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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.

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 [

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 [

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 [

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.

The interaction potential called Lennard-Jones potential,

where; a_{0}, b_{0} are constant scientific parameters; m & n are integers and r is distance between atoms [

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 _{0} is the fitting parameter

determined under the equilibrium condition for minimum system energy [

Developing the interaction potential model second and third order elastic constants (SOECs and TOECs) can be calculated by the equations [

where p = c/a is called axial ratio;^{ }

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 [

where _{ }and

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 (a)_{Akh} (Akhieser type loss) due to phonon-phonon interaction mechanism is given by following expression [

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 [_{V} is the specific heat per unit volume of the material [

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 _{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 [

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

For InAs NWs, the basal plane distance (a), axial ratio (p) and density (r) are 4.284 Å, 1.633 and 5667 kg∙m^{−3 }respectively [

(

(_{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 _{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 [

The ultrasonic attenuation coefficients for all sized nanowires and their variation with size are shown in

SOECs | C_{11} | C_{33} | C_{44} | C_{66} | C_{12} | C_{13} | ||||
---|---|---|---|---|---|---|---|---|---|---|

10.65 | 11.07 | 2.74 | 4.01 | 2.61 | 2.29 | |||||

10.03^{*} | 11.07* | 2.30^{*} | 2.90^{*} | 4.22^{*} | 3.18^{*} | |||||

TOECs | C_{111} | C_{222} | C_{333} | C_{112} | C_{113} | C_{123} | C_{133} | C_{144} | C_{155} | C_{344} |

−173.61 | ?137.36 | −141.43 | −27.53 | −5.87 | −7.45 | −37.39 | −8.69 | −5.79 | −35.05 |

E_{0} (10^{8} J∙m^{−3}) | C_{V} (10^{5} J∙K^{−1}∙m^{−3}) | ^{3} m∙s^{−1}) | ^{3}m∙s^{−1}) | ^{3}m∙s^{−1}) | D_{L} | D_{S} |
---|---|---|---|---|---|---|

1.68 | 7.29 | 4.42 | 2.20 | 2.44 | 10.38 | 0.47 |

d (nm) | K (W∙m^{−1}∙K^{−1}) | t (10^{−12} s) | (a/f^{2})_{Th} (10^{−18} Np∙s^{2}∙m^{−1}) | (a/f^{2})_{Akh.Long} (10^{−18} Np∙s^{2}∙m^{−1}) | (a/f^{2})_{Akh.Shear} (10^{−18} Np∙s^{2}∙m^{−1}) | (a/f^{2})_{Total} (×10^{−18} Np∙s^{2}∙m^{−1}) |
---|---|---|---|---|---|---|

2.57 | 4.21 | 2.91 | 0.17 | 273.11 | 50.20 | 323.48 |

3.42 | 3.91 | 2.71 | 0.16 | 253.65 | 46.62 | 300.43 |

4.28 | 3.79 | 2.62 | 0.15 | 245.86 | 45.19 | 291.20 |

5.13 | 3.84 | 2.66 | 0.16 | 249.11 | 45.79 | 295.06 |

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 [

It is clear from

Thus on the basis of experimental/calculated data and perusal of

Also, activation energy/crystallinity of nanosized hexagonal structural material decreases/increases with size [

However the crystallinity increases beyond 4.5 nm diameter. The size dependency and order of t approve the semiconducting nature of InAs NWs. [

Thus ultrasonic properties are well correlated with the structure based materials properties depending upon the diameter of the NWs.

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.

The authors are thankful to the UGC New Delhi, India for the financial support.

MohitGupta,Punit K.Dhawan,Raja RamYadav,Satyendra KumarVerma, (2015) Diameter Dependent Ultrasonic Characterization of InAs Semiconductor Nanowires. Open Journal of Acoustics,05,218-225. doi: 10.4236/oja.2015.54017