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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 15 — Jul. 16, 2012
  • pp: 16611–16617
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Femtosecond excitation of radial breathing mode in 2-D arrayed GaN nanorods

Hung-Pin Chen, Yueh-Chun Wu, Pierre Adrien Mante, Shang-Ju Tu, Jinn-Kong Sheu, and Chi-Kuang Sun  »View Author Affiliations


Optics Express, Vol. 20, Issue 15, pp. 16611-16617 (2012)
http://dx.doi.org/10.1364/OE.20.016611


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Abstract

Radial breathing oscillation of 2-D arrayed GaN nanorods was successfully excited in rods with different diameters by using femtosecond transient reflectivity measurement. Through analyzing thus measured diameter dependent oscillation frequency, we discovered that modification of the mechanical property appeared in the 2-D arrayed piezoelectric GaN nanorods, fabricated on top of a bulk substrate, when the rod diameter was on the order of or less than 50 nm. Our measurement observed a much reduced elastic stiffness constant (C11) of 193 ± 24 GPa in 35nm diameter nanorods, compared with the 365 ± 2 GPa in bulk GaN. This size-reduction induced mechanical modification would be a critical factor to be considered for future sensing and energy applications. Our study also provides a new spectroscopic method to explore the size-reduction-induced softening effect through the measurement of the radial breathing oscillations.

© 2012 OSA

1. Introduction

2. Sample preparation

The scanning electron microscopy (SEM) images of the fabricated 2-D arrayed GaN nanorods are shown in Fig. 1
Fig. 1 Top SEM views of the fabricated 2D-arrayed GaN nanorods on GaN substrates. The average rod diameters are (a) 324nm, (b) 250nm, (c) 230nm, (d) 183nm, (e) 135nm, (f) 52nm, (g) 40nm, (h) 35nm. (i) The side view SEM image of (e).
. In order to pattern the nano-diameter nanorods with a short-period and a high-density, E-beam lithography, thermal evaporation, and inductive coupled plasma reactive ion etching (ICP RIE) were utilized. Metal masks were used for RIE to achieve a high AR of rods. First e-beam lithography resist (Zep-520A) was coated on top of a 3μm-thick GaN film and the thickness of resist was 120nm. The 2-D arrayed pattern was fabricated on the resist by the e-beam lithography. The sample was then loaded into an evaporation chamber for the deposition of 30-nm metal film and the subsequent lift-off process of metal was performed by dimethylsulfoxide. Next, we used the ICP RIE system to dry etch the 2-D arrayed GaN nanorods. The etching gas and RF power were Cl2/BCl3 mixture and 200W, respectively [17

17. C.-C. Yu, C.-F. Chu, J.-Y. Tsai, H.-W. Huang, T.-H. Hsueh, C.-F. Lin, and S.-C. Wang, “Gallium nitride nanorods fabricated by inductively coupled plasma reactive ion etching,” Jpn. J. Appl. Phys. 41(Part 2, No. 8B), L910–L912 (2002). [CrossRef]

]. After the ICP RIE process, the residual metal mask on top was removed with the selective chemical wet etching and potassium hydroxide (KOH) was utilized to decrease the defect density on the side walls of fabricated rods [18

18. D. Zhaung and J. H. Edgar, “Wet etching of GaN, AlN, and SiC: a review,” Mater. Sci. Eng. Rep. 48(1), 1–46 (2005). [CrossRef]

]. Finally, the geometry of 2-D arrayed GaN nanorods was observed by the field emission SEM (FEI Nova 600i). The diameters of the fabricated GaN nanorods were from 35 nm to 350nm nm and the heights were between 100nm to 1080nm.

3. Experimental details

For the femtosecond transient reflectivity measurement, a degenerate femtosecond pump-probe system was adopted with a Ti:sapphire laser. The frequency-doubled 360-nm UV femtosecond laser pulse was generated by a beta barium-borate (BBO) crystal and was divided into a pump beam and a probe beam. The average powers of the pump and probe beams were 30mW and 5mW, respectively. Finally, the pump and probe light were incident into the 2-D arrayed GaN nanorods with the same aspheric lens and the reflectivity of the probe light was detected by a biased Si detector. A pump-probe spot size of 20μm diameter (full width at half-maximum intensity) can be estimated at the focal plane by measuring the ratio of the focused pump and probe powers through a 10μm-diameter pin-hole. The spot size covered more than forty nanorod periods in its diameter.

4. Results and Discussion

The crystal class of the metal-organic-chemical-vapor-deposition-grown GaN film is of a hexagonal wurtzite structure with a reported elastic stiffness constant (C11) of 365 ± 2 GPa [19

19. M. Yamaguchi, T. Yagi, T. Azuhata, T. Sota, K. Suzuki, S. Chichibu, and S. Nakamura, “Brillouin scattering study of gallium nitride: elastic stiffness constants,” J. Phys. Condens. Matter 9(1), 241–248 (1997). [CrossRef]

] while the high quality GaN film was grown on a c-plane sapphire substrate. Based on the hexagonal structure, the sound velocity can be deduced from the slowness of curve with the density, stiffness constant and Poisson ratio of the well-known GaN crystal. The frequency f of the radial breathing mode can then be predicted by [16

16. M. Hu, X. Wang, G. V. Hartland, P. Mulvaney, J. P. Juste, and J. E. Sader, “Vibrational response of nanorods to ultrafast laser induced heating: theoretical and experimental analysis,” J. Am. Chem. Soc. 125(48), 14925–14933 (2003). [CrossRef] [PubMed]

]
f=τnvrπd,
(1)
where vr is the longitudinal sound velocity along the radial direction and d is the diameter of nanorod. The dimensionless τn constant could be calculated from an eigenvalue equation dependent on Poisson’s ratio of bulk GaN. For 2-D arrayed GaN nanorods, radial breathing mode frequencies would be expected to be inversely proportional to diameter if sound velocity and Poisson’s ratio keep the same values as the bulk GaN.

One example trace of the experimentally measured transient reflection change on the 2-D arrayed nanorod sample with a 135 nm diameter was shown in Fig. 2(a)
Fig. 2 (a) Measured probe reflection change as a function of time delay. The sample is with a 135nm diameter and a 220nm period. The inset scheme shows the geometry of a nanorod. D, d1, d2 and L are average diameter, top diameter, bottom diameter and height of a nanorod, determined by top and side SEM views. AR: aspect ratio. (b) An oscillation frequency of 33GHz was revealed in the fast-Fourier-transformed spectrum. The inset figure illustrates the observed oscillatory signal after removing the carrier dynamics background of (a).
. On top of the carrier-dynamics background, an oscillatory signal was observed to last for 500ps after the excitation of the pump light (inset of Fig. 2(b)). Through the Fourier transform analysis, the observed 33 GHz central frequency is in agreement with the prediction of the radial breathing mode model shown in Eq. (1). The oscillation bandwidth was 5 GHz, reflecting a slow damping process.

The periods and aspect ratios of 2-D arrayed GaN nanorods could be designed to be different to study the possible frequency shift contributed from surface acoustics effects [20

20. J.-F. Robillard, A. Devos, I. Roch-Jeune, and P. A. Mante, “Collective acoustic modes in various two-dimensional crystals by ultrafast acoustics: Theory and experiment,” Phys. Rev. B 78(6), 064302 (2008). [CrossRef]

]. We first studied the effect of rod periodicity on the observed oscillatory frequency for 150 ± 10 nm-diameter and 240 ± 10 nm-diameter GaN nanorods. As shown in Fig. 3(a)
Fig. 3 (a) The observed oscillatory frequency as a function of rod periodicity for 150 ± 10 nm-diameter (black solid squares) and 240 ± 10 nm-diameter (red solid triangles) GaN nanorods. (b) The observed oscillatory frequency as a function of aspect ratio for 240 ± 10 nm diameter GaN nanorods. The observed frequency is 21 ± 1.5GHz while the aspect ratio is higher than 2.
, our study indicates that the observed oscillatory frequency does not have a significant dependency on the periodicity within our studied range. We also investigated the aspect ratio (AR) dependency. The result for 240 ± 10 nm diameter GaN nanorods is shown in Fig. 3(b). The shift of the observed central frequency would be less than 1 GHz as the AR of rod was greater than 2. It is important to notice that with a 600ps scanning period in the transient measurement, our corresponding frequency resolution was only 1.8 GHz. In order to reduce the effect of the frequency shift due to different AR, in the following diameter-dependency study we selected only the samples with AR greater than 2.

Figure 4(a)
Fig. 4 (a) The observed oscillatory frequency versus the diameter of nanorod. Rod diameter ranges from 324nm to 35 nm. The horizontal error bars represent the diameter nonuniformity based on the SEM images and the vertical error bars represents full-width half-maximum of the amplitude in Fourier-transformed oscillatory spectra. The black curve is the result of the theoretical calculation on the radial breathing mode based on Eq. (1) with vr = 7744 m/s and τn = 2.05. (b) Elastic stiffness constant (C11) as a function of GaN rod diameter. The black dashed line represents the bulk C11 value.
summarizes the observed oscillatory frequency as a function of rod diameter. For nanorods with a diameter ranging between 135nm and 324nm, our observed central oscillatory frequency agrees well with the characteristic frequency of radial breathing mode following Eq. (1) while the vr of bulk GaN and τn are taken as 7744 m/s and 2.05 [19

19. M. Yamaguchi, T. Yagi, T. Azuhata, T. Sota, K. Suzuki, S. Chichibu, and S. Nakamura, “Brillouin scattering study of gallium nitride: elastic stiffness constants,” J. Phys. Condens. Matter 9(1), 241–248 (1997). [CrossRef]

,21

21. I. Vurgaftman and J. R. Meyer, “Band parameters for nitrogen-containing semiconductors,” J. Appl. Phys. 94(6), 3675–3696 (2003). [CrossRef]

,22

22. A. Polian, M. Grimsditch, and I. Grzegory, “Elastic constants of gallium nitride,” J. Appl. Phys. 79(6), 3343–3344 (1996). [CrossRef]

], respectively. We assumed that τn was fixed for different rod diameters due to similar excitation and boundary conditions. The excellent agreement between the measured diameter-dependent oscillation frequency and Eq. (1) confirms our suggestion that the observed oscillatory signals are contributed from the radial breathing oscillations of the fabricated nanorods. When the diameters of the 2D arrayed GaN nanorods were below 50 nm, the vibrational frequency of the observed breathing mode was found to be lower than the theoretical prediction. With τn as a dimensionless constant in Eq. (1), the relatively low breathing mode frequency would indicate a lower sound velocity in nanorods with a diameter less than 50nm. Therefore, the elastic stiffness constant (C11) can then be deduced from the measured results with the formula, vr = (c11/ρ)1/2 where ρ is the density of GaN. Thus derived C11 elastic stiffness constant of nanorod as a function of rod diameter is shown in Fig. 4(b). We can notice that the C11 elastic constant is significantly lower for GaN nanorods with a diameter smaller than 50 nm. With a 35nm rod diameter, our experimental method found that the C11 value decreases by 48% in comparison with the bulk counterpart.

4. Conclusion

In summary, we not only successfully excited and observed the radial breathing mode in 2D-arrayed GaN nanorods by femtosecond transient reflectivity spectroscopy, but also found that a slow sound velocity existed for rods with a diameter smaller than 50nm. The corresponding elastic stiffness constant of the 2D arrayed GaN nanorods was acquired in our measurement and was found to be much reduced. Our observation further implied that the observed softening effect appeared in the radial direction of rods and could be a critical parameter modification to be considered in the development of future nanoelectromechanical system.

Acknowledgments

References and links

1.

K.-Q. Peng and S.-T. Lee, “Silicon Nanowires for Photovoltaic Solar Energy Conversion,” Adv. Mater. (Deerfield Beach Fla.) 23(2), 198–215 (2011). [CrossRef] [PubMed]

2.

R. Yan, D. Gargas, and P. Yang, “Nanowire photonics,” Nat. Photonics 3(10), 569–576 (2009). [CrossRef]

3.

C.-T. Huang, J. Song, W. F. Lee, Y. Ding, Z. Gao, Y. Hao, L. J. Chen, and Z. L. Wang, “GaN nanowire arrays for high-output Nanogenerators,” J. Am. Chem. Soc. 132(13), 4766–4771 (2010). [CrossRef] [PubMed]

4.

R. Agrawal, B. Peng, and H. D. Espinosa, “Experimental-computational investigation of ZnO nanowires strength and fracture,” Nano Lett. 9(12), 4177–4183 (2009). [CrossRef] [PubMed]

5.

J. Heo, W. Guo, and P. Bhattacharya, “Monolithic single GaN nanowire laser with photonic crystal microcavity on silicon,” Appl. Phys. Lett. 98(2), 021110 (2011). [CrossRef]

6.

Y. Chen, I. Stevenson, R. Pouy, L. Wang, D. N. McIlroy, T. Pounds, M. G. Norton, and D. E. Aston, “Mechanical elasticity of vapour-liquid-solid grown GaN nanowires,” Nanotechnology 18(13), 135708 (2007). [CrossRef] [PubMed]

7.

P. Yu, C. H. Chiu, Y.-R. Wu, H. H. Yen, J. R. Chen, C. C. Kao, H. W. Yang, H. C. Kuo, T. C. Lu, W. Y. Yeh, and S. C. Wang, “Strain relaxation induced microphotoluminescence characteristics of a single InGaN-based nanopillar fabricated by focused ion beam milling,” Appl. Phys. Lett. 93(8), 081110 (2008). [CrossRef]

8.

R. A. Bernal, R. Agrawal, B. Peng, K. A. Bertness, N. A. Sanford, A. V. Davydov, and H. D. Espinosa, “Effect of growth orientation and diameter on the elasticity of GaN nanowires. A combined in situ TEM and atomistic modeling investigation,” Nano Lett. 11(2), 548–555 (2011). [CrossRef] [PubMed]

9.

J. J. Brown, A. I. Baca, K. A. Bertness, D. A. Dikin, R. S. Ruoff, and V. M. Bright, “Tensile measurement of single crystal gallium nitride nanowires on MEMS test stages,” Sens. Actuators A Phys. 166(2), 177–186 (2011). [CrossRef]

10.

C.-Y. Nam, P. Jaroenapibal, D. Tham, D. E. Luzzi, S. Evoy, and J. E. Fischer, “Diameter-dependent electromechanical properties of GaN nanowires,” Nano Lett. 6(2), 153–158 (2006). [CrossRef] [PubMed]

11.

T. Henry, K. Kim, Z. Ren, C. Yerino, J. Han, and H. X. Tang, “Directed growth of horizontally aligned gallium nitride nanowires for nanoelectromechanical resonator arrays,” Nano Lett. 7(11), 3315–3319 (2007). [CrossRef] [PubMed]

12.

S. O. Mariager, D. Khakhulin, H. T. Lemke, K. S. Kjaer, L. Guerin, L. Nuccio, C. B. Sørensen, M. M. Nielsen, and R. Feidenhans’l, “Direct observation of acoustic oscillations in InAs nanowires,” Nano Lett. 10(7), 2461–2465 (2010). [CrossRef] [PubMed]

13.

H. Lange, M. Mohr, M. Artemyev, U. Woggon, and C. Thomsen, “Direct observation of the radial breathing mode in CdSe Nanorods,” Nano Lett. 8(12), 4614–4617 (2008). [CrossRef] [PubMed]

14.

C. Guillon, P. Langot, N. Del Fatti, F. Vallée, A. S. Kirakosyan, T. V. Shahbazyan, T. Cardinal, and M. Treguer, “Coherent acoustic vibration of metal nanoshells,” Nano Lett. 7(1), 138–142 (2007). [CrossRef] [PubMed]

15.

A. Amziane, L. Belliard, F. Decremps, and B. Perrin, “Ultrafast acoustic resonance spectroscopy of gold nanostructures: Towards a generation of tunable transverse waves,” Phys. Rev. B 83(1), 014102 (2011). [CrossRef]

16.

M. Hu, X. Wang, G. V. Hartland, P. Mulvaney, J. P. Juste, and J. E. Sader, “Vibrational response of nanorods to ultrafast laser induced heating: theoretical and experimental analysis,” J. Am. Chem. Soc. 125(48), 14925–14933 (2003). [CrossRef] [PubMed]

17.

C.-C. Yu, C.-F. Chu, J.-Y. Tsai, H.-W. Huang, T.-H. Hsueh, C.-F. Lin, and S.-C. Wang, “Gallium nitride nanorods fabricated by inductively coupled plasma reactive ion etching,” Jpn. J. Appl. Phys. 41(Part 2, No. 8B), L910–L912 (2002). [CrossRef]

18.

D. Zhaung and J. H. Edgar, “Wet etching of GaN, AlN, and SiC: a review,” Mater. Sci. Eng. Rep. 48(1), 1–46 (2005). [CrossRef]

19.

M. Yamaguchi, T. Yagi, T. Azuhata, T. Sota, K. Suzuki, S. Chichibu, and S. Nakamura, “Brillouin scattering study of gallium nitride: elastic stiffness constants,” J. Phys. Condens. Matter 9(1), 241–248 (1997). [CrossRef]

20.

J.-F. Robillard, A. Devos, I. Roch-Jeune, and P. A. Mante, “Collective acoustic modes in various two-dimensional crystals by ultrafast acoustics: Theory and experiment,” Phys. Rev. B 78(6), 064302 (2008). [CrossRef]

21.

I. Vurgaftman and J. R. Meyer, “Band parameters for nitrogen-containing semiconductors,” J. Appl. Phys. 94(6), 3675–3696 (2003). [CrossRef]

22.

A. Polian, M. Grimsditch, and I. Grzegory, “Elastic constants of gallium nitride,” J. Appl. Phys. 79(6), 3343–3344 (1996). [CrossRef]

23.

Z. Wang, X. Zu, L. Yang, F. Gao, and W. J. Weber, “Molecular dynamics simulation on the buckling behavior of GaN nanowires under uniaxial compression,” Physica E 40(3), 561–566 (2008). [CrossRef]

24.

A. Gulans and I. Tale, “Ab initio calculation of wurtzite-type GaN nanowires,” Phys. Status Solidi., C Curr. Top. Solid State Phys. 4(3), 1197–1200 (2007). [CrossRef]

25.

H. Liang, M. Upmanyu, and H. Huang, “Size-dependent elasticity of nanowires: Nonlinear effects,” Phys. Rev. B 71(24), 241403 (2005). [CrossRef]

OCIS Codes
(130.5990) Integrated optics : Semiconductors
(320.5390) Ultrafast optics : Picosecond phenomena
(320.7100) Ultrafast optics : Ultrafast measurements

ToC Category:
Integrated Optics

History
Original Manuscript: April 18, 2012
Revised Manuscript: May 31, 2012
Manuscript Accepted: June 3, 2012
Published: July 9, 2012

Citation
Hung-Pin Chen, Yueh-Chun Wu, Pierre Adrien Mante, Shang-Ju Tu, Jinn-Kong Sheu, and Chi-Kuang Sun, "Femtosecond excitation of radial breathing mode in 2-D arrayed GaN nanorods," Opt. Express 20, 16611-16617 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-15-16611


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References

  1. K.-Q. Peng and S.-T. Lee, “Silicon Nanowires for Photovoltaic Solar Energy Conversion,” Adv. Mater. (Deerfield Beach Fla.)23(2), 198–215 (2011). [CrossRef] [PubMed]
  2. R. Yan, D. Gargas, and P. Yang, “Nanowire photonics,” Nat. Photonics3(10), 569–576 (2009). [CrossRef]
  3. C.-T. Huang, J. Song, W. F. Lee, Y. Ding, Z. Gao, Y. Hao, L. J. Chen, and Z. L. Wang, “GaN nanowire arrays for high-output Nanogenerators,” J. Am. Chem. Soc.132(13), 4766–4771 (2010). [CrossRef] [PubMed]
  4. R. Agrawal, B. Peng, and H. D. Espinosa, “Experimental-computational investigation of ZnO nanowires strength and fracture,” Nano Lett.9(12), 4177–4183 (2009). [CrossRef] [PubMed]
  5. J. Heo, W. Guo, and P. Bhattacharya, “Monolithic single GaN nanowire laser with photonic crystal microcavity on silicon,” Appl. Phys. Lett.98(2), 021110 (2011). [CrossRef]
  6. Y. Chen, I. Stevenson, R. Pouy, L. Wang, D. N. McIlroy, T. Pounds, M. G. Norton, and D. E. Aston, “Mechanical elasticity of vapour-liquid-solid grown GaN nanowires,” Nanotechnology18(13), 135708 (2007). [CrossRef] [PubMed]
  7. P. Yu, C. H. Chiu, Y.-R. Wu, H. H. Yen, J. R. Chen, C. C. Kao, H. W. Yang, H. C. Kuo, T. C. Lu, W. Y. Yeh, and S. C. Wang, “Strain relaxation induced microphotoluminescence characteristics of a single InGaN-based nanopillar fabricated by focused ion beam milling,” Appl. Phys. Lett.93(8), 081110 (2008). [CrossRef]
  8. R. A. Bernal, R. Agrawal, B. Peng, K. A. Bertness, N. A. Sanford, A. V. Davydov, and H. D. Espinosa, “Effect of growth orientation and diameter on the elasticity of GaN nanowires. A combined in situ TEM and atomistic modeling investigation,” Nano Lett.11(2), 548–555 (2011). [CrossRef] [PubMed]
  9. J. J. Brown, A. I. Baca, K. A. Bertness, D. A. Dikin, R. S. Ruoff, and V. M. Bright, “Tensile measurement of single crystal gallium nitride nanowires on MEMS test stages,” Sens. Actuators A Phys.166(2), 177–186 (2011). [CrossRef]
  10. C.-Y. Nam, P. Jaroenapibal, D. Tham, D. E. Luzzi, S. Evoy, and J. E. Fischer, “Diameter-dependent electromechanical properties of GaN nanowires,” Nano Lett.6(2), 153–158 (2006). [CrossRef] [PubMed]
  11. T. Henry, K. Kim, Z. Ren, C. Yerino, J. Han, and H. X. Tang, “Directed growth of horizontally aligned gallium nitride nanowires for nanoelectromechanical resonator arrays,” Nano Lett.7(11), 3315–3319 (2007). [CrossRef] [PubMed]
  12. S. O. Mariager, D. Khakhulin, H. T. Lemke, K. S. Kjaer, L. Guerin, L. Nuccio, C. B. Sørensen, M. M. Nielsen, and R. Feidenhans’l, “Direct observation of acoustic oscillations in InAs nanowires,” Nano Lett.10(7), 2461–2465 (2010). [CrossRef] [PubMed]
  13. H. Lange, M. Mohr, M. Artemyev, U. Woggon, and C. Thomsen, “Direct observation of the radial breathing mode in CdSe Nanorods,” Nano Lett.8(12), 4614–4617 (2008). [CrossRef] [PubMed]
  14. C. Guillon, P. Langot, N. Del Fatti, F. Vallée, A. S. Kirakosyan, T. V. Shahbazyan, T. Cardinal, and M. Treguer, “Coherent acoustic vibration of metal nanoshells,” Nano Lett.7(1), 138–142 (2007). [CrossRef] [PubMed]
  15. A. Amziane, L. Belliard, F. Decremps, and B. Perrin, “Ultrafast acoustic resonance spectroscopy of gold nanostructures: Towards a generation of tunable transverse waves,” Phys. Rev. B83(1), 014102 (2011). [CrossRef]
  16. M. Hu, X. Wang, G. V. Hartland, P. Mulvaney, J. P. Juste, and J. E. Sader, “Vibrational response of nanorods to ultrafast laser induced heating: theoretical and experimental analysis,” J. Am. Chem. Soc.125(48), 14925–14933 (2003). [CrossRef] [PubMed]
  17. C.-C. Yu, C.-F. Chu, J.-Y. Tsai, H.-W. Huang, T.-H. Hsueh, C.-F. Lin, and S.-C. Wang, “Gallium nitride nanorods fabricated by inductively coupled plasma reactive ion etching,” Jpn. J. Appl. Phys.41(Part 2, No. 8B), L910–L912 (2002). [CrossRef]
  18. D. Zhaung and J. H. Edgar, “Wet etching of GaN, AlN, and SiC: a review,” Mater. Sci. Eng. Rep.48(1), 1–46 (2005). [CrossRef]
  19. M. Yamaguchi, T. Yagi, T. Azuhata, T. Sota, K. Suzuki, S. Chichibu, and S. Nakamura, “Brillouin scattering study of gallium nitride: elastic stiffness constants,” J. Phys. Condens. Matter9(1), 241–248 (1997). [CrossRef]
  20. J.-F. Robillard, A. Devos, I. Roch-Jeune, and P. A. Mante, “Collective acoustic modes in various two-dimensional crystals by ultrafast acoustics: Theory and experiment,” Phys. Rev. B78(6), 064302 (2008). [CrossRef]
  21. I. Vurgaftman and J. R. Meyer, “Band parameters for nitrogen-containing semiconductors,” J. Appl. Phys.94(6), 3675–3696 (2003). [CrossRef]
  22. A. Polian, M. Grimsditch, and I. Grzegory, “Elastic constants of gallium nitride,” J. Appl. Phys.79(6), 3343–3344 (1996). [CrossRef]
  23. Z. Wang, X. Zu, L. Yang, F. Gao, and W. J. Weber, “Molecular dynamics simulation on the buckling behavior of GaN nanowires under uniaxial compression,” Physica E40(3), 561–566 (2008). [CrossRef]
  24. A. Gulans and I. Tale, “Ab initio calculation of wurtzite-type GaN nanowires,” Phys. Status Solidi., C Curr. Top. Solid State Phys.4(3), 1197–1200 (2007). [CrossRef]
  25. H. Liang, M. Upmanyu, and H. Huang, “Size-dependent elasticity of nanowires: Nonlinear effects,” Phys. Rev. B71(24), 241403 (2005). [CrossRef]

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