Finite-size effects in the phonon density of states of nanostructured germanium: Acomparative study of nanoparticles, nanocrystals, nanoglasses and bulk phases

D. Sopu,∗ and J. Kotakoski,1 K. AlbeInstitut fur Materialwissenschaft, Technische Universitat Darmstadt,

Petersenstr. 32, D-64287 Darmstadt, Germany(Dated: April 21, 2011)

Finite-size effects in the phonon density of states (PDOS) of nanostructured materials (nanopar-ticles, nanocrystals, embedded nanoparticles and nanoglasses) are systematically studied by meansof molecular dynamics simulations, where germanium was used as representative reference material.By comparing with the PDOS of single crystalline and amorphous structures the physical origins ofadditional or vanishing vibrational modes or frequency shifts were identified. Our findings are dis-cussed in terms of phonon confinement effects, structural disorder and surface stresses and providea general view on the interplay of nanostructural features and lattice vibrations.

PACS numbers: Valid PACS appear here

I. INTRODUCTION

Phonons generally affect the thermal, optical, mechan-ical and electrical properties of materials. While thephonon density of states (PDOS) is primarily a functionof the local atomic structure, it is also sensitive to atomiclevel stresses and the microstructure. If the characteristiclength scales get down to the nanometer regime, devia-tions of the PDOS from the corresponding bulk structurehave been observed, both in experiment and theory.2–7

The microscopic origins of the observed size-effects are,however, very often not clear.

Size effects due to the variation of the surface to vol-ume ratio are expected to scale with D−α (α ≈ 1),8

where D describes the intrinsic structural length, likegrain or particle diameter. Phonon confinement due tofinite particle size10 or quantum phenomena originatingfrom the discreteness of electron states and band-gapvariation with nano-structures size,11 are both scalingwith D−2. The presence of grain boundaries (GBs)2,5,12

or surfaces8 leads to a locally lower atomic density andhigher structural disorder in comparison to the bulk ma-terial resulting in additional vibrational modes at low andhigh frequencies.6 Confinement effects affect the opticalmodes at the Brillouin zone center (q = 0), since morephonons with q 6= 0 (away from Brillouin zone center)contribute to the PDOS and lead to a red shift in Ramanspectra.3,10,18 This has been reported for different nano-scale materials like nanocrystals,3,18,19 nanoparticles em-bedded in glass4,20 or other nanostructures21. The red-shift observed in Raman spectra, however, has also beenexplained by tensile surfaces stresses,16 while the blue-shift has been attributed to compressive surface stresses(or pressure).17 This interpretation was supported byatomistic simulations8 that revealed the influence of sur-face stresses on the PDOS, resulting in a shift of theentire PDOS towards higher frequencies for the case ofmetals.

In view of the partially controversial interpretationof the PDOS of nanostructured materials, a detailedstudy revealing the role of the various size effects ap-

pears to be a worthwhile task. In this study, we calcu-late the PDOS separately for nanoparticles, nanocrystals,nanoparticles embedded in a glass and a nanoglass, whichis a class of material that is synthesized by consolidat-ing glassy nanoparticles.22 We determine how the vibra-tional modes are affected by changes in the nanostructureas compared to crystalline and amorphous bulk materi-als. Moreover, we show how the PDOS changes withnanoparticle size through surface stress and confinementeffects. Germanium is chosen as a prototype of a cova-lently bonded material with crystalline and amorphousmodifications. After a short description of the computa-tional methods we start with the case of a free crystallinenanoparticle. Then a particle embedded in a glass struc-ture is studied, before we investigate a nanocrystallinemicrostructure. Finally, we discuss the case of a glasswith microstructure, a so called nanoglass, where inter-nal planar defects are present.

II. METHODS AND COMPUTATIONALDETAILS

Classical molecular dynamics (MD) simulations werecarried using the MD code PARCAS.23 Tersoff’s inter-atomic potential for germanium was used in this studyto represent the interactions between the atoms.24 Forcomparison, some of the results were repeated using theStillinger-Weber potential.26

The phonon densities of states (PDOS) were computedfrom the Fourier transform of the velocity autocorrelationfunction (VAC):31

g(ω) =

∫dteiωt

〈~v(t) � ~v(0)〉〈~v(0)2〉

(1)

where v(0) is the average velocity vector of a particle atinitial time, v(t) is the average velocity at time t, and ωis the frequency. In order to exclude the contribution ofatomic rearrangements or diffusional jumps to the VAC,the system was equilibrated in an isobaric-isothermal

2

0

1

2

3

4

5

6

7

8

9

10

L Γ K X W L K W X Γ

Fre

qu

ency

[T

Hz]

FIG. 1. Calculated phonon dispersion for diamond Ge crystaland its calculated density of states (in arbitrary units).

ensemble. In consequence, all PDOS approach zero atω = 0. We calculated the average of Eg. (1) over all par-ticles and for ten different independent cases. All PDOScalculations were performed at 50 K and for times longenough to obtain a good frequency resolution.

Using this technique, we first calculated the PDOS ofGe in a single-crystalline diamond structure (see Fig. 1right). In order to understand the features of the PDOS,we also calculated the phonon dispersion for the diamondstructure (Fig. 1 left) using the frozen phonon method.27

Four crystalline peaks are identified in the PDOS, namelythe transverse acoustic (TA), longitudinal acoustic (LA),longitudinal optical (LO) and transverse optical (TO)modes, in the order of ascending energy (see Fig. 1 right).

The calculated phonon dispersion and PDOS for theGe crystal is in fair agreement with experimental and the-oretical data.28,30 However, Tersoff-type potentials tendto overestimate the frequencies of the transverse acous-tic (TA) modes at the zone boundary and also the op-tical modes, since this model is too stiff for angulardistortions.29 Despite of this fact, the Tersoff potentialis suitable for studying finite-size effects in the PDOSof nanostructured Ge on qualitatively level, since it isknown to reproduce the structural properties of crys-talline, liquid and amorphous states of germanium.25

The PDOS can be used for determining thermody-namic properties of a material. For instance, the specificheat at constant volume Cv can be directly calculatedwithin the harmonic approximation as

Cv(T ) = 3kB

ωMAX∫0

(~ωkBT

)2e

~ωkBT

(e~ω

kBT − 1)2g(ω)dω (2)

where g(ω) is the PDOS (see Eq. 1), ~ the Plack’s con-stant divided by 2π, and kB the Bolzmann’s constant.

III. RESULTS AND DISCUSSIONS

A. Free Nanoparticles

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0 2 4 6 8 10

Ph

on

on

DO

S

THz

Tersoff potential

Single-crystalline bulk

Nanoparticles:

d=7.5 nmd=3.9 nm

(a)

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0 2 4 6 8 10

Ph

on

on

DO

S

Inner atomsSurface atoms

THz

Single-crystalline bulk

Nanoparticle d=3.9 nm:

(b)

Phonon D

OS

THz

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0.50

0.60

8 9 10 8 9 10

Tersoff Stillinger-Weber

Single-crystalline bulk

Nanoparticles:

d=7.5 nmd=3.9 nm

0.40

(c)

FIG. 2. Phonon density of states for crystalline Ge-nanoparticles: (a) PDOS of a single crystal compared withthe PDOS of nanoparticles with a diameter of d=3.9 nm andd=7.5 nm. (b) PDOS of a single crystal compared with thepartial PDOS of the surface and center atoms of a nanopar-ticle with a diameter of d=3.9 nm. (c) Partial PDOS for thecenter atoms in nanoparticles with diameter of d=3.9 nm andd=7.5 nm The left panel shows the calculation for a Tersoff-potential with surface stresses, while the right panel showsthe same calculation using a Stillinger-Weber potential with-out surface stresses and therefore no frequency shift.

3

In the first set of calculations, free non-supported par-ticles were studied, which for simplicity were created asspherical cuts of the bulk phase and relaxed at 50K tem-perature before the PDOS was analyzed. The result isshown in Fig. 2(a), where the PDOS of the single crys-tal is compared with the PDOS of nanoparticles withdiamaters of 3.9 nm and 7.5 nm. Essentially, two ma-jor features can be observed, namely the broadening ofthe acoustical branch and a red-shift of the entire distri-bution which gets more pronounced as the particle sizedecreases. The enhancement of the acoustical modes atlower frequency can be directly attributed to the pres-ence of weakly bonded surface atoms, characterized by alower coordination.

By removing all atoms with a diamond-like configu-ration from the PDOS calculation through a commonneighbor analysis (CNA),32 it becomes obvious that ad-ditional acoustic modes are due to surface atoms (see Fig.2(b)), only, while the presence of surface atoms also leadsto a slight blue shift in the optical modes.

The atoms located in the inner part of the particle, incontrast, exhibit a clear red-shift over the entire phononfequency range. Here, in principle surface stress or con-finement effects could serve as an explanation.33,34 In or-der to discriminate between both, we have recalculatedthe PDOS using another interatomic potential for Ge,namely the Stillinger-Weber format. By construction,this potential has no surface stress and therefore we onlyexpect to see the result of phonon confinement effects. In-deed, the result (see Fig. 2(c)) shows no variation of thepeak maximum of the optical modes. Only the shoulderat high frequencies is shifted and can therefore be at-tributed to a confinement effect due to the lack of vibra-tional modes close to the Γ point. This can be understoodby considering the wave vector |~q | ≈ π/D, where D isthe particle diameter. Small values for q are only possi-ble, if D is very large. If D is in the nanometer regime,however, the limiting case q → 0 can not be reached.Therefore, phonons with q close to zero are missing fromthe PDOS (see Fig. 9).

In return, the red-shift of the optical modes ob-served with the Tersoff potential can unambigously beattributed to strain effects due to surface stresses. Incase of Tersoff’s Ge potential surface stresses are ten-sile, leading to a slight lattice expansion correspondingto an increase in interatomic bond length of about 0.1 %,which indicates a softening of interatomic force constant.All following calculations are conducted with the Ter-soff potential only, since this potential provides a non-negligible and tensile surface stress in line with experi-mental data.14,15,35

B. Embedded Nanoparticles

In a next step we have studied the role of interfacestresses by studying the case of a nanoparticle embeddedin a glassy matrix. For this setup we expect to see an

overlapp of the contributions of the glassy matrix withthe modes of the embedded nanoparticle, on which inter-face stresses are acting.

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0 2 4 6 8 10

Bulk glass

Ph

on

on

DO

S

THz

Single-crystalline bulk

d=7.8 nmd=6.7 nm

d=8.9 nmEmbedded Nanoparticle:

(a)

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0 2 4 6 8 10

Pho

non

DO

S

Interface atoms

Bulk glass

Free vs. embedded nanoparticle (d=7.8 nm) :

THz

Surface atoms(free nanoparticle)

(embedded nanoparticle)

(b)

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0 2 4 6 8 10

Pho

non

DO

S

Free nanoparticle

THz

Free vs. embedded nanoparticle (d=7.8 nm) :(inner atoms)

Embedded nanoparticle

(c)

FIG. 3. (a) Total PDOS for three different nanoparticles withdiameter of d=6.7 nm, d=7.8 nm and d=8.9 nm embeddedin a glass, as compared to the perfect crystal and the bulkglass. (b) Partial PDOS for interface and surface atoms of anembedded and free nanoparticle, respectively, in comparisonto the bulk glass. (c) Partial PDOS for inner atoms of a freeand an embedded nanoparticle. In both cases the nanoparticlesize is d=7.8 nm.

The bulk glass was prepared by rapid quenching ofa melt (quench rate 5 K/ps) leading to an amorphousstructure. In the resulting glass individual particles ofdifferent sizes were introduced. This time cubic shapes

4

were chosen, in order to better control the thickness ofcrystal-glass interfaces. Prior to the PDOS calculations,each structure was again relaxed at 50K in order to formphysically reasonable interfaces between the glass and theparticles.

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0 2 4 6 8 10

re d s h ift

con n e m e n t effe ct

Phonon D

OS

THz

Single-crystalline bulk

d=7.8 nmd=6.7 nm

d=8.9 nmEmbedded Nanoparticle:(inner atoms)

FIG. 4. Partial PDOS for different nanoparticles embeddedin the glass (only diamond atoms) as compared to the perfectcrystal. The inset shows a magnification of the peak of theoptical modes.

In Fig. 3(a), the PDOS for three different cubicnanoparticles immersed in the glass are displayed andcompared with the PDOS of the perfect crystal and thebulk glass. The total number of atoms for all three struc-tures is about 64000. By decreasing the particle size, wefind an increased PDOS in the low frequency and a de-creased PDOS at the high frequency peak. The increaseof the acoustic modes can be easily explained by the in-crease of the relative amount of the glassy phase with de-creasing crystallite size. The PDOS of a glass displays acharacteristic enhancement of the low frequency phononmodes due to the higher volume when compared with thecrystaline phase.

Next, we checked if the vibrational modes of theglassy matrix extend into the inner part of the embed-ded nanoparticle. For this, the PDOS of an embeddednanoparticle and a free nanoparticle, both with the samegeometry and diameter of d=7.8 nm, were calculated.First, we studied how the glassy matrix affects the PDOSof the crystal-glass interface and compared to the PDOSof the surface and bulk glass. In Fig. 3(b), it can be seenthat the partial PDOS of the interface atoms of the em-bedded nanoparticle differs strongly from the PDOS ofthe surface atoms of a free nanoparticle, and resemblesmore the PDOS of a bulk glass. This can be explainedby the fact that glassy vibrational modes extend in thecrystal-glass interfaces. Interface atoms are those whichhave nearest neighbor atoms in a diamond-like and glassyconfiguration, respectively. The calculated partial PDOSof the inner atoms of the embedded and free nanoparticleshow no significant difference (Fig. 3(c)). Therefore, we

can conclude that vibrational modes of the glassy matrixdo not significantly affect the inner part of the embeddednanoparticle.

In fact, the nanoparticles which are embedded into theglassy structure show also the effect of homogeneous lat-tice expansion (shift of the PDOS towards lower frequen-cies). This can be seen from the partial PDOS of onlyinner atoms of the nanoparticles and as a function ofsize (Fig. 4). In analogy to the nanoparticle in vacuum,under free surface conditions, the nanoparticles in glassexpand due to the interface stress which also in this caseis tensile. This behavior can be rationalized by consid-ering that the glassy phase has a higher volume whencompared with the crystalline phase.

The disappearing shoulder on the main optical peak isagain attributable to the confinement effect, which hasbeen discussed in detail in the previous section (see insetin Fig. 4).

C. Nanocrystal

The next setup studied was a nanocrystalline mi-crostructure which can be understood as an ensemble ofnanoparticles connected by ”glassy-like” grain boundaryareas. All structures were build by means of the Voronoitesselation method.36 Samples were prepared with 16grains of 7.8 nm size, 54 grains of 5.2 nm size, 128 grainsof 3.9 nm size and finally 5488 grains of 1.1 nm size. Allnanocrystals were initially relaxed at 500 K for 100 psand then cooled to 50 K prior PDOS calculations.

In Fig. 5(a) we present the PDOS as calculated forthese nanocrystals and for the perfect crystal as well asthe bulk glass. With decreasing particle size, the maxi-mum of the low frequency region shifts slowly to lower fre-quencies and the peaks become broader. These featuresare connected to the ratio between the number of atomsin the grain boundaries and the number of atoms in thebulk-like regions. Due to the higher atomic disorder, theGBs show a signature similar to a glass. Additionally,GBs are characterized by an enhanced free volume pref-erentially leading to low frequency contributions to thePDOS.

In order to check the validity of our interpretation, wecalculated the PDOS separately for atoms in GBs and inthe grains. Fig. 6(a) shows the partial PDOS for grainswith different sizes. As can be seen, the changes of PDOSfor the grains in this nanocrystaline structure are essen-tially identical to the case of the nanoparticles immersedin the glass. Also in this case (inset of Fig. 6(a)), onecan easily see the general shift towards lower frequencieswith decreasing grain size. As mentioned in the previoussection this behavior is indicative for a tensile interfacestress. Previous simulation studies on Cu and Ni 5,12

could not capture shifts in the PDOS with grain size.This is because fcc metals do not exhibit optical modeswhere finite-size effects due to surface stresses and con-finement are most prominent. Moreover, the interatomic

5

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Phonon D

OS

THz

Nanocrystal with grains of size:

d=5.2 nmd=3.9 nm

d=7.8 nm

d=1.1 nm

Single-crystalline bulkBulk glass

(unstable)

(a)

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0 100 200 300 400 500 600 700 800 900

Exce

ss h

c[k

B]

T[K]

Nanocrystal with grains of size:

d=5.2 nmd=3.9 nm

d=7.8 nm

(b)

FIG. 5. (a) Total PDOS for nanocrystals with different grainsizes as compared to the perfect crystal and a bulk glass. (b)Excess specific heat for three nanocrystals with grain sizes ofd=3.9 nm, d=5.2 nm and d=7.8 nm as compared to a bulkcrystal.

potentials used in these studies exhibit only a weak com-pressive surface stress, which has a minor influence onthe sparsely populated acoustic modes. This is also thereason why a recent experimental study reported a closesimilarity of the PDOS of nanograins in nanocrystallineFe90Zr7B3 and α-Fe single crystals.6

On the other hand, it can clearly be seen that theGBs are the strongest contributors to the increase in theacoustic modes of the PDOS (see Fig. 6(b)) due to ageneral broadening of the PDOS in those regions. En-hancement of the phonon modes both at low and highenergy for GB atoms has been reported also earlier.6,9

The increase of the acoustic low frequency modes inthe PDOS will cause a direct change in the specific heatand the related thermal properties. In Fig. 5(b), to-gether with the PDOS, we plot the excess specific heatfor the three different nanocrystals as compared to the

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Phonon D

OS

re d sh ift

conne m e nt effe ct

THz

Single-crystalline bulk

Nanocrystal with grains of size: d=5.2 nmd=3.9 nm

d=7.8 nm(excluding GB)

(a)

start

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Phonon D

OS

THz

Nanocrystal with grains of size:d=7.8 nmd=5.2 nmd=3.9 nmd=1.1 nm

Bulk glass

(only GB)

(unstable)

(b)

FIG. 6. (a) Partial PDOS for three nanocrystals: a) for grainsas compared to the perfect crystal (the inset shows a magnifi-cation of the peak of the optical modes), and (b) for GB andresulting amorphous structure of NC with 1.1 nm grains ascompared to the the bulk glass.

perfect crystal. With decreasing the grain size, or di-rectly increasing the GB fraction, excess low frequencymodes will be added, causing an increase in the excessspecific heat.

Finally, it should be noted that grain sizes below1.1 nm lead to unstable grains and result in an amor-phous structure. This is in a good agreement with thesmallest possible grain size for this material (1-2 nm) ob-served in experiments.37,38 The PDOS of this amorphousstructure is different from the one calculated for a bulkglass; the acoustic branch is shifted to a lower frequencyand is broader. This is due the excess free volume inthe amorphous state. This new structure has an atomicvolume of VNC ≈24.86 A3, whereas for the bulk glassthe atomic volume is Vglass ≈23.03 A3. The differenceis about 8% and in good agreement with our previousresults,22 where we have shown that it is possible to in-

6

ject an excess free volume up to 8% at room temperaturein a Ge glass.

D. Nanoglass

Finally, we have studied the case of a nanoglass. Thisnew type of non-crystalline solid is generated by consol-idating nanometer glassy powder, and is characterizedby interfaces with an excess free volume.22 Because thisprocess involves the use of high pressure for powder com-paction, also the atomic structure of the inner part ofglassy droplets is affected, and in consequence its PDOSis changed compared to the case of bulk glass preparedby rapid quenching of a melt. Therefore, in order tostudy the size effect on the PDOS as a function of theinterface fraction and free volume localized in the inter-faces, a simplified glass cubic model (64000 atoms) witha parallel set of glass-glass interfaces of identical thick-ness was studied. The interfaces were randomly dilutedby 10%, and relaxed at 300 K. The density distributionthroughout the sample was monitored during the simu-lation. After relaxation, interfaces with 8% dilution werefound. The calculated interface energy is 1.65 J/m2.

We calculated the PDOS for two different nanoglasssystems. The first one had one interface and the otherone three interfaces with the same thickness (1 nm) andthe same percentage of 8% free volume. In Fig. 7(a),we see how the acoustic peak shifts to a lower frequencywith increasing number of interfaces. The free volumefrom the interfaces acts as a low frequency source for thephonon modes. In other words, an increase in free vol-ume leads to an increase of low frequency modes in thePDOS. No shifting was observed in the optical modes,only a decrease in the optical peak amplitude due to thePDOS normalization (i.e. with respect to other peaks).This can be explained by a lower interface stress in ananoglass compared to the interface stress calculated toa nanocrystalline structure. We found no considerablelattice expansion under interfaces stress, and, therefore,we expect no significant change in the PDOS of a nano-glass.

Fig. 8(b) shows the related excess specific heat withrespect to the bulk glass. It exhibits a pronounced max-imum at 80 K, which arises from the acoustic modeswith lower frequencies2 and grows with the number ofinterfaces. The optical modes provide only a minor con-tribution to specific heat.

In order to check if phonon confinement depends onthe interface thickness in nanoglasses, a similar procedurewas followed for two cubic nanoglass structures with thesame size; one with one interface of 3 nm, and anotherone with 3 interfaces of 1 nm. The total free volumewas approximately the same in both cases. It can beseen in Fig. 8, that no considerable difference in PDOSis observed for these cases. Furthermore, the excess spe-cific heat of the two structures nearly superpose with ex-tremely small differences (see Fig. 7(b)). This indicates

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DO

S

1 interface of 3nm3 interfaces of 1nm

THz

Bulk glassNanoglass:

(a)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 100 200 300 400 500 600 700 800 900

Exce

ss h

c[k

B]

3 interfaces of 1 nm

T[K]

1 interface of 1nmNanoglasses:

1 interface of 3 nm

(b)

FIG. 7. (a) PDOS of a bulk glass compared with the PDOSof two nanoglasse: one with one interface of 1 nm and anotherwith three interfaces of 1 nm. All interfaces have the samepercentage of 8% free volume. (b) Excess specific heat forthese two nanoglasses and a third nanoglass with one interfaceof 3 nm as compared to the bulk glass.

that no extra phonon modes arise from the size effect. Insummary, the origin of low frequency phonon modes isonly due to the enhanced free volume in the interfaces.

IV. CONCLUSIONS

In this study, we have investigated the lattice vibra-tions for different nanostructures for Ge as a prototypeof a covalently bonded material. We identified whichnanostructural discontinuities affect the phonon modes,realized a complete description of size effects on PDOS,and displayed the specific heat anomaly. We find threesize effects (see Fig. 9) which change the PDOS as follow:

1. Nanostructural discontinuities (surface atoms,GBs, interfaces) characterized by a lower density

7

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Pho

non

DO

S

THz

Bulk glassNanoglasses:

1 interface of 1 nm3 interfaces of 1 nm

FIG. 8. PDOS for bulk glass compared with the PDOS of twonanoglasses, first with one interface of 3 nm, and second withthree interfaces of 1 nm. All interfaces have the same per-centage of 8% free volume. The total amount of free volumeis the same for both nanoglasses.

in comparison with the bulk crystal cause an en-hanced population of acoustical modes with lowfrequency, and a general broadening of PDOS dueto the higher structural disorder.

2. Tensile (compresive) surface stresses result in ashift of the entire PDOS to lower (higher) frequen-cies.

3. Confinement due to the finit particle size causedthe disappearance of several optical modes at theBrillouin zone center (q = 0).

We systematically studied how these three size effectsinfluence the PDOS of different nanostructured materi-als (nanoparticles, nanocrystals, embedded nanoparticlesand nanoglasses).

Overall, all discontinuities (e.g. GBs, interfaces, sur-faces) in nanostructures will introduce vibrational modeswith low frequencies that directly affect the thermalproperties of the material. Beside those discontinuities,the PDOS changes also with nanoparticle size throughthe influence of surface stresses and confinement due tothe particle.

V. ACKNOWLEDGMENT

The authors acknowledge the financial support ofthe Deutsche Forschungsgemeinschaft (DFG) throughproject Al-578-6. A DAAD-PPP travel grant is alsoacknowledged. Computing time was made available atHHLR Frankfurt and Darmstadt, as well as by CSCJulich. The authors acknowledge Peter Agoston for fruit-ful discussions.

LΓ

KX

WL

KW

XΓ

Bulk

FIG. 9. Size effects on PDOS: a) Nanostructural discontinu-ities, b) surface stress, c) confinement due to the particle.

VI. REFERENCES

8

∗ sopu@mm.tu-darmstadt.de1 Present address: Department of Physics, University ofHelsinki, PO Box 43, 00014 Helsinki, Finland.

2 D. Wolf, J. Wang, S. R. Phillpot, and H. Gleiter, Phys.Rev. Lett. 74, 4686 (1995).

3 C. C. Yang and S. Li, The Journal of Phys. Chem. B 112,14193 (2008).

4 W. S. O. Rodden, C. M. S. Torres, and C. N. Ironside,Semicond. Sci. Technol. 10, 807 (1995).

5 P. M. Derlet, R. Meyer, L. J. Lewis, U. Stuhr, and H.Van Swygenhoven, Phys. Rev. Lett. 87, 205501 (2001).

6 S. Stankov et al., Phys. Rev. Lett. 100, 235503 (2008).7 A. Valentin, J. See, S. Galdin-Retailleau, and P. Dollfus,

J. Phys.: Condens. Matter 20, 145213 (8pp) (2008).8 A. Kara and T. S. Rahman, Phys. Rev. Lett. 81, 1453

(1998).9 A. Kara, A. N. Al-Rawi and T.S. Rahman, J. Comput.

Theor. Nanosci.1, 216 (2004).10 H. Richter, Z. P. Wang, and L. Ley, Solid State Commun.

39, 625 (1981).11 L. Pizzagalli, G. Galli, J. E. Klepeis, and F. Gygi, Phys.

Rev. B 63, 165324 (2001).12 C. Hudon, R. Meyer, and L. J. Lewis, Phys. Rev. B (Con-

densed Matter and Materials Physics) 76, 045409 (2007).13 R. Shuttleworth, Proceedings Of The Physical Society Of

London Section A 63, 444 (1950).14 S. Tsunekawa, K. Ishikawa, Z. Q. Li, Y. Kawazoe, and A.

Kasuya, Phys. Rev. Lett. 85, 3440 (2000).15 D. Shreiber and W. A. Jesser, Surf. Sci. 600, 4584 (2006).16 A. Korotcov, H. P. Hsu, Y. S. Huang, D. S. Tsai, and K. K.

Tiong, Crystal Growth & Design 6, 2501 (2006).17 J. Groenen, C. Priester, and R. Carles, Phys. Rev. B 60,

16013 (1999).18 S. K. Gupta and P. K. Jha, Solid State Commun. 149,

1989 (2009).

19 Y.-T. Nien et al., Mater. Lett. 62, 4522 (2008).20 D. S. Chuu, C. M. Dai, W. F. Hsieh, and C. T. Tsai, J.

Appl. Phys. 69, 8402 (1991).21 O. O. Mykhaylyk, Y. M. Solonin, D. N. Batchelder, and

R. Brydson, J. Appl. Phys. 97, 074302 (2005).22 D. Sopu, K. Albe, Y. Ritter, and H. Gleiter, Appl. Phys.

Lett. 94, 191911 (2009).23 K. Nordlund, Comput. Mater. Sci. 3, 448 (1995).24 J. Tersoff, Phys. Rev. B 39, 5566 (1989).25 J. K. Bording and J. Taftoo, Phys. Rev. B 62, 8098 (2000).26 T. A. Weber and F. H. Stillinger, Phys. Rev. B 31, 1954

(1985).27 R. Yu, D. Singh, and H. Krakauer, Phys. Rev. B 43, 6411

(1991).28 S. Wei and M. Y. Chou, Phys. Rev. B 50, 2221 (1994).29 L. J. Porter, J. F. Justo, and S. Yip, J. Appl. Phys. 82,

5378 (1997).30 H. Wang, Chem. Phys. 344, 299 (2008)31 J. M. Dickey and A. Paskin, Phys. Rev. 188, 1407 (1969).32 J. D. Honeycutt and H. C. Andersen, J. Phys. Chem 91,

4950 (1987).33 D. C. Wallace, Thermodynamics of Crystals (Dover Publi-

cations, 31 East 2nd Street, Mineola, N.Y. 11501, 1998).34 P. Agoston and K. Albe, Phys. Chem. Chem. Phys. 11,

3226 (2009).35 M. I. Ahmad and S. S. Bhattacharya, Appl. Phys. Lett.

95, 191906 (2009).36 V. Yamakov, D. Wolf, S. R. Phillpot, and H. Gleiter, Acta

Mater. 50, 61 (2002).37 T. Nieh and J. Wadsworth, Scr. Metall. Mater. 25, 955

(1991).38 L. L. Araujo et al., Phys. Rev. B 78, 094112 (2008).