Drude Review model (free electron approximation) + (independent electron approximation) + (Maxwell Boltzmann statistics) Sommerfeld model (free electron approximation) + (independent electron approximation) + (Fermi Dirac statistics) Next: Band structure (crystal potential) + (independent electron approximation) + (Fermi Dirac statistics) Crystal potential arises from the crystal structure made of atoms -- important for understanding the differences among metals, semiconductors, and insulators.
Crystal Structure Crystal structure is evident from naturally formed facets of many mineral samples. Based on x-ray diffraction: Crystal structure = periodic,(b) infinite(a) repetition of structural units(c)
(a) Infinite: N 1023 , bulk properties are independent of surfaces. (b) Periodic: translational symmetry Define Bravais lattice with lattice vector R n1a1 n2 a 2 n3 a 3 ; n1,2,3 = integers
a1,2,3 primitive lattice vectors (noncoplanar); they generate the lattice.
2d example:
a2 a1
(c) Structural unit: basis = a set of atoms associated with each lattice point Summarize :
crystal structure = Bravais lattice + basis
Theorem: A crystal looks the same from every Bravais lattice point.
TH T 1 H (system is invariant under translation)
Theorem: Primitive vectors are not unique; there are ways of choosing them. 2d example:
a2
a2' a1
a2"
Definition: primitive unit cell = a region in space, when translated using all lattice vectors R, fills neatly the whole space (w/o overlapping regions or voids) 2d examples: 2 possible choices out of
3d example: (obvious choice) a3 a2 a1 Theorem: volume of primitive cell c a1 a 2 a 3 lattice points One unit cell contains one lattice point (on average). Definition: Wigner-Seitz unit cell = the region of space that is closer to a selected lattice point (origin O) than to any other lattice points; it is a primitive unit cell. Method of construction: 1. Draw lines from O to all nearby lattice points. 2. Draw bisectors (planes in 3-d). 3. Find the smallest volume enclosed.
V ; V = volume of crystal, N = # of N
2d example:
Proof: Every point in space can be uniquely assigned to a W-S cell associated with a lattice point (except for those on the boundaries, but those have zero volume). Translational symmetry all W-S cells are the same. Since each point in space (other than those on boundaries) is accounted for once and only once, we have no overlapping regions or voids.
W-S cell is a primitive cell. It does not depend on the choice of a1,2,3 . As symmetric as the original lattice (e.g., 3-fold, 4-fold, mirror planes, ...).
Definition: conventional unit cells (non-primitive cells): volume n volume of primitive cell; filling the whole space when translated using a subset of R ; containing n lattice points per cell; often chosen for convenience. Examples of Bravais lattices:
1. Simple cubic (sc)
a1,2,3 mutually perpendicular; a 1 a 2 a 3 a a = lattice constant or lattice parameter Primitive cell = cube = conventional cell W-S cell is also cubic. Po, the only one in the periodic table (under typical conditions)
a3 a2 a1
2. Body centered cubic (bcc) Conventional cell = sc + another atom at center of cube body Primitive lattice vectors (symmetric form): z a a ^ ^ ^ a1 x y z 1,1,1 a 2 2 a a ^ ^ ^ a 2 x y z 1, 1,1 2 2 a2 a a a1 ^ ^ ^ a 3 x y z 1,1, 1 2 2 a = lattice constant a3
x
Make sure all points are generated: a a R n1 a1 n2 a 2 n3 a 3 n2 n3 n1 , n3 n1 n2 , n1 n2 n3 1 , 2 , 3 2 2
1,2,3 = integers but not arbitrary; n1,2,3 arbitrary. 1 2 2 n3 even, etc. So, 1,2,3 either all even or all odd. a R 0,0,0 corner; 2,0,0 corner; 0, 2,0 corner; ... 1,1,1 body-center; ... 2
Wigner-Seitz cell: "truncated octahedron" (see textbook)
3 Conventional cell: v a
a3 v = a1 a 2 a 3 Primitive cell: 8
1 1
a3 1 1 1 2 1 1 1
1
Conventional cell = 2 x primitive cell bcc = sc + a two-atom basis (useful for X-ray diffraction).
a1 a 1,0,0 , a 2 a 0,1,0 , a 3 a 0,0,1 a 1,1,1 2
Basis: d0 a 0,0,0 and d1
Examples: Ba, Cr, Li, etc.
3. Face centered cubic (fcc) Conventional cell = sc + 6 atoms at faces centers
Bravais lattice a a1 0,1,1 2 a a 2 1,0,1 2 a a 3 1,1,0 2
R ni a i generates the lattice (verify yourself).
3 Conventional cell: v a
a a1
a2 a3
3 Primitive cell: v a1 a 2 a 3 a 4
There are 4 lattice points per conventional cell Counting Lattice points 8 corner atoms 6 face atoms Weight 1/8 1/2 Total 1 3 4
Or, could one offset the cube slightly to avoid shared points. Examples: Ni, Cu, Ag, Au, Al, Pt, ...
4. Simple hexagonal (sh) 60 triangular net in 2d, stacked vertically directly atop a1 a2 a a3 c a 3 ^ ^ ^ ^ a1 a x , a 2 x a y , a3 c z a, c: lattice constants; z-axis = c-axis 2 2
c a a
Examples of lattice + nontrivial basis, one kind of atom: 1. Diamond structure
a 1,1,1 4
fcc lattice (lattice constant a) + 2-atom basis at 0,0,0 and
Coordination number = 4; tetrahedral bonding Examples: diamond, Si, Ge, grey-Sn
2.
Hexagonal close packed (hcp)
hcp = (simple hexagonal lattice) 1 1 1 + 2-atom basis at 0,0,0 and a1 a 2 a3 3 3 2 For close packed hard spheres, c / a 8 (ideal structure); 3 but most are not ideal.
c a a
A B
Green: first layer. Gold: second layer. The third layer repeats the first layer positions.
Stacking sequence: ABAB Examples: Ru, Cd, Re, Y, Hf , Gd, etc.
Changing the stacking sequence to ABCABC yields the fcc structure.
A B C
First layer at A sites; second layer at B sites, third layer at C sites; fourth layer at A sites; etc.
Looking along the body diagonal, the fcc structure can be viewed as a close stacking of atomic layers. fcc and hcp have the same packing density (close packed).
Examples of lattice + nontrivial basis, different kinds of atoms 1. Zinc blende (ZnS)
a 1,1,1 4 Replacing Zn and S by C diamond structure
fcc lattice + 2-atom basis, Zn at 0,0,0 , S at
Coordination number = 4
2. NaCl structure fcc lattice + 2-atom basis, Na at 0,0,0 , Cl at Two interpenetrating fcc lattices.
a 1,1,1 2
3. CsCl structure sc lattice + 2-atom basis, Cs at 0,0,0 , Cl at Replacing Cl by Cs bcc Cs Coordination # = 8 4. Perovskite structure (CaTiO3, or ABO3 generally) sc lattice + 5-atom basis Ca2+ at A site, Ti4+ at B site, and 3 O atoms Key feature: oxygen cage. Many closely related variations. Examples: complex oxides, high-temp superconductors, colossal magnetoresistive materials, ferroelectrics , ....
a 1,1,1 2
Symmetry (Brief review in terms of group theory) Space group: all symmetry operations {PT} that leave the crystal invariant, where P is a point operation (leaving one point fixed) including rotation, inversion, reflection, and combos, and T is a translation. T might be a translation by R n1a1 n2 a 2 n3a 3 , but it could also involve a nontrivial fraction of R in more complicated cases: Glide planes mirror reflection + translation by, for example, of a primitive lattice vector Screw axes rotation by + translation by, for example, of a primitive lattice vector Translation group: all translations by R n1a1 n2 a 2 n3a 3 ; an invariant subgroup of the space group (invariant under all symmetry transformations). Note {IT} (I being the identity operator) is not necessary the translation group, as there can be additional translations involving a nontrivial fraction of R associated with glide planes and screw axes. Point group: all operations {PI}. Not necessarily a subgroup of the space group because of glide planes and screw axes. It leaves the point lattice & Wigner Seitz cell invariant. Factor group: (Space group) (Translation group); isomorphic to the point group. Bravais lattice 7 (crystal systems) 14 (lattice structures) Crystal structure 32 230
Point group Space group
Listed in M.J. Berger, Elementary Crystallography, Wiley, NY, 1963.
See textbook or http://en.wikipedia.org/wiki/Crystal_structure
7 crystal systems Triclinic Monoclinic Simple Orthorhombic Simple Hexagonal Rhombohedral Tetragonal (trigonal) Simple Cubic Simple
Base-centered
Base-centered
Body-centered
Body-centered
14 Bravais lattices
Body-centered
Face-centered
Face-centered
Bravais lattices: Consider the highest symmetry, the cubic point group Oh (full symmetry group of a cube); there are 3 Bravais lattices having this point group symmetry: sc, fcc, and bcc. Distorting the system in various ways results in additional crystal systems and lattice structures that have lower symmetries (fewer symmetry operations). The triclinic structure has the lowest symmetry. For systems with one atom per unit cell, there are only 14 possible structures (Bravais lattices).
Crystal structures: The number of possible structures explodes to 230 as you add a basis (which tends to reduce the number of symmetry operations). For enumerating all possible structures, the exercise is to reduce the symmetry in a systematic manner. Make sure that all structures are distinct. It is an interesting but timeconsuming exercise.