Résumé :

If a force is applied to the unit cell of a crystalline solid, then the cell is deformed. This deformation is described by a strain. The force divided by the area of the face of the unit cell to which the force is applied is called stress, which has the same units as pressure (force per unit area). To result in a vanishing net force and net torque, the forces applied to the cell must be balanced. This makes the stress and strain symmetric second rank tensors. Stress and strain are connected through the fourth rank elasticity and compliance tensors by Hooke’s Law in three dimensions. In principle, the elasticity and compliance tensors have 81 elements, but the stress and strain tensor symmetry reduces them to symmetric 6x6 second rank tensors with no more than 21 independent elements. As described by Nye, crystalline symmetry further reduces that number further. A good introduction into these continuum mechanics topics is given by the Elasticity Theory volume of Landau and Lifshitz.

Examples of stress are hydrostatic pressure (equal stress along all three cartesian directions), uniaxial stress (a thin rod compressed by a vice), or biaxial stress (a thin film on a substrate with a different lattice constant). The resulting strain is always three-dimensional. Group theory allows the unique decomposition of the strain in irreducible representations. For example, in a cubic system, it is sufficient to consider the effects of hydrostatic compression as well as a pure uniaxial shear strain along the (100) and (111) directions. Strain in thin layers can be measured using high-resolution x-ray diffraction, especially asymmetric reciprocal space maps.

A hydrostatic compression will not change the symmetry of electron and phonon dispersions but will raise or lower electronic and vibrational energies. Uniaxial stress reduces the symmetry of the Brillouin zone and therefore lifts degeneracies or removes the equivalency of high-symmetry points. For example, a (100) stress will split the three X-points in the diamond or zinc blende crystal structure into a singlet and a doublet without lifting the degeneracy of the four L-points. Modern semiconductor devices use such stress techniques to lower the effective mass of carriers and to reduce intervalley scattering. My own interest is primarily in the impact of stress and strain on the optical spectra of semiconductors obtained with photoluminescence, Raman spectroscopy, and spectroscopic ellipsometry. For example, biaxial stress in silicon lifts the three-fold degeneracy of the optical phonon, which can be detected with Raman spectroscopy. The phonon shifts and splittings can then be used to determine the magnitude of the stress. Similarly, biaxial stress changes the direct and indirect band gaps of pseudomorphic silicon-germanium and germanium-tin alloy layers, which can be measured with photoluminescence spectroscopy. Finally, spectroscopic ellipsometry measures the E1 and E1+Δ1 critical point energies with very high accuracy. These are also affected by alloy composition and stress.

Contact : patrick.bruno@esrf.fr