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Encyclopedia ofNanoscience andNanotechnology

Phonon Connement inNanostructured Materials

Akhilesh K. Arora, M. Rajalakshmi, T. R. Ravindran

Indira Gandhi Centre for Atomic Research, Kalpakkam, India

CONTENTS

1. Introduction2. Optical Phonons3. Results4. Acoustic Phonons5. Summary

GlossaryReferences

1. INTRODUCTIONThere is considerable current interest in the physics ofnanostructured materials in view of their numerous tech-nological applications [1, 2] in a variety of areas such ascatalysis [3], magnetic data storage [4], ferrouids [5], softmagnets [6], machinable ceramics and metallurgy [7], non-linear optical and optoelectronic devices [8], and sensors[9]. In addition, obtaining an understanding of the proper-ties of nanostructured materials is of interest from a funda-mental point of view. Only a proper understanding of thedependence of a given property on the grain/particle size canlead to design/tailoring of the nanostructured material forthe related application. It is also important to understandwhen a material could be considered as nanostructured.Although one can in principle classify materials with grainsize less than 1000 nm as nanostructured, several propertiessuch as optical [10] and vibrational [11] properties do notdiffer much from the corresponding bulk value unless thegrain/particle size is less than typically 20 nm. In view of thisit is reasonable to treat a material with a grain size smallerthan a certain value as nanostructured only if the propertyof interest differs from the bulk value at least by a few per-cent. It is also possible that a material with nanometer grainsize may behave as nanostructured for a specic propertywhile it could act like bulk for other properties. In additionto the grain size, the properties of the nanostructured mate-rials may sometimes depend on the method of their synthe-sis. Generally nanostructured materials are synthesized in

one of the three forms: (a) as isolated or loosely connectednanoparticles in the form of powder [12], (b) as compos-ites of nanoparticles dispersed in another host [13], or (c)as compact collection of nanograins as pellets [14] or thinlms [15]. The last form is also called nanophase material.This chapter reviews the vibrational properties of the vari-ous forms of nanostructured materials.Phonons are quanta of atomic vibrations in crystalline

solids. In a simple monoatomic solid with only one atom perprimitive cell (for example, -iron, copper), one can haveonly three acoustic phonon branches corresponding to thethree degrees of freedom of atomic motion. On the otherhand, for monoatomic solids with two atoms per primitivecell such as diamond, magnesium, or diatomic compoundssuch as GaAs, one also has three optic phonon branches inaddition to the three acoustic phonons [16]. In compoundswith a greater number of atoms and complex crystal struc-tures, the number of optic phonons is more than three. Ifthe crystal unit cell contains N atoms, then 3N degrees offreedom result in 3 acoustic phonons and 3N 3 opticalphonons. These phonons can propagate in the lattice of asingle crystal as a wave and exhibit dispersion dependingon their wavelength or equivalently their wavevector in theBrillouin zone [17]. Phonon propagation is interrupted whena grain boundary is encountered in a polycrystalline mate-rial. In an isolated grain the phonon can get reected fromthe boundaries and remain conned within the grain. How-ever, from the point of view of phonons, a well-crystallizedpolycrystalline sample with several micrometer grain size canbe treated as a bulk/innite crystal for all practical purposes.The consequences of phonon connement are noticeable inthe vibrational spectra only when the grain size is smallerthan typically 20 lattice parameters.The atomic vibrational frequencies in crystalline solids

range from zero to about 100 THz. A more commonunit to describe the vibrational frequencies is wavenumber(cm1 obtained by dividing the actual frequency by thevelocity of light or by inverting the wavelength. Acousticphonons have frequencies from zero to about a few hundred

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Encyclopedia of Nanoscience and NanotechnologyEdited by H. S. Nalwa

Volume 8: Pages (499512)

500 Phonon Connement in Nanostructured Materials

wavenumbers while the optic phonons have higher frequen-cies. Hence the vibrational spectra could be probed usinginfrared absorption/reectivity or using Raman spectroscopy[18]. The wavevector of the IR photon for these energies isof the order of 102103 cm1. On the other hand, in a Ramanscattering experiment the magnitude of scattering vector is2k0 sin(/2), where k0 is the wavevector of the incident lightand is the scattering angle. Thus the maximum value of thescattering vector could at best be 2k0 (corresponding to thebackscattering geometry), which has a value 5104 cm1for visible light. Hence the wavevector probed by either ofthese two techniques is much smaller than the wavevectorq of the full phonon dispersion curve, which extends up tothe boundary of the Brillouin zone (/a 108 cm1, wherea is the lattice parameter). Thus these techniques sampleonly the optical phonons close to the zone center (q 0).This q 0 selection rule is essentially a consequence of theinnite periodicity of the crystal lattice [19]. However, if theperiodicity of the crystal lattice is interrupted, as in the caseof nanocrystalline materials, this rule is relaxed and phononsaway from the Brillouin zone center also contribute to thephonon lineshape observed in spectroscopic measurements.This can be qualitatively explained in the following manner.Consider the phonon dispersion curve of a typical diatomicsolid as shown in Figure 1. For a particle/grain of size d,the phonon wavefunction must decay to a very small valueclose to the boundary. This restriction on the spatial extentof the wavefunction, via a relationship of the uncertaintyprinciple type, results in an uncertainty q /d in thewavevector of the zone-center optical phonon and a cor-responding uncertainty 2 in its frequency [20]. Now thelight scattering can take place from quasi-zone-center opti-cal phonons with wavevectors q up to /d. This results inasymmetric broadening of the phonon lineshape. In additionto the changes in the optical phonon lineshape, connementof acoustic phonons also leads to other interesting changesin the spectra. A quantitative formalism of phonon conne-ment will be discussed in subsequent sections.

1.1. Dimensionality of Connement

It is important to distinguish between the dimensionality ofthe system and the dimensionality/degree of connement.A bulk material is a 3D system and is unconned; thatis, dimensionality/degree of connement is zero. The rst

q0 2/a

0 2

(q

)

q/d

Figure 1. Schematic representation of the optical phonon dispersioncurve and the range q of the wavevectors probed in Raman scatteringfrom a nanoparticle of diameter d. 2 is the corresponding range ofphonon frequencies that contribute to the rst-order Raman scattering.

level of connement occurs in single- and multilayer thinlms grown using layered deposition on substrates. Semi-conductor superlattices, single-quantum well structures, andmultiple-quantum well structures [21] are well-known exam-ples of 2D systems because the phonons and charge carriersare conned within a plane, say, the x-y plane; however,the degree of connement is 1D because phonons andcharge carriers are restricted along the z direction. Simi-larly, 2D connement occurs in nanowires [22] and in carbonnanotubes [23], whereas the dimensionality of the systemreduces to 1D. The highest degree of connement (3D)occurs in quantum dots [24] and nanoparticles [25, 26] wherethe propagation is restricted in all three directions. Here thedimensionality of the system is zero.

2. OPTICAL PHONONSAs mentioned in the Introduction, only zone-center opti-cal phonons can be observed in ideal single crystals usingoptical techniques such as Raman spectroscopy. However,this q = 0 selection rule is relaxed due to interruption oflattice periodicity in a nanocrystalline material. In this sec-tion we present a phenomenological model of phonon con-nement in an isolated nanoparticle. The case of sphericalnanoparticle is considered rst.

2.1. Phenomenological Connement Model

A quantitative formalism for the conned-phonon lineshape[27] involves taking into account the contributions of thephonons over the complete Brillouin zone with appropriateweight factors. Consider a spherical nanoparticle of diame-ter d as shown in Figure 2. A plane-wave-like phonon wave-function cannot exist within the particle because the phononcannot propagate beyond the crystal surface. In view of this,one must multiply the phonon wavefunction with a conne-ment function or envelope function Wr, which decays toa very small value close to the boundary.The wavefunction of a conned phonon of wavevector q0

can be written as

q0 r = Wruq0 r expiq0 r (1)q0 r = q0 ruq0 r (2)

W (r)

d

r

Figure 2. The Gaussian connement function Wr for a sphericalnanocrystal of diameter d.

Phonon Connement in Nanostructured Materials 501

where uq0 r has the periodicity of the lattice. In orderto calculate the effect on the Raman spectrum, we expand q0 r in Fourier series:

q0 r =Cq0 q expiq rd3q (3)

where the Fourier coefcients Cq0 q are given by

Cq0 q = 23 q0 r expiq rd3r (4)

The particle (nanocrystal) phonon wavefunction is a super-position of plane waves with q vectors centered at q0. Gaus-sian connement functions have been extensively used as theconnement function [2729]. One can write Wr as

Wr = expr2/d2 (5)where the value of decides how rapidly the wavefunc-tion decays as one approaches the boundary. This givesCq0 q as

Cq0 q = expd2q q02/4 (6)Richter et al. (RWL model) [27] used the boundary condi-tion that the phonon amplitude 2 W 2r reduced to 1/eat the boundary r = d/2 of the particle. This correspondsto = 2. Other values of such as 82 used by Campbelland Fauchet (CF model) [28] and 9.67 (bond polarizabilitymodel) [29] have also been proposed. Thus are the eigen-functions of the phonons with wavevectors q in an intervalqq0< /d centered at q0 and the weight factors Cq0 qalso have a Gaussian distribution. As one-phonon Ramanscattering probes zone-center phonons, one can set q0 = 0.This yields

C0 q2 = expq2d2/2 (7)The rst-order Raman spectrum is then given as

I = C0 q2d3q q2 + 0/22

(8)

where q is the phonon dispersion curve and 0 is thenatural linewidth of the zone-center optical phonon in thebulk. In order to simplify the calculations, one can assume aspherical Brillouin zone and consider the phonon dispersioncurve to be isotropic. These assumptions are valid when onlya small region of the Brillouin zone, centered at point,contributes to the scattering. The optical phonon dispersioncurve could then be approximated to an analytical functionof the type

q = 0 sin2qa/4 (9)where a is the lattice parameter, 0 is the zone-center opti-cal phonon frequency, and is the width of the phonondispersion curve.The calculated Raman lineshapes of 4- and 8-nm GaAs

nanoparticles are compared in Figure 3 with that of thebulk. One can see that as the particle size reduces, the

240

0

10

20

30

40

50

280 320 360

Bulk

8 nm

4 nm

Raman shift (cm-1)

Inte

nsity

(arb

units

)

Figure 3. Calculated Raman spectra of conned LO phonon in GaAsnanoparticles. The bulk spectrum is also shown for comparison. Notethe asymmetric broadening of the lineshape and also the shift of thepeak towards the low-frequency side.

Raman spectra develop marked asymmetry towards the low-frequency side and exhibit marginal shift in the peak posi-tion also towards the same side. As the optical phonondispersion curves in most solids have negative disper-sion, that is, phonon frequency decreases as a function ofwavenumber, the increased intensity in the wing of theRaman spectra on the low-frequency side basically arisesfrom the contribution from the phonon branch away fromthe zone center. The dependence of the peak shift and theline broadening on the particle size is shown in Figure 4 forthe longitudinal optic phonon in GaAs. Note that both peakshift and the linewidth increase as the particle size reduces.However, the changes are marginal if the particle size islarger than 10 nm.It is sometimes useful to combine the results of Figure 4

into a single curve of peak shift versus line broadening. Thisis particularly useful while analyzing data on nanocrystallinesystems where information about the particle size is notavailable. For ion-implanted GaAs, Tiong et al. [30] arguedthat crystallite size reduced due to irradiation-induced dam-age in the lattice; however, spatial correlation (size ofcrystalline region) persisted over a small length. In view ofthis the changes in the peak shift and line broadening asa function of uence were ascribed to the residual spatialcorrelation over the nanocrystalline grains in the implantedsample. Figure 5 shows peak shift as a function of linebroadening for GaAs nanoparticles.In addition to the Gaussian function, other analytic func-

tions such as sinc and exponential have also been consideredas connement functions. In analogy with the ground-statewavefunction of an electron in a spherical potential well, asinc sinx/x function was considered [28]:

WSr = sin2r/d/2r/d if r < d/2= 0 otherwise (10)

Here the wavefunction becomes zero at the boundary of theparticle. The Fourier coefcient in this case becomes [29]

CS0 q

sinqd/2

q42 q2d2 (11)

Similarly, in analogy with the propagation of a wave in alossy medium, an exponential decay of phonon amplitude

502 Phonon Connement in Nanostructured Materials

20

16

12

8

4

0

10

10 12

8

8

6

6

4

4

DIAMETER (nm)

PEA

KSH

IFT

(cm

1)

LIN

EW

IDT

H(c

m1

)

2

2

0

0

(a)

(b)

Figure 4. The dependence of linewidth (a) and peak shift (b) of thelongitudinal optic phonon on the particle size for GaAs nanoparticles.Dashed line is the linewidth of the bulk phonon.

10 20

EXP

SINCGAUSS

(cm1)0

0

4

8

(c

m1

)

Figure 5. The relationship between peak shift and line broadening for the three different connement functions. Adapted with permissionfrom [28], I. H. Campbell and P. M. Fauchet, Solid State Commun. 58,739 (1986). 1986, Elsevier Science.

has also been considered [28]. The connement functionused in this case was

WEr = exp42r/d (12)The Fourier coefcient is then given by [28]

CE0 q

1

164 q2d22 (13)

These connement functions yield different dependence ofpeak shift on the line broadening. Figure 5 also comparesthe dependencies for the sinc and exponential connementfunctions with that of Gaussian. Note that each connementfunction has a different shape of versus curve. Fromthe analysis of their data and other reported results [27, 31],Campbell and Fauchet [28] have shown that a Gaussian con-nement with = 82 ts best to the data. This corre-sponds to a strong/rigid connement of phonons within thenanoparticle with zero amplitude near the boundary.We now consider other connement geometries. As men-

tioned earlier, rodlike shape corresponds to 2D conne-ment and platelike (thin-lm) shape gives 1D connement.A rodlike particle is characterized by two length scales, itsdiameter d1 being much smaller than its length d2. Againusing = 82 in Gaussian connement, the expressions forthe Fourier coefcients C(0, q1q2 have been obtained [28]:

C0 q1 q22 expq21d21/162 expq22d22/162

1 erf

(iq2d232

)2

(14)

On the other hand, a thin lm has only its thickness d as theconning dimension. For a thin lm, the Fourier coefcientis given by [28]

C0 q2 expq2d2/1621 erf

(iqd32

)2

(15)

The changes in the Raman lineshape, quantied in terms ofline broadening and peak shift, are compared in Figure 6for 1-, 2-, and 3D conned systems within the frameworkof Gaussian connement. Note that as the dimensionality ofconnement reduces, the magnitude of peak shift and linebroadening reduce dramatically. The departure from bulk ismaximum for a particle while it is least for a thin lm of thesame dimension.

2.2. Bond Polarizability Model

In addition to the phenomenological models of phononconnement, there have been some attempts to theoreti-cally obtain [29] the Raman spectrum of nanocrystals usingbond polarizability model [32] within the framework ofpartial density approximation [33]. In this method eigen-values and eigenvectors are obtained by diagonalizing thedynamical matrix, while the force constants are obtainedby using partial density approach. The eigenvectors thusobtained are analyzed to give vibrational amplitude as afunction of distance from the center of the particle. Thephonon amplitude in a Si nanosphere was shown to closely

Phonon Connement in Nanostructured Materials 503

THIN FILM

ROD

SPHERE

0

4

8

010 20

(cm1)

(c

m1

)

Figure 6. The relationship between peak shift and line broadening for three different shapes of the nanoparticles. In all the three curvesthe open circles correspond to a size of 4 nm. Adapted with permissionfrom [28], I. H. Campbell and P. M. Fauchet, Solid State Commun. 58,739 (1986). 1986, Elsevier Science.

resemble a sinc function or a Gaussian with = 967 [29].Amplitude at the boundary was calculated to be around3.6%. This is much smaller than 1/e used by Richter et al.[27] and much larger than exp42 used by Campbelland Fauchet (CF) [28]. The Gaussian connement func-tions and the corresponding Fourier coefcients correspond-ing to three different (2, 9.67, and 82 are compared inFigure 7. Note that weight factor Cq2 for the Ramanintensity drops too rapidly for = 2 and too slowly for =82 as one moves away from the Brillouin zone center. Con-sequently the RWL model predicts only a marginal changein the Raman spectra while that of CF model causes maxi-mum departure from the bulk for the same particle size. Onthe other hand, the calculations of Zi et al. suggest an effectintermediate between the two limiting cases. It is importantto point out that a large number of results have been sat-isfactorily explained on the basis of Gaussian connementmodel using = 82 [20, 3438].

3. RESULTSIn this section we discuss various results reported on phononconnement in 1-, 2-, and 3D conned systems.

3.1. 1D Conned Systems

Superlattices consisting of alternate thin layers of a pair ofsemiconducting materials such as GaAs and AlAs, grownon a substrate using molecular beam epitaxy, have beenextensively studied [21] in view of their applications in light-emitting diodes and diode lasers [39]. In these superlatticesGaAs layer constitutes the quantum well while the AlAslayer forms the barrier layer. It is important to point outthat the range of optical phonon frequencies of GaAs doesnot overlap with that of AlAs. Hence the phonons of GaAslayer cannot propagate into the neighboring AlAs layers andvice versa. Thus phonons in each of the GaAs and AlAs

0.0

0.0

(3) (2)

(1)

(a)

(b)

0.1 0.3r/d

0.4

0.4

0.6

0.8

1.0

0.50.2

0.2

0.0

0.0

(1) (2)(3)

0.4

0.4

0.6

0.6

(qa/)

0.8

0.8

1.0

1.0

0.2

0.2|C

(q)|2

W2 (

r)

Figure 7. Squares of the Gaussian connement functions Wr (a) andthe corresponding Fourier transform Cq (b) for different values of .Curve (1) = 20 (RWL model), (2) = 967 (bond-polarizabilitymodel), and (3) = 82 (CF model). The phonon amplitude W 2 isplotted up to the boundary of the particle (r = d/2) and C2 up to theBrillouin zone boundary q = /a.

layers are conned within those layers. The conned opticalphonons in such superlattices can be described as modes ofa thin slab, arising from the standing wave pattern formedwithin each slab. A set of modes at discrete wavevectorsqj = j/d1, where d1 is the thickness of the GaAs layer, areallowed. The conned phonon frequencies j then corre-spond to the discrete qj points on the dispersion curve ofGaAs [40]. Similarly the conned optical phonons of AlAslayer of thickness d2 correspond to the qj = j/d2 discretepoints on the AlAs dispersion curve. In GaAs/AlAs superlat-tices the conned optical phonons in the GaAs layers havebeen observed only under resonant conditions, that is, whenthe incident photon energy is close to that of an electronicexcitation of the GaAs quantum well [40]. Under nonreso-nant conditions the intensities of these modes are weak.In many cases one of the layers is an alloy AlxGa1xAs

[21]. This mixed crystal system exhibits a two-mode behav-ior [41]; that is, it exhibits both GaAs-like and AlAs-likemodes. Hence AlAs-like phonons remain conned in thebarrier layer (AlxGa1xAs) in a GaAs/AlxGa1xAs super-lattice. On the other hand, the GaAs-like modes of thequantum-well layer (GaAs) can propagate in the barrierlayer and vice versa. In view of this one expects zone fold-ing to take place with new periodicity of (d1 + d2 at qj =j/(d1 + d2; however, this effect has not been observed forpropagating optical phonons in GaAs/AlxGa1xAs superlat-tices because of the highly dispersive character of the optical

504 Phonon Connement in Nanostructured Materials

modes [42]. It may be mentioned that the acoustic phonons,whose dispersion curves overlap over a wide range of fre-quencies, propagate through both the layers exhibiting zone-folding effects due to new periodicity [43].Conned optical phonons have been found also in single

GaAs quantum wells under resonant conditions [44]. Res-onance was achieved at a xed photon energy by exploit-ing the temperature dependence of electronic excitations inthe quantum well. Recently, IR absorption measurementshave been used for studying conned optical phonons in(PbTe)m/(EuTe)n superlattices [45]. The conned phononsmanifest themselves as minima in the transmission spec-trum. In the normal incidence only transverse optic (TO)phonons are observed, while in oblique incidence both TOand longitudinal optic (LO) phonons are seen. From thefrequencies of these conned phonons the dispersion curvealong the 111 direction could be deduced. In contrastto GaAs/AlxGa1xAs superlattices, zone-folding effects havebeen observed in GaN/AlxGa1xN superlattices [46]. As thismixed crystal system exhibits one-mode behaviour, it isargued that the overlap between the density of states in thetwo layers is signicant. In superlattices and in quantum-well structures, interface optical phonons have also beenobserved [47, 48]. Phenomenological models [49] predictthat these modes have frequencies between TO and LOphonons of the constituent layers. If the interfaces are sharp,the interface phonons are found to be weak [46].

3.2. 2D Conned Systems

Recently, several tens of micrometers long nanowires of avariety of materials such as Si [22, 50], Ge [51], GaAs [52],SiC [5355], and TiC [56] have been synthesized using laserablation [57] and other methods. The diameter of thesenanowires ranges from 5 to 50 nm. Their optical propertiesare strongly inuenced by the connement of electrons andholes in these 1D systems. In view of their unique properties,they nd applications in several devices [56]. In analogy withelectron connement, phonon connement has also beenfound to give rise to interesting changes in the vibrationalspectra.Raman spectra of the F2g optical phonon in Si nanowires

show broadening and peak shifts [50] similar to thosepredicted by Gaussian connement model. For a 10-nm-diameter nanowire the peak is found to shift to 505 cm1

from 519 cm1 and also broaden to 13 cm1 from 3.5 cm1.Additional peaks at 302 and 964 cm1 have also beenreported. These were assigned to overtones of the zone-boundary phonons [50]. Appearance of zone-boundaryphonons in crystals with large density of defects [58] or inmixed crystals [59] has often been reported. This arises dueto the relaxation of the q = 0 selection rule due to thepresence of disorder in the crystal. Similarly, overtones andcombinations constitute the second-order spectra and thesealso are enhanced in the presence of disorder. On the otherhand, Wang et al. [50] apply the phonon connement modelalso to the overtones of zone-boundary phonon and try tointerpret their shifts and broadening. In fact, the changes inthe Raman spectra of overtones, etc., as a consequence ofreducing the nanowire diameter should only be ascribed tohigher defect density resulting in the appearance of peaks

corresponding to one- and two-phonon density of states.Quite interestingly, Wang et al. also introduce a new termphonon connement length (in analogy with exciton con-nement length). By this they imply a size of nanostructurebelow which the phonon connement effects are noticeablein the Raman spectra. In this context it is important to pointout that for a given material the connement effects maybe different for phonons of different symmetries [37], mak-ing such terms lose their physical signicance. This will bediscussed further in a subsequent section.Germanium nanowires with an oxide layer coating have

been synthesized using laser ablation technique [51]. Asexpected, larger core diameters in the range 2051 nm donot exhibit any noticeable change in the Raman spectra. Onthe other hand, nanowires with 617 nm core show asym-metric broadening; however, no quantitative analysis hasbeen carried out. Gallium arsenide nanowires with a GeOxsheath have exhibited broad TO and LO phonon modes[52]; however, the broadening was found to be nearly sym-metric. Surprisingly, the red shift of the LO phonon was verylarge, 40 cm1 for nanowires with diameters in the range10120 nm with an average diameter of 60 nm. Such a largeshift cannot be accounted for based on phonon conne-ment effect alone. Other factors such as defects and residualstresses have been argued to contribute to the decrease ofLO phonon frequency. Silicon carbide nanowires of averagediameter 80 nm with a coating of 17 nm SiOx [53] have alsoshown very broad Raman spectra that resemble those arisingfrom phonon density of states rather than from phonon con-nement effects. The red shifts of 12 to 34 cm1 for the TOand LO phonons were attributed to connement effects andinternal stresses [53]. CdSe nanolaments incorporated inbrous magnesium silicate (chrysotile asbestos) have shownpolarized Raman spectrum [60].Carbon nanotubes are unique one-dimensional systems

[61, 62], whose diameters are typically 12 nm and whoselengths are up to several tens of micrometers. Vibrationalproperties of these hollow tubes are quite different fromthose of solid nanorods discussed earlier. A single-wall car-bon nanotube (SWNT) can be described as a single atomiclayer of graphite rolled up into a seamless cylinder. A SWNTis specied by a pair of indices (m, n) that represent thenumber of unit vectors na1 and ma2 on the 2D hexagonalhoneycomb lattice contained in the chiral vector. Folding ofthe graphite sheet is done such that the chiral vector is per-pendicular to the axis of the nanotube (see, e.g., Fig. 1 of[63]). The magnitude of the chiral vector gives its circumfer-ence. The diameter of the nanotube is related to the (m, n)indices as

d = 31/2m2 +mn+ n21/2acc/where acc is the nearest-neighbor C-C distance (1.421 in graphite). The phonon dispersion relations in a carbonnanotube are obtained from those of an isolated 2D graphitelayer (graphene sheet) by using the zone-folding approach[64]. Zone folding of acoustic branches leads to several low-frequency modes, whose frequencies depend strongly on thediameter of the nanotube. Notable among these are the E2gmode, the E1g , mode, and the A1g radial breathing mode.For a (10, 10) nanotube of diameter 1.36 nm, these modes

Phonon Connement in Nanostructured Materials 505

have frequencies of 17, 118, and 165 cm1 [63]. For tubediameters ranging between 0.6 and 1.4 nm, a power lawdependence of the mode frequencies has been found [65].The exponent for the E1g and A1g modes is close to 1 whilethat for the lowest energy E2g mode is close to 2. Figure 8shows the dependence of the frequencies of several Ramanactive modes on the index n for (n, n armchair nanotubes[66]. The inverse dependence of the radial breathing modefrequency on the tube diameter serves as a secondary char-acterization to estimate the mean diameter of SWNTs. Inaddition to these features, the internal modes associatedwith the hexagonal ring stretching vibration of the graphitesheet around 1581 cm1 exhibits splitting into A1g + Eg .This splitting arises due to the curvature of the nanotubegraphene sheet. A novel feature of the Raman spectra ofSWNTs is the diameter selective scattering at different exci-tation energies, arising from the 2D quantum connementof electrons. Resonance Raman spectroscopic studies in theenergy range 14.8 eV have proved to be a powerful probeof these quasi-1D materials [6769]. Depending on its chi-rality (i.e., n and m), an individual SWNT could be semicon-ducting or metallic. Any sample of SWNTs is a mixture of1/3 metallic and 2/3 semiconducting tubes. Raman exci-tations at different energies could selectively probe either ofthese sets of tubes.Specic-heat measurements at low temperature have

shown evidence of quantized phonon spectrum of SWNT[70]. The data show the expected linear T dependence andwere found to be signicantly different from that of 3Dgraphite and 2D graphene. The analysis also yielded anenergy of 4.3 meV for the lowest quantized phonon subband.The nanotubes that are produced in either an electric arc orpulsed laser vaporization are mostly in the form of bundles,where nanotubes are aligned in a close-packed triangularlattice. Intertube interactions that arise in the lattice are usu-ally weak and are approximated by van der Waals interac-tion. This is similar to coupling between adjacent graphenelayers in 3D crystalline graphite. This coupling causes aslight increase (7%) in the frequency of radial breathing

0

0

200

MO

DE

FRE

QU

EN

CY

(cm

-1)

400

600

800

1000

1200

1400

1600

5 10 15 20

n

Figure 8. Dependence of the vibrational frequencies of (n, n) armchairsingle-walled carbon nanotubes on the index n. The diameter of thenanotube is dn = 1357n . The frequencies of the A1g radial breathingmode and other low-frequency modes depend strongly on the diameter.Adapted with permission from [66], P. C. Eklund et al., Carbon 33, 959(1995). 1995, Elsevier Science.

mode, independent of the tube diameter. This arises fromadditional restoring force due to the nearest-neighbor tube-tube interaction [71].

3.3. 3D Conned Systems

Isolated or loosely connected nanoparticles as self-supporting powders and nanoparticles dispersed in otherhosts have been the most extensively studied nanostructuredsystems. In many investigations quantitative tting ofphonon lineshape has also been carried out [20, 38, 72].The extent of peak shift and line broadening is expected todepend on the shape of the dispersion curve. For phononbranches with less dispersion the effects are expected to besmall. On the other hand, the phonons that exhibit largedispersion would get modied signicantly. This was demon-strated for the rst time [37] in the case of zinc oxidenanoparticles by examining the phonons of different symme-tries (irreducible representations). Zinc oxide has wurtzitestructure and consequently has phonons of symmetries A1,E1, and E2 at 393, 591, and 465 cm1, respectively. Fora 4-nm particle size, E1-LO mode exhibited a change inlinewidth from 18 to 38 cm1 whereas E2 mode showedan increase of only 2 cm1. Figure 9 shows the tted line-shape for 4-nm particles along with the data. Because ofinsufcient intensity, A1 mode was not analyzed in detail.The widely different behavior of E1 and E2 phonons couldbe understood when the widths of the corresponding dis-persion curves were taken into account. Table 1 showsthe peak shift and line broadening data for these modes in4-nm particles.There are a number of Raman spectroscopic studies on

nanocrystalline powders which exhibit broadening and peakshifts similar to those expected for phonon connement[7375]; however, quantitative analyses have not been car-ried out. In the case of composites synthesized either as thinlms using co-sputtering [15] or by doping melt-quenchedoxide glasses [76], nanocrystalline precipitates form duringannealing at elevated temperatures [77, 78]. A departureof LO phonon frequency from the expected behavior for

400

4

8

12

500

E2

E1A1

600 700

Raman Shift (cm1)

Inte

nsity

(arb

.uni

ts)

Figure 9. Raman spectrum of 4-nm ZnO nanoparticles. The continuouscurve is the calculated spectrum from Gaussian connement model.Adapted with permission from [37], M. Rajalakshmi et al., J. Appl. Phys.87, 2445 (2000). 2000, American Institute of Physics.

506 Phonon Connement in Nanostructured Materials

Table 1. Peak shift and line broadening data for optical phononsof different symmetries for 4-nm zinc oxide nanoparticles.

Phonon (cm1 0 (cm1 (cm1 (cm1

E2 12 12 14 1E1-LO 60 18 38 7

Note: is the width of the dispersion curve and 0 is the natural linewidthof the phonon.

CdS nanoparticles smaller than 5 nm dispersed in GeO2matrix has been attributed to defects [79]. Evidence of thepresence of CdO surface capping layer on pulsed laserdeposited CdS nanoparticles in SiO2 matrix has been foundfrom the presence of its characteristic peak in the Ramanspectrum [80]. A comparison of the Raman spectra of Sidoped SiO2 lms with those of theoretically calculated vibra-tional density of states of Si33 and Si45 clusters suggestedthe presence of such clusters in SiO2 [81]. Semiconduc-tor mixed crystals such as CdSxSe1x [82] and Cd1xZnxS[77] dispersed in oxide glasses as nanocrystalline precipi-tates have been extensively studied in view of their interest-ing optical properties and applications as long-pass opticallters. The system CdSxSe1x exhibits two-mode behaviorand both CdSe-like and CdS-like conned LO phononsare observed [20]. On the other hand, Cd1xZnxS systemexhibits single-mode behavior. The shift of the LO phononfrequency during late stage of annealing of Cd1xZnxSnanoparticles dispersed in oxide glass host containing 20%ZnO suggested a change in the stoichiometry (x in thenanoparticle [77]. A few monolayers of AlSb deposited onGaAs substrate using molecular beam epitaxy are found toself-assemble in the form of platelike quantum dots duringannealing at 500 C [83]. In addition to phonon connementeffects, sometimes compressive stresses also play a role indetermining Raman lineshapes [84]. Nanopores of zeoliteshave also been used to capture nanoparticles of Se and Te[85]. Raman spectra showed evidence of trapping of eithera molecular Se8 or Te8 or formation of an irregular arrayof chains and clusters depending on the size and connectiv-ity of pores. The lifetime of phonons in nanocrystalline Sihas also been measured and found to be more than that inamorphous Si [86].As pointed out earlier, in most of the systems the opti-

cal phonon frequency decreases as one moves away fromthe Brillouin zone center; that is, the optical phonon branchexhibits negative dispersion. In this context, thorium oxideis a unique system, whose optical phonon branch splitsinto two components; one exhibits a negative dispersionwith = 50 cm1, while the other undergoes a positivedispersion of = +160 cm1 [87]. For nanocrystallinethorium oxide both the branches are expected to con-tribute to the Raman lineshape. Recently, Raman spec-tra of nanocrystalline ThO2 have been reported, which arefound to be less asymmetric as compared to other crystals[88]. This is attributed to the broadening arising on the leftand the right side of the peak from the contributions frombranches of the dispersion curve with negative and positivedispersions, respectively.In addition to the conned optic phonons, the presence

of surface phonons in the Raman spectra of nanostructuredmaterials has been reported in a number of systems

[72, 8991]. Surface phonons are expected to have frequen-cies between TO and LO phonons [72]. As the fraction ofsurface atoms increases as the grain size of a nanostructuredmaterial reduces, the surface phonons are observed withnoticeable intensity for small size particles. The dependenceof surface phonon frequency on the dielectric constant ofthe surrounding medium has also been examined [90]. Inthe nanoparticles of mixed crystals such as CdSxSe1x, twosurface phonons, one each of CdSe-like and CdS-like, havebeen reported [89].Porous-silicon (p-Si), obtained from electrochemical etch-

ing of Si [92], has been a subject of considerable interest inview of its efcient photo- and electroluminescence at ambi-ent temperature [93]. The pore diameter and consequentlythe size of interconnected Si-nanostructure depends on theelectrochemical conditions [94]. Raman spectrum of p-Siconsists of an asymmetrically broadened F2g phonon linecharacteristic of nanocrystalline Si and an overlapping broadpeak at 480 cm1 associated with amorphous Si [95, 96]. Fit-ting of the Raman spectrum to a conned phonon lineshapehas frequently been carried out to estimate the average par-ticle size [97]. Conned phonons of p-Si have been foundto be responsible for the photoluminescence arising fromradiative recombination of carriers across the indirect tran-sition [98] similar to that found in crystalline Si.As mentioned earlier, if the phonon spectrum of the

particle overlaps signicantly with that of the surround-ing medium, phonons of the particles can propagate intothe surrounding medium. In such cases a strong conne-ment model of the Gaussian type is not expected to besatisfactory. This was indeed found to be true [99] in thecase of nanocrystalline diamond particles surrounded byamorphous-carbon region. Figure 10 shows the Raman spec-trum of nanocrystalline diamond embedded in amorphous-carbon matrix. The observed linewidth was found to bemuch more than expected for a Gaussian connementmodel. In order to understand these results, an alternateconnement model was proposed, which took into accountthe reection of the phonon from the dielectric/elasticboundary of the particle. This leads to the existence of a

1280

0

100

200

300

400

500

1320 1340 1360 13801300

Raman shift (cm1)

Inte

nsity

(arb

.uni

ts)

Figure 10. Raman spectrum of 26-nm diamond particles embedded inamorphous-carbon host. The continuous curve is the calculated spec-trum based on the discrete model of phonon connement. Adaptedwith permission from [99], A. K. Arora et al., Diamond Relat. Mater. 10,1477 (2001). 2001, Elsevier Science.

Phonon Connement in Nanostructured Materials 507

standing wave pattern in the particle with phonon wavevec-tor sampling the Brillouin zone at discrete points qn =nQB/N (1 nN, where Na is the size of the particle. Theintensities In from the discrete phonons qn were taken tovary according to a power law In bn (b < 1). The discretemodel of the phonon connement yielded a satisfactory tto the experimental phonon lineshape.In order to probe the changes in the electron-phonon

interaction, resonance Raman scattering from connedoptical phonons has been investigated in a number ofsystems. Effect of valence band mixing, arising from thedegeneracy at point in the Brillouin zone of zinc-blendesemiconductors, on the electron phonon coupling has beenstudied [100] in GaP nanoparticles. Surface phonons withangular momentum l = 2 were found to participate in theresonance Raman scattering, while in another study [101]only the transitions with l = 0 2 have been argued to con-tribute to the resonance Raman scattering. In addition to theLO phonons, zone-boundary TO phonons and their over-tones have also been observed [102] in the nanoparticles ofindirect gap semiconductor AgBr under resonant conditions.Using a phenomenological model, the ratio of intensitiesof the overtone and the fundamental Raman spectra havebeen analyzed [103] and the results suggested that electron-phonon coupling decreased as the particle size reduced.Interference effects in the resonance Raman efciency pro-le [104] of 1- and 2-LO conned phonons in Cd1xZnxSmixed crystal nanoparticles, arising from a nonresonant con-tribution to the polarizability, have also been reported.A detailed theory [105] of the one-phonon resonanceRaman scattering from spherical nanoparticles has shownthat in the dipole approximation, only l = 0 phonon modescouple to the photon. On the other hand, in the elec-tric quadrupole approximation l = 1 phonon modes can beexcited and their polarizability amplitude is proportional tothe wavevector of the photon.In most of the analyses of the phonon lineshapes, the

bulk phonon dispersion curves have been assumed to be stillapplicable. However, this is not guaranteed for very smallparticles. Recent studies have shown that use of bulk phonondispersion in Gaussian connement model gives a goodagreement for 6-nm CdS particle; on the other hand, for1.6-nm particles the predicted lineshape is more asymmetric(Fig. 11) than that observed [106]. This disagreement hasbeen attributed to the inapplicability of the bulk phonon dis-persion curves. It may be pointed out that the phonon den-sity of states (DOS) of nanocrystalline iron [107] measuredusing neutron scattering exhibits smearing of sharp featuresand broadening on both low- and high-frequency side ascompared to the bulk. Molecular dynamics simulations showthat the increased density of states at low energies arisesfrom the vibrations of atoms at surface/grain boundaries[108], whereas the increase in the DOS at high frequencieshas been attributed to shortening of bond length [109] andlifetime broadening [110] due to anharmonic effects.

4. ACOUSTIC PHONONSSimilar to the optical phonons, the acoustic phonons alsoget conned within the particles. In the elastic continuumlimit, the connement of long-wavelength acoustic phonons

Diameter = 1.6 nm

Diameter = 6 nm

1.5

0.5

1.5

0.5

0260 280 300 320 340

Raman Shift (cm1)

1

0

1

Experimental

experimental

calculated

calculated

Nor

mal

ized

Inte

nsity

Nor

mal

ized

Inte

nsity

Figure 11. Raman spectra of CdS nanoparticles of diameter 1.6 and6 nm. The disagreement of the calculated lineshape and the data forthe 1.6-nm particles suggests inapplicability of bulk phonon dispersioncurves. Reprinted with permission from [106], P. Nandakumar et al.,Physica E 11, 377 (2001). 2001, Elsevier Science.

(sound waves) leads to the emergence of discrete modesof particle which depend on the elastic properties throughthe longitudinal and transverse sound velocities [111]. Theseare spheroidal and torsional modes of the particle and theirfrequencies depend on the angular momentum associatedwith the vibration. We now briey describe the procedure toobtain the frequencies of these modes.

4.1. Vibrational Modes of a Small Particle

By considering a spherical particle to be a homogeneouselastic body, its free vibrations can be obtained by solvingthe equation of motion [111113],

*+2D/+t2 = .+ /00 D+ /0 2D (16)

508 Phonon Connement in Nanostructured Materials

where D is the displacement, * is the density, and . and /are Lames constants. The solutions are obtained by intro-ducing scalar and vector potentials and by using appropri-ate boundary conditions for spheroidal and torsional modes.The eigenvalue equation for the spheroidal modes is [111]

212 + l 1l + 21jl+11/jl1 l + 1 2jl+12/jl2 1214 + l 12l + 112+ 12 2ll 1l + 21jl+11/jl1 = 0 (17)

where l is the angular momentum, 1 = R/ct and 2 =1ct/cl are dimensionless variables, is the spheroidal modefrequency, R is the radius of the particle, and ct and clare the transverse and longitudinal sound velocities, respec-tively. jl(1) is the spherical Bessel function of the rst kind.The angular momentum quantum number l can take val-ues 0 1 2 The eigenvalue equation for torsional modesis [114]

jl+11l 1

1jl1 = 0 for l 1 (18)

Solving these equations for discrete values of l results ina set of eigenvalues for each l, labeled as 1Sl n and 1

Tl n

for spheroidal and torsional modes, respectively. The indexn represents the branch number. It may be mentioned thatl = 0 torsional mode has null displacement. In addition,the eigenvalues of the torsional modes do not depend onthe material, whereas those of spheroidal modes are com-pletely determined by the ratio cl/ct . The eigenvalues ofthese modes have been reported for many materials withwidely different values of the velocity ratio, such as 1.54(MgAl2O4) [115], 2.00 (Se) [38], 2.28 (CdS) [116, 114], 2.32(CdSe) [117], 2.51 (Pb) [111], and 2.77 (In) [111]. The low-est eigenvalues for n = 0 for both spheroidal and torsionalmodes correspond to the surface modes. These modes havelarge amplitude near the surface. The subsequent eigenval-ues (n 1) correspond to the inner modes. These discretemodes for different values of l are essentially similar to theacoustic phonons at discrete q-points in the Brillouin zonegiven by l+ 1/2/R up to a maximum value /a at the zoneboundary [111]. These modes modify the density of states ofthe bulk [118] and have been argued to be responsible forexcess specic heat of small particles at low temperatures[119, 120].The symmetries of the particle vibrations correspond to

the irreducible representations of the rotation inversiongroup O(3) of the sphere. The spheroidal modes transformaccording to the irreducible representations Dg1 D

u2 D

g3

[121]. The subscripts represent the angular momentum, andthe superscripts g and u imply symmetry and antisymmetrywith respect to inversion. In addition, the components ofelectric dipole moment transform according to Du1 . On theother hand, the components of the symmetric polarizabil-ity tensor for Raman scattering will transform according tothe irreducible representations resulting from the symmetricproduct Du1 Du1 sym =Dg0 +Dg2 . Therefore the only allowedRaman active modes are the spherical mode Dg0 and thequadrupolar mode Dg2 . The torsional modes are not Ramanactive [121]. This is different from the assignment by other

researchers [114, 122] based only on the parity of the wave-functions, where all spheroidal modes with even l and alltorsional modes of odd l were argued to be Raman active.In addition to the elastic continuum model, a microscopic

lattice dynamical calculation of conned acoustic phononsin nanocrystalline Si has been carried out [123]. For thisthe bond polarizability model within the partial densityapproximation has been used, similar to the conned opticalphonon calculation [29]. The disagreement of the calculatedconned acoustic phonon frequencies with the experimen-tal data [124] was attributed to the fact that the calculationswere carried out for free Si particles while the data wasfor Si nanoparticles dispersed in SiO2 matrix. On the otherhand, studies of the effect of host matrix on the spheroidalmode frequencies [111, 125] suggest only marginal changes.In a recent formalism the linewidth of the conned acous-tic phonon has also been taken into account by making theeigenvalue complex [126].

4.2. Low-Frequency Raman Scattering

The frequencies of the spheroidal and torsional modes canbe calculated from the eigenvalues 1Sl n and 1

Tl n, respec-

tively, as

Sl n = 1Sl nct/R (19a)and

Tl n = 1Tl nct/R (19b)

In order to express the vibrational frequencies in cm1, onecan divide Eqs. (19) by the velocity of light in vacuum c. Inmost materials, for particles of diameter less than 10 nm,these frequencies lie in the range of 5 to 50 cm1. Henceit is possible to observe the modes, allowed by the selectionrules, in the low-frequency Raman scattering. It is notewor-thy that Brillouin spectroscopy is extensively used for prob-ing the acoustic phonon branch close to the zone center incrystalline solids. The Brillouin shifts of the order of 1 cm1

are conveniently measured by interferometric techniquesusing FabryPerot etalon. However, the range of connedacoustic phonon frequencies makes Brillouin spectroscopyunsuitable for this purpose. On the other hand, the particlemodes were observed for the rst time by Duval et al. [115]in the spinal (MgAl2O4) particles dispersed in oxide glassin the low-frequency Raman scattering. The inverse depen-dence of the mode frequencies on the particle size was alsoestablished experimentally. Subsequently, conned acousticphonons have been reported in metal [127129] and semi-conductor [124, 130, 131] nanoparticles.Often the assignment of the peaks observed in the low-

frequency Raman spectrum in terms of angular momentumquantum number l and branch number n may be tricky andnonunique. In order to obtain unambiguous assignment, itis useful to know the size of the particle from independentmeasurements such as high-resolution transmission electronmicroscopy (HRTEM) [122], X-ray diffraction (XRD) [132],or small-angle X-ray scattering (SAXS) [117] and plot theobserved mode frequencies as a function of inverse diameteralong with the theoretically expected linear dependencies

Phonon Connement in Nanostructured Materials 509

[72, 114]. In addition, making a polarized measurement isuseful in the assignment, as the spheroidal mode appearsonly in the polarized geometry while the quadrupolar modeoccurs in both polarized and depolarized geometry.Silver nanoparticles dispersed in different hosts such as

alkali halides [128], soda-lime glass [129], SiO2 [122, 127],and ZrO2 [132] have been studied in great detail. Fuji et al.found the frequencies of Ag particles of sizes between 2 and5 nm close to those expected for l = 0 and l = 2 spheroidalmodes [122]. The strong Raman signal was attributedto resonance associated with localized surface plasmons.The growth of Ag nanoparticles in SiO2 during isochronalannealing has been recently examined [127]. In view of itsdepolarized characteristics, the Raman peak was assignedto the l=2 quadrupolar mode. The average size thus esti-mated was found to be consistent with that obtained fromthe width of the surface plasmon absorption. Quadrupolarmode has been observed also for Ag nanoparticles dispersedin soda-lime glass [129]. Silver nanoparticles synthesized inZrO2 by annealing a polycrystalline pellet coated with Ag,at 1073 K exhibited as many as four peaks in the Ramanspectra [132] which were assigned to the quadrupolar mode.These were interpreted as arising from a multimodal dis-tribution of particle sizes ranging between 3.5 and 7 nm.However, the average particle diameters obtained from thewidth of the XRD peak and TEM were of the order of23 nm. The quadrupolar mode frequency corresponding tosuch a large average size is much smaller and not expectedto be observed in the low-frequency Raman spectrum. Thismakes their assignment to quadrupolar modes doubtful. Theaverage size of gold clusters found in SiO2 due to ion beammixing and subsequent annealing has been estimated to bearound 4 nm from low-frequency Raman scattering [133].Semiconductor nanoparticles dispersed in oxide glass

[77, 116, 134], SiO2 [135], and GeO2 [114] have been exten-sively studied. Many of these composites are synthesizedby doping the glass by the semiconductor up to about2% in the melt and rapidly cooling to room temperature.The solid solution thus formed is supersaturated and semi-conductor nanoparticles nucleate and grow upon anneal-ing the semiconductor doped glass above 500 C [136].Synthesis of composites in the form of thin lms is car-ried out using the technique of co-sputtering. There havebeen several studies on commercial long-pass optical l-ters such as GG495, GG475, and RG630 from SCHOTT,which contain CdSxSe1x mixed crystals dispersed in oxideglass [137]. In the early studies the lowest energy surfacemode and the lowest energy breathing mode were identi-ed in the Raman spectra [116]. The frequencies of thesemodes were found to decrease during annealing, suggestinggrowth of particles [117]. Figure 12 shows the dependenceof spheroidal mode frequency and the estimated particlesize on the annealing temperature in Cd1xZnxS095Se005nanoparticles dispersed in oxide glass. During annealing thezinc concentration x in the nanoparticle was reported toincrease, if the host glass contained ZnO as one of theconstituents [77]. Some studies have revealed very largewidth of the conned acoustic phonon 2 2 in semicon-ductor doped glass GG495 [134]. This has been attributedto the existence of a log-normal particle size distribution.In addition, resonance enhancement of the quadrupolar

ANNEALING TEMP (C)M

OD

EF

RE

QU

EN

CY

(cm

1)

PAR

TIC

LES

IZE

(nm

)

500

12

14

16

600 700

6

7

8

9

(a)

(b)

Figure 12. Dependence of conned acoustic phonon frequencies(a) and average particle size (b) of CdS1xSex nanoparticles dispersedin borosilicate glass (GG475) on annealing temperature. Symbols and represent the corresponding values for the unannealed lter.Reprinted with permission from [77], M. Rajalakshmi et al., J. Phys.:Condens. Matter 9, 9745 (1997). 1997, IOP Publishing.

mode was found for excitation with green wavelength butnot with blue line. For CdS nanoparticles dispersed in GeO2glass, a linear dependence of conned phonon frequencieson inverse diameter has been reported [114]. From this anal-ysis the mode was identied as the l = 0 spheroidal mode.In addition to the conned acoustic phonons, another broadpeak called boson peak arising from the glassy host hasbeen observed in the low-frequency Raman spectra of semi-conductor doped glasses [72]. Conned acoustic phononshave also been reported in selenium nanoparticles dispersedin a polymer host [38].

5. SUMMARYIn this chapter we have examined the consequences ofconnement of the optical and acoustic phonons within agrain of nanostructured materials on the vibrational spectra.These are the shift and asymmetric broadening of the opti-cal phonon lineshape and the appearance of spheroidal andquadrupolar modes of the nanoparticle in the low-frequencyRaman spectra. Among the conned acoustic phonons, onlythe spherical mode with angular momentum l = 0 andquadrupolar mode with l = 2 are Raman active. The inversedependence of the particle mode frequencies on its diameteris useful as a secondary method for the estimation of par-ticle size. Changes in the optical phonon Raman lineshapearise from the contribution of phonon with nite wavevectorwhose magnitude is of the order of inverse diameter of the

510 Phonon Connement in Nanostructured Materials

grain. Noticeable differences in the spectra are found onlywhen the particle/grain size is smaller than about 20 latticeparameters. Furthermore, the effect of connement is leastfor 1D connement and maximum for 3D connement. Forvery small grain size the phenomenological models that usethe bulk phonon dispersion curves as inputs fail because forthese sizes, phonon density of states differs signicantly fromthat of the bulk. Detailed theoretical studies are required toobtain a full understanding of the vibrational spectra of verysmall size particles.

GLOSSARYAcoustic phonons Phonons created by atoms vibrating inphase with each other.Anharmonic effects Effects such as thermal expansion,arising from non-parabolic nature of interaction potentialbetween atoms. Parabolic atomic potential leads to har-monic atomic vibrations.Brillouin scattering Inelastic scattering of photons byacoustic phonons in a solid.Brillouin zone Unit cell formed in the reciprocal space byreciprocal lattice vectors.Carbon nanotube A graphene layer rolled up into a tubeof diameter of the order of a few nanometers and lengthseveral micrometer.Chiral vector A vector formed from the superposition ofintegral multiples of unit vectors of a 2D-hexagonal planeof graphite. This vector is perpendicular to the axis of thecarbon nano-tube, and its magnitude determines the circum-ference of the tube.Dispersion curve Plot of phonon frequency versuswavevector for a phonon of specic symmetry.Long-pass lter An optical lter that allows light longerthan a given wavelength to pass through.Nanowire Wire/rod of a few nanometers diameter and sev-eral micrometer length.Optical phonon Phonons in the high frequency region cre-ated by atoms in a crystal vibrating out-of-phase with eachother.Phonon Quantum of atomic vibrations in crystalline solids.Quantum dot A small particle surrounded by vacuum or amaterial of larger band gap, capable of exhibiting effects ofquantum mechanical connement of charge carriers.Quantum well A thin layer of a material of smaller bandgap sandwiched between adjacent layers of another materialwith larger band gap, capable of exhibiting effects of quan-tum mechanical connement of charge carriers.Raman scattering Inelastic scattering of photons byatomic vibrations or other elementary excitations insolids/molecules.Resonance Raman scattering Enhancement of efciencyof Raman scattering when the incident photon energy isclose to one of the electronic transitions in a solid/molecule.Scattering vector Wavevector transferred in the scatteringof light by elementary excitations.Superlattice A periodic arrangement formed by depositingthin alternate layers of two materials on a substrate.

Wavevector A vector in the direction of propagation ofphonon with a magnitude that is inverse of wavelength ofthe phonon.

ACKNOWLEDGMENTSIt is a pleasure to acknowledge very fruitful collaborationswith Professors S. Mahamuni, D. S. Misra, B. R. Mehta, andC. Vijayan. We also thank Dr. B. Viswanathan for interest inthe work, Dr. Baldev Raj for support, and Mr. S. B. Bhojefor encouragement.

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