Review Article Prediction of Spectral Phonon Mean Free Path ...obtained the phonon relaxation times...

Review ArticlePrediction of Spectral Phonon Mean Free Path andThermal Conductivity with Applications to Thermoelectricsand Thermal Management A Review

Tianli Feng and Xiulin Ruan

School of Mechanical Engineering and the Birck Nanotechnology Center Purdue University West Lafayette IN 47907-2088 USA

Correspondence should be addressed to Xiulin Ruan ruanpurdueedu

Received 12 November 2013 Accepted 16 January 2014 Published 31 March 2014

Academic Editor Urszula Narkiewicz

Copyright copy 2014 T Feng and X RuanThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We give a review of the theoretical approaches for predicting spectral phonon mean free path and thermal conductivity of solidsThemethods can be summarized into two categories anharmonic lattice dynamics calculation andmolecular dynamics simulationIn the anharmonic lattice dynamics calculation the anharmonic force constants are used first to calculate the phonon scatteringrates and then the Boltzmann transport equations are solved using either standard single mode relaxation time approximationor the Iterative Scheme method for the thermal conductivity The MD method involves the time domain or frequency domainnormal mode analysis We present the theoretical frameworks of the methods for the prediction of phonon dispersion spectralphonon relaxation time and thermal conductivity of pure bulk materials layer and tube structures nanowires defective materialsand superlattices Several examples of their applications in thermal management and thermoelectric materials are given Thestrength and limitations of these methods are compared in several different aspects For more efficient and accurate predictionsthe improvements of those methods are still needed

1 Introduction

In recent years increasing attention has been focused onseeking novel structures and materials with desired thermalproperties especially thermal conductivity High thermalconductivity can help remove heat rapidly and reduce devicetemperatures so as to improve performance of nanoelectron-ics and optoelectronics while low thermal conductivity isdesired in thermoelectrics for improving the figures of merit119885119879 [1] of the material119885119879 = 119878

2120590119879119896 where 119878 120590 119879 and 119896 are

Seebeck coefficient electronic conductivity temperature andthermal conductivity respectively The thermal conductivity119896 is a summation of the lattice contribution 119896

119897and electron

contribution 119896119890 Since in most thermoelectric materials

the phonon mean free path is much longer than that ofelectrons one major strategy to enhance 119885119879 is to reduce 119896

119897

without much affecting 119896119890 This is made possible by the rapid

development of nanofabrication techniquesGaining a deeper physical insight into the spectral

phonon properties for example the spectral phonon

relaxation time and mean free path is necessary to correctlyexplain experimental results and accurately predict andguide the further designs and applications Analyticalmodels have been used by Balandin and Wang to estimatefrequency-dependent phonon group velocity and variousphonon scattering rates including phonon-phonon phonon-impurity and phonon-boundary scattering processes Theyused this approach to observe the strong modification ofacoustic phonon group velocity and enhanced phononscattering rate due to boundary scattering in semiconductorquantumwells so as to successfully explain their significantlyreduced lattice thermal conductivity [2] This effect ofphonon confinement was then extended to nanowiresand quantum dot superlattices [3ndash5] Analytical modelsof spectral phonon properties are advantageous in theirclear physical insights but they usually contain empiricalfitting parameters and this limitation has motivated thedevelopment of numerical methods based on first principlesand molecular dynamics that can predict these spectralproperties from their atomic structure without fitting

Hindawi Publishing CorporationJournal of NanomaterialsVolume 2014 Article ID 206370 25 pageshttpdxdoiorg1011552014206370

2 Journal of Nanomaterials

parameters and with greater accuracy This review will befocused on these predictive simulation methods

The methods of predicting spectral phonon relaxationtimes and mean free paths become increasingly importantfor predicting the thermal properties of numerous novelmaterials For instance superlattice structure is found tobe an effective way to suppress the thermal conductivitybecause of the interface mass mismatch scattering [6ndash8] butthe phenomenon in which the short-period superlattice canhave even higher thermal conductivity still needs the deepinsight of phonon relaxation time Doping and alloying arewidely used to explore novel high performance materials[9] and the natural materials are rarely pure thus the studyof impurity scattering contributing to phonon relaxationtime is important Layer and tube structured materials forexample graphene [10ndash13] and carbon nanotube (CNT) [14ndash17] are proved to have unusual phonon transport featuressuch as high thermal conductivity [18ndash20] which still needto be further understood Comprehensive reviews of theirthermal transport can be found in [21ndash23] Nanowires aremost commonly studied and used in both theoretical andexperimental research [24ndash32] and the accurate predictionof thermal conductivity needs the knowledge of spectralphonon scattering by boundaries

Manymethods have been proposed and applied to predictspectral phonon relaxation time in the last half centuryAt the earliest Klemens and other researchers obtainedthe frequency-dependent phonon relaxation time mostly bylong-wave approximation (LWA) andDeybemodel Klemensobtained the phonon relaxation times by Umklapp (119880)three-phonon scattering [33 34] and defect scattering [35]Herring studied normal (119873) three-phonon scattering [36]Holland extended the results to dispersive transverse moderange [37] and Casimir studied boundary scattering [38ndash40] A more accurate method the third-order anharmoniclattice dynamics (ALD) calculation which can predict theintrinsic spectral phonon relaxation times without LWAwas presented by Maradudin and the coworkers [41 42]ALD methods were then applied to silicon and germaniumby ab initio approach first by Debernardi et al [43] andDeinzer et al [44] Beyond the standard ALD calculationOmini and Sparavigna [45 46] proposed an Iterative Schemewhich gives exact solutions to the linearized Boltzmanntransport equation (BTE) The Iterative Scheme has beensuccessfully applied tomany structures in the recent ten yearsby Broido Lindsay Ward and so forth [47ndash65] Other thanthe lattice dynamics calculation a time domain normalmodeanalysis (NMA)method based onmolecular dynamics (MD)simulation was proposed by Ladd et al [66] and extendedby McGaughey and Kaviany [67] Another version of normalmode analysis is implemented in frequency domain so calledspectral energy density (SED) analysis The normal modeanalysis was early implemented by Wang et al [68ndash75] toobtain the relaxation times of a few phonon modes and thenextended by de Koker [76] and Thomas et al [77 78] tocalculate lattice thermal conductivity

In this work we present a review of the methods ofpredicting spectral phonon properties discuss the applica-tions to each method and compare them in different aspects

Section 2 gives an overview of thermal conductivity and thefrequency-dependent relaxation time predicted from earlylong-wave approximation (LWA) andDebyemodel Section 3presents the ALD calculation which is divided into threesubsections Section 31 covers the standard single moderelaxation time approximation (SMRTA) Section 32 givesthe Iterative Scheme and Section 33 reviews examples ofthe applications to pure bulk layer and tube structuresnanowires defective materials and superlattice In Section 4we introduce the time domain NMA and frequency NMAmethods based on MD simulations and their applicationsThe summary is presented in Section 5 The appendix pro-vides some derivations of ALD methods

2 Theory Overview

Spectral phonon mean free path (MFP) determined byphonon scattering rate dominates the behavior of thermalproperties especially the thermal conductivity 119896 Based onBTEunder the relaxation time approximation (RTA) thermalconductivity is determined by the spectral phonon relaxationtime 120591

120582 phonon group velocity k

120582= nablak120596 and phonon

specific heat 119888120582[34]

119896119911=1

119881sum

120582

(k120582sdot )

2

119888120582120591120582 (1)

where denotes the transport direction 120582 is the shorthandof phonon mode (k ]) with k representing the phonon wavevector and ] labeling phonon dispersion branch 119881 is thevolume of the domain and the summation is done over theresolvable phononmodes in the domainThe specific heat permode is 119888

120582= ℏ120596

120582120597119899

0

120582120597119879 = 119896

119861119909

2119890119909(119890

119909minus 1)

2 where 1198990

120582is

phonon occupation number of the Bose-Einstein distribution1198990

120582= (119890

119909minus 1)

minus1 and 119909 is the shorthand of ℏ120596119896119861119879 Equation

(1) can also be expressed in terms of phonon mean free pathrarr

Λ 120582= k

120582120591120582 The continuous form of (1) is with the help of

sumk = (119881(2120587)3) int 119889k

119896119911=

1

(2120587)3sum

]int (k

120582sdot )

2

119888120582120591120582119889k (2)

If isotropic heat transport is assumed the integration of|V

120582119911|2 in (2) gives V2

1205823 and int119889k gives int 41205871198962

119889119896 we get thecommonly used formula

119896119911=4120587

3

1

(2120587)3sum

]int 119888

120582V120582Λ

1205821198962119889119896 (3)

The early theoretical predictions of phonon relaxationtimes for different scattering processes are briefly summa-rized in Table 1 [37 79] 119879 is temperature subscripts 119873 119880119879 and 119871 indicate the Umklapp scattering normal scatteringtransverse wave and longitudinal wave respectively 119860 119861rsquosand 119862rsquos are constants 120579 is Debye temperature 120572 is numericalconstant in [34] Low119879means119879 ≪ 120579 and high119879means119879 ≫

120579 1205961is the transverse mode frequency at which the group

velocity starts to decrease and 1205962is the maximum transverse

Journal of Nanomaterials 3

Table 1 Analytical models of inverse relaxation time for differentscattering processes

Scattering process Inverse relaxation timeIntrinsic Three-Phonon119873 process

Herringa 120591minus1

119871119873= 119861

119871120596

2119879

3 low 119879

120591minus1

119879119873= 119861

119879120596119879

4 low 119879

120591minus1

119871119873= 119861

1015840

119871120596

2119879 high 119879

120591minus1

119879119873= 119861

1015840

119879120596119879 high 119879

Callawayb 120591minus1

119873= 119861

119873120596

2119879

3

119880 process

Klemensc 120591minus1

119880= 119861

119880120596

2119879

3 exp(minus 120579

120572119879) low 119879

Klemensd 120591minus1

119880= 119861

119880120596119879

3 exp(minus 120579

120572119879) low 119879

120591minus1

119880= 119861

1015840120596

2119879 high 119879


119880= 119861

119880120596

2119879

3

Hollande 120591minus1

119879119880=

119861119879119880120596

2

sinh (119909) 120596

1le 120596 le 120596

2

0 120596 lt 1205961

Asen-Palmer et alf 120591minus1

119879119880= 119861

1015840

119879119880120596

2119879 exp(119862119879

119879)

120591minus1

119871119880= 119861

1015840

119871119880120596

2119879 exp(119862119871

119879)

Boundaryg 120591minus1

119887=V120582

119897119865119897 diameter 119897 = 2radic119897

11198972120587

119865 surface roughmess

Impurityh 120591minus1

im =120587

2119892120596

2119863(120596) asymp

1198810119892

4120587 ⟨V3⟩120596

4sim 119860120596

4

References a[36] b[80] c[33] d[34] e[37] f[81] g[38ndash40] and h[35]

frequency The intrinsic three-phonon scattering rates arederived mostly in LWA or linear dispersion approximation

Boundary scattering 120591minus1

119887exists anywhere since every

sample has a finite size 119865 captures the boundary scatteringcharacteristic of the sample with 119865 = 1 representingcompletely diffusive and 119865 rarr infin meaning specular 119897 =

2radic119878119888120587 is ameasure of the size perpendicular to the transport

direction with 119878119888being the area of cross section V

120582is often

replaced by the average phonon speed of the three acousticbranches Vave for simplicity [37]

Vave = [1

3

3

sum

]

1

V]]

minus1

= [1

3(2

V119879

+1

V119871

)]

minus1

(4)

The last equation in Table 1 takes into account the impu-rity scattering rate where

119892 = sum

119894

119891119894(1 minus

119898119894

119898)

2

(5)

is a measure of mass disorder 119863(120596) is phonon density ofstates normalized to unity 119891

119894is the concentration of the

impurity species 119894 and 119898119894and 119898 are the mass of 119894 and

average mass for the given composition respectively Theexact expression for ⟨V3⟩ is found in [82] while in longwave approximation ⟨V3⟩ approximates the cube of acoustic

phonon speed of the material This equation was derived byKlemens for isotope scattering with only mass disorder Forcrystal defects other than isotope doping such as vacancyinterstitial and antisite defects the impurity scattering comesfromnot only themass disorder but also the interatomic forcechange and link break Klemens took into account such effectby adding a modification to 119892

119892 = 119892 + 2sum

119894

119891119894(Δ120601

119894

120601minus64120574Δ119903

119894

119903)

2

(6)

where 120601119894120601 and Δ119903

119894119903 describe the average relative variations

of the local force constants and atomic displacements [35 83ndash85] respectively Some consider the dislocations by adding ascattering term 120591

minus1

119863sim 120596 to the total phonon scattering rate

[84] predicted from single dislocation assumption by [34 3586 87] Although 120591

minus1

im = 1198601205964 is derived for low frequency

phonons many works use it to predict thermal conductivityor explain data from experiments for alloys and crystals withimpurities [24ndash26 37 81 84 85 88ndash91] In Section 334 wewill give more precise expressions for isotope scattering

For the system that contains several scattering mecha-nisms the Matthiessen rule is often used to evaluate the totalscattering rate

120591minus1= sum

119894

120591minus1

119894 (7)

In most cases the Matthiessen rule gives reasonable resultsalthough it is found to be not accurate in some cases recently[58 92 93]

These frequency dependent relaxation time expressionsin Table 1 have been used in many works for thermalconductivity prediction and analysis and the choice of thoseexpressions looks quite arbitrary For instance in the choiceof intrinsic phonon relaxation time in the thermal conductiv-ity analysis of silicon Glassbrenner and Slack [94] used 120591minus1

sim

1205962119879 while Asen-Palmer et al [81] and Mingo et al [24 25]

used 120591minus1sim 120596

2119879 exp(119862119879) for all phononmodes Martin et al

[26] used 120591minus1

119871sim 120596

2119879

3 for longitudinal mode while Hollandadded 120591minus1

119879sim 120596

2 sinh(119909) to dispersive transverse range The

thermal conductivity results predicted by these expressionscan be reasonable due to the adjustable fitting parametersTherefore it becomes important to accurately predict spectralphonon relaxation time without any fitting parameter whichallows us to understand thermal transport and examine (a)the validity of low-frequency approximation or the Debyemodel (b) the importance of optical branch to thermal trans-port (c) the contributions of phonons with different meanfree path or different wavelength to thermal conductivity (d)the relative importance of different scattering mechanisms ina given material and so forth

3 Anharmonic Lattice Dynamics Methods

In perturbation theory the steady-state phonon BTE [3479 95] describes the balance of phonon population betweendiffusive drift and scattering as

k120582sdot nabla119899

120582=120597119899

120582

120597119905

10038161003816100381610038161003816100381610038161003816119904

(8)


where 119899120582= 119899

0

120582+ 119899

1015840

120582is the total phonon occupation number

with 1198991015840

120582representing the deviation from the equilibrium

phonon distribution 1198990

120582 With nabla119899

120582= (120597119899

120582120597119879)nabla119879 and

assuming that 1198991015840

120582is independent of temperature (120597119899

120582120597119879) ≃

(1205971198990

120582120597119879) we have


1205971198990

120582

120597119879=120597119899

1015840

120582

120597119905

100381610038161003816100381610038161003816100381610038161003816119904

(9)

The RTA assumes that deviation of single phonon modepopulation decays exponentially with time

1198991015840

120582sim exp(minus 119905

120591120582

) (10)

where 120591120582is the relaxation time Therefore the collision term

in BTE (9) becomes

1205971198991015840

120582

120597119905

100381610038161003816100381610038161003816100381610038161003816119904

≃ minus1198991015840

120582

120591120582

(11)

Generally the value of 120591120582is considered as the average time

between collisions of the phonon mode 120582 with other modeswhereby 120591

120582= 1Γ

120582 where Γ

120582denotes the scatting rate

Considering only three-phonon scattering (9) becomes[95]


1205971198990

120582

120597119879

= minussum

120582101584012058210158401015840

[1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840]L

+

+1

2[119899

120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840]L

minus

(12)

where the summation is done over all the phonon modes 1205821015840

and 12058210158401015840 that obey the energy conservation 120596

120582plusmn 120596

1205821015840 = 120596

12058210158401015840

and quasimomentum conservation k plusmn k1015840= k10158401015840

+G withG =

0 for 119873 processes and G = 0 for 119880 processes where G is areciprocal-lattice vectorL

plusmnis the probability of 120582plusmn1205821015840

rarr 12058210158401015840

scattering occurrence determined via Fermirsquos golden rule

Lplusmn=

ℏ120587

41198730

10038161003816100381610038161003816119881

(3)

plusmn

10038161003816100381610038161003816

2 120575 (120596120582plusmn 120596

1205821015840 minus 120596

12058210158401015840)

120596120582120596

1205821015840120596

12058210158401015840

(13)

119881(3)

plusmn= sum

119887119897101584011988710158401198971015840101584011988710158401015840

sum

120572120573120574

Φ120572120573120574

0119887119897101584011988710158401198971015840101584011988710158401015840

119890120582

120572119887119890plusmn1205821015840

1205731198871015840 119890

minus12058210158401015840

12057411988710158401015840

radic119898119887119898

1198871015840119898

11988710158401015840

119890plusmn119894k1015840 sdotr

1198971015840

119890minus119894k10158401015840sdotr

11989710158401015840

(14)

where 119887rsquos and 119897rsquos are the indexes of basis atoms and unit cellsrespectively 120572 120573 and 120574 represent coordinate directions 119898

119887

is the mass of basis atom 119887 considering that some dopingmaterial 119898

119887is the average mass in the 119887th basis sites 119890120582

119887120572

is the 120572 component of the 119887th part of the mode 120582 =

(k ])rsquos eigenvector andΦ is the third-order interatomic forceconstant (IFC) The factor ldquo12rdquo in (12) accounts for the

double counting in the summation of 1205821015840 and 12058210158401015840 for the ldquominusrdquoprocess In (14) the factor 119890119894ksdotr119897 is often omitted since it is aconstant in the summation and thus contributes nothing to|119881

(3)

plusmn|2

31 Standard Single Mode Relaxation Time ApproximationThe Standard SMRTA assumes that the system is in itscomplete thermal equilibrium except that one phononmode120582 has its occupation number 119899

120582= 119899

0

120582+ 119899

1015840

120582differing a

small amount from its equilibrium value 1198990

120582 Therefore on

the right hand side of (12) replacing 119899120582by 1198990

120582+ 119899

1015840

120582 whilst

1198991205821015840 and 119899

12058210158401015840 by 1198990

1205821015840 and 119899

0

12058210158401015840 respectively one can obtain the

phonon relaxation time 1205910

120582of mode 120582 (for the derivation see

Appendix A1)

1

1205910

120582

=

+

sum

120582101584012058210158401015840

Γ+

120582120582101584012058210158401015840 +

1

2

minus

sum

120582101584012058210158401015840

Γminus

120582120582101584012058210158401015840 +sum

1205821015840

Γext1205821205821015840 (15)

where the first two terms on the right hand side are intrinsicthree-phonon scattering rates (120596

120582plusmn 120596

1205821015840 = 120596

12058210158401015840)

Γplusmn

120582120582101584012058210158401015840 =

ℏ120587

41198730

1198990

1205821015840 minus 119899

0

12058210158401015840

1198990

1205821015840 + 119899

0

12058210158401015840 + 1

times10038161003816100381610038161003816119881

(3)

plusmn

10038161003816100381610038161003816

2 120575 (120596120582plusmn 120596

1205821015840 minus 120596

12058210158401015840)

120596120582120596

1205821015840120596

12058210158401015840

(16)

The last term Γext1205821205821015840 represents the extrinsic scattering such as

boundary scattering and impurity scattering

32 Iterative Scheme Exact Solution to Linearized BTEDifferent from the Standard SMRTA the other method tosolve the phonon BTE allows all the modes to be in theirthermal nonequilibrium states at the same time By replacingthe occupation numbers 119899

120582 119899

1205821015840 and 119899

12058210158401015840 by 1198990

120582+ 119899

1015840

120582 1198990

1205821015840 + 119899

1015840

1205821015840

and 1198990

12058210158401015840 + 119899

1015840

12058210158401015840 respectively on the right hand side of (12) the

relaxation time 120591120582of mode 120582 is obtained (for the derivation

see Appendix A2)

120591120582= 120591

0

120582(1 + Δ

120582) (17)

Δ120582=

(+)

sum

120582101584012058210158401015840

Γ+

120582120582101584012058210158401015840 (1205851205821205821015840101584012059112058210158401015840 minus 12058512058212058210158401205911205821015840)

+

(minus)

sum

120582101584012058210158401015840

1

2Γminus

120582120582101584012058210158401015840 (1205851205821205821015840101584012059112058210158401015840 + 12058512058212058210158401205911205821015840)

+sum

1205821015840

Γext120582120582101584012058512058212058210158401205911205821015840

(18)

where 1205851205821205821015840 = V

1205821015840119911120596

1205821015840V

120582119911120596

120582 and V

119911is phonon group velocity

component along the transport directionEquation (17) is solved iteratively because both the left

and the right hand sides contain the unknown variable 120591120582

and thus the method is called Iterative Scheme This schemeis also based on RTA thus (10) and (11) are still valid (one canreach this by substituting (A3) (A4) (A11) (A12) (A13)and (A14) into (9)) The last summation in (18) is done over120582

1015840 with 1205821015840= 120582


33 Discussions and Applications ALD methods can bedivided into classical method and ab initiomethod differingin how to calculate the harmonic and anharmonic IFCswhich are the only inputs to these methods The classicalapproach relies on empirical interatomic potential whose 119899thorder derivatives are taken as the 119899th order IFCs

Φ1205721sdotsdotsdot120572119899

11989711198871119897119899119887119899

=120597119899Φ

1205971199061(119897

11198871) 120597119906

119899(119897

119899119887119899) (19)

In contrast the ab initio approach is a first principle calcu-lation in the framework of density functional perturbationtheory (DFPT) [43 96 97] using norm-conserving pseu-dopotentials in the local density approximation (LDA) with-out introducing any adjustable parameters The formulism ofthe IFCs using first principle method can be found in [44]and realized by for example the QUANTUM ESPRESSOpackage [98] Compared to the classical method this methodcan deal with new materials whose empirical interatomicpotentials are unknown Further this method can be moreaccurate since the empirical interatomic potentials cannotalways represent the exact nature of interatomic force

In (16) the delta function 120575(120596 plusmn 1205961015840minus 120596

10158401015840) is typically

approximated by 120575(119909) = lim120576rarr0+

(1120587)(120576(1199092+ 120576

2)) To

accurately evaluate (16) the choice of 120576 value is critical itmust be small but larger than the smallest increment indiscrete 119909 which results from the use of finite grid of 119896 pointsin Brillouin zone The general practice is as follows pickthe densest grid possible and start with a sufficiently smallguess and increase it gradually until the final results reachconvergence

To calculate the relaxation time one can use StandardSMRTA scheme [43 44 99ndash111] or Iterative Scheme [47ndash65 112] and in each of them one can choose empiricalinteratomic potential approach [47ndash49 52ndash56 58ndash60 99ndash102] or ab initio-derived IFC IFC [43 44 50 51 57 61ndash65103ndash112] The methods can be used on pure bulk nanowiresdoped bulk doped nanowires alloys and so forth

One way to predict thermal conductivity 119896 withoutworking out all the phonon modes relaxation times is theMonte Carlo integration technique [101 113] The protocolof this technique is as follows (1) randomly sample somephonon modes 120582 (2) for each of these modes randomlychoose two other modes 1205821015840 and 120582

10158401015840 that interact with 120582 tocalculate the relaxation time and (3) select as many points asnecessary to ensure that the statistical error is small enoughin both cases Monte Carlo technique only works for theStandard SMRTA scheme since the Iterative Scheme requiresthe relaxation times of all the phonon modes to do iterationMonte Carlo technique reduces the computational cost butlowers the accuracy

In addition to intrinsic phonon scattering Γplusmn extrinsic

scattering 1120591ext120582

plays an important role in nanostructures

1

120591ext120582

= sum

1205821015840

Γext1205821205821015840 (20)

such as boundary scattering 1120591bs120582

and impurity scattering1120591

imp120582

1000

800

600

400

200

001 1

120596 (THz)

Erro

r (

)

Rela

xatio

n tim

e (s

)

10minus7

10minus8

10minus9

10minus10

10minus11

10minus12

120591(U)(120596)

120591(N)(120596)120591(0)(120596)

0 10 20 30 40 50 60 70 80

120596eff (THz)

Figure 1 Percent error (color online) in |119881(3)

plusmn|2 from the LWA

compared to first principle for silicon at 300K Insert showsthe normal (blue dashed curve) Umklapp (green dotted curve)and total (red solid curve) relaxation times for the LA phononscalculated from Standard SMRTA by ab initio approach Adaptedwith permission from [110] Copyrighted by the American PhysicalSociety

331 Intrinsic Phonon Scattering Bulk Materials Withoutany fitting parameters Standard SMRTA with ab initioapproach can accurately predict spectral phonon relaxationtimes and thermal conductivities Ward and Broido [110]checked the validity of some old approximations introducedin Section 2 (1) long-wave approximation for three-phononscattering and (2) ignoring optical phonons using siliconand germanium as examples First the values of matrixelement |119881(3)

plusmn|2 which govern the scattering strength Γ from

ab initio calculation for acoustic phonons are compared tothose given by LWAThe percentage error of |119881(3)

plusmn|2 is shown

in Figure 1 We note that the LWA only works for the verylow frequency 120596eff lt 08THz while for most part 08 lt

120596eff lt 12THz the LWA gives large discrepancy where120596eff equiv (120596

120582120596

1205821015840120596

12058210158401015840)

13 is the geometric average of the three-phonon frequencies Second the relaxation times of opticalmodes are found to only contribute less than 10 to the totalthermal conductivity of silicon However ignoring opticalmodes is erroneous since the optical phonons are essential toprovide channels for acoustic phonon scatteringThe explicitcalculation of millions of three-phonon scattering shows thatoptical phonons are involved in 50ndash60 of the total acousticphonon-scattering processes in Si and Ge Last beyond the120596 119879 dependencies of 120591 listed in Table 1 which rely on manyapproximations the ALD calculation can give more precise120591 120596 119879 dependenceThis is illustrated in the inset of Figure 1the relaxation times of the LA phonons in Si for 119879 = 300KBy decomposing the total scattering into 119880 process and 119873

process we find the 119880 process has a stronger frequency


dependence 120591(119880)(120596) sim 120596

minus4 than 119873 process 120591(119873)(120596) sim 120596

minus2The results also show that normal scattering governs thetotal relaxation time 120591(0)

(120596) at low frequency while Umklappscattering dominates at high frequency Such sim120596minus4 relation isnot expected in the analytical models in Table 1

One flaw of the Standard SMRTA is that it does notgrasp the interplay between the 119873 process and 119880 processThe right hand side of (15) can be decomposed as Γ(119873)

120582+

Γ(119880)

120582(only consider intrinsic phonon scattering) according

to whether they are 119873 or 119880 scattering events The StandardSMRTA scheme treats the 119873 process and 119880 process as twoindependent scattering events and use Matthiessenrsquos rule toaccount for the total relaxation time 11205910

120582= 1120591

(119873)

120582+ 1120591

(119880)

120582

where 120591(119873119880)

120582is defined as 120591(119873119880)

120582equiv 1Γ

(119873119880)

120582 However it is

well know that 119873 process does not contribute to thermalresistance directly Instead it affects the 119880 process (low-frequency 119873 scattering produces high-frequency phononswhich boosts 119880 process) and then the 119880 process producesthermal resistanceThis error can be remedied in the IterativeScheme by doing the iteration in (17)Therefore the StandardSMRTA scheme only works for the system where 119880 processdominates so that the 119873 scattering makes little difference to119880 process as well as to thermal resistance [51 110]

For Si and Ge at room temperature where the 119880 processis strong the thermal conductivity predicted by StandardSMRTA scheme is only 5ndash10 smaller than that by IterativeScheme [110] the latter shows excellent agreement withexperiment (see Figure 1 of [61])

In contrast the 119880 scattering in diamond is much weaker[114ndash116] due to the much smaller phase space [117] As aresult the thermal conductivity given by these two methodscan differ by 50 at room temperature [110] As shown inFigure 2 this discrepancy increases with decreasing tem-perature since the Umklapp scattering is weakened whentemperature decreasesThe thermal conductivity of diamondpredicted by Iterative Scheme with ab initio approach agreesexcellently with experiment as shown in [110] It is also notedthat the Standard SMRTA scheme always underpredictsthe thermal conductivity because it treats 119873 process as anindependent channel for thermal resistance On the otherhand if the relaxation time for 119880 process only is used in thecalculation the thermal conductivity is always overpredictedThis again confirms that the 119873 process has an indirect andpartial contribution to the thermal resistance

One important application of ALD calculation is topredict and understand the thermal conductivity of thermo-electric materials and help to design higher thermoelectricperformance structures Based on first principle calculationShiga et al [104] obtain the frequency-dependent relaxationtimes of pristine PbTe bulk at 300K as shown in Figure 3At low-frequency region TA phonons have longer relaxationtimes than LA phonons with 120591rsquos exhibitingsim120596minus2 dependenceSeparating the scattering rates into those of normal andUmklapp processes they find the relations 120591Normal sim 120596

minus2

and 120591Umklapp sim 120596minus3 which again indicate that the normal

process dominates low-frequency region while the Umklappdominates high-frequency part By further studying the

10000

1000

Error ()

Iterative scheme

Standard SMRTA

80

70

60

50

40

30

20

10

01000500300200

Temperature (K)

er

mal

con

duct

ivity

(Wm

-K)

Erro

r (

)

Figure 2 The calculated intrinsic lattice thermal conductivity ofdiamond for the Standard SMRTA (dashed line) and the IterativeScheme (solid line) both by ab initio approach Dotted line showspercent error of the Standard SMRTA result compared to theIterative Scheme solution Reprinted with permission from [51]Copyrighted by the American Physical Society

participation of each phonon mode to the total scatteringrates they find that the low thermal conductivity of PbTeis attributed to the strong scattering of LA phonons byTO phonons and the small group velocity of TA phononsFigure 4 compares phonon relaxation times of PbTe andPbSe [106] Although the anharmonicity of PbSe is normallyexpected to be larger due to the larger average Gruneisenparameter reported from experiments [121] in this work itis found that for TA mode the relaxation times of PbSeare substantially longer than those of PbTe Surprisingly theoptical phonons are found to contribute as much as 25 forPbSe and 22 for PbTe to the total thermal conductivity atthe temperature range 300ndash700K Motivated by the questionthat phonons with what kind of MFP contribute the most tothe total thermal conductivity the cumulative 119896rsquos as functionsof phonon MFP are calculated by ALD method with firstprinciple approach as shown in Figure 5 Silicon is foundto have phonon MFPs which span 6 orders of magnitude(0ndash106 nm) while the thermal transport in diamond isdominated by the phonon with narrow range of MFP (04ndash2 120583m) It is found that the phonons with MFP below 4 120583mfor silicon 16 120583m for GaAs 120 nm for ZrCoSb 20 nm forPbSe and 10 nm for PbTe contribute 80 of total thermalconductivity GaAsAlAs superlattice is found to have similarphonon MFP with bulk GaAs The curves of the alloyMg

2Si

06Sn

04and its pure phases Mg

2Si and Mg

2Sn cross at

the intermediate MFPs These results provide great guidancefor experimental works For example the PbTe-PbSe alloyswith size of nanoparticle below 10 nm are synthesized andfound to lead to as much as 60 reduction to the thermal


100

10

1

01 1

TATALA

TOTOLO

Rela

xatio

n tim

e (p

s)

120591 prop 120596minus2

Frequency 120596 (THz)

(a)

100

10

1

01 1

Normal

120591 prop 120596minus2


TATALA

TOTOLO

Rela

xatio

n tim

e (p

s)

(b)

100

10

1

01 1

Umklapp

TATALA

TOTOLO

120591 prop 120596minus3


Rela

xatio

n tim

e (p

s)

(c)

Figure 3 (a) Spectral phonon relaxation times of pristine PbTe bulk at 300K by Standard SMRTA schemewith first principle IFCs Relaxationtimes of (b) normal and (c)Umklapp processes respectivelyThe solid lines plot (a) 120591 = 6times10

12120596

minus2 (b) 120591 = 8times1012120596

minus2 and (c) 120591 = 8times1024120596

minus3Reprinted with permission from [104] Copyrighted by the American Physical Society

conductivity which provides large space for improving ZT[122]

332 Single and Few-Layer 2D Materials Nanoribbons andNanotubes For single- and multilayer 2D materials theboundary scattering from the sides perpendicular to thetransport direction ismuchweaker than for 3D systems [123]making the boundary scattering expression in Table 1 unsuit-able Instead when studying single-multilayer graphene(SLGMLG) and graphite [54 55] single-wall carbon nan-otubes (SWCNTs) [52 53] single-multilayer boron nitride(SLBNMLBN) and boron nitride nanotubes (BNNTs) [5658] Lindsay and Broido only consider the boundary scatter-ing from the two ends in the transport direction and showthat

1

120591bs=21003816100381610038161003816k120582

sdot 1003816100381610038161003816

119871

(21)

works well in accounting for the boundary scattering with 119871being the length between boundaries in the transport direc-tion Such formula has been shown to give correct thermalconductivity values of nanotubes [124] and nanoribbons [125]in the ballistic limit (119871 rarr 0) and diffusive limit (119871 rarr infin)

Vibrations in 2D lattices are characterized by two typesof phonons those vibrating in the plane of layer (TA andLA) and those vibrating out of plane so called flexuralphonons (ZA and ZO) Lindsay et al [54] find the selection

rule for all orders in anharmonic phonon-phonon scatteringin the 2D crystals only even numbers (including zero) offlexural phonons can be involved arising from the reflectionsymmetry perpendicular to the plane of layer This selectionrule has forbidden about 60 of both 119873 and 119880 three-phonon scattering phase space of ZA phonons for singlelayer graphene They show that such suppressed scatteringyields long relaxation time and mean free path for ZAphonons leading to ZA phonons contributing most of thethermal conductivity of SLG about 70 at room temperature(another cause being the large density of states and occu-pation number of ZA modes) However this conclusion isstill under debate since this approach does not include thefourth- and higher-order phonon scattering rates which arenot necessarily low since the reflection symmetry allowsmore4-phonon processes than 3-phonon processes Actually themethod of spectral energy analysis based on MD (discussedin Section 4) indicates that only 25ndash30 of the total 119896 iscontributed by ZA mode at room temperature [126ndash128] Itshould be noted that MD has its own drawback of not repro-ducing the Bose-Einstein distribution for graphene phononsat room temperature Hence the discrepancies between thetwo methods still need further study

The selection rule mentioned above does not hold formultilayer graphene twisted graphene graphite (becauseof the interlayer coupling) CNT (due to the curvature)graphene nanoribbon (GNR) (due to boundary scattering)


103

102

101

100

10010minus1

Life

time

(ps)

TA

Frequency (THz)

PbSePbTe

120591 sim 120596minus2

(a)

103

102

101

100

10010minus1

Life

time

(ps)

LA

Frequency (THz)

PbSePbTe

120591 sim 120596minus2

(b)

101100

101

100

10minus1

Life

time

(ps)

TO

Frequency (THz)

PbSePbTe

(c)

101100

101

100

10minus1

Life

time

(ps)

LO

Frequency (THz)

PbSePbTe

(d)

Figure 4 Spectral phonon relaxation times of PbSe bulk (squares) and PbTe bulk (crosses) at 300K by Standard SMRTA scheme with IFCsfrom first principle calculation (a) TA (b) LA (c) TO and (d) LO Reprinted with permission from [106] Copyrighted by the AmericanPhysical Society

substrate-supported graphene (due to scattering with thesubstrate) and defective graphene (due to defective scatter-ing) Therefore the thermal conductivity of these structuresis typically lower than that of single layer graphene andthe contribution of each phonon mode changes [54 129ndash132] In Figure 6 single-layer graphene GNR and SWCNTare compared where graphene has an infinite width andfinite length 119871 SWCNT has a finite diameter 119889 and length119871 and GNR has a finite width 119882 = 120587119889 with artificialperiodic boundary condition applied As expected 119896RTAunderpredicted thermal conductivity SWCNT is found tohave a lower thermal conductivity than graphene with aminimum value of 77 of 119896graphene at a critical diameter 119889 asymp

15 nm From this critical diameter 119896SWCNT increases withincreasing diameter and reaches 90 of 119896graphene at 119889 asymp 4 nmOn the other hand if 119889 goes small enough phonon-phononscattering decreases and the thermal conductivity increasesAt this short limit of 119889 the system becomes more like a 1Dchain which generally has much larger thermal conductivitythan 2D and 3D systems The increasing trend of 119896GNR withdecreasing 119889 comes from the reason that the decrease of thewidth 120587119889 pushes the optical modes to higher frequencies andthus the 119880 scattering by optical phonons becomes weaker

For 2D materials and nanotube structures the119873 scatter-ing is usually strong For example for CNT Lindsay et al[52] find that all the three-acoustic-phonon scatterings are119873


10minus1 100 101 102 103 104 105 106

PbTe05Se05Mg2SiMg2SnMg2Si06Sn04

DiamondSi (iterative)Si (standard)GaAsGaAsAlAs SLPbTe

PbSe

ZrCoSb

Phonon MFP (nm)

Acc

umul

ated

ther

mal

con

duct

ivity

()

100

80

60

40

20

0

Figure 5 The accumulated thermal conductivity of different bulkmaterials as a function of phonon mean free path at room temper-ature calculated from ALD by first principle approach (Diamond[64] Si (iterative) [64] Si (standard) [107] GaAs [118] FaAsAlAs[119] PbTe [106] PbSe [106] PbTe

05Se

05[106] Mg

2Si [63] Mg

2Sn

[63] Mg2Si

06Sn

04[63] ZrCoSb [120])

processes so that the119880 process is respectively weak becauseit must involve optical phonons which are less likely to bethermally excited Thus the Iterative Scheme can be used tolayer and tube structure rather than Standard SMRTAwhoseresults are less accurate Figure 7 shows the ratio betweenthermal conductivities 119896

119871(from Iterative Scheme) and 119896RTA

(from Standard SMRTA) where the discrepancy is typicallylarger than 100The ZAmode shows the largest divergencewhich can reach 8-fold at length of 10 120583m because the flexuralphonons have lower frequencies than other modes and thusstronger119873 process than 119880 process

333 Boundary Scattering Nanowires For nanowires theCasimir model (Table 1) has been applied to predict thermalconductivity in many works [24ndash27 27ndash32 133] recentlyGenerally 119896 decreases with decreasing nanowire diame-ter however at some point as the diameter continues todecrease 119896 will increase due to the 3D-1D transition Themain problems are that the results strongly rely on fittingparameters and that the use of Matthiessen approximation isstill questioned Instead Ziman [95] presents an approach ofsolving space-dependent BTE (Peierls-BTE [134]) The finalresult of this Peierls-BTE approach gives according to the

12

1

08

06

04

02

0 1 2 3 4 5 6 7 8

Diameter d (nm)

kGNR kgraphene

kRTA

kk

grap

hene k

Figure 6 Thermal conductivity 119896 versus diameter 119889 (color online)for single wall carbon nanotubes Solid red circles blue squares andgreen triangles represent zigzag armchair and chiral predicted byIterative Scheme open red circles blue squares and green trianglesare those from Standard SMRTA Black line shows 119896graphene while theblack open squares give 119896GNR For all cases length 119871 = 3 120583m and119879 = 300K Reprinted with permission from [53] Copyrighted bythe American Physical Society

ZA

TA

LA

10

11

10

L (120583m)

kLk

RTA

Figure 7 The ratio (color online) between thermal conductivitypredicted from Iterative Scheme (119896

119871) and Standard SMRTA (119896RTA)

of graphene as a function of length 119871 for temperature 300K 119896ZA(solid red) 119896TA (blue dashed) and 119896LA (green dotted) being thecontributions from different branches Reprinted with permissionfrom [54] Copyrighted by the American Physical Society


007

006

005

004

003

002

001

00 01 02 03 04 0 01 02 03 04 0 01 02 03 04 05

0 01 02 03 04 05

Scat

terin

g ra

te(c

mminus1)

Scat

terin

g ra

te(c

mminus1)

120585 120585 120585

1

120591TAasymp

1

120591aTA+

1

120591bTA+

1

120591cTA

Optical Optical

Optical

Acoustic

Acoustic Acoustic

Si28AverageSiGe[001]1+1 superlattice

02

015

01

005

0

120585

1

120591aTA

1

120591bTA

1

120591cTA

1

120591TA

TATATA

(a) (b) (c)

(d)

Figure 8 Scattering rate (color online) of TA mode at 300K due to (a) absorption of an acoustic phonon to yield another acoustic phonon(b) absorption of an acoustic phonon to yield an optical phonon (c) absorption of an optical phonon to yield another optical phonon and (d)total scattering rate along (120585 120585 120585) in 28Si ldquoaverage materialrdquo and SiGe[001]

1+1superlattice 120585 is reduced wave vector with k = (120585 120585 120585) sdot 2120587119886

Reprinted with permission from [108] Copyright (2011) American Chemical Society

simplification by Li et al [63 64] the position-dependentspectral phonon relaxation time

120591r120582 = 1205910

120582(1 + Δ

120582) 1 minus 119890

minus|(rminusr119887)1205910

120582V120582|119866r120582 (22)

where Δ120582is the average value of Δ

120582over the cross section

r119887is the point on the surface with r minus r

119887being the same

directionwith group velocity vector k120582 and119866r120582 describes the

boundary condition with 119866r120582 = 1 for completely diffusiveand 119866r120582 = 0 for mirror like The average of 120591r120582 over crosssection is

120591120582= 120591

0

120582(1 + Δ

120582) [

1

119878119888

int119878119888

1 minus 119890minus|(rminusr

119887)1205910

120582V120582|119866r120582 119889s] (23)

So far the calculation for nanowire still needs an adjustableparameter to account for the boundary scattering

334 Impurity-Isotope Scattering Doping and Alloys Fromsecond-order perturbation theory [34 95 135] assuming thatthe isotopes are distributed randomly the single scatteringrate by the isotopes [82 136] is given by

Γiso1205821205821015840 =

120587

2119873119888

120596120582120596

1205821015840

119899

sum

119887

119892119887

10038161003816100381610038161003816e119887120582sdot e119887lowast

1205821015840

10038161003816100381610038161003816

2

120575 (120596120582minus 120596

1205821015840) (24)

where 119873119888and 119899 stand for the number of unit cells and

the number of atoms per unit cell respectively e119887120582is the


1

011 15 2 25 3

1 times 1

2 times 2

4 times 4

kSLk

T

M1M2

Figure 9 Thermal conductivities calculated by Standard SMRTAmethod (dashed lines) and Iterative Scheme (solid lines) of 1+1 2+2and 4 + 4 SiGe-based SLs as a function of119872

1119872

2 the mass ratio of

constituent atomsThe thin vertical line shows the GeSi mass ratioReprinted with permission from [50] Copyrighted by the AmericanPhysical Society

eigenvector of 120582 mode in the basis atom 119887 part lowast denotescomplex conjugate and

119892119887= sum

119894

119891119894119887(1 minus

119898119894119887

119898119887

)

2

(25)

characterizes the magnitude of mass disorder where 119894 indi-cates isotope types 119891

119894119887is the fraction of isotope 119894 in lattice

sites of basis atom 119887119898119894119887is the mass of isotope 119894 and119898

119887is the

average atom mass of basis 119887 sites Sum over all the modes 1205821015840

with 1205821015840= 120582 the total scattering rate or the inverse relaxation

time of mode 120582 is

1

120591iso120582

= sum

1205821015840

Γiso1205821205821015840 =

120587

2119873119888

1205962

120582

119899

sum

119887

sum

1205821015840

119892119887

10038161003816100381610038161003816e119887120582sdot e119887lowast

1205821015840

10038161003816100381610038161003816

2

120575 (120596120582minus 120596

1205821015840)

(26)

For cubic symmetry system [82] such as Si Ge and Ar thesummation of eigenvectors in (26) can be reduced to

1

120591iso120582

=120587

6119899119873119888

1198921205962

120582sum

1205821015840

120575 (120596120582minus 120596

1205821015840)

=120587119881

0

6119892120596

2119863 (120596)

(27)

=120587

2119892120596

2119863 (120596) (28)

where 119892 is given by (5)1198810= 119881(119899119873

119888) is the volume per atom

119863(120596) = sum1205821015840 120575(120596120582

minus1205961205821015840)119881 is density of states per unit volume

and 119863(120596) is density of states normalized to unity noticingthat the total amount of states is 3119899119873

119888and the total volume

10

08

06

04

02

00 2 4 6 8 10

t (ps)

⟨Ei(t)Ei(0)⟩⟨E

i(0)Ei(0)⟩

Total energy (kinetic and potential)Potential energy

120587

120596i

i (T = 50K 120578 = 4 klowast = 05 transverse polarization)

Figure 10 Autocorrelation functions of potential and total energiesof time-dependent normal modes of TAmode at 119896lowast

= 05 in [1 0 0]direction of argon at 50K Reprinted with permission from [67]Copyrighted by the American Physical Society

is 119881 = 1198810119899119873

119888 Rewriting the 119863(120596) [82] one can obtain the

formula in Table 1 From (27) the relaxation time of mode120582 only depends on the frequency rather than the phononbranches

Equation (24) or (26) combined with the StandardSMRTA or Iterative Scheme have been used to predict thespectral phonon relaxation times of doped materials andeven alloys For isotope-doped system this method can bedirectly applied such as silicon and germanium [47 6091] hexagonal boron nitride layers and nanotube [56 58]GaN [57] and SiC [48] For alloy the disordered crystal istreated as an ordered one of the average atomic mass latticeparameter and force constants This approach the so-calledvirtual crystal approach first introduced by Abeles [137]has been applied to Si-Ge alloys [65 109] and PbTe

(1minusx)Sexalloys [106] by ab initio IFCs (Bi

(1minusx)Sbx)2Te3 alloys [138]with classical potential and Ni

055Pd

045alloys [139] for

comparison with experiment It turns out that the second-order perturbation ((24) or (26)) can give good predictioneven for large mass disorder

335 Superlattices Superlattices (SLs) composed of period-ically arranged layers of two or more materials have beenextensively investigated in the aspect of thermal transportBecause of the heat transport suppression by interfaces andmass mismatch superlattice has been designed to have alower thermal conductivity than pure bulk SLs are classifiedinto two categories diffuse and specular interfaces Thephonons in the first case are diffusively scattered by interfaceswhile the phonons in the latter one propagate through thewhole structure as if in one material so-called coherentphonon transport [119] Although proposed in theoreticalstudies long ago the coherent phonon transport in SLs was


10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Empty CNT


278ps(429ps)

80ps(138ps)

25ps(26ps)

263ps(268ps)

47ps(50ps)

0 1 2 3 4 5

Φ(kz=0120596

) (J s

)

(a)

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11


Water-filled CNT

24ps 22ps 23ps

250ps

46ps

0 1 2 3 4 5

Φ(kz=0120596

) (J s

)

(b)

Figure 11 SED functions (color online) of (a) empty CNT and (b) water-filled CNT along 119896119911= 0 for frequencies below 5 THz at 119879 = 298K

Reprinted with permission from [77] Copyrighted by the American Physical Society

20

15

10

05

000806040200 10

Longitudinal

1120591

(1p

s)

20K LD50K LD

20K MD50K MD

klowast (a2120587)

(a)

08

06

04

02

000806040200 10

Transverse

1120591

(1p

s)

20K LD50K LD

20K MD50K MD

klowast (a2120587)

(b)

Figure 12 The inverse relaxation times (color online) of LA and TA mode of argon as functions of wave vector in [109] direction predictedby Standard SMRTA method at 20K (solid curve) and 50K (dashed curve) and time domain NMA at 20K (open triangle) and 50K (solidtriangle) Adapted with permission from [102] Copyrighted by the American Physical Society

not observed in experiments until Luckyanova et al studiedfinite-thickness GaAsAlAs SLs by time-domain thermore-flectance measurements [119] It is found that the cross-plane119896 increases linearly with the number of periods when keepingthe periods constants Such phenomenon suggests that thephonon MFPs are equal to the sample thickness and thephonons do not ldquoseerdquo interfaces Luckyanova et al performeda first principle calculation (Standard SMRTA scheme) ofGaAsAlAs SLs to support their experimental results Theyfound that the anharmonic scattering rates and interfacescattering rates within the low-frequency region had thefrequency dependence as sim 120596minus2 and sim 120596minus4 respectively Thehigh-frequency phonons are scattered by interfaces whilethe low-frequency phonons have long MFPs and thus canpropagate though the entire SLs Another evidence that thephonons in SL do not ldquoseerdquo interfaces is the fact that theaccumulated 119896 of GaAsALAs SL is similar to bulk GaAsas shown in Figure 5 All the calculated results support theexperimental finding of coherent phonon propagations Inthe following discussion of SLs we only consider coherentphonon transport

Generally 119896 increases with increasing period length 119875119871

(at the limit 119875119871

rarr infin 119896 increases to that of the pure

bulk material) However it is found that for extremely shortperiod length 119896 even increases with decreasing 119875

119871 This leads

to a phenomenon that 119896 as a function of 119875119871reaches its

minimum at a critical 119875119871 and calculation of such value of 119875

119871

is crucially important for designing low thermal conductivitymaterials For instance Yang et al found that the isotopesilicon superlattice isoSi28Si nanowire had its lowest 119896 at119875119871asymp 1 nm [6] Hu and Poulikakos noticed that the SiGe

superlattice nanowire with 307 nm diameter had its lowest 119896at119875

119871asymp 4 nm [7] 119896 ofGaAsAlAs superlattice [8] as a function

of periodic length also obeys this principleThe exact phononrelaxation time explanation for such phenomenon is notavailable until the ALD method is explored [50 59 101 103108]

Garg et al [108] studied short-period (03 nm)SiGe[001]

1+1superlattice using Standard SMRTA with

ab initio IFCs They find that the thermal conductivity andphonon relaxation time of such superlattice are even greaterthan those of the two composition materials pristine Siand Ge bulks To understand this unusual behavior theinverse relaxation time of TA mode is calculated and shownin Figure 8 Also plotted are the detailed three-phononscattering rates for (a) TA + ArarrA (b) TA + ArarrO and


50

40

30

20

10

050403020100

Phon

on li

fetim

e (p

s)

Phonon frequency (THz)

ZA sTA sLA sZO sTO sLO s

ZA pTA pLA pZO pTO pLO p

Figure 13 The relaxation time (color online) of suspended (ldquosrdquo)and SiO

2supported (ldquoprdquo) single-layer graphene as a function of

phonon frequency for different phonon branches at temperature300K Reprinted with permission from [140] Copyright 2012 AIPPublishing LLC

(c) TA + OrarrO where A and O stand for acoustic andoptical respectively The ldquoaverage materialrdquo is an imaginarymaterial with averaged mass and potential of Si and GeBulk for comparison with SiGe[001]

1+1superlattice We

note that only the (a) component can provide scattering forTA phonons and that both (b) and (c) which affiliate withoptical modes are almost completely absent This indicatesthat the gap between optical and acoustic modes becomesso larger that the acoustic phonon can hardly be scatteredby optical phonons Such reduced scattering makes therelaxation times and thermal conductivity much larger thanthe two composition bulk materials

More generally Broido and Reinecke [59] and Ward andBroido [50] studied 119872

1119872

2[119899+119899]superlattice (the diamond

structure with periodical 119899 layers of mass 1198721atoms and

mass1198722atoms in [0 0 1] direction) using Iterative Scheme

In these two works the IFCs are determined using Keatingmodel [144 145] and adiabatic bond charge (ABC) model[146 147] respectively In such 119872

1119872

2[119899+119899]superlattice the

high thermal conductivity is also found for 119899 = 1 (Figure 9)When 119899 and mass ratio 119872

1119872

2are increasing 119896 is deter-

mined by the competition between the decease of phonongroup velocity and the increase of phonon relaxation time Itturns out that fromabout119872

1119872

2= 23 the latter competitor

dominates and thus 119896 increase with increasing mass ratioIn Figure 9 as expected the 119896rsquos from Iterative Scheme aregenerally larger than those from Standard SMRTA methodThis difference increases with increasing mass ratio because

the occurrence of 119873 process increases when the acoustic-optical gap gets larger

4 MD Simulation

41 Time Domain Normal Mode Analysis The time domainnormal mode analysis based onMD simulation was first pro-posed by Ladd et al [66] and then modified by McGaugheyand Kaviany [67] From (10) a result of SMRTA the relax-ation time 120591

120582can be obtained by

120591120582=

intinfin

0⟨119899

1015840

120582(119905) 119899

1015840

120582(0)⟩ 119889119905

⟨11989910158402

120582(0)⟩

(29)

According to the analysis by Ladd et al [66] the fluctuation1198991015840

120582in (29) can be replaced by the total phonon occupation

number 119899120582 which does not influence the calculation of

thermal conductivity when considering that the ensemble-average heat current is zero From lattice dynamics [34 148]the occupation number 119899

120582is proportional to the energy

of single phonon mode 120582 described by the normal modeamplitude

119864120582=1

2( 119902

120582119902lowast

120582+ 120596

2

120582119902120582119902lowast

120582) (30)

where 119902120582is the normal mode coordinate Thus (29) is

transformed to

120591120582=

intinfin

0⟨119864

120582(119905) 119864

120582(0)⟩ 119889119905

⟨1198642

120582(0)⟩

(31)

which is exactly what McGaughey and Kaviany [67] got withthe equivalent form ⟨119864

120582(119905)119864

120582(0)⟩⟨119864

2

120582(0)⟩ = exp(minus119905120591

120582)

Originally Ladd et al [66] only considered the potentialenergy and assumed 119864

120582sim 119902

120582119902lowast

120582 which does not influence

the result since the normal mode has the form [66 149]

119902120582(119905) = 119902

1205820exp [119894 (120596119860

120582+ 119894120578

120582) 119905] (32)

where 1199021205820

is the vibration amplitude a constant for a givenmode 120582 120596119860

120582= 120596

120582+ Δ

120582is the anharmonic frequency and 120578

120582

is linewidth With the help of this equation both 119864120582sim 119902

120582119902lowast

120582

and 119864120582sim 119902

120582119902lowast

120582give the equivalent value of 120591

120582= 12120578

120582

The calculation of normal mode coordinate 119902120582(119905) is

required to evaluate 119864120582in (30) and further predict 120591

120582in (31)

From lattice dynamics [148]

119902120582(119905) =

3

sum

120572

119899

sum

119887

119873119888

sum

119897

radic119898

119887

119873119888

119906119897119887

120572(119905) 119890

120582lowast

119887120572exp [119894k sdot r119897

0]

=

3

sum

120572

119899

sum

119887

119890120582lowast

119887120572119902119887

120572(k 119905)

(33)

where 120572 indicates 119909 119910 119911 directions 119906119897119887

120572(119905) is 120572 component

of the displacement of the 119887th atom in 119897th unit cell from itsequilibrium position r119897

0is the equilibrium position of unit

cell 119897 the star denotes complex conjugate and 119902119887

120572(k 119905) denotes


Temperature (K)300 400 500 600

300

200

100

0

ZA sTA sLA sZO s

ZA pTA pLA pZO p

er

mal

con

duct

ivity

(Wm

K)

(a)

Temperature (K)

()

300 400 500 600

40

30

20

10

ZA sTA sLA sZO s

ZA pTA pLA pZO p

(b)

Figure 14 (a) Contributions (color online) and (b) the corresponding percentages of thermal conductivity from ZA TA LA and TOmodesin suspended (ldquosrdquo) and supported (ldquoprdquo) SLG at different temperatures Reprinted with permission from [140]

the contribution of the 119887th basis atom in 120572 direction to thetotal normal mode with

119902119887

120572(k 119905) =

119873119888

sum

119897

radic119898

119887

119873119888

119906119897119887

120572(119905) exp [119894k sdot r119897

0] (34)

In (33) the time history of the atomic position displacement119906(119905) is extracted from MD simulation and the eigenvector 119890is obtained from LD calculations

42 Frequency Domain Normal Mode Analysis Here thefrequency domain normal mode analysis is demonstrated bya simplified version for detailed derivation see [76 77 150]Staring from (32) we have the spectral energy density (SED)

Φ120582(120596) =

1003816100381610038161003816 119902120582(120596)

1003816100381610038161003816

2

=

10038161003816100381610038161003816100381610038161003816

int

+infin

0

119902120582(119905) 119890

minus119894120596119905119889119905

10038161003816100381610038161003816100381610038161003816

2

=119862

120582

(120596 minus 120596119860

120582)2

+ 1205782

120582

(35)

where119862120582= (120596

119860

120582

2

+Γ2

120582)119902

2

1205820is a constant related to120582 Physically

Φ120582(120596) is the kinetic energy of single-phonon mode 120582 in the

frequency domain in contrast to (30) which is the energy intime domain Equation (35) is actually a Lorentzian functionwith peak position 120596119860

k] and full width at half maximum 2120578120582

By fitting this SED function as Lorentzian form the relaxationtime 120591

120582= 12120578

120582can be obtained

In some works the total SED function for a given wavevector Φ(k 120596) = sum]Φ120582

(120596) which is the summation of theSEDs of phonons with the same 119896 but from different phononbranches is evaluated instead of that of each mode Thomaset al [77 150] and Feng et al [151] pointed out that theeigenvectors are unnecessary due to the orthogonality thus

Φ (k 120596) =3119899

sum

]Φ

120582(120596)

=1

41205871205910

3119899

sum

120572119887

119898119887

119873119888

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119897

1205910

int

0

119897119887

120572(119905) exp [119894k sdot r119897

0minus 119894120596119905]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

=

3

sum

120572

119899

sum

119887

10038161003816100381610038161003816119902119887

120572(k 120596)10038161003816100381610038161003816

2

(36)

where 119902119887

120572(k 120596) is time derivative and Fourier Transform of

(34) From (33) to (36) the eigenvector has been abandonedand the mathematical proof of this is presented in [151]

According to Ong et al [152] the expression (36) isequivalent to the SED functions in [68ndash76] In some of theworks the mass 119898

119887and unit cell number 119873

119888in (34) are

discarded for single-mass system since the constants do notinfluence the fitting results of the Lorentzian function in (35)so that only atomic velocities are needed

43 Discussion and Applications Figures 10 and 11 showtwo examples of time-domain NMA and frequency-domain


Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)100

10

1

100

10

10 1 2 0 1 2

Frequency (THz) Frequency (THz)

300K 600K

[100] LA[110] LA[111] LA

[100] LA[110] LA[111] LA

(a)

[100] TA[110] TA1[110] TA2[111] TA

[100] TA[110] TA1[110] TA2[111] TA

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

300K 600K

(b)

1

0135 40

1

0135 40

300K 600K

[100] LO[110] LO[111] LO

[100] LO[110] LO[111] LO

Rela

xatio

n tim

e (p

s)

Frequency (THz)

Rela

xatio

n tim

e (p

s)

Frequency (THz)

(c)

30

300K 600K

[100] TO[110] TO1[110] TO2[111] TO

[100] TO[110] TO1[110] TO2[111] TO

1

01

1

0135 40 30 35 40

Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)


(d)

Figure 15 Spectral phonon relaxation time bulk PbTe (color online) at 300K and 600K Reprinted with permission from [141] Copyright(2011) with permission from Elsevier

NMA methods Figure 10 presents the autocorrelation func-tions of total energy and potential energy of normal modeas functions of time of the TA mode of argon at 50KThe oscillation of the potential energy indicates that thephonon frequency and the decay rate of total energy gives therelaxation time Figure 11 shows the SED functions of emptyCNT and water-filled CNT From the fitting of these peaks asLorentzian functions the phonon frequencies and relaxationtimes are obtained The linewidth broadening caused by thewater filled is clear from Figure 11(b)

The relaxation times predicted from MD simulationincludes the effects of three- four- and higher-order phononscattering processes in contrast ALD calculation only con-siders the lowest one Thus the ALD calculation may lose

its accuracy when temperature increases since the higher-order anharmonicity of lattice becomes greater for highertemperature due to thermal expansion For instance Turneyet al [102] compared the relaxation times of argon bulkpredicted from the Standard SMRTA ALD calculation andthe time-domain NMA at different temperatures Figure 12shows the inverse relaxation times of LA and TA phononmodes for argon at 20K and 50KWe note that at 20K thesetwo methods give reasonable agreement whilst at 50 K theALD calculation underpredicts the scattering rate by asmuchas 2 or more times

Compared to ALD calculation MD simulation is a bettertool for predicting the phonon properties of complex systemssuch as theCNTfilledwithwater and the graphene supported


80

60

40

20

00 1 2 3 4

46

23

26

300K

Frequency (THz)

TA1TA2LA

TO1TO2LO

Perc

enta

ge c

ontr

ibut

ion

(au

)

(a)

80

60

40

20

00 1 2 3 4

50

2124

TA1TA2LA

TO1TO2LO

600K

Frequency (THz)

Perc

enta

ge c

ontr

ibut

ion

(au

)

(b)

Figure 16 Contribution (color online) of each phonon mode to total thermal conductivity of PbTe bulk at 300K and 600K Reprinted withpermission from [141] Copyright (2011) with permission from Elsevier

150

100

50

0 1 2 3 4 5

Frequency (THz)

L = 12

L = 9

L = 6

Acoustic phonon

Low-frequencyoptical phonon

optical phononHigh-frequency

Rela

xatio

n tim

e (p

s)

(a)


0

150

100

50

02 04 06 08

Predicted by NMA

Rela

xatio

n tim

e (p

s)

Fitting with Aminus2

(b)

Figure 17 (a) Phonon relaxation times (color online) of of Bi2Te

3along the Γ-119885 direction computed using time-domain NMA 119871 denotes

the number of cells along axis at 300K (b) Phonon relaxation times of low frequency acoustic phonons along Γ-119885 and the power law fittingReprinted with permission from [142] Copyright 2013 by ASME

by substrate So far it is hard for ALD method to handlethe extrinsic phonon scattering processes other than theUmklapp scattering without fitting parameters However inthe MD simulation the surrounding influence is reflectedby the atomic vibrating trajectory of the studied system Qiuand Ruan [127 128 140] studied the phonon transport insuspended and silicon dioxide supported SLG by frequencydomain NMA with the results shown in Figures 13 and14 We note that the flexural phonon modes (ZA and ZO)have much longer relaxation times than the other modesfor suspended SLG which qualitatively agree with the ALDcalculation results discussed in Section 332 The MD resultindicates that ZA mode contributes about 29 to the total119896 for suspended SLG while TA and LA modes contribute33 and 26 respectively Chen and Kumar [126] performedthe same NMA method and obtained the similar results

that ZA TA and LA modes contribute 23 21 and 41respectivelyThe relaxation times of supported SLG are foundgenerally shorter by about 10 ps than suspended SLG Thisindicates that the SiO

2substrate provides strong phonon

scattering by the interface and breaks down the reflectionsymmetry in suspended SLG As a result the percentagethermal conductivity contribution from ZA mode decreasesabout 10 while those of TA and LA modes increase about3 and 8 respectively

Due to the low computational complexity the NMAmethods have been applied to many cases Time-domainNMAwas used for Ar [66 67 102] Si [143 153] Ge [154] andpolyethylene [155 156] in the meanwhile frequency domainNMA has been applied to Ar [150] Ge [151] MgO [76] CNT[77 78 157] supported CNT [152] suspended and supportedgraphene [140] and thermoelectric materials such as PbTe


10minus1 100 101 102 103 104 105

Si (MD)Si (ALD)PbTe (MD)

PbTe (MD 600K)PbTe (ALD)Bi2Te3 (MD)

MFP (nm)

Acc

umul

ated

ther

mal

con

duct

ivity

() 100

80

60

40

20

0

Figure 18 The normalized accumulated thermal conductivity(color online) of several bulk materials at room temperature as afunction of phononMFP (Si (MD) [143] Si (ALD) [107] PbTe (MD)[141] PbTe (MD600K) [141] PbTe (ALD) [106] Bi

2Te

3(MD) [142])

[141] and Bi2Te

3[142] So far only few works applied NMA

to defective bulk nanowires [153] and nanoribbons As arepresentative application of frequency domainNMA to bulkmaterial the spectral phonon relaxation times of pristinePbTe bulk at different temperatures in different directionsare given in Figure 15 The results reveal typical featuresof phonon relaxation time in bulk materials (a) acousticphonons generally have much higher relaxation times thanoptical phonons (b) for acoustic modes the relaxation timesalways decrease with increasing frequency except for thehigh-frequency ranges which often showopposite trend suchphenomenon is also found in other materials such as argon[67 102] silicon [143] and germanium [151] (c) the valueof 120572 in frequency dependence relation 120591 sim 120596

minus120572 of theacoustic phonon often deviates from 2 and ranges from 05to 4 (d) 120591 of optical mode has weak frequency dependenceand (e) increasing temperature typically shortens the phononrelaxation time and mean free path The order of relaxationtime amplitudes of PbTe bulk at 300Kobtained by frequency-domain NMA agrees well with those obtained from ALDcalculation in previous sections [104 106] It is importantto note that NMA methods do not distinguish between 119873

scattering and 119880 scattering but give a total scattering ratejust as the method of ALD calculation based on StandardSMRTA Figure 16(a) gives the contribution of each phononmode to total thermal conductivity of PbTe bulk at 300K and600K The results show that optical modes only contribute5 to the total 119896 different with 20 given by first principleALD calculation The discrepancy may come from the igno-rance of higher-order phonon scattering in ALD calculationFigure 16(b) gives the accumulated 119896rsquos as functions phononMFP for Silicon at 300K and PbTe at 300K and 600K 80

of the total 119896 of PbTe is contributed by the phonons withMFP below 50 nm different from the value of 10 nm in ALDcalculation [106] This suggests that the relaxation times oflow-frequency phonons predicted from ALD are longer thanthose from NMA since both ALD and NMA results givereasonable total thermal conductivity The MFPs of phononsof PbTe decrease roughly by a factor of 2 when temperatureincreases from 300K to 600K It is found that the phononswith MFP below 10 nm contribute about 32 of 119896 at 300Kwhile about 65 of 119896 at 600K

The phonon properties of Bi2Te

3are studied by time-

domain NMA [142] The relaxation times and power lawfitting of the low-frequency range are presented in Figure 17The phonons with wavelength of 125 nm have relaxationtime 169 ns which indicate that those phonons do notexperience obvious scatteringwhen traveling for about 400 psin Bi

2Te

3 consistent with experimental measurements [158]

The normalized accumulated thermal conductivity of Bi2Te

3

as a function of phonon MFP is plotted in Figure 18 It isfound that 90 and 50 of total thermal conductivity arecontributed by the phonons with MFPs shorter than 10 nmand 3 nm respectively Also shown in Figure 18 is comparisonbetween the results from ALD calculation and MD simula-tion The two curves for Si agree well with each other whilea discrepancy is found for bulk PbTe This discrepancy maycome from the inaccuracy of the interatomic potential usedin performingMD simulationThese results are useful for thenanodesign of Bi

2Te

3PbTeSi based thermoelectricmaterials

in the future

5 Summary

The three methods anharmonic lattice dynamics based onStandard SMRTA iterative anharmonic lattice dynamics andnormalmode analysis can all predict thermal conductivity bycalculating the velocities relaxation times and specific heatsof all phonon modes The applications are listed in Table 2and the features of these methods are compared and listed inTable 3

All the three methods are based on phonon BoltzmannTransport Equation and relaxation time approximation Toobtain the spectral phonon relaxation time the first twomethods calculate three-phonon scattering rates from anhar-monic interatomic force constants while the last method cal-culate the linewidth of spectral energy in frequency domainor the decay rate of spectral energy in time domain frommolecular dynamics Since the first two methods ignore the4th- and higher-order phonon scattering processes they areonly valid at low temperature The first two methods differwith each other at solving the phonon BTE the first methodassumes single mode RTA while the second one solves thelinearized BTE iteratively instead As a result the firstmethodtreats 119873 scattering and 119880 scattering as two independentprocesses that provide thermal resistance individually How-ever it is well known that the 119873 scattering only contributeto thermal resistance by influencing the 119880 scattering rateThe Iterative ALD remedies this error by recording all thephonon scattering processes step by step and evaluates the


Table 2 Applications of the numerical methods in predicting spectral phonon properties and thermal conductivity

Materials Methodslowast FPdagger Reference YearAr 1 and 3 [66] 1986Ar 2 [46] 1995Ar and Kr 2 [45] 1996Ar 3 [67] 2004Ar 1 and 3 [102] 2009Ar Si thin films 1 [92] 2010C Si and Ge 1 radic [43] 1995C isotope-doped C Si and Ge 1 and 2 radic [51] 2009C (Pure and natural) (extreme pressure) 1 and 2 [62] 2012C nanowire 1 and 2 radic [64] 2012Si (isotope-doped) 1 [136] 1999Si (isotope-doped) 1 [91] 2001SiC 2 [48] 2002Si and Ge 1 radic [44] 2003Si 1 and 2 [60] 2005Si and Ge 2 radic [61] 2007Si 3 [143] 2008Si (isotope-doped) 2 [47] 2009Si 1 radic [107] 2011Si 1 [99] 2012Si Nanowire 3 [153] 2009Si Ge 1 radic [110] 2010SiGe119872

1119872

2SLs 1 and 2 [59] 2004

SiGe SLs 1 radic [108] 2011SiGe SLs 1 [101] 2013SiGe SLs 1 radic [103] 2013SiGe alloys with embedded nanoparticles 2 radic [65] 2011SiGe alloys 1 radic [109] 2011Si Ge and Si05Ge05 1 [100] 2012SiGe GaAsAlAs and119872

1119872

2SLs 1 and 2 radic [50] 2008

Ge 3 [154] 2010Ge 4 [151] 2013Semiconductors (Groups IV IIIndashV and IIndashVI) 1 [117] 2008Graphene 1 and 2 [54] 2010Graphene and graphite 1 and 2 [55] 2011Graphene (supported and suspended) 4 [140] 2012Graphene (free-standing and strained) 1 radic [105] 2012Graphene and graphite 1 radic [111] 2013CNT to graphene (diameter dependence) 1 and 2 [53] 2010CNT 4 [157] 2006CNT 1 and 2 [52] 2009CNT (empty and water-filled) 4 [77] 2010CNT (on amorphous silica) 4 [152] 2011BN (pristine and isotope-doped) 1 and 2 [56] 2011BN (multilayer and nanotubes) (pristine and isotope-doped) 2 [58] 2012Mg2Si119909Sn1minus119909

alloys (bulk and nanowire) 2 [63] 2012Compound semiconductors (Si Ge GaAs Al-V Ga-V In-V SiC AlN etc) 1 and 2 radic [112] 2013Ionic solids (MgO UO

2 and SrTiO3) 2 [49] 2011

GaN (GaAs GaSb and GaP) (pristine and isotope-doped) 1 and 2 radic [57] 2012


Table 2 Continued

Materials Methodslowast FPdagger Reference YearPbTe 1 radic [104] 2012PbTe 4 [141] 2011PbTe PbSe and PbTe

1minus119909Se

1199091 radic [106] 2012

Bi2Te

34 [142] 2013

Heusler 3 [120] 2011MgO 4 radic [76] 2009Polyethylene 3 [155] 2009Polyethylene 3 [156] 2009GaAs 1 radic [118] 2013GaAsAlAs SLs 1 radic [119] 2012lowastMethods 1 Standard SMRTA scheme 2 Iterative Scheme 3 time-domain NMA 4 frequency-domain NMAdaggerFP first principle

Table 3 Comparison of different methods for predicting spectral phonon relaxation time and thermal conductivity

Methods Analytical model ALD calculation MD simulationStandard SMRTA Iterative Scheme Time NMA Frequency NMA

Equations Table 1 Equations (14)(15) and (16)

Equations (14)(15) (16) (17)

and (18)

Equations (30)(31) and (33)

Equations (33)and (35)


CharacteristicsLots ofapproximationsneed fittingparameters

119873 and 119880processes areindependent

thermal resistantsources

119873 process doesnot provide

thermal resistantitself

Eigenvectors needed Eigenvectors notneeded

Need 2nd- and 3rd-order IFCs Need interatomic potential (or ab initioMD)

Suitable for Long wavelengthDebye model

Low temperature higher-orderanharmonicity not large Higher than Debye temperature

Temperature not too low quantum effect negligibleAccuracy Low Medium Higher HigherComputationalcomplexity High Higher Low

Applications sofar

Some thermalconductivityanalysis andprediction

Pure and isotope-doped bulk alloysuperlattice nanostructures Pure lattice Materials with surrounding influences

Further research Temperature dependent IFCs 4th-and higher-order phonon scattering

Accurate interatomic potential large domain firstprinciple MD defects boundaries

119880 scattering rates in the end Compared to Green-Kubo MD(GK-MD) and Nonequilibrium MD (NEMD) these threemethods give deeper insight into the thermal conductivitythe spectral phonon velocity relaxation time and mean freepath and the contribution of each phonon mode to thermalconductivity which can guide the nanodesign For accuracyand capability the ab initio ALD calculations are better thanGK-MD and NEMD since calculating ab initio 3rd-orderIFCs is much easier than implementing ab initio MD Thelimitations of the normal mode analysis are as follows (1) itcannot distinguish119880 and119873 processes and (2) it is of classicalnature so it cannot accurately capture the quantum distribu-tion function (Bose-Einstein distribution) for high Debye-temperature materials at relatively low temperatures (such asgraphene and CNT at room temperature) The disadvantageof these three methods is the much computational costCompared to analytical models these methods do not rely

on adjustable fitting parameters and thus give more reliableand accurate predictions

These numericalmethods have been applied to numerousmaterials and structures and revealed lots of physical naturethat has never been reached before The acoustic phononsare verified to have the sim 120596

minus120572 frequency dependence whichagrees with earlier analytical models while the facts thatthe value of 120572 varies from 0 to 4 at low frequency and thatthe frequency dependence becomes weak and abnormal athigh frequency were not observed clearly before The opticalmodes are found to carry very little heat but contribute muchto the scattering of acoustic phonons and thus are essentialto thermal transport In layer-tube-structured materials thestrict selection rule of phonon scattering because reflectionsymmetry severely blocks the scattering of flexural acousticphonons and thus causes extremely high relaxation time andthen high thermal conductivity In short-period superlattice


the large gaps between acoustic and optical phonon branchesmake the scattering rarely happen and thus lead to high ther-mal conductivity even higher than its corresponding purematerials These methods are also applied to defected andalloy materials using virtual crystal approach Despite theseapplications further work is still needed to predict spectralphonon properties more accurately and efficiently such asconsidering the temperature-dependent IFCs and higher-order anharmonicities in ALD calculations implementinglarge domain ab initiomolecular dynamics for normal modeanalysis

Appendix

A

Themathematic preparations are

120596 + 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) (119899

0

1205821015840 + 1) 119899

0

12058210158401015840 = 119899

0

1205821198990

1205821015840 (119899

0

12058210158401015840 + 1) (A1)

120596 minus 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) 119899

0

1205821015840119899

0

12058210158401015840 = 119899

0

120582(119899

0

1205821015840 + 1) (119899

0

12058210158401015840 + 1) (A2)

taking advantage of 1198990= [exp(ℏ120596119896

119861119879) minus 1]

minus1Relaxation time approximation assumes

1198991015840

120582= minusΨ

120582

1205971198990

120582

120597 (ℏ120596120582)= Ψ

120582sdot

1

1198961198611198791198990

120582(119899

0

120582+ 1) (A3)

The expression of Ψ120582is obtained from the single-mode

approximation (11)

Ψ120582= 120591

120582

ℏ120596120582

119879k120582sdot nabla119879 (A4)

A1 Standard SMRTA The Derivation from (12) to (15) InStandard SMRTA only 120582mode has perturbation

119899120582= 119899

0

120582+ 119899

1015840

120582

1198991205821015840 = 119899

0

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840

(A5)

Substituting (A5) and (A3) into (12) with the help of(A1) we get

1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

= 1198991205821015840 [119899

0

1205821015840 (1 + 119899

0

12058210158401015840) minus (1 + 119899

0

1205821015840) 119899

0

12058210158401015840]

= 1198991205821015840 (119899

0

1205821015840 minus 119899

0

12058210158401015840)

(A6)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (119899

0

1205821015840 minus 119899

0

12058210158401015840)

119896119861119879

(A7)

And with the help of (A2) we get

119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

= 1198991205821015840 [(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840) minus 119899

0

1205821015840119899

0

12058210158401015840]

= 1198991205821015840 (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

(A8)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

119896119861119879

(A9)

From (A6) and (A8) we reach the relation 1205971198991205821015840120597119905 sim 119899

1205821015840

and comparedwith (11) we obtain (15) From (A7) and (A9)the expression of Ψ

120582is obtained the same with (A4)

A2 Iterative Scheme The Derivation from (12) to (17) TheIterative Scheme solves phonon BTE (12) by assuming

119899120582= 119899

0

120582+ 119899

1205821015840

1198991205821015840 = 119899

0

1205821015840 + 119899

1015840

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840 + 119899

1015840

12058210158401015840

(A10)

where 1198991015840

1205821015840 and 1198991015840

12058210158401015840 have the same form as 1198991015840

120582

1198991015840

1205821015840 = minusΨ

1205821015840

1205971198990

1205821015840

120597 (ℏ1205961205821015840)= Ψ

1205821015840 sdot

1

1198961198611198791198990

1205821015840 (119899

0

1205821015840 + 1) (A11)

1198991015840

12058210158401015840 = minus Ψ

12058210158401015840

1205971198990

12058210158401015840

120597 (ℏ12059612058210158401015840)

= Ψ12058210158401015840 sdot

1

1198961198611198791198990

12058210158401015840 (119899

0

12058210158401015840 + 1)

(A12)

Ψ1205821015840 = 120591

1205821015840

ℏ1205961205821015840

119879k1205821015840 sdot nabla119879 (A13)

Ψ12058210158401015840 = 120591

12058210158401015840

ℏ12059612058210158401015840

119879k12058210158401015840 sdot nabla119879 (A14)

Substituting (A3) (A10) (A11) and (A12) into (12) withthe help of (A1) abandoning the higher order terms Ψ

120582Ψ

1205821015840

Ψ120582Ψ

12058210158401015840 and Ψ

1205821015840Ψ

12058210158401015840 we have

1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

=

(Ψ120582+ Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

1205821198990

1205821015840 (1 + 119899

0

12058210158401015840)

119896119861119879

(A15)

and with the help of (A2) we have

119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

=

(Ψ120582minus Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

120582(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840)

119896119861119879

(A16)

Substituting (A4) (A13) and (A14) into (A15) and (A16)we obtain the results (17) and (18)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


Acknowledgments

Theauthorswould like to thank theNational Science Founda-tion Air Force Office of Scientific Research and the PurdueNetwork for Computational Nanotechnology (NCN) for thepartial support

References

[1] R Venkatasubramanian E Siivola T Colpitts and B OrsquoQuinnldquoThin-film thermoelectric devices with high room-temperaturefigures of meritrdquo Nature vol 413 no 6856 pp 597ndash602 2001

[2] A Balandin and K L Wang ldquoSignificant decrease of the latticethermal conductivity due to phonon confinement in a free-standing semiconductor quantum wellrdquo Physical Review BCondensedMatter andMaterials Physics vol 58 no 3 pp 1544ndash1549 1998

[3] A Khitun A Balandin and K L Wang ldquoModification ofthe lattice thermal conductivity in silicon quantum wires dueto spatial confinement of acoustic phononsrdquo Superlattices andMicrostructures vol 26 no 3 pp 181ndash193 1999

[4] J Zou andA Balandin ldquoPhonon heat conduction in a semicon-ductor nanowirerdquo Journal of Applied Physics vol 89 no 5 pp2932ndash2938 2001

[5] O L Lazarenkova and A A Balandin ldquoElectron and phononenergy spectra in a three-dimensional regimented quantumdot superlatticerdquo Physical Review B Condensed Matter andMaterials Physics vol 66 Article ID 245319 2002

[6] N Yang G Zhang and B Li ldquoUltralow thermal conductivity ofisotope-doped silicon nanowiresrdquoNano Letters vol 8 no 1 pp276ndash280 2008

[7] M Hu and D Poulikakos ldquoSiGe superlattice nanowires withultralow thermal conductivityrdquo Nano Letters vol 12 no 11 pp5487ndash5494 2012

[8] W S Capinski H J Maris T Ruf M Cardona K Ploogand D S Katzer ldquoThermal-conductivity measurements ofGaAsAlAs superlattices using a picosecond optical pump-and-probe techniquerdquo Physical Review B vol 59 no 12 pp 8106ndash8113 1999

[9] H Fang T Feng H Yang X Ruan and Y Wu ldquoSynthesis andthermoelectric properties of compositional-modulated leadtelluride-bismuth telluride nanowire heterostructuresrdquo NanoLetters vol 13 no 5 pp 2058ndash2063 2013

[10] K S Novoselov A K Geim S V Morozov et al ldquoElectric fieldin atomically thin carbon filmsrdquo Science vol 306 no 5696 pp666ndash669 2004

[11] K S Novoselov A K Geim S V Morozov et al ldquoTwo-dimensional gas of massless Dirac fermions in graphenerdquoNature vol 438 no 7065 pp 197ndash200 2005

[12] Y Zhang J W Tan H L Stormer and P Kim ldquoExperimentalobservation of the quantum Hall effect and Berryrsquos phase ingraphenerdquo Nature vol 438 no 7065 pp 201ndash204 2005

[13] A K Geim and K S Novoselov ldquoThe rise of graphenerdquo NatureMaterials vol 6 no 3 pp 183ndash191 2007

[14] C Yu L Shi Z Yao D Li and A Majumdar ldquoThermalconductance and thermopower of an individual single-wallcarbon nanotuberdquo Nano Letters vol 5 no 9 pp 1842ndash18462005

[15] E Pop D Mann Q Wang K Goodson and H Dai ldquoThermalconductance of an individual single-wall carbon nanotubeabove room temperaturerdquoNano Letters vol 6 no 1 pp 96ndash1002006

[16] N Hamada S-I Sawada and A Oshiyama ldquoNew one-dimensional conductors graphitic microtubulesrdquo PhysicalReview Letters vol 68 no 10 pp 1579ndash1581 1992

[17] S Reich C Thomsen and J Maultzsch Carbon Nan-OtubesBasic Concepts and Physical Properties Wiley-VCHWeinheimGermany 2004

[18] J Hu X Ruan and Y P Chen ldquoThermal conductivity andthermal rectification in graphene nanoribbons a moleculardynamics studyrdquoNano Letters vol 9 no 7 pp 2730ndash2735 2009

[19] J H Seol I Jo A L Moore et al ldquoTwo-dimensional phonontransport in supported graphenerdquo Science vol 328 no 5975 pp213ndash216 2010

[20] A A Balandin S Ghosh W Bao et al ldquoSuperior thermalconductivity of single-layer graphenerdquo Nano Letters vol 8 no3 pp 902ndash907 2008

[21] A A Balandin ldquoThermal properties of graphene and nanos-tructured carbon materialsrdquoNature Materials vol 10 no 8 pp569ndash581 2011

[22] D Nika and A A Balandin ldquoTwo-dimensional phonon trans-port in graphenerdquo Journal of Physics CondensedMatter vol 24no 23 Article ID 233203 2012

[23] A A Balandin and D Nika ldquoPhononics in low-dimensionalmaterialsrdquoMaterials Today vol 15 no 6 pp 266ndash275 2012

[24] N Mingo ldquoCalculation of Si nanowire thermal conductivityusing complete phonon dispersion relationsrdquo Physical Review BCondensed Matter and Materials Physics vol 68 no 11 ArticleID 113308 2003

[25] N Mingo L Yang D Li and A Majumdar ldquoPredicting theThermal Conductivity of Si and Ge Nanowiresrdquo Nano Lettersvol 3 no 12 pp 1713ndash1716 2003

[26] P Martin Z Aksamija E Pop and U Ravaioli ldquoImpact ofphonon-surface roughness scattering on thermal conductivityof thin Si nanowiresrdquo Physical Review Letters vol 102 no 12Article ID 125503 2009

[27] Y Chen D Li J R Lukes and A Majumdar ldquoMonte Carlosimulation of silicon nanowire thermal conductivityrdquo Journal ofHeat Transfer vol 127 no 10 pp 1129ndash1137 2005

[28] I Ponomareva D Srivastava and M Menon ldquoThermal con-ductivity in thin silicon nanowires phonon confinement effectrdquoNano Letters vol 7 no 5 pp 1155ndash1159 2007

[29] T Markussen A-P Jauho and M Brandbyge ldquoHeat conduc-tance is strongly anisotropic for pristine silicon nanowiresrdquoNano Letters vol 8 no 11 pp 3771ndash3775 2008

[30] P N Martin Z Aksamija E Pop and U Ravaioli ldquoReducedthermal conductivity in nanoengineered rough Ge and GaAsnanowiresrdquo Nano Letters vol 10 no 4 pp 1120ndash1124 2010

[31] C W Padgett O Shenderova and D W Brenner ldquoThermalconductivity of diamond nanorods molecular simulation andscaling relationsrdquoNano Letters vol 6 no 8 pp 1827ndash1831 2006

[32] J FMoreland J B Freund andGChen ldquoThedisparate thermalconductivity of carbon nanotubes and diamond nanowiresstudied by atomistic simulationrdquo Microscale ThermophysicalEngineering vol 8 no 1 pp 61ndash69 2004

[33] P G Klemens ldquoThe thermal conductivity of dielectric solids atlow temperatures (theoretical)rdquo Proceedings of the Royal SocietyA vol 208 no 1092 pp 108ndash133 1951

[34] P G Klemens ldquoThermal conductivity and lattice vibrationalmodesrdquo in Solid State Physics F Seitz and D Turnbull Eds vol7 pp 1ndash98 Academic Press New York NY USA 1958

[35] P G Klemens ldquoThe scattering of low-frequency lattice waves bystatic imperfectionsrdquo Proceedings of the Physical Society A vol68 no 12 article 303 pp 1113ndash1128 1955


[36] C Herring ldquoRole of low-energy phonons in thermal conduc-tionrdquo Physical Review vol 95 no 4 pp 954ndash965 1954

[37] M G Holland ldquoAnalysis of lattice thermal conductivityrdquo Phys-ical Review vol 132 no 6 pp 2461ndash2471 1963

[38] H B G Casimir ldquoNote on the conduction of heat in crystalsrdquoPhysics vol 5 p 495 1938

[39] R Berman F E Simon and J M Ziman ldquoThe thermalconductivity of diamond at low temperaturesrdquo Proceedings ofthe Royal Society A vol 220 no 1141 pp 171ndash183 1953

[40] R Berman E L Foster and J M Ziman ldquoThermal conductionin artificial sapphire crystals at low temperatures 1 nearlyperfect crystalsrdquo Proceedings of the Royal Society A vol 231 no1184 pp 130ndash144 1955

[41] A A Maradudin and A E Fein ldquoScattering of neutrons by ananharmonic crystalrdquo Physical Review vol 128 no 6 pp 2589ndash2608 1962

[42] A A Maradudin A E Fein and G H Vineyard ldquoOn theevaluation of phonon widths and shiftsrdquo Physica Status SolidiB vol 2 no 11 pp 1479ndash1492 1962

[43] A Debernardi S Baroni and E Molinari ldquoAnharmonicphonon lifetimes in semiconductors from density-functionalperturbation theoryrdquo Physical Review Letters vol 75 no 9 pp1819ndash1822 1995

[44] G Deinzer G Birner and D Strauch ldquoAb initio calculationof the linewidth of various phonon modes in germanium andsiliconrdquo Physical Review B Condensed Matter and MaterialsPhysics vol 67 no 14 Article ID 144304 pp 1443041ndash14430462003

[45] M Omini and A Sparavigna ldquoBeyond the isotropic-modelapproximation in the theory of thermal conductivityrdquo PhysicalReview B Condensed Matter and Materials Physics vol 53 no14 pp 9064ndash9073 1996

[46] M Omini and A Sparavigna ldquoAn iterative approach to thephonon Boltzmann equation in the theory of thermal conduc-tivityrdquo Physica B Physics of CondensedMatter vol 212 no 2 pp101ndash112 1995

[47] J A Pascual-Gutierrez J Y Murthy and R Viskanta ldquoThermalconductivity and phonon transport properties of silicon usingperturbation theory and the environment-dependent inter-atomic potentialrdquo Journal of Applied Physics vol 106 no 6Article ID 063532 2009

[48] A Sparavigna ldquoLattice thermal conductivity in cubic siliconcarbiderdquo Physical Review B Condensed Matter and MaterialsPhysics vol 66 no 17 Article ID 174301 2002

[49] A Chernatynskiy J E Turney A J H McGaughey C HAmon and S R Phillpot ldquoPhonon-mediated thermal con-ductivity in ionic solids by lattice dynamics-based methodsrdquoJournal of the American Ceramic Society vol 94 no 10 pp3523ndash3531 2011

[50] A Ward and D A Broido ldquoIntrinsic lattice thermal conduc-tivity of SiGe and GaAsAlAs superlatticesrdquo Physical Review BCondensed Matter and Materials Physics vol 77 no 24 ArticleID 245328 2008

[51] A Ward D A Broido D A Stewart and G Deinzer ldquoAb initiotheory of the lattice thermal conductivity in diamondrdquo PhysicalReview B Condensed Matter and Materials Physics vol 80 no12 Article ID 125203 2009

[52] L Lindsay D A Broido and N Mingo ldquoLattice thermalconductivity of single-walled carbon nanotubes beyond therelaxation time approximation and phonon-phonon scatteringselection rulesrdquo Physical Review B Condensed Matter andMaterials Physics vol 80 no 12 Article ID 125407 2009

[53] L Lindsay D A Broido and N Mingo ldquoDiameter dependenceof carbon nanotube thermal conductivity and extension tothe graphene limitrdquo Physical Review B Condensed Matter andMaterials Physics vol 82 no 16 Article ID 161402 2010

[54] L Lindsay D A Broido and N Mingo ldquoFlexural phonons andthermal transport in graphenerdquo Physical Review B CondensedMatter and Materials Physics vol 82 no 11 Article ID 1154272010

[55] L Lindsay D A Broido and N Mingo ldquoFlexural phononsand thermal transport in multilayer graphene and graphiterdquoPhysical Review B CondensedMatter andMaterials Physics vol83 no 23 Article ID 235428 2011

[56] L Lindsay and D A Broido ldquoEnhanced thermal conductivityand isotope effect in single-layer hexagonal boron nitriderdquoPhysical Review B CondensedMatter andMaterials Physics vol84 no 15 Article ID 155421 2011

[57] L Lindsay D A Broido and T L Reinecke ldquoThermal con-ductivity and large isotope effect in GaN from first principlesrdquoPhysical Review Letters vol 109 no 9 Article ID 095901 2012

[58] L Lindsay and D A Broido ldquoTheory of thermal transport inmultilayer hexagonal boron nitride and nanotubesrdquo PhysicalReview B Condensed Matter and Materials Physics vol 85 no3 Article ID 035436 2012

[59] D A Broido and T L Reinecke ldquoLattice thermal conductivityof superlattice structuresrdquo Physical Review B CondensedMatterand Materials Physics vol 70 no 8 Article ID 081310 4 pages2004

[60] D A Broido A Ward and N Mingo ldquoLattice thermal conduc-tivity of silicon from empirical interatomic potentialsrdquo PhysicalReview B Condensed Matter and Materials Physics vol 72 no1 Article ID 014308 2005

[61] D A Broido M Malorny G Birner N Mingo and D A Stew-art ldquoIntrinsic lattice thermal conductivity of semiconductorsfrom first principlesrdquo Applied Physics Letters vol 91 no 23Article ID 231922 2007

[62] D A Broido L Lindsay and A Ward ldquoThermal conductivityof diamond under extreme pressure a first-principles studyrdquoPhysical Review B CondensedMatter andMaterials Physics vol86 no 11 Article ID 115203 2012

[63] W Li L Lindsay D A Broido D A Stewart and N MingoldquoThermal conductivity of bulk and nanowire Mg

2Si

119909Sn

1minus119909

alloys from first principlesrdquo Physical Review B CondensedMatter and Materials Physics vol 86 no 17 Article ID 1743072012

[64] W Li N Mingo L Lindsay D A Broido D A Stewart andN A Katcho ldquoThermal conductivity of diamond nanowiresfrom first principlesrdquo Physical Review B Condensed Matter andMaterials Physics vol 85 no 19 Article ID 195436 2012

[65] A Kundu N Mingo D A Broido and D A Stewart ldquoRoleof light and heavy embedded nanoparticles on the thermalconductivity of SiGe alloysrdquo Physical Review B CondensedMatter and Materials Physics vol 84 no 12 Article ID 1254262011

[66] A J C Ladd B Moran and W G Hoover ldquoLattice thermalconductivity a comparison of molecular dynamics and anhar-monic lattice dynamicsrdquo Physical Review B Condensed Matterand Materials Physics vol 34 no 8 pp 5058ndash5064 1986

[67] A McGaughey and M Kaviany ldquoQuantitative validation of theBoltzmann transport equation phonon thermal conductivitymodel under the single-mode relaxation time approximationrdquoPhysical Review B CondensedMatter andMaterials Physics vol69 no 9 Article ID 094303 2004


[68] C ZWang C T Chan and KM Ho ldquoEmpirical tight-bindingforce modelfor molecular-dynamics simulation of Sirdquo PhysicalReview B Condensed Matter and Materials Physics vol 39 no12 pp 8586ndash8592 1989

[69] C Z Wang C T Chan and K M Ho ldquoMolecular-dynamicsstudy of anharmonic effects in siliconrdquo Physical Review BCondensedMatter andMaterials Physics vol 40 no 5 pp 3390ndash3393 1989

[70] C Z Wang C T Chan and K M Ho ldquoTight-bindingmolecular-dynamics study of phonon anharmonic effects insilicon and diamondrdquo Physical Review B CondensedMatter andMaterials Physics vol 42 no 17 pp 11276ndash11283 1990

[71] C ZWang C T Chan and KM Ho ldquoStructure and dynamicsof C60 and C70 from tight-binding molecular dynamicsrdquoPhysical Review B CondensedMatter andMaterials Physics vol46 no 15 pp 9761ndash9767 1992

[72] C ZWang KM Ho and C T Chan ldquoStructure and dynamicsof liquid carbonrdquo Physical Review B Condensed Matter andMaterials Physics vol 47 no 22 pp 14835ndash14841 1993

[73] C Z Wang C T Chan and K M Ho ldquoTight-bindingmolecular-dynamics study of defects in siliconrdquo Physical ReviewLetters vol 66 no 2 pp 189ndash192 1991

[74] C Z Wang K M Ho and C T Chan ldquoTight-bindingmolecular-dynamics study of amorphous carbonrdquo PhysicalReview Letters vol 70 no 5 pp 611ndash614 1993

[75] C Z Wang and K M Ho ldquoStructure dynamics and electronicproperties of diamondlike amorphous carbonrdquo Physical ReviewLetters vol 71 no 8 pp 1184ndash1187 1993

[76] N de Koker ldquoThermal conductivity of MgO periclase fromequilibrium first principles molecular dynamicsrdquo PhysicalReview Letters vol 103 no 12 Article ID 125902 2009

[77] J A Thomas J E Turney R M Iutzi C H Amon and A JH McGaughey ldquoPredicting phonon dispersion relations andlifetimes from the spectral energy densityrdquo Physical Review BCondensed Matter and Materials Physics vol 81 no 8 ArticleID 081411 2010

[78] J A Thomas J E Turney R M Iutzi A J H McGaugheyand C H Amon ldquoPredicting the phonon properties of carbonnanotubes using the spectral energy densityrdquo in Proceedings ofthe 2010 14th International Heat Transfer Conference (IHTC rsquo10)IHTC14-22262 pp 305ndash312 August 2010

[79] M KavianyHeat Transfer Physics Cambridge University PressNew York NY USA 2008

[80] J Callaway ldquoModel for lattice thermal conductivity at lowtemperaturesrdquo Physical Review vol 113 no 4 pp 1046ndash10511959

[81] M Asen-Palmer K Bartkowski E Gmelin et al ldquoThermalconductivity of germaniumcrystals with different isotopic com-positionsrdquo Physical Review B Condensed Matter and MaterialsPhysics vol 56 no 15 pp 9431ndash9447 1997

[82] S Tamura ldquoIsotope scattering of dispersive phonons in GerdquoPhysical Review B CondensedMatter andMaterials Physics vol27 no 2 pp 858ndash866 1983

[83] G P Srivastava The Physics of Phonons Adam Higer BristolUK 1990

[84] A P Zhernov and A V Inyushkin ldquoKinetic coefficients inisotopically disordered crystalsrdquo Physics-Uspekhi vol 45 no 5pp 527ndash552 2002

[85] A V Inyushkin ldquoThermal conductivity of isotopicallymodifiedsilicon current status of research1rdquo Inorganic Materials vol 38no 5 pp 427ndash433 2002

[86] R Berman Thermal Conductivity in Solids Oxford UniversityPress Oxford UK 1976

[87] P Carruthers ldquoTheory of thermal conductivity of solids at lowtemperaturesrdquo Reviews of Modern Physics vol 33 no 1 pp 92ndash138 1961

[88] MCardona andM LWThewalt ldquoIsotope effects on the opticalspectra of semiconductorsrdquo Reviews of Modern Physics vol 77no 4 pp 1173ndash1224 2005

[89] N Mingo D Hauser N P Kobayashi M Plissonnier and AShakouri ldquoNanoparticle-in-alloy approach to efficient thermo-electrics silicides in SiGerdquoNano Letters vol 9 no 2 pp 711ndash7152009

[90] Z Zhou C Uher A Jewell and T Caillat ldquoInfluence of point-defect scattering on the lattice thermal conductivity of solidsolution Co(Sb

1minus119909As

119909)3rdquo Physical Review B Condensed Matter

and Materials Physics vol 71 no 23 Article ID 235209 2005[91] A P Zhernov ldquoThe solution of the kinetic equation for phonon

heat conductivity by the method of momenta and the influenceof isotopic disorder on phonon heat conductivity of germaniumand silicon crystals at T = 300 Krdquo Journal of Experimental andTheoretical Physics vol 93 no 5 pp 1074ndash1081 2001

[92] J E Turney A J H McGaughey and C H Amon ldquoIn-planephonon transport in thin filmsrdquo Journal of Applied Physics vol107 no 2 Article ID 024317 2010

[93] Y Takeda and T P Pearsall ldquoFailure of Matthiessenrsquos rule inthe calculation of carrier mobility and alloy scattering effects inGa047In053Asrdquo Electronics Letters vol 17 no 16 pp 573ndash5741981

[94] C J Glassbrenner and G A Slack ldquoThermal conductivity ofsilicon and germanium from 3∘ to the melting pointrdquo PhysicalReview vol 134 no 4 article A1058 1964

[95] J M Ziman Electrons and Phonons Clarendon Press 1962[96] S Baroni P Giannozzi and A Testa ldquoGreens-function

approach to linear response in solidsrdquo Physical Review Lettersvol 58 no 18 pp 1861ndash1864 1987

[97] X Gonze ldquoPerturbation expansion of variational principlesat arbitrary orderrdquo Physical Review A Atomic Molecular andOptical Physics vol 52 no 2 pp 1086ndash1095 1995

[98] P Giannozzi S Baroni N Bonini et al ldquoQUANTUMESPRESSO a modular and open-source software project forquantum simulations of materialsrdquo Journal of Physics Con-densed Matter vol 21 no 39 Article ID 395502 2009

[99] C Ni and J Y Murthy ldquoPhonon transport modeling usingBoltzmann transport equation with anisotropic relaxationtimesrdquo Journal of Heat Transfer vol 134 no 8 Article ID082401 2012

[100] Y He I Savic D Donadio and G Galli ldquoLattice thermalconductivity of semiconducting bulkmaterials atomistic simu-lationsrdquo Physical Chemistry Chemical Physics vol 14 no 47 pp16209ndash16222 2012

[101] I Savic D Donadio F Gygi and G Galli ldquoDimensionality andheat transport in Si-Ge superlatticesrdquo Applied Physics Lettersvol 102 no 7 Article ID 073113 2013

[102] J E Turney E S Landry A J H McGaughey and CH Amon ldquoPredicting phonon properties and thermal con-ductivity from anharmonic lattice dynamics calculations andmolecular dynamics simulationsrdquoPhysical ReviewB CondensedMatter and Materials Physics vol 79 no 6 Article ID 0643012009

[103] I O Thomas and G P Srivastava ldquoThermal conductivity ofgraphene and graphiterdquo Physical Review B Condensed Matterand Materials Physics vol 87 no 11 Article ID 085410 2013


[104] T Shiga J Shiomi J Ma et al ldquoMicroscopic mechanism oflow thermal conductivity in lead telluriderdquo Physical Review BCondensed Matter and Materials Physics vol 85 no 15 ArticleID 155203 2012

[105] N Bonini J Garg and N Marzari ldquoAcoustic phonon lifetimesand thermal transport in free-standing and strained graphenrdquoNano Letters vol 12 no 6 pp 2673ndash2678 2012

[106] Z Tian J Garg K Esfarjani T Shiga J Shiomi and G ChenldquoPhonon conduction in PbSe PbTe and PbTe

1minus119909Se

119909from first-

principles calculationsrdquo Physical Review B Condensed Matterand Materials Physics vol 85 no 18 Article ID 184303 2012

[107] K Esfarjani G Chen and H T Stokes ldquoHeat transport insilicon from first-principles calculationsrdquo Physical Review BCondensed Matter and Materials Physics vol 84 no 8 ArticleID 085204 2011

[108] J Garg N Bonini and N Marzari ldquoHigh thermal conductivityin short-period superlatticesrdquo Nano Letters vol 11 no 12 pp5135ndash5141 2011

[109] J Garg N Bonini B Kozinsky and N Marzari ldquoRole ofdisorder and anharmonicity in the thermal conductivity ofsilicon-germanium alloys a first-principles studyrdquo PhysicalReview Letters vol 106 no 4 Article ID 045901 2011

[110] A Ward and D A Broido ldquoIntrinsic phonon relaxation timesfrom first-principles studies of the thermal conductivities of Siand Gerdquo Physical Review B Condensed Matter and MaterialsPhysics vol 81 no 8 Article ID 085205 2010

[111] L Paulatto F Mauri and M Lazzeri ldquoAnharmonic propertiesfrom a generalized third-order ab initio approach yheory andapplications to graphite and graphenerdquo Physical Review BCondensed Matter and Materials Physics vol 87 no 21 ArticleID 214303 2013

[112] L Lindsay D A Broido and T L Reinecke ldquoAb initio thermaltransport in compound semiconductorsrdquo Physical Review BCondensed Matter and Materials Physics vol 87 no 16 ArticleID 165201 2013

[113] WH Press S A TeukolskyW T Vetterling and B P FlanneryNumerical Recipes The Art of Scientific Computing CambridgeUniversity Press Cambridge UK 2007

[114] J R Olson R O Pohl J W Vandersande A Zoltan TR Anthony and W F Banholzer ldquoThermal conductivity ofdiamond between 170 and 1200 K and the isotope effectrdquoPhysical Review B CondensedMatter andMaterials Physics vol47 no 22 pp 14850ndash14856 1993

[115] L Wei P K Kuo R L Thomas T R Anthony and WF Banholzer ldquoThermal conductivity of isotopically modifiedsingle crystal diamondrdquo Physical Review Letters vol 70 no 24pp 3764ndash3767 1993

[116] R Berman ldquoThermal conductivity of isotopically enricheddiamondsrdquo Physical Review B Condensed Matter and MaterialsPhysics vol 45 no 10 pp 5726ndash5728 1992

[117] L Lindsay and D A Broido ldquoThree-phonon phase spaceand lattice thermal conductivity in semiconductorsrdquo Journal ofPhysics Condensed Matter vol 20 no 16 Article ID 1652092008

[118] T F Luo J Garg K Esfarjani J Shiomi and G Chen ldquoGalliumarsenide thermal conductivity and optical phonon relaxationtimes from first-principles calculationsrdquo Europhysics Lettersvol 101 no 1 article 16001 2013

[119] MN Luckyanova J Garg K Esfarjani et al ldquoCoherent phononheat conduction in superlatticesrdquo Science vol 338 no 6109 pp936ndash939 2012

[120] J Shiomi K Esfarjani and G Chen ldquoThermal conductivityof half-Heusler compounds from first-principles calculationsrdquoPhysical Review B CondensedMatter andMaterials Physics vol84 no 10 Article ID 104302 2011

[121] O Delaire J Ma K Marty et al ldquoGiant anharmonic phononscattering in PbTerdquo Nature Materials vol 10 no 8 pp 614ndash6192011

[122] S N Girard J He C Li et al ldquoIn situ nanostructure generationand evolution within a bulk thermoelectric material to reducelattice thermal conductivityrdquo Nano Letters vol 10 no 8 pp2825ndash2831 2010

[123] Z Wang and N Mingo ldquoAbsence of Casimir regime in two-dimensional nanoribbon phonon conductionrdquo Applied PhysicsLetters vol 99 no 10 Article ID 101903 2011

[124] N Mingo and D A Broido ldquoLength dependence of carbonnanotube thermal conductivity and the lsquoproblemof longwavesrsquordquoNano Letters vol 5 no 7 pp 1221ndash1225 2005

[125] E Munoz J Lu and B Yakobson ldquoBallistic thermal conduc-tance of graphene ribbonsrdquoNano Letters vol 10 no 5 pp 1652ndash1656 2010

[126] L Chen and S Kumar ldquoThermal transport in graphene sup-ported on copperrdquo Journal of Applied Physics vol 112 no 4Article ID 043502 2012

[127] B Qiu and X Ruan ldquoMechanism of thermal conductivityreduction from suspended to supported graphene a quantita-tive spectral analysis of phonon scatteringrdquo in Proceedings ofthe ASME 2011Mechanical Engineering Congress and ExpositionASME Paper IMECE2011-62963 November 2011

[128] B Qiu and X Ruan ldquoMolecular dynamics simulations of ther-mal conductivity and spectral phonon relaxation time in sus-pended and supported graphenerdquo httparxivorgabs11114613

[129] A Alofi and G P Srivastava ldquoPhonon conductivity ingraphenerdquo Journal of Applied Physics vol 112 Article ID 0135172012

[130] Z Aksamija and I Knezevic ldquoThermal transport in graphenenanoribbons supported on SiO

2rdquo Physical Review B vol 86

no 16 Article ID 165426 2012[131] A Y Serov Z Ong and E Pop ldquoEffect of grain boundaries on

thermal transport in graphenerdquoApplied Physics Letters vol 102no 3 Article ID 033104 2013

[132] A I Cocemasov D L Nika and A A Balandin ldquoPhonons intwisted bilayer graphenerdquoPhysics ReviewB vol 88 no 3 ArticleID 035428 2013

[133] Z Tian K Esfarjani J Shiomi A S Henry and G Chen ldquoOnthe importance of optical phonons to thermal conductivity innanostructuresrdquo Applied Physics Letters vol 99 no 5 ArticleID 053122 2011

[134] R E PeierlsQuantumTheory of Solids OxfordUniversity PressLondon UK 1955

[135] P A M Dirac ldquoOn the theory of quantum mechanicsrdquoProceedings of the Royal Society A vol 112 no 762 pp 661ndash6771926

[136] W S Capinski H JMaris and S Tamura ldquoAnalysis of the effectof isotope scattering on the thermal conductivity of crystallinesiliconrdquo Physical Review B Condensed Matter and MaterialsPhysics vol 59 no 15 pp 10105ndash10110 1999

[137] B Abeles ldquoLattice thermal conductivity of disordered semicon-ductor alloys at high temperaturesrdquo Physical Review vol 131 no5 pp 1906ndash1911 1963


[138] N A Katcho N Mingo and D A Broido ldquoLattice thermalconductivity of (Bi

1minus119909Sb

119909)2Te

3alloys with embedded nanopar-

ticlesrdquo Physical Review B Condensed Matter and MaterialsPhysics vol 85 no 11 Article ID 115208 2012

[139] W A Kamitakahara and B N Brockhouse ldquoVibrations ofa mixed crystal neutron scattering from Ni55Pd45rdquo PhysicalReview B Condensed Matter and Materials Physics vol 10 no4 pp 1200ndash1212 1974

[140] B Qiu and X Ruan ldquoReduction of spectral phonon relaxationtimes from suspended to supported graphenerdquo Applied PhysicsLetters vol 100 no 19 Article ID 193101 2012

[141] B Qiu H Bao G Zhang Y Wu and X Ruan ldquoMoleculardynamics simulations of lattice thermal conductivity and spec-tral phonon mean free path of PbTe bulk and nanostructuresrdquoComputational Materials Science vol 53 no 1 pp 278ndash2852011

[142] Y G Wang B Qiu A McGaughey X L Ruan and X F XuldquoMode-wise thermal conductivity of bismuth telluriderdquo Journalof Heat Transfer vol 135 no 9 Article ID 091102 6 pages 2013

[143] A S Henry and G Chen ldquoSpectral phonon transport proper-ties of silicon based on molecular dynamics simulations andlattice dynamicsrdquo Journal of Computational and TheoreticalNanoscience vol 5 no 2 pp 141ndash152 2008

[144] P N Keating ldquoEffect of invariance requirements on the elasticstrain energy of crystals with application to the diamondstructurrdquo Physical Review vol 145 no 2 pp 637ndash645 1966

[145] P N Keating ldquoTheory of the third-order elastic constants ofdiamond-like crystalsrdquo Physical Review vol 149 no 2 pp 674ndash678 1966

[146] W Weber ldquoAdiabatic bond charge model for the phonons indiamond Si Ge and 120572-Snrdquo Physical Review B CondensedMatter andMaterials Physics vol 15 no 10 pp 4789ndash4803 1977

[147] K C Rustagi and W Weber ldquoAdiabatic bond charge model forthe phonons in A3B5 semiconductorsrdquo Solid State Communica-tions vol 18 no 6 pp 673ndash675 1976

[148] M T Dove Introduction to Lattice Dynamics CambridgeUniversity Press New York NY USA 1993

[149] D C Wallace Thermodynamics of Crystals Wiley New YorkNY USA 1972

[150] J E Turney J AThomas A J HMcGaughey andCH AmonldquoPredicting phonon properties from molecular dynamics sim-ulations using the spectral energy densityrdquo in Proceedings ofthe ASMEJSME 2011 8thThermal Engineering Joint ConferenceAJTEC2011-44315 March 2011

[151] T Feng BQiu andXRuan ldquoRole ofharmonic and anharmonicphonon eigenvectors in the phonon normal mode analysisrdquo

[152] Z Y Ong E Pop and J Shiomi ldquoReduction of phonon lifetimesand thermal conductivity of a carbon nanotube on amorphoussilicardquo Physical Review B Condensed Matter and MaterialsPhysics vol 84 no 16 Article ID 165418 2011

[153] D Donadio and G Galli ldquoAtomistic simulations of heat trans-port in silicon nanowiresrdquo Physical Review Letters vol 102 no19 Article ID 195901 2009

[154] J V Goicochea and B Michel ldquoPhonon relaxation times ofgermanium determined by molecular dynamics at 1000 Krdquo inProceedings of the IEEECPMT 26th Semiconductor ThermalMeasurement Modeling amp Management Symposium (SEMI-THERM rsquo10) pp 278ndash282 February 2010

[155] A Henry and G Chen ldquoAnomalous heat conduction inpolyethylene chains theory and molecular dynamics simu-lationsrdquo Physical Review B Condensed Matter and MaterialsPhysics vol 79 no 14 Article ID 144305 2009

[156] AHenry andGChen ldquoExplicit treatment of hydrogen atoms inthermal simulations of polyethylenerdquoNanoscale andMicroscaleThermophysical Engineering vol 13 no 2 pp 99ndash108 2009

[157] J Shiomi and S Maruyama ldquoNon-Fourier heat conduction ina single-walled carbon nanotube classical molecular dynamicssimulationsrdquo Physical Review B Condensed Matter and Materi-als Physics vol 73 no 20 Article ID 205420 2006

[158] YWang C Liebig X Xu and R Venkatasubramanian ldquoAcous-tic phonon scattering in Bi

2Te

3Sb

2Te

3superlatticesrdquo Applied

Physics Letters vol 97 no 8 Article ID 083103 2010

Submit your manuscripts athttpwwwhindawicom

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materials


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parameters and with greater accuracy This review will befocused on these predictive simulation methods

The methods of predicting spectral phonon relaxationtimes and mean free paths become increasingly importantfor predicting the thermal properties of numerous novelmaterials For instance superlattice structure is found tobe an effective way to suppress the thermal conductivitybecause of the interface mass mismatch scattering [6ndash8] butthe phenomenon in which the short-period superlattice canhave even higher thermal conductivity still needs the deepinsight of phonon relaxation time Doping and alloying arewidely used to explore novel high performance materials[9] and the natural materials are rarely pure thus the studyof impurity scattering contributing to phonon relaxationtime is important Layer and tube structured materials forexample graphene [10ndash13] and carbon nanotube (CNT) [14ndash17] are proved to have unusual phonon transport featuressuch as high thermal conductivity [18ndash20] which still needto be further understood Comprehensive reviews of theirthermal transport can be found in [21ndash23] Nanowires aremost commonly studied and used in both theoretical andexperimental research [24ndash32] and the accurate predictionof thermal conductivity needs the knowledge of spectralphonon scattering by boundaries

Manymethods have been proposed and applied to predictspectral phonon relaxation time in the last half centuryAt the earliest Klemens and other researchers obtainedthe frequency-dependent phonon relaxation time mostly bylong-wave approximation (LWA) andDeybemodel Klemensobtained the phonon relaxation times by Umklapp (119880)three-phonon scattering [33 34] and defect scattering [35]Herring studied normal (119873) three-phonon scattering [36]Holland extended the results to dispersive transverse moderange [37] and Casimir studied boundary scattering [38ndash40] A more accurate method the third-order anharmoniclattice dynamics (ALD) calculation which can predict theintrinsic spectral phonon relaxation times without LWAwas presented by Maradudin and the coworkers [41 42]ALD methods were then applied to silicon and germaniumby ab initio approach first by Debernardi et al [43] andDeinzer et al [44] Beyond the standard ALD calculationOmini and Sparavigna [45 46] proposed an Iterative Schemewhich gives exact solutions to the linearized Boltzmanntransport equation (BTE) The Iterative Scheme has beensuccessfully applied tomany structures in the recent ten yearsby Broido Lindsay Ward and so forth [47ndash65] Other thanthe lattice dynamics calculation a time domain normalmodeanalysis (NMA)method based onmolecular dynamics (MD)simulation was proposed by Ladd et al [66] and extendedby McGaughey and Kaviany [67] Another version of normalmode analysis is implemented in frequency domain so calledspectral energy density (SED) analysis The normal modeanalysis was early implemented by Wang et al [68ndash75] toobtain the relaxation times of a few phonon modes and thenextended by de Koker [76] and Thomas et al [77 78] tocalculate lattice thermal conductivity

In this work we present a review of the methods ofpredicting spectral phonon properties discuss the applica-tions to each method and compare them in different aspects

Section 2 gives an overview of thermal conductivity and thefrequency-dependent relaxation time predicted from earlylong-wave approximation (LWA) andDebyemodel Section 3presents the ALD calculation which is divided into threesubsections Section 31 covers the standard single moderelaxation time approximation (SMRTA) Section 32 givesthe Iterative Scheme and Section 33 reviews examples ofthe applications to pure bulk layer and tube structuresnanowires defective materials and superlattice In Section 4we introduce the time domain NMA and frequency NMAmethods based on MD simulations and their applicationsThe summary is presented in Section 5 The appendix pro-vides some derivations of ALD methods

2 Theory Overview

Spectral phonon mean free path (MFP) determined byphonon scattering rate dominates the behavior of thermalproperties especially the thermal conductivity 119896 Based onBTEunder the relaxation time approximation (RTA) thermalconductivity is determined by the spectral phonon relaxationtime 120591

120582 phonon group velocity k

120582= nablak120596 and phonon

specific heat 119888120582[34]

119896119911=1

119881sum

120582

(k120582sdot )

2

119888120582120591120582 (1)

where denotes the transport direction 120582 is the shorthandof phonon mode (k ]) with k representing the phonon wavevector and ] labeling phonon dispersion branch 119881 is thevolume of the domain and the summation is done over theresolvable phononmodes in the domainThe specific heat permode is 119888

120582= ℏ120596

120582120597119899

0

120582120597119879 = 119896

119861119909

2119890119909(119890

119909minus 1)

2 where 1198990

120582is

phonon occupation number of the Bose-Einstein distribution1198990

120582= (119890

119909minus 1)

minus1 and 119909 is the shorthand of ℏ120596119896119861119879 Equation

(1) can also be expressed in terms of phonon mean free pathrarr

Λ 120582= k

120582120591120582 The continuous form of (1) is with the help of

sumk = (119881(2120587)3) int 119889k

119896119911=

1

(2120587)3sum

]int (k

120582sdot )

2

119888120582120591120582119889k (2)

If isotropic heat transport is assumed the integration of|V

120582119911|2 in (2) gives V2

1205823 and int119889k gives int 41205871198962

119889119896 we get thecommonly used formula

119896119911=4120587

3

1

(2120587)3sum

]int 119888

120582V120582Λ

1205821198962119889119896 (3)

The early theoretical predictions of phonon relaxationtimes for different scattering processes are briefly summa-rized in Table 1 [37 79] 119879 is temperature subscripts 119873 119880119879 and 119871 indicate the Umklapp scattering normal scatteringtransverse wave and longitudinal wave respectively 119860 119861rsquosand 119862rsquos are constants 120579 is Debye temperature 120572 is numericalconstant in [34] Low119879means119879 ≪ 120579 and high119879means119879 ≫

120579 1205961is the transverse mode frequency at which the group

velocity starts to decrease and 1205962is the maximum transverse





119871119873= 119861

119871120596

2119879

3 low 119879

120591minus1

119879119873= 119861

119879120596119879

4 low 119879

120591minus1

119871119873= 119861

1015840

119871120596

2119879 high 119879

120591minus1

119879119873= 119861

1015840

119879120596119879 high 119879


119873= 119861

119873120596

2119879

3

119880 process


119880= 119861

119880120596

2119879

3 exp(minus 120579

120572119879) low 119879


119880= 119861

119880120596119879

3 exp(minus 120579

120572119879) low 119879

120591minus1

119880= 119861

1015840120596

2119879 high 119879


119880= 119861

119880120596

2119879

3


119879119880=

119861119879119880120596

2

sinh (119909) 120596

1le 120596 le 120596

2

0 120596 lt 1205961


119879119880= 119861

1015840

119879119880120596

2119879 exp(119862119879

119879)

120591minus1

119871119880= 119861

1015840

119871119880120596

2119879 exp(119862119871

119879)


119887=V120582

119897119865119897 diameter 119897 = 2radic119897

11198972120587



im =120587

2119892120596

2119863(120596) asymp

1198810119892

4120587 ⟨V3⟩120596

4sim 119860120596

4








120582is often


Vave = [1

3

3

sum

]

1

V]]

minus1

= [1

3(2

V119879

+1

V119871

)]

minus1

(4)


119892 = sum

119894

119891119894(1 minus

119898119894

119898)

2

(5)






119892 = 119892 + 2sum

119894

119891119894(Δ120601

119894

120601minus64120574Δ119903

119894

119903)

2

(6)

where 120601119894120601 and Δ119903



minus1



minus1




120591minus1= sum

119894

120591minus1

119894 (7)



sim





119871sim 120596

2119879


119879sim 120596






120582=120597119899

120582

120597119905

10038161003816100381610038161003816100381610038161003816119904

(8)


where 119899120582= 119899

0

120582+ 119899

1015840


with 1198991015840



120582 With nabla119899

120582= (120597119899

120582120597119879)nabla119879 and



120582120597119879) ≃

(1205971198990

120582120597119879) we have


1205971198990

120582

120597119879=120597119899

1015840

120582

120597119905

100381610038161003816100381610038161003816100381610038161003816119904

(9)


1198991015840


120591120582

) (10)


in BTE (9) becomes

1205971198991015840

120582

120597119905

100381610038161003816100381610038161003816100381610038161003816119904

≃ minus1198991015840

120582

120591120582

(11)



120582= 1Γ

120582 where Γ




1205971198990

120582

120597119879

= minussum

120582101584012058210158401015840

[1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840]L

+

+1

2[119899

120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840]L

minus

(12)



120582plusmn 120596

1205821015840 = 120596

12058210158401015840


+G withG =



rarr 12058210158401015840


Lplusmn=

ℏ120587

41198730

10038161003816100381610038161003816119881

(3)

plusmn

10038161003816100381610038161003816

2 120575 (120596120582plusmn 120596

1205821015840 minus 120596

12058210158401015840)

120596120582120596

1205821015840120596

12058210158401015840

(13)

119881(3)

plusmn= sum

119887119897101584011988710158401198971015840101584011988710158401015840

sum

120572120573120574

Φ120572120573120574

0119887119897101584011988710158401198971015840101584011988710158401015840

119890120582

120572119887119890plusmn1205821015840

1205731198871015840 119890

minus12058210158401015840

12057411988710158401015840

radic119898119887119898

1198871015840119898

11988710158401015840

119890plusmn119894k1015840 sdotr

1198971015840

119890minus119894k10158401015840sdotr

11989710158401015840

(14)


119887



119887120572




(3)

plusmn|2


120582= 119899

0

120582+ 119899

1015840

120582differing a


120582 Therefore on


120582+ 119899

1015840

120582 whilst

1198991205821015840 and 119899

12058210158401015840 by 1198990

1205821015840 and 119899

0




Appendix A1)

1

1205910

120582

=

+

sum

120582101584012058210158401015840

Γ+

120582120582101584012058210158401015840 +

1

2

minus

sum

120582101584012058210158401015840

Γminus

120582120582101584012058210158401015840 +sum

1205821015840

Γext1205821205821015840 (15)


120582plusmn 120596

1205821015840 = 120596

12058210158401015840)

Γplusmn

120582120582101584012058210158401015840 =

ℏ120587

41198730

1198990

1205821015840 minus 119899

0

12058210158401015840

1198990

1205821015840 + 119899

0

12058210158401015840 + 1

times10038161003816100381610038161003816119881

(3)

plusmn

10038161003816100381610038161003816

2 120575 (120596120582plusmn 120596

1205821015840 minus 120596

12058210158401015840)

120596120582120596

1205821015840120596

12058210158401015840

(16)




120582 119899

1205821015840 and 119899

12058210158401015840 by 1198990

120582+ 119899

1015840

120582 1198990

1205821015840 + 119899

1015840

1205821015840

and 1198990

12058210158401015840 + 119899

1015840



see Appendix A2)

120591120582= 120591

0

120582(1 + Δ

120582) (17)

Δ120582=

(+)

sum

120582101584012058210158401015840

Γ+

120582120582101584012058210158401015840 (1205851205821205821015840101584012059112058210158401015840 minus 12058512058212058210158401205911205821015840)

+

(minus)

sum

120582101584012058210158401015840

1

2Γminus

120582120582101584012058210158401015840 (1205851205821205821015840101584012059112058210158401015840 + 12058512058212058210158401205911205821015840)

+sum

1205821015840

Γext120582120582101584012058512058212058210158401205911205821015840

(18)

where 1205851205821205821015840 = V

1205821015840119911120596

1205821015840V

120582119911120596

120582 and V





1015840 with 1205821015840= 120582




11989711198871119897119899119887119899

=120597119899Φ

1205971199061(119897

11198871) 120597119906

119899(119897

119899119887119899) (19)



10158401015840) is typically


(1120587)(120576(1199092+ 120576

2)) To








1

120591ext120582

= sum

1205821015840

Γext1205821205821015840 (20)



imp120582

1000

800

600

400

200

001 1

120596 (THz)

Erro

r (

)

Rela

xatio

n tim

e (s

)

10minus7

10minus8

10minus9

10minus10

10minus11

10minus12

120591(U)(120596)

120591(N)(120596)120591(0)(120596)

0 10 20 30 40 50 60 70 80

120596eff (THz)







plusmn|2 is shown



120582120596

1205821015840120596

12058210158401015840)




dependence 120591(119880)(120596) sim 120596





120582+

Γ(119880)



120582= 1120591

(119873)

120582+ 1120591

(119880)

120582

where 120591(119873119880)

120582is defined as 120591(119873119880)

120582equiv 1Γ

(119873119880)






minus2



10000

1000

Error ()

Iterative scheme

Standard SMRTA

80

70

60

50

40

30

20

10

01000500300200

Temperature (K)

er

mal

con

duct

ivity

(Wm

-K)

Erro

r (

)



2Si

06Sn


2Si and Mg

2Sn cross at



100

10

1

01 1

TATALA

TOTOLO

Rela

xatio

n tim

e (p

s)

120591 prop 120596minus2


(a)

100

10

1

01 1

Normal

120591 prop 120596minus2


TATALA

TOTOLO

Rela

xatio

n tim

e (p

s)

(b)

100

10

1

01 1

Umklapp

TATALA

TOTOLO

120591 prop 120596minus3


Rela

xatio

n tim

e (p

s)

(c)


12120596

minus2 (b) 120591 = 8times1012120596

minus2 and (c) 120591 = 8times1024120596




1

120591bs=21003816100381610038161003816k120582

sdot 1003816100381610038161003816

119871

(21)






103

102

101

100

10010minus1

Life

time

(ps)

TA

Frequency (THz)

PbSePbTe

120591 sim 120596minus2

(a)

103

102

101

100

10010minus1

Life

time

(ps)

LA

Frequency (THz)

PbSePbTe

120591 sim 120596minus2

(b)

101100

101

100

10minus1

Life

time

(ps)

TO

Frequency (THz)

PbSePbTe

(c)

101100

101

100

10minus1

Life

time

(ps)

LO

Frequency (THz)

PbSePbTe

(d)






10minus1 100 101 102 103 104 105 106



PbSe

ZrCoSb

Phonon MFP (nm)

Acc

umul

ated

ther

mal

con

duct

ivity

()

100

80

60

40

20

0


05Se

05[106] Mg

2Si [63] Mg

2Sn

[63] Mg2Si

06Sn

04[63] ZrCoSb [120])





12

1

08

06

04

02

0 1 2 3 4 5 6 7 8

Diameter d (nm)

kGNR kgraphene

kRTA

kk

grap

hene k


ZA

TA

LA

10

11

10

L (120583m)

kLk

RTA





007

006

005

004

003

002

001

00 01 02 03 04 0 01 02 03 04 0 01 02 03 04 05

0 01 02 03 04 05

Scat

terin

g ra

te(c

mminus1)

Scat

terin

g ra

te(c

mminus1)

120585 120585 120585

1

120591TAasymp

1

120591aTA+

1

120591bTA+

1

120591cTA

Optical Optical

Optical

Acoustic

Acoustic Acoustic


02

015

01

005

0

120585

1

120591aTA

1

120591bTA

1

120591cTA

1

120591TA

TATATA

(a) (b) (c)

(d)





120591r120582 = 1205910

120582(1 + Δ

120582) 1 minus 119890


120582V120582|119866r120582 (22)







120591120582= 120591

0

120582(1 + Δ

120582) [

1

119878119888

int119878119888


119887)1205910

120582V120582|119866r120582 119889s] (23)



Γiso1205821205821015840 =

120587

2119873119888

120596120582120596

1205821015840

119899

sum

119887

119892119887

10038161003816100381610038161003816e119887120582sdot e119887lowast

1205821015840

10038161003816100381610038161003816

2

120575 (120596120582minus 120596

1205821015840) (24)




1

011 15 2 25 3

1 times 1

2 times 2

4 times 4

kSLk

T

M1M2


1119872

2 the mass ratio of



119892119887= sum

119894

119891119894119887(1 minus

119898119894119887

119898119887

)

2

(25)




119887is the




1

120591iso120582

= sum

1205821015840

Γiso1205821205821015840 =

120587

2119873119888

1205962

120582

119899

sum

119887

sum

1205821015840

119892119887

10038161003816100381610038161003816e119887120582sdot e119887lowast

1205821015840

10038161003816100381610038161003816

2

120575 (120596120582minus 120596

1205821015840)

(26)


1

120591iso120582

=120587

6119899119873119888

1198921205962

120582sum

1205821015840

120575 (120596120582minus 120596

1205821015840)

=120587119881

0

6119892120596

2119863 (120596)

(27)

=120587

2119892120596

2119863 (120596) (28)

where 119892 is given by (5)1198810= 119881(119899119873


119863(120596) = sum1205821015840 120575(120596120582




10

08

06

04

02

00 2 4 6 8 10

t (ps)


i(0)Ei(0)⟩


120587

120596i




is 119881 = 1198810119899119873






055Pd

045alloys [139] for




10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Empty CNT


278ps(429ps)

80ps(138ps)

25ps(26ps)

263ps(268ps)

47ps(50ps)

0 1 2 3 4 5

Φ(kz=0120596

) (J s

)

(a)

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11


Water-filled CNT

24ps 22ps 23ps

250ps

46ps

0 1 2 3 4 5

Φ(kz=0120596

) (J s

)

(b)



20

15

10

05

000806040200 10

Longitudinal

1120591

(1p

s)

20K LD50K LD

20K MD50K MD

klowast (a2120587)

(a)

08

06

04

02

000806040200 10

Transverse

1120591

(1p

s)

20K LD50K LD

20K MD50K MD

klowast (a2120587)

(b)




(at the limit 119875119871



119871 This leads



119871









50

40

30

20

10

050403020100

Phon

on li

fetim

e (p

s)








1+1superlattice We



1119872





1119872



1119872



1119872




4 MD Simulation



120591120582=

intinfin

0⟨119899

1015840

120582(119905) 119899

1015840

120582(0)⟩ 119889119905

⟨11989910158402

120582(0)⟩

(29)







119864120582=1

2( 119902

120582119902lowast

120582+ 120596

2

120582119902120582119902lowast

120582) (30)


transformed to

120591120582=

intinfin

0⟨119864

120582(119905) 119864

120582(0)⟩ 119889119905

⟨1198642

120582(0)⟩

(31)


120582(119905)119864

120582(0)⟩⟨119864

2

120582(0)⟩ = exp(minus119905120591

120582)


120582sim 119902

120582119902lowast



119902120582(119905) = 119902

1205820exp [119894 (120596119860

120582+ 119894120578

120582) 119905] (32)

where 1199021205820


120582= 120596

120582+ Δ


120582


120582119902lowast

120582

and 119864120582sim 119902

120582119902lowast


120582= 12120578

120582



120582in (31)


119902120582(119905) =

3

sum

120572

119899

sum

119887

119873119888

sum

119897

radic119898

119887

119873119888

119906119897119887

120572(119905) 119890

120582lowast

119887120572exp [119894k sdot r119897

0]

=

3

sum

120572

119899

sum

119887

119890120582lowast

119887120572119902119887

120572(k 119905)

(33)


120572(119905) is 120572 component




120572(k 119905) denotes



300

200

100

0

ZA sTA sLA sZO s

ZA pTA pLA pZO p

er

mal

con

duct

ivity

(Wm

K)

(a)

Temperature (K)

()

300 400 500 600

40

30

20

10

ZA sTA sLA sZO s

ZA pTA pLA pZO p

(b)



119902119887

120572(k 119905) =

119873119888

sum

119897

radic119898

119887

119873119888

119906119897119887

120572(119905) exp [119894k sdot r119897

0] (34)



Φ120582(120596) =

1003816100381610038161003816 119902120582(120596)

1003816100381610038161003816

2

=

10038161003816100381610038161003816100381610038161003816

int

+infin

0

119902120582(119905) 119890

minus119894120596119905119889119905

10038161003816100381610038161003816100381610038161003816

2

=119862

120582

(120596 minus 120596119860

120582)2

+ 1205782

120582

(35)

where119862120582= (120596

119860

120582

2

+Γ2

120582)119902

2






120582= 12120578




Φ (k 120596) =3119899

sum

]Φ

120582(120596)

=1

41205871205910

3119899

sum

120572119887

119898119887

119873119888

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119897

1205910

int

0

119897119887

120572(119905) exp [119894k sdot r119897

0minus 119894120596119905]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

=

3

sum

120572

119899

sum

119887

10038161003816100381610038161003816119902119887

120572(k 120596)10038161003816100381610038161003816

2

(36)

where 119902119887





119888in (34) are




Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)100

10

1

100

10

10 1 2 0 1 2


300K 600K

[100] LA[110] LA[111] LA

[100] LA[110] LA[111] LA

(a)

[100] TA[110] TA1[110] TA2[111] TA

[100] TA[110] TA1[110] TA2[111] TA

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

300K 600K

(b)

1

0135 40

1

0135 40

300K 600K

[100] LO[110] LO[111] LO

[100] LO[110] LO[111] LO

Rela

xatio

n tim

e (p

s)

Frequency (THz)

Rela

xatio

n tim

e (p

s)

Frequency (THz)

(c)

30

300K 600K

[100] TO[110] TO1[110] TO2[111] TO

[100] TO[110] TO1[110] TO2[111] TO

1

01

1

0135 40 30 35 40

Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)


(d)







80

60

40

20

00 1 2 3 4

46

23

26

300K

Frequency (THz)

TA1TA2LA

TO1TO2LO

Perc

enta

ge c

ontr

ibut

ion

(au

)

(a)

80

60

40

20

00 1 2 3 4

50

2124

TA1TA2LA

TO1TO2LO

600K

Frequency (THz)

Perc

enta

ge c

ontr

ibut

ion

(au

)

(b)


150

100

50

0 1 2 3 4 5

Frequency (THz)

L = 12

L = 9

L = 6

Acoustic phonon



Rela

xatio

n tim

e (p

s)

(a)


0

150

100

50

02 04 06 08

Predicted by NMA

Rela

xatio

n tim

e (p

s)


(b)










10minus1 100 101 102 103 104 105



MFP (nm)

Acc

umul

ated

ther

mal

con

duct

ivity

() 100

80

60

40

20

0


2Te

3(MD) [142])

[141] and Bi2Te









2Te



3


2Te


in the future

5 Summary






1119872

2SLs 1 and 2 [59] 2004


1119872

2SLs 1 and 2 radic [50] 2008



2 and SrTiO3) 2 [49] 2011



Table 2 Continued


1minus119909Se

1199091 radic [106] 2012

Bi2Te

34 [142] 2013





Equations (14)(15) (16) (17)

and (18)














Applications sofar











Appendix

A


120596 + 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) (119899

0

1205821015840 + 1) 119899

0

12058210158401015840 = 119899

0

1205821198990

1205821015840 (119899

0

12058210158401015840 + 1) (A1)

120596 minus 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) 119899

0

1205821015840119899

0

12058210158401015840 = 119899

0

120582(119899

0

1205821015840 + 1) (119899

0

12058210158401015840 + 1) (A2)


119861119879) minus 1]


1198991015840

120582= minusΨ

120582

1205971198990

120582

120597 (ℏ120596120582)= Ψ

120582sdot

1

1198961198611198791198990

120582(119899

0

120582+ 1) (A3)


approximation (11)

Ψ120582= 120591

120582

ℏ120596120582

119879k120582sdot nabla119879 (A4)


119899120582= 119899

0

120582+ 119899

1015840

120582

1198991205821015840 = 119899

0

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840

(A5)


1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

= 1198991205821015840 [119899

0

1205821015840 (1 + 119899

0

12058210158401015840) minus (1 + 119899

0

1205821015840) 119899

0

12058210158401015840]

= 1198991205821015840 (119899

0

1205821015840 minus 119899

0

12058210158401015840)

(A6)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (119899

0

1205821015840 minus 119899

0

12058210158401015840)

119896119861119879

(A7)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

= 1198991205821015840 [(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840) minus 119899

0

1205821015840119899

0

12058210158401015840]

= 1198991205821015840 (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

(A8)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

119896119861119879

(A9)


1205821015840




119899120582= 119899

0

120582+ 119899

1205821015840

1198991205821015840 = 119899

0

1205821015840 + 119899

1015840

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840 + 119899

1015840

12058210158401015840

(A10)

where 1198991015840

1205821015840 and 1198991015840


120582

1198991015840

1205821015840 = minusΨ

1205821015840

1205971198990

1205821015840

120597 (ℏ1205961205821015840)= Ψ

1205821015840 sdot

1

1198961198611198791198990

1205821015840 (119899

0

1205821015840 + 1) (A11)

1198991015840

12058210158401015840 = minus Ψ

12058210158401015840

1205971198990

12058210158401015840

120597 (ℏ12059612058210158401015840)

= Ψ12058210158401015840 sdot

1

1198961198611198791198990

12058210158401015840 (119899

0

12058210158401015840 + 1)

(A12)

Ψ1205821015840 = 120591

1205821015840

ℏ1205961205821015840

119879k1205821015840 sdot nabla119879 (A13)

Ψ12058210158401015840 = 120591

12058210158401015840

ℏ12059612058210158401015840

119879k12058210158401015840 sdot nabla119879 (A14)


120582Ψ

1205821015840

Ψ120582Ψ

12058210158401015840 and Ψ

1205821015840Ψ

12058210158401015840 we have

1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

=

(Ψ120582+ Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

1205821198990

1205821015840 (1 + 119899

0

12058210158401015840)

119896119861119879

(A15)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

=

(Ψ120582minus Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

120582(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840)

119896119861119879

(A16)





Acknowledgments


References

































































2Si

119909Sn

1minus119909






























1minus119909As




















1minus119909Se

119909from first-





































1minus119909Sb

119909)2Te























2Te

3Sb

2Te










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials







119871119873= 119861

119871120596

2119879

3 low 119879

120591minus1

119879119873= 119861

119879120596119879

4 low 119879

120591minus1

119871119873= 119861

1015840

119871120596

2119879 high 119879

120591minus1

119879119873= 119861

1015840

119879120596119879 high 119879


119873= 119861

119873120596

2119879

3

119880 process


119880= 119861

119880120596

2119879

3 exp(minus 120579

120572119879) low 119879


119880= 119861

119880120596119879

3 exp(minus 120579

120572119879) low 119879

120591minus1

119880= 119861

1015840120596

2119879 high 119879


119880= 119861

119880120596

2119879

3


119879119880=

119861119879119880120596

2

sinh (119909) 120596

1le 120596 le 120596

2

0 120596 lt 1205961


119879119880= 119861

1015840

119879119880120596

2119879 exp(119862119879

119879)

120591minus1

119871119880= 119861

1015840

119871119880120596

2119879 exp(119862119871

119879)


119887=V120582

119897119865119897 diameter 119897 = 2radic119897

11198972120587



im =120587

2119892120596

2119863(120596) asymp

1198810119892

4120587 ⟨V3⟩120596

4sim 119860120596

4








120582is often


Vave = [1

3

3

sum

]

1

V]]

minus1

= [1

3(2

V119879

+1

V119871

)]

minus1

(4)


119892 = sum

119894

119891119894(1 minus

119898119894

119898)

2

(5)






119892 = 119892 + 2sum

119894

119891119894(Δ120601

119894

120601minus64120574Δ119903

119894

119903)

2

(6)

where 120601119894120601 and Δ119903



minus1



minus1




120591minus1= sum

119894

120591minus1

119894 (7)



sim





119871sim 120596

2119879


119879sim 120596






120582=120597119899

120582

120597119905

10038161003816100381610038161003816100381610038161003816119904

(8)


where 119899120582= 119899

0

120582+ 119899

1015840


with 1198991015840



120582 With nabla119899

120582= (120597119899

120582120597119879)nabla119879 and



120582120597119879) ≃

(1205971198990

120582120597119879) we have


1205971198990

120582

120597119879=120597119899

1015840

120582

120597119905

100381610038161003816100381610038161003816100381610038161003816119904

(9)


1198991015840


120591120582

) (10)


in BTE (9) becomes

1205971198991015840

120582

120597119905

100381610038161003816100381610038161003816100381610038161003816119904

≃ minus1198991015840

120582

120591120582

(11)



120582= 1Γ

120582 where Γ




1205971198990

120582

120597119879

= minussum

120582101584012058210158401015840

[1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840]L

+

+1

2[119899

120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840]L

minus

(12)



120582plusmn 120596

1205821015840 = 120596

12058210158401015840


+G withG =



rarr 12058210158401015840


Lplusmn=

ℏ120587

41198730

10038161003816100381610038161003816119881

(3)

plusmn

10038161003816100381610038161003816

2 120575 (120596120582plusmn 120596

1205821015840 minus 120596

12058210158401015840)

120596120582120596

1205821015840120596

12058210158401015840

(13)

119881(3)

plusmn= sum

119887119897101584011988710158401198971015840101584011988710158401015840

sum

120572120573120574

Φ120572120573120574

0119887119897101584011988710158401198971015840101584011988710158401015840

119890120582

120572119887119890plusmn1205821015840

1205731198871015840 119890

minus12058210158401015840

12057411988710158401015840

radic119898119887119898

1198871015840119898

11988710158401015840

119890plusmn119894k1015840 sdotr

1198971015840

119890minus119894k10158401015840sdotr

11989710158401015840

(14)


119887



119887120572




(3)

plusmn|2


120582= 119899

0

120582+ 119899

1015840

120582differing a


120582 Therefore on


120582+ 119899

1015840

120582 whilst

1198991205821015840 and 119899

12058210158401015840 by 1198990

1205821015840 and 119899

0




Appendix A1)

1

1205910

120582

=

+

sum

120582101584012058210158401015840

Γ+

120582120582101584012058210158401015840 +

1

2

minus

sum

120582101584012058210158401015840

Γminus

120582120582101584012058210158401015840 +sum

1205821015840

Γext1205821205821015840 (15)


120582plusmn 120596

1205821015840 = 120596

12058210158401015840)

Γplusmn

120582120582101584012058210158401015840 =

ℏ120587

41198730

1198990

1205821015840 minus 119899

0

12058210158401015840

1198990

1205821015840 + 119899

0

12058210158401015840 + 1

times10038161003816100381610038161003816119881

(3)

plusmn

10038161003816100381610038161003816

2 120575 (120596120582plusmn 120596

1205821015840 minus 120596

12058210158401015840)

120596120582120596

1205821015840120596

12058210158401015840

(16)




120582 119899

1205821015840 and 119899

12058210158401015840 by 1198990

120582+ 119899

1015840

120582 1198990

1205821015840 + 119899

1015840

1205821015840

and 1198990

12058210158401015840 + 119899

1015840



see Appendix A2)

120591120582= 120591

0

120582(1 + Δ

120582) (17)

Δ120582=

(+)

sum

120582101584012058210158401015840

Γ+

120582120582101584012058210158401015840 (1205851205821205821015840101584012059112058210158401015840 minus 12058512058212058210158401205911205821015840)

+

(minus)

sum

120582101584012058210158401015840

1

2Γminus

120582120582101584012058210158401015840 (1205851205821205821015840101584012059112058210158401015840 + 12058512058212058210158401205911205821015840)

+sum

1205821015840

Γext120582120582101584012058512058212058210158401205911205821015840

(18)

where 1205851205821205821015840 = V

1205821015840119911120596

1205821015840V

120582119911120596

120582 and V





1015840 with 1205821015840= 120582




11989711198871119897119899119887119899

=120597119899Φ

1205971199061(119897

11198871) 120597119906

119899(119897

119899119887119899) (19)



10158401015840) is typically


(1120587)(120576(1199092+ 120576

2)) To








1

120591ext120582

= sum

1205821015840

Γext1205821205821015840 (20)



imp120582

1000

800

600

400

200

001 1

120596 (THz)

Erro

r (

)

Rela

xatio

n tim

e (s

)

10minus7

10minus8

10minus9

10minus10

10minus11

10minus12

120591(U)(120596)

120591(N)(120596)120591(0)(120596)

0 10 20 30 40 50 60 70 80

120596eff (THz)







plusmn|2 is shown



120582120596

1205821015840120596

12058210158401015840)




dependence 120591(119880)(120596) sim 120596





120582+

Γ(119880)



120582= 1120591

(119873)

120582+ 1120591

(119880)

120582

where 120591(119873119880)

120582is defined as 120591(119873119880)

120582equiv 1Γ

(119873119880)






minus2



10000

1000

Error ()

Iterative scheme

Standard SMRTA

80

70

60

50

40

30

20

10

01000500300200

Temperature (K)

er

mal

con

duct

ivity

(Wm

-K)

Erro

r (

)



2Si

06Sn


2Si and Mg

2Sn cross at



100

10

1

01 1

TATALA

TOTOLO

Rela

xatio

n tim

e (p

s)

120591 prop 120596minus2


(a)

100

10

1

01 1

Normal

120591 prop 120596minus2


TATALA

TOTOLO

Rela

xatio

n tim

e (p

s)

(b)

100

10

1

01 1

Umklapp

TATALA

TOTOLO

120591 prop 120596minus3


Rela

xatio

n tim

e (p

s)

(c)


12120596

minus2 (b) 120591 = 8times1012120596

minus2 and (c) 120591 = 8times1024120596




1

120591bs=21003816100381610038161003816k120582

sdot 1003816100381610038161003816

119871

(21)






103

102

101

100

10010minus1

Life

time

(ps)

TA

Frequency (THz)

PbSePbTe

120591 sim 120596minus2

(a)

103

102

101

100

10010minus1

Life

time

(ps)

LA

Frequency (THz)

PbSePbTe

120591 sim 120596minus2

(b)

101100

101

100

10minus1

Life

time

(ps)

TO

Frequency (THz)

PbSePbTe

(c)

101100

101

100

10minus1

Life

time

(ps)

LO

Frequency (THz)

PbSePbTe

(d)






10minus1 100 101 102 103 104 105 106



PbSe

ZrCoSb

Phonon MFP (nm)

Acc

umul

ated

ther

mal

con

duct

ivity

()

100

80

60

40

20

0


05Se

05[106] Mg

2Si [63] Mg

2Sn

[63] Mg2Si

06Sn

04[63] ZrCoSb [120])





12

1

08

06

04

02

0 1 2 3 4 5 6 7 8

Diameter d (nm)

kGNR kgraphene

kRTA

kk

grap

hene k


ZA

TA

LA

10

11

10

L (120583m)

kLk

RTA





007

006

005

004

003

002

001

00 01 02 03 04 0 01 02 03 04 0 01 02 03 04 05

0 01 02 03 04 05

Scat

terin

g ra

te(c

mminus1)

Scat

terin

g ra

te(c

mminus1)

120585 120585 120585

1

120591TAasymp

1

120591aTA+

1

120591bTA+

1

120591cTA

Optical Optical

Optical

Acoustic

Acoustic Acoustic


02

015

01

005

0

120585

1

120591aTA

1

120591bTA

1

120591cTA

1

120591TA

TATATA

(a) (b) (c)

(d)





120591r120582 = 1205910

120582(1 + Δ

120582) 1 minus 119890


120582V120582|119866r120582 (22)







120591120582= 120591

0

120582(1 + Δ

120582) [

1

119878119888

int119878119888


119887)1205910

120582V120582|119866r120582 119889s] (23)



Γiso1205821205821015840 =

120587

2119873119888

120596120582120596

1205821015840

119899

sum

119887

119892119887

10038161003816100381610038161003816e119887120582sdot e119887lowast

1205821015840

10038161003816100381610038161003816

2

120575 (120596120582minus 120596

1205821015840) (24)




1

011 15 2 25 3

1 times 1

2 times 2

4 times 4

kSLk

T

M1M2


1119872

2 the mass ratio of



119892119887= sum

119894

119891119894119887(1 minus

119898119894119887

119898119887

)

2

(25)




119887is the




1

120591iso120582

= sum

1205821015840

Γiso1205821205821015840 =

120587

2119873119888

1205962

120582

119899

sum

119887

sum

1205821015840

119892119887

10038161003816100381610038161003816e119887120582sdot e119887lowast

1205821015840

10038161003816100381610038161003816

2

120575 (120596120582minus 120596

1205821015840)

(26)


1

120591iso120582

=120587

6119899119873119888

1198921205962

120582sum

1205821015840

120575 (120596120582minus 120596

1205821015840)

=120587119881

0

6119892120596

2119863 (120596)

(27)

=120587

2119892120596

2119863 (120596) (28)

where 119892 is given by (5)1198810= 119881(119899119873


119863(120596) = sum1205821015840 120575(120596120582




10

08

06

04

02

00 2 4 6 8 10

t (ps)


i(0)Ei(0)⟩


120587

120596i




is 119881 = 1198810119899119873






055Pd

045alloys [139] for




10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Empty CNT


278ps(429ps)

80ps(138ps)

25ps(26ps)

263ps(268ps)

47ps(50ps)

0 1 2 3 4 5

Φ(kz=0120596

) (J s

)

(a)

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11


Water-filled CNT

24ps 22ps 23ps

250ps

46ps

0 1 2 3 4 5

Φ(kz=0120596

) (J s

)

(b)



20

15

10

05

000806040200 10

Longitudinal

1120591

(1p

s)

20K LD50K LD

20K MD50K MD

klowast (a2120587)

(a)

08

06

04

02

000806040200 10

Transverse

1120591

(1p

s)

20K LD50K LD

20K MD50K MD

klowast (a2120587)

(b)




(at the limit 119875119871



119871 This leads



119871









50

40

30

20

10

050403020100

Phon

on li

fetim

e (p

s)








1+1superlattice We



1119872





1119872



1119872



1119872




4 MD Simulation



120591120582=

intinfin

0⟨119899

1015840

120582(119905) 119899

1015840

120582(0)⟩ 119889119905

⟨11989910158402

120582(0)⟩

(29)







119864120582=1

2( 119902

120582119902lowast

120582+ 120596

2

120582119902120582119902lowast

120582) (30)


transformed to

120591120582=

intinfin

0⟨119864

120582(119905) 119864

120582(0)⟩ 119889119905

⟨1198642

120582(0)⟩

(31)


120582(119905)119864

120582(0)⟩⟨119864

2

120582(0)⟩ = exp(minus119905120591

120582)


120582sim 119902

120582119902lowast



119902120582(119905) = 119902

1205820exp [119894 (120596119860

120582+ 119894120578

120582) 119905] (32)

where 1199021205820


120582= 120596

120582+ Δ


120582


120582119902lowast

120582

and 119864120582sim 119902

120582119902lowast


120582= 12120578

120582



120582in (31)


119902120582(119905) =

3

sum

120572

119899

sum

119887

119873119888

sum

119897

radic119898

119887

119873119888

119906119897119887

120572(119905) 119890

120582lowast

119887120572exp [119894k sdot r119897

0]

=

3

sum

120572

119899

sum

119887

119890120582lowast

119887120572119902119887

120572(k 119905)

(33)


120572(119905) is 120572 component




120572(k 119905) denotes



300

200

100

0

ZA sTA sLA sZO s

ZA pTA pLA pZO p

er

mal

con

duct

ivity

(Wm

K)

(a)

Temperature (K)

()

300 400 500 600

40

30

20

10

ZA sTA sLA sZO s

ZA pTA pLA pZO p

(b)



119902119887

120572(k 119905) =

119873119888

sum

119897

radic119898

119887

119873119888

119906119897119887

120572(119905) exp [119894k sdot r119897

0] (34)



Φ120582(120596) =

1003816100381610038161003816 119902120582(120596)

1003816100381610038161003816

2

=

10038161003816100381610038161003816100381610038161003816

int

+infin

0

119902120582(119905) 119890

minus119894120596119905119889119905

10038161003816100381610038161003816100381610038161003816

2

=119862

120582

(120596 minus 120596119860

120582)2

+ 1205782

120582

(35)

where119862120582= (120596

119860

120582

2

+Γ2

120582)119902

2






120582= 12120578




Φ (k 120596) =3119899

sum

]Φ

120582(120596)

=1

41205871205910

3119899

sum

120572119887

119898119887

119873119888

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119897

1205910

int

0

119897119887

120572(119905) exp [119894k sdot r119897

0minus 119894120596119905]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

=

3

sum

120572

119899

sum

119887

10038161003816100381610038161003816119902119887

120572(k 120596)10038161003816100381610038161003816

2

(36)

where 119902119887





119888in (34) are




Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)100

10

1

100

10

10 1 2 0 1 2


300K 600K

[100] LA[110] LA[111] LA

[100] LA[110] LA[111] LA

(a)

[100] TA[110] TA1[110] TA2[111] TA

[100] TA[110] TA1[110] TA2[111] TA

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

300K 600K

(b)

1

0135 40

1

0135 40

300K 600K

[100] LO[110] LO[111] LO

[100] LO[110] LO[111] LO

Rela

xatio

n tim

e (p

s)

Frequency (THz)

Rela

xatio

n tim

e (p

s)

Frequency (THz)

(c)

30

300K 600K

[100] TO[110] TO1[110] TO2[111] TO

[100] TO[110] TO1[110] TO2[111] TO

1

01

1

0135 40 30 35 40

Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)


(d)







80

60

40

20

00 1 2 3 4

46

23

26

300K

Frequency (THz)

TA1TA2LA

TO1TO2LO

Perc

enta

ge c

ontr

ibut

ion

(au

)

(a)

80

60

40

20

00 1 2 3 4

50

2124

TA1TA2LA

TO1TO2LO

600K

Frequency (THz)

Perc

enta

ge c

ontr

ibut

ion

(au

)

(b)


150

100

50

0 1 2 3 4 5

Frequency (THz)

L = 12

L = 9

L = 6

Acoustic phonon



Rela

xatio

n tim

e (p

s)

(a)


0

150

100

50

02 04 06 08

Predicted by NMA

Rela

xatio

n tim

e (p

s)


(b)










10minus1 100 101 102 103 104 105



MFP (nm)

Acc

umul

ated

ther

mal

con

duct

ivity

() 100

80

60

40

20

0


2Te

3(MD) [142])

[141] and Bi2Te









2Te



3


2Te


in the future

5 Summary






1119872

2SLs 1 and 2 [59] 2004


1119872

2SLs 1 and 2 radic [50] 2008



2 and SrTiO3) 2 [49] 2011



Table 2 Continued


1minus119909Se

1199091 radic [106] 2012

Bi2Te

34 [142] 2013





Equations (14)(15) (16) (17)

and (18)














Applications sofar











Appendix

A


120596 + 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) (119899

0

1205821015840 + 1) 119899

0

12058210158401015840 = 119899

0

1205821198990

1205821015840 (119899

0

12058210158401015840 + 1) (A1)

120596 minus 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) 119899

0

1205821015840119899

0

12058210158401015840 = 119899

0

120582(119899

0

1205821015840 + 1) (119899

0

12058210158401015840 + 1) (A2)


119861119879) minus 1]


1198991015840

120582= minusΨ

120582

1205971198990

120582

120597 (ℏ120596120582)= Ψ

120582sdot

1

1198961198611198791198990

120582(119899

0

120582+ 1) (A3)


approximation (11)

Ψ120582= 120591

120582

ℏ120596120582

119879k120582sdot nabla119879 (A4)


119899120582= 119899

0

120582+ 119899

1015840

120582

1198991205821015840 = 119899

0

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840

(A5)


1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

= 1198991205821015840 [119899

0

1205821015840 (1 + 119899

0

12058210158401015840) minus (1 + 119899

0

1205821015840) 119899

0

12058210158401015840]

= 1198991205821015840 (119899

0

1205821015840 minus 119899

0

12058210158401015840)

(A6)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (119899

0

1205821015840 minus 119899

0

12058210158401015840)

119896119861119879

(A7)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

= 1198991205821015840 [(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840) minus 119899

0

1205821015840119899

0

12058210158401015840]

= 1198991205821015840 (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

(A8)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

119896119861119879

(A9)


1205821015840




119899120582= 119899

0

120582+ 119899

1205821015840

1198991205821015840 = 119899

0

1205821015840 + 119899

1015840

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840 + 119899

1015840

12058210158401015840

(A10)

where 1198991015840

1205821015840 and 1198991015840


120582

1198991015840

1205821015840 = minusΨ

1205821015840

1205971198990

1205821015840

120597 (ℏ1205961205821015840)= Ψ

1205821015840 sdot

1

1198961198611198791198990

1205821015840 (119899

0

1205821015840 + 1) (A11)

1198991015840

12058210158401015840 = minus Ψ

12058210158401015840

1205971198990

12058210158401015840

120597 (ℏ12059612058210158401015840)

= Ψ12058210158401015840 sdot

1

1198961198611198791198990

12058210158401015840 (119899

0

12058210158401015840 + 1)

(A12)

Ψ1205821015840 = 120591

1205821015840

ℏ1205961205821015840

119879k1205821015840 sdot nabla119879 (A13)

Ψ12058210158401015840 = 120591

12058210158401015840

ℏ12059612058210158401015840

119879k12058210158401015840 sdot nabla119879 (A14)


120582Ψ

1205821015840

Ψ120582Ψ

12058210158401015840 and Ψ

1205821015840Ψ

12058210158401015840 we have

1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

=

(Ψ120582+ Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

1205821198990

1205821015840 (1 + 119899

0

12058210158401015840)

119896119861119879

(A15)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

=

(Ψ120582minus Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

120582(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840)

119896119861119879

(A16)





Acknowledgments


References

































































2Si

119909Sn

1minus119909






























1minus119909As




















1minus119909Se

119909from first-





































1minus119909Sb

119909)2Te























2Te

3Sb

2Te










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




where 119899120582= 119899

0

120582+ 119899

1015840


with 1198991015840



120582 With nabla119899

120582= (120597119899

120582120597119879)nabla119879 and



120582120597119879) ≃

(1205971198990

120582120597119879) we have


1205971198990

120582

120597119879=120597119899

1015840

120582

120597119905

100381610038161003816100381610038161003816100381610038161003816119904

(9)


1198991015840


120591120582

) (10)


in BTE (9) becomes

1205971198991015840

120582

120597119905

100381610038161003816100381610038161003816100381610038161003816119904

≃ minus1198991015840

120582

120591120582

(11)



120582= 1Γ

120582 where Γ




1205971198990

120582

120597119879

= minussum

120582101584012058210158401015840

[1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840]L

+

+1

2[119899

120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840]L

minus

(12)



120582plusmn 120596

1205821015840 = 120596

12058210158401015840


+G withG =



rarr 12058210158401015840


Lplusmn=

ℏ120587

41198730

10038161003816100381610038161003816119881

(3)

plusmn

10038161003816100381610038161003816

2 120575 (120596120582plusmn 120596

1205821015840 minus 120596

12058210158401015840)

120596120582120596

1205821015840120596

12058210158401015840

(13)

119881(3)

plusmn= sum

119887119897101584011988710158401198971015840101584011988710158401015840

sum

120572120573120574

Φ120572120573120574

0119887119897101584011988710158401198971015840101584011988710158401015840

119890120582

120572119887119890plusmn1205821015840

1205731198871015840 119890

minus12058210158401015840

12057411988710158401015840

radic119898119887119898

1198871015840119898

11988710158401015840

119890plusmn119894k1015840 sdotr

1198971015840

119890minus119894k10158401015840sdotr

11989710158401015840

(14)


119887



119887120572




(3)

plusmn|2


120582= 119899

0

120582+ 119899

1015840

120582differing a


120582 Therefore on


120582+ 119899

1015840

120582 whilst

1198991205821015840 and 119899

12058210158401015840 by 1198990

1205821015840 and 119899

0




Appendix A1)

1

1205910

120582

=

+

sum

120582101584012058210158401015840

Γ+

120582120582101584012058210158401015840 +

1

2

minus

sum

120582101584012058210158401015840

Γminus

120582120582101584012058210158401015840 +sum

1205821015840

Γext1205821205821015840 (15)


120582plusmn 120596

1205821015840 = 120596

12058210158401015840)

Γplusmn

120582120582101584012058210158401015840 =

ℏ120587

41198730

1198990

1205821015840 minus 119899

0

12058210158401015840

1198990

1205821015840 + 119899

0

12058210158401015840 + 1

times10038161003816100381610038161003816119881

(3)

plusmn

10038161003816100381610038161003816

2 120575 (120596120582plusmn 120596

1205821015840 minus 120596

12058210158401015840)

120596120582120596

1205821015840120596

12058210158401015840

(16)




120582 119899

1205821015840 and 119899

12058210158401015840 by 1198990

120582+ 119899

1015840

120582 1198990

1205821015840 + 119899

1015840

1205821015840

and 1198990

12058210158401015840 + 119899

1015840



see Appendix A2)

120591120582= 120591

0

120582(1 + Δ

120582) (17)

Δ120582=

(+)

sum

120582101584012058210158401015840

Γ+

120582120582101584012058210158401015840 (1205851205821205821015840101584012059112058210158401015840 minus 12058512058212058210158401205911205821015840)

+

(minus)

sum

120582101584012058210158401015840

1

2Γminus

120582120582101584012058210158401015840 (1205851205821205821015840101584012059112058210158401015840 + 12058512058212058210158401205911205821015840)

+sum

1205821015840

Γext120582120582101584012058512058212058210158401205911205821015840

(18)

where 1205851205821205821015840 = V

1205821015840119911120596

1205821015840V

120582119911120596

120582 and V





1015840 with 1205821015840= 120582




11989711198871119897119899119887119899

=120597119899Φ

1205971199061(119897

11198871) 120597119906

119899(119897

119899119887119899) (19)



10158401015840) is typically


(1120587)(120576(1199092+ 120576

2)) To








1

120591ext120582

= sum

1205821015840

Γext1205821205821015840 (20)



imp120582

1000

800

600

400

200

001 1

120596 (THz)

Erro

r (

)

Rela

xatio

n tim

e (s

)

10minus7

10minus8

10minus9

10minus10

10minus11

10minus12

120591(U)(120596)

120591(N)(120596)120591(0)(120596)

0 10 20 30 40 50 60 70 80

120596eff (THz)







plusmn|2 is shown



120582120596

1205821015840120596

12058210158401015840)




dependence 120591(119880)(120596) sim 120596





120582+

Γ(119880)



120582= 1120591

(119873)

120582+ 1120591

(119880)

120582

where 120591(119873119880)

120582is defined as 120591(119873119880)

120582equiv 1Γ

(119873119880)






minus2



10000

1000

Error ()

Iterative scheme

Standard SMRTA

80

70

60

50

40

30

20

10

01000500300200

Temperature (K)

er

mal

con

duct

ivity

(Wm

-K)

Erro

r (

)



2Si

06Sn


2Si and Mg

2Sn cross at



100

10

1

01 1

TATALA

TOTOLO

Rela

xatio

n tim

e (p

s)

120591 prop 120596minus2


(a)

100

10

1

01 1

Normal

120591 prop 120596minus2


TATALA

TOTOLO

Rela

xatio

n tim

e (p

s)

(b)

100

10

1

01 1

Umklapp

TATALA

TOTOLO

120591 prop 120596minus3


Rela

xatio

n tim

e (p

s)

(c)


12120596

minus2 (b) 120591 = 8times1012120596

minus2 and (c) 120591 = 8times1024120596




1

120591bs=21003816100381610038161003816k120582

sdot 1003816100381610038161003816

119871

(21)






103

102

101

100

10010minus1

Life

time

(ps)

TA

Frequency (THz)

PbSePbTe

120591 sim 120596minus2

(a)

103

102

101

100

10010minus1

Life

time

(ps)

LA

Frequency (THz)

PbSePbTe

120591 sim 120596minus2

(b)

101100

101

100

10minus1

Life

time

(ps)

TO

Frequency (THz)

PbSePbTe

(c)

101100

101

100

10minus1

Life

time

(ps)

LO

Frequency (THz)

PbSePbTe

(d)






10minus1 100 101 102 103 104 105 106



PbSe

ZrCoSb

Phonon MFP (nm)

Acc

umul

ated

ther

mal

con

duct

ivity

()

100

80

60

40

20

0


05Se

05[106] Mg

2Si [63] Mg

2Sn

[63] Mg2Si

06Sn

04[63] ZrCoSb [120])





12

1

08

06

04

02

0 1 2 3 4 5 6 7 8

Diameter d (nm)

kGNR kgraphene

kRTA

kk

grap

hene k


ZA

TA

LA

10

11

10

L (120583m)

kLk

RTA





007

006

005

004

003

002

001

00 01 02 03 04 0 01 02 03 04 0 01 02 03 04 05

0 01 02 03 04 05

Scat

terin

g ra

te(c

mminus1)

Scat

terin

g ra

te(c

mminus1)

120585 120585 120585

1

120591TAasymp

1

120591aTA+

1

120591bTA+

1

120591cTA

Optical Optical

Optical

Acoustic

Acoustic Acoustic


02

015

01

005

0

120585

1

120591aTA

1

120591bTA

1

120591cTA

1

120591TA

TATATA

(a) (b) (c)

(d)





120591r120582 = 1205910

120582(1 + Δ

120582) 1 minus 119890


120582V120582|119866r120582 (22)







120591120582= 120591

0

120582(1 + Δ

120582) [

1

119878119888

int119878119888


119887)1205910

120582V120582|119866r120582 119889s] (23)



Γiso1205821205821015840 =

120587

2119873119888

120596120582120596

1205821015840

119899

sum

119887

119892119887

10038161003816100381610038161003816e119887120582sdot e119887lowast

1205821015840

10038161003816100381610038161003816

2

120575 (120596120582minus 120596

1205821015840) (24)




1

011 15 2 25 3

1 times 1

2 times 2

4 times 4

kSLk

T

M1M2


1119872

2 the mass ratio of



119892119887= sum

119894

119891119894119887(1 minus

119898119894119887

119898119887

)

2

(25)




119887is the




1

120591iso120582

= sum

1205821015840

Γiso1205821205821015840 =

120587

2119873119888

1205962

120582

119899

sum

119887

sum

1205821015840

119892119887

10038161003816100381610038161003816e119887120582sdot e119887lowast

1205821015840

10038161003816100381610038161003816

2

120575 (120596120582minus 120596

1205821015840)

(26)


1

120591iso120582

=120587

6119899119873119888

1198921205962

120582sum

1205821015840

120575 (120596120582minus 120596

1205821015840)

=120587119881

0

6119892120596

2119863 (120596)

(27)

=120587

2119892120596

2119863 (120596) (28)

where 119892 is given by (5)1198810= 119881(119899119873


119863(120596) = sum1205821015840 120575(120596120582




10

08

06

04

02

00 2 4 6 8 10

t (ps)


i(0)Ei(0)⟩


120587

120596i




is 119881 = 1198810119899119873






055Pd

045alloys [139] for




10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Empty CNT


278ps(429ps)

80ps(138ps)

25ps(26ps)

263ps(268ps)

47ps(50ps)

0 1 2 3 4 5

Φ(kz=0120596

) (J s

)

(a)

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11


Water-filled CNT

24ps 22ps 23ps

250ps

46ps

0 1 2 3 4 5

Φ(kz=0120596

) (J s

)

(b)



20

15

10

05

000806040200 10

Longitudinal

1120591

(1p

s)

20K LD50K LD

20K MD50K MD

klowast (a2120587)

(a)

08

06

04

02

000806040200 10

Transverse

1120591

(1p

s)

20K LD50K LD

20K MD50K MD

klowast (a2120587)

(b)




(at the limit 119875119871



119871 This leads



119871









50

40

30

20

10

050403020100

Phon

on li

fetim

e (p

s)








1+1superlattice We



1119872





1119872



1119872



1119872




4 MD Simulation



120591120582=

intinfin

0⟨119899

1015840

120582(119905) 119899

1015840

120582(0)⟩ 119889119905

⟨11989910158402

120582(0)⟩

(29)







119864120582=1

2( 119902

120582119902lowast

120582+ 120596

2

120582119902120582119902lowast

120582) (30)


transformed to

120591120582=

intinfin

0⟨119864

120582(119905) 119864

120582(0)⟩ 119889119905

⟨1198642

120582(0)⟩

(31)


120582(119905)119864

120582(0)⟩⟨119864

2

120582(0)⟩ = exp(minus119905120591

120582)


120582sim 119902

120582119902lowast



119902120582(119905) = 119902

1205820exp [119894 (120596119860

120582+ 119894120578

120582) 119905] (32)

where 1199021205820


120582= 120596

120582+ Δ


120582


120582119902lowast

120582

and 119864120582sim 119902

120582119902lowast


120582= 12120578

120582



120582in (31)


119902120582(119905) =

3

sum

120572

119899

sum

119887

119873119888

sum

119897

radic119898

119887

119873119888

119906119897119887

120572(119905) 119890

120582lowast

119887120572exp [119894k sdot r119897

0]

=

3

sum

120572

119899

sum

119887

119890120582lowast

119887120572119902119887

120572(k 119905)

(33)


120572(119905) is 120572 component




120572(k 119905) denotes



300

200

100

0

ZA sTA sLA sZO s

ZA pTA pLA pZO p

er

mal

con

duct

ivity

(Wm

K)

(a)

Temperature (K)

()

300 400 500 600

40

30

20

10

ZA sTA sLA sZO s

ZA pTA pLA pZO p

(b)



119902119887

120572(k 119905) =

119873119888

sum

119897

radic119898

119887

119873119888

119906119897119887

120572(119905) exp [119894k sdot r119897

0] (34)



Φ120582(120596) =

1003816100381610038161003816 119902120582(120596)

1003816100381610038161003816

2

=

10038161003816100381610038161003816100381610038161003816

int

+infin

0

119902120582(119905) 119890

minus119894120596119905119889119905

10038161003816100381610038161003816100381610038161003816

2

=119862

120582

(120596 minus 120596119860

120582)2

+ 1205782

120582

(35)

where119862120582= (120596

119860

120582

2

+Γ2

120582)119902

2






120582= 12120578




Φ (k 120596) =3119899

sum

]Φ

120582(120596)

=1

41205871205910

3119899

sum

120572119887

119898119887

119873119888

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119897

1205910

int

0

119897119887

120572(119905) exp [119894k sdot r119897

0minus 119894120596119905]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

=

3

sum

120572

119899

sum

119887

10038161003816100381610038161003816119902119887

120572(k 120596)10038161003816100381610038161003816

2

(36)

where 119902119887





119888in (34) are




Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)100

10

1

100

10

10 1 2 0 1 2


300K 600K

[100] LA[110] LA[111] LA

[100] LA[110] LA[111] LA

(a)

[100] TA[110] TA1[110] TA2[111] TA

[100] TA[110] TA1[110] TA2[111] TA

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

300K 600K

(b)

1

0135 40

1

0135 40

300K 600K

[100] LO[110] LO[111] LO

[100] LO[110] LO[111] LO

Rela

xatio

n tim

e (p

s)

Frequency (THz)

Rela

xatio

n tim

e (p

s)

Frequency (THz)

(c)

30

300K 600K

[100] TO[110] TO1[110] TO2[111] TO

[100] TO[110] TO1[110] TO2[111] TO

1

01

1

0135 40 30 35 40

Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)


(d)







80

60

40

20

00 1 2 3 4

46

23

26

300K

Frequency (THz)

TA1TA2LA

TO1TO2LO

Perc

enta

ge c

ontr

ibut

ion

(au

)

(a)

80

60

40

20

00 1 2 3 4

50

2124

TA1TA2LA

TO1TO2LO

600K

Frequency (THz)

Perc

enta

ge c

ontr

ibut

ion

(au

)

(b)


150

100

50

0 1 2 3 4 5

Frequency (THz)

L = 12

L = 9

L = 6

Acoustic phonon



Rela

xatio

n tim

e (p

s)

(a)


0

150

100

50

02 04 06 08

Predicted by NMA

Rela

xatio

n tim

e (p

s)


(b)










10minus1 100 101 102 103 104 105



MFP (nm)

Acc

umul

ated

ther

mal

con

duct

ivity

() 100

80

60

40

20

0


2Te

3(MD) [142])

[141] and Bi2Te









2Te



3


2Te


in the future

5 Summary






1119872

2SLs 1 and 2 [59] 2004


1119872

2SLs 1 and 2 radic [50] 2008



2 and SrTiO3) 2 [49] 2011



Table 2 Continued


1minus119909Se

1199091 radic [106] 2012

Bi2Te

34 [142] 2013





Equations (14)(15) (16) (17)

and (18)














Applications sofar











Appendix

A


120596 + 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) (119899

0

1205821015840 + 1) 119899

0

12058210158401015840 = 119899

0

1205821198990

1205821015840 (119899

0

12058210158401015840 + 1) (A1)

120596 minus 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) 119899

0

1205821015840119899

0

12058210158401015840 = 119899

0

120582(119899

0

1205821015840 + 1) (119899

0

12058210158401015840 + 1) (A2)


119861119879) minus 1]


1198991015840

120582= minusΨ

120582

1205971198990

120582

120597 (ℏ120596120582)= Ψ

120582sdot

1

1198961198611198791198990

120582(119899

0

120582+ 1) (A3)


approximation (11)

Ψ120582= 120591

120582

ℏ120596120582

119879k120582sdot nabla119879 (A4)


119899120582= 119899

0

120582+ 119899

1015840

120582

1198991205821015840 = 119899

0

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840

(A5)


1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

= 1198991205821015840 [119899

0

1205821015840 (1 + 119899

0

12058210158401015840) minus (1 + 119899

0

1205821015840) 119899

0

12058210158401015840]

= 1198991205821015840 (119899

0

1205821015840 minus 119899

0

12058210158401015840)

(A6)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (119899

0

1205821015840 minus 119899

0

12058210158401015840)

119896119861119879

(A7)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

= 1198991205821015840 [(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840) minus 119899

0

1205821015840119899

0

12058210158401015840]

= 1198991205821015840 (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

(A8)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

119896119861119879

(A9)


1205821015840




119899120582= 119899

0

120582+ 119899

1205821015840

1198991205821015840 = 119899

0

1205821015840 + 119899

1015840

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840 + 119899

1015840

12058210158401015840

(A10)

where 1198991015840

1205821015840 and 1198991015840


120582

1198991015840

1205821015840 = minusΨ

1205821015840

1205971198990

1205821015840

120597 (ℏ1205961205821015840)= Ψ

1205821015840 sdot

1

1198961198611198791198990

1205821015840 (119899

0

1205821015840 + 1) (A11)

1198991015840

12058210158401015840 = minus Ψ

12058210158401015840

1205971198990

12058210158401015840

120597 (ℏ12059612058210158401015840)

= Ψ12058210158401015840 sdot

1

1198961198611198791198990

12058210158401015840 (119899

0

12058210158401015840 + 1)

(A12)

Ψ1205821015840 = 120591

1205821015840

ℏ1205961205821015840

119879k1205821015840 sdot nabla119879 (A13)

Ψ12058210158401015840 = 120591

12058210158401015840

ℏ12059612058210158401015840

119879k12058210158401015840 sdot nabla119879 (A14)


120582Ψ

1205821015840

Ψ120582Ψ

12058210158401015840 and Ψ

1205821015840Ψ

12058210158401015840 we have

1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

=

(Ψ120582+ Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

1205821198990

1205821015840 (1 + 119899

0

12058210158401015840)

119896119861119879

(A15)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

=

(Ψ120582minus Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

120582(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840)

119896119861119879

(A16)





Acknowledgments


References

































































2Si

119909Sn

1minus119909






























1minus119909As




















1minus119909Se

119909from first-





































1minus119909Sb

119909)2Te























2Te

3Sb

2Te










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials






11989711198871119897119899119887119899

=120597119899Φ

1205971199061(119897

11198871) 120597119906

119899(119897

119899119887119899) (19)



10158401015840) is typically


(1120587)(120576(1199092+ 120576

2)) To








1

120591ext120582

= sum

1205821015840

Γext1205821205821015840 (20)



imp120582

1000

800

600

400

200

001 1

120596 (THz)

Erro

r (

)

Rela

xatio

n tim

e (s

)

10minus7

10minus8

10minus9

10minus10

10minus11

10minus12

120591(U)(120596)

120591(N)(120596)120591(0)(120596)

0 10 20 30 40 50 60 70 80

120596eff (THz)







plusmn|2 is shown



120582120596

1205821015840120596

12058210158401015840)




dependence 120591(119880)(120596) sim 120596





120582+

Γ(119880)



120582= 1120591

(119873)

120582+ 1120591

(119880)

120582

where 120591(119873119880)

120582is defined as 120591(119873119880)

120582equiv 1Γ

(119873119880)






minus2



10000

1000

Error ()

Iterative scheme

Standard SMRTA

80

70

60

50

40

30

20

10

01000500300200

Temperature (K)

er

mal

con

duct

ivity

(Wm

-K)

Erro

r (

)



2Si

06Sn


2Si and Mg

2Sn cross at



100

10

1

01 1

TATALA

TOTOLO

Rela

xatio

n tim

e (p

s)

120591 prop 120596minus2


(a)

100

10

1

01 1

Normal

120591 prop 120596minus2


TATALA

TOTOLO

Rela

xatio

n tim

e (p

s)

(b)

100

10

1

01 1

Umklapp

TATALA

TOTOLO

120591 prop 120596minus3


Rela

xatio

n tim

e (p

s)

(c)


12120596

minus2 (b) 120591 = 8times1012120596

minus2 and (c) 120591 = 8times1024120596




1

120591bs=21003816100381610038161003816k120582

sdot 1003816100381610038161003816

119871

(21)






103

102

101

100

10010minus1

Life

time

(ps)

TA

Frequency (THz)

PbSePbTe

120591 sim 120596minus2

(a)

103

102

101

100

10010minus1

Life

time

(ps)

LA

Frequency (THz)

PbSePbTe

120591 sim 120596minus2

(b)

101100

101

100

10minus1

Life

time

(ps)

TO

Frequency (THz)

PbSePbTe

(c)

101100

101

100

10minus1

Life

time

(ps)

LO

Frequency (THz)

PbSePbTe

(d)






10minus1 100 101 102 103 104 105 106



PbSe

ZrCoSb

Phonon MFP (nm)

Acc

umul

ated

ther

mal

con

duct

ivity

()

100

80

60

40

20

0


05Se

05[106] Mg

2Si [63] Mg

2Sn

[63] Mg2Si

06Sn

04[63] ZrCoSb [120])





12

1

08

06

04

02

0 1 2 3 4 5 6 7 8

Diameter d (nm)

kGNR kgraphene

kRTA

kk

grap

hene k


ZA

TA

LA

10

11

10

L (120583m)

kLk

RTA





007

006

005

004

003

002

001

00 01 02 03 04 0 01 02 03 04 0 01 02 03 04 05

0 01 02 03 04 05

Scat

terin

g ra

te(c

mminus1)

Scat

terin

g ra

te(c

mminus1)

120585 120585 120585

1

120591TAasymp

1

120591aTA+

1

120591bTA+

1

120591cTA

Optical Optical

Optical

Acoustic

Acoustic Acoustic


02

015

01

005

0

120585

1

120591aTA

1

120591bTA

1

120591cTA

1

120591TA

TATATA

(a) (b) (c)

(d)





120591r120582 = 1205910

120582(1 + Δ

120582) 1 minus 119890


120582V120582|119866r120582 (22)







120591120582= 120591

0

120582(1 + Δ

120582) [

1

119878119888

int119878119888


119887)1205910

120582V120582|119866r120582 119889s] (23)



Γiso1205821205821015840 =

120587

2119873119888

120596120582120596

1205821015840

119899

sum

119887

119892119887

10038161003816100381610038161003816e119887120582sdot e119887lowast

1205821015840

10038161003816100381610038161003816

2

120575 (120596120582minus 120596

1205821015840) (24)




1

011 15 2 25 3

1 times 1

2 times 2

4 times 4

kSLk

T

M1M2


1119872

2 the mass ratio of



119892119887= sum

119894

119891119894119887(1 minus

119898119894119887

119898119887

)

2

(25)




119887is the




1

120591iso120582

= sum

1205821015840

Γiso1205821205821015840 =

120587

2119873119888

1205962

120582

119899

sum

119887

sum

1205821015840

119892119887

10038161003816100381610038161003816e119887120582sdot e119887lowast

1205821015840

10038161003816100381610038161003816

2

120575 (120596120582minus 120596

1205821015840)

(26)


1

120591iso120582

=120587

6119899119873119888

1198921205962

120582sum

1205821015840

120575 (120596120582minus 120596

1205821015840)

=120587119881

0

6119892120596

2119863 (120596)

(27)

=120587

2119892120596

2119863 (120596) (28)

where 119892 is given by (5)1198810= 119881(119899119873


119863(120596) = sum1205821015840 120575(120596120582




10

08

06

04

02

00 2 4 6 8 10

t (ps)


i(0)Ei(0)⟩


120587

120596i




is 119881 = 1198810119899119873






055Pd

045alloys [139] for




10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Empty CNT


278ps(429ps)

80ps(138ps)

25ps(26ps)

263ps(268ps)

47ps(50ps)

0 1 2 3 4 5

Φ(kz=0120596

) (J s

)

(a)

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11


Water-filled CNT

24ps 22ps 23ps

250ps

46ps

0 1 2 3 4 5

Φ(kz=0120596

) (J s

)

(b)



20

15

10

05

000806040200 10

Longitudinal

1120591

(1p

s)

20K LD50K LD

20K MD50K MD

klowast (a2120587)

(a)

08

06

04

02

000806040200 10

Transverse

1120591

(1p

s)

20K LD50K LD

20K MD50K MD

klowast (a2120587)

(b)




(at the limit 119875119871



119871 This leads



119871









50

40

30

20

10

050403020100

Phon

on li

fetim

e (p

s)








1+1superlattice We



1119872





1119872



1119872



1119872




4 MD Simulation



120591120582=

intinfin

0⟨119899

1015840

120582(119905) 119899

1015840

120582(0)⟩ 119889119905

⟨11989910158402

120582(0)⟩

(29)







119864120582=1

2( 119902

120582119902lowast

120582+ 120596

2

120582119902120582119902lowast

120582) (30)


transformed to

120591120582=

intinfin

0⟨119864

120582(119905) 119864

120582(0)⟩ 119889119905

⟨1198642

120582(0)⟩

(31)


120582(119905)119864

120582(0)⟩⟨119864

2

120582(0)⟩ = exp(minus119905120591

120582)


120582sim 119902

120582119902lowast



119902120582(119905) = 119902

1205820exp [119894 (120596119860

120582+ 119894120578

120582) 119905] (32)

where 1199021205820


120582= 120596

120582+ Δ


120582


120582119902lowast

120582

and 119864120582sim 119902

120582119902lowast


120582= 12120578

120582



120582in (31)


119902120582(119905) =

3

sum

120572

119899

sum

119887

119873119888

sum

119897

radic119898

119887

119873119888

119906119897119887

120572(119905) 119890

120582lowast

119887120572exp [119894k sdot r119897

0]

=

3

sum

120572

119899

sum

119887

119890120582lowast

119887120572119902119887

120572(k 119905)

(33)


120572(119905) is 120572 component




120572(k 119905) denotes



300

200

100

0

ZA sTA sLA sZO s

ZA pTA pLA pZO p

er

mal

con

duct

ivity

(Wm

K)

(a)

Temperature (K)

()

300 400 500 600

40

30

20

10

ZA sTA sLA sZO s

ZA pTA pLA pZO p

(b)



119902119887

120572(k 119905) =

119873119888

sum

119897

radic119898

119887

119873119888

119906119897119887

120572(119905) exp [119894k sdot r119897

0] (34)



Φ120582(120596) =

1003816100381610038161003816 119902120582(120596)

1003816100381610038161003816

2

=

10038161003816100381610038161003816100381610038161003816

int

+infin

0

119902120582(119905) 119890

minus119894120596119905119889119905

10038161003816100381610038161003816100381610038161003816

2

=119862

120582

(120596 minus 120596119860

120582)2

+ 1205782

120582

(35)

where119862120582= (120596

119860

120582

2

+Γ2

120582)119902

2






120582= 12120578




Φ (k 120596) =3119899

sum

]Φ

120582(120596)

=1

41205871205910

3119899

sum

120572119887

119898119887

119873119888

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119897

1205910

int

0

119897119887

120572(119905) exp [119894k sdot r119897

0minus 119894120596119905]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

=

3

sum

120572

119899

sum

119887

10038161003816100381610038161003816119902119887

120572(k 120596)10038161003816100381610038161003816

2

(36)

where 119902119887





119888in (34) are




Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)100

10

1

100

10

10 1 2 0 1 2


300K 600K

[100] LA[110] LA[111] LA

[100] LA[110] LA[111] LA

(a)

[100] TA[110] TA1[110] TA2[111] TA

[100] TA[110] TA1[110] TA2[111] TA

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

300K 600K

(b)

1

0135 40

1

0135 40

300K 600K

[100] LO[110] LO[111] LO

[100] LO[110] LO[111] LO

Rela

xatio

n tim

e (p

s)

Frequency (THz)

Rela

xatio

n tim

e (p

s)

Frequency (THz)

(c)

30

300K 600K

[100] TO[110] TO1[110] TO2[111] TO

[100] TO[110] TO1[110] TO2[111] TO

1

01

1

0135 40 30 35 40

Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)


(d)







80

60

40

20

00 1 2 3 4

46

23

26

300K

Frequency (THz)

TA1TA2LA

TO1TO2LO

Perc

enta

ge c

ontr

ibut

ion

(au

)

(a)

80

60

40

20

00 1 2 3 4

50

2124

TA1TA2LA

TO1TO2LO

600K

Frequency (THz)

Perc

enta

ge c

ontr

ibut

ion

(au

)

(b)


150

100

50

0 1 2 3 4 5

Frequency (THz)

L = 12

L = 9

L = 6

Acoustic phonon



Rela

xatio

n tim

e (p

s)

(a)


0

150

100

50

02 04 06 08

Predicted by NMA

Rela

xatio

n tim

e (p

s)


(b)










10minus1 100 101 102 103 104 105



MFP (nm)

Acc

umul

ated

ther

mal

con

duct

ivity

() 100

80

60

40

20

0


2Te

3(MD) [142])

[141] and Bi2Te









2Te



3


2Te


in the future

5 Summary






1119872

2SLs 1 and 2 [59] 2004


1119872

2SLs 1 and 2 radic [50] 2008



2 and SrTiO3) 2 [49] 2011



Table 2 Continued


1minus119909Se

1199091 radic [106] 2012

Bi2Te

34 [142] 2013





Equations (14)(15) (16) (17)

and (18)














Applications sofar











Appendix

A


120596 + 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) (119899

0

1205821015840 + 1) 119899

0

12058210158401015840 = 119899

0

1205821198990

1205821015840 (119899

0

12058210158401015840 + 1) (A1)

120596 minus 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) 119899

0

1205821015840119899

0

12058210158401015840 = 119899

0

120582(119899

0

1205821015840 + 1) (119899

0

12058210158401015840 + 1) (A2)


119861119879) minus 1]


1198991015840

120582= minusΨ

120582

1205971198990

120582

120597 (ℏ120596120582)= Ψ

120582sdot

1

1198961198611198791198990

120582(119899

0

120582+ 1) (A3)


approximation (11)

Ψ120582= 120591

120582

ℏ120596120582

119879k120582sdot nabla119879 (A4)


119899120582= 119899

0

120582+ 119899

1015840

120582

1198991205821015840 = 119899

0

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840

(A5)


1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

= 1198991205821015840 [119899

0

1205821015840 (1 + 119899

0

12058210158401015840) minus (1 + 119899

0

1205821015840) 119899

0

12058210158401015840]

= 1198991205821015840 (119899

0

1205821015840 minus 119899

0

12058210158401015840)

(A6)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (119899

0

1205821015840 minus 119899

0

12058210158401015840)

119896119861119879

(A7)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

= 1198991205821015840 [(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840) minus 119899

0

1205821015840119899

0

12058210158401015840]

= 1198991205821015840 (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

(A8)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

119896119861119879

(A9)


1205821015840




119899120582= 119899

0

120582+ 119899

1205821015840

1198991205821015840 = 119899

0

1205821015840 + 119899

1015840

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840 + 119899

1015840

12058210158401015840

(A10)

where 1198991015840

1205821015840 and 1198991015840


120582

1198991015840

1205821015840 = minusΨ

1205821015840

1205971198990

1205821015840

120597 (ℏ1205961205821015840)= Ψ

1205821015840 sdot

1

1198961198611198791198990

1205821015840 (119899

0

1205821015840 + 1) (A11)

1198991015840

12058210158401015840 = minus Ψ

12058210158401015840

1205971198990

12058210158401015840

120597 (ℏ12059612058210158401015840)

= Ψ12058210158401015840 sdot

1

1198961198611198791198990

12058210158401015840 (119899

0

12058210158401015840 + 1)

(A12)

Ψ1205821015840 = 120591

1205821015840

ℏ1205961205821015840

119879k1205821015840 sdot nabla119879 (A13)

Ψ12058210158401015840 = 120591

12058210158401015840

ℏ12059612058210158401015840

119879k12058210158401015840 sdot nabla119879 (A14)


120582Ψ

1205821015840

Ψ120582Ψ

12058210158401015840 and Ψ

1205821015840Ψ

12058210158401015840 we have

1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

=

(Ψ120582+ Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

1205821198990

1205821015840 (1 + 119899

0

12058210158401015840)

119896119861119879

(A15)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

=

(Ψ120582minus Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

120582(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840)

119896119861119879

(A16)





Acknowledgments


References

































































2Si

119909Sn

1minus119909






























1minus119909As




















1minus119909Se

119909from first-





































1minus119909Sb

119909)2Te























2Te

3Sb

2Te










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




dependence 120591(119880)(120596) sim 120596





120582+

Γ(119880)



120582= 1120591

(119873)

120582+ 1120591

(119880)

120582

where 120591(119873119880)

120582is defined as 120591(119873119880)

120582equiv 1Γ

(119873119880)






minus2



10000

1000

Error ()

Iterative scheme

Standard SMRTA

80

70

60

50

40

30

20

10

01000500300200

Temperature (K)

er

mal

con

duct

ivity

(Wm

-K)

Erro

r (

)



2Si

06Sn


2Si and Mg

2Sn cross at



100

10

1

01 1

TATALA

TOTOLO

Rela

xatio

n tim

e (p

s)

120591 prop 120596minus2


(a)

100

10

1

01 1

Normal

120591 prop 120596minus2


TATALA

TOTOLO

Rela

xatio

n tim

e (p

s)

(b)

100

10

1

01 1

Umklapp

TATALA

TOTOLO

120591 prop 120596minus3


Rela

xatio

n tim

e (p

s)

(c)


12120596

minus2 (b) 120591 = 8times1012120596

minus2 and (c) 120591 = 8times1024120596




1

120591bs=21003816100381610038161003816k120582

sdot 1003816100381610038161003816

119871

(21)






103

102

101

100

10010minus1

Life

time

(ps)

TA

Frequency (THz)

PbSePbTe

120591 sim 120596minus2

(a)

103

102

101

100

10010minus1

Life

time

(ps)

LA

Frequency (THz)

PbSePbTe

120591 sim 120596minus2

(b)

101100

101

100

10minus1

Life

time

(ps)

TO

Frequency (THz)

PbSePbTe

(c)

101100

101

100

10minus1

Life

time

(ps)

LO

Frequency (THz)

PbSePbTe

(d)






10minus1 100 101 102 103 104 105 106



PbSe

ZrCoSb

Phonon MFP (nm)

Acc

umul

ated

ther

mal

con

duct

ivity

()

100

80

60

40

20

0


05Se

05[106] Mg

2Si [63] Mg

2Sn

[63] Mg2Si

06Sn

04[63] ZrCoSb [120])





12

1

08

06

04

02

0 1 2 3 4 5 6 7 8

Diameter d (nm)

kGNR kgraphene

kRTA

kk

grap

hene k


ZA

TA

LA

10

11

10

L (120583m)

kLk

RTA





007

006

005

004

003

002

001

00 01 02 03 04 0 01 02 03 04 0 01 02 03 04 05

0 01 02 03 04 05

Scat

terin

g ra

te(c

mminus1)

Scat

terin

g ra

te(c

mminus1)

120585 120585 120585

1

120591TAasymp

1

120591aTA+

1

120591bTA+

1

120591cTA

Optical Optical

Optical

Acoustic

Acoustic Acoustic


02

015

01

005

0

120585

1

120591aTA

1

120591bTA

1

120591cTA

1

120591TA

TATATA

(a) (b) (c)

(d)





120591r120582 = 1205910

120582(1 + Δ

120582) 1 minus 119890


120582V120582|119866r120582 (22)







120591120582= 120591

0

120582(1 + Δ

120582) [

1

119878119888

int119878119888


119887)1205910

120582V120582|119866r120582 119889s] (23)



Γiso1205821205821015840 =

120587

2119873119888

120596120582120596

1205821015840

119899

sum

119887

119892119887

10038161003816100381610038161003816e119887120582sdot e119887lowast

1205821015840

10038161003816100381610038161003816

2

120575 (120596120582minus 120596

1205821015840) (24)




1

011 15 2 25 3

1 times 1

2 times 2

4 times 4

kSLk

T

M1M2


1119872

2 the mass ratio of



119892119887= sum

119894

119891119894119887(1 minus

119898119894119887

119898119887

)

2

(25)




119887is the




1

120591iso120582

= sum

1205821015840

Γiso1205821205821015840 =

120587

2119873119888

1205962

120582

119899

sum

119887

sum

1205821015840

119892119887

10038161003816100381610038161003816e119887120582sdot e119887lowast

1205821015840

10038161003816100381610038161003816

2

120575 (120596120582minus 120596

1205821015840)

(26)


1

120591iso120582

=120587

6119899119873119888

1198921205962

120582sum

1205821015840

120575 (120596120582minus 120596

1205821015840)

=120587119881

0

6119892120596

2119863 (120596)

(27)

=120587

2119892120596

2119863 (120596) (28)

where 119892 is given by (5)1198810= 119881(119899119873


119863(120596) = sum1205821015840 120575(120596120582




10

08

06

04

02

00 2 4 6 8 10

t (ps)


i(0)Ei(0)⟩


120587

120596i




is 119881 = 1198810119899119873






055Pd

045alloys [139] for




10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Empty CNT


278ps(429ps)

80ps(138ps)

25ps(26ps)

263ps(268ps)

47ps(50ps)

0 1 2 3 4 5

Φ(kz=0120596

) (J s

)

(a)

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11


Water-filled CNT

24ps 22ps 23ps

250ps

46ps

0 1 2 3 4 5

Φ(kz=0120596

) (J s

)

(b)



20

15

10

05

000806040200 10

Longitudinal

1120591

(1p

s)

20K LD50K LD

20K MD50K MD

klowast (a2120587)

(a)

08

06

04

02

000806040200 10

Transverse

1120591

(1p

s)

20K LD50K LD

20K MD50K MD

klowast (a2120587)

(b)




(at the limit 119875119871



119871 This leads



119871









50

40

30

20

10

050403020100

Phon

on li

fetim

e (p

s)








1+1superlattice We



1119872





1119872



1119872



1119872




4 MD Simulation



120591120582=

intinfin

0⟨119899

1015840

120582(119905) 119899

1015840

120582(0)⟩ 119889119905

⟨11989910158402

120582(0)⟩

(29)







119864120582=1

2( 119902

120582119902lowast

120582+ 120596

2

120582119902120582119902lowast

120582) (30)


transformed to

120591120582=

intinfin

0⟨119864

120582(119905) 119864

120582(0)⟩ 119889119905

⟨1198642

120582(0)⟩

(31)


120582(119905)119864

120582(0)⟩⟨119864

2

120582(0)⟩ = exp(minus119905120591

120582)


120582sim 119902

120582119902lowast



119902120582(119905) = 119902

1205820exp [119894 (120596119860

120582+ 119894120578

120582) 119905] (32)

where 1199021205820


120582= 120596

120582+ Δ


120582


120582119902lowast

120582

and 119864120582sim 119902

120582119902lowast


120582= 12120578

120582



120582in (31)


119902120582(119905) =

3

sum

120572

119899

sum

119887

119873119888

sum

119897

radic119898

119887

119873119888

119906119897119887

120572(119905) 119890

120582lowast

119887120572exp [119894k sdot r119897

0]

=

3

sum

120572

119899

sum

119887

119890120582lowast

119887120572119902119887

120572(k 119905)

(33)


120572(119905) is 120572 component




120572(k 119905) denotes



300

200

100

0

ZA sTA sLA sZO s

ZA pTA pLA pZO p

er

mal

con

duct

ivity

(Wm

K)

(a)

Temperature (K)

()

300 400 500 600

40

30

20

10

ZA sTA sLA sZO s

ZA pTA pLA pZO p

(b)



119902119887

120572(k 119905) =

119873119888

sum

119897

radic119898

119887

119873119888

119906119897119887

120572(119905) exp [119894k sdot r119897

0] (34)



Φ120582(120596) =

1003816100381610038161003816 119902120582(120596)

1003816100381610038161003816

2

=

10038161003816100381610038161003816100381610038161003816

int

+infin

0

119902120582(119905) 119890

minus119894120596119905119889119905

10038161003816100381610038161003816100381610038161003816

2

=119862

120582

(120596 minus 120596119860

120582)2

+ 1205782

120582

(35)

where119862120582= (120596

119860

120582

2

+Γ2

120582)119902

2






120582= 12120578




Φ (k 120596) =3119899

sum

]Φ

120582(120596)

=1

41205871205910

3119899

sum

120572119887

119898119887

119873119888

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119897

1205910

int

0

119897119887

120572(119905) exp [119894k sdot r119897

0minus 119894120596119905]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

=

3

sum

120572

119899

sum

119887

10038161003816100381610038161003816119902119887

120572(k 120596)10038161003816100381610038161003816

2

(36)

where 119902119887





119888in (34) are




Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)100

10

1

100

10

10 1 2 0 1 2


300K 600K

[100] LA[110] LA[111] LA

[100] LA[110] LA[111] LA

(a)

[100] TA[110] TA1[110] TA2[111] TA

[100] TA[110] TA1[110] TA2[111] TA

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

300K 600K

(b)

1

0135 40

1

0135 40

300K 600K

[100] LO[110] LO[111] LO

[100] LO[110] LO[111] LO

Rela

xatio

n tim

e (p

s)

Frequency (THz)

Rela

xatio

n tim

e (p

s)

Frequency (THz)

(c)

30

300K 600K

[100] TO[110] TO1[110] TO2[111] TO

[100] TO[110] TO1[110] TO2[111] TO

1

01

1

0135 40 30 35 40

Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)


(d)







80

60

40

20

00 1 2 3 4

46

23

26

300K

Frequency (THz)

TA1TA2LA

TO1TO2LO

Perc

enta

ge c

ontr

ibut

ion

(au

)

(a)

80

60

40

20

00 1 2 3 4

50

2124

TA1TA2LA

TO1TO2LO

600K

Frequency (THz)

Perc

enta

ge c

ontr

ibut

ion

(au

)

(b)


150

100

50

0 1 2 3 4 5

Frequency (THz)

L = 12

L = 9

L = 6

Acoustic phonon



Rela

xatio

n tim

e (p

s)

(a)


0

150

100

50

02 04 06 08

Predicted by NMA

Rela

xatio

n tim

e (p

s)


(b)










10minus1 100 101 102 103 104 105



MFP (nm)

Acc

umul

ated

ther

mal

con

duct

ivity

() 100

80

60

40

20

0


2Te

3(MD) [142])

[141] and Bi2Te









2Te



3


2Te


in the future

5 Summary






1119872

2SLs 1 and 2 [59] 2004


1119872

2SLs 1 and 2 radic [50] 2008



2 and SrTiO3) 2 [49] 2011



Table 2 Continued


1minus119909Se

1199091 radic [106] 2012

Bi2Te

34 [142] 2013





Equations (14)(15) (16) (17)

and (18)














Applications sofar











Appendix

A


120596 + 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) (119899

0

1205821015840 + 1) 119899

0

12058210158401015840 = 119899

0

1205821198990

1205821015840 (119899

0

12058210158401015840 + 1) (A1)

120596 minus 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) 119899

0

1205821015840119899

0

12058210158401015840 = 119899

0

120582(119899

0

1205821015840 + 1) (119899

0

12058210158401015840 + 1) (A2)


119861119879) minus 1]


1198991015840

120582= minusΨ

120582

1205971198990

120582

120597 (ℏ120596120582)= Ψ

120582sdot

1

1198961198611198791198990

120582(119899

0

120582+ 1) (A3)


approximation (11)

Ψ120582= 120591

120582

ℏ120596120582

119879k120582sdot nabla119879 (A4)


119899120582= 119899

0

120582+ 119899

1015840

120582

1198991205821015840 = 119899

0

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840

(A5)


1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

= 1198991205821015840 [119899

0

1205821015840 (1 + 119899

0

12058210158401015840) minus (1 + 119899

0

1205821015840) 119899

0

12058210158401015840]

= 1198991205821015840 (119899

0

1205821015840 minus 119899

0

12058210158401015840)

(A6)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (119899

0

1205821015840 minus 119899

0

12058210158401015840)

119896119861119879

(A7)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

= 1198991205821015840 [(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840) minus 119899

0

1205821015840119899

0

12058210158401015840]

= 1198991205821015840 (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

(A8)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

119896119861119879

(A9)


1205821015840




119899120582= 119899

0

120582+ 119899

1205821015840

1198991205821015840 = 119899

0

1205821015840 + 119899

1015840

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840 + 119899

1015840

12058210158401015840

(A10)

where 1198991015840

1205821015840 and 1198991015840


120582

1198991015840

1205821015840 = minusΨ

1205821015840

1205971198990

1205821015840

120597 (ℏ1205961205821015840)= Ψ

1205821015840 sdot

1

1198961198611198791198990

1205821015840 (119899

0

1205821015840 + 1) (A11)

1198991015840

12058210158401015840 = minus Ψ

12058210158401015840

1205971198990

12058210158401015840

120597 (ℏ12059612058210158401015840)

= Ψ12058210158401015840 sdot

1

1198961198611198791198990

12058210158401015840 (119899

0

12058210158401015840 + 1)

(A12)

Ψ1205821015840 = 120591

1205821015840

ℏ1205961205821015840

119879k1205821015840 sdot nabla119879 (A13)

Ψ12058210158401015840 = 120591

12058210158401015840

ℏ12059612058210158401015840

119879k12058210158401015840 sdot nabla119879 (A14)


120582Ψ

1205821015840

Ψ120582Ψ

12058210158401015840 and Ψ

1205821015840Ψ

12058210158401015840 we have

1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

=

(Ψ120582+ Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

1205821198990

1205821015840 (1 + 119899

0

12058210158401015840)

119896119861119879

(A15)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

=

(Ψ120582minus Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

120582(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840)

119896119861119879

(A16)





Acknowledgments


References

































































2Si

119909Sn

1minus119909






























1minus119909As




















1minus119909Se

119909from first-





































1minus119909Sb

119909)2Te























2Te

3Sb

2Te










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




100

10

1

01 1

TATALA

TOTOLO

Rela

xatio

n tim

e (p

s)

120591 prop 120596minus2


(a)

100

10

1

01 1

Normal

120591 prop 120596minus2


TATALA

TOTOLO

Rela

xatio

n tim

e (p

s)

(b)

100

10

1

01 1

Umklapp

TATALA

TOTOLO

120591 prop 120596minus3


Rela

xatio

n tim

e (p

s)

(c)


12120596

minus2 (b) 120591 = 8times1012120596

minus2 and (c) 120591 = 8times1024120596




1

120591bs=21003816100381610038161003816k120582

sdot 1003816100381610038161003816

119871

(21)






103

102

101

100

10010minus1

Life

time

(ps)

TA

Frequency (THz)

PbSePbTe

120591 sim 120596minus2

(a)

103

102

101

100

10010minus1

Life

time

(ps)

LA

Frequency (THz)

PbSePbTe

120591 sim 120596minus2

(b)

101100

101

100

10minus1

Life

time

(ps)

TO

Frequency (THz)

PbSePbTe

(c)

101100

101

100

10minus1

Life

time

(ps)

LO

Frequency (THz)

PbSePbTe

(d)






10minus1 100 101 102 103 104 105 106



PbSe

ZrCoSb

Phonon MFP (nm)

Acc

umul

ated

ther

mal

con

duct

ivity

()

100

80

60

40

20

0


05Se

05[106] Mg

2Si [63] Mg

2Sn

[63] Mg2Si

06Sn

04[63] ZrCoSb [120])





12

1

08

06

04

02

0 1 2 3 4 5 6 7 8

Diameter d (nm)

kGNR kgraphene

kRTA

kk

grap

hene k


ZA

TA

LA

10

11

10

L (120583m)

kLk

RTA





007

006

005

004

003

002

001

00 01 02 03 04 0 01 02 03 04 0 01 02 03 04 05

0 01 02 03 04 05

Scat

terin

g ra

te(c

mminus1)

Scat

terin

g ra

te(c

mminus1)

120585 120585 120585

1

120591TAasymp

1

120591aTA+

1

120591bTA+

1

120591cTA

Optical Optical

Optical

Acoustic

Acoustic Acoustic


02

015

01

005

0

120585

1

120591aTA

1

120591bTA

1

120591cTA

1

120591TA

TATATA

(a) (b) (c)

(d)





120591r120582 = 1205910

120582(1 + Δ

120582) 1 minus 119890


120582V120582|119866r120582 (22)







120591120582= 120591

0

120582(1 + Δ

120582) [

1

119878119888

int119878119888


119887)1205910

120582V120582|119866r120582 119889s] (23)



Γiso1205821205821015840 =

120587

2119873119888

120596120582120596

1205821015840

119899

sum

119887

119892119887

10038161003816100381610038161003816e119887120582sdot e119887lowast

1205821015840

10038161003816100381610038161003816

2

120575 (120596120582minus 120596

1205821015840) (24)




1

011 15 2 25 3

1 times 1

2 times 2

4 times 4

kSLk

T

M1M2


1119872

2 the mass ratio of



119892119887= sum

119894

119891119894119887(1 minus

119898119894119887

119898119887

)

2

(25)




119887is the




1

120591iso120582

= sum

1205821015840

Γiso1205821205821015840 =

120587

2119873119888

1205962

120582

119899

sum

119887

sum

1205821015840

119892119887

10038161003816100381610038161003816e119887120582sdot e119887lowast

1205821015840

10038161003816100381610038161003816

2

120575 (120596120582minus 120596

1205821015840)

(26)


1

120591iso120582

=120587

6119899119873119888

1198921205962

120582sum

1205821015840

120575 (120596120582minus 120596

1205821015840)

=120587119881

0

6119892120596

2119863 (120596)

(27)

=120587

2119892120596

2119863 (120596) (28)

where 119892 is given by (5)1198810= 119881(119899119873


119863(120596) = sum1205821015840 120575(120596120582




10

08

06

04

02

00 2 4 6 8 10

t (ps)


i(0)Ei(0)⟩


120587

120596i




is 119881 = 1198810119899119873






055Pd

045alloys [139] for




10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Empty CNT


278ps(429ps)

80ps(138ps)

25ps(26ps)

263ps(268ps)

47ps(50ps)

0 1 2 3 4 5

Φ(kz=0120596

) (J s

)

(a)

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11


Water-filled CNT

24ps 22ps 23ps

250ps

46ps

0 1 2 3 4 5

Φ(kz=0120596

) (J s

)

(b)



20

15

10

05

000806040200 10

Longitudinal

1120591

(1p

s)

20K LD50K LD

20K MD50K MD

klowast (a2120587)

(a)

08

06

04

02

000806040200 10

Transverse

1120591

(1p

s)

20K LD50K LD

20K MD50K MD

klowast (a2120587)

(b)




(at the limit 119875119871



119871 This leads



119871









50

40

30

20

10

050403020100

Phon

on li

fetim

e (p

s)








1+1superlattice We



1119872





1119872



1119872



1119872




4 MD Simulation



120591120582=

intinfin

0⟨119899

1015840

120582(119905) 119899

1015840

120582(0)⟩ 119889119905

⟨11989910158402

120582(0)⟩

(29)







119864120582=1

2( 119902

120582119902lowast

120582+ 120596

2

120582119902120582119902lowast

120582) (30)


transformed to

120591120582=

intinfin

0⟨119864

120582(119905) 119864

120582(0)⟩ 119889119905

⟨1198642

120582(0)⟩

(31)


120582(119905)119864

120582(0)⟩⟨119864

2

120582(0)⟩ = exp(minus119905120591

120582)


120582sim 119902

120582119902lowast



119902120582(119905) = 119902

1205820exp [119894 (120596119860

120582+ 119894120578

120582) 119905] (32)

where 1199021205820


120582= 120596

120582+ Δ


120582


120582119902lowast

120582

and 119864120582sim 119902

120582119902lowast


120582= 12120578

120582



120582in (31)


119902120582(119905) =

3

sum

120572

119899

sum

119887

119873119888

sum

119897

radic119898

119887

119873119888

119906119897119887

120572(119905) 119890

120582lowast

119887120572exp [119894k sdot r119897

0]

=

3

sum

120572

119899

sum

119887

119890120582lowast

119887120572119902119887

120572(k 119905)

(33)


120572(119905) is 120572 component




120572(k 119905) denotes



300

200

100

0

ZA sTA sLA sZO s

ZA pTA pLA pZO p

er

mal

con

duct

ivity

(Wm

K)

(a)

Temperature (K)

()

300 400 500 600

40

30

20

10

ZA sTA sLA sZO s

ZA pTA pLA pZO p

(b)



119902119887

120572(k 119905) =

119873119888

sum

119897

radic119898

119887

119873119888

119906119897119887

120572(119905) exp [119894k sdot r119897

0] (34)



Φ120582(120596) =

1003816100381610038161003816 119902120582(120596)

1003816100381610038161003816

2

=

10038161003816100381610038161003816100381610038161003816

int

+infin

0

119902120582(119905) 119890

minus119894120596119905119889119905

10038161003816100381610038161003816100381610038161003816

2

=119862

120582

(120596 minus 120596119860

120582)2

+ 1205782

120582

(35)

where119862120582= (120596

119860

120582

2

+Γ2

120582)119902

2






120582= 12120578




Φ (k 120596) =3119899

sum

]Φ

120582(120596)

=1

41205871205910

3119899

sum

120572119887

119898119887

119873119888

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119897

1205910

int

0

119897119887

120572(119905) exp [119894k sdot r119897

0minus 119894120596119905]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

=

3

sum

120572

119899

sum

119887

10038161003816100381610038161003816119902119887

120572(k 120596)10038161003816100381610038161003816

2

(36)

where 119902119887





119888in (34) are




Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)100

10

1

100

10

10 1 2 0 1 2


300K 600K

[100] LA[110] LA[111] LA

[100] LA[110] LA[111] LA

(a)

[100] TA[110] TA1[110] TA2[111] TA

[100] TA[110] TA1[110] TA2[111] TA

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

300K 600K

(b)

1

0135 40

1

0135 40

300K 600K

[100] LO[110] LO[111] LO

[100] LO[110] LO[111] LO

Rela

xatio

n tim

e (p

s)

Frequency (THz)

Rela

xatio

n tim

e (p

s)

Frequency (THz)

(c)

30

300K 600K

[100] TO[110] TO1[110] TO2[111] TO

[100] TO[110] TO1[110] TO2[111] TO

1

01

1

0135 40 30 35 40

Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)


(d)







80

60

40

20

00 1 2 3 4

46

23

26

300K

Frequency (THz)

TA1TA2LA

TO1TO2LO

Perc

enta

ge c

ontr

ibut

ion

(au

)

(a)

80

60

40

20

00 1 2 3 4

50

2124

TA1TA2LA

TO1TO2LO

600K

Frequency (THz)

Perc

enta

ge c

ontr

ibut

ion

(au

)

(b)


150

100

50

0 1 2 3 4 5

Frequency (THz)

L = 12

L = 9

L = 6

Acoustic phonon



Rela

xatio

n tim

e (p

s)

(a)


0

150

100

50

02 04 06 08

Predicted by NMA

Rela

xatio

n tim

e (p

s)


(b)










10minus1 100 101 102 103 104 105



MFP (nm)

Acc

umul

ated

ther

mal

con

duct

ivity

() 100

80

60

40

20

0


2Te

3(MD) [142])

[141] and Bi2Te









2Te



3


2Te


in the future

5 Summary






1119872

2SLs 1 and 2 [59] 2004


1119872

2SLs 1 and 2 radic [50] 2008



2 and SrTiO3) 2 [49] 2011



Table 2 Continued


1minus119909Se

1199091 radic [106] 2012

Bi2Te

34 [142] 2013





Equations (14)(15) (16) (17)

and (18)














Applications sofar











Appendix

A


120596 + 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) (119899

0

1205821015840 + 1) 119899

0

12058210158401015840 = 119899

0

1205821198990

1205821015840 (119899

0

12058210158401015840 + 1) (A1)

120596 minus 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) 119899

0

1205821015840119899

0

12058210158401015840 = 119899

0

120582(119899

0

1205821015840 + 1) (119899

0

12058210158401015840 + 1) (A2)


119861119879) minus 1]


1198991015840

120582= minusΨ

120582

1205971198990

120582

120597 (ℏ120596120582)= Ψ

120582sdot

1

1198961198611198791198990

120582(119899

0

120582+ 1) (A3)


approximation (11)

Ψ120582= 120591

120582

ℏ120596120582

119879k120582sdot nabla119879 (A4)


119899120582= 119899

0

120582+ 119899

1015840

120582

1198991205821015840 = 119899

0

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840

(A5)


1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

= 1198991205821015840 [119899

0

1205821015840 (1 + 119899

0

12058210158401015840) minus (1 + 119899

0

1205821015840) 119899

0

12058210158401015840]

= 1198991205821015840 (119899

0

1205821015840 minus 119899

0

12058210158401015840)

(A6)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (119899

0

1205821015840 minus 119899

0

12058210158401015840)

119896119861119879

(A7)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

= 1198991205821015840 [(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840) minus 119899

0

1205821015840119899

0

12058210158401015840]

= 1198991205821015840 (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

(A8)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

119896119861119879

(A9)


1205821015840




119899120582= 119899

0

120582+ 119899

1205821015840

1198991205821015840 = 119899

0

1205821015840 + 119899

1015840

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840 + 119899

1015840

12058210158401015840

(A10)

where 1198991015840

1205821015840 and 1198991015840


120582

1198991015840

1205821015840 = minusΨ

1205821015840

1205971198990

1205821015840

120597 (ℏ1205961205821015840)= Ψ

1205821015840 sdot

1

1198961198611198791198990

1205821015840 (119899

0

1205821015840 + 1) (A11)

1198991015840

12058210158401015840 = minus Ψ

12058210158401015840

1205971198990

12058210158401015840

120597 (ℏ12059612058210158401015840)

= Ψ12058210158401015840 sdot

1

1198961198611198791198990

12058210158401015840 (119899

0

12058210158401015840 + 1)

(A12)

Ψ1205821015840 = 120591

1205821015840

ℏ1205961205821015840

119879k1205821015840 sdot nabla119879 (A13)

Ψ12058210158401015840 = 120591

12058210158401015840

ℏ12059612058210158401015840

119879k12058210158401015840 sdot nabla119879 (A14)


120582Ψ

1205821015840

Ψ120582Ψ

12058210158401015840 and Ψ

1205821015840Ψ

12058210158401015840 we have

1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

=

(Ψ120582+ Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

1205821198990

1205821015840 (1 + 119899

0

12058210158401015840)

119896119861119879

(A15)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

=

(Ψ120582minus Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

120582(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840)

119896119861119879

(A16)





Acknowledgments


References

































































2Si

119909Sn

1minus119909






























1minus119909As




















1minus119909Se

119909from first-





































1minus119909Sb

119909)2Te























2Te

3Sb

2Te










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




103

102

101

100

10010minus1

Life

time

(ps)

TA

Frequency (THz)

PbSePbTe

120591 sim 120596minus2

(a)

103

102

101

100

10010minus1

Life

time

(ps)

LA

Frequency (THz)

PbSePbTe

120591 sim 120596minus2

(b)

101100

101

100

10minus1

Life

time

(ps)

TO

Frequency (THz)

PbSePbTe

(c)

101100

101

100

10minus1

Life

time

(ps)

LO

Frequency (THz)

PbSePbTe

(d)






10minus1 100 101 102 103 104 105 106



PbSe

ZrCoSb

Phonon MFP (nm)

Acc

umul

ated

ther

mal

con

duct

ivity

()

100

80

60

40

20

0


05Se

05[106] Mg

2Si [63] Mg

2Sn

[63] Mg2Si

06Sn

04[63] ZrCoSb [120])





12

1

08

06

04

02

0 1 2 3 4 5 6 7 8

Diameter d (nm)

kGNR kgraphene

kRTA

kk

grap

hene k


ZA

TA

LA

10

11

10

L (120583m)

kLk

RTA





007

006

005

004

003

002

001

00 01 02 03 04 0 01 02 03 04 0 01 02 03 04 05

0 01 02 03 04 05

Scat

terin

g ra

te(c

mminus1)

Scat

terin

g ra

te(c

mminus1)

120585 120585 120585

1

120591TAasymp

1

120591aTA+

1

120591bTA+

1

120591cTA

Optical Optical

Optical

Acoustic

Acoustic Acoustic


02

015

01

005

0

120585

1

120591aTA

1

120591bTA

1

120591cTA

1

120591TA

TATATA

(a) (b) (c)

(d)





120591r120582 = 1205910

120582(1 + Δ

120582) 1 minus 119890


120582V120582|119866r120582 (22)







120591120582= 120591

0

120582(1 + Δ

120582) [

1

119878119888

int119878119888


119887)1205910

120582V120582|119866r120582 119889s] (23)



Γiso1205821205821015840 =

120587

2119873119888

120596120582120596

1205821015840

119899

sum

119887

119892119887

10038161003816100381610038161003816e119887120582sdot e119887lowast

1205821015840

10038161003816100381610038161003816

2

120575 (120596120582minus 120596

1205821015840) (24)




1

011 15 2 25 3

1 times 1

2 times 2

4 times 4

kSLk

T

M1M2


1119872

2 the mass ratio of



119892119887= sum

119894

119891119894119887(1 minus

119898119894119887

119898119887

)

2

(25)




119887is the




1

120591iso120582

= sum

1205821015840

Γiso1205821205821015840 =

120587

2119873119888

1205962

120582

119899

sum

119887

sum

1205821015840

119892119887

10038161003816100381610038161003816e119887120582sdot e119887lowast

1205821015840

10038161003816100381610038161003816

2

120575 (120596120582minus 120596

1205821015840)

(26)


1

120591iso120582

=120587

6119899119873119888

1198921205962

120582sum

1205821015840

120575 (120596120582minus 120596

1205821015840)

=120587119881

0

6119892120596

2119863 (120596)

(27)

=120587

2119892120596

2119863 (120596) (28)

where 119892 is given by (5)1198810= 119881(119899119873


119863(120596) = sum1205821015840 120575(120596120582




10

08

06

04

02

00 2 4 6 8 10

t (ps)


i(0)Ei(0)⟩


120587

120596i




is 119881 = 1198810119899119873






055Pd

045alloys [139] for




10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Empty CNT


278ps(429ps)

80ps(138ps)

25ps(26ps)

263ps(268ps)

47ps(50ps)

0 1 2 3 4 5

Φ(kz=0120596

) (J s

)

(a)

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11


Water-filled CNT

24ps 22ps 23ps

250ps

46ps

0 1 2 3 4 5

Φ(kz=0120596

) (J s

)

(b)



20

15

10

05

000806040200 10

Longitudinal

1120591

(1p

s)

20K LD50K LD

20K MD50K MD

klowast (a2120587)

(a)

08

06

04

02

000806040200 10

Transverse

1120591

(1p

s)

20K LD50K LD

20K MD50K MD

klowast (a2120587)

(b)




(at the limit 119875119871



119871 This leads



119871









50

40

30

20

10

050403020100

Phon

on li

fetim

e (p

s)








1+1superlattice We



1119872





1119872



1119872



1119872




4 MD Simulation



120591120582=

intinfin

0⟨119899

1015840

120582(119905) 119899

1015840

120582(0)⟩ 119889119905

⟨11989910158402

120582(0)⟩

(29)







119864120582=1

2( 119902

120582119902lowast

120582+ 120596

2

120582119902120582119902lowast

120582) (30)


transformed to

120591120582=

intinfin

0⟨119864

120582(119905) 119864

120582(0)⟩ 119889119905

⟨1198642

120582(0)⟩

(31)


120582(119905)119864

120582(0)⟩⟨119864

2

120582(0)⟩ = exp(minus119905120591

120582)


120582sim 119902

120582119902lowast



119902120582(119905) = 119902

1205820exp [119894 (120596119860

120582+ 119894120578

120582) 119905] (32)

where 1199021205820


120582= 120596

120582+ Δ


120582


120582119902lowast

120582

and 119864120582sim 119902

120582119902lowast


120582= 12120578

120582



120582in (31)


119902120582(119905) =

3

sum

120572

119899

sum

119887

119873119888

sum

119897

radic119898

119887

119873119888

119906119897119887

120572(119905) 119890

120582lowast

119887120572exp [119894k sdot r119897

0]

=

3

sum

120572

119899

sum

119887

119890120582lowast

119887120572119902119887

120572(k 119905)

(33)


120572(119905) is 120572 component




120572(k 119905) denotes



300

200

100

0

ZA sTA sLA sZO s

ZA pTA pLA pZO p

er

mal

con

duct

ivity

(Wm

K)

(a)

Temperature (K)

()

300 400 500 600

40

30

20

10

ZA sTA sLA sZO s

ZA pTA pLA pZO p

(b)



119902119887

120572(k 119905) =

119873119888

sum

119897

radic119898

119887

119873119888

119906119897119887

120572(119905) exp [119894k sdot r119897

0] (34)



Φ120582(120596) =

1003816100381610038161003816 119902120582(120596)

1003816100381610038161003816

2

=

10038161003816100381610038161003816100381610038161003816

int

+infin

0

119902120582(119905) 119890

minus119894120596119905119889119905

10038161003816100381610038161003816100381610038161003816

2

=119862

120582

(120596 minus 120596119860

120582)2

+ 1205782

120582

(35)

where119862120582= (120596

119860

120582

2

+Γ2

120582)119902

2






120582= 12120578




Φ (k 120596) =3119899

sum

]Φ

120582(120596)

=1

41205871205910

3119899

sum

120572119887

119898119887

119873119888

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119897

1205910

int

0

119897119887

120572(119905) exp [119894k sdot r119897

0minus 119894120596119905]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

=

3

sum

120572

119899

sum

119887

10038161003816100381610038161003816119902119887

120572(k 120596)10038161003816100381610038161003816

2

(36)

where 119902119887





119888in (34) are




Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)100

10

1

100

10

10 1 2 0 1 2


300K 600K

[100] LA[110] LA[111] LA

[100] LA[110] LA[111] LA

(a)

[100] TA[110] TA1[110] TA2[111] TA

[100] TA[110] TA1[110] TA2[111] TA

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

300K 600K

(b)

1

0135 40

1

0135 40

300K 600K

[100] LO[110] LO[111] LO

[100] LO[110] LO[111] LO

Rela

xatio

n tim

e (p

s)

Frequency (THz)

Rela

xatio

n tim

e (p

s)

Frequency (THz)

(c)

30

300K 600K

[100] TO[110] TO1[110] TO2[111] TO

[100] TO[110] TO1[110] TO2[111] TO

1

01

1

0135 40 30 35 40

Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)


(d)







80

60

40

20

00 1 2 3 4

46

23

26

300K

Frequency (THz)

TA1TA2LA

TO1TO2LO

Perc

enta

ge c

ontr

ibut

ion

(au

)

(a)

80

60

40

20

00 1 2 3 4

50

2124

TA1TA2LA

TO1TO2LO

600K

Frequency (THz)

Perc

enta

ge c

ontr

ibut

ion

(au

)

(b)


150

100

50

0 1 2 3 4 5

Frequency (THz)

L = 12

L = 9

L = 6

Acoustic phonon



Rela

xatio

n tim

e (p

s)

(a)


0

150

100

50

02 04 06 08

Predicted by NMA

Rela

xatio

n tim

e (p

s)


(b)










10minus1 100 101 102 103 104 105



MFP (nm)

Acc

umul

ated

ther

mal

con

duct

ivity

() 100

80

60

40

20

0


2Te

3(MD) [142])

[141] and Bi2Te









2Te



3


2Te


in the future

5 Summary






1119872

2SLs 1 and 2 [59] 2004


1119872

2SLs 1 and 2 radic [50] 2008



2 and SrTiO3) 2 [49] 2011



Table 2 Continued


1minus119909Se

1199091 radic [106] 2012

Bi2Te

34 [142] 2013





Equations (14)(15) (16) (17)

and (18)














Applications sofar











Appendix

A


120596 + 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) (119899

0

1205821015840 + 1) 119899

0

12058210158401015840 = 119899

0

1205821198990

1205821015840 (119899

0

12058210158401015840 + 1) (A1)

120596 minus 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) 119899

0

1205821015840119899

0

12058210158401015840 = 119899

0

120582(119899

0

1205821015840 + 1) (119899

0

12058210158401015840 + 1) (A2)


119861119879) minus 1]


1198991015840

120582= minusΨ

120582

1205971198990

120582

120597 (ℏ120596120582)= Ψ

120582sdot

1

1198961198611198791198990

120582(119899

0

120582+ 1) (A3)


approximation (11)

Ψ120582= 120591

120582

ℏ120596120582

119879k120582sdot nabla119879 (A4)


119899120582= 119899

0

120582+ 119899

1015840

120582

1198991205821015840 = 119899

0

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840

(A5)


1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

= 1198991205821015840 [119899

0

1205821015840 (1 + 119899

0

12058210158401015840) minus (1 + 119899

0

1205821015840) 119899

0

12058210158401015840]

= 1198991205821015840 (119899

0

1205821015840 minus 119899

0

12058210158401015840)

(A6)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (119899

0

1205821015840 minus 119899

0

12058210158401015840)

119896119861119879

(A7)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

= 1198991205821015840 [(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840) minus 119899

0

1205821015840119899

0

12058210158401015840]

= 1198991205821015840 (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

(A8)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

119896119861119879

(A9)


1205821015840




119899120582= 119899

0

120582+ 119899

1205821015840

1198991205821015840 = 119899

0

1205821015840 + 119899

1015840

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840 + 119899

1015840

12058210158401015840

(A10)

where 1198991015840

1205821015840 and 1198991015840


120582

1198991015840

1205821015840 = minusΨ

1205821015840

1205971198990

1205821015840

120597 (ℏ1205961205821015840)= Ψ

1205821015840 sdot

1

1198961198611198791198990

1205821015840 (119899

0

1205821015840 + 1) (A11)

1198991015840

12058210158401015840 = minus Ψ

12058210158401015840

1205971198990

12058210158401015840

120597 (ℏ12059612058210158401015840)

= Ψ12058210158401015840 sdot

1

1198961198611198791198990

12058210158401015840 (119899

0

12058210158401015840 + 1)

(A12)

Ψ1205821015840 = 120591

1205821015840

ℏ1205961205821015840

119879k1205821015840 sdot nabla119879 (A13)

Ψ12058210158401015840 = 120591

12058210158401015840

ℏ12059612058210158401015840

119879k12058210158401015840 sdot nabla119879 (A14)


120582Ψ

1205821015840

Ψ120582Ψ

12058210158401015840 and Ψ

1205821015840Ψ

12058210158401015840 we have

1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

=

(Ψ120582+ Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

1205821198990

1205821015840 (1 + 119899

0

12058210158401015840)

119896119861119879

(A15)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

=

(Ψ120582minus Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

120582(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840)

119896119861119879

(A16)





Acknowledgments


References

































































2Si

119909Sn

1minus119909






























1minus119909As




















1minus119909Se

119909from first-





































1minus119909Sb

119909)2Te























2Te

3Sb

2Te










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




10minus1 100 101 102 103 104 105 106



PbSe

ZrCoSb

Phonon MFP (nm)

Acc

umul

ated

ther

mal

con

duct

ivity

()

100

80

60

40

20

0


05Se

05[106] Mg

2Si [63] Mg

2Sn

[63] Mg2Si

06Sn

04[63] ZrCoSb [120])





12

1

08

06

04

02

0 1 2 3 4 5 6 7 8

Diameter d (nm)

kGNR kgraphene

kRTA

kk

grap

hene k


ZA

TA

LA

10

11

10

L (120583m)

kLk

RTA





007

006

005

004

003

002

001

00 01 02 03 04 0 01 02 03 04 0 01 02 03 04 05

0 01 02 03 04 05

Scat

terin

g ra

te(c

mminus1)

Scat

terin

g ra

te(c

mminus1)

120585 120585 120585

1

120591TAasymp

1

120591aTA+

1

120591bTA+

1

120591cTA

Optical Optical

Optical

Acoustic

Acoustic Acoustic


02

015

01

005

0

120585

1

120591aTA

1

120591bTA

1

120591cTA

1

120591TA

TATATA

(a) (b) (c)

(d)





120591r120582 = 1205910

120582(1 + Δ

120582) 1 minus 119890


120582V120582|119866r120582 (22)







120591120582= 120591

0

120582(1 + Δ

120582) [

1

119878119888

int119878119888


119887)1205910

120582V120582|119866r120582 119889s] (23)



Γiso1205821205821015840 =

120587

2119873119888

120596120582120596

1205821015840

119899

sum

119887

119892119887

10038161003816100381610038161003816e119887120582sdot e119887lowast

1205821015840

10038161003816100381610038161003816

2

120575 (120596120582minus 120596

1205821015840) (24)




1

011 15 2 25 3

1 times 1

2 times 2

4 times 4

kSLk

T

M1M2


1119872

2 the mass ratio of



119892119887= sum

119894

119891119894119887(1 minus

119898119894119887

119898119887

)

2

(25)




119887is the




1

120591iso120582

= sum

1205821015840

Γiso1205821205821015840 =

120587

2119873119888

1205962

120582

119899

sum

119887

sum

1205821015840

119892119887

10038161003816100381610038161003816e119887120582sdot e119887lowast

1205821015840

10038161003816100381610038161003816

2

120575 (120596120582minus 120596

1205821015840)

(26)


1

120591iso120582

=120587

6119899119873119888

1198921205962

120582sum

1205821015840

120575 (120596120582minus 120596

1205821015840)

=120587119881

0

6119892120596

2119863 (120596)

(27)

=120587

2119892120596

2119863 (120596) (28)

where 119892 is given by (5)1198810= 119881(119899119873


119863(120596) = sum1205821015840 120575(120596120582




10

08

06

04

02

00 2 4 6 8 10

t (ps)


i(0)Ei(0)⟩


120587

120596i




is 119881 = 1198810119899119873






055Pd

045alloys [139] for




10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Empty CNT


278ps(429ps)

80ps(138ps)

25ps(26ps)

263ps(268ps)

47ps(50ps)

0 1 2 3 4 5

Φ(kz=0120596

) (J s

)

(a)

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11


Water-filled CNT

24ps 22ps 23ps

250ps

46ps

0 1 2 3 4 5

Φ(kz=0120596

) (J s

)

(b)



20

15

10

05

000806040200 10

Longitudinal

1120591

(1p

s)

20K LD50K LD

20K MD50K MD

klowast (a2120587)

(a)

08

06

04

02

000806040200 10

Transverse

1120591

(1p

s)

20K LD50K LD

20K MD50K MD

klowast (a2120587)

(b)




(at the limit 119875119871



119871 This leads



119871









50

40

30

20

10

050403020100

Phon

on li

fetim

e (p

s)








1+1superlattice We



1119872





1119872



1119872



1119872




4 MD Simulation



120591120582=

intinfin

0⟨119899

1015840

120582(119905) 119899

1015840

120582(0)⟩ 119889119905

⟨11989910158402

120582(0)⟩

(29)







119864120582=1

2( 119902

120582119902lowast

120582+ 120596

2

120582119902120582119902lowast

120582) (30)


transformed to

120591120582=

intinfin

0⟨119864

120582(119905) 119864

120582(0)⟩ 119889119905

⟨1198642

120582(0)⟩

(31)


120582(119905)119864

120582(0)⟩⟨119864

2

120582(0)⟩ = exp(minus119905120591

120582)


120582sim 119902

120582119902lowast



119902120582(119905) = 119902

1205820exp [119894 (120596119860

120582+ 119894120578

120582) 119905] (32)

where 1199021205820


120582= 120596

120582+ Δ


120582


120582119902lowast

120582

and 119864120582sim 119902

120582119902lowast


120582= 12120578

120582



120582in (31)


119902120582(119905) =

3

sum

120572

119899

sum

119887

119873119888

sum

119897

radic119898

119887

119873119888

119906119897119887

120572(119905) 119890

120582lowast

119887120572exp [119894k sdot r119897

0]

=

3

sum

120572

119899

sum

119887

119890120582lowast

119887120572119902119887

120572(k 119905)

(33)


120572(119905) is 120572 component




120572(k 119905) denotes



300

200

100

0

ZA sTA sLA sZO s

ZA pTA pLA pZO p

er

mal

con

duct

ivity

(Wm

K)

(a)

Temperature (K)

()

300 400 500 600

40

30

20

10

ZA sTA sLA sZO s

ZA pTA pLA pZO p

(b)



119902119887

120572(k 119905) =

119873119888

sum

119897

radic119898

119887

119873119888

119906119897119887

120572(119905) exp [119894k sdot r119897

0] (34)



Φ120582(120596) =

1003816100381610038161003816 119902120582(120596)

1003816100381610038161003816

2

=

10038161003816100381610038161003816100381610038161003816

int

+infin

0

119902120582(119905) 119890

minus119894120596119905119889119905

10038161003816100381610038161003816100381610038161003816

2

=119862

120582

(120596 minus 120596119860

120582)2

+ 1205782

120582

(35)

where119862120582= (120596

119860

120582

2

+Γ2

120582)119902

2






120582= 12120578




Φ (k 120596) =3119899

sum

]Φ

120582(120596)

=1

41205871205910

3119899

sum

120572119887

119898119887

119873119888

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119897

1205910

int

0

119897119887

120572(119905) exp [119894k sdot r119897

0minus 119894120596119905]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

=

3

sum

120572

119899

sum

119887

10038161003816100381610038161003816119902119887

120572(k 120596)10038161003816100381610038161003816

2

(36)

where 119902119887





119888in (34) are




Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)100

10

1

100

10

10 1 2 0 1 2


300K 600K

[100] LA[110] LA[111] LA

[100] LA[110] LA[111] LA

(a)

[100] TA[110] TA1[110] TA2[111] TA

[100] TA[110] TA1[110] TA2[111] TA

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

300K 600K

(b)

1

0135 40

1

0135 40

300K 600K

[100] LO[110] LO[111] LO

[100] LO[110] LO[111] LO

Rela

xatio

n tim

e (p

s)

Frequency (THz)

Rela

xatio

n tim

e (p

s)

Frequency (THz)

(c)

30

300K 600K

[100] TO[110] TO1[110] TO2[111] TO

[100] TO[110] TO1[110] TO2[111] TO

1

01

1

0135 40 30 35 40

Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)


(d)







80

60

40

20

00 1 2 3 4

46

23

26

300K

Frequency (THz)

TA1TA2LA

TO1TO2LO

Perc

enta

ge c

ontr

ibut

ion

(au

)

(a)

80

60

40

20

00 1 2 3 4

50

2124

TA1TA2LA

TO1TO2LO

600K

Frequency (THz)

Perc

enta

ge c

ontr

ibut

ion

(au

)

(b)


150

100

50

0 1 2 3 4 5

Frequency (THz)

L = 12

L = 9

L = 6

Acoustic phonon



Rela

xatio

n tim

e (p

s)

(a)


0

150

100

50

02 04 06 08

Predicted by NMA

Rela

xatio

n tim

e (p

s)


(b)










10minus1 100 101 102 103 104 105



MFP (nm)

Acc

umul

ated

ther

mal

con

duct

ivity

() 100

80

60

40

20

0


2Te

3(MD) [142])

[141] and Bi2Te









2Te



3


2Te


in the future

5 Summary






1119872

2SLs 1 and 2 [59] 2004


1119872

2SLs 1 and 2 radic [50] 2008



2 and SrTiO3) 2 [49] 2011



Table 2 Continued


1minus119909Se

1199091 radic [106] 2012

Bi2Te

34 [142] 2013





Equations (14)(15) (16) (17)

and (18)














Applications sofar











Appendix

A


120596 + 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) (119899

0

1205821015840 + 1) 119899

0

12058210158401015840 = 119899

0

1205821198990

1205821015840 (119899

0

12058210158401015840 + 1) (A1)

120596 minus 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) 119899

0

1205821015840119899

0

12058210158401015840 = 119899

0

120582(119899

0

1205821015840 + 1) (119899

0

12058210158401015840 + 1) (A2)


119861119879) minus 1]


1198991015840

120582= minusΨ

120582

1205971198990

120582

120597 (ℏ120596120582)= Ψ

120582sdot

1

1198961198611198791198990

120582(119899

0

120582+ 1) (A3)


approximation (11)

Ψ120582= 120591

120582

ℏ120596120582

119879k120582sdot nabla119879 (A4)


119899120582= 119899

0

120582+ 119899

1015840

120582

1198991205821015840 = 119899

0

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840

(A5)


1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

= 1198991205821015840 [119899

0

1205821015840 (1 + 119899

0

12058210158401015840) minus (1 + 119899

0

1205821015840) 119899

0

12058210158401015840]

= 1198991205821015840 (119899

0

1205821015840 minus 119899

0

12058210158401015840)

(A6)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (119899

0

1205821015840 minus 119899

0

12058210158401015840)

119896119861119879

(A7)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

= 1198991205821015840 [(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840) minus 119899

0

1205821015840119899

0

12058210158401015840]

= 1198991205821015840 (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

(A8)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

119896119861119879

(A9)


1205821015840




119899120582= 119899

0

120582+ 119899

1205821015840

1198991205821015840 = 119899

0

1205821015840 + 119899

1015840

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840 + 119899

1015840

12058210158401015840

(A10)

where 1198991015840

1205821015840 and 1198991015840


120582

1198991015840

1205821015840 = minusΨ

1205821015840

1205971198990

1205821015840

120597 (ℏ1205961205821015840)= Ψ

1205821015840 sdot

1

1198961198611198791198990

1205821015840 (119899

0

1205821015840 + 1) (A11)

1198991015840

12058210158401015840 = minus Ψ

12058210158401015840

1205971198990

12058210158401015840

120597 (ℏ12059612058210158401015840)

= Ψ12058210158401015840 sdot

1

1198961198611198791198990

12058210158401015840 (119899

0

12058210158401015840 + 1)

(A12)

Ψ1205821015840 = 120591

1205821015840

ℏ1205961205821015840

119879k1205821015840 sdot nabla119879 (A13)

Ψ12058210158401015840 = 120591

12058210158401015840

ℏ12059612058210158401015840

119879k12058210158401015840 sdot nabla119879 (A14)


120582Ψ

1205821015840

Ψ120582Ψ

12058210158401015840 and Ψ

1205821015840Ψ

12058210158401015840 we have

1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

=

(Ψ120582+ Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

1205821198990

1205821015840 (1 + 119899

0

12058210158401015840)

119896119861119879

(A15)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

=

(Ψ120582minus Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

120582(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840)

119896119861119879

(A16)





Acknowledgments


References

































































2Si

119909Sn

1minus119909






























1minus119909As




















1minus119909Se

119909from first-





































1minus119909Sb

119909)2Te























2Te

3Sb

2Te










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




007

006

005

004

003

002

001

00 01 02 03 04 0 01 02 03 04 0 01 02 03 04 05

0 01 02 03 04 05

Scat

terin

g ra

te(c

mminus1)

Scat

terin

g ra

te(c

mminus1)

120585 120585 120585

1

120591TAasymp

1

120591aTA+

1

120591bTA+

1

120591cTA

Optical Optical

Optical

Acoustic

Acoustic Acoustic


02

015

01

005

0

120585

1

120591aTA

1

120591bTA

1

120591cTA

1

120591TA

TATATA

(a) (b) (c)

(d)





120591r120582 = 1205910

120582(1 + Δ

120582) 1 minus 119890


120582V120582|119866r120582 (22)







120591120582= 120591

0

120582(1 + Δ

120582) [

1

119878119888

int119878119888


119887)1205910

120582V120582|119866r120582 119889s] (23)



Γiso1205821205821015840 =

120587

2119873119888

120596120582120596

1205821015840

119899

sum

119887

119892119887

10038161003816100381610038161003816e119887120582sdot e119887lowast

1205821015840

10038161003816100381610038161003816

2

120575 (120596120582minus 120596

1205821015840) (24)




1

011 15 2 25 3

1 times 1

2 times 2

4 times 4

kSLk

T

M1M2


1119872

2 the mass ratio of



119892119887= sum

119894

119891119894119887(1 minus

119898119894119887

119898119887

)

2

(25)




119887is the




1

120591iso120582

= sum

1205821015840

Γiso1205821205821015840 =

120587

2119873119888

1205962

120582

119899

sum

119887

sum

1205821015840

119892119887

10038161003816100381610038161003816e119887120582sdot e119887lowast

1205821015840

10038161003816100381610038161003816

2

120575 (120596120582minus 120596

1205821015840)

(26)


1

120591iso120582

=120587

6119899119873119888

1198921205962

120582sum

1205821015840

120575 (120596120582minus 120596

1205821015840)

=120587119881

0

6119892120596

2119863 (120596)

(27)

=120587

2119892120596

2119863 (120596) (28)

where 119892 is given by (5)1198810= 119881(119899119873


119863(120596) = sum1205821015840 120575(120596120582




10

08

06

04

02

00 2 4 6 8 10

t (ps)


i(0)Ei(0)⟩


120587

120596i




is 119881 = 1198810119899119873






055Pd

045alloys [139] for




10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Empty CNT


278ps(429ps)

80ps(138ps)

25ps(26ps)

263ps(268ps)

47ps(50ps)

0 1 2 3 4 5

Φ(kz=0120596

) (J s

)

(a)

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11


Water-filled CNT

24ps 22ps 23ps

250ps

46ps

0 1 2 3 4 5

Φ(kz=0120596

) (J s

)

(b)



20

15

10

05

000806040200 10

Longitudinal

1120591

(1p

s)

20K LD50K LD

20K MD50K MD

klowast (a2120587)

(a)

08

06

04

02

000806040200 10

Transverse

1120591

(1p

s)

20K LD50K LD

20K MD50K MD

klowast (a2120587)

(b)




(at the limit 119875119871



119871 This leads



119871









50

40

30

20

10

050403020100

Phon

on li

fetim

e (p

s)








1+1superlattice We



1119872





1119872



1119872



1119872




4 MD Simulation



120591120582=

intinfin

0⟨119899

1015840

120582(119905) 119899

1015840

120582(0)⟩ 119889119905

⟨11989910158402

120582(0)⟩

(29)







119864120582=1

2( 119902

120582119902lowast

120582+ 120596

2

120582119902120582119902lowast

120582) (30)


transformed to

120591120582=

intinfin

0⟨119864

120582(119905) 119864

120582(0)⟩ 119889119905

⟨1198642

120582(0)⟩

(31)


120582(119905)119864

120582(0)⟩⟨119864

2

120582(0)⟩ = exp(minus119905120591

120582)


120582sim 119902

120582119902lowast



119902120582(119905) = 119902

1205820exp [119894 (120596119860

120582+ 119894120578

120582) 119905] (32)

where 1199021205820


120582= 120596

120582+ Δ


120582


120582119902lowast

120582

and 119864120582sim 119902

120582119902lowast


120582= 12120578

120582



120582in (31)


119902120582(119905) =

3

sum

120572

119899

sum

119887

119873119888

sum

119897

radic119898

119887

119873119888

119906119897119887

120572(119905) 119890

120582lowast

119887120572exp [119894k sdot r119897

0]

=

3

sum

120572

119899

sum

119887

119890120582lowast

119887120572119902119887

120572(k 119905)

(33)


120572(119905) is 120572 component




120572(k 119905) denotes



300

200

100

0

ZA sTA sLA sZO s

ZA pTA pLA pZO p

er

mal

con

duct

ivity

(Wm

K)

(a)

Temperature (K)

()

300 400 500 600

40

30

20

10

ZA sTA sLA sZO s

ZA pTA pLA pZO p

(b)



119902119887

120572(k 119905) =

119873119888

sum

119897

radic119898

119887

119873119888

119906119897119887

120572(119905) exp [119894k sdot r119897

0] (34)



Φ120582(120596) =

1003816100381610038161003816 119902120582(120596)

1003816100381610038161003816

2

=

10038161003816100381610038161003816100381610038161003816

int

+infin

0

119902120582(119905) 119890

minus119894120596119905119889119905

10038161003816100381610038161003816100381610038161003816

2

=119862

120582

(120596 minus 120596119860

120582)2

+ 1205782

120582

(35)

where119862120582= (120596

119860

120582

2

+Γ2

120582)119902

2






120582= 12120578




Φ (k 120596) =3119899

sum

]Φ

120582(120596)

=1

41205871205910

3119899

sum

120572119887

119898119887

119873119888

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119897

1205910

int

0

119897119887

120572(119905) exp [119894k sdot r119897

0minus 119894120596119905]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

=

3

sum

120572

119899

sum

119887

10038161003816100381610038161003816119902119887

120572(k 120596)10038161003816100381610038161003816

2

(36)

where 119902119887





119888in (34) are




Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)100

10

1

100

10

10 1 2 0 1 2


300K 600K

[100] LA[110] LA[111] LA

[100] LA[110] LA[111] LA

(a)

[100] TA[110] TA1[110] TA2[111] TA

[100] TA[110] TA1[110] TA2[111] TA

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

300K 600K

(b)

1

0135 40

1

0135 40

300K 600K

[100] LO[110] LO[111] LO

[100] LO[110] LO[111] LO

Rela

xatio

n tim

e (p

s)

Frequency (THz)

Rela

xatio

n tim

e (p

s)

Frequency (THz)

(c)

30

300K 600K

[100] TO[110] TO1[110] TO2[111] TO

[100] TO[110] TO1[110] TO2[111] TO

1

01

1

0135 40 30 35 40

Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)


(d)







80

60

40

20

00 1 2 3 4

46

23

26

300K

Frequency (THz)

TA1TA2LA

TO1TO2LO

Perc

enta

ge c

ontr

ibut

ion

(au

)

(a)

80

60

40

20

00 1 2 3 4

50

2124

TA1TA2LA

TO1TO2LO

600K

Frequency (THz)

Perc

enta

ge c

ontr

ibut

ion

(au

)

(b)


150

100

50

0 1 2 3 4 5

Frequency (THz)

L = 12

L = 9

L = 6

Acoustic phonon



Rela

xatio

n tim

e (p

s)

(a)


0

150

100

50

02 04 06 08

Predicted by NMA

Rela

xatio

n tim

e (p

s)


(b)










10minus1 100 101 102 103 104 105



MFP (nm)

Acc

umul

ated

ther

mal

con

duct

ivity

() 100

80

60

40

20

0


2Te

3(MD) [142])

[141] and Bi2Te









2Te



3


2Te


in the future

5 Summary






1119872

2SLs 1 and 2 [59] 2004


1119872

2SLs 1 and 2 radic [50] 2008



2 and SrTiO3) 2 [49] 2011



Table 2 Continued


1minus119909Se

1199091 radic [106] 2012

Bi2Te

34 [142] 2013





Equations (14)(15) (16) (17)

and (18)














Applications sofar











Appendix

A


120596 + 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) (119899

0

1205821015840 + 1) 119899

0

12058210158401015840 = 119899

0

1205821198990

1205821015840 (119899

0

12058210158401015840 + 1) (A1)

120596 minus 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) 119899

0

1205821015840119899

0

12058210158401015840 = 119899

0

120582(119899

0

1205821015840 + 1) (119899

0

12058210158401015840 + 1) (A2)


119861119879) minus 1]


1198991015840

120582= minusΨ

120582

1205971198990

120582

120597 (ℏ120596120582)= Ψ

120582sdot

1

1198961198611198791198990

120582(119899

0

120582+ 1) (A3)


approximation (11)

Ψ120582= 120591

120582

ℏ120596120582

119879k120582sdot nabla119879 (A4)


119899120582= 119899

0

120582+ 119899

1015840

120582

1198991205821015840 = 119899

0

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840

(A5)


1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

= 1198991205821015840 [119899

0

1205821015840 (1 + 119899

0

12058210158401015840) minus (1 + 119899

0

1205821015840) 119899

0

12058210158401015840]

= 1198991205821015840 (119899

0

1205821015840 minus 119899

0

12058210158401015840)

(A6)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (119899

0

1205821015840 minus 119899

0

12058210158401015840)

119896119861119879

(A7)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

= 1198991205821015840 [(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840) minus 119899

0

1205821015840119899

0

12058210158401015840]

= 1198991205821015840 (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

(A8)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

119896119861119879

(A9)


1205821015840




119899120582= 119899

0

120582+ 119899

1205821015840

1198991205821015840 = 119899

0

1205821015840 + 119899

1015840

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840 + 119899

1015840

12058210158401015840

(A10)

where 1198991015840

1205821015840 and 1198991015840


120582

1198991015840

1205821015840 = minusΨ

1205821015840

1205971198990

1205821015840

120597 (ℏ1205961205821015840)= Ψ

1205821015840 sdot

1

1198961198611198791198990

1205821015840 (119899

0

1205821015840 + 1) (A11)

1198991015840

12058210158401015840 = minus Ψ

12058210158401015840

1205971198990

12058210158401015840

120597 (ℏ12059612058210158401015840)

= Ψ12058210158401015840 sdot

1

1198961198611198791198990

12058210158401015840 (119899

0

12058210158401015840 + 1)

(A12)

Ψ1205821015840 = 120591

1205821015840

ℏ1205961205821015840

119879k1205821015840 sdot nabla119879 (A13)

Ψ12058210158401015840 = 120591

12058210158401015840

ℏ12059612058210158401015840

119879k12058210158401015840 sdot nabla119879 (A14)


120582Ψ

1205821015840

Ψ120582Ψ

12058210158401015840 and Ψ

1205821015840Ψ

12058210158401015840 we have

1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

=

(Ψ120582+ Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

1205821198990

1205821015840 (1 + 119899

0

12058210158401015840)

119896119861119879

(A15)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

=

(Ψ120582minus Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

120582(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840)

119896119861119879

(A16)





Acknowledgments


References

































































2Si

119909Sn

1minus119909






























1minus119909As




















1minus119909Se

119909from first-





































1minus119909Sb

119909)2Te























2Te

3Sb

2Te










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




1

011 15 2 25 3

1 times 1

2 times 2

4 times 4

kSLk

T

M1M2


1119872

2 the mass ratio of



119892119887= sum

119894

119891119894119887(1 minus

119898119894119887

119898119887

)

2

(25)




119887is the




1

120591iso120582

= sum

1205821015840

Γiso1205821205821015840 =

120587

2119873119888

1205962

120582

119899

sum

119887

sum

1205821015840

119892119887

10038161003816100381610038161003816e119887120582sdot e119887lowast

1205821015840

10038161003816100381610038161003816

2

120575 (120596120582minus 120596

1205821015840)

(26)


1

120591iso120582

=120587

6119899119873119888

1198921205962

120582sum

1205821015840

120575 (120596120582minus 120596

1205821015840)

=120587119881

0

6119892120596

2119863 (120596)

(27)

=120587

2119892120596

2119863 (120596) (28)

where 119892 is given by (5)1198810= 119881(119899119873


119863(120596) = sum1205821015840 120575(120596120582




10

08

06

04

02

00 2 4 6 8 10

t (ps)


i(0)Ei(0)⟩


120587

120596i




is 119881 = 1198810119899119873






055Pd

045alloys [139] for




10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Empty CNT


278ps(429ps)

80ps(138ps)

25ps(26ps)

263ps(268ps)

47ps(50ps)

0 1 2 3 4 5

Φ(kz=0120596

) (J s

)

(a)

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11


Water-filled CNT

24ps 22ps 23ps

250ps

46ps

0 1 2 3 4 5

Φ(kz=0120596

) (J s

)

(b)



20

15

10

05

000806040200 10

Longitudinal

1120591

(1p

s)

20K LD50K LD

20K MD50K MD

klowast (a2120587)

(a)

08

06

04

02

000806040200 10

Transverse

1120591

(1p

s)

20K LD50K LD

20K MD50K MD

klowast (a2120587)

(b)




(at the limit 119875119871



119871 This leads



119871









50

40

30

20

10

050403020100

Phon

on li

fetim

e (p

s)








1+1superlattice We



1119872





1119872



1119872



1119872




4 MD Simulation



120591120582=

intinfin

0⟨119899

1015840

120582(119905) 119899

1015840

120582(0)⟩ 119889119905

⟨11989910158402

120582(0)⟩

(29)







119864120582=1

2( 119902

120582119902lowast

120582+ 120596

2

120582119902120582119902lowast

120582) (30)


transformed to

120591120582=

intinfin

0⟨119864

120582(119905) 119864

120582(0)⟩ 119889119905

⟨1198642

120582(0)⟩

(31)


120582(119905)119864

120582(0)⟩⟨119864

2

120582(0)⟩ = exp(minus119905120591

120582)


120582sim 119902

120582119902lowast



119902120582(119905) = 119902

1205820exp [119894 (120596119860

120582+ 119894120578

120582) 119905] (32)

where 1199021205820


120582= 120596

120582+ Δ


120582


120582119902lowast

120582

and 119864120582sim 119902

120582119902lowast


120582= 12120578

120582



120582in (31)


119902120582(119905) =

3

sum

120572

119899

sum

119887

119873119888

sum

119897

radic119898

119887

119873119888

119906119897119887

120572(119905) 119890

120582lowast

119887120572exp [119894k sdot r119897

0]

=

3

sum

120572

119899

sum

119887

119890120582lowast

119887120572119902119887

120572(k 119905)

(33)


120572(119905) is 120572 component




120572(k 119905) denotes



300

200

100

0

ZA sTA sLA sZO s

ZA pTA pLA pZO p

er

mal

con

duct

ivity

(Wm

K)

(a)

Temperature (K)

()

300 400 500 600

40

30

20

10

ZA sTA sLA sZO s

ZA pTA pLA pZO p

(b)



119902119887

120572(k 119905) =

119873119888

sum

119897

radic119898

119887

119873119888

119906119897119887

120572(119905) exp [119894k sdot r119897

0] (34)



Φ120582(120596) =

1003816100381610038161003816 119902120582(120596)

1003816100381610038161003816

2

=

10038161003816100381610038161003816100381610038161003816

int

+infin

0

119902120582(119905) 119890

minus119894120596119905119889119905

10038161003816100381610038161003816100381610038161003816

2

=119862

120582

(120596 minus 120596119860

120582)2

+ 1205782

120582

(35)

where119862120582= (120596

119860

120582

2

+Γ2

120582)119902

2






120582= 12120578




Φ (k 120596) =3119899

sum

]Φ

120582(120596)

=1

41205871205910

3119899

sum

120572119887

119898119887

119873119888

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119897

1205910

int

0

119897119887

120572(119905) exp [119894k sdot r119897

0minus 119894120596119905]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

=

3

sum

120572

119899

sum

119887

10038161003816100381610038161003816119902119887

120572(k 120596)10038161003816100381610038161003816

2

(36)

where 119902119887





119888in (34) are




Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)100

10

1

100

10

10 1 2 0 1 2


300K 600K

[100] LA[110] LA[111] LA

[100] LA[110] LA[111] LA

(a)

[100] TA[110] TA1[110] TA2[111] TA

[100] TA[110] TA1[110] TA2[111] TA

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

300K 600K

(b)

1

0135 40

1

0135 40

300K 600K

[100] LO[110] LO[111] LO

[100] LO[110] LO[111] LO

Rela

xatio

n tim

e (p

s)

Frequency (THz)

Rela

xatio

n tim

e (p

s)

Frequency (THz)

(c)

30

300K 600K

[100] TO[110] TO1[110] TO2[111] TO

[100] TO[110] TO1[110] TO2[111] TO

1

01

1

0135 40 30 35 40

Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)


(d)







80

60

40

20

00 1 2 3 4

46

23

26

300K

Frequency (THz)

TA1TA2LA

TO1TO2LO

Perc

enta

ge c

ontr

ibut

ion

(au

)

(a)

80

60

40

20

00 1 2 3 4

50

2124

TA1TA2LA

TO1TO2LO

600K

Frequency (THz)

Perc

enta

ge c

ontr

ibut

ion

(au

)

(b)


150

100

50

0 1 2 3 4 5

Frequency (THz)

L = 12

L = 9

L = 6

Acoustic phonon



Rela

xatio

n tim

e (p

s)

(a)


0

150

100

50

02 04 06 08

Predicted by NMA

Rela

xatio

n tim

e (p

s)


(b)










10minus1 100 101 102 103 104 105



MFP (nm)

Acc

umul

ated

ther

mal

con

duct

ivity

() 100

80

60

40

20

0


2Te

3(MD) [142])

[141] and Bi2Te









2Te



3


2Te


in the future

5 Summary






1119872

2SLs 1 and 2 [59] 2004


1119872

2SLs 1 and 2 radic [50] 2008



2 and SrTiO3) 2 [49] 2011



Table 2 Continued


1minus119909Se

1199091 radic [106] 2012

Bi2Te

34 [142] 2013





Equations (14)(15) (16) (17)

and (18)














Applications sofar











Appendix

A


120596 + 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) (119899

0

1205821015840 + 1) 119899

0

12058210158401015840 = 119899

0

1205821198990

1205821015840 (119899

0

12058210158401015840 + 1) (A1)

120596 minus 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) 119899

0

1205821015840119899

0

12058210158401015840 = 119899

0

120582(119899

0

1205821015840 + 1) (119899

0

12058210158401015840 + 1) (A2)


119861119879) minus 1]


1198991015840

120582= minusΨ

120582

1205971198990

120582

120597 (ℏ120596120582)= Ψ

120582sdot

1

1198961198611198791198990

120582(119899

0

120582+ 1) (A3)


approximation (11)

Ψ120582= 120591

120582

ℏ120596120582

119879k120582sdot nabla119879 (A4)


119899120582= 119899

0

120582+ 119899

1015840

120582

1198991205821015840 = 119899

0

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840

(A5)


1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

= 1198991205821015840 [119899

0

1205821015840 (1 + 119899

0

12058210158401015840) minus (1 + 119899

0

1205821015840) 119899

0

12058210158401015840]

= 1198991205821015840 (119899

0

1205821015840 minus 119899

0

12058210158401015840)

(A6)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (119899

0

1205821015840 minus 119899

0

12058210158401015840)

119896119861119879

(A7)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

= 1198991205821015840 [(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840) minus 119899

0

1205821015840119899

0

12058210158401015840]

= 1198991205821015840 (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

(A8)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

119896119861119879

(A9)


1205821015840




119899120582= 119899

0

120582+ 119899

1205821015840

1198991205821015840 = 119899

0

1205821015840 + 119899

1015840

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840 + 119899

1015840

12058210158401015840

(A10)

where 1198991015840

1205821015840 and 1198991015840


120582

1198991015840

1205821015840 = minusΨ

1205821015840

1205971198990

1205821015840

120597 (ℏ1205961205821015840)= Ψ

1205821015840 sdot

1

1198961198611198791198990

1205821015840 (119899

0

1205821015840 + 1) (A11)

1198991015840

12058210158401015840 = minus Ψ

12058210158401015840

1205971198990

12058210158401015840

120597 (ℏ12059612058210158401015840)

= Ψ12058210158401015840 sdot

1

1198961198611198791198990

12058210158401015840 (119899

0

12058210158401015840 + 1)

(A12)

Ψ1205821015840 = 120591

1205821015840

ℏ1205961205821015840

119879k1205821015840 sdot nabla119879 (A13)

Ψ12058210158401015840 = 120591

12058210158401015840

ℏ12059612058210158401015840

119879k12058210158401015840 sdot nabla119879 (A14)


120582Ψ

1205821015840

Ψ120582Ψ

12058210158401015840 and Ψ

1205821015840Ψ

12058210158401015840 we have

1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

=

(Ψ120582+ Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

1205821198990

1205821015840 (1 + 119899

0

12058210158401015840)

119896119861119879

(A15)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

=

(Ψ120582minus Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

120582(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840)

119896119861119879

(A16)





Acknowledgments


References

































































2Si

119909Sn

1minus119909






























1minus119909As




















1minus119909Se

119909from first-





































1minus119909Sb

119909)2Te























2Te

3Sb

2Te










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Empty CNT


278ps(429ps)

80ps(138ps)

25ps(26ps)

263ps(268ps)

47ps(50ps)

0 1 2 3 4 5

Φ(kz=0120596

) (J s

)

(a)

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11


Water-filled CNT

24ps 22ps 23ps

250ps

46ps

0 1 2 3 4 5

Φ(kz=0120596

) (J s

)

(b)



20

15

10

05

000806040200 10

Longitudinal

1120591

(1p

s)

20K LD50K LD

20K MD50K MD

klowast (a2120587)

(a)

08

06

04

02

000806040200 10

Transverse

1120591

(1p

s)

20K LD50K LD

20K MD50K MD

klowast (a2120587)

(b)




(at the limit 119875119871



119871 This leads



119871









50

40

30

20

10

050403020100

Phon

on li

fetim

e (p

s)








1+1superlattice We



1119872





1119872



1119872



1119872




4 MD Simulation



120591120582=

intinfin

0⟨119899

1015840

120582(119905) 119899

1015840

120582(0)⟩ 119889119905

⟨11989910158402

120582(0)⟩

(29)







119864120582=1

2( 119902

120582119902lowast

120582+ 120596

2

120582119902120582119902lowast

120582) (30)


transformed to

120591120582=

intinfin

0⟨119864

120582(119905) 119864

120582(0)⟩ 119889119905

⟨1198642

120582(0)⟩

(31)


120582(119905)119864

120582(0)⟩⟨119864

2

120582(0)⟩ = exp(minus119905120591

120582)


120582sim 119902

120582119902lowast



119902120582(119905) = 119902

1205820exp [119894 (120596119860

120582+ 119894120578

120582) 119905] (32)

where 1199021205820


120582= 120596

120582+ Δ


120582


120582119902lowast

120582

and 119864120582sim 119902

120582119902lowast


120582= 12120578

120582



120582in (31)


119902120582(119905) =

3

sum

120572

119899

sum

119887

119873119888

sum

119897

radic119898

119887

119873119888

119906119897119887

120572(119905) 119890

120582lowast

119887120572exp [119894k sdot r119897

0]

=

3

sum

120572

119899

sum

119887

119890120582lowast

119887120572119902119887

120572(k 119905)

(33)


120572(119905) is 120572 component




120572(k 119905) denotes



300

200

100

0

ZA sTA sLA sZO s

ZA pTA pLA pZO p

er

mal

con

duct

ivity

(Wm

K)

(a)

Temperature (K)

()

300 400 500 600

40

30

20

10

ZA sTA sLA sZO s

ZA pTA pLA pZO p

(b)



119902119887

120572(k 119905) =

119873119888

sum

119897

radic119898

119887

119873119888

119906119897119887

120572(119905) exp [119894k sdot r119897

0] (34)



Φ120582(120596) =

1003816100381610038161003816 119902120582(120596)

1003816100381610038161003816

2

=

10038161003816100381610038161003816100381610038161003816

int

+infin

0

119902120582(119905) 119890

minus119894120596119905119889119905

10038161003816100381610038161003816100381610038161003816

2

=119862

120582

(120596 minus 120596119860

120582)2

+ 1205782

120582

(35)

where119862120582= (120596

119860

120582

2

+Γ2

120582)119902

2






120582= 12120578




Φ (k 120596) =3119899

sum

]Φ

120582(120596)

=1

41205871205910

3119899

sum

120572119887

119898119887

119873119888

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119897

1205910

int

0

119897119887

120572(119905) exp [119894k sdot r119897

0minus 119894120596119905]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

=

3

sum

120572

119899

sum

119887

10038161003816100381610038161003816119902119887

120572(k 120596)10038161003816100381610038161003816

2

(36)

where 119902119887





119888in (34) are




Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)100

10

1

100

10

10 1 2 0 1 2


300K 600K

[100] LA[110] LA[111] LA

[100] LA[110] LA[111] LA

(a)

[100] TA[110] TA1[110] TA2[111] TA

[100] TA[110] TA1[110] TA2[111] TA

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

300K 600K

(b)

1

0135 40

1

0135 40

300K 600K

[100] LO[110] LO[111] LO

[100] LO[110] LO[111] LO

Rela

xatio

n tim

e (p

s)

Frequency (THz)

Rela

xatio

n tim

e (p

s)

Frequency (THz)

(c)

30

300K 600K

[100] TO[110] TO1[110] TO2[111] TO

[100] TO[110] TO1[110] TO2[111] TO

1

01

1

0135 40 30 35 40

Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)


(d)







80

60

40

20

00 1 2 3 4

46

23

26

300K

Frequency (THz)

TA1TA2LA

TO1TO2LO

Perc

enta

ge c

ontr

ibut

ion

(au

)

(a)

80

60

40

20

00 1 2 3 4

50

2124

TA1TA2LA

TO1TO2LO

600K

Frequency (THz)

Perc

enta

ge c

ontr

ibut

ion

(au

)

(b)


150

100

50

0 1 2 3 4 5

Frequency (THz)

L = 12

L = 9

L = 6

Acoustic phonon



Rela

xatio

n tim

e (p

s)

(a)


0

150

100

50

02 04 06 08

Predicted by NMA

Rela

xatio

n tim

e (p

s)


(b)










10minus1 100 101 102 103 104 105



MFP (nm)

Acc

umul

ated

ther

mal

con

duct

ivity

() 100

80

60

40

20

0


2Te

3(MD) [142])

[141] and Bi2Te









2Te



3


2Te


in the future

5 Summary






1119872

2SLs 1 and 2 [59] 2004


1119872

2SLs 1 and 2 radic [50] 2008



2 and SrTiO3) 2 [49] 2011



Table 2 Continued


1minus119909Se

1199091 radic [106] 2012

Bi2Te

34 [142] 2013





Equations (14)(15) (16) (17)

and (18)














Applications sofar











Appendix

A


120596 + 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) (119899

0

1205821015840 + 1) 119899

0

12058210158401015840 = 119899

0

1205821198990

1205821015840 (119899

0

12058210158401015840 + 1) (A1)

120596 minus 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) 119899

0

1205821015840119899

0

12058210158401015840 = 119899

0

120582(119899

0

1205821015840 + 1) (119899

0

12058210158401015840 + 1) (A2)


119861119879) minus 1]


1198991015840

120582= minusΨ

120582

1205971198990

120582

120597 (ℏ120596120582)= Ψ

120582sdot

1

1198961198611198791198990

120582(119899

0

120582+ 1) (A3)


approximation (11)

Ψ120582= 120591

120582

ℏ120596120582

119879k120582sdot nabla119879 (A4)


119899120582= 119899

0

120582+ 119899

1015840

120582

1198991205821015840 = 119899

0

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840

(A5)


1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

= 1198991205821015840 [119899

0

1205821015840 (1 + 119899

0

12058210158401015840) minus (1 + 119899

0

1205821015840) 119899

0

12058210158401015840]

= 1198991205821015840 (119899

0

1205821015840 minus 119899

0

12058210158401015840)

(A6)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (119899

0

1205821015840 minus 119899

0

12058210158401015840)

119896119861119879

(A7)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

= 1198991205821015840 [(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840) minus 119899

0

1205821015840119899

0

12058210158401015840]

= 1198991205821015840 (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

(A8)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

119896119861119879

(A9)


1205821015840




119899120582= 119899

0

120582+ 119899

1205821015840

1198991205821015840 = 119899

0

1205821015840 + 119899

1015840

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840 + 119899

1015840

12058210158401015840

(A10)

where 1198991015840

1205821015840 and 1198991015840


120582

1198991015840

1205821015840 = minusΨ

1205821015840

1205971198990

1205821015840

120597 (ℏ1205961205821015840)= Ψ

1205821015840 sdot

1

1198961198611198791198990

1205821015840 (119899

0

1205821015840 + 1) (A11)

1198991015840

12058210158401015840 = minus Ψ

12058210158401015840

1205971198990

12058210158401015840

120597 (ℏ12059612058210158401015840)

= Ψ12058210158401015840 sdot

1

1198961198611198791198990

12058210158401015840 (119899

0

12058210158401015840 + 1)

(A12)

Ψ1205821015840 = 120591

1205821015840

ℏ1205961205821015840

119879k1205821015840 sdot nabla119879 (A13)

Ψ12058210158401015840 = 120591

12058210158401015840

ℏ12059612058210158401015840

119879k12058210158401015840 sdot nabla119879 (A14)


120582Ψ

1205821015840

Ψ120582Ψ

12058210158401015840 and Ψ

1205821015840Ψ

12058210158401015840 we have

1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

=

(Ψ120582+ Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

1205821198990

1205821015840 (1 + 119899

0

12058210158401015840)

119896119861119879

(A15)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

=

(Ψ120582minus Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

120582(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840)

119896119861119879

(A16)





Acknowledgments


References

































































2Si

119909Sn

1minus119909






























1minus119909As




















1minus119909Se

119909from first-





































1minus119909Sb

119909)2Te























2Te

3Sb

2Te










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




50

40

30

20

10

050403020100

Phon

on li

fetim

e (p

s)








1+1superlattice We



1119872





1119872



1119872



1119872




4 MD Simulation



120591120582=

intinfin

0⟨119899

1015840

120582(119905) 119899

1015840

120582(0)⟩ 119889119905

⟨11989910158402

120582(0)⟩

(29)







119864120582=1

2( 119902

120582119902lowast

120582+ 120596

2

120582119902120582119902lowast

120582) (30)


transformed to

120591120582=

intinfin

0⟨119864

120582(119905) 119864

120582(0)⟩ 119889119905

⟨1198642

120582(0)⟩

(31)


120582(119905)119864

120582(0)⟩⟨119864

2

120582(0)⟩ = exp(minus119905120591

120582)


120582sim 119902

120582119902lowast



119902120582(119905) = 119902

1205820exp [119894 (120596119860

120582+ 119894120578

120582) 119905] (32)

where 1199021205820


120582= 120596

120582+ Δ


120582


120582119902lowast

120582

and 119864120582sim 119902

120582119902lowast


120582= 12120578

120582



120582in (31)


119902120582(119905) =

3

sum

120572

119899

sum

119887

119873119888

sum

119897

radic119898

119887

119873119888

119906119897119887

120572(119905) 119890

120582lowast

119887120572exp [119894k sdot r119897

0]

=

3

sum

120572

119899

sum

119887

119890120582lowast

119887120572119902119887

120572(k 119905)

(33)


120572(119905) is 120572 component




120572(k 119905) denotes



300

200

100

0

ZA sTA sLA sZO s

ZA pTA pLA pZO p

er

mal

con

duct

ivity

(Wm

K)

(a)

Temperature (K)

()

300 400 500 600

40

30

20

10

ZA sTA sLA sZO s

ZA pTA pLA pZO p

(b)



119902119887

120572(k 119905) =

119873119888

sum

119897

radic119898

119887

119873119888

119906119897119887

120572(119905) exp [119894k sdot r119897

0] (34)



Φ120582(120596) =

1003816100381610038161003816 119902120582(120596)

1003816100381610038161003816

2

=

10038161003816100381610038161003816100381610038161003816

int

+infin

0

119902120582(119905) 119890

minus119894120596119905119889119905

10038161003816100381610038161003816100381610038161003816

2

=119862

120582

(120596 minus 120596119860

120582)2

+ 1205782

120582

(35)

where119862120582= (120596

119860

120582

2

+Γ2

120582)119902

2






120582= 12120578




Φ (k 120596) =3119899

sum

]Φ

120582(120596)

=1

41205871205910

3119899

sum

120572119887

119898119887

119873119888

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119897

1205910

int

0

119897119887

120572(119905) exp [119894k sdot r119897

0minus 119894120596119905]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

=

3

sum

120572

119899

sum

119887

10038161003816100381610038161003816119902119887

120572(k 120596)10038161003816100381610038161003816

2

(36)

where 119902119887





119888in (34) are




Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)100

10

1

100

10

10 1 2 0 1 2


300K 600K

[100] LA[110] LA[111] LA

[100] LA[110] LA[111] LA

(a)

[100] TA[110] TA1[110] TA2[111] TA

[100] TA[110] TA1[110] TA2[111] TA

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

300K 600K

(b)

1

0135 40

1

0135 40

300K 600K

[100] LO[110] LO[111] LO

[100] LO[110] LO[111] LO

Rela

xatio

n tim

e (p

s)

Frequency (THz)

Rela

xatio

n tim

e (p

s)

Frequency (THz)

(c)

30

300K 600K

[100] TO[110] TO1[110] TO2[111] TO

[100] TO[110] TO1[110] TO2[111] TO

1

01

1

0135 40 30 35 40

Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)


(d)







80

60

40

20

00 1 2 3 4

46

23

26

300K

Frequency (THz)

TA1TA2LA

TO1TO2LO

Perc

enta

ge c

ontr

ibut

ion

(au

)

(a)

80

60

40

20

00 1 2 3 4

50

2124

TA1TA2LA

TO1TO2LO

600K

Frequency (THz)

Perc

enta

ge c

ontr

ibut

ion

(au

)

(b)


150

100

50

0 1 2 3 4 5

Frequency (THz)

L = 12

L = 9

L = 6

Acoustic phonon



Rela

xatio

n tim

e (p

s)

(a)


0

150

100

50

02 04 06 08

Predicted by NMA

Rela

xatio

n tim

e (p

s)


(b)










10minus1 100 101 102 103 104 105



MFP (nm)

Acc

umul

ated

ther

mal

con

duct

ivity

() 100

80

60

40

20

0


2Te

3(MD) [142])

[141] and Bi2Te









2Te



3


2Te


in the future

5 Summary






1119872

2SLs 1 and 2 [59] 2004


1119872

2SLs 1 and 2 radic [50] 2008



2 and SrTiO3) 2 [49] 2011



Table 2 Continued


1minus119909Se

1199091 radic [106] 2012

Bi2Te

34 [142] 2013





Equations (14)(15) (16) (17)

and (18)














Applications sofar











Appendix

A


120596 + 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) (119899

0

1205821015840 + 1) 119899

0

12058210158401015840 = 119899

0

1205821198990

1205821015840 (119899

0

12058210158401015840 + 1) (A1)

120596 minus 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) 119899

0

1205821015840119899

0

12058210158401015840 = 119899

0

120582(119899

0

1205821015840 + 1) (119899

0

12058210158401015840 + 1) (A2)


119861119879) minus 1]


1198991015840

120582= minusΨ

120582

1205971198990

120582

120597 (ℏ120596120582)= Ψ

120582sdot

1

1198961198611198791198990

120582(119899

0

120582+ 1) (A3)


approximation (11)

Ψ120582= 120591

120582

ℏ120596120582

119879k120582sdot nabla119879 (A4)


119899120582= 119899

0

120582+ 119899

1015840

120582

1198991205821015840 = 119899

0

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840

(A5)


1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

= 1198991205821015840 [119899

0

1205821015840 (1 + 119899

0

12058210158401015840) minus (1 + 119899

0

1205821015840) 119899

0

12058210158401015840]

= 1198991205821015840 (119899

0

1205821015840 minus 119899

0

12058210158401015840)

(A6)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (119899

0

1205821015840 minus 119899

0

12058210158401015840)

119896119861119879

(A7)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

= 1198991205821015840 [(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840) minus 119899

0

1205821015840119899

0

12058210158401015840]

= 1198991205821015840 (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

(A8)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

119896119861119879

(A9)


1205821015840




119899120582= 119899

0

120582+ 119899

1205821015840

1198991205821015840 = 119899

0

1205821015840 + 119899

1015840

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840 + 119899

1015840

12058210158401015840

(A10)

where 1198991015840

1205821015840 and 1198991015840


120582

1198991015840

1205821015840 = minusΨ

1205821015840

1205971198990

1205821015840

120597 (ℏ1205961205821015840)= Ψ

1205821015840 sdot

1

1198961198611198791198990

1205821015840 (119899

0

1205821015840 + 1) (A11)

1198991015840

12058210158401015840 = minus Ψ

12058210158401015840

1205971198990

12058210158401015840

120597 (ℏ12059612058210158401015840)

= Ψ12058210158401015840 sdot

1

1198961198611198791198990

12058210158401015840 (119899

0

12058210158401015840 + 1)

(A12)

Ψ1205821015840 = 120591

1205821015840

ℏ1205961205821015840

119879k1205821015840 sdot nabla119879 (A13)

Ψ12058210158401015840 = 120591

12058210158401015840

ℏ12059612058210158401015840

119879k12058210158401015840 sdot nabla119879 (A14)


120582Ψ

1205821015840

Ψ120582Ψ

12058210158401015840 and Ψ

1205821015840Ψ

12058210158401015840 we have

1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

=

(Ψ120582+ Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

1205821198990

1205821015840 (1 + 119899

0

12058210158401015840)

119896119861119879

(A15)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

=

(Ψ120582minus Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

120582(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840)

119896119861119879

(A16)





Acknowledgments


References

































































2Si

119909Sn

1minus119909






























1minus119909As




















1minus119909Se

119909from first-





































1minus119909Sb

119909)2Te























2Te

3Sb

2Te










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





300

200

100

0

ZA sTA sLA sZO s

ZA pTA pLA pZO p

er

mal

con

duct

ivity

(Wm

K)

(a)

Temperature (K)

()

300 400 500 600

40

30

20

10

ZA sTA sLA sZO s

ZA pTA pLA pZO p

(b)



119902119887

120572(k 119905) =

119873119888

sum

119897

radic119898

119887

119873119888

119906119897119887

120572(119905) exp [119894k sdot r119897

0] (34)



Φ120582(120596) =

1003816100381610038161003816 119902120582(120596)

1003816100381610038161003816

2

=

10038161003816100381610038161003816100381610038161003816

int

+infin

0

119902120582(119905) 119890

minus119894120596119905119889119905

10038161003816100381610038161003816100381610038161003816

2

=119862

120582

(120596 minus 120596119860

120582)2

+ 1205782

120582

(35)

where119862120582= (120596

119860

120582

2

+Γ2

120582)119902

2






120582= 12120578




Φ (k 120596) =3119899

sum

]Φ

120582(120596)

=1

41205871205910

3119899

sum

120572119887

119898119887

119873119888

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119897

1205910

int

0

119897119887

120572(119905) exp [119894k sdot r119897

0minus 119894120596119905]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

=

3

sum

120572

119899

sum

119887

10038161003816100381610038161003816119902119887

120572(k 120596)10038161003816100381610038161003816

2

(36)

where 119902119887





119888in (34) are




Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)100

10

1

100

10

10 1 2 0 1 2


300K 600K

[100] LA[110] LA[111] LA

[100] LA[110] LA[111] LA

(a)

[100] TA[110] TA1[110] TA2[111] TA

[100] TA[110] TA1[110] TA2[111] TA

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

300K 600K

(b)

1

0135 40

1

0135 40

300K 600K

[100] LO[110] LO[111] LO

[100] LO[110] LO[111] LO

Rela

xatio

n tim

e (p

s)

Frequency (THz)

Rela

xatio

n tim

e (p

s)

Frequency (THz)

(c)

30

300K 600K

[100] TO[110] TO1[110] TO2[111] TO

[100] TO[110] TO1[110] TO2[111] TO

1

01

1

0135 40 30 35 40

Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)


(d)







80

60

40

20

00 1 2 3 4

46

23

26

300K

Frequency (THz)

TA1TA2LA

TO1TO2LO

Perc

enta

ge c

ontr

ibut

ion

(au

)

(a)

80

60

40

20

00 1 2 3 4

50

2124

TA1TA2LA

TO1TO2LO

600K

Frequency (THz)

Perc

enta

ge c

ontr

ibut

ion

(au

)

(b)


150

100

50

0 1 2 3 4 5

Frequency (THz)

L = 12

L = 9

L = 6

Acoustic phonon



Rela

xatio

n tim

e (p

s)

(a)


0

150

100

50

02 04 06 08

Predicted by NMA

Rela

xatio

n tim

e (p

s)


(b)










10minus1 100 101 102 103 104 105



MFP (nm)

Acc

umul

ated

ther

mal

con

duct

ivity

() 100

80

60

40

20

0


2Te

3(MD) [142])

[141] and Bi2Te









2Te



3


2Te


in the future

5 Summary






1119872

2SLs 1 and 2 [59] 2004


1119872

2SLs 1 and 2 radic [50] 2008



2 and SrTiO3) 2 [49] 2011



Table 2 Continued


1minus119909Se

1199091 radic [106] 2012

Bi2Te

34 [142] 2013





Equations (14)(15) (16) (17)

and (18)














Applications sofar











Appendix

A


120596 + 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) (119899

0

1205821015840 + 1) 119899

0

12058210158401015840 = 119899

0

1205821198990

1205821015840 (119899

0

12058210158401015840 + 1) (A1)

120596 minus 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) 119899

0

1205821015840119899

0

12058210158401015840 = 119899

0

120582(119899

0

1205821015840 + 1) (119899

0

12058210158401015840 + 1) (A2)


119861119879) minus 1]


1198991015840

120582= minusΨ

120582

1205971198990

120582

120597 (ℏ120596120582)= Ψ

120582sdot

1

1198961198611198791198990

120582(119899

0

120582+ 1) (A3)


approximation (11)

Ψ120582= 120591

120582

ℏ120596120582

119879k120582sdot nabla119879 (A4)


119899120582= 119899

0

120582+ 119899

1015840

120582

1198991205821015840 = 119899

0

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840

(A5)


1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

= 1198991205821015840 [119899

0

1205821015840 (1 + 119899

0

12058210158401015840) minus (1 + 119899

0

1205821015840) 119899

0

12058210158401015840]

= 1198991205821015840 (119899

0

1205821015840 minus 119899

0

12058210158401015840)

(A6)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (119899

0

1205821015840 minus 119899

0

12058210158401015840)

119896119861119879

(A7)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

= 1198991205821015840 [(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840) minus 119899

0

1205821015840119899

0

12058210158401015840]

= 1198991205821015840 (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

(A8)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

119896119861119879

(A9)


1205821015840




119899120582= 119899

0

120582+ 119899

1205821015840

1198991205821015840 = 119899

0

1205821015840 + 119899

1015840

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840 + 119899

1015840

12058210158401015840

(A10)

where 1198991015840

1205821015840 and 1198991015840


120582

1198991015840

1205821015840 = minusΨ

1205821015840

1205971198990

1205821015840

120597 (ℏ1205961205821015840)= Ψ

1205821015840 sdot

1

1198961198611198791198990

1205821015840 (119899

0

1205821015840 + 1) (A11)

1198991015840

12058210158401015840 = minus Ψ

12058210158401015840

1205971198990

12058210158401015840

120597 (ℏ12059612058210158401015840)

= Ψ12058210158401015840 sdot

1

1198961198611198791198990

12058210158401015840 (119899

0

12058210158401015840 + 1)

(A12)

Ψ1205821015840 = 120591

1205821015840

ℏ1205961205821015840

119879k1205821015840 sdot nabla119879 (A13)

Ψ12058210158401015840 = 120591

12058210158401015840

ℏ12059612058210158401015840

119879k12058210158401015840 sdot nabla119879 (A14)


120582Ψ

1205821015840

Ψ120582Ψ

12058210158401015840 and Ψ

1205821015840Ψ

12058210158401015840 we have

1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

=

(Ψ120582+ Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

1205821198990

1205821015840 (1 + 119899

0

12058210158401015840)

119896119861119879

(A15)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

=

(Ψ120582minus Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

120582(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840)

119896119861119879

(A16)





Acknowledgments


References

































































2Si

119909Sn

1minus119909






























1minus119909As




















1minus119909Se

119909from first-





































1minus119909Sb

119909)2Te























2Te

3Sb

2Te










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)100

10

1

100

10

10 1 2 0 1 2


300K 600K

[100] LA[110] LA[111] LA

[100] LA[110] LA[111] LA

(a)

[100] TA[110] TA1[110] TA2[111] TA

[100] TA[110] TA1[110] TA2[111] TA

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

Rela

xatio

n tim

e (p

s) 100

10

10 1 2

Frequency (THz)

300K 600K

(b)

1

0135 40

1

0135 40

300K 600K

[100] LO[110] LO[111] LO

[100] LO[110] LO[111] LO

Rela

xatio

n tim

e (p

s)

Frequency (THz)

Rela

xatio

n tim

e (p

s)

Frequency (THz)

(c)

30

300K 600K

[100] TO[110] TO1[110] TO2[111] TO

[100] TO[110] TO1[110] TO2[111] TO

1

01

1

0135 40 30 35 40

Rela

xatio

n tim

e (p

s)

Rela

xatio

n tim

e (p

s)


(d)







80

60

40

20

00 1 2 3 4

46

23

26

300K

Frequency (THz)

TA1TA2LA

TO1TO2LO

Perc

enta

ge c

ontr

ibut

ion

(au

)

(a)

80

60

40

20

00 1 2 3 4

50

2124

TA1TA2LA

TO1TO2LO

600K

Frequency (THz)

Perc

enta

ge c

ontr

ibut

ion

(au

)

(b)


150

100

50

0 1 2 3 4 5

Frequency (THz)

L = 12

L = 9

L = 6

Acoustic phonon



Rela

xatio

n tim

e (p

s)

(a)


0

150

100

50

02 04 06 08

Predicted by NMA

Rela

xatio

n tim

e (p

s)


(b)










10minus1 100 101 102 103 104 105



MFP (nm)

Acc

umul

ated

ther

mal

con

duct

ivity

() 100

80

60

40

20

0


2Te

3(MD) [142])

[141] and Bi2Te









2Te



3


2Te


in the future

5 Summary






1119872

2SLs 1 and 2 [59] 2004


1119872

2SLs 1 and 2 radic [50] 2008



2 and SrTiO3) 2 [49] 2011



Table 2 Continued


1minus119909Se

1199091 radic [106] 2012

Bi2Te

34 [142] 2013





Equations (14)(15) (16) (17)

and (18)














Applications sofar











Appendix

A


120596 + 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) (119899

0

1205821015840 + 1) 119899

0

12058210158401015840 = 119899

0

1205821198990

1205821015840 (119899

0

12058210158401015840 + 1) (A1)

120596 minus 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) 119899

0

1205821015840119899

0

12058210158401015840 = 119899

0

120582(119899

0

1205821015840 + 1) (119899

0

12058210158401015840 + 1) (A2)


119861119879) minus 1]


1198991015840

120582= minusΨ

120582

1205971198990

120582

120597 (ℏ120596120582)= Ψ

120582sdot

1

1198961198611198791198990

120582(119899

0

120582+ 1) (A3)


approximation (11)

Ψ120582= 120591

120582

ℏ120596120582

119879k120582sdot nabla119879 (A4)


119899120582= 119899

0

120582+ 119899

1015840

120582

1198991205821015840 = 119899

0

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840

(A5)


1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

= 1198991205821015840 [119899

0

1205821015840 (1 + 119899

0

12058210158401015840) minus (1 + 119899

0

1205821015840) 119899

0

12058210158401015840]

= 1198991205821015840 (119899

0

1205821015840 minus 119899

0

12058210158401015840)

(A6)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (119899

0

1205821015840 minus 119899

0

12058210158401015840)

119896119861119879

(A7)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

= 1198991205821015840 [(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840) minus 119899

0

1205821015840119899

0

12058210158401015840]

= 1198991205821015840 (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

(A8)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

119896119861119879

(A9)


1205821015840




119899120582= 119899

0

120582+ 119899

1205821015840

1198991205821015840 = 119899

0

1205821015840 + 119899

1015840

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840 + 119899

1015840

12058210158401015840

(A10)

where 1198991015840

1205821015840 and 1198991015840


120582

1198991015840

1205821015840 = minusΨ

1205821015840

1205971198990

1205821015840

120597 (ℏ1205961205821015840)= Ψ

1205821015840 sdot

1

1198961198611198791198990

1205821015840 (119899

0

1205821015840 + 1) (A11)

1198991015840

12058210158401015840 = minus Ψ

12058210158401015840

1205971198990

12058210158401015840

120597 (ℏ12059612058210158401015840)

= Ψ12058210158401015840 sdot

1

1198961198611198791198990

12058210158401015840 (119899

0

12058210158401015840 + 1)

(A12)

Ψ1205821015840 = 120591

1205821015840

ℏ1205961205821015840

119879k1205821015840 sdot nabla119879 (A13)

Ψ12058210158401015840 = 120591

12058210158401015840

ℏ12059612058210158401015840

119879k12058210158401015840 sdot nabla119879 (A14)


120582Ψ

1205821015840

Ψ120582Ψ

12058210158401015840 and Ψ

1205821015840Ψ

12058210158401015840 we have

1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

=

(Ψ120582+ Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

1205821198990

1205821015840 (1 + 119899

0

12058210158401015840)

119896119861119879

(A15)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

=

(Ψ120582minus Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

120582(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840)

119896119861119879

(A16)





Acknowledgments


References

































































2Si

119909Sn

1minus119909






























1minus119909As




















1minus119909Se

119909from first-





































1minus119909Sb

119909)2Te























2Te

3Sb

2Te










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




80

60

40

20

00 1 2 3 4

46

23

26

300K

Frequency (THz)

TA1TA2LA

TO1TO2LO

Perc

enta

ge c

ontr

ibut

ion

(au

)

(a)

80

60

40

20

00 1 2 3 4

50

2124

TA1TA2LA

TO1TO2LO

600K

Frequency (THz)

Perc

enta

ge c

ontr

ibut

ion

(au

)

(b)


150

100

50

0 1 2 3 4 5

Frequency (THz)

L = 12

L = 9

L = 6

Acoustic phonon



Rela

xatio

n tim

e (p

s)

(a)


0

150

100

50

02 04 06 08

Predicted by NMA

Rela

xatio

n tim

e (p

s)


(b)










10minus1 100 101 102 103 104 105



MFP (nm)

Acc

umul

ated

ther

mal

con

duct

ivity

() 100

80

60

40

20

0


2Te

3(MD) [142])

[141] and Bi2Te









2Te



3


2Te


in the future

5 Summary






1119872

2SLs 1 and 2 [59] 2004


1119872

2SLs 1 and 2 radic [50] 2008



2 and SrTiO3) 2 [49] 2011



Table 2 Continued


1minus119909Se

1199091 radic [106] 2012

Bi2Te

34 [142] 2013





Equations (14)(15) (16) (17)

and (18)














Applications sofar











Appendix

A


120596 + 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) (119899

0

1205821015840 + 1) 119899

0

12058210158401015840 = 119899

0

1205821198990

1205821015840 (119899

0

12058210158401015840 + 1) (A1)

120596 minus 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) 119899

0

1205821015840119899

0

12058210158401015840 = 119899

0

120582(119899

0

1205821015840 + 1) (119899

0

12058210158401015840 + 1) (A2)


119861119879) minus 1]


1198991015840

120582= minusΨ

120582

1205971198990

120582

120597 (ℏ120596120582)= Ψ

120582sdot

1

1198961198611198791198990

120582(119899

0

120582+ 1) (A3)


approximation (11)

Ψ120582= 120591

120582

ℏ120596120582

119879k120582sdot nabla119879 (A4)


119899120582= 119899

0

120582+ 119899

1015840

120582

1198991205821015840 = 119899

0

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840

(A5)


1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

= 1198991205821015840 [119899

0

1205821015840 (1 + 119899

0

12058210158401015840) minus (1 + 119899

0

1205821015840) 119899

0

12058210158401015840]

= 1198991205821015840 (119899

0

1205821015840 minus 119899

0

12058210158401015840)

(A6)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (119899

0

1205821015840 minus 119899

0

12058210158401015840)

119896119861119879

(A7)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

= 1198991205821015840 [(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840) minus 119899

0

1205821015840119899

0

12058210158401015840]

= 1198991205821015840 (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

(A8)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

119896119861119879

(A9)


1205821015840




119899120582= 119899

0

120582+ 119899

1205821015840

1198991205821015840 = 119899

0

1205821015840 + 119899

1015840

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840 + 119899

1015840

12058210158401015840

(A10)

where 1198991015840

1205821015840 and 1198991015840


120582

1198991015840

1205821015840 = minusΨ

1205821015840

1205971198990

1205821015840

120597 (ℏ1205961205821015840)= Ψ

1205821015840 sdot

1

1198961198611198791198990

1205821015840 (119899

0

1205821015840 + 1) (A11)

1198991015840

12058210158401015840 = minus Ψ

12058210158401015840

1205971198990

12058210158401015840

120597 (ℏ12059612058210158401015840)

= Ψ12058210158401015840 sdot

1

1198961198611198791198990

12058210158401015840 (119899

0

12058210158401015840 + 1)

(A12)

Ψ1205821015840 = 120591

1205821015840

ℏ1205961205821015840

119879k1205821015840 sdot nabla119879 (A13)

Ψ12058210158401015840 = 120591

12058210158401015840

ℏ12059612058210158401015840

119879k12058210158401015840 sdot nabla119879 (A14)


120582Ψ

1205821015840

Ψ120582Ψ

12058210158401015840 and Ψ

1205821015840Ψ

12058210158401015840 we have

1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

=

(Ψ120582+ Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

1205821198990

1205821015840 (1 + 119899

0

12058210158401015840)

119896119861119879

(A15)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

=

(Ψ120582minus Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

120582(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840)

119896119861119879

(A16)





Acknowledgments


References

































































2Si

119909Sn

1minus119909






























1minus119909As




















1minus119909Se

119909from first-





































1minus119909Sb

119909)2Te























2Te

3Sb

2Te










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




10minus1 100 101 102 103 104 105



MFP (nm)

Acc

umul

ated

ther

mal

con

duct

ivity

() 100

80

60

40

20

0


2Te

3(MD) [142])

[141] and Bi2Te









2Te



3


2Te


in the future

5 Summary






1119872

2SLs 1 and 2 [59] 2004


1119872

2SLs 1 and 2 radic [50] 2008



2 and SrTiO3) 2 [49] 2011



Table 2 Continued


1minus119909Se

1199091 radic [106] 2012

Bi2Te

34 [142] 2013





Equations (14)(15) (16) (17)

and (18)














Applications sofar











Appendix

A


120596 + 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) (119899

0

1205821015840 + 1) 119899

0

12058210158401015840 = 119899

0

1205821198990

1205821015840 (119899

0

12058210158401015840 + 1) (A1)

120596 minus 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) 119899

0

1205821015840119899

0

12058210158401015840 = 119899

0

120582(119899

0

1205821015840 + 1) (119899

0

12058210158401015840 + 1) (A2)


119861119879) minus 1]


1198991015840

120582= minusΨ

120582

1205971198990

120582

120597 (ℏ120596120582)= Ψ

120582sdot

1

1198961198611198791198990

120582(119899

0

120582+ 1) (A3)


approximation (11)

Ψ120582= 120591

120582

ℏ120596120582

119879k120582sdot nabla119879 (A4)


119899120582= 119899

0

120582+ 119899

1015840

120582

1198991205821015840 = 119899

0

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840

(A5)


1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

= 1198991205821015840 [119899

0

1205821015840 (1 + 119899

0

12058210158401015840) minus (1 + 119899

0

1205821015840) 119899

0

12058210158401015840]

= 1198991205821015840 (119899

0

1205821015840 minus 119899

0

12058210158401015840)

(A6)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (119899

0

1205821015840 minus 119899

0

12058210158401015840)

119896119861119879

(A7)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

= 1198991205821015840 [(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840) minus 119899

0

1205821015840119899

0

12058210158401015840]

= 1198991205821015840 (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

(A8)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

119896119861119879

(A9)


1205821015840




119899120582= 119899

0

120582+ 119899

1205821015840

1198991205821015840 = 119899

0

1205821015840 + 119899

1015840

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840 + 119899

1015840

12058210158401015840

(A10)

where 1198991015840

1205821015840 and 1198991015840


120582

1198991015840

1205821015840 = minusΨ

1205821015840

1205971198990

1205821015840

120597 (ℏ1205961205821015840)= Ψ

1205821015840 sdot

1

1198961198611198791198990

1205821015840 (119899

0

1205821015840 + 1) (A11)

1198991015840

12058210158401015840 = minus Ψ

12058210158401015840

1205971198990

12058210158401015840

120597 (ℏ12059612058210158401015840)

= Ψ12058210158401015840 sdot

1

1198961198611198791198990

12058210158401015840 (119899

0

12058210158401015840 + 1)

(A12)

Ψ1205821015840 = 120591

1205821015840

ℏ1205961205821015840

119879k1205821015840 sdot nabla119879 (A13)

Ψ12058210158401015840 = 120591

12058210158401015840

ℏ12059612058210158401015840

119879k12058210158401015840 sdot nabla119879 (A14)


120582Ψ

1205821015840

Ψ120582Ψ

12058210158401015840 and Ψ

1205821015840Ψ

12058210158401015840 we have

1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

=

(Ψ120582+ Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

1205821198990

1205821015840 (1 + 119899

0

12058210158401015840)

119896119861119879

(A15)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

=

(Ψ120582minus Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

120582(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840)

119896119861119879

(A16)





Acknowledgments


References

































































2Si

119909Sn

1minus119909






























1minus119909As




















1minus119909Se

119909from first-





































1minus119909Sb

119909)2Te























2Te

3Sb

2Te










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials






1119872

2SLs 1 and 2 [59] 2004


1119872

2SLs 1 and 2 radic [50] 2008



2 and SrTiO3) 2 [49] 2011



Table 2 Continued


1minus119909Se

1199091 radic [106] 2012

Bi2Te

34 [142] 2013





Equations (14)(15) (16) (17)

and (18)














Applications sofar











Appendix

A


120596 + 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) (119899

0

1205821015840 + 1) 119899

0

12058210158401015840 = 119899

0

1205821198990

1205821015840 (119899

0

12058210158401015840 + 1) (A1)

120596 minus 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) 119899

0

1205821015840119899

0

12058210158401015840 = 119899

0

120582(119899

0

1205821015840 + 1) (119899

0

12058210158401015840 + 1) (A2)


119861119879) minus 1]


1198991015840

120582= minusΨ

120582

1205971198990

120582

120597 (ℏ120596120582)= Ψ

120582sdot

1

1198961198611198791198990

120582(119899

0

120582+ 1) (A3)


approximation (11)

Ψ120582= 120591

120582

ℏ120596120582

119879k120582sdot nabla119879 (A4)


119899120582= 119899

0

120582+ 119899

1015840

120582

1198991205821015840 = 119899

0

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840

(A5)


1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

= 1198991205821015840 [119899

0

1205821015840 (1 + 119899

0

12058210158401015840) minus (1 + 119899

0

1205821015840) 119899

0

12058210158401015840]

= 1198991205821015840 (119899

0

1205821015840 minus 119899

0

12058210158401015840)

(A6)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (119899

0

1205821015840 minus 119899

0

12058210158401015840)

119896119861119879

(A7)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

= 1198991205821015840 [(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840) minus 119899

0

1205821015840119899

0

12058210158401015840]

= 1198991205821015840 (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

(A8)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

119896119861119879

(A9)


1205821015840




119899120582= 119899

0

120582+ 119899

1205821015840

1198991205821015840 = 119899

0

1205821015840 + 119899

1015840

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840 + 119899

1015840

12058210158401015840

(A10)

where 1198991015840

1205821015840 and 1198991015840


120582

1198991015840

1205821015840 = minusΨ

1205821015840

1205971198990

1205821015840

120597 (ℏ1205961205821015840)= Ψ

1205821015840 sdot

1

1198961198611198791198990

1205821015840 (119899

0

1205821015840 + 1) (A11)

1198991015840

12058210158401015840 = minus Ψ

12058210158401015840

1205971198990

12058210158401015840

120597 (ℏ12059612058210158401015840)

= Ψ12058210158401015840 sdot

1

1198961198611198791198990

12058210158401015840 (119899

0

12058210158401015840 + 1)

(A12)

Ψ1205821015840 = 120591

1205821015840

ℏ1205961205821015840

119879k1205821015840 sdot nabla119879 (A13)

Ψ12058210158401015840 = 120591

12058210158401015840

ℏ12059612058210158401015840

119879k12058210158401015840 sdot nabla119879 (A14)


120582Ψ

1205821015840

Ψ120582Ψ

12058210158401015840 and Ψ

1205821015840Ψ

12058210158401015840 we have

1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

=

(Ψ120582+ Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

1205821198990

1205821015840 (1 + 119899

0

12058210158401015840)

119896119861119879

(A15)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

=

(Ψ120582minus Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

120582(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840)

119896119861119879

(A16)





Acknowledgments


References

































































2Si

119909Sn

1minus119909






























1minus119909As




















1minus119909Se

119909from first-





































1minus119909Sb

119909)2Te























2Te

3Sb

2Te










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




Table 2 Continued


1minus119909Se

1199091 radic [106] 2012

Bi2Te

34 [142] 2013





Equations (14)(15) (16) (17)

and (18)














Applications sofar











Appendix

A


120596 + 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) (119899

0

1205821015840 + 1) 119899

0

12058210158401015840 = 119899

0

1205821198990

1205821015840 (119899

0

12058210158401015840 + 1) (A1)

120596 minus 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) 119899

0

1205821015840119899

0

12058210158401015840 = 119899

0

120582(119899

0

1205821015840 + 1) (119899

0

12058210158401015840 + 1) (A2)


119861119879) minus 1]


1198991015840

120582= minusΨ

120582

1205971198990

120582

120597 (ℏ120596120582)= Ψ

120582sdot

1

1198961198611198791198990

120582(119899

0

120582+ 1) (A3)


approximation (11)

Ψ120582= 120591

120582

ℏ120596120582

119879k120582sdot nabla119879 (A4)


119899120582= 119899

0

120582+ 119899

1015840

120582

1198991205821015840 = 119899

0

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840

(A5)


1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

= 1198991205821015840 [119899

0

1205821015840 (1 + 119899

0

12058210158401015840) minus (1 + 119899

0

1205821015840) 119899

0

12058210158401015840]

= 1198991205821015840 (119899

0

1205821015840 minus 119899

0

12058210158401015840)

(A6)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (119899

0

1205821015840 minus 119899

0

12058210158401015840)

119896119861119879

(A7)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

= 1198991205821015840 [(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840) minus 119899

0

1205821015840119899

0

12058210158401015840]

= 1198991205821015840 (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

(A8)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

119896119861119879

(A9)


1205821015840




119899120582= 119899

0

120582+ 119899

1205821015840

1198991205821015840 = 119899

0

1205821015840 + 119899

1015840

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840 + 119899

1015840

12058210158401015840

(A10)

where 1198991015840

1205821015840 and 1198991015840


120582

1198991015840

1205821015840 = minusΨ

1205821015840

1205971198990

1205821015840

120597 (ℏ1205961205821015840)= Ψ

1205821015840 sdot

1

1198961198611198791198990

1205821015840 (119899

0

1205821015840 + 1) (A11)

1198991015840

12058210158401015840 = minus Ψ

12058210158401015840

1205971198990

12058210158401015840

120597 (ℏ12059612058210158401015840)

= Ψ12058210158401015840 sdot

1

1198961198611198791198990

12058210158401015840 (119899

0

12058210158401015840 + 1)

(A12)

Ψ1205821015840 = 120591

1205821015840

ℏ1205961205821015840

119879k1205821015840 sdot nabla119879 (A13)

Ψ12058210158401015840 = 120591

12058210158401015840

ℏ12059612058210158401015840

119879k12058210158401015840 sdot nabla119879 (A14)


120582Ψ

1205821015840

Ψ120582Ψ

12058210158401015840 and Ψ

1205821015840Ψ

12058210158401015840 we have

1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

=

(Ψ120582+ Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

1205821198990

1205821015840 (1 + 119899

0

12058210158401015840)

119896119861119879

(A15)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

=

(Ψ120582minus Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

120582(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840)

119896119861119879

(A16)





Acknowledgments


References

































































2Si

119909Sn

1minus119909






























1minus119909As




















1minus119909Se

119909from first-





































1minus119909Sb

119909)2Te























2Te

3Sb

2Te










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





Appendix

A


120596 + 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) (119899

0

1205821015840 + 1) 119899

0

12058210158401015840 = 119899

0

1205821198990

1205821015840 (119899

0

12058210158401015840 + 1) (A1)

120596 minus 1205961015840= 120596

10158401015840 process

(1198990

120582+ 1) 119899

0

1205821015840119899

0

12058210158401015840 = 119899

0

120582(119899

0

1205821015840 + 1) (119899

0

12058210158401015840 + 1) (A2)


119861119879) minus 1]


1198991015840

120582= minusΨ

120582

1205971198990

120582

120597 (ℏ120596120582)= Ψ

120582sdot

1

1198961198611198791198990

120582(119899

0

120582+ 1) (A3)


approximation (11)

Ψ120582= 120591

120582

ℏ120596120582

119879k120582sdot nabla119879 (A4)


119899120582= 119899

0

120582+ 119899

1015840

120582

1198991205821015840 = 119899

0

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840

(A5)


1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

= 1198991205821015840 [119899

0

1205821015840 (1 + 119899

0

12058210158401015840) minus (1 + 119899

0

1205821015840) 119899

0

12058210158401015840]

= 1198991205821015840 (119899

0

1205821015840 minus 119899

0

12058210158401015840)

(A6)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (119899

0

1205821015840 minus 119899

0

12058210158401015840)

119896119861119879

(A7)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

= 1198991205821015840 [(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840) minus 119899

0

1205821015840119899

0

12058210158401015840]

= 1198991205821015840 (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

(A8)

= Ψ120582sdot

1198990

120582(119899

0

120582+ 1) (1 + 119899

0

1205821015840 + 119899

0

12058210158401015840)

119896119861119879

(A9)


1205821015840




119899120582= 119899

0

120582+ 119899

1205821015840

1198991205821015840 = 119899

0

1205821015840 + 119899

1015840

1205821015840

11989912058210158401015840 = 119899

0

12058210158401015840 + 119899

1015840

12058210158401015840

(A10)

where 1198991015840

1205821015840 and 1198991015840


120582

1198991015840

1205821015840 = minusΨ

1205821015840

1205971198990

1205821015840

120597 (ℏ1205961205821015840)= Ψ

1205821015840 sdot

1

1198961198611198791198990

1205821015840 (119899

0

1205821015840 + 1) (A11)

1198991015840

12058210158401015840 = minus Ψ

12058210158401015840

1205971198990

12058210158401015840

120597 (ℏ12059612058210158401015840)

= Ψ12058210158401015840 sdot

1

1198961198611198791198990

12058210158401015840 (119899

0

12058210158401015840 + 1)

(A12)

Ψ1205821015840 = 120591

1205821015840

ℏ1205961205821015840

119879k1205821015840 sdot nabla119879 (A13)

Ψ12058210158401015840 = 120591

12058210158401015840

ℏ12059612058210158401015840

119879k12058210158401015840 sdot nabla119879 (A14)


120582Ψ

1205821015840

Ψ120582Ψ

12058210158401015840 and Ψ

1205821015840Ψ

12058210158401015840 we have

1198991205821198991205821015840 (1 + 119899

12058210158401015840) minus (1 + 119899

120582) (1 + 119899

1205821015840) 119899

12058210158401015840

=

(Ψ120582+ Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

1205821198990

1205821015840 (1 + 119899

0

12058210158401015840)

119896119861119879

(A15)


119899120582(1 + 119899

1205821015840) (1 + 119899

12058210158401015840) minus (1 + 119899

120582) 119899

1205821015840119899

12058210158401015840

=

(Ψ120582minus Ψ

1205821015840 minus Ψ

12058210158401015840) 119899

0

120582(1 + 119899

0

1205821015840) (1 + 119899

0

12058210158401015840)

119896119861119879

(A16)





Acknowledgments


References

































































2Si

119909Sn

1minus119909






























1minus119909As




















1minus119909Se

119909from first-





































1minus119909Sb

119909)2Te























2Te

3Sb

2Te










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




Acknowledgments


References

































































2Si

119909Sn

1minus119909






























1minus119909As




















1minus119909Se

119909from first-





































1minus119909Sb

119909)2Te























2Te

3Sb

2Te










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials
































2Si

119909Sn

1minus119909






























1minus119909As




















1minus119909Se

119909from first-





































1minus119909Sb

119909)2Te























2Te

3Sb

2Te










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



























1minus119909As




















1minus119909Se

119909from first-





































1minus119909Sb

119909)2Te























2Te

3Sb

2Te










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials







1minus119909Se

119909from first-





































1minus119909Sb

119909)2Te























2Te

3Sb

2Te










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





1minus119909Sb

119909)2Te























2Te

3Sb

2Te










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



Review Article Prediction of Spectral Phonon Mean Free Path ...obtained the phonon relaxation times...

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Transcript of Review Article Prediction of Spectral Phonon Mean Free Path ...obtained the phonon relaxation times...