Download Part VI - TTU Physics

3-Dimensional
Crystal
Structure
3-Dimensional Crystal Structure
3-D Crystal Structure
BW, Ch. 1; YC, Ch. 2; S, Ch. 2
(see lectures from Physics 4309-5304 for MANY more details!
http://www.phys.ttu.edu/~cmyles/Phys4309-5304/lectures1.html)
• 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.
• The lattice is generated 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).
• 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 of the
DIRECT LATTICE
Diamond & Zincblende Structures
• 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: 2 atoms are the same. Zincblende: 2 atoms are
different. The Cubic Unit Cell looks like
Zincblende/Diamond Lattices
Diamond Lattice
The Cubic Unit Cell
Zincblende Lattice
The Cubic Unit Cell
Other views of the cubic unit cell
Diamond Lattice
Diamond Lattice
The Cubic Unit Cell
Zincblende (ZnS) Lattice
Zincblende Lattice
The Cubic Unit Cell.
• View of tetrahedral coordination & 2 atom basis:
Zincblende/Diamond
 face centered cubic (fcc)
lattice with a 2 atom basis
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
Wurtzite Lattice
Wurtzite  hexagonal close
packed (hcp) lattice,
2 atom basis
View of tetrahedral coordination
& 2 atom basis.
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!
Primitive fcc lattice
cubic unit cell
Diamond:
2 identical atoms per fcc point
Zincblende:
2 different atoms per fcc point
Wurtzite Crystals
primitive lattice points
• 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
Primitive hcp lattice
orthogonal!
hexagonal unit cell
• Wurtzite Crystals
2 atoms per hcp point
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
• 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 the 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)
• The reciprocal lattice geometry clearly depends on direct lattice
geometry!
• The reciprocal lattice is generated by forming all
possible reciprocal lattice vectors: (ℓ1, ℓ2, ℓ3 = integers)
K = ℓ1b1+ ℓ2b2 + ℓ3b3
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. It can easily be shown that:
The reciprocal lattice to the fcc direct lattice
is the body centered cubic (bcc) lattice.
• It can also 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
• 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
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
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!
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.
• 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 from one band to another
when discussing optical & other properties (later in the
course), can be shown by symmetry to vanish:
So, some transitions are forbidden. This gives
OPTICAL & other SELECTION RULES
Math of High Symmetry
• The Math tool for all of this is
GROUP THEORY
This is 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.
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.
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.
Rotational Symmetries of the CH4 Molecule
The Td Point Group. The same as for diamond & zincblende crystals
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
Group Theory
• Applications:
It is used to simplify the computational
effort necessary in the highly
computational electronic bandstructure
calculations.