Mechanisms controlling the hardness of Si and Ge
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Mechanisms controlling the hardness of Si and Ge L.J. Vandeperre, F. Giuliani, S.J. Lloyd, W.J. Clegg Department of Materials Science and Metallurgy, University of Cambridge Cambridge, CB2 3QZ, United Kingdom. ABSTRACT It is generally accepted that the hardness of silicon and germanium is limited by a phase transformation. Observations of the deformation under indents using transmission electron microscopy indicate that, in addition to the phase transformation, there is also plastic flow both in the transformed and in the untransformed material. These observations are consistent with those made elsewhere and with predictions based on the spherical cavity model for indentation, modified to quantify the effect of a phase transformation on the measured hardness. The analysis predicts that the hardness can only be approximately equal to the transformation pressure provided the yield strength of the transformed material is low, and that there is a region where plastic deformation in the untransformed material can occur in addition to the phase transformation. These predictions are consistent with the experimental observations of substantial plastic deformation of the transformed phase, as well as with estimates of its Peierls stress. INTRODUCTION It is well established that the indentation of elements such as Si or Ge causes the initial diamond structure to transform to that of the tetragonal β-Sn structure on loading, giving a reduction in volume of 22%. Si and Ge do not transform reversibly on unloading, but instead an amorphous material and various crystalline phases have been observed [1,2]. The measured hardness is approximately equal to the pressures required for these transformations in a diamond anvil cell. As these pressures are less than those required for dislocation flow at low temperatures, it is commonly assumed that the hardness is determined entirely by the pressure required to initiate the phase transformation. This view is certainly consistent with the almost complete lack of any temperature dependence to the hardness [3] whereas for materials where the lattice resistance is the greatest obstacle to dislocation motion, the flow stress and hardness are expected to decrease with temperature and such a decrease is observed in the compressive flow stress. However, dislocation motion is often observed outside the transformation zone, which is perhaps surprising if it is thought that the transformation is occurring at lower stresses than those required for dislocation motion. For materials such as Si and Ge, where the ratio of the yield strength to the Young modulus is higher than that for many other materials, it is expected that the plastic impression of the indenter is accommodated by the radial flow of material away from the indenter. Such flow has been likened to the way in which material flows when a cavity is expanded within an infinite body, where the half-volume of the cavity is equal to the volume of the plastic impression [4]. If it is assumed that the shape change due to the transformatio
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