In this video, we will discuss crystal structure. Depending on the arrangement of atoms that make up the material, we can classify solids into several different kinds. If the atoms are arranged in a random fashion as shown in the left figure here, this type of solid is called amorphous solid. If on the other hand, atoms are arranged in a perfectly regular pattern as shown in the right figure here, that type of solid is called crystalline solid or sometimes to emphasize that the entire crystal, entire material has a single pattern, we call it single crystalline solid. We could have an intermediate case where you have regular arrangement locally, but globally, the make material is made up of different domains of the single crystal in crystallites arranged in a random fashion. This type of solid as shown in the middle is called poly-crystalline solid. In semiconductor devices, single crystalline semiconductor materials are used predominantly. So, we will focus on single crystalline material for the rest of the course. At this point, however, I just want to point out that amorphous semiconductor and poly-crystalline semiconductors are also used in some devices. Most notably, the thin-film transistor used in liquid crystal displays typically use amorphous silicon or sometimes poly-crystalline silicon. So, what is crystal? Crystal can be understood as a Bravais lattice with basis, and we're going to define these two in sequence. Bravais lattice is a repeating arrangement of points that completely filled up space. Alternatively, you can think of it as a repeating arrangement of tiles of different shape that completely fill up space. So, in the example shown in the figure below, you have an arrangement of these dots. Each represents a lattice point, and they are in a regular pattern, repeating pattern, and they completely fill up space, two-dimensional space. So, this is an example of two-dimensional Bravais lattice. To characterize this lattice, we use lattice vectors. Lattice vector is a vector that define the type of translational symmetry this Bravais lattice possesses. So, simply, the lattice vector can be understood as a vector that connects any two arbitrary lattice points. For example, you pick this lattice point here. You pick this lattice point here and this lattice point there, and then you join the two. This factor is a lattice vector. You can do this with any two arbitrarily chosen points. If you translate this entire lattice by this lattice vector, then you will see that the Bravais lattice original lattice maps onto itself. So, this lattice vector represents a translational operation which keeps the lattice invariant, and that's the translational symmetry that we're talking about here. Now, the smallest of these lattice vectors are called a unit vector. So, in this example here, there are two unit vectors here, a1 and a2. We have two unit vectors because this example is a 2D Bravais lattice. If you have a 3D lattice, then you would have three independent unit vectors. Once you identify unit vector, you can represent any arbitrary lattice point by simply a linear combination of these unit vectors. So, for example, if you have a three-dimensional Bravais lattice, then any arbitrary lattice vector R can be represented by the linear combination of three unit vectors, a1, a2, a3. C1, C2, C3 are some arbitrary integer coefficients that you can choose. In two-dimensional space, there are only five Bravais lattices as shown here. So, there is a square lattice, and there are two different types of rectangular lattice as shown here, and then there is a hexagonal lattice, and then there is an oblique lattice. This is an example that I showed you in the previous slide. Each lattice is characterized by the size of the unit vectors and the angles between them. In 3D, there are more lattices. There are total of 14 Bravais lattices, and they are shown here. The most important lattices for semiconductor is face-centered cubic lattice shown here and hexagonal lattice shown here. Now, before I move on, I want to introduce Miller indices which are used to specify crystal plane. So, suppose that I want to specify a particular crystal plane as shown here. This triangle is a section. The shaded triangle is a section of the crystal plane that I want to specify. Now, in order to do that, we use Miller indices. Miller indices are, basically, the vector perpendicular to the plane. So, in order to find Miller indices, you first identify the intersection of the plane with the crystallography axis. So, in this example, we're considering a cubic lattice. So, the crystallographic axes are the same as the Cartesian axes x, y, and z here. In this example, the intersections are p, q, and r. So, find the intersection p, q, r, and then the Miller indices are found by simply taking the reciprocal of these numbers. Now, in general, you will get some sort of a refraction. So, to make a readily understandable, easily readable indices, you just basically multiply the least common multiplier to turn these reciprocals into simple small integer values. Now, with basis, let's consider this example shown here in a two-dimensional honeycomb crystal structure. So, this looks at first glance quite similar to the hexagonal lattice that I shown in an earlier slide. However, you notice that it is different from the hexagonal lattice in that it is missing these lattice points that's supposed to be there in the case of hexagonal lattice at the center of these each hexagon. So it is not hexagonal lattice, but it is a regular pattern that fills up the space. So, how do you characterize or specify this type of crystal? Well, to do that, you first have to notice that if you consider this pair, this pair of atoms as one unit, if you consider this as one unit, then you will notice that these pair of atoms are arranged in a hexagonal Bravais lattice. So, to highlight that, I changed the color of one of these atoms in the pair. So, in this case, the atomic pair is made of one blue and one orange. Now, if you join up only the blue part of these two atom pair, then you will see that the blue atoms are perfectly situated in the hexagonal Bravais lattice. So, honeycomb crystal is viewed as hexagonal Bravais lattice with two atom basis. This pair that makes a unit that is placed in a Bravais lattice is called the basis. So, this particular example, I have two atom basis. You could have a more complex basis, three atom basis, four atom basis arranged in a more complex pattern, and those make more complex crystal structures. Now, why is this important? Because you could have a similar situation in a 3D case. So, in the left here is shown a face-centered cubic crystal structure. If you place a two atom basis on each of these lattice point of the FCC crystal, you could get a diamond crystal structure as shown here. So, in this figure in the right, you can see that these blue and green atom pair is the two atom basis, and these blue atoms are placed on a FCC, face-centered cubic, lattice points, and each blue atom has one additional green atom paired to it. It's a two atom basis. Now, these other blue atoms also should have these green atom that pairs up with these blue atoms to make up a basis, but they are outside the cubic cell. They are located on the outside of this cubic unit cell. So there are not shown here. So, this type of crystal structure is called a diamond crystal structure. Carbon atoms forms this type of crystal structure. This carbon atom crystallize in the diamond crystal structure is called diamond. But more importantly for our class, elemental semiconductors such as silicon and germanium crystallize into this diamond crystal structure. So this is the crystal structure of silicon and germanium. Now, you could have a situation where these two atom basis that forms the diamond crystal structure is made of two different atomic species. In other words, in the case of carbon, diamond, and silicon, and germanium crystal, these two atom basis, blue and green here, are actually two of the same atoms, all carbon, all silicon, or all germanium. But you could have a situation where these two atom basis are made of two different atoms, gallium and arsenic, indium and phosphorus. In that case, you have a zinc blende crystal structure. So, in this figure here, you can see that these atomic placements are identical to the diamond crystal structure as shown in the previous slide except that, in this case, you now have a two atom basis made of two different atom. So, they are shown as two different colors and two different sizes here in this schematic. So, this smaller atom is gallium atom, for example, and this larger sphere is an arsenic atom. In that case, you have a gallium arsenite, one of the representative, III-V compound semiconductors. So, many of the III-V compound semiconductors such as gallium arsenide, indium phosphide crystallize into zinc blende crystal structure. Some of the II-VI compound semiconductors such as zinc selenide and cadmium telluride also crystallize into zinc blende crystal structure. You could have a similar situation with hexagonal Bravais lattice as well. So, in this figure, you see these gray spheres form a hexagonal crystal structure, hexagonal Bravais lattice, and each of these gray atoms have a yellow atom pair to it. So this is the two atom basis that is mapped onto hexagonal Bravais lattice. Hexagonal Bravais lattice with two atom basis in this geometry is called the wurtzite structure. Some of the III-V compounds such as gallium nitride crystallizes into wurtzite structure. Also, some of the II-VI compound such as zinc sulfide, when prepared at high-temperature, forms this wurtzite crystal structure.