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Description of a Crystal: Lattice and Unit-Cell
Beside the natural crystals, thanks to the advent of semiconductor technologies, man-made nano-structured materials may be fabricated with different dimensionalities. Modulating the physical parameters of a homogeneous material « in only one » dimension leads to the so-called 1-D materials. A typical example is the stratified media illustrated in Figure 1-(a) by two layers of different materials alternatively stacked on top of each other. In the same way Figure 1-(b) show an example of 2-D inhomogeneity designed by modulating the material parameter in two directions (say x and y) while the material still homogeneous in the third direction z. In this example, pillars arranged in air (or holes drilled in a buck material), the 2-D material description may be limited to any cross section parallel to the ( , ) plane. Figure (1-c) gives an example of a 3-D structure formed by spheres stacked on top of each other.Whatever the crystal dimensionality and/or symmetry is, it may be described using a lattice associated to a unit-cell. The choice of the lattice is not unique. An illustrative example is shown in Figure 2-(a) for the special case of a 2-D centered rectangular lattice. The red dots represent the nodes, the black arrows stand for the basis vectors. The associated unit-cells are colored in yellow. Any of these unit-cells if successively translated by any linear combination of the associated basis vectors will reconstitute the 2-D lattice. While, the three lower row unit-cells cover the same area and comprise only one node (4 nodes shared by 4 neighbor unit-cells), the upper row unit-cells exhibit twice this area and contain two nodes. Another difference resides in the fact that the basis vectors of the upper row will not address all the nodes by successive translation. Finally, as illustrated at the right of the upper row in Figure 2- (a). The unit-cell is not necessarily included in the basis vectors parallelogram, but it may be shifted to embrace the nodes it owns.
Primitive, Conventional, and Wigner-Seitz Cells
A lattice and its associated unit-cell are said to be primitive if the latter encloses a unique node and represents the smallest volume of unit-cell. It is worth noting that a unit-cell contains all the structural and symmetry information to build up the macroscopic structure of the lattice by successive translations without neither blanks nor overlaps.
Continuing the preceding example of Figure 2-(a), all the unit-cells have been chosen as the area defined by the parallelogram delimited by the basis vectors. But these chosen unit-cells may not explicitly exhibit the inherent symmetry properties of the lattice. So, in the field of crystallography, one usually defines a conventional unit-cell which is not necessarily primitive for lattice classification. In the above example only, the Centered Rectangle shaped unit-cell (on the left of the upper row) exhibit the overall symmetry properties. This is the reason why it is this « basis vectors / unit-cell » combination that has been selected as the conventional one and that the representation and this lattice symmetry is named after it.
In the field of symmetry group analysis, it is not too troublesome to use a non-primitive unit-cell. But, in the field of wave propagation and its special case X-ray diffraction, the use of a non-primitive unit-cell will lead to band folding effects and complicates the interpretation of the theoretical obtain results. So, when wave propagation in periodic media is addressed, one prefers the use of primitive cells. In Figure 2-(b), we redraw the conventional centered rectangular cell in the upper row. In the lower row, we represent the Wigner-Seitz cell defined as the inner area delimited by the planes which perpendicularly bisect the straight lines joining the nearest neighbors of the lattice node. The Wigner-Seitz offers the advantage to be a primitive cell and to exhibit the lattice symmetries as well.
Reciprocal Space and Brillouin Zones
Initially the reciprocal space, more specifically, its reciprocal basis vectors have been introduced in geometrical crystallography. They appear to be a very useful tool to handle non-orthogonal basis. Indeed, they are introduced as a set of three vectors in such a way that each one of them is orthogonal to the plane sustained by two of the basis vectors of the direct lattice. So, their use facilitates the dot product and thus makes easier the writing of the analytical expression of lattice planes. The reciprocal lattice vectors are direct measures of the spacing distance between 2 planes of the same family; an essential parameter for X-ray analysis in modern crystallography. Here we will introduce the reciprocal lattice following another approach based on wave propagation in periodic material which is concerned in the present thesis. We also present the concept of Brillouin zones.
Table of contents :
General Introduction
Chapter Ⅰ State of the Art
1.1 Introduction
1.2 A Short Introduction to Waves in Artificial Periodic Media
1.2.1 From Diffraction to Photonic Crystals
1.2.2 Generalization to Phononic Crystals
1.2.3 The Emergence of PhoXonic Crystals
1.3 Active PhoXonic Cavities
1.4 Conclusion
Chapter Ⅱ Introduction to Periodic Media Theory
2.1 Introduction
2.2 Description of a Crystal: Lattice and Unit-Cell
2.3 Primitive, Conventional, and Wigner-Seitz Cells
2.4 3-D Analytic Expression of the Lattice and Unit-Cell
2.5 Bravais Lattices
2.5.1 2-D Bravais Lattice
2.5.2 3-D Bravais Lattice
2.6 Crystal Structure : Lattice + Unit-Cell
2.7 Reciprocal Space and Brillouin Zones
2.7.1 Reciprocal Space Basis Vectors
2.7.2 Brillouin Zones
2.8 Irreducible Brillouin Zones
2.8.1 Hexagonal Lattice
2.8.2 Square Lattice
2.8.3 Transformation of the Reciprocal Lattice Vectors
2.9 The Band Diagram
2.9.1 Bloch Propagating Modes
2.9.2 Bloch Waves in the Reciprocal Domain
2.9.3 Periodicity of the Dispertion Relation
2.9.4 Bandgap: From a Perturbation Point of View
2.9.5 Bandgap: From a Scattering Point of View
2.9.6 Band Structures in 2-D Crystals
2.10 Conclusion
Chapter Ⅲ PhoXonic Crystals
3.1 Introduction
3.2 Wave Propagations in Homogeneous Media
3.2.1 Electromagnetic Wave in Homogeneous Dielectric
3.2.2 Acoustic Wave in Homogeneous Media
3.3 Wave Propagations in Periodic Media
3.3.1 Eigenproblems and Generalized Eigenproblems
3.3.2 Solving the Wave Equation
3.3.3 The Unit-Cell Concept
3.4 Specific Features of Wave Propagations in 2-D Periodic Structures
3.5 Principle Parameters of 2-D Crystals Design
3.5.1 Periodic Structure Parameters of the Studied Structure
3.5.2 Dispersion Diagrams of the Periodic Structure Considered
3.6 Photonic Crystals Cavities
3.6.1 Cavity Confinement and Quality Factor
3.6.2 Concept of Supercell
3.7 Mechanisms Behind Photon-Phonon Interaction
3.7.1 Photo-Elastic Effect
3.7.2 Opto-Mechanical Effect
3.8 Conclusion
Chapter Ⅳ Evaluation of the Opto-Mechanical Coupling
4.1 Introduction
4.2 Perturbation Approach of Opto-Mechanical Coupling
4.3 Opto-Mechanical Cavity Characteristics
4.4 Perturbative Modulation Induced by Acoustical Perturbation
4.5 Potentialities of the Perturbation Approach
4.6 Conclusion
General Conclusion
Appendix A
TM and TE Polarizations
References
