Lattice vs Superlattice What is the difference between lattice and superlattice ? Can anyone describe with a schematic figure? I have encountered superlattice in the context of bose hubbard model. Also what are the advantages or disadvantages in superlattice compared to lattices in this context?
 A: @Inmaurer gave a nice explanation of superlattices in general, so I will concentrate on the specific example of the Bose-Hubbard model. A typical experimental implementation consists of holding a gas of ultracold bosons (often rubidium-87) in a magneto-optical trap, and then applying an optical lattice potential. This consists of a standing wave of laser light, and depending on the frequency of the light, the bosons are either attracted or repelled from the maxima of this standing wave. This provides the underlying lattice potential.
To produce a superlattice structure, an additional standing wave is applied, with a different wavelength. This will modulate the lattice potential on a longer length-scale. For example, if the second laser has twice the wavelength of the original, it will produce this net potential:

You see that the unit cell of the superlattice - the supercell - contains two lattice sites, which have different potentials. By choosing different wavelengths, you can obtain supercells with different number of sites within them. It is even possible to combine several additional potentials to create even more complicated lattice potentials, with different periodicities.
A: In solid state physics, a (Bravais) lattice is an orderly arrangement of atoms into crystals. For example, Silicon forms a face-centered cubic lattice (with a two-atom basis). A super lattice is a repeating orderly arrangement of crystals. E.g. you have a couple atomic layers of crystalline Si followed by a few atomic layers of crystalline Ge, and this arrangement repeats. As for a figure, try google image search. There are tons of them. Here's one

The right side shows how the super lattice is made of alternating crystals.
I don't know much about superlattices in the Bose-Hubbard model, but it would be a serious abuse of terminology unless it describes something similar. (But who knows, stranger things have happened.) I have no idea what use they are in a Bose-Hubbard model, but they are very useful in making actual semiconductor devices.
