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DTIC ADA541665: Exponential Suppression of Thermal Conductance using Coherent Transport and Heterostructures PDF

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Preview DTIC ADA541665: Exponential Suppression of Thermal Conductance using Coherent Transport and Heterostructures

PHYSICALREVIEWB82,113105(cid:2)2010(cid:1) Exponential suppression of thermal conductance using coherent transport and heterostructures Wah Tung Lau (留華東(cid:1),* Jung-Tsung Shen (沈榮聰(cid:1),† and Shanhui Fan (范汕洄(cid:1)‡ EdwardL.GinztonLaboratory,StanfordUniversity,Stanford,California94305,USA (cid:2)Received25August2010;published27September2010(cid:1) We consider coherent thermal conductance through multilayer photonic crystal heterostructures, consisting of a series of cascaded nonidentical photonic crystals.We show that thermal conductance can be suppressed exponentiallywiththenumberofcascadedcrystalsduetothemismatchbetweenphotonicbandsofallcrystals intheheterostructure. DOI:10.1103/PhysRevB.82.113105 PACSnumber(cid:2)s(cid:1): 42.70.Qs,63.22.Np,44.40.(cid:1)a,05.60.(cid:2)k Coherent thermal transport, using thermal channels with parallel to the layers, respectively.17 Summing over all pho- length smaller than the mean-free path of the thermal carri- ton states, we have10 (cid:5) ers, is important for fundamental study of thermal processes (cid:4) kdk andfornewdeviceopportunitiesinthermalmanagement.1–11 G (cid:2)T(cid:1)= (cid:3) (cid:3)G(cid:2)T,k,(cid:4)(cid:1). (cid:2)1(cid:1) 3D 2(cid:5) (cid:3) InthisBriefReport,weconsideracoherentthermalchannel (cid:4) formed by cascading a series of nonidentical photonic crys- Here tals.Weshowthatitsthermalconductancecanbesuppressed (cid:5) exponentially with respect to the channel length l. This is G(cid:2)T,k,(cid:4)(cid:1)=k (cid:6) d(cid:3)(cid:6)(cid:7)(cid:3)/(cid:2)kBT(cid:1)(cid:7)2e(cid:7)(cid:3)/(cid:2)kBT(cid:1)(cid:8)(cid:2)(cid:3),k,(cid:4)(cid:1) (cid:2)2(cid:1) fundamentally different from the incoherent process, where (cid:3) B 2(cid:5) (cid:6)e(cid:7)(cid:3)/(cid:2)kBT(cid:1)−1(cid:7)2 (cid:3) the conductance typically decreases as 1/l.12 We therefore 0 demonstrate that coherent processes can be very effective in is the thermal conductance of a one-dimensional thermal suppressing thermal conductance. This result also indicates channel of a given parallel wave number k and polarization (cid:3) that an aperiodic coherent thermal channel is qualitatively (cid:4). k is the Boltzmann constant and (cid:7)=h/(cid:2)2(cid:5)(cid:1) is the re- B different from all previously considered coherent thermal duced Planck constant. The factor (cid:8)(cid:2)(cid:3),k ,(cid:4)(cid:1) measures the (cid:3) channels1–10 including periodic multilayer photonic crystals contribution from photon states at (cid:2)(cid:3),k ,(cid:4)(cid:1). For a single (cid:3) as we previously considered,9,10 where the intrinsic thermal crystal(cid:2)i.e.,N=1(cid:1),(cid:8)(cid:2)(cid:3),k ,(cid:4)(cid:1)=1ifthefrequency(cid:3)liesina (cid:3) conductance of the channels were all independent of the channel length l. (a) infinitenumberofperiods N periods N periods infinitenumberofperiods P P As a concrete implementation, we utilize the concept of photoniccrystalheterostructure,13–15andconsidermultilayer ... ... systemsconsistingofatotalofN differentphotoniccrystals, as illustrated in Fig. 1(cid:2)a(cid:1). All crystals consist of alternate periodiclayersofvacuumanddielectric.Theuseofvacuum crystal1 crystal2 crystal3 crystalN=4 ensures that heat transfer is carried only by photons. The (b) dielectric can be silicon (cid:2)n=n =3.42(cid:1). For photon frequen- Si crystal1 cieswithintheblackbodyspectrumatroomtemperature,sili- ds=0.8297a con has very little dispersion and dissipation, with typical attenuation length exceeding millimeter.16 Consequently crystal2 photonic thermal transport should be coherent across the en- ds=0.1633a tire structure.9,10 For the structure shown in Fig. 1(cid:2)a(cid:1), we assume that all crystal3 ds=0.6427a crystals have the same period a. The mth crystal has a di- electric layer thickness d and a vacuum layer thickness sm crystal4 dvm. At either ends of the structure, we have two semi- ds=0.2327a infinite photonic crystals. In between these two ends, there canbeaseriesofcrystalscascadedtogether,eachhavingthe overlapped bands same number of periods N and hence the same total thick- ness. (cid:2)Thus the length ofP the channel is proportional to 0 0.5 1 ω(21π.5c/a) 2 2.5 3 N−2.(cid:1) We also assume that the two crystals at the ends are FIG.1. (cid:2)a(cid:1)Aphotoniccrystalheterostructurewithfourcrystals. maintainedattemperaturesofTandT+dT,respectively.The Eachcrystalhasalternatesiliconandvacuumlayers,andaperioda. three-dimensional (cid:2)3D(cid:1) thermal conductance per unit area is Thetwocrystalsateitherendsareassumedtobeinfinite.(cid:2)b(cid:1)Inthe then defined as G (cid:2)T(cid:1)=dQ(cid:2)T(cid:1)/dT, where dQ(cid:2)T(cid:1) is the heat topfourpanels,grayregionscorrespondtofrequencyrangeswhere 3D flux per unit area. thecorrespondingcrystal,assumedtobeinfinite,hasapropagating In such multilayer structures, each photon state is charac- state along the normal incident direction. White regions are the terizedbythreeparameters:frequency(cid:3),wavenumberk in band gaps. d is the thickness of the silicon layers. In the bottom (cid:3) s the direction parallel to the layers, and polarization (cid:4)=s,p. panel,theblackregionsrepresentfrequencyrangeswherethereare The s and p polarizations have electric and magnetic fields photonicbandsinallcrystals. 1098-0121/2010/82(cid:2)11(cid:1)/113105(cid:2)4(cid:1) 113105-1 ©2010TheAmericanPhysicalSociety Report Documentation Page Form Approved OMB No. 0704-0188 Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. 1. REPORT DATE 3. DATES COVERED AUG 2010 2. REPORT TYPE 00-00-2010 to 00-00-2010 4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER Exponential suppression of thermal conductance using coherent 5b. GRANT NUMBER transport and heterostructures 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION Stanford University,Edward L. Ginzton Laboratory,Stanford,CA,94305 REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S) 11. SPONSOR/MONITOR’S REPORT NUMBER(S) 12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release; distribution unlimited 13. SUPPLEMENTARY NOTES 14. ABSTRACT 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF ABSTRACT OF PAGES RESPONSIBLE PERSON a. REPORT b. ABSTRACT c. THIS PAGE Same as 4 unclassified unclassified unclassified Report (SAR) Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18 BRIEFREPORTS PHYSICALREVIEWB82,113105(cid:2)2010(cid:1) (a) 1 bandstructuresasawhole,ratherthanthedetailedproperties N=2 0.75 N=3 of individual bands, become important. From Eq. (cid:2)2(cid:1), we G(T)/G(T)0 0.5 N=4 thus olbimtainG(cid:2)T(cid:1) = lim 1(cid:5)(cid:11)d(cid:3)(cid:8)(cid:2)(cid:3)(cid:1)(cid:8)(cid:12), for N=1. (cid:2)3(cid:1) 0.25 T→(cid:6)G0(cid:2)T(cid:1) (cid:11)→(cid:6)(cid:11) 0 0 4 8 12 16 20 Here(cid:12)(cid:13)1 denotestheproportionoffrequenciesinthepho- T [hc/(ka)] B tonicbands.Usingergodictheory,weshowedinRef.10that (b) except for a zero-measure set of parameter choices, all crys- 10-1 N=3 (cid:9) T)] tals satisfy (cid:5) (cid:10) (cid:11) G(0 1 (cid:5) −2+(cid:2)(cid:15)+1(cid:1)cos(cid:14) m[G(T)/∞10-2 (cid:12)= (cid:5)2 0(cid:10) d(cid:14) Re cos−1 (cid:15)(cid:11)(cid:12)−1 liT N=8 10-30 2 4 6 8 10 12 −Re cos−12+(cid:2)(cid:15)+1(cid:1)cos(cid:14) , (cid:2)4(cid:1) numberofperiods(N) (cid:15)−1 (c) 0.5 P 0.4 NN==23 where (cid:15)=0.5(cid:2)n+1/n(cid:1). We emphasize that (cid:12), and hence the T)] N=4 ergodic limit of the normalized conductance, is independent G(0 0.3 G(T)/ 0.2 of tIhnealahyeetrertohsictrkuncetussreeswiniththNe ucnryitstcaellsl.(cid:6)Fig. 1(cid:2)a(cid:1)(cid:7), photonic m[∞ liT 0.1 band gaps are located at different frequencies for different 0 crystals (cid:6)Fig. 1(cid:2)b(cid:1)(cid:7). To determine its thermal conductance in 0 1000 2000 3000 4000 5000 samplenumber the ergodic limit, we first assume that thermal transport oc- FIG.2. (cid:2)Coloronline(cid:1)(cid:2)a(cid:1)Normalizedone-dimensionalthermal cursonlyatfrequenciesthatfallwithinthebandsofallcrys- conductanceG(cid:2)T(cid:1)/G (cid:2)T(cid:1)asafunctionoftemperatureTforhetero- tals, and set (cid:8)(cid:2)(cid:3)(cid:1)=0 for all other frequencies. We then as- 0 structureswithdifferentnumberofcrystalsN.ForeachN,different sume that the frequency distributions of the photonic bands curvesrepresentstructuresofdifferentthicknessesofsilicond and between different crystals are independent. From these two s vacuumd .Thebrown(cid:2)darkgray(cid:1)dashedcurveshowsacommen- assumptions,thetotalproportionoffrequenciesthatcontrib- v suratecaseofN=2,whered =0.7851aandd =0.2262asuchthat utes to thermal conductance is then (cid:12)N, with (cid:12)given by Eq. s1 s2 n d /(cid:2)a−d (cid:1)=1. (cid:2)b(cid:1) High-T conductance for heterostructures of (cid:2)4(cid:1). Finally, for each frequency (cid:3)that does contribute, the si s2 s2 different number of periods NP at the intermediate crystals. Each power transmission coefficient (cid:8)(cid:2)(cid:3)(cid:1) is typically less than line corresponds to structures with the same number of crystals N. unity,duetoimpedancemismatchbetweendifferentcrystals. 3(cid:16)N(cid:16)8.DatapointsexistonlyatintegerNP.(cid:2)c(cid:1)High-T conduc- Combining all these considerations, we have tanceforatotalof15000heterostructureswithrandomlygenerated (cid:5) thicknessesofsiliconlayers.0.05a(cid:16)d (cid:16)0.95a forallcrystals. G(cid:2)T(cid:1) 1 (cid:11) s lim = lim d(cid:3)(cid:8)(cid:2)(cid:3)(cid:1)(cid:13)(cid:12)N, for N(cid:9)2. (cid:2)5(cid:1) photonicband,and(cid:8)(cid:2)(cid:3),k ,(cid:4)(cid:1)=0ifthefrequency(cid:3)liesina T→(cid:6)G0(cid:2)T(cid:1) (cid:11)→(cid:6)(cid:11) 0 (cid:3) band gap region.10 For the heterostructure (cid:6)Fig. 1(cid:2)a(cid:1)(cid:7), where Equation (cid:2)5(cid:1) is the main result of this paper: the thermal there are at least two crystals (cid:2)i.e., N(cid:9)2(cid:1), (cid:8)(cid:2)(cid:3),k ,(cid:4)(cid:1) van- conductance of a heterostructure in the ergodic limit has an (cid:3) ishes unless, at a given (cid:2)(cid:3),k ,(cid:4)(cid:1), propagating eigenmodes upper bound (cid:12)N that decreases exponentially with the num- (cid:3) exist for the semi-infinite crystals at both ends. In such a berofcrystalsN.Moreover,suchanupperboundisindepen- case, (cid:8)(cid:2)(cid:3),k ,(cid:4)(cid:1) is the power transmission coefficient dent of the detailed geometry such as layer thickness and (cid:3) through the entire structure when an incident wave is one of lattice constant but instead depends only on the refractive these eigenmodes, and can be determined using the transfer indices of the layers. matrixmethod.17Wewillfirstconsiderthebehaviorsofone- We now support the theoretical analysis above with ex- dimensional conductance at normal incidence, and suppress tensive numerical simulations. We first verify that the er- the labels k and (cid:4). godic limit indeed exists. In Fig. 2(cid:2)a(cid:1), we plot the tempera- (cid:3) For any one-dimensional channel consisting of homoge- ture dependence of G(cid:2)T(cid:1)/G (cid:2)T(cid:1) for various heterostructures. 0 neous dielectric material, (cid:8)(cid:2)(cid:3)(cid:1)=1 for all frequencies. Equa- G(cid:2)T(cid:1)/G (cid:2)T(cid:1) all decrease rapidly at small T and converge to 0 tion (cid:2)2(cid:1) then leads to the universal quantized thermal con- specific values when T(cid:10)hc/(cid:2)k a(cid:1). These specific values: B ductance G0(cid:2)T(cid:1)=(cid:5)kBT/(cid:2)6(cid:7)(cid:1).1,2,4 For any one-dimensional limT→(cid:6)(cid:6)G(cid:2)T(cid:1)/G0(cid:2)T(cid:1)(cid:7),whichwereferasthe“high-Tconduc- channelingeneral,wedefineitsnormalizedthermalconduc- tance,” define the ergodic limit for the conductance in a tance as G(cid:2)T(cid:1)/G (cid:2)T(cid:1). givenstructure.Below,wewillonlyconsiderconductanceat 0 We now briefly review the property of thermal conduc- this ergodic limit. tance suppression of a single photonic crystal at normal In a heterostructure, except for the two crystals at the incidence.10 The suppression is particularly effective at the ends, all intermediate crystals have a finite number of peri- ergodic limit of k Ta/(cid:2)hc(cid:1)(cid:10)1, where the thermal conduc- ods N .At frequencies in the band gaps of any intermediate B P tance has contribution from a large number of photonic crystal, photons can still transmit through the crystal, result- bands.10Inthislimit,thestatisticalpropertiesofthephotonic ing in a nonzero (cid:8)(cid:2)(cid:3)(cid:1). Such a finite-size effect, however, 113105-2 BRIEFREPORTS PHYSICALREVIEWB82,113105(cid:2)2010(cid:1) should be negligible if N is sufficiently large. In Fig. 2(cid:2)b(cid:1), 100 P we consider heterostructures with different number of crys- tals N. For each N, we vary N . The high-T conductance all P 10-1 converges when N (cid:9)10. Thus, with a choice of N =10, all intermediate crystaPls can essentially be treated asPinfinite, G(T)]0 andthus(cid:8)(cid:2)(cid:3)(cid:1)=0 unlessatthefrequency(cid:3)therearephoton T)/ 10-2 states in the band structures for all crystals, validating the G( m[∞ first assumption made in our analysis. Regarding the second liT 10-3 assumption, one can in fact prove that frequency distribu- tions of photon states in different crystals are independent, exceptforazero-measuresetofparameterchoices,usingan 10-4 0 2 4 6 8 10 analysis similar to Ref. 10. numberofcrystals(N) Equation(cid:2)5(cid:1)predictsthatanupperboundthatisuniversal FIG.3. One-dimensionalhigh-Tconductanceforphotoniccrys- in the sense that it is independent of layer thicknesses. Nu- tal heterostructures as a function of number of crystals (cid:2)N(cid:1). Data mericalcalculationsestablishanevenstrongerresult.InFig. points exist only at integer N. N=0 corresponds to vacuum. N=1 2(cid:2)c(cid:1), we consider a total of 15000 structures, with the num- corresponds to an infinite crystal. N=2 corresponds to two semi- ber of crystals N between 2 and 4. In these structures, the infinitecrystalsplacedtogether.ForN(cid:9)3,eachintermediatecrystal thickness of the dielectric layers is randomly generated. For hasN =10numberofperiods.Thesolidlineistheactualconduc- P thevastmajorityofstructures,thehigh-Tconductancethem- tance.Thedashedlineisthecorrespondingtheoreticalupperbound selves have universal values that depend only on N and are (cid:12)N(cid:8)exp(cid:2)−(cid:18)N(cid:1) fromEq.(cid:2)5(cid:1),where(cid:18)=0.62. independent of the layer thicknesses. This numerical result mensurate cases hereafter since their occurrences require providesastrongsupportofourstatisticalanalysisthatleads control of layer thicknesses with high accuracy.10 to Eq. (cid:2)5(cid:1). We now provide a direct numerical confirmation of Eq. Examining Fig. 2(cid:2)c(cid:1), we also see few cases for which the (cid:2)5(cid:1).InFig.3,weplotthehigh-Tconductanceofheterostruc- high-Tconductanceofastructuredeviatesfromtheuniversal tures with different number of crystals N. We consider only values. (cid:6)One of these cases is also shown in Fig. 2(cid:2)a(cid:1).(cid:7) Such the incommensurate cases where the high-T conductance deviations occur if one of the crystals belongs to the com- takes the universal values. The numerically determined mensuratecaseseitherasdiscussedinRef.10,orwhenpho- high-T conductanceclearlyfallsbelowtheexponentiallyde- tonic bands of two or more crystals become correlated with creasing bound as set by Eq. (cid:2)5(cid:1). each other. These cases, in the ergodic limit, form the zero- For three-dimensional thermal conductance, using Eqs. measure set as discussed above. We will ignore these com- (cid:2)1(cid:1) and (cid:2)2(cid:1), we have10 (cid:9) (cid:12) (cid:5) (cid:5) (cid:4) n (cid:6) (cid:3)2 (cid:6)(cid:7)(cid:3)/(cid:2)k T(cid:1)(cid:7)2e(cid:7)(cid:3)/(cid:2)kBT(cid:1) G (cid:2)T(cid:1)= udu d(cid:3) k B (cid:8)(cid:2)(cid:3),u,(cid:4)(cid:1) , (cid:2)6(cid:1) 3D (cid:4)=s,p 0 0 4(cid:5)2c2 B (cid:6)e(cid:7)(cid:3)/(cid:2)kBT(cid:1)−1(cid:7)2 where u(cid:8)kc/(cid:3). At the ergodic limit, for each channel la- dence, numerical results (cid:2)not shown here(cid:1) further indicate (cid:3) beledbyu,werepeatthesamestatisticalargumentsaboveto that 3D conductance themselves in the ergodic limit are in obtain fact universal. (cid:5) (cid:5) We now confirm Eq. (cid:2)7(cid:1) with numerical results. For the s G (cid:2)T(cid:1) (cid:4) 1 1 (cid:11) polarization, the band gaps persist in the entire range of 0 Tl→im(cid:6)Gv3aDc(cid:2)T(cid:1) =(cid:4)=s,p (cid:5)0 udu·(cid:11)li→m(cid:6)(cid:11) 0 d(cid:3)(cid:8)(cid:2)(cid:3),u,(cid:4)(cid:1) (cid:13)bouun(cid:13)d1o.n9,1t0heHheingche-T(cid:12)c(cid:2)oun,d(cid:4)u=csta(cid:1)n(cid:13)ce1.foFrrtohmesEpqo.la(cid:2)r7iz(cid:1),attihoentuhpepreer- (cid:4) 1 fore decreases exponentially with respect to the number of (cid:16) (cid:6)(cid:12)(cid:2)u,(cid:4)(cid:1)(cid:7)Nudu. (cid:2)7(cid:1) crystals N, as supported by the numerical results in Fig. 4. (cid:13) (cid:4)=s,p 0 For the p polarization, u = n2/(cid:2)n2+1(cid:1) corresponds to the B Here (cid:12)(cid:2)u,(cid:4)(cid:1) is obtained by replacing (cid:15) in Eq. (cid:2)4(cid:1) with Brewster angle where the reflection at the vacuum-dielectric (cid:13) interface vanishes.18,19 At u=u there is no photonic band (cid:15)(cid:2)(cid:13)u,(cid:4)(cid:1)=0.5(cid:6)na/(cid:2)nbP(cid:1)+(cid:2)nbP(cid:1)/na(cid:7), where na= n2−u2, nb gap at any frequency and (cid:12)(cid:2)uBB,(cid:4)=p(cid:1)=1. As a result, the = 1−u2, and P=1 and n2 for the s and p polarizations, re- high-T conductance from the p polarization, and hence the spectively. total3Dhigh-T conductance,showmoregradualdecaywith In Eq. (cid:2)7(cid:1), the equality holds only when N=1. Also, in respect to N (cid:2)Fig. 4(cid:1). derivingEq.(cid:2)7(cid:1),weusethefactthatintheergodiclimit,the Inourstructure,exponentialreductionin3Dthermalcon- states with u(cid:17)1, where photons are evanescent in the ductancecanbeaccomplishedusingasimpleanglefilterthat vacuum layers, do not contribute.10 Hence the range of inte- blocks out thermal radiation with a large angle of incidence. gration on u is between 0 and 1. Finally, Eq. (cid:2)7(cid:1) indicates a Asasimpleillustration,restrictingtheintegrationrangeofu universal upper bound that depends only on the index and inEq.(cid:2)6(cid:1)to0(cid:16)u(cid:16)0.5issufficienttogeneratetheexponen- the number of crystals. Similar to the case of normal inci- tial reduction (cid:2)Fig. 4(cid:1).Alternatively, for heterostructures be- 113105-3 BRIEFREPORTS PHYSICALREVIEWB82,113105(cid:2)2010(cid:1) 100 to Refs. 21 and 22, where thermal contacts are made to the ends of the structure, and thermal conductance is measured 10−1 by recording the amount of power transferred. Our results may also be practically significant for ultimate thermal insu- G(T)]vac 10−2 ltaiotinonu.tiClizuersremntulylt,ipalneelaffyeecrtsivoefmmeectahlaannisdmvafcouruthme.r2m3,2a4lIinnssuulcah- G(T)/3D 10−3 aduscttrauncctuersec,atlheesrmasal1/trNans,fwerheisreinNcohiesretnhte.nItusmtbheerrmoaflmceotna-l m[∞ M M liT10−4 layers.23,24 Our results here indicate a qualitatively different mechanism for achieving thermal insulation using dielectric systems,whichmaypresentnewopportunities.Forexample, 10−5 0 2 4 6 8 10 the structure in Fig. 1 allows high transmission through nar- numberofcrystals(N) row band of frequencies, which may be important for com- munication through such a thermally insulating medium. FIG. 4. (cid:2)Color online(cid:1) 3D high-T thermal conductance versus We end by highlighting closely related works. The con- number of crystals (cid:2)N(cid:1) in the structures. Data points exist only at cept of using a single phononic crystal structure to suppress integerN.Thethickblacklinesarethetotalconductance.Thethick coherent phononic thermal conductance was discussed in green(cid:2)gray(cid:1)linesarethetotalconductancewithananglefilter.The thin dark blue (cid:2)dark gray(cid:1) and the thin red (cid:2)light gray(cid:1) lines are Ref.25,andwasextendedtothephotoniccaseinRefs.9and 10. In these structures, the thermal conductance is indepen- contributionsfromthesandppolarizations,respectively.Thesolid dent of the channel length. Meanwhile, mismatch between lines are direct numerical calculations of the thermal conductance. ThedashedlinesarethetheoreticalupperboundsfromEq.(cid:2)7(cid:1).For two different phononic band structures had been used to N(cid:9)3,eachintermediatecrystalhasN =10numberofperiods. achieve very small interfacial thermal conductance.26,27 Our P result, showing exponential suppression of thermal conduc- tween two semi-infinite vacuum regions, we can achieve ex- tance using a large number of crystals, is a step further and ponential reduction using omnidirectional reflectors as the points to a new regime of coherent thermal transport. While intermediatecrystals.18Finally,Ref.20showsthattheBrew- we have considered photonic thermal transport, the general ster angle can be made imaginary using films with large bi- concept in this paper may be applicable to other thermal refringence.Multilayerstructuresconsistingofsuchbirefrin- carriers as well provided that the coherent effect is signifi- gent films and vacuum therefore could also be useful for cant. Finally, in analogy to electron transport, where an ex- demonstrating exponential reduction in 3D thermal conduc- ponentialreductioninelectronicconductancewithrespectto tance. distance is a direct signature of electron localization, our We now briefly comment on the experimental feasibility resultindicatesanintriguingnotionthatheatcanbelocalized in demonstrating our predictions. Multilayer structures can as well. befabricatedbyavarietyoftechniques.18,20Ourtheorycalls for a measurement of the photonic thermal conductance in This work was supported by AFOSR-MURI programs these structures. For this purpose we notice the exciting re- (cid:2)GrantsNo.FA9550-08-1-0407andNo.FA9550-09-1-0704(cid:1) cent experimental developments in measuring photonic ther- and byAFRL. 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