japplphys_thermal conductivity of silicon bulk and nanowires effects of isotopic

Thermal conductivity of silicon bulk and nanowires: Effects of isotopiccomposition, phonon confinement, and surface roughness

M. Kazan,1,a� G. Guisbiers,2 S. Pereira,2 M. R. Correia,3 P. Masri,4 A. Bruyant,1

S. Volz,5 and P. Royer1

1Laboratoire de Nanotechnologie et d’Instrumentation Optique, ICD, CNRS (FRE2848), Université deTechnologie de Troyes, 10010 Troyes, France2Department of Physics, CICECO, University of Aveiro, Aveiro 3810-193, Portugal3Department of Physics, I3N, University of Aveiro, Aveiro 3810-193, Portugal4Groupe d’Etude des Semiconducteurs, CNRS-UMR 5650, University of Montpellier II, Montpellier 34095,France5Laboratoire d’Energie Moléculaire et Macroscopique, Combustion CNRS UPR 288, Ecole Centrale Paris,Grande Voie des Vignes, F-92295 Châtenay-Malabry Cedex, France

�Received 24 March 2009; accepted 30 January 2010; published online 21 April 2010�

We present a rigorous analysis of the thermal conductivity of bulk silicon �Si� and Si nanowires �SiNWs� which takes into account the exact physical nature of the various acoustic and optical phononmechanisms. Following the Callaway solution for the Boltzmann equation, where resistive andnonresistive phonon mechanisms are discriminated, we derived formalism for the lattice thermalconductivity that takes into account the phonon incidence angles. The phonon scattering processesare represented by frequency-dependent relaxation time. In addition to the commonly consideredacoustic three-phonon processes, a detailed analysis of the role of the optical phonon decay intoacoustic phonons is performed. This optical phonon decay mechanism is considered to act asacoustic phonon generation rate partially counteracting the acoustic phonon scattering rates. Wehave derived the analytical expression describing this physical mechanism which should be includedin the general formalism as a correction to the resistive phonon-point-defects and phonon-boundaryscattering expressions. The phonon-boundary scattering mechanism is taken as a function of thephonon frequency, incidence angles, and surface roughness. The importance of all the mechanismswe have involved in the model is demonstrated clearly with reference to reported data regarding theisotopic composition effect in bulk Si and Si NW samples. Namely, our model accounts forpreviously unexplained experimental results regarding �i� the isotope composition effect on thethermal conductivity of bulk silicon reported by Ruf et al. �Solid State Commun. 115, 243 �2000��,�ii� the size effect on ��T� of individual Si NWs reported by Li et al. �Appl. Phys. Lett. 83, 2934�2003��, and �iii� the dramatic decrease in the thermal conductivity for rough Si NWs reported byHochbaum et al. �Nature �London� 451, 163 �2008��. © 2010 American Institute of Physics.�doi:10.1063/1.3340973�

I. INTRODUCTION

Experimental results show that the thermal conductivity�� of bulk semiconductor materials has a general qualitativebehavior as a function of the temperature �T�.1–9 At very lowT �few Kelvins�, � depends on the size of the crystal andincreases with T mirroring the temperature dependence ofthe lattice specific heat ��T3�, to reach a maximum at aroundTmax=0.05�D, where �D is the acoustic Debye temperature.Near this maximum, � is T independent but sensitive to thecrystal imperfections and isotopic composition. At T slightlyabove Tmax, ��T� decreases with T due to phonon scatteringvia normal �N� processes. At T approaching 0.1�D, ��T� de-creases with T due to Umklapp �U� processes.

For the case of semiconductor nanowires �NWs�, preciseexperiments have shown that ��T� has a different behaviorfrom what has been previously described. As such, Li et al.10

measured ��T� of silicon NWs �Si NWs� with diameters of22, 37, 56, and 115 nm. They reported on ��T� values two

orders of magnitude lower than bulk Si, with a clear devia-tion from the Debye T3 law at low T. The authors suggestedthat these experimental observations could be due to an in-creased phonon-boundary scattering and possible phononspectrum modification due to spatial phonon confinement.More recently, Boukai et al.11 demonstrated that by varyingthe Si NW size and doping levels, Seebeck coefficient12 val-ues representing an approximately 100-folded improvementover bulk Si can be achieved over a wide T range. Moreover,Hochbaum et al.13 reported on rough Si NW having Seebeckcoefficient and electrical resistivity values that are the sameas doped bulk Si but exhibit 100-folded reduction in ��T�. Sofar, the remarkable differences regarding the experimentallyobserved ��T� behavior in bulk Si and NWs with differentsizes and surface conditions could not be explained by cur-rent theories.

It is highly desirable to have a theoretical approachwhich can describe the physical phenomena responsible forthe experimental observations in the full T range. With thistheoretical tool in hand, one can gain insight into the effectsa�Electronic mail: [email protected].

JOURNAL OF APPLIED PHYSICS 107, 083503 �2010�

0021-8979/2010/107�8�/083503/14/$30.00 © 2010 American Institute of Physics107, 083503-1

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http://dx.doi.org/10.1063/1.3340973

http://dx.doi.org/10.1063/1.3340973

http://dx.doi.org/10.1063/1.3340973

of the size and temperature on the harmonic and anharmonicterms of the interatomic forces and rationally tailor materialsfor efficient design of fast heat removals, or alternatively,thermoelectric devices. The usual approach to calculate ��T�in semiconductor materials is to consider a Boltzmann equa-tion with a relaxation time approximation. Then, the scatter-ing cross section can be calculated by perturbation theories.14

In such a treatment, the temperature and frequency depen-dences of anharmonic three-phonon processes are stronglyaffected by the details of the phonon branch and anharmoniccoefficients, which makes the expressions derived valid foronly specific phonons and limited T range. In order to obtainthe expressions for the three-phonon processes in a wide Trange, an ab initio approach involving the combination ofBoltzmann formalism with density functional calculation ofthe harmonic and anharmonic interatomic constants was pro-posed recently.15 This calculation method showed the possi-bility to obtain good agreement with the experimental dataon ��T� of bulk Si and germanium �Ge� without the use ofany fitting parameters. However, it is limited to the case ofbulk materials because for the case of nanostructures suchcalculation becomes computationally prohibitive. On theother hand, the formalism for ��T� established by Callaway16

for low T and modified later by several authors17–19 to fit��T� in a wide T range has been widely used to describeseveral effects on ��T�. Although the modified-Callaway for-malisms allow including, in addition to the phonon normalprocesses, various resistive processes �such as the scatteringby lattice imperfections and sample boundaries� their formsestablished for the case of bulk semiconductors fail to de-scribe the isotopic composition effect on ��T� of bulk Si andto provide a reasonable representation of even the qualitativebehavior of the experimentally measured Si NW ��T�.18,19

Mingo et al.20,21 attributed the poor predictions obtained withCallaway model to the use of the approximation of linearizeddispersion relations and inadequate forms of the anharmonicscattering rates. They attempted to calculate the Si NW ��T�by using the real dispersion relations of Si, omitting the nor-mal processes, and considering, in addition to the resistivephonon mechanisms arisen from Umklapp processes andcrystal imperfections scattering, a boundary scattering ex-pression which depends on a constant F related to the pho-non specularity. When the reflection from the surface is com-pletely specular, F is zero. When the boundary scattering iscompletely diffusive, F is one. Whenever partial specularreflection occurs, F takes values between zero and one. Theapproach of Mingo requires only bulk data as inputs andthus, can be used to estimate theoretically the NWs ��T� withhigher accuracy than that provided by the modified-Callawaymodels. Nevertheless, even with the use of the exact phonondispersion relation, the best representation of ��T� for the SiNWs of cross section �37 is obtained with F values largerthan one.20,21 The authors attributed the physically unreason-able value of F obtained to possible errors on the measure-ments of the NW cross section and to a very diffusive bound-ary scattering. However, it is well known that the specularreflection rate from the surface decreases, or alternatively thediffuse scattering rate increases, with increasing the surfaceroughness �see below in Sec. II the physics behind�, what

leads to a decrease in ��T�.8,22–30 Thus, the dramatic decreasein ��T� for rough Si NWs reported recently by Hochbaum etal.13 clearly indicates that the boundary scattering in the so-considered “smooth” single Si NWs initially measured by Liet al.10 and modeled by Mingo et al.20,21 were far from com-pletely diffusive and the corresponding F value should bewell below one. Consequently, even with the use of the exactphonon dispersion relation, the ��T� behavior in Si NWs as afunction of size could not be well described with the com-monly considered phonon physical mechanisms. This, to-gether with the fact that the experimental ��T� curve for SiNW of 22 nm cross section could not at all be representedwith the exact bulk Si dispersion spectrum, strongly suggestthat some phonon mechanisms contributing significantly to��T� have not been taken into account in the previous ap-proaches.

Here, we propose a theoretical approach based on thesolution of Boltzmann equation by taking into account thephysical aspects of the relevant phonon mechanisms. Themodel considers the normal three-phonon processes which,in contrast to the resistive processes, tend to displace thePlanck distribution. These normal phonon processes do nothave a complete resistive nature and, therefore, should not beneglected even when the resistive processes rates are toohigh. Moreover, in order to approach to realistic and reason-able phonon mechanisms at the surface we consider that theprobability for phonon specularity and diffuse scattering isjudged by the phonon incidence angles to the surface and thephonon wavelength compared to the surface roughness. Therelevance of such effect is supported by reflection measure-ments of heat pulses from free solid surfaces which haveshown that the assumption of specular phonon reflectionbreaks down for high frequency phonon ��100 GHz�, un-less the surfaces are cleaved in situ or laser annealed.31 Thus,in our approach the parameter F in the expression of theboundary scattering is no longer a constant but depends onthe specific characteristics of each phonon. In the case ofNWs we also take into account the phonon confinement andthe consequent quantization of the phonon spectrum. An-other key development presented is that the decay of opticalphonons into acoustic phonons is considered in detail. Theimportance of this physical mechanism has been previouslyshown in the description of the effect of crystallographicorientation and isotopic composition on ��T� of bulk alumi-num nitride �Ref. 32� and Ge.33 Here we show that the opti-cal phonon decay effect on ��T� is particularly relevant forthe case of NWs.

As a result we show that, even when using linearizeddispersion relations, a rigorous and appropriate analysis ofthe phonon physical mechanisms allows to consistently ex-plain the effects observed on ��T� for bulk and NW semi-conductor structures.

II. THEORIES

A. General formalism for the lattice thermalconductivity

The phonon heat current density �1 /A�JQ is the sum ofphonons with given frequency, � j, and incidence angles �

083503-2 Kazan et al. J. Appl. Phys. 107, 083503 �2010�


and � that are incident on a unit area A per unit time, timesthe phonon energy �� j. Here j refers to a particular phononbranch, � is the angle between the phonon wave vector andthe normal to the unit area A and � is the azimuthal angle.Let v j,q . cos � be the normal component of velocity, d�

=sin �d�d�, q� the phonon wave vector, and N=Nq� ,j0 −Nq� ,j the

deviation in the phonon distribution Nq� ,j from its equilibriumvalue Nq� ,j

0 . Then the phonon heat current density can be writ-ten as

�1/A�JQ = − �j

�q�

0

2 �0

�� j,qv j,qcos � sin �

4Nd�d� .

�1�

Assuming small T variation along well-defined direction thethermal conductivity ��T� is −�1 /A�JQ /�T and thus its ex-pression can be written as

��T� =1

4�

j�

q�

0

2 �0

�� j,qv j,q cos��

sin��N

�Td�d� . �2�

Assuming now linearized dispersion relations �v j�q��=� j /q�and using the Debye density of states approximation for oneunit volume, i.e., �q=1 / �2�3�d3q=1 /22�q2dq=1 /22��2 /v3d�, we can write

��T� =1

83�j�

0

2 �0

�0

�D,j

�� j3v j

−2 cos��

sin��N

�Td�d�d� , �3�

where �D,j is the Debye frequency �or the cut-off frequency�for the jth phonon branch.

In principle, ��T� can be calculated if N is known. Inorder to do this we need to solve the Boltzmann equation for

N

�N

�t

v− v � T

dN

dT= 0. �4�

In the Boltzmann equation �Eq. �4�� the second term de-scribes the changes in the distribution function due to tem-perature gradient. Since we assumed a small T variation, N inthe second term can be replaced by N0. The first term de-scribes the change in the phonon distribution function duethe various phonon collision processes. Here, one should dis-criminate between two types of processes. The normal pro-cesses in which the momentum is conserved and the resistiveprocesses in which the momentum is not conserved. Thenormal processes tend to displace the Planck distribution,while the resistive processes tend to bring back the Planckdistribution to its initial position. Thus, the strong differencebetween the nature of the normal processes and that of theresistive processes should be taken into account while solv-ing the Boltzmann equation. Following Callawayapproximation16 we can write the first term of the Boltzmann

equation in the following form which is physically valid forsmall T variation and can maintain the difference in thephysical aspects of the collision processes described above

�N

�t

v=

N�� − N

�N+

N0 − N

�R. �5�

Here �N is a relaxation time for normal processes and �R is arelaxation time for the overall resistive processes. The termN�� is the displaced Planck distribution and � is a constantvector in the direction of the temperature gradient.

The displaced Planck distribution N�� can be written asa function of the Planck distribution at the equilibrium stateN0 in the following form16,34

N�� = N0 +�q�

kBT

exp��/kBT��exp��/kBT� − 1�2 , �6�

where kB is the Boltzmann constant. Substituting Eq. �6� inEq. �4� we can write

−��

kBTv � T

exp��/kBT��exp��/kBT� − 1�2

+�q�

�NkBT

exp��/kBT��exp��/kBT� − 1�

+ N 1

�N+

1

�R = 0. �7�

Thus, we can have an expression for N that depends on thenormal and resistive relaxation times. Its substitution in Eq.�3�, after performing mathematical arrangements similar tothose in Ref. 16, allows writing an incidence-angle-dependent expression for the thermal conductivity

��T� = �j

kB

83v j kBT

�3

�I1,j + I2,j� , �8a�

where

I1,j = �0

�D,j/T �0

2 �0

�c,jx4ex

�ex − 1�2cos2 � sin �d�d�dx ,

�8b�

and

I2,j = �0

�D,j/T�02�0

�c,j

�N,j

x4ex

�ex − 1�2cos2 � sin �d�d�dx�2

�0�D,j/T�0

2�0 �c,j

�N,j�R,j

x4ex

�ex − 1�2cos2 � sin �d�d�dx

.

�8c�

Thus, ��T� can be described as the sum of two components;�1�T� and �2�T� with

�1�T� = �j

kB/83v j�kBT/��3I1,j and �2�T�

= �j

kB/83v j�kBT/��3I2,j .

Here we define a combined relaxation time �c,j−1=�N,j

−1 +�R,j−1 ,

and a dimensionless parameter x=�� /kBT.



B. Expressions for the various phonon relaxationtimes

After writing an expression for ��T� in which the pho-non incidence angles are taken into account and the normaland resistive processes are fully considered, let us now dis-cuss the phonon physical mechanisms which contribute tothe normal and resistive processes and approximate theircontribution to ��T� by phonon relaxation times.

III. PHONON-POINT-DEFECTS SCATTERING

The rate of the scattering of the phonon by point-defectsor mass different from that of the host crystal in an otherwiseperfect lattice is provided by Klemens.35 For a given phononbranch j the scattering rate can be written as

�I,j−1 =

VkB4

4�4v j3x4T4, �9a�

where V is the volume per atom and is the mass-fluctuation-phonon scattering parameter defined by

=�i�ciMi�2 − ��iciMi�2

��iciMi�2 . �9b�

Here ci and Mi denote the concentration and the mass of theconstituent isotopes and impurities.

IV. PHONON-BOUNDARY SCATTERING

The phonon-boundary scattering has been first investi-gated by Casimir.28 Later, Berman et al.29 have extended theCasimir theory to include the effect of finite size l and non-zero specularity factor of the phonon P. The authors pro-posed for the phonon-boundary scattering rate the followingform

�B,j−1 = v j 1

L

�1 − Pj��1 + Pj�

+1

l� , �10�

where L is the effective diameter of the sample which isequal twice the radius for a cylindrical cross section and thesquare root of the side lengths product for a square or rect-angular cross section. If we replace the term ��1− Pj� / �1+ Pj�� by 1 /F with Pj a constant independent from the pho-non branch we get the expression for the phonon-boundaryscattering used by Mingo et al.20,21 However, Zimann30 andlater Soffer36 developed a statistical model for the reflectionof plane wave from a rough surface. It has been shown thatthe reflectivity of a plane wave from a rough surface leads toa plane wave in the specular direction and to a diffused planewave in a direction that depends on the incident plane wave,the surface roughness, and the tangential correlation of thesurface asperities. Thus, reducing the surface specularity anddiffuse scattering to a constant would be inadequate for de-scribing accurately the phonon-boundary scattering. A morerealistic expression is thus essential to investigate the experi-mentally observed effects. Soffer described the average scat-tered flux density by the following expression:

��F��,�,��2�� = 42 p��0,��

+��0��1 − ps��0��

c��0�c��0,�� , �11�

where �0= ��0 ,�0� and �= �� ,�� refer to the directions ofthe incident and emerging wave vectors, respectively,

��0,�� = �2/��sin �0 cos �0 − sin � cos �� ,

��0,�� = �2/��sin �0 sin �0 − sin � sin �� ,

��0,�� = cos �0 + cos � ,

p��0,�� = exp�− �2��2�� ,

��x,y� = �2/�� . z�x,y� ,

ps��0� = p��0,�0� ,

��0� =� ��d�

c��0,�� = L2/4 exp�− �2��2� �n=1

�

��2��2�n/n!n

exp�− �R2/4n��2 + �2��, and c��0�

=� c��0,��d� .

The first term on the right hand in Eq. �11� is the specularpart and the second term is the diffuse part, with � the wave-length and R the tangential correlation length. In the case ofcylindrical-like NW, the term ��x ,y� which is present in bothspecular and diffuse part is the one way phase shift causedby a diameter fluctuation around the value L. The quantity��2� can thus be considered as the rms height deviation �h�.Only the diffuse part depends on the tangential correlationlength R. Let P��0 ,�� be the transition probability of pho-non wave per unit solid angle, from �0 to � �after an inter-action with the boundary�. By definition, P��0 ,�� is theaverage flux density of Eq. �11� normalized to unity transi-tion probability for all allowed angles. Thus, it can be ex-pressed as

P��0,�� = �1/��0��P��0,�� + ��0�

��1 − P��0��c��0,��/c��0�� . �12�

Now, the total transition probability is

� d�0P��0,�� . �13�

Substituting Eq. �12� in Eq. �13� we find

� d�0P��0,�� = P�� +� d�0�1 − P��0��

�c��0,��/c��0�� , �14�

with a specular reflection probability P�� equal to exp�



−cos2 ��2��. In order to express the specular reflectionprobability as a function of measurable physical parameterswe write

Pj��,�,h� = exp − 2h� cos �

v j2� . �15�

In the case of a temperature gradient along the length of thesample, the angle � should be replaced by � /2−�� in theexpression of the specularity probability.

Substituting Eq. �15� in Eq. �10�, one can certainly ob-tain a more realistic expression for the phonon-boundaryscattering.

V. NORMAL THREE-PHONON SCATTERING

The normal three-phonon processes can be understoodfrom the cubic anharmonic term in the Hamiltonian

H�3��q�� = �q,j,q�,j�,q�,j�

V�3��q� , j,q��, j�,q��, j��

a�q� , j�a�q��, j��a+�q��, j�� , �16�

where a and a+ are the annihilation and creation operators,respectively. As shown from Eq. �16�, in the normal three-phonon processes, two acoustic phonons of polarizations jand j� and wave vectors q and q� are annihilated and onehigher energy phonon �optical phonon� of polarization j� andwave vector q� is created. The term V�3� is the coefficient ofthe cubic anharmonicity. In the case of normal processes,V�3� vanishes unless

�� = �0 − � − �� = 0, �17a�

and

q�0 + q� + q�1 = 0. �17b�

In a pure and relatively large samples �larger than the phononmean free path in the full T range�, where the phonon-point-defects scattering and the phonon-boundary scattering can beneglected, ��T� at T slightly above Tmax=0.05�D, can be de-termined by the normal three-phonon processes alone. Notethat in the T range around Tmax, the higher order normalprocesses, as well as Umklapp processes are unlikely. Thisled to an experimental determination of analytical expressionfor the normal three-phonon scattering rates.2,3,24 For theclass of cubic semiconductor materials such as Ge, diamond,and Si, it has been found that the expression of the normalthree phonon scattering rate for longitudinal phonons is ofthe form

�N,L−1 = BN,L�2T3, �18a�

and that for transversal phonons is of the form

�N,T−1 = BN,T�T4. �18b�

The magnitudes of the coefficients BN are chosen to satisfythe following dependence on phonon velocity19,37

BN,L =kB

3�L2V

M�2vL5 , and BN,T =

kB4�T

2V

M�3vT2 , �18c�

where �L�T� is the longitudinal �transversal� Grüneisen con-stants.

VI. UMKLAPP THREE-PHONON SCATTERING

The Umklapp three-phonon processes can be describedby the same cubic anharmonic term given by Eq. �16�. Nev-ertheless, in the Umklapp processes V�3� vanishes unless

q�0 + q� + q�1 = b� , �19�

where b� is a nonzero reciprocal lattice vector. Thus, in theUmklapp three-phonon scattering processes two acousticphonons are annihilated and one higher energy phonon iscreated without conserving the total momentum.

At temperatures in the region of T�0.1�D, the dominantphonon scattering mechanism is the Umklapp scattering. Thethermal conductivity of several high purity, isotopically cleancrystals have been measured and the results for T�0.1�D

could be approximated by a universal curve.38–40 This uni-versal curve is the ratio of � at T to � at �D and is assumedto depend only on the reduced temperature �T /�D�. This as-sumption is founded on the same basis as those used to de-rive the Debye specific-heat function Cv�T /�D�. It has beendemonstrated that the expression for the Umklapp processeswhich can fit ��T� of several pure crystals at T�0.1�D is ofthe form18,19,38–40

�U,j−1 = BU,j�

2T exp�− �D,j/3T� , �20a�

with

BU,j =�� j

2

Mv j2�D

. �20b�

Several other expressions for the Umklapp scattering rate orattempts to ignore the Umklapp phonon mechanism haveappeared in the literature.41–45 However, in the fitting of thetheoretical expression for ��T� to the experimentally mea-sured data, they lead to physically unreasonable values forthe phonon dispersion curves characteristics. Since Eq. �20�has been derived by a physically reasonable experimentalmethod and described the Umklapp processes in several highpurity crystals belonging to the class of materials to which Sibelongs, we choose here to use Eq. �20� as an analyticalexpression describing the Umklapp scattering rate in Si.

VII. DECAY OF OPTICAL PHONONS INTO ACOUSTICPHONONS

After describing the annihilation of acoustic phononsand the creation of higher energy phonons via the variousacoustic phonon scattering processes, we should consider thereverse process, i.e., the decay of high energy phonons, suchas the optical phonons, into acoustic phonons. In fact, due toits low group velocity, an optical phonon by itself cannotconduct a noticeable amount of heat. However, in order tomaintain the lattice thermal equilibrium, it should decay andinterchange energy with low energy phonons which conduct



the heat. The relevancy of this phonon process has beenwidely observed experimentally.46–48 Thus, the rate of thedecay of optical phonon into acoustic phonon of a givenfrequency � should be considered as an acoustic phonongeneration rate partially counteracting the scattering rate ofthe acoustic phonons of the same frequency.

There is no expression for the anharmonic optical pho-non scattering rates deduced from systematic experimentalmeasurements. The reason is the incapability of detecting theoptical phonons lifetimes of nonvanishing wave vectors.Therefore, we will approximate analytical expression for theoptical phonon decay from the total potential energy of Sicrystalline lattice.

The unperturbed harmonic Hamiltonian for each latticenormal-mode takes the form

H0 = 1/2M�2�a+�j,q��a�j,q�� + a�j,q��a+�j,q�� , �21�

where M is the atomic mass. The term in the cubic anhar-monic Hamiltonian HA�3� which describes the strength of theinteraction between one optical phonon of frequency �0 andwave vector q�0 with two acoustic phonons of frequencies �and �1 and wave vectors q� and q�1 takes the form

HA�3� = �q�0,j0,q� ,j,q�1,j1

V3�q�0, j0,q� , j,q�1, j1�a�j0,q�0�

a+�j,q��a+�j1,q�1� . �22a�

The term in the quartic anharmonic Hamiltonian HA�4� whichdescribes the strength of the interaction between one opticalphonon of frequency �0 and wave vector q�0 with threeacoustic phonons of frequencies �, �1, and �2 and wavevectors q� ,q�1 and q�2 takes the form

HA�4� = �q�0,j0,q� ,j,q�1,j1,q�2,j2

V4�q�0, j0,q� , j,q�1, j1,q�2, j2�

a�j0,q�0�a+�j,q�a+�j1,q�1�a+�j2,q�2� . �22b�

Here, the cubic and quartic anharmonic coefficients are de-fined by

V3�q�0, j0,q� , j,q�1, j1� = g3�0��1��q�0 + q� + q�1� , �23a�

and

V4�q�0,j0,q� , j,q�1, j1,q�2, j2�

= g4�0��1�2��q�0 + q� + q�1 + q�2� , �23b�

where g3 and g4 are constants and ��q��=1 if q� =b� and 0otherwise. The Hamiltonians HA�3� and HA�4� give rise, infirst-order perturbation, to three-phonon and four-phononprocesses, respectively. If we consider NOP the occupationnumber of the optical phonon �0,q�0,j0

equal to the sum of its

deviation from the initial value NOP and its thermal equilib-rium value NOP

0 , and N0 ,N10 ,N2

0 the occupation numbers ofthe created acoustic phonons �q� ,j, �1,q�1,j1

, and �2,q�2,j2at ther-

mal equilibrium, the rate of change in the optical phononoccupation number due to three-phonon processes can bewritten as

td

dt�NOP��3� = 2 �

Mr3

�j,j1,q� ,b�

�V3�21

�0��1

1 − cos ��t

�2��2 ��NOP + NOP0 ��N0 + 1��N1

0 + 1�

− �NOP + NOP0 + 1�N0N1

0� , �24a�

and that due to four-phonon processes as

td

dt�NOP��4� = 2 �

Mr4

�j,j1,j2,q� ,q�1,b�

�V4�21

�0��1�2

1 − cos ��t

�2��2 ��NOP + NOP0 ��N0 + 1��N1

0 + 1�

�N20 + 1� − �NOP + NOP

0 + 1�N0N10N2

0� . �24b�

The resonance factor ensures that the principal contributionsin the summation arise from processes for which the energyis conserved ��=0�.

At the thermal equilibrium

NOP0 �N0 + 1��N1

0 + 1� − �NOP0 + 1�N0N1

0 = 0, �25a�

and

NOP0 �N0 + 1��N1

0 + 1��N20 + 1� − �NOP

0 + 1�N0N10N2

0 = 0.

�25b�

Equations �25a� and �25b� can be used to reduce the term

between brackets in Eq. �24a� to NOP�1+N0+N10� and that in

Eq. �24b� to NOP�1+N0+N10+N2

0+N0N10+NN2

0+N10N2

0�.On the other hand, we define the phonon scattering rate

�−1 by

�−1 = −1

NOP

d

dt�NOP� . �26�

Thus, we can write the expression of the optical phononscattering rate due to three-phonon processes as

�A�3�−1 = − 2 �

M31

t�

j,j1,q� ,b��V3�2

1

�0��1

1 − cos ��t

�2��2 �1 + N0 + N10� , �27a�

and that due to four-phonon processes as

�A�4�−1 = − 2 �

M41

t�

j,j1,j2,q� ,q�1,b��V4�2

1

�0��1�2

1 − cos ��t

�2��2

�1 + N0 + N10 + N2

0 + N0N10 + N0N2

0 + N10N2

0� .

�27b�

One can notice that Eqs. �27a� and �27b� depend on T onlythrough the occupation numbers which have the form�1 / �e��/kBT−1��. At T=0, the occupation numbers are omit-ted and the expressions of the optical phonon scattering ratecan be reduced to



�A�3�−1 = − 2 �

M31

t�

j,j1,q� ,b��V3�2

1

�0��1

1 − cos ��t

�2��2 , �27c�

and

�A�4�−1 = − 2 �

M41

t�

j,j1,j2,q� ,q�1,b��V4�2

1

�0��1�2

1 − cos ��t

�2��2 .

�27d�

Thus, one can perform the calculation of �A�3�−1 and �A�4�

−1 atT=0 �independently from T� and then add to the result the Tdependent factor.

Let us consider now Eq. �27c�. Assuming that J and Bare integers representing, respectively, the number of thephonon branches and the number of the reciprocal latticevectors for which the total energy is conserved and the totalmomentum is conserved or equal to a reciprocal lattice vec-tor, � j,j1

can be replaced by J and �b� can be replaced by B.On the other hand, the concept of phonon density of statesallows writing

�q�

=GV

�2�3� d3q =GV

�2�34� q2dq , �28�

where G is the number of atoms and V is the volume peratom. By using Eqs. �22a�, �27c�, and �28� we can write thatfor T=0

�A�3�−1 = − 2 �

Mr31

t

1

�2

g32JBGV

�2�3 4� �0��1

1 − cos ��t

��2 q2dq . �29�

For the three-phonon processes, within the approximation ofa linearized dispersion relations, we can write

�� = � − vqq − vq1q1 = 0, �30a�

where vq and vq1are the velocities of the waves q and q1,

respectively. However, we can find a constant � such asvq1

q1=�vqq. Then, we can write

�� = � − �1 + ��vqq , �30b�

and

d��

dq= − �1 + ��vq. �30c�

Indeed an optical phonon �0j0,q�0cannot decay into any two

acoustic phonons � j,q� and �1j1,q�1or any three acoustic

phonons � j,q�, �1j1,q�1, and �2j2,q�2

. It should decay only toother phonons where the total energy is conserved and thetotal momentum is conserved or equal to a reciprocal latticevector. The all allowed decay channels can be approximatedfrom several approaches, such as Debye isotropic continuummodel,49 anisotropic continuum model, and full lattice dy-namical model.50 However, such a treatment would make thenumerical calculation of ��T� extremely cumbersome, if notimpossible. To avoid this problem one can assume the fol-lowing. The optical phonon of frequency �0 decays into twoacoustic phonons of lower frequencies, � and �1 with �

=�1�0 and �1= �1−�1��0, where �1 is a constant numberbelow unity. This comes from the principle of energy con-servation. We can notice that �1= �1−�1� /�1�. Since �1 isa positive constant which can take some positive values be-low unity, we can reasonably assume for it an average valueof 0.5 and then, consider the Klemens channel,51 which as-sumes that the optical phonon decays into acoustic phononsof equal energies �i.e., �=�1=�0 /2 for the three-phononprocesses and �=�1=�2=�0 /3 for the four-phonon pro-cesses�. The use of Klemens approximation and Eqs.�30a�–�30c� in Eq. �29� allows expressing the optical three-phonon processes at T=0 as

�A�3�−1 =

1

t

2�

Mr

g3JBGV

2v3�1 + ��5� 1 − cos ��t

��2 d�� . �31�

We note that �1−cos ��t /��2d��=t. Thus, the opticalphonon scattering rate due to the three-phonon processes atT=0 can be written as

�A�3�−1 =

2�

Mr

g3JBGV

v3�1 + ��5. �32a�

We can introduce now the T dependent factor to describe theoptical phonons decay as function of T

�A�3�−1 �T� =

2�

Mr

g3JBGV

v3�1 + ��5�1 + N0 + N1

0� . �32b�

Since we are assuming for the optical phonon decay the Kle-mens channel ��=�1 and consequently N0=N1

0�, Eq. �32b�can be reduced to

�A�3�−1 �T� = CA�3��

5 coth ��

2kBT , �33�

where CA�3� is a temperature and frequency independent con-stant equal to

2�/Mr g3JBGA3/v3�1 + �� .

Equation �33� describes only the anharmonic decay of opti-cal phonon into two acoustic phonons. However, in the mostof the materials, the decay of the optical phonons is due tothe combined effects of anharmonic decay, which determinesthe intrinsic decay, together with scattering of the opticalphonon by point defects. Let us now introduce the additionaleffect of defects scattering on the optical phonon scatteringrate. The best convenient method to do so is to convolute thepoint-defect scattering with the anharmonic process.52–54 Theexperimentally observed52,55,56 enhancement in the opticalphonon scattering rate due point-defects scattering was con-sidered as creation of additional favorable channels for theoptical phonon decay with introducing point-defects. We as-sume here that these additional channels consist on anhar-monic optical phonon scattering into a neighbor mode q�followed by point-defect scattering into acoustic modes qand q1. According to the coherent potential approximation,57

if there is overlap between an optical phonon mode q�0 offrequency �0 and other optical phonon mode q�� of frequency��, the total optical phonon relaxation rate, which we willrefer to as a generation rate of two acoustic phonons of fre-quency �, can be written as



�G�3�−1 = �A+I,�3�

−1 = �0

2

12Nd, �34�

where Nd is the density of modes q�. Within the isotropicmedium approximation Nd can be considered as being thenumber of modes in the sphere of radius q�, divided by thepure anharmonic scattering rate. Thus, it can be written as

Nd =GV

�2�3

4

33 q�3

�A�3�−1 . �35�

Let �0C be the optical phonon frequency at the zone centerfor the phonon branch branch j, and Kq0

the curvature of thephonon branch j around the wave vector q�0. We can thenwrite

�0 = �0C1 −q0

2

Kq0

2 . �36�

Indeed, Eq. �36� is valid only when the quantity q02 /Kq0

2 issmaller than unity. The Si phonon dispersion spectrum pre-sented in Fig. 1 was calculated by using the full Weber adia-batic bond charge model58,59 which is based on the theoriesof Phillips60 and Martin.61 This spectrum shows that the cur-vature of the phonon branches is sufficiently large in thewhole Brillouin zone, so that Eq. �36� is valid in the wholeBrillouin zone and thus can be adopted to describe the point-defects scattering of an average optical phonon. Now, taking�A�3�

−1 =�0C−�0, the expression of Nd takes the form

Nd =4GV

3�2�3Kq0

3��A�3�

−1 �1/2

�0C3/2 . �37�

After substituting in Eq. �34�, we obtain

�G�3�−1 =

�2GVKq0

3

72�23/2 �1/2��A

−1�1/2 = CI�3� �1/2��A−1�1/2, �38�

where, CI�3�=�2GVKq0

3 /72�23/2 and �2=�0 /�0C. The term

�2 is a constant below unity, introduced to express an aver-age optical phonon frequency as a function of the zone centeroptical phonon frequency. The factor �2 comes from the as-sumption of Klemens channel for the decay of the opticalphonon �i.e., �0=2��. Thus, we can write the total expres-

sion of the optical phonon decay into two acoustic phonons,or the expression of the two acoustic phonon generation rateas

�G�3�−1 = �A+I,�3�

−1 = CG�3� �3 coth ��

2kBT�1/2

. �39�

Similar treatment for Eq. �27d� allows us to write the expres-sion of the generation rate of three acoustic phonons of fre-quency � as

�G�4�−1 = CG�4� �5 1 +

3

e��/kBT − 1+

3

�e��/kBT − 1�2�1/2.

�40�

When the scattering processes are independent, the scatteringprobabilities are additive. Since the scattering processes ofthe optical phonons are independent from those of the acous-tic phonons and since the rates of the decay of the opticalphonons is considered as a rate of generation of acousticphonons partially counteracting the resistive processes wecan reasonably consider that the total resistive scatteringprobability is the sum over the various acoustic phononsscattering probabilities minus the probability of the decay ofthe optical phonons

�R,j−1 = ��B,j

−1 + �I,j−1 − �G�3�,j

−1 − �G�4�,j−1 � + �U,j

−1 . �41�

The phonon polarization or branch j can be either longitudi-nal �L� or transversal �T�. Thus, in Eq. �41� we separate thedecay of optical phonon into longitudinal acoustic phononsfrom that into transversal acoustic phonons. This separationsplits the two free adjustable parameters of the model, CG�3�and CG�4� to four free adjustable parameters CG�3�,L, CG�3�,T,CG�4�,L, and CG�4�,T, which is physically reasonable becausethese constants depend on phonon branches of different char-acteristics. Thus, they should be different for different pho-non branches. Since the four-phonon processes are notice-able only at high temperatures, CG�3�,L and CG�3�,T should beadjusted to describe ��T� at relatively low T, while CG�4�,Land CG�4�,T should be adjusted to describe ��T� at higher T.

Indeed, an optical phonon can combine with otherphonons and produces optical phonons of higher energies.These created optical phonons will certainly have higher en-ergies and thus relatively very low velocities. Consequently,this event cannot influence the thermal conductivity and,therefore, is not taken into account.

VIII. COMPARISION WITH EXPERIMENTS

A. Isotopic composition effect on the thermalconductivity of bulk Si

Experimental measurements regarding the isotope com-position effect on ��T� of bulk Si showed an enhancement in��T� of the isotopically enriched over the natural abundanceSi of almost 60%. The most recent modifications brought tothe model of Callaway �model of Morelli et al.19�, could notaccount for this effect. Thus, it is interesting to apply theapproach proposed to describe ��T� for natural abundance Si�naturalSi� and isotopically enriched Si �99.8588% 28Si � in thefull T range.

FIG. 1. Phonon dispersion curves for silicon calculated from the full Weberadiabatic bond charge model.



The symbols in Fig. 2 present the experimental data on��T� of naturalSi and 99.8588% 28Si. The data on naturalSi and99.8588% 28Si are obtained from Refs. 62 and 63, respec-tively.

For the sake of comparison, we plot these experimentalresults together with the calculated ��T� curves obtained byusing the model of Morelli et al., and those obtained byusing the theoretical approach proposed. In our calculationswe have used the same values for the transversal and longi-tudinal acoustic phonon Debye temperatures and phonon ve-locities given by Morelli et al. These values are presented inTable I. The theoretical results obtained from the model ofMorelli et al. for ��T� of naturalSi and 9.8588% 28Si are plot-ted in Fig. 2 by using dashed lines. Although the model ofMorelli et al. provides a good description of ��T� of naturalSiat low T and T�200 K, it underestimates ��T� near thetemperature corresponding to the maximum thermal conduc-tivity, Tmax, i.e., 20–120 K. For enriched Si, the disagreementbetween the data and the theoretical curves becomes muchmore important; the model of Morelli et al. significantly un-derestimates ��T� when T exceeds Tmax. It is believed by thepresent authors that their model does not account for theisotopic composition effect on ��T� of Si.

Several changes in the scattering rates involved in themodel of Morelli et al. were attempted in order to improvethe fit. Namely, we tried to decrease and bring the theoret-ical curves closer to the experimental data at T around Tmax

but this leads to a larger discrepancy at lower T. Also, wehave considered longitudinal and transversal normal scatter-ing rate having the general form ��2V�a+b−2�/3 /Mv�a+b��KB /��b�aTb and tuned a between 1.0 and 2.0 for eachvalue of b between 2.0 and 4.0. In spite of using severalreported forms for the three-phonon processes,16–19,43,44,64,65

we did not find any combination of a and b which couldfurther improve the fit.

The theoretical ��T� curves for naturalSi and 9.8588% 28Siobtained with our approach are plotted in Fig. 2 by usingsolid lines. Since the deduced expressions for the scatteringrates are all independent from the azimuthal angle �, theintegral �0

2d� in the formalism for ��T� proposed in thiswork contributed 2. Then, the numerical calculation of theremaining double integral was carried out by solving in afirst step the inner integral, using recursive adaptive Simpsonquadrature,66 and in a second step the outer integral by usingthe same numerical method. Since the three-phonon pro-cesses are likely at relatively low T and the four phonon-processes are likely at higher T, the free adjustable param-eters CG�3�,L and CG�3�,T were tuned to obtain the best fit tothe experimental data at low T, while the free adjustableparameters CG�4�,L and CG�4�,T were tuned to obtain the bestfit to the experimental data at higher T. The values of thesefree adjustable parameters, as obtained from the least-squaresmethod, are shown in Table I. The excellent agreement be-tween the theoretical approach proposed and the experimen-tal results regarding the isotopic composition effects on ��T�of Si highlights the importance of taking into account theexpression for the decay of optical phonons into acousticphonons as correction to the expression for the phonon-point-defects scattering rate. Moreover, these results showthat, in contrast to the common accepted view, the opticalphonons have an indirect, yet crucial role in the thermal con-ductivity.

In order to illustrate the magnitude of the acoustic pho-non generation rate relatively to the different scattering rateswe plot in Fig. 3 �1�T� and �2�T�, the two components of thefull lattice thermal conductivity formalism, for the case of

FIG. 2. �Color online� Thermal conductivity of natural abundance and iso-topically enriched silicon bulk samples. Symbols: experimental data.Dashed lines: theoretical results obtained from the conventional full Calla-way model and single mode approximation. Solid lines: theoretical resultsobtained from the theoretical approach described in the present work whichincludes the contribution of the decay of the optical phonon into acousticphonons to the lattice thermal conductivity.

TABLE I. Parameters obtained from the least-squares best fit to the experimental data on the lattice thermal conductivity of Si bulk samples differing in theirisotope composition and NWs differing in their cross section and surface roughness.

SampleL

�m�h

�m�vT

�m/s�vL

�m/s��D,T

�K��D,L

�K� C�3�,L C�3�,T C�4�,L C�4�,T

naturalSi 410−3¯ 5840 8430 240 586 29.75104 1.25107 2.625 9.375

99.8588% 28Si 1410−3¯ 5840 8430 240 586 29.75104 1.25107 2.625 9.375

naturalSi NW 11510−9 110−9 5759 8228 125 311 7.33106 2.4107 64.68 18naturalSi NW 5610−9 110−9 5575 7900 121 302 7.33106 2.4107 64.68 18naturalSi NW 3710−9 110−9 5436 7782 118 294 7.33106 2.4107 64.68 18naturalSi NW 2210−9 110−9 5344 7646 116 289 7.33106 2.4107 64.68 18naturalSi NW 11510−9 610−9 5759 8228 125 311 7.33106 2.4107 64.68 18naturalSi NW 9810−9 510−9 5574 8122 123 307 7.33106 2.4107 64.68 18naturalSi NW 5010−9 5.710−9 5483 7858 119 297 7.33106 2.4107 64.68 18



naturalSi. As we can notice, the introduction of the acousticphonon generation rate has influence on �2�T� �in the tem-perature range where the model of Morelli et al. underesti-mates the thermal conductivity� but not on �1�T�. From theexpressions of �1�T� and �2�T�, one can readily notice thatthe acoustic phonon generation rate is much less than thecombined scattering rate �c,j

−1=�N,j−1 +�R,j

−1 in the full tempera-ture range. However, it becomes comparable to and counter-acts the overall resistive scattering rate �R,j

−1 in the tempera-ture range where the model of Morelli underestimates thethermal conductivity.

B. Thermal conductivity of individual silicon NWs

Let us now apply the theoretical approach proposed todescribe available experimental data on ��T� of individual SiNWs. As mentioned earlier, for the case of NWs the phonon-boundary scattering is expected to be the most dominantscattering mechanism in a wide T range. This can be wellnoticed if we compare the T dependent intrinsic mean freepath of the acoustic phonon �A�T� and that of the opticalphonon �O�T� to the NW cross section L. In Fig. 4, we illus-trate �A�T� and �O�T� which are calculated by using the data

of Si phonon dispersion spectrum shown in Fig. 1 within thesingle mode relaxation time model.49,67,68 It can be observedthat in spite the fast decrease in �A�T� with T, �A�T� remainslarger than the cross section of a Si NW in a wide T range�e.g., �A�T� remains larger than 100 nm up to 450 K�. Thisclearly shows that for Si NWs the phonon-boundary scatter-ing is the dominant scattering mechanism in a wide T range.Thus, an expression for the phonon-boundary scattering ratedescribing all the phonon mechanisms at the surface is cru-cial to explain all the effects which are observed experimen-tally.

An additional mechanism might also contribute to ��T�of NWs. It is the enhanced scattering of optical phonons dueto an additional scattering by the sample boundary. This phe-nomenon is likely when the optical phonon mean free path�O�T� is comparable to the sample cross section. However,as can be noticed from Fig. 4, for low T ��300 K�, �O�T� iscomparable to the cross sections of some of the NWs inves-tigated ��22 nm� but �A�T� is orders of magnitude largerthan it. Consequently, at relatively low T the enhancement inthe optical phonon scattering due to boundary scattering can-not have a noticeable influence on the acoustic-phonon-boundary scattering rate. At higher T, �A�T� is only one orderof magnitude higher than �O�T� but �O�T� becomes smallerthan the cross sections of the NWs investigated. For thesereasons, in the present investigation we do not consider theenhancement in the acoustic phonon generation rate due tothe scattering of optical phonons by the sample boundary

IX. SIZE EFFECT ON THE THERMAL CONDUCTIVITYOF SI NWS

In order to describe ��T� of Si NWs, in addition to anaccurate phonon-boundary scattering expression, we shouldtake into account the modification of the acoustic phonondispersion due to confinement induced by boundary. Forsmall cross section NW, typically below �8 nm, calculationof the size dependent phonon spectrum can be carried out byusing molecular dynamics simulation.69–71 For larger NWcross sections the molecular dynamics simulation becomescomputationally prohibitive and the only way for calculatingthe size dependent phonon dispersion spectrum is the use ofthe elastic continuum approximation.72–80 Although the elas-tic continuum approximation is valid only for phonons ofwavelengths much larger than the interatomic spacing, anumber of experiments confirm its usefulness in describingqualitatively the effect of size on the Debye temperatures andthe phonons velocities,81–84 which are the only phonon spec-trum characteristics involved in the present formalism for��T�. Within the elastic continuum approximation,76,80,85 fora longitudinal wave propagating along the axis of a NW ofcross section L, the dispersion relations are given by

�q2 − qT2�2 �qLL/2�J0�qLL/2�

J1�qLL/2�− 2qL

2�q2 + qT2�

+ 4q2qL2 �qTL/2�J0�qTL/2�

J1�qTL/2�= 0, �42a�

where J0 and J1 are the ordinary Bessel functions and qL andqT are two parameters related by

FIG. 3. �Color online� Contribution of �1�T� and �2�T� components to theoverall lattice conductivity of natural bulk silicon. Dashed lines: when theoptical phonon decay is not taken into account. Solid lines: when the opticalphonon decay is taken into account.

FIG. 4. �Color online� Calculated mean free paths for the optical, �O, andacoustic, �A, phonons as a function of temperature.



qL�T�2 =

�L,n2

vL�T�2 − q2. �42b�

The dispersion relations for a shear wave or transversal waveare given by

�T,n = vT�qz,n

2 + q2, �43�

where qz,n= �2n /L� and n is a quantization integer.By numerically solving Eqs. �42� and �43� we can obtain

the dispersion relations of confined acoustic phonon modesin the single Si NWs investigated. At each q one can findmany solutions for qL, qT, and qz. The index n is used asquantization index to indicate the different solutions so that�n is the phonon frequency for the nth branch. In Fig. 5 weshow the dispersion relations of the seven lowest confinedlongitudinal and transversal acoustic phonon branches in thesingle Si NW investigated, together with the longitudinal andtransversal bulk dispersion relations. Because of the quanti-zation and manifold nature of a confined phonon spectrumand because the elastic continuum model is limited tophonons of wave vectors very close to the zone center, it isnot possible to provide quantitative averaged values for thetransversal and longitudinal velocities vL�T� and Debye tem-

peratures �D,L�T� of confined phonons from the size depen-dent Si acoustic phonon spectrum. However, their qualitativetrends with the NW cross section can be clearly understood.As can be noticed, the slopes of the confined acoustic pho-non branches decrease with decreasing the cross section ofthe NW, which implies a decrease in the transversal and lon-gitudinal averaged phonon velocities. Furthermore, withinthe elastic continuum approximation the longitudinal andtransversal Debye temperatures are given by86

�D,L�T�2 = �2�/kB�3�3NA/4V�vL�T�

3 , �44�

where NA is the Avogadro number. They are proportional tothe averaged phonon velocities. Thus, they should also de-crease with decreasing the NW cross section. The best way

to take into account the decrease in vL�T� and �D,L�T� when theNW cross section decreases is to set them as free adjustableparameters with permitted values below those of vL�T� and�D,L�T� for bulk Si. As can be noticed from Fig. 4, at low Tthe anharmonic effect is very weak. This allows us to con-sider that, at low T, the acoustic phonon generation prob-

abilities almost vanish and hence vL�T� and �D,L�T� can be setas the only free adjustable parameters. Furthermore, the De-

bye temperatures �D,L�T� are involved only in the expressionsof the acoustic phonon scattering rate due to Umklapp pro-cesses, which does not have noticeable influence on ��T� atvery low T �below 30 K for Si�. Thus, vL�T� can be obtainedfrom the least-squares best fit to the experimental data below

30 K, whereas �D,L�T� can be obtained from that between 30and 50 K.

The experimental ��T� data for individual Si NWs withdiameters 115, 56, 37, and 22 nm reported in Ref. 10 areplotted in Fig. 6. Assuming an rms value of 1 nm for all theSi NWs investigated in Ref. 10 �see Fig. 2 in Ref. 10� wehave plotted in Fig. 6 the calculated ��T� curves obtainedfrom the formalism proposed with �solid lines� and without�dashed lines� taking into account the effect of opticalphonons decay into acoustic phonons. The averaged valuesfor the phonon velocities vL�T� and Debye temperatures

�D,L�T� obtained from the least-squares best fit to the experi-mental data are given in Table I. At T close to 0 K theanharmonic processes �optical phonons decay� are negligibleand both the theoretical curves coincide. At higher T, withouttaking into account the optical phonons decay into acousticphonons as acoustic phonon generation rate, the theoreticalmodel clearly underestimates ��T� for all the NWs investi-

gated. Any readjustment of vL�T� and/or �D,L�T� to bring thetheoretical curve closer to the experimental data at high Tcauses strong deviation between the theoretical curve and the

FIG. 5. �Color online� Elastic continuum approximation prediction of thedispersion relations for longitudinal �solid lines� and transversal �dashedlines� modes in silicon NWs of cross sections 115, 56, 37, and 22 nm.

FIG. 6. �Color online� Thermal conductivity of individual silicon NWs.Symbols: experimental data. Solid lines: theoretical results obtained fromthe theoretical approach described in the present work which considers theoptical phonons decay into acoustic phonons as acoustic phonon generationrate. Dashed lines: theoretical results obtained from the theoretical approachdescribed in the present work without accounting for the optical phonondecay.



experimental data at low T and around Tmax. However, whenthe optical phonon decay into acoustic phonons is intro-duced, an accurate description of the size dependent ��T� inthe full T range is obtained. It is worth to note here that thecalculations of ��T� were carried out with a unique set ofCG�3�,L, CG�3�,T, CG�4�,L, and CG�4�,T �listed in Table I� in pur-pose to provide approximate values for these constants thatcan be used to predict ��T� for Si NWs of cross sections notinvestigated in the present work. The adjustment of theseconstants for a particular cross section certainly leads to abetter agreement between the theoretical and the experimen-tal results. Although our approach gives satisfactory descrip-tion of ��T� of the NW of 22 nm cross section, we can noticefrom the experimental curve a light positive curvature above150 K, while the theoretical curve in this T region shows alight negative curvature. This small disagreement is probablydue to an alteration in the expression of the normal three-phonon processes caused by strong phonon confinement.

An additional confinement effect can be observed fromTable I. The Cs are enhanced for the confined structure,which implies an enhancement in the optical phonon decay.To understand the physical reasons for this enhancement,consider the decay conditions for an optical phonon. An op-tical phonon of wave vector q�0 and frequency �0 can decayinto, for instance, two other phonons of wave vectors q� andq�1 and frequencies � and �1 if the total energy is conservedand the total momentum is conserved or equal to a reciprocallattice vector. In the case of a bulk material, for one wavevector q� we can associate one longitudinal acoustic phononfrequency and two transversal acoustic frequencies. How-ever, for the case of a NW, as demonstrated previously, forone wavevector q� we can associate many longitudinal andmany transversal acoustic phonon frequencies and thus thedecay conditions become satisfied for much higher numberof wavevectors q� . In other words, due to the manifold natureof a confined phonon dispersion spectrum the number of thedecay channels for optical phonons increases significantly,which can explain the enhancement in the optical phonondecay in NWs and the enhancement in Cs. On the otherhand, it has been reasonably suggested that in nanostructuredmaterials, in addition to the short range mechanical force�and the long range Coulomb force in the case of polar ma-terials�, there are long range mechanical forces �in nonpolarand polar materials� which give rise to spatial dispersion anddefine the periodicity conditions for the displacementfields.87,88 Due to these forces, the radial wavevectors forlongitudinal optical phonons and those for transversal opticalphonons are quantized according to two different formal-isms. This fact results in raising the frequency of an opticallongitudinal phonon over that of an optical transversal pho-non. From the electrodynamics point of view this splittingcan be seen as a confinement induced macroscopic electricfield associated with the longitudinal phonon. This electricfield serves to stiffen the force constant of the phonon andthereby raise the frequency of the longitudinal phonon overthat of the transversal phonon. Since the longitudinal opticalphonon has a frequency higher than that of the transversalphonon, its decay channels number in manifold phonon spec-trum must be higher than that for the transversal phonon.

This effect can explain the fact that the Cs for a longitudinalphonon have undergone two orders of magnitude enhance-ment, while the Cs for the transversal phonon have increasedby a factor of two only.

X. SURFACE ROUGHNESS EFFECT ON THETHERMAL CONDUCTIVITY OF SI NWS

Considering that the expression for the phonon-boundaryscattering involved in the present formalism for ��T� im-poses criteria for specular reflection and diffuse scatteringmechanisms, it should enable providing accurate descriptionof the experimental results which show dramatic decrease in��T� for rough Si NWs.13 According to Ref. 13 the meanroughness height of these NWs varies from a wire to a wirebut is typically 1–5 nm with a roughness period of the orderof several nanometers. This roughness is attributed in Ref. 13to randomness of the lateral oxidation and etching in thecorrosive aqueous solution or slow HF etching and facetingof the lattice during synthesis.

The experimental results showing the dramatic decreasein ��T� for rough Si NWs are plotted in Fig. 7 together withthe calculated ��T� curves obtained from the model proposedwith: �i� keeping the same set of CG�3�,L, CG�3�,T, CG�4�,L, and

CG�4�,T, �ii� estimating vL�T� and �D,L�T� from their trends inTable I, and �iii� setting the surface roughness �rms� value asthe only free adjustable parameter since there is no preciseexperimental data on the rms values for the Si NWs surfacesinvestigated. The rms value obtained for each Si NW is givenin Table I. The very good agreement between the theoreticaland experimental results for these rather complex nanostruc-tures adds support to the wide scope and applicability of themodel proposed. Now, the strong decrease in ��T� of Si NWswhen the surface roughness increases can be explained. Thiseffect can be understood with reference to Eqs. �8�, �10�, and�15�. When the surface roughness parameter, h, increases, thesurface specularity, P�� ,� ,h�, decreases exponentially. Thisleads to a strong enhancement in the phonon-boundary scat-tering rate �B

−1 and consequently, to a significant decrease in��T�.

FIG. 7. �Color online� Thermal conductivity of rough silicon NWs. Sym-bols: experimental results. Solid lines: theoretical results obtained from thetheoretical approach described in the present work.



Although the present approach is more computationallydemanding than the commonly used Botlzmann transportequation approach which includes mainly phonon scatteringat the wire surfaces and neglects the nonresistiveprocesses,77,89,90 it allows understanding the effects of thevarious phonon mechanisms on ��T�, especially in circum-stances where the phonon-boundary and phonon-point-defects scattering are the dominant mechanisms. In Fig. 8 wepresent both the contributions of �1�T�=� jkB /83v j�kBT /��3I1,j and �2�T�=� jkB /83v j�kBT /��3I2,j to theoverall lattice thermal conductivity ��T� for the 115 nmrough Si NW. As can be noticed, at low T �say below�70 K� the I2�T� integrals may be omitted and thus thenumerical complexity for the calculation of ��T� of NWs canbe significantly reduced.

It is worth to note here that the model proposed in thissection for the size effect on the thermal conductivity ofNWs, which is based on the elastic continuum approxima-tion, holds only for NWs of diameters much larger than theatomic spacing. For the case of ultrathin NWs �L�4 nm�,recent atomistic simulation showed that, in contrast to whatwould be predicted from our model, the thermal conductivitybecomes insensitive to the NW diameter and phonon con-finement may lead to an increase in ��T� at low temperaturesdue to the occurrence of long wavelength acoustic phononsof very long mean free paths.91 On the other hand, the ato-mistic simulation is consistent with our model in describingthe roughness effect on the thermal conductivity of Si NWs.

XI. CONCLUSION

In summary, we have used the Boltzmann equation so-lution of Callaway, which discriminates between the nonre-sistive nature of the normal three-phonon processes and theresistive nature of the other phonon mechanisms, to establishformalism for the thermal conductivity that takes into ac-count the phonons incidence angles. The acoustic phononscattering processes have been represented by frequency-dependent relaxation time and the optical phonons decay intoacoustic phonons has been considered as acoustic phonons

generation rate counteracting the acoustic phonon scatteringrates. In addition, the commonly used expression for thephonon-boundary scattering has been corrected by includinga phonon specularity probability depending on the phononfrequency, incidence angle, and surface roughness. We havedemonstrated the importance of the corrections brought andwide scope of the formalism proposed with reference to re-ported data regarding the effect of the isotope compositionon the thermal conductivity of bulk Si samples and the ef-fects of size and surface roughness on the thermal conduc-tivity of Si NWs. The experimental thermal conductivitycurves, for all cases could be well described in the full tem-perature range. The formalism proposed for the thermal con-ductivity is flexible to include additional phonon scatteringmechanisms to describe the thermal conductivity of evenmore complex semiconductor structures with technologicalinterest.

ACKNOWLEDGMENTS

M. Kazan would like to acknowledge financial supportof “Région-Champagne Ardene” and “Agent National de laRecherche �ANR�.” S. Pereira and M. R. Correia would liketo acknowledge financial support by FCT, Portugal �GrantNo. PTDC/FIS/65233/2006�.

1I. Pomeranchuk, J. Phys. �Moscow� 6, 237 �1942�.2T. H. Geballe and G. W. Hull, Phys. Rev. 110, 773 �1958�.3J. R. Olson, R. O. Pohl, J. W. Vandersande, A. Zoltan, T. R. Anthony, andW. F. Banholzer, Phys. Rev. B 47, 14850 �1993�.

4L. Wei, P. K. Kuo, R. L. Thomas, T. R. Anthony, and W. F. Banholzer,Phys. Rev. Lett. 70, 3764 �1993�.

5J. E. Berman, J. B. Hastings, D. P. Siddons, M. Koike, V. Strojanoff, andS. Sharma, Synchrotron Radiat. News 6, 21 �1993�.

6S. R. Bakhchieva, N. E. Kekelidze, and M. G. Kekua, Phys. Status SolidiA 83, 139 �1984�.

7B. Abeles, Phys. Rev. 125, 44 �1962�.8J. W. Vandersande, Phys. Rev. B 15, 2355 �1977�.9J. E. Graebner, M. E. Reiss, L. Seibles, T. M. Hartnett, R. P. Miller, and C.J. Robinson, Phys. Rev. B 50, 3702 �1994�.

10D. Li, Y. Wu, P. Kim, L. Shi, P. Yang, and A. Majumdar, Appl. Phys. Lett.83, 2934 �2003�.

11A. I. Boukai, Y. Bunimovich, J. Tahir-Kheli, J.-K. Yu, W. A. Goddard III,and J. R. Heath, Nature �London� 451, 168 �2008�.

12R. Venkatasubramanian, E. Siivola, T. Colpitts, and B. O’Quinn, Nature�London� 413, 597 �2001�.

13A. I. Hochbaum, R. Chen, R. D. Delgado, W. Liang, E. C. Garnett, M.Najarian, A. Majumdar, and P. Yang, Nature �London� 451, 163 �2008�.

14P. Carruthers, Rev. Mod. Phys. 33, 92 �1961�.15D. A. Broido, M. Malorny, G. Birner, N. Mingo, and D. A. Stewart, Appl.

Phys. Lett. 91, 231922 �2007�.16J. Callaway, Phys. Rev. 113, 1046 �1959�.17M. G. Holland, Phys. Rev. 132, 2461 �1963�.18M. Asen-Palmer, K. Bartkowski, E. Gmelin, M. Carona, A. P. Zhernov, A.

V. Inyushkin, A. Taldenkov, V. I. Ozhogin, K. M. Itoh, and E. E. Haller,Phys. Rev. B 56, 9431 �1997�.

19D. T. Morelli, J. P. Heremans, and G. A. Slack, Phys. Rev. B 66, 195304�2002�.

20N. Mingo, L. Yang, D. Li, and A. Majumdar, Nano Lett. 3, 1713 �2003�.21N. Mingo, Phys. Rev. B 68, 113308 �2003�.22P. D. Thacher, Phys. Rev. 156, 975 �1967�.23R. Berman, Thermal Conductivity in Solids �Oxford University Press, Ox-

ford, 1976�.24W. S. Hurst and D. R. Frankl, Phys. Rev. 186, 801 �1969�.25G. J. Campisi and D. R. Frankl, Phys. Rev. B 10, 2644 �1974�.26D. P. Singh and Y. P. Foshi, Phys. Rev. B 19, 3133 �1979�.27J. M. Ziman, Can. J. Phys. 34, 1256 �1956�.28H. B. G. Casimir, Physica 5, 495 �1938�.

FIG. 8. �Color online� Contribution of �1�T� and �2�T� components to theoverall lattice thermal conductivity of ��T�. Symbols: experimental data onrough silicon nanowire with diameter of 115 nm. Dashed lines: contributionof �1�T� and �2�T�. Solid line: the total thermal conductivity ��T�, i.e.,�1�T�+�2�T�.



http://dx.doi.org/10.1103/PhysRev.110.773

http://dx.doi.org/10.1103/PhysRevB.47.14850

http://dx.doi.org/10.1103/PhysRevLett.70.3764

http://dx.doi.org/10.1002/pssa.2210830114

http://dx.doi.org/10.1002/pssa.2210830114




http://dx.doi.org/10.1063/1.1616981

http://dx.doi.org/10.1038/nature06458

http://dx.doi.org/10.1038/35098012

http://dx.doi.org/10.1038/35098012

http://dx.doi.org/10.1038/nature06381

http://dx.doi.org/10.1103/RevModPhys.33.92

http://dx.doi.org/10.1063/1.2822891

http://dx.doi.org/10.1063/1.2822891





http://dx.doi.org/10.1021/nl034721i






http://dx.doi.org/10.1139/p56-139

http://dx.doi.org/10.1016/S0031-8914(38)80162-2

29R. Berman, F. E. Simon, and J. M. Ziman, Proc. R. Soc. London, Ser. A220, 171 �1953�.

30J. M. Ziman, Electrons and Phonons �Oxford University Press, New York,1967�.

31W. Eisenmenger, in Phonon Scattering in Condensed Matter V, edited byA. C. Anderson and J. P. Wolfe �Springer, Berlin, 1986�, p. 194.

32M. Kazan, S. Pereira, M. R. Correia, and P. Masri, Phys. Rev. B 77,180302�R� �2008�.

33M. Kazan, S. Pereira, J. Coutinho, M. R. Correia, and P. Masri, Appl.Phys. Lett. 92, 211903 �2008�.

34P. G. Klemens, Encyclopedia of Physics, edited by S. Flügge �Springer-Verlag, Berlin, 1956�, Vol. 14, p. 198.

35P. G. Klemens, Proc. R. Soc. London, Ser. A 68, 1113 �1955�.36S. B. Soffer, J. Appl. Phys. 38, 1710 �1967�.37T. D. Ositinskaya, A. P. Podoba, and S. V. Shmegaera, Sverkhtverd. Mater.

14, 27 �1992�.38G. A. Slack, Phys. Rev. 122, 1451 �1961�.39G. A. Slack, Phys. Rev. 133, A253 �1964�.40G. Leibfried and E. Schlömann, Nachr. Akad. Wiss. Goett. II, Math.-Phys.

Kl. IIa, 71 �1954�.41R. O. Pohl, Phys. Rev. 118, 1499 �1960�.42B. K. Agrawal and G. S. Verma, Phys. Rev. 128, 603 �1962�.43R. Berman and J. C. F. Brock, Proc. R. Soc. London, Ser. A 289, 66

�1965�.44R. M. Kimber and S. J. Roger, J. Phys. C 6, 2279 �1973�.45B. Abeles, Phys. Rev. 131, 1906 �1963�.46M. Balkanski, R. F. Wallis, and E. Haro, Phys. Rev. B 28, 1928 �1983�.47B. Song, J. K. Jian, G. Wang, H. Q. Bao, and X. L. Chen, J. Appl. Phys.

101, 124302 �2007�.48M. Kazan, Ch. Zgheib, E. Moussaed, and P. Masri, Diamond Relat. Mater.

15, 1169 �2006�.49G. P. Srivastava, The Physics of Phonons �Adam Hilger, Bristol, 1990�.50H. M. Tütüncü and G. P. Srivastava, Phys. Rev. B 65, 035319 �2001�.51P. G. Klemens, Phys. Rev. 148, 845 �1966�.52J. M. Zhang, M. Giehler, A. Göbel, T. Ruf, M. Cardona, E. E. Haller, and

K. Itoh, Phys. Rev. B 57, 1348 �1998�.53P. G. Klemens, Physica B 316–317, 413 �2002�.54F. Widulle, T. Ruf, A. Göbel, I. Silier, E. Schönherr, M. Cardona, J. Ca-

machio, A. Cantarero, W. Kriegeis, and V. I. Ozhogin, Physica B 263–264,381 �1999�.

55L. Bergman, D. Alexson, P. L. Murphy, R. J. Nemanich, M. Dutta, M. A.Stoscio, C. Balakas, H. Shin, and R. F. Davis, Phys. Rev. B 59, 12977�1999�.

56M. Kazan, B. Rufflé, Ch. Zgheib, and P. Masri, Diamond Relat. Mater. 15,1525 �2006�.

57R. J. Elliott, J. A. Krumhansel, and P. L. Leath, Rev. Mod. Phys. 46, 465�1974�.

58W. Weber, Phys. Rev. B 15, 4789 �1977�.59M. Hofmann, A. Zywietz, K. Karch, and F. Bechstedt, Phys. Rev. B 50,

13401 �1994�.60J. C. Phillips, Phys. Rev. 166, 832 �1968�.61R. M. Martin, Phys. Rev. 186, 871 �1969�.62C. J. Glassbrenner and G. A. Slack, Phys. Rev. 134, A1058 �1964�.63T. Ruf, R. W. Henn, M. Asen-Palmer, E. Gmelin, M. Cardona, H.-J. Pohl,

G. G. Devyatych, and P. G. Sennikov, Solid State Commun. 115, 243�2000�.

64C. Mion, J. F. Muth, E. A. Preble, and D. Hanser, Appl. Phys. Lett. 89,092123 �2006�.

65N. V. Novikov, A. P. Podoba, S. V. Shmegara, A. Witek, A. M. Zaitsev, A.B. Denisenko, W. R. Fahrner, and M. Werner, Diamond Relat. Mater. 8,1602 �1999�.

66W. Gander and W. Gautschi, BIT 40, 84 �2000�.67A. Alshaikhi and G. P. Srivastava, Diamond Relat. Mater. 16, 1413 �2007�.68A. Alshaikhi and G. P. Srivastava, Phys. Rev. B 76, 195205 �2007�.69I. Ponomareva, D. Srivastava, and M. Menon, Nano Lett. 7, 1155 �2007�.70N. Yang, G. Zhang, and B. Li, Nano Lett. 8, 276 �2008�.71W. Cheng and S.-F. Ren, Phys. Rev. B 65, 205305 �2002�.72N. Nishiguchi, Phys. Rev. B 52, 5279 �1995�.73N. Bannov, V. Aristov, V. Mitin, and M. A. Stroscio, Phys. Rev. B 51,

9930 �1995�.74S. Yu, K. W. Kim, M. A. Stroscio, and G. J. Iafrate, Phys. Rev. B 51, 4695

�1995�.75S. G. Walkauskas, D. A. Broido, K. Kempa, and T. L. Reinecke, J. Appl.

Phys. 85, 2579 �1999�.76A. Khitun, A. Balandin, and K. L. Wang, Superlattices Microstruct. 26,

181 �1999�.77A. Svizhenko, A. Balandin, S. Bandyopadhyay, and M. A. Stroscio, Phys.

Rev. B 57, 4687 �1998�.78S. Yu, K. W. Kim, M. A. Stroscio, G. J. Iafrate, and A. Ballato, Phys. Rev.

B 50, 1733 �1994�.79X. Lü, J. H. Chu, and W. Z. Shen, J. Appl. Phys. 93, 1219 �2003�.80J. Zou and A. Balandin, J. Appl. Phys. 89, 2932 �2001�.81Z. V. Popvic, J. Spitzer, T. Ruf, M. Cardona, R. Notzel, and K. Ploog,

Phys. Rev. B 48, 1659 �1993�.82R. Bhadra, M. Grimsditch, I. K. Schuller, and F. Nizzoli, Phys. Rev. B 39,

12456 �1989�.83M. Grimsditch, R. Bhadra, and I. K. Schuller, Phys. Rev. Lett. 58, 1216

�1987�.84A. Tanaka, S. Onari, and T. Arai, Phys. Rev. B 47, 1237 �1993�.85C. Lanczos, The Vibrational Principles of Mechanics �University of Tor-

onto Press, Toronto, 1964�.86N. M. Wolcott, J. Chem. Phys. 31, 536 �1959�.87R. Enderlein, Phys. Rev. B 47, 2162 �1993�.88R. Enderlein, Phys. Rev. B 43, 14513 �1991�.89P. Hyldgaard and G. D. Mahan, Thermal Conductivity �Technomic, Lan-

caster, 1996�, Vol. 23, p. 172.90X. Lü, W. Z. Shen, and J. H. Chu, J. Appl. Phys. 91, 1542 �2002�.91D. Donadio and G. Galli, Phys. Rev. Lett. 102, 195901 �2009�.



http://dx.doi.org/10.1098/rspa.1953.0180


http://dx.doi.org/10.1063/1.2937113

http://dx.doi.org/10.1063/1.2937113

http://dx.doi.org/10.1063/1.1709746


http://dx.doi.org/10.1103/PhysRev.133.A253



http://dx.doi.org/10.1098/rspa.1965.0249

http://dx.doi.org/10.1088/0022-3719/6/14/008



http://dx.doi.org/10.1063/1.2749282

http://dx.doi.org/10.1016/j.diamond.2005.11.014




http://dx.doi.org/10.1016/S0921-4526(02)00530-6

http://dx.doi.org/10.1016/S0921-4526(98)01390-8



http://dx.doi.org/10.1103/RevModPhys.46.465





http://dx.doi.org/10.1103/PhysRev.134.A1058

http://dx.doi.org/10.1016/S0038-1098(00)00172-1

http://dx.doi.org/10.1063/1.2335972

http://dx.doi.org/10.1016/S0925-9635(99)00040-0

http://dx.doi.org/10.1023/A:1022318402393



http://dx.doi.org/10.1021/nl062823d

http://dx.doi.org/10.1021/nl0725998





http://dx.doi.org/10.1063/1.369576

http://dx.doi.org/10.1063/1.369576

http://dx.doi.org/10.1006/spmi.1999.0772





http://dx.doi.org/10.1063/1.1531810

http://dx.doi.org/10.1063/1.1345515





http://dx.doi.org/10.1063/1.1730391



http://dx.doi.org/10.1063/1.1427134


japplphys_thermal conductivity of silicon bulk and nanowires effects of isotopic

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