Semiconducting nanowires represent interesting solid state systems with unique geometry offering great possibility for further development of optoelectronic devices and sensors applications with numerous possibilities for studying exciting physical phenomena arising from carrier confinement and the large surface-to-volume ratio (Hayden 2008, Pauzauskie & Yang, 2006; Yang et al., 2010; Lu & Leiber, 2007; Patolsky et al., 2007; Cao et al., 2009; Algra et al., 2008 & Wang et al., 2001). However, the growth of nanowires free of contaminants, controlled surface states and structural defects is still one of the key issues.
Studies comparing Raman scattering experiments of bulk and nanostructured materials have been reported in literature for several different kind of systems with extensive use as a primary characterization tool for study of phase and optical properties of nanowires and nanoparticles. (Zeng et al., 2006; Lin et al., 2003; Mahan et al., 2003; Adu et al., 2006; Hartschuh et al., 2003; Zardo et al., 2009; Cao et al., 2006; Shan et al., 2006; Pauzauskie et al., 2005; Xiong et al., 2006).With respect to one-dimensional (1-D) structures, several new phenomena have been reported to date, e.g., the high surface-to-volume ratio enabling the measurement of surface optical (SO) phonon modes (Gupta et al., 2003; Xiong et al., 2004; Spirkoska et al., 2008; Sahoo et al., 2009a; Sahoo et al., 2010; Adu et al., 2006). A increase in the scattered intensity is reported for nanoscale structures with respect to their bulk counterpart, describing the effect denominated as ‘Raman antenna effect’ (Xiong et al., 2004; Xiong et al., 2006; Cao et al., 2007). In addition, polarization dependent experiments on single carbon nanotubes and nanowires have shown that the physics behind Raman scattering of such 1-D nanostructures can differ significantly from the bulk (Frechette et al., 2006; Pauzauskie et al., 2005; Livneh et al., 2006; Cao et al., 2006). As a matter of fact, the highly anisotropic shape of the nanowires can lead to angular dependencies of the phonon modes which otherwise would not be expected from selection rules (Pauzauskie et al., 2005; Nobile et al., 2007). In general, Raman spectroscopy is an efficient tool to provide qualitative information from semiconductor by probing SO modes.
1.1. Surface optical phonons
SO phonons are similar in concept to bulk phonons, except that the atomic amplitudes are confined to the near-surface region of the material. The amplitudes of these phonons decay exponentially with distance measured away from the sample SO modes. In general, two different types of SO phonon modes, depending on the penetration depths, can be distinguished as macroscopic and microscopic ones. Macroscopic SO modes are classified as optical (e.g. Fuchs-Kliewer; Fuchs & Kliewer, 1965) and acoustic (e.g. Rayleigh) SO phonon frequencies. Acoustic SO modes in isotropic elastic media propagate along the surface or interface with displacements in the sagittal plane which is defined by the normal to the surface and the direction of propagation (Wallis, 1994). These modes decay exponentially into the bulk and the penetration depth is proportional to the wavelength in the long-wave limit. Contrary to an acoustic mode the Fuchs-Kliewer phonon is connected to a macroscopic electric field (Fuchs & Kliewer, 1965). The frequency of a Fuchs-Kliewer phonon can be determined by solving the Laplace equation for the electrostatic potential of an ionic crystal under appropriate electromagnetic boundary conditions. Then the frequency is found to lie in between the frequency of the transverse optical (TO) and longitudinal optical (LO) bulk phonon. The corresponding pictorial representation of the atomic motion in a phonon mode has been illustrated in figure 1 (Richter et al., 2000).
Experimental techniques for analyzing SO phonon properties have so far been based on SO phonon sensitive probes such as low-energy electrons and atoms, i.e. high resolution electron energy loss spectroscopy and He-atom scattering (Ibach & Mills, 1982; Benedek & Toennies, 1994). Due to the strong interaction of low-energy electrons and atoms with matter, the penetration depth of these probes is limited to the few outermost atomic layers giving rise to the sensitivity of SO modes in these techniques. Raman spectroscopy, in contrast, is based on the comparably weak interaction of photons with matter. Even under strong absorbing conditions the penetration depth of photons is no less than approximately fifty atomic layers and thus the Raman signal is bulk rather than SO mode related. However, the resolution and sensitivity of modern Raman equipment allows also for the observation of Raman signals generated by SO phonons (Gupta et al., 2003; Xiong et al., 2004). The reasons are, firstly the vibrational frequencies of SO phonons are different from those of the bulk, since the top layer is free and bonding atoms are missing. Secondly, the energies of electronic states corresponding to the surface are different from bulk states and thus the resonance condition of SO phonons is different from that of bulk phonons. These differences in SO and bulk phonon scattering are pronounced for a perturbed surface or for hetero-termination i.e., when a surface is terminated with atoms not being present in the bulk.
SO phonons are reported in polar semiconductors exhibiting a formal charge separation between cation and anion sub-lattices. These SO modes show dispersion, i.e. their frequency depends on the wave vector
Normally, SO modes are observed when the translational symmetry of the SO potential is broken. This break down of symmetry can activate a larger wave vector SO mode whose frequency is sufficiently separated from the other Raman-active optical phonons, to be observed experimentally. An interplay of a strong Fourier component of the perturbed SO potential acts as a source of wave vector supplied by the symmetry breaking (Sernelius 2001). This can be achieved in several ways, e.g., by surface roughness, or by formation of a grating along the surface. Another method for detecting SO modes involves placing a prism at the surface of the sample. The evanescent wave from the prism–sample interface can probe the surface and the desired SO mode (Falge et al., 1974)
Overall, Raman spectroscopy of nanostructures represents an extremely active and exciting field for the benefit of science and technology at the nanoscale. The arising new phenomena and technical possibilities not only open new vistas for the characterization of materials but also for the understanding of fundamental processes at nanoscale. In this chapter, we provide a review of SO modes on nanowires using Raman spectroscopy depicting overview of the appearance of new modes and their effect on geometry of samples.
2. Surface optical phonons from semiconductor nanowires
There are geometry and size-related phonons appearing for several different kind of systems when dealing generally with nanowires and nanoparticles, such as the SO and breathing modes arising from acoustic confinement. In case of nanowires, the geometry is one of the long filaments. The cross-sectional shape of nanowire matters for the SO phonon dispersion. It is usually observed that the TO and the LO modes have a position in energy close to that observed in bulk. When scaling down the size and the dimensionality of the structures, the position can change along with the appearance of new Raman modes due to breakdown of translational symmetry in the finite size. Moreover, effects related to the shape of the system can become significant. The existence of boundary conditions at the nanoscale gives rise to electric and polarization forces. The surface represents a new mechanical boundary, since the surface atoms are less bound and experience a different local field than that from the bulk. This has consequences even in the propagation of an optical phonon.
Several works have reported the presence SO modes in Raman spectra of semiconductor nanowires which have been assigned to SO phonons (Gupta et al., 2003; Shan et al., 2006; Sahoo et al., 2008a, 2010b; Sahoo et al., 2010; Lin et al., 2003; Zeng et al., 2006; Spirkoska et al., 2008). The SO phonons are generated at the interface between different materials with different dielectric functions and propagate along the interface. The atoms involved in their propagation are those close to the surface, so that the amplitude of the oscillations decays exponentially with the distance from the surface. This mode is activated by breakdown of the translational symmetry of the SO potential, which in the case of the nanowire can be addressed to the presence of roughness, saw tooth faceting on the nanowire sidewall or to a diameter oscillation along the nanowire length. There are two characteristics which are distinctive of the SO modes and can therefore allow a reliable assignment of the mode: the dependence of the phonon frequency (1) on the dielectric constant of the medium surrounding the wires and (2) on the diameter or on the period of the diameter oscillation of the wires. Indeed, it was already observed that blue shift of the SO mode value increases with the dielectric constant of the surrounding optical medium and decreases with the nanowire diameter (Gupta et al., 2003; Sahoo et al., 2009). Furthermore, the frequency of the SO modes at the center of the Brillouin zone is located between those of the TO and the LO and also the long-wavelength (
2.1. SO phonons in cylindrical geometry
The SO modes dispersion at the interface between a semiconductor and a dielectric material can be calculated taking into account of the geometrical constraint and surrounding dielectric medium, by imposing the condition:
Where the dielectric function of the semiconductor and is the dielectric constant of the medium. For a cylindrical interface, the SO mode frequency can be obtained by solving the following equation,
for a semiconductor, the SO mode frequency is
where is the screened ion plasma frequency given by +,
This is exactly the SO phonon frequency of an infinite semiconductor flat surface expressed by (5). Therefore, equations (6) and (7) establishes the dependency of the SO phonon energy on the external medium and on the size of the wire, since the position of the SO phonon can be related to the dielectric constant of the surrounding medium as well as to the nanowire radius. Furthermore, values of
2.1.1. GaP nanowires
The SO potential along the axis of the GaP nanowire is perturbed by a strong component with wave vector
Figure 2 shows the Raman spectra of the optical phonons for GaP nanowires with diameter of 50 nm in air (
It is assumed that the LO and TO phonon branches are dispersion less as shown by horizontal lines. The nanowire diameter is given by 2r, and the thin lines are the result of a Lorentzian line shape analysis. The decomposition of the band into individual Lorentzians is also shown. The lowest and highest frequency bands, respectively, are identified with the first-order modes of TO at 367 cm−1 and LO at 401 cm−1. A third Raman band lies midway between the TO and LO bands (Fig. 2). This band is assigned to SO modes because of its sensitivity to the dielectric constant of an external medium (εm) e.g. air, dichloromethane, and aniline in contact with the nanowire. The SO band show red shift (Fig. 2), as the dielectric constant of the surrounding medium is increased. The peak positions of the LO and TO bands, however, do not change. Based on this observation, the intermediate frequency band has been unambiguously assigned to SO modes.
From the calculated SO dispersion and the SO obtained from experiments with the nanowires in various dielectric media (e.g. the crossing points between the solid curve and the dashed lines in Fig. 3), the wavelength λ = 2π/q of the SO potential (perturbation) which is important for activating the SO mode scattering has been estimated. For cylindrical cross section GaP nanowires, λ = 2π/q is ~ 40 nm. In case of d= 20 nm wire, the maximum dispersion takes place at q* wire ~ 2 107/m by using equation (6). The q* value marking maximum dispersion is much larger than its counterpart for SO modes at a plane surface (i.e., qplaneLO/c =2.5105/m). Thus, it is clearly important to use the correct (cylindrical) geometry while describing the SO phonons in small diameter nanowires (Adu et al., 2006; Spirkoska et al., 2008).
It appears that values of the SO band frequency, as calculated for x= 4, correspond to the experimentally obtained values for the sample of d= 50 nm, while x= 1.5 values are close to those SO mode observed for the sample with a diameter, d= 20 nm. Approximately the same value of q = 1.5108/m is found for the two sets of wires d= 20 or 50 nm. Therefore, the length scale which is responsible for excitation of the SO mode is given by 2/q = 40 nm. Since this length scale is similar in both cases, there must be an inherent length scale in the nanowires responsible for the activation and observation of the SO band. The symmetry breakdown mechanism responsible for the observation of the SO band is due to the presence of the wire diameter modulation along the nanowire growth direction. This modulation can indeed be observed in TEM images (Fig. 4) where it has been shown that four different images with different wire diameters with modulation wavelength λ. The diameter modulation wavelengths, λ (30-70 nm), for the small subset of nanowires (Fig. 4) is consistent with the length scale (40 nm).
Again, it is already discussed that the dispersion for a SO mode calculated in a cylindrical geometry is crucial for estimating the wavelength of the important Fourier component of the SO potential that breaks the symmetry and “activates” the SO modes in a Raman scattering event. It is also possible that SO Raman mode from larger nanowire diameters (d> 40 nm) in the lognormal diameter distribution with a more prominent diameter modulation (Fig. 5) might dominate the SO spectrum for the nanowire ensemble. This assumption can be proved if SO mode frequency can be measured from individual wires with known diameter modulation wavelength.
The SO band position does not include any dynamics of the SO phonons, and therefore it is inadequate to describe the variation of the SO mode line width as a function of εm as shown in figure 5 (top). The line width measured in air for the SO band in the 20 nm diameter sample is twice of that observed in the 50 nm sample, even if the line widths of the LO mode in the two nanowire samples with different diameter distributions are comparable to within 10% in all overlaying media. This implies that the SO modes in the 20 nm nanowires have a lifetime that is half of the 50 nm nanowire batch. This may results due to the difference in the decay channel phase space available to the SO phonons for the two samples as the dielectric constant varies.
2.1.2. GaAs nanowires
Figure 6 represents scanning electron microscopy image of the GaAs nanowires, grown by molecular beam epitaxy (Spirkoska et al., 2008). The Raman spectra have been recorded in ensembles of GaAs nanowires having diameters 160 and 69 nm (Fig. 7a). The existence of a SO mode is confirmed by comparing the spectra of the nanowires in an environment of air and PMMA with higher value of dielectric constant.As expected, the SO mode shifts to lower wave numbers for smaller diameters, as it is observed in the spectra obtained for nanowires with an average diameter of 69 nm. The entire trend of the position as a function of the diameter is shown in figure 7b. The SO mode is barely observed as the position is very close to the LO phonon for the nanowire with the largest diameter. Continuous line indicates data for nanowires with a circular section. GaAs nanowires exhibit a hexagonal section explaining the discrepancy with the experimental data.
2.1.3. GaN nanowires
In our latest study,(Sahoo et al., 2010) the SO phonon modes has been identified in GaN nanowires grown by vapor liquid solid (VLS) technique showing a strong correlation between the surface morphology and Raman phonon spectra. The SO phonons associated with A1(TO) at 533 cm–1 and E1(TO) at 560 cm–1 are calculated for GaN by taking the qr value of 1.07, which yields SO phonon frequencies (ωso) pertaining to A1 (654 cm–1) and E1 (688 cm–1) characters (Fig. 8). Hence, the reported peaks around 652 and 691 cm–1 has been accounted for SO phonon modes in GaN.
It can also be seen that the intensities of the observed SO phonons in the present study are comparable to other phonons. The magnitude of surface roughness determines the SO peak intensity owing to the breakdown of the translational symmetry of the SO potential. The wavelength λ = 2π/q (qr = 1.07; r~250) corresponding to the perturbation of SO potential has been estimated as ~1500 nm for the GaN nanostructures (using the dispersion relations in Fig. 8b for typical nanostructure diameter of ~500 nm). Surface roughness with a modulation of ~150–300 nm is observed (indicated by arrows in the inset of Fig. 8b) with any integral multiple equivalent to the λ (10×150 nm = 5×300 nm = 1500 nm) will be sufficient to initiate breakdown of translational symmetry in contributing SO potential toward SO modes. The process makes the intensity of the SO mode comparable to that of the other phonons.
SO mode in a single GaN nanowire grown by molecular beam epitaxy has also been reported by Hsiao and co-workers(Hsiao et al.,2007) during angle dependent Raman spectroscopy measurement with the z axis pointing along the crystallographic c axis of the rod. The direction of different angles between the laser polarization and the rod length (c axis) are shown in figure 9 (a). The reference frame (x, y polarization) is rotated against the rod. The angle between the laser polarization and the z axis (x axis) is represented by (Fig. 9a). In the Porto notation, (Arguello et al., 2969) the representations of = 0º and = 90º scattering configurations with =90º are x(z, z)and x(y, y), respectively. The hexagonal geometry shape with vibrational modes are resolved in scattering configuration to obtain all main Raman peaks which further demands for application as angle selective nanosensors. However, a new peak at 708.5 cm−1 have been ascribed to the SO mode (Fig. 9b) which is absent in GaN compact film (also shown in lower spectra of figure 9b). With high surface-to-volume ratio and the high perfection of the hexagonal surface with highly crystalline structure, the peak is assigned to the SO mode of GaN nanocrystals rather than the disorder-active Raman mode which refers to the effect of defects in crystal or a-GaN and agrees well with the calculated value.
2.1.4. InN nanowires
An optical mode of one-dimensional nanostructures of InN, such as nanowires and nanobelts grown by chemical vapor deposition, has been studied in our earliest report (Sahoo et al., 2008). Typical Raman spectra of nanowires and nanobelts are shown in figures 10(a) and (b), respectively and SO phonons modes have been reported taking into consideration of cylindrical geometry to both. The range of qr is determined from the dispersion relation of SO as a function (Fig. 11a) by the spread in diameter (d~75 to 150 nm for nanowires and ~50 to 200 nm for nanobelts). For d=100 nm the value of qr = 1.18 is shown as a vertical dot-dashed line in figure 11. Based on the above agreement the modes around 528–560 cm−1 have been assigned to SO modes with A1 and E1 characters, respectively. A further confirmation has been clearly reported by recording the spectra in a higher dielectric medium m=2.24 for CCl4) which shows inherently a red shift of 4 cm−1 to SO modes. The spread in SO phonon essentially arises due to the wide range of diameter and dimension of the nanostructures.
The wavelength =2/q, corresponding to the SO potential, can be estimated as ~135 nm for nanowires and ~110 nm for nanobelts using the dispersion relations in figure 11 with maximum qr=1.65 for nanowires and~1.4 for nanobelts. Surface roughness with modulation of ~65 nm for nanowires (Fig. 11b) and ~55 nm for nanobelts in figure 11b have been observed. However, the intensity of SO modes found to be higher than that for the TO modes. Defects related to abrupt termination of translational symmetry in the nanostructures may reduce the intensities of fundamental phonon modes; on the other hand, rough surface and interface may result in intense SO modes.
2.1.5. AlN nanowires
In one of our recent studies (Sahoo et al., 2010) we have studied typical Raman spectra for optical phonons in AlN nanotips (Fig. 12a). Apart from the allowed optical phonons in AlN, an additional phonon mode at 850 cm−1 has been assigned to SO phonons, in view of its strong response to change in dielectric medium and good agreement with calculated SO phonon frequency. A prominent red shift of SO mode (23 cm-1 in CCl4 medium) has been reported (inset of Fig. 12a).
The typical analysis of the nanotip surface (as directed by two doted arrows) shows that the steps (labeled as A, B, and C in Fig. 12b)are systematically decreasing along the growth direction. The width (d=2r) of the steps are 120, 90, and 50 nm with a corresponding length of 350, 260, and 140 nm, respectively. The wavelengths (λ=2/q) of perturbation have been calculated taking into account of the corresponding r and x=1.2 value of each step. The values of wavelength corresponding to A, B, and C are reported to be 314, 235, and 130 nm, respectively. Thus these perturbation values, which are in close agreement with the length of the steps (Fig. 12b), absorb the required phonon momentum for the observation of the SO mode.
2.1.6. InGaN/GaN multi-quantum-well nanopillars
In another study (Wu et al,. 2009) we have reported the fabrication and the observation of SO phonon in aligned GaN nanopillars with spatial control of embedded InGaN/GaN multi-quantum-well (MQWs) using focused ion beam milling. Figure 13 (a) shows the SEM image of resulting InGaN/GaN MQWsstructure milled with a serpentine-scanning beam and subsequently wet-etched with KOH solution. The pillar base diameter (Fig. 13a) is around 95 nm with aspect ratio of 7: 1. The pillar size is further reduced to 30 nm with enhanced high aspect ratio of 16: 1, with an extended KOH treatment time. The presence of surface roughness or even surface modulations of ~ 25 nm at two different regions of these 1-D MQW nanopillars have been observed (Fig. 13b).
Figure 14 displays distinct Raman features of KOH etched nanopillars with 632.8 nm (Fig. 14a) and 514 nm (Fig. 14b) laser excitations. Along with all allowed Raman modes, the fitted broad peak in the range 665–699 cm−1 (inset in Fig. 14b) have been examined carefully to elucidate the possibility of SO phonon being responsible. The Raman spectra for nanopillars in a higher dielectric medium (εm= 2.56 for aniline) have been recorded and compared with those in air for the same excitation of 532 nm laser line (Fig.15a). It is reported that the peak positions of the A1(LO)GaN and A1(LO)InGaN modes did not change, whereas an apparent red shift of ∼15 cm−1 for SO modes for GaN has been reported for the spectrum collected from the sample surrounded with a higher value of dielectric constant than that of air (εm= 1). These unidentified peaks had been assigned to SO phonon modes originating from surface roughness or even surface modulation in 1-D MQW a nanopillars (Fig. 13b).
The plots of ωSO for GaN as a function of qr dispersionrelation of SO modesfor these 1-D MQW nanopillars, taking into consideration of cylindrical geometry, is shown in figure 15b.
The wavelength λ = 2π/q, corresponding to the SO potential (perturbation), has been estimated as ∼250–300 nm for the nanopillars (using the dispersion relations in Fig. 15b with various excitations). Surface roughness with a modulation of∼25 nm is observed (Fig. 14b) in these nanostructures. Modulation length with any integral multiple equivalent to the λ (10×25 nm=250 nm or 12×25 nm=300 nm) will be sufficient to initiate breakdown of translational symmetry for the contribution of the SO potential towards the SO phonons and to make the intensity of the SO modes comparable to that of other phonons. The origin of the broad new peak in the range of 665–699 cm−1 can be assigned to the SO modes corresponding to 1-D GaN and In0
2.2. SO phonons in rectangular geometry
The theoretical prediction for the SO modes varies from circular cross section wires to rectangular. However, there is no analytical expression for SO modes in rectangular cross section wires. Nevertheless, the dielectric continuum (DC) model approach that has been used for cylindrical nanowires remains the most elegant method and provides analytical expressions for SO(q) vs m for rectangular wires with further approximation,. An approximate DC model for rectangular wires has been introduced by Stroscio and co-workers (Stroscio et al,. 1990; Stroscio et al., 2001) that neglects the exponentially decaying electrostatic fields emanating from the corner regions. This assumption makes the problem separable in the plane perpendicular to the wire axis. The following dispersion relations is obtained for the SO phonons upon imposition of the usual electrostatic boundary conditions;
Equation 8 is the symmetric (S) mode and equation 9 is the asymmetric (AS) mode, w(m) is the dielectric function inside (outside) the wire, L
where equation 11 is the requirement that potentials of optic phonons in the x and y directions should have the sameparity. Neglecting the damping and crystal anisotropy, the dielectric function w() can be expressed as,
and the Lyddane-Sachs-Teller (Yu and Cardona, 1999) relation gives
for nanowires with infinite edge length, i.e. L→∞, tanh (qL/2)→1, and coth (qL/2)→1; both symmetric (14) and asymmetric modes (13) recover again the SO phonon frequency of an infinite flat semiconductor surface expressed by (5).
2.2.1. ZnS nanowire
High crystalline, rectangular cross-sectional uniaxial wurtzite ZnS nanowires is grown along either a-axis () or the c axis () and details phonon spectra have been discussed by Xiong et al.,(2004).The crystal anisotropy and the two growth orientations make the calculations even more complicated. Isotropic equation has been employed to capture the essential physical ideas without too much complication. An exact treatment for the dielectric function and the Lyddane–Sachs–Teller relation for uniaxial wurtzite crystals can be found. For wurtzite ZnS, ε11(∞) and ε33(∞) vary from 8.25 to 8.76. As an isotropic approximation of ε0= 8.3 and ε∞ = 5.11 from and ωLO = 346.5 cm−1 are made.
Raman spectrum of rectangular cross section ZnS nanowires collected at room temperature in air is show in figure 16. Strong first-order LO scattering has been observed at 346 cm−1and a TO doublet has been reported at 269 and 282 cm−1. Some broad continuum scattering intensity has been observed in the range 350–450 cm−1 along with other features due to combination and overtone modes. The insets in figure 16 show the Lorentzian line shape analysis of the TO doublet and the LO–SO bands. Magnified spectrum is shown (Fig. 16) for the line shape analysis of the LO-SO region. As shown in figure 16 an additional Raman band have been reported at 335 cm−1 (in air) for square cross section ZnS nanowires. It has also been identified with SO modes, as it red shifts in dielectric media (dichloromethane and aniline). In air, the SO band found at 335 cm−1 red shifts to 328 cm−1 in dichloromethane and 326 cm−1 in aniline. The variation of experimental SO frequencies ZnS nanowires with the dielectric constants for the surrounding liquids are plotted in figure 16b. A linear decrease of the SO frequency has been reported as the dielectric constant of the overlaying medium is increased.
Figure 17(a) shows the SO dispersions for both S and ASasymmetric modes of ZnS for three overlaying dielectric media (εm). The S and AS modes exhibit different dispersion at small qL, but they converge to the same frequency as qL approaches very large values. Three horizontal dashed lines represent the actual experimental SO phonon frequencies in the three dielectric media. The three experimental frequencies connect to approximately the same wave vector q, showing an important self-consistency proof of the validity of the dielectric continuum model (Givargizov et al., 1973; Johansson et al., 2006). The surface modulation in TEM images (inset in Fig. 17a) along the wire axis, however, appears to be not as clearly periodic as in cylindrical nanowires. Nevertheless, still it confirms that the theoretical calculations based on the dielectric continuum model are consistent with TEM images; i.e., the wavelength of the perturbation that breaks the symmetry and activates SO mode is λ~70 nm.
The effect of the nanowire morphology, looking into the cross-sectional shape, on the SO phonon dispersions has been made by computing the SO mode dispersion for ZnS cylindrical nanowires with diameter (d) against that of square cross section nanowires with sides of length L = d. The dashed curve shows the calculated SO dispersion for cylindrical ZnS nanowires in air (Fig. 17b). From the measured SO frequency in air, e.g., SO= 334.7 cm-1 and using the SO dispersion (Fig. 17a), the important wave vector q responsible for activating the SO modes is estimated. It has been reported that qd = 25 predicts the experimental SO frequency in air. The wavelength for the perturbation is therefore λ=2/q =13 nm. This λ differs by a factor of about 5 from the λ obtained from the rectangular cross section SO analysis. So it is obvious that the correct nanowire shape must be used in the analysis. The importance of the effect of the nanowire shape on the SO analysis is apparent from figure 17b.
2.2.2. GaN nanoribbons
The higher wave number fitted Raman spectrum (Fig. 18a) has been obtained from the GaN nanoribon grown inside the nanochannels of Na-4 in our study (Bhattacharya et al., 2011). The dispersion relation ωSO(q) for a GaN nanoribbon encapsulated in Na-4 mica (Fig. 18b) shows a plot of both the S and AS modes of SO phonon dispersion curves for A1 and E1 symmetries. Horizontal lines corresponding to the experimentally observed wavenumbers at 633 and 678 cm−1 intersect the dispersion curves of ASSO modes at q
The observed SO phonons intensities are comparable to those of the other phonons. The wavelengths for the SO potential perturbation (with the average nanoribbon cross-section dimensions as 50 × 1 (L
The interaction potentials for the SO modes scales as (q
2.3. SO phonons in spheroidal geometry
It is convenient to use cylindrical coordinates () for a spheroidal nanorod with semi-axes a and c and the dispersions of the anisotropic dielectric function and in the following denotations (Fonoberov & Baladin, 2004, 2005)
the phonon potential inside (in) and outside (out) of the nanorod as
Here are associated Legendre functions of the first and second kind, respectively. The integers L (L) and M (L) are quantum numbers of the phonon mode, and the eigen frequencies of the phonon modes (Fonoberov & Baladin; 2005).
2.3.1. CdSe nanorod
The optical phonon excitations of randomly oriented and laterally aligned nanorod arrays (Fig. 19a) have been studied using resonant Raman scattering by Nobile and co-authors (Nobile et al., 2007). Colloidal wurtzite CdSe nanorods are fabricated by chemical synthesis in a hot mixture of surfactants. Nanorods are aligned by drop-caste on to an interdigitated electrode device and applying an AC electric field of 105 V.cm-1 at a frequency of 10 MHz during solvent evaporation (Fig. 19b).
Raman spectra of randomly oriented (dotted) and closely packed laterally aligned (solid) nanorods (diameter 10 nm, and length 50 nm), which are taken on regions outside and within the interdigitated electrodes, respectively, on the same sample are shown in figure 19a. In the spectrum of the randomly oriented rods, the LO phonon at 214 cm-1, and at the low-energy side, a broad peak centered around 185 cm-1 originating from the SO phonon modes are observed. The spectrum of the laterally aligned nanorods (with c-axis perpendicular to the incident light polarization) shows a very small blue-shift of the LO phonon peak and a significant decrease in the SO phonon intensity. In this contest, the experimental data is compared with first principle calculations of nanorods with corresponding aspect ratio (c/a of CdSe nanorods taken around 5). The theory predicts that only phonon modes with even L and with M =0 can be active in a resonant Raman spectrum.
According to the formalism reported above, figure 20 shows the calculated (phenomenological) potential of the SO phonon modes with M = 0 and even angular quantum numbers L. The energy of these modes is depicted in figure 19a by the black bars with full heads (energy increasing with higher angular momentum number). The calculated Raman active energies with the data obtained from the randomly oriented rods have shown excellent agreement. The phonon potentials of the SO modes with low L extend significantly outside of the rod surface as shown in figure 20.
The SO or interface phonon modes are stimulated by the change in dielectric medium at the interface. In a close-packed 2D array of nanorods, the space adjacent to the rods is occupied by other nanorods. Thus the dielectric medium surrounding the individual rods is severely modified, leading to the observed suppression of the SO phonon modes. The fact that SO phonon modes with higher angular momentum extend less outside the rod region accounts for the low-energy shoulder of the LO phonon peak around 200 cm-1. The observation agrees well with the spatial distribution of potential of the SO phonon modes with long axis orientation of nanowires.
The essential for understanding SO phonons is presented in this report for a wide range of compound semiconductor nanowires. The experimental analysis, as well as, theoretical calculations concluded that the SO mode in semiconductor nanowires is primarily associated with the surface modulation and geometry. The SO mode is clearly identified by its shift in the Raman spectrum as the overlaying medium dielectric constant is varied. The characteristic wave length of perturbation of the SO potential can be readily measured by the position of SO mode in Raman spectra. We believe that growth instabilities leading to the appearance of SO modes in the phonon spectra should be a general indicator of the presence of a corrugated surface. The instability of surface modulation mostly arises during vapour-liquid-solid growth of most nanowires system, and strongly depends on growth conditions, such as temperature, impurities, and supersaturation. In general, Raman scattering from SO phonons not only enlighten us on the atomic dynamics on surface but allows also help exploring the interaction of SO phonons with electronic surface states, and hence, can be exploited as a primary tool for surface analysis of one- (nanowire) and two-dimensional systems.
We acknowledge the contributions from S. Sahoo and A. K. Arora of MSG, IGCAR; M. S. Hu, K. H. Chen and L. C. Chen of Center for Condensed Matter Sciences, NTU, Taiwan; S. E. Wu, and C. P. Liu of National Cheng Kung University, Taiwan for their active involvement in the study of surface optical modes. We also thank S. Dash of SND, IGCAR for his general support and encouragement.