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Bravais lattices are the most fundamental building blocks of crystallography. They are classified into groups according to their translational, rotational, and inversion symmetries. In computational analysis of Bravais lattices, fulfilment of symmetry conditions is usually determined by analysis of the metric tensor, using either a numerical tolerance to produce a binary (i.e. yes or no) classification, or a distance function which quantifies the deviation from an ideal lattice type. The metric tensor, though, is not scale-invariant, which complicates the choice of threshold and the interpretation of the distance function. Here, we quantify the distance of a lattice from a target Bravais class using strain. For an arbitrary lattice, we find the minimum-strain transformation needed to fulfil the symmetry conditions of a desired Bravais lattice type; the norm of the strain tensor is used to quantify the degree of symmetry breaking. The resulting distance is invariant to scale and rotation, and is a physically intuitive quantity. By symmetrizing to all Bravais classes, each lattice can be placed in a 14 dimensional space, which we use to create a map of the space of Bravais lattices and the transformation paths between them. A software implementation is available online under a permissive license.
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Calculation of band structure of three dimensional photonic crystals amounts to solving large-scale Maxwell eigenvalue problems, which are notoriously challenging due to high multiplicity of zero eigenvalue. In this paper, we try to address this problem in such a broad context that band structure of three dimensional isotropic photonic crystals with all 14 Bravais lattices can be efficiently computed in a unified framework. We uncover the delicate machinery behind several key results of our work and on the basis of this new understanding we drastically simplify the derivations, proofs and arguments in our framework. In this work particular effort is made on reformulating the Bloch boundary condition for all 14 Bravais lattices in the redefined orthogonal coordinate system, and establishing eigen-decomposition of discrete partial derivative operators by systematic use of commutativity among them, which has been overlooked previously, and reducing eigen-decomposition of double-curl operator to the canonical form of a 3x3 complex skew-symmetric matrix under unitary congruence. With the validity of the novel nullspace free method in the broad context, we perform some calculations on one benchmark system to demonstrate the accuracy and efficiency of our algorithm.
We investigate the effect of strain on the etching rate of two SiGe wet etchants, namely NH4OH:H2O2 and H2O2. For both etchants, we found that there is no appreciable strain selectivity, i.e. the etching rates do not depend on the actual strain state in the SiGe films. Instead, for the NH4OH:H2O2 solution, the rates are primarily determined by the Ge content. Finally, we show that both etchants are isotropic with no preferential etching of particular facets.
Quasi-two-dimensional (quasi-2D) materials hold promise for future electronics because of their unique band structures that result in electronic and mechanical properties sensitive to crystal strains in all three dimensions. Quantifying crystal strain is a prerequisite to correlating it with the performance of the device, and calls for high resolution but spatially resolved rapid characterization methods. Here we show that using fly-scan nano X-ray diffraction we can accomplish a tensile strain sensitivity below 0.001% with a spatial resolution of better than 80 nm over a spatial extent of 100 $mu$m on quasi 2D flakes of 1T-TaS2. Coherent diffraction patterns were collected from a $sim$ 100 nm thick sheet of 1T-TaS2 by scanning 12keV focused X-ray beam across and rotating the sample. We demonstrate that the strain distribution around micron and sub-micron sized bubbles that are present in the sample may be reconstructed from these images. The experiments use state of the art synchrotron instrumentation, and will allow rapid and non-intrusive strain mapping of thin film samples and electronic devices based on quasi 2D materials.
Recently, hexagonal boron nitride (hBN) layers have generated a lot of interest as ideal substrates for 2D stacked devices. Sapphire-supported thin hBN films of different thicknesses are grown using metalorganic vapour phase epitaxy technique by following a flow modulation scheme. Though these films of relatively large size are potential candidates to be employed in designing real devices, they exhibit wrinkling. The formation of wrinkles is a key signature of strain distribution in a film. Raman imaging has been utilized to study the residual strain distribution in these wrinkled hBN films. An increase in the overall compressive strain in the films with an increase in the layer thickness has been observed. To find whether the residual lattice strain in the films can be removed by a thermal treatment, temperature dependent Raman measurements of these films are carried out. The study demonstrates that the thermal rate of strain evolution is higher in the films of lower thickness than in the thicker films. This observation further provides a possible explanation for the variation of strain in the as-grown films. An empirical relation has been proposed for estimating the residual strain from the morphology of the films. We have also shown that the residual strain can be partially released by the delamination of the films.
Ferroelectric hafnia is being explored for next generation electronics due to its robust ferroelectricity in nanoscale samples and its compatibility with silicon. However, its ferroelectricity is not understood. Other ferroelectrics usually lose their ferroelectricity for nanoscopic samples and thin films, and the hafnia ground state is non-polar baddeleyite. Here we study hafnia with density functional theory (DFT) under epitaxial strain, and find that strain not only stabilizes the ferroelectric phases, but also leads to unstable modes and a downhill path in energy from the high temperature tetragonal structure. We find that under tensile epitaxial strain $eta$ the tetragonal phase will distort to one of the two ferroelectric phases: for $eta > 1.5$%, the $Gamma^{-}_{5}$ mode is unstable and leads to oII , and at $eta > 3.75$% coupling between this mode and the zone boundary M1 mode leads to oI. Furthermore, under compressive epitaxial strain $eta < 0.55$% the ferroelectric oI is most stable, even more stable than baddeleyite.
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