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 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.

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

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.

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

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.

Group: Create Wigner-Seitz cell of the graphene lattice X 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)

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.

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

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)]

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.

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

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

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.

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. 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.

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

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.

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.

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?

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.

Octahedrals connect interstitial sites

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

AX Crystal Structures AX–Type Crystal Structures include NaCl, CsCl, and zinc blende Cesium Chloride structure:  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 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

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?

Octahedral Tilting

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

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.