Shape and orientation effects on the ballistic phonon thermal properties of ultra-scaled Si nanowires
SShape and orientation effects on the ballistic phonon thermal properties ofultra-scaled Si nanowires
Abhijeet Paul, a) Mathieu Luisier, and Gerhard Klimeck School of Electrical and Computer Engineering, Network for ComputationalNanotechnology, Purdue University,West Lafayette, Indiana, USA, 47907. (Dated: 5 December 2018)
The effect of geometrical confinement, atomic position and orientation of Siliconnanowires (SiNWs) on their thermal properties are investigated using the phonondispersion obtained using a Modified Valence Force Field (MVFF) model. The spe-cific heat ( C v ) and the ballistic thermal conductance ( κ ball ) shows anisotropic variationwith changing cross-section shape and size of the SiNWs. The C v increases with de-creasing cross-section size for all the wires. The triangular wires show the largest C v due to their highest surface-to-volume ratio. The square wires with [110] orientationshow the maximum κ ball since they have the highest number of conducting phononmodes. At the nano-scale a universal scaling law for both C v and κ ball are obtainedwith respect to the number of atoms in the unit cell. This scaling is independentof the shape, size and orientation of the SiNWs revealing a direct correlation of thelattice thermal properties to the atomistic properties of the nanowires. Thus, en-gineering the SiNW cross-section shape, size and orientation open up new ways oftuning the thermal properties at the nanometer regime. a) Electronic mail: [email protected] a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug . INTRODUCTION The increasing variety of application of Silicon nanowires (SiNWs) ranging from MOSFETs to rechargeable batteries , thermoelectric (TE) devices , bio-sensors , solar cells , etc., re-quires a good understanding of the lattice thermal properties. Thermal requirements canbe very contrasting depending on the type of applications, for eg. MOSFETs will requireheat dissipation for better performance whereas thermoelectric devices require low thermalconductivity to maintain a good temperature gradient for higher TE efficiency .Improvement in the process technologies have led to the fabrication of SiNWs with differ-ent shapes, sizes and channel orientations . SiNWs with channel orientations along [100],[110] and [111] are the most commonly manufactured ones. At the nanometer scale, stronggeometrical confinement, atomic positions and increased surface-to-volume ratio (SVR) playsignificant roles in determining the thermal properties of the SiNWs .Thermal transport measurements using techniques like the 3 ω method and thermo-reflectance have led to a good understanding of the heat flow in large nanowires (diameter > .The phonon spectra of SiNWs can provide a theoretical estimate of heat transport throughultra-scaled structured since phonons (lattice vibrations) are responsible for carrying mostof the heat in semiconductors .In this paper we theoretically explore the effect of (i) cross-section geometry, (ii) cross-section size, and (iii) wire orientation of ultra-scaled SiNWs on the thermal properties suchas the ballistic thermal conductance ( κ ball ) and the specific heat ( C v ). Furthermore, analyt-ical expressions for the variation of these physical quantities with size for each cross-sectionshape and orientation are provided to allow for a compact modeling representation of ther-mal properties to be used in simulators like Thermal-Spice and Themoelectric modulesimulator .The thermal properties of SiNWs differ considerably from bulk Si . The transition fromthe particle to wave nature of heat transport with structural miniaturization calls for im-proved heat transport models. The traditional continuum models for thermal conductivityby Callaway and Holland are based on the Debye limit for phonons, sound velocity andmany other parameters, which render these models quite cumbersome at the nanoscale, asdiscussed by Mingo et al . The dependence of continuum models on a large set of fit-2ing parameters make them unsuitable for predicting the lattice thermal properties withdimensional scaling. This limitation can be overcome by atomistic models which automati-cally take into account the effects of structural miniaturization like geometrical confinement,orientation effects, cross-sectional shape and surface-to-volume ratio effects .Theoretical estimates of the thermal conductivity and the specific heat using phononspectra have been addressed in the literature in the past. Calculations of thermal conduc-tivity using the full phonon spectra of large to medium sized SiNWs along certain specificorientations have been performed . The influence of surface roughness on nanowires hasbeen studied in Refs. . A study of specific heat in [111] SiNWs is reported in Ref. .Mingo et. al theoretically bench-marked the thermal conductivity of large SiNWs (diameter ≥ and also studied the effect of amorphous coating on the thermal conductivityof SiNWs . A complete study of orientation effects on the thermal conductivity of SiNWhas been provided in Ref. . The thermal conductivity of Si nano-clusters and hollow Sinanowires has also been studied.In this work we utilize a semi-empirical phonon model to perform thermal calculations.The phonon dispersion is obtained using a Modified Valence Force Field (MVFF) model .The continuum models though computationally simpler and faster, lack the properphysics to extend them to nanostructures due to the use of open fitting parameters .Thefirst principles based models are limited to extremely small structures (W < . At thesame time the MVFF model also explains the experimental phonon and elastic propertiesin zinc-blende materials very well .The emphasis of this work is to purely understand the atomistic effects on the thermalproperties of ultra-scaled SiNWs such as the effect of the surfaces and atomic coordinationnumber. This understanding is further translated into analytical equations allowing easyusage for other thermal modeling like thermo-electric circuit simulations which require C v and κ l as input . Scattering effects are dominating at the nanometer scale in determiningthe lattice thermal conductivity. However, scattering effects have been neglected in thepresent study in order to understand the geometric and atomistic effects which will be3therwise convoluted by scattering. Also the ballistic phonon transport can be supportedin SiNWs with width smaller than 20nm as pointed in Ref. . The scattering effects couldbe studied as a future work.The paper is organized as follows. Section II provides a brief description of the MVFFmodel for the calculation of phonons in SiNWs, the calculation of thermal properties(Sec. II B) and details of the SiNWs used for the study (Sec. II C). The effect of cross-section shape, size and wire orientation on the specific heat and thermal conductance arediscussed in Sec. III A and Sec. III B, respectively. This is followed by a discussion on theatomistic effects on the thermal properties in Sec. III C. Conclusions are summarized inSec. IV. II. THEORY AND APPROACHA. MVFF Phonon model
In the MVFF model, the phonon frequencies are calculated from the forces acting onatoms produced by finite displacements of the atoms from their equilibrium positions in acrystal . First the total potential energy of the solid (U) is estimated from the restoringforce(F). In the MVFF model, U is approximated as , U ≈ (cid:88) i ∈ N A (cid:34) (cid:88) j ∈ nn ( i ) U ijbs + j (cid:54) = k (cid:88) j,k ∈ nn ( i ) (cid:0) U jikbb + U jikbs − bs + U jikbs − bb (cid:1) + j (cid:54) = k (cid:54) = l (cid:88) j,k,l ∈ COP i U jiklbb − bb (cid:35) , (1)where N A , nn ( i ), and COP i represent the total number of atoms in one unit cell, numberof nearest neighbors for atom ‘i’, and the coplanar atom groups for atom ‘i’, respectively.The first two terms U ijbs and U jikbb represent the elastic energy obtained from bond stretchingand bending between atoms connected to each other . The terms U jikbs − bs , U jikbs − bb , and U jiklbb − bb represent the cross bond stretching , the cross bond bending-stretching , and the coplanarbond bending interactions, respectively. The detailed procedure for obtaining the phononspectra in bulk Si and NWs are outlined in Ref. .4 . Lattice thermal properties The complete phonon dispersion provides information about the thermal properties ofnanostructures . The constant volume temperature (T) dependent specific heat ( C v ( T ))can be evaluated using the following relation , C v ( T ) = (2)( k B m uc ) · (cid:88) n,q (cid:104) ( (cid:126) ω n,q k B T ) · exp( − (cid:126) ω n,q k B T )[1 − exp( − (cid:126) ω n,q k B T )] (cid:105) [ J/kgK ] , where k B , (cid:126) and m uc are the Boltzmann’s constant, reduced Planck’s constant, and themass of the SiNW unit cell in kg, respectively. The quantity ω n,q is the phonon frequencyassociated with the branch ‘n’ and crystal momentum vector ‘q’.For a semiconductor slab/wire with a small temperature difference ∆ T between its twoextremities, the thermal conductance ( κ ball ) is obtained using Landauer’s method as , κ ball ( T ) = (cid:126) (cid:90) ω max T ( ω ) · M ( ω ) · ω · (3) ∂∂T (cid:104) (exp( (cid:126) ωk B T ) − − (cid:105) · dω [ W/K ] , The term M ( ω ) is the number of phonon modes at frequency ω and T ( ω ) is the trans-mission for each mode. For ballistic conductance each mode transmits with a probability of1 while for coherent scattering dominated conductance the transmission value is less than 1. C. Si Nanowire details
In this study, four types of cross-section shapes for [100] SiNWs have been considerednamely, (a) circular, (b) hexagonal, (c) square and (d) triangular (Fig. 1). Square SiNWswith [110] and [111] channel orientations have been studied too (Fig. 2). The feature sizeis determined by the width parameter W. The value of W is varied from 2 to 6 nm. Theconfinement direction of the SiNWs are along Y and Z. The heat transport direction isalong X. The surface atoms are allowed to vibrate freely without any passivating species.These wires are assumed to have a tetrahedral geometry. It has been shown that wires with5iameter below 2nm tend to lose the tetrahedral structure due to surface pressure andinternal strain.
III. RESULTS AND DISCUSSION
In this section the results on the effect of cross-section shape, size and orientation onthe thermal properties of SiNWs are presented and discussed. All the thermal quantitiesare calculated at 300K. However, the analysis holds for any temperature (T) where theanharmonic phonon effects are small. For T < T
Debye the anharmonic interactions are quitesmall . For bulk Si, T Debye ∼ and it further increases for SiNWs . A. Specific heat in SiNWs
Influence of the shape and size on C v : The C v of SiNWs increases with decreasing cross-section size in all the wire shapes (Fig. 3a). The size dependence of C v can be approximatedby the following relation , C v ( W ) = C bulkv + AW , (4)where, A is a fitting parameter extracted from the linear fit of the numerical simulations.Table I shows the value of C bulkv and A for each geometry. As W → ∞ (increasing cross-section size), the C v of all the SiNWs converges to a fixed value of ∼
681 J/kg.K whichis reasonably close to the experimental C v value for bulk Si ( ∼
682 J/Kg.K as provided inRefs. ). The triangular wires show the maximum C v for all the W value, whereas theother shapes show similar C v values at any given cross-section size (Fig. 3a).The plot of ∆ C v (= C wirev − C bulkv ) vs. SVR (SVR = Total surface Atoms/Total atoms inunit cell) shows a linear behavior (Fig. 3b), which can be represented as, C wirev ≈ m c × SVR + C bulkv , (5)where m c describes the additional contribution to the C v of the SiNWs with increasingsurface-to-volume ratio. The value of m c is positive for all the wire shapes (Fig. 3b) whichcorroborates the fact that specific heat increases with increasing surface area . Different6oordination number of surface atoms for the various cross-section shapes result in different m c values which depict the atomistic effect on the C v value in ultra-scaled SiNWs.The C v increase with decreasing W can be attributed to two phenomena, (i) phononconfinement due to small cross-section size and (ii) an increased surface-to-volume ratio(SVR) in smaller wires . With increasing geometrical confinement (smaller cross-sectionsize) the phonon bands are more separated in energy which makes only the few lowerenergy bands active at a given temperature (see Eq. 2). Thus, more energy is needed toraise the temperature of the smaller wires.The shape dependence of the C v can be understood from Eq. 5. The variation of (i) SVR,and (ii) m c with W for different shapes decide the eventual C v order. Figure 4(a) showsthat triangular wires have the maximum SVR while the other shapes have similar SVR at afixed W. The increasing SVR results in a higher phonon density of states (DOS) associatedwith the wire surface, which further enhances the specific heat with decreasing wire cross-section . The square wires provide the largest surface contribution to C v as depicted bythe variation in m c (∆ C v /SVR) (Fig. 4b). An optimal value of SVR and m c in SiNWswill maximize the C v . The SV R × m c value has the following order, triangle ( ∼ > hexagonal ( ∼ > square ( ∼ > circular ( ∼ C v due to the highest SVR. However, the trends for SVR ( SV R sq < SV R hex < SV R ci )and m c ( m sqc > m hexc > m cic ) are opposite for the other shapes, hence resulting in almostsimilar C v values. The C v values have the following shape order in [100] SiNWs: triangular > hexagonal ≈ square ≈ circular.Influence of orientation on C v : The specific heat is also a function of the SiNW orientation(Fig. 5a). The C v varies inversely with W, similar to the trend extracted from differentshapes (Eq. 4). The width parameters (Eq. 4) for the variation of C v with orientation areprovided in Table II. The ∆ C v value again shows a linear variation with SVR for differentwire orientations (Fig. 5b). The C v has the following trend with orientation for differentcross-section sizes, C v > C v > C v . This trend can be explained again by lookingat the impact of (i) SVR and (ii) m c ( Eq. 5) on the overall C v value. The SVR shows thefollowing order with W, SVR > SVR > SVR (inc. of ∼ × from [100] to [111]) asillustrated in Fig. 6a. However, m c shows the following order with W, m c > m c > m c (dec. of ∼ × from [100] to [111]) (Fig. 6b). The larger surface to volume ratio plays themain role in deciding the C v trend for SiNWs with different orientations. . Ballistic thermal conductance of SiNWs The thermal conductance indicates how a structure can carry heat. For high thermo-electric efficiency (ZT) a small thermal conductance is needed whereas for CMOStransistors a high κ l is preferred to evacuate the heat. The cross-section shape, size andwire orientation of SiNWs can be used to tune their the thermal conductance. Influence of the shape and size on κ ball : The ballistic thermal conductance ( κ ball ) for 4different shapes (Fig. 1) of [100] SiNWs are calculated as shown in Fig. 7a. The κ ball can befitted according to the following size (W) relation, κ ball ( W ) = κ (cid:16) Wa (cid:17) d , (6)where a is the silicon lattice constant (0.5431 nm), d is a power exponent, and κ is aconstant of proportionality. The values of ‘d’ and κ for different wires shapes are providedin Table III. The value of d varies between 1.92 and 2.011 which implies that κ ball has asimilar size dependence for all the wire shapes. However, the pre-factor value ( κ ) reflectsthe shape dependence. This pre-factor has the same ordering as the thermal conductanceordering (Fig. 7a and Table III).The ballisitic thermal conductance exhibits a linear behavior with the number of atomsper unit cell (NA) as depicted in Fig. 7b. This linear relation can be approximated by thefollowing equation, κ ball ≈ m k × N A + κ ball (0) , (7)where the slope m k represents the average contribution from each atom in the unit cell to κ ball and κ ball (0) is the thermal conductance at NA = 0. The value of κ ball (0) is zero withinnumerical error ( κ ball (0) ≈ e −
7) which is expected for NA = 0. The value of m k takes intoaccount the surface, shape and atomic effects since the calculation procedure involves thecomplete phonon dispersion. This relation shows a direct correlation of the atomistic effectsto the ballistic thermal conductance.The size dependence can be explained by the fact that the larger wires have (i) morephonon sub-bands resulting in higher number of modes ( M ( ω ) in Eq. 3) and (ii) a higheracoustic sound velocity which is responsible for a larger heat conduction in these SiNWs .The shape dependence can be explained as an interplay of two effects, (i) the total numberof atoms (NA) present in the unit cell of SiNWs, and (ii) the average contribution of every8tom towards κ ball . For a fixed cross-section size W, the NA ordering is N A sq > N A ci >N A hex > N A T eri (Fig. 8a). The total number of phonon branches are 3 × NA, due tothe three degrees of freedom associated with each atom . The number of phonon modes( M ( ω )) is directly proportional to the phonon energy sub-bands. The contribution per atomto the thermal conductance ( m k ) stays almost constant with the wire cross-section size fora given shape (Fig. 8b). Since, the values of m k are quite similar for all the cross-sectionshapes , the ordering of NA with shape governs the dependence of κ ball on the cross-sectionshape.Influence of wire orientation on κ ball : The thermal conductance is anisotropic in SiNWswith the following order, κ l > κ l > κ l (Fig. 9a). This result is similar to the onereported in Ref. . The κ l value exhibits a linear variation with NA for all the wire orientations(Fig. 9b). The width parameters for the thermal conductance (Eq. 6 ) for different wireorientations are provided in Table IV. The order of the ballistic thermal conductance withW for different orientations can be understood by the product of NA × m k ( P nm ). The NAhas the following variation, N A > N A
N A (Fig. 10a), whereas m k depicts thefollowing order m k > > m k > m k (Fig.10b). These two orders are opposite to eachother. However, the product shows the following order, P nm > P nm > P nm ). Thus, [110]wires give the highest κ l due to the optimal value of NA and m k .An important point to note is that κ ball is expected to decrease further in smaller wires dueto phonon scattering by other phonons, interfaces and boundaries which are neglected inthis present study. The main idea here is to understand the geometrical effects on the phonondispersion and the lattice thermal properties of these small nanowires which is attributedto (i) the modification of the phonon dispersion, and (ii) phonon confinement effects in thecoherent phonon transport regime. C. Discussion
In this work, all the thermal properties are shown to scale with W. Also NA depends onW as follows,
N A ∝ W γ , (8)where γ >
0. So using Eq. (8), (4) and (6) can be recasted in terms of NA as,9 C v ( N A ) = C · ( N A ) − γ (9)= C · ( N A ) − η κ ball ( N A ) = K · ( N A ) dγ (10)= K · ( N A ) ρ , where, C and K are the pre-factors. Thus, a universal power law can be derived for thethermal properties depending on the number of atoms per unit cell (NA) which representsthe atomistic effect on the thermal quantities. In these SiNWs, 1.98 ≤ γ ≤ ρ and η , ρ ∈ [0 . , .
03] (11) η ∈ [0 . , .
50] (12)The variation in the thermal conductance with NA is illustrated on a log-log scale in Fig.11a. All the SiNWs depict almost the same power law with an average exponent value of ∼ C v with NA isplotted on a log-log scale in Fig. 11b. All the SiNWs used in this study show the samepower law (Eq. 9) with an average exponent of -0.51, which is in the limit derived in Eq.12. Thus, the thermal quantities show a universal power law behavior with the number ofatoms in the unit cell (NA) irrespective of the details of the unit cell. The details of shapeand orientation are embedded in the pre-factors C and K (Eq. 9 and 10).The relationship of C v and κ l to the shape, size and orientation of SiNWs have beenprovided explicitly in Eqs. (4), (5), (6), and (7). These closed form analytical expressionsare very handy for the compact modeling of the thermal and thermoelectric properties ofSiNWs . Since these expression are derived from physics-based model, they capture theimportant geometrical and atomistic effects, thus enabling fast modeling of realistic systems. IV. CONCLUSIONS
We have shown the application of the MVFF model for the calculation of the thermalproperties of SiNWs. It has been shown that at the nanometer scale these thermal prop-10rties are quite sensitive to the wire cross-section size, shape, and orientation. Analyticalexpressions for the size dependence of the thermal properties of SiNWs of different cross-section shape and channel orientation have been provided. They can be used as componentfor the compact modeling of the thermal properties of ultra-scaled SiNWs. It has beendemonstrated that all the SiNWs follow a universal power law for the specific heat and thethermal conductance which reveals the impact of the atomistic details on these properties.The triangular SiNWs show a high C v and low κ l , thus making them good candidates forthermoelectric devices. The [110] oriented square Si nanowires are better in terms of heatdissipation due to their high thermal conductance and are therefore good candidates fortransistors from a heat management point of view. ACKNOWLEDGMENTS
The authors acknowledge financial support from MSD Focus Center, under the FocusCenter Research Program (FCRP), a Semiconductor Research Corporation (SRC) entity,Nanoelectronics Research Initiative (NRI) through the Midwest Institute for NanoelectronicsDiscovery (MIND), NSF (Grant No. OCI-0749140) and Purdue University. Computationalsupport from nanoHUB.org, an NCN operated and NSF (Grant No. EEC-0228390) fundedproject is also gratefully acknowledged.
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TABLE I. Width Parameter for C v in [100] SiNWs at T = 300K.Shape C bulkv Aunits → ( J/kg · K ) ( J · nm/kg · K )Circular 681.4 47.82Hexagon 681.2 53.58Square 680.9 52.66Triangular 681.0 73.43TABLE II. Width Parameter for C v in SiNWs at T = 300K.Orientation C bulkv Aunits → ( J/kg · K ) ( J · nm/kg · K )[100] 680.9 52.66[110] 681.5 43.03[111] 681.2 39.4TABLE III. Width Parameter for κ ball in [100] SiNWs at T = 300K.Shape κ ( nW/K ) dCircular 0.133 1.92Square 0.141 1.96Triangular 0.062 1.97Hexagon 0.097 2.01TABLE IV. Width Parameter for κ ball in SiNWs at T = 300K.Orientation κ ( nW/K ) d[100] 0.141 1.96[110] 0.204 1.87[111] 0.129 1.88 IGURES
FIG. 1. Projected unit cell structures of free-standing [100] oriented silicon nanowires with (a)Circular, (b) Hexagonal, (c) Square, and (d) Triangular cross-section shapes. Width and height ofthe cross-section are defined using a single width variable W (width = height). These structuresare at W = 2nm.FIG. 2. Projected unit cells of square free-standing SiNWs with (a) [100], (b) [110], and (c) [111]wire axis orientation. The width and the height of the SiNWs are defined using W. Here, W =2nm. IG. 3. (a) Dependence of the specific heat ( C v ) on cross-section shape and size in [100] SiNWs.(b) Variation in ∆ C v ( = C v − C bulkv ) with SVR for all the [100] SiNW shapes. C bulkv for eachshape is given in Table. I.FIG. 4. (a) Surface-to-volume ratio (SVR) for different cross-section shape and size [100] SiNWs.(b) Incremental contribution to the specific heat with SVR for different cross-section shape [100]SiNWs.FIG. 5. (a) Dependence of the specific heat ( C v ) on the orientation of square SiNWs. (b) Variationin ∆ C v ( = C v − C bulkv ) with SVR for all the SiNW orientations. C bulkv are taken from Table. II. IG. 6. (a) Surface-to-volume ratio (SVR) for different wire orientations of square SiNWs. (b)Incremental contribution to the specific heat with SVR for different orientations of SiNWs.FIG. 7. (a) Effect of cross-section shape and size on the ballistic thermal conductance of [100]SiNWs. (b) Variation in κ ball with the total number of atoms per unit cell (NA) for differentcross-section shapes.FIG. 8. (a) The number of atoms (NA) in [100] SiNW unit cell for different cross-section size andshapes. (b) Contribution to κ l per atom for different cross-section shapes. IG. 9. (a) Effect of size on the ballistic thermal conductance in SiNWs with different orientations.(b) Variation in κ ball with the total number of atoms in the unit cell (NA) for different wireorientations.FIG. 10. (a) The number of atoms (NA) with W in one SiNW unit cell for different orientations.(b) Contribution to κ l per atom for different SiNW orientations. IG. 11. (a) The number of atoms (NA) in one [100] SiNW unit cell for different cross-section sizeand shapes. (b) Contribution to κ l per atom for different cross-section shapes.per atom for different cross-section shapes.