Structure of Solids
Objectives
By the end of this section you should be able to:
• Understand typical ionic crystal structure
• Be able to define the primitive unit cell for both
graphene and graphite
• Differentiate wurtzite and zinc blende structures
• Define the perovskite crystal structure & why it’s
the most commonly studied oxide structure
Metallic Crystal Structures
Tend to be densely packed.
Several reasons for dense packing:
- Metallic bonding is not directional.
- Nearest neighbor distances tend to be
small in order to lower bond energy.
- The “electron cloud” shields cores from
each other
Metals have the simplest crystal structures.
FCC
FCC and HCP have very similar lattice energies
No clear cut trends
Why Aren’t All Systems Close-Packed?
• Even with marbles, closed packing is not the only
consideration. (Here: confined space)
• In crystals: bond distances, types and strengths
• Also not all atoms have spherical symmetry
Other non close-packed structures
• In covalently bonded materials, bond direction is
more important than packing
What is the atomic # of C?
diamond (only 34 % packing)
graphite
How many
bonds would
you expect?
Not all bonds shown
Group: Create Wigner-Seitz cell of this
hexagonal lattice
How else might you
define the unit cell?
1. Pick an origin
2. Draw perp. bisector to all neighboring lattice points
3. Draw smallest area enclosed by bisectors
Group: Create Wigner-Seitz cell
of the graphene lattice
Graphene
y
α
a2
O
a) Situation of atoms at the
corners of regular hexagons
a1
x
b) Crystal lattice obtained by
identifying all the atoms in (a)
8
Group Exercise
• How many atoms are in the primitive
unit cell of graphite? Identify a unit cell.
Close-packed structures: fcc and hcp
hcp
ABABAB...
fcc
ABCABCABC...
In groups, build these two
differing crystal structures.
HCP vs FCC
In both the (a) ABA and (b) ABC close-packed arrangements,
the coordination number of each atom is 12.
Diamond & Zincblende Crystal
Structure
• Basis set: 2 atoms. Lattice  face centered cubic (fcc).
• The fcc primitive lattice is generated by r = n1a1+n2a2+n3a3
with lattice vectors:
a1 = a(0,1,0)/2, a2 = a(1,0,1)/2, a3 = a(1,1,0)/2
NOTE: The ai’s are NOT mutually orthogonal!
Diamond:
2 identical atoms in basis (e.g. 2 C)
fcc lattice
Zincblende:
2 different atoms in basis and fcc lattice
For FCC 2 atom ABCABC stacking, it is called zinc blende
For ABAB… stacking it is called wurzite structure (fcc zincblende was ABCABC…)
Some compounds can have either structure (i.e., GaN, SiC)
Many semiconductors have the
Wurtzite Structure
Tetrahedral coordination: Each atom has 4 nearest-neighbors (nn).
Basis set: 2 atoms. Lattice  hexagonal close packed (hcp).
A Unit Cell looks like
hcp primitive lattice vectors :
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)]
Different planes in FCC
Top views
Surface unit cells
Holes in Close Packed Crystals
Two types of
holes created by a
close-packed
arrangement.
Octahedral holes
lie within two
staggered triangular
planes of atoms.
The coordination number of an atom
occupying an octahedral hole is 6.
Holes in Close Packed Crystals
Tetrahedral holes are formed by a
planar triangle of atoms, with a 4th
atom covering the indentation in the
center. The resulting hole has a
coordination number of 4.
Surface relaxation
• Once the initial slab geometry is set, the
system is then subjected to geometry
optimization, i.e., the atoms within the
supercell are allowed to adjust their positions
such that the atomic forces are close to zero
• Surface relaxation: a general phenomenon, in
which the interplanar distances normal to the
free surface change with respect to the bulk
value.
Surface reconstruction
• Relaxation: movement of atoms normal to the
surface plane
• Reconstruction: movement of atoms along the
surface plane
The unreconstructed Si (001) surface
What kind of crystal structure is this?
Surface unit cell
Si Surface Reconstruction
Reconstructed (001) surface
Unreconstructed (001) surface
Why does this reconstruction happen?
To “passivate” dangling bonds
Does the surface have as many nearest neighbors?
Ionic materials
(Transferred Electron)
• In ionic materials, different
considerations can be important
(electrostatics, different size of ions)
• Figure shows the crystal structure of Cs+Cl-. The
lattice constant is 4.12 Å and all the bonds shown
have the same length. The grey atoms are Cs and
the green ones are Cl.
• Group: Define crystal structure: meaning what is
the primitive Bravais lattice and the associated
basis for this crystal (including the locations of
these atoms in terms of lattice parameter a)?
• What is the angle between the chemical bonds?
Cesium Chloride Structure Cs+Cl• Simple cubic lattice with a basis consisting of a
cesium ion at the origin 0 and a chlorine ion
at the cube center



a / 2( x  y  z )
• CsBr and CsI crystallize in this structure. The
lattice constants are in the order of 4
angstroms.
NaCl (Rock Salt) Structure
• In NaCl the small Na are in
interstitial positions
between the Cl ions
• Group: Define the crystal
structure
Cs+Cl-
For comparison
Octahedrals connect interstitial sites
Simple Crystal Structures
NaCl
• NaCl: interpenetrating fcc
structures
– One atom at (0,0,0)
– Second atom displaced by
(1/2,0,0)
• Majority of ionic crystals
prefer NaCl structure despite
lower coordination (fewer NN)
– Radius of cations much smaller
than anions typically
– For very small cations, anions
can not get too close in the
other typical structure (CsCl)
– This favors NaCl structure where
anion contact does not limit
structure as much
Predicting Crystal Structures
unreliable
r cation
Coordination # increases with
r anion
unreliable
rNa/rCl = 0.564
moderately
reliable
quite reliable
CsCl ion radius ratio
Cesium Chloride structure:
rCs 
rCl 
0.170

 0.939
0.181
 Since 0.732 < 0.939 < 1.0,
cubic sites preferred
So each Cs+ has 8 neighbor Cl-
Define the Crystal Structure of Perovskites
A-site (Ca)
Oxygen
• Superconductors
• Ferroelectrics
CaTiO3
(BaTiO3)
• Colossal Magnetoresistance (LaSrMnO3)
B-site (Ti)
eg
• Multiferroics
(BiFeO3)
• High εr Insulators
(SrTiO3)
• Low εr Insulators
(LaAlO3)
• Conductors
(Sr2RuO4)
• Thermoelectrics
(doped SrTiO3)
• Ferromagnets
(SrRuO3)
t2g
 Perovskite formula – ABO3
A atoms at the corners
 B atoms (smaller) at the body-center
 O atoms at the face centers
PEROVSKITES
A-site (Ca)
Oxygen
CaTiO3
B-site (Ti)
• Lattice: Simple Cubic (idealized cubic structure)
• 1 CaTiO3 per unit cell
• Cell Motif: Ti at (0, 0, 0); Ca at (1/2, 1/2, 1/2); 3 O at
(1/2, 0, 0), (0, 1/2, 0), (0, 0, 1/2) could label differently
• Ca 12-coordinate by O, Ti 6-coordinate by O, O
distorted octahedral
Is this cube a primitive lattice?
Octahedral Tilting
How could you measure this?
Test 1: Oct 6 (Chapters 1-6 & 8)
One page of notes is strongly encouraged.
• Test questions will not be as hard as the Ashcroft
homework problems, more similar to others
• I will not make you solve super hard integrals
• Look back at learning objectives from each class
I like to test a range of skills, so:
• Expect to have to calculate some numbers
• Expect to derive some things
• Expect some of it to be conceptual
• Expect to need to be able to define a crystal
structure and/or lattice and its reciprocal