1. 3-Dimensional Crystal Structure

2. 3-Dimensional Crystal Structure

3. 3-D Crystal Structure
General: A crystal structure is defined by primitive lattice vectors a1, a2, a3.
a1, a2, a3: Depend on geometry. Once specified, the primitive lattice structure is specified.
Generate lattice by translating through a direct lattice vector: r = n1a1+n2a2+n3a3. (n1,n2,n3) are integers. r generates the lattice points. Each lattice point corresponds to a set of (n1,n2,n3).

4. Primitive lattice structure + basis.
Basis (or basis set) 
The set of atoms which, when placed at each lattice point, generates the crystal structure.
Crystal Structure 
Primitive lattice structure + basis.
Translate the basis through all possible lattice vectors
r = n1a1+n2a2+n3a3 to get the crystal structure or the
DIRECT LATTICE

5. Diamond & Zincblende Structures
We’ve seen: Many common semiconductors have
Diamond or Zincblende crystal structures
Tetrahedral coordination: Each atom has 4 nearest-neighbors (nn). Basis set: 2 atoms. Primitive lattice  face centered cubic (fcc).
Diamond or Zincblende  2 atoms per fcc lattice point.
Diamond: The 2 atoms are the same. Zincblende: The 2 atoms are different.
The Cubic Unit Cell looks like

6. Zincblende/Diamond Lattices
The Cubic Unit Cell
Zincblende Lattice
The Cubic Unit Cell
Other views of the cubic unit cell

7. Diamond Lattice
Diamond Lattice
The Cubic Unit Cell

8. Zincblende (ZnS) Lattice
Zincblende Lattice
The Cubic Unit Cell.

9.  face centered cubic (fcc) lattice with a 2 atom basis
View of tetrahedral coordination & 2 atom basis:
Zincblende/Diamond
 face centered cubic (fcc)
lattice with a 2 atom basis

10. Wurtzite Structure Wurtzite Structure
We’ve also seen: Many semiconductors have the
Wurtzite Structure
Tetrahedral coordination: Each atom has 4 nearest-neighbors (nn). Basis set: 2 atoms. Primitive lattice  hexagonal close packed (hcp).
2 atoms per hcp lattice point
A Unit Cell looks like

11. Wurtzite Lattice Wurtzite  hexagonal close packed (hcp) lattice,
2 atom basis
View of tetrahedral coordination & 2 atom basis.

12. Diamond & Zincblende crystals
The primitive lattice is fcc. The fcc primitive lattice is generated by r = n1a1+n2a2+n3a3.
The fcc primitive lattice vectors are:
a1 = (½)a(0,1,0), a2 = (½)a(1,0,1), a3 = (½)a(1,1,0)
NOTE: The ai’s are NOT mutually orthogonal!
Diamond:
2 identical atoms per fcc point
Zincblende:
2 different atoms per fcc point
Primitive fcc lattice
cubic unit cell

13. NOTE! These are NOT mutually
primitive lattice points
Wurtzite Crystals
The primitive lattice is hcp. The hcp primitive lattice is generated by
r = n1a1 + n2a2 + n3a3.
The hcp primitive lattice vectors are:
a1 = c(0,0,1)
a2 = (½)a[(1,0,0) + (3)½(0,1,0)]
a3 = (½)a[(-1,0,0) + (3)½(0,1,0)]
NOTE! These are NOT mutually
orthogonal!
2 atoms per hcp point
Primitive hcp lattice
hexagonal unit cell

14. bandstructure calculations.
Group Theory
Applications:
It is used to simplify the computational effort necessary in the highly computational electronic
bandstructure calculations.