Raman spectroscopy of optical phonon confinement in nanostructured materials

Transcription

1 JOURNAL OF RAMAN SPECTROSCOPY J. Raman Spectrosc. 2007; 38: Published online 11 April 2007 in Wiley InterScience ( Review Raman spectroscopy of optical phonon confinement in nanostructured materials Akhilesh K. Arora, M. Rajalakshmi, T. R. Ravindran and V. Sivasubramanian Materials Science Division, Indira Gandhi Centre for Atomic Research, Kalpakkam , India Received 26 July 2006; Accepted 24 November 2006 If the medium surrounding a nano-grain does not support the vibrational wavenumbers of a material, the optical and acoustic phonons get confined within the grain of the nanostructured material. This leads to interesting changes in the vibrational spectrum of the nanostructured material as compared to that of the bulk. Absence of periodicity beyond the particle dimension relaxes the zone-centre optical phonon selection rule, causing the Raman spectrum to have contributions also from phonons away from the Brillouin-zone centre. Theoretical models and calculations suggest that the confinement results in asymmetric broadening and shift of the optical phonon Raman line, the magnitude of which depends on the widths of the corresponding phonon dispersion curves. This has been confirmed for zinc oxide nanoparticles. Microscopic lattice dynamical calculations of the phonon amplitude and Raman spectra using the bond-polarizability model suggest a power-law dependence of the peak-shift on the particle size. This article reviews recent results on the Raman spectroscopic investigations of optical phonon confinement in several nanocrystalline semiconductor and ceramic/dielectric materials, including those in selenium, cadmium sulphide, zinc oxide, thorium oxide, and nano-diamond. Resonance Raman scattering from confined optical phonons is also discussed. Copyright 2007 John Wiley & Sons, Ltd. KEYWORDS: phonon confinement; nanostructured materials; nanoparticles; vibrational spectroscopy INTRODUCTION The twenty-first century is rapidly emerging as an era of nanotechnology because of the near exponential growth of applications of nanostructured materials in a wide range of areas such as catalysis, magnetic data storage, ferrofluids, soft magnets, machinable ceramics and metallurgy, non-linear optical and optoelectronic devices, and sensors. 1 7 Although a prediction about the potential of nanotechnology was made as early as 1959, 8 the technological applications grew only when some understanding of the fundamental properties of the materials as a function of grain/particle size evolved during the 1980s and the 1990s. Nanostructured materials are generally synthesized in a number of forms: as isolated or loosely connected nanoparticles in the form of powder, 9 as composites of nanoparticles dispersed in a host, 10 or as a compact collection of nano-grains as pellets 11 or thin films. 12 Though in principle one can define a nanostructured Ł Correspondence to: Akhilesh K. Arora, Materials Science Division, Indira Gandhi Centre for Atomic Research, Kalpakkam , India. aka@igcar.gov.in material as one in which at least one of the dimensions is less than 1000 nm, it is often found that several properties do not differ from those of the bulk unless the dimension is less than typically 20 nm. 13,14 In view of this, it is appropriate to define a critical size below which the property of interest differs from that of the bulk value by at least a few percent. The critical size is thus governed by the property and sometimes also by the method of synthesis. This is due to the defect structures and their concentration. Monoatomic solids with two atoms per primitive cell, such as diamond, magnesium, or diatomic compounds such as GaAs, have three optic phonon branches in addition to the three acoustic phonons. 15 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. 16 The momentum conservation selection rule determines the region of the Brillouin zone that can be accessed in the Raman scattering/infrared absorption process. The wavevector of the IR photon for these energies is of the order of cm 1. On the other hand, in a Raman Copyright 2007 John Wiley & Sons, Ltd.

2 Optical phonon confinement in nanostructured materials 605 scattering experiment the magnitude of scattering vector is 2k 0 sin /2, wherek 0 is the wavevector of the incident light and the scattering angle. Therefore the maximum value of the scattering vector could at best be 2k 0 (corresponding to the backscattering geometry), which has a value ¾5 ð 10 4 cm 1 for visible light. Hence, the wavevector probed by either of these 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 ¾ 10 8 cm 1,wherea is the lattice parameter). Furthermore, as the wavenumber of the acoustic phonons corresponding to q ¾ 5 ð 10 4 cm 1 is just about 1 cm 1, it is not straightforward to investigate the acoustic phonons using conventional Raman equipment. Because of this, the acoustic phonons are most conveniently probed using a Brillouin spectrometer consisting of a Fabry Perot interferometer. Thus IR and Raman techniques sample only the optical phonons close to the zone centre (q ³ 0). This q ³ 0 selection rule is essentially a consequence of the infinite periodicity of the crystal lattice. 17,18 However, if the periodicity of the crystal is interrupted, as in the case of nanocrystalline materials, this rule is relaxed. This will be discussed in detail in the next section. This is because the phonon propagation gets 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 a grain size of several micrometres 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. Optical as well as acoustical phonons get confined within the particle. In the case of spherical nanoparticles, this confinement results in the appearance of several discrete spheroidal and torsional modes of the nanoparticle, which are labelled using the angular momentum quantum number and the branch number. The wavenumbers of the confined acoustical phonons depend inversely on the size of the nanoparticle. Owing to the selection rules, only the spheroidal modes are Raman active and these modes are found at low wavenumbers between 10 and 50 cm 1 for particle sizes less than 10 nm. Wavenumbers of these modes have often been used for the estimation of the average particle size. The confined acoustical phonons are beyond the scope of this article, as these are the subjects of other reviews in this Special Issue of the journal. The degree/dimensionality of confinement of phonons is essentially governed by the dimensionality of the system. These two quantities are complementary to each other; for example, a bulk material is a 3D system and is unconfined, i.e. dimensionality/degree of confinement is zero. The first level of confinement occurs in single- and multi-layer thin films grown using layered deposition on substrates. Semiconductor superlattices, single-quantumwell structures and multiple-quantum-well structures 19 are well-known examples of 2D systems because the phonons and charge carriers are confined to remain 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 20 and in carbon nanotubes, 21 whereas the dimensionality of the system reduces to 1D, leading to unique quantum conductance 22 and van Hove singularities in the density of states (DOS). 23 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. OPTICAL PHONON CONFINEMENT As mentioned earlier, in ideal single crystals only the zone-centre optical phonons can be observed using optical techniques such as Raman spectroscopy. However, this q D 0 selection rule is relaxed owing to interruption of lattice periodicity in a nanocrystalline material. This can be qualitatively explained in the following manner: For a particle 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, leads to discrete values of wave vector q,ofwhichthesmallestq is /d,anditsmultiples.this allows phonons corresponding to these selected wave vectors to be sampled by techniques such as Raman spectroscopy. The additional contribution to the Raman intensity from these phonons results in asymmetric broadening of the line shape. Strictly speaking, phonons of several q s over the complete Brillouin zone contribute to the spectral line shape; however, their relative contributions gradually diminish as q approaches the Brillouin-zone boundary. The conceptual visualization of phonon confinement in an isolated nanoparticle, as discussed above, is straightforward; however, several nanostructured systems exist as composites, i.e. nanoparticles embedded in a host matrix or as a thin 2D layer sandwiched between layers of another material, e.g. semiconductor heterostructures. In order to understand the phonon confinement in such cases, one needs to consider the phonon spectra of the particle/layer as well as that of the host. If the optical phonon dispersion curves of the particle/layer do not overlap with that of the host, then these phonons cannot propagate into the host material and the confinement effects are similar to those of an isolated particle, e.g. a GaAs particle embedded in an AlAs matrix. On the other hand, if the phonon dispersion curves of the particle and that of the host overlap partially or completely, as in the case of AlAs and InP, then the phonon confinement can be weak, and partially confined/propagating phonons can exist in such systems. This aspect will be discussed in more detail in subsequent sections.

3 606 A. K. Arora et al. MODELS AND CALCULATIONS Three different types of approaches have been used to theoretically investigate the consequences of confinement on phonon spectra. In analogy with the spatial correlation model originally developed for disordered 27 and amorphous materials, 28 a Gaussian confinement model was proposed by Richter et al. 29 and generalized by Campbell and Fauchet 30 to take into account the contribution from phonons away from the zone centre. Subsequently, Roca et al. 31 presented a rigorous continuum theory, which took into account the coupling between the vibrational amplitude and electrostatic potential to obtain the optical vibrational modes in a spherical quantum dot, and applied it to several systems. 32,33 Microscopic lattice dynamical calculations for nanoparticles containing up to a few thousand atoms have also been carried out and the changes in the phonon spectra as compared to the bulk have been discussed This section briefly reviews all three approaches. The Gaussian confinement model Motivated by the spatial correlation model, it has been argued that the restriction of the phonon wave function within a grain of size d results in an uncertainty 1q ¾ /d in the wave vector of a zone-centre optical phonon and a corresponding uncertainty υω in its wavenumber. 37 Consequently light scattering takes place from quasi-zonecentre optical phonons with wave vectors q up to /d. This is in fact valid only for weak phonon localization and not for completely confined phonons. The Gaussian confinement model due to Richter et al. 29 (RWL model) takes into account the contributions of the phonons away from the zone centre to the Raman phonon line shape and has been extensively used by researchers. Here we describe this model very briefly and give only the final results. For details of the derivation, the reader may refer to the original article. The case of a spherical nanoparticle was considered in the RWL model, which was subsequently extended to rod-like and plate-like geometries by Campbell and Fauchet 30 (CF model). Consider a spherical nanoparticle of diameter d. 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 9 q 0, r with a confinement function or an envelope function W r, which decays to a very small value close to the boundary. Gaussian confinement functions have been extensively used as confinement functions One can write W r as: W r D exp r 2 /d 2 where the value of decides how rapidly the wavefunction decays as one approaches the boundary. RWL showed that for one-phonon Raman scattering the q-dependent weight factor C q for estimating the contributions of phonons away 1 from the zone centre is essentially the Fourier transform of the confinement function. For a Gaussian confinement function, C q becomes: jc q j 2 D exp q 2 d 2 /2 Richter et al. 29 used the boundary condition that the phonon amplitude, which is proportional to W 2 r, reduced to 1/e at the boundary r D d/2 of the particle. This corresponds to D 2. Other values of, suchas8 2 used by Campbell and Fauchet 30 and 9.67 (bond-polarizability model), 34 have also been proposed. The first-order Raman spectrum is then obtained by integrating these contributions over the complete Brillouin zone, as: I ω D 2 jc q j 2 d 3 q [ω ω q ] 2 C 0 0 /2 2 3 where ω q is the phonon dispersion curve and 0 0 is the natural linewidth of the zone-centre optical phonon in the bulk. The calculated Raman line shapes of 4- and 8-nm GaAs nanoparticles are compared in Fig. 1 with that of the bulk. One can notice that as the particle size reduces, the Raman spectra develop marked asymmetry towards the lowwavenumber side and exhibit a marginal shift in the peak position also towards the same side. The increased intensity in the wing of the Raman spectra on the low-wavenumber side basically arises from the contribution due to the negative Raman Intensity / Arbitr. units nm 8 nm Bulk Wavenumber / cm -1 Figure 1. Calculated Raman spectra of confined LO phonon in GaAs nanoparticles. The bulk spectrum is also shown for comparison. Note the asymmetric broadening of the line shape and also the shift of the peak towards the low-wavenumber side. (Reprinted with permission from Ref. 45, Arora et al., Encyclopedia of Nanoscience and Nanotechnology Ed. H.S. Nalwa, Vol. 8, 499 (2004). 2004, American Scientific Publishers).

4 Optical phonon confinement in nanostructured materials 607 dispersion of the phonon branch away from the zone centre. Note that the changes are marginal if the particle size is larger than 10 nm. In order to simplify the calculations, often the dispersion curves have been assumed to be isotropic 38 and approximated to certain analytic forms such as: 39 ω q ¾ A C B cos qa, ω q D ω 0 1ω sin 2 qa/2 (Ref. 40), and the parabolic ω q D ω 0 ˇq 2 (Ref. 31). Here a is the lattice parameter, ω 0 the zone-centre phonon wavenumber, and 1ω the width of the dispersion curve. The zone boundary corresponds to qa D. A, B, andˇ are parameters specific to the material. Empirically fitted more complex functions have also been used. 41 The isotropy assumption may be reasonable in the case of some materials, while it may yield inaccurate results in some other systems. 39,41 It has been shown that the calculated shift and broadening of the Raman peak, when all dispersion branches are separately considered, turn out to be nearly the same as those with the isotropic assumption in nano-diamond, 39 whereas in nano-tio 2, the isotropy assumption gives inaccurate results. The dependence of the peak-shift υω and the line broadening 0 on the particle size has been reported. 30 Other confinement functions such as sinc (sin x /x) (in analogy with the ground state wavefunction of an electron in a spherical potential well) and exponentially decaying function (in analogy with the propagation of a wave in a lossy medium) have also been considered and analytic functions for C q have been obtained. 30 The results suggest that line-broadening is least for a sinc function and largest for an exponential confinement function. 30 Other confinement geometries such as rod-like (2D confinement) and plate-like (1D confinement) have also been considered. It is found that as the dimensionality of confinement reduces, the magnitude of peak-shift and line-broadening reduces dramatically. The departure from bulk is a maximum for a particle, while it is least for a thin film of the same dimension. It is often useful to combine/collapse υω vs d and 0 vs d curves into a single peak-shift vs line-broadening (υω vs. 0) curve for all sizes. The shape of such a curve is found to be unique to the choice of the confinement function. From the analysis of their data and other reported results, 29,42 Campbell and Fauchet 30 found that a Gaussian confinement with D 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. The υω vs 0 curve is particularly useful while analysing data on nanocrystalline systems for which information about the particle size is not available. For example, in the case of ion-implanted GaAs, Tiong et al. 27 argued that the crystallite size reduced owing to irradiation-induced damage in the lattice; however, spatialcorrelation (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. Continuum theory A macroscopic continuum theory that considers the coupling between the mechanical vibrational amplitude and electrostatic potential has been developed by Roca et al. 31 to obtain the vibrational modes of spherical quantum dots. The discrete, allowed wave vectors for a spherical particle are given by q n D 2 n /d, where n is the nth mode of the spherical Bessel function j 1. Thus discrete phonon wavenumbers ω q n are the allowed modes of the nanoparticle. For the Raman scattering process, only the Frohlich interaction is considered. It is shown that in the dipole approximation only the phonons with l D 0 orbital quantum number contribute to the intensity when valence band mixing is neglected. As the deformation potential interaction is neglected, the formalism is applicable only for the Raman scattering close to resonance. The theoretical spectra have been compared with those of CdSe dispersed in glass 32 and CdS embedded in fused quartz. 33 The fitting was found satisfactory only when the intrinsic linewidth of the phonon 0 was assumed to increase as the particle size was reduced. 32 Furthermore, the origin of additional intensity lying above the theoretical curve in the wings on the low-wavenumber side is not well understood. The particle size distribution of CdS crystallites in fused quartz has also been taken in to account while attempting to fit the experimental spectra to the theoretical line shape. However, in order to account for the additional shoulder in the wing on the low-wavenumber side, it was necessary to invoke additional effects such as random variation of CdS bond wavenumber, cluster shape irregularities, and fluctuation of the nearest-neighbour interaction constant. 33 These results suggest that real composite systems are far from ideal spherical particles for which the theory is well established. Microscopic lattice dynamical calculations Lattice dynamics of an Si sphere consisting of up to 657 atoms has been carried out by direct diagonalization of the dynamical matrix. 34 The force constants were obtained using the partial density approach, 43 in which the ion ion interaction contribution is calculated using the Ewald method and the electronic part is obtained from a pseudopotential calculation. The eigen wavenumbers and eigen vectors thus obtained were used for calculating the Raman spectra in the framework of the bond-polarizability model. 44 The polarizability of the whole system is calculated as a sum of contributions from each bond. From the calculations, the vibrational amplitude was found to drop to about 3.6% at the boundary. 34 The shift of the Raman peak wavenumber was found to exhibit an empirical power-law behaviour on the particle size as υω ¾ d Microscopic lattice dynamical calculations have also been carried out for GaP quantum dots 35 and Ge nanocrystals. 36 The phonon spectrum obtained using the atomistic potential has been separated into bulklike (dot interior) or surface-like depending on the spatial localization. The wavenumbers of the bulk-like zone-centre

5 608 A. K. Arora et al. longitudinal optic (LO) and transverse optic (TO) phonons were found to decrease with decreasing dot size. 35 In Ge nanocrystals of size up to 6.8 nm (7289 atoms), surface modes existed both at low wavenumbers (<50 cm 1 )and at high wavenumbers (>260 cm 1 ). The high-wavenumber mode was identified to be a resonant mode, whereas the low-wavenumber modes were essentially confined acoustic phonons predicted by the Lamb theory. A deviation from the Lamb model was also found for particles of size less than 4 nm. Thus microscopic calculations have been able to provide further insight about the nature of the vibrations of the nanoparticles. Phonon amplitudes from microscopic lattice dynamical calculation have also been attempted to fit to a Gaussian confinement function, and a value of 9.67 for the exponent wasobtainedasthebestfit. 34 It may be pointed out that the phonon amplitude at the boundary of the nanoparticles (¾3.6% of that at the centre) found from the calculation 34 is much smaller than 1/e used by RWL 29 and much larger than exp ( 4 2 ) used by CF. 30 A comparison between the Gaussian confinement functions and the corresponding Fourier coefficients for the three different values of (2, 9.67, and 8 2 ) has been made recently 45 and the results are shown in Fig. 2. One can notice that the weight factor jc q j 2 for W 2 (r) C(q) (3) (2) (a) (1) r/d (1) (2) (3) (b) (qa/π) Figure 2. (a) Squares of the Gaussian confinement functions W r and (b) the corresponding Fourier transform C q for different values of. (1) D 2.0 (RWL model), (2) D 9.67 (bond-polarizability model) and (3) D 8 2 (CF model). The phonon amplitude W 2 is plotted up to the boundary of the particle (r D d/2) and C 2 up to the Brillouin-zone boundary q D /a. Reprinted with permission from Ref. 45, Arora et al., Encyclopedia of Nanoscience and Nanotechnology Ed. H.S. Nalwa, Vol. 8, 499 (2004). 2004, American Scientific Publishers. the Raman intensity drops too rapidly for D 2andtoo slowly for D 8 2 as one moves away from the Brillouinzone centre. Consequently, the RWL model predicts only a marginal change in the Raman spectra, while the CF model causes maximum departure from the bulk for the same particle size. On the other hand, the calculations of Zi et al. 34 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 37,40,46 49 on the basis of the Gaussian confinement model using D 8 2. The unique feature of Raman spectra from nanocrystalline systems is the broadening and asymmetry of the line shape on the low-wavenumber side. In the Gaussian confinement model, asymmetry is intrinsically linked to the negative dispersion. On the other hand, within the framework of the rigorous continuum theory, the asymmetry can arise because of higher-order, less intense discrete modes. 32 Microscopic lattice dynamics has also shown the existence of modes of wavenumbers smaller than the zone-centre optical phonons, 35 consistent with the dispersion curve. Thus barring a few exceptions 50 where the asymmetry is marginal, the asymmetries in most of the systems are understandable on the basis of phonon dispersion of the bulk. However, often broadening and asymmetry more than theoretically expected are reported. The excess broadening is attributed to life-time broadening due to the presence of defects. On the other hand, excess asymmetry has been tentatively assigned to several factors such as particle size distribution, random variations in the bond wavenumber, and particle shape irregularity. SURVEY OF EXPERIMENTAL RESULTS As mentioned earlier, a variety of semiconductor, ceramic/dielectric, and metallic materials have been synthesized as systems with different dimensionality ranging from 3 to 0. The consequences of one or more dimensions being in the nanometre range on the vibrational spectra have extensively been investigated by researchers. This section reviews the reported results on nanoparticles, nanocomposites, and nanostructured films. For the sake of completeness, a few selected results on 1D and 2D confined systems are also included and discussed. Single- and multi-layers and superlattices 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 19 in view of their applications in light emitting diodes and diode lasers. 51 In these superlattices, the 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 wavenumbers of GaAs does not overlap with that of AlAs. Hence the phonons of the GaAs layer cannot propagate into the neighbouring AlAs layers and vice versa. Thus phonons in each of the GaAs and AlAs

6 Optical phonon confinement in nanostructured materials 609 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 q j D j/d 1,whered 1 is the thickness of the GaAs layer, are allowed. The confined phonon wavenumbers ω j then correspond to the discrete q j points on the dispersion curve of GaAs. 52 Similarly, the confined optical phonons of the AlAs layer of thickness d 2 correspond to the q j D j/d 2 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, i.e. when the incident photon energy is close to that of an electronic excitation of the GaAs quantum well. 52 Under non-resonant conditions, the intensities of these modes are weak. In many cases, 19 one of the layers is an alloy Al x Ga 1 x As. This mixed-crystal system exhibits a two-mode behaviour, 53 i.e. it exhibits both GaAs-like and AlAs-like modes. Hence AlAs-like phonons remain confined in the barrier layer (Al x Ga 1 x As) in a GaAs/Al x Ga 1 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 (d 1 C d 2 )atq j D j/ d 1 C d 2 ; however, this effect has not been observed for propagating optical phonons in GaAs/Al x Ga 1 x As superlattices because of the highly dispersive character of the optical modes. 54 In contrast to GaAs/Al x Ga 1 x As superlattices, zone-folding effects have been observed in GaN/Al x Ga 1 x N superlattices. 55 As this mixed-crystal system exhibits one-mode behaviour, it is argued that the overlap between the DOS in the two layers is significant. It may be mentioned that the acoustic phonons, whose dispersion curves overlap over a wide range of wavenumbers, propagate through both the layers exhibiting zone-folding effects due to the new periodicity. 56 Confined optical phonons have been found also in single GaAs quantum wells under resonant conditions. 57 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. 58 The confined phonons manifest themselves as minima in the transmission spectrum. Under normal incidence only TO phonons are observed, while in oblique incidence both TO and LO phonons are seen. From the wavenumbers of these confined phonons, the dispersion curve along the <111> direction could be deduced. In superlattices and in quantum-well structures interface optical phonons have also been observed. 59,60 Phenomenological models 61 predictthatthesemodeshavewavenumbers between the TO and LO phonons of the constituent layers. If the interfaces are sharp, the interface phonons are found to be weak. 55 Nanowires and nanorods Recently, nanowires of several tens of micrometre length of a variety of materials such as Si, 20,62 Ge, 63 GaAs, 64 SiC, ZnO, and TiC 71 have been synthesized using laser ablation 72 and other methods. The diameters of these nanowires range 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. 71 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 F 2g optical phonon in Si nanowires show broadening and peak-shifts 62 similar to that predicted by the Gaussian confinement model. For a 10-nm diameter nanowire, the peak is found to shift to 505 cm 1 from 519 cm 1 and also to broaden to 13 cm 1 from 3.5cm 1. Additional peaks at 302 and 964 cm 1 have also been reported. These were assigned to overtones of the zone-boundary phonons. 62 Appearance of zone-boundary phonons in crystals with a large density of defects 73 or in mixed crystals 74 has often been reported. This arises because of the relaxation of the q D 0 selection rule due to the presence of disorder in the crystal. Similarly, overtones and combinations constitute the second-order spectra and these are also enhanced in the presence of disorder. On the other hand, Wang et al. 62 apply the phonon confinement model also to the overtones of the 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 be ascribed only to higher defect density, resulting in the appearance of peaks corresponding to oneand two-phonon DOS. 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, 49 making such terms lose their physical significance. This will be discussed further in a subsequent section. In another study on 60-nm diameter aligned Si nanowires synthesized using CVD, a 20 cm 1 broad Raman spectrum centred at 506 cm 1 was found. 75 Though the authors attribute the shift to the phonon confinement, its magnitude appears too large for a 60-nm nanowire. ZnO nanowires exhibit Raman spectra arising from LO phonons and their overtones. 69,70 Raman peaks up to 4-A 1 -LO have been reported in 150-nm diameter ZnO nanowires grown on sapphire substrates. 70 This essentially corresponds to a bulk-like sample. An increase of the electron phonon interaction as a function of diameter, inferred from the ratio I 2LO /I 1LO, has been found in the case of nanowires of diameters nm 69 similar to that reported in nanoparticles. Raman measurements on 4 14 nm diameter SnO 2 nanowires

7 610 A. K. Arora et al. have revealed new modes at 691, 514, and 358 cm 1,which are assigned to the activation of Raman-forbidden A 2u LO modes and defects in small-sized nanowires. 76 Germanium nanowires with an oxide layer coating have been synthesized using the laser ablation technique. 63 As expected, larger core diameters in the range of 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 GeO x sheath have exhibited broad TO and LO phonon modes; 64 however, the broadening was found to be nearly symmetric. Surprisingly, the red-shift of the LO phonon was very large, ¾40 cm 1 for nanowires with diameters in the range nm with an average diameter of 60 nm. Such a large shift cannot be accounted for on the basis of the phonon confinement effect alone. Other factors such as defects and residual stresses have been argued to contribute to the decrease of the LO phonon wavenumber. Silicon carbide nanowires of average diameter 80 nm with a coating of 17 nm SiO x have also shown 65 very broad Raman spectra that resemble those arising from phonon DOS rather than from phonon confinement effects. The redshifts of 12 to 34 cm 1 for the TO and LO phonons were attributed to confinement effects and internal stresses. 65 CdSe nano-filaments incorporated in fibrous magnesium silicate (chrysotile asbestos) have shown a polarized Raman spectrum. 77 Isolated and compacted nanoparticles and nanocomposites Isolated or loosely connected nanoparticles such as selfsupporting powders and nanoparticles dispersed in other hosts have been the most extensively studied nanostructured systems. In many investigations, quantitative fitting of the phonon line shape has also been carried out. 37,40,78 The results are grouped into four subsections. Oxides and compounds 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 49 in the case of zinc oxide nanoparticles by examining phonons of different symmetries (irreducible representations). Zinc oxide has the wurtzite structure and, consequently, has phonons of symmetries A 1, E 1,andE 2 at 393, 591, and 465 cm 1 respectively. Figure 3 compares the Raman spectrum of 4-nm ZnO nanoparticles with that of the bulk. One can notice that for the nanoparticles the E 1 -LO mode exhibits a much larger broadening as compared to that for the E 2 mode. Table 1 gives the peak-shift and line-broadening data for these modes in 4-nm particles. Note that the widely different broadening of E 1 and E 2 Raman Intensity / Arbitr. units nm Bulk Wavenumber / cm -1 Figure 3. Raman spectrum of 4-nm ZnO nanoparticles. The continuous curve is the calculated spectrum based on the Gaussian confinement model. A spectrum of polycrystalline ZnO powder representing bulk is also included in the figure for the sake of comparison. Adapted with permission from Ref. 49, M. Rajalakshmi etal.j.appl.phys.87, 2445 (2000). 2000, American Institute of Physics. Table 1. Peak-shift υω and line-broadening 0 for optical phonons of different symmetries for 4-nm zinc oxide nanoparticles. 0 0 is the natural linewidth of the phonon and 1ω is the width of the dispersion curve Phonon υω cm 1 0 cm cm 1 1ω cm 1 E E 1 -LO phonons arises because of widely different widths 1ω of the corresponding dispersion curves. Thus, the broader Raman line shape of the E 1 phonon for a given size of the nanoparticle is understandable on the basis of the width of its dispersion curve. Subsequently, several studies on ZnO nanoparticles, synthesized by a variety of methods, have been reported Thin ZnO films grown on SiO 2 opal structure have shown resonance enhancement of the 3-LO Raman peak along with evidence of Frohlich vibrational modes. 79 Excitation of 20- nm ZnO nanoparticles (synthesized using a wet chemical method) with the 325-nm UV line exhibited a red-shift (as much as 14 cm 1 ) of 1-LO phonon wavenumber as a function of laser power. 81 This was attributed to the laser heating of the nanoparticles to temperatures as high as 1000 K, as estimated form the anharmonicity of the phonons. 80 In another investigation on ZnO nanoparticles, a new peak at 523 cm 1, assigned to the surface phonons, 82 has been found. On the other hand, surface optical phonons in Zn/ZnO core shell structures, with wavenumbers ranging from 565 to 548 cm 1 depending on the dielectric constant

8 Optical phonon confinement in nanostructured materials 611 of the surrounding medium (air, ethanol, benzene), have been identified. 83 The mode wavenumber is also found to depend on the shell thickness. There are also a number of Raman spectroscopic studies on nanocrystalline powders that exhibit broadening and peak-shifts similar to those expected for phonon confinement; however, quantitative analyses have not been carried out. As pointed out earlier, in most of the systems the optical phonon wavenumber decreases as one moves away from the Brillouin-zone centre, i.e. 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 1ω D 50 cm 1, while the other undergoes 90 a positive dispersion of 1ω DC160 cm 1. For nanocrystalline thorium oxide both the branches are expected to contribute to the Raman line shape. Recently, Raman spectra of nanocrystalline ThO 2 have been reported, 91 which are found to be less asymmetric as compared to other crystals. Figure 4 shows 92 the Raman spectrum of a nanocrystalline sample arising from the triply degenerate F 2g mode of ThO 2.Ascomparedtothe bulk, the Raman spectrum broadens from 7 to 26 cm 1. The average particle size was 7.5 nm from transmission electron microscopy (TEM) and 6 nm from X-ray diffraction (XRD). The calculated spectrum was obtained by adding the intensities arising form the two phonon branches after appropriate scaling of their relative contributions. 92 The calculated spectrum still remained significantly below the data on the high-wavenumber side of the spectrum. To account for this additional intensity, a Lorentzian peak centred at 475 cm 1 and of width 42 cm 1 was also added to Raman Intensity / Arbitr. units Wavenumber / cm -1 Figure 4. Measured and calculated Raman spectrum of nanocrystalline powder of ThO 2. Dashed curve: 1ω DC160 cm 1 contribution; dot-dash curve: 1ω D 50 cm 1 contribution; dotted curve: 475 cm 1 Lorentzian; and continuous curve: the total calculated spectrum. Reprinted with permission from Ref. 92, Rajalakshmi et al., J. Nanosci. Nanotechnol. 3, 420 (2003). 2003, American Scientific Publishers. the total spectrum. This additional peak was attributed to a contribution from an overtone of a zone-boundary phonon at the point M in the Brillouin zone. The less pronounced asymmetry 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. 92 There have been several investigations on nanocrystalline TiO 2 synthesized as thin films 93 as well as powders. 38,41 Raman spectroscopy has also been used extensively for distinguishing between the anatase and rutile phases of TiO 2 from the positions of their characteristic peaks. In cluster-deposited films, the large clusters were found to be anatase, while the smaller ones had the rutile structure. 93 However, annealing of these films above 800 C resulted in transformation to the rutile phase. It has been possible to synthesize nanocrystalline anatasepowderwithgrain size as small as 5 nm using the aqueous precipitation method. 38 In order to estimate the particle size from the peak position and line-broadening of the 144 cm 1 Raman peak, their dependencies on the particle size have been calculated using the Gaussian confinement model by assuming a spherical/isotropic Brillouin zone. Empirical relations of the type ω D ω 0 C Ad 1.5 and 0 D 0 0 C Bd 1.5,whereAand B are empirical parameters, were suggested for the peak position and line-broadening, respectively. 38 Subsequently, Bassi et al. 41 have emphasized that the isotropic dispersion curve model is inapplicable in the case of TiO 2 and that it leads to incorrect predictions of peak-shift and linebroadening. In fact the dispersion curves along the 0 Z and 0 N symmetry directions are nearly flat, and strong positive dispersion exists only along 0 X direction. 94 Good agreement for TiO 2 nanopowders was found when the actual dispersion curves were taken into account properly. 41 Furthermore, the 144 cm 1 Raman peak position was also shown to depend strongly on the oxygen stoichiometry. In recent studies on nanocrystalline CeO 2 films, the role of defects such as oxygen non-stoichiometry and that of grain size in determining the Raman line shape have been examined. 95,96 The linewidth of the 466 cm 1 Ce O stretching mode was found to increase linearly with the inverse grain size. 96 It has been further argued that the correlation length, estimated by fitting the Raman line shape to the Gaussian confinement model, can be identified with the average interdefect distance (oxygen vacancy) instead of the grain size. The effect of disorder has been discussed in other systems also recently. 97 Several GaAs nanostructures have been created using electrochemical 98,99 and photochemical methods. 100 An orthorhombic structure of a nano-gaas film on an indium tin oxide substrate, inferred from TEM, 98 exhibits a new Raman peak at 250 cm 1, in addition to the TO and LO phonons. The new peak has been assigned to a point defect such as an arsenic vacancy. 99 The particle size distribution (Gaussian) has also been taken into account while fitting

9 612 A. K. Arora et al. the calculated line shape to the Raman spectra of GaAs nanostructures. 100 Temperature dependence of the Raman spectra of nanocrystalline ZnSe has been analysed on the basis of the anharmonicity of the LO phonons. 101 These results suggest a smaller anharmonicity for the nanoparticles as compared to the bulk and the surface phonon modes to be the main source of broadening/asymmetry of the spectra. A few monolayers of AlSb deposited on a GaAs substrate using molecular beam epitaxy are found to self-assemble in the form of plate-like quantum dots during annealing at 500 C. 102 In addition to phonon confinement effects, sometimes compressive stresses also play a role in determining Raman line shapes. 39 Monoatomic systems Several investigations on nanostructured silicon have been reported Paillard et al. 103 have shown that the size of Si nanoparticles can be estimated accurately from the peak-shift data if the asymmetry of the phonon dispersion curves is properly taken into account in the confinement model. They used a sinc confinement function to calculate the peak-shift as a function of the particle size and found good agreement for size-selected, cluster-assembled Si nanofilms. In another study on a cluster-deposited Si film, evidence of hydrogenation was found from in situ infrared spectroscopy. 105 Silicon nanoparticles have also been synthesized by ball-milling SiO 2 and aluminium powders together for about 12 h. 104 A particle size of about 4 nm was estimated by fitting the asymmetric Raman line shape to the Gaussian confinement model. Raman investigations on a 2D periodic array of Si quantum dots (feature size ¾70 nm) using different wavelengths show peak-shift and broadening as a function of laser power of the 532-nm line, and not with 785-nm excitation. 106 This was attributed to laser-induced heating of the quantum dots to temperatures as high as 700 K, estimated from the anti-stokes to Stokes intensity ratio. As the feature size was too large to exhibit a phonon confinement effect, the authors assigned the broadening and peak-shifts found for the 785-nm excitation to the nanocrystalline domain substructure at the interface of 2D nanostructured Si. The life-time of phonons in nanocrystalline Si has also been measured and found 107 to be more than that in amorphous Si. A comparison of the Raman spectra of Si-doped SiO 2 films with those of theoretically calculated vibrational DOS of Si 33 and Si 45 clusters suggested 108 the presence of such clusters in SiO 2. In another study on deposited Ge-doped SiO 2 films, a broad Raman peak at 270 cm 1, characteristic of amorphous- Ge, was found, which evolved into an asymmetric sharper peak at 300 cm 1 upon annealing at above 700 C. 109 The nanocrystallite size, estimated from the fitting of the Raman spectra, was found to increase from 4 to 11 nm upon annealing at different temperatures. Porous silicon (p-si), obtained 110 from electrochemical etching of Si, has been a subject of considerable interest in view of its efficient photo and electroluminescence at ambient temperature. 111 The pore diameter and consequently the size of interconnected Si nanostructure depends on the electrochemical conditions. 112 The Raman spectrum of p-si consists of an asymmetrically broadened F 2g phonon line characteristic of nanocrystalline Si and an overlapping broad peak at 480 cm 1 associated with amorphous Si. 113,114 Fitting of the Raman spectrum to a confined phonon line shape has frequently been carried out to estimate the average particle size. 115 Confined phonons of p-si have been found to be responsible for the photoluminescence arising from radiative recombination of carriers across the indirect transition 116 similar to that found in crystalline Si. Often it has also been possible to stabilize metastable phases of compounds in the form of nanocrystalline powders 38,117 and composites. 40 Metastable monoclinic phase of selenium dispersed in a polyacrylamide host has exhibited phonon confinement effects. 40 The Raman line shape for the E 1 phonon was calculated using Eqn (3) and the phonon wavenumber of the bulk Se. The intrinsic linewidth of the phonon was taken as 13 cm 1, whereas the width of the dispersion curve was taken as 16 cm 1 on the basis of the reported dispersion curve. The good agreement of the calculated profile with the spectrum suggests that the peakshift and line-broadening are consistent with the Gaussian confinement model. Nanopores of zeolites have also been used to capture nanoparticles of Se and Te. 118 Raman spectra showed evidence of trapping of either a molecular Se 8 or Te 8 or formation of an irregular array of chains and clusters depending on the size and connectivity of pores. Natural graphite samples, retrieved from geological rocks in the Dharwar region of India 119 and from east Antarctica, 120 have been found to be nanocrystalline. The rocks contained approximately 1% graphite. Raman spectroscopic studies on these samples have revealed, in addition to the first and the second-order Raman peaks, the existence of a disorder peak (D-band) of varying intensity at ¾1350 cm 1 as shown in Fig. 5. Note the broadening of the main peak (1580 cm 1 ) arising from the E 2g phonon of graphite for the 5-nm grain size. The D-band was initially assigned to the zone-boundary phonons activated in the first-order Raman spectrum owing to the presence of defects/disorder. 121 Subsequently, on the basis of its dependence on the excitation photon energy, it has been attributed to a double-resonance arising from the upper, fully symmetric phonon (TO) branch near the K- point in the Brillouin zone. 122,123 The intensity of the disorder peak relative to the main peak has been reported to vary inversely with the intra-planar grain size (La) of the graphite samples, 124 which was estimated from the broadening of (002) XRD peak. The grain sizes ranged from 5 to 20 nm depending on the geothermal (pressure temperature) conditions in the interior of the earth. 119,120 Figure 6 shows the inverse correlation, which is also valid for pyrolytic graphite. As mentioned earlier, if the phonon spectrum of the particle overlaps significantly with that of the surrounding

10 Optical phonon confinement in nanostructured materials 613 zone at discrete points q n D nq B /N 1 n N, wherena is the size of the particle. The intensities I n from the discrete phonons ω q n were taken to vary according to a power law I n ¾ b n b < 1. The discrete model of the phonon confinement yielded a satisfactory fit to the experimental phonon line shape. 125 Figure 5. First-order Raman spectra of natural graphite samples of different grain sizes. Adapted with permission from Ref. 119, Sharma et al., J. Geol. Soc. India 55, 413 (2000) Geological Society of India. Figure 6. Correlation of the intensity (area) ratio of the disorder to main peak (D/O) with the inverse grain size La 1. Reprinted with permission from Ref. 119, Sharma et al., J. Geol. Soc. India 55, 413 (2000) Geological Society of India. 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 125 in the case of nanocrystalline diamond particles surrounded by amorphous carbon region. 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 standing wave pattern in the particle with the phonon wavevector sampling the Brillouin Semiconductor-doped glasses and mixed crystals In the case of composites synthesized either as thin films using co-sputtering 12 or by doping melt-quenched oxide glasses, 126 nanocrystalline precipitates form during annealing at elevated temperatures. 127,128 The departure of the LO phonon wavenumber from the expected behaviour for CdS nanoparticles smaller than 5 nm dispersed in a GeO 2 matrix has been attributed to defects. 129 Evidence of thepresenceofacdosurface-cappinglayeronpulsedlaser-deposited CdS nanoparticles in SiO 2 matrix has been found from the presence of its characteristic peak in the Raman spectrum. 130 The peak-shifts with respect to the bulk have been reported to be rather small in the case of CdS nanoparticles, even for 3 nm size, 131 whereas the broadening was attributed to the decay of LO phonons into acoustic phonons. The small peak-shift is essentially a consequence of the rather narrow dispersion curve of hexagonal CdS. 16 Semiconductor mixed crystals such as CdS x Se 1 x (Ref. 132) and Cd 1 x Zn x S (Ref. 127) dispersed in oxide glasses as nanocrystalline precipitates have beenextensively studied in view of their interesting optical properties and applications as long-pass optical filters. The system CdS x Se 1 x exhibits a two-mode behaviour, and both CdSe-like and CdS-like confined LO phonons are observed. 37 On the other hand, the Cd 1 x Zn x S system exhibits the one-mode behaviour. The blue-shift of the LO phonon wavenumber during the late stage of annealing of Cd 1 x Zn x S nanoparticles dispersed in oxide glass host that contained 20% ZnO has been attributed to a change in the stoichiometry (x) from 0.16 to 0.23 in the nanoparticle. 127 A CdSe/CdS core shell structure, synthesized using an aqueous precipitation method, has exhibited CdSe and CdS LO phonons and also a CdSelike surface phonon 133 at ¾182 cm 1. A Raman line shape calculated using the Gaussian confinement model, including the surface phonon, was used for fitting the spectra. In addition to the confined optic phonons, the presence of surface phonons in the Raman spectra of nanostructured 78, materials has been reported in a number of systems. Surface phonons are expect to have wavenumbers between TO and LO phonons. 38 Since the fraction of surface atoms increases as the grain size of a nanostructured material reduces, surface phonons are observed with noticeable intensity for small-size particles. The dependence of surface phonon wavenumber on the dielectric constant of the surrounding medium has also been examined. 135 In the nanoparticles of mixed crystals such as CdS x Se 1 x, two surface phonons, one each of CdSe-like and CdS-like, have been reported. 134