{'fraction_non_alphanumeric': 0.025903326154582434, 'fraction_numerical': 0.018006700167504188, 'mean_word_length': 4.802499132245748, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Size dependent Acoustic Phonon Dynamics of CdTe0.68Se0.32 Nanoparticles in Borosilicate glass

Low frequency acoustic vibration and phonon linewidth for CdTe0.68Se0.32 nanoparticle embedded in borosilicate glass are calculated using two different approaches by considering the elastic continuum model and fixed boundary condition. The presence of medium significantly affects the phonon peaks and results into the broadening of the modes. The linewidth is found to depend inversely on the size, similar to that reported experimentally. The damping time and quality factor have also been calculated. The damping time that is of the order of picoseconds decreases with the decrease in size. High value of quality factor for l=2 normal mode suggests the less loss of energy for this mode.


I. Introduction
Semiconductor nanoparticles or quantum dots have been extensively studied in the past several years due to their wide range of application and an important model system in the study of three dimensional quantum confinements. 1 As size is reduced of a nanoparticle, the electronic structure is modified as compared to the bulk with the oscillator strength concentrated into a few discrete transitions. 2,3 Recently, the semiconductor nanoparticles have received attention due to its importance in biological applications as a replacement for the dye molecules. 4,5 In addition, the semiconductor nanoparticles doped glasses have long attracted attention for their potential uses in optoelectronic devices. Since the glass matrix is transparent in visible range, the optical properties of these composite materials are determined by those of the nanoparticles. As the emission of phonon is one of the most important electronic dephasing mechanisms, there has been much interest in the vibrational properties of semiconductor nanoparticles. Apart from modifying the electronic structure, confinement also affects coupling of electrons to the optical and acoustic phonons. In the case of nanoparticle, the coupling is enhanced due to the localized acoustic vibrational motion. In addition, the low frequency phonon modes (acoustic phonons) of nanoparticles bear a unique signature of their structural and mechanical properties directly reflecting the impact of confinement on the ionic movement. This is in contrast to the optical frequency domain whose features are determined by the electronic response. Optical phonons, [6][7][8][9][10][11][12][13][14][15] confined acoustic phonons, 8,[9][10][11][12][13] surface 16,17 and disorder-activated phonons 18,19 have been extensively investigated recently by both theoretical and experimental means. The acoustic phonons are found to be accountable for the width of the lowest exciton transition in nanocrystals. 20,21 These transitions in nanoparticles often have a linear temperature dependence attributed to the thermal excitation of acoustic phonons. 22 The acoustic phonon sidebands encountered in the photoluminescence (PL) are found to affect energy relaxation in nanoparticles. In a recent paper, Bragas et al 23 performed the spontaneous Raman scattering (RS) and pump-probe measurements for CdTe 0.68 Se 0.32 quantum dots embedded in borosilicate glass matrix and ascribed them to the spheroidal l=0, 2 and torsional l=1 modes of a spherical particle though the torsional l=1 mode is not a Raman active mode and should be absent in Raman scattering. 24 The CdTe 0.68 Se 0.32 nanoparticle is of special importance due to its high quantum efficiencies. Moreover, by changing the size of the CdTe 0.68 Se 0.32 nanoparticle, it can be made to radiate throughout the visible spectrum. Bragas et al 23 found that the frequency of the symmetric acoustic mode (l=0), only one to be experimentally observed in the standard pump-probe experiments exhibits a strong dependence on the laser energy in the pump-probe but not in the RS.
However, there is a large variation in the frequency of fully symmetrical mode (SPH, l=0) from these two experiments. To further correlate the observed oscillations with the nanoparticle vibrations, they also calculated theoretically the low frequency modes for l=0 and n=0, l=2, n=0 by using the classical Lamb's model with free boundary condition, 25  . This will not only make actual comparison but also will be helpful in understanding the actual role of borosilicate glass on the low frequency modes and coherent acoustic phonon modes 28 observed in the experimental study.
In the present paper, we report a theoretical calculation of the low frequency acoustic phonon modes (LFM) for the CdTe 0.68 Se 0.32 nanoparticles embedded in borosilicate glass by applying the approaches that also considers the interaction of the nanoparticles with the surrounding borosilicate glass. These approaches are the complex frequency model (CFM) 16,28 and core shell model (CSM) [29][30][31] and have been used to describe the low frequency acoustic phonon modes for several systems. In addition, the parameters such as damping time, linewidth quality factor etc. associated with the damping of acoustic oscillation resulting due to the presence of the surrounding medium are also calculated. The paper is organized as follows: in sec II, we describe the methodology to obtain the CFM and CSM models; in section III, we present results and discussion followed by the conclusion in section IV.

II. Methodology and Computation
In the present study, Lamb's theory of vibrations of a homogenous, free-standing elastic body of spherical shape is used for the calculation of the low frequency phonon modes of borosilicate embedded CdTe 0.68 Se 0.32 nanoparticles. However, to understand the key features, the vibrations of the CdTe 0.68 Se 0.32 nanoparticle embedded in borosilicate glass are calculated by using three different methods based on classical Lamb's theory, in the present paper: (i) Eigen value equation of free isotropic sphere (Lamb's model) 25 (ii) Complex frequency model applicable to embedded nanoparticle [28][29] and (iii) Core-shell model applicable to both free as well embedded nanoparticle. 31 The displacement field ( , ) u r t of an elastic medium of density ( ) r r is governed by Navier's equation Where ( ) ijkl c r is the fourth rank elastic constant tensor field. However, the present equation of motion for a spherical elastic body is written as 25,26 ( Where, l and mare Lame's constants and r is the mass density of nanoparticle. Introducing proper scalar and vector potential, the solution of Eq. (2) gives two vibrational modes, namely a spheroidal mode and a torsional mode. The eigenvalue equations for spheroidal modes were derived as 11,26 Where h is the dimensionless eigenvalues and j l (η) or j l (ζ) are the spherical Bessel's function of first kind and z = h (V t /V l ).The frequencies of these modes are indexed with integers n and l, known as the harmonic numbers and angular quantum number respectively. The Raman peak frequencies of the spheroidal mode can be obtained from the expression as 26 Where V l and t V are the average longitudinal and transverse velocities. The torsional modes are purely transverse in nature and independent of material property and are defined for l ≥1 and are orthogonal to the spheroidal modes. 24 The spheroidal modes are characterized by l ≥0, where l=0 is the symmetric breathing mode, l=1 is the dipolar mode and l=2 is the quadrupole mode. Duval 24 has shown that for a spherical particle only the l=0 and 2 modes are Raman active. The l=0 mode is of particular interest as this is observable in both pumpprobe and Raman experiments. 33 In the complex frequency model (CFM), 28-31 the nanoparticle is surrounded by a homogeneous and isotropic matrix of density ρ m and speeds of sound V lm and V tm . The CFM is the result of fixed boundary condition and is different from the free boundary condition in the sense that it takes into account the stresses at the nanoparticles boundary, continuity of ū and force balance at the nanoparticle-matrix interface due to the existence of the boundaries between particle and matrix interface. The boundaries between the particle and matrix modify the confined phonon modes and even some times responsible for the appearance of new modes. The boundary condition at large R is that ū is an outgoing travelling wave. This model yields complex frequency, the real part Re(ω) represents the frequency of the free oscillations. The imaginary part Im(ω), which is always positive, stands for the attenuation coefficient of the oscillations because the amplitude of the displacement is proportional to e iRe(ω)t-Im(ω)t at any place and therefore Im(ω) yields damping. The real frequency, Re(ω) in the present case are shifted from the frequency ω obtained from Lamb equations. 28 Therefore, the resulting complex nature of the frequency due to radiative decay enables to estimate both frequencies and their damping. Unlike the free vibrations of an elastic body the scalar and vector potentials include first kind of spherical Hankel functions, whose variable is purely imaginary in the region outside the nanoparticle in embedded systems. Verma et al 9 have also used a similar approach, to calculate the low frequency phonon modes of nanoparticle embedded in matrix. This approach also yields complex frequency similar to the present CFM and show the shifting of frequency in the presence of surrounding medium. In experimental observations of such modes a broadened peak is observed with its centre at real frequency.
Though CFM model is able to predict the real frequency i.e the actual peak position and broadening (FWHM) of the peak due to the damping actually observed in the Raman spectra but contains several inherent limitations. In these the most important one is the absence of valid wave functions required for the Raman scattering spectra calculations. In addition, the correspondence to quantum theory is also unclear due to the modes not being orthonormalizable and blowing up exponentially with the radial co-ordinate. 35 To overcome this, Portalés et al 32 proposed an approach namely core-shell model (CSM) considering nanoparticle as core surrounded by a macroscopically large spherical matrix as shell which leads to real valued mode frequencies. 31 Therefore for calculating the absolute intensity or shape of a low frequency Raman spectrum or for the comparison with the experimental Raman spectra, the CSM is the right approach.
In CSM, the motion of nanoparticle, the mean square displacement 2 p u á ñ which is nothing but the measure of the internal motion of the nanoparticle is obtained as a function of mode's frequency and can be expressed as 31 Where, v p is the volume of the particle. The mean-square displacement 2 proportional to a quantity which modifies the electron-phonon interaction and therefore will be responsible for the change in the profile of the Raman spectra. 33

III. Results and Discussion
In the following, we present results on the investigation of low frequency vibrational modes of CdTe 0.68 Se 0. 32 38 The sound velocities so obtained for the CdTe 0.68 Se 0.32 are 3.219x10 5 cm/s and 1.764x10 5 cm/s for l V and t V respectively. Fig. 1 presents the size dependence of the spheroidal (l=0, 1, 2 and n=0) modes for the free boundary condition approach (Lamb's theory). The inverse size dependence of the frequency of low frequency phonon modes is evident. It is seen from the Fig. 1 and table I that   also includes the data from experimental pump-probe and Raman measurements. 23 One can see that while the Raman data follows both l=0 and l=2 modes the pump-probe data follows only l=0 mode. This is due to the fact that standard pump-probe configuration detect only the fully symmetric modes such as l=0 mode. 33 One can see that the frequencies obtained for l=2 matches reasonably well with the experimental Raman data (l=2) while for l=0, it is close to the pump-probe data in particular near 1 0.2 R = nm -1 . The poor agreement of calculated and experimental Raman data, particularly for l=0 seems to be due to the large error bars in experimental Raman data. In view of the large scatter in the data (l=0) the inverse size dependence is not clear. Moreover, the usual trend of frequency variation with size is only seen for resonantly selected radius in pump-probe measurement. The present CSM calculation is performed for the embedded nanoparticles while the theoretical calculation performed in ref. 23 23 The linewidth calculated from the CSM are comparable with the linewidth obtained from Raman scattering. 23 However, for Raman data the 1 R dependence is not clear due to the large error bars, but the pump-probe data show clear 1 R dependence for the available data. It is seen from figure 5 that the calculated linewidth obtained in the present study and that found in the Raman measurement is larger than the pump-probe data. This is due the fact that the acoustic phonon signal originating in pump-probe is from smaller nanoparticles and very narrow distribution of nanoparticles size. 23 The present calculations use the average size of the nanoparticles in the distribution. The Calculations of strain dependent acoustic phonons for InAs sphere embedded in GaAs material show the similar trend for energy dependence of phonon amplitudes. 36 The oscillating behavior of phonon amplitude here resembles the eigenmodes of an isolated sphere 36 and peaks may be the overtones, which are not investigated in the present study. The Fig. 6 reveals that the damping time for CdTe 0.68 Se 0.32 nanoparticle is of the order of ps and decreases for the smallar nanoparticles. This implies that the energy is transferred faster to the surrounding medium in the case of smaller particle, which is consistent with other results. 31 In addition to the role of surrounding medium there may be a contribution from the surface of the nanoparticles to the damping of the phonon modes as the damping constant (opposite of damping time) variation with size of the nanoparticle is similar to that of the size dependency of the surface atoms (surface/volume ratio) for a nanoparticle. On the basis of these two effects it appears that the damping of the phonon modes occurs through surface mechanical coupling of the vibrations of nanoparticle to the matrix and therefore the consideration of surrounding medium in the description of vibrations of borosilicate embedded CdTe 0.68 Se 0.32 nanoparticle is important. The damping time can be used to determine the quality factor. The quality factor is a quantity which measures sharpness of the response of a resonating system to external excitation and can be expressed as where D t is damping time and w is the real frequency obtained in CFM approach. The values of quality factor Q for the 3.9 nm CdTe 0.68 Se 0.32 are 1.982 and 0.634 for l=0 and l=2 phonon mode respectively which is independent of size for any particular mode. This suggests that there is less loss of energy for the l=0 mode. It is found that the quality factor Q is always lower for matrix-embedded nanoparticles and independent of the size, 12 This independent behavior of Q on the size may have some other applications. [30][31] The energy loss, which determine Q for a particular mode, also determine the maximum amplitude of the oscillation when the resonance condition is exactly satisfied, as well as the width of the resonance i.e.
how far off the resonant frequency the system can be driven and still yield significant oscillation amplitude.

IV. Conclusion
Low frequency acoustic phonon modes have been calculated for the CdTe 0.68 Se 0.32 nanoparticle embedded in the borosilicate glass by using the two models namely the complex frequency approach and core shell model which considers the presence of surrounding medium based on the classical Lamb's theory. These models establish a relation between the particle size, the frequencies, and widths of confined acoustic phonons. A reasonable well agreement between theoretical and experimental results is obtained. The phonon energies are size dependent and are lower for embedded nanoparticles. It is observed that the phonon line width varies similar to the phonon frequency with size of the particle for low frequency phonon modes. The reduction in size as well as the surrounding medium both contributes to the phonon line broadening. The damping time and quality factor are also calculated. The damping time is of the order of picoseconds, which decreases with the decrease in size. The high value of quality factor for l=2 mode suggests less loss of energy for this normal model.     l=2, Expt. Raman data Ref. [23] l=0, Expt. Raman data Ref. [23] Pump-Probe Ref. [23] l=0, 2 CSM l=0 l=2