1. 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

2. 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
Densely packed means often close packed or close to it.
Typically, only one element is present, so all atomic radii are the same.
Directional bonding means some electrons are shared and/or exchanged. Those electrons require space to sit in, which can push the bonded atoms part a bit more than compared to if the electrons are free to move through the crystal like a metal.
Metals have the simplest crystal structures.

3. FCC and HCP have very similar lattice energies
Remember that simple cubic is quite rare.
FCC is also called cubic close packed (CCP).
FCC
FCC and HCP have very similar lattice energies
No clear cut trends

4. 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
If we took a bunch of marbles, we wouldn’t expect to see them stack this way.

5. 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)
How many bonds would you expect?
graphite
CH2 only has two neighbors!
The excited state promotes one 2s electron to 2p and this energy can be more than offset through bonding. Why do we get energy from food? From breaking bonds. So these bonds have some energy associated with them.
If all bonds shown on diamond structure, the figure gets a little confusing. Instead, I highlight the three where you can see all four bonds when drawing only a single cube. However, all atoms have four neighbors.
sp2 and sp3 bonds are shown as lines
Diamond structure likes to have 4 nearest neighbors. Note even worse packing than simple cubic, which wasn’t very good.
Graphite prefers 3.
Note that these both can be made of the same element! For why this happens, we’ll have to wait until later in the class.
Not all bonds shown

6. Group: Create Wigner-Seitz cell of this hexagonal lattice
In 3-d, think about a polyhedron
Before we do graphene, this remind ourselves of a simple hexagonal lattice with one atom in the basis.

7. Group: Create Wigner-Seitz cell of the graphene latticeX
Group: Create Wigner-Seitz cell of the graphene lattice
Graphene α a1 a2 O y x
a) Situation of atoms at the corners of regular hexagons
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.
Here we see another option for the unit cell in one dimension (just a mirror reflection). Still other options.
Can see this through a rotation of the layers by 120 degrees.
M1.4: Graphite = a staggered arrangement of stacked hexagonal layers
Answer: 4
Hard to see. Not one from corner on bottom and one inside, then halfway up, one from corners and one inside.

9. 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

10. Many semiconductors have the
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)]

11. Close-packed structures: fcc and hcp
ABABAB...
fcc
ABCABCABC...
Let’s start in a plane. There it’s really simple
fcc face centred cubic
hcp hexagonal close-packed
If time allows:
In groups, build these two differing crystal structures.

12. HCP vs FCC
In both the (a) ABA and (b) ABC close-packed arrangements, the coordination number of each atom is 12.

13. Different planes in FCC
Top views
If you were an electron moving through the crystal, which one of these planes do you think you’d prefer to move along and in what direction? Where would you get the least resistance?
Why direction can strongly affect the properties along certain directions

14. 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.
For n atoms in a close-packed structure, there are n octahedral holes.
The coordination number of an atom occupying an octahedral hole is 6.

15. 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.

16. 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.
We will talk more about surface effects later, but I thought this was good spot to introduce the idea of why the structure can deviate at the surface.

17. Surface reconstruction
Relaxation: movement of atoms normal to the surface plane
Reconstruction: movement of atoms along the surface plane

18. The unreconstructed Si (001) surface What kind of crystal structure is this?
Surface unit cell

19. Si Surface Reconstruction
Reconstructed (001) surface
Unreconstructed (001) surface
Why does this reconstruction happen?
To “passivate” dangling bonds

20. 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 are 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?
(1V3) simple cubic with two atom basis: (0 0 0) and (½ ½ ½). Need to identify which atom is at which. Though, it doesn’t matter which you pick, as you could do either, but a full definition of the crystal structure defines the lattice vectors, the atoms positions and atom types.

21. 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
CsBr and CsI crystallize in this structure. The lattice constants are in the order of 4 angstroms.

22. NaCl (Rock Salt) Structure
Cs+Cl-
In NaCl the small Na are in interstitial positions between the Cl ions
Group: Define the crystal structure
For comparison
A more common ionic structure
The negative ions are in general much bigger than the positive ions because of the smaller nuclear charge.
So the positive ions can be fit into a close-packed lattice of negative ions as an interstitial alloy.
Can you see the fcc crystal / two fcc into each other?

23. 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
Anions have extra electrons which make them bigger than a neutral atom of the same element.

24. Octahedrals connect interstitial sites

25. Predicting Crystal Structures
c03tf03
unreliable r
cation
anion
Coordination # increases with
unreliable
moderately reliable
rNa/rCl = 0.564
quite reliable

26. AX Crystal Structures
AX–Type Crystal Structures include NaCl, CsCl, and zinc blende
Cesium Chloride structure:
 Since < < 1.0, cubic sites preferred
So each Cs+ has 8 neighbor Cl-

27. Define the Crystal Structure of Perovskites
A-site (Ca)
Oxygen
B-site (Ti)
CaTiO3
Superconductors
Ferroelectrics (BaTiO3)
Colossal Magnetoresistance (LaSrMnO3)
Multiferroics (BiFeO3)
High εr Insulators (SrTiO3)
Low εr Insulators (LaAlO3)
Conductors (Sr2RuO4)
Thermoelectrics (doped SrTiO3)
Ferromagnets (SrRuO3) eg
t2g
In addition to being among the most abundant minerals on earth, complex oxides give some of the most varied and interesting properties. These include their use as dielectric and superconducting materials. Yet, only recently has the research in the field of complex oxides flourished because they were long thought to be …well complex. This complexity comes from the strong coupling with charge, spin, and lattice dynamics, which often results in very full phase diagrams.
Though the coupling may lead to complex behaviors, the structure of these materials can be quite simple, such as the perovskite form shown here, where the A and B sites are typically different cations and X is an anion that bonds to both. This octahedral arrangement (imagine connecting the oxygens) gives rise to a crystal field potential, hinders the free rotation of the electrons and quenches the orbital angular momentum by introducing the crystal field splitting of the d orbitals. Even among only perovskite structures, we can see these varied behaviors.
Perovskite formula – ABO3
A atoms at the corners
B atoms (smaller) at the body-center
O atoms at the face centers

28. Is this cube a primitive lattice?
PEROVSKITES
A-site (Ca)
Oxygen
B-site (Ti)
CaTiO3
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
Motif is another word for basis.
Is it a bravais lattice? Yes, because it’s simple cubic. Don’t confuse atoms with lattice points.
Yes primitive—only one lattice point per unit cell. Don’t confuse lattice points with atoms, of which there are many.
Anion doesn’t have to be oxygen, even though that is common. For example, fluorides are also of interest. Current theory suggests that the properties in fluorides might be even better than oxides.
Is this cube a primitive lattice?

29. Octahedral Tilting

30. Test 1: February 26 (Chapters 1-6) One page of notes is strongly encouraged.
Test questions will not be as hard as the homework from Chapter 1
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

31. Octahedral vs Tetrahedral Environment
Only makes sense to use this slide if going to show spinel crystal structure, which I decided might be too complicated.