Phonon Confinement in Nanostructured Materials

Encyclopedia of Nanoscience and Nanotechnology

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Phonon Confinement in Nanostructured Materials Akhilesh K. Arora, M. Rajalakshmi, T. R. Ravindran Indira Gandhi Centre for Atomic Research, Kalpakkam, India

CONTENTS 1. Introduction 2. Optical Phonons 3. Results 4. Acoustic Phonons 5. Summary Glossary References

1. INTRODUCTION There is considerable current interest in the physics of nanostructured materials in view of their numerous technological applications [1, 2] in a variety of areas such as catalysis [3], magnetic data storage [4], ferrofluids [5], soft magnets [6], machinable ceramics and metallurgy [7], nonlinear optical and optoelectronic devices [8], and sensors [9]. In addition, obtaining an understanding of the properties of nanostructured materials is of interest from a fundamental point of view. Only a proper understanding of the dependence of a given property on the grain/particle size can lead to design/tailoring of the nanostructured material for the related application. It is also important to understand when a material could be considered as nanostructured. Although one can in principle classify materials with grain size less than 1000 nm as nanostructured, several properties such as optical [10] and vibrational [11] properties do not differ much from the corresponding bulk value unless the grain/particle size is less than typically 20 nm. In view of this it is reasonable to treat a material with a grain size smaller than a certain value as nanostructured only if the property of interest differs from the bulk value at least by a few percent. It is also possible that a material with nanometer grain size may behave as nanostructured for a specific property while it could act like bulk for other properties. In addition to the grain size, the properties of the nanostructured materials may sometimes depend on the method of their synthesis. Generally nanostructured materials are synthesized in ISBN: 1-58883-064-0/$35.00 Copyright © 2004 by American Scientific Publishers All rights of reproduction in any form reserved.

one of the three forms: (a) as isolated or loosely connected nanoparticles in the form of powder [12], (b) as composites of nanoparticles dispersed in another host [13], or (c) as compact collection of nanograins as pellets [14] or thin films [15]. The last form is also called nanophase material. This chapter reviews the vibrational properties of the various forms of nanostructured materials. Phonons are quanta of atomic vibrations in crystalline solids. In a simple monoatomic solid with only one atom per primitive cell (for example,
-iron, copper), one can have only three acoustic phonon branches corresponding to the three degrees of freedom of atomic motion. On the other hand, for monoatomic solids with two atoms per primitive cell such as diamond, magnesium, or diatomic compounds such as GaAs, one also has three optic phonon branches in addition to the three acoustic phonons [16]. In compounds with a greater number of atoms and complex crystal structures, the number of optic phonons is more than three. If the crystal unit cell contains N atoms, then 3N degrees of freedom result in 3 acoustic phonons and 3N − 3 optical phonons. These phonons can propagate in the lattice of a single crystal as a wave and exhibit dispersion depending on their wavelength or equivalently their wavevector in the Brillouin zone [17]. Phonon propagation is interrupted when a grain boundary is encountered in a polycrystalline material. In an isolated grain the phonon can get reflected from the boundaries and remain confined within the grain. However, from the point of view of phonons, a well-crystallized polycrystalline sample with several micrometer grain size can be treated as a bulk/infinite crystal for all practical purposes. The consequences of phonon confinement are noticeable in the vibrational spectra only when the grain size is smaller than typically 20 lattice parameters. The atomic vibrational frequencies in crystalline solids range from zero to about 100 THz. A more common unit to describe the vibrational frequencies is wavenumber (cm−1  obtained by dividing the actual frequency by the velocity of light or by inverting the wavelength. Acoustic phonons have frequencies from zero to about a few hundred Encyclopedia of Nanoscience and Nanotechnology Edited by H. S. Nalwa Volume 8: Pages (499–512)

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Phonon Confinement in Nanostructured Materials

wavenumbers while the optic phonons have higher frequencies. Hence the vibrational spectra could be probed using infrared absorption/reflectivity or using Raman spectroscopy [18]. The wavevector of the IR photon for these energies is of the order of 102 –103 cm−1 . On the other hand, in a Raman scattering experiment the magnitude of scattering vector is 2k0 sin(/2), where k0 is the wavevector of the incident light and  is the scattering angle. Thus the maximum value of the scattering vector could at best be 2k0 (corresponding to the backscattering geometry), which has a value ∼5×104 cm−1 for visible light. Hence the wavevector probed by either of these two techniques is much smaller than the wavevector q of the full phonon dispersion curve, which extends up to the boundary of the Brillouin zone (/a ∼ 108 cm−1 , where a is the lattice parameter). Thus these techniques sample only the optical phonons close to the zone center (q ∼ 0). This q ∼ 0 selection rule is essentially a consequence of the infinite periodicity of the crystal lattice [19]. However, if the periodicity of the crystal lattice is interrupted, as in the case of nanocrystalline materials, this rule is relaxed and phonons away from the Brillouin zone center also contribute to the phonon lineshape observed in spectroscopic measurements. This can be qualitatively explained in the following manner. Consider the phonon dispersion curve of a typical diatomic solid as shown in Figure 1. For a particle/grain of size d, the phonon wavefunction must decay to a very small value close to the boundary. This restriction on the spatial extent of the wavefunction, via a relationship of the uncertainty principle type, results in an uncertainty q ∼ /d in the wavevector of the zone-center optical phonon and a corresponding uncertainty 2 in its frequency [20]. Now the light scattering can take place from quasi-zone-center optical phonons with wavevectors q up to /d. This results in asymmetric broadening of the phonon lineshape. In addition to the changes in the optical phonon lineshape, confinement of acoustic phonons also leads to other interesting changes in the spectra. A quantitative formalism of phonon confinement will be discussed in subsequent sections.

1.1. Dimensionality of Confinement It is important to distinguish between the dimensionality of the system and the dimensionality/degree of confinement. A bulk material is a 3D system and is unconfined; that is, dimensionality/degree of confinement is zero. The first ∆q∼π/d ω0

level of confinement occurs in single- and multilayer thin films grown using layered deposition on substrates. Semiconductor superlattices, single-quantum well structures, and multiple-quantum well structures [21] are well-known examples of 2D systems because the phonons and charge carriers are confined within a plane, say, the x-y plane; however, the degree of confinement is 1D because phonons and charge carriers are restricted along the z direction. Similarly, 2D confinement occurs in nanowires [22] and in carbon nanotubes [23], whereas the dimensionality of the system reduces to 1D. The highest degree of confinement (3D) occurs in quantum dots [24] and nanoparticles [25, 26] where the propagation is restricted in all three directions. Here the dimensionality of the system is zero.

2. OPTICAL PHONONS As mentioned in the Introduction, only zone-center optical phonons can be observed in ideal single crystals using optical techniques such as Raman spectroscopy. However, this q = 0 selection rule is relaxed due to interruption of lattice periodicity in a nanocrystalline material. In this section we present a phenomenological model of phonon confinement in an isolated nanoparticle. The case of spherical nanoparticle is considered first.

2.1. Phenomenological Confinement Model A quantitative formalism for the confined-phonon lineshape [27] involves taking into account the contributions of the phonons over the complete Brillouin zone with appropriate weight factors. Consider a spherical nanoparticle of diameter d as shown in Figure 2. A plane-wave-like phonon wavefunction cannot exist within the particle because the phonon cannot propagate beyond the crystal surface. In view of this, one must multiply the phonon wavefunction with a confinement function or envelope function W r, which decays to a very small value close to the boundary. The wavefunction of a confined phonon of wavevector q0 can be written as  q0  r = W ruq0  r exp−iq0 · r 

 q0  r =  q0  ruq0  r

(1) (2)

d

∆ω

ω(q)

2δω

W (r)

0

q

2π/a

Figure 1. Schematic representation of the optical phonon dispersion curve and the range q of the wavevectors probed in Raman scattering from a nanoparticle of diameter d. 2 is the corresponding range of phonon frequencies that contribute to the first-order Raman scattering.

r

Figure 2. The Gaussian confinement function W r for a spherical nanocrystal of diameter d.

501

Phonon Confinement in Nanostructured Materials

where the Fourier coefficients Cq0  q are given by
Cq0  q = 2−3   q0  r exp−iq · rd 3 r

(5)

where the value of
decides how rapidly the wavefunction decays as one approaches the boundary. This gives Cq0  q as Cq0  q = exp−d 2 q − q0 2 /4


(6)

Richter et al. (RWL model) [27] used the boundary condition that the phonon amplitude  2 ∝ W 2 r reduced to 1/e at the boundary r = d/2 of the particle. This corresponds to
= 2. Other values of
such as 8 2 used by Campbell and Fauchet (CF model) [28] and 9.67 (bond polarizability model) [29] have also been proposed. Thus   are the eigenfunctions of the phonons with wavevectors q in an interval q − q0  < /d centered at q0 and the weight factors Cq0  q also have a Gaussian distribution. As one-phonon Raman scattering probes zone-center phonons, one can set q0 = 0. This yields C0 q2 = exp−q 2 d 2 /2


(7)

The first-order Raman spectrum is then given as I  =




C0 q2 d 3 q  − q2 + 0 /22

(8)

where q is the phonon dispersion curve and 0 is the natural linewidth of the zone-center optical phonon in the bulk. In order to simplify the calculations, one can assume a spherical Brillouin zone and consider the phonon dispersion curve to be isotropic. These assumptions are valid when only a small region of the Brillouin zone, centered at  point, contributes to the scattering. The optical phonon dispersion curve could then be approximated to an analytical function of the type q = 0 − sin2 qa/4

(9)

where a is the lattice parameter, 0 is the zone-center optical phonon frequency, and is the width of the phonon dispersion curve. The calculated Raman lineshapes of 4- and 8-nm GaAs nanoparticles are compared in Figure 3 with that of the bulk. One can see that as the particle size reduces, the

40

4 nm

30 20

8 nm

10 Bulk

0 240

(4)

The particle (nanocrystal) phonon wavefunction is a superposition of plane waves with q vectors centered at q0 . Gaussian confinement functions have been extensively used as the confinement function [27–29]. One can write W r as W r = exp−
r 2 /d 2 

50

Intensity (arb units)

where uq0  r has the periodicity of the lattice. In order to calculate the effect on the Raman spectrum, we expand   q0  r in Fourier series:
  q0  r = Cq0  q expiq · rd 3 q (3)

280

320

360 -1

Raman shift (cm )

Figure 3. Calculated Raman spectra of confined LO phonon in GaAs nanoparticles. The bulk spectrum is also shown for comparison. Note the asymmetric broadening of the lineshape and also the shift of the peak towards the low-frequency side.

Raman spectra develop marked asymmetry towards the lowfrequency side and exhibit marginal shift in the peak position also towards the same side. As the optical phonon dispersion curves in most solids have negative dispersion, that is, phonon frequency decreases as a function of wavenumber, the increased intensity in the wing of the Raman spectra on the low-frequency side basically arises from the contribution from the phonon branch away from the zone center. The dependence of the peak shift and the line broadening on the particle size is shown in Figure 4 for the longitudinal optic phonon in GaAs. Note that both peak shift and the linewidth increase as the particle size reduces. However, the changes are marginal if the particle size is larger than 10 nm. It is sometimes useful to combine the results of Figure 4 into a single curve of peak shift versus line broadening. This is particularly useful while analyzing data on nanocrystalline systems where information about the particle size is not available. For ion-implanted GaAs, Tiong et al. [30] argued that crystallite size reduced due to irradiation-induced damage in the lattice; however, spatial correlation (size of crystalline region) persisted over a small length. In view of this the changes in the peak shift and line broadening as a function of fluence were ascribed to the residual spatial correlation over the nanocrystalline grains in the implanted sample. Figure 5 shows peak shift as a function of line broadening for GaAs nanoparticles. In addition to the Gaussian function, other analytic functions such as sinc and exponential have also been considered as confinement functions. In analogy with the ground-state wavefunction of an electron in a spherical potential well, a sinc sinx/x function was considered [28]: WS r = sin2r/d/2r/d =0

if r < d/2 otherwise

(10)

Here the wavefunction becomes zero at the boundary of the particle. The Fourier coefficient in this case becomes [29] CS 0 q

sinqd/2 q4 2 − q 2 d 2 

(11)

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

502

Phonon Confinement in Nanostructured Materials

has also been considered [28]. The confinement function used in this case was

20

LINE WIDTH (cm–1)

WE r = exp−4 2 r/d

(a)

16

The Fourier coefficient is then given by [28] 12

CE 0 q

0 10

(b)

PEAK SHIFT (cm–1)

8

1 16 4 − q 2 d 2 2

(13)

These confinement functions yield different dependence of peak shift on the line broadening. Figure 5 also compares the dependencies for the sinc and exponential confinement functions with that of Gaussian. Note that each confinement function has a different shape of  versus  curve. From the analysis of their data and other reported results [27, 31], Campbell and Fauchet [28] have shown that a Gaussian confinement with
= 8 2 fits best to the data. This corresponds to a strong/rigid confinement of phonons within the nanoparticle with zero amplitude near the boundary. We now consider other confinement geometries. As mentioned earlier, rodlike shape corresponds to 2D confinement and platelike (thin-film) shape gives 1D confinement. A rodlike particle is characterized by two length scales, its diameter d1 being much smaller than its length d2 . Again using
= 8 2 in Gaussian confinement, the expressions for the Fourier coefficients C(0, q1 q2  have been obtained [28]:

8

4

6

4

C0 q1  q2 2 exp−q12 d12 /16 2  exp−q22 d22 /16 2    2  iq2 d2   (14) × 1 − erf √ 32 

2

0 0

2

4

6

8

10

12

DIAMETER (nm)

Figure 4. The dependence of linewidth (a) and peak shift (b) of the longitudinal optic phonon on the particle size for GaAs nanoparticles. Dashed line is the linewidth of the bulk phonon. 8

GAUSS

SINC

δω (cm–1)

(12)

4

On the other hand, a thin film has only its thickness d as the confining dimension. For a thin film, the Fourier coefficient is given by [28]   2  iqd  C0 q2 exp−q 2 d 2 /16 2 1 − erf √ 32 

(15)

The changes in the Raman lineshape, quantified in terms of line broadening and peak shift, are compared in Figure 6 for 1-, 2-, and 3D confined systems within the framework of Gaussian confinement. Note that as the dimensionality of confinement reduces, the magnitude of peak shift and line broadening reduce dramatically. The departure from bulk is maximum for a particle while it is least for a thin film of the same dimension.

2.2. Bond Polarizability Model EXP

0

0

10

Γ(cm–1)

20

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

In addition to the phenomenological models of phonon confinement, there have been some attempts to theoretically obtain [29] the Raman spectrum of nanocrystals using bond polarizability model [32] within the framework of partial density approximation [33]. In this method eigenvalues and eigenvectors are obtained by diagonalizing the dynamical matrix, while the force constants are obtained by using partial density approach. The eigenvectors thus obtained are analyzed to give vibrational amplitude as a function of distance from the center of the particle. The phonon amplitude in a Si nanosphere was shown to closely

503

Phonon Confinement in Nanostructured Materials 8 1.0

(a)

SPHERE

W2(r)

δω (cm–1)

0.8 0.6 (1) (2)

(3)

0.4

4 ROD

0.2 0.0 0.0

0.1

0.2

THIN FILM

r/d

0.3

0.4

0.5

1.0

0

Γ (cm–1)

(b)

20

Figure 6. The relationship between peak shift  and line broadening  for three different shapes of the nanoparticles. In all the three curves the open circles correspond to a size of 4 nm. Adapted with permission from [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 around 3.6%. This is much smaller than 1/e used by Richter et al. [27] and much larger than exp−4 2  used by Campbell and Fauchet (CF) [28]. The Gaussian confinement functions and the corresponding Fourier coefficients corresponding to three different
(2, 9.67, and 8 2  are compared in Figure 7. Note that weight factor Cq2 for the Raman intensity drops too rapidly for
= 2 and too slowly for
= 8 2 as one moves away from the Brillouin zone center. Consequently the RWL model predicts only a marginal change in the Raman spectra while that of CF model causes maximum departure from the bulk for the same particle size. On the other hand, the calculations of Zi et al. suggest an effect intermediate between the two limiting cases. It is important to point out that a large number of results have been satisfactorily explained on the basis of Gaussian confinement model using
= 8 2 [20, 34–38].

3. RESULTS In this section we discuss various results reported on phonon confinement in 1-, 2-, and 3D confined systems.

3.1. 1D Confined Systems Superlattices consisting of alternate thin layers of a pair of semiconducting materials such as GaAs and AlAs, grown on a substrate using molecular beam epitaxy, have been extensively studied [21] in view of their applications in lightemitting diodes and diode lasers [39]. In these superlattices GaAs layer constitutes the quantum well while the AlAs layer forms the barrier layer. It is important to point out that the range of optical phonon frequencies of GaAs does not overlap with that of AlAs. Hence the phonons of GaAs layer cannot propagate into the neighboring AlAs layers and vice versa. Thus phonons in each of the GaAs and AlAs

0.8

2

10

|C(q)|

0

0.6 0.4 0.2

(3)

(2)

(1)

0.0 0.0

0.2

0.4

0.6

0.8

1.0

(qa/π)

Figure 7. Squares of the Gaussian confinement functions W r (a) and the corresponding Fourier transform Cq (b) for different values of
. Curve (1)
= 20 (RWL model), (2)
= 967 (bond-polarizability model), and (3)
= 8 2 (CF model). The phonon amplitude W 2 is plotted up to the boundary of the particle (r = d/2) and C 2 up to the Brillouin zone boundary q = /a.

layers are confined within those layers. The confined optical phonons in such superlattices can be described as modes of a thin slab, arising from the standing wave pattern formed within each slab. A set of modes at discrete wavevectors qj = j/d1 , where d1 is the thickness of the GaAs layer, are allowed. The confined phonon frequencies j then correspond to the discrete qj points on the dispersion curve of GaAs [40]. Similarly the confined optical phonons of AlAs layer of thickness d2 correspond to the qj = j/d2 discrete points on the AlAs dispersion curve. In GaAs/AlAs superlattices the confined optical phonons in the GaAs layers have been observed only under resonant conditions, that is, when the incident photon energy is close to that of an electronic excitation of the GaAs quantum well [40]. Under nonresonant conditions the intensities of these modes are weak. In many cases one of the layers is an alloy Alx Ga1−x As [21]. This mixed crystal system exhibits a “two-mode” behavior [41]; that is, it exhibits both GaAs-like and AlAs-like modes. Hence AlAs-like phonons remain confined in the barrier layer (Alx Ga1−x As) in a GaAs/Alx Ga1−x As superlattice. On the other hand, the GaAs-like modes of the quantum-well layer (GaAs) can propagate in the barrier layer and vice versa. In view of this one expects “zone folding” to take place with new periodicity of (d1 + d2  at qj = j/(d1 + d2 ; however, this effect has not been observed for propagating optical phonons in GaAs/Alx Ga1−x As superlattices because of the highly dispersive character of the optical

504 modes [42]. It may be mentioned that the acoustic phonons, whose dispersion curves overlap over a wide range of frequencies, propagate through both the layers exhibiting zonefolding effects due to new periodicity [43]. Confined optical phonons have been found also in single GaAs quantum wells under resonant conditions [44]. Resonance was achieved at a fixed photon energy by exploiting the temperature dependence of electronic excitations in the quantum well. Recently, IR absorption measurements have been used for studying confined optical phonons in (PbTe)m /(EuTe)n superlattices [45]. The confined phonons manifest themselves as minima in the transmission spectrum. In the normal incidence only transverse optic (TO) phonons are observed, while in oblique incidence both TO and longitudinal optic (LO) phonons are seen. From the frequencies of these confined phonons the dispersion curve along the 111 direction could be deduced. In contrast to GaAs/Alx Ga1−x As superlattices, zone-folding effects have been observed in GaN/Alx Ga1−x N superlattices [46]. As this mixed crystal system exhibits “one-mode behaviour,” it is argued that the overlap between the density of states in the two layers is significant. In superlattices and in quantumwell structures, interface optical phonons have also been observed [47, 48]. Phenomenological models [49] predict that these modes have frequencies between TO and LO phonons of the constituent layers. If the interfaces are sharp, the interface phonons are found to be weak [46].

3.2. 2D Confined Systems Recently, several tens of micrometers long nanowires of a variety of materials such as Si [22, 50], Ge [51], GaAs [52], SiC [53–55], and TiC [56] have been synthesized using laser ablation [57] and other methods. The diameter of these nanowires ranges from 5 to 50 nm. Their optical properties are strongly influenced by the confinement of electrons and holes in these 1D systems. In view of their unique properties, they find applications in several devices [56]. In analogy with electron confinement, phonon confinement has also been found to give rise to interesting changes in the vibrational spectra. Raman spectra of the F2g optical phonon in Si nanowires show broadening and peak shifts [50] similar to those predicted by Gaussian confinement model. For a 10-nmdiameter nanowire the peak is found to shift to 505 cm−1 from 519 cm−1 and also broaden to 13 cm−1 from 3.5 cm−1 . Additional peaks at 302 and 964 cm−1 have also been reported. These were assigned to overtones of the zoneboundary phonons [50]. Appearance of zone-boundary phonons in crystals with large density of defects [58] or in mixed crystals [59] has often been reported. This arises due to the relaxation of the q = 0 selection rule due to the presence of disorder in the crystal. Similarly, overtones and combinations constitute the second-order spectra and these also are enhanced in the presence of disorder. On the other hand, Wang et al. [50] apply the phonon confinement model also to the overtones of zone-boundary phonon and try to interpret their shifts and broadening. In fact, the changes in the Raman spectra of overtones, etc., as a consequence of reducing the nanowire diameter should only be ascribed to higher defect density resulting in the appearance of peaks

Phonon Confinement in Nanostructured Materials

corresponding to one- and two-phonon density of states. Quite interestingly, Wang et al. also introduce a new term “phonon confinement length” (in analogy with exciton confinement length). By this they imply a size of nanostructure below which the phonon confinement effects are noticeable in the Raman spectra. In this context it is important to point out that for a given material the confinement effects may be different for phonons of different symmetries [37], making such terms lose their physical significance. This will be discussed further in a subsequent section. Germanium nanowires with an oxide layer coating have been synthesized using laser ablation technique [51]. As expected, larger core diameters in the range 20–51 nm do not exhibit any noticeable change in the Raman spectra. On the other hand, nanowires with 6–17 nm core show asymmetric broadening; however, no quantitative analysis has been carried out. Gallium arsenide nanowires with a GeOx sheath have exhibited broad TO and LO phonon modes [52]; however, the broadening was found to be nearly symmetric. Surprisingly, the red shift of the LO phonon was very large, ∼40 cm−1 for nanowires with diameters in the range 10–120 nm with an average diameter of 60 nm. Such a large shift cannot be accounted for based on phonon confinement effect alone. Other factors such as defects and residual stresses have been argued to contribute to the decrease of LO phonon frequency. Silicon carbide nanowires of average diameter 80 nm with a coating of 17 nm SiOx [53] have also shown very broad Raman spectra that resemble those arising from phonon density of states rather than from phonon confinement effects. The red shifts of 12 to 34 cm−1 for the TO and LO phonons were attributed to confinement effects and internal stresses [53]. CdSe nanofilaments incorporated in fibrous magnesium silicate (chrysotile asbestos) have shown polarized Raman spectrum [60]. Carbon nanotubes are unique one-dimensional systems [61, 62], whose diameters are typically 1–2 nm and whose lengths are up to several tens of micrometers. Vibrational properties of these hollow tubes are quite different from those of solid nanorods discussed earlier. A single-wall carbon nanotube (SWNT) can be described as a single atomic layer of graphite rolled up into a seamless cylinder. A SWNT is specified by a pair of indices (m, n) that represent the number of unit vectors na1 and ma2 on the 2D hexagonal honeycomb lattice contained in the chiral vector. Folding of the graphite sheet is done such that the chiral vector is perpendicular to the axis of the nanotube (see, e.g., Fig. 1 of [63]). The magnitude of the chiral vector gives its circumference. The diameter of the nanotube is related to the (m, n) indices as d = 31/2 m2 + mn + n2 1/2 ac−c / where ac−c is the nearest-neighbor C-C distance (1.421 Å in graphite). The phonon dispersion relations in a carbon nanotube are obtained from those of an isolated 2D graphite layer (graphene sheet) by using the zone-folding approach [64]. Zone folding of acoustic branches leads to several lowfrequency modes, whose frequencies depend strongly on the diameter of the nanotube. Notable among these are the E2g mode, the E1g , mode, and the A1g radial breathing mode. For a (10, 10) nanotube of diameter 1.36 nm, these modes

505

Phonon Confinement in Nanostructured Materials

have frequencies of 17, 118, and 165 cm−1 [63]. For tube diameters ranging between 0.6 and 1.4 nm, a power law dependence of the mode frequencies has been found [65]. The exponent for the E1g and A1g modes is close to −1 while that for the lowest energy E2g mode is close to −2. Figure 8 shows the dependence of the frequencies of several Raman active modes on the index n for (n, n armchair nanotubes [66]. The inverse dependence of the radial breathing mode frequency on the tube diameter serves as a secondary characterization to estimate the mean diameter of SWNTs. In addition to these features, the internal modes associated with the hexagonal ring stretching vibration of the graphite sheet around 1581 cm−1 exhibits splitting into A1g + Eg . This splitting arises due to the curvature of the nanotube graphene sheet. A novel feature of the Raman spectra of SWNTs is the diameter selective scattering at different excitation energies, arising from the 2D quantum confinement of electrons. Resonance Raman spectroscopic studies in the energy range 1–4.8 eV have proved to be a powerful probe of these quasi-1D materials [67–69]. Depending on its chirality (i.e., n and m), an individual SWNT could be semiconducting or metallic. Any sample of SWNTs is a mixture of ∼1/3 metallic and ∼2/3 semiconducting tubes. Raman excitations at different energies could selectively probe either of these sets of tubes. Specific-heat measurements at low temperature have shown evidence of quantized phonon spectrum of SWNT [70]. The data show the expected linear T dependence and were found to be significantly different from that of 3D graphite and 2D graphene. The analysis also yielded an energy of 4.3 meV for the lowest quantized phonon subband. The nanotubes that are produced in either an electric arc or pulsed laser vaporization are mostly in the form of bundles, where nanotubes are aligned in a close-packed triangular lattice. Intertube interactions that arise in the lattice are usually weak and are approximated by van der Waals interaction. This is similar to coupling between adjacent graphene layers in 3D crystalline graphite. This coupling causes a slight increase (∼7%) in the frequency of radial breathing

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

3.3. 3D Confined Systems Isolated or loosely connected nanoparticles as selfsupporting powders and nanoparticles dispersed in other hosts have been the most extensively studied nanostructured systems. In many investigations quantitative fitting of phonon lineshape has also been carried out [20, 38, 72]. The extent of peak shift and line broadening is expected to depend on the shape of the dispersion curve. For phonon branches with less dispersion the effects are expected to be small. On the other hand, the phonons that exhibit large dispersion would get modified significantly. This was demonstrated for the first time [37] in the case of zinc oxide nanoparticles by examining the phonons of different symmetries (irreducible representations). Zinc oxide has wurtzite structure and consequently has phonons of symmetries A1 , E1 , and E2 at 393, 591, and 465 cm−1 , respectively. For a 4-nm particle size, E1 -LO mode exhibited a change in linewidth from 18 to 38 cm−1 whereas E2 mode showed an increase of only 2 cm−1 . Figure 9 shows the fitted lineshape for 4-nm particles along with the data. Because of insufficient intensity, A1 mode was not analyzed in detail. The widely different behavior of E1 and E2 phonons could be understood when the widths of the corresponding dispersion curves were taken into account. Table 1 shows the peak shift and line broadening data for these modes in 4-nm particles. There are a number of Raman spectroscopic studies on nanocrystalline powders which exhibit broadening and peak shifts similar to those expected for phonon confinement [73–75]; however, quantitative analyses have not been carried out. In the case of composites synthesized either as thin films using co-sputtering [15] or by doping melt-quenched oxide glasses [76], nanocrystalline precipitates form during annealing at elevated temperatures [77, 78]. A departure of LO phonon frequency from the expected behavior for

1400

E2

1200

12

1000

Intensity (arb. units)

MODE FREQUENCY (cm-1)

1600

800 600 400

E1 A1

8

200 0

4 0

5

10

15

20

n 400

Figure 8. Dependence of the vibrational frequencies of (n, n) armchair single-walled carbon nanotubes on the index n. The diameter of the nanotube is dn = 1357n Å. The frequencies of the A1g radial breathing mode 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.

600

500

700

Raman Shift (cm–1)

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

506

Phonon Confinement in Nanostructured Materials

Phonon

(cm−1 

0 (cm−1 

 (cm−1 

 (cm−1 

E2 E1 -LO

12 60

12 18

14 38

1 7

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

CdS nanoparticles smaller than 5 nm dispersed in GeO2 matrix has been attributed to defects [79]. Evidence of the presence of CdO surface capping layer on pulsed laser deposited CdS nanoparticles in SiO2 matrix has been found from the presence of its characteristic peak in the Raman spectrum [80]. A comparison of the Raman spectra of Si doped SiO2 films with those of theoretically calculated vibrational density of states of Si33 and Si45 clusters suggested the presence of such clusters in SiO2 [81]. Semiconductor mixed crystals such as CdSx Se1−x [82] and Cd1−x Znx S [77] dispersed in oxide glasses as nanocrystalline precipitates have been extensively studied in view of their interesting optical properties and applications as long-pass optical filters. The system CdSx Se1−x exhibits two-mode behavior and both CdSe-like and CdS-like confined LO phonons are observed [20]. On the other hand, Cd1−x Znx S system exhibits single-mode behavior. The shift of the LO phonon frequency during late stage of annealing of Cd1−x Znx S nanoparticles dispersed in oxide glass host containing 20% ZnO suggested a change in the stoichiometry (x in the nanoparticle [77]. A few monolayers of AlSb deposited on GaAs substrate using molecular beam epitaxy are found to self-assemble in the form of platelike quantum dots during annealing at 500  C [83]. In addition to phonon confinement effects, sometimes compressive stresses also play a role in determining Raman lineshapes [84]. Nanopores of zeolites have also been used to capture nanoparticles of Se and Te [85]. Raman spectra showed evidence of trapping of either a molecular Se8 or Te8 or formation of an irregular array of chains and clusters depending on the size and connectivity of pores. The lifetime of phonons in nanocrystalline Si has also been measured and found to be more than that in amorphous Si [86]. As pointed out earlier, in most of the systems the optical phonon frequency decreases as one moves away from the Brillouin zone center; that is, the optical phonon branch exhibits negative dispersion. In this context, thorium oxide is a unique system, whose optical phonon branch splits into two components; one exhibits a negative dispersion with = −50 cm−1 , while the other undergoes a positive dispersion of = +160 cm−1 [87]. For nanocrystalline thorium oxide both the branches are expected to contribute to the Raman lineshape. Recently, Raman spectra of nanocrystalline ThO2 have been reported, which are found to be less asymmetric as compared to other crystals [88]. This is attributed to the broadening arising on the left and the right side of the peak from the contributions from branches of the dispersion curve with negative and positive dispersions, respectively. In addition to the confined optic phonons, the presence of surface phonons in the Raman spectra of nanostructured materials has been reported in a number of systems

[72, 89–91]. Surface phonons are expected to have frequencies between TO and LO phonons [72]. As the fraction of surface atoms increases as the grain size of a nanostructured material reduces, the surface phonons are observed with noticeable intensity for small size particles. The dependence of surface phonon frequency on the dielectric constant of the surrounding medium has also been examined [90]. In the nanoparticles of mixed crystals such as CdSx Se1−x , two surface phonons, one each of CdSe-like and CdS-like, have been reported [89]. Porous-silicon (p-Si), obtained from electrochemical etching of Si [92], has been a subject of considerable interest in view of its efficient photo- and electroluminescence at ambient temperature [93]. The pore diameter and consequently the size of interconnected Si-nanostructure depends on the electrochemical conditions [94]. Raman spectrum of p-Si consists of an asymmetrically broadened F2g phonon line characteristic of nanocrystalline Si and an overlapping broad peak at 480 cm−1 associated with amorphous Si [95, 96]. Fitting of the Raman spectrum to a confined phonon lineshape has frequently been carried out to estimate the average particle size [97]. Confined phonons of p-Si have been found to be responsible for the photoluminescence arising from radiative recombination of carriers across the indirect transition [98] similar to that found in crystalline Si. As mentioned earlier, if the phonon spectrum of the particle overlaps significantly with that of the surrounding medium, phonons of the particles can propagate into the surrounding medium. In such cases a strong confinement model of the Gaussian type is not expected to be satisfactory. This was indeed found to be true [99] in the case of nanocrystalline diamond particles surrounded by amorphous-carbon region. Figure 10 shows the Raman spectrum of nanocrystalline diamond embedded in amorphouscarbon matrix. The observed linewidth was found to be much more than expected for a Gaussian confinement model. In order to understand these results, an alternate confinement model was proposed, which took into account the reflection of the phonon from the dielectric/elastic boundary of the particle. This leads to the existence of a 500

400

Intensity (arb. units)

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

300

200

100

0 1280

1300

1320

1340

1360

1380

Raman shift (cm–1)

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

507

Phonon Confinement in Nanostructured Materials

4. ACOUSTIC PHONONS Similar to the optical phonons, the acoustic phonons also get confined within the particles. In the elastic continuum limit, the confinement of long-wavelength acoustic phonons

1.5

Diameter = 1.6 nm Experimental

Normalized Intensity

calculated

1

0.5

0 1.5

Diameter = 6 nm calculated experimental

Normalized Intensity

standing wave pattern in the particle with phonon wavevector sampling the Brillouin zone at discrete points qn = nQB /N (1 ≤ n ≤ N , where Na is the size of the particle. The intensities In from the discrete phonons qn  were taken to vary according to a power law In ∼ b n (b < 1). The discrete model of the phonon confinement yielded a satisfactory fit to the experimental phonon lineshape. In order to probe the changes in the electron-phonon interaction, resonance Raman scattering from confined optical phonons has been investigated in a number of systems. Effect of valence band mixing, arising from the degeneracy at  point in the Brillouin zone of zinc-blende semiconductors, on the electron phonon coupling has been studied [100] in GaP nanoparticles. Surface phonons with angular momentum l = 2 were found to participate in the resonance Raman scattering, while in another study [101] only the transitions with l = 0 2 have been argued to contribute to the resonance Raman scattering. In addition to the LO phonons, zone-boundary TO phonons and their overtones have also been observed [102] in the nanoparticles of indirect gap semiconductor AgBr under resonant conditions. Using a phenomenological model, the ratio of intensities of the overtone and the fundamental Raman spectra have been analyzed [103] and the results suggested that electronphonon coupling decreased as the particle size reduced. Interference effects in the resonance Raman efficiency profile [104] of 1- and 2-LO confined phonons in Cd1−x Znx S mixed crystal nanoparticles, arising from a nonresonant contribution to the polarizability, have also been reported. A detailed theory [105] of the one-phonon resonance Raman scattering from spherical nanoparticles has shown that in the dipole approximation, only l = 0 phonon modes couple to the photon. On the other hand, in the electric quadrupole approximation l = 1 phonon modes can be excited and their polarizability amplitude is proportional to the wavevector of the photon. In most of the analyses of the phonon lineshapes, the bulk phonon dispersion curves have been assumed to be still applicable. However, this is not guaranteed for very small particles. Recent studies have shown that use of bulk phonon dispersion in Gaussian confinement model gives a good agreement for 6-nm CdS particle; on the other hand, for 1.6-nm particles the predicted lineshape is more asymmetric (Fig. 11) than that observed [106]. This disagreement has been attributed to the inapplicability of the bulk phonon dispersion curves. It may be pointed out that the phonon density of states (DOS) of nanocrystalline iron [107] measured using neutron scattering exhibits smearing of sharp features and broadening on both low- and high-frequency side as compared to the bulk. Molecular dynamics simulations show that the increased density of states at low energies arises from the vibrations of atoms at surface/grain boundaries [108], whereas the increase in the DOS at high frequencies has been attributed to shortening of bond length [109] and lifetime broadening [110] due to anharmonic effects.

1

0.5

0 260

280

300

320

340

Raman Shift (cm–1)

Figure 11. Raman spectra of CdS nanoparticles of diameter 1.6 and 6 nm. The disagreement of the calculated lineshape and the data for the 1.6-nm particles suggests inapplicability of bulk phonon dispersion curves. 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 modes of particle which depend on the elastic properties through the longitudinal and transverse sound velocities [111]. These are spheroidal and torsional modes of the particle and their frequencies depend on the angular momentum associated with the vibration. We now briefly describe the procedure to obtain the frequencies of these modes.

4.1. Vibrational Modes of a Small Particle By considering a spherical particle to be a homogeneous elastic body, its free vibrations can be obtained by solving the equation of motion [111–113], *+ 2 D/+t 2 = . + /0 0 · D + /0 2 D

(16)

508

Phonon Confinement in Nanostructured Materials

where D is the displacement, * is the density, and . and / are Lame’s constants. The solutions are obtained by introducing scalar and vector potentials and by using appropriate boundary conditions for spheroidal and torsional modes. The eigenvalue equation for the spheroidal modes is [111] 212 + l − 1l + 21jl+1 1/jl 1 − l + 1 × 2jl+1 2/jl 2 − 21 14 + l − 12l + 112 + 12 − 2ll − 1l + 21jl+1 1/jl 1 = 0

(17)

where l is the angular momentum, 1 = R/ct and 2 = 1ct /cl are dimensionless variables, is the spheroidal mode frequency, R is the radius of the particle, and ct and cl are the transverse and longitudinal sound velocities, respectively. jl (1) is the spherical Bessel function of the first kind. The angular momentum quantum number l can take values 0 1 2     The eigenvalue equation for torsional modes is [114] jl+1 1 −

l − 1 jl 1 = 0 1

for l ≥ 1

(18)

Solving these equations for discrete values of l results in S T a set of eigenvalues for each l, labeled as 1l n and 1l n for spheroidal and torsional modes, respectively. The index n represents the branch number. It may be mentioned that l = 0 torsional mode has null displacement. In addition, the eigenvalues of the torsional modes do not depend on the material, whereas those of spheroidal modes are completely determined by the ratio cl /ct . The eigenvalues of these modes have been reported for many materials with widely different values of the velocity ratio, such as 1.54 (MgAl2 O4 ) [115], 2.00 (Se) [38], 2.28 (CdS) [116, 114], 2.32 (CdSe) [117], 2.51 (Pb) [111], and 2.77 (In) [111]. The lowest eigenvalues for n = 0 for both spheroidal and torsional modes correspond to the surface modes. These modes have large amplitude near the surface. The subsequent eigenvalues (n ≥ 1) correspond to the inner modes. These discrete modes for different values of l are essentially similar to the acoustic phonons at discrete q-points in the Brillouin zone given by l + 1/2/R up to a maximum value /a at the zone boundary [111]. These modes modify the density of states of the bulk [118] and have been argued to be responsible for excess specific heat of small particles at low temperatures [119, 120]. The symmetries of the particle vibrations correspond to the irreducible representations of the rotation inversion group O(3) of the sphere. The spheroidal modes transform g g according to the irreducible representations D1  D2u  D3     [121]. The subscripts represent the angular momentum, and the superscripts g and u imply symmetry and antisymmetry with respect to inversion. In addition, the components of electric dipole moment transform according to D1u . On the other hand, the components of the symmetric polarizability tensor for Raman scattering will transform according to the irreducible representations resulting from the symmetric g g product D1u × D1u sym = D0 + D2 . Therefore the only allowed g Raman active modes are the spherical mode D0 and the g quadrupolar mode D2 . The torsional modes are not Raman active [121]. This is different from the assignment by other

researchers [114, 122] based only on the parity of the wavefunctions, where all spheroidal modes with even l and all torsional modes of odd l were argued to be Raman active. In addition to the elastic continuum model, a microscopic lattice dynamical calculation of confined acoustic phonons in nanocrystalline Si has been carried out [123]. For this the bond polarizability model within the partial density approximation has been used, similar to the confined optical phonon calculation [29]. The disagreement of the calculated confined acoustic phonon frequencies with the experimental data [124] was attributed to the fact that the calculations were carried out for free Si particles while the data was for Si nanoparticles dispersed in SiO2 matrix. On the other hand, studies of the effect of host matrix on the spheroidal mode frequencies [111, 125] suggest only marginal changes. In a recent formalism the linewidth of the confined acoustic phonon has also been taken into account by making the eigenvalue complex [126].

4.2. Low-Frequency Raman Scattering The frequencies of the spheroidal and torsional modes can S T be calculated from the eigenvalues 1l n and 1l n , respectively, as S Sl n = 1l n ct /R

(19a)

T Tl n = 1l n ct /R

(19b)

and

In order to express the vibrational frequencies in cm−1 , one can divide Eqs. (19) by the velocity of light in vacuum c. In most materials, for particles of diameter less than 10 nm, these frequencies lie in the range of 5 to 50 cm−1 . Hence it is possible to observe the modes, allowed by the selection rules, in the low-frequency Raman scattering. It is noteworthy that Brillouin spectroscopy is extensively used for probing the acoustic phonon branch close to the zone center in crystalline solids. The Brillouin shifts of the order of 1 cm−1 are conveniently measured by interferometric techniques using Fabry–Perot etalon. However, the range of confined acoustic phonon frequencies makes Brillouin spectroscopy unsuitable for this purpose. On the other hand, the particle modes were observed for the first time by Duval et al. [115] in the spinal (MgAl2 O4 ) particles dispersed in oxide glass in the low-frequency Raman scattering. The inverse dependence of the mode frequencies on the particle size was also established experimentally. Subsequently, confined acoustic phonons have been reported in metal [127–129] and semiconductor [124, 130, 131] nanoparticles. Often the assignment of the peaks observed in the lowfrequency Raman spectrum in terms of angular momentum quantum number l and branch number n may be tricky and nonunique. In order to obtain unambiguous assignment, it is useful to know the size of the particle from independent measurements such as high-resolution transmission electron microscopy (HRTEM) [122], X-ray diffraction (XRD) [132], or small-angle X-ray scattering (SAXS) [117] and plot the observed mode frequencies as a function of inverse diameter along with the theoretically expected linear dependencies

509

Phonon Confinement in Nanostructured Materials

8

7

PARTICLE SIZE (nm)

9

(b) 6

∆ MODE FREQUENCY (cm–1)

[72, 114]. In addition, making a polarized measurement is useful in the assignment, as the spheroidal mode appears only in the polarized geometry while the quadrupolar mode occurs 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 and 5 nm close to those expected for l = 0 and l = 2 spheroidal modes [122]. The strong Raman signal was attributed to resonance associated with localized surface plasmons. The growth of Ag nanoparticles in SiO2 during isochronal annealing has been recently examined [127]. In view of its depolarized characteristics, the Raman peak was assigned to the l=2 quadrupolar mode. The average size thus estimated was found to be consistent with that obtained from the width of the surface plasmon absorption. Quadrupolar mode has been observed also for Ag nanoparticles dispersed in soda-lime glass [129]. Silver nanoparticles synthesized in ZrO2 by annealing a polycrystalline pellet coated with Ag, at 1073 K exhibited as many as four peaks in the Raman spectra [132] which were assigned to the quadrupolar mode. These were interpreted as arising from a multimodal distribution of particle sizes ranging between 3.5 and 7 nm. However, the average particle diameters obtained from the width of the XRD peak and TEM were of the order of 23 nm. The quadrupolar mode frequency corresponding to such a large average size is much smaller and not expected to be observed in the low-frequency Raman spectrum. This makes their assignment to quadrupolar modes doubtful. The average size of gold clusters found in SiO2 due to ion beam mixing and subsequent annealing has been estimated to be around 4 nm from low-frequency Raman scattering [133]. Semiconductor nanoparticles dispersed in oxide glass [77, 116, 134], SiO2 [135], and GeO2 [114] have been extensively studied. Many of these composites are synthesized by doping the glass by the semiconductor up to about 2% in the melt and rapidly cooling to room temperature. The solid solution thus formed is supersaturated and semiconductor nanoparticles nucleate and grow upon annealing the “semiconductor doped glass” above 500  C [136]. Synthesis of composites in the form of thin films is carried out using the technique of co-sputtering. There have been several studies on commercial long-pass optical filters such as GG495, GG475, and RG630 from SCHOTT, which contain CdSx Se1−x mixed crystals dispersed in oxide glass [137]. In the early studies the lowest energy surface mode and the lowest energy breathing mode were identified in the Raman spectra [116]. The frequencies of these modes were found to decrease during annealing, suggesting growth of particles [117]. Figure 12 shows the dependence of spheroidal mode frequency and the estimated particle size on the annealing temperature in Cd1−x Znx S095 Se005 nanoparticles dispersed in oxide glass. During annealing the zinc concentration x in the nanoparticle was reported to increase, if the host glass contained ZnO as one of the constituents [77]. Some studies have revealed very large width of the confined acoustic phonon 2 2 in semiconductor doped glass GG495 [134]. This has been attributed to the existence of a log-normal particle size distribution. In addition, resonance enhancement of the quadrupolar

16

14

(a)

12 500

600

700

ANNEALING TEMP (°C)

Figure 12. Dependence of confined acoustic phonon frequencies (a) and average particle size (b) of CdS1−x Sex nanoparticles dispersed in borosilicate glass (GG475) on annealing temperature. Symbols ‘•’ and ‘ ’ represent the corresponding values for the unannealed filter. 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 but not with blue line. For CdS nanoparticles dispersed in GeO2 glass, a linear dependence of confined phonon frequencies on inverse diameter has been reported [114]. From this analysis the mode was identified as the l = 0 spheroidal mode. In addition to the confined acoustic phonons, another broad peak called “boson peak” arising from the glassy host has been observed in the low-frequency Raman spectra of semiconductor doped glasses [72]. Confined acoustic phonons have also been reported in selenium nanoparticles dispersed in a polymer host [38].

5. SUMMARY In this chapter we have examined the consequences of confinement of the optical and acoustic phonons within a grain of nanostructured materials on the vibrational spectra. These are the shift and asymmetric broadening of the optical phonon lineshape and the appearance of spheroidal and quadrupolar modes of the nanoparticle in the low-frequency Raman spectra. Among the confined acoustic phonons, only the spherical mode with angular momentum l = 0 and quadrupolar mode with l = 2 are Raman active. The inverse dependence of the particle mode frequencies on its diameter is useful as a secondary method for the estimation of particle size. Changes in the optical phonon Raman lineshape arise from the contribution of phonon with finite wavevector whose magnitude is of the order of inverse diameter of the

510 grain. Noticeable differences in the spectra are found only when the particle/grain size is smaller than about 20 lattice parameters. Furthermore, the effect of confinement is least for 1D confinement and maximum for 3D confinement. For very small grain size the phenomenological models that use the bulk phonon dispersion curves as inputs fail because for these sizes, phonon density of states differs significantly from that of the bulk. Detailed theoretical studies are required to obtain a full understanding of the vibrational spectra of very small size particles.

GLOSSARY Acoustic phonons Phonons created by atoms vibrating in phase with each other. Anharmonic effects Effects such as thermal expansion, arising from non-parabolic nature of interaction potential between atoms. Parabolic atomic potential leads to harmonic atomic vibrations. Brillouin scattering Inelastic scattering of photons by acoustic phonons in a solid. Brillouin zone Unit cell formed in the reciprocal space by reciprocal lattice vectors. Carbon nanotube A graphene layer rolled up into a tube of diameter of the order of a few nanometers and length several micrometer. Chiral vector A vector formed from the superposition of integral multiples of unit vectors of a 2D-hexagonal plane of graphite. This vector is perpendicular to the axis of the carbon nano-tube, and its magnitude determines the circumference of the tube. Dispersion curve Plot of phonon frequency versus wavevector for a phonon of specific symmetry. Long-pass filter An optical filter that allows light longer than a given wavelength to pass through. Nanowire Wire/rod of a few nanometers diameter and several micrometer length. Optical phonon Phonons in the high frequency region created by atoms in a crystal vibrating out-of-phase with each other. Phonon Quantum of atomic vibrations in crystalline solids. Quantum dot A small particle surrounded by vacuum or a material of larger band gap, capable of exhibiting effects of quantum mechanical confinement of charge carriers. Quantum well A thin layer of a material of smaller band gap sandwiched between adjacent layers of another material with larger band gap, capable of exhibiting effects of quantum mechanical confinement of charge carriers. Raman scattering Inelastic scattering of photons by atomic vibrations or other elementary excitations in solids/molecules. Resonance Raman scattering Enhancement of efficiency of Raman scattering when the incident photon energy is close to one of the electronic transitions in a solid/molecule. Scattering vector Wavevector transferred in the scattering of light by elementary excitations. Superlattice A periodic arrangement formed by depositing thin alternate layers of two materials on a substrate.

Phonon Confinement in Nanostructured Materials

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

ACKNOWLEDGMENTS It is a pleasure to acknowledge very fruitful collaborations with Professors S. Mahamuni, D. S. Misra, B. R. Mehta, and C. Vijayan. We also thank Dr. B. Viswanathan for interest in the work, Dr. Baldev Raj for support, and Mr. S. B. Bhoje for encouragement.

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