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Light Scattering of Semiconducting Nanoparticles G. Irmer, J. Monecke Institute of Theoretical Physics, Freiberg University of Mining and Technology, D-09596 Freiberg, Germany P. Verma Department of Applied Physics, Osaka University, Osaka 565-0871, Japan CONTENTS 1. Introduction 2. Quantum Confinement in Semiconductors 3. Electron–Phonon Coupling 4. Acoustic Phonons 5. Surface Phonons 6. Optical Phonons 7. New Instrumentation and Methods Glossary Acknowledgments References 1. INTRODUCTION The electronic and optical properties of semiconducting crystallites with a size of a few nanometers (often called nanocrystals or quantum dots) differ considerably from those of the corresponding bulk material. Systems with nanoparticles represent a new class of materials with promising properties due to their nonlinear optical, photoconductivity, photoemission, and electroluminescence behaviors. They have been thoroughly investigated during the last two decades for their promising practical applications, for instance, in solar energy conversion, optical processing devices, and photocatalytic processes. The bulk crystalline structure is preserved in nanocrystals. However, due to quantum confinement nanocrystals have molecule-like discrete electronic states which exhibit strong size dependence. Semiconductor nanocrystal systems are often synthesized in oxide glass hosts or prepared in the form of dispersed colloids. Among numerous other methods that have been developed we mention the preparation of ISBN: 1-58883-001-2/$35.00 Copyright © 2003 by American Scientific Publishers All rights of reproduction in any form reserved. nanocrystals in polymers, zeolites, and micelles. By electrochemical anodic oxidation of crystalline semiconductor substrates porous nanostructures or free standing columns can be produced. Most of the light scattered by a particle is elastically scattered without an energy shift. A small part, a few orders weaker in intensity, is scattered inelastically. This inelastic scattering carries information about energetic levels of the particle. In a Raman scattering process with incident photon energy L the scattered photon S has lost (Stokes scattering) or gained (anti-Stokes scattering) the energy of an elementary excitation (phonons, electronic transitions, etc.) of the particle: S = L ∓ (1) The Raman scattering is sensitive for probing the local atomic arrangement. Crystalline and amorphous phases or the influence of strain can be detected in the Raman vibrational spectra. Light couples to phonons intermediate through electronic exitations by either the deformation potential interaction or the Fröhlich interaction. The deformation interaction is present in polar crystals such as the compound semiconductors as well as in nonpolar crystals such as Si and Ge. In polar crystals the electric fields induced by vibrations interact with electronic excitations via the Fröhlich mechanism. When the Raman scattering process involves real intermediate electronic states by excitation with energy near gaps of the electronic band structure, the scattering cross section may be resonantly enhanced [1]. The Raman scattering probes the electron–phonon interaction as well as the confinement of the phonons and contains information about the confined vibrational and electronic states. The light-scattering process in an infinite crystal obeys the pseudo-momentum conservation relation = k L ∓ q k S (2) Encyclopedia of Nanoscience and Nanotechnology Edited by H. S. Nalwa Volume X: Pages (1–26) 2 S k L denote the wavevectors of the plane waves where k of the scattered (incident) light, q is the wavevector transfered to the excitation. The energies of crystalline vibrations probed by Raman scattering are about <01 eV and those of the exciting visible laserlight are about 2.5 eV (corresponding to the wavelength L ≈ 500 nm). Therefore, from Eq. (1) we get S ≈ L and the wavevector in a typical backscat ≈ 4n/L and tering experiment will be on the order of q much smaller than the wavevector q = 2/a at the boundary of the Brillouin zone, a is the lattice constant, and n denotes the refractive index of the sample. That means that only excitations near the center of the Brillouin zone ( -point) will contribute to the Raman scattering. If the process of light scattering by excitations is limited to the finite space occupied by the nanoparticle, an uncertainty of the transferred wavevector q occurs and excitations with wavevectors from the whole Brillouin zone contribute to the scattering. This effect will vary with size as well as with the eigenfrequencies of acoustic and optical phonons and the electron–phonon interaction in small particles. The vibrational Raman spectra of nanoparticles can therefore be used for size determinations. This is important because one of the major goals in the field of preparation is the synthesis of monodisperse nanocrystals with a narrow size distribution and with well-defined surfaces. Size effects will be discussed in Sections 3, 4, and 6. In addition to acoustic and optical phonon modes, surface-related modes appearing in polar crystals will be discussed in Section 5. They are observable for particle sizes smaller than the wavelength of the exciting laser light inside the particle. The pioneering work was the first observation of confined acoustic modes with low-frequency Raman scattering in nanocrystals embedded in glasses by Duval et al. [2] and the unambiguous observation of surface-related modes in semiconductor nanoparticles by Hayashi and Kanamori [3]. It is relatively easy to produce nanocrystals of II–VI compounds embedded in different media such as glasses, polymers, or solutions. Besides potential applications of such systems the possibility of growing isolated nanoparticles in glasses with desired size and narrow size distribution and nearly spherically shaped particles makes such systems interesting for light-scattering investigations in comparison with theoretical calculations. The observation of strong visible photoluminescence at room temperature from porous Si [4] with the possibility of optoelectronic applications has initiated intensive research on nanostructured silicon (e.g., reviewed in [5]). Intensive light emission from Ge nanocrystals has also been found. Within the last few years there has also been progress in the fabrication of nanostructured compound semiconductors with desired structures, especially porous III–V semiconductors. Nanostructuring provides important degrees of freedom for phonon engineering. It may change considerably the density of phonon states, induce surface-related vibrations, and spatially confine the bulk phonon modes. Although not expected by use of the term “quantum dots” for nanoparticles, many properties of confined vibrations Light Scattering of Semiconducting Nanoparticles and surface modes can be understood by applying classic physics developed about 100 years ago [6, 7]. Inelastic light scattering also has been very intensively investigated in two-dimensionally layered semiconductor structures exhibiting one-dimensional confinement in growth direction. Readers may refer to reviews in Refs. [8, 9]. 2. QUANTUM CONFINEMENT IN SEMICONDUCTORS Semiconductor crystals with sizes of a few nanometers typically contain several hundreds to several thousands of atoms; they are too small to have bulk electronic properties. Quantum confinement in such crystals arises due to the finite size of the crystal, which limits the motion of electrons, holes, and excitons. In a small nanocrystal with size comparable to the Bohr radius of the corresponding bulk material, spatial confinement effects on the electron–hole system become significant. At very small crystal sizes, the character of the exciton starts disappearing and the electrons and holes become individually confined, because the kinetic energies of the electrons and holes become dominant and the Coulomb interaction between electrons and holes starts losing its significance. Theoretical models predict [10] that the energy spectrum of such a material consists of a series of discrete lines, which change their positions with particle size, and a blueshift of the band gap with respect to the corresponding bulk material is observed. This blueshift gives a measure of the confinement, which can be observed, for example, in absorption [11] or photoluminescence (PL) [12] experiments. For a bulk crystal, the electron wavelength can extent to take all values including zero to infinity, facilitating k and, hence, the band structures are parabolic. Electrons (holes) can occupy any position on the parabolic conduction (valence) band. The band gap is defined as the energy difference between the conduction and the valence band at = 0. On the other hand, in the case of a nanocrystal, the k electron wavelength is confined to the size of the crystal and, takes only nonzero discrete values, depending on hence, k the particle size. The electrons and the holes with minimum = 0, and the band energy can only take a position where k gap for a nanocrystal has to be defined as the energy difference between the conduction and the valence band at this which is higher than the band gap of nonzero value of k, the corresponding bulk. This can be understood from the illustration presented in Figure 7 of Ref. [13]. The band gap for nanocrystals can be expressed as E = Eg + E (3) where Eg is the bulk band gap. The energy difference E is referred as the confinement energy. The value of E depends on the crystal size and becomes zero for bulk. The confinement can be classified into two categories, depending upon the size of the crystal. If the Bohr radius of the exciton is aB and R is the particle radius, then the system is said to have strong confinement if R aB and weak confinement if 3 Light Scattering of Semiconducting Nanoparticles R aB . In the simplest approach, the confinement energy in the case of strong confinement is given by [10, 14–17] E = 2 2 2R2 with 1 1 1 + = me mh (4) In the case of weak confinement the confinement energy is E = 2 2 2MR2 with M = me + m h (5) Here, me and mh are the electron and the hole mass, respectively, and are roots of the spherical Bessel functions; the first root for the lowest energy transition is = . The simple effective mass model was improved by empirical pseudopotential [18] and tight-binding [19, 20] methods with application to II–VI semiconductors. The first-principles pseudopotential method was also applied to silicon [21]. For more details see, for example, the review in [22]. The II–VI nanoparticles are the most investigated materials, and PL spectroscopy is widely used to measure the confined energy [12, 23]. A typical PL spectrum from nanoparticles contains a sharp structure corresponding to the band edge, and some broad structures related to shallow and deep traps. The structure corresponding to the band edge is usually asymmetrically broadened, partially due to the temperature and the shallow traps very close to the band edge. However, under careful experimental conditions, the luminescence at the band edge can still be used with sufficient accuracy to estimate the band gap of the material. As discussed above, the effective band gap of a nanoparticle system is increased due to confinement. This can be easily observed in PL spectra, in the form of a blueshift of the band-edge luminescence. Because this shift depends on particle size, and the samples usually have an asymmetric particle size distribution, the band-edge luminescence is also partially asymmetrically broadened due to the particle size distribution. It has been observed [13] that in some cases the peak position of the band-edge luminescence strongly depends on the probing laser power and, hence, it is important to measure the luminescence at very low probing power-density to avoid other causes, which may shift the luminescence peak. By measuring the peak position of this band-edge luminescence, one can estimate the effective band gap E = Eg + E of the material. By substituting the bulk band gap Eg and by using the above-mentioned equation for the confinement energies, one can estimate the particle size. Although the absolute particle size estimation using PL experiments alone may not give a very accurate value of the particle size, it gives a good estimation of the confinement energy, especially when one wants to compare the confinements due to changing particle sizes. A comparison with Raman scattering also shows satisfactory agreement for the particle size estimation. 3. ELECTRON–PHONON COUPLING Electron–longitudinal optical (LO) phonon coupling, governed by the Fröhlich interaction, is very important in the electronic and optical properties of a nanoparticle system, and its investigation has attracted a lot of attention. The coupling between free charge carriers and vibrational excitations plays a central role in determining the transport properties and the energy relaxation rates of the excited carriers. One of the interesting points is that the electron– phonon interaction in small nanoparticles can be investigated experimentally beyond the bulk approximations. In this system, the confinement effects influence not only the electronic and vibrational states, but also the coupling between them. Although the effect of size dependence on the electron–phonon coupling has inspired a lot of work, the initial studies have produced contradicting results. In the early work, Schmitt-Rink et al. [24] theoretically suggested that, if electron and hole charge distributions are identical, then exciton–LO phonon coupling should vanish in small nanoparticles. Klein et al. [25] predicted size-independent coupling, if the dimensions of the charge distribution scales as the particle size. However, other authors [26–28] suggested that electron–phonon coupling increases with decreasing particle size for small particles. Here, the electron–hole correlation, the valence band degeneracy, the conduction band nonparabolicity, and proper confined phonon wavefunctions were considered. Experimentally, absorption spectroscopy [29] and photoluminescence [30] measurements indicated that the electron– phonon coupling increases with decreasing particle size. On the other hand, in some resonant Raman scattering experiments, in which in principle the electron–phonon interaction can be directly probed, either a size-independent coupling [25] or a decrease of electron–phonon coupling with decreasing particle size was detected [31, 32]. Later, Scamarcio et al. [33] unambiguously demonstrated that the electron–LO phonon coupling increases with increasing confinement in the strong confinement region. This was experimentally demonstrated through the size dependence of the ratio between the two-phonon and the one-phonon Raman cross sections in the resonant Raman scattering of a prototype system. These experiments were done under improved conditions, with constant resonance conditioning being maintained for all the particle sizes investigated. It was experimentally shown that, as the nanoparticle size changes from about 3 to 2 nm, the electron–phonon interaction increases more than two times. They also suggested that the electron–phonon coupling governed by the Fröhlich interaction increases stronger than that governed by the deformation potential with decreasing particle size. Resonant Raman scattering up to the third order was studied by Rodríguez-Suárez et al. [34]. The observed relative intensities of the overtones were found to be very sensitive to the particle size, in accordance with calculations considering the Fröhlich interaction between excitons and phonons. For exciton–acoustic phonon interaction the deformation potential coupling mechanism was shown to predominate in semiconductor nanocrystals [35, 36]. 4. ACOUSTIC PHONONS 4.1. Confinement The finite size of nanoparticles restricts the motion of phonons inside a nanocrystal; hence the confinement effects on the phonons can be observed. For a bulk 4 2 s/t 2 = vt2 s + vl2 − vt2 grad div s (6) under the corresponding boundary conditions. Here, s is the lattice displacement vector, and vt and vl are the transverse and the longitudinal sound velocities, respectively. By classifying the eigenfrequencies according to the symmetry group of the sphere, two types of vibrational modes can be obtained, spheroidal modes S and torsional modes T . The torsional modes are dilatation-free vibrations without a volume change of the sphere and the spheroidal modes are curl-free. The eigenvalues are characterized by the angular quantum number and by another quantum number p, which gives the order of the zeroth of the radial part of the wavefunction. The torsional mode is only defined for ≥ 1 because the mode with = 0 has no displacement. The eigenvalues determine the quantized vibrational frequencies Sp and Tp for the particle with the radius R from the relations Sp = S p R vl Tp = T p R vt (7) where and are the eigenvalues of Eq. (6). The “wavevectors” of acoustic phonons in nanoparticles S are also quantized and are described by the two sets p /R T /R. Figure 1a illustrates the deformations of the and p vibrating sphere for the eigenfrequencies with lowest indices. The total symmetric spheroidal mode or the “breathing” mode is denoted by S01 , S21 indicates the quadrupolar symmetric spheroidal mode, and T21 is a torsional mode. Only the lowest modes Sp with p = 1 have large amplitudes s 2x2 S ω01 S ω02 S ω21 0 T a S ω03 Proofs Only s2x2 crystal, the phonons can extend up to infinity, which allows q to take any value in the reciprocal lattice, including zero. Owing to the scattering selection rules, phonons at the -point (q = 0) are the only observable phonons in the firstorder Raman scattering. Therefore, the acoustic phonons, with vanishing energy at the -point, are not observed in the first-order Raman scattering of a bulk crystal. On the other hand, because of the uncertainty of momentum transfer for small particles, acoustic phonons in nanocrystals become observable in first-order Raman spectra. They are called “confined acoustic phonons.” The vibrational frequencies of these phonons are very close to zero. They appear in the low-frequency range of the Raman spectra, with frequencies inversely proportional to the diameter of the nanoparticles. One way to understand these acoustic phonons is to consider the possible vibrations of the whole particle. To calculate these vibrations, the particle can be considered as an elastic sphere and elastic continuum theory can be used, along with the proper boundary conditions at the surface of the nanoparticle. More than 100 years ago, Lamb [6] discussed the vibrations of a homogeneous elastic body of spherical shape under stress-free boundary conditions. This model, which was extended later (see, e.g., Tamura et al. [37]), is valid if the wavelengths of the acoustic phonons sufficiently exceed those of the lattice constants. It has been proved to be useful as an approximation for the calculations of vibrational confinement in nanoparticles. The eigenfrequencies of the elastic body can be obtained by solving the differential equation s 2x2 Light Scattering of Semiconducting Nanoparticles ω2 2 b S ω01 0 .2 0 .4 0 .6 0 .8 x /R 1 Figure 1. Some vibrational modes of an elastic isotropic sphere. (a) Spheroidal “breathing” mode S01 with = 0 and p = 1, spheroidal quadrupolar mode S21 , and torsional mode T21 . (b) Radial distribution of the energy density s 2 x2 of the spheroidal breathing mode S01 (surface mode) and the first two overtones S02 and S03 (inner modes) for spherical particles with free surface. near the surface of the nanocrystals; the higher modes with p > 1 correspond to inner modes. Figure 1b shows the radial distribution of the energy density s 2 R2 of the spheroidal breathing mode S01 (also called surface mode) and the first two overtones S02 and S03 (inner modes). As shown by Duval [38], only the spheroidal modes with angular quantum numbers = 0 and = 2 are Raman active. The total symmetric mode = 0 is polarized, and the quadrupolar symmetric mode = 2 is depolarized. The simplest case = 0 with the eigenvalue equation sin = 42 j1 (8) can be analyzed easily. Here, j1 x = sinx − x cosx/x2 is the spherical Bessel function of first order. Eigenvalue equations for higher quantum numbers are given, for example, in Refs. [37, 39, 40]. For a free particle, the parameter in Eq. (8) is given by the ratio of the transverse and longitudinal sound velocities vt /vl . In the case of an embedded particle, the relations for the continuity of the displacement and of the stress vectors at the spherical surface result in a generalized eigenvalue Eq. (41). The parameter then additionally depends on the sound velocities in the matrix and on the ratio of the mass densities of the particle and the matrix. Further, is obtained as a complex number, sound emission into the matrix occurs, and therefore the particle vibrations are damped. 4.2. Raman Scattering 4.2.1. Scattering Efficiency The Raman scattering is caused by fluctuations in the dielectric susceptibility due to the elastic waves. The Raman scattering efficiency can be expressed as a function of the 5 Light Scattering of Semiconducting Nanoparticles elasto-optic coefficients [1, 42]. The wavevector dependence of an acoustic mode is given by V L −k S − q· r dV exp −ik 2 (9) where n = exp/kT − 1−1 is the Bose–Einstein population factor. For bulk materials, corresponding to large dimensions of the scattering volume V in comparison with 1/q, the integral in Eq. (9) can be replaced by the function − q. L − k This expresses the momentum conservation: k S S and of the excitThe wavevectors of the scattered light k ing light kL are very small. Therefore, only phonons near the center of the first Brillouin zone with wavevectors q = ∼ 0 contribute to the light scattering which hinders L − k k S the observation of Raman scattering by acoustic phonons in bulk material. By assuming nanoparticles to be spherically shaped with radius R and by integrating over the sphere volume, the Raman scattering efficiency results in d/d ∼ n + 1qSq R (10) Intensity (arb. units) d/d ∼ n+1q 2 1 -1 00 3 0 1 00 4 Proofs Only c b a 0 40 80 120 Raman shift (cm -1) with Sq R = 3j1 qR qR 2 (11) The scattering function Sq R replaces the function. In this case, the wavevector selection rules are relaxed. Phonons with larger wavevectors contribute to the Raman scattering, and the observation of the confined acoustic phonons is allowed. For particles embedded in a matrix, the elastic excitations are exponentially damped due to complex eigenvalues. Therefore, we assume for the phonon modes Sp Lorentzian bands L p bp ∼ 1 + 2 − p /bp 2 −1 with the maximum at p and half-width blp describing the damping. Further, we take into account the density of states ∼q 2 and a size distribution cR. Then the Raman scattering intensity is obtained as d/d ∼ n+13 SqRcRV RL Sp bi dR (12) with the wavevector q = Sp /vl and the particle volume V R. The homogeneous line broadening due to the matrix effect is considerable and must be taken into account if one wants to deduce the particle size distribution from the Raman spectra [41]. 4.2.2. II–VI Semiconductors Figure 2 represents as an example low-frequency Raman spectra of CdSx Se1−x nanoparticles with a mean radius of R = 31 nm embedded in a silicate glass matrix. The spectra were excited with a Ti:sapphire laser operating at a wavelength far from resonance. The curves (a) and (b) correspond to polarized and depolarized configurations, respectively. For comparison, curve (c) shows an unpolarized spectrum of the base material, which contains no nanoparticles. The broad band with maximum at about 50 cm−1 , which Figure 2. Low-frequency Raman scattering spectra of CdS067 Se033 nanocrystals embedded in borosilicate glass excited at the wavelength = 857 nm below the absorption band. The mean particle radius is 3.1 nm. The quadrupolar mode is indicated by 1; the phonon modes 3 and 4 are overtones of the breathing mode 2. (a) Polarized spectrum; (b) depolarized spectrum; (c) base material of the sample. In the inset the polarized spectra in the anti-Stokes and in the Stokes regions are shown. is typical for glass vibrations and is known as the Boson peak [43, 44], is also present in the spectra (a) and (b). Band 1 in curve (a) at 8.5 cm−1 corresponds to the depolarized spheroidal vibration with = 2 and p = 1 and band 2 at 17.5 cm−1 corresponds to the polarized spheroidal vibration with = 0 and p = 1. Weaker bands 3 and 4 at about 40 and 55 cm−1 can be assigned to the inner modes with = 0 and p = 2 and p = 3, respectively. The frequencies of the bands scale with 1/R, as can be seen in Figure 3 where results for CdSx Se1−x nanoparticles with different sizes are shown. Such doped semiconductor glasses are used as sharp cutoff filters, the absorption edge being adjusted by changing the particle size. The value obtained experimentally agree well with calculations based on the continuum theory. The influence of the composition x is small as can be seen by the three curves for each phonon mode calculated for x = 08 05, and 0.2. Nanocrystals of II–VI semiconductors are conventionally synthesized either in oxide glass hosts or by colloidal precipitation. The synthesis of quantum dots in polymer films has also been reported. For more information see, for example, Woggon [22], Herron [45], Kamat et al. [46], and references therein. In the case of semiconductor-doped oxide glasses, precipitation of the semiconducting dots from the solid solution is obtained by annealing the glass after quenching. Silicate glasses containing CdSx Se1−x mixed crystals have been extensively investigated. Particle sizes have been obtained by Raman scattering of acoustic modes in 6 Light Scattering of Semiconducting Nanoparticles R (n m ) 10 5 4 3 ω03 2 ω02 R a m a n s h ift (c m - 1 ) 50 Proofs Only ω01 30 ω22 ω21 0 .0 0 .2 0 .4 1 /R (n m - 1 ) O G 5 30 O G 5 15 O G 55 0 R G 63 0 R G 66 5 R G 645 R G 695 O G 590 10 0 .6 Figure 3. Measured low-frequency peaks in dependence on 1/R for various filter glass samples. p indicates various polarized modes, depolarized modes, and their overtones. The experimental values for the corresponding phonons are shown by full circles. The dashed, solid, and dotted lines correspond to the theoretical values calculated with x = 08 05, and 0.2, respectively. It can be seen that the influence of the composition x on the phonon frequencies is small. Reprinted with permission from [41], P. Verma et al. Phys. Rev. B 60, 5778 (1999). © 1999, American Physical Society. CdSx Se1−x [41, 47, 49–52], CdS [49, 53, 54], CdSe [39, 55], Znx Cd1−x S [49] crystals in silicate glass matrix, CdS [56], and CdSe [39] crystals in GeO2 glass matrix. Other methods used to determine the particle sizes were transmission electron microscopy (TEM) [23, 47, 56–61], optical absorption [47, 49, 51, 59, 62], small angle scattering of neutrons (SANS) [63], small angle scattering of X-rays (SAXS) [13, 50, 54, 64, 65], and photoluminescence [58, 66]. In addition to size-dependent shifts of the photoluminescence peaks, the discrete acoustic phonon modes could also be observed in photoluminescence spectra [67]. Saviot et al. [53, 54] observed size and excitation dependence of resonant low-frequency Raman scattering in CdS nanocrystals embedded in glass. They reported size-selective excitation, when the exciting laser wavelength was changed within the absorption band. With variation of the excitation within the absorption band, they observed dependence of the low-frequency band position and changes of its depolarization. The low-frequency band was 100% polarized for excitation below the absorption edge while it was depolarized when excited above the absorption. etching in hydrofluoric acid solution of heavily boron-doped silicon wavers. With the assumption of spherical particles, on the basis of Eq. (7), 2R = 65 nm is obtained. However, due to the columnar morphology of the porous structure a model with particles of elongated nonspherical shapes is more realistic. For long cylinders this model yields 2R ≈ 41 nm, 2R corresponds to the diameter of the columns. A similar value was deduced from the observed downshift and broadening of the 520 cm−1 one-phonon peak (see Section 6). Liu et al. [65] succeeded in preparing Si clusters embedded in a porous Si skeleton. The Si clusters were produced by evaporation and inert gas condensation [69, 70], and the cluster size was controlled by adjusting the evaporation temperature, the Ar gas pressure, and the distance between the substrate and the W boat containing the Si powder. The cluster size was determined with TEM measurements. The strongest peaks in the low-frequency region of the Raman spectra were observed at 29 cm−1 (particle size 3.8 nm) and at 28 cm−1 (particle size 3.9 nm), respectively. Raman scattering from acoustic phonons confined in Si nanocrystals which were dispersed in SiO2 thin films have been observed by Fujii et al. [71]. The samples were prepared by a radiofrequency cosputtering method [72, 73]. The sizes of the particles were determined from high-resolution electron microscopic observations. It was found that the Si nanocrystals are single crystals with good crystallinity. Raman spectra are shown in Figure 4. As the size decreases, the peaks shift to higher frequencies. The intensity ratio of the peaks in the polarized and depolarized spectra is about 0.25 for samples with d = 31 and 3.5 nm; for d = 55 nm a distinct peak was not observed in the depolarized spectrum. However, the quantitative comparison between the experiment and the theory based on the continuum model is not successful. This will be discussed in Section 4.3. Low-frequency Raman measurements on Ge microcrystals with sizes of 5 nm and greater embedded in a GeO2 glass matrix have been performed by Ovsyuk and Novikov [40, 74]. The crystals were grown using the reaction 2GeO(gas) = Ge + GeO2 [74]. The low-frequency Raman scattering data 4.2.3. Si and Ge Low-frequency Raman modes of nanometric Si crystallites were first observed in porous Si [65, 68]. Gregora et al. [68] found a well-defined peak at 37 cm−1 in the low-frequency spectra of porous silicon layers produced by electrochemical Figure 4. Low-frequency Raman scattering of Si nanocrystals embedded in SiO2 matrices. (a) Polarized spectra; (b) depolarized spectra. Reprinted with permission from [71], M. Fujii et al. Phys. Rev. B 54, R8373 (1996). © 1996, American Physical Society. 7 Light Scattering of Semiconducting Nanoparticles were used to show that the vibrations of the Ge nanocrystals differ considerable from those of free particles and that the influence of the matrix is considerable. Using the theory of Tamura et al. [37] the influence of the matrix on the spheroidal and torsional modes of nanocrystals was analyzed. It was found that surface vibrations were caused by the restoring force which limits the free rotation of microcrystals. 4.2.4. Other Compounds While in semiconducting nanoparticles the spheroidal breathing mode with = 0 and p = 1 is the strongest in the Raman spectra, the scattering intensity of this mode is very low for metallic nanocrystals, for example, consisting of Ag or Au, and the light scattering of the quadrupolar mode dominates the Raman spectra. The reason is that in the latter case the inelastic light scattering is induced by the phonon–plasmon coupling. The electric dipole moment of the surface plasmon is modulated by the vibrational modes. As noted by Gersten et al. [75], the breathing mode ( = 0) will produce no significant modulation; however, the quadrupolar vibrations ( = 2) will be strongly coupled to the dipolar plasmon. A depolarization ratio of 0.75 was determined for the = 2 modes [76]. Light scattering is sensitive to the size and the shape of the nanoparticles. For ellipsoidally distorted shapes, the threefold degenerate dipolar plasmon splits into a lower-energy plasmon oscillating along the long axis of the ellipsoid and a twofold degenerate higher-energy plasmon oscillating perpendicular to this axis. Further, the fivefold degeneracy of the = 2 mode (m = 0, ±1, ±2) is lifted. The observed large inhomogeneous width of the plasmon absorption of Ag clusters may be explained by ellipsoidal distortions because the plasmon energy very weakly depends on the size [77]. The shape of the clusters can even be selected by Raman scattering: Palpant et al. [78] investigated thin films consisting of small Ag clusters embedded in a porous alumina matrix. Most clusters were roughly spherical, but a noticeable amount of clusters, generally the larger ones, were found with a prolate ellipsoidal shape, as shown by TEM investigations. By excitation with the 457.9 nm Ar laser line close to the absorption maximum at 420 nm, the low-frequency peaks of spherical nanoclusters were observed with a depolarization ratio close to 34 . By shifting the excitation to the red region, the low-frequency peak was shifted to lower frequency with a depolarization ratio close to 13 . This can be explained by selective enhanced excitation of the nondegenerate plasmon along the long axis of the ellipsoidally distorted particles by excitation in the low-energy wing of the absorption band. Silver nanocrystals embedded in SiO2 [79], alkali halide [76], and porous alumina matrix [78] were investigated too. It was shown that the observed low-frequency peaks agree fairly well with calculations based on Lamb’s theory if the particle size is less than about 4 nm [79]. Courty et al. [80] performed low-frequency Raman measurements on silver nanocrystals self-organized on a hexagonal compact array on highly oriented pyrolytic graphite substrate. Besides the quadrupolar vibration mode, two weaker bands were observed at larger frequency shifts. They were attributed to the spheroidal breathing modes = 0, and p = 1 2, and their observation was made possible by the high crystallinity and the narrow size distribution of the particles. Recently, low-frequency Raman measurements of Au particles supported on CeO2 and Fe2 O3 catalytic matrixes were performed [81, 82] with the aim of measuring their grain size. Supported by measurements of the depolarization ratio, it was concluded that the observed bands are due to the = 0 and = 2 spheroidal modes [82]. Confined acoustic phonons were investigated in Sn nanoparticles embedded in a SiOx thin film, the inelastic scattered light was measured by a Sandercock interferometer [83]. The observed low-frequency modes scale with the inverse of the particle size in accordance with Lamb’s theory. The vibrational modes of the smallest particles were overdamped. A similar result was obtained with femtosecond pump-and-probe spectroscopy measurements, where oscillations with comparable frequencies were detected for the big particles but not for the smaller ones. By increasing the temperature, clear effects of particle melting on the acoustic and the optical modes could be observed [84]. For SnO2 nanoparticles grain size and distribution deduced from Raman scattering were found in agreement with TEM measurements, especially for the smaller particles [85]. 4.3. Size Determination In this section we will discuss other methods that are used for particle size determination and often correlated with Raman scattering measurements. The average sizes calculated from the frequencies at the low-frequency peak maxima in the Raman spectra are overestimated if a size distribution is present because the scattering intensity is proportional to the particle volume. This is true for perfect lattice structure; however, lattice defects may reduce the scattering intensity due to spatial coherence effects as discussed by Duval et al. [86]. (a) Transmission Electron Microscopy. TEM is often used to deduce a mean particle size and size distributions from a limited number of particle images. Preparation techniques are often pretentious and may affect the observation; the particle sizes are underestimated. The lattice images of nanocrystals observed with high-resolution TEM are produced by interference of diffracted beams; different types of lattice images are found, depending on the orientation of the particle relative to the electron beam. They are visible when the crystal thickness exceeds a threshold value t0 . For the case of CdSe in glass, Champagnon et al. [47] estimated a threshold thickness of t0 ≈ 3 nm. Therefore, the lattice is visible only in particles larger than the threshold thickness. Further, for larger particles the lattice image is produced from the central parts but may be invisible (below a threshold intensity) near the surface over a width w ≈ t02 /4R, depending on the particle radius R. The deviation of the apparent size from the real size, therefore, is much more important at small sizes. This effect could be one possible reason for discrepancies in the observed sizes of very small 8 Light Scattering of Semiconducting Nanoparticles waves. The scattered waves are all in phase when the scattering direction is the same as that of the incident beam. As the scattering angle increases, the difference in phase between the various scattered waves also increases and the intensity of the resultant scattered wave decreases because of destructive interference. The intensity becomes zero when there are as many waves between 0 and as between and 2; this will occur for a scattering angle of about 2 ≈ /L, where L is an average size of the particle. The scattering cross section for spherical particles with radius R and size distribution cR embedded in a matrix can be expressed as [94, 95] d q E ∼ AE Sq RcRV 2 R dR d (13) where q = 4/ sin is the magnitude of the scattering vector, E is the X-ray energy, and the scattering function Sq R was defined in Eq. (11). The expression A depends on the electronic contrast between the particle and the matrix. Figure 5 shows the scattering curve of a sample with CdS067 Se033 nanocrystals. The corresponding Raman spectrum is given in Figure 2. The solid line has been fitted by assuming spherically shaped particles with asymmetric size distribution as shown in the inset. The dashed line was calculated for a size distribution with a -function-like profile. From Eq. (13) it can be seen that particles with larger radius contribute much more to the scattering cross section than smaller ones. Therefore, average size values could be overestimated if Guinier plots and not a complete fit procedure are used. In Figure 6 results of Raman measurements of nanocrystals in glass matrices are plotted against the inverse radius A S A X S intensity (e.u./S iO 2 m ol.) particles determined with different methods [47]; see Figures 6 and 7. (b) Absorption, Luminescence. The energetically lowest optical transition can be used to determine the particle size. The corresponding absorption is often not a well-defined narrow peak. Some authors used the second derivative of the absorption spectrum [61] to locate the band edge; other authors took the energy at which the absorption appears to extrapolate to zero [87]. A clear inverse square size dependence in both the absorption edge and in the photoluminescence was observed in most cases, at least for II–VI compounds. Potter and Simmons [58] studied CdS crystals at sizes between 4 and 40 nm and obtained good agreement with the case of strong confinement [Eq. (4)]. The bulk Bohr radius in CdS is 2.8 nm. In general, the determination of size values from absorption and luminescence measurements depends on the model used and requires knowledge of parameters such as exciton masses. (c) X-Ray Diffraction (XRD). The finite crystallite size leads to a broadening of the peaks in the XRD spectra. This can be understood if one considers that in a finite crystal the number of rays reflected from successive lattice planes that add up in constructive or destructive interference is finite. Therefore, they do not reinforce or cancel out completely. The determination of the crystalline size from the XRD peaks is discussed in several books [88–90], mostly the Scherrer formula is used. The peak broadening gives a weighted average of the mean grain size, but the grain size may be considerably less than the particle size [91]. Variations in the lattice constant, for instance, from one crystallite to another and structural defects will also broaden the peaks. Several methods have been described to separate crystalline size- and distortion-induced broadenings. Further the XRD spectra can be influenced by particle shape effects. For prolate nanoparticles with more planes contributing to the diffraction in the direction of the prolate axis the corresponding XRD peaks can be increased in intensity and reduced in width as shown for CdSe nanocrystals [92] with sizes of about 8 nm and for Si nanocrystals [93] with sizes of about 50 nm. In the last case the peakdependent linewidth broadening was used as evidence for pressure-induced diamond-to-hexagonal structure phase transition accompanied by shape changes from spherical to prolate. (d) Small Angle Scattering (SAXS and SANS). The size as well as the size distribution can be obtained with SAXS or SANS. We will discuss here as an example SAXS. The central scattering of X-rays due to the presence of nanoparticles having dimensions from several tens to several hundred times the X-ray wavelength is analogous to the well-known phenomenon of optical diffraction, where a halo is produced by the passage of a light ray in a powder with grain dimensions on the order of a hundred times the light wavelength. Let us consider a particle in a beam of X-rays; then all the electrons are sources of scattered 10 3 1 0 .8 10 2 Proofs Only 0 .6 0.8 nm 0 .4 10 1 0 .2 0 10 0 0 2 4 6 R (n m ) 0.1 1 q (1/nm ) 10 Figure 5. Small angle X-ray scattering curve of the sample with CdS067 Se033 nanocrystals described in Figure 2. The solid line has been fitted by assuming spherically shaped particles with asymmetric size distribution as shown in the inset. The dashed line was calculated for a size distribution with a -function-like profile. Reprinted with permission from [95], G. Irmer et al. J. Appl. Phys. 88, 1873 (2000). © 2000, American Institute of Physics. 9 Light Scattering of Semiconducting Nanoparticles R (nm ) 10 5 4 3 R (n m ) 2 10 5 4 3 30 a 2 1 80 R a m a n s h if t (c m -1 ) 20 R a m a n s h ift (c m - 1 ) 10 0 b 30 20 60 Proofs Only 40 20 10 0 0 c 0 0 .4 0 .8 1 .2 1/R (n m -1 ) 30 20 10 0 0 0 .2 0 .4 1 /R (nm - 1 ) 0 .6 Figure 6. Position of the spheroidal mode S01 as a function of the inverse of the particle radius as observed by other scattering methods or TEM, respectively. (a) Nucleated cordierite glass: •, SANS [2]. (b) CdS in glass matrix: , SAXS [53]; , [55]; , [241]. (c) CdSx Se1−x in glass matrix: , anomalous SAXS [13]; , SANS [62]; ×, SAXS [47]; +, TEM [47]. as determined by other scattering methods (SAXS, anomalous SAXS, or SANS) or TEM, respectively. The straight full lines are calculated with Lamb’s theory for spheres with free surfaces, and the dotted line in Figure 6c takes into account the matrix effect. The agreement between various scattering methods is good; however, for smaller particles deviations from results of TEM measurements can be observed [47]. Large discrepancies between experimental results and the theory based on the continuum model were obtained for Si nanocrystals, as shown in Figure 7. Because the formulae in Section 4.1 are based on the isotropic elastic medium approximation, mean values for the sound velocities vl and vt were used for the calculation of the size dependence of the polarized spheroidal mode 01 and the depolarized spheroidal mode 21 of a free particle. The calculated values are much larger than those of the measured peaks, even if the large anisotropy of the elastic constants in Si is taken into account. The influence of the SiO2 matrix [72] is discussed as one possible reason for the deviation between experiment and theory. However, quantitative estimations are still lacking and further theoretical studies are necessary. For instance, opposite predictions were made for the influence of SiO2 matrices on Ag particles. Montagna and Dusi [96] predicted very small and negligible influences, while Ovsyuk and Novikov [40] reported that the phonon modes Figure 7. Position of the spheroidal mode S01 as a function of the inverse of the particle radius of Si nanocrystals. The radii were obtained by •, TEM measurements [71]; , TEM measurements [65]; , estimation of the size analyzing the shift and shape of the optical phonon band assuming a columnar structure (column radius R) [64]. The straight lines were calculated based on Lamb’s theory for particles with free surfaces: solid line, spheres; dotted line, columns, assuming isotropic elastic medium. The hatched region indicates the range of theoretical values when the elastic anisotropy of Si is taken into account. For comparison theoretical results (+) obtained with lattice dynamical calculations on Si clusters are included [166]. due to matrix effects shift to lower frequencies by considerable amounts. 5. SURFACE PHONONS 5.1. Surface Modes of Small Particles The classical electromagnetic theory for the absorption and scattering of a sphere was given a century ago by Mie [7]. About 60 years later Mie’s theory was applied to spherical ionic crystallites [99–102] characterized by the frequencydependent dielectric function which also describes the optical behavior of a polar semi-insulating cubic semiconductors in the infrared (IR) region: 2 − 2 − i = 2LO (14) TO − 2 − i Here, TO and LO are the frequencies of the transverse and longitudinal optical bulk phonons, respectively, is the dielectric constant at high frequencies, and the constant takes damping into account. The bulk phonons in small particles have properties similar to those of the corresponding phonons in an infinite crystal; however their wavefunctions are adapted to the geometry of the small particle. For spherical geometry, for example, the phonon wavefunctions can be described by orthogonal eigenfunctions consisting of products of spherical Bessel functions j qn r with spherical harmonics Ym , classified by the quantum numbers 10 Light Scattering of Semiconducting Nanoparticles m, and n. The radial wavevectors qn are size-quantized due to the boundary conditions with qn = n /R, where n are the th order spherical Bessel function roots [103]. In addition to the bulk phonons in small particles, there exist surface phonon modes, which have no counterpart in infinite crystals. Their frequencies are intermediate between TO and LO . They can be obtained by solving the equation + 1 · + 1/ = 0 = 1 2 3 (15) where 1 is the dielectric constant of the embedding medium and is an angular quantum number. Unlike bulk phonons, the surface phonons have nonvanishing electrical fields outside the particle; inside the electrical field changes with r −1 radial dependence. The surface mode with the lowest index = 1 has a constant amplitude inside the sphere (therefore, the restriction of the notation “surface mode” to modes with > 1 is also in use). This mode of homogeneous polarization has a frequency F given by the solution of + 21 = 0 and is often called the Fröhlich mode [104]. With each surface mode a surface charge distribution is associated, which generates the polarization field inside the sphere. The Fröhlich mode with = 1 corresponds to a dipolar charge distribution, the mode with = 2 to a quadrupolar, and the inner modes with higher to higher multipolar surface charge distributions. For particles that are small compared with the wavelength, the electrostatic approximation can be used and the mode with = 1 dominates [105]. Up to now we have considered a spherical particle. However, the shape has an important influence on the surface mode frequencies. As an example, we consider an ellipsoid of homogeneous polarization (Fröhlich mode). Three frequencies are obtained instead of one, given by the solutions of the equations + 1 · 1/ni − 1 = 0 i = 1 2 3 of longitudinal type are purely radial with the mechanical displacement vector s=0 n = Aj1 n r/Rer (17) where n is the nth zero of the spherical Bessel function j1 . Assuming a quadratic negative bulk LO phonon dispersion, the eigenfrequencies of the modes with = 0 are given by 2n = 2LO − 2L n /R2 (18) Here, 2L is a parameter describing the dispersion of the LO phonon in the bulk [109, 111]. For large R, n → LO is obtained. For example, for GaAs quantum dots embedded in AlAs, the n = 1 mode was calculated to occur basically at the LO frequency of bulk GaAs for radii >25 nm. ≥ 1: The frequencies of these surface modes are more difficult to obtain [109]. However, for large R the frequencies are again solutions of Eq. (15), meaning that the effect of mechanical boundary conditions becomes important only for small R. The theories mentioned above concern the optical response of an isolated particle embedded in a homogeneous medium characterized by a dielectric constant 1 ; particle interactions are not included. However, in practice the particles are often not separated and light-scattering experiments have to be performed on collections of particles like powders or nanosized porous media. As mentioned above, the surface modes in polar semiconductors are accompanied by electrical fields. For instance, the Fröhlich mode of an particle with its dipole field will interact with those of other particles via dipole–dipole interaction. A theory that accounts approximately for the interactions is the effective medium theory. 5.2. Effective Medium Theory (16) where ni are the so-called depolarization factors corresponding to the ith axis of the ellipsoid. The depolarization factors obey the relations 0 ≤ ni ≤ 1 and n1 + n2 + n3 = 1. The inverse of ni is approximately proportional to the ith axis. Note, that for n1 = n2 = n3 = 13 the case + 21 = 0 of the sphere is obtained; for an infinite long cylinder with n1 = n2 = 21 n3 = 0 the surface frequency is determined by + 1 = 0. The electromagnetic theory was extended to cylindrical [106], cubic [107], and other nonspherical particles; reviews are given in Refs. [102, 105, 108]. Roca et al. [109] and Chamberlain et al. [110] extended the treatment of polar optical phonons in quantum dots by including mechanical boundary conditions. Phonon dispersion up to quadratic terms in the wavevector and the coupling between the mechanical displacement and the electrostatic potential were taken into account. Their model gives (1) uncoupled transverse optical (TO) modes with purely transverse character and (2) coupled modes with in general mixed LO–TO character. = 0: The most important contribution to Raman scattering corresponds to the modes with = 0. These LO modes For visible light interacting with semiconducting nanoparticles (characteristic size L and dielectric function 2 ) distributed in a medium with the dielectric constant 1 in the limit L the heterogeneous composite can be treated as a homogeneous medium, and the so-called effective medium theory applies. The effective dielectric function eff is defined by = eff E D (19) and E denote spatial averages of the dielectric where D displacement and the electric field, respectively. For a composite of two materials with the dielectric functions 1 and 2 we get 1 − f 1 E 1 + f2 E 2 = eff 1 − f E 1 + f E 2 (20) where the averages are taken over the volumes occupied by the material with the dielectric function 1 or 2 , respectively. The parameter f is the relative volume V2 /V occupied by the material with 2 . If the fields have the same directions in the two media (which has to be proved), the 11 Light Scattering of Semiconducting Nanoparticles dielectric function can be expressed as a function of the field ratio = E2 /E1 : eff = 1 − f 1 + f2 1 − f + f (21) The field ratio can be obtained by solving the electro = 0 under the approstatic equations rot E = 0 and div D priate boundary conditions. Exact solutions exist only in a few cases: (a) The medium consists of parallel slabs of both materials; the applied electrical field is parallel to the slabs (perpendicular to their common normal). Continuity of the tangential electric field components requires = 1 and we get from Eq. (21) eff = upp = 1 − f 1 + f2 (22) the so-called upper Wiener limit [112], because it can be proven to be the absolute upper limit for eff for a two-material composite. (b) The medium consists of parallel slabs of both materials; the applied electrical field is perpendicular to the slabs (parallel to their common normal). Continuity of the normal displacement field components requires = 1 /2 and we get from Eq. (21) the socalled lower Wiener limit [112] as the absolute lowest limit for eff for a two-material composite: 1 1 1 − f f = = + eff low 1 2 (23) (c) An exact solution is also possible for the case of an external electrical field applied along an axis of an ellipsoid (e.g., dielectric function 2 ) embedded into an infinite matrix (of dielectric function 1 ); see, for example, Ref. [105]. Here, we restrict ourselves to the special case of a sphere. The field ratio is then = 31 /2 + 21 , often derived in textbooks of electrodynamics. From Eq. (21) we obtain the Maxwell–Garnett (MG) I result [113]; see Table 1, three-dimensional case. The result is exact for the f → 0 limit (and for the trivial case f = 1). Therefore, the application of the formula is justified if the matrix contains only a few well-separated spheres (f → 0), but often it is used as interpolation formula for all f . The formula, for example, describes the case of semiconducting spheres with concentration f and dielectric function 2 = embedded in air (1 ). The replacements 1 ↔ 2 and f ↔ 1 − f result in the Maxwell–Garnett II formula eff = MGII (see Table 1) corresponding to the “swiss cheese” case of spherical air inclusions (1 ) in the semiconducting material with 2 = . It can be shown that (if 1 < 2 MGI and MGII are the absolute lowest and the absolute upper limit, respectively, for the effective dielectric function eff for isotropic or cubic two-material composites [114]: low < MGI ≤ eff ≤ MGII < upp The replacements 1 ↔ 2 and f ↔ 1 − f do not result in MGI = MGII ; such a topology is called a matrix topology. If one of the phases is strongly diluted, the Maxwell–Garnett formulae are good approximations. Examples are semiconductor doped glasses with filling factors f of about 0.01 discussed in the next sections. If phases 1 and 2 are equivalent, a so-called aggregate topology has to be described. An example is the Bruggeman expression [115] (see Table 1, first row). The solution eff of the Bruggeman formula is invariant under the replacements 1 ↔ 2 and f ↔ 1 − f , characteristic of an aggregate topology. The Bruggeman formula can be derived by generalizing Eq. (21) for three materials, spherical inclusions 1 and 2 with volume fractions f1 = V1 /V and f2 = V2 /V embedded in a matrix m : eff − m − m − m = f1 1 + f2 2 2m + eff 2m + 1 2m + 2 Table 1. Effective dielectric functions. Topology Matrix Aggregate 3D 2D Maxwell–Garnett I 21 − f 1 + 1 + 2f 2 eff = MGI = 1 2 + f 1 + 1 − f 2 Maxwell–Garnett I 1 − f 1 + 1 + f 2 ⊥eff = 1 1 + f 1 + 1 − f 2 Maxwell–Garnett II 3 − 2f 1 + 2f2 eff = MGII = 2 f1 + 3 − f 2 Maxwell–Garnett II 2 − f 1 + f2 ⊥eff = 2 f1 + 2 − f 2 Bruggeman Bruggeman − eff − eff 0 = 1 − f 1 +f 2 2eff + 1 2eff + 2 Monecke + 4f1 2 − 1 /21 + 2 eff = 1 1 + f 2 − 1 /21 + 2 2f 2 2 − 1 2 /21 + 2 + 1 + f 2 − 1 /21 + 2 (24) 0 = 1 − f 1 − eff − eff +f 2 eff + 1 eff + 2 (25) 12 Light Scattering of Semiconducting Nanoparticles If, in a self-consistent way as the matrix material m the effective medium eff itself is choosen, with f1 = 1 − f and f2 = f , the Bruggeman expression is obtained. It is often considered to be the best possible interpolation formula for an aggregate topology. It has, however, the following essential shortcoming. Let us consider a few semiconducting inclusions with the dielectric function 2 in a matrix of constant 1 . In the IR region between the transverse and the longitudinal optical phonon, 2 is negative and 21 + 2 may become zero at the Fröhlich frequency F . For f → 0 the Maxwell–Garnett I formula is correct and results in a pole at 21 + 2 = 0, giving rise to the Fröhlich mode [104]. This exact result is not reproduced for f → 0 by the Bruggeman formula. To overcome this deficiency, another interpolation formula for an aggregate topology has to be found. An isotropic aggregate topology is given, for example, by spherical 2 inclusions in a matrix 1 for f → 0 interpolated to spherical 1 inclusions in a matrix 2 for f → 1. In both limits the field ratios are exactly known as f →0 = 31 /21 + 2 or f →1 = 32 /22 + 1 −1 , respectively. The simplest interpolation for is given by = 31 − f 1 + f 22 + 1 3f2 + 1 − f 21 + 2 (26) By inserting into Eq. (16), a new expression for eff is obtained [116, 117]; see Table 1. Often two-dimensional (2D) effective functions more adequately describe materials with nanosized structures. Examples are porous semiconductor structures produced by electrochemical etching with cylindrical pores, porous membranes with columnar semiconductor structures, and materials with quantum wires. A typical porous polar semiconductor can be described as a heterogeneous material consisting of, for example, airfilled (1 ≈ 1) cylindrical pores in the z direction, which are randomly distributed in the xy plane and embedded in the semiconducting matrix with 2 = . The symmetry of the material is reduced. For a diatomic semiconductor with zincblende structure, the cubic point group Td is changed to an uniaxial one [D2d for a (100) surface and C3v for a (111) surface]. In both cases the effective dielectric function becomes a tensor ⊥ 0 eff 0 0 ⊥ 0 ˜ eff = (27) eff 0 0 eff II formula. Single columns (f → 0) would then result in a Fröhlich mode given by F + 1 = 0. Any dielectric function has to obey the Bergman spectral representation [118]. It can be shown that the effective dielectric functions given above fulfill the corresponding conditions. Additionally, in the two-dimensional case the Keller theorem [119] has to be fulfilled. This is the case for both Maxwell–Garnett formulae and for the Bruggeman expression. The behavior of the effective dielectric function in the region of the optical phonons is illustrated in Figure 8. Parameters for GaP (TO = 366 cm−1 , LO = 402 cm−1 , = 1 cm−1 , = 85, and f = 07) and the model interpolating between I and II were used. Phonons of transverse (longitudinal) character appear at the poles (zeros) of the real part Re() of the effective dielectric function; see Figure 8c (the poles are smoothed due to the finite damping constant used). Between the pole corresponding to the TO phonon and the zero corresponding to the LO phonon an additional pole and an additional zero appear. They can be assigned to the Fröhlich mode. The unusual sequence, the longitudinal Fröhlich mode being below the transverse one, results from the fact that in any dielectric function poles should alternate with zeros in the sequence T → L → T → L, etc., with increasing frequency. The Raman scattering efficiency is ∝ Im() for TO phonons and ∝ Im(−1/) for LO phonons; the corresponding imaginary parts are plotted in Figure 8a and Figure 8b in arbitrary units. Although a strong Fröhlich mode phonon of longitudinal character can be seen, the transverse Fröhlich mode at F−T is very weak (not shown in Fig. 8b; its intensity is <1% of that of the TO c b Im (-1 / ε ) R e ( ε) Proofs Only x100 LO F -T F -L d R (ω ) For the electric field component parallel to the pores, eff is given exactly by the upper Wiener limit [Eq. (22)]. For ⊥ eff a two-dimensional variant of the Maxwell–Garnet model I can be used (see Table 1). The formula describes the morphology of the porous structure as an interpolation between two boundary cases: cylindrical voids in the semiconducting matrix (large material concentration) and a skeleton of intercrossing semiconductor plates (large void concentration). Being correct in both limits f → 1 and f → 0, it can be used in the entire range f ∈ 0 1. For columnar nanostructures it is more appropriate to use the two-dimensional equivalent of the Maxwell–Garnett Im ( ε) a TO 340 360 380 400 420 ω (c m -1 ) Figure 8. Effective dielectric function model: interpolation between three-dimensional (3D) MGI and 3D MGII [117]. Parameters: TO = 367 cm−1 , LO = 402 cm−1 , = 1 cm−1 , and f = 074 (closely packed spheres). (a) Imaginary part Im(; (b) Im(−1/; (c) real part Re(; (d) reflectivity R(. Light Scattering of Semiconducting Nanoparticles phonon) and should be not observable in the Raman spectra. However, the L–T splitting of the Fröhlich mode gives rise to a reflectivity minimum; see Figure 8d. The reflectivity is easy to calculate according to the standard formulae n − 12 + 2 (28) n + 12 + 2 √ √ with n = a2 + b 2 + a, = a2 + b 2 − a, a = Re⊥ , and b = Im⊥ . R = 5.3. Raman Scattering 5.3.1. III–V Semiconductors Hayashi and Kanamori [3] were the first to observe Raman spectra by surface modes in polar semiconductors unambiguously. The GaP microcrystals they investigated were prepared by a gas evaporation technique. Figure 9 shows Raman spectra of samples with different particle sizes in the range between 51 and 430 nm. The surface peak can be clearly seen between the TO and the LO phonon peaks, when the microcrystals are about 1 order of magnitude smaller than the wavelength of the incident laser. For the samples used the surrounding medium was air. A frequency shift of the surface mode for different sizes was not reported. The frequency shifted to lower frequencies when the dielectric constant of the surrounding medium was increased. Proofs Only Figure 9. Light-scattering spectra of the surface mode of GaP nanoparticles. d¯ is the average particle size. Reprinted with permission from [3], S. Hayashi and H. Kanamori, Phys. Rev. B 26, 7079 (1982). © 1982, American Physical Society. 13 This was realized by embedding the microcrystals in the liquids nujol (1 = 200), aniline (1 = 256), and methylene iodide (1 = 310). The lineshape of the peaks attributed to an ellipsoidal particle shape was calculated using effectivemedium theory [120]. It was observed that the intensity of the surface phonon peak increases as the average particle size decreases. An explanation was given based on calculations of Ruppin [103] for Raman scattering efficiencies of bulk and surface phonons in very small spherical crystals. Watt et al. [121] investigated Raman spectra from arrays of GaAs cylinders fabricated by lithographic techniques with radii in the range of 30–100 nm and heights of 140–570 nm. Those with radius ≤ 40 nm showed a well-defined feature between the TO and LO phonons. The position of the additional peak was found to be in agreement with calculations based on the model developed by Ruppin and Englman [102] for the case of an infinite cylinder embedded in a dielectric medium. Wang et al. [122] reported Raman measurements of GaAs dots, fabricated by electron beam lithography and dry etching. It was found that the surface phonon intensity increased dramatically with decreasing dot sizes. da Silva et al. [123] observed the surface phonon at 287.6 cm−1 in 6–10 m long wire-like crystals of GaAs with radii of 30 nm. The GaAs crystals were epitaxially grown on porous Si. They did not observe a dependence of the Raman intensity on the angle of the incident light, in contradiction to the observations of Watt et al. [121]. In recent years, in addition to porous silicon, increasing attention has been paid to porous III–V semiconductor materials [124–130] due to their potential applications in the field of electronics and photonics [131]. The nanosized structures of such polar porous media (not in the case of the nonpolar Si) allow investigations of the surface modes. Their properties strongly depend on the morphology (i.e., symmetry, structural geometry, effective size, etc.) and on the constituents. The analysis of these modes helps to characterize the porous structure by the comparison of experimental results with theoretical calculations based on the effective-medium theory. Although for powders or nanocrystals embedded in homogeneous media 3D models of the effective-medium theory are successfully applied, 2D models are more appropriate to describe the morphology of porous semiconductors. The morphology of the porous structures fabricated by electrochemical etching of III–V materials depends mainly on the doping level of the bulk material, on the defect density on the surface and on the crystal orientation for given etching conditions [126, 127, 132–134]. A surface-related phonon mode in porous GaP at 397 cm−1 observed by Tiginyanu et al. [135] was interpreted in terms of a Fröhlich mode on the basis of an effective medium approach [128]. It was shown that its frequency decreases with increasing anodization current. The fabrication of free-standing porous membranes, detached from the substrate, described in [136], enabled the investigation of the surface modes in porous membranes filled with liquids [137] and infrared reflectance measurements [134]. Figure 10 shows infrared reflectance spectra (a) and Raman scattering spectra (b) of a porous (111) GaP membrane. The pores of about 30 nm diameter exhibit a column-like shape, stretching into the sample perpendicularly to the initial crystal surface. The experimental spectra (upper row) 14 a 1 por-G aP G aP bulk 0 .8 0 .6 R e fle c tiv ity 0 .4 0 .2 0 0 .8 b phonon and a surface mode approximately 10 times smaller in intensity were observed, in agreement with calculations based on the Fröhlich interaction and taking into account a degenerate valence band. Varying the excitation energy below the direct gap the LO phonon band decreases dramatically, whereas the intensity of the surface mode band is not much changed. This could be evidence that besides the Fröhlich interaction other scattering mechanisms like deformation potential scattering contribute to the surface mode scattering. However, up to now calculations on the electron– phonon interaction concentrated only upon the Fröhlich interaction. por-GaP G aP bulk Proofs Only surface m ode 0 .6 0 .4 0 .2 0 R a m a n s c a tte r in g in t e n s it y ( a r b . u n its ) Light Scattering of Semiconducting Nanoparticles ωΤ O ω LO 300 400 300 ω( c m ) -1 400 surface m ode ωL O 380 400 5.3.2. II–VI Semiconductors 380 400 420 ω( c m ) -1 Figure 10. Surface modes in GaP with columnar nanosized pores. (a) Fourier transform infrared reflectivity spectra and (b) Raman spectra of porous and of bulk material. The lower row shows results of calculations based on the effective dielectric function corresponding to the 2D MGII model. can be interpreted theoretically (lower row) using a twodimensional effective dielectric function 2DII; see Table 1, which describes the morphology of the porous structures as an interpolation between two cases: cylindrical voids in the semiconducting matrix (f → 1) and a skeleton of intercrossing semiconductor plates (f → 0). The wave-like structures in the reflectance spectra of the porous membrane are interferences of light reflected at the membrane boundaries. The effect of hydrostatic pressure on the surface modes in porous GaP was investigated in [138] and the temperature influence in [139]. Sarua et al. [133, 134] studied the influence of free charge carrier in porous GaAs and porous InP on the surface phonons. It could be shown that the Fröhlich mode, like the LO phonon mode, couples to the plasmon modes induced by vibrations of the free carrier plasma of electrons. As the LO phonons, the Fröhlich modes were shifted to higher frequencies with increasing carrier density. In porous InP this effect could be observed both by light-scattering and by infrared reflectance measurements; in porous GaAs the F-T–F-L splitting is too small to be observed in the Fourier transform infrared experiment. In GaN columnar nanostructures a new mode near 716 cm−1 was found by Raman scattering measurements [140]. The columnar structures with the thickness of about 1.5 m were fabricated by electrochemical dissolution of bulk material. Scanning electron microscope images proved that the columns with transverse dimensions of about 50 nm or less were oriented perpendicular to the initial surface. By Raman lineshape analysis based on the effective dielectric function for a composite of GaN columns in an air matrix (2D MGI; see Table 1) the new mode could be attributed to Fröhlich vibrational modes. Electron–phonon coupling due to Fröhlich interaction has been considered by Efros et al. [26]. They investigated resonance Raman scattering of GaP nanocrystals embedded in glass. With laser excitation near the direct gap a strong LO Surface modes in II–VI compounds were first observed by Scott and Damen [141] in cylindrical microcrystallites ( 1 m) of CdS in a polycrystalline film using Raman scattering. Pan et al. [142] studied Raman scattering of CdS nanocrystals (size <30 nm) by surface modes in different organic media. Most Raman scattering experiments were performed with nanocrystals embedded in a glassy matrix: CdS [143], CdSe [25], CdSx Se1−x [34, 144–146], and ZnS [147]. As an example, we discuss here the case of CdSx Se1−x nanoparticles. It is known that in the mixed crystal CdSx Se1−x the optical phonons show the so-called two-mode behavior, meaning that a CdS-like mode (TO1 and LO1 ) and a CdSe-like mode (TO2 and LO2 ) can be attributed to two oscillators which contribute with different oscillator strengths depending on the composition x. The dielectric function of the mixed crystal can be described by 2L1 x − 2 − i1 2T1 x − 2 − i1 2 L2 x − 2 − i2 × 2T2 x − 2 − i2 x = x (29) with L1 x = 266 + 55x − 19x2 , T1 x = 266 − 28x, L2 x = 211 − 25x, T2 x = 168 + 17x, and x = 53x + 631 − x. The parameters were taken from bulk measurements [148]. The phonon frequencies that depend on the composition x are plotted in Figure 11. The symbols correspond to measurements on CdSx Se1−x nanoparticles embedded in glass. Below both LO phonons additional modes can be observed (Fig. 12). In comparison with calculations on the basis of the effective dielectric function according to the 3D MGI model (dotted curve in Fig. 11), they can be attributed to Fröhlich modes. To study the influence of the particle size on the optical phonons and the surface modes, a semiconductor doped glass slab was annealed in a furnace with an applied temperature gradient with temperatures between 850 and 1100 K. With increasing temperature the CdS06 Se04 particles grow to larger sizes. The composition x = 06 was estimated from the position of the LO phonons [148]. The positions of the LO phonons and the surface mode were obtained by fitting with Lorentzian bands. The dotted lines in Figure 13a correspond to the calculated frequency of the Fröhlich modes 15 Light Scattering of Semiconducting Nanoparticles C d S 0.6 Se 0.4 5 4 320 a TO1 24 0 L O2 240 LO 2 200 F-L 2 20 0 Proofs Only F -L2 T O2 160 40 b 0 0 .2 0 .4 0 .6 0 .8 1 x Figure 11. Dependence of the optical phonons and surface modes on the composition x of CdSx Se1−x nanoparticles embedded in borosilicate glass. The lines were calculated with the effective dielectric function of the 3D MGI model. 30 ω (c m - 1 ) 16 0 LO1 F -L 1 T (K ) ω (c m - 1 ) ω (cm -1 ) 280 2 875 F -L 1 28 0 R (nm ) 3 1000 975 LO1 C d SxS e1-x 92 5 900 850 32 0 20 10 0 0 assuming spherical particles. Contrary to the acoustic vibrations (Fig. 13b), the optical phonons and the Fröhlich modes are nearly not size dependent. The effect of small shifts due to confinement effects for small particle sizes is discussed in Section 6.2.2. The electron–phonon coupling in the case of the Fröhlich interaction between the field induced by the vibrational motion and the charge distribution was studied by several authors theoretically with different approximations and partly contradicting results (see Section 3). Klein et al. [25] observed surface modes in CdSe. Their calculations were based on a classical dielectric model. They concentrated their investigations on the 1s–1s electronic transition and assumed that only the electron wavefunction is confined Cd S 0.6 Se 0.4 R = 3.7 nm In tensity ( a rb. units) LO1 F-L 1 LO 2 F-L 2 1 60 2 00 2 40 2 80 3 20 3 60 -1 R ama n sh ift (cm ) Figure 12. Surface modes in CdS06 Se04 nanoparticles embedded in borosilicate glass. 0 .2 0 .4 0 .6 1 /R (n m -1 ) Figure 13. Size dependence of the (a) optical phonons and surface modes and (b) acoustic modes of CdS06 Se04 nanoparticles embedded in borosilicate glass. whereas the hole resides near the center of the sphere. Their result is that the radial charge distribution in the spherical box does not couple to surface modes. As a possible reason for the observation of surface modes in the Raman spectra they assume deviations from the spherical particle shape. Efros et al. [26] included holes in a four-fold degenerate valence band in their calculation and showed that surface modes with = 2 are allowed due to a mixing of the valence bands even for perfectly spherical nanocrystals. Chamberlain et al. [110] considered the mechanical as well as the electrostatic boundary conditions in their model. In addition to the dipole approximation they considered the quadrupole contribution of the Fröhlich interaction, proportional to the wavevector. They showed that this term dominates for intersubband scattering and occurs only via = 1 phonons. Because they used a one-band effective mass model, the phonon modes are restricted to = 0 when the dipole approximation is used. The published calculations are based on the Fröhlich interaction, but other scattering mechanisms could contribute to the light scattering by surface modes, too. More detailed theoretical investigations are necessary and more experiments should be performed, analyzing the dependence on the excitation wavelength (near and far of resonance), on the polarization of the scattered light, and on the particle size to bring to light the properties of the surface modes. The selection rules for Raman scattering of optical phonons in a homogeneous sphere are similar to those obtained for the acoustic vibrations discussed by Duval [38]. The vibrational modes must belong to the irreducible reprel l sentations Dg and Du of the three-dimensional rotation– inversion group O(3) (g and u mean even or odd upon inversion). The coupled optical (and the spheroidal acoustic) 16 Light Scattering of Semiconducting Nanoparticles 0 2 modes transform according to Dg , Du , and the uncou1 pled optical (torsional) modes transform according to Dg , 2 3 Du , Du , . The Raman transition operator for dipoleallowed scattering is the polarizability tensor. Its compo0 2 nents transform like Dg and Dg . The IR absorption is electric dipolar; the dipole operator transforms according to 1 Du . Therefore, the uncoupled transverse optical (torsional acoustic) modes are not expected to be optically active. It should be noted that the considerations above do not reflect the crystal symmetry inside the sphere. For zincblende-type materials belonging to the point group Td or wurtzite-type materials with point group C6v the inversion symmetry is absent. For binary compounds such as GaAs, InP, and GaP, upon lowering of the symmetry from O3 0 to Td the representation Dg becomes 1 [109]. Scattering by optical phonons due to the so-called dipole forbidden Fröhlich interaction with diagonal Raman tensor belongs to this representation and will be visible. However, both repre2 1 sentations Dg and Du convert into 15 . The scattering by optical phonons due to deformation potential and electro1 optic mechanisms belongs to this Raman activity and Du surface modes are thus Raman active. The question of when, upon increasing R, the sphere will behave as a piece of bulk material, is discussed in more detail in [109], considering the coherence length of phonons. particles nearly spherically shaped with an average diameter of about 30 nm as proved by XRD measurements. In the Raman spectra from these particles in air below the LO phonon a band at 937 cm−1 was observed (Fig. 14c). Its position was found to be in agreement with calculations based on the effective medium theory. The observed half-width of the band is much larger than the half-width of the LO phonon. One reason could be local changes in the particle packing density. The theoretical curves (dotted lines in Fig. 14) were obtained with the 3D Mo model (Table 1), assuming a distribution of f values around f = 07 with standard deviation = 03. The sensitivity of the surface mode to the environment of the particles was proved by embedding them into amorphous silica (1 ≈ 2). This was achieved by heating the powder in a laser beam. The temperature was controlled by measurement of the intensity ratio of the Stokes and antiStokes TO and LO phonons and by the frequency shift of the phonons. At about 1200 C the oxidation of the SiC crystals starts, resulting in a SiO2 oxide layer around the particles [154, 155]. The SiO2 observed in the Raman spectra could be assigned to amorphous silica [156]. The corresponding peaks at about 500 and 600 cm−1 are shown in Figure 14a. The small peak below the strong TO phonon is a weak phonon of TO character, indicating the 6H polytype of SiC. 5.3.3. Other Compounds LO TO S iC 93 7 LO In te n s ity (a r b . u n its ) A reflectivity gap corresponding to surface modes in a polar porous semiconductor was first observed by Danishevskii et al. [149] in porous SiC and interpreted by McMillan et al. [150] on the basis of the effective medium theory. Sasaki et al. [50] measured Raman spectra from SiC fibers consisting of microcrystalline SiC about 50 nm in size embedded in amorphous SiC and carbon. They attributed features in the spectra between the TO and LO phonon frequencies to surface modes. The features observed were very weak and very broad, perhaps due to a high free carrier concentration and high plasmon damping of the SiC microcrystals and therefore difficult to interprete. DiGregorio and Furtak [151] investigated commercially available 6H-SiC particles with an average size of about 3 m. The particles were irregular in shape with an average aspect ratio of about 2. They obtained a weak band in the Raman spectra below the LO phonon. However, scattering by surface modes is expected to be negligible in crystals that are larger than the wavelength of the exciting light. The appearance of weak surface modes was qualitatively explained as being caused by a decrease of the effective particle radius in irregularly shaped particles, enhancement of the intrinsic electric fields, and localization of surface modes near sharp corners and facets on the particles [152]. The observed mode frequencies 947 cm−1 for the particles in air and 928 cm−1 for particles embedded in KBr (1 = 233) are near the solutions 932.3 and 948.8 cm−1 of the equation + 1 = 0 for cylinder-like particles. Raman scattering spectra in [50, 151] were obtained from SiC microcrystals of irregular shape and/or large size and, hence, weak surface phonon structures. This suggests the presentation here of Raman measurements of nanosized c F -L Proofs Only 9 09 b F -L 840 900 960 S iO 2 a 0 400 800 ω (c m -1 ) Figure 14. Influence of the embedding medium on the surface modes of SiC nanoparticles. (a) Raman spectrum of SiC nanoparticles embedded in an amorphous silica matrix. (b) Detail of spectrum (a) with surface mode F-L and phonon. (c) Raman spectrum of the SiC nanoparticles embedded in air. Calculations (dotted lines) were based on the effective dielectric function interpolated between MGI 3D and MGII 3D [153]. 17 Light Scattering of Semiconducting Nanoparticles The surface mode is located at about 909 cm−1 between the TO and LO phonons of SiC, shown in Figure 14a and in the inset (Fig. 14b). The theoretical curve in Figure 14b was calculated with f = 065 and = 015. In the optical phonon Raman spectra of SiC nanorods (diameters range from 3 to 30 nm; lengths are >1 m) with a high density of defects, bands at 791, 864, and 924 cm−1 were observed [157]. By comparison with calculations it was found that the Fröhlich enhanced effective density of states of the optical phonons, which becomes important because of the broken translation symmetry induced by defects, is responsible for the features in the Raman spectra. Observations of surface phonon modes have also been reported for ZnO [158], SiO2 [159], and Si [65]. (30) has the periodicity of the lattice. The vibrawhere uq0 r tional wavefunction of the nanocrystal is approximated by 6.1. Confinement Models In the Raman spectra of nanocrystalline materials with particle sizes of a few nanometers, peak-position shifts, broadenings, and asymmetries of optical phonon bands appear. In most cases the Raman peak shifts to progressively lower energies and the lineshape gets progressively broader and asymmetric (on the low-energy side) as the particle size gets smaller. This effect can be used to determine the particle size. A typical Raman spectrum of the optical phonons of nanocrystals with about 3 nm diameter is shown in Figure 15, Si nano cluster 509 R a m a n in te n s ity (a r b . u n it s ) 5 09 Proofs Only 520 c-S i 0 Confinement Model CM-I, Phenomenological Phonon Confinement Model To characterize and interpret the observed bands, a phenomenological phonon-confinement model is very widely used; therefore, it is briefly outlined below. This model, originally proposed by Richter et al. [160], was later extended by Campbell et al. [161]. In a Stokes scattering process in an infinite crystal the wavevec S is transferred to a phonon with =k L − k tor difference k wavevector q0 . The wavefunction of a phonon with wavevector q0 in an infinite crystal is = uq0 re −iq0 r q0 r 6. OPTICAL PHONONS 440 together with the Raman spectrum of a single crystal. The upper dashed curve in the inset shows a calculated profile. Different confinement models are in use to determine particle sizes and size distributions from the measured spectra. 480 520 400 800 12 0 0 R a m an s hift (c m - 1 ) 16 0 0 Figure 15. Raman scattering of confined optical phonons in nanocluster Si with crystals of about 3 nm size. The inset shows for comparison the Raman spectrum of an optical phonon in bulk Si. The dashed curve was calculated with the confinement model CM-II for spheres. = W r Lq0 r = q0 ru q0 r q0 r (31) where the localization of the phonon in a microcrystal is taken into account by the weighting function W r L. A Gaussian weighting function W r R ∼ exp− r 2 /R2 , introduced in [160] is mostly used in the traditional approach to bandshape analysis. The parameter is fixed arbitrarily to get a small value at the sphere boundary R. Some other weighting functions were also applied [161–163]. can be expanded in a Fourier series, the Fourier q0 r coefficients being 1 3 = −iq0 r (32) d r q0 re Cq0 q 2 3 The Raman scattering intensity is then described by a continuous superposition of Lorentzian curves with bandwidth centered at the wavenumbers q of the phonon disper 2 factor. By assumsion curve and weighted by the Cq0 q ing a spherical Brillouin zone, isotropic dispersion curve, L − k S ≈ 0 for one-phonon scattering, the Raman and q0 = k intensity can be written as 2 Cq (33) I ∝ d 3 q − q2 − /22 The Fourier coefficients for the Gaussian weighting function are Cq ∼ exp−q 2 L2 /4. This formula has been often used to fit Raman spectra to obtain mean particle sizes. However, there is no physical reason to use a Gaussian weighting function or in the assumption that the phonon wavefunction amplitude differs from zero at the crystalline boundary. For the fit parameter the value 21 was originally used by Richter et al. [160]. Campbell et al. [160] found the best agreement with experiments setting = 2 2 ; this value has been used by the most authors afterwards. With W 2 r R ∼ exp−4 2 r 2 /R2 strong confinement of the phonons to the core is obtained: W 2 r R decreases from 100% at r = 0 to 5% at r = 028R which means that the phonon energy is concentrated in only 2% of the sphere volume. Although the phonon confinement model is practicable to estimate particle sizes, it is theoretically not well founded. 18 Light Scattering of Semiconducting Nanoparticles Confinement Model CM-II This model uses the scattering function, which is well established in the analysis of SAXS experiments on small particles [164] (see Section 4.3). The application of similar expressions to light-scattering processes on small particles was also proposed in [48, 49, 96]. To derive the structure factor we follow Nemanich et al. [165], who presented an evaluation of the susceptibility function over a limited spatial extent. The results indicate that spectral changes are related to the phonon dispersion, and the photon wavevector uncertainty is accounted for. The light-scattering cross section is proportional to the space-time Fourier transform of the correlation function of the fluctuations in the polarizability r t [42]: ∝ d 3 r d 3 r dt ∗ r t r 0 I k · r − r − it × exp−ik (34) The fluctuations r t can be expressed by the atomic displacements r t = ur t. u In the absence of interaction between different phonon eigenmodes the displacements of modes with different q are uncorrelated, and we obtain uq u∗ r t ur 0 = 1/V u∗ q q × expiq · r − r + i0 t (35) describes the dispersion of the phonon mode. where 0 q The correlation function in Eq. (35) depends only on the difference r − r . Therefore, the integrals can be factorized. Further, the displacement correlation function is expressed in terms of the Bose–Einstein factor n, and Eq. (34) can be written ∝ I k n + 1 − 0 q Sq k Aq q (36) where the scattering function S is obtained by 2 = 1/V d 3 r expiq − k · r k Sq scattering [10] in opaque semiconductors. However, because we have to deal with very small scattering volumes of particles with characteristic lengths L 1/, the absorption can be neglected in general. For finite scattering volumes the coupling of light to exci is tation modes with wavevectors q in a range about k k describes the range of allowed and the function Sq wavevectors that take part in the light-scattering process. For optical phonons we assume small frequency changes along the dispersion curves and we will use the approxima . The finite lifetime of the tion of a constant term Aq phonons is taken into account by replacing the -function by a Lorentzian function and the sum is approximated by =k an integral over a spherical Brillouin zone. With k L S ≈ 0 we obtain −k (37) V is a measure of the uncertainty of k The function Sq For an infinite transparent crystal with the wavevector q. ∼ q − k wavevector conservation is obtained. k Sq Before we discuss the scattering function for small particles, some remarks concerning absorption are needed. For a crystal that is opaque to light, the effective scattering volume is limited by the penetration depth, which is on the order of 1/, where is the absorption coefficient. The effect of absorption can be described by assuming complex = k + ik and k S = k + ik . By intewavevectors k L L L S S gration of Eq. (37) over the half-space z ≤ 0 we obtain ∝ qz − kL − kS 2 + 2 −1 , a Lorentzian function Sq where we have used ≈ kL + kS . This means that the wavevector component qz has a Lorentzian distribution centered at k = kL − kS with a half-width of 2. Lorentzian distributions were successfully used to interpret Raman scattering of LO phonon–plasmon excitations [9] and Brillouin ∝ I k d3 q Sq − q2 − /22 (38) instead of an expression similar to Eq. (33) but with Sq 2 . Table 2 presents the function Sq for some differCq ently shaped particles and for a layer. In Figure 16 these functions are shown in dependence on the dimensionless parameter : = q · 2R for spheres with radius R, = q · 2R for long cylinders (length L radius R), and = qx · L for a cube (edge length L). The function S describes the range of wavevectors that take part in the light-scattering process. For example, for spheres the curve Sq diminishes from the value 1 at = 0 (q = 0) to a weight of 21 at about = 4, corresponding to a wavevector of q = 2/R; wavevectors with less weight contribute up to about = 8 corresponding to q = 4/R. The wavevector corresponds to a momentum p = q; therefore, we can discuss these equations in terms of Heisenberg’s uncertainty relation x · p ≥ /2 correlating the uncertainties of the coordinate x and the momentum p. A Gaussian distribution of x values corresponds to a Gaussian distribution of p values. For Gaussian distributions only the lower limit p = /2x of Heisenberg’s uncertainty relation is obtained [168]. In our case of x values strongly restricted to the particle volume the uncertainty is larger. Using the example of the sphere we estimate that p ≈ 4/x, setting x = 2R and q ≈ 2/R. For comparison two Gaussian weighting functions used in the phonon confinement models CM-I are also shown ( = 1 Table 2. Scattering functions Sq. Shape Sphere Sq 9 j12 qR qR2 R: radius j1 : spherical Bessel function Cylinder Rhomboeder 4J q R sin qz L/2 · R: radius q R2 qz L/22 L: length in z direction J1 : Bessel function i=x y z Layer 2 2 1 sin2 Li qi /2 Li qi /22 sin2 Lqz /2 Lqz /22 Li : lengths of the edges L: layer thickness in z direction 19 Light Scattering of Semiconducting Nanoparticles Raman intensities was applied to different porous Si structures with connected or isolated Si structures [174]. Comparing the experimental spectra with those simulated by models the results were consistent with cylindrical nanocrystals of 3:1 length-to-diameter ratio and diameters of ∼ 5 nm, measured by TEM. 1 .0 s phe re 0 .6 c ylind er 0 .4 e xp (-κ2 ) 0 .2 6.2. Size Determination by Raman Scattering cu b e e xp(-κ /2 ) 2 0 .0 6.2.1. Si and Ge 2 4 6 8 10 κ Figure 16. Scattering functions S for light scattering of differently shaped particles. The parameter is the dimensionless product of a characteristic length L (2R for spheres and cylinders; edge length for the cube) times the corresponding wavevector component. and = 1/4. The two formulae [Eq. (33) and Eq. (38)] look similar, but their interpretations are different. It should be noted that the model CM-II does not involve arbitrary fitting parameters. To take the phonon confinement into account, we should consider that the largest possible wavelength of the phonons in the particle is restricted by their size. For example, in a sphere the corresponding smallest wavevector is 4.49/L, the wavevectors allowed are approximately 4.49/L, 7.72/L, and 10.90/L · · · 2/a, where a is the lattice constant. For L a the approximation of the sum over q values by an integral in Eq. (33) or (38) is justified. The phonon dispersion curves often are approximated on the basis of a linear chain model by q = −A + A2 − B1 − cosaq. The parameters A and B are chosen to fit neutron scattering data or theoretical calculations, and a is the lattice constant. Other approximation formulae are given in Ref. [109, 111, 145, 169]. Other Confinement Models Raman shifts of nanocrystals versus size were studied theoretically by a bond polarizability model [170]. In the first step the vibrational properties of clusters of some hundred atoms were calculated. The force constants and the atom positions were taken to be the same as those in the bulk. The force constants were considered up to the fifth order; a partial density approach [171] was adopted to calculate them. The vibrational eigenfrequencies and eigenfunctions were calculated for Si spheres and Si columns consisting of up to 657 atoms. Then the Raman spectra were calculated by a bond polarizability model [172, 173] as a sum of independent contributions from each bond. It was found that the Raman shifts due to the confinement effect could be described by = 0 − L = Aa/L , where L is the phonon frequency of the nanocrystal with size L, 0 is the frequency of the optical phonon in bulk material at the Brillouin zone center, and a is the lattice constant. The fitted parameters were A = 47412092 cm−1 and = 144108 for spheres (cylinders), respectively. A simple microscopic calculation using a one-dimensional linear chain model with bond polarizabilities to calculate the The confinement of optical phonons has been investigated by Raman scattering in Si nanocrystals prepared by the plasma transport method [159, 175], the gas-evaporation technique [176], the reactive sputtering technique [177], molecular beam deposition [178], laser annealing of a-Si:H samples [179], plasma-enhanced chemical vapor deposition [180], and surface modification by implantation [181] and in porous Si layers [64, 162, 163, 169, 174, 181–187]. From the observed phonon shifts the particle sizes have been estimated using the confinement model CM-I [Eq. (33)] with Gaussian weighting functions [169, 179, 180, 181, 184, 187] or with the weighting function W r L = sinr/r, by analogy with the ground state of an electron in a hard sphere [162, 163]. In some cases the particle sizes were determined independently by other methods. In Figure 17 observed frequency shifts and half-width broadenings are plotted versus the particle size determined by TEM [169] or XRD [177, 180]. For comparison calculated results were included, obtained with the phonon confinement model CM-I using Si 100 a 1 2 ,s ∆Γ (c m - 1 ) 0 2,c 10 Proofs Only 1 b ∆Γ (c m - 1 ) S ( κ) 0 .8 3 ,s 10 1 3 ,c 1 2 ,c 1 2 ,s 10 L (n m ) Figure 17. Observed frequency shifts and half-widths of the Raman bands of the optical phonon in nanocluster Si ([188], [177], [180]) and porous Si (• [169]) depending on the particle size L. The curves are results of calculations: (1) confinement model CM-I, parameters taken from [169]; (2) confinement model CM-II; s, spheres; c, long cylinders; (3) microscopic calculations [170]; s: spheres; c: cylinders. 20 a Gaussian weighting function (parameters taken from Ref. [169]), with the model CM-II (spheres and cylinders), and results calculated with the bond polarizability model [170] (spheres and cylinders). For the following reasons the confinement models cannot be used without taking some care. (i) The particle shapes are often not well defined and the particles have a size distribution. A lineshape analysis of Raman spectra of porous Si by assuming that the crystallite sizes obey a lognormal distribution has been performed in [163], assuming two nanocrystalline components with Gaussian distributions in [187]. (ii) Strain-induced changes in the phonon frequencies have to be taken into account [161, 180, 184]. It is known that in nanoporous Si the Raman peak can be shifted by strain to lower energies (≤2 cm−1 [189, 190]. Stress inhomogeneities on a nanometer scale can be expected to broaden the Raman signal. (iii) The structure of the particles is not perfect crystalline. In some cases spectral features were observed which consist of two or more bands [169, 176, 180, 187, 191–193]. Although a broad band at about 475 cm−1 can be assigned to amorphous Si, we supposed that the origin of two bands between this band and 520 cm−1 (bulk phonon frequency) is due to a splitting of the optical phonon (degenerate for q = 0 in the Brillouin zone center) in TO and LO modes for larger q values [191] or to a combined contribution of two types of nanocrystallites [184, 187, 193, 194] or to a shell structure of the particles with modified vibrational contributions of the near-surface atoms. Xia et al. [180] observed one band for crystalline sizes L ≤ 2.2 nm and L ≥ 5.3 nm in hydrogenated nanocrystalline Si films. For samples with 2.2 nm ≤ L ≤ 5.3 nm the Raman spectra were composed of two bands: one peaked at 505–509 cm−1 and the other at 512–517 cm−1 . A spherically shaped cluster with two shells was used to model the nanocrystals, and the observed Raman spectra were analyzed by combining the confinement model CM-I and strain effects. The size dependence of the relative intensity of the two bands corresponds to the ratio of the near-surface region and the crystalline core. It should be noted that a hexagonal phase of Si with Raman peak at about 500 cm−1 has also been found in pressure-treated Si [196] and in wear debris from dicing of silicon wafers [197]. For L ≤ 1.5 nm (corresponding to about 200 atoms) reported Raman spectra of Si clusters are to some extent similar to those of amorphous Si films. A new aspect is the observation of Raman bands around 550 cm−1 extending up to 600 cm−1 , well above the highest peaks of bulk Si in Si-rich SiO2 films with clusters [198]. The observed spectral shapes are in qualitative agreement with the density of states spectra calculated by Feldman et al. [199] for Si33 and Si45 clusters. Raman spectra of molecular clusters Si4 , Si6 , and Si7 have been successfully measured by Honea et al. [200] by applying a technique of surface-enhanced Raman scattering. Si cluster ions were produced by laser vaporization, size-selected in a quadrupole mass spectrometer and co-deposited with an N2 matrix onto a liquid helium-cooled Light Scattering of Semiconducting Nanoparticles substrate. By comparison of the observed very sharp vibrational lines with ab initio calculations they could determine the cluster structures. In comparison with Si nanoparticles of the same size the observed frequency shifts and half-width broadenings in Ge nanoparticles are much smaller. A smaller shift is expected due to the position of the optical phonon in c-Ge at the lower frequency of about 300 cm−1 , the main reason could be the smaller dispersion of the optical phonons, which was observed by neutron scattering for Ge [201] compared with measurements for Si [202–204]. Kanata et al. [205] investigated gas-evaporated Ge particles with sizes of 10–60 nm, as determined by Raman measurements and by XRD measurements. Good agreement was obtained by applying the CM-I model ( = 25/8) and assuming spheres with a singlecrystalline core and a coating amorphous shell. The confinement effect in Ge dots in the size range L = 2.6–13 nm has been studied systematically by Bottani et al. [206] with Raman scattering, TEM, and absorption measurements. The samples were grown by an evaporation-condensation selforganization technique that allows obtaining nearly spherically shaped nanoparticles with a narrow size dispersion [207]. As the size decreases an inhomogeneous broadening, a redshift and a softening of the optical phonon peak were observed. The measurements were found to be in agreement with confinement model calculations (CM-I), taking into account the size distribution as measured by TEM. Sasaki and Horie [208] investigated gas–evaporated Ge nanocrystals in the size range 2.6–13 nm with resonant Raman scattering. They found that the size as deduced from the frequency shift and half-width broadening is about half of that obtained from the lowest electronic excitation energy measured in the resonant Raman scattering. Additional anharmonicity is assumed to be associated with a decay process including surface vibrational states. They used a confinement model with phonon waves confined to a cube and vanishing amplitudes at the surface. Intensive photoluminescence around 2 eV has been observed by formation of Ge nanocrystals in SiO2 [209–211]. The phonon confinement measured by Raman scattering was used to show that the crystalline phase formed consists of Ge nanocrystals. If Si substrates are used, the analysis of the experimental results has to be done with extreme care, because an observed peak at about 300 cm−1 could originate from the two-phonon peak of the Si substrate and not from Ge ([211–213]; see also Fig. 15). To tackle this problem resonant Raman scattering with excitation laser energy at 2.4 eV and polarized Raman scattering can be used [214]. 6.2.2. II–VI Semiconductors In polar semiconductors the confinement effect of the optical phonons in particles of a few nanometers in size can be more difficult to observe due to the simultaneous appearance of surface modes between the LO and the TO phonons [145]. Small frequency shifts (up to about 3 cm−1 of the LO phonon of small quantum dots of II–VI compounds in comparison with the bulk value were observed for decreasing radii (down to ≈2 nm) in glass matrices. Redshifts for CdSe dots [39, 54, 110, 146] blueshifts [60] as well as redshifts [52, 145] for CdSx Se1−x dots, and size-independent LO Light Scattering of Semiconducting Nanoparticles phonon frequency for CdS dots [32] are reported. Obviously the shifts depend also on the glass matrix used [215]. To explain the observed phonon shifts and broadenings some authors, in addition to the confinement models, introduced lattice contraction caused by the mismatch of the thermal expansion between the quantum dots and their host material and surface tension increasing with decreasing dot size [60, 215]. Shiang et al. [32] estimated with a confinement model the observed asymmetric band shape, the increase in bandwidth, and the frequency downshift with decreasing size. They assumed that surface pressure effects nearly compensate for the confinement effect in such a way that no size dependence of the LO frequency results. For small CdSe nanoparticles embedded in glass, an asymmetrical broadening of the LO phonon band toward the lowfrequency side for nanocrystals with R = 21 and 1.8 nm and a decrease to lower frequencies was observed [111]. This was attributed to confinement of the LO phonons. The dispersion of the observed LO phonon frequencies was compared with ab initio calculations [111]. 6.2.3. Other Compounds Phonon confinement effects were also studied on selfassembled GaN quantum dots of 2–3 nm height [216], crystalline As precipitates in GaAs [217], in CeO2−y nanoparticles [218], diamond nanoparticles [219, 220] and in BN micro- and nanoparticles [165]. The latter case is interesting because an upward shift of the high frequency mode E2g was observed with decreasing particle size similar to that for small particles of graphite. Calculations were performed with the confinement model CM-II using phonon dispersion curves reported in [221]. Redshift or blueshift in dependence on the coating was observed in coated TiO2 nanoparticles [222]. 7. NEW INSTRUMENTATION AND METHODS High-Throughput Devices Many applications of Raman spectroscopy benefit from developments to measure the usually weak signals more efficiently. For recording scattered light, charge-coupled devices (CCD) have become very widespread during the last 10 years and are now used in the most multichannel Raman spectrometers. They have a very high quantum efficiency over a broad spectral range and a very low dark count level when cooled. The separation of the Raman signals from the much stronger elastically scattered light has been achieved in the past by use of double or triple monochromators. Another technical solution, now used more and more, are Raman spectrometers consisting of a holographic notch filter for stray light suppression and a single grating spectrograph. Such spectrometers have up to 10 times larger optical throughput, are more compact with fewer optical components, and are less expensive than triple monochromators. The notch filters suppress the incident laser light by a factor of more than 106 while the rest of the spectrum is transmitted with a high efficiency of about 80%. The small edgewidth of less than 4 nm allows measurements of frequencies as low as 50 cm−1 . The application of this new generation of Raman spectrometers is 21 attractive for measurements on nanoparticles with low signals, too. However, for low-frequency Raman measurements on acoustic modes closer to the laser line, better stray light rejection is required. This can be realized by application of a triple monochromator and a photomultiplier as detector or by using a double monochromator in the configuration as a cutoff filter followed by a single spectrograph and CCD as detector. Surface-Enhanced Raman Scattering An enhancement of Raman scattering intensity was observed for molecules deposited on silver surfaces. A large increase in the local electric field intensity occurs when the wavevector of the surface plasmon mode of the silver film is matched to that of the incident laser [223–226]. Vibrational Raman spectra of small Si clusters, size-selected in a quadrupole mass spectrometer and co-deposited with an N2 matrix onto a liquid helium-cooled silver film were measured with an signal enhancement of about 50 at the surface plasmon–polariton resonance [200]. The Raman scattering intensity of molecules can be enhanced in SERS processes by 5 to 6 orders of magnitude when they are adsorbed on a roughened silver surface. Although all details of the origin of the signal enhancement are not yet understood, two main mechanisms are usually assumed to be responsible. The so-called electromagnetic effect is considered by most authors to be the major contributor to SERS. The electromagnetic effect was explained in terms of the excitation of localized surface plasmons in silver nanostructures (and nanostructures of a few other metals), which enhances the electromagnetic fields near the nanoparticle’s surface. Resonance of the exciting/scattered fields with the surface plasmons is assumed. The second mechanism discussed, the so-called chemical enhancement effect, is attributed to resonant Raman processes involving charge transfer between the adsorbed molecule and the metal surface. Although the average enhancement factor has been normally measured to lie in the 105 –106 range, it has been shown recently that the enhancement is laterally highly inhomogeneous and concentrated at hot spots at the surface of the metallic nanostructures where the SERS enhancement can be as large as 1010 [227, 228]. The lateral resolution is thus not determined by the diffraction limit, but by the spatial confinement of the local fields. Over the last 5 years it was shown that SERS can be observed even from single molecules [229–233]. The fact that enhancements large enough to allow single-molecule spectra to be detected have only been observed for some adsorbates (generally dye molecules) suggests that resonance processes are involved. The target species are in local electromagnetic fields of high intensity and, in particular, also very strong field gradients. These effects could explain some observed results of SERS measurements on single carbon nanotubes, not seen with conventional Raman scattering, for example, exchange in scattering intensity between phonon modes and changes of the depolarization ratios [234, 235]. Near-Field Optical Microscopy In conventional Raman microscopy a pixel size no smaller than 1 m is common. Resolutions of 100 nm or even substantially smaller can be realized by combining Raman scattering with the near-field 22 optical microscopy [236]. A laser beam is coupled into an optical fiber that is sharpened to a very small tip with conical shape at the end. The dimensions of the tip are much smaller than the wavelength of the exciting laser light. Initially, the major obstacles for obtaining Raman spectra were arising from the weak light intensity given by the excitation and the detection through small apertures and by the Raman scattering process. The principle of the method was demonstrated, for example, by the measurement of the phonon spectra of diamond with a fiber probe of 100 nm diameter [237]. In measurements of Raman spectra from liquid CCl4 the sampling volume of 70 nm diameter and 10 nm depth contained only about 250,000 molecules [238]. For applications with weaker scatterers a considerable improvement of the signals obtained is necessary. One way to enhance the Raman cross section is resonant excitation, which was already used in some cases. Another technique is the combination of near-field optical microscopy with SERS, with which strong enhancement of the Raman signals by many orders of magnitude was obtained [239, 240]. Tip-Enhanced Near-Field Raman Spectroscopy Tipenhanced near-field Raman scattering is one of the forthcoming characterization techniques, which can be effectively used to understand the vibronic and electronic properties of a material at nanoscale. This technique can also be used even for a single molecule detection, because it allows for the detection of molecular vibration spectra with a resolution smaller than the diffraction limit of the light. It has recently been demonstrated [241] that the nearfield detection of Raman scattering using a metallized tip achieves high spatial resolution beyond the known limits of surface-enhanced Raman scattering and reveals new weak vibrational modes, which cannot be seen in usual far-field measurements. It has been demonstrated that because of the local field enhancement, the Raman scattering is enhanced by a factor of 40 and the spectra could be measured at a spatial resolution of several tens of nanometers. By using this technique, single nanoparticle detection could be possible. Also, single molecule detection could be much more precise and accurate compared to SERS, where the spatial resolution is diffraction limited. For near-field detection, it is possible to observe the dynamics of a molecule with spatial resolution on a molecular scale. In these experiments, a metallic tip with a size of a few tens of nanometers is approached to the sample where the source light is tightly focused. The localized surface plasmon–polariton is excited at the tip apex. The area of the field localized at the tip corresponds to the diameter of the tip, which works as the nearfield light source and attends the super-resolving capability. Molecules soaked in this localized field around the tip scatter inelastically through Raman scattering. The Raman spectra recorded contain useful information about molecular vibrations at the nanoscale level, which cannot be detected through other techniques. In this way, it is possible to understand the special behavior of molecules, excitons, and electron–phonon interaction inside a nanoparticle, which can reveal new understanding about the confinement and the effect of confinement on the electron–phonon system. Light Scattering of Semiconducting Nanoparticles GLOSSARY Effective medium theory The properties (e.g., the dielectric function) of a composite are described by averaging over local inhomogeneities introducing suitable effective media models. Quantum confinement In nanometer-sized structures (quantum wells, wires, and dots) the allowed energy states and the density of states are modified in comparison with the corresponding bulk material. Onset occurs when one or more dimensions of a structure become comparable to the characteristic length scale of an elementary excitation. Raman scattering process Inelastic light scattering. 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