Emergent Periodicity in 2D Crystals

In the previous section, there are a few clear behaviors that we observed as we assembled many Lennard-Jones-obeying atoms together:

1. The atoms will cluster together in an attempt to maximize their nearest neighbors - i.e. maximize the number of bonds.
2. All atoms are approximately equidistant from their nearest neighbors, although they vibrate around depending on the system temperature.
3. As we add more atoms, we could see that the new atoms positioned themselves in predictable, regular locations with respect to their nearest neighbors.
4. As we add more atoms, we observed variation in shapes (hexagons, tetrahedrons, etc.) of the assemblies.

The fact that these regular arrangements actually emerge from our simple bonding model is a central phenomenon in matter, and is a structural feature that is critical to many materials systems: crystallinity. A crystal is a periodic, ordered arrangement of atoms or molecules. Crystals posses symmetry with respect to rotation and reflection (among other operations).

Undoubtedly, you've encountered crystals before. Some of the most stunning examples of crystals arise when you get enough atoms (and the right geological conditions) such that they form gems. Figure 5.4.1 Shows some examples from the Chicago Field Museum's Grainger Hall of Gems. These gems are pretty - in part because of the symmetry they possess (I particularly like the faceting in the topaz), but they aren't very often useful in technological applications in their current form. We'll show many more examples in the class, but considering that crystals are a fundamental structural feature of many materials, we'll first focus on how to describe them.

To effectively describe crystals, we must define a number of central concepts in structural regularity. We'll expound on each, but for now, we'll define them, focusing on simple 2D patterns for simplicity. Note - while most materials we come into contact with are three-dimensional, 2D crystals were first produced in 2004 (Novoselov, Geim et al., 2004), but has been followed by dozens upon dozens of single-layer materials. These materials may someday have application in electronics, medicine, energy, and other fields.

Quantifying a Crystal in 2D

To build a crystal, we're going to essentially put atoms and molecules into a shape, and then tesselate that shape in a space-filling pattern. We saw the first of these patterns when observing the assembly of atoms that arise due to Lennard-Jones interactions in the previous section - in which atoms packed closely together is a sort of hexagonal pattern. It was clear that we could draw a shape containing a few atoms that, when repeated in space, produced a periodic, ordered array of atoms over long distances. This is a crystal.

The study of crystals, or crystallography, is an extensive topic. Our goal here is to identify and quantify the smallest repeating unit of a crystal so that we can concisely communicate a crystal structure. To do that, we need to define two things: the motif the object that gets repeated (i.e., a group of atoms), and the unit cell, which is a box that contains the motif an can be translated (but not rotated) to tile in space. There is a third feature that all crystals possess called a Bravais lattice. The Bravais lattice is an array of points in space that all have identical surroundings.

The Bravais Lattice

The Bravais lattice is an infinite arrangement of symmetry-equivalent points in space which define the periodicity of the crystal. This means that if you look at any point in a periodic, ordered array, there is an infinite array of points that have the exact same surroundings. Amazingly, there are only five different Bravais lattices in 2D crystals (Bravais, 1850). There are fourteen Bravais lattices in three dimensions. We care about 2D and 3D crystals in MSE, but it turns out that this idea of a finite array of infinite repeating patterns is generalizable to arbitrary $n>3$ dimensions - a concept of particular interest to physicists and mathematicians who are thinking in higher dimensions (think 4D spacetime or 10D superstring theory). Luckily, in MSE, we can limit ourselves to 2D and 3D.

The 5 Bravais lattices in 2D are shown in Figure 5.4.2. Note, conventionally the rhombus is shown in a way in which some lattice points are connected to form a rectangle. Can you show an "alternate" lattice for the rhombic cell which Can you think of why crystallographers might prefer that representation?

The Unit Cell

Each Bravais lattice has a unit cell, which is a fundamental geometric entity (square, cube, rhombus, etc.) that, when repeated in space via translation alone (no rotating!), will tile to completely fill space. In 2D, the unit cell is always a parallelogon. (In 3D it is always a parallelohedron.) The unit cell is defined by its lattice vectors which tell us the size and shape of the geometric entity by connecting the adjacent lattice points. This include the lattice parameters, which are the length of the sides of parallelogon (or parallelohedron in 3D), and the angles between these axes.

The unit cells for the Bravais lattices in 2D are shown in Figure 5.4.2. denoted by lengths $a$ and $b$, and angle $\alpha$. Note - in the picture below we have an array of points repeating in space., or a lattice. There is only one lattice point in each of the unit cell in Figure 5.4.2. Why's this? Each lattice lattice point is only fractionally contained in the unit cell. For example, the square and rectangular lattice have 1/4 of a lattice point within the unit cell at each corner. This will be important when we define crystal structures (like face-centered cubic, or body-centered cubic), because we need to correctly count the number of atoms in the unit cell - and atoms may only be partially positioned in a unit cell.

Very importantly, the unit cell can be "tiled" to fill in space by translating it integer values of the lattice vectors. This allows us to periodically tile lattice points or atoms arbitrarily in any direction. Let's say we want to visualize the longer-range periodicity of the rectangular lattice. We can tile the lattice points in the rectangular unit cell in Figure 5.4.2 by translating the contents of the $n=0, m=0$ unit cell by $\mathbf{T} = n\mathbf{a} + m\mathbf{b}$. We demonstrate this in Figure 5.4.3.

The Motif

For each unit cell, we will have can have one or more atoms or molecules positioned within the cell. These atoms form the motif of the crystal structure. The motif is associated with a lattice point. Different Bravais lattices can have different motifs associated with each lattice point to produce different crystals!

So, for example, you could have a 2D square lattice as shown in Figure 5.4.4(a.), in which we have a single organic molecule located at corner (represented by a titled oval-shape). Or, we could have two molecules within the unit cell - one rotated from the other as shown in Figure 5.4.4(b.). These two structures have the same unit cell, but different motifs, or repeating entities. Figure 5.4.4(c.) and (d.) show images of the two motifs, respectively, that are associated with the square Bravais lattice unit cells shown in (a.) and (b.).

So, for every crystal we have:

1. A Bravais Lattice - an infinite array of symmetry-equivalent points.
2. A Unit Cell - a geometric entity which tiles to fill space (i.e., only translates). In 2D these can only be parallelogons.
3. A Motif - an pattern of atoms which is associated with each lattice point.

With the unit cell and a motif, you can describe a crystal very simply! You take the motif and place it at the appropriate position in the unit cell. Then, you reproduce the motif ad infinitum by tiling it in space as defined by the lattice vectors (i.e. reproduce the atoms at $n\mathbf{a}+m\mathbf{b}$, where $n$ and $m$ are integer values). You've made a crystal!

Now - before we get ahead of ourselves, we've described crystals with a bare-bones description of only a few paragraphs... there are books upon books published on crystal structures. A single Morfli page is just a beginning. Crystal structure is much more complicated than the what is described above, and requires more complex classifications founded on crystal symmetry, which describes how crystals are invariant (i.e. unchanging) to certain types of spatial operations (rotation, translation, reflection, inversion, etc.).

Crystal symmetry is important when we ask the questions: "Why does this material have different properties (mechanical, thermal, electrical, etc) when it's is oriented differently?" and "Why does this this material have this property (piezoelectricity, ferromagnetism, etc.) while others don't? "This is the topic of an upper-division course required for MSE undergraduates and graduate students in most programs. Regardless, we'll need to address symmetry a bit because it will be useful in assisting us in navigating crystal structures and understanding properties, but we won't be exhaustive.

There are many books written on the topic, including the monstrous but indispensable International Tables for Crystallography.

Let's use what we've learned to look at a perhaps familiar 2D crystal structure to figure out its unit cell and motif. Graphene (Figure 5.4.5). Let's practice with Exercise 5.4.1.

Terms Summary

There's a fair number of new terms to keep track of here. Here's a rundown:

1. Crystal: A periodic ordered array of atoms or molecules.
2. Motif: The actual, physical entities that are repeated in space to make up the crystal: most relevantly: atoms and molecules.
3. Bravais Lattice: An infinite array of symmetry-equivalent points within a crystal (or other repeating structure) that define its periodicity.
4. Unit Cell: The fundamental geometric entity which contains the motif, and which - when tesselated (tiled) throughout space only via translation, not rotation - fills space an produces the crystal structure.