Reduction of Thermal Conductivity by Nanoscale 3D Phononic Crystal.

1.9 MB·English

by  Yang, Lina

Reduction of Thermal Conductivity by
Nanoscale 3D Phononic Crystal
SUBJECTAREAS:
LinaYang1,NuoYang2&BaowenLi1,2,3
STRUCTURALPROPERTIES
THERMOELECTRICS
1DepartmentofPhysicsandCentreforComputationalScienceandEngineering,NationalUniversityofSingapore,Singapore
STRUCTUREOFSOLIDSAND
117542,RepublicofSingapore,2CenterforPhononicsandThermalEnergyScience,SchoolofPhysicalScienceandEngineering,
LIQUIDS
TongjiUniversity,200092Shanghai,People’sRepublicofChina,3NUSGraduateSchoolforIntegrativeSciencesandEngineering,
MECHANICALENGINEERING NationalUniversityofSingapore,Singapore117456,RepublicofSingapore.
Received Westudiedhowtheperiodlengthandthemassratioaffectthethermalconductivityofisotopicnanoscale
9October2012 three-dimensional(3D)phononiccrystalofSi.Simulationresultsbyequilibriummoleculardynamicsshow
isotopicnanoscale3DphononiccrystalscansignificantlyreducethethermalconductivityofbulkSiathigh
Accepted temperature(1000 K),whichleadstoalargerZTthanunity.Thethermalconductivitydecreasesasthe
6December2012 periodlengthandmassratioincreases.Thephonondispersioncurvesshowanobviousdecreaseofgroup
velocitiesin3Dphononiccrystals.Thephonon’slocalizationandbandgapisalsoclearlyobservedinspectra
Published
ofnormalizedinverseparticipationratioinnanoscale3Dphononiccrystal.
31January2013
Thermoelectricmaterialsareimportantforgeneratingelectricityfromwasteheatandbeingusedassolid-state
Correspondenceand Peltiercoolers.Theperformanceofthermoelectricmaterialsdependsonthefigureofmerit(ZT)1,ZT5
requestsformaterials S2sT/k,whereS,T,s,andkaretheSeebeckcoefficient,absolutetemperature,electricalconductivityand
totalthermalconductivity,respectively.ZTcanbeincreasedbyincreasingSors,ordecreasingk.However,itis
shouldbeaddressedto
difficult to improve ZT in conventional materials. First, simple increase S forgeneral materials will leadto a
N.Y.(imyangnuo@
simultaneousdecreaseins1,2.Also,anincreaseinsleadstoacomparableincreaseintheelectroniccontribution
tongji.edu.cn)orB.L.
tok1,2.
([email protected])
AnalternativewaytoincreaseZTistoreducethethermalconductivitywithoutaffectingelectronicproperty1,3.
Moreover,ultra-lowthermalconductivityisalsorequiredtopreventtheback-flowofheatfromhotendtocool
end.Therefore,reductionofthermalconductivityiscrucialinthermoelectricapplication.
Phononiccrystalsareconstructedbyaperiodicarrayofscatteringinclusionsdistributedinahostmaterial.Due
toitsperiodicchangeofthedensityand/orelasticconstants,phononiccrystalsexhibitphononicbandgaps4.This
remarkablepropertyisverydifferentfromthoseoftheconventionalmaterialsandcanbeengineeredtoachieve
new functionalities. Aspecialone-dimensional phononiccrystalis thesuperlattice, onedimensional periodic
arrangementoftwodifferentmaterials.Itisdemonstratedthatsuperlatticecrystalsareeffectivetoachievevery
lowthermalconductivity5–11.Superlatticeshavebeenextensivelystudiedtodesignthermoelectricmaterialswith
high ZT. Preliminary works show that there is a minimum value of thermal conductivity in the direction
perpendiculartotheplanesofsuperlatticewhentheperiodlengthisreducedtonanoscale12–14.
Recently, it is demonstrated experimentally that the Si nanomesh film, a two-dimensional (2D) phononic
crystal,exhibitedlowthermalconductivity15bymodificationofphononbandstructure.SinglecrystallineSiby
phononiccrystalpatterningin2Dhasasmallervalueofthermalconductivity(,6 W/m-K)thanbulkSibecause
ofthelowgroupvelocitiesandthecoherentphononiceffects16.Itispredictedthatatomic-scale3Dphononic
crystalofGequantum-dotinSihasverylowthermalconductivityinallthreespatialdirections17.Thethermal
conductivity is reduced by several orders of magnitude compared with bulk Si. This reduction of thermal
conductivityisduetothereductioningroupvelocitiesandmultiplescatteringofparticle-likephonons.
In this letter, we study the thermal conductivity of nanoscale 3D silicon phononic crystal. The 3D crystal
consistsof28Siatomsand‘‘isotopes’’MSiatomswhichhavethesamepropertiesas28Siexceptthemass,whereMis
theatomicmassoftheisotopeofSi.Themassratio,R,isdefinedasR5M/28.The3Disotopicphononiccrystal
couldalsobenamedas3Dsuperlatticebecausedifferentmaterialarrangedperiodicallyinthreespatialdirections.
Wefindthe3Disotopicphononiccrystalhastheabilitytoflattenphonondispersioncurvescomparedwith
thatofbulkSianditcouldshowbandgapswhenproperlyarranged.Westudiedhowtheperiodlengthandthe
massratioaffectthethermalconductivityofthe3Dphononiccrystal.Thephonondispersioncurvesandinverse
participationratioarealsocomputedtounderstandthemechanismofthereductionofthermalconductivity.
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Ontheotherhand,thescatteringsofisotopicdopingcouldsig-
nificantly reduce the lattice thermal conductivity without much
reduction of the electrical conductivity1,14. As a result, the thermal
conductivityof3Disotopicphononiccrystalhasaquitelowvalue,
whichcanleadtoalargerZTthanunity.
Results
Fig. 1showsthestructures oftheisotopicnanoscale 3Dphononic
crystal,where28SiandMSiatomsareassembledperiodicallyinthree
spatialdirections.Fig.1(a)–(d)showsthestructuresof3Dphononic
crystals with different period lengths, corresponding to 1.09 nm,
2.17 nm,3.26 nmand6.52 nm,respectively.Thevolumeofsimu-
lation cell is 12 3 12 3 12 unit3 (1 unit is 0.543 nm), which has
13,824 atoms.
Thereisfinitesizeeffectincalculatingthermalconductivitywhen
the simulation cell is not big enough18,19. As shown in Fig. 2, we
calculatedthermalconductivityofisotopicnanoscale3Dphononic
Figure2|Thermalconductivityversusthesidelengthofsimulationcell.
crystalwithdifferentsizeofsimulationcellbyequilibriummolecular
Themassratioofthe3Disotopicphononiccrystalis2andtheperiod
dynamic (EMD) method at 1000 K. The period length is set as
lengthis2units.Theerrorbarsarecalculatedfrom16simulationswith
2 unitsandthemassratioissetas2.Thecalculatedvalueofthermal
differentinitialconditions.Allvaluesarecalculatedat1000Kwhichis
conductivityconvergeswhenthesidelengthofcubicsimulationcell
largerthantheDebyetemperature,T ,ofSi(,658K).
is larger than 10 units. To overcome the finite size effect on the D
calculatedthermalconductivity,weusethesidelengthofsimulation
mayincreasetheamountofbandfoldinganddecreasetheaverage
cellas12 unitsinthefollowingsimulations.
velocityinthesuperlattice, resulting inadecreaseofthermalcon-
For comparison with the thermal conductivity of pure 28Si at
ductivity. The phonon mean free path of Si is around 60 nm at
1000 K,wealsocalculateitsvalueas5062 W/m-K(thedashdot
1000 K18,whichismuchlongerthantheperiodlengthinoursimu-
lineinFig.3(a)).OurresultiscomparabletoSchellingetal.’sresults
lation.ThetendencyofthermalconductivityinFig.3(a)isconsistent
of MD simulation, 61 W/m-K18. However, MD results can not
withSimkinandManhan9sresultswhenthephononmeanfreepath
exactly coincide with the experimental value of 28Si at 1000 K20,
islargerthanperiodlength12.
around 30 W/m-K, because of the inaccuracies of semi-empirical
Anotherwaytomodulatethephonontransportinthe3Dpho-
potentials and the impurity of the sample in measurements. This
noniccrystalistovarythemass.Themassofimpurityatomscould
non-coincidencehaslittleeffectonthecomparingMDresultscal-
perturbthephonondensityofstateandphonondispersioncurves,
culatedusingsamepotentialparameters.
whichcanaffectthegroupvelocities.
We calculated the thermal conductivity of 3D phononic crystal
Fig.3(b)showsthedependenceofthermalconductivityonthemass
withdifferentperiodlength,wherethemassratioRis2.Asshownin
ratio, where the period length is 12units. Our results indicate that
Fig. 3(a), the thermal conductivity rapidly decreases as the period
thermal conductivity rapidly decreases as the mass ratio increases
length increases. The smallest value of thermal conductivity is
from1to6.TheheaviestSiisotopeatomsproducedis43Si21.where
2.14 W/m-K, which is only 4.3% of pure 28Si calculated by EMD
themassratioRcorrespondsto1.5.Thevalueofthermalconductivity
method. Simkin and Mahan show12 that increasing period length
is4.2W/m-K,thatis,8.4percentofpure28Si(50W/m-K).
Figure1|Thestructuresoftheisotopicnanoscale3Dphononiccrystals- Figure3|(a)Thermalconductivityversustheperiodlengthofisotopic
threedimensionalperiodicarrangementsof28SiandMSiatoms. From(a) nanoscale3DphononiccrystalofSi.Themassratiois2.Thedashdotline
to(d),theperiodlengthsofthose3Dphononiccrystalsare2,4,6and correspondstothemoleculardynamicresultofthermalconductivityof
12units,respectively.Thelatticeconstantis0.543nm,thatis,1unit pure28Si.(b)Thermalconductivityversusthemassratioofisotopic
represents0.543nm.Insimulations,theperiodicboundaryconditionis nanoscale3DphononiccrystalsofSi.Theperiodlengthis12units.All
appliedinallthreedirections. valuesarecalculatedat1000K.
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Artificial Si isotopic atoms are used here to explore the mass Fig.4(a)showsacousticbranchesandpartialopticalbranchesof
influence on thermal transport and show the trend of large mass the dispersion curves of the 3D phononic crystal, where period
effects. The smallest value of thermal conductivity is 0.54 W/m-K lengthsaredifferent,andthemassratioisthesame.TheBrillouin
(whenRis6),whichisonly1.1%ofpure28Si.ArtificialMSiatomscan zonewith2 unitsinperiodlengthiseighttimesaslargeasthatwith
belookedasotheratoms,suchas54Fe22.Whenthereareotherkind 4 unitsinperiodlength.Incalculatingthedispersioncurvesofstruc-
atoms,thesystemismorecomplicated.Thatis,themassisnotthe turewith2 unitsinperiodlength,weusealargerunitcellwhosesize
onlyfactorinvolved.Thebondstrengthandlatticerelaxationsmust isasthesameasthatofstructurewith4 unitsinperiodlength.
playarole,whichisnotstudiedinthisletter. Astheopticalphononscontributelesstothethermalconductivity
Ina28Si MSi quasi-1Dsupercellwith10MSiatoms(5%)ran- duetothelowergroupvelocities,wefocusedontheacousticpho-
182 10
domlydistributed,GibbonsandEstreicher22foundthethermalcon- nons.ItisclearlyshowninFig.4(a)closetoRpointthatthegroup
ductivity decreased first and reached a minimum when the mass velocities decrease as the period length increase, which causes the
ratio was ‘‘two’’, and then the thermal conductivity increased as reductionofthethermalconductivity(showninFig.3(a)).Fig.4(b)
theincreaseofM.However,theystatedthattheycannotcomment showsacousticbranchesofthe3Dphononiccrystalwithdifferent
aboutthereasonsforthisminimumanddonotknowifthefactor massratios,wheretheperiodlengthis2 units.Thedispersioncurves
‘‘two’’ remains valid for concentrations other than 5%. Different areaffectedbythemassofimpurityatomsandthegroupvelocities
fromrandomly distributed, MSi in3Dphononic crystalis periodic decreaseasthemassofimpurityatomsincrease,whichcontributesto
distributed in 28Si. The concentration of MSi (50%) is much bigger thedecreaseofthermalconductivity(showninFig.3(b)).
than5%.Ourresultsshowthatthethermalconductivityof3Dpho- Tounderstandmoreabouttheunderlyingphysicalmechanismof
nonic crystal decreases monotonously with increase of M. This is thermalconductivityreduction,wecarryoutavibrationeigen-mode
coincidence of the monotonous decrease of group velocities analysis on 3D phononic crystals. The mode localization can be
(Fig.4(b)). qualitatively characterized by the normalized inverse participation
As shown above, changing the mass of impurity atoms and the ratio(NIPR)25.TheNIPRforphononmodekisdefinedthroughthe
period lengths are two effective ways in modulating the thermal normalizedeigenvectoru
k
conductivity. To find the mechanism in the decrease of thermal !2
N 3
conductivityof3Dphononiccrystal,thephonondispersioncurves P(cid:2)1~N:X Xu2 ð1Þ
k ia,k
arecalculatedthroughclassicallatticedynamics.Wecalculatedthe i~1 a~1
dispersion curves by general utility lattice program (GULP)23, and
Stillinger-Weberpotential24whichisthesameatominteractionasin
ourMDsimulation.
Figure4|(a)Acousticandpartialopticalbranchesalongthe[1,1,1]
direction.Dispersioncurvesinred(black)correspondtothenanoscale3D
phononiccrystalwith2(4)unitcellsinperiodlength.Themassratiosare
2.ItisclearlyshownclosetoRpointthatthegroupvelocitiesdecreaseas Figure5|Thenormalizedinverseparticipationratio(NIPR)spectra.
theperiodlengthincreases,whichcausesthereductionofthethermal NIPRiscalculatedbasedonEq.(1).ThelargerofthevalueofNIPRthe
conductivity.(b)Acousticbranchesalongthe[1,1,1]direction.Themass morelocalizedofaphononmode.(a)NIPRspectraof3Dphononic
ratioof3Dphononiccrystalschangesfrom1to2.5.Differentcoloris crystalswithdifferentperiodlength.Theleftandrightpanelsare
referredtodifferentmassratio.Theperiodlengthsof3Dphononiccrystals correspondingto3Dphononiccrystalswith2unitand4unitperiod
arekeptsame,2units.Thedispersioncurvesareaffectedbythemassof length,respectively.Themassratiosarethesameas2.(b)NIPRspectraof
impurityatomsandthegroupvelocitiesdecreaseasthemassofimpurity 3Dphononiccrystalswithdifferentmassratio,R.Theperiodlengthsare
atomsincrease,whichcontributestothedecreaseofthermalconductivity. thesameas2units.Theupperleftpanel(R51)correspondstopureSi.
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where N is the total number of atoms. When there are less atoms Insimulations,theperiodicboundaryconditionisappliedinallthreedirections.
participating in the motion, the phonon mode has a larger NIPR Toderivetheforceterm,weuseStillinger-Weber(SW)potentialforSi24,which
includesbothtwo-bodyandthree-bodypotentialterms.TheSWpotentialhasbeen
value.Forexample,NIPRisNwhenthereisonlyoneatomvibrates
usedwidelytostudythethermalpropertiesofSibulkmaterial14,26,29foritsaccuratefit
in the localized mode. When all atoms participate in the motion, forexperimentalresultsonthethermalexpansioncoefficients.Theheatcurrentis
NIPRiscalculatedoutas1.Thatis,thelargerofthevalueofNIPR definedas18
themorelocalizedofaphononmode.
twoFigd.i5ff(ear)enshtowpesritohde NleInPgRthssp,ecwtrhaeroef 3thDepmhoanssonricaticorysRtaliwsit2h. ~JlðtÞ~Xi ~vieiz12ijXi=j~rij(cid:2)~Fij:~vi(cid:3)zXijk~rij(cid:2)~FjðijkÞ:~vj(cid:3) ð2Þ
Obviously, the NIPR for 3D phononic crystal with 4 units in per- ? ?
iodlengthhaslargervaluesthanthatwith2 unitsinperiodlength. whereFijandFijkdenotethetwo-bodyandthree-bodyforce,respectively.Thermal
conductivityiscalculatedfromtheGreen-Kuboformula30
Twhitaht4isu,nthitesrienapreerimodorleenmgtohd,ewshliochcaalilzseodleiands3Dtoaphroednuocntiiconcroyfstiatsl k~ 1 ð?v~J(t):~J(0)wdt ð3Þ
3k T2V
thermal conductivity. Fig. 5(b) shows the NIPR spectra of 3D B 0
phononic crystal with different mass ratio R, where the period wherekisthermalconductivity,kBistheBoltzmannconstant,Visthesystem
length is 2 units. The values of NIPR for R 5 1.5, R 5 2 and R volume,Tisthetemperature,andtheangularbracketdenotesanensembleaverage.
Generally,thetemperatureinMDsimulation,T ,iscalculatedfromthekinetic
5 2.5 are close to each other and larger than pure Si (R 5 1), MD
energyofatomsaccordingtotheBoltzmanndistribution:
whichshowthattheisotopicatomscouldcausemorelocalizations.
I(n.1F2igT.H5z()bo),f sbpaencdtragfaoprsRa5pp2eaarndinRt5he2.h5i,gahndfrtehqeuebnacnyd gpaaprst hEi~XN 12mvi2~32NkBTMD ð4Þ
i~1
for R 52.5 are wider than those for R 52. That is, the phonons
wherehEiisthetotalkineticenergy,v isthevelocity,mistheatomicmass,Nisthe
with frequency in the range of band gaps cannot exist in the 3D i
phononic crystal. When R is smaller than 1.5 (corresponding to nvaulmidbaetrhoifgphatretmiclpeesriantuthrees(yTs?temT,,anTdkisBtihsethDeeBboyletztmemanpnercaotunrset)a.nt.Thisequationis
D D
43Si), band gaps are not observed in 3D isotopic phononic crystal. Numerically,velocityVerletalgorithmisemployedtointegrateequationsof
So,bandgapsareminoreffectswhenthemassdifferenceisnotbig motion,andeachMDstepissetas1.0fs.Firstly,canonicalensembleMDwith
enough. langevinheatreservoirrunsfor220stepstoequilibratethewholesystemat1000K.
Then,microcanonicalensemble(NVE)MDrunsforanother225steps(33.5ns).
Besidesdecreasinggroupvelocityandphononmodeslocalization,
Meanwhile,heatcurrentisrecordedateachstep.Attheend,thethermalconductivity
interfacedisorderisanotherfactorinreducingthethermalconduc-
iscalculatedbyEq.(3).Inthecalculationofthermalconductivity,theintegrationis
tivity for superlattices11 and phononic crystal16. In our simulation, fromzerotoacut-offtimewhichisdeterminedby‘‘firstavalanche’’method31.The
28SiandMSiatomshasaperfectinterfacewhichdonotincludethe finalresultisaveragedoversixteenrealizationswithdifferentinitialconditionsto
disordereffect.Therewillbeafurtherdeductioninthermalconduc- satisfyergodicity.
tivityafterconsideringtheinterfacedisorder.So,thedisordereffect
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