thermal conductivity modeling of periodic two-dimensional nano composites

PHYSICAL REVIEW B 69, 195316 ~2004!

Thermal conductivity modeling of periodic two-dimensional nanocomposites

Ronggui Yang and Gang Chen*Mechanical Engineering Department, Massachusetts Institute Technology, Cambridge, Massachusetts 02139, USA

~Received 22 December 2003; revised manuscript received 11 March 2004; published 27 May 2004!

In this paper, the phonon Boltzmann equation model is established to study the phonon thermal conductivityof nanocomposites with nanowires embedded in a host semiconductor material. Special attention has been paidto cell–cell interaction using periodic boundary conditions. The simulation shows that the temperature profilesin nanocomposites are very different from those in conventional composites due to ballistic phonon transportat nanoscale. Such temperature profiles cannot be captured by existing models in literature. The generalapproach is applied to study silicon wire/germanium matrix nanocomposites. We predict the thermal conduc-tivity dependence on the size of the nanowires and the volumetric fraction of the constituent materials. Atconstant volumetric fraction the smaller the wire diameter, the smaller is the thermal conductivity of periodictwo-dimensional nanocomposites. For fixed silicon wire dimension, the lower the atomic percentage of ger-manium, the lower the thermal conductivity of the nanocomposites. Comparison is also made with the thermalconductivity of superlattices. The results of this study can be used to direct the development of high efficiencythermoelectric materials.

DOI: 10.1103/PhysRevB.69.195316 PACS number~s!: 68.65.2k, 44.10.1i, 63.22.1m

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I. INTRODUCTION

The efficiency and energy density of thermoelectric dvices are determined by the dimensionless thermoelefigure of meritZT5S2sT/k, whereS is the Seebeck coefficient, s is the electrical conductivity,k is the thermal con-ductivity, andT is the absolute temperature.1 Significant ad-vances for increasingZT have been made based on new ideto engineer electron and phonon transport.2 One particularlyfruitful and exciting approach has been the use of nanosttures, so that the electron performance can be improvemaintained concurrently with a reduction of phonon thermconductivity.3–5 Nanostructure-based materials suchBi2Te3 /Sb2Te3 superlattices and PbTe/PbSeTe quantumsuperlattices have shown significant increases inZT valuescompared to their bulk counterparts6,7 due to mainly reducedphonon thermal conductivity of these structures. Nanocoposites may realize a similar thermal conductivity reductand provide a pathway to scale-up the nanoscale effectsserved in superlattices to thermoelectric material in bform.

Most of the previous studies on thermal transport in nastructures have focused on thin films, semiconductor sulattices, and nanowires. One key question in modeling ofthermal conductivity in nanostructures is when the wavefect, i.e., the phonon dispersion change, shouldconsidered.8,9 For example, models for phonon transportsuperlattices generally fall into two groups. One gro~‘‘wave models’’! assumes that phonons form superlattbands and calculates the modified phonon dispersion ulattice dynamics or other methods.10–13 The other group ofmodels~‘‘particle models’’! assumes that the major reasfor the thermal conductivity reduction is the sequential sctering of phonons at interfaces.14–17In these models, phonotransport falls in the totally incoherent regime and supertices are treated as inhomogeneous multilayer structures.fuse interface scattering is usually incorporated into the Bzmann equation~BE! as boundary conditions. These mode

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succeeded in fitting experimental values of several supetice systems in the thick period range. However, becausewave nature of phonons is ignored, these models fail toplain the thermal conductivity recovery in the short perilimit that is observed in some superlattices. More recenmodels9,18 and direct simulations19 combining the abovetwo pictures have been presented. These models and simtions further confirm the importance of diffuse interfascattering for thermal conductivity reduction. The diffuse iterface scattering cannot only reduce the phonon mfree path~MFP! but also destroy the coherence of phonoDue to the loss of coherence, the phonon dispersion chain nanostructures predicted by ideal lattice dynamapproximation cannot be realized. Previous studies onthermal conductivity of superlattices demonstrated that csical size effect models are expected to be applicablewide range of nanostructures. This has been furtconfirmed by a recent study on developing the classicaleffect model for phonon transport in nanowires and supertice nanowires.20 The model, assuming gray and diffusphonon scattering at interfaces and side walls, has succfully explained the thermal conductivity reduction effemeasured from nanowires. Based on previous studies,will apply in this work the phonon Boltzmann equationstudy the classical size effect on the thermal conductivitynanocomposites.

Another field of related research is effective thermal coductivity of composites. The effective thermal conductiviof composites in macroscale has been studied since Max~for a review, see Milton21! and a variety of methods havbeen proposed to estimate physical properties of heterneous media. In most of these research works, the interfbetween two heteromedia are treated as nonresistive toflow. The interface thermal resistance, or Kapitresistance,22 has been considered only recently. The first twtheoretical analyses that included the interface thermal retance were conducted by Hasselman and Johnson23 and byBenvensite,24 respectively. Hasselman and Johnson exten

©2004 The American Physical Society16-1

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RONGGUI YANG AND GANG CHEN PHYSICAL REVIEW B69, 195316 ~2004!

the classical work of Maxwell and Rayleigh to derivMaxwell–Garnett-type effective medium approximatio~EMA! for calculating effective thermal conductivity of simply spherical particulate and cylindrical fiber reinforced mtrix composites in which interface effect and particle seffect are included. Benvensite and Miloh24 developed a general framework incorporating thermal boundary resistanceaveraging all pertinent variables like heat flux and intensover the composite medium viewed as a continuum consing of a matrix with inclusions. Everyet al.25 refined theeffective medium theory and presented an asymmeBruggeman type EMA, corrected for Kapitza resistance, asimple solution for high volumetric fraction of inclusionbased on Bruggeman’s integration-embedding principlenoticeable work was by Nanet al.,26 who adopted the mul-tiple scattering theory27 to develop a more general EMAformulation for the effective thermal conductivity of arbtrary particulate composites with interfacial thermal restance. They considered the properties of the matrix and rforcement, particle size and size distribution, volumfraction, interface resistance, and the effect of shape. Omodels have also been proposed including the boundmodel28,29 and thermal resistance network theory. Numerisimulation of thermal conductivity of composites caalso be found in literature.30–33 However, these macroscopmodels are developed based on Fourier heat conductheory that is not valid at nanoscale due to the ballisphonon transport.

So far, there are not many theoretical studies on the tmal conductivity of nanocomposites despite their importain both thermoelectrics and thermal management of electics ~especially the development of thermal interface matals!. Closely related works are done by Khitunet al.34 andBalandin and Lazarenkova35 to explain theZT enhancemenof Ge quantum dot structures~where Ge quantum dots ar;4 nm and can be thought of as nanoparticles!. The modelby Khitun et al. calculates the reduced thermal conductivthrough the relaxation time change due to the nanopartiembedded using the Mathiessen rule.36 They used the Mietheory for acoustic wave scattering to calculate the scattecross section of a single particle and thus the additionallaxation time due to single particle scattering. Their methis valid if there is no inelastic scattering inside the partiand the interface scattering must be specular.37 This approachdoes not recover to bulk material properties of compossince the thermal conductivity of nanoparticle material donot get into the final picture. Similar to the wave modelsuperlattices, Balandin and Lazarenkova35 assumed that anew homogeneous material is formed and calculates theelectron and phonon dispersion relation thus the grouplocity reduction. This approach requires phonon cohereover several unit cells and does not apply to diffuse interfscattering as observed in previous studies on superlattic

The motivation of this work is to develop a microscopframework for thermal conductivity of nanocompositesterms of the phonon particle transport model. For simplicwe will study the thermal conductivity of periodic twodimensional~2D! nanocomposites. The model will help tunderstand some basic physical phenomena that exis

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nanocomposites and serve as a prototype to be further deoped to characterize more complex nanocomposites, sucnanoparticle composites. Some fundamental questisuch as~a! how is phonon transport in nanocompositdifferent from that in macroscale composites,~b! what is thebehavior of the size effect in nanowire-embeded composiand ~c! can nanocomposites have lower effective thermconductivity than a superlattice, will be addressed ininvestigation.

II. THEORETICAL MODEL

Based on the reasoning that the phonon particle mocan be a good predictive tool for thermal conductivitynanocomposites, we focus our work on the phonon transin nanowire-embedded composites for the case whereheat-flow direction is perpendicular to the wire axis. Ashown in Fig. 1~a!, there is no heat flow along the wire axthus the problem is simplified to two-dimensional althouthe nature of phonon transport is three-dimensional in naas shown in Fig. 1~c!. The unit cell to be simulated is showin Fig. 1~b!. The detail about the interface and boundacondition will be presented in a later section. To makecomparison, we also calculated the cross-plane thermalductivity of a simple one-dimensional Si–Ge layered struture, which is often called superlattices when the thickneseach layer is tens of nanometers, as shown in Fig. 1~d!. Toestablish the phonon particle model the following assumtions are made:~1! the phonon wave effect can be exclude~2! the frequency-dependent scattering rate in the bulk mdium is approximate by an average phonon MFP; and~3! theinterface scattering is diffuse.

A. Phonon Boltzmann equation

With the above assumptions, the phonon BE canexpressed in terms of the total phonon intensity definas38,39

I i51

4p (m

E0

vmaxunmiu f \vDmi~v!dv, ~1!

where subscripti ~51,2! denotes properties of the host anwire material ~or alternative material of a 1D layerestructure, in this worki 51 is for germanium andi 52for silicon!, D(v) is the density of states per unit volume,fthe phonon distribution function,\ the Planck constant,uvmiuthe magnitude of the phonon group velocity,v the phononfrequency, andvmax is the maximum frequency of eacpolarization. The summation indexm is over the threephonon polarizations. The purpose of introducing the phonintensity concept38 is for the mathematical convenience bcause the concept of photon intensity is widely used in thmal radiation literature and many results in radiative transmight be used for phonon transport studies.40,41

The 2D phonon BE under the single mode relaxation tiapproximation can be written as

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THERMAL CONDUCTIVITY MODELING OF PERIODIC . . . PHYSICAL REVIEW B 69, 195316 ~2004!

FIG. 1. ~a! Heat flow across periodic 2D composite with silicon wires embedded in germanium host,~b! the unit cell to be simulated,~c!local coordinate used in phonon Boltzmann equation simulation, and~d! heat flow across 1D Si–Ge layered structure~superlattice!.

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I i2I oi

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where u and f, as shown in Fig. 1~c!, are the polar andthe azimuthal angles, respectively, andL is the averagephonon MFP.I oi is determined by the Bose-Einstein distbution of phonons and depends on the local equilibritemperature. In nanostructures, however, local equilibricannot be established and thus the temperature obtashould not be treated the same as the case of local theequilibrium. An energy balance shows thatI oi can be calcu-lated from16,42

I oi~x,y!51

4p E4p

I i~r ,V!dV

51

4p E0

2pE0

p

I i~x,y,u,f! sinududf) ~3!

and the corresponding temperature obtained is a measuthe local energy density. We note the equations for thelayered structure are much simpler because the intensitynot depend on thef direction andy position.

A rigorous phonon transport simulation should incorprate the frequency dependence of phonon relaxation time

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group velocity, and thus account for interactions amophonons of different frequencies. This will relax the assumtion ~2!. However, it requires solution of the phonon BE fmany different frequencies. Previous works show that anerage MFP is a good approximation for thermal conductivmodeling across the interfaces~cross-plane transport!.16 Fortransport parallel to the interface, frequency dependent reation time gives a better solution.14 However, the existingtheories for the frequency dependence of relaxation tcontain large uncertainties because they are based on mapproximations and rely on the fitting parameters fromperimental data.43 Therefore we will use a frequency independent phonon MFP for simplicity. Often the phonon MFL is estimated from the thermal conductivity, the speciheat, and the speed of sound, according to the standarnetic theory expression,

k5 13 CvL, ~4!

whereC is the volumetric specific heat. This method, hoever, neglects the facts that at room temperature most actic phonons are populated close to the zone boundary wthe phonon group velocity is significantly smaller than tsound velocity and that the optical phonons contribute anificant portion of specific heat but little to thermal condu

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TABLE I. Room-temperature parameters used in the calculation.

C ~J/m3 K! v ~m/s! k ~W/m K! a ~nm!

Silicon 0.933106 1804 150 0.5431Germanium 0.873106 1042 60 0.5658

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tivity due to their near zero group velocity. A better estimtion of the phonon MFP and the group velocity canobtained by approximating the dispersion of the transveand the longitudinal-acoustic phonons with simple sine futions and subtracting the optical phonon contribution tospecific heatC in calculating the phonon MFP. This estimtion leads to a mean free path in silicon of the order250–300 nm at room temperature. Experiments and moing from Goodson’s group also lead to similar numbers44

The method gave a better fitting of the experimental therconductivity data on superlattices than that based on bspecific heat forC and the speed of sound forv in Eq.~4!.14,16In our analysis, some parameters used are taken fRef. 16 as listed in Table I.

B. Boundary and interface conditions

Mathematically, the structure of the BE demandsboundary conditions to be specified on those parts of bouaries where the characteristics point into the domain.45 Thisboundary condition is termed as emitted temperaturewall.16 Most of the previous works on phonon transposimulation used a specified emitted temperature boundcondition. This can give an artificial temperature jump atboundaries due to the additional nonphysical scattering aboundaries.46 A more physical boundary condition idesirable to simulate phonon transport in realistic structuIn this work, the periodic boundary condition is used bason the underlying physics for phonon transport in periostructures. As shown in Fig. 1~b!, heat is enforced toflow in thex direction. The specular reflected boundary coditions are enforced aty50 andy5LGe boundaries due tosymmetry,

I ~x,LGe,u,f!5I ~x,LGe,u,2f!, ~5!

I ~x,0,u,f!5I ~x,0,u,2f!. ~6!

In the x direction, we applied periodic boundary conditionAt equilibrium, the phonon distribution, i.e., the phonointensity @see Eq.~1!#, is isotropic. If heat is enforced toflow, the phonon distribution/intensity will be distortePhysically the periodic boundary condition means thatdistortion of phonon intensity in each direction at each po(0,y) at thex50 boundary is the same as the distortionthe corresponding direction at the corresponding po(LGe,y) at the x5LGe boundary. The equation can be epressed as

I ~LGe,y,u,f!2I 0~LGe,y!5I ~0,y,u,f!2I 0~0,y!. ~7!

With this boundary condition, the total surface heat fluxconserved in thex direction and the cell–cell interaction itaken into account. Because of the periodicity, the pho

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transport of the whole structure can be representedthat in the unit cell. Boundary conditions thus defined requan iterative scheme to solve the equation, which implthat it is more complicated than the front-to-end schemeis usually adopted to solve the emitted temperature boundcondition problems. This periodic boundary condition hbeen used in studying the cross-plane thermal conductiof superlattices~which is a 1D structure! before.16 It can beproven that the temperature difference across the unitshould be independent ofy. That is,

I o~0,y!2I o~LGe,y!5C1v1@T~0,y!2T~LGe,y!#

4p5const.

~8!

In our simulation, we superimposedT(0,y)2T(LGe,y)51 K in the above equation. If we do not superimpose sua temperature difference in the program, Eq.~7! will auto-matically converge to a constant temperature differeT(0,y)2T(LGe,y) value. The converged value varies wisimulated structures. But the final results of thermal condtivity value do not depend on whether the temperatureference is superimposed. However, the calculation is mfaster when the temperature difference value is superposed. We should note that superimposing the temperadifference across the unit cell is physically different frosuperimposing temperature~either emitted or Fourier-limittemperature! at each boundary.

The interface scattering between the nanowire andhost material is assumed to diffuse. Ziman36 proposed thefollowing expression for estimating the interface specularparameterp:

p5expS 216p3d2

l2 D , ~9!

whered is the characteristic interface roughness andl is thecharacteristic phonon wavelength. At room temperature,characteristic phonon wavelengthl5hv/kBT is about 1 nm,whereh is the Planck constant,kB is the Boltzmann constantandv is the sound velocity in the material. Clearly even omonolayer roughnessd;0.3 nm gives an interface speculaity parameterp50, and allows for totally diffuse interfaceassumption. Determining the phonon reflectivity atransmissivity at interface is difficult as in the treatmeof the classical thermal boundary resistance problem.47–49

One rather crude model is called the diffuse mismamodel,49 which assumes that phonons emerging frominterface do not really bear any relationship to its origin, i.one cannot tell which side they come from. This assumptimplies that

Td215Rd12512Td12, ~10!

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where the second equation comes from the energy consetion identity Rd121Td1251. We remind again that subscrip12 means from medium 1 into medium 2 and vice verDames and Chen20 obtained the following equation foTd12, which is expected to be valid over a wide temperatrange:

Td12~T!5U2v2

U1~T!v11U2~T!v2, ~11!

whereU is the volumetric internal energy.With the given interface transmissivity and reflectivity, w

can write down the phonon intensity at interfaces. Asexample, the phonon intensity foru.0 at the x5(LGe2LSi)/2, (LGe2LSi)/2<y<(LGe1LSi)/2 interface can bewritten from the energy balance equation.

E0

2pE0

p/2

I S LGe2LSi

2,y,u,f D cosu•sinududf

5Rd21E0

2pEp/2

p

I S LGe2LSi

2,y,u,f D cosu•sinududf

1Td12E0

2pE0

p/2

I S LGe2LSi

2,y,u,f D cosu•sinududf.

~12!

Because the phonons are scattered diffusely at interfacesphonons leaving an interface are isotropically distributand Eq.~11! can be written as

I S LGe2LSi

2,y,u,f D5

Rd21

p E0

2pEp/2

p

I S LGe2LSi

2,y,u,f D

3cosu•sinududf

1Td12

p E0

2pE0

p/2

I S LGe2LSi

2,y,u,f D

3cosu•sinududf for u.0. ~13!

Phonons leaving foru,0 and other interfaces can be simlarly written.

C. Method of numerical solution

Equation~2! is similar to the photon radiative transpoequation~RTE!.42 The key is to solve for the intensity distrbution I (r ,V). Once the intensity is found by solving Eq~2!, the heat fluxqx(x,y), qy(x,y) and the effective temperature T(x,y) can be determined through the numerical ingration over the whole solid angle 4p. A variety of solutionmethod is available in the thermal radiative transliterature.40,41 The most often used methods are the discrordinates method~DOM!50 and the finite volume method.51

The discrete ordinate method is a tool to transform the eqtion of radiative transfer into a set of simultaneous pardifferential equations. This is based on a discrete represetion of directional variation of intensity. A solution to thtransport problem is found by solving the equation of rad

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tive transfer for a set of discrete directions spanning thetire solid angle. The integrals over the solid angle areproximated by numerical quadratures. For phonon transin nanostructures, the challenge is to reduce the ‘‘ray effewhich often happens similarly in thermal radiation in thoptical thin limit. In our previous work,46 double Gauss-Legender quadratures have been used to replace the contional SN quadratures for the discrete ordinate method ademonstrated to successfully resolve the ray effect probof phonon transport simulation in nanostructures. We briediscuss the calculation method here. The method separadiscretizes the integrating points in them5cosu ~the angleu! and the anglef using Gauss-Legender quadrature. To otain high accuracy,m is discretized into 120 points from21to 1 andf is discretized into 24 points for 0;p ~not 0;2p due to symmetry!. Then Eq.~3! can be written in dis-crete form as

I 0~x,y!52

4p (m

(n

I ~x,y,mn ,fm!wnwm8 . ~14!

The weights satisfy(m(nwnwm8 52p. In order to accuratelycapture the physics of the transport phenomena and mmize the calculation time, a nonuniform grid system is usThe step scheme is used for spatial discretization. Total stial grids of 1023102 are used in the calculation. Fewpoints in both spatial and angular coordinates can be usedfaster calculation. The equation is solved by iteration othe value of the equivalent equilibrium intensityI 0(x,y)52/4p(m(nI (x,y,mn ,fm)wnwm8 . When the relative errorof the calculated value of the equivalent equilibrium intesity between two iteration steps is less than 231026, theprogram is assumed to converge and the effective tempture and heat flux are calculated.

Although at nanoscale, temperature cannot be definedmeasure of equilibrium, we can use the effective temperato reflect the local energy density inside the medium.36,39

Assuming constant specific heat over a wide temperarange, we can write the effective temperature as

T~x,y!54pI 0~x,y!

Cuvu5

1

Cuvu (n(m

~x,y,mn ,fm!wnwm8 .

~15!

The heat fluxes at every point can be accordingly writas

qx~x,y!5(m

(n

I ~x,y,mn ,fm!mnwnwm8 , ~16!

qy~x,y!5(m

(n

I ~x,y,mn ,fm!A12mn2 cosfm•wnwm8 ,

~17!

where qx and qy are heat flux in thex and y directions,respectively. After local effective temperature distributioand heat flux are obtained, the thermal conductivity calcution is straightforward, taking advantage of the unit cell cocept. The surface heat flux in thex direction can be calcu-lated as

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Qx~x!5LZE0

LGeqx~x,y!dy, ~18!

where Lz is the unit length in thez direction. Equations~5! and ~6!, i.e., the specular reflected boundary conditioenforced at y50 and y5LGe boundaries have ensureQx(x)5const. The average energy density~average temperature! at eachy-z plane along thex direction can be writtenas

T̄~x!51

LGeE

0

LGeT~x,y!dy. ~19!

Therefore the effective thermal conductivityk of the com-posite can be obtained as

k5Qx

Lz@ T̄~LGe!2T̄~0!#~20!

from

Qx5k~LzLGe!@ T̄~LGe!2T̄~0!#

LGe. ~21!

The following dimensionless parameters have been induced to present temperature and heat flux results:

q* 5q

C1v1, Qx* 5

Qx

LGeLZC1v1, x* 5

x

LGe, y* 5

y

LGe.

~22!

III. RESULTS AND DISCUSSIONS

A. Nonequilibrium temperature and heat flux distribution

Figures 2~a! and 2~c! show the effective temperature ditribution in the composite structures with silicon wire dimesion of LSi5268 nm and LSi510 nm, respectively. Theatomic percentage is 20% for Si and 80% for Ge. Simcalculation gives the geometric ratioLGe/LSi52.35. Thechoosing ofLSi5268 nm is based on the fact that the MFvalue is around 268 nm calculated from Eq.~4! by usingthe parameters listed in Table I. Figure 2~a! is very close tothe temperature we expect in macroscale composwith interface thermal resistance. Therefore, for a wdimension larger than 268 nm, the effective temperaturetribution is expected to be similar to that plotted in Fig. 2~a!.The comparison of Figs. 2~a! and 2~c! shows that thetemperature or energy density distribution at nanoscaleperiodic 2D composites can be very different from thatmacroscale due to the ballistic nature of phonons. To beunderstand the effect of interface thermal resistance,plot the temperature distribution along thex direction at cer-tain fixedy positions of the two structures in Figs. 2~b! and2~d!. Apparent temperature jumps at the wire–host mateinterfaces are clearly shown in Figs. 2~b! and 2~d!. There arealso temperature jumps along they direction as indicated inFigs. 2~a! and 2~c!. The larger the wire dimension, the lowethe temperature jump relative to the total temperature difence across the interface, and thus the lower the contribu

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of the interface resistance to the effective thermal resistaof the composite. When the nanowire dimension is musmaller than the phonon MFP, sayLSi510 nm, the tempera-ture gradient along thex direction can be negative in somlocal regions. The heat fluxes in thex direction, however, arealways positive as shown in Figs. 3~a! and 3~b!. This phe-nomenon was not observed before in macroscale compoand cannot be predicted by Fourier heat conduction batheories. Another question that may raise is whether thesults shown in Figs. 2~c! and 2~d! violate the second law othermodynamic because the temperature inside the celarger than the cell boundary. To answer this question,should first remind, as pointed out before, that temperaturdefined is not the same as the case of thermal equilibriumlocal thermal equilibrium. When ballistic transport domnates, no local thermal equilibrium can be establishedtemperature calculated represents the local energy denFigures 4~a! and 4~b! illustrate the mechanisms of the obsevation in Fig. 2. WhenLSi is much smaller than the phonoMFP, the internal scattering in the medium~both host mate-rial and the wire! is negligible. We further assume that thphonon reflectivity is unity at the wire and host materinterface. Then the scenario can be simplified as thermadiation in vacuum with opaque wire inclusions~hostmaterial–vacuum, interface–solid wall, wire-opaque sobody!. Referring to Fig. 4~a!, we are interested in knowingthe temperature distribution of pointsA–F when the heat isenforced to flow in thex direction. We can qualitatively calthe left half of the region shown as the ‘‘hot’’ region and thright half as the ‘‘cold’’ region. As shown in the figure, poinD andF ‘‘see’’ the hot region and pointsA andC ‘‘see’’ thecold region, thus pointsD and F locally receive higher en-ergy phonons and have higher effective temperature tpointsA andC. Moving from pointA to F ~or from C to D!,more hot area is seen than cold area, thus the effectiveperature increases. Comparing toD and F, point E has alower temperature due to a small view factor from the hregion. Similarly pointB has a higher effective temperatuthan A and C. Figure 2 shows the temperature distributioonly in one unit cell. To visualize the temperature distribtion, one needs to stitch several periods of Fig. 2. Figure 4~b!shows the temperature distribution alongx at severaly pointsover three periods. The energy over those regions with ehigher temperature than the unit cell boundary comfrom the much higher temperature region in their previocells. The results do not violate the second law of thermdynamics.

B. Effect of wire dimension

To calculate the effective thermal conductivity of compoites, the surface heat flux and the average temperature~aver-age energy density! at eachy-zplane along thex direction iscalculated. As examples, Fig. 5 shows the dimensionlesserage energy density distribution along thex direction in aSi0.2– Ge0.8 composite with a silicon wire dimension ofLSi5268 and 10 nm, respectively. The dimensionless surfheat flux is conserved toQx* (x)50.088 for anLSi5268 nmcomposite andQx* (x)50.037 for anLSi510 nm composite.Again, the surface heat flux is conserved and the tempera

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FIG. 2. Effective temperature (T2Tref) distribution in the unit cell of Si0.2– Ge0.8 composites withT(0,y)2T(LGe,y)51 K applied fordifferent wire dimensions:~a! temperature contour forLSi5268 nm,~b! the temperature distribution alongx* at y* 50.5, 0.7, and 0.85 forLSi5268 nm, ~c! temperature contour forLSi510 nm, and~d! the temperature distribution alongx* at y* 50.5, 0.7, and 0.85 forLSi

510 nm. The temperature discontinuity at the interface is clearly shown. The temperature distribution inLSi510 nm nanocomposite is verdifferent from macroscale composites due to the ballistic phonon transport at nanoscale and cannot be captured by Fourier heattheory.

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jump appears at the interface. The smaller the wire dimsion, the larger the average temperature jump and thuslarger is the interface resistance contribution to the effecthermal resistance of the composite. We can expthat when the wire dimension is 2 to 3 times or even larthan the silicon MFP, the contribution of the interfathermal resistance will be negligible and the results wrecover the Fourier limit. Figure 6 shows the thermconductivity of Si0.2– Ge0.8 composites as a function of thsilicon wire dimension. To make a comparison, we incluthe results of cross-plane~perpendicular to the interfaces!thermal conductivity of a simple one-dimensional Si–Glayered structure~superlattices!. Simple calculation showsthat the thickness of the germanium layer should be 4times the thickness of the silicon layer in 1D stacks withgermanium atomic percentage of 80%. Figure 6 clea

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shows that at constant volumetric fraction~or atomicpercentage! the smaller the characteristic length of silico~the silicon wire dimension in composites and the thicknof the silicon layer in superlattices!, the smaller the thermaconductivity. The simple 1D layered structure has a lowthermal conductivity than periodic nanowire compositesthis atomic percentage. We point out that the thermconductivity of superlattices calculated here is lower thanexperimental data because the interface scattering in sulattices may not fall into a totally diffuse scattering limit.14,16

The comparison shown in this paper is just for theoreticonsistency.

C. Effect of atomic percentage

Some other questions of interests are:~1! can the thermalconductivity of nanowire composites be lower than that

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.


FIG. 3. The dimensionless heat flux distribution in thex direction qx* : ~a! LSi5268 nm composite, and~b! LSi510 nm composite. Itshows that thex-directional heat flux is always positive even in the localized negative temperature gradient region shown in Fig. 2

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simple 1D stacks~superlattice!? ~2! How the thermal con-ductivity changes with the atomic percentage? Figureshows the thermal conductivity of Si12xGex composites as afunction of atomic percentagex of germanium for wire di-mensionsLSi of 50 and 10 nm, respectively. By changing tatomic percentage, the geometric ratio of the unit cell, ithe dimension of germanium, is changed in the numersimulation. It shows that for a fixed silicon wire dimensiothe lower the atomic percentage of germanium; the lowethe thermal conductivity of the nanocomposites. This is vdifferent from macroscale composites and nanoparticle-fipolymers, in which thermal conductivity of the compositincreases with the decreasing volumetric fraction of the lthermal conductivity component. This is caused by the blistic transport of phonons in both the host material andnanowires, and the interface resistance between the hosterial and the nanowires. In polymer nanocomposites,thermal conductivity of the host polymer is usually very loand the thermal transport in polymers is diffusive. Thusthermal conductivity of polymer nanocomposites increawith the volumetric fraction of high thermal conductivitnanoparticle fillers. Figure 7 also shows that the thermal cductivity of the periodic 2D nanocomposites is lower ththat of the superlattice with corresponding characterilength when the atomic percentagex of germanium is lowerthan 35%. Simple calculation shows that the geometric rLGe/LSi is around 1.182 whenx50.35. For a simple 1Dlayered structure as shown in Fig. 8~a!, phonons experiencecross-plane interface scattering in all the cross-sectionalz-y when the heat is enforced to flow in thex direction.Comparing Figs. 8~a! and 8~b!, we know that phonons caflow through a fraction of (LGe2LSi)/LGe open area withoutexperiencing cross-plane interface scattering in nanocomites. However, phonons must experience an additional ftion of LSi /LGe interface scattering parallel to the heat flodirection ~in-plane scattering!. When the thermal conductivity of a simple 1D layered structure is the same as that ofperiodic 2D nanocomposites atx50.35, we can approxi-mately infer that a fraction ofLSi /LGe

19531

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eFIG. 4. ~a! Illustration to show the mechanisms of negative te

perature gradient in the localized regions using thermal radiaanalogy.~b! The temperature distribution alongx* at y* 50.5, 0.7,and 0.85 forLSi510 nm over three periods.

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in-plane interface scattering is equivalent to a fraction(LGe2LSi)/LGe cross-interface scattering. In other words, tefficiency of cross-interface scattering to reduce thermal cductivity is around five times as effective as the scatterparallel to the interface. This result is consistent wprevious experiments and modeling of the in-plane across-plane thermal conductivity of superlattices.14,16 It alsosuggests that the anisotropic nanocomposites mightmore effective in reducing thermal conductivity of nanocoposites.

IV. CONCLUSIONS

We studied theoretically the phonon thermal conductivof periodic two-dimensional nanocomposites with nanowi

FIG. 5. The dimensionless average temperature distribualong thex direction in Si0.2– Ge0.8 composite with silicon wiredimension ofLSi5268 nm andLSi510 nm, respectively.

FIG. 6. Thermal conductivity of Si0.2– Ge0.8 composites as afunction of the silicon wire dimension or layer thickness. Tsmaller the characteristic length of silicon~the silicon wire dimen-sion in composites and the thickness of the silicon layer in sulattices!, the smaller is the thermal conductivity.

19531

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embedded in a host semiconductor material using the phoBE. Special attention has been paid to cell–cell interactusing periodic boundary conditions. The simulatioshows that the temperature profiles in nanocompositesvery different from those in conventional composites dto ballistic phonon transport at nanoscale. Such temperaprofiles cannot be captured by existing models in literatuWe predict the thermal conductivity dependence oninterface conditions, the size of the nanowires, andvolumetric fraction of the constituent materials. The smalthe wire diameter, the smaller is the thermal conductivof periodic two-dimensional nanocomposites. The thermconductivity of 2D Si–Ge composites is predicted to befunction of the atomic percentage of germanium for wdimensions of 50 and 10 nm. It shows that for fixesilicon wire dimension, the lower the atomic percentagegermanium, the lower the thermal conductivity of th

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FIG. 7. Thermal conductivity of Si12x– Gex composites as afunction of atomic percentagex of germanium. For fixed siliconwire dimension, the lower the atomic percentage of germanium,lower is the thermal conductivity of the nanocomposites. The reis very different from the bulk material due to the ballistic naturephonon transport at nanoscale and interface effect.

FIG. 8. Illustration to show that phonons experience less crointerface scattering in periodic 2D composites than that in 1D lered structures but additional scattering parallel to the interface.efficiency of cross-interface scattering to reduce thermal conducity is around five times as effective as scattering parallel tointerface.

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nanocomposites. This is very different from bulk composibecause the interface dominates the ballistic transpornanocomposites. The periodic 2D nanocomposites can ha lower thermal conductivity than their 1D counterpart whthe Ge atomic percentage is lower than 35%. This suggthat the anisotropic nanocomposites might be more effecin reducing thermal conductivity of nanocomposites. Resof this study can be used to direct the development of bhigh efficiency thermoelectric materials and thermal int

*Electronic address: [email protected]. J. Golsmid,Thermoelectric Refrigeration~Plenum, New York,

1964!.2Special Issues on Thermoelectrics, edited by T. M. Tritt, Se

cond. Semimetals69–71 ~2001!.3M. S. Dresselhaus, Y. M. Lin, S. B. Cronin, O. Rabin, M. R

Black, G. Dresselhaus, and T. Koga, Semicond. Semimetals71,1 ~2001!.

4G. Chen, Semicond. Semimetals71, 2003~2001!.5R. Venkatasubramanian, Semicond. Semimetals71, 175 ~2001!.6T. C. Harman, P. J. Taylor, M. P. Walsh, and B. E. LaFor

Science297, 2229~2002!.7R. Venkatasubramanian, E. Silvona, T. Colpitts, and B. O’Qui

Nature~London! 413, 597 ~2001!.8G. Chen, D. Borca-Tasciuc, and R. G. Yang, inEncyclopedia of

Nanoscience and Nanotechnology, edited by H. S. Nalwa~American Scientific Publishers, Stevenson Ranch, CA, 20!,Vol. 7, N68.

9B. Yang and G. Chen, Phys. Rev. B67, 195311~2003!.10P. Hyldgaard and G. D. Mahan, Phys. Rev. B56, 10754~1997!.11S. Tamura, Y. Tanaka, and H. J. Maris, Phys. Rev. B60, 2627

~1999!.12W. E. Bies, R. J. Radtke, and H. Ehrenreich, J. Appl. Phys.88,

1498 ~2000!.13B. Yang and G. Chen, Microscale Thermophys. Eng.5, 107

~2001!.14G. Chen, ASME J. Heat Transfer119, 220 ~1997!.15G. Chen and M. Neagu, Appl. Phys. Lett.71, 2761~1997!.16G. Chen, Phys. Rev. B57, 14958~1998!.17P. Hyldgaard and G. D. Mahan, Therm. Conduct.23, 172 ~1996!.18M. V. Simkin and G. D. Mahan, Phys. Rev. Lett.84, 927 ~2000!.19B. Daly, H. Maris, K. Imamura, and S. Tamura, Phys. Rev. B66,

024301~2002!.20C. Dames and G. Chen, J. Appl. Phys.95, 682 ~2004!.21G. W. Milton, The Theory of Composites~Cambridge University

Press, New York, 2002!.22P. L. Kapitza, J. Phys.4, 181 ~1941!; for a review on thermal

boundary resistance, see Ref. 49.23D. P. H. Hasselman and L. F. Johnson, J. Compos. Mater.21, 508

~1987!.24Y. Benvensite, J. Appl. Phys.61, 2840~1987!; Y. Benvensite and

T. Miloh, ibid. 69, 1337~1991!.25A. G. Every, Y. Tzou, D. P. H. Hasselman, and R. Raj, Ac

Metall. Mater.40, 123 ~1992!.26C.-W. Nan, R. Birringer, D. R. Clarke, and H. Gleiter, J. App

Phys.81, 6692~1997!.27C.-W. Nan and F. S. Jin, Phys. Rev. B48, 8578 ~1993!; C.-W.

19531

sinve

tse

tsth-

face material with high thermal conductivity particle or wiinclusions.

ACKNOWLEDGMENTS

The authors would like to thank Professor M.S. Dresshaus, W. L. Liu, and C. Dames for comments and usediscussions. R.Y. would like to acknowledge the help froYong Li. The work was financially supported by DOE BE~DE-FG02-02ER45977! and NSF~CTS-0129088!.

i-

,

,

Nan, J. Appl. Phys.76, 1155~1994!.28P. A. Smith and S. Torquato, J. Appl. Phys.65, 893 ~1989!; S.

Torquato and M. D. Rintoul, Phys. Rev. Lett.75, 4067~1995!.29R. Lipton and B. Vernescu, J. Appl. Phys.79, 8964~1996!.30See, for example, M. Jiang, I. Jasiuk, and M. Ostoja-Starzew

Comput. Mater. Sci.25, 329 ~2002!; F. W. Jones and F. PascaGeophysics60, 1038~1995!.

31S. Graham and D. L. McDowell, ASME J. Heat Transfer125, 389~2003!.

32K. Ramani and A. Vaidyanathan, J. Compos. Mater.29, 1725~1995!.

33M. R. Islam and A. Pramila, J. Compos. Mater.33, 1699~1999!.34A. Khitun, A. Balandini, J. L. Liu, and K. L. Wang, J. Appl. Phys

88, 696 ~2000!; Superlattices Microstruct.30, 1 ~2001!; A. Khi-tun, K. L. Wang, and G. Chen, Nanotechnology11, 327 ~2000!;J. L. Liu et al., Phys. Rev. B67, 165333~2003!.

35O. L. Lazarenkova and A. A. Balandin, J. Appl. Phys.89, 5509~2001!; Phys. Rev. B66, 245319~2002!; A. A. Balandin and O.L. Lazarenkova, Appl. Phys. Lett.82, 415 ~2003!.

36J. M. Ziman,Electrons and Phonons~Oxford University Press,London, 1985!.

37R. Prasher, Appl. Phys. Lett.83, 48 ~2003!; ASME J. Heat Trans-fer 125, 1156~2003!.

38A. Majumdar, ASME J. Heat Transfer115, 7 ~1993!.39G. Chen and C. L. Tien, J. Thermophys. Heat Transfer7, 311

~1993!.40R. Siegel and J. R. Howell,Thermal Radiation Heat Transfer, 4th

ed. ~Taylor & Francis, Washington, D.C., 2001!.41M. F. Modest,Radiative Heat Transfer, 2nd ed.~Academic Press,

New York, 2003!.42A. A. Joshi and A. Majumdar, J. Appl. Phys.74, 31 ~1993!.43P. G. Klemens, Solid State Phys.7, 1 ~1958!.44K. E. Goodson and Y. S. Ju, Annu. Rev. Mater. Sci.29, 261

~1999!.45P. Markowich, C. Ringhofer, and C. Schmeiser,Semiconductor

Equations~Springer, Wien, 1990!.46R. G. Yang, G. Chen, M. Laroche, and Y. Taur, ASME J. He

Transfer~to be published!.47R. M. Costescu, M. A. Wall, and D. G. Cahill, Phys. Rev. B67,

054302~2003!.48D. G. Cahill, W. K. Ford, K. E. Goodson, G. D. Mahan, A. Ma

jumdar, H. J. Maris, R. Merlin, and S. R. Phillpot, J. Appl. Phy93, 793 ~2003!.

49E. T. Swartz and R. O. Pohl, Rev. Mod. Phys.61, 605 ~1989!.50W. A. Fiveland, ASME J. Heat Transfer109, 809 ~1987!; J. S.

Truelove,ibid. 109, 1048~1987!.51G. D. Raithby, Numer. Heat Transfer, Part B35, 389 ~1994!.

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