ^ constant
(cid:10) ^ Gru«neisen constant
M ^ average atomic mass
V ^ volume per atom
mi ^ mass of atom
fi ^ fraction of atoms with mass mi
mm(cid:1) ^ average mass of all atoms
d ^ sample size for single crystal or grain size for a polycrystalline sample
ND ^ number dislocation lines per unit area
r ^ core radius
BD ^ Burgers vector of dislocation
C ^ constant proportional to concentration of resonant defects
(cid:11) ^ electron-phonon interaction constant or deformation potential
d ^ mass density
G ^ proportionality constant
(cid:2) ^ chemical shift related to splitting of electronic states
r ^ mean radius of localized state
0
Le ^ mean free path of electron
(cid:8)p ^ wavelength of phonon
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LIST OF SYMBOLS IX
Ch. 1.2 Uher
(cid:1) ^ total thermal conductivity
(cid:1)e ^ thermal conductivity of electrons
(cid:1)p ^ thermal conductivity of phonons
(cid:2) ^ electrical conductivity
(cid:3) ^ resistivity
L ^ constant that relates thermal conductivity of pure metals to their electrical
conductivity
Je ^ electrical current density
n ^ number of electrons per unit volume
v ^ velocity
t ^ time
(cid:4) ^ electrical ¢eld
(cid:5) ^ relaxation time/ collision time
le ^ mean free path between electrons
m ^ electron mass
JQ ^ thermal current density
cv ^ electronic speci¢c heat per unit volume
cM ^ molar speci¢c heat
(cid:6) ^ chemical potential/ Fermi level/ Fermi energy (EF)
E ^ particular energy state
D ^ density
U ^ internal energy of electron
kB ^ Boltzmann constant
TF ^ Fermi temperature
NA ^ Avagardo’s number
Z ^ valence of electron
R ^ universal gas constant
(cid:7) ^ coe⁄cient of linear speci¢c heat of metals#Sommer¢eld constant
(cid:1)h ^ Planck’s constant
k ^ Bloch wave vector of an electron
E(k) ^ Bloch energy of an electron
(cid:2)k ^ wavefunction
Uk ^ internal energy
H’ ^ perturbation Hamiltonian
(cid:3)ðEÞ ^ varying function of E ?
S ^ Seebeck coe⁄cient#thermoelectric power
(cid:4) ^ Peltier coe⁄cient
m* ^ e¡ective mass
A(E) ^ vector quantity
(cid:3)imp ^ impurity resistivity
Wkk0 ^ collision probability
V ^ volume
kTF ^ Thomas-Fermi screening parameter
kF ^ Fermi wave-vector (momentum)
a ^ Bohr radius
0
ni ^ density of impurities
q ^ wave vector
!q ^ frequency of vibration
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X LIST OF SYMBOLS
!D ^ Debye frequency
V0 ^ sample volume
k ^ incoming Bloch state
k’ ^ outgoing Bloch state
G ^ reciprocal lattice vector
r ^ point of electron
Ri ^ point of lattice ion
He(cid:1)ion^ potential energy of an electron in the ¢eld of all ions of lattice^interaction
Hamiltonian
e“q(cid:1) ^ unit polarization vectors
c ^ speed of sound
(cid:2)D ^ Debye temperature
(cid:1)s ^ temperature related to longitudinal sound velocity
d ^ dimensionality of metallic system
Lexp ^ Wiedemann-Franz ratio
Wexp ^ total measured electronic thermal resistivity
(cid:3)exp ^ electrical resistivity
(cid:2)U ^ umklapp electron-electron value of the e¡ectiveness parameter (cid:2)
We(cid:1)p ^ ideal thermal resistivity of metals
M ^ mass of an ion
C2 ^ coupling constant
j
Ma ^ average atomic mass
nc ^ number of atoms in a primitive cell
(cid:4) ^ high temperature Gru«neisen constant
(cid:2)0 ^ low temperature heat capacity
ND ^ dislocation density
B ^ Burgers vector of dislocation
TL:T: ^ low temperature domain
TH:T: ^ high temperature regime (T = 1)
k ^ electron wave vector
q ^ phonon small wave vector
C ^ constant
D ^ constant
Ch. 1.3 Murashove & White
(cid:5) ^ thermal conductivity
C ^ heat capacity per unit volume
v ^ average phonon group velocity
(cid:5) ^ heat transfer in gases
m ^ polarization branch
!D ^ Debye cut-o¡ frequency
l ^ mean free path of phonons
t ^ relaxation time
k ^ wave vectors
n ^ number of atoms
nD ^ Debye cut-o¡ frequency
! ^ frequency
G ^ reciprocal lattice vector
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