Crystallographic data concerning geometric properties of hexagonal lattices of C,~, Zn, B%, Ti,~, Zr~, Mg and Cd are obtained from two different computation techniques. These properties are related to the relative orientations of identical hexagonal lattices 1 and 2 which superimpose two multiple cells M1 and M2 within a given small deformation. These orientations are listed for ratios 27 = Ivolume of cells M1 (or M2)/volume of the unit cell l varying from 1 to 25. Their number are limited by choosing all the principal strains transforming M1 into M2 less than or equal to 1%. IntroductionThree techniques have been used to determine the relative orientations of identical hexagonal crystals which give rise to a near coincidence of two cells of the two crystal lattices 1 and 2, cells denoted hereinafter M 1 and M2 respectively. [This has been referred to by previous workers as a 'coincidence' or 'near-coincidence site lattice' or 'orientation de macle',* where the number 271 (or 2~2) is defined by the ratio volume of M1 (or M2)/volume of the unit cell.] Two techniques depend either upon searching for vectors of common length arising from rational values of (c/a) 2 (Fortes, 1973;Warrington, 1975) or in searching for coincidences arising from rotation about specific axes, chosen a priori, of (low) crystallographic index (Bruggeman, Bishop & Hartt, 1972). The first method derives from that used by Warrington & Bufalini * We note for the benefit of our readers whose mother tongue is English that the term 'orientation de macle' as defined by Friedel (1964) is more general a concept than the nearest English equivalent of'twin' used in its restrictive sense.0567-7394/81/020184-06501.00 (1971) for cubic crystals; the second from the use of a 'generation function' typified by Ranganathan (1966) and Goux (1961). The third technique (Bonnet & Cousineau, 1977), tested on Zn/Zn and NiaA1 (cubic)/ Ni3Nb(orthorhombic ), depends on a numerical method of calculation capable of treating the case of general lattices and envisaged in part by Santoro & Mighell (1973). It takes into account the experimental [rather than idealised or rational values of (c/a) 2] values of the lattice parameters. The technique determines relative orientations that, with additional imposed constraints (which may be chosen arbitrarily small) on M1, will give full or true coincidence with M2. Determination of relative orientationsIn searching for a 'constrained coincidence' the worker must compromise and set limits on the deviation from exact coincidence that is to be allowed. In the numerical method (Bonnet & Cousineau, 1977) this is represented by a maximum value of S = It, ll + l e21 + lea1 where e 1, e 2, e a are the principal strains of the pure deformation D transforming lattice 1 into lattice 2. D -~ transforms lattice 2 into a fictitious lattice denoted lattice 2', which can be exactly superposed onto lattice 1 (Bonnet & Durand, 1975). The unit cell of this CSL (coincidence site lattice) is defined either by M1 or by the deformed cell M2, d...

A computation method is presented for determining: (i) pairs of non-primitive cells M1 and M2, constructed on three translation vectors of a lattice 1 and three vectors of a lattice 2 respectively, such that the sizes of M1 and M2 are (almost) identical; (ii) Z~ (E2), defined by the number of primitive cells of lattice 1 (lattice 2) contained in M1 (M2); (iii) a characteristic relative orientation of the two lattices for which M1 and M2 coincide exactly or approximately, for which the transformation relating M1 to M2 (denoted A in general) is a pure deformation, whose principal strains are calculated; (iv) base vectors for the DSC-1 and DSC-2 lattices, so that the Burgers vectors of intrinsic phase (or grain) boundary dislocations are determined. The DSC-1 lattice is constructed by summing the vectors of lattice 1 and lattice 2', deduced from lattice 2 by A-1. The DSC-2 lattice is derived from the DSC-1 lattice by A. Tables of results are presented for a lattice 1/lattice 2 of Zn/Zn, up to ZI = Z2 = 25, and for Ni3A1 (cubic)/Ni3Nb (orthorhombic), up to E1 =21 and Z 2 = 10.

Group IV semiconductors, such as GeSn and SiGeSn, are of growing interest for silicon-compatible electronic and photonic applications, notably due to the possibility of tuning the bandgap directness and energy to cover a broad range from the short-wave infrared (SWIR) to the mid-infrared (MIR) [1]. While germanium is an indirect-bandgap semiconductor, an indirect-to-direct transition in GeSn alloys is expected for Sn incorporation higher than 9%, but the residual compressive strain typical to epitaxial layers increases the concentration of Sn to obtain a direct-bandgap [2]. Therefore, an understanding of the interplay of strain and composition on the band structure of GeSn semiconductors is of compelling importance in order to precisely control their physical properties. This calls for accurate analysis of the as-grown material properties and its correlation with the measured optical and electrical performance. Raman spectroscopy is a non-destructive method commonly used to assess the role of the effects of composition and strain in shaping the lattice vibrations. The use of a 633 nm excitation enables a clear detection of all Raman modes in GeSn layers independently of their composition, comparatively to the more restricted results associated with the use of 488 nm[3] or 532 nm [4] excitation. This is plausibly attributed to the fact that this wavelength might be close to resonance with the alloy’s E1 gap [5]. There are very few reports on the identification of the Ge-Sn mode [6] and the disorder-activated mode coinciding with the maximum in the one-phonon density of states in Ge [7], but quantitative analyses of the effects of composition and strain on these modes remain conspicuously missing in literature. With this perspective, this work presents a detailed study of Raman vibrational modes in Sn-rich GeSn semiconductors and how their behavior is affected by temperature. The use of samples of various degrees of Sn composition and in-plane compressive strain allows the decoupling of these effects on the Ge-Ge, DA and Ge-Sn vibrational modes, thus providing an effective platform to test the current predictive models. Samples were grown in a chemical vapor deposition (CVD) reactor using monogermane (GeH4) and tin-tetrachloride (SnCl4) precursors [8]. The GeSn multi-layer heterostuctures with a Sn content in the 4-16 at.% range were grown on a Ge virtual substrate (VS) on a Si wafer. Using X-ray diffraction (XRD), reciprocal space mappings (RSM) were performed on all samples to retrieve the Sn composition and the strain. Temperature-dependent Raman measurements were performed from room temperature (300K) and down to 77 K. In GeSn, the asymmetrical profile of the observed Raman modes is due to alloying as substitutional Sn atoms break the translational symmetry and lead to a relaxation of the momentum selection rule [9]. Therefore, the use of exponentially modified gaussian functions (EMG) as a fitting curve can reflect the asymmetrical profile of the different analyzed Raman modes with much better accuracy [10]. The measurement temperature has a clear effect on the fitting parameters, such as the integrated area of the peak , the full width at half maximum (FWHM), the asymmetry, and its position. For example, the Ge-Sn and Ge-Ge peaks increase in wavenumber with decreasing temperature, while the DA peak shows the opposite trend. Ongoing work focuses on establishing the behavior of the characteristics of each mode. Based on these detailed Raman studies, an exhaustive discussion of the influence of temperature, lattice strain and Sn content on GeSn vibrational modes will be presented. Acknowledgements The authors thank J. Bouchard for the technical support, and NSERC Canada (Discovery, SPG, and CRD Grants), Canada Research Chairs, Canada Foundation for Innovation, Mitacs. Reference [1] S. Gupta, et al., J. Appl. Phys. 113, 7 (2013). [2] A. Attiaoui and O. Moutanabbir, J. Appl. Phys., 116, 6 (2014). [3] R. R. Lieten et al., ECS J. Solid State Sci. Technol., 3, 12, pp. P403–P408 (2014). [4] A. Gassenq et al., Appl. Phys. Lett., 110, 11, p. 112101 (2017). [5] R. Chen et al., ECS J. Solid State Sci. Technol., 2, 4, pp. P138–P145 (2013). [6] D. Zhang et al., J. Alloys Compd., 684, pp. 643–648 (2016). [7] V. D’Costa et al., Solid State Communications, 144, 5-6, pp. 240-244 (2007). [8] S. Assali, J. Nicolas and O. Moutanabbir, Journal of Applied Physics, 125, 2, p. 025304 (2019). [9] P. Parayanthal and F. H. Pollak, Phys. Rev. Lett., 52, 20, pp. 1822–1825 (1984). [10] É. Bouthillier et al., “Decoupling the effects of composition and strain on the vibrational modes of GeSn”, https://arxiv.org/abs/1901.00436, 2019.