3-Dimensional
Crystal Structure
3-Dimensional Crystal Structure
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.
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
Weve 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
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
Weve 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
View of tetrahedral coordination & 2
atom basis.
Wurtzite hexagonal close
packed (hcp) lattice,
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 = ()a(0,1,0), a2 = ()a(1,0,1), a3 = ()a(1,1,0)
NOTE: The ais are NOT mutually orthogonal!
Diamond:
2 identical atoms per fcc point
Zincblende:
2 different atoms per fcc point
Primitive fcc lattice
cubic unit cell
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 = ()a[(1,0,0) + (3)(0,1,0)]
a3 = ()a[(-1,0,0) + (3)(0,1,0)]
NOTE! These are NOT mutually
orthogonal!
Wurtzite Crystals
2 atoms per hcp point
Primitive hcp lattice
hexagonal unit cell
Reciprocal Lattice
Review? BW, Ch. 2; YC, Ch. 2; S, Ch. 2
Motivations: (More discussion later).
The Schrdinger Equation & wavefunctions k(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:
k(r) = eikr uk(r), where uk(r) = uk(r+R).
k(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 2(aj ak)/
i,j,k, = 1,2,3 in cyclic permutations, = direct lattice
primitive cell volume 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: (1, 2, 3 = integers)
K = 1b1+ 2b2 + 3b3
The First Brillouin Zone (BZ)
The region in k space which is the smallest
polyhedron confined
by planes bisecting the bis
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 bis for this are
b1 = 2(-1,1,1)/a
b2 = 2(1,-1,1)/a
b3 = 2(1,1,1)/a
The 1st BZ for the fcc lattice (the primitive cell
for the bcc k space lattice) looks like:
b1 = 2(-1,1,1)/a
b2 = 2(1,-1,1)/a
b3 = 2(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 (0,0,0)
The symmetry directions:
[100] X , [111] L , [110] K
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]
K
[100] X
[111] L
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 ks
Using symmetry can save computational effort!!
Consequences of BZ symmetries for bandstructures
Wavefunctions k(r) can be expressed such that they have
definite transformation properties under crystal symmetry
operations.
QM Matrix elements of some operators O:
such as <k(r)|O|k(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
Math of High Symmetry
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!
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 (2/n) radians
C2 = (180), C3 = () (120), C4 = () (90), C6 = () (60)
Reflection symmetry through a plane
i Inversion symmetry
Sn Cn rotation, followed by a reflection through a plane to
the rotation axis
, 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: , we need to specify reflection plane
We usually use Miller indices (from SS physics)
k, , n integers
For Planes: (k,,n) or (kn): The plane containing
the origin & is to the vector [k,,n] or [kn]
For Vector directions: [k,,n] or [kn]: The
vector to the plane (k,,n) or (kn)
Also: k (bar on top) - k, (bar on top) -, etc.
Rotational Symmetries of the CH4 Molecule
The Td Point Group! The same as for diamond & zincblende crystals!
Diamond & Zincblende Symmetries ~ CH 4
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.
