Transcript Crystal Structure

II. Crystal Structure
A. Lattice, Basis, and the Unit Cell
B. Common Crystal Structures
C. Miller Indices for Crystal Directions and
Planes
D. The Reciprocal Lattice
A. Lattice, Basis, and Unit Cell
An ideal crystalline solid is an infinite repetition of identical
structural units in space. The repeated unit may be a single atom
or a group of atoms.
An important concept:
crystal structure
=
=
lattice
+
+
basis
lattice: a periodic array of points in space. The environment
surrounding each lattice point is identical.
basis: the atom or group of atoms “attached” to each lattice
point in order generate the crystal structure.
The translational symmetry of a lattice
is given by the base
  
vectors or lattice vectors a, b , c . Usually these vectors
are chosen either:
1. to be the shortest possible vectors, or
2. to correspond to a high symmetry unit cell
Example: a 2-D lattice
These two choices of lattice vectors illustrate two types of unit cells:

b

a
Conventional (crystallographic) unit cell:
larger than primitive cell; chosen to
display high symmetry unit cell

b

a
Primitive unit cell: has minimum volume
and contains only one lattice point
A lattice translation vector connects two points in the lattice that have
identical symmetry: 



r  n1a  n2 b  n3c
n1 n2 n3  integers
 
a b


a  2b
In our 2-D lattice:

b

a
B. Common Crystal Structures
2-D
only 5 distinct point lattices that can fill all space
3-D
only 14 distinct point lattices (Bravais lattices)
The 14 Bravais lattices can be subdivided into 7 different “crystal classes”,
based on our choice of conventional unit cells (see text, handout).
Attaching a basis of atoms to each lattice point introduces new types of
symmetry (reflection, rotation, inversion, etc.) based on the arrangement of
the basis atoms. When each of these “point groups” is combined with the
14 possible Bravais lattices, there are a total of 230 different possible
“space groups” in 3-D. We will focus on the few that are common for
metals, semiconductors, and simple compounds.
Crystal Structure Diagrams
(a) NaCl
(b) CsCl
(c) fluorite
(d) perovskite
(e) Laves phase
(f) A15
Crystal Structure Diagrams
(continued)
hexagonal close
packed (Be, Mg, Zn)
diamond structure
(C, Si, Ge)
Analysis of Common Crystal Structures
1. NaCl structure (many ionic solids)
lattice: face-centered cubic (fcc)
basis: Na at 000, Cl at ½½½
2. CsCl structure (some ionic solids and intermetallic alloys)
lattice: simple cubic (sc)
basis: Cs at 000, Cl at ½½½
Common Crystal Structures, cont’d
3. hexagonal-close-packed (divalent metals)
lattice: hexagonal
basis: 000, 2/3 1/3 1/2
(see text for an alternate choice of lattice and basis)
4. diamond structure (C, Si, Ge)
lattice: face-centered-cubic (fcc)
basis: 000, ¼¼¼
5. zincblende structure (ZnS, GaAs, InP, compound semicond’s)
lattice: face-centered-cubic (fcc)
basis: Zn at 000, S at ¼¼¼
C. Miller Indices for Crystal Directions & Planes
Because crystals are usually anisotropic (their properties differ
along different directions) it is useful to regard a crystalline
solid as a collection of parallel planes of atoms.
Crystallographers and CM physicists use a shorthand notation
(Miller indices) to refer to such planes.
z
1. Determine intercepts (x, y, z)
of the plane with the coordinate
axes
z=3
y=2
y
x
x=1
C. Miller Indices,cont’d.
2. Express the intercepts as multiples of the base vectors of the lattice

In this example, let’s assume that the lattice is given by: a  1iˆ
Then the intercept ratios become:
3. Form reciprocals:
a 1
 1
x 1
x 1
 1
a 1
b 1

y 2
y 2
 2
b 1

b  1 ˆj

c  3kˆ
z 3
 1
c 3
c 1
 1
z 1
4. Multiply through by the factor that allows you to express these indices as
the lowest triplet of integers:
2  (1 12 1)  (212)
We call this the (212) plane.
Another example
z
Find the Miller indices of
the shaded plane in this
simple cubic lattice:
a
a
x
Intercepts:
Intercept ratios:
Reciprocals:
x
y
ya

a  aiˆ

b  aˆj

c  akˆ
a
z
x

a
y
1
a
z

a
a
0
x
a
1
y
a
0
z
non-intersecting  intercept at 
We call this the (010) plane.
Note: (hkl) = a single plane; {hkl} = a family of symmetry-equivalent planes
Crystal Planes and Directions
Crystal directions are
specified [hkl] as the
coordinates of the lattice
point closest to the origin
along the desired direction:
z
[001]
y
x
Note: [hkl] = a specific direction;
<hkl> = a family of symmetryequivalent directions
[100]
[001]
Note that for cubic lattices, the direction
[hkl] is perpendicular to the (hkl) plane
[010]
D. The Reciprocal Lattice
Crystal planes (hkl) in the real-space or direct lattice are characterized by the
normal vector nˆ hkl and the interplanar spacing d hkl :
z
y
nˆ hkl
x
d hkl
Long practice has shown CM physicists the usefulness of defining a different
lattice in reciprocal space whose points lie at positions given by the vectors

2nˆhkl
Ghkl 
d hkl
This vector is parallel to the
[hkl] direction but has
magnitude 2/dhkl, which is a
reciprocal distance
The Reciprocal Lattice, cont’d.

The reciprocal lattice is composed of all points lying at positions Ghkl from
the origin, so that there is one point in the reciprocal lattice for each set of
planes (hkl) in the real-space lattice.
This seems like an unnecessary abstraction. What is the payoff for defining such
a reciprocal lattice?
1.
The reciprocal lattice simplifies the interpretation of x-ray diffraction from
crystals (coming soon in chapter 3)
2.
The reciprocal lattice facilitates the calculation of wave propagation in
crystals (lattice vibrations, electron waves, etc.)
The Reciprocal Lattice: An Analogy
In the analysis of electrical signals that are periodic in time, we use Fourier
analysis to express such a signal in the frequency domain:
f (t )   C eit

If f(t) has period T, then the coefficient C is
nonzero only for frequencies given by

2n
T
n = integer
Waves of lattice vibrations or electron waves
moving through a crystal with a
  
periodicity specified by base vectors a b c can likewise be decomposed into a
sum of plane waves:

Here, the coefficient Ck is nonzero only

i ( k r t )
 (r , t )   Ck e
when the vector k is a reciprocal lattice
k
translation vector:

 
  
k  Ghkl  hA  kB  lC
A, B, and C are the base vectors of the
reciprocal lattice (some books use a*, b*, c*)
Definition of Reciprocal Lattice Base Vectors
These reciprocal lattice base vectors are defined:
 
 2 b  c
A   
a  b c


 2 c  a
B   
a  b c




 2 a  b
C   
a  b c


Which have the simple dot products with the direct-space lattice vectors:
     
A  a  B  b  C  c  2
   
       
A b  A c  0  B  c  B  a  C  a  C  b
 
Ghkl  a  2h
 
So compare, for example: T  2n
Ghkl  b  2k
 
frequency  time
Ghkl  c  2l
Reciprocal lattice direct lattice
Remember:
Problems worthy
of attack
Prove their worth
by hitting back
--Piet Hein