1. 3-Dimensional Crystal Structure

2. 3-Dimensional Crystal Structure

3. 3-D Crystal Structure BW, Ch. 1; YC, Ch. 2; S, Ch. 2
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. Reciprocal Lattice Review? BW, Ch. 2; YC, Ch. 2; S, Ch. 2
Motivations: (More discussion later).
The Schrödinger Equation & wavefunctions ψk(r). The solutions for electrons in a periodic potential.
In a 3d periodic crystal lattice, the electron potential has the form:
V(r)  V(r + R) R is the lattice periodicity
It can be shown that, for this V(r), wavefunctions have the form:
ψk(r) = eikr uk(r), where uk(r) = uk(r+R).
ψk(r)  Bloch Functions
It can also be shown that, for r  points on the direct lattice, the wavevectors k  points on a lattice also
 Reciprocal Lattice

15. bi  2π(aj  ak)/Ω K = ℓ1b1+ ℓ2b2 + ℓ3b3
Reciprocal Lattice: A set of lattice points defined in terms of the (reciprocal) primitive lattice vectors b1, b2, b3.
b1, b2, b3 are defined in terms of direct primitive lattice vectors a1, a2, a3 as
bi  2π(aj  ak)/Ω
i,j,k, = 1,2,3 in cyclic permutations, Ω = direct lattice primitive cell volume Ω  a1(a2  a3)
Reciprocal lattice geometry clearly depends on direct lattice geometry!
Generate the reciprocal lattice by forming all possible reciprocal lattice vectors: (ℓ1, ℓ2, ℓ3 = integers)
K = ℓ1b1+ ℓ2b2 + ℓ3b3

16. The First Brillouin Zone (BZ)
 The region in k space which is the smallest polyhedron confined
by planes bisecting the bi’s
The symmetry of the 1st BZ is determined by the symmetry of direct lattice.
The reciprocal lattice to the fcc direct lattice is the body centered cubic (bcc) lattice.
It can be easily shown that the bi’s for this are
b1 = 2π(-1,1,1)/a b2 = 2π(1,-1,1)/a
b3 = 2π(1,1,1)/a

17. The 1st BZ for the fcc lattice (the primitive cell for the bcc k space lattice) looks like:
b1 = 2π(-1,1,1)/a
b2 = 2π(1,-1,1)/a
b3 = 2π(1,1,1)/a

18. [100]  ΓΔX , [111]  ΓΛL , [110]  ΓΣK
For the energy bands: Now discuss the labeling conventions for the high symmetry BZ points
Labeling conventions
The high symmetry points on the
BZ surface  Roman letters
The high symmetry directions
inside the BZ  Greek letters
The BZ Center  Γ  (0,0,0)
The symmetry directions:
[100]  ΓΔX , [111]  ΓΛL , [110]  ΓΣK
We need to know something about these to understand how to interpret energy bandstructure diagrams: Ek vs k

19. Detailed View of BZ for Zincblende Lattice
 [110]  ΓΣK
[100]  ΓΔX 
 [111]  ΓΛL
To understand & interpret bandstructures, you need to be familiar with the high symmetry directions in this BZ!

20.  Using symmetry can save computational effort!!
The fcc 1st BZ: Has High Symmetry! A result of the high symmetry of direct lattice
The consequences for the bandstructures:
If 2 wavevectors k & k in the BZ can be transformed into each other by a symmetry operation  They are equivalent!
e.g. In the BZ figure: There are 8 equivalent BZ faces  When computing Ek one need only compute it for one of the equivalent k’s
 Using symmetry can save computational effort!!

21. OPTICAL & other SELECTION RULES
Consequences of BZ symmetries for bandstructures
Wavefunctions ψk(r) can be expressed such that they have definite transformation properties under crystal symmetry operations.
QM Matrix elements of some operators O:
such as <ψk(r)|O|ψk(r)>, used in calculating probabilities for transitions rom one band to another when discussing optical & other properties (later in the course), can be shown by symmetry to vanish:
 Some transitions are forbidden. This gives
OPTICAL & other SELECTION RULES

22. Math of High Symmetry 22 pages in YC (Sect. 2.3)!
The Math tool for all of this is GROUP THEORY
An extremely powerful & important tool for understanding & simplifying the properties of crystals of high symmetry!
22 pages in YC (Sect. 2.3)!
Read on your own!
Most is not needed for this course!
However, we will now briefly introduce some simple group theory notation & discuss some simple, relevant symmetries!

23. C2 = π (180°), C3 = (⅔)π (120°), C4 = (½)π (90°), C6 = (⅓)π (60°)
Group Theory Notation: Crystal symmetry operations (which transform the crystal into itself)
Operations Relevant for the diamond & zincblende lattices
E  Identity operation
Cn  n-fold rotation  Rotation by (2π/n) radians
C2 = π (180°), C3 = (⅔)π (120°), C4 = (½)π (90°), C6 = (⅓)π (60°)
σ  Reflection symmetry through a plane
i  Inversion symmetry
Sn  Cn rotation, followed by a reflection through a plane  to the rotation axis
σ, I, Sn  “Improper rotations”
Also: All of these have inverses!

24. Crystal Symmetry Operations
For Rotations: Cn, we need to specify the rotation axis
For Reflections: σ, we need to specify reflection plane
We usually use Miller indices (from SS physics)
k, ℓ, n  integers
For Planes: (k,ℓ,n) or (kℓn): The plane containing
the origin & is  to the vector [k,ℓ,n] or [kℓn]
For Vector directions: [k,ℓ,n] or [kn]: The
vector  to the plane (k,ℓ,n) or (kℓn)
Also: k (bar on top)  - k, ℓ (bar on top)  -ℓ, etc.

25. Rotational Symmetries of the CH4 Molecule The Td Point Group
Rotational Symmetries of the CH4 Molecule The Td Point Group! The same as for diamond & zincblende crystals!

26. Diamond & Zincblende Symmetries ~ CH4
HOWEVER, diamond has even more symmetry, since the 2 atom basis is made from 2 identical atoms.
The diamond lattice has more translational symmetry than the zincblende lattice

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