Fellows,
I am now a doctorate in inorganic/physical chemistry and am working on a presentation that will be done at the American Physics Society Meeting, in Dallas, about the eigenstates of graphene. Although my presentation is in good shape right now, I have a question regarding a basic tenet of information, the wavelength of eigenstates that relate to crystal momentum.
For example, at Gamma, the wavelength is taken to be infinite, as there is no momentum to speak of. I use Gamma eigenstates of regular and larger unit cells to describe the eigenstates in graphene for different k-points, transferring the wavelengths of the original eigenstates onto the Gamma eigenstates of the computed unit cells (1x1, 2x2, or 3x3). This simplifies description of eigenstates significantly, and is the central point of my presentation. The wavelength is used to describe phase relationships, and in Gamma eigenstates, there is no phase change anywhere other than individual orbitals inside the unit cell. Going across crystal boundaries does not produce a phase change in all of the orbitals present inside the unit cell. This makes the wavelength infinite in Gamma eigenstates, as the unit cell contents are used to represent the phase relationships for wavelength in crystal momentum. In other k-points, such as M and K, however, this changes considerably, as the orbitals acquire a finite wavelength as the crystal boundary is crossed over. This is what produces the crystal momentum in M and K, the phase relationship is modulated across crystal boundaries.
Using larger unit cells causes zone-folding in energy diagrams, but is quite useful for creating wavefunctions without momentum, so you obtain the phase relationships as if they were Gamma eigenstates!
How do we use the expression for crystal momentum to describe wavelength of eigenstates, given that we know the coordinates for each k-point? I would like to do this so I can correct some minor details involving wavelengths of eigenstates. I need the wavelengths that the k-points, M and K, represent, in terms of the lattice constant, a. To put this in other words, if M is (0.5, 0) or (0, 0.5), and K is (1/3, 1/3), then what is the wavelength of the eigenstates in terms of the hexagonal lattice constant, a? Or is there a quick way to use coordinates of a k-point to obtain the wavelength a 1x1 unit cell eigenstate would have at that point?
