The effect of the electron–phonon coupling on the thermal conductivity of silicon nanowires

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IOP PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 24 (2012) 295402 (7pp) doi:10.1088/0953-8984/24/29/295402

The effect of the electron–phononcoupling on the thermal conductivity ofsilicon nanowiresWenhui Wan1, Bangguo Xiong2, Wenxing Zhang3, Ji Feng2 andEnge Wang2

1 Institute of Physics, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China2 School of Physics, Peking University, Beijing 100871, People’s Republic of China3 Department of Physics, Taiyuan University of Technology, Taiyuan 030024, People’s Republic ofChina

E-mail: egwang@pku.edu.cn

Received 23 April 2012, in final form 4 June 2012Published 22 June 2012Online at stacks.iop.org/JPhysCM/24/295402

AbstractThe thermal conductivity of free-standing silicon nanowires (SiNWs) with diameters from1–3 nm has been studied by using the one-dimensional Boltzmann’s transport equation. Ourmodel explicitly accounts for the Umklapp scattering process and electron–phonon couplingeffects in the calculation of the phonon scattering rates. The role of the electron–phononcoupling in the heat transport is relatively small for large silicon nanowires. It is found that theeffect of the electron–phonon coupling on the thermal conduction is enhanced as the diameterof the silicon nanowires decreases. Electrons in the conduction band scatter low-energyphonons effectively where surface modes dominate, resulting in a smaller thermalconductivity. Neglecting the electron–phonon coupling leads to overestimation of the thermaltransport for ultra-thin SiNWs. The detailed study of the phonon density of states from thesurface atoms and central atoms shows a better understanding of the nontrivial sizedependence of the heat transport in silicon nanowire.

(Some figures may appear in colour only in the online journal)

1. Introduction

Silicon nanowires (SiNWs), as a unique class of quasi-one-dimension material, have attracted much attention inthe past two decades because of their potential applicationsin many areas, ranging from thermoelectric materials tomicro-/nano-electronic devices [1, 2]. Surface chemistrycan modify SiNWs such that they have a sufficiently lowthermal conductivity to make them a potential candidate forthermoelectric materials [3–6]. On the other hand, surfacepassivated SiNWs are always semiconductor, and devicesmade of SiNWs are compatible with traditional Si-basedelectronic devices. For these reasons, SiNWs are of wideinterest for their applications in next generation electronicdevices [2].

Heat conduction is a critical issue in the study of SiNWs,for a few reasons. As on-going miniaturization of electronic

devices continually challenges Moore’s law, dissipation ofheat is vital to the efficiency of devices. In terms of synthesisand device fabrication, surface adsorption and roughnessshould be minimized to decrease phonon scattering. However,despite the successful synthesis of ultra-thin SiNWs withsizes of less than 10 nm [7], the fundamental mechanismsdetermining the phonon lifetimes in ultra-thin SiNWs, whichare significant for the improvement of thermal conductivity,are still unclear.

While experimental data on heat transport for ultra-thinSiNWs are lacking, theory and simulations give contradictingresults regarding the size dependence of thermal conductivityfor thick and thin SiNWs. Many theoretical works on thermalproperties of SiNWs have been published during the lasttwo decades. The main theoretical methods involved includethe Boltzmann transport equation (BTE) [8–13] or molecular

10953-8984/12/295402+07$33.00 c© 2012 IOP Publishing Ltd Printed in the UK & the USA

J. Phys.: Condens. Matter 24 (2012) 295402 W Wan et al

dynamics (MD) simulation [14–17]. The MD method canexplicitly simulate the lattice thermal transport of materialswhen the anharmonic effect is included. However, the sizeof system amenable to MD simulation is severely limitedby computational cost. For thicker SiNWs with diameters(d) above 35 nm, BTE calculations and experimental resultsare consistent with each other, confirming that the thermalconductivities of SiNWs, for example 25 W m−1 K−1 for56 nm thick SiNWs at room temperature [3], are muchlower than the one of bulk silicon (145 W m−1 K−1 [18]),and decrease continuously with their diameter because ofthe change of phonon dispersion due to the quantumconfinement effect and increasing phonon scattering dueto surface roughness [9, 12]. For ultra-thin SiNWs (d <10 nm) with perfect surfaces, it is found that a reductionof diameter does not necessarily lead to a lower thermalconductivity. Quite to the contrary, theoretical simulationpredicted that the thermal conductivity of ultra-thin SiNWscan be comparable to that of bulk Si [15, 17]. Theseanomalous behaviors have not been fully understood in MDsimulation. Besides, in real measurements, the charge carrier(electrons in conduction bands in our works) concentrationof a device is always nonzero at finite temperatures. Theinfluence of electron–phonon (e–ph) coupling on the thermalconductivity of SiNWs has never been considered before. Theeffect of e–ph coupling in electronic devices made of SiNWswith d smaller than 10 nm has proved to be the main constraintto the electronic mobility at room temperature [19]. It is,therefore, natural to suspect that the strong e–ph coupling mayalso play a nontrivial role in the heat transport in SiNWs.Our calculations show that the effect of electron–phononcoupling on the thermal conduction increases as the diameterof the SiNW decreases. Electrons in the conduction bandscatter low-energy phonons effectively where surface modesdominate. Neglecting e–ph coupling leads to overestimationthe thermal transport for ultra-thin SiNWs.

2. Theoretical model

When a temperature gradient is applied, the phonon distri-bution deviates from thermodynamic equilibrium, namely theBose–Einstein distribution. The phonon Boltzmann transportequation (BTE) is an equation describing the time evolutionof the phonon distribution function in the one-particle phasespace [20]. When the system reaches steady state, thespatial gradient of the phonon distribution in the presenceof a temperature gradient compensates the change arisingfrom collisions. Then we can determine the departure ofthe phonon distribution function from the equilibrium onewithin the relaxation time approximation to derive the thermalconductivity.

For a quasi-one-dimensional system, the thermal conduc-tivity at temperature T is calculated by BTE methods withinthe standard relaxation time approximation (RTA) [21],

κ =kB

2πS

∑s

∫x2ex

(ex − 1)2υ2

q,sτq,s dq. (1)

Here we define x = hωq,skBT , where ωq,s, υq,s and τq,s,

respectively, are the frequency, group velocity and lifetimeof the phonon mode with wavevector q and polarization s; hand kB are the Planck and Boltzmann constants, respectively;T is the temperature; S is the cross-sectional area of theSiNWs. The diffusive contribution of the phonons to theheat transport is ignored [22]. In calculating the phononlifetimes, we consider both phonon Umklapp scattering andelectron–phonon coupling processes, τ−1

= τ−1U +τ

−1e−ph. The

assumption is justified as there are no defects and no dopedatoms inside or around the surfaces of the SiNWs.

For the Umklapp scattering process [23] we treat theacoustic and optical modes separately. For an acousticphonon with a linear dispersion relation of ω ∼ q, we knowthe scattering rate of the acoustic phonons has a relationτ−1

U ∝ ω2 [24]. The second-order three-phonon scatteringprocesses [25] must be taken into account and give anextra scattering rate τ−1

U ∝ T2 to prevent divergence of thethermal conductivity in equation equation (1). As shownbelow, acoustic phonons with the quadratic relation ω ∼ q2

exist in SiNWs. It is verified that for these acoustic phononsthe scattering rate has a relation τ−1

U ∝ ω3/4 [21]. Therefore,the scattering rates for the acoustic modes are parameterizedsemi-empirically as, for Umklapp processes,

τ−1U =

{ATω2

+ CT2 for ω ∼ q

BTω3/4 for ω ∼ q2.(2)

For optical modes we adopt the standard Klemensformula [24]

τ−1U = ATω2. (3)

The A, B and C are parameters which can be obtained byfitting with MD results available in the literature [16, 17].

The scattering rate of phonons due to e–ph couplingcan be obtained from the standard first-order time-dependentperturbation theory. Here we assume that the electrons followthe equilibrium Fermi–Dirac distribution in the presence ofa small temperature gradient in a first-order treatment. Theexpression for τ−1

e−ph is given as [26]

τ−1e−ph =

4πh

∑k,b,b′|Mk,b,b′

q,s |2

· [f 0k,b − f 0

k+q,b′ ] · δ(εk+q,b′ − εk,b − hωq,s) (4)

where k is the electronic wavevector, b is the band index ofthe electron state of |k, b〉, εk,b and f 0

k,b are the correspondingelectronic energy and Fermi–Dirac distribution functionrespectively; Mk,b,b′

q,s is the matrix element for the scattering ofan electron from a state |k, b〉 to |k+ q, b′〉 with absorption ofa phonon of pseudo-momentum q with polarization s. δ is theDirac theta function which ensures the conservation of energy.The detailed expression of Mk,b,b′

q,s in equation (4) can be foundin [19]. An interpolation method related to the calculation ofτ−1

e−ph is adopted, as presented in the appendix.In contrast to former works, our efforts are focused on

increasing the thermal conduction of SiNWs as much aspossible to improve the dissipation of heat. In this work,

2

J. Phys.: Condens. Matter 24 (2012) 295402 W Wan et al

Figure 1. (a) Cross sections and side views of [100]-oriented 1 nm SiNWs with five primitive cells. The yellow spheres are silicon atoms,and the white ones are hydrogen atoms. (b) Some low frequencies of the phonon spectrum for [001]-oriented 1 nm SiNWs; q refers to thereduced wavevector in the first Brillouin zone of the SiNWs. Four acoustic modes are clearly visible. The two linear modes correspond tothe longitudinal and torsional modes. The two softer quadratic modes correspond to flexural modes, which are characteristic forquasi-one-dimensional systems. (c) The band structure of [100]-oriented 1 nm SiNWs with a direct bandgap of 3.07 eV.

atomistic model SiNWs are produced by carving a rod-shapedpart out of bulk Si (the cell parameter of the cubic unitcell is 5.431 A) along a specific crystal axis. These SiNWshave perfect surface facets such that we can neglect phononscattering due to surface roughness. SiNWs can be passivatedwith oxygen, hydrogen, or other chemical species [1, 7]. Inour models, the Si atoms on the surface are terminated withH atoms, keeping the coordination number of Si atoms asfour. We investigate the thermal conductivity of crystallineSiNWs grown along the [100] direction with d = 1–3 nm.The atomistic model of the SiO2 surface is more complex,so that we choose a simple hydrogen passivation mode toensure the semiconducting property of the SiNWs. Furtherab initio calculation finds that the length of the Si–Si bondsnear the surface has a small change compared with thatof the Si–Si bonds in the core zone of the SiNWs afterhydrogen passivation and no surface reconstruction occurs.So we consider SiNWs with perfect surface facets in ourcalculation.

3. The phonon dispersion and band structure ofSiNWs

A generalized Keating model proposed by Vanderbiltet al [27] is employed to describe the interactions betweenthe atoms of the SiNWs. This model captures both theharmonic and the anharmonic elastic properties of silicon,and provides a very good fit to the bulk phonon-dispersioncurves [18]. We have calculated the phonon-dispersion

relations of [100]-oriented cylindrical SiNWs with d =1–3 nm and an example of 1 nm thick SiNWs is displayed.Figure 1(a) shows the atomic structure of 1 nm thick SiNWsand figure 1(b) shows the corresponding phonon dispersionat low energy. Four acoustic modes in SiNWs are displayedin figure 1(b), in agreement with previous works [28]. Thetwo usual acoustic phonon modes with linear dispersioncorrespond to the dilatational mode along the wire axisand the torsional mode around the wire axis. Owing tothe quasi-one-dimensional nature of SiNWs, there are twoflexural modes whose energies are proportional to q2 near thezone center with polarization vectors perpendicular to the wireaxis.

The electronic band structures of the SiNWs aredescribed by a nearest-neighbor sp3d5s∗ tight-binding (TB)model developed by Niquet [29]. By introducing extrastrain-dependent on-site parameters in the sp3d5s∗ TB model,the influence of geometric distortion on the electron bandenergies and effective masses is considered over the entireBrillouin zone. The strain-dependent treatment ensures a goodtransferability of the TB parameters from bulk Si to SiNWs.[100]-oriented SiNWs passivated with hydrogen atoms aredirect-gap semiconductors. The bandgap of 1 nm thick SiNWsis 3.07 eV (figure 1(c)), much larger than that of bulksilicon (about 1.18 eV) in the present TB model [29]. Thebandgap decreases as the diameter of the SiNWs increases.The electron bands around the conduction band minimum(CBM) are flat. This means that the effective mass of electronsis large, leading to much stronger e–ph coupling in [001]SiNWs.

3

J. Phys.: Condens. Matter 24 (2012) 295402 W Wan et al

Figure 2. The temperature dependence of κph for SiNWs withd = 1 nm (black line), 2 nm (red line) and 3 nm (blue line) from200 to 600 K. The open black squares, red circles and blue trianglesare corresponding MD reference data for SiNWs with d = 1, 2 and3 nm, respectively in [17].

Table 1. The coefficients A, B and C used in the Umklappscattering rate.

d (nm) A (10−18 s K−1) B (10−2 s−0.25 K−1) C (106 s−1 K−2)

1 2.520 1.922 2.2222 3.150 2.403 2.5003 2.974 2.227 2.667

4. The thermal conductivity of SiNWs

Substituting the phonon dispersion into equation (1), weobtain the parameters A, B and C by fitting the MD resultsin [16, 17]. Since the MD simulation of thermal conductivityonly considered Umklapp scattering, we take τ−1

= τ−1U .

Hereafter, we shall call the thermal conductivity without e–phcoupling the intrinsic phonon thermal conductivity, κph. Theparameters A, B and C are adjusted to reproduce the κph curvefor each SiNW with different diameter. The fitting curvesare displayed in figure 2 and the coefficients A, B, C areshown in table 1. The good fitting confirms the validity of oursemi-empirical model.

We calculated the thermal conductivities of SiNWsconsidering Umklapp scattering and the electron–phononcoupling process κph+e−ph by substituting equations (2)–(4)into equation (1), using the numerically evaluated phonon andelectronic band structures and e–ph matrix elements. In orderto examine the effect of e–ph coupling we define a ratio, r, as

r = (κph − κph+e−ph)/κph × 100%. (5)

Here, we do not consider dopant scattering in SiNWs,as the average distance between dopants is estimated to be100 nm at an electron concentration (ne) of 1018 cm−3. Fromtable 2, we estimate the average group velocity of a phononas 104 m s−1, and the typical lifetime of a phonon is foundto be 1 ps from equation (2) to (4). The mean free path ofthe phonons is of the order of 10 nm. Moreover, the phononscattering rate of the dopants [9] has a relation τ−1

= ω4,

Figure 3. (a) Ratio of decline between the κph and κph+e−ph of thinSiNWs at 200 K, as a function of the ne from 1014 to 1019 cm−3.(b) Ratio of decline between the κph and κph+e−ph of thin SiNWs atan ne of 1018 cm−3 as a function of temperature from 200 to 600 K.In panels (a) and (b), the filled squares, circles and triangles refer toSiNWs with d = 1, 2 and 3 nm, respectively. (c) Diameterdependence of κph and κph+e−ph for thin SiNWs at 200 K withne = 1019 cm−3.

Table 2. Transverse (υT) and longitudinal (υL) acoustic soundvelocities of SiNWs from 1 to 3 nm.

d (nm) υT (km s−1) υL (km s−1)

1 3.084 5.2372 4.391 6.3783 4.858 6.764

compared with τ−1= ω for low-energy phonons in e–ph

coupling which can be derived from equation (4). Therefore,the dopant scattering can reasonably be ignored for the systemunder scrutiny in this work.

Figure 3(a) displays the r ratio at 200 K as a function ofelectron doping concentration, ne. It is found that the e–phcoupling is insignificant to the thermal conductivity at a lowne, but r becomes larger as ne increases. As mentioned above,the ne in SiNWs produced in experiment is about 1018 cm−3.At such a concentration, we find that neglect of e–ph

4

J. Phys.: Condens. Matter 24 (2012) 295402 W Wan et al

coupling will overestimate κ by about 5%–15%, as shownin figure 3(a). At the same ne, the e–ph coupling is strongerin 1 nm thick SiNWs than that in 2 and 3 nm thick SiNWs,which suggests a more notable e–ph coupling effect on κ inthinner SiNWs. Figure 3(b) displays r at ne = 1018 cm−3 asfunction of temperature. It is found that r decreases as thetemperature increases. The role of e–ph coupling in κ reducescompared with Umklapp scattering at higher temperatures. Inthis calculation the electronic thermal conductivity, verified byother electronic structure calculations, remains smaller than1 W m−1 K−1 for even 1019 cm−3 concentrations [30]. It isexpected that the temperature dependence of κph+e−ph regainsthe 1/T form at higher temperatures, where κph dominates asshown in figure 2. The value of r decreases with increasingtemperature more quickly in 3 nm thick SiNWs than that in2 nm ones, though they possess much the same κph. This againshows that the e–ph coupling is stronger in thinner SiNWs.

Although the e–ph coupling effect has been includedas one kind of phonon scattering mechanism, the κph+e−phof SiNWs with d = 1 nm is still larger than for biggerSiNWs which can be seen in figure 3(c). A similar resultcan also be found at other electron concentrations and othertemperatures. It should be noted that different positions of theaxis produce SiNWs with the same d but with different surfacestructures. We calculated the κph and κph+e−ph of SiNWs withd = 1–3 nm for various [100]-oriented SiNWs with differentpositions of the axis. In all cases we found that both κph andκph+e−ph of SiNWs with d = 1 nm are larger than for biggerSiNWs, so the d dependence of κph and κph+e−ph in otherultra-thin SiNWs will be the same as for the case mentionedabove. The anomalous size dependence of κph in SiNWs withd < 3 nm is universal.

In the size range of the nanowires considered in thiswork, the thermal conductivity increases as the diameterdeclines. This is opposite to the size dependence of thethermal conductivity in SiNWs with d > 35 nm. To explainthe results, we divided the κph into the contributions from thephonons with various energies at 300 K temperature, which isshown as dis(ω) in figure 4 as well as the density of states,g(ω). It is shown that only phonons with energy less than30 meV have nontrivial contributions to κph. Higher energyphonons can be safely ignored until higher temperature. Ifwe convert the summation in equation (1) into an integralover the phonon frequency, then it contains a factor of thephonon density of states (DOS), g(ω). To better understandthe evolution of g(ω) as function of the wire diameter, wecalculated the g(ω) by substituting the sum of delta functionsby the sum of Gaussian smearing functions with a full widthat half-maximum of 0.05 meV. g(ω) was normalized withthe number of phonon bands. It is clear that a large κph in1 nm thick SiNWs arises from a larger g(ω) in the rangeof 0–10 meV. We find, as a result of quantum confinement,that the phonon dispersion of 1 nm thick SiNWs is flatterthan that for thicker SiNWs, making a larger g(ω). The longwavelength modes present in thinner SiNWs give rise to largervalues of κph. As the diameter of the SiNW increases, theg(ω) of the low-energy phonon declines while the g(ω) of thehigh-energy phonon is found to increase.

Figure 4. (a) The thermal conductivity distribution functionwithout e–ph coupling as the phonon energy dis(ω) at 300 Ktemperature and (b) the density of states g(ω). In panels (a) and (b),the black, red and blue lines refer to SiNWs with d = 1, 2 and 3 nm,respectively.

However, the group velocity of the phonons is smallas a result of the flat phonon dispersion. The phonons willtravel more slowly although they have a longer lifetimewhich reduces the thermal conductivity for smaller diameterwires. We calculated the transverse (υT) and longitudinal (υL)acoustic sound velocities of SiNWs with phonon dispersionof the acoustic phonon modes with linear dispersion from 1 to3 nm, as displayed in table 2. It is found that the transverseacoustic and longitudinal acoustic sound velocities fall by31% and 22% as the diameters of SiNWs decrease from 3to 1 nm.

The e–ph coupling is also stronger in 1 nm SiNWs thanin thicker ones. For more detailed analysis, we present infigure 5(a) the dis(ω) of κph and κph+e−ph for 2 nm thickSiNWs with a doping concentration of ne = 1018 cm−3 andT = 300 K. The dis(ω) of SiNWs with other diametershave similar profiles. It is found that the electrons in theconduction band mainly shorten the lifetime of low-energyphonons, resulting in a reduction of κph. One-nanometer thickSiNWs have a larger g(ω) in the low-energy zone so that e–phcoupling has a stronger effect on the thermal conductivity. Itis therefore concluded that the effect of large g(ω) exceedsthe effect of a decrease in phonon group velocity and strongere–ph coupling, which results in a larger κph in thinner SiNWs.

5. Phonon density of states analysis

To explain the large g(ω) in thin SiNWs, we calculated thepartial density of states (PDOS) of SiNWs with d = 2 nm,to differentiate the contributions to the phonon DOS fromsurface atoms and central atoms. The PDOSs of SiNWs

5

J. Phys.: Condens. Matter 24 (2012) 295402 W Wan et al

Figure 5. (a) The distribution function dis(ω) of κph and κph+e−phrelative to the phonon energy of 2 nm thick SiNWs with ne of1018 cm−3 at 300 K. (b) The partial density of states gp(ω) and totalDOS g(ω) of 2 nm thick SiNWs. The black solid line refers to thetotal DOS, the red solid line refers to the PDOS from the centralatoms and the blue solid line refers to the PDOS from the surfaceatoms. The atomic vibration amplitude of one surface mode whichmakes a main contribution to the peak of the PDOS around 8 meVis shown in the inset. The darker atom has a larger amplitude ofvibration compared to the others.

with other diameters have similar profiles. Here we definedthe surface atoms as the hydrogen atoms and silicon atomsconnected with one or two hydrogen atoms; the other siliconatoms were regarded as central atoms. The PDOS is definedas [31, 32]

gp(ω) =1V

∑q,s

∑j

∑α=x,y,z

|εαq,s(j)|2δ(ω − ωq,s) (6)

where εαq,s(j) is the α component of displacement for a given jatom in the vibrational mode ωq,s. The sum over j includesatoms belonging to either the surface atoms or the rest ofthe atoms, i.e., the central atoms. The sum of the PDOSs forcentral and surface parts gives the total DOS.

One can see, from figure 5(b), that the PDOS for theatoms at the surface makes the principal contribution to thetotal DOS for phonon energies less than 10 meV. Analysisof the vibrational eigenvector of the modes in this energyzone shows that these modes are mainly surface modes. Theoscillation amplitudes for each atom of one surface modewhich makes a main contribution to the peak of the PDOSaround 8 meV are displayed in the inset of figure 5(b). It isfound the vibration amplitude decreases quickly away fromthe surfaces of the SiNWs [33, 34]. The PDOS from thecenter atoms is notable only in the range from 20 to 30 meV.Phonons with energy less than 20 meV play a primary rolein heat transport at 300 K in figure 5(a). It is remarkablethat the heat is mainly transported through the surfaces ofSiNWs with very small diameters. Then we can expect thatchemical modification of the cores of ultra-thin SiNWs mayhave a smaller influence on κph, but it changes the transport of

electrons more dramatically. This finding points to a possibledirection of micro-technology of electronic devices.

It is concluded that the large surface-to-volume ratio isthe likely reason for a larger κph for SiNWs with smaller d.Long wavelength modes with long lifetime present in SiNWswith d = 1 nm give rise to a better thermal conductor thanthicker SiNWs with d > 1 nm.

As the d of the SiNWs increases continuously, thesurface-to-volume ratio decreases and the contribution to thetotal DOS from surface modes declines. Vibrational modesconcentrated in the interior atoms begin to dominate the totalDOS. These modes exhibit a PDOS akin to that of bulksilicon. The role of e–ph coupling in thermal conductivity willproportionally decrease.

When d increases from 3 nm, κph should not descendcontinuously. It is found that g(ω) in the 0–10 meV energyrange decreases as d rises from 1 nm. g(ω) for high energyincreases and an increasing portion of the optical modesbegin to contribute to the heat transport. Meanwhile, thegroup velocities increase for not only acoustic modes but alsooptical modes continuously as the diameter increases. Thegroup velocity and DOS of high-energy phonons will havea positive effect in equation equation (1). We expect thatκph will increase when d goes over a critical value (whichcan be inferred from experimental results [3]). We noticethat the parameters used in Umklapp scattering do not varymuch for d = 2 and 3 nm. If we assume that SiNWs withd > 3 nm have almost the same Umklapp scattering parameteras for d = 3 nm, then substituting the phonon dispersion intoequation equation (1) we find that κph indeed increases whend is above 4 nm. A further MD simulation of SiNWs is neededto confirm the expectation.

We expect that the strong e–ph coupling in ultra-thinSiNWs will be reflected in Raman spectroscopy in the low-energy zone. The width of the peak in Raman spectroscopyis inversely proportional to the lifetime of the correspondingphonon modes. The width of the peak of low-energy phononsshould be away from the value calculated from Umklappscattering while the width of the peak of high-energy phononsdoes not change much compared to the intrinsic value if weprepare the sample with high purity.

6. Conclusion

In conclusion, we have studied the thermal conductivity ofSiNWs with diameters of d < 3 nm including Umklappscattering and e–ph coupling. We find that the role ofe–ph coupling in determining the thermal conductivity isvery important in ultra-thin SiNWs. Electrons can lowerthe intrinsic thermal conductivity further by scattering thesurface modes with low energy. The effect of e–ph couplingis more important at relatively high electron concentrationsand relatively low temperatures. In the future, SiNWs alongother directions and with different surface conditions will beconsidered. The e–ph coupling may be directional anisotropy.Moreover, in the present work, we consider the e–ph couplingbetween propagating phonons and electrons for SiNWswith perfect structures. The influence of lattice defects and

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J. Phys.: Condens. Matter 24 (2012) 295402 W Wan et al

impurities on the e–ph coupling effect will also be aninteresting topic for further study. We hope that our resultswill provide guidance in the manufacture of microelectronicSiNW-based devices in the future.

Acknowledgments

We gratefully acknowledge the support from the NationalScience Foundation of China (NSFC) through Projects10974238, 91021007 and 11174009, and the MOST throughProject 2009DFA01290.

Appendix

The scattering rate of phonons due to e–ph coupling, τ−1e−ph, is

given as

τ−1e−ph =

4πh

∑k,b,b′|Mk,b,b′

q,s |2· [f 0

k,b − f 0k+q,b′ ]q

· δ(εk+q,b′ − εk,b − hωq,s). (A.1)

The evaluation of equation (A.1) is difficult because of thepresence of the delta functions. The usual procedure is toreplace the delta function with a Gaussian function, but wehave found that the convergence of the calculations is slow.So we rewrite equation (A.1) as

τ−1e−ph =

4πh

∑k,b,b′|Mk,b,b′

q,s |2· [f 0

k,b − f 0k+q,b′ ]

· δ(εk+q,b′ − εk,b − hωq,s)

=4πh

∫ πl

−πl

Nl

2πdk

∑b,b′|Mk,b,b′

q,s |2

· [f 0k,b − f 0

k+q,b′ ] · δ(εk+q,b′ − εk,b − hωq,s)

=2Nl

h

∑{k,b,b′}

|Mk,b,b′q,s |

2· [f 0

k,b − f 0k+q,b′ ]

|∂(εk+q,b′−εk,b−hωq,s)

∂k |

. (A.2)

Here l is the length of the primitive cell of the SiNWs andN is the number of cells. The sum {k, b, b′} is the collection of|k, b〉 and |k + q, b′〉 states that are resolutions of the functionεk+q,b′ − εk,b − hωq,s = 0 for a given phonon |q, s〉.

In equation (A.2) we take advantage of the followingproperty of the delta function:∫ b

af (x)δ[g(x)] dx =

∑i

f (xi)

|g′(xi)|(A.3)

where the xi are the solutions of g(x) = 0 in the interval [a, b],and g′(xi) is the derivative of g(x). In simulation, we calculatethe electron band structure and phonon dispersion on a regulargrid of q and k points in the first Brillouin zone. The equationof εk+q,b′ − εk,b − hωq,s = 0 for a given phonon |q, s〉 is

solved by using a linear interpolation between successive qpoints. Typically, 300 q points for the thermal conductivityare sufficient for convergence and 300 k points to get a goodinterpolation in our simulation.

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