Syntheses and Realization of Oriented Plates
As described in details in ref. 6, transparent, well-facetted, and millimeter-sized single crystals of GeO2 were grown by spontaneous nucleation with a high-temperature flux-growth process by using the slow-cooling method. Oriented platelets were cut and polished from as-grown samples presenting no visible imperfections such as trapped gas-bubbles or flux inclusions.
To identify the crystalline orientation, we used the dielectric system, also known as the orthogonal system. In this system, the Z-axis of α-GeO2 coincided with the crystallographic c-axis; the X-axis matched the crystallographic a-axis, and the Y-axis was normal to the X- and Z-axes (with the b-axis in the XY plane at 120° from the a-axis). The average size of the flux-grown GeO2 single crystals, 3–5 mm in length, allowed us to prepare plates with simple orientations. Three platelets were obtained, defined in the orthogonal system as X-plate, Y-plate, and Z-plate, which corresponded to the (10.0), (2–1.0), and (00.3) hkl-crystallographic planes, respectively.
As for α-quartz, there is a possibility of the existence of optical and electrical twins in these α-GeO2 plates. However, following the procedure described in ref. 4, we were able to determine that no Brazil (optical) twins were present in the measured samples.
Brillouin scattering is caused by the inelastic interaction of an incident photon with acoustic phonons of the medium. It results in a frequency shift, ±, of the scattered light relative to the laser frequency, depending on its velocity of propagation and, hence, on the elastic tensor, Cijkl, of the medium. For piezoelectric media, such as α-GeO2, a stress component is also produced by the internal electric field, E, so that the usual relationship between the stress tensor, Tij, and the strain tensor, Skl, is modified according to Equation (9):26
in which e is the piezoelectric tensor and summation over repeated indices is implied. For long waves, it can be shown that it results in an effective elastic tensor, C(e), that depends on the propagation direction [Eq. (10)]:
in which ε is the dielectric tensor at the frequency of the elastic wave. For the propagation direction, , the three eigenvalues of C(s) are given by the usual Christoffel matrix [Eq. (11)]:
in which is the unit tensor.
These eigenvalues, C(s), are associated to a phase velocity, v(s), of the sound wave according to Equation (12):
in which ρ is the material mass density.
Finally, the Brillouin frequency shift measured in the backscattering geometry are related to the sound velocity through Equation (13):
in which λ is the wavelength of the incident laser light, ni and ns are the refractive indices of the crystal for incident and scattered light, respectively.
More details concerning this technique and the different relationships established between the frequency shift, the density, and the wavelength of the incident wave, according to the propagation vector, can be found elsewhere.26–28
A high-resolution Brillouin spectrometer (resolving power of about 107) operating with an Ar-laser (λ=514.5 nm) was used in the backscattering geometry to measure the sound velocities necessary to determine the elastic constant. The spectrometer was composed of a fixed planar Fabry–Perot (FP) interferometer [1.5 mm spacing, free spectral range (FSR) of ca. 100 GHz, resolution ca. 1.7 GHz full-width at half-maximum (FWHM)], working as a band-pass filter. The thickness and the parallelism of this FP were dynamically adjusted to maximize the transmission of one of the two Brillouin lines of the acoustic mode under study. To that effect, an electro-optically modulated signal at the Brillouin frequency shift was periodically sent into the spectrometer. The high-resolution Brillouin spectrum was obtained by scanning a thick confocal FP interferometer (25 mm spacing, FSR ca. 3 GHz, resolution ca. 60 MHz at FWHM) placed after the planar FP. A detailed description of the apparatus and its principles can be found elsewhere.29, 30 According to the narrow band-pass of the first FP, only one acoustic mode was generally recorded at a time with this set-up.
The backscattered signal from the sample was collected by a short-focal-length lens (f=25 mm) through a numerical aperture (NA) of approximately 0.18 and sent to the Brillouin spectrometer. The finite aperture-induced Brillouin frequency downshift was taken into account in the data analysis, assuming an isotropic sound velocity in the probed -region. The single-crystals were mounted on a Linkam TS 1500 heating stage and heated in air from ambient temperature to 1273 K, with a heating rate of 20 K min−1. The measurements were started after a waiting time of 300 s, ensuring thermal equilibrium. Approximately 30 scans, corresponding to an acquisition time of approximately 8 min, were needed to achieve sufficient statistics. After cooling to room temperature, it was systematically verified that the room-temperature Brillouin frequency shift was recovered.
High-temperature XRD was performed on a D8 Avance diffractometer (Bruker) equipped with a Ge monochromator [crystallographic plane (111)] and working with the CuKα1 radiation (λ=1.54059 Å). Samples, sieved to 20 μm, were placed on a platinum surface in an HTK 1200N oven (Anton Paar) and heated to around 1323 K in air, with temperature steps of 100 K. The thermal expansion of the platinum plate was automatically compensated. The temperature ramp was 10 K min−1 (heating) or 30 K min−1 (cooling). Temperatures were fixed for at least 600 s before recording the diffractogram from 10° to 110° (2 θ values). From the XRD experiments, the temperature dependence of the mass density and the bulk thermal expansion coefficients were deduced.
Conoscopy figures were recorded on a polarizing microscope composed of two crossed-polarizing filters (polarizer and analyzer) into the optical path. The refractive indices were measured on a SOPRA GES5E spectroscopic ellipsometer equipped with a xenon lamp. The measurements were made at λ=514.5 nm on a α-GeO2 Y-plate. Two orientations (Z-axis of the Y-plate placed parallel or perpendicular to the polarization of the electric field) were used to deduce both extraordinary and ordinary refractive indices.
Computed Cij values were calculated within the density functional theory framework, as implemented in the ABINIT package.31 Details about the computation of the elastic, piezoelectric, and internal force-response tensors can be found in ref. 32. The exchange-correlation energy functional was evaluated within the local density approximation (LDA) as proposed by Perdew and Wang.33 Relaxations of the lattice parameters and the atomic positions were performed by using the Broyden–Fletcher–Goldfarb–Shanno algorithm until the maximum residual forces on the atoms and stress were less than 1×10−6 Ha Bohr−1 and 1×10−4 GPa, respectively. Convergence was reached for a 70 Ha plane-wave kinetic energy cutoff and a 8×8×8 mesh of special k-points.34