Diameter Dependent Ultrasonic Characterization of InAs Semiconductor Nanowires

In this paper, we report the diameter dependent ultrasonic characterization of wurtzite structured InAs semiconductor nanowires at the room temperature. In this work, we have calculated the non-linear higher order elastic constants of InAs nanowires validating the interaction potential model. The ultrasonic attenuation and velocity in the nanowires are determined using the elastic constants for different diameters of the nanowires. Where possible, the results are compared with the experiments. Finally, we have established the correlation between the size dependent thermal conductivity and the ultrasonic attenuation of the nanowires.


Introduction
The small band gap of semiconducting Indium Arsenide nanowires (InAs NWs) is 0.35 eV and its electron mobility is high due to its small electron effective mass.InAs NWs have the highest thermoelectric figure of merit (ZTs) of all III-V NWs.Due to all these properties, InAs NWs have attracted a tremendous amount of interest and also have been a hot topic in recent years.Much of it is due to their potential applications in various fields such as biosensors [1] [2], field-effect transistors [3]- [6], light-emitting diodes [7], diode lasers and infrared detectors [8]- [10], logic gates [11], photo detectors [12] [13], solar cells [14], ultrahigh-density memory and logic devices [5] and thermoelectric devices.Additionally, heterostructures in one-dimensional geometries such as NWs have been to act as high performance function building blocks in devices.
A special feature is that it is possible to grow InAs NWs with hexagonal wurtzite (WZ) structure, which is non-existing in the corresponding bulk materials where the cubic zinc blende (ZB) structural modification prevails.Thus InAs NWs exhibits both a WZ phase as well as a ZB phase while InAs crystals typically exhibit ZB phase [15] [16].For InAs NWs, in the WZ phase, each atom is tetrahedrally coordinated with ABAB stacking sequence, which is similar to the ZB phase.The nearest neighbor arrangement for each atom in the WZ phase is the same as that for the ZB phase, which is achieved when the c/a is 1.633.In fact, the <111> direction for the ZB phase is equivalent to the <001> direction of the WZ phase [17].
The previous studies concerning crystal structure also suggest that the InAs NWs with the smallest diameters have the WZ phase while the NWs with larger diameters have the ZB phase.For the intermediate diameters, the NWs consist of alternating segments of the two crystal phases [18].However, the growth of WZ phase InAs NWs has been demonstrated using chemical beam epitaxy (CBE) [19] [20] and other techniques.The synthesis of these WZ phase InAs NWs resulted in extensive interest within the research fields to determine how their electrical and thermal transport properties are compared to those of ZB phase InAs NWs.However, while a significant amount of theoretical and experimental work has been focused on investigating mechanical, thermal, electrical and optical properties of InAs NWs in their WZ phase but none of the work reported in the literature so far is focused on the ultrasonic characterization of the InAs NWs.The ultrasonic evaluation method is very simple, non-radiative and non-destructive characterization method to predict the microstructural behavior and the thermophysical properties.
In view of this background, in the present work, we have calculated second and third order elastic constants, ultrasonic attenuation, and ultrasonic velocity along with related non-linear parameters in InAs NWs at 300 K aiming to the nondestructive characterization of the materials and establishment of the theories for the calculation.

Higher-Order Elastic Constants
The interaction potential called Lennard-Jones potential, ( ) r ϕ , used to calculate the higher-order elastic con- stants of WZ structured materials is given as: where; a 0 , b 0 are constant scientific parameters; m & n are integers and r is distance between atoms [21].The meaning may be understood as follows: ( ) where η is the lagrangian strain, M is the mass of the atom and D = a 0 is the nearest neighbour distance in the basal plane.K 2 is the harmonic parameter which is related to η as ( ) Further, b 0 is the fitting parameter determined under the equilibrium condition for minimum system energy [22].
Developing the interaction potential model second and third order elastic constants (SOECs and TOECs) can be calculated by the equations [23] [24] as: where p = c/a is called axial ratio; ; and c is height of the unit cell.The harmonic and anharmonic parameters (χ and ψ) are calculated using one reasonable value of SOECs.The basal plane distance, axial ratio collectively called lattice parameters [25] are a = 4.284 Å and p = 1.633.

Ultrasonic Velocities
The anisotropic properties of a material are related to its ultrasonic velocities as they are related to higher-order elastic constants.If ultrasonic wave is propagating along the length of the NWs then there are two types of ultrasonic velocities: one longitudinal and other shear wave velocities [26] [27] which are given by following equations: ( ) ( ) where L v and S v are the longitudinal and shear wave velocities while ρ is the mass density of the material.

Ultrasonic Attenuation and Allied Parameters
The main causes for the ultrasonic attenuation in solid are electron-phonon interaction, phonon-phonon interaction, grain boundary loss or scattering loss, Bardoni relaxational loss and thermoelastic loss.The electron mean free path is not comparable to phonon wavelength at high temperature; therefore attenuation due to electron-phonon interaction will be absent.Scattering loss is prominent for polycrystalline material and it has no role in case of single crystals.Bardoni relaxational loss has been found to be effective at low temperature for metals.So, two dominant processes responsible for the ultrasonic attenuation at high temperature are phonon-phonon interaction also known as Akhieser loss and thermo-elastic attenuation.The ultrasonic attenuation coefficient (α) Akh (Akhieser type loss) due to phonon-phonon interaction mechanism is given by following expression [27]: where f is the frequency of the ultrasonic wave; V is the velocity for longitudinal and shear waves as defined in the set of Equations ( 3); E 0 is the thermal energy density [28] and C V is the specific heat per unit volume of the material [28]; T is the temperature and j i γ is the Grüneisen number; i and j are the mode and direction of the propagation.The Grüneisen number for a hexagonal WZ structured crystal along the <001> orientation is a direct consequence of the SOECs and TOECs.The acoustic coupling constant "D" is the measure of the acoustic energy converted into thermal energy given by the equation as: ( ) when an ultrasonic wave propagates through a crystalline material, the equilibrium of phonon distribution is disturbed.The time taken for re-establishment of equilibrium of the thermal phonons is called the thermal relaxation time "τ" and it is given as: where L τ is the thermal relaxation time for the longitudinal wave; S τ is the thermal relaxation time for the shear wave; and K is the thermal conductivity [29]; V D is the Debye average velocity calculated by the equation as: The propagation of the longitudinal ultrasonic wave results in the thermoelastic loss "(α) Th " due to creation of compressions and rarefactions throughout the lattice and it is calculated by the equation [27] as: The thermoelastic loss for the shear wave has no physical significance because the average of the Grüneisen number for each mode and direction of propagation is equal to zero for the shear wave.Only the longitudinal wave is responsible for thermoelastic loss because it causes variation in entropy along the direction of propagation.The total attenuation is given as: where ( )

Results and Discussion
For InAs NWs, the basal plane distance (a), axial ratio (p) and density (ρ) are 4.284 Å, 1.633 and 5667 kg•m −3 respectively [25].The calculated values of elastic constants using Equation ( 2) in present work are shown in Table 1.The bulk modulus ( ( ) ), shear modulus ( ( ) ) of the chosen material are also determined using the SOECs are respectively 51.94 GPa, 35.75 GPa, 87.24 GPa and 0.35.The calculated values of SOECs and the values obtained from the literature [25] are compared in Table 1.Thus our calculations of SOECs/TOECs based on only lattice parameters at 300 K are validated.The calculated values of longitudinal, shear and Debye average velocities of ultrasonic wave propagating along the unique axis of the InAs NWs using Equations ( 3), (4), and ( 8) and the calculated values of thermal energy density "E 0 ", specific heat per unit volume "C V ", acoustic coupling constants (D L and D S ) related with higher order elastic constants through Grüneisen numbers which are functions of SOECs and TOECs using Equation ( 6) are shown in Table 2.It is also clear from Table 2 that D L > D S .This implies that the conversion of acoustical energy to thermal energy will be large for wave propagating along the length of wire than the surface wave.
Thermal conductivities of InAs NWs at different diameters (d) taken from the literature [30] are given in Table 3.The size (diameter) dependent thermal relaxation time and ultrasonic attenuation coefficients over frequency square under the condition 1 ωτ  are evaluated using Equation (7) and Equations ( 5), (9), and (10) respec- tively.They are also listed in Table 3.The plots of ( ) versus diameter of NWs is shown in Figure 1.
The plot of τ and K versus diameter of NWs is shown in Figure 2.
The ultrasonic attenuation coefficients for all sized nanowires and their variation with size are shown in Figure 1.The dominant mechanism for total ultrasonic attenuation is p − p interaction.The attenuation due to p − p interaction is directly proportional to the acoustic coupling constant and thermal relaxation time.The ultrasonics velocity is directly related to second order elastic constants while they are related to lattice parameter.The slight Table 2. Thermal energy density (E 0 : in 10 8 J•m −3 ), specific heat per unit volume (C V : in 10 5 J•K −1 •m −3 ), ultrasonic velocities ( L v , S v and D v : in 10 3 m•s −1 ), and acoustic coupling constants (D L and D S ) for InAs NWs at 300 K.      change in velocities, E 0 and C v for the InAs NW is due to slight change in lattice parameter.In comparison to bulk material the thermal conductivity is lowered at room temperature in the NWs at all diameters.Accordingly the thermal relaxation time is also lowered.It means that the time taken is less for regaining the thermal equilibrium after interaction of the ultrasonic wave with the NWs.Consequently the ultrasonic attenuation in the NWs at frequency ~200 MHz is smaller in comparison to bulk materials [30] [31].
It is clear from Figure 2 that the variation of the total ultrasonic attenuation coefficient over frequency squares with the diameter of NWs. Figure 1 follows the same nature as the thermal conductivity or thermal relaxation time.Furthermore, loss of ultrasonic energy for longitudinal wave due to Akhieser damping (phononphonon interaction mechanism) is larger than that of shear and other Table 3.
Thus on the basis of experimental/calculated data and perusal of Figure 2 following the above analysis.It is clear that the NWs diameter dependence of the total ultrasonic attenuation is predominantly affected by the unusual low thermal conduction of the NWs.The diameter dependent ultrasonic attenuation curve follows the same pattern as the diameter dependence of thermal conductivity curve at 300 K.
Also, activation energy/crystallinity of nanosized hexagonal structural material decreases/increases with size [32].The activation energy is well related to resistivity while resistivity is connected with thermal conductivity.
Hence decrease in τ with size (diameter) of NWs is attributed to the decrease in crystallinity and increase in activation energy.
However the crystallinity increases beyond 4.5 nm diameter.The size dependency and order of τ approve the semiconducting nature of InAs NWs.[33].There is deviation from the decreasing behavior of ultrasonic attenuation after 4.5 nm diameter.Thus above the 4.5 nm diameter InAs NW may tend towards its bulk material property.
Thus ultrasonic properties are well correlated with the structure based materials properties depending upon the diameter of the NWs.

Conclusion
The simple interaction potential model for the calculations of second and third order elastic constants is validated for the NWs nanowires of different diameters.Theoretical approach for the determination of ultrasonic attenuation in NWs at 300 K with different diameters is established.There is a strong correlation between size dependent ultrasonic attenuation and size dependent thermal conductivity of the NWs.Thus an ultrasonic attenuation mechanism is established to extract the important information about the microstructural phenomena like p − p interaction in the NWs and thermal conductivity behavior with respect to diameters of NWs at 300 K.
attenuation coefficients due to Akhieser loss for the longitudinal and shear waves respectively.

Figure 1 .
Figure 1.Ultrasonic attenuation of InAs NWs versus diameter at 300 K.

Figure 2 .
Figure 2. Dependence of thermal conductivity and thermal relaxation time on diameter of InAs NW.

Table 3 .
Size (diameter) dependent thermal conductivity (K), thermal relaxation time (τ) and (α/f 2 ) for longitudinal and shear wave due phonon-phonon interaction and thermoelastic relaxation mechanisms for InAs NWs at 300 K.