Floer homology in symplectic geometry and in mirror symmetry

Layered materials draw attention in recent years in response to the world-wide drive to discover new functional materials. High-Tc superconducting oxide is one example. Internal interfaces in such layered materials differ significantly from those of cubic metals. They are often parallel to the layer of the neighboring crystals in sintered samples(layer plane boundary), while periodically ordered interfaces with the two neighboring crystals in mirror symmetry to each other are relatively rare. Consequently, the atomistic features of the interface differ significantly from those of cubic metals. In this paper grain boundaries in sintered high-Tc superconducting oxides, joined interfaces between engineering ceramics with metals, and polytype interfaces in vapor-deposited bicrystal are examined to collect atomic information of the interfaces in layered materials. The analysis proved that they are not neccessarily more complicated than that of simple grain boundaries in cubic metals. The interfaces are majorly layer plane type which is parallel to the compound layer. Secondly, chemical information is often available, which helps the interpretation of the interface atomic structure.

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The limits of strain and lattice parameter measurements by CBED

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100154561 ◽

1989 ◽

Vol 47 ◽

pp. 518-519

Author(s):

Hamish L. Fraser

Keyword(s):

Electron Beam ◽

Mirror Symmetry ◽

Local Symmetry ◽

Lattice Parameter ◽

Growth Direction ◽

Surface Relaxation ◽

Strained Layer ◽

Axis Parallel ◽

Local Lattice ◽

Strained Layer Superlattice

The topic of strain and lattice parameter measurements using CBED is discussed by reference to several examples. In this paper, only one of these examples is referenced because of the limitation of length. In this technique, scattering in the higher order Laue zones is used to determine local lattice parameters. Work (e.g. 1) has concentrated on a model strained-layer superlattice, namely Si/Gex-Si1-x. In bulk samples, the strain is expected to be tetragonal in nature with the unique axis parallel to [100], the growth direction. When CBED patterns are recorded from the alloy epi-layers, the symmetries exhibited by the patterns are not tetragonal, but are in fact distorted from this to lower symmetries. The spatial variation of the distortion close to a strained-layer interface has been assessed. This is most readily noted by consideration of Fig. 1(a-c), which show enlargements of CBED patterns for various locations and compositions of Ge. Thus, Fig. 1(a) was obtained with the electron beam positioned in the center of a 5Ge epilayer and the distortion is consistent with an orthorhombic distortion. When the beam is situated at about 150 nm from the interface, the same part of the CBED pattern is shown in Fig. 1(b); clearly, the symmetry exhibited by the mirror planes in Fig. 1 is broken. Finally, when the electron beam is positioned in the center of a 10Ge epilayer, the CBED pattern yields the result shown in Fig. 1(c). In this case, the break in the mirror symmetry is independent of distance form the heterointerface, as might be expected from the increase in the mismatch between 5 and 10%Ge, i.e. 0.2 to 0.4%, respectively. From computer simulation, Fig.2, the apparent monocline distortion corresponding to the 5Ge epilayer is quantified as a100 = 0.5443 nm, a010 = 0.5429 nm and a001 = 0.5440 nm (all ± 0.0001 nm), and α = β = 90°, γ = 89.96 ± 0.02°. These local symmetry changes are most likely due to surface relaxation phenomena.

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Structural Transformation of a Germanium Σ=13 (510) [001] Tilt Grain Boundary under the Electron Beam

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100179014 ◽

1990 ◽

Vol 48 (1) ◽

pp. 52-53 ◽

Cited By ~ 1

Author(s):

Jean-Luc Rouvière ◽

Alain Bourret

Keyword(s):

Electron Microscopy ◽

High Resolution ◽

Grain Boundary ◽

Structural Transformation ◽

Mirror Symmetry ◽

High Resolution Electron Microscopy ◽

Structural Transformations ◽

Covalent Bonds ◽

Resolution Electron ◽

Tilt Grain Boundary

The possible structural transformations during the sample preparations and the sample observations are important issues in electron microscopy. Several publications of High Resolution Electron Microscopy (HREM) have reported that structural transformations and evaporation of the thin parts of a specimen could happen in the microscope. Diffusion and preferential etchings could also occur during the sample preparation.Here we report a structural transformation of a germanium Σ=13 (510) [001] tilt grain boundary that occurred in a medium-voltage electron microscopy (JEOL 400KV).Among the different (001) tilt grain boundaries whose atomic structures were entirely determined by High Resolution Electron Microscopy (Σ = 5(310), Σ = 13 (320), Σ = 13 (510), Σ = 65 (1130), Σ = 25 (710) and Σ = 41 (910), the Σ = 13 (510) interface is the most interesting. It exhibits two kinds of structures. One of them, the M-structure, has tetracoordinated covalent bonds and is periodic (fig. 1). The other, the U-structure, is also tetracoordinated but is not strictly periodic (fig. 2). It is composed of a periodically repeated constant part that separates variable cores where some atoms can have several stable positions. The M-structure has a mirror glide symmetry. At Scherzer defocus, its HREM images have characteristic groups of three big white dots that are distributed on alternatively facing right and left arcs (fig. 1). The (001) projection of the U-structure has an apparent mirror symmetry, the portions of good coincidence zones (“perfect crystal structure”) regularly separate the variable cores regions (fig. 2).

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Symplectic Geometry

10.1017/cbo9780511526343 ◽

1994 ◽

Cited By ~ 10

Keyword(s):

Symplectic Geometry

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Symplectic Topology and Floer Homology

10.1017/cbo9781139680462 ◽

2015 ◽

Author(s):

Yong-Geun Oh

Keyword(s):

Floer Homology ◽

Symplectic Topology

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