Size Dependent Elastic and Thermophysical Properties of Zinc Oxide Nanowires
Size Dependent Elastic and Thermophysical Properties of Zinc Oxide Nanowires
Semiconductor materials characterisation for high temperature applications
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 (ZnO-NWs) of different diameters at high temperatures. The higher order elastic constants of ZnO-NWs 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 ZnO-NWs were also determined. The ultrasonic attenuation due to the phonon-viscosity mechanism is predominant for the total ultrasonic attenuation for ZnO-NWs. The correlation among the ultrasonic parameters, thermal conductivity and size of ZnO-NWs is established leading to potential industrial applications.
Semiconductor oxides, especially zinc oxide nanowires (ZnO-NWs), 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 ZnO-NWs 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 Zn2+ and O2– ions stacked in the <001> direction (3). Figure 1(b) shows the cross-sectional view of ZnO-NWs 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 C6ν4 space group and in the Hermann-Mauguin notation it is P63mc. The size dependent mechanical properties, especially stress distribution, of ZnO-NWs are important for many practical applications.
The Young’s modulus enhancement in atmosphere has been investigated using a core-shell model to compare with the model in vacuum (5). Improved mechanical behaviour of ZnO-NWs was obtained through embedding various nanowires of different materials with various cross-sectional models (6). Another study reveals that Young’s modulus and fracture strain of ZnO-NWs varies with diameter. The electric-field induced resonance method showed that Young’s modulus increases with decreasing diameter (7). Pan et al. (8) synthesised single ZnO-NWs with thermal evaporation and measured the mechanical properties by the micromanipulator nanoprobe system installed in the scanning electron microscopy (SEM) chamber. Different non-linear 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 ZnO-NWs 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 non-destructive 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 electron-phonon interactions or phonon-phonon 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 ZnO-NWs 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.
2.1 Theoretical Approach for the Computation of SOECs and TOECs
The chosen material for study i.e. ZnO-NWs has a hexagonal close-packed (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 ζKL(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 p0, q0, m, n and is given as Equation (ii) (18):
where p0 and q0 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 (CKL) and TOECs (CKLM) 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 = nb0 (n – m)/8an+4, Ψ = –χ/6a2(m + n + 6) represent the lattice parameters, the axial ratio, the harmonic and anharmonic parameters respectively. Also C′ = χa/p5; C″ = ψa3/p3 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 Lennard-Jones parameter (bo) for ZnO-NWs.
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 ZnO-NWs. The ultrasonic velocity helps to determine the anisotropic behaviour of the material. Thus, a brief discussion of the ultrasonic wave propagation in wurtzite ZnO-NWs is necessary. The ultrasonic wave velocity in ZnO-NWs 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 (VL) and two shear wave velocities (i.e. quasi-shear (VS1) and shear (VS2)) 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 phonon-phonon 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, E0, V represent the thermal relaxation time, thermal energy density and the longitudinal or shear modes ultrasonic velocity respectively. The acoustic coupling constant D = 3ΔC/E0 gives the measure of energy transformation from acoustical to thermal, where the applied strain causes change in elastic modulus ΔC, which depends on E0, Grüneisen number (γ ij) and specific heat per unit volume (CV). The deviation in elastic modulus ΔC is given by Equation (xi):
where Grünesien number γ ij for ZnO-NWs 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 ij 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/f2)Akh.Long, (α/f2)Akh.Shear with respect to square of frequency for longitudinal and shear wave due to Akhiezer type loss. The re-establishment time for thermal phonons termed as thermal relaxation time (τ) is given as Equation (xiv):
Here, VD shows the Debye average velocity, τl and τs represent relaxation time for the longitudinal wave and shear wave respectively. The Debye average velocity (VD) is related to the elastic constants via ultrasonic velocities and is defined as Equation (xv) (16, 26):
where VL is the longitudinal wave velocity, VS1 is the quasi-shear wave velocity and VS2 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 ZnO-NWs (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 Lennard-Jones parameter (bo) 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 Lennard-Jones parameter (bo) under equilibrium condition is 2.095 × 10–64 erg cm7. 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 ZnO-NWs 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.
|Temperature, K||C222, GPa||C333, GPa||Y, GPa||B, GPa||G, GPa||ν||–||–|
Higher values of C11 and C33 indicate that for the chosen material, the compressibility of the c-axis is higher than that of the a-axis. The large value of the elastic constant C44 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 ZnO-NWs 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 ZnO-NWs. 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 ZnO-NWs 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 exchange-correlation functional form proposed by Perdew, Burke and Ernzerhof (PAW-PBE) (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 VD maxima occur at 45° with unique axis of the crystal at room temperature. Since VD depends on VL, VS1, VS2 and at 45° significant growth in the value of longitudinal, shear wave velocity is observed while the quasi-shear wave velocity decreases. Hence, the change in angle of VD is affected by the respective ultrasonic velocities. It shows that when an acoustic wave propagates at 45° with a unique ZnO-NWs axis then the average acoustic velocity is maximum. The longitudinal velocity reported in the literature (34) of 6.356 × 103 m s–1 is in good agreement with our computed value of 6.081 × 103 m s–1. The magnitude of shear wave velocity reported in the literature (35) is 2.735 × 103 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 ZnO-NWs is slightly higher than that in indium nitride nanowires (InN-NWs) (37), which leads to better elastic and mechanical behaviour in comparison. Hence our approach to compute the ultrasonic velocity is justified.
Table III depicts the values of ultrasonic velocities, acoustic coupling constants (DL, DS), thermal energy density (E0) and specific heat per unit volume (CV) within the temperature range 100–300 K. CV and E0 were determined using the θD/T tables in the “American Institute of Physics Handbook” (38). It is obvious from Table III that DL > DS, 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.
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 ZnO-NWs is given by the polynomial Σ2i = 0Pidi. 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 ZnO-NWs 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 VD. Thus, if the ultrasonic wave is propagating along a unique axis at 45° the re-establishment 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.
The diameter dependent ultrasonic attenuation (α/f2)Akh for longitudinal and shear waves and the thermoelastic loss (α/f2)Th of ZnO-NWs 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 ZnO-NWs, hence ultrasonic attenuation is found to increase with temperature and diameter. The Akhiezer loss (α/f2)Akh in ZnO-NWs 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 ZnO-NWs was found to be increasing with temperature and size of the nanowire. Therefore, the phonon-viscosity mechanism is the major cause of ultrasonic attenuation for ZnO-NWs 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 ZnO-NWs, the ultrasonic attenuation also increases. Thus, ZnO-NWs of large diameter have high κ and large attenuation. This characteristic of ZnO-NWs 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 ZnO-NWs at different diameters, evaluated using Equation (xiii). It is clear from Figure 4 that the total ultrasonic attenuation (α/f2)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 ZnO-NWs. Since the SOECs, TOECs and ultrasonic velocities show very minor variation with the diameter of ZnO-NWs, the elastic and similar parameters other than τ do not affect (α/f2)Total with variations in diameter of the nanowire. As shown in Figure 1(b), the cross-sectional view of ZnO-NWs 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 (DL) and ultrasonic attenuation (α/f2)Akh.Long for a longitudinal wave is larger than that of a shear wave at any frequency. As the diameter of ZnO-NWs increased from 40 nm to 180 nm, the increase in thermal conduction leads to higher phonon-phonon interaction, thus (α/f2)Total i.e. the total ultrasonic attenuation characteristics indicate the semiconducting nature of the nanowire.
We have computed nonlinear elastic properties as well as ultrasonic properties for wurtzite structured ZnO-NWs using the Lennard-Jones 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 ZnO-NWs. 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 ZnO-NWs. 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 ZnO-NWs with relation to other well-known properties might be used in characterisation and in industrial applications.
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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 non-destructive 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).