Raman Spectroscopy of nanomaterials - institut de chimie et des ...

ARTICLE IN PRESSG. Gouadec, Ph. Colomban / Progress in Crystal Growth and Characterization of Materialsxx (2007) 1e56In an isotropic material under hydrostatic compression:P h ¼ dVV 0¼ 33 l+ MODEL35ð46ÞThe combination of Eqs. (41), (45) and (46) then leads to:g ¼ a þ r þ 3ð47Þ6The Grüneisen parameter thus becomes a direct characteristic of the interatomic potentials(parameters a and r were introduced in Eq. (39)).5.1.3. Raman Spectroscopy of single crystals under general stressIf a deformation is applied to a single crystal, the vibrational mode independence is lostand phononephonon couplings occur [434]. The fundamental equation of Dynamics thenbecomes:m v2 r ivt 2 ¼¼X jlm2vk ijv3 lm3 lm r j ¼64k ii r i þ X jlmjsiX jlmK ijlm r j375;K ijlm r j8>:i; j; l; m ¼ x; y or zr i ¼ component of normal coordinater in direction im ¼ reduced massThe wavenumber shifts can then be directly linked to any macroscopic stress/strain via theelastic compliance tensor 28 and the K ijlm coefficients that describe the strain-induced deformationof the bonding potentials, in other words: anharmonicity [434]. Few of these coefficientsare independent in high-symmetry crystals (see [435] in cubic crystals or [436,437] for hexagonalcorundum). The mathematical expressions for the stress dependency of optical modesin cubic crystals can be derived for given crystallographic directions [89,434] and planes[438e442] 29 . Ager and Drory [440] reported less than 15% disagreement between the biaxialresidual stress measured by RS in a diamond film (deposited on a titanium alloy) using linearcombination of uniaxial stresses perpendicular to the growth direction, and the theoretical valueexpected from elasticity theory. The same kinds of measurements are common for semiconductors[443,444].5.1.4. The case of non-crystalline materialsIn polycrystalline or amorphous materials, the tensor formalism is irrelevant (stress directionsare unknown). We must therefore adopt a ‘‘statistical’’ approach. Let q b be the initial anglebetween a bond and the solicitation axis in a material macroscopically strained to D3 % M . The28 Compliance terms S ij are defined according of Voigt’s notation by 3 i ¼ S ij s j (3 ¼ strain; s ¼ stress; i ¼ 1, 2, 3, 4, 5,6 correspond to directions xx, yy, zz, yz, xz and xy, respectively, with 3 4 ¼ 2 3 yz , 3 5 ¼ 2 3 xz and 3 6 ¼ 2 3 xy ).29 Some errors in formulas from Ager et al. [440] and Cheong et al. [441] were listed in Ref. [378].Please cite this article in press as: G. Gouadec, Ph. Colomban, Prog. Cryst. Growth Charact. Mater. (2007),doi:10.1016/j.pcrysgrow.2007.01.001