A lattice is a periodic array of points in space. All lattice points in 3-D space can be represented by R = n1a1 + n2a2 + n2a2, where n1, n2, n3 are arbitrary integers, and a1, a2, a3 are three translational vectors. Because a point in space is actually a mathematical abstraction, a lattice (with many lattice points) is a mathematical abstraction. A crystal structure is built upon a lattice by attaching a basis that might be one atom, a set of atoms, or some molecules to each lattice point in exactly the same way. The parallelepiped defined by the three translational vectors a1, a2, a3 is called a unit cell. The unit cell is the building block of the crystal that is described by the lattice with a basis. If the three translational vectors a1, a2, a3 define the smallest unit cell, these three translational vectors are called primitive translational vectors, and the unit cell they defined is called primitive unit cell.
Three translational vectors perpendicular to each other with the same length characterize a cubic unit cell. All of the six faces of a cubic unit cell are squares, and the volume of a cubic unit cell is V = a3. A cubic unit cell is the simplest unit cell and a crystal structure with such a unit cell is called a cubic crystal.
Crystal Structures
There are three common cubic crystal structures, and they are discussed below.
Simple Cubic (SC) - In the SC structure, there is one lattice point, as represented by the blue spheres, at each of the eight corners of the cubic unit cell and each lattice point is shared by eight unit cells. So, there is one lattice point in each of the unit cell (8 * 1/8 = 1). If we use the letter Z to represent the number of lattice points per cubic unit cell, Z=1 for simple cubic structure. If each lattice point is occupied by one atom, it is the simplest simple cubic crystal structure.
Body-Centered Cubic (BCC) — In the BCC structure, in addition to one lattice point, as represented by the blue spheres, at each of the eight corners of the cubic unit cell, there is an additional lattice point at the center of the cubic unit cell. So there are two lattice points in each unit cell (8 * 1/8 + 1 = 2, Z = 2). For BCC structure, a cubic unit cell contains two primitive unit cells and it is called a conventional unit cell.
Face-Centered Cubic (FCC) — In the FCC structure, in addition to one lattice point, again represented as blue spheres, at each of the eight corners of the cubic unit cell, there is a lattice point at the center of each of the six faces of the cubic unit cell. Because two unit cells on the two sides of the face share each of the point at the face center, each cubic unit cell has 4 lattice points (8 * 1/8 + 6 * 1/2 = 4, Z=4). For FCC, the cubic unit cell is also a conventional unit cell and it contains 4 primitive unit cells. One FCC structure that we are going to discuss frequently is a Zincblende structure, or a Diamond structure, which is a special case of Zincblende structure. The Zincblende structure is composed of atoms with tetrahedral bonds. The tetrahedron is a geometric figure shaped like a pyramid with four triangular faces and all the sides of these triangles has the same length. The tetrahedral (Td) molecule is a molecule that has four atoms extending out from a central atom, and the four atoms are located at the vertices of a tetrahedron. All the bond angles from the center atom are 109.5°. The Zincblende structure is a FCC lattice with two atoms associated with each lattice point, one atom at (0,0,0) and another at (1/4,1/4,1/4). These two atoms form a basis of the Zincblende crystals. For diamond structure, these two atoms are the same. Many semiconductors, such as silicon (Si), Germanium (Ge), and Gallium Arsenide (GaAs), have a Zincblende/diamond structure.