Keywords: Superlattice nanowire; Molecular dynamics simulation; Thermal conductivity; Interface scattering effect

ARTICLE IN PRESS*Corresponding author. Tel.: 86253792178; fax: 86253791414.

Email-address: yunfeichen@yahoo.com (Y. Chen).

0921-4526/$ - see front matter r 2004 Elsevier B.V. All rights reserved.doi:10.1016Molecular dynamics study of the lattice thermal conductivityof Kr/Ar superlattice nanowires

Yunfei Chena,*, Deyu Lib, Juekuan Yangc, Yonghua Wuc,Jennifer R. Lukesd, Arun Majumdare

aDepartment of Mechanical Engineering and China Education Council Key Laboratory of MEMS,

Southeast University, Nanjing, 210096, PR ChinabDepartment of Mechanical Engineering, University of California, Berkeley, CA 94720, USA

cDepartment of Mechanical Engineering and China Education Council Key Laboratory of MEMS,

Southeast University, Nanjing, 210096, PR ChinadDepartment of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104-6315, USA

eDepartment of Mechanical Engineering, University of California, Berkeley, CA 94720, USA

Received 4 December 2003; received in revised form 16 March 2004; accepted 17 March 2004

Abstract

The nonequilibrium molecular dynamics (NEMD) method has been used to calculate the lattice thermal

conductivities of Ar and Kr/Ar nanostructures in order to study the effects of interface scattering, boundary

scattering, and elastic strain on lattice thermal conductivity. Results show that interface scattering poses signicant

resistance to phonon transport in superlattices and superlattice nanowires. The thermal conductivity of the Kr/Ar

superlattice nanowire is only about 13of that for pure Ar nanowires with the same cross-sectional area and total length

due to the additional interfacial thermal resistance. It is found that nanowire boundary scattering provides signicant

resistance to phonon transport. As the cross-sectional area increases, the nanowire boundary scattering decreases,

which leads to increased nanowire thermal conductivity. The ratio of the interfacial thermal resistance to the total

effective thermal resistance increases from 30% for the superlattice nanowire to 42% for the superlattice lm. Period

length is another important factor affecting the effective thermal conductivity of the nanostructures. Increasing the

period length will lead to increased acoustic mismatch between the adjacent layers, and hence increased interfacial

thermal resistance. However, if the total length of the superlattice nanowire is xed, reducing the period length will lead

to decreased effective thermal conductivity due to the increased number of interfaces. Finally, it is found that the

interfacial thermal resistance decreases as the reference temperature increases, which might be due to the inelastic

interface scattering.

r 2004 Elsevier B.V. All rights reserved.

PACS: 63.20.e; 63.22.+m, 68.65.gPhysica B 349 (2004) 270280/j.physb.2004.03.247

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ca B1. Introduction

Semiconductor superlattices (SL) are of greatinterest due to their potential applications inthermoelectric and optoelectronic devices [1,2].Superlattice structures provide the possibility ofdecreasing materials thermal conductivity whileretaining their electrical conductivity, thus achiev-ing a high thermoelectric gure-of-merit andimproving the performance of thermoelectricdevices. A large amount of experimental andtheoretical work [38] has been carried out tostudy the effects of lattice period and interface onthe thermal conductivity of various kinds ofsuperlattice lms. Results showed that the super-lattice may have a much lower thermal conductiv-ity than the value for each of the two materialscomposing the superlattice structure along boththe in-plane and the cross-plane directions. Dif-ferent mechanisms have been proposed to explainthe reduction of the thermal conductivity, includ-ing interface phonon scattering due to acousticimpedance mismatch, phonon scattering by crystalimperfections at the interface, phonon spectrummismatch, and mini-band formation. However,quantitative analysis of the relative importance ofthese different mechanisms has not been com-pleted yet and it is not clear which mechanismcontributes most to the reduction of the thermalconductivity under different conditions. Experi-mental results [6,7] indicate that as the period ofthe superlattice decreases, the thermal conductivityalong the cross-plane direction also decreases, andfor very short period superlattice, its thermalconductivity could even fall below the alloy limit,i.e. if the two materials mixed homogeneously intoan alloy. The in-plane thermal conductivityfollows a similar pattern but it jumps to highervalues for some very short period superlattice [8].This phenomenon cannot be explained solely bydiffuse mismatch theory about phonon scatteringat the interface. Other mechanisms such asacoustic impedance mismatch, mini-band forma-tion or phonon spectrum mismatch have to betaken into account.Theoretically, lattice dynamics and particle

transport models are usually used to study the

Y. Chen et al. / Physithermal conductivity of superlattice structures.Some recent lattice dynamics work was carriedout by Hyldgaad and Mahan [9] and Tamura andTanaka [10]. In their work, a simple cubic latticemodel [9] and a face-centered cubic model [10]were used to calculate the group velocity ofacoustic phonons in the cross-plane direction.Their results suggested that reduction of thephonon group velocity in superlattices could leadto a reduced thermal conductivity. Consideringtheir lattice models were too simple to providedetails of phonon dispersion spectra, Kiselev et al.[11] used a diatomic unit cell model to simulate thedispersion spectra of the Si/Ge superlattice. Due tothe large mass of Ge atoms in comparison to Si,the most probable acoustic phonons in Si layers atroom temperature have no counterpart in thephonon spectra of the Ge layers. In other words, aphonon at a given frequency in the Si layer maynot be able to proceed into the Ge layer withoutscattering with, or into, one or more phonons ofdifferent frequency. This leads to highly efcienttrapping of high-energy phonons in the Si layerand a drastic reduction of the superlattice thermalconductivity. Although qualitative agreement canbe obtained through lattice dynamics, it is difcultto compare those results quantitatively withexperimental results for different materials underdifferent temperatures due to the fact that latticedynamics can only model very simple systems. Theparticle transport model treats phonons as indivi-dual particles and solves the Boltzmann transportequation (BTE) to study the phonon transport inmicrostructures. Simkin and Mahan [12] showedthat for layers thinner than the mean free path(mfp) of phonons, the wave aspect of phononsmust be taken into account and wave theory mustbe applied, while for layers thicker than the mfp ofphonons, the particle treatment of phonons wasacceptable and BTE could be used to study thephonon transport. The rst study on the thermalconductivity of superlattices was carried out byRen and Dow [13]. They modeled the thermalconductivity of ideal superlattice by combining theBTE with a quantum mechanical treatment of theadditional scattering process caused by the mini-bands. However, their predicted results could notmatch the experimental data. Based on the BTE

349 (2004) 270280 271and the assumption of partially specular and

layer thickness. This conclusion supported the

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ca Bpartially diffuse boundary scattering, some inter-esting results were obtained by Chen et al. [1417]for superlattice thermal conductivity along bothin-plane and cross-plane directions. In his modelthe reduction of the superlattice thermal conduc-tivity was mainly attributed to diffuse scattering atthe interface.Because some assumptions must be introduced

to get closed form solutions, the theoretical resultsusually deviate signicantly from the experimentaldata and sometimes the thermal conductivityreduction mechanisms cannot be readily explainedfrom those theoretical studies. Classical MolecularDynamics (MD) simulation provides an alterna-tive approach to investigate heat transport innanostructures. Given the interaction potentialbetween atoms, the force acting on each atom canbe calculated. Based on Newtons second law, themotion of a large number of atoms can bedescribed. Without any further assumptions,statistical physical properties can be derived fromthe ensemble of atoms. In particular, if the size ofthe nanostructure is smaller than the phonon mfp,it is questionable to use the BTE to describephonon transport, while MD can be convenientlyused to analyze the effects of size connement onlattice thermal conductivities. Volz et al. [18]demonstrated by MD simulation that Si nanowirethermal conductivity could be two orders ofmagnitude smaller than the corresponding bulkvalue and they further argued that by adjusting thespecularity parameter, results from the solution ofthe BTE could t their MD simulation. Liang et al.[19] applied MD simulation to investigate theeffects of atomic mass ratio in the alternatinglayers of a superlattice on the lattice thermalconductivity. Their results indicated that thethermal conductivity has a minimum value forsome specic atomic mass ratio. Abramson et al.[20] studied the effects of interface number andelastic strain on the lattice thermal conductivity ofKr/Ar superlattices with MD simulation. It wasargued that increase of the interface number perunit length does not necessarily result in decreasedlattice thermal conductivity from their simulationresults. Daly et al. [21,22] reported MD simulationof a classical face centered cubic (FCC) lattice

Y. Chen et al. / Physi272model to study the effects of interface roughnesshypothesis that zone folding was the dominanteffect on lattice thermal conductivity in theshort period superlattice. However, it should benoted that no lattice mismatch between thealternative materials of the superlattice wasconsidered in their model. Volz [24] introducedthe conjugate gradient method to minimize thepotential energy of Si/Ge superlattices in order torelax the elastic strain on the alternating layers.Simulation results predicted an increasing trend ofthe superlattice thermal conductivity with the layerthickness.Recently, experimental investigation on the

thermal conductivity of Si and Si/SiGe superlatticenanowires has been carried out [25,26]. Theirresults showed that for Si nanowires, the thermalconductivity could be greatly reduced comparedwith that of bulk Si because of the strong nanowireboundary scattering. For Si/SiGe nanowires, thethermal conductivity was below that of the two-dimensional Si/SiGe superlattice lms, which wasascribed to the additional scattering mechanismprovided by the nanowire boundary. In this paper,we apply molecular dynamics to study the latticethermal conductivity of Kr/Ar superlattice nano-wires in order to investigate the effects of interfacescattering, nanowire boundary scattering, andperiod length.

2. Theoretical model and analysis

Nonequilibrium molecular dynamics (NEMD)was used to calculate the lattice thermal conduc-tivities of solid Ar and Kr/Ar superlattice nano-wires in the present study. The Lennard-Jonesand isotope scattering on thermal conductivities.Simulation results predicted the similar trends forthe lattice thermal conductivities of GaAs/AlAssuperlattice along both in-plane and cross-planedirections compared with the experimental data[23]. In their model, it was also demonstratedthat the lattice thermal conductivity of GaAs/AlAssuperlattice had a minimum value with different

349 (2004) 270280(L-J) potential was used to represent the interac-

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ca BTable 1

Simulation parameters

Variables Parameters

KrKr e1 (J) 2:25 1021

s1 (m) 3:65 1010

Lattice constant A1 (m) 5:69 1010

Atom mass M1 (Kg) 1:39 1025

ArAr e2 (J) 1:67 1021

s2 (m) 3:4 1010

Lattice constant A2 (m) 5:3 1010

Atom mass M2 (Kg) 6:63 1026

KrAr e12 (J)e1e2

ps12 (m) s1 s2=2

Y. Chen et al. / Physition between two particles

Vijr 4eijsijr

12sijr

6 ; 1

where subscripts i and j stand for either argon orkrypton particles, eij and sij represent the energyand length scale of the potential, and r denotes thedistance between the two particles. Table 1 givesthe selected parameters and the simulation domainfor the Kr/Ar superlattice nanowire is shown inFig. 1.The simulation domain consisted of a hot bath

with a high-constant temperature TH; a cold bathwith a low-constant temperature TL and alternat-ing Kr/Ar layers. Constant heat ux was added tothe hot bath and the same amount of heat ux wassubtracted from the cold bath at the same time. In

HT

Kr Ar Kr ArL

T

x

y

z

o L

Lp

Fig. 1. Physical model of Kr/Ar superlattice nanowire.this way, heat ux was introduced into thesimulation domain and a temperature gradientwas set up along the heat ux direction. From theheat ux J and the temperature gradient, thethermal conductivity of the nanowires can becalculated from the Fourier law

K J

ArT; 2

where A is the cross-sectional area and rT is thetemperature gradient. The temperature gradientcan be obtained according to the equation

rT TH TLLp N

; 3

where Lp and N are the length and the number ofperiods for the superlattice nanowire shown in Fig.1. TH and TL temperatures of the bath atoms incontact with either side of the nanowire. Phononscattering at the interfaces between the hot andcold baths and the superlattice structure leads totemperature jumps between the thermal bathlayers nearest the nanowire and the rst adjacentlayer of the nanowire [27,28], which results in asmaller temperature gradient within the wire thanthat imposed externally. In our simulation, Kratoms were selected to form the hot bath and Aratoms were selected as the cold bath material inorder to minimize the interfacial thermal resistanceon either side. Fixed boundary conditions on theouter surface of the nanowire were introduced toavoid evaporation from the nanowire domainduring the simulation process. In the xedboundary regions, the atoms were xed in theirequilibrium positions. They interacted with theatoms in the nanowire through the L-J potential.The parameters are listed in Table 1. In Fig. 1 thexed boundary regions in the x direction aredepicted by the dashed lines. The thickness of thexed boundary region was set as 1 unit cell (UC).It has been shown [18] that for Si nanowires, thedifference between xed boundary condition andfree boundary condition was less than 10%.According to Lukes et al. [27], two importantcriteria for the simulation are that the effectivethermal ux in each atomic layer is nearly the sameas that imposed externally, and that the tempera-

349 (2004) 270280 273ture proles are reasonably close to linear. The

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ca Bproles should not be expected to be completelylinear, since the thermal conductivity is tempera-ture dependent. In order to satisfy the two criteria,computation time should be selected as long aspossible for the system to reach steady-state. Inthis paper, to avoid undue computational burden,values roughly 20 times of the characteristicdiffusion time were found sufcient to satisfy thecriteria. For the thermal conductivity simulationsreported here, the values of the effective thermalux were all within 4% of the imposed ux and thetemperature proles were reasonably linear.Local temperature calculation is another issue

to be concerned in MD simulation, which relatesto the size of the region over which temperature isdened. To dene a local temperature, localthermodynamic equilibrium must be set up, whichmeans that the temperature must be dened over aregion with characteristic length larger than thephonon mfp. However, it has been shown that forclassical MD simulation, local temperature can bedened for each layer of atoms as long as there areenough atoms in each layer [30]. In this paper, Tpin the prole was evaluated from the equation

Tp maXNpi1

v2i =3NpkB; 4

where ma; Np are the mass and the number ofatoms in a local plane, vi the velocity of each atom,and kB the Boltzmann constant. The criteria forEq. (4) to be valid are that local thermodynamicequilibrium is established in each layer, and thesimulation temperature is above the Debye tem-perature of the material so that the heat capacity istemperature independent. According to Hafskjold[30], at least 30 atoms are needed in each layer inorder to reach local equilibrium and to dene aninstantaneous temperature. So for a superlatticenanowire with a square cross-section, the width ofthe cross-section should be up to at least 4UC. Onthe other hand, Lukes et al. [27] demonstrated thatprolonging the simulation time was more efcientfor a system to reach equilibrium compared withincreasing the number of atoms in the simulationsystem. So in our simulation, the lower limit of thecross-sectional area of the superlattice nanowires

Y. Chen et al. / Physi274was selected as 4UC 4UC: As the simulationtemperature in this paper was below the Debyetemperatures of Kr and Ar (72, 92 K; respec-tively), quantum modication should be intro-duced to correct both the MD temperature and thethermal conductivity. However, it has been sug-gested [31] that the classical MD model could giveacceptable prediction for lattice thermal conduc-tivity as long as the temperature is higher thanone-fourth of the Debye temperature. Since thereference temperature in our simulation system isabove one-fourth of the Debye temperature forboth Kr and Ar, for simplicity, we still used Eq. (4)to estimate the local temperature.Temperature discontinuity appeared in the

temperature prole at interfaces between Kr andAr layers along the superlattice nanowires, whichwas caused by the interfacial thermal resistance.The interfacial thermal resistance can be calculatedfrom the following equation [32]:

Rint DTJ=A

; 5

where A is the cross-sectional area, DT thetemperature discontinuity at the interface and Jthe heat ux. The interfacial thermal resistancedepends on the number of the phonons incident onthe interface, the energy carried by each phononand the probability that each phonon is trans-mitted across the interface. Two limiting modelshave been proposed to describe phonon transportacross an interface: the acoustic mismatch model,which assumes specular reection of phonons atthe interface, and the diffuse mismatch model,which assumes that all phonons incident on theinterface will be diffusively scattered. However, itis difcult to argue if the interfacial thermalresistance can be calculated from these two modelsbecause of the strong assumptions made in thesetwo theories. On the other hand, in MD simula-tion, the interfacial thermal resistance can bereadily calculated from Eq. (5) according to theclassical denition of interfacial thermal resis-tance. The total thermal resistance along thenanowire can be described as

Rtot nRint NRA RB 6

where n is the number of the interface, N the

349 (2004) 270280number of periods and n 2N 1: RA; RB are

As we know, under atmospheric pressure, for an

sink. If the length of the simulation domain isshorter than the phonon mfp, then the nanowirethermal conductivity should be length dependent,i.e. the thermal conductivity increases with thelength of the simulation domain. The phonon mfpfor Ar [31,33] is estimated to be approximately5 nm (1016UC for Ar and Kr) at 30 K: However,low frequency phonons have longer mfps whilehigh frequency phonons have shorter mfps, whichmeans that even for a system of a length longerthan the phonon mfp, the effects of phononscattering at the heat source and heat sink maystill have some effect on the total thermalconductivity. In our calculation, the thermalconductivity of Ar nanowire increases with thenanowire length until the length exceeds 40UC,which we think comes from the scattering of long

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ca Binterface between two bulk materials, if there is nointerface imperfection, the interfacial thermalresistance only depends on temperature and thematerials that constitute the interface. Once thematerials and the temperature are determined, theinterfacial thermal resistance should be a constant.Thus, from Eq. (7), it can be concluded that theeffective thermal conductivity of superlattice na-nowire will depend on the length and number ofperiods. It predicts that the thermal conductivityof the superlattice nanowire increases with theperiod length. However, for superlattice nano-wires, due to the fact that the period length may beless than the phonon mean free path, the thermalresistance in the alternating layers and the inter-facial thermal resistance are all layer thicknessdependent, so the relation between the effectivethermal conductivity and the period length mightbe nonlinear.

3. Simulation results and discussion

3.1. Effect of the number of interfaces

NEMD was used to predict lattice thermalconductivities of pure Ar and Kr/Ar superlatticenanowires. Fig. 2 shows the thermal conductivitiesof Ar and Kr/Ar superlattice nanowires versusnanowire length for a xed period length of 8UC.For the Kr/Ar superlattice nanowire, one period iscomposed of two layers and the thickness of eachlayer is 4UC. The total length of the Ar nanowirethe thermal resistance of each Ar and Kr layer,respectively. Eq. (6) is only an approximateexpression for the total thermal resistance. In fact,due to the difference of the reference temperaturein different layers along the superlattice nanowire,the thermal resistances RA; RB should not beexactly the same for different sections. Assumingthat RA; RB are the same along the wholenanowire, the effective thermal conductivity ofthe superlattice nanowire can be approximated as

Keff Lp

RA RB 2Rint Rint=N: 7

Y. Chen et al. / Physiwas chosen to be the same as that of the Kr/Arsuperlattice nanowire. Also, the two kinds ofnanowires are of the same cross-sectional area as4UC 4UC: The temperatures in the hot bath andthe cold bath are 50 and 10 K; respectively, andthe reference temperature is 30 K: Fig. 2 showsthat the thermal conductivity of the Kr/Arnanowire is only about one-third of that of theAr nanowire, which we believe is due to theadditional interface scattering in the Kr/Ar super-lattice nanowire. For Ar nanowire, its thermalconductivity increases as the length of the nano-wire increases up to 40UC. We believe that this isdue to the scattering from the heat source and heat

2 4 860.15

0.20

0.25

0.30

0.35

0.40

0.45

Ther

mal

con

duct

ivity

(W/K

.m)

Period number

Ar Nanowire Ar/Kr superlattice nanowire

Fig. 2. Thermal conductivities of Ar and Kr/Ar superlattice

nanowires.

349 (2004) 270280 275wavelength phonons at the heat source and heat

Kr atoms) or expand (for Ar atoms) to align withadjacent atoms because of the lattice mismatchbetween these two materials. For short periodsuperlattice nanowires, the elastic strain causes thelattice in the adjacent layers to match better witheach other, which poses less resistance to phonontransport. The difference of elastic strain fordifferent period length is shown in Fig. 5. Thecross-sectional areas for the two nanowires are5:19 nm2 and their period lengths are 4.4 and13:2 nm; respectively. Two and three periods weresimulated for the 13:2 nm and 4:4 nm periodlength nanowires, respectively, so the nanowiresare labeled as 2 13:2 nm 5:19 nm2; 34:4 nm 5:19 nm2: The two dashed lines labeled

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6 9 12 15 18 21 240.10

0.12

0.14

0.16

0.18

0.20

0.22

0.15

0.18

0.21

0.24

0.27

0.30

0.33R

int (K

.m2 /W

)

Period length (UC)

Rint

R

int/R

Rint/R

Fig. 4. Interfacial thermal resistance of Kr/Ar superlattice

nanowires.

ca Bsink. Typically for simulations on bulk materialsone should not see a size dependence and for thosecases size effects would be an unphysical artifact.However for materials with characteristic dimen-sions on or below the order of a phonon mfp, therewill be a size dependence and it is a true, physicaleffect [27,29]. Thermal conductivity in that case isnot intrinsic but is dominated by boundary effects.Nonetheless, this size dependent phenomenoncannot be observed for Kr/Ar superlattice nano-wire, as shown in Fig. 2. With the increase ofnanowire length, the thermal conductivities of Kr/Ar superlattice nanowires remain almost constantand a very slight decreasing trend appears whenthe period number exceeds 5. The reason might bethat the effects of phonon scattering at theboundaries of the hot and cold baths are maskedby the interface scattering. The effects of phononscattering on the heat source and heat sinkboundaries play the same role as the interfacescattering so for superlattice nanowires, the dataof the thermal conductivities for different length ofthe nanowire superlattice seem to be consistentwith formula (7), i.e. for a xed period lengthsuperlattice nanowire, its thermal conductivitykeeps constant.

3.2. Effect of period length on the thermal

conductivity and the interfacial thermal resistance

In order to study the effects of period length,Kr/Ar superlattice and superlattice nanowires ofdifferent period length were investigated with axed total length of 48UC and a cross-sectionalarea of 4UC 4UC: Fig. 3 gives thermal con-ductivity versus period length, which reveals theeffects of the interface and nanowire boundary onthe lattice thermal conductivity. For nanowires ofxed total length, the lattice thermal conductivityincreases with the period length because fewerinterfaces are present in the longer period super-lattice nanowire. In addition, the interfacialthermal resistance is not a constant as the periodlength changes. It increases as the period lengthincreases, as shown in Fig. 4. This phenomenon isattributed to the interface strain effects on phonontransport [20]. The atomic spacing on either side of

Y. Chen et al. / Physi276the interface tends to either slightly contract (for6 9 12 15 18 21 24

0.15

0.18

0.21

0.24

0.27

0.30

0.33

K (W

/K.m

)

Period length (UC)

superlattice SL nanowire

Fig. 3. Thermal conductivities of Kr/Ar superlattice and

superlattice nanowires.

349 (2004) 270280Kr and Ar stand for the distance between two

nanowire, the nanowire boundary scattering posesignicant resistance to the phonon transport.

3.3. Effects of cross-sectional area on the thermal

conductivity and the interfacial thermal resistance

At low temperature, thermal conductivity in-creases with the sample size due to the fact thatphonon-boundary scattering dominates the ther-mal resistance. For larger cross section, theprobability of phonon-boundary scattering re-duces, which leads to higher lattice thermalconductivity. To explore the effects of differentcross-sectional area on the thermal conductivity ofsuperlattice nanowires, we simulated thermal

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ca B 349 (2004) 270280 277neighboring atomic planes along the x-axis forpure Kr and pure Ar, respectively. For the longerperiod nanowire, the atomic planes far away fromthe interface tend to maintain the equilibriumspacing found in the pure material. For the shorterperiod nanowire, however, most of the atoms inthe two layers deviate from their equilibriumspacing, as shown in Fig. 5. We believe that asthe period length increases, the acoustic impedancemismatch between the two different material layersincreases, which causes the larger interfacialthermal resistance as shown in Fig. 4. However,because the total length of the nanowire is xed,the ratio of the total interface resistance to thetotal effective thermal resistance decreases for

0 20 40 60 80 1000.72

0.74

0.76

0.78

0.80

0.82

Ar

Kr

Atomic plane number along X Axis

Dim

ensio

nles

s dist

ance

N=3, Lp=4.4nm,A=5.19nm2

N=2,Lp=13.2nm,A=5.19nm2

Fig. 5. Elastic strain in Kr/Ar superlattice nanowires.

Y. Chen et al. / Physiincreasing period length. This is due to the factthat the number of interfaces is reduced for longerperiod superlattice nanowires if the total length ofthe nanowire is xed.Fig. 3 also gives the thermal conductivity of a

Kr/Ar superlattice lm in order to compare theeffects of cross-sectional area on the thermaltransport. The period length for the superlatticeis the same as that of Kr/Ar superlattice nano-wires. Periodic boundary conditions were imposedin y and z directions to stand for innite cross-sectional area. Simulation results show that thethermal conductivity of superlattice lms is muchhigher than that of superlattice nanowires for thesame period length, which is consistent as theexperimental observations [26]. This means that inaddition to the interface scattering in superlatticetransport in nanowires of different cross-sectionalarea as 4UC 4UC; 5UC 5UC; 6UC 6UCand 7UC 7UC: To save computational time, thetotal length of the nanowire was chosen as 24UC,the period length as 8UC, and the thickness ofeach layer as 4UC. The simulation results areshown in Fig. 6. It can be seen that the thermalconductivity increases nearly linearly with the sizeof the cross section. If we assume constant specicheat for all four nanowires, kinetic theory predictsthat the thermal conductivity is proportional tothe phonon mfp. The equivalent phonon mfp canbe written as

1

l

1

lphonon

1

linterface

1

lboundary; 8

4 6 8 10 12 14 16

0.15

0.18

0.21

0.24

0.27

0.30

K(W

/K.m

)

Section area (nm2)

SL nanowireSL

Fig. 6. Thermal conductivity of Kr/Ar superlattice and super-lattice nanowires.

where the subscripts phonon, interface andboundary stand for the effective phonon mfpdetermined by the three phonon scattering process,the interface scattering, and nanowire boundaryscattering, which is related to the cross-sectionsize. If we do not take into account the modica-tion of the phonon dispersion relation due to thedifferent diameters of the nanowires, the phononmfp due to anharmonic scattering can be approxi-mated as constant. From Eq. (8), it can be deducedthat the effective thermal resistance of the nano-wire decreases for increasing cross section, which is

dispersion relations among the nanowires isneglected [34,35]. The reduced inuence of bound-ary scattering processes for the thicker nanowiresis the main reason that the value of thermalconductivity increases with the cross sectionalarea.

3.4. The dependence of interfacial thermal

resistance on temperature

The total thermal resistance of a superlatticenanowire equals the sum of the thermal resistancein each layer and the interfacial thermal resistance.

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Y. Chen et al. / Physica B 349 (2004) 270280278conrmed by the results shown in Fig. 6. Fig. 7shows that for superlattice nanowires, the ratio ofthe interfacial thermal resistance to the totalthermal resistance increases as the cross-sectionalarea increases. If the cross-sectional area increasedto innity, i.e. the superlattice nanowire becomesthe superlattice lm, the ratio increased to 42%.For Kr/Ar superlattice lms shown in Figs. 6

and 7, the period length and period number are thesame as that of the superlattice nanowires. Thedifference between the superlattice lms of differ-ent simulation domain is only marginal due to theperiodic boundary conditions for superlatticelms. The value of the thermal conductivity forsuperlattice lms is the upper limit of the thermalconductivity for superlattice nanowires. This isalso true for the ratio of the interface resistance tothe total thermal resistance, as shown in Fig. 7.Considering the minor difference among thenanowires diameter for the four types of nano-wires, any resulting differences in dispersionrelation will be slight so the change of the phonon

864 10 12 14 160.31

0.32

0.33

0.34

0.35

0.38

0.39

0.40

0.41

0.42

Rin

t/R fo

r SL

nano

wire

SLSL nanowire

R

int/R

for S

L

Fig. 7. Interfacial thermal resistance of Kr/Ar superlattice andsuperlattice nanowires.From both the acoustic mismatch model and thediffuse mismatch model, at very low temperature,the interfacial thermal resistance decreases withtemperature at a rate proportional to T3; due tothe temperature dependence of specic heat. Asthe temperature rises, the specic heat approachesa constant value, so the interfacial thermalresistance also approaches a constant value. Fig.8 plots the interfacial thermal resistance and thethermal conductivity of superlattice nanowiresversus temperature, which shows that as thetemperature increases, both the interfacial thermalresistance and the nanowire thermal conductivitydecrease in the temperature range of 3555 K: Thereason for the thermal conductivity reduction canbe ascribed to the increased thermal resistance ofeach layer due to phononphonon scattering.However, the reason for the interfacial thermalresistance decrease is not fully understood. From

35 40 45 50 55

0.19

0.20

0.21

0.22

0.23

0.16

0.18

0.20

0.22

Temperature T(K)

KR

K (W

/K. m

)

R (K

m2 /W

)

Fig. 8. Thermal conductivity and interfacial thermal resistanceof Kr/Ar superlattice nanowire.

4. Conclusion

42% for Kr/Ar superlattice lms. It is also found

support of the Chinese Natural Science Founda-

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ca BIn summary, the lattice thermal conductivity ofAr nanowires and Kr/Ar nanowires were studiedby NEMD. The interfacial thermal resistance inthe Kr/Ar nanowires contributes signicantly tothe thermal resistance and results in a latticethermal conductivity that is only one-third of thatof pure Ar nanowires. Interfacial thermal resis-tance increases with the length of the superlatticeperiod for the Kr/Ar superlattice nanowires, whichis attributed to increased acoustic impedancemismatch between adjacent layers for long periodnanowires. For short period nanowires, theacoustic impedance in the alternating layers besidethe interface matches better due to elastic strainthroughout each layer. The phonons can passthrough the interface more easily, which results insmaller interfacial thermal resistance. On the otherhand, for longer period superlattice nanowires, theatoms located far away from the interface main-tain their pure crystalline lattice positions so thatthe strain is localized near the interface. It is morethe acoustic mismatch model, the thermal bound-ary resistance can be approximated as 1

4Cvsa

1;where, C; vs; and a are the specic heat, the Debyephonon group velocity, and the averaged phonontransmission probability. The Debye phonongroup velocity can be regarded as a constant.The phonon transmission probability is typicallyassumed in the literature to be independent oftemperature [32]. The above equation predicts thatthe thermal boundary resistance is independent oftemperature. The decrease of the interfacialthermal resistance cannot be derived from thediffuse mismatch model either. One possiblereason might be that inelastic scattering occurs atthe interface [32]. At higher temperature, the high-frequency phonons might break into two or morelow-frequency phonons, which will be predomi-nantly transmitted. This inelastic scattering pro-cess leads to higher phonon transmissionprobabilities, thus the interfacial thermal resis-tance decreases.

Y. Chen et al. / Physidifcult for phonons to pass through the interfacetion (Project No. 50276011, 50275026) and the 863High Technology Program (Project No.2003AA404160).

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Acknowledgements

Y.C. would like to acknowledge the nancialdue to the larger acoustic impedance mismatchbetween the two layers.If the total length of the superlattice nanowire is

xed, increasing the interface number per unitlength will lead to decreased interfacial thermalresistance. However, the effective thermal con-ductivity for the superlattice nanowire decreasesdue to the fact that the number of interfacesincreases. Nanowire boundary scattering alsocontributes to the total thermal resistance. In-creasing the cross-sectional area of superlatticenanowires leads to higher thermal conductivity.The ratio of the interfacial thermal resistanceincreases with the cross-sectional area, and as anupper limit, the ratio of the interfacial thermalresistance to the total thermal resistance reaches

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ARTICLE IN PRESS

Y. Chen et al. / Physica B 349 (2004) 270280280

Molecular dynamics study of the lattice thermal conductivity of Kr/Ar superlattice nanowiresIntroductionTheoretical model and analysisSimulation results and discussionEffect of the number of interfacesEffect of period length on the thermal conductivity and the interfacial thermal resistanceEffects of cross-sectional area on the thermal conductivity and the interfacial thermal resistanceThe dependence of interfacial thermal resistance on temperature

ConclusionAcknowledgementsReferences