Johnson Matthey Technol. Rev., 2019, 63, (3), 166
doi:10.1595/205651319x15514400132039
氧化锌纳米线尺寸依赖的弹性和热物理特性
在高温情况下应用的半导体材料的表征
尺寸依赖性表征对于高温模式下纳米机电系统、纳米发电机、生物传感器和其他相关领域的应用很重要。在此篇文章中，我们探索了高温条件下不同直径氧化锌纳米线的弹性、机械特性、热特性与超声波特性。我们利用简单的相互作用势模型计算出了氧化锌纳米线更高阶的弹性常数，并根据算出的弹性常数确定了体积模量、杨氏模量、剪切模量和泊松比等机械特性。我们通过弹性常数和密度获得了超声波速度、超声波Grüneisen参数和超声波衰减等各类超声波参数。我们还使用弹性常数计算了从不同方向沿纳米线长度方向传播的、依赖于温度的超声波速度，从而确定了其各向异性表现。氧化锌纳米线与直径有关的超声波损失和热特性也得到了确定。声子黏度机理造成的超声波衰减在氧化锌纳米线的所有超声波衰减中居于主导地位。我们通过确定氧化锌纳米线的超声波参数、导热性和尺寸之间的关系，从而确定了其在工业应用中的潜力。
Size Dependent Elastic and Thermophysical Properties of Zinc Oxide Nanowires
Semiconductor materials characterisation for high temperature applications

 Sudhanshu Tripathi*
 University School of Information, Communication and Technology, Guru Gobind Singh Indraprastha University, Sector16C, Dwarka, New Delhi110078, India
 Rekha Agarwal
 Department of Electronics and Communication Engineering, Amity School of Engineering and Technology, Amity Campus, Sector125, Noida201313, India
 Devraj Singh
 Amity Institute of Applied Sciences, Amity University Uttar Pradesh, Noida201313, India
Article Synopsis
Size dependent characterisation is important for applications in nanoelectromechanical systems (NEMS), nanogenerators, biosensors and other related areas at higher temperature regimes. In this paper we have computed elastic, mechanical, thermal and ultrasonic properties of zinc oxide nanowires (ZnONWs) of different diameters at high temperatures. The higher order elastic constants of ZnONWs were computed using a simple interaction potential model. The mechanical properties such as bulk modulus, Young’s modulus, shear modulus and Poisson’s ratio were determined based on the formulated elastic constants. Various ultrasonic parameters such as ultrasonic wave velocities, ultrasonic Grüneisen parameter and ultrasonic attenuation were obtained with the help of elastic constants and density. The temperature dependent ultrasonic wave velocities propagating along the length of the nanowire at different orientations were calculated using elastic constants to determine anisotropic behaviour. The diameter dependent ultrasonic losses and thermal characteristics of ZnONWs were also determined. The ultrasonic attenuation due to the phononviscosity mechanism is predominant for the total ultrasonic attenuation for ZnONWs. The correlation among the ultrasonic parameters, thermal conductivity and size of ZnONWs is established leading to potential industrial applications.
1. Introduction
Semiconductor oxides, especially zinc oxide nanowires (ZnONWs), find their utility because of their structural, mechanical, electronic and thermal properties. The large piezoelectricity of ZnO makes it a suitable candidate for electroacoustic devices. Thus, the ultrasonic investigation of wurtzite ZnONWs is of great importance. ZnO is a II–VI compound semiconductor with ionicity lying at the border between covalent and ionic semiconductors. Its thermodynamically stable phase under ambient conditions is of wurtzite symmetry (B4) and because of its thermal, elastic and piezoelectric properties in the B4 phase, it is most suitable for making various electronic devices such as actuators and sensors (1, 2). Figure 1(a) shows the hexagonal wurtzite structure of ZnO formed due to stacking of ZnO bilayers in the <001> direction. Each ZnO bilayer consists of Zn^{2+} and O^{2–} ions stacked in the <001> direction (3). Figure 1(b) shows the crosssectional view of ZnONWs which indicates that very few ZnO atoms lie on the surface of the nanowire in comparison to the volume (4). In the Schoenflies notation, the wurtzite phase belongs to the C_{6ν}^{4} space group and in the HermannMauguin notation it is P6_{3}mc. The size dependent mechanical properties, especially stress distribution, of ZnONWs are important for many practical applications.
Fig. 1.
The Young’s modulus enhancement in atmosphere has been investigated using a coreshell model to compare with the model in vacuum (5). Improved mechanical behaviour of ZnONWs was obtained through embedding various nanowires of different materials with various crosssectional models (6). Another study reveals that Young’s modulus and fracture strain of ZnONWs varies with diameter. The electricfield induced resonance method showed that Young’s modulus increases with decreasing diameter (7). Pan et al. (8) synthesised single ZnONWs with thermal evaporation and measured the mechanical properties by the micromanipulator nanoprobe system installed in the scanning electron microscopy (SEM) chamber. Different nonlinear effects and surface elasticity were used to explain the elastic response of the nanostructures. Desai et al. (9) reported experimental observations on elastic modulus and fractural strain. The measured value of Young’s modulus was reported as approximately 21 GPa. The fracture strain increases with decrease in size of the nanowire. Roy et al. (10) tested eleven nanowires of different diameters and reported that elastic moduli are independent of size variation whereas the surface defects depends directly on the diameter. The fracture strength of the nanowire increases with decrease in diameter. Xu et al. (11) studied the elastic and failure properties of ZnONWs and revealed that elasticity size effects occur due to surface stiffness. Size and orientation dependent elastic response of ZnO nanobelts has been reported by Kulkarni et al. (12), which shows the inverse relationship between compressive stress and lateral size of nanobelts. Wu et al. (13) predicted and compared the thermal conductivity (κ) of wurtzite ZnO with wurtzite gallium nitride (GaN). They also reported size dependent κ combining phonon scattering due to anharmonicity and size effects. Koutu et al. (14) performed a size and temperature dependent study of ZnO nanoparticles and advocated their suitability for applications in nanoelectronic devices, sensors and transducers. They had also discussed the electrical and thermal properties of nanoparticles. Kumar et al. (15) had proposed equations of state based on molecular dynamics simulation used to study the isothermal compression and pressure dependence of bulk modulus of various nanomaterials.
The detail analysis of the elastic and mechanical behaviour is still required, as the elastic, mechanical and thermophysical characteristics of nanowires depends on measurement technique and size, as well as structural and surface effects respectively. To broaden the scope of semiconductor oxide characterisation and applications, we followed an ultrasonic based investigation method. Ultrasonic characterisation is popular among various other characterisation methods because of its nondestructive nature. The ultrasonic velocity is well related to the structural, elastic parameters and the density of the material. Information about the mechanical and anisotropic behaviour of the material can be obtained via allowing the ultrasonic wave to propagate through it along different axes. The observed loss in the energy of the ultrasonic wave is due to different mechanisms such as absorption, scattering electronphonon interactions or phononphonon interactions during its propagation through the material, leading to ultrasonic attenuation. The ultrasonic attenuation is well correlated with the other physical parameters. Thus, materials can be characterised using ultrasonic parameters under varying physical conditions (16). In this paper, we describe the temperature and size dependent theoretical detailed analysis of thermophysical, mechanical and lattice properties using ultrasonic theory. Firstly, the second order elastic constants (SOECs) and third order elastic constants (TOECs) of ZnONWs were estimated in the temperature range 100–300 K. Further, these elastic constants were utilised to estimate various elastic and mechanical properties, Grüneisen parameter and ultrasonic velocity. These evaluated parameters were used to find out the ultrasonic attenuation. The computed results are discussed in correlation with available theoretical and experimental data.
2. Theory
2.1 Theoretical Approach for the Computation of SOECs and TOECs
The chosen material for study i.e. ZnONWs has a hexagonal closepacked (hcp) structure. The determination of elastic constants is an essential subject to those areas where mechanical stability and durability is of concern. Also, with the help of the SOECs and TOECs, the mechanical and acoustical properties of the material can be analysed. The elastic energy density (U) in terms of Lagrangian strain components can be given as per Equation (i):
where ζ_{KL}(K, L = x, y, z) represents the strain tensor component and F represents the free energy density of the material (17). The elastic energy density of the material is related to the interaction potential φ(r) defined in terms of constants p_{0}, q_{0}, m, n and is given as Equation (ii) (18):
where p_{0} and q_{0} are constants. The value of r depends on lattice parameters (a, c) of the hexagonal wurtzite structured material and represents the position vector for the two atoms lying above and below the basal plane. The higher order elastic constants i.e. SOECs (C_{KL}) and TOECs (C_{KLM}) for the material in terms of elastic energy density are given as Equation (iii) (19):
The defined elastic constants in Equation (iii) under symmetric and equilibrium conditions with interaction potential (Equation (ii)) leads to the formulation of six SOECs and ten TOECs given as Equations (iv)–(v) (18, 19):
where (a, c), p = c/a, χ = nb_{0} (n – m)/8a^{n+4}, Ψ = –χ/6a^{2}(m + n + 6) represent the lattice parameters, the axial ratio, the harmonic and anharmonic parameters respectively. Also C′ = χa/p^{5}; C″ = ψa^{3}/p^{3} are constants. The values of χ and ψ strongly depend on the values of (a, c), positive values of integers (m, n) and on the value of the LennardJones parameter (b_{o}) for ZnONWs.
SOECs were applied to determine the values of mechanical parameters such as the Young’s modulus (Y), the bulk modulus (B), the shear modulus (G) and the Poisson’s ratio (ν) given as Equation (vi):
2.2 Theoretical Approach to Compute the Ultrasonic Velocity and Related Parameters
Ultrasonic velocity depends on the SOECs and density of ZnONWs. The ultrasonic velocity helps to determine the anisotropic behaviour of the material. Thus, a brief discussion of the ultrasonic wave propagation in wurtzite ZnONWs is necessary. The ultrasonic wave velocity in ZnONWs depends on its stiffness parameter. As the ultrasonic wave propagates along the length of the NWs, it forms the three types of velocity: one longitudinal (V_{L}) and two shear wave velocities (i.e. quasishear (V_{S1}) and shear (V_{S2})) given by Equations (vii)–(ix) (20, 21):
Here, θ and ρ are the angle of propagation with the unique axis of the crystal and the material’s density respectively. The principal reason for the ultrasonic attenuation in solids is acoustic scattering by inhomogeneities in the medium. The frequencies of thermal phonons are modulated by ultrasonic waves, which in turn relax toward a new equilibrium distribution largely by a phononphonon entropy producing process and results in ultrasonic attenuation of the chosen material. The Akhiezer loss (α)_{Akh} is the dominant factor to give rise to considerable ultrasonic attenuation. The Akhiezer type loss is given by Equation (x) (22, 23):
where ω = 2πf is the angular frequency of the ultrasonic wave. Here, f is the frequency of the ultrasonic wave. τ, E_{0}, V represent the thermal relaxation time, thermal energy density and the longitudinal or shear modes ultrasonic velocity respectively. The acoustic coupling constant D = 3ΔC/E_{0} gives the measure of energy transformation from acoustical to thermal, where the applied strain causes change in elastic modulus ΔC, which depends on E_{0,} Grüneisen number (γ _{i}^{j}) and specific heat per unit volume (C_{V}). The deviation in elastic modulus ΔC is given by Equation (xi):
where Grünesien number γ _{i}^{j} for ZnONWs is a direct consequence of SOECs and TOECs (24). As the ultrasonic longitudinal wave propagates throughout the crystal lattice, it creates a rarefied and compressed region within the crystal. The flow of heat energy occurs between these two regions because of their temperature difference. This flow of heat energy results in the thermoelastic loss (25, 26) given by Equation (xii):
where κ indicates the thermal conductivity of the material, T is the absolute temperature. The thermoelastic loss for the shear mode is insignificant as the average of γ _{i}^{j} for each direction of propagation and each mode is zero for the shear wave. Hence, the thermoelastic loss occurs because of the variation in entropy of the longitudinal wave along the direction of propagation. The total ultrasonic attenuation with respect to square of frequency is given by Equation (xiii):
where (α/f^{2})_{Akh.Long}, (α/f^{2})_{Akh.Shear} with respect to square of frequency for longitudinal and shear wave due to Akhiezer type loss. The reestablishment time for thermal phonons termed as thermal relaxation time (τ) is given as Equation (xiv):
Here, V_{D} shows the Debye average velocity, τ_{l} and τ_{s} represent relaxation time for the longitudinal wave and shear wave respectively. The Debye average velocity (V_{D}) is related to the elastic constants via ultrasonic velocities and is defined as Equation (xv) (16, 26):
where V_{L} is the longitudinal wave velocity, V_{S1} is the quasishear wave velocity and V_{S2} is the shear wave velocity respectively.
3. Results and Discussion
3.1 Higher Order Elastic Constants
The value of the lattice parameter ‘a’ and ‘c’ for ZnONWs (13) are taken as 3.249 Ångström (Å) and 5.2068 Å at room temperature (300 K) respectively. The value of axial ratio (p) obtained at 300 K is 1.602. The LennardJones parameter (b_{o}) was determined for minimum system energy under equilibrium conditions with suitable values of constants (m = 6, n = 7) and lattice constants. The obtained value of the LennardJones parameter (b_{o}) under equilibrium condition is 2.095 × 10^{–64} erg cm^{7}. The basal plane distance and axial ratio are not constant with temperature. The elastic constants are important parameters to analyse the mechanical and dynamic behaviour of the ZnONWs relative to the nature of the force acting on it. The resistance to linear compression along the planar axis can be measured with the help of SOECs. The evaluated values of SOECs and TOECs using Equations (iv) and (v) at different temperatures are depicted in Table I and Table II respectively and compared with previously available theoretical and experimental results at nano and bulk scales.
Table I
Table II
Higher values of C_{11} and C_{33} indicate that for the chosen material, the compressibility of the caxis is higher than that of the aaxis. The large value of the elastic constant C_{44} represents its capacity to oppose monoclinic shear distortion in the <100> plane. Table II depicts TOECs and the mechanical parameters i.e. Young’s modulus, bulk modulus, shear modulus and Poisson’s ratio of the selected materials as determined using Equations (v) and (vi). The calculated bulk modulus value is in good agreement with the values reported by Xu and Ching (27). The Young’s modulus value for ZnONWs reported in the literature (3) lies within 142–198 GPa which matches with our computation. The hcp structural Born stability criterion (28): is satisfied by the elastic constants, which indicates the mechanical stability of the ZnONWs. Using SOECs and TOECs, the stress and deformation state under large stress or strain may be calculated. It is found that TOECs have negative values.
It is obvious from Table I that the observed values of SOECs are in good agreement with results obtained by others (29–32) and hence our computational method is justified for these calculations of wurtzite structured nanowires. However, TOECs values of ZnONWs are not available in the literature therefore the comparison is not possible. The negative values of TOECs for bulk ZnO were found based on density functional theory (DFT) with projector augmented wave (PAW) and the exchangecorrelation functional form proposed by Perdew, Burke and Ernzerhof (PAWPBE) (33).
3.2 Ultrasonic Velocity and Thermoacoustic Parameters
The velocity of ultrasonic waves in a semiconductor depends on the stiffness parameters. The longitudinal wave propagation through the material causes compression and rarefaction throughout the lattice and creates regions of different temperatures. The orientation dependent ultrasonic velocities with unique axis (θ) of the crystal i.e. longitudinal, shear and Debye average velocities of the material under consideration were determined using Equations (vii)–(ix) within the temperature range 100–300 K and are shown in Figure 2. It can be seen that the minima of longitudinal velocity occur at 35° and maxima of quasi shear velocity occur at 45° respectively, while shear wave velocity increases with orientation. This abnormal behaviour of angle dependencies is because of the integrated impact of elastic constants and materials density. As it is clear from Figure 2(d) the V_{D} maxima occur at 45° with unique axis of the crystal at room temperature. Since V_{D} depends on V_{L}, V_{S1}, V_{S2} and at 45° significant growth in the value of longitudinal, shear wave velocity is observed while the quasishear wave velocity decreases. Hence, the change in angle of V_{D} is affected by the respective ultrasonic velocities. It shows that when an acoustic wave propagates at 45° with a unique ZnONWs axis then the average acoustic velocity is maximum. The longitudinal velocity reported in the literature (34) of 6.356 × 10^{3} m s^{–1} is in good agreement with our computed value of 6.081 × 10^{3} m s^{–1}. The magnitude of shear wave velocity reported in the literature (35) is 2.735 × 10^{3} m s^{–1} at 300 K which is approximately equal to our computed value. The nature of the velocities obtained for the nanowire was found to be very similar to that of III–V group semiconductor nanowires (19, 36). However, it has been observed that the ultrasonic wave velocity in ZnONWs is slightly higher than that in indium nitride nanowires (InNNWs) (37), which leads to better elastic and mechanical behaviour in comparison. Hence our approach to compute the ultrasonic velocity is justified.
Fig. 2.
Table III depicts the values of ultrasonic velocities, acoustic coupling constants (D_{L}, D_{S}), thermal energy density (E_{0}) and specific heat per unit volume (C_{V}) within the temperature range 100–300 K. C_{V} and E_{0} were determined using the θ_{D}/T tables in the “American Institute of Physics Handbook” (38). It is obvious from Table III that D_{L} > D_{S,} which indicates that more transformation of energy (acoustic to thermal) occurs when the wave propagation is along the length rather than across the surface of the nanowire.
Table III
Figure 3 shows the τ and κ variation with nanowire diameter at room temperature. The curve fit analysis indicates that the expression for τ and κ with the diameter (d) of ZnONWs is given by the polynomial Σ^{2}_{i = 0}P_{i}d^{i}. The size (diameter) dependent τ values (with the assumption ωτ less than unity) for the nanowire at different temperatures were computed using Equation (xiv) and are given in Table IV. Thermal conductivities of ZnONWs for different diameters at various temperatures were obtained from the literature (39). Figure 3 shows that the variation of τ with diameter follows the same trend as that of κ, as τ is directly proportional to κ and inversely proportional to square V_{D}. Thus, if the ultrasonic wave is propagating along a unique axis at 45° the reestablishment time for thermal phonons will be minimum. The order of τ is picoseconds which shows that after passing the ultrasonic wave, the thermal phonon distribution comes back to its equilibrium position in ~10^{–12} s. Hence, τ for nanowires of different diameters are governed by their κ.
Table IV
Fig. 3.
The diameter dependent ultrasonic attenuation (α/f^{2})_{Akh} for longitudinal and shear waves and the thermoelastic loss (α/f^{2})_{Th} of ZnONWs were calculated using Equations (x) and (xii) within the temperature range 100–300 K and are listed in Table IV. It is obvious from Table IV that τ and κ at any temperature increases with the size of the ZnONWs, hence ultrasonic attenuation is found to increase with temperature and diameter. The Akhiezer loss (α/f^{2})_{Akh} in ZnONWs is primarily governed by κ and its thermal energy density. It has been observed that with increases in temperature the thermal energy density increases and the ultrasonic velocity decreases as a result of which the Akhiezer loss in ZnONWs was found to be increasing with temperature and size of the nanowire. Therefore, the phononviscosity mechanism is the major cause of ultrasonic attenuation for ZnONWs hence the loss of ultrasonic energy for longitudinal and shear waves is greater than the loss due to the thermo relaxation mechanism.
The increase in the diameter of NWs leads to an increase in mean free phonon paths which in turn increases the ultrasonic attenuation. Also, it can be observed from Table IV that as τ and κ increase with increase in diameter of the ZnONWs, the ultrasonic attenuation also increases. Thus, ZnONWs of large diameter have high κ and large attenuation. This characteristic of ZnONWs reveals that the ultrasonic attenuation and κ in hcp crystalline semiconductors are closely correlated.
Figure 4 shows the characteristics of temperature dependent total ultrasonic attenuation for ZnONWs at different diameters, evaluated using Equation (xiii). It is clear from Figure 4 that the total ultrasonic attenuation (α/f^{2})_{Total} increases with increase in NWs diameter. A similar phenomenon has been observed and is presented in Table IV for τ and κ with variations in NWs diameter at different temperatures. Therefore, because of the reduction in κ with increase in temperature, as depicted in Table IV, total ultrasonic attenuation is affected and found to increase with temperature and the diameter of the nanowire. This behaviour of total ultrasonic attenuation with diameter is due to the variation of τ and κ depending on the diameter of the ZnONWs. Since the SOECs, TOECs and ultrasonic velocities show very minor variation with the diameter of ZnONWs, the elastic and similar parameters other than τ do not affect (α/f^{2})_{Total} with variations in diameter of the nanowire. As shown in Figure 1(b), the crosssectional view of ZnONWs indicates that very few ZnO atoms lie on the surface of the nanowire in comparison to the volume. As a result, during the propagation of ultrasonic waves more longitudinal phonons will interact with the thermal phonons of the medium in comparison to that of shear phonons. Hence, the acoustic coupling constant (D_{L}) and ultrasonic attenuation (α/f^{2})_{Akh.Long} for a longitudinal wave is larger than that of a shear wave at any frequency. As the diameter of ZnONWs increased from 40 nm to 180 nm, the increase in thermal conduction leads to higher phononphonon interaction, thus (α/f^{2})_{Total} i.e. the total ultrasonic attenuation characteristics indicate the semiconducting nature of the nanowire.
Fig. 4.
4. Conclusion
We have computed nonlinear elastic properties as well as ultrasonic properties for wurtzite structured ZnONWs using the LennardJones potential model. Trends of SOECs and TOECs are comparable to other hcp structured materials. The hcp structured stability criterion for mechanical stability is satisfied for ZnONWs. The τ for the equilibrium distribution of thermal phonons is lowest for wave propagation along 45°. The order of relaxation is of picoseconds, which confirms the semiconducting nature of ZnONWs. The ultrasonic behaviour discussed above indicates important microstructural characteristic features which are well bridged to the thermoelastic properties of the materials. All the characteristic features related to elastic constants and ultrasonic properties of ZnONWs with relation to other wellknown properties might be used in characterisation and in industrial applications.
References
 1.
A. Dev, A. Elshaer and T. Voss, IEEE J. Sel. Top. Quantum Electron., 2011, 17, (4), 896 LINK https://doi.org/10.1109/jstqe.2010.2082506  2.
A. Gupta, B. C. Kim, D. Li, E. Edwards, C. Brantley and P. Ruffin, ‘Zinc Oxide Nanowires for Biosensing Applications’, 11th IEEE International Conference on Nanotechnology, Portland, USA, 15th–18th August, 2011, IEEE, Piscataway, USA, pp. 1615–1618 LINK https://doi.org/10.1109/nano.2011.6144446  3.
T. S. Herng, A. Kumar, C. S. Ong, Y. P. Feng, Y. H. Lu, K. Y. Zeng and J. Ding, Sci. Rep., 2012, 2, 587 LINK https://doi.org/10.1038/srep00587  4.
W. Wang, Z. Pi, F. Lei and Y. Lu, AIP Adv., 2016, 6, (3), 35111 LINK https://doi.org/10.1063/1.4944499  5.
S. Fan, S. Bi, Q. Li, Q. Guo, J. Liu, Z. Ouyang, C. Jiang and J. Song, Nanotechnol., 2018, 29, (12), 125702 LINK https://doi.org/10.1088/13616528/aaa929  6.
A. Vazinishayan, S. Yang, D. R. Lambada and Y. Wang, Results Phys., 2018, 9, 218 LINK https://doi.org/10.1016/j.rinp.2018.02.048  7.
C. Q. Chen, Y. Shi, Y. S. Zhang, J. Zhu and Y. J. Yan, Phys. Rev. Lett., 2006, 96, (7), 075505 LINK https://doi.org/10.1103/physrevlett.96.075505  8.
J. Pan, Y. Zhou, C. Zhao, Y. Zheng and C. Li, Mater. Res. Express, 2018, 6, (2), 25012 LINK https://doi.org/10.1088/20531591/aaeb6d  9.
A. V Desai and M. A. Haque, Sensors Actuators A: Phys., 2007, 134, (1), 169 LINK https://doi.org/10.1016/j.sna.2006.04.046  10.
A. Roy, J. Mead, S. Wang and H. Huang, Sci. Rep., 2017, 7, 9547 LINK https://doi.org/10.1038/s41598017098435  11.
F. Xu, Q. Qin, A. Mishra, Y. Gu and Y. Zhu, Nano Res., 2010, 3, (4), 271 LINK https://doi.org/10.1007/s1227401010304  12.
A. J. Kulkarni, M. Zhou and F. J. Ke, Nanotechnol., 2005, 16, (12), 2749 LINK https://doi.org/10.1088/09574484/16/12/001  13.
X. Wu, J. Lee, V. Varshney, J. L. Wohlwend, K. A. Roy and T. Luo, Sci. Rep., 2016, 6, 22504 LINK https://doi.org/10.1038/srep22504  14.
V. Koutu, O. Subohi, L. Shastri and M. M. Malik, Adv. Powder Technol., 2018, 29, (9), 2061 LINK https://doi.org/10.1016/j.apt.2018.05.012  15.
R. Kumar and M. Kumar, Indian J. Pure Appl. Phys., 2013, 51, (2), 87 LINK http://nopr.niscair.res.in/handle/123456789/15911  16.
D. K. Pandey, and S. Pandey, ‘Ultrasonics: A Technique of Material Characterization’, in “Acoustic waves”, ed. D. W. Dissanayak, Ch. 18, InTech, Rijeka, Croatia, 2010, pp. 398–430 LINK https://www.intechopen.com/books/acousticwaves/ultrasonicsatechniqueofmaterialcharacterization  17.
S. Mori and Y. Hiki, J. Phys. Soc. Japan, 1978, 45, (5), 1449 LINK https://doi.org/10.1143/jpsj.45.1449  18.
A. K. Yadav, R. R. Yadav, D. K. Pandey and D. Singh, Mater. Lett., 2008, 62, (17–18), 3258 LINK https://doi.org/10.1016/j.matlet.2008.02.036  19.
D. K. Pandey, D. Singh and R. R. Yadav, Appl. Acoust., 2007, 68, (7), 766 LINK https://doi.org/10.1016/j.apacoust.2006.04.004  20.
G. A. Alers and J. R. Neighbours, J. Phys. Chem. Solids, 1958, 7, (1), 58 LINK https://doi.org/10.1016/00223697(58)90180x  21.
M. Rosen and H. Klimker, Phys. Rev. B, 1970, 1, (9), 3748 LINK https://doi.org/10.1103/physrevb.1.3748  22.
W. P. Mason and A. Rosenberg, J. Acoust. Soc. Am., 1969, 45, (2), 470 LINK https://doi.org/10.1121/1.1911397  23.
C. P. Yadav, D. K. Pandey and D. Singh, Indian J. Phys., 2019, Online First LINK https://doi.org/10.1007/s12648019013898  24.
M. Nandanpawar and S. Rajagopalan, J. Acoust. Soc. Am., 1982, 71, (6), 1469 LINK https://doi.org/10.1121/1.387844  25.
W. P. Mason, ‘Effect of Impurities and Phonon Processes on the Ultrasonic Attenuation of Germanium, Crystal Quartz, and Silicon’, in “Physical Acoustics”, ed. W. P. Mason, Ch. 6, Vol. 3, Part B, Academic Press Inc, New York, USA, 1965, pp 235–286 LINK https://doi.org/10.1016/b9780123956699.500138  26.
C. Tripathy, D. Singh and R. Paikaray, Can. J. Phys., 2018, 96, (5), 513 LINK https://doi.org/10.1139/cjp20170491  27.
Y.N. Xu and W. Y. Ching, Phys. Rev. B, 1993, 48, (7), 4335 LINK https://doi.org/10.1103/physrevb.48.4335  28.
M. Born and K. Huang, “Dynamical Theory of Crystal Lattices”, Clarendon Press, Oxford, UK, 1954, 420 pp  29.
P. Gopal and N. A. Spaldin, J. Electron. Mater., 2006, 35, (4), 538 LINK https://doi.org/10.1007/s116640060096y  30.
G. Carlotti, D. Fioretto, G. Socino and E. Verona, J. Phys.: Condens. Matter, 1995, 7, (48), 9147 LINK https://doi.org/10.1088/09538984/7/48/006  31.
I. B. Kobiakov, Solid State Commun., 1980, 35, (3), 305 LINK https://doi.org/10.1016/00381098(80)905025  32.
R. Chowdhury, S. Adhikari and F. Scarpa, Phys. E: Lowdimensional Syst. Nanostructures, 2010, 42, (8), 2036 LINK https://doi.org/10.1016/j.physe.2010.03.018  33.
X. Wang, Y. Gu, X. Sun, H. Wang and Y. Zhang, J. Appl. Phys., 2014, 115, (21), 213516 LINK https://doi.org/10.1063/1.4881775  34.
S. M. Galagali, N. S. Sankeshwar and B. G. Mulimani, J. Phys. Chem. Solids, 2015, 83, 8 LINK https://doi.org/10.1016/j.jpcs.2015.03.016  35.
B. Guo, U. Ravaioli and M. Staedele, Comput. Phys. Commun., 2006, 175, (7), 482 LINK https://doi.org/10.1016/j.cpc.2006.06.008  36.
D. K. Pandey, P. K. Yadawa and R. R. Yadav, Mater. Lett., 2007, 61, (30), 5194 LINK https://doi.org/10.1016/j.matlet.2007.04.028  37.
S. K. Verma, D. K. Pandey and R. R. Yadav, Phys. B: Condens. Matter, 2012, 407, (18), 3731 LINK https://doi.org/10.1016/j.physb.2012.05.052  38.
“American Institute of Physics Handbook”, 3rd Edn., ed. D. E. Gray, McGrawHill, New York, USA, 1972  39.
J. AlvarezQuintana, E. Martínez, E. PérezTijerina, S. A. PérezGarcía and J. RodríguezViejo, J. Appl. Phys., 2010, 107, (6), 063713 LINK https://doi.org/10.1063/1.3330755
The Authors
Sudhanshu Tripathi obtained his Master of Technology from Panjab University, Chandigarh, India. Currently he is pursuing his PhD at University School of Information, Communication and Technology, Guru Gobind Singh, Indraprastha University, New Delhi, India. His research interests are nanomaterial characterisation using ultrasonic techniques. He is a life member of the Ultrasonics Society of India (USI).
Rekha Agarwal received her BEng (Electronics and Communication) from Madhav Institute of Technology and Science, Jiwaji University, Gwalior, Madhya Pradesh, India; MEng (Communications) from Malaviya National Institute of Technology, Jaipur, Rajasthan, India and PhD from Guru Gobind Singh Indraprastha University, New Delhi, India. At present she is working as Professor, Department of Electronics and Communication Engineering at Amity School of Engineering and Technology, New Delhi, India. Her research interest lies in the area of wireless communications, coding techniques and antenna arrays.
Devraj Singh is Assistant Professor of the Amity Institute of Applied Sciences at Amity University Uttar Pradesh, Noida, India. His research interests are in the ultrasonic nondestructive characterisation of condensed materials. Presently, he is working on mechanical and thermophysical properties of advanced materials. He is a fellow of the USI and the Associate Editor of MAPAN–Journal of Metrology Society of India (Springer).