Narration script

Unit three, crystal structures. In a crystal, atoms repeat in a perfect three-dimensional pattern. Silicon's pattern, called diamond cubic, follows directly from its four tetrahedral bonds. Gallium arsenide and cadmium telluride use the same geometry with two kinds of atom, called zinc blende. C I G S doubles it, called chalcopyrite. And halide perovskites use a different, cage-like structure. We will also learn Miller indices, the naming system for crystal planes, which matters because wafers are cut on one particular plane and etching reveals another.

You will learn the vocabulary of lattices and unit cells, describe the diamond structure and compute silicon's density from it, recognise the other solar crystal structures, and use Miller indices to find angles and spacings between planes.

Crystal orientation decides how ingots are grown, how wafers break, how they etch into pyramids, and how films grow on them. And X-ray diffraction, a standard measurement tool, reads crystal structure directly.

A crystal is a lattice, a regular array of points, plus a basis, the group of atoms placed at each point. The smallest repeating box showing the full symmetry is the unit cell, and its edge length is the lattice constant. Many structures here are built on the face-centred cubic lattice, with points at the cube corners and at the centre of each face.

Silicon's structure is face-centred cubic with a two-atom basis: one atom at each lattice point, and a second shifted a quarter of the way along the cube's body diagonal. Each atom ends up with four nearest neighbours at the corners of a tetrahedron, exactly the bonding we saw in unit two.

Here is the diamond cubic unit cell of silicon, computed from the real geometry. The blue atoms sit on the face-centred cubic sites, the orange atoms on the shifted sites, and the lines are covalent bonds. The cell edge is five point four three one angstroms. Counting shared atoms carefully, eight corners each shared by eight cells, six faces each shared by two, and four fully inside, gives eight atoms per cell.

From that, we can predict silicon's properties. The cell volume is about one point six times ten to the minus twenty-two cubic centimetres. Eight atoms in that volume gives about five times ten to the twenty-two atoms per cubic centimetre, the number used everywhere in this field. Multiply by the atomic mass and you get a density of two point three two nine grams per cubic centimetre, matching the measured value. The nearest neighbours sit two point three five angstroms apart. And the diamond structure is surprisingly empty: only about thirty-four percent of space is filled, because directional bonds, not packing, set the geometry.

Now put one element on the face-centred sites and a different element on the shifted sites. That is zinc blende. Every gallium bonds to four arsenics and vice versa. Gallium arsenide, cadmium telluride, indium phosphide and zinc selenide all use it. The average is still four valence electrons per atom, so the tetrahedral bonding is preserved.

Replace the group two atoms of zinc blende alternately with copper and with indium or gallium, in an ordered way, and the cell doubles along one axis. That is chalcopyrite, the structure of C I G S. Changing the gallium fraction tunes the band gap from about one point oh to about one point seven electron-volts.

Perovskites have the formula A B X three. A small B cation, lead or tin in solar perovskites, sits inside an octahedron of halide anions. The octahedra share corners, and a large A cation fills the cages between them. A simple ratio of ionic radii, the Goldschmidt tolerance factor, predicts whether a combination will form a stable perovskite.

Here is the cubic perovskite. The A cations sit at the corners, lead at the centre, and the halides at the face centres, forming the shaded octahedron. Methylammonium lead iodide has a band gap of about one point five five to one point six electron-volts, and mixing in bromine raises it toward the one point six five to one point seven five wanted for a silicon tandem top cell.

This chart plots band gap against lattice constant for diamond and zinc blende semiconductors. Materials with matching lattice constants, like gallium arsenide and germanium, can be grown on top of each other as perfect layers. That is the foundation of the multi-junction cells used in space.

Now Miller indices. To name a plane, find where it cuts the three axes in units of the cell edge, take the reciprocals, and clear fractions to the smallest whole numbers. A plane parallel to an axis never cuts it, so that index is zero. Round brackets name a single plane, curly brackets a family of equivalent planes, and square brackets a direction.

Here are the three planes you will use most. The one zero zero plane is a cube face. The one one zero plane cuts diagonally through two axes. And the one one one plane slices across three cube corners.

Here is why this matters. The angle between the one zero zero and one one one planes is fifty-four point seven four degrees. Alkaline etching attacks one zero zero surfaces much faster than one one one surfaces, because a surface atom on a one one one plane has three bonds into the crystal and only one exposed. So a one zero zero wafer etches into square-based pyramids whose four faces are one one one planes, tilted at fifty-four point seven degrees. That is the random pyramid texture on every monocrystalline cell.

Another use: X-ray diffraction. With copper X-rays and the one one one plane spacing of three point one four angstroms, Bragg's law predicts a peak at two theta equals twenty-eight point four four degrees, the textbook silicon reference peak.

Silicon cleaves most easily along one one one planes, and also along one one zero. When you scribe and snap a one zero zero wafer, it breaks along straight, perpendicular lines, which you will see yourself in Lab B.

Misconceptions. A unit cell is not a molecule; corner and face atoms are shared with neighbours. Pyramids are not etched through a mask; they form on their own because of etch-rate differences. A plane and a direction are not the same thing. And perovskite is not one material; it is a structure type and a family of compositions.

For the job: ingots are pulled from seeds oriented along one zero zero so wafers texture properly. Wafer breakage and half-cell cutting follow crystal planes. X-ray diffraction characterises thin films. And lattice matching governs the growth of tandem and compound cells.

Try the review questions. Next unit, we look at real materials, which are rarely perfect crystals.