CRYSTALLINE SOLIDS IN 3D
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1 CRYSTALLINE SOLIDS IN 3D Andrew Baczewski PHY 491, October 7th, 2011
2 OVERVIEW First - are there any questions from the previous lecture? Today, we will answer the following questions: Why should we care about crystal structure? How many Bravais lattices are there in 3D? How do we describe them mathematically? What are some common examples of these structures? What are some not-so common examples?
3 REVIEW OF 2D BRAVAIS LATTICES In 2D, we saw that there are 5 distinct Bravais lattices. Characterized by 2 lattice vectors (2 magnitudes + 1 angle between vectors) 1 = oblique 2 = rectangular 3 = centered rectangular 4 = triangular net/hexagonal 5 = square net Recall that must assign a basis to the lattice to describe a real solid. Image source: Wikipedia
4 3D BRAVAIS LATTICES In 3D, lattices are characterized by 3 lattice vectors (3 magnitudes, 3 angles) We find that there are 14 distinct Bravais lattices. These 14 can be divided further into 7 lattice classes There are 2 common conventions for labeling lattice vectors: Kittel (your book) uses: Vectors: (a1, a2, a3) Angles: (α12, α23, α31) α12 is the angle b/w a1 and a2 Perhaps more common: Vectors: (a, b, c) Angles: (α, β, Ɣ) α is the angle b/w b and c Image source: Kittel
5 3D BRAVAIS LATTICES (CONT.) 7 lattice classes, 14 distinct Bravais lattices You will also see volume centered structures referred to as body centered There is an error in this table: trigonal refers to a crystal system, NOT a Bravais lattice. The Bravais lattice is actually called rhombohedral. For now, you will be most interested in the cubic lattice classes If you continue in condensed matter physics, you will probably encounter some of the more exotic structures in your research
6 CUBIC BRAVAIS LATTICES Left to right: simple, body/volume centered, face centered You will often see the abbreviations sc, bcc, and fcc for these lattices. Assigning lattice vectors is, as in 2D, non-unique, but not all choices are equally useful. Let s consider a bcc example.
7 BODY CENTERED CUBIC EXAMPLE We can conceptualize the bcc structure as 2 interpenetrating sc structures Vertices of a cube in one lattice lie at the center of the other Consider 2 choices of lattice vectors: Origin at an A atom: 2 vectors to other A atoms, 1 to a B atom All 3 vectors to B atoms Let s codify this mathematically
8 BODY CENTERED CUBIC EXAMPLE (CONT). a1 = ax a2 = ay a3 = a/2 (x+y+z) a1 = a/2 (y+z-x) a2 = a/2 (z+x-y) a3 = a/2 (x+y-z)
9 PRIMITIVE CELLS The unit cell is a volume that can fill all of space, without gaps, when translated by lattice vectors The choice of unit cell is also non-unique, so it is convenient to have a standard for assigning them. This is called the Wigner-Seitz cell, it is an example of a primitive cell A primitive cell is the minimum volume of space the can be used to characterize a Bravais lattice that contains exactly 1 lattice point. This is, of course, also non-unique: Notice that many of the choices have symmetries that differ from the underlying lattice...
10 PRIMITIVE CELLS (CONT.) Things can look even weirder in 3D... The unshaded area is the conventional fcc unit cell. The shaded area is one choice of the primitive cell This may seem counter-intuitive: parallelepiped has 8 vertices, each of which contains 1/8 of a lattice point - 8 x 1/8 = 1 point per cell Parallelepiped does NOT have the cubic symmetry of the original Bravais lattice...
11 WIGNER-SEITZ CELL We want a convenient standard for constructing primitive cells that preserve the symmetry of the original Bravais lattice. Along came these 2 gentlemen... Eugene Wigner Hungarian theoretical physicist Nobel laureate (1963) Fun fact: Paul Dirac s brother-in-law Frederick Seitz Wigner s student Started big lab at UIUC Fun facts: Global warming skeptic (before it was cool) Image source: Wikipedia
12 WIGNER-SEITZ CELL (CONT.) For a given lattice point, the Wigner-Seitz cell is the volume of space that is closer to that point than any other in the lattice Algorithm for construction Choose a lattice point Find its nearest neighbors Draw a line from the point to its neighbors Bisect this line with a plane Planes bound the Wigner-Seitz cell We will find this same construction useful when we learn about Brillouin Zones (Left) bcc Wigner-Seitz cell (Right) fcc Wigner-Seitz cell If you ever wind up working in applied math/computer science/topology, a generalization of this concept is frequently found (Voronoi polyhedra) that actually goes all the way back to Descartes!
13 UNIT CELL VOLUME The volume of the unit cell, when occupied by a basis of atoms, will have some bearing upon the physical properties of the crystal. The volume of a conventional, cubic unit cell is simply a1 3 = a2 3 = a3 3 More generally, you can use the lattice vectors to calculate the volume of the unit cell: a 1 (a 2 a 3 ) = a 2 (a 3 a 1 ) = a 3 (a 1 a 2 ) So far we have only considered the conventional unit cell... What about the primitive unit cell? Conventional unit cell for fcc and bcc have 4 and 2 atoms per cell, respectively. What is the volume of the primitive cells for fcc and bcc structures?
14 EXEMPLARY CRYSTAL STRUCTURES Remember, to create an actual crystal, we need to assign a basis of atoms to the Bravais lattice. What are some of the more common crystal structures that arise in solid state physics? CsCl: simple cubic with a basis of 2 atoms NaCl (halite, rock salt): fcc lattice with a basis of 2 atoms Diamond: fcc lattice with a basis of 2 atoms of the same species ZnS (zincblende): fcc lattice with a basis of 2 atoms of different species Hexagonal close packed (HCP): simple hexagonal lattice with a two atom basis Let s look at these individually, in more detail...
15 CESIUM CHLORIDE Simple cubic lattice with a basis of 2 atoms: Green sites are Chlorine Blue sites are Cesium Simple choice of basis: (0,0,0) and (a/2,a/2,a/2) Each site has 8 nearest neighbors of the opposite species. Alkali halide, so bonding is similar to NaCl, but here the atoms have very different sizes...
16 SODIUM CHLORIDE Face centered cubic lattice with a basis of 2 atoms: Green sites are Chlorine Red sites are Sodium Same choice of basis: (0,0,0) and (a/2,a/2,a/2) Centering is different here! Each site has 6 nearest neighbors of the opposite species. Again an alkali halide, so bonding will be similar, but the atoms have similar sizes, so the crystal structure is different.
17 DIAMOND Face centered cubic lattice with a basis of 2 atoms of the same species (Carbon for diamond diamond) New choice of basis: (0,0,0) and (a/4,a/4,a/4) Centering is different here! Each site has 4 nearest neighbors of the same species. Diamond structures typically arise in covalently bonded crystals Of great technological importance: Silicon solidifies in the diamond crystal structure
18 ZINCBLENDE Face centered cubic lattice with a basis of 2 atoms of the different species (Zinc and Sulfur) Same choice of basis: (0,0,0) and (a/4,a/4,a/4) Each site has 4 nearest neighbors of the opposite species. Zincblende structures also arise in covalently bonded crystals Also of great technological importance: Commonly arises in III-V semiconductors GaAs is a good example
19 HEXAGONAL CLOSE PACKED (HCP) Simple hexagonal lattice with a basis of 2 atoms Think of this as stacking layers of 2D triangular nets Basis: (0,0,0) and (a/3,a/3,a/2) The name comes from packing hard spheres (bottom) 30 elemental solids crystallize in HCP (Mg and Be are the most common) Many different types of close-packing arrangements Think of it like close-packing 2D layers... HCP is not Bravais lattice, but FCC close-packing is...
20 MODERN APPLICATIONS The topological insulating state is a topic of considerable contemporary interest. In a conventional band insulator, the electrons in a crystalline solid are immobile throughout the entire crystal (i.e., even at the surface) In a topological insulator, symmetries of the underlying crystal, along with spin-orbit coupling effects give rise to materials where the bulk insulates, but the surfaces have electronic states that can participate in conduction. These surface states are very exotic : Electron spin is locked perpendicular to its (quasi)-momentum Scattering channels are suppressed by symmetry, so the conduction is ballistic (free of scattering) Anomalous magnetoelectric effect (non-maxwellian electrodynamics) Some of the associated crystal structures are quite complex...
21 MODERN APPLICATIONS On the left is Bi2Se3, a confirmed topological insulator Rhombohedral Bravais lattice On the right is Tl2Hg3Te4, a potential topological insulator Monoclinic Bravais lattice I am studying the latter, trying to figure out whether or not this state will arise in this particular material
22 CONCLUSIONS Take home points from today s lecture: There are 14 distinct Bravais lattices in 3D. Each lattice is characterized by 3 lattice vectors (3 magnitudes, 3 angles) Primitive cell can be constructed using Wigner-Seitz method In Monday s lecture, we will discuss: What type of experimental probes exist for characterizing crystalline solids? What is a reciprocal lattice and what is its relationship to the lattices we have already discussed? What is a Brillouin Zone, and why is it theoretically useful? Any questions? Before leaving, take out a piece of paper - here is a 5 minute quiz.
23 QUIZ Consider the simple cubic, face centered cubic, and body centered cubic lattices. Assuming each lattice site is occupied by a basis consisting of a single sphere at (0,0,0), with a radius such that it is touching its nearest neighbor, what percentage of the conventional unit cell is filled by spheres?
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