Influence of Order–Disorder Transition on Thermal Conductivity of Solids

\

Chaos\ Solitons + Fractals Vol[ 09\ No[ 01\ pp[ 1976Ð1987\ 0888 Þ 0888 Elsevier Science Ltd[ All rights reserved 9859Ð9668:88:, ! see front matter

Pergamon

PII] S9859!9668"87#99149!0

In~uence of OrderÐDisorder Transition on Thermal Conductivity of Solids JERZY BODZENTA Institute of Physics\ Silesian Technical University\ Krzywoustego 1\ 33!099 Gliwice\ Poland

Abstract*The relation between the crystalline structure and thermal properties of solids is analyzed[ The brief description of the theories of heat conduction in polycrystalline solids "Callaway model# and amorph! ous solids "CahillÐPohl# model is carried out[ It is shown that transition from the heat transport by phonons in polycrystals to the random energy transfer between localized oscillators in glasses leads to the meaningful reduction of the thermal conductivity[ This conclusion is illustrated by experimental data[ The limitations of obtaining thin _lms with good thermal properties are underlined[ Þ 0888 Elsevier Science Ltd[ All rights reserved[

0[ INTRODUCTION

The transition from ideally ordered monocrystals through partially disordered polycrystals to completely disordered amorphous solids is connected with a radical change of physical properties[ This change is also apparent for the thermal conductivity[ Let us restrict further considerations within dielectrics and the lattice thermal conductivity[ Its theory is based on the assumption\ that heat is transported by phonons\ quasi!particles describing quantized lattice vibrations[ Using the formula for the kinetic theory of gases\ the thermal conductivity is ð0Ł 0 k Cul\ 2

"0#

where C is the speci_c heat\ u is the velocity of the phonons "the sound velocity# and l is the phonon mean free path[ The mean free path is the mean distance between subsequent scattering events[ There are two mechanisms of phonon scattering] scattering by another phonon and scattering by crystal imperfections "geometrical scattering#[ The orderÐdisorder transition causes an increase in the concentration of scattering centers "grain boundaries\ dislocations\ etc[#[ It leads to a decrease of the thermal conductivity[ At present\ thin layers of di}erent materials have many practical applications[ Thin dielectric _lms of SiO1 and SiNx are commonly used in microelectronics as interconductor dielectrics and passivation layers[ A continuous rise of switching speed and circuit densities causes heat transport in these layers to becomes critical for device operation[ The same situation takes place for the optical coatings using in the power laser systems[ These facts are re~ected in a growing interest in layers with high thermal conductivity[ From this point of view\ diamond and diamond!like carbon "DLC# layers seem to be very promising[ On the one hand\ the thermal conductivity of

 Author for correspondence[ E!mail] bodzentaÝtytan[mat_z[polsl[gliwice[pl[ 1976

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high!quality single crystals of diamond "type IIa# is about 1199 W:"m=K# at room temperature ð1Ł and it is the highest known value[ But on the other hand\ measured thermal conductivities of thin DLC layers are much smaller\ for instance\ 84 W:"m=K# for a 0 mm thick layer ð2Ł and only 13 W:"m=K# for a 39 nm thick layer "calculated from data published in ð3Ł and ð4Ł#[ So\ the thermal conductivity of such layers is even smaller than the one measured in silicon\ 037 W:"m=K# ð5Ł[ It follows\ from these examples\ that the thermal properties of bulk materials and poly! crystalline or amorphous layers are completely di}erent[ The aim of this paper is the analysis of the physical reasons for this[ Only knowledge of the sources of the limitations of heat transport in disordered solids gives us the possibility of answering two fundamental questions] 0[ What is the higher possible thermal conductivity of polycrystalline or amorphous layers and 1[ What should be done to obtain layers with high thermal conductivity[ In the next section\ theoretical models of heat transport in polycrystals and glasses are brie~y described[ Then the methods most often used for determining the thermal conductivity of layers are characterized[ Based upon experimental data\ the in~uence of the inner structure of the layer on the heat transport is analyzed[ The paper ends with conclusions on the limitations of using thin layers for heat management[

1[ THEORETICAL MODELS

As mentioned above\ the heat transport in dielectrics is connected with the ~ow of phonons[ The ~ux of phonons is caused by the temperature gradient[ Phonons move at the velocity of sound and are scattered by other phonons and crystal imperfections[ Equation "0# may only be used as the simplest approximation of the thermal conductivity[ The proper theory of the thermal conductivity of solids is elaborated by Debye and leads to the formula ð6Ł kB3 k 1 2 T 2 1p : u

uD:T

t"x#x3 ex

9

"ex −0#1

g

dx\

"1#

where kB and : are Boltzmann|s and Planck|s constants\ uD"5p1n#0:2:u:kB is the Debye tempera! ture\ n is the concentration of atoms\ x:v:kBT and t"x# is the total phonon relaxation time for a phonon of frequency v\ at temperature T[ In the presence of N scattering processes\ the total phonon scattering rate t−0 is calculated using the formula N

t−0  s ti−0 \

"2#

i0

where ti−0 is the scattering rate of the ith scattering process[ In eq[ "2#\ scattering processes which do not conserve the momentum of the crystal are taken into account[ Only these processes tend to return the phonon system to an equilibrium Planck distribution and introduce thermal resistance[ Such processes are called umklapp processes[ In the case of polycrystalline solids\ the following umklapp processes with respective scattering rates are considered ð7Ð01Ł] 0[ phononÐphonon umklapp scattering] tu−0 Bv1 T exp"uD :bT#\ where B and b are constants^ 1[ phononÐboundary scattering] tb−0 "0−p#:"0¦p# = "u:ad#\ where a is the constant of order unity "which depends on the shape of the sample#\ d is the sample size in the direction perpendicular to the heat ~ux\ p is the probability of specular re~ection of the phonon at the surface^ 2[ phononÐpoint defect scattering] tp−0 Pv3 \ where P is a constant related to the concentration of point defects^

In~uence of orderÐdisorder transition on thermal conductivity of solids

1978

3[ phononÐextended defect scattering]

6

te−0 

pne a1 u:3 for a:l×1p pne a5 v3 :3u2

for a:l³1p\

where ne is the extended defect concentration\ a is the radius of the defect and l1pu:v is the the phonon wavelength^ 4[ phononÐdislocation scattering] td−0 Dv\ where D is a constant related to the concentration of dislocations[ Taking into consideration all _ve scattering processes mentioned above\ the total scattering rate is t−0 tu−0 ¦tb−0 ¦tp−0 te−0 ¦td−0 [ It is stated above that only umklapp scattering processes have a direct in~uence on the thermal conductivity[ But phonons are also scattered in so!called normal processes\ in which the crystal momentum is conserved[ Normal processes do not introduce thermal resistance by themselves but a}ect the heat transport indirectly[ Normal processes may convert low frequency phonons with little scattering to higher frequencies\ for which the scattering is stronger[ The in~uence of normal scattering processes on the thermal conductivity is considered in the phenomenological model developed by Callaway ð02Ł[ The Callaway model is often used for interpretation of the thermal conductivity dependence on temperature\ especially at low temperatures ð09\ 00Ł[ The scattering rate of normal phononÐphonon scattering processes is typically taken as tN−0 AvT 2 \ where A is a constant[ Taking into account the normal and umklapp processes\ the thermal conductivity is ð09\ 00\ 02Ł

F 3 2 kB 0H k 1 2 T 2 s g j0 uj 5p : H f

g

uDj:T

9

tCj "x#

3 x

x e x

"e −0#

1

dx¦

$g g

uDj:T

9

uDj:T

9

tCj "x# x3 ex dx tNj "x# "ex −0#1

1

% Hh J

x3 ex tCj "x# dx H tj "x#tNj "x# "ex −0#1 j

\

"3#

where the separate contributions of transverse and longitudinal modes are added[ Quantities associated with particular modes are distinguished by index j[ The combined scattering rate is −0 −0 "tj−0 ¦tNj # "of the the sum of the scattering rates of both umklapp and normal processes tCj jth mode#[ The Callaway model can be applied for analysis of the thermal conductivity of polycrystalline samples[ This model is based on the idea of phonons and may be used only in materials with the long!range order in the arrangement of their atoms[ The situation is qualitatively di}erent in amorphous materials "glasses#[ There is no long!range order in glasses[ Only short!range order\ embracing a few nearest neighboring atoms exists[ The vibrations of atoms in such situations cannot be described by phonons[ The phonon mean free path calculated from eq[ "0#\ using the thermal conductivity of amorphous solids above 099 K\ is on the order of the phonon wavelength and interatomic distance ð0Ł[ It means that vibrations of the atoms in glasses are localized and should be described as damped localized oscillators\ not waves[ Einstein developed the theory of such a model trying to describe the heat capacity of solids "see\ e[g[\ ð0Ł#[ In Einstein|s theory\ every atom is considered as an harmonic oscillator with the same frequency of vibration[ In the theory of thermal conductivity\ each atom is coupled to its _rst\ second and third nearest neighbors[ Energy di}uses from oscillator to oscillator during approximately half of the period of oscillation[ The thermal conductivity results from this random walk of energy and can be expressed by the formula ð03Ł

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x1 ex kB1 n0:2 \ uE x p: "e −0#1

kE 

"4#

where uE is the Einstein temperature\ xuE:T[ Based on his model\ Einstein wanted to understand the thermal conductivity of crystalline solids but he did not achieve agreement with experimental data[ The thermal conductivity of crystalline solids must be described by the Debye model eq[ "1#[ But there is a good agreement between data calculated using eq[ "4# and experimental dependencies of the thermal conductivity on temperature measured above 099 K in amorphous solids[ The drawback of the Einstein model is some freedom in choosing uE[ Cahill and Pohl proposed a modi_cation of this theory ð04Ł[ They introduced larger oscillating entities than single atoms*the sample regions of size\ l:1\ whose frequencies of oscillation are v1pu:l[ The relaxation time of each oscillator is one half of the period of vibration "as in the Einstein model#[ The thermal conductivity resulting from the random energy walk between these localized excitations is kCP 

0:2

01 p 5

2

kB n1:2 s uj j0

T uDj

g

uDj:T

9

x2 ex "ex −0#1

dx[

"5#

the main di}erences from the Einstein model are that the localized oscillators have varying sizes and frequencies\ and that uDj is clearly de_ned[ The Cahill and Pohl model may be used for a description of the thermal conductivity of amorphous and highly disordered solids above 49 K\ and gives better agreement with the experimental data than the Einstein model[ To summarize\ the transition from crystalline solids with long!range order in the arrangement of their atoms to highly disordered glasses is connected with a change in the mechanism of the heat transport[ Vibrations of atoms in crystals are correlated\ which leads to the creation of elastic waves in the lattice[ These waves are described through phonons[ The heat is transferred between di}erent points by phonons[ The heat resistance arises because of the scattering of phonons[ In the case of amorphous solids\ description of the heat transport in terms of phonons is no longer valid[ The heat conduction is a result of a random energy walk between localized\ heavily damped oscillators[ It is evident that the second mechanism is less e}ective and leads to lower thermal conductivity[ Dependencies of the thermal conductivity on temperature\ calculated for two crystalline solids with di}erent grain sizes and an amorphous solid using eq[ "3# and eq[ "5#\ respectively\ are shown in Fig[ 0[ The thermal conductivity of the amorphous sample is two orders of magnitude smaller than that for the polycrystalline sample[

2[ METHODS FOR DETERMINING THE THERMAL CONDUCTIVITY OF SOLIDS

The problem of experimental determination of the thermal conductivity is not a trivial task[ Particular di.culties arise when the investigated sample is either a very good or a very poor thermal conductor[ There are many problems connected with the measurement of the thermal conductivity of thin _lms deposited on thick substrates[ As this paper is not devoted to measuring methods\ the characteristics of the main methods applied in determining thermal conductivity are discussed only brie~y[ This section is based on review papers ð05Ð08Ł[ The heat transport in a homogeneous medium is described by the FourierÐKirchho} equation ð19Ł\ Q"rł\t# 0 1T"rł\t# −DT"rł\t# \ b 1t k

"6#

where bk:rC is the thermal di}usivity\ r is the mass density\ C is the heat capacity\ Q"rł\t# is

In~uence of orderÐdisorder transition on thermal conductivity of solids

1980

Fig[ 0[ The thermal conductivity dependence on temperature calculated for two polycrystalline solids with di}erent grain size "solid line*d09 mm\ line with ¦*d9[0 mm# and an amorphous solid "line with ×#[ The calculations are carried out based on the Callaway ð02Ł and CahillÐPohl ð04Ł models[ Data for the calculations are taken from ð8Ł[

the density of volume heat sources[ Based on the dependence of Q"rł\t# on time measuring techniques for thermal properties\ determination can be divided into three groups] 0[ the steady!state methods where the heat sources and\ consequently\ the temperature do not depend on time] Qf "rł#\ 1T:1t9^ 1[ pulsed methods\ where the time dependence of heat sources are described by the Dirac or Heavyside distribution] Qf "rł#d"t# or Qf "rł#H"t#^ 2[ modulated methods\ where the heat sources are periodic in time] Qf "rł#eivt [ The steady!state methods demand quantitative evaluation of temperature gradients and cor! respondence heat ~uxes[ These methods allow direct determination of the thermal conductivity k[ Practical application of steady!state measuring techniques is\ in many cases\ very complicated\ especially for thin layers[ In such situations\ sophisticated structures of heaters and thermocouples are used ð09\ 10Ð16Ł[ The pulse method of determining the thermal di}usivity is proposed by Parker et al[ ð17Ł and is called the {~ash method|[ The method is based on a simple principle[ The front sample surface is heated by a short light pulse and the temperature evolution of the back surface in time is registered[ The characteristic time of this process is directly connected with the thermal di}usivity of the sample[ The ~ash method has been improved by many researchers\ its modi_cations are used in investigations of very di}erent samples "layered samples\ liquids\ composite materials\ porous media\ etc[#[ A representative review of pulsed methods has been done by Balageas ð18Ł[ Two relatively new pulse methods should be mentioned here[ The _rst one is the method of the converging thermal wave\ proposed by Cielo et al[ ð29Ł[ A pulsed laser beam\ focused on the sample surface\ heats an annular!shaped area and the surface temperature is monitored at the center of the annulus[ The thermal di}usivity of the sample is determined from the temperature dependence on time[ The main bene_t of this method is the increase of the signal!to!noise ratio\ thanks to the focussing e}ect of the annulus[ The second\ very elegant\ pulsed method is the

1981

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transient thermal grating method\ proposed by Eichler et al[ ð20Ł[ The thermal grating at the sample surface is created by the absorption of light of two interfering pulsed laser beams[ The decay time of the grating can be used to calculate the thermal di}usivity[ This method allows good accuracy and high spatial resolution ð21Ł[ The transient thermal grating method is especially useful in investigations of thin _lms ð3\ 22\ 23Ł[ In the modulated method\ the sample is periodically heated[ In this case\ the solution of eq[ "6# looks like the equation describing heavily damped wavesÐthermal waves "see\ e[g[\ ð19Ł#[ The wave number of the thermal wave\ kTW zv:1b\ is equal to its absorption coe.cient[ Measuring the propagation of the thermal wave thus permits determination of the thermal di}usivity[ Measuring techniques utilizing thermal wave propagation can be realized in di}erent con! _gurations and can be based on 0D ð24Ð39Ł\ 1D ð30Ł or 2D ð31Ð35Ł theoretical models[ Detailed analysis of the most often used thermal wave techniques is carried out in ð05\ 06Ł[ A very smart concept for measuring the periodic temperature changes caused by periodic heating for determining the thermal conductivity is utilized in the so!called {2v method| ð36Ł[ The single metal line is used as both the heater and the thermometer[ An a[c[ driving current\ of frequency\ v\ heats the surface of the sample at frequency\ 1v[ Consequently\ the voltage drop across the metal line has a small component at 2v that can be used for evaluation of the temperature oscillations and\ therefore\ the thermal response of the sample[ The 2v method is used for the examination of dielectric _lms ð37\ 38Ł[ As mentioned above\ the characteristics of the main measuring techniques used in the deter! mination of thermal properties are only brie~y described in this section[ Steady!state techniques allow determination of the thermal conductivity\ while pulsed and modulated techniques give the possibility of determining the thermal di}usivity[ Depending on the experimental geometry\ these thermal parameters can be measured for the direction parallel "k>\ b># or perpendicular "k_\ b_# to the sample surface[ 3[ EXPERIMENTAL RESULTS

The thermal properties of materials are widely investigated because of their practical import! ance in many applications[ This is especially true in the case of synthetic diamond and diamond! like carbon[ There are many papers devoted to the analysis of heat transport in such materials and their structural limitations[ The phonon scattering processes limiting the thermal conductivity are enumerated in Section 1[ Four of them] phononÐpoint defect scattering\ phononÐboundary scattering\ phononÐextended defect scattering and phononÐdislocation scattering\ are directly connected with the internal structure of the sample[ In this section\ the in~uence on the thermal conductivity of the processes mentioned above are analyzed\ based on experimental data[ Data for synthetic diamond are mainly used because it is the material whose thermal properties have been very intensively investigated[ The phononÐpoint defect scattering in~uence on the thermal conductivity can be illustrated by measurements carried out for isotopically enriched diamond ð49Ð41Ł[ Natural diamond con! tains about 0) 02C isotopic {impurity|[ Heavier carbon atoms in the lattice are additional scattering centers and should decrease the thermal conductivity^ experimental results con_rm this fact[ The thermal conductivity of isotopically pure diamond "88[8) 01C# is up to 49) higher than that of the sample with natural abundance[ Such an increase cannot be explained only by decreasing the concentration of scattering centers[ Detailed analysis of the sample structures shows that the isotopically enriched samples have substantially less defect structure than natural abundance samples ð41Ł[ The in~uence of the isotopic {impurity| on the thermal conductivity of synthetic diamond can be observed only in samples with very low concentration of other defects[ In the case of thin

In~uence of orderÐdisorder transition on thermal conductivity of solids

1982

Fig[ 1[ Dependence of the thermal conductivity of CVD diamond layers on its thickness at room temperature[ Data are taken from] *ð44Ł\ ž*ð42Ł\ ×*ð15Ł\ ¦*ð2Ł\ *ð45Ł\ Ž*ð10Ł\ r*ð46Ł[

diamond and DLC layers\ the main scattering mechanism is phononÐboundary scattering[ It is known that chemical!vapor!deposited "CVD# diamond grows in a columnar structure with a grain size which starts out very small and increases with increasing _lm thickness "see\ e[g[\ ð42\ 43Ł#[ Also\ the crystalline quality enhances with an increase in _lm height ð43Ł[ These two facts lead to the dependence of thermal properties on _lm thickness[ Such dependence is described in several papers ð2\ 10\ 15\ 45\ 46Ł[ Data from mentioned works are collected in Fig[ 1[ In spite of this\ that investigated layers are deposited in di}erent conditions and thus\ have di}erent inner structures\ the tendency is clear[ The e}ective thermal conductivity of a layer increases with the layer thickness[ A similar dependence is also obtained for P!SiC layers ð47Ł and thin metallic _lms ð48Ł[ The thermal conductivity dependence on the grain size at room temperature is shown in Fig[ 2[ The thermal conductivity reaches a value comparable to single crystals for grains larger than about 19 mm[ Analysis of the in~uence of the two others scattering processes is more complicated[ Besides grain boundaries\ the main type of defects in CVD diamond samples is the inclusion of non! diamond carbon ð13\ 14\ 59Ł[ A strong correlation between thermal parameters and the con! centration of non!diamond carbon is found[ But there are also a lot of other defects such as clusters of nitrogen ð50Ł\ clusters of vacancies\ voids\ microcracks ð51Ł\ etc[ All these defects are mainly located at or near the grain boundaries and cause additional thermal resistance of these boundaries ð51Ð54Ł[ From this point of view\ the 0D model for the e}ective conductivity of polycrystalline diamond\ proposed in ð55\ 56Ł\ is very interesting[ According to this model\ the sample contains grains of ideal diamond "k1299 W:"m=K## separated by boundaries with high thermal resistance "Rth1=09−8 m1K:W#[ The thermal resistance for the heat ~ow across the sample is the sum of the thermal resistance of the bulk material and the additional resistance from the grain boundaries[ There is a good agreement between this simple approach and the experimental data[

1983

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Fig[ 2[ Dependence of the thermal conductivity of CVD diamond on the grain size at room temperature[ Data are taken from] *ð42Ł\ ž*ð15Ł\ ×*ð2Ł\ ¦*ð10Ł\ *ð46Ł[

The thermal conductivities of di}erent amorphous samples "SiO1\ SiNx ð38Ł\ TiO1 ð57Ł\ Si2N3 ð11Ł\ Si]H ð58Ł# have similar values\ about 0Ð1 W:"m=K#[ It means that the mechanism of heat transport in all these materials is very similar[ Interesting results are obtained in investigations of thin _lms of polycrystalline DLC and amorphous diamond ð69Ł[ The thermal conductivity of amorphous diamond is more than an order higher in comparison with polycrystalline DLC[ It is also 2Ð4 times larger than the thermal conductivity predicted from the CahillÐPohl theory[ This fact can be explained as a result of the appearance of a medium!range order in amorphous diamond because of bonding constraints[ The in~uence of the orderÐdisorder transition on the thermal conductivity is clearly visible for natural diamond single crystal irradiated by neutrons ð8\ 60Ł[ Low levels of irradiation create vacancies and small regions of disordered carbon thus create phonon scattering centers[ The thermal conductivity gradually decreases with increasing level of irradiation[ Heavy doses\ above 0919 neutrons:cm1\ initiate vitri_cation of the sample and the sample has the same thermal properties as glass[ As shown in Fig[ 3\ changes in the inner structure of diamond result in a decrease of its thermal conductivity by more than two orders\ from 1099 W:"m=K# to 3 W:"m=K#[ 4[ CONCLUSIONS

Thermal properties are very sensitive to the material|s inner structure[ In the polycrystalline dielectrics\ heat is transported by phonons*quasi!particles describing quantized and correlated vibrations of atoms in a lattice[ Scattering of phonons by other phonons and imperfections of lattice causes the thermal resistance[ In polycrystals built of large grains of good crystalline quality\ the thermal conductivity is limited by umklapp phononÐphonon scattering at room temperature[ With decreasing grain size\ the concentration of lattice imperfections\ such as grain boundaries\ point defects\ dislocations\ microcraks\ voids\ etc[\ increases^ thus\ the concentration

In~uence of orderÐdisorder transition on thermal conductivity of solids

1984

Fig[ 3[ Changes in the thermal conductivity of diamond caused by neutron irradiation[

of scattering centers also increases[ These scattering centers limit the mean free path of phonons and the thermal conductivity of the material[ Lattice imperfections are accumulated near grain boundaries\ then a polycrystalline solid can be treated as a set of crystalline grains\ whose properties are similar to those of bulk material\ separated by boundaries with high thermal resistance[ The orderÐdisorder transition causes qualitative change in the mechanism of heat transport and leads to a meaningful reduction of the thermal conductivity[ The heat transport cannot be understood as a ~ux of phonons in an amorphous solid[ Glass samples can be treated as a set of localized\ damped\ uncorrelated oscillators[ The heat conductivity is a result of random energy di}usion between these oscillators[ This mechanism of heat transport is less e}ective and glasses are generally worse heat conductors than polycrystalline solids[ When the thermal conductivity is critical for an application\ the {crystal quality| of the sample should be as good as possible[ It means that\ in the case of polycrystalline solids\ the material should consist of large grains and grain boundaries should be smooth*without microcracks\ voids or inclusions of impurities[ This condition can be satis_ed for samples thicker than 49Ð 099 mm\ although it is very di.cult to obtain thin _lms with satisfactory thermal parameters[ Thin _lms are built of small crystallites\ often separated by regions of non!crystalline material[ It leads to very low thermal conductivity of thin polycrystalline _lms\ sometimes even lower than the thermal conductivity of amorphous _lms[ Considering possible applications of thin _lms in heat management\ one should also remember that there is an interfacial layer between the _lm and the substrate with a high concentration of defects[ This layer is an additional barrier for the heat ~ow from the substrate to the _lm[ As a result\ experimentally determined e}ective thermal conductivity of a thin _lm in the direction perpendicular to the _lms may be very low[ For instance\ the measured e}ective thermal conductivity of thin DLC _lms "½0 mm# deposited on silicon is about 9[1Ð9[2 W:"m=K#\ four orders of magnitude lower than for diamond ð39\ 61Ł[

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Acknowled`ement*I gratefully acknowledge the Polish State Committee for Scienti_c Research "KBN# for support under grant No[ 6!T97C!914!01[

REFERENCES 0[ Kittel\ C[\ Introduction to Solid State Physics[ John Wiley and Sons\ New York\ 0845[ 1[ Bermann\ R[\ In Properties of Diamond\ ed[ J[ E[ Field[ Academic Press\ London\ 0868[ 2[ von Kaenel\ Y[\ Stiegler\ J[\ Blanck\ E[\ Chauvet\ O[\ Hellwig\ Ch[\ Plamann\ K[\ Microstructure evolution and defect incorporation in highly oriented and textured CVD diamond _lms[ Phys[ Stat[ Sol[ "A#\ 0885\ 043\ 108Ð127[ 3[ Shen\ Q[\ Harata\ A[\ Sawada\ T[\ Analysis of the thermal and acoustic properties of ion!implanted diamondlike carbon _lms using the transient re~ecting grating technique[ J[ Appl[ Phys[\ 0884\ 66\ 0377Ð0380[ 4[ Graebner\ J[ E[\ Measurement of speci_c heat and mass density in CVD diamond[ Diamond Relat[ Mater[\ 0885\ 4\ 0255Ð0269[ 5[ Rhode\ M[\ Photoacoustic characterization of thermal transport properties in thin _lms and microstructures[ Thin Solid Films\ 0883\ 127\ 088Ð193[ 6[ Bermann\ R[\ Thermal Conductivity in Solids[ Oxford University Press\ Oxford\ 0865[ 7[ Anthony\ T[ R[\ Fleischer\ J[ L[\ Olson\ J[ R[\ Cahill\ D[ G[\ The thermal conductivity of isotopically enriched polycrystalline diamond _lms[ J[ Appl[ Phys[\ 0880\ 58\ 7011Ð7014[ 8[ Morelli\ D[ T[\ Perry\ T[ A[\ Farmer\ J[ W[\ Phonon scattering in lightly neutron!irradiated diamond[ Phys[ Rev[ B\ 0882\ 36\ 020Ð028[ 09[ Graebner\ J[ E[\ Reiss\ M[ E[\ Seibles\ L[\ Hartnett\ T[ M[\ Miller\ R[ P[\ Robinson\ C[ J[\ Phonon scattering in chemical!vapor!deposited diamond[ Phys[ Rev[ B\ 0883\ 49\ 2691Ð2602[ 00[ Graebner\ J[ E[\ Jin\ S[\ Herb\ J[ A[\ Gardinier\ C[ F[\ Local thermal conductivity in chemical!vapor!deposited diamond[ J[ Appl[ Phys[\ 0883\ 65\ 0441Ð0445[ 01[ Touzelbaev\ M[ N[\ Goodson\ K[ E[\ Application of micron!scale passive diamond layers for the integrated circuits and microelectromechanical systems industries[ Diamond Relat[ Mater[\ 0887\ 6\ 0Ð03[ 02[ Callaway\ J[\ Model for lattice thermal conductivity at low temperatures[ Phys[ Rev[\ 0848\ 002\ 0935Ð0940[ 03[ Einstein\ A[ Ann[ Phys[\ 0800\ 24\ 568[ 04[ Cahill\ D[ G[\ Pohl\ R[ O[\ Heat ~ow and lattice vibrations in glasses[ Solid State Commun[\ 0878\ 69\ 816Ð829[ 05[ Fournier\ D[\ Plamann\ K[\ Thermal measurements on diamond and related materials[ Diamond Relat[ Mater[\ 0885\ 3\ 798Ð708[ 06[ Plamann\ K[\ Fournier\ D[\ Thermal conductivity of CVD diamond] methods and results[ Phys[ Stat[ Sol[ "A#\ 0885\ 043\ 240Ð258[ 07[ Graebner\ J[ E[\ Measurement techniques and results for the thermal conductivity of CVD diamond[ Electrochemical Soc[ Proc[\ 0884\ 84!3\ 368Ð378[ 08[ Bodzenta\ J[\ Determination of thermal properties of solids by photothermal methods[ In Proceedin`s of 1nd National Conference on {Physical Principles of NDT|\ Gliwice\ 0886\ pp[ 14Ð25 "in Polish#[ 19[ Carslaw\ H[ S[ and Jaeger\ J[ C[\ Conduction of Heat in Solids[ Oxford University Press\ Oxford\ 0867[ 10[ Graebner\ J[ E[\ Mucha\ J[ A[\ Seibles\ L[\ Kammlott\ G[ W[\ The thermal conductivity of chemical!vapor!deposited diamond _lms on silicon[ J[ Appl[ Phys[\ 0881\ 60\ 2032Ð2035[ 11[ Gri.n\ A[ J[ Jr[\ Brotzen\ F[ R[\ Loos\ P[ J[\ The e}ective transverse thermal conductivity of amorphous Si2N3 thin _lms[ J[ Appl[ Phys[\ 0883\ 65\ 3996Ð3900[ 12[ Belay\ K[\ Etzel\ Z[\ Onn\ D[ G[\ Anthony\ T[ R[\ The thermal conductivity of polycrystalline diamond _lms] e}ects of isotope content[ J[ Appl[ Phys[\ 0885\ 68\ 7225Ð7239[ 13[ Jansen\ E[\ Obermeier\ E[\ Measurements of the thermal conductivity of CVD diamond _lms using micromechanical devices[ Phys[ Stat[ Sol[ "A#\ 0885\ 043\ 284Ð391[ 14[ Jansen\ E[\ Dorsh\ O[\ Obermeier\ E[\ Kulish\ W[\ Thermal conductivity measurements on diamond _lms based on micromechanical devices[ Diamond Relat[ Mater[\ 0885\ 4\ 533Ð537[ 15[ Worner\ E[\ Wild\ C[\ Muller!Sebert\ W[\ Locher\ R[\ Koidl\ P[\ Thermal conductivity of CVD diamond _lms] high! precision\ temperature!resolved measurements[ Diamond Relat[ Mater[\ 0885\ 4\ 577Ð581[ 16[ Wagner\ C[\ Krotz\ G[\ Thermal properties of b!SiC epitaxial layers between 049>C and 499>C measured by using microstructures[ Diamond Relat[ Mater[\ 0886\ 5\ 0227Ð0230[ 17[ Parker\ W[ J[\ Jenkins\ R[ J[\ Butler\ C[ P[\ Abbott\ G[ L[\ Flash method for determining thermal di}usivity\ heat capacity and thermal conductivity[ J[ Appl[ Phys[\ 0850\ 21\ 0568Ð0573[ 18[ Balageas\ D[ L[\ Thermal di}usivity measurement by pulsed methods[ Hi`h TemperaturesÐHi`h Pressure\ 0878\ 10\ 74Ð85[ 29[ Cielo\ P[\ Utracki\ L[ A[\ Lamontagne\ M[\ Thermal di}usivity measurements by the converging!thermal!wave technique[ Can[ J[ Phys[\ 0875\ 53\ 0061Ð0066[ 20[ Eichler\ H[ J[\ Guenter\ P[\ and Pohl\ D[ W[\ Laser!induced dynamic gratings[ In Sprin`er Ser[ Opt[ Sci[\ Vol[ 49[ Springer\ Berlin\ 0875[ 21[ Kading\ O[ W[\ Skurk\ H[\ Maznev\ A[ A[\ Matthias\ E[\ Transient thermal gratings ar surfaces for thermal characterization of bulk materials and thin _lms[ Appl[ Phys[ A\ 0884\ 50\ 142Ð150[ 22[ Marshall\ C[ D[\ Fishman\ I[ M[\ Fayer\ M[ D[\ Ultrasonic wave propagation and barrier!limited heat ~ow in thin _lms of YBa1Cu2O6−x[ Phys[ Rev[ B\ 0889\ 32\ 1585Ð1588[ 23[ Graebner\ J[ E[\ Ralchenko\ V[ G[\ Smolin\ A[ A[\ Obraztsova\ E[ D[\ Korotushenko\ K[ G[\ Konov\ V[ I[\ Thermal conductivity of thin diamond _lms grown from d[c[ discharge[ Diamond Relat[ Mater[\ 0885\ 4\ 582Ð587[

In~uence of orderÐdisorder transition on thermal conductivity of solids

1986

24[ Adams\ M[ J[\ Kirkbright\ G[ F[\ Thermal di}usivity and thickness measurements for solid samples utilising the optoacoustic e}ect[ Analyst\ 0866\ 091\ 567Ð571[ 25[ Lachaine\ A[\ Poulet\ P[\ Photoacoustic measurement of thermal properties of polyester _lm[ Appl[ Phys[ Lett[\ 0873\ 34\ 842Ð843[ 26[ Leite\ N[ F[\ Cella\ N[\ Vargas\ H[\ Miranda\ L[ C[ M[\ Photoacoustic measurement of thermal di}usivity of polymer foils[ J[ Appl[ Phys[\ 0876\ 50\ 2914Ð2916[ 27[ Zhang\ Z[ C[\ Roger\ J[ P[\ Fournier\ D[\ Boccara\ A[ C[\ Wang\ J[ C[\ Thermal di}usivity of amorphous semiconductor superlattice _lms[ Thin Solid Films\ 0889\ 075\ 250Ð255[ 28[ Bodzenta\ J[\ Mazur\ J[\ Bukowski\ R[\ Kleszczewski\ Z[\ Photothermal measurements for plates[ Proc[ SPIE\ 0884\ 1532\ 175Ð181[ 39[ Bodzenta\ J[\ Mazur\ J[\ Kleszczewski\ Z[\ Photothermal measurements of thermal conductivity of thin amorphous _lms[ J[ Chem[ Vap[ Deposit[\ 0886\ 4\ 177Ð186[ 30[ Hurler\ W[\ Pietralla\ M[\ Hammerschmidt\ A[\ Determination of thermal properties of hydrogenated amorphous carbon _lms via mirage e}ect measurements[ Diamond Relat[ Mater[\ 0884\ 3\ 843Ð846[ 31[ Kuo\ P[ K[\ Lin\ M[ J[\ Reyes\ C[ B[\ Favro\ L[ D[\ Thomas\ R[ L[\ Kim\ D[ S[\ Zhang\ S[!Y[\ Inglehart\ L[ J[\ Fournier\ D[\ Boccara\ A[ C[\ Mirage!e}ect measurement of thermal di}usivity\ Part I] Experiment[ Can[ J[ Phys[\ 0875\ 53\ 0054Ð0056[ 32[ Kuo\ P[ K[\ Sendler\ E[ D[\ Favro\ L[ D[\ Thomas\ R[ L[\ Mirage!e}ect measurement of thermal di}usivity\ Part II] Theory[ Can[ J[ Phys[\ 0875\ 53\ 0057Ð0060[ 33[ Figari\ A[\ Analytical relations between the phase of the photothermal signal and the thermal wavelength[ J[ Appl[ Phys[\ 0881\ 60\ 2027Ð2031[ 34[ Rantala\ J[\ Wei\ L[\ Kuo\ P[ K[\ Jaarinen\ J[\ Luukkala\ M[\ Thomas\ R[ L[\ Determination of thermal!di}usivity materials using the mirage method with multiparameter _tting[ J[ Appl[ Phys[\ 0882\ 62\ 1603Ð1612[ 35[ Salazar\ A[\ Sanchez!Lavega\ A[\ Thermal di}usivity measurements using linear relations from photothermal wave experiments[ Rev[ Sci[ Instrum[\ 0883\ 54\ 1785Ð1899[ 36[ Cahill\ D[ G[\ Thermal conductivity measurement from 29 to 649 K] the 2v method[ Rev[ Sci[ Instrum[\ 0889\ 50\ 791Ð797[ 37[ Anthony\ T[ R[\ Fleischer\ J[ L[\ Olson\ J[ R[\ Cahill\ D[ G[\ The thermal conductivity of isotopically enriched polycrystalline diamond _lms[ J[ Appl[ Phys[\ 0880\ 58\ 7011Ð7014[ 38[ Lee\ S[!M[\ Cahill\ D[ G[\ Heat transport in thin dielectric _lms[ J[ Appl[ Phys[\ 0886\ 70\ 1489Ð1484[ 49[ Anthony\ T[ R[\ Banholzer\ W[ F[\ Fleischer\ J[ F[\ Wei\ L[\ Kuo\ P[ K[\ Thomas\ R[ L[\ Pryor\ R[ W[\ Thermal di}usivity of isotopically enriched 01C diamond[ Phys[ Rev[ B\ 0889\ 31\ 0093Ð0000[ 40[ Wei\ L[\ Kuo\ P[ K[ and Thomas\ R[ L[\ Thermal di}usivity measurement of diamond materials[ In Sprin`er Ser[ Opt[ Sci[\ Vol[ 58[ Springer\ Berlin\ 0881\ pp[ 140Ð142[ 41[ Tokmako}\ A[\ Banholzer\ W[ F[\ Fayer\ M[ D[\ Thermal di}usivity of natural and isotopically enriched diamond by picosecond infrared transient grating experiments[ Appl[ Phys[ A\ 0882\ 45\ 76Ð89[ 42[ Graebner\ J[ E[\ Jin\ S[\ Kammlott\ G[ W[\ Herb\ J[ A[\ Gardinier\ C[ F[\ Unusually high thermal conductivity in diamond _lms[ Appl[ Phys[ Lett[\ 0881\ 59\ 0465Ð0467[ 43[ Kading\ O[ W[\ Rosler\ M[\ Zachai\ R[\ Fu)er\ H[!J[\ Matthias\ E[\ Lateral thermal di}usivity of epitaxial diamond _lms[ Diamond Relat[ Mater[\ 0883\ 2\ 0067Ð0071[ 44[ Chen\ Z[\ Mandelis\ A[\ Thermal!di}usivity measurements of ultrahigh thermal conductors with use of scanning photothermal rate!window spectrometry] chemical!vapor!deposition diamonds[ Phys[ Rev[ B\ 0881\ 35\ 02415Ð02427[ 45[ Petrovsky\ A[ N[\ Salnick\ A[ O[\ Mukhin\ D[ O[\ Spitsyn\ B[ V[\ Thermal conductivity measurements of synthetic diamond _lms using the photothermal beam de~ection technique[ Materials Science and En`ineerin`\ 0881\ B00\ 242Ð243[ 46[ Goodson\ K[ E[\ Kading\ O[ W[\ Rosler\ M[\ Zachai\ R[\ Experimental investigation of thermal conduction normal to diamondÐsilicon boundaries[ J[ Appl[ Phys[\ 0884\ 66\ 0274Ð0281[ 47[ Collins\ A[ K[\ Pickering\ M[ A[\ Taylor\ R[ L[\ Grain size dependence of the thermal conductivity of polycrystalline chemical vapor deposited b!SiC at low temperatures[ J[ Appl[ Phys[\ 0889\ 57\ 5409Ð5401[ 48[ Yamane\ T[\ Mori\ Y[\ Katayama\ S[!I[\ Todoki\ M[\ Measurement of thermal di}usivities of thin metallic _lms using the a[c[ calorimetric method[ J[ Appl[ Phys[\ 0886\ 71\ 0042Ð0045[ 59[ Bachmann\ P[ K[\ Hagemann\ H[ J[\ Lade\ H[\ Leers\ D[\ Wiechert\ D[ U[\ Wilson\ H[\ Fournier\ D[\ Plamann\ K[\ Thermal properties of C:H!\ C:H:O!\ C:H:N! and C:H:X!grown polycrystalline CVD diamond[ Diamond Relat[ Mater[\ 0884\ 3\ 719Ð715[ 50[ Boudina\ A[\ Fitzer\ E[\ Netzelmann\ U[\ Reiss\ H[\ Thermal di}usivity of diamond _lms synthesized from methane by arc discharge plasma jet CVD[ Diamond Relat[ Mater[\ 0881\ 1\ 741Ð747[ 51[ Graebner\ J[ E[\ Mucha\ J[ A[\ Baiocchi\ F[ A[\ Sources of the thermal resistance in chemically vapor deposited diamond[ Diamond Relat[ Mater\ 0885\ 4\ 571Ð576[ 52[ Verhoeven\ H[\ Hartmann\ J[\ Rechling\ M[\ Muller!Sebert\ W[\ Zachai\ R[\ Structural limitations to local thermal di}usivities of diamond _lms[ Diamond Relat[ Mater[\ 0885\ 4\ 0901Ð0905[ 53[ McNamara Rutledge\ K[ M[\ Scruggs\ B[ E[\ Gleason\ K[ K[\ In~uence of hydrogenated defects and voids on the thermal conductivity of polycrystalline diamond[ J[ Appl[ Phys[\ 0884\ 66\ 0348Ð0351[ 54[ Nepsha\ V[ I[\ Grinberg\ V[ R[\ Klyuev\ Yu[ A[\ Kolchemanv\ N[ A[\ Naletov\ A[ M[\ Thermal conductivity and structure of diamond _lms[ Surface and Coatin`s Technolo`y\ 0880\ 36\ 277Ð280[ 55[ Hartmann\ J[\ Reichling\ M[\ Matthias\ E[\ Thermal resistance of thermal barriers in polycrystalline diamond[ Pro`ress in Natural Science\ 0885\ 5\ S186ÐS299\ "Suppl[#[ 56[ Reichling\ M[\ Microscopic heat transfer in high quality polycrystalline diamond[ In Proceedin`s of 15th Winter School on Molecular and Quantum Acoustic\ Ustron\ 0886\ pp[ 76Ð84[

1987

J[ BODZENTA

57[ Lee\ S[!M[\ Cahill\ D[ G[\ Allen\ T[ H[\ Thermal conductivity of sputtered oxide _lms[ Phys[ Rev[ B\ 0884\ 41\ 142Ð 146[ 58[ Cahill\ D[ G[\ Katiyar\ M[\ Abelson\ J[ R[\ Thermal conductivity of a!Si]H thin _lms[ Phys[ Rev[ B\ 0883\ 49\ 5966Ð 5970[ 69[ Morath\ Ch[ J[\ Maris\ H[ J[\ Cuomo\ J[ J[\ Pappas\ D[ L[\ Grill\ A[\ Patel\ V[ V[\ Doyle\ J[ P[\ Saenger\ K[ L[\ Picosecond optical studies of amorphous diamond and diamondlike carbon] Thermal conductivity and longitudinal sound velocity[ J[ Appl[ Phys[\ 0883\ 65\ 1525Ð1539[ 60[ Morelli\ D[ T[\ Perry\ T[ A[\ Vandersande\ J[ W[\ Uher\ C[\ Glasslike thermal transport in heavily irradiated diamond[ Phys[ Rev[ B\ 0882\ 37\ 2926Ð2930[ 61[ Mazur\ J[\ Bodzenta\ J[\ Kleszczewski\ Z[\ Photothermal measurements of the thermal conductivity of superhard thin _lms[ In Diamond Based Composites\ ed[ M[ A[ Prelas et al[ NATO ASI Series 2\ Hi`h Technolo`y\ Vol[ 27\ 0886\ pp[ 150Ð156[