Research Papers

Constitutive Relations for Modeling Single Crystal GaN at Elevated Temperatures

[+] Author and Article Information
Antoinette Maniatty

Fellow ASME
Department of Mechanical, Aerospace, and
Nuclear Engineering,
Rensselaer Polytechnic Institute,
Troy, NY 12180-3590
e-mail: maniaa@rpi.edu

Payman Karvani

Department of Mechanical, Aerospace, and
Nuclear Engineering,
Rensselaer Polytechnic Institute,
Troy, NY 12180-3590
e-mail: paymaan@gmail.com

1Corresponding author.

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received April 28, 2014; final manuscript received August 8, 2014; published online September 24, 2014. Assoc. Editor: Irene Beyerlein.

J. Eng. Mater. Technol 137(1), 011002 (Sep 24, 2014) (7 pages) Paper No: MATS-14-1094; doi: 10.1115/1.4028441 History: Received April 28, 2014; Revised August 08, 2014

Thermal–mechanical constitutive relations for bulk, single-crystal, wurtzite gallium nitride (GaN) at elevated temperatures, suitable for modeling crystal growth processes, are presented. A crystal plasticity model that considers slip and the evolution of mobile and immobile dislocation densities on the prismatic and basal slip systems is developed. The experimental stress–strain data from Yonenaga and Motoki (2001, “Yield Strength and Dislocation Mobility in Plastically Deformed Bulk Single-Crystal GaN,” J. Appl. Phys., 90(12), pp. 6539–6541) for GaN is analyzed in detail and used to define model parameters for prismatic slip. The sensitivity to the model parameters is discussed and ranges for parameters are given. Estimates for basal slip are also provided.

Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.


Strite, S., and Morkoc, H., 1992, “GaN, AlN, and InN: A Review,” J. Vac. Sci. Technol. B, 10(4), pp. 1237–1266. [CrossRef]
Nakamura, S., 1997, “First III-V-Nitride-Based Violet Laser Diodes,” J. Cryst. Growth, 170(1-4), pp. 11–15. [CrossRef]
Wang, J., Ryu, H.-B., Park, M.-S., Lee, W.-J., Choi, Y.-J., and Lee, H.-Y., 2013, “Epitaxy of GaN on Si(111) Substrate by the Hydride Vapor Phase Epitaxy Method,” J. Cryst. Growth, 370, pp. 249–253. [CrossRef]
Motoki, K., Okahisa, T., Nakahata, S., Matsumoto, N., Kimura, H., Kasai, H., Takemoto, K., Uematsu, K., Ueno, M., Kumagai, Y., Koukitu, A., and Seki, H., 2002, “Growth and Characterization of Freestanding GaN Substrates,” J. Cryst. Growth, 237(2), pp. 912–921. [CrossRef]
Andre, Y., Trassoudaine, A., Tourret, J., Cadoret, R., Gil, E., Castelluci, D., Aoude, O., and Disseix, P., 2007, “Low Dislocation Density High-Quality Thick Hydride Vapour Phase Epitaxy (HVPE) GaN Layers,” J. Cryst. Growth, 306(1), pp. 86–93. [CrossRef]
Yoshikawa, A., Ohshima, E., Fukuda, T., Tsuji, H., and Oshima, K., 2004, “Crystal Growth of GaN by Ammonothermal Method,” J. Cryst. Growth, 260(1), pp. 67–72. [CrossRef]
Dwiliński, R., Doradziński, R., Garczyński, J., Sierzputowski, L., Puchalski, A., Kanbara, Y., Yagi, K., Minakuchi, H., and Hayashi, H., 2008, “Excellent Crystallinity of Truly Bulk Ammonothermal GaN,” J. Cryst. Growth, 310(17), pp. 3911–3916. [CrossRef]
Dwiliński, R., Doradziński, R., Garczyński, J., Sierzputowski, L., Kucharski, R., Zajac, M., Rudziński, M., Kudrawiec, R., Strupiński, W., and Misiewicz, J., 2011, “Ammonothermal GaN Substrates: Growth Accomplishments and Applications,” Phys. Status Solidi, 208(7), pp. 1489–1493. [CrossRef]
Yonenaga, I., and Motoki, K., 2001, “Yield Strength and Dislocation Mobility in Plastically Deformed Bulk Single-Crystal GaN,” J. Appl. Phys., 90(12), pp. 6539–6541. [CrossRef]
Leszczynski, M., Suski, T., Teisseyre, H., Perlin, P., Grzegory, I., Jun, J., Porowski, S., and Moustakas, T. D., 1994, “Thermal Expansion of Gallium Nitride,” J. Appl. Phys., 76(8), pp. 4909–4911. [CrossRef]
Reeber, R. R., and Wang, K., 2000, “Lattice Parameters and Thermal Expansion of Important Semiconductors and Their Substrates,” MRS Proc., 622(1), p. T6.35.1. [CrossRef]
Wang, H.-Y., Hui, X., Huang, T.-T., and Deng, C.-S., 2008, “Thermodynamics of Wurtzite GaN From First-Principle Calculation,” Eur. Phys. J. B, 62(1), pp. 39–43. [CrossRef]
Yadav, R. R., and Pandey, D. K., 2006, “Ultrasonic Characterisation of Gallium Nitride,” Mater. Res. Innov., 10(4), pp. 402–407. [CrossRef]
Reeber, R. R., and Wang, K., 2001, “High Temperature Elastic Constant Prediction of Some Group III-Nitrides,” MRS Internet J. Nitride Semicond. Res., 6, p. e3. [CrossRef]
Fujikane, M., Yokogawa, T., Nagao, S., and Nowak, R., 2012, “Nanoindentation Study on Insight of Plasticity Related to Dislocation Density and Crystal Orientation in GaN,” Appl. Phys. Lett., 101(20), p. 201901. [CrossRef]
Lu, J.-Y., Ren, H., Deng, D.-M., Wang, Y., Chen, K. J., Lau, K.-M., and Zhang, T.-Y., 2012, “Thermally Activated Pop-In and Indentation Size Effects in GaN Films,” J. Phys. D, 45(8), p. 085301. [CrossRef]
Yonenaga, I., Hoshi, T., and Usui, A., 2000, “Hardness of Bulk Single-Crystal Gallium Nitride at High Temperatures,” Jpn. J. Appl. Phys., 39(3A/B), pp. L200–L201. [CrossRef]
Lloyd, S. J., Molina-Aldareguia, J. M., and Clegg, W. J., 2001, “Deformation Under Nanoindents in Si, Ge, and GaAs Examined Through Transmission Electron Microscopy,” J. Mater. Res., 16(12), pp. 3347–3350. [CrossRef]
Wheeler, J. M., Niederberger, C., Tessarek, C., Christiansen, S., and Michler, J., 2013, “Extraction of Plasticity Parameters of GaN With High Temperature, In Situ Micro-Compression,” Int. J. Plast., 40, pp. 140–151. [CrossRef]
Alexander, H., and Haasen, P., 1968, “Dislocations in the Diamond Structure,” Advances in Solid State Physics, Vol. 22, F.Sietz, D.Turnbull, and H.Ehrenreich, eds., Academic Press, New York, pp. 27–158.
Yonenaga, I., and Sumino, K., 1978, “Dislocation Dynamics in the Plastic Deformation of Silicon Crystals,” Phys. Status Solidi A, 50(2), pp. 685–693. [CrossRef]
Yonenaga, I., 1997, “Mechanical Properties and Dislocation Dynamics in III-V Compounds,” Journal de Physique III, 7(7), pp. 1435–1450. [CrossRef]
Garofalo, F., 1963.,“Empirical Relation Defining Stress Dependence of Minimum Creep Rate in Metals,” Metall. Soc. AIME, 227(2), pp. 351–356.
Moosbrugger, J. C., 1995, “Continuum Slip Viscoplasticity With the Haasen Constitutive Model: Application to CdTe Single Crystal Inelasticity,” Int. J. Plast., 11(7), pp. 799–826. [CrossRef]
Kalan, R. J., and Maniatty, A. M., 2001, “Micromechanical Constitutive Relations for Modeling the Bulk Growth of Single Crystal InP,” J. Cryst. Growth, 233(4), pp. 645–659. [CrossRef]
Cochard, J., Yonenaga, I., Gouttebroze, S., M'Hamdi, M., and Zhang, Z. L., 2010, “Constitutive Modeling of Intrinsic Silicon Monocrystals in Easy Glide,” J. Appl. Phys., 107(3), p. 033512. [CrossRef]
Bower, A. F., 2010, Applied Mechanics of Solids, CRC Press, Boca Raton, FL.
Orowan, E., 1940, “Problems of Plastic Gliding,” Proc. Phys. Soc. London, 52(1), pp. 8–22. [CrossRef]
Frost, H. J., and Ashby, M. F., 1982, Deformation-Mechanism Maps, Pergamon Press, New York.
Utsumi, W., Saitoh, H., Kaneko, H., Watanuki, T., Aoki, K., and Shimomura, O., 2003, “Congruent Melting of Gallium Nitride at 6 GPa and Its Application to Single-Crystal Growth,” Nat. Mater., 2(11), pp. 735–738. [CrossRef] [PubMed]
Maniatty, A. M., Dawson, P. R., and Lee, Y. S., 1992, “A Time Integration Algorithm for Elasto-Viscoplastic Cubic Crystals Applied to Modeling Polycrystalline Deformation,” Int. J. Numer. Methods Eng., 35(8), pp. 1565–1588. [CrossRef]
Hutchinson, J. W., 1976, “Bounds and Self-Consistent Estimates for Creep of Polycrystalline Materials,” Proc. R. Soc. A, 348(1652), pp. 101–127. [CrossRef]
Johnston, W. G., and Gilman, J. J., 1959, “Dislocation Velocities, Dislocation Densities, and Plastic Flow in Lithium Fluoride Crystals,” J. Appl. Phys., 30(2), pp. 129–143. [CrossRef]
Dew-Hughes, D., 1961, “Dislocations and Plastic Flow in Germanium,” IBM J. Res. Dev., 5(4), pp. 279–286. [CrossRef]
Peissker, E., Haasen, P., and Alexander, H., 1962, “Anisotropic Plastic Deformation of Indium Antimonide,” Philos. Mag., 7(80), pp. 1279–1303. [CrossRef]
Kubin, L. P., and Estrin, Y., 1990, “Evolution for Dislocation Densities and the Critical Conditions for the Portevin–Le Chatelier Effect,” Acta Metall. Mater., 38(5), pp. 697–708. [CrossRef]
Huang, J., Xu, K., Gong, X. J., Wang, J. F., Fan, Y. M., Liu, J. Q., Zeng, X. H., Ren, G. Q., Zhou, T. F., and Yang, H., 2011, “Dislocation Cross-Slip in GaN Single Crystals Under Nanoindentation,” Appl. Phys. Lett., 98(22), p. 221906. [CrossRef]
Weingarten, N. S., and Chung, P. W., 2013, “A-Type Edge Dislocation Mobility in Wurtzite GaN Using Molecular Dynamics,” Scr. Mater., 69(4), pp. 311–314. [CrossRef]
Dasilva, Y. A. R., Ruterana, P., Lahourcade, L., Monroy, E., and Nataf, G., 2010, “Extended Crystallographic Defects in Gallium Nitride,” Mater. Sci. Forum, 644, pp. 117–122. [CrossRef]
Bai, J., Dudley, M., Raghothamachar, B., Gouma, P., Skromme, B. J., Chen, L., Hartlieb, P. J., Michaels, E., and Kolis, J. W., 2004, “Correlated Structural and Optical Characterization of Ammonothermally Grown Bulk GaN,” Appl. Phys. Lett., 84(17), pp. 3289–3291. [CrossRef]
Beyerlein, I. J., and Tomé, C. N., 2008, “A Dislocation-Based Constitutive Law for Pure Zr Including Temperature Effects,” Int. J. Plast., 24(5), pp. 867–895. [CrossRef]
Hosford, W. F., 1993, The Mechanics of Crystals and Textured Polycrystals, Oxford University Press, New York.
Patel, J. R., and Chaudhuri, A. R., 1963, “Macroscopic Plastic Properties of Dislocation-Free Germanium and Other Semiconductor Crystals. I. Yield Behavior,” J. Appl. Phys., 34(9), p. 2788. [CrossRef]
Yonenaga, I., and Sumino, K., 1993, “Effects of Dopants on Dynamic Behavior of Dislocations and Mechanical Strength in InP,” J. Appl. Phys., 74(2), pp. 917–924. [CrossRef]
Reppich, B., Reiger, K., and Muller, G., 1990, “Dynamische Verformung von InP-Einkristallen Bei Hochsten Temperaturen Mittels Liquid-Encapsulation (LE)-Technik,” Z. Metallkd., 81, pp. 166–173.
Samant, A. V., Zhou, W. L., and Pirouz, P., 1998, “Effect of Test Temperature and Strain Rate on the Yield Stress of Monocrystalline 6H-SiC,” Phys. Status Solidi A, 166(1), pp. 155–169. [CrossRef]
Anderssen, R., and Hegland, M., 1999, “For Numerical Differentiation, Dimensionality Can Be a Blessing!” Math. Comput., 68(227), pp. 1121–1142. [CrossRef]
Simo, J. C., and Hughes, T. J. R., 1998, Computational Inelasticity, Springer, New York.


Grahic Jump Location
Fig. 1

Orientation of the GaN hexagonal lattice relative to the compression experiment, where the x1, x2 are coordinates of the global reference frame with x2 aligned with the compressive axis, and x¯1,x¯2 are the lattice reference frame coordinates

Grahic Jump Location
Fig. 2

Stress–strain curves of GaN bulk single crystals, based on data from Yonenaga and Motoki [9]

Grahic Jump Location
Fig. 3

Stress–strain curves of GaN bulk single crystals at temperatures of 900, 950, and 1000 °C, based on data from Yonenaga and Motoki [9], corrected for machine compliance

Grahic Jump Location
Fig. 4

Computed plastic strain rate for Yonenaga and Motoki [9] experiment at temperatures of 900, 950, and 1000 °C

Grahic Jump Location
Fig. 5

Stress–strain curves from proposed model for GaN versus experiment data [9], at temperatures of 900, 950, and 1000 °C. Model A assumes Eq. (20) and model B assumes Eq. (21).

Grahic Jump Location
Fig. 6

Computed dislocation density for the experiment [9], at temperatures of 900, 950, and 1000 °C



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In