Crystal Structure123454321ghjklmnbvc.ppt

EEE539 Solid State Electronics
1. Crystal Structure
Issues that are addressed in this chapter
include:
 Periodic array of atoms
 Fundamental types of lattices
 Index system for crystal planes
 Simple crystal structures
 Imaging of atomic structure
 Non-ideal structures

1.1 Periodic Array of Atoms
Crystals are composed of a periodic array of
atoms:
 The structure of all crystals can be described in
terms of a lattice, with a group of atoms attached
to each lattice point called basis:
basis + lattice = crystal structure
+ =

 The lattice and the lattice translation vectors a1,
a2, and a3 are primitive if any two points satisfy:
where u1, u2 and u3 are integers.
 The primitive lattice translation vectors specify
unit cell of smallest volume.
 A lattice translation operator is defined as a
displacement of a crystal with a crystal
translation operator.
 To describe a crystal, it is necessary to specify
three things:
1. What is the lattice
2. What are the lattice translation vectors
3. What is the basis
3
3
2
2
1
1 a
a
a
r
r' u
u
u 



3
3
2
2
1
1 a
a
a
T u
u
u 



 A basis of atoms is attached to a lattice point
and each atom in the basis is specified by:
where 0  xj, yj, zj  1.
The basis consists of one or several atoms.
 The primitive cell is a parallelepiped specified
by the primitive translation vectors. It is a
minimum volume cell and there is one lattice
point per primitive cell.
 The volume of the primitive cell is:
 Basis associated with a primitive cell is called a
primitive basis and contains the least # of
atoms.
3
2
1 a
a
a
r j
j
j
j z
y
x 


 
3
2
1 a
a
a
V 


c

1.2 Fundamental types of lattices
To understand the various types of lattices, one
has to learn elements of group theory:
 Point group consists of symmetry operations in
which at least one point remains fixed and
unchanged in space. There are two types of
symmetry operations: proper and improper.
 Space group consists of both translational and
rotational symmetry operations of a crystal. In
here:
T = group of all translational symmetry opera-
tions
R = group of all symmetry operations that involve
rotations.

The most common symmetry operations are
listed below:
 C2 = two-fold rotation or a rotation by 180°
 C3 = three-fold rotation or a rotation by 120°
 C4 = four-fold rotation or a rotation by 90°
 C6 = six-fold rotation or a rotation by 180°
  = reflection about a plane through a lattice point
 i = inversion, I.e. rotation by 180° that is followed
by a reflection in a plane normal to rotation axis.
Two-dimensional lattices, invariant under C3, C4 or C6
rotations are classified into five categories:
Oblique lattice
Special lattice types
(square, hexagonal, rectangular and
centered rectangular)

Three-dimensional lattices – the point symmetry
operations in 3D lead to 14 distinct types of lattices:
 The general lattice type is triclinic.
 The rest of the lattices are grouped based on the
type of cells they form.
 One of the lattices is a cubic lattice, which is
further separated into:
- simple cubic (SC)
- Face-centered cubic (FCC)
- Body-centered cubic (BCC)
 Note that primitive cells by definition contain one
lattice point, but the primitive cells of FCC lattice
contains 4 atoms and the primitive cell of BCC
lattice contains 2 atoms.

 The primitive translation vectors for cubic lattices
are:
simple cubic
face-centered cubic
body-centered cubic








z
a
y
a
x
a
a
a
a
3
2
1
 
 
 











x
z
a
z
y
a
y
x
a
a
a
a
2
1
3
2
1
2
2
1
1
 
 
 















z
y
x
a
z
y
x
a
z
y
x
a
a
a
a
2
1
3
2
1
2
2
1
1

1.3 Index System for Crystal Planes – Miler Indices
The orientation of a crystal plane is determined
by three points in the plane that are not
collinear to each other.
It is more useful to specify the orientation of a
plane by the following rules:
 Find the intercepts of the axes in terms of lattice
constants a1, a2 and a3.
 Take a reciprocal of these numbers and then
reduce to three integers having the same ratio.
The result (hkl) is called the index of a plane.
 Planes equivalent by summetry are denoted in
curly brackets around the indices {hkl}.

Miller indices for direction are specified in the
following manner:
 Set up a vector of arbitrary length in the direction
of interest.
 Decompose the vector into its components along
the principal axes.
 Using an appropriate multiplier, convert the
component values into the smallest possible
whole number set.
[hkl] – square brackets are used to designate
specific direction within the crystal.
<hkl> - triangular brackets designate an
equivalent set of directions.

The calculation of the miller indices using
vectors proceeds in the following manner:
 We are given three points in a plane for which
we want to calculate the Miller indices:
P1(022), P2(202) and P3(210)
 We now define the following vectors:
r1=0i+2j+2k, r2=2i+0j+2k, r3=2i+j+0k
and calculate the following differences:
r - r1=xi + (2-y)j + (2-z)k
r2 - r1=2i - 2j + 0k
r3-r1 = 2i – j - 2k
 We then use the fact that:
(r-r1).
[(r2-r1) ×(r3-r1)] =A.
(B×C)= 0

 We now use the following matrix representation,
that gives
The end result of this manipulation is an equation of
the form:
4x+4y+2z=12
 The intercepts are located at:
x=3, y=3, z=6
 The Miller indices of this plane are then:
(221)
0
2
1
2
0
2
2
2
2
)
(
3
2
1
3
2
1
3
2
1










z
y
x
C
C
C
B
B
B
A
A
A
C
B
A

The separation between adjacent planes in a
cubic crystal is given by:
The angle between planes is given by:
2
2
2
l
k
h
a
d



  
2
2
2
2
2
2
2
1
2
1
2
1
2
1
2
1
2
1
cos
l
k
h
l
k
h
l
l
k
k
h
h









1.4 Simple Crystal Structures
There are several types of simple crystal
structures:
 Sodium Chloride (NaCl) -> FCC lattice, one Na
and one Cl atom separated by one half the body
diagonal of a unit cube.
 Cesium Chloride -> BCC lattice with one atom of
opposite type at the body center
 Hexagonal Closed packed structure (hcp)
 Diamond structure -> Fcc lattice with primitive
basis that has two identical atoms
 ZnS -> FCC in which the two atoms in the basis
are different.

1.5 Imaging of Atomic Structure
The direct imaging of lattices is accomplished with TEM.
One can see, for example, the density of atoms along
different crystalographic directions.

1.6 Non-ideal Crystal Structures
There are two different types of non-idealities in
the crystalline structure:
Random stacking – The structure differs in stacking
sequence of the planes. For example FCC has the
sequence ABCABC …, and the HCP structure has the
sequence ABABAB … .
Polytypism – The stacking sequence has long repeat
unit along the stacking axis.
 Examples include ZnS and SiC with more than 45
stacking sequences.
 The mechanisms that induce such long range
order are associated with the presence of spiral
steps due to dislocations in the growth nucleus.

Crystal Structure123454321ghjklmnbvc.ppt

More Related Content

Similar to Crystal Structure123454321ghjklmnbvc.ppt (20)

More from TibyanKhan (9)

Recently uploaded (20)

Crystal Structure123454321ghjklmnbvc.ppt