Skip to main content
×
×
Home

The size effect in the mechanical strength of semiconductors and metals: Strain relaxation by dislocation-mediated plastic deformation

  • David J. Dunstan (a1)
Abstract

The Ph.D. work of Jan H. van der Merwe in 1949 established a new paradigm for the understanding of dislocation dynamics in restricted volumes. This led to a comprehensive understanding of plasticity, or strain relaxation, in the context of strained-layer semiconductor structures. However, this understanding was largely overlooked in the context of traditional metallurgy and micromechanics. We identify four reasons for this, the apparent need for an unstrained substrate in van der Merwe’s theory, the supposed inapplicability to strain gradients, the supposed inapplicability to the Hall–Petch effect (dependence of strength on the inverse square root of grain size), and an emphasis on understanding strain hardening rather than the yield point. Addressing these four points in particular, here it is shown how the insights of van der Merwe and of the earlier work by Lawrence Bragg lead to a coherent and unified view of the size-effect phenomena ranging from the Hall–Petch effect to the strain-gradient plasticity theory.

Copyright
Corresponding author
a) Address all correspondence to this author. e-mail: d.dunstan@qmul.ac.uk
Footnotes
Hide All

Contributing Editor: Artur Braun

This section of Journal of Materials Research is reserved for papers that are reviews of literature in a given area.

Footnotes
References
Hide All
1. Volterra, V.: Sur l’équilibre des corps élastiques multiplement connexes. Ann. Sci. École Norm. Sup. 24, 401 (1907).
2. Taylor, G.I.: The mechanism of plastic deformation of crystals. Part I. Theoretical. Proc. R. Soc. London, Ser. A 145, 362 (1934).
3. Orowan, E.: Zur Kristallplastizität. III. Über den Mechanismus des Gleitvorganges. Z. Phys. 89, 634 (1934).
4. Polanyi, M.: Über eine Art Gitterstörung, die einen Kristall plastisch machen könnte. Z. Phys. 89, 660 (1934).
5. Frank, F.C. and van der Merwe, J.H.: One-dimensional dislocations. II. Misfitting monolayers and oriented overgrowth. Proc. R. Soc. London, Ser. A 198, 216 (1949).
6. van der Merwe, J.H.: Misfitting monolayers and oriented overgrowths. Discuss. Faraday Soc. 5, 201 (1949).
7. Li, Y., Bushby, A.J., and Dunstan, D.J.: The Hall–Petch effect as a manifestation of the general size effect. Proc. R. Soc. A 472, 20150890 (2016).
8. Bragg, W.L.: A theory of the strength of metals. Nature 149, 511 (1942).
9. Hall, E.O.: The deformation and ageing of mild steel: III discussion of results. Proc. R. Soc. B 64, 747 (1951).
10. Petch, N.J.: The cleavage strength of polycrystals. J. Iron Steel Inst. 174, 25 (1953).
11. Kelly, A.: Lawrence Bragg’s interest in the deformation of metals and 1950–1953 in the Cavendish—A worm’s-eye view. Acta Crystallogr., Sect. A: Found. Crystallogr. 69, 16 (2013).
12. Bragg, W.L.: The strength of metals. Math. Proc. Cambridge Philos. Soc. 45, 125 (1949).
13. Kuhlmann-Wilsdorf, D.: The impact of F.R.N. Nabarro on the LEDS theory of workhardening. Prog. Mater. Sci. 54, 707 (2009).
14. Matthews, J.W. and Crawford, J.L.: Accomodation of misfit between single-crystal films of nickel and copper. Thin Solid Films 5, 187 (1970).
15. Matthews, J.W. and Jesser, W.A.: Experimental evidence for pseudomorphic growth of platinum on gold. Acta Metall. 15, 595 (1967).
16. Matthews, J.W. and Klokholm, E.: Fracture of brittle epitaxial films under the influence of misfit stress. Mater. Res. Bull. 7, 213 (1972).
17. Matthews, J.W. and Blakeslee, A.E.: Defects in epitaxial multilayers: I. Misfit dislocations. J. Cryst. Growth 27, 118 (1974).
19. People, R. and Bean, J.C.: Calculation of critical layer thickness versus lattice mismatch for Ge x Si1−x /Si strained-layer heterostructures. Appl. Phys. Lett. 47, 322 (1985).
20. Drigo, A.V., Aydinli, A., Carnera, A., Genova, F., Rigo, C., Ferrari, C., Franzosi, P., and Salviati, G.: On the mechanisms of strain release in molecular-beam-epitaxy-grown In x Ga1−x As/GaAs single heterostructures. J. Appl. Phys. 66, 1975 (1989).
21. Dunstan, D.J.: Strain and strain relaxation in semiconductors. J. Mater. Sci.: Mater. Electron. 8, 337 (1997).
22. Eshelby, J.D., Frank, F.C., and Nabarro, F.R.N.: The equilibrium of linear arrays of dislocations. Philos. Mag. 42, 351 (1951).
23. van der Merwe, J.H.: Strains in crystalline overgrowths. Philos. Mag. 7, 1433 (1962).
24. Kuhlmann-Wilsdorf, D. and van der Merwe, J.H.: Theory of dislocation cell sizes in deformed metals. Mater. Sci. Eng. 55, 79 (1982).
25. Nix, W.D.: Mechanical properties of thin films. Metall. Trans. A 20, 2217 (1989).
26. Thompson, C.V.: The yield stress of polycrystalline thin films. J. Mater. Res. 8, 237 (1993).
27. Unwin, W.C.: The Testing of Materials of Construction (Longmans, Green, and Co., London, 1888).
28. Brenner, S.S.: Tensile strength of whiskers. J. Appl. Phys. 27, 1484 (1956).
29. Uchic, M.D., Dimiduk, D.M., Florando, J.N., and Nix, W.D.: Sample dimensions influence strength and crystal plasticity. Science 305, 986 (2004).
30. Korte, S. and Clegg, W.J.: Discussion of the dependence of the effect of size on the yield stress in hard materials studied by microcompression of MgO. Philos. Mag. 91, 1150 (2010).
31. Fleck, N.A., Muller, G.M., Ashby, M.F., and Hutchinson, J.W.: Strain gradient plasticity: Theory and experiment. Acta Metall. Mater. 42, 475 (1994).
32. Stölken, J.S. and Evans, A.G.: A microbend test method for measuring the plasticity length scale. Acta Mater. 46, 5109 (1998).
33. Ashby, M.F.: The deformation of plastically non-homogeneous materials. Philos. Mag. 21, 399 (1970).
34. Li, J.M.C.: Petch relation and grain boundary sources. Trans. TMS 227, 239 (1963).
35. Conrad, H.: Effect of grain size on the lower yield and flow stress on iron and steel. Acta Metall. 11, 75 (1963).
36. Zhang, X. and Aifantis, K.: Interpreting the internal length scale in strain gradient plasticity. Rev. Adv. Mater. Sci. 41, 72 (2015).
37. Nix, W.D. and Gao, H.J.: Indentation size effects in crystalline materials: A law for strain gradient plasticity. J. Mech. Phys. Solids 46, 411 (1998).
38. Evans, A.G. and Hutchinson, J.W.: A critical assessment of theories of strain gradient plasticity. Acta Mater. 57, 1675 (2009).
39. Beanland, R.: Dislocation multiplication mechanisms in low-misfit strained epitaxial layers. J. Appl. Phys. 77, 6217 (1995).
40. Dunstan, D.J., Ehrler, B., Bossis, R., Joly, S., P’ng, K.M.Y., and Bushby, A.J.: Elastic limit and strain-hardening of thin wires in torsion. Phys. Rev. Lett. 103, 155501 (2009).
41. Shan, Z.W., Mishra, R.K., Syed Asif, S.A., Warren, O.L., and Minor, A.M.: Mechanical annealing and source-limited deformation in submicrometre-diameter Ni crystals. Nat. Mater. 7, 115 (2007).
42. Greer, J.R. and Nix, W.D.: Nanoscale gold pillars strengthened through dislocation starvation. Phys. Rev. B 73, 245410 (2006).
43. Tersoff, J.: Dislocations and strain relief in compositionally graded layers. Appl. Phys. Lett. 62, 693 (1993).
44. Motz, C. and Dunstan, D.J.: Observation of the critical thickness phenomenon in dislocation dynamics simulation of microbeam bending. Acta Mater. 60, 1603 (2012).
45. Fleck, N.A. and Hutchinson, J.W.: A phenomenological theory for strain gradient effects in plasticity. J. Mech. Phys. Solids 41, 1825 (1993).
46. Aifantis, E.C.: The physics of plastic deformation. Int. J. Plast. 3, 211 (1987).
47. Dunstan, D.J.: Validation of a phenomenological strain-gradient plasticity theory. Philos. Mag. Lett. 96, 305 (2016).
48. Liu, D. and Dunstan, D.J.: Material length scale of strain gradient plasticity: A physical interpretation. Int. J. Plast. (2017). (in press).
49. Dunstan, D.J. and Bushby, A.J.: The scaling exponent in the size effect of small scale plastic deformation. Int. J. Plast. 40, 152 (2013).
50. Baldwin, W.M.: Yield strength of metals as a function of grain size. Acta Metall. 6, 139 (1958).
51. Aldrich, J.W. and Armstrong, R.W.: The grain size dependence of the yield, flow and fracture stress of commercial purity silver. Metall. Trans. 1, 2547 (1970).
52. Dunstan, D.J. and Bushby, A.J.: Grain size dependence of the strength of metals: The Hall–Petch effect does not scale as the inverse square root of grain size. Int. J. Plast. 53, 56 (2014).
53. Li, Y., Bushby, A.J., and Dunstan, D.J.: Quantitative explanation of the reported values of the Hall–Petch parameter. (unpublished).
54. Nissen, S.B., Magidson, T., Gross, K., and Bergstrom, C.T.: Publication bias and the canonization of false facts. eLife 5, e21451 (2016).
55. Narutani, T. and Takamura, J.: Grain-size strengthening in terms of dislocation density measured by resistivity. Acta Metall. Mater. 39, 2037 (1991).
56. Brown, L.M.: Indentation size effect and the Hall–Petch ‘law’. Mater. Sci. Forum 662, 13 (2011).
57. Argon, A.S.: Strengthening Mechanisms in Crystal Plasticity (Oxford University Press, Oxford, U.K. 2008); ch. 8.8.
58. Edwards, A.W.F.: Likelihood (Cambridge University Press, 1972; John Hopkins University Press, Baltimore, 1992).
59. Dong, D., Dunstan, D.J., and Bushby, A.J.: Plasticity and thermal recovery of thin copper wires in torsion. Philos. Mag. 95, 1739 (2015).
60. Walter, M. and Kraft, O.: A new method to measure torsion moments on small-scaled specimens. Rev. Sci. Instrum. 82, 035109 (2011).
61. Liu, D., He, Y., Dunstan, D.J., Zhang, B., Gan, Z., Hu, P., and Ding, H.: Anomalous plasticity in the cyclic torsion of micron scale metallic wires. Phys. Rev. Lett. 110, 244301 (2013).
62. Ehrler, B., Hou, X.D., Zhu, T.T., P’ng, K.M.Y., Walker, C.J., Bushby, A.J., and Dunstan, D.J.: Grain size and sample size interact to determine strength in a soft metal. Philos. Mag. 88, 3043 (2008).
63. Douthwaite, R.M.: Relationship between the hardness, flow stress, and grain size of metals. J. Iron Steel Inst. 208, 265 (1970).
64. Bei, H., Shim, S., Pharr, G.M., and George, E.P.: Effects of pre-strain on the compressive stress-strain response of Mo-alloy single-crystal micropillars. Acta Mater. 56, 4762 (2008).
65. Weiss, J., Ben Rhouma, W., Richeton, T., Dechanel, S., Louchet, F., and Truskinovsky, L.: From mild to wild fluctuations in crystal plasticity. Phys. Rev. Lett. 114, 105504 (2015).
66. Weinberger, C.R. and Cai, W.: Plasticity of metal wires in torsion: Molecular dynamics and dislocation dynamics simulations. J. Mech. Phys. Solids 58, 1011 (2010).
67. Brenchley, M.E., Hopkinson, M., Kelly, A., Kidd, P., and Dunstan, D.J.: Coherency strain as an athermal strengthening mechanism. Phys. Rev. Lett. 78, 3912 (1997).
68. Kümmel, F., Kreuz, M., Hausöl, T., Höppel, H.W., and Göken, M.: Microstructure and mechanical properties of accumulative roll-bonded AA1050A/AA5005 laminated metal composites. Metals 6, 56 (2016).
69. Ritchie, R.O.: The conflicts between strength and toughness. Nat. Mater. 10, 817 (2011).
70. Gillin, W.P., Dunstan, D.J., Homewood, K.P., Howard, L.K., and Sealy, B.J.: Interdiffusion in InGaAs/GaAs quantum well structures as a function of depth. J. Appl. Phys. 73, 3782 (1993).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Materials Research
  • ISSN: 0884-2914
  • EISSN: 2044-5326
  • URL: /core/journals/journal-of-materials-research
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed